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Elements of Chemical Reaction Engineering Elements of Chemical Reaction Engineering Third Edition By H. Scott Fogle
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Elements of Chemical Reaction Engineering
Elements of Chemical Reaction Engineering
Third Edition By H. Scott Fogler University of Michigan Ann Arbor, Michigan
Welcome! Welcome to the CD-ROM that accompanies the Third Edition of Elements of Chemical Reaction Engineering by H. Scott Fogler. Follow the links (below) to learn how to get the most out of this CDROM. Introduction Begin: Chapter 1
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Elements of Chemical Reaction Engineering
APPENDICES
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Introduction
Welcome Navigation Components Usage
Welcome This CD-ROM is intended to be used as a learning resource; the material on this CD-ROM supports the chemical reaction engineering concepts covered in the text. You are encouraged to use the CD-ROM to supplement and expand upon your own studies. We are certain that you will find the extra knowledge you gain will be worthy of the time you invest to obtain it.
Warnings! Macintosh Users The majority of the files on the CD-ROM are HTML files. Great care was taken in trying to insure that these files would work on both PCs and Macs. However, many of the non-HTML files on the CD-ROM (i.e., Polymath, all of the Interactive Computer Modules, and most of the plug-ins included on the CD) are Windows/DOS-based programs, for which there are (unfortunately) no Macintosh equivalents. You may still use these files, if you have a PC emulator program on your Mac, such as Virtual PC. ICMs Some users have experienced problems, trying to run the Interactive Computer Modules directly from the ICM directory on the CD-ROM. If you have trouble with being able to run the Interactive Computer Modules from the CD-ROM, then try installing them on your hard drive. Hidden Files To clear up some of the confusion about which files to use in certain directories (e.g., Polymat4 and ICMs), some files and folders were hidden. You may find it easier to navigate the CD-ROM, if you make certain that you are NOT viewing hidden files.
Recommended Software Before you begin, we advise you to download or install the following software programs and plug-ins on your computer, if they are not already present on your system:
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Introduction
Adobe Acrobat Reader and Plug-in The Appendices (and certain other files on the CD-ROM) are in PDF format. You must have Adobe Acrobat Reader installed to access these files. You will also need the Adobe Acrobat Reader Plug-in to read these files from your web browser. Note: The PC version of Adobe Acrobat Reader 3.02 has been included on the CD-ROM in a directory called Software\Adobe. If you run the ar302.exe program, it will install the reader software and the browser plug-in on your computer. Apple QuickTime Plug-in There are a few QuickTime movies on the CD-ROM, which will require the QuickTime Plug-in to view them. Note: QuickTime 3.0 (for PCs) has been included on the CD-ROM in a directory called Software\Quick. If you run the quick3.exe program, it will install the movie viewer and the plug-in on your computer and in your browser, respectively. WinZip (Optional) Some of the Polymath files for the Living Example Problems are archived as Zip files. You may need an unzip utility, such as WinZip to access them. See the Polymath section of the CD-ROM for more information. MATLAB (Optional) In addition to Polymath, you may use MATLAB to access the Living Example Problems. See the MATLAB section of the CD-ROM for more information. IMPORTANT! Different browsers and font sizes may affect the alignment and general appearance of the HTML content of the CD-ROM. To ensure that items are aligning properly, you may need to adjust your browser's font size. The HTML content of this CD-ROM is also available at the University of Michigan's Chemical Reaction Engineering Website: file:///H:/htmlmain/intro.htm[05/12/2011 16:54:01]
Introduction
http://www.engin.umich.edu/~cre
Next Step Once you have downloaded and/or installed this software, you should proceed to the section on Navigation, to learn how to get around this CD. The Components section will give you information on the various "modules" that are available on this CD. The Usage section will give you information on the best way to integrate the information on this CD with the information in your book.
Welcome Navigation Components Usage Begin: Chapter 1
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Elements of Chemical Reaction Engineering, Credits
The following people have conspired to bring you this CD-ROM:
The University of Michigan Team H. Scott Fogler, Author Dieter Andrew Schweiss, Media Project Manager Ellyne Buckingham, Artist, ToPS Scott Conaway, Wetlands Susan Fugett, MATLAB Examples Anuj Hasija, HTML Designer Lisa Ingall, ToPS Brad Lintner, ToPS Timothy Mashue, Reactive Distillation James Piana, ICMs Susan Stagg, Cobra Problem Author Gavin Sy, Cobra Problem (ToPS = Thoughts on Problem Solving) Special thanks to: Nicholas Abu-Absi John Bell Michael Cutlip, Polymath Sean Connors Anurag P. Mairal Professor Susan Montgomery Mordechai Shacham, Polymath Probjot Singh Ibrahim "Abe" Sendajarevic Mayur Valanju
The Team at Prentice-Hall PTR: Bernard Goodwin, Executive Editor Diane Spina, Assistant to the Executive Editor Sophie Papanikolaou, Director of Production Lisa Iarkowski, Manager, Production Yvette Raven, Media Project Manager Talisman Desktop Productions, Developer Scholar's Net Academic Multimedia, Design/Programming file:///H:/htmlmain/credits.htm[05/12/2011 16:54:02]
Elements of Chemical Reaction Engineering, Credits
Additional Credits Membrane Reactors Parts of this site was originally presented as an Open-Ended Problem in the Winter 1997 Chemical Reaction Engineering Class at the University of Michigan. The students who developed this module were Kim Dillon, Namrita Kumar, Amy Miles, and Lynn Zwica. The module was further expanded and improved by Ellyne Buckingham, Dieter Andrew Schweiss, Anurag Mairal, and H. Scott Fogler for use with the Chemical Reaction Engineering Web Site and CD-ROM.
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Easter Egg
Easter Egg This is the only easter egg that I know of on the CD-ROM (but there may be more). My name is Dieter Andrew Schweiss, and after devoting a year of my life to this project, I had to put something hidden on this thing! I've really enjoyed working with Scott, the University of Michigan Team, and the people from Prentice-Hall on this CD-ROM project. It has helped me fulfill a long-time dream of contributing to the accumulated knowledge that is the field of Chemical Engineering. (That, and the fact that nothing quite like this CD had ever been done before in ChE!) Anyways, good luck with your classes. Be sure to use the resources available to you on this CD, especially the Lecture Notes and the worked example problems. They'll really come in handy. (Trust me, I know!) I like what this symbol represents: life is a balance between order and chaos, yet pure order still retains some element of chaos and pure chaos retains some element of order (kind of like my desk...)
Special Recognition Professor H. Scott Fogler would like to extend a special thanks to Dieter Andrew Schweiss, without whom the Elements of Chemical Reaction Engineering CD-ROM would never have been possible. Dieter worked countless days and nights to bring this project to completion, coordinating his efforts with both the University of Michigan Team and the Prentice-Hall Team. Thank you, Dieter!
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Chapter One
CHAPTER 1
1
Mole Balances After completing Chapter 1 of the text and associated CD-ROM material the reader will be able to: Define the rate of chemical reaction. Apply the mole balance equations to a batch reactor, CSTR, PFR, and PBR. Describe two industrial reaction engineering systems. Describe photos of real reactors. Describe how to surf the CD-ROM attached with this text.
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Legal Information
About Prentice Hall PRENTICE HALL CD-ROM/WEB SITE: LEGAL DISCLAIMER Unless otherwise indicated, this CD-ROM/Web Site and its contents are the property of Prentice-Hall, Inc. ("Prentice Hall") and are protected, without limitation, pursuant to U.S. and foreign copyright and trademark laws. PRENTICE HALL MAKES NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THIS CD-ROM/WEB SITE OR ITS CONTENTS, WHICH ARE PROVIDED FOR USE "AS IS." PRENTICE HALL DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING WITHOUT LIMITATION THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, WITH RESPECT TO THE CD-ROM/WEB SITE AND ANY CD-ROM/WEB SITE WITH WHICH IT IS LINKED. PRENTICE HALL ALSO MAKES NO REPRESENTATIONS OR WARRANTIES AS TO WHETHER THE INFORMATION ACCESSIBLE VIA THIS CD-ROM/WEB SITE, OR ANY CD-ROM/WEB SITE WITH WHICH IT IS LINKED, IS ACCURATE, COMPLETE, OR CURRENT. Price information is subject to change without notice. In no event shall Prentice Hall or its employees, agents, suppliers, or contractors be liable for any damages of any kind or character, including without limitation any compensatory, incidental, direct, indirect, special, punitive, or consequential damages, loss of use, loss of data, loss of income or profit, loss of or damage to property, claims of third parties, or other losses of any kind or character, even if Prentice Hall has been advised of the possibility of such damages or losses, arising out of or in connection with the use of this CD-ROM/WEB Site or any CD-ROM/WEB Site with which it is linked. Portions of this Site (such as "chat rooms" or "bulletin boards") may provide users an opportunity to post and exchange information, ideas and opinions (the "Postings"). BE ADVISED THAT PRENTICE HALL DOES NOT SCREEN, EDIT, OR REVIEW POSTINGS PRIOR TO THEIR APPEARANCE ON THIS SITE, and Postings do not necessarily reflect the views of Prentice Hall. In no event shall Prentice Hall assume or have any responsibility or liability for the Postings or for any claims, damages or losses resulting from their use and/or appearance on this Site. You hereby represent and warrant that you have all necessary rights in and to all Postings you provide and all information they contain and that such Postings shall not infringe any proprietary or other rights of third parties or contain any libelous, tortious, or otherwise unlawful information. You hereby authorize Prentice Hall to use and/or authorize others to use your Postings in any manner, format or medium that Prentice Hall sees fit.
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Legal Information
Removing pieces of our Companion CD-ROM/Websites and incorporating them into class homepages always requires formal permission in writing. We are very cooperative in arranging permission for use of our components by faculty who have adopted our book. Note that the creation of a class homepage linked to Companion CD-ROM/Websites using Syllabus Builder does not require permission, if you are using our textbook for your course. Please read our legal statement for details. To contact our permissions department, please fax a request to Attention: CD-ROM/Website Permissions Dept fax 201-236-3290 Note that your fax must include a printed copy of the CD-ROM/Webpages that contain the material you'd like to re-use, with a clear indication of the pieces you desire. You must also include a short written description of your intended use for these components. This MUST include: the intended audience, an estimate of how many persons will view the content you've requested, and the dates that you expect the material to be available to your audience. a description of access control methods (is a password required to access the site) a description of the cost to access your site if possible, an example of how you would display our content and copyright notice © Prentice-Hall, Inc. A Simon & Schuster Company Upper Saddle River, New Jersey 07458
* if possible, an example of how you would display our content and copyright notice © Prentice-Hall, Inc. A Simon & Schuster Company Upper Saddle River, New Jersey 07458
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Chapter Two
CHAPTER 2
2
Conversion and Reactor Sizing After completing Chapter 2 of the text and associated CD-ROM material the reader will be able to: Define conversion and space time. Write the mole balances in terms of conversion for a batch reactor, CSTR, PFR, and PBR. Size reactors either alone or in series once given the rate of reaction, -r A, as a function of conversion, X. Write relationship between the relative rates of reaction.
BEGIN
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Chapter Three
CHAPTER 3
3
Rate Law and Stoichiometry After completing Chapter 3 of the text and associated CD-ROM material the reader will be able to: Write a rate law and define reaction order and activation energy. Set up a stoichiometric table for both batch and flow systems and express concentration as a function or conversion. Calculate the equilibrium conversion for both gas and liquid phase reactions. Write the combined mole balance and rate law in measures other than conversion. Set up a stoichiometric table for reactions with phase change.
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Chapter Four
CHAPTER 4
4
Isothermal Reactor Design After completing Chapter 4 of the text and associated CD ROM material the reader will be able to: Describe the algorithm that allows the reader to solve chemical reaction engineering problems through logic rather than memorization. Size batch reactors, semibatch reactors, CSTRs, PFRs, and PBRs for isothermal operation given the rate law and feed conditions. Discuss solutions to problems taken from the California Professional Engineers Registration Examination. Account for the effects of pressure drop on conversion in packed bed tubular reactors and in packed bed spherical reactors.
BEGIN
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Chapter Four
APPENDICES
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Chapter Five
CHAPTER 5
5
Collection and Analysis of Rate Data After completing Chapter 5 of the text and associated CD-ROM material the reader will be able to: Determine the reaction order and specific reaction rate from experimental data obtained from either batch or flow reactors. Describe how to use equalarea differentiation, polynomial fitting, numerical difference formulas and regression to analyze experimental data to determine the rate law. Describe how the methods of half lives, and of initial rate, are used to analyze rate data. Describe two or more types of laboratory reactors used to obtain rate law data along with their advantages and disadvantages. Describe how to plan an experiment.
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© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Chapter Five
APPENDICES
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Chapter Six
CHAPTER 6
6
Multiple Reactions After completing Chapter 6 of the text and associated CD-ROM material the reader will be able to: Define different types of selectively and yield. Choose a reaction system that would maximize the selectivity of the desired product given the rate laws for all the reactions occurring in the system. Describe the algorithm used to design reactors with multiple reactions. Size reactors to maximize the selectivity and to determine the species concentrations in a batch reactor, semibatch reactor, CSTR, PFR, and PBR, systems.
BEGIN
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Chapter Seven
CHAPTER 7
7
Nonelementary Reaction Kinetics After completing Chapter 7 of the text and associated CD-ROM material the reader will be able to: Discuss the pseudo-steadystate-hypothesis and explain how it can be used to solve reaction engineering problems. Discuss different types of polymerization reactions and rate laws. Describe Michealis-Menton enzyme kinetics and enzyme inhibition. Write material balances on cells, substrates, and products in bioreactors.
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Chapter Eight
CHAPTER 8
8
Steady-State Nonisothermal Reactor Design After completing Chapter 8 of the text and associated CD-ROM material the reader will be able to: Describe the algorithm for CSTRs, PFRs, and PBRs that are not operated isothermally. Size adiabatic and nonadiabatic CSTRs, PFRs, and PBRs. Use reactor staging to obtain high conversions for highly exothermic reversible reactions. Carry out an analysis to determine the Multiple Steady States (MSS) in a CSTR along with the ignition and extinction temperatures. Analyze multiple reactions carried out in CSTRs, PFRs, and PBRs which are not operated isothermally in order to determine the concentrations and temperature as a function of position (PFR/PBR) and operating variables.
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© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8
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Chapter Eight Legal Statement
APPENDICES
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Chapter Nine
CHAPTER 9
9
Unsteady-State Nonisothermal Reactor Design After completing Chapter 9 of the text and associated CD-ROM material the reader will be able to: Analyze batch reactors and semibatch not operated isothermally. Analyze the start up of nonisothermal CSTRs. Analyze perturbations in temperature and presence for CSTRs being operated at steady state and describe under what conditions the reactors can be unsafe (safety). Describe the effects of adding a controller to a CSTR. Analyze multiple reactions in batch and semibatch reactors not operated isothermally.
BEGIN
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Chapter Nine
APPENDICES
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Chapter Ten
CHAPTER 10
10
Catalysis and Catalytic Reactors After completing Chapter 10 of the text and associated CD-ROM material the reader will be able to: Define a catalyst, a catalytic mechanism and a rate limit step. Describe the steps in a catalytic mechanism and how one goes about deriving a rate law and a mechanism and rate limiting step consistent with the experimental data. Size isothermal reactors for reactions with LangmuirHinschelwood kinetics. Discuss the different types of catalyst deactivation and the reactor types and describe schemes that can help offset the deactivation. Analyze catalyst decay and conversion for CSTRs and PFRs with temperaturetime trajectories, moving bed reactors, and straight through transport reactors. Describe the steps in Chemical Vapor Deposition(CVD). Analyze moving bed reactors that are not operated isothermally.
BEGIN
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Chapter Ten
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APPENDICES
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Chapter Eleven
CHAPTER 11
11
External Diffusion Effects on Heterogeneous Reactions After completing Chapter 11 of the text and associated CD-ROM material the reader will be able to: Define the mass transfer coefficient, explain what it is function of and how it is measured or calculated. Analyze PBRs in which mass transfer limits the rate of reaction. Discuss how one goes form a region mass transfer limitation to reaction limitation. Describe how catalyst monoliths and wire gauze reactors are analyzed. Apply the shrinking core model to analyze catalyst regeneration.
BEGIN
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Chapter Eleven
APPENDICES
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Chapter Twelve
CHAPTER 12
12
Diffusion and Reaction in Porous Catalysts After completing Chapter 12 of the text and associated CD-ROM material the reader will be able to: Define the Thiele modules and the effectiveness factor. Describe the regions of reaction limitations and internal diffusion limitations and the conditions that affect them. Determine which resistance is controlling in a slurry reactor. Analyze trickle bed reactors. Analyze fluidized bed reactors. Describe the operation of a CVD Boat Reactor.
BEGIN
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Chapter Twelve
APPENDICES
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Chapter Thirteen
CHAPTER 13
13
Distributions of Residence Times for Chemical Reactors After completing Chapter 13 of the text and associated CD-ROM material the reader will be able to: Define a residence time distribution RTD [E(t), F(t)] and the mean residence time. Determine E(t) form tracer data. Write the RTD functions (E(t), F(t), I(t)) for ideal CSTRs, PFRs, and laminar flow reactors. Predict conversions from RTD data using the segregation and maximum mixedness models. Predict effluent concentrations for multiple reactions using the segregation and maximum mixedness models.
BEGIN
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Chapter Thirteen
APPENDICES
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Chapter Fourteen
CHAPTER 14
14
Models for Nonideal Reactors After completing Chapter 14 of the text and associated CD-ROM material the reader will be able to: Describe the tanks-in-series and dispersion one parameter models. Describe how to obtain the mean residence time and variance to calculate the number of tanks-in-series and the Peclet number. Calculate the conversion for a first order reaction taking place in a tubular reactor with dispersion Describe how to use combinations of ideal rectors to model a real reactor and how to use tracer data to determine the model parameters.
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Appendices
APPENDICES
The Appendices are in PDF format. You must have Adobe Acrobat Reader installed to access these files. You will also need the Adobe Acrobat Reader Plug-in to read these files from your browser. (See the CDROM Introduction for more information.) Appendix D: Measurement of Slopes on Semilog Paper Appendix E: Software Packages Appendix H: Open-Ended Problems Appendix J: Use of Computational Chemistry Software Packages
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Navigation
Welcome Navigation Components Usage
Navigation The Objectives
At the opening of every chapter is the Objectives. Clicking on BEGIN takes you to the Chapter Outline -- the contents for that chapter.
Chapter Outline - The Main Interface
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Navigation
The Main Interface lists the contents for that chapter. There are links to the entire chapter's contents from this page. Return to this page by using the left-hand navigation bar and click on Chapter Outline. In the Professional Reference Shelf and Learning Resources sections, there are Examples that link from within these sections. When you click on an Example, a new browser window will open. Some Examples can also be accessed from the Chapter Outline, where a new browser window will not open.
An Example in a New Browser Window
This new browser window is not the main interface. To return to the previous page, close this window by clicking on the top-left button of the new browser window (for Macs), or on the top-right button of the new browser window (for PCs). file:///H:/htmlmain/intro2.htm[05/12/2011 16:54:33]
Navigation
Throughout this material there will be Footnotes. By clicking on a footnote, you are going to open a new browser window.
Footnotes in a New Browser Window
Again, this new browser window is not the main interface. To return to the previous page, close this window by clicking on the top-left button of the new browser window (for Macs), or on the top-right button of the new browser window (for PCs).
Left Navigation Bar
Chapter Number takes you to the Objectives page for that chapter, while Chapter Outline takes you to Contents page for that chapter. Software Toolbox takes you to the Software Toolbox, e.g. Polymath. Interactive Computer Modules takes you to the ICMs main menu. Thoughts on Problem Solving takes you to the main problem solving menu.
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Navigation
Updates & FAQs takes you to updates and corrections for the book. Representative Syllabi takes you to the sample 3- and 4-credit course syllabi. Help takes you to the help menu.
Except for Chapter Number and Chapter Outline, information you access from this navigation bar is not specific to any chapter. You can go directly to the Thoughts on Problem Solving section to see examples of the 10 types of home problems or visit the Interactive Computer Modules, without leaving the chapter that you are in. All the material is ordered by chapter. The lower navigation bar will take you to a specific Chapter or the Appendices. The HOME button will take you to the welcome screen for this CD.
Lower Navigation Bar
To find out more, go to the Components section of this CD.
Welcome Navigation Components Usage file:///H:/htmlmain/intro2.htm[05/12/2011 16:54:33]
Navigation
Begin: Chapter 1
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Components
Welcome Navigation Components Usage
Components Components of the CD-ROM Each chapter is broken into the following bulleted components: Learning Resources Summary Notes Web Modules Interactive Computer Modules Solved Problems Living Example Problems Professional Reference Shelf Additional Homework Problems
Learning Resources These resources give an overview of the material in each chapter and provide extra explanations, examples, and applications to reinforce the basic concepts of chemical reaction engineering. The learning resources on the CDROM include: 1. Summary Notes The Summary Notes of the lectures given at the University of Michigan will serve as an overview of each chapter. They contain a logical flow of the equations being derived, along with additional examples and material that can be viewed either before or after reading the text. The first 26 lectures are covered in a four-credit hour undergraduate course. The last 11 (27-37) are taken from the graduate course at the University of Michigan. 2. Web Modules These modules show how key concepts of chemical reaction engineering can be applied to nonstandard problems (e.g. the use of Wetlands to degrade toxic chemicals). Current modules focus on Chapters 4 and 6 and include Wetlands, Cobra Bites, Membrane Reactors and Reactive Distillation modules. Additional web modules (http://www.engin.umich. edu/~cre) are expected to be added over the next several years. 3. Interactive Computer Modules Most chapters have one or more interactive computer modules (ICMs) to accompany them as a learning resource. For these chapters, students can use the corresponding ICM(s) to review the important material and then apply it to real problems in a unique and entertaining fashion. Each module contains: Menu Review of concepts Interactive problem
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Components
Solution to the problem For example, in the Murder Mystery module students take on the role of assistant sleuth as they use basic chemical engineering principles to solve the strange disappearance of several of the Nutmega Spice Company's employees. This particular module has long been a favorite with students across the nation. 4. Solved Problems A number of solved problems are presented along with problem solving heuristics. Problem solving strategies and additional worked example problems and are available in the Thoughts on Problem Solving section of the CD-ROM. The Ten Types of Homework Problems section contains two worked examples for each of the ten homework problem types. These examples are based on the material from Chapter 4, and they provide useful information on how one can attack homework problems. The section on Getting Unstuck is especially helpful. Living Example Problems The example problems that use an ODE solver (e.g., POLYMATH) are referred to as "living example problems," because the students can load the program directly on their own computers in order to study it. Students are then encouraged to "play" with the example's key variables and assumptions. Students can change parameter values, such as the reaction rate constants to learn to deduce trends or predict the behavior of a given reaction system, and gain a better understanding of the concepts being studied. Using the living example problems provides students with the opportunity to practice critical and creative thinking skills as they explore the problem and ask "what if...?" questions. Professional Reference Shelf This section of the CD-ROM contains: (1) material that is important to the practicing engineer, although it is typically not included in the majority of chemical reaction engineering courses. A short synopsis of each of the following topics is given at the appropriate point in the text. These sections are: i. ii. iii. iv. v. vi. vii. viii. ix. x. xi. xii. xiii.
Photographs of real reactors Recycle reactors Weighted least squares Experimental planning Laboratory reactors Inhibition and cofactors in enzymatic reactors Bifurcation analysis Wet and dry etching of semiconductors Catalytic monoliths Wire gauze reactors Trickle bed reactors Fluidized bed reactors CVD boat reactors (2) material that gives a more detailed explanation of derivations that were abbreviated in the text. The intermediate steps to these derivations are given on the CD-ROM: a) First order reaction in a semibatch reactor b) Temperature-conversion relationship for an adiabatic reactor c) Aris-Taylor dispersion Additional Homework Problems New problems were developed for this edition that provide a greater opportunity to use today's
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Components
computing power to solve realistic problems. In addition, parts of problems were designed to promote and develop critical thinking skills. Many instructors alternate the homework problems they assign from year to year, ususally taken from a limited number of problems at the end of each chapter. Consequently, some of the more traditional, yet excellent problems of previous editions were placed on the CD-ROM and they can serve as practice problems along with those unassigned problems in the text. Table CDI-1 gives the resources available in each chapter. TABLE CDI-1 Chapter: Learning Resources Summary Notes
1
Web Modules
2
3
4
7
8
9
10 11 12 13 14
Solved Problems Professional Reference Shelf Additional Homework Problems
6
Interactive Computer Modules Living Example Problems
5
In addition to the components listed at the end of each chapter the following components are also included on the CD-ROM Software Toolbox Instructions on how to use the different software packages to solve examples are described for: POLYMATH MatLab ASPEN PLUS All living example problems on the CD are in both POLYMATH and MatLab Format. The POLYMATH program can either be loaded to a computer and executed directly from the CD-ROM. The POLYMATH examples may also be loaded on a computer or run directly from the CD-ROM. In order to execute MatLab examples, MatLab must be available on a server with a site license or the student version of MatLab must be purchased. Similarly, in regard to ASPEN, the CD shows an example of how to use ASPEN to solve chemical reaction engineering problems, however, a site license must be available to actually use ASPEN to solve the homework problems. Representative Syllabi for both 3 and 4 Credit Courses The syllabi give a sample pace at which the courses could be taught as well as suggested homework problems. Virtual Reality Module This module provides an opportunity to move inside a catalyst pellet to observe surface reactions and coking. This module also allows students to navigate through a catalyst pore and see the catalytic steps of diffusion, adsorption, surface reaction, and coking occurring on a catalyst pellet.
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Components
Credits Special recognition to the students who contributed so much to the CD-ROM. In particular, Dieter Schweiss, Anuj Hasija, and Susan Fugett. In addition, Gavin Sy, Scott Conaway, Tim Mashue, and Ellyne Buckingham also worked on the CD-ROM. Clicking on the topic you wish to view will bring up the following information: Learning Resources A. Summary Notes 1. Click on this hot button and a listing of all the lectures in pairs of two (e.g. Lectures 1 and 2, Lectures 3 and 4, etc.) will appear on the screen. Under each lecture pair will be a short listing of the topics covered in those two lectures at the University of Michigan along with the chapters that the lectures are based upon. 2. Click on particular lecture pairs of interest to view the Summary Notes.
B. Additional Homework Problems 1. Click on the topic you choose under Web Modules (e.g. Ch 6 - Pharmacokinetics of Cobra Bites) and the module will appear on your screen. 2. Click on the Interactive Computer Modules title (e.g. Ch 4 - Mystery Theater) and a description of that modules specific will appear. Next click on (2) the instructions that describe how to install the module on to your computer. Finally, load and run the interactive computer module. C. Living Example Problems If you wish to run the POLYMATH examples you can run them directly. If you wish to run the examples on MatLab you will have to purchase the student edition of MatLab or have MatLab available on the server and use an interface to load and run them on your own computer. All the examples are in the POLYMATH directory "POLYMATH/EXAMPLES." To access them, you can run POLYMATH dorectly or install POLYMATH on your computer. If you want to study the examples which use the ODE solver for example enter 1 when the blue POLYMATH screen appears. Type F9, and F9 again and the list of examples should appear. D. Web Modules Click on the web module of interest (e.g. Wetlands (Ch 4)) to pull up the module. These modules provide supplementary Examples on how the principles of chemical reaction engineering can be applied to nontraditional situations. E. Software Toolbox 1. Click on this hot button and the following menu will appear POLYMATH MatLab ASPEN 2. Click on the hot button of your choice (e.g. POLYMATH) and instructions will appear on the screen describing how to use the software to solve the homework problems. F. Syllabi 1. Click on Syllabi and the menu
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Components
a. 3 credit hour course b. 4 credit hour course will appear on your screen. 2. Click on the Syllabus of your choice and a week by week (lecture by lecture) listing of the topics and chapter pages covered along with assigned homework problems will appear. G. Thoughts on Problem Solving (1) Click on this hot button and the following menu appears Closed-Ended Problems (CEP) Open-Ended Problems (OEP) Ten Types of Home Problems (10 types) Strategies for Problem Solving H. Credits Click on this hot button (on the HOME Screen) to learn about the people who helped develop the web page and the CD for this text. To find out about the ways to use this CD, go to the Usage section of this CD.
Welcome Navigation Components Usage Begin: Chapter 1
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Navigation
Welcome Navigation Components Usage
Usage How to Use the CD-ROM The primary purpose of the CD-ROM is to serve as an enrichment resource. The objectives are fourfold: (1) To provide the option/opportunity for further study or clarification on a particular concept or topic through Lecture Notes, additional examples, interactive computing modules and web modules, (2) To provide the opportunity to practice critical thinking skills, creative thinking skills, and problem solving skills through the use of "What if" questions and "living example problems," (3) To provide additional technical material for the professional bookshelf, (4) To provide other tutorial information, such as additional homework problems, thoughts on problem solving, how to use computational software in chemical reaction engineering, and a representative course structure. There are a number of ways one can use the CD in conjunction with the text. The general guideline is that the CD provides enrichment resources for the reader. Pathways on how to use the materials to learn chemical reaction engineering are shown in Figure I-1 and I-2.
The keys to the CRE learning flow sheet are: Squares = Primary Resources Circles/Ovals = Enrichment Resources I. University Student
Figure CDI-1 "A" Student Pathway to Integral Class, Text, CD.
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Navigation
II. Practicing Engineer
Figure CDI-2 "A" Problem Solving Pathway to Integrate text, CD.
III. Instructor
Figure CDI-3 Resources for Instructors
The author recommends that instructors use the living example problems before assigning home problems, but they may be by-passed if time is not available. This is, of course, true for all of the enrichment resources. Please note however, that class testing had shown that the enrichment resources not only aid students in learning the material, but they also motivate students by the novel use of CRE principles. Possible Implementation Strategies I. Learning Resources A. Lecture Notes: This material could be reviewed before reading the chapter get an overview of the material. B. Interactive Computer Modules - (ICM): Each module requires approximately 30 minutes to complete. If a module is not assigned or required, the student could quickly go on through the Review of Fundamentals Section to get an overview or to review (ca. 10 min.). The complete modules could be used by the student as a self test to check their level of understanding. A number of schools assign one either every week or every other week. C. Web Modules: This material can be used to motivate students by showing them the wide range of CRE applications of or as a basis for special projects or open-ended problems. The general problem solving algorithm could be one of the first modules to be reviewed.
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Navigation
D. Solved Examples: After reading the material in the chapters and studying the example problems, students have the opportunity to see one or two more solved problems before embarking on solutions of the homework problems. II. Living Example Problems These examples are meant to be used in conjunction with the second problem of every chapter, beginning with Chapter 4 (i.e. P4-2). Typically one might assign a living example problem (e.g. P6-2) as one of the first problems assigned in a chapter to get students familiar and comfortable with the material. III. Professional Reference Shelf This material is important to the practicing engineer, but is not included in the majority of undergraduate or graduate courses in chemical reaction engineering. Consequently, instructors my pick and choose from this material along the lines of special topics. Material from Chapters 5, 8, 9 and 12 are used in the graduate course at the University of Michigan (i.e. Experimental Planning, Bifurcation Analysis, Control of Chemical Reactors, and the K. L. Model of Fluidized Beds, respectively). IV. The Web (http://www.engin.umich.edu/~cre) The web will be used to update the CD-ROM and text material, provide new examples and more solved problems, and correct of typographical errors from the first printing of the 3rd Edition.
Welcome Navigation Components Usage Begin: Chapter 1
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Elements of Chemical Reaction Engineering
The following Interactive Computer Modules (ICMs) are contained on the Elements of Chemical Reaction Engineering CD-ROM: Kinetics Challenge 1 -- Quiz Show Introduction to Kinetics Learning Resource for Chapter One Staging -- Reactor Sequencing Optimization Game Learning Resource for Chapter Two Kinetics Challenge 2 -- Quiz Show Stoichiometry and Rate Laws Learning Resource for Chapter Three Murder Mystery CSTR Volume Algorithm Learning Resource for Chapter Four Tic Tac Isothermal Reactor Design: Ergun, Arrhenius, and Van't Hoff Equations Learning Resource for Chapter Four Ecology A Wetlands Problem Collection and Analysis of Rate Date: Ecological Engineering Learning Resource for Chapter Five Heat Effects 1 Basketball Challenge Mole and Energy Balances in a CSTR Learning Resource for Chapter Eight Heat Effects 2 Effect of Parameter Variation on a PFR Mole and Energy Balances in a PFR Learning Resource for Chapter Eight Heterogeneous Catalysis Learning Resource for Chapter Ten Some users have experienced problems, trying to run the Interactive Computer Modules directly from the ICM directory on the CD-ROM. If you have trouble with being able to run the Interactive Computer Modules from the CD-ROM, then try installing them on file:///H:/htmlmain/interac.htm[05/12/2011 16:54:35]
Elements of Chemical Reaction Engineering
your hard drive. Instructions for installing the ICMs and for using the ICMs are available.
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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CRE -- Appendices
The Appendices are in PDF format. You must have Adobe Acrobat Reader installed to access these files. You will also need the Adobe Acrobat Reader Plug-in to read these files from your browser. (See the CDROM Introduction for more information.) Appendix D: Measurement of Slopes on Semilog Paper Appendix E: Software Packages Appendix H: Open-Ended Problems Appendix J: Use of Computational Chemistry Software Packages
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Polymath
Polymath Polymath is a DOS-based program that can help you solve differential equations, analyze rate data (with non-linear regression, etc.), and more. The program is fairly straight-forward, but you will want to read through the Polymath Manual before you start. Then we recommend that you take the Polymath Short Course to start you on your way.
Polymath-Related Files Polymath Program Files Polymath has been included on your CD-ROM in a directory called Polymat4. You can run Polymath directly from your CD-ROM, or you can install Polymath on your hard drive.
Polymath Examples (aka, Living Example Problems) You may have noticed that certain chapters have links to Living Example Problems. The example problems are actually in the text for the 3rd edition of Elements of Chemical Reaction Engineering, not on the CD-ROM. The chapter links for these examples direct you to the Polymath code for these problems. The Polymath code for the Living Example Problems from Chapters 2-10 and Chapters 13-14 is in the Html\Toolbox\Polymath\Examples directory on your CD-ROM. Each chapter is represented by a folder named Ch#, which is short for chapter number, of course. The Polymath code for these examples has been included on the CD-ROM for your convenience, so you won't have to waste time duplicating the examples from the text. (See the section on accessing the example problems or the Polymath Short Course for more information.) Once you load up an example, you are encouraged to "play around with it" by modifying the values of constants, varying key parameters, etc. In this way, you can get a feel for how modifying different variables will affect your results.
Using Polymath Polymath is easy to use -- once you know how! See the section on Using Polymath for more information, and don't forget to take a look at the Polymath Manual for instructions on how to run and generally use Polymath:
On-Line Information Polymath 4.0.2 Manual in PDF format. Polymath 4.1 Manual in PDF format.
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Polymath
NOTE: You will need the Adobe Acrobat Reader Plug-in to read this file. (See the CD-ROM Introduction for more information.)
Why Two Versions of Polymath? This is the second printing of the text and the CD. Polymath 4.1 has better printing features than Polymath 4.0.2, but it was not available for the first printing of the CD. We could have replaced Polymath 4.0.2 with Polymath 4.1, but we decided to make both versions available instead.
Polymath Short Course You can also take a look at the Polymath Short Course for a quick-and-dirty introduction to Polymath. You won't learn everything about Polymath, but you will learn enough to get you started, so you can "play around" with the Living Example Problems. Polymath Main | Using Polymath | Installing Polymath | Short Course
References Polymath was created by Mordechai Shacham and Michael B. Cutlip. They make use of Polymath in their own text, Problem Solving in Chemical Engineering with Numerical Methods, also from Prentice-Hall. These pages on Polymath were created by Dieter Andrew Schweiss. Many thanks to Tim Hubbard and Jessica Hamman for proof-reading and error-testing them.
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index.htm
MATLAB Information by Susan Fugett, D.A. Schweiss, and Mayur Valanju
Ordering MATLAB In order to use the MATLAB programs included on this CD-ROM, you must have your own copy of MATLAB. The latest version, MATLAB 5, is available from: The MathWorks, Inc. University Sales Department 24 Prime Park Way Natick, MA 01760-1500. Phone: (508) 653-1415 Fax: (508) 653-2997 Email: [email protected] Web: http://www.mathworks.com A Student Edition is also available.
On the CD-ROM Appendix E Appendix E contains detailed instructions for using MATLAB to solve the problems from the text. It is included in the Appendices section of the CD-ROM as an Adobe Acrobat Reader file (PDF format, see the CD-ROM Introduction for more information). It is also available in Word 6 format (for the PC) in the Html\Toolbox\Matlab\Word directory. Even if you are an experienced MATLAB user, we encourage you to read Appendix E to learn how to use the m-files in MATLAB. Otherwise, you may have difficulty using them.
M-Files
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Included on this CD-ROM are the m-files for all the example problems that were solved in the text using POLYMATH. For PC users, these files can be found in the directory Html\Toolbox\Matlab\m-files. These files may be copied onto the hard drive of your computer or used directly from the CD-ROM. Mac users will be able to open these files, but they may need to edit the contents slightly.
MATLAB Notebook, M-book The Microsoft Word 6.0 Notebook files for each example problem have also been included on the CD-ROM for users who wish to use this utility. These files may be found in directory Html\Toolbox\Matlab\Word on the CD-ROM. The Notebook option provides an interface with the Microsoft Word 6.0 program, the M-book. Please note, however, that the M-book files are not included on the CD-ROM, but are included with the MATLAB software. The Notebook interface allows you to run MATLAB within Word, enabling you to fully explain and document your MATLAB operations. By typing Control + Enter at the end of a line of text, the user instructs MATLAB to perform the commands written on that line. Input into MATLAB is then changed to a different font from the text and appears green on the screen. The output from MATLAB is in another font which is blue. Plots generated in MATLAB are also added to the Word file using this interface. The M-book function is a convenient tool to prepare a detailed presentation of your MATLAB work.
Using the M-Files See the page on Using and Modifying the M-Files for more information.
Remember! We included Appendix E on the CD-ROM, because it contains detailed instructions for using MATLAB with our m-files to solve the problems from the text. It is in the Appendices section of the CD-ROM. If you plan to use MATLAB to solve these problems, then read Appendix E first!
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CRE -- Chapter One-Objectives
1
Mole Balances After completing Chapter 1 of the text and associated CD-ROM material the reader will be able to: Define the rate of chemical reaction. Apply the mole balance equations to a batch reactor, CSTR, PFR, and PBR. Describe two industrial reaction engineering systems. Describe photos of real reactors. Describe how to surf the CD-ROM attached with this text.
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CRE -- Chapter One
1
Learning Resources 1. Summary Notes for Lectures 1 and 2 2. Web Modules A. Problem Solving Algorithm for Closed-Ended Problems B. Hints for Getting Unstuck on a Problem 3. Interactive Computer Modules A. Quiz Show I 4. Solved Problems A. CDP1-AB Batch Reactor Calculations: A Hint of Things to Come Professional Reference Shelf 1. Photographs of Real Reactors Additional Homework Problems CDP1-A Calculate the time to consume 99% and 80% of species A in a constant-volume batch reactor for a first order and for a second order reaction, respectively. Solution Included CDP1-B Derive the differential mole balance equation for a foam reactor.
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Elements of Chemical Reaction Engineering
How to Create and Simulate Chemical Reaction Models with: Polymath Two versions of Polymath have been included on your CD-ROM, 4.0.2 and 4.1. You can run version 4.0.2 directly from the CD, so that you examine and modify the Living Example Problems specific to each Chapter. You can also install it on your computer. To use version 4.1, you will have to install it on your computer. Polymath 4.1 has better printing capabilities than Polymath 4.0.2. MATLAB MATLAB m-files have been included on your CD-ROM, but you will have to purchase your own copy of MATLAB, or MATLAB must be available on your school's computers, to be able to use them. Aspen Plus Read these pages to learn how to use Aspen Plus to design chemical engineering reaction systems. You will have to purchase (or acquire a site license for) your own copy of the Aspen Plus software, since it is not included on this CD.
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Elements of Chemical Reaction Engineering
The Thoughts on Problem Solving area of this CD-ROM offers students step-by-step instruction for the purpose of further developing problem solving skills.
Closed-Ended Problems - these single answer homework problems include two example problems along with techniques for getting unstuck when stopped along the solution path.
Open-Ended Problems - solution heuristic to problems that may be ill-posed, have no solution as posed, or allow for the possibility of multiple solutions.
Ten Types of Home Problems - describes how different types of home problems can be used to improve critical and creative thinking skills.
Strategies for Creative Problem Solving - the award winning book on developing creative problem solving skills
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Elements of Chemical Reaction Engineering, Updates
3rd Edition Updates & FAQs This area contains updates to the 3rd Edition of Elements of Chemical Reaction Engineering as things like typographical errors, etc. are found. This information is current, as of just prior to the 2nd printing of the text and CD-ROM. You are encouraged to visit the Chemical Reaction Engineering Web Site every few months for new updates as more typos are found, or as new problems, activities, etc. are added to the web site. This is also the location our reaction engineering Frequently Asked Questions (FAQs) page.
Links Typos in the First Printing Which Printing Do I Have? Frequently Asked Questions (FAQs)
On the CD Updates to the 3rd Edition can be viewed in PDF format. NOTE:You must have Adobe Acrobat Reader installed to access PDF files. You will also need the Adobe Acrobat Reader Plug-in to read these files from your browser. (See the CDROM Introduction for more information.)
On the CRE Web Site CD-ROM users are encouraged to check the Updates Section of the Chemical Reaction Engineering Web Site every few months for new material.
© 1999 Prentice-Hall PTR Prentice Hall, Inc.
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Elements of Chemical Reaction Engineering, Updates
ISBN 0-13-531708-8 Legal Statement
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Elements of Chemical Reaction Engineering
Representative Syllabus for a 3 Credit Hour Course from the University of Illinois, ChE 381, Fall 1998 Professor Richard Braatz
Representative Syllabus for a 4 Credit Hour Course from the University of Michigan, ChE 344, Winter 1998 Professor H. Scott Fogler
Representative Syllabus for a 4 Credit Hour Course from the University of Michigan, ChE 344, Winter 1999 Professor H. Scott Fogler
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Elements of Chemical Reaction Engineering
1. Welcome 2. Navigation 3. Components 4. Usage 5. Downloading Software Interactive Computer Modules Polymath, MATLAB, Aspen 6. About this CD
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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CRE -- Chapter Two-Objectives
2
Conversion and Reactor Sizing After completing Chapter 2 of the text and associated CD-ROM material the reader will be able to: Define conversion and space time. Write the mole balances in terms of conversion for a batch reactor, CSTR, PFR, and PBR. Size reactors either alone or in series once given the rate of reaction, -r A, as a function of conversion, X. Write relationship between the relative rates of reaction.
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CRE -- Chapter Two
2
Learning Resources 1. Summary Notes for Lectures 1 and 2 3. Interactive Computer Modules A. Reactor Staging 4. Solved Problems A. CD P2-AB More CSTR and PFR Calculations -- No Memorization Additional Homework Problems CDP2-AB Use Levenspiel plots to calculate PFR and CSTR reactor volumes given -r A = f(X). Solution Included CDP2-BA An ethical dilemma as to how to determine the reactor size in a competitor's chemical plant. CDP2-CA Use Levenspiel plots to calculate PFR and CSTR volumes. CDP2-DA Use Levenspiel plots to calculate CSTR and PFR volumes for the reaction A+B
C
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CRE -- Chapter Three-Objectives
3
Rate Law and Stoichiometry After completing Chapter 3 of the text and associated CD-ROM material the reader will be able to: Write a rate law and define reaction order and activation energy. Set up a stoichiometric table for both batch and flow systems and express concentration as a function or conversion. Calculate the equilibrium conversion for both gas and liquid phase reactions. Write the combined mole balance and rate law in measures other than conversion. Set up a stoichiometric table for reactions with phase change.
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CRE -- Chapter Three
3
Learning Resources 1. Summary Notes for Lectures 3 and 4 Summary Notes for Lectures 5 and 6 3. Interactive Computer Module A. Quiz Show II 4. Solved Problems A. CD P3-AB Activation Energy for a Beetle Pushing a Ball of Dung B. CD P3-BB Microelectronics Industry and the Stoichiometric Table Additional Homework Problems CDP3-AB Data on the tenebrionid beetle whose body mass is 3.3g shows it can push a 35g ball of dung at 6.5 cm/s at 27 C, 13 cm/s at 37 C and 18 cm/s at 40 C. How fast can it push at 41.5 C (Heinrich, B., The Hot-Blooded Insects. Harvard Press, Cambridge, 1993). Solution Included CDP3-BB Silicon is used in the manufacture of microelectronic devices. Set up a stoichiometric table for the reaction: SiHCl 3 (g) + H2 (g)
Si(s) + HCl(g) + Si x Hg Cl z(g)
Solution Included CDP3-CB The elementary reaction A(g) + B(g) C(g) takes place in a square duct containing liquid B, which evaporates into the gas phase to react with A. CDP3-DB Condensation occurs in the gas phase reaction: C2 H4 (g) + 2Cl 2 (g) CDP3-EB Set up a stoichiometric table for the reaction:
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CH 2 Cl 2 (g,l) + 2HCl(g)
CRE -- Chapter Three
C6 H5 COCH + 2NH 5
C6 H5 ONH2 + NH 2 Cl
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CRE -- Chapter Four-Objectives
4
Isothermal Reactor Design After completing Chapter 4 of the text and associated CD ROM material the reader will be able to: Describe the algorithm that allows the reader to solve chemical reaction engineering problems through logic rather than memorization. Size batch reactors, semibatch reactors, CSTRs, PFRs, and PBRs for isothermal operation given the rate law and feed conditions. Discuss solutions to problems taken from the California Professional Engineers Registration Examination. Account for the effects of pressure drop on conversion in packed bed tubular reactors and in packed bed spherical reactors.
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CRE -- Chapter Four
4
Learning Resources 1. Summary Notes for Lectures 3 and 4 Summary Notes for Lectures 5 and 6 Summary Notes for Lectures 7 and 8 Summary Notes for Lectures 9 and 10 2. Web Modules A. Wetlands B. Membrane Reactors C. Reactive Distillation 3. Interactive Computer Modules A. Murder Mystery B. Tic Tac -- A Game of Reaction Engineering Tic-Tac-Toe 4. Solved Problems A. CD P4-AB A Sinister Gentleman Messing with a Batch Reactor B. Solution to a California Registration Exam Problem C. Ten Types of Home Problems: 20 Solved Problems 5. Analogy of CRE Algorithms to a Menu in a Fine French Restaurant 6. Algorithm for Gas Phase Reaction Living Example Problems The following examples can be accessed through the Software Toolbox. 1. Example 4-7 Pressure Drop with Reaction -- Numerical Solution 2. Example 4-8 Dehydrogenation in a Spherical Reactor 3. Example 4-9 Working in Terms of Molar Flow Rate in a PFR 4. Example 4-10 Membrane Reactor 5. Example 4-11 Isothermal Semibatch Reactor with a Second-Order Reaction Professional Reference Shelf
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CRE -- Chapter Four
1. Time to Reach Steady State for a First Order Reaction in a CSTR 2. Recycle Reactors 3. Critiquing Journal Articles Additional Homework Problems CDP4-AB A sinister looking gentleman is interested in producing methyl perchlorate in a batch reactor. The reactor has a strange and unsettling rate law. [2nd Ed. P4-28] Solution Included CDP4-BC Ecological Engineering. A much more complicated version of problem 4-17 uses actual pond (CSTR) sizes and flow rates in modeling the site with CSTRs for the Des Plaines river experimental wetlands site (EW3) in order to degrade atrazine. CDP4-CB The rate of binding ligands to receptors is studied in this application of reaction kinetics to bioengineering. The time to bind 50% of the ligands to the receptros is required. [2nd Ed. P4-34] CDP4-DB A batch reactor is used for the bromination of p-chlorophenyl isopropyl ether. Calculate the batch reaction time. [2nd Ed. P4-29] CDP4-EB California Professional Engineers Exam Problem, in which the reaction B + H2
A
is carried out in a batch reactor. [2nd Ed. P4-15] CD P4-FA The gas-phase reaction A + 2B
2D
has the rate law -r A = 2.5 CA 0.5 CB. Reactor volumes of PFRs and CSTRs are required in this mulitpart problem. [2nd Ed. P4-21] CD P4-GB What type and arrangement of flow reactors should you use for a decomposition reaction with the rate law -r A = k1 CA 0.5 / (1 + k2 CA )? [2nd Ed. P4-14] CD P4-HA Verify that the liquid-phase reaction of 5, 6-benzoquinoline with hydrogen is psuedo-first-order. [2nd Ed. P4-7]
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CRE -- Chapter Four
CD P4-I B The liquid-phase reaction 2A + B
C+D
is carried out in a semibatch reactor. Plot the conversion, volume, and species concentrations as a function of time. Reactive distillation is also considered in part (e). [2nd Ed. P4-27] CD P4-J B The reaction A B is catalyzed by H2 SO 4 . The reaction is carried out in a semibatch reactor, in which A is fed continuously to H 2 SO 4 . Here plots of concentrations as a function of time are required. [2nd Ed. P4-27] CD P4-KB Calculate the overall conversion for a PFR with recycle. [2nd Ed. P4-28] CD P4-LB The overall conversion is required in a packed-bed reactor with recycle. [2nd Ed. P4-22] CD P4-M B A recycle reactor is used for the reaction A+B
C
in which species C is partially condensed. The PFR reactor volume is required for a 50% conversion. [2nd Ed. P4-32] CD P4-NB Radical flow reactors can be used to good advantage for exothermic reactions with large heats of reaction. The radical velocity varies as:
Vary the parameters and plot X as a function of r. [2nd Ed. P4-31] CD P4-OB The growth of a bacterium is to be carried out in excess nutrient. nutrient + cells
more cells + product
The growth rate law is:
CD P4-PB California Registration Examination Problem. Second-order reaction in different CSTR and PFR arrangements. CD P4-QB
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CRE -- Chapter Four
An unremarkable semibatch reactor problem, but it does require assessing which equation to use.
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CRE -- Chapter Five-Objectives
5
Collection and Analysis of Rate Data After completing Chapter 5 of the text and associated CD-ROM material the reader will be able to: Determine the reaction order and specific reaction rate from experimental data obtained from either batch or flow reactors. Describe how to use equalarea differentiation, polynomial fitting, numerical difference formulas and regression to analyze experimental data to determine the rate law. Describe how the methods of half lives, and of initial rate, are used to analyze rate data. Describe two or more types of laboratory reactors used to obtain rate law data along with their advantages and disadvantages. Describe how to plan an experiment.
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CRE -- Chapter Five
5
Learning Resources 1. Summary Notes for Lectures 9 and 10 2. Interactive Computer Module A. Ecology -- A Wetlands Problem 3. Solved Problems A. CD P5-B B Oxygenating Blood B. Example CD 5-1 Integral Method of Analysis of Pressure-Time Data Living Example Problems The following examples can be accessed through the Software Toolbox. 1. Example 5-6 Hydrogenation of Ethylene to Ethane Professional Reference Shelf 1. Weighted Least-Squares Analysis 2. Experimental Planning 3. Laboratory Reactors Additional Homework Problems CDP5-AB The reaction of penicillin G with NH2OH is carried out in a batch reactor. A colorimeter was used to measure the absorbency as a function of time. [1st Ed. P5-10] CDP5-B B
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CRE -- Chapter Five
The kinetics of the deoxygenation of hemoglobin in the blood were studied with the aid of a tubular reactor. [1st Ed. P5-3] Solution Included CDP5-C C The kinetics of the formulation of an important propellant ingredient, triaminoguandine, were studied in a batch reactor where the ammonia concentration was measured as a function of time. [1st Ed. P5-6] CDP5-DB The half-life of one of the pollutants, NO, in automotive exhaust is required. [1st Ed. P5-11] CDP5-EB The kinetics of a gas phase reaction A2 2A were studied in a constant-pressure batch reactor, in which the volume was measured as a function of time. [1st Ed. P5-6]
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CRE -- Chapter Six-Objectives
6
Multiple Reactions After completing Chapter 6 of the text and associated CD-ROM material the reader will be able to: Define different types of selectively and yield. Choose a reaction system that would maximize the selectivity of the desired product given the rate laws for all the reactions occurring in the system. Describe the algorithm used to design reactors with multiple reactions. Size reactors to maximize the selectivity and to determine the species concentrations in a batch reactor, semibatch reactor, CSTR, PFR, and PBR, systems.
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CRE -- Chapter Six
6
Learning Resources 1. Summary Notes for Lectures 11 and 12 Summary Notes for Lectures 13 and 14 2. Web Module A. Cobra Bites 3. Solved Problems A. CDP6-B B All You Wanted to Know About Making Maleic Anhydride and More 4. Clarification: PFR with feed streams along the length of the reactor Living Example Problems The following examples can be accessed through the Software Toolbox. 1. Example 6-6 Hydrodealkylation of Mesitylene in a PFR 2. Example 6-7 Hydrodealkylation of Mesitylene in a CSTR 3. Example 6-8 Calculating Concentrations as a Function of Position for NH 3 Oxidation in a PFR Additional Homework Problems CDP6-AB Suggest a reaction system and conditions to minimize X and Y for the parallel reactions A X, A B, and A Y. [2nd Ed. P9-5] CDP6-B B Rework the maleic anhydride problem, P6-14, for the case when reaction 1 is second order. [2nd Ed. P9-8]
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CRE -- Chapter Six
Solution Included CDP6-C B The reaction sequence A CSTR. [2nd Ed. P9-12]
B, B
C, and B
D is carried out in a batch reactor and in a
CDP6-DB Isobutylene is oxidized to methacrolum, CO, and CO 2. [1st Ed. P9-16] CDP6-EB Given a batch reactor with A Ed. P9-11]
B
D
, calculate the composition after 6.5 hours. [1st
CDP6-F B Chlorination of benzene, CO, monochlorobenzene, and dichlorobenzene in a CSTR. [1st Ed. P9-14] CDP6-GC Determine the number of independent reactions in the oxidation of ammonia. [1st Ed. P9-17] CDP6-HB Oxidation of formaldehyde: HCOOH HCHO + 1/2 O2 HCOOCH 3 2HCHO CDP6-IB Continuation of CDP6-H: HCOOH HCOOH
CO 2 + H2 CO + H2O
CDP6-J B Continuation of CDP6-H and -I: HCOOCH 3
CH 3OH + HCOOH
CDP6-KC Design a reactor for the alkylation of benzene with propylene to maximize the selectivity of isopropylbenzene. [Proc. 2nd Joint China/USA Chem. Eng. Conf. III, 51, (1997).] CDP6-LD Reactions between paraffins and olefins to form highly branched paraffins are carried out in a slurry reactor to increase the octane number in gasoline. [Chem. Eng. Sci. 51, 10, 2053 (1996).]
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CRE -- Chapter Six
CDP6-M A Design a reaction system to maximize the production of alkyl choride. [1st Ed. P9-19] CDP6-NC Design a reaction system to maximize the selectivity of p-xylene from methanol and toluene over a HZSM-8 zeolite catalyst. [2nd Ed. P9-17] CDP6-OB Rework the maleic anhydride problem (P6-14) for the case when reaction 1 is second order. [2nd Ed. P9-8] CDP6-P C The oxidation of propylene to acrolein [Chem. Eng. Sci., 51, 2189 (1996)].
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CRE -- Chapter Seven-Objectives
7
Nonelementary Reaction Kinetics After completing Chapter 7 of the text and associated CD-ROM material the reader will be able to: Discuss the pseudo-steadystate-hypothesis and explain how it can be used to solve reaction engineering problems. Discuss different types of polymerization reactions and rate laws. Describe Michealis-Menton enzyme kinetics and enzyme inhibition. Write material balances on cells, substrates, and products in bioreactors.
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CRE -- Chapter Seven
7
Learning Resources 1. Summary Notes for Lectures 25 and 26 Summary Notes for Lectures 36 and 37 Summary Notes for Lectures 38 and 39 4. Solved Problems A. Hydrogen Bromide Example CD7-1 Deducing the Rate Law Example CD7-2 Deriving the Rate Law from the Reaction Mechanism Living Example Problems The following example can be accessed through the Software Toolbox. 1. Example 7-2 PSSH Applied to Thermal Cracking of Ethane Professional Reference Shelf 1. Enzyme Inhibition A. Competitive Example CD7-3 Derive a Rate Law for Competitive Inhibition B. Uncompetitive C. Non-Competitive Example CD7-4 Derive a Rate Law for Non-Competitive Inhibition Example CD7-5 Match Eadie Plots to the Different Types of Inhibition 2. Multiple Enzymes and Substrate Systems A. Enzyme Regeneration Example CD7-6 Construct a Lineweaver-Burke Plot for Different Oxygen Concentrations
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CRE -- Chapter Seven
B. Enzyme Co-factors Example CD7-7 Derive an Initial Rate Law for Alcohol Dehydrogenates C. Multiple Substrate Systems Example CD7-8 Derive a Rate Law for a Multiple Substrate System Example CD7-9 Calculate the Initial Rate of Formation of Ethanol in the Presence of Propanediol D. Multiple Enzyme Systems 3. Oxidation-Limited Fermentation 4. Fermentation Scale-up Additional Homework Problems CD P7-AB Determine the rate law and mechanism for the reaction: 2GCH 3
2G + CH 2 + H2
[2nd Ed. P7-6A] CD P7-B B Suggest a mechanism for the reaction: I- + OCl -
OI- + Cl -
[2nd Ed. P7-8B ] CDP7-C A Develop a rate law for substrate inhibition of an enzymatic reaction. [2nd Ed. P7-16A] CDP7-DB Use Polymath to analyze an enzymatic reaction. [2nd Ed. P7-19B ] CDP7-EB Redo Problem P7-17 to include chain transfer. [2nd Ed. P7-23B ] CDP7-F B Determine the rate of diffusion of oxygen to cells. [2nd Ed. P12-12B ] CDP7-GB Determine the growth rate of amoeba predatory on a bacteria. [2nd Ed. P12-15C ] file:///H:/html/07chap/html/seven.htm[05/12/2011 16:54:47]
CRE -- Chapter Seven
CDP7-HC Plan the scale-up of an oxygen fermentor. [2nd Ed. P12-16B ] CDP7-IB Assess the effectiveness of bacteria used for denitrification in a batch reactor. [2nd Ed. P1218B ] CDP7-J A Determine rate law parameters for the Monod equation. [2nd Ed. P12-19A]
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CRE -- Chapter Eight-Objectives
8
Steady-State Nonisothermal Reactor Design After completing Chapter 8 of the text and associated CD-ROM material the reader will be able to: Describe the algorithm for CSTRs, PFRs, and PBRs that are not operated isothermally. Size adiabatic and nonadiabatic CSTRs, PFRs, and PBRs. Use reactor staging to obtain high conversions for highly exothermic reversible reactions. Carry out an analysis to determine the Multiple Steady States (MSS) in a CSTR along with the ignition and extinction temperatures. Analyze multiple reactions carried out in CSTRs, PFRs, and PBRs which are not operated isothermally in order to determine the concentrations and temperature as a function of position (PFR/PBR) and operating variables.
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CRE -- Chapter Eight-Objectives Legal Statement
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CRE -- Chapter Eight
8
Learning Resources 1. Summary Notes for Lectures 13 and 14 Summary Notes for Lectures 15 and 16 Summary Notes for Lectures 17 and 18 Summary Notes for Lecture 35A 3. Interactive Computer Modules A. Heat Effects I B. Heat Effects II 4. Solved Problems for Heat Capacities Expressed as Quadratic Functions of A. Example CD 8-1 Temperature B. Example CD 8-2 Second Order Reaction Carried Out Adiabatically in a CSTR 5. PFR/PBR Solution Procedure for a Reversible Gas-Phase Reaction Living Example Problems The following examples can be accessed through the Software Toolbox. 1. Example 8-5 CSTR with a Cooling Coil 2. Example 8-6 Liquid Phase Isomerization of Normal Butene 3. Example 8-7 Production of Acetic Anhydride 4. Example 8-10 Oxidation of SO2 5. Example 8-11 Parallel Reaction in a PFR with Heat Effects 6. Example 8-12 Multiple Reactions in a CSTR
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CRE -- Chapter Eight
Professional Reference Shelf 1. Steady State Bifurcation Analysis A. Fundamentals B. Example CD 8-3 Determine the Parameters That Give Multiple Steady States (MSS) Additional Homework Problems CD P8-AB The exothermic reaction A
2B
is carried out in both a plug-flow reactor and a CSTR with heat exchange. You are requested to plot conversion as a function of reactor length for both adiabatic and nonadiabatic operation, as well as to size a CSTR. [2nd Ed. P8-16] CD P8-B B Use bifurcation theory (Section 8.6.5 on the CD-ROM) to determine the possible regions for multiple steady states for the gas reaction with the rate law:
[2nd Ed. P8-26] CD P8-C B In this problem, bifurcation theory (CD-ROM Section 8.6.5) is used to determine if multiple steady states are possible for each of three types of catalyst. [2nd Ed. P8-27] CD P8-DB In this problem, bifurcation theory (CD-ROM Section 8.6.5) is used to determine the regions of multiple steady states for the autocatalytic reaction: A+B
2B
[2nd Ed. P8-28] CD P8-EC This problem concerns the SO 2 reaction with heat losses. [2nd Ed. P8-33] CD P8-F C This problem concerns the use of interstage cooling in SO 2 oxidation. [2nd Ed. P8-34(a)]
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CRE -- Chapter Eight
CD P8-GB This problem is a continuation of the SO 2 oxidation example problem. Reactor costs are considered in the analysis cooling. [2nd Ed. P8-34(b and c)] CD P8-HB Parallel reactions taking place in a CSTR with heat effects. [1st Ed. P9-21] CD P8-IB This problem concerns multiple steady states for the second-order, reversible, liquid-phase reaction [Old exam problem] CD P8-J B Series reactions take place in a CSTR with heat effects. [1st Ed. P9-23] CD P8-KB A drug intermediate is produced in a batch reactor with heat effects. The reaction sequence is: 2A + B C+A+B
C+D E+D
The desired product is C. CD P8-LB In the multiple steady state for A
B
the phase plane of C A vs. T shows a separatrix. [2nd Ed. P8-22] CD P8-NB A second-order reaction with multiple steady states is carried out in different solvents. CD P8-OC Multiple reactions 2B A C 2A + B are carried out adiabatically in a PFR. CDP8-P B An exothermic 2nd order reversible reaction is carried out in a packed bed reactor.
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CRE -- Chapter Eight
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CRE -- Chapter Nine-Objectives
9
Unsteady-State Nonisothermal Reactor Design After completing Chapter 9 of the text and associated CD-ROM material the reader will be able to: Analyze batch reactors and semibatch not operated isothermally. Analyze the start up of nonisothermal CSTRs. Analyze perturbations in temperature and presence for CSTRs being operated at steady state and describe under what conditions the reactors can be unsafe (safety). Describe the effects of adding a controller to a CSTR. Analyze multiple reactions in batch and semibatch reactors not operated isothermally.
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CRE -- Chapter Nine
9
Learning Resources 1. Summary Notes for Lectures 17 and 18 Summary Notes for Lecture 35B 4. Solved Problems A. Example CD 9-1 Startup of a CSTR B. Example CD 9-2 Falling Off the Steady State C. Example CD 9-3 Proportional Integral (PI) Control
Living Example Problems The following examples can be accessed through the Software Toolbox. 1. Example 9-1 Adiabatic Batch Reactor 2. Example 9-2 Safety in Chemical Plants with Exothermic Reactions 3. Example 9-3 Heat Effects in a Semibatch Reactor 4. Example 9-4 Startup of a CSTR 5. Example 9-5 Falling Off the Steady State 6. Example 9-6 Integral Control of a CSTR 7. Example 9-7 Proportional Integral Control of a CSTR 8. Example 9-8 Multiple Reactions in a Semibatch Reactor
Professional Reference Shelf file:///H:/html/09chap/html/nine.htm[05/12/2011 16:54:49]
CRE -- Chapter Nine
1. Intermediate Steps in the Adiabatic Batch Reactor Derivation 2. Approach to Steady-State Phase-Plane Plots and Trajectories of Concentration versus Temperature 3. Unsteady Operation of Plug Flow Reactors
Additional Homework Problems CD P9-AB The production of propylene glycol (discussed in Examples 8-4, 9-4, 9-5, 9-6, and 9-7) is carried out in a semibatch reactor. [2nd Ed. P8-14] CD P9-B C Reconsider problem P9-14 when a PI controller is added to the coolant stream.
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CRE -- Chapter Ten-Objectives
10
Catalysis and Catalytic Reactors After completing Chapter 10 of the text and associated CD-ROM material the reader will be able to: Define a catalyst, a catalytic mechanism and a rate limit step. Describe the steps in a catalytic mechanism and how one goes about deriving a rate law and a mechanism and rate limiting step consistent with the experimental data. Size isothermal reactors for reactions with LangmuirHinschelwood kinetics. Discuss the different types of catalyst deactivation and the reactor types and describe schemes that can help offset the deactivation. Analyze catalyst decay and conversion for CSTRs and PFRs with temperaturetime trajectories, moving bed reactors, and straight through transport reactors. Describe the steps in Chemical Vapor Deposition(CVD). Analyze moving bed reactors that are not operated isothermally.
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CRE -- Chapter Ten-Objectives
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CRE -- Chapter Ten
10
Learning Resources 1. Summary Notes for Lectures 19 and 20 Summary Notes for Lectures 21 and 22 Summary Notes for Lectures 23 and 24 3. Interactive Computer Module A. Heterogeneous Catalysis 4. Solved Problems A. Example CD10-1 Analysis of Heterogeneous Data [Class Problem, Winter 1997] B. Example CD10-2 Least-Squares to Determine Rate Law Parameters k, KT, and KB (Example 6-2 in 2nd Edition) C. Example CD10-3 Hydrodemethylation of Toluene in a PBR without Pressure Drop [2nd Ed. Example 6-3] D. Example CD10-4 Cracking of Texas Gas-Oil in a STTR [2nd Ed. Example 6-5] Living Example Problems The following examples can be accessed through the Software Toolbox. 1. Example 10-5 Catalyst Decay in a Fluidized Bed Modeled as a CSTR 2. Example 10-6 Catalytic Cracking in a Moving-Bed Reactor 3. Example 10-7 Decay in a Straight Through Transport Reactor Professional Reference Shelf 1. Hydrogen Adsorption A. Molecular Adsorption B. Dissociative Adsorption 2. Catalyst Poisoning in a Constant Volume Batch Reactor 3. Differential Method of Analysis to Determine the Decay Law 4. Etching of Semiconductors A. Dry Etching B. Wet Etching
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CRE -- Chapter Ten
C. Dissolution Catalysis Additional Homework Problems CDP10-AA Suggest a rate law and mechanism for the catalytic oxidation of ethanol over tantalum oxide when the adsorption of ethanol and oxygen takes place on different sites. [2nd Ed. P6-17] CDP10-BB Analyze the data for the vapor-phase esterification of acetic acid over a resin catalyst at 118 C. CDP10-CB Silicon dioxide is grown by CVD according to the reaction SiH2 Cl 2 (g) + 2N2 0(g)
SiO2 (s) + 2N2 (g) + 2HCL(g)
Use the rate data to determine the rate law, reaction mechanism, and rate law parameters. [2nd Ed. P6-13] CDP10-DB The autocatalytic reaction A + B 2B is carried out in a moving-bed reactor. The decay law is firstorder in B. Plot the activity and the concentrations of A and B as a function of catalyst weight. CDP10-EB Determine the rate law and rate law parameters for the wet etching of an aluminum silicate. CDP10-FB Titanium films are used in decorative coatings as well as wear-resistant tools because of their thermal stability and low electrical resistivity. TiN is produced by CVD from a mixture of TiCl 4 and NH 3 TiN. Develop a rate law, a mechanism, and a rate-limiting step, and evaluate the rate law parameters. CDP10-GC The decomposition of cumene is carried out over a LaY zeolite catalyst, and deactiviation is found to occur by coking. Determine the decay law and rate law, and use these to design a STTR. [2nd Ed. P6-27] CDP10-HB The dehydrogenation of ethylbenzene is carried out over a Shell catalyst. From the data provided, find the cost of the catalyst required to produce a specified amount of styrene. [2nd Ed. P6-20] CDP10-I B A second-order reaction over a decaying catalyst takes place in a moving-bed reactor. [Final Exam, Winter 1994] CDP10-J B A first-order reaction A
B + C takes place in a moving-bed reactor.
CDP10-KB For the cracking of normal paraffins (P n ), the rate has been found to increase with increasing temperature up to a carbon number of 15 (i.e., n < 16) and to decrease with increasing temperature for a carbon number greater than 16. [J. Wei, Chem. Eng. Sci., 51, 2995 (1996)]
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CRE -- Chapter Ten
CDP10-LB The formation of CH 4 from CO and H2 is studied in a differential reactor. CDP10-M B The reaction A + B
C + D is carried out in a moving-bed reactor.
CDP10-NA Determine the rate law and mechanism for the reaction A + B
C.
CDP10-OB Determine the rate law from data where the pressures are varied in such a way that the rate is constant. [2nd Ed. P6-18] CDP10-PB Determine the rate law and mechanism for the vapor phase dehydration of ethanol. [2nd Ed. P6-21] CDP10-QA Second order reaction and zero order decay in a batch reactor. CDP10-RB First order decay in a moving bed reactor for the series reaction A
B
C
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CRE -- Chapter Eleven-Objectives
11
External Diffusion Effects on Heterogeneous Reactions After completing Chapter 11 of the text and associated CD-ROM material the reader will be able to: Define the mass transfer coefficient, explain what it is function of and how it is measured or calculated. Analyze PBRs in which mass transfer limits the rate of reaction. Discuss how one goes form a region mass transfer limitation to reaction limitation. Describe how catalyst monoliths and wire gauze reactors are analyzed. Apply the shrinking core model to analyze catalyst regeneration.
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CRE -- Chapter Eleven
11
Learning Resources 1. Summary Notes for Lectures 27 and 28 2. Solved Problems A. Example CD11-1 Calculating Steady State Fluxes B. Example CD11-2 Relating the Fluxes W A, BA, and J A C. Example CD11-3 Diffusion Through a Stagnant Gas Professional Reference Shelf 1. Mass Transfer Limited Reactions on Metallic Surfaces A. Catalyst Monoliths (Catalytic Converter for Autos) B. Wire Gauzes Additional Homework Problems CDP11-AA An isomerization reaction that follows Langmuir-Hinshelwood kinetics is carried out on a monolith catalyst. [2nd Ed. P10-11] CDP11-BB A parameter sensitivity analysis is required for this problem in which an isomerization is carried out over a 20-mesh gauze screen. [2nd Ed. P10-12] CDP11-CC This problem examines the effect on temperature in a catalyst monolith. [2nd Ed. P10-13] CDP11-DB A second-order catalytic reaction is carried out in a catalyst monolith. [2nd Ed. P10-14] CDP11-EC Fracture acidizing is a technique to increase the productivity of oil wells. Here acid is injected at high pressures to fracture the rock and form a channel that extends out from the well bore. As the acid flows through the channel, it etches the sides of the channel to make it larger, and thus less resistant to the flow of oil. Derive equations for the concentration profile of acid and the channel width, each as a function of distance from the well bore. [2nd Ed. P10-15] CDP11-FC The solid-gas reaction of silicon to form SiO is an important process in microelectronics fabrication. The file:///H:/html/11chap/html/eleven.htm[05/12/2011 16:54:51]
CRE -- Chapter Eleven
2
oxidation occurs at the Si-SiO2 interface. Derive an equation for the thickness of the SiO2 layer as a fucntion of time. [2nd Ed. P10-17] CDP11-GB Mass transfer limitations in CVD processing to product material with ferroelectric and pezoelectric properties. [2nd Ed. P10-17] CDP11-HB Calculate multicomponent properties. [2nd Ed. P10-17] CDP11-I B Application of the shrinking core model to FeS2 rock samples in acid mine drainage. [2nd Ed. P10-18]
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CRE -- Chapter Twelve-Objectives
12
Diffusion and Reaction in Porous Catalysts After completing Chapter 12 of the text and associated CD-ROM material the reader will be able to: Define the Thiele modules and the effectiveness factor. Describe the regions of reaction limitations and internal diffusion limitations and the conditions that affect them. Determine which resistance is controlling in a slurry reactor. Analyze trickle bed reactors. Analyze fluidized bed reactors. Describe the operation of a CVD Boat Reactor.
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CRE -- Chapter Twelve
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Learning Resources 1. Summary Notes for Lectures 27 and 28 Summary Notes for Lectures 29 and 30 Professional Reference Shelf 1. Trickle Bed Reactors A. Fundamentals B. Limiting Situations C. Evaluating the Transport Coefficients 2. Fluidized Bed Reactors A. Overview B. Mechanics of Fluidized Beds C. Descriptive Behavior of the Kunii-Levenspiel Bubbling Bed Model D. Mass Transfer in Fluidized Beds E. Reaction in a Fluidized Bed F. Mole Balances on the (1) Bubble, (2) Cloud, and (3) Emulsion G. Solution to the Balance Equations for a First Order Reaction Example CD12-3 Catalytic Oxidation of Ammonia H. Limiting Situations Example CD12-4 Calculating the Resistances Example CD12-5 Effect of Particle Size on Catalyst Weight for a Slow Reaction Example CD12-6 Effect of Catalyst Weight for a Rapid Reaction I. Summary 3. CVD Boat Reactors A. Fundamentals B. Examples Additional Homework Problems CDP12-AB file:///H:/html/12chap/html/twelve.htm[05/12/2011 16:54:52]
CRE -- Chapter Twelve
Determine the catalyst size that gives the highest conversion in a packed bed reactor. CDP12-B D Determine the temperature profiles to achieve a uniform thickness. [2nd Ed. P11-18] CDP12-C B Explain how varying a number of the parameters in the CVD boat reactor will affect the wafer shape. [2nd Ed. P11-19] CDP12-DB Determine the wafer shape in a CVD boat reactor for a series of operating conditions. [2nd Ed. P11-20] CDP12-EC Model the buildup of a silicon wafer on parallel sheets. [2nd Ed. P11-21] CDP12-F C Rework CVD boat reactor accounting for the reaction: SiH 2 + H2 SiH 4 [2nd Ed. P11-22] CDP12-GB Hydrogenation of an unsaturated organic is carried out in a trickle bed reactor. [2nd Ed. P127] CDP12-HH The oxidation of ethanol is carried out in a trickle bed reactor. [2nd Ed. P12-9] CDP12-J B The hydrogenation of aromatics in a napthenic lube oil distillate takes place in a trickle bed reactor. CDP12-KC Compare the models discussed in the article with the Kunii-Levenspiel model--e.g. list major difference, the advantages, disadvantages, and strong and weak points of each model. CDP12-LC Reconsider example problem 12-2, using the correlations of C. Chavarie and J. R. Grace [Ind. Engrg Chem. Fundamentals, Vol. 14. No. 2 pp. 75-79 (1975)] CDP12-M A If the temperature were increased from 273°K to 546°K, the minimum fluidization velocity of the gas would approximately:
(a) Increase by a factor of a) 1.4, b) 2.8, c) 2.0. (b) Decrease by a factor of a) 1.4, b) 2.8, c) 2.0. file:///H:/html/12chap/html/twelve.htm[05/12/2011 16:54:52]
CRE -- Chapter Twelve
(c) Remain the same. (d) Cannot be calculated from the information given. (e) None of the above. CDP12-NA At the very top of the column, what is the rate of reaction in the emulsion phase in [gmole/(cm3 of
bubble)(s)] as predicted by the Kunii-Levenspiel model? CDP12-OB When a fluidized column is operated at a superficial velocity of 10 cm/sec which is 5 times the minimum fluidization velocity, it was found the bubble size at the mid-point in the column was 10 cm and that bubbles occupied exactly 10.6% of the column. CDP12-P B The irreversible gas phase isomerization reaction
CDP12-QB When operated at twice the minimum fluidization velocity 86.5% conversion is realized in a of 0.5. It is proposed to increase the pressure from 1 fluidized bed reactor which has a atm. to 4.0 atm. and decrease the temperature from 427°C to 227°C. CDP12-R B The irreversible catalytic isomerization CDP12-S B The production of methane from carbon monoxide and hydrogen over a 1/8´ nickel catalyst is carried out at 500°F. CDP12-TB Using tracer studies to characterize fluidized bed reactors. CDP12-UB The hydrogenation of aromatics in a napthenic lube oil distillate takes place in a trickle bed reactor. CDP12C-1 Using values of the minimum fluidization velocity provided by Kurian and Raja Rao (Ind. J. of Tech., 8, 275 (1970)).
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CRE -- Chapter Twelve
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CRE -- Chapter Thirteen-Objectives
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Distributions of Residence Times for Chemical Reactors After completing Chapter 13 of the text and associated CD-ROM material the reader will be able to: Define a residence time distribution RTD [E(t), F(t)] and the mean residence time. Determine E(t) form tracer data. Write the RTD functions (E(t), F(t), I(t)) for ideal CSTRs, PFRs, and laminar flow reactors. Predict conversions from RTD data using the segregation and maximum mixedness models. Predict effluent concentrations for multiple reactions using the segregation and maximum mixedness models.
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CRE -- Chapter Thirteen
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Learning Resources 1. Summary Notes for Lectures 31 and 32 2. Web Modules A. The Attainable Region Analysis 4. Solved Problems A. Example CD13-1 Calculate the exit concentrations for the series reaction
using the segregation model and the maximum mixedness model. B. Example CD13-2 Determination of the effect of variance on the exit concentrations for the series reaction
Living Example Problems The following examples can be accessed through the Software Toolbox. 1. Example 13-8 Using Software to Make Maximum Mixedness Model Calculations 2. Example 13-9 RTD and Complex Reactions Reactor with Asymmetric RTD Segregation Model, Maximum Mixedness Model Reactor with Bimodal RTD Maximum Mixedness Model, Segregation Model Professional Reference Shelf file:///H:/html/13chap/html/thirteen.htm[05/12/2011 16:54:53]
CRE -- Chapter Thirteen
1. Attainable Region Analysis 2. Comparison of Conversion for Segregation and Maximum Mixedness Models for Reaction Orders Between 0 and 1 Additional Homework Problems CD P13-AC After showing that E(t) for two CSTRs in series having different values is
you are aksed to make a number of calculations. [2nd Ed. P13-11] CD P13-B B Determine E(t) and from data taken, form a pluse test in which the pulse is not perfect and the inlet concentration varies with time. [2nd Ed. P13-15] CD P13-C B Derive the E(t) curve for a Bingham plastic flowing in a cylindrical tube. [2nd Ed. P13-16]. CD P13-DB The order of a CSTR and PFR in series is investigated for a third-order reaction. [2nd Ed. P13-10] CD P13-EB Review the Muphree pilot plant data when a second-order reaction occurs in the reactor. [1st Ed. P13-15] CD P13-F A Gasoline shortages in the United States have produced long lines of motorists at service stations. CD P13-GB Vary the parameters to learn their effects on conversion. CD P13-HB The reactions described in Problem 6-16 are to be carried out in the reactor whose RTD is described in Example 13-7.
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CRE -- Chapter Thirteen
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CRE -- Chapter Fourteen-Objectives
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Models for Nonideal Reactors After completing Chapter 14 of the text and associated CD-ROM material the reader will be able to: Describe the tanks-in-series and dispersion one parameter models. Describe how to obtain the mean residence time and variance to calculate the number of tanks-in-series and the Peclet number. Calculate the conversion for a first order reaction taking place in a tubular reactor with dispersion Describe how to use combinations of ideal rectors to model a real reactor and how to use tracer data to determine the model parameters.
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CRE -- Chapter Fourteen
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Learning Resources 1. Summary Notes for Lectures 33 and 34 4. Solved Problems A. Example CD14-1 Two CSTRs with Interchange Professional Reference Shelf 1. Derivation of Equation for Taylor-Aris Dispersion 2. Real Reactor Modeled as an Ideal CSTR with Exchange Volume Additional Homework Problems CDP14-AC A real reactor is modeled as:
[2nd Ed. P14-5] CD P14-B B A batch reactor is modeled as:
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CRE -- Chapter Fourteen
[2nd Ed. P14-13] CDP14-C C Develop a model for a real reactor with RTD data obtained from a step input. [2nd Ed. P1410] CDP14-DB Calculate Da and X from sloppy tracer data. [2nd Ed. P14-6] CDP14-EB Use RTD data from Oak Ridge National Laboratory to calculate the conversion from the tanks-in-series and dispersion models. [2nd Ed. P14-7]
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CD-ROM
D
CD.1 Using Semilog Plots in Rate Data Analysis Semilog graph paper is used when dealing with either exponential growth or decay, such as y 5 bemx
(CD-1)
For the first-order elementary reaction A → products which is carried out at constant volume, the rate of the disappearance of A is given by 2dCA --------------- 5 kCA dt
(CD-2)
When t 5 0: CA 5 CA0 , where the units of CA are g mol/dm3, t is expressed in minutes, and k is expressed in reciprocal minutes. Integrating the rate equation, we obtain CA - 5 2kt ln -------CA0
(CD-3)
We wish to determine the specific reaction rate constant, k. A plot of ln CA versus t should produce a straight line whose slope is 2k. We may eliminate the calculation of the log of each concentration data point by plotting our data on semilog graph paper. The points in Table CDD-1 are plotted on the semilog graph shown in Figure CDD-1. 1
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t (min) CA (gmol/dm3 )
0
2
4
8
14
2.0
1.64
1.38
0.95
0.60
10 8
CA (g mol/liter)
6 5 4 3 2 ∆y
1.0 0.8 0.6 0.5 0.4
∆x
0.3 0.2
0.1 0
2
4
6
8
10
12
14
16
t (min) Figure CDD-1
Fogler/PrenHall/CDD.1 S/S
Algebraic Method Draw the best straight line through your data points. Choose two points on this line, t1 and t2, and the corresponding concentrations CA1 and CA2 at these times: CA1 - 5 2kt1 ln -------CA0
CA2 - 5 2kt2 ln -------CA0
ln CA2 2 ln CA1 5 2k (t2 2 t1 )
(CDD-4)
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Rearranging yields ln CA2 2 ln CA1 ln ( CA1 / CA2 ) - 5 -----------------------------k 5 2----------------------------------t2 2 t1 t2 2 t1
(CDD-5)
When t 5 8, CA 5 1.05; when t 5 12, CA 5 0.75. Substituting into Equation (CDD-5) gives us 0.336 ln ( 1.05 / 0.75 ) k 5 ---------------------------------- 5 -------------4 min ( 12 2 8 ) min 5 0.084 min21 Graphical Technique In the preceding example, we had CA - 5 2kt ln -------CA0
(CDD-6)
Dividing by 2.3, we convert to log base 10: ln ( CA / CA0 ) 2kt ---------------------------- 5 log ( CA / CA0 ) 5 --------2.3 2.3 The slope of a plot of log CA versus time should be a straight line with slope 2k/2.3. Referring to Figure CDD-1, we draw a right triangle with the acute angles located at points CA 5 1.6, t 5 2.8 and CA 5 0.7, t 5 12.8. Next, the distances x and y are measured with a ruler. These measured lengths in y and x are 1.35 and 4.65 cm, respectively: 1 cycle Dy 5 21.35 cm 3 ----------------- 5 20.35 cycle 3.9 cm 14 min Dx 5 4.65 cm 3 ----------------- 5 9.7 min 6.7 cm 20.35 slope 5 --------------- 5 20.0361 9.7 k 5 22.3 (slope) 5 22.3 ( 20.0361 ) min21 5 0.083 min21 A modification of the algebraic method is possible by drawing a line on semilog paper so that the dependent variable changes by a factor of 10. From Equation (CDD-5) in the form ln ( CA1 / CA2 ) k 5 -----------------------------t2 2 t1 2.3 log ( CA1 / CA2 ) 5 ------------------------------------------t2 2 t1
(CDD-7)
choose the points (CA1 , t1 ) and (CA2 , t2 ) so that CA2 5 0.1CA1 : 2.3 2.3 log 10 k 5 ------------------------ 5 -------------t2 2 t1 t2 2 t1 This modification is referred to as the decade method.
(CDD-8)
MATLAB
E
Susan A. Fugett MATLAB ® version 5 is a very powerful tool useful for many kinds of mathematical tasks. For the purposes of this text, however, MATLAB 5 will be used only to solve four types of problems: polynomial curve fitting, system of algebraic equations, system of ordinary differential equations, and nonlinear regression. This appendix serves as a quick guide to solving such problems. The solutions were all prepared using the Student Edition of MATLAB 5. Please note that the MATLAB 5 software must be purchased independently of the CD-ROM accompanying this book.
E.1 A Quick Tour When MATLAB is opened, the command window of the MATLAB Student Edition appears: To get started, type one of these commands: helpwin, helpdesk, or demo EDU» You may then type commands at the EDU» prompt. Throughout this appendix, different fonts are used to represent MATLAB input and output, and italics are used to explain function arguments.
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E.1.1 MATLAB’s Method
MATLAB can range from acting as a calculator to performing complex matrix operations and more. All of MATLAB’s operations are performed as matrix operations, and every variable is stored in MATLAB’s memory as a matrix even if it is only 1x1 in size. Therefore, all MATLAB input and output will be in matrix form. E.1.2 Punctuation
MATLAB is case sensitive and recognizes the difference between capital and lowercase letters. Therefore, it is possible to work with the variables “X” and “x” at the same time. The semicolon and period play very important roles in performing calculations using MATLAB. A semicolon placed at the end of a command line will suppress restatement of that output. MATLAB will still perform the command, but will not display the answer. For example, type beta=1+4 and MATLAB will display the answer beta =5, but if you then type alpha = 30/2;, MATLAB will not tell you the answer. To see the value of a variable, simply type the name of the variable, alpha and MATLAB will display its value, alpha =15. The command who can also be used to view a list of current variables: Your variables are: alpha beta The period is used when element-by-element matrix multiplication is performed. To perform standard matrix multiplication of two matrices, say “A” and “B,” type A*B. To multiply every element of matrix “A” by 2 type A*2. However, to multiply every element of “A” with the corresponding element of “B,” one must type A.*B. This element-by-element matrix multiplication will be used for the purposes of this text. To learn more about MATLAB, type demo at the command prompt; to see a demo about matrix manipulations, type matmanip. E.1.3 Help
MATLAB has an extensive on-line help program that can be accessed through the help command, the lookfor command, and by the helpwin command (or by choosing help from the menu bar). By typing “help topic,” for example help log, MATLAB will give an explanation of the topic. LOG Natural logarithm. LOG(X) is the natural logarithm of the elements of X. Complex results are produced if X is not positive. See also LOG2, LOG10, EXP, LOGM. It is likely that in many instances you will not know the exact name of the topic for which you need help. (By typing helpwin, you will open the help window, which houses a list of help topics.)
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The lookfor command can be used to search through the help topics for a key word. For example you could type lookfor logarithm and receive the following list: LOGSPACE LOG LOG10 LOG2 BETALN GAMMALN LOGM
Logarithmically spaced vector. Natural logarithm. Common (base 10) logarithm. Base 2 logarithm and dissect floating point number. Logarithm of beta function. Logarithm of gamma function. Matrix logarithm.
from which the search can be narrowed. Please note that all built-in MATLAB commands are lower case, although in help they are displayed in uppercase letters. It is strongly recommended that students take time to explore the demo before attempting to solve problems using MATLAB. E.1.4 M-files
Many of the commands in MATLAB are really a combination of commands and manipulations that are stored in an m-file. Users can also write their own m-files with their own commands and data. The m-files are simply text files that have an “m” extension (e.g., example1.m). The name of the file can then be called upon later to execute the commands in the m-file as though they were being entered line-by-line by the user at the EDU» prompt. The m-file saves time by relieving the user of the need to type lines of commands over and over and by enabling him or her to change values of one or more variables easily and repeatedly.
E.2 Examples Examples of each of the four types of problems listed above will now be explained. Please refer to the examples in the book that were solved using POLYMATH. It may be wise to type the command clear before starting any new problems to clear the values from all variables in MATLAB’s memory. E.2.1 Polynomial Curve Fitting: Example 2-3
In this example, a third-order polynomial is fit to conversion-rate data.
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Step 1: First, the data have to be entered as matrices by listing them between brackets, leaving a space between each entry. X=[0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.85]; ra=[0.0053 0.0052 0.005 0.0045 0.004 0.0033 0.0025 0.0018 0.00125 0.001]; Step 2: Next, the function polyfit is used to fit the data to a third order polynomial. To learn more about the function polyfit, type help polyfit at the command prompt. p=polyfit(X,ra,3) (matrix of coefficients 5 polyfit(ind. variable, dep. variable, order of polynomial) p =0.0092
-0.0153
0.0013
0.0053
The coefficients are arranged in decreasing order of the independent variable of the polynomial, therefore, ra 5 0.0092X 3 2 0.0153X 2 1 0.0013X 1 0.0053 Please note that the typical mathematical convention for ordering coefficients is y 5 a0 1 a1 x 1 a2 x 2 1 ??? 1 an x n whereas, MATLAB returns the solution ordered from an to a0 . Step 3: Next, a variable “f” is assigned to evaluate the polynomial at the data points (i.e., “f” holds the “ra” values calculated from the equation of the polynomial fit.) Since “X” is a 1x10 matrix, “f” will also be 1x10 in size. f=polyval(p,X); f=polyval(matrix of coefficients, ind. variable) To learn more about the function polyval, type help command prompt.
polyval at the
Step 4: Finally, a plot is prepared to show how well the polynomial fits the data. plot(X,ra,'o',X,f,'-') plot(ind. var., dep. var., ‘symbol’, ind. var., f, ‘symbol’) where ‘symbol’ denotes how the data are to be plotted. In this case, the data set is plotted as circles and the fitted polynomial is plotted as a line. The following commands label and define the scale of the axes. xlabel('X'); ylabel('ra "o", f "-"'); axis([0 0.85 0 0.006]); xlabel(‘text’); ylabel(‘text’); axis([xmin xmax ymin ymax]) Please refer to help plot for more information on preparing plots.
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x 10
-3
6
5
4
ra
3
, f "-" 2
1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x The variance, or the sum of the squares of the difference between the actual data and the values calculated from the polynomial, can also be calculated. variance=sum((ra-f).^2) variance =1.6014e-008 The command (ra-f)creates a matrix the same size as “ra” and “f” and contains the element-by-element subtraction of “f” from “ra .” Every element of this new matrix is then squared to create a third matrix. Then, by summing all of the elements of this third matrix, the result is a 1x1 matrix, a scalar, equal to the variance. E.2.2 Solving a System of Algebraic Equations: Example 6-7
In this example a system of three algebraic equations 1/2
1/2
0 5 C H 2 C H0 1 ( k 1C H C M 1 k 2C H C X ) t 1/2
0 5 C M 2 C M0 1 k 1C H C M t 1/2
1/2
0 5 (k1CH CM 1 k2CH C X ) t 2 C X is solved for three variables, CH , CM , and CX . Step 1: To solve these equations using MATLAB the constants are declared to be symbolic, the values for the constants are entered in the equations and the equations are entered as eq1, eq2, and eq3 in the following form: (eq1=symbolic equation).
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It is very important that the three variables, CH , CM , and CX , be represented by variables of only one character in length (e.g., h, m, and x)
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h 5 CH
CH0 5 .021
k1 5 55.2
m 5 CM
CM0 5 .0105
k2 5 30.2
x 5 CX
t 5 0.5
syms h m x; eq1=h-.021+(55.2*m*h^0.5+30.2*x*h^0.5)*0.5; eq2=m-0.0105+(55.2*m*h^0.5)*0.5; eq3=(55.2*m*h^0.5-30.2*x*h^0.5)*0.5-x; Step 2: Next, to solve this system of equations, we type S=solve(eq1,eq2,eq3); The answers can be displayed by typing the following commands: S.h ans = .89435804499169139775064976230242e-2 S.m ans = .29084696757170701507538493259810e-2 S.x ans = .31266410984827736759987989710624e-2 Therefore, CH 5 0.00894, CM 5 0.00291, and CX 5 0.00313. E.2.3 Solving a System of Ordinary Differential Equations: Example 4-7
In Example 4-7, a system of two differential equations and one supplementary equation d ( X ) rate ------------ 5 ---------- ; d (W ) fa0
d ( y) ( 1 1 eps ? X ) ------------ 5 2alpha ----------------------------- ; 2y d (W )
1 1 eps ? X f 5 ------------------------y
was solved using POLYMATH. Using MATLAB to solve this problem requires two steps: (1) Create an m-file containing the equations, and (2) Use the MATLAB ode45 command to numerically integrate the equations stored in the m-file created in step 1. Part 1: Solving for X and y Step 1: To begin, choose New from the File menu and select M-file. A new text editor window will appear; the commands of the m-file are to be written there. Step 2: Write the m-file. The m-file for this example may be divided into four parts.
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Part 1: The first part contains only comments and information for the user or future users. Each comment line begins with a percent sign since MATLAB ignores the rest of a line following %. Part 2: The second part is the function command which must be the first line in the m-file that is not a comment line. This command assigns a new function to the name of the m-file. The new function is composed of any combination of existing commands and functions from MATLAB. The information and commands that define the new function must be saved in a file whose name is the same as that of the new function. Part 3: The third part of the m-file contains all other information and auxiliary equations used to solve the differential equations. It may also include the global command that allows the value for variables to be passed into or out of the m-file. Part 4: The final part of the m-file contains the differential equations to be solved. MATLAB requires that the variables of the ODEs be the elements of a single column vector. Therefore, a vector x is defined such that, for N variables, x=[var1; var2; var3; ...; varN] or x(1)=var1, x(2)=var2, x(N)=varN. In the case of Example 4-7, var1=X and var2=y. Step 3: Save the m-file under the name “ex4_7.m.” This file must be saved in a directory in MATLAB's path. The path is the list of places MATLAB looks to find the files it needs. To see the current path, to temporarily add a directory to the path, or to permanently change the path, use the pathtool command. Step 4: To see the m-file we type type ex4_7 This command tells MATLAB to type the m-file named “ex4_7.m.” Step 5: Now to solve the problem, the initial conditions need to be entered from the command window. A matrix called “ic” is defined to hold the initial conditions of x(1) and x(2), respectively, and “wspan” is used to define the range of the independent variable. ic=[0;1]; wspan = [0 60]; Step 6: The global command is also repeated from the command window. global eps kprime Step 7: Finally, we will use the ode45 built-in function. This function numerically integrates the set of differential equations saved in an m-file. [w,x]=ode45('ex4_7',wspan,ic); [ind. var., dep. var.] = ode45(‘m-file’, range of ind. variable, initial conditions of dep. variables)
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Lines beginning with % are comments and are ignored by MATLAB. The comment lines are used to explain the variables in the m-file. This line assigns the function xdot to the m-file ex4_7.m (in this case w is the independent variable and x is the dependent variable). This line tells MATLAB to allow the value for the variables "eps" and "kprime" to be passed outside the m-file. These lines provide important information necessary to solve the problem.
% % % % % % %
Part 1 "ex4_7" m-file to solve example 4-7 x(1)=X x(2)=y xdot(1)=dX/dW, xdot(2)=dy/dW
%Part 2 function xdot=ex4_7(w,x) global eps kprime %Part 3 kprime=0.0266; eps=-0.15; alpha=0.0166; rate=kprime*((1-x(1))/(1+eps*x(1)))*x(2); fa0=1; %Part 4 xdot(1, :)=rate/fa0; xdot(2, :)=-alpha*(1+eps*x(1))/(2*x(2)); These lines are the equations for the ODEs to be solved. MATLAB requires that the variables of the ODE's be assigned to one column vector. Therefore, a vector x is defined such that x(1)=X and x(2)=y. Also, xdot is the derivative of x.
For more information, type help ode45 at the command prompt. Part 2: Evaluating Variables not Contained in the Solution Matrix Step 1: We want to solve for “f,” which is not contained in the solution matrix, “x,” but is a function of part of the solution matrix. To see the size of the matrix “x,” we type size(x). This returns the following: ans = 57 2. Therefore, “x” is a 57 by 2 matrix of the form: X (1) y(1) x1( 1 ) x2( 1 ) X (2) y(2) x1( 2 ) x2( 2 ) x 5 x1( 3 ) x2( 3 ) 5 X ( 3 ) y( 3 ) Ú Ú Ú Ú x 1( 57 ) x 2( 57 ) X ( 57 ) y( 57 ) Step 2: Next we need to write the equation for “f” in terms of the “x” matrix. Using MATLAB notation, x(1:z,1:y) represents x(row 1:row n, column 1:column n) rows 1 through z and columns 1 through y of X=x(1:57,1:1) = x(1:57,1) y=x(1:57,2:2) = x(1:57,2) the matrix “x.” Similarly, x(1:57,1) represents all the rows in the first column of the “x” matrix, which in our case is X. Similarly x(1:57,2) defines the second column, y.
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The notation x( : , 1) also defines all the rows in the first column of the “x” matrix. This is usually more convenient than sizing the matrix, but at times, only part of the solution matrix may be needed. For example, you may want to plot only part of the solution. So, we can write the formula (f=(1+eps*X)/y) Multiplication and division signs in the following way: are preceded by a period to denote element-by-element operations as described in the Quick Tour. (The operation is performed on every element in the matrix.)
f=(1+eps.*x(:,1))./x(:,2);
And we can write the formula for “rate” as follows: rate=kprime.*((1-x(:,1))./(1+eps.*x(:,1))) .*x(:,2); Note: This is why we used the global command. We needed the values for “eps” and “kprime” to solve for “rate” and “f.” Step 3: A plot can then be made displaying the results of the computation. To plot “X”, “y,” and “f” as a function of “w”: plot(w,x,w,f); plot(ind. var., dep. var., ind. var., dep. var.); title('Example 4.7');xlabel('w (lb)');ylabel('X,y,f') Since the solution matrix “x” contains two sets of data (two columns) and “f” contains one column, the plot should display three lines. Example 4–7 3.5
3
2.5
x y f
2
1.5
1
0.5
0 0
10
20
30
40
50
60
w (lb)
To plot the rate: plot(w,rate);title('Example 4.7');xlabel('w (lb)');ylabel('rate');
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Example 4–7 0.03
0.025
rate
0.02
0.015
0.01
0.005
0 0
10
20
30
40
50
60
w (lb) E.2.4 Solving a System of Ordinary Differential Equations: Example 4-8
To review what you learned about Example 4-7, please examine Example 4-8. type ex4_8 % % % % % %
"ex4_8" m-file to solve example 4.8 x(1)=X x(2)=y xdot(1)=dX/dz, xdot(2)=dy/dz
function xdot=ex4_8(z,x) Fa0=440; P0=2000; Ca0=0.32; R=30; phi=0.4; kprime=0.02; L=27; rhocat=2.6; m=44; Ca=Ca0*(1-x(1))*x(2)/(1+x(1)); Ac=pi*(R^2-(z-L)^2); V=pi*(z*R^2-1/3*(z-L)^3-1/3*L^3);
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G=m/Ac; ra=-kprime*Ca*rhocat*(1-phi); beta=(98.87*G+25630*G^2)*0.01; W=rhocat*(1-phi)*V; xdot(1,:)=-ra*Ac/Fa0; xdot(2,:)=-beta/P0/x(2)*(1+x(1)); Now, from the command window enter ic=[0;1]; zspan = [0 54]; [z,x]=ode45('ex4_8',zspan,ic); plot(z,x);title('Example4.8');xlabel('z(dm)') ;ylabel('X,y'); axis([0 54 0 1.2]) Example 4–8
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x y
0.6
0.4
0.2
0 0
10
20
30
40
50
z (dm) E.2.5 Nonlinear Regression: Example 5-6
In this example, rate-pressure data are fit to four rate equations to evaluate the rate constants. These fits are then compared to determine the best rate equation for the data. To accomplish this, an m-file is required to compute the least-squares regression for the data. Only part (a) of Example 5-6 will be demonstrated here. The rate equation for part (a) is kP E P H ra 5 ---------------------------------------------1 1 K A P EA 1 K E P E
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Step 1: Write the m-file. The structure of this m-file is very similar to the m-file for Example 4-7. Please refer to Example 4-7 for comparison and further explanation. The function "f" is assigned to the m-file "rate_a" f must have an initial value. Therefore, it is given a value of 0 before the “for” loop to initiate the sum at zero.
% "rate_a" % m-file to perform least-squares regression % x(1)=k; x(2)=Ke; x(3)=Ka function f=rate_a(x) global ra pe pea ph2 n f=0; for i=1:n f=f+(ra(i)-(x(1)*pe(i)*ph2(i))/(1+x(3)*pea(i)+x(2)*pe(i)))^2; end This "for" loop calculates the square of the difference between the actual rate and the proposed rate equation. The result of the loop is the sum of the squares we are trying to minimize and is saved in the variable "f." This equation will be different for each rate law. n
f 5 ^ ( ra 2 ra )2 actual calculated 1
Step 2: From the command window, the global command is repeated. In this case it allows the values for variables to be passed into the m-file. global ra pe pea ph2 n Step 3: The data are entered. ra=[1.04 3.13 5.21 3.82 4.19 2.391 3.867 2.199 0.75]; pe=[1 1 1 3 5 0.5 0.5 0.5 0.5]; pea=[1 1 1 1 1 1 0.5 3 5]; ph2=[1 3 5 3 3 3 5 3 1]; Also, the value for “n” is assigned. Since there are nine data points, n=9; Step 4: To perform the least-squares regression for the data, the fmins command is used to find the values of the constants that minimize the value of “f.” Type help fmins for more information. xo=[1 1 1]; x=fmins('rate_a',xo) dep. variable = fmins(‘m-file’,[matrix of initial guesses])
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The solution: x =3.3479
2.2111
0.0428
Therefore, k 5 3.35; KE 5 2.21; and KA 5 0.043. Step 5: To see how close the solution fits the data, look at the sum of the squares to see the final value of “f” for the solution in step 4. Assign to the variable “residual” the final value of “f,” residual=rate_a(x) residual =0.0296 The sum of the squares (s 2) is 0.0296.
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H.1 AIChE National Student Chapter Competition This exercise was given as an open-ended problem to 160 students (working in groups of 4) in the University of Michigan Chemical Reaction Engineering class Winter Term 1997. However, the students were to only design the experiment and not build the experiment as is the case with the National Competition. Each group of students presented their design on poster boards to an outside panel of judges. The same judging criteria applied for the posters as for the National Competition. MEMORANDUM TO: Student Chapters RE: National Student Chapter Competition For the past several years we have all seen the esprit de corps, excitement, and learning that has been generated among undergraduates from engineering disciplines engaging in national competitions. The civil engineers have the “concrete canoe race,” the mechanical engineers the “egg-drop competition,” and there in the interdisciplinary “solar car race.” Many students, faculty, and practicing engineers would like to give chemical engineers a similar experience, one that would educate others about our profession and receive similar publicity (e.g., newspapers, perhaps even TV coverage). It has been suggested that the latter would be more probable if the competition involved topical issues: environment, energy reduction, world-wide food production, and the like. In any case, safety should be a primary concern (e.g., no explosive or toxic chemicals). 1
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To celebrate the newest division of the American Institute of Chemical Engineers, the Chemical Reaction Engineering Division, the first competition will be concerned with chemical reaction engineering. THE 1998 COMPETITION
Design, build, and operate an apparatus for an undergraduate laboratory experiment that demonstrates chemical reaction engineering principles and that is novel or perhaps strives to do the improbable (e.g., won't a concrete canoe sink?). The experiment should be bench scale and of the type currently found in undergraduate laboratory courses. It is also possible that the experiment could be used for a lecture demonstration. The experiment should cost less than $500 in purchased parts to build. The first year’s, competition could include experiments that would either produce a product (e.g., yogurt or something that would find use in the feeding of starving nations) or demonstrate how an environmental problem might be solved (e.g., wetlands to degrade toxic chemicals). The winners (perhaps second place also) of the regional competition will be invited to bring their experiments to the annual AIChE meeting, where the national winners will be selected. General Mills has agreed to sponsor the competition and the following prizes will be awarded to the student chapters: 1st prize 2nd prize 3rd prize
$2000 $1000 $500
In addition, a description of the winning experiment will be published in Chemical Engineering Education. The first regional competition will be held at the 1998 Student Chapter Regional Conferences and the finals at the 1998 annual meeting in Miami Beach, Florida. The rules and judging criteria are attached. Rules 1. “Design, build, and operate an apparatus for an undergraduate laboratory experiment that demonstrate, the principles of reaction engineering principles and that is novel or perhaps strives to do the improbable (e.g, won't a concrete canoe sink?).” The experiment should be bench scale and of the type currently found in most undergraduate laboratory courses. It is also acceptable that the experiment be of the type that would be used for a lecture demonstration. 2. The experiments should encompass “simplicity/ease of communication to nontechnical people.” The process should be one easily understood by people outside the profession. Either the object of the process, such as the manufacture of yogurt, the importance of the
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project, such as feeding many people from increasingly scarce resources, or the process itself should be easily communicated to people without a background in chemical engineering. Media coverage (newspapers/television/radio) is one way to show success. The competition will be conducted on the honor system. The faculty and graduate students cant act only as sounding boards to student queries. The faculty cannot be idea generators for the project. The student chapter advisor or department chair must write a cover letter stating to the best of his or her knowledge that students have abided by the rules. Students who work on the project must also sign a statement stating that they have abided by the rules. The competition is to be a team competition with at least 20% of the team being composed of members from each of the junior and senior chemical engineering classes. The minimum number of participants is 5 and the maximum is 15 per university. Associated measuring equipment (e.g., pH meter) must be of the type that is readily available at most universities through department ownership or borrowing from other departments in the university. Purchased parts must cost less than $500. This price does not include a PC for data acquisition, or associated measuring or other (e.g., pumps, fittings, vessels) equipment that exists in most chemical engineering undergraduate laboratories. The experiments will be displayed at the regional meeting. A poster board should accompany the apparatus as well as a 5- to 10-page report describing how the idea for the experiment was generated, the underlying principles, the experimental procedure, and sample results. In the event that the apparatus may not be physically brought to the meeting, videotape or other means may be used to assist understand the experiment. The top one or two winners of the regional student chapters will be eligible to compete in the finals to be held at the annual meeting. Safety regarding assembling and operating the experiment must be addressed. The student chapter advisor or department chair at the host chapter at the regional conference will select a panel of three judges. The judges can be from industry, faculty, or students. The judges cannot be affiliated with any organization that has an entry. The number of winners selected to go to the finals will depend the number of regional entries. If there are six or fewer entries, one winner will be selected to advance to the national competition. If there are seven or more entries, two winners will be selected. The decision of the regional and national judges shall be final.
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Judging Criteria 1. Creativity/novelty/originality. 2. Statement of the principle to be demonstrated and clarity in demonstrating that principle. 3. Proper description of the safety issues associated with building and operating the experiment. 4. Simplicity/ease of communication/media coverage. 5. Quality of communication: Introduction: How was the idea generated? What principle does the experiment demonstrate and why is it important? Discussion: Explain the fundamentals. Procedure: Discuss safety concerns. Results: Describe what you found. 6. Opportunity for subsequent laboratory groups to study different variables or outcomes using the same apparatus. 7. Ease, desirability, and feasibility of being replicated by another student chapter. 8. Physical appearance. 9. Participation (more than 16 hours) by someone who is not a chemical engineering major) (3 points for each non-chemical engineering major) and/or participation by chemical engineering sophomores (2 points for each sophomore) (10 points maximum).
40 points 30 points 15 points 15 points 10 points
10 points 10 points 10 points
10 points 150 points
H.2 Effective Lubricant Design* Background Lubricants are often applied at the interface between rubbing surfaces to reduce friction and prevent wear by disallowing direct surface to surface contact. An automobile engine has many contacting metal parts, such as the pistons and cylinders, and the cam lobes and cam followers. Without adequate lubrication, the sliding metal parts within the engine would wear appreciably, leading to engine failure. A typical consumer may expect to drive more than 100,000 miles before experiencing severe engine problems resulting from wear within the engine. To meet this expectation, lubricant manufacturers, in close collaboration with automobile manufacturers, continue to develop improved lubricant formulations. Lubricants are formulated by blending a base oil with additives to yield a mixture with the desirable physical and chemical properties dictated by the application environment. Base oils are typically derived from petroleum and * Problem provided by General Motors Research Laboratories, Warren, Michigan.
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are complex mixtures of aliphatic and aromatic hydrocarbons. However, some lubricants are blended using synthetic base oils. Examples of synthetic base oils include esters, polyphenyl ethers, polyalphaolefins, and perfluoroalkylethers. Lubricant additives are classified according to their function, including antioxidants, viscosity index (VI) improvers (to maintain desired viscosity over a wide temperature range), antiwear additives, friction modifiers, dispersants, detergents, pour point depressants, and antifoaming agents. A fully formulated lubricant typically consists of 80–90% base oil and 10–20% additives. Lubricant development requires an understanding of the specific problems and needs associated with lubrication such as: the need for automotive lubricants with enhanced oxidation stability, antiwear properties, and physical properties under severe operating condition. A kinetics model may be used to try and predict the oxidative degradation behavior of lubricants under differing conditions. Lubricant Degradation Model: A fresh lubricant may have physical properties ideally suited to its application, but as the lubricant degrades its physical properties can change markedly. This transformation can lead to increased friction and wear at lubricated surfaces. After significant degradation takes place, sludge and varnish deposits may form on lubricated surfaces to further hinder the smooth operation of lubricated components. There is general agreement in the literature that under normal service conditions a major portion of lubricant degradation is due to oxidation of the lubricant base oil. Consequently, a great deal of lubricant research has focused on base oil degradation and the inhibition of oxidation through the use of antioxidant additives. The oxidation of a lubricant base oil follows the hydroperoxide chain mechanism for hydrocarbon oxidation. Some of the major steps of this mechanism are listed in Reactions (1)–(10). Low Temperatures, No Antioxidants INITIATION ( 2 ) I?1 RH → R? 1 HI k0
( 1 ) I2 → 2I? ki
kP1
PROPAGATION ( 4 ) RO2? 1 RH → ROOH 1 R?
( 3 ) R? 1 O2 → RO2? kP2
kt ( 5 ) RO2? 1 RO2? → INACTIVE PRODUCTS TERMINATION High Temperatures, No Antioxidants ki ( 6 ) ROOH → RO? 1 ?OH INITIATION 3
P ( 7 ) RO? 1 RH → ROH 1 R? PROPAGATION kP ( 8 ) ?OH 1 RH → HOH 1 R?
k
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Low and High Temperatures With Antioxidants k
A1 ( 9 ) RO2? 1 AH → ROOH 1 A? kA2 ( 10 ) A? 1 RO2? → inactive products
Also hydroperoxide decomposing antioxidants can transform hydroperoxides to stable products as shown in Reaction (11) below: k
A3 ( 11 ) ROOH 1 A? → inactive products
This antioxidant is effective provided the products are stable, and the rate is much faster than reaction (6). An example of a hydroperoxide decomposing antioxidant is phenothiazine. In the absence of an antioxidant (or after antioxidant additive depletion), significant quantities of hydroperoxides may accumulate as a result of extensive base oil oxidation. Resulting secondary oxidation reactions may occur, leading to the formation of alcohols, ketones, carboxylic acids, and esters. Extensive oxidation can also lead to the formation of high-molecular-weight material which may form deposits on lubricated surfaces. A dramatic increase in viscosity generally results from extensive oxidation of a lubricant, which may also lead to poor lubricant performance. Clearly, extensive lubricant oxidation leads to a rapid deterioration of lubricant effectiveness, and at the point of antioxidant depletion, an automobile engine lubricant should generally be considered ineffective and should be replaced with fresh lubricant. Problem Statement 1. Consider the possible attributes of currently available, general purpose engine lubricants. List them in what you consider their order of importance. (You may wish to examine sales displays, advertisements, cans/bottles of motor oil, etc. for help with this information.) 2. If you could design an ideal engine lubricant to totally dominate the marketplace, what characteristics would you give it? 3. In the future, a priority will most likely be to make automobile maintenance easier for the owner. What kind of creative ideas/inventions can you conceive of for streamlining oil change maintenance for the consumer? After you’ve generated some ideas (by brainstorming perhaps) critique them from the standpoint of practicality, cost, availability of technology, etc. 4. Degradation by oxidation is a major cause for having to replace engine lubricants. Using your knowledge of reaction kinetics, analyze the degradation of engine lubricants due to oxidation using the reactions shown in the introduction (i.e. find an expression for the rate of degradation). Consider the following four cases. In the absence of antioxidants, examine base oil (RH) degradation a) at low temperatures (258C)
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b) In c) d)
at high temperatures (1008C) the presence of antioxidants, examine base oil (RH) degradation at low temperatures (258C) at high temperatures (1008C)
Estimated Values for the Kinetic Parameters (258C) [ RH ]0 5 3 M
[ I2 ]0 5 0.10 M
k 0 5 1026 ( Eact 5 7.5)
k A1 5 103 ( Eact 5 10)
k P2 5 1.58 ( Eact 5 8.5)
k i3 5 1026 ( Eact 5 9)
k t 5 107 ( Eact 5 0)
k A2 5 103 ( Eact 5 9)
k A3 5 3.33 3 1023 ( Eact 5 10)
The units on the above rate constants are sec21 for first order and dm3 /mole ? sec for second order. The units for the activation energies are kcal/mole. You may assume for the purposes of this investigation that 10% conversion of the base oil (RH) is the point at which the lubricant will have to be replaced. From your analysis, what kind of recommendations can you make for the improvement of the engine lubricants? If you were making suggestions to the R&D division of a lubricant manufacturer, what would you have them investigate to make the most impact on the retardation of oil degradation by oxidation? What kind of experiments should they do? What are the drawbacks of this type of kinetic model for oil degradation? Do you have a better modeling suggestion? 5. Some suggestions that have been turned in to the suggestion box of Synthoil, an up and coming, newly formed lubricant manufacturer, are: a) New cars should be equipped up with a feed and bleed system for the oil. Every so often, a quart of used oil should be drained and a quart of new oil should be added to the automobile. This should lengthen the necessary time between complete oil changes and save the consumer money. b) We should design and market an inhibitor feed system for automobiles that would allow us to maintain a minimum inhibitor concentration in the engine oil, thereby protecting it from excessive oxidative breakdown. As head of the R&D division of this progressive company, it is your job to investigate the technical feasibility of these suggestions and report on them at the next Board of Directors meeting. Investigate one of these suggestions, or substitute an equally good one of you own and investigate it. Be creative! Problem Information Synthoil Cost Data for Evaluation Purposes: Complete Oil Change - $29.95 (includes 5 qts. oil, filter, and labor) Quart of Oil - $1.50 (typical engine 5 5 qt. capacity) Inhibitor (Antioxidant, AH) - $0.10/gram, Approx. MW 5 100 g
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Additional Information Foulers H. Scott. Elements of Chemical Reaction Engineering. 2nd Ed., Englewood Cliffs, N.J.: Prentice-Hall, 1992. Given as an OEP at The University of Michigan, Winter 1991.
H.3 Peach Bottom Nuclear Reactor* Background The Peach Bottom nuclear reactor, located in Georgia, has been built and is almost operational. The reactor is a boiling water nuclear reactor that produces 1100 MW of power and cost approximately 2 billion dollars to build. The effluent from the reactor contains cooling air and isotopes of Krypton and Xenon. The major constituents of the stream are Kr-83m and Xe-135 which are present in the exit gas at concentrations of 3.19 3 106 mCi/dm3 and 1.4 3 1023 mCi/dm3, respectively. The volumetric gas flow rate exiting the reactor is 2.75 m3 /hr at a temperature of 308C. The Nuclear Regulatory Commission (NRC) issued emission limits for Kr-83m and Xe-135 in the Federal Register, Vol. 56, No. 98, Part VI, dated Tuesday, May 21, 1991. The emission standards as issued are .05 mCi/dm3 for Kr-83m and 7.0 3 1025 mCi/dm3 for Xe-135.
Peach Bottom Nuclear Reactor
Effluent 2.75 m3/hr Air Kr-83m Xe-135
* Problem developed by Susan Stagg, University of Michigan, from a problem suggested by Octave Levenspiel, Oregon State University.
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Problem Statement Propose two or more solutions that will enable the Peach Bottom nuclear reactor to go on line with the 2.75 m3 /hr of exit gas meeting the NRC emission standards. Please show all calculations and include any diagrams necessary to thoroughly explain your solutions. Additional information and properties for Kr-83m and Xe-135 can be obtained from the Handbook of Chemistry and Physics. Problem Information Ci is the symbol for a curie. A curie is the unit of radioactivity equivalent to 3.70 3 1010 disintegrations per second. This unit is named after Marie Curie. Effluent data: Constituent in Effluent Kr-83m Xe-135
Concentration (mCi/dm3) 3.19 3 106 1.4 3 1023
NRC Emission Standard (mCi/dm3) 0.05 7.0 3 1025
Additional Information Fogler, H. Scott. Elements of Chemical Reaction Engineering. 2nd Ed., Englewood Cliffs, N.J.: Prentice-Hall, 1992. Lide, David R. CRC Handbook of Chemistry and Physics. Ann Arbor: CRC Press, 1993. Given as OEP at The University of Michigan, Winter 1994.
H.4 Underground Wet Oxidation* Background Several of your company’s chemical processes generate aqueous waste streams containing a large number of hazardous compounds that are presently being destroyed by incineration. The Chief Executive Officer (CEO) of your company saw the attached article in a local journal. He asked the Director of the Engineering Service Division (ESD) if the technology would be useful for treating the aqueous waste streams from your plant. After assuring the CEO that he would investigate the possibilities, the ESD Director asked the Manager of the Reaction Engineering Group to check it out. The manager, who is also your supervisor, handed the assignment to you. Problem Statement Your mission is to evaluate the technology, size a reactor system, and specify appropriate operating conditions for oxidizing the components of the aqueous waste streams. The Engineering Economics group will then compare * Problem developed by Professor Phillip E. Savage, University of Michigan.
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the costs of your artesian with incineration to assess the relative financial merits of using the new technology. A few questions to provide some initial direction in your evaluation are given below. • At what temperature and pressure should the reactor operate? • Is an underground reactor better than the conventional above-ground reactor? • What safety considerations do you need to include in your design for this high-temperature, high-pressure process involving hazardous chemicals? • Can you ethically recommend this technology to your management? Is it sufficiently proven? • How confident are you that your reactor will be able to destroy the hazardous chemicals and meet the design specifications? • Are the products of incomplete oxidation also hazardous? • What material should be used to construct the reactor? Will corrosion be a problem? • Should the reactor operate isothermally, adiabatically, or with heat transfer? Problem Information To complete your work you will need more information than the article provides. Fortunately, your company’s Technical Service Division can conduct experiments for you (for a fee, of course). To request their services, you need only send the Technical Services Manager a written memo explaining what you want them to do. They will let you know the cost and time required to do the work. Then, if you still want the work done, they will provide the results. Additional Information: Fogler, H. Scott. Elements of Chemical Reaction Engineering. 2nd Ed., Englewood Cliffs, N.J.: Prentice-Hall, 1992. Given as an OEP at The University of Michigan, Winter 1990. (Note: The following journal article is based upon one found in the December 7, 1988 edition of the New York Times.)
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UNDERGROUND WET OXIDATION OF WASTE MATERIAL Oxygen
Clean Water Wet Waste Material
Five Layer View
Reaction
When wet waste material is mixed with oxygen under high pressures, the wet oxidation process produces sterile ash and clean water. The process is ideal for degrading sewage sludge and other waste material. However, frequently the high pressure requires special equipment and large process plants. It has been suggested that gravity might provide the high pressures, as indicated by the above set-up. The reaction is carried out 5,000 ft. underground. The waste material and oxygen are transported by pipes to the bottom of the vessel. Falling waste material provides the needed excess pressure. The reaction typically occurs at approximately 550 degrees Fahrenheit and 2,000 psi. The products are drawn to the surface via a third pipe.
H.5 Hydrodesulfurization Reactor Design* Background Just as you arrive at work one morning, your supervisor, Dr. Jones, says he needs to speak with you and your design group. He seems concerned about something, so you locate your group members and hurry into his office. As the
* Problem developed by John T. Santini, Jr., University of Michigan.
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last member of your group shuffles in, you turn to your supervisor and ask, “O.K, we are all here. What did you want to talk to us about?” “Well, one of the processes recently proposed by the Process R&D group produces a by-product stream consisting of nearly pure benzothiophene. Because benzothiophene contains sulfur and an aromatic ring, we cannot vent this stream into the atmosphere. The engineers in Catalyst Development believe that we can use a hydrodesulfurization reaction to convert benzothiophene into ethylbenzene with a cobalt-molybdenum catalyst supported on alumina. If we can design a reactor to do this efficiently, we could sell the ethylbenzene as a commodity chemical. The other product, hydrogen sulfide, could be sent to the sulfur treatment facilities in Building 12.” “I know that your group’s specialty is reactor design, so I’m assigning the hydrodesulfurization reactor project to you. I would like a progress report in three weeks and a final design report four weeks after that. I’ve compiled a list of items that you should include in each of these reports. I know it has been a while since most of you designed a reactor from start to finish, so I’ve included a partial list of references that may help you. They will be especially helpful with the selection of the materials of construction. I know this assignment is open-ended and requires a lot of engineering judgment, but just remember to use your common sense and BE CREATIVE! Any questions?” “No. We’ll get started right away,” you reply as you and your group leave the office. Problem Statement As your supervisor told you, the engineers in Catalyst Development think that benzothiophene could be converted to ethylbenzene by a hydrodesulfurization reaction. Before you commit the company’s time and money to design a reactor for this reaction, you may want to attempt to verify that the production of ethylbenzene is economically feasible. In other words, are the products worth more than the reactants and energy required to make them? If you discover the answer is no, you may have saved your company thousands of dollars in design fees. Research this issue and discuss your findings in your progress report. (See the Additional Information section for some references that may help in answering these questions.) If you knew that your supervisor supported the design of the new reactor and you discovered that producing ethylbenzene from benzothiophene was not cost effective, how would you inform your supervisor of this? The progress report may consist of a maximum of five pages, excluding figures and appendices. In the progress report, be sure that your group provides support for your choice of: • reactor (i.e. PER, PER, fluidized CSTR, etc.), • adiabatic vs. isothermal reactor operation, • reactor temperatures and pressures (Hint—single phase reactions are less complicated than multiple phase reactions), • feed ratio of hydrogen to benzothiophene, • effluent conditions and compositions, and • the weight of catalyst required.
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Also include: • a qualitative discussion of the effect that operating conditions and the method of operation have on capital and operating costs, • justification for any assumptions made, and • appendices summarizing your calculations. The final design report may consist of a maximum of ten pages, excluding figures and appendices. In the final design report, you should provide support for your choice of: • materials of construction for the reactor (Hint—is the material susceptible to accelerated corrosion due to the presence of sulfur or hydrogen?) • reactor shape and dimensions, and • reactor wall thickness. Also include: • support for any changes in the initial reactor design presented in your progress report, • any environmental or safety concerns that may be relevant to your design, • diagram of the reactor, • justification for any assumptions made, and • appendices summarizing ALL calculations. Problem Information The reaction is:
+
3H2
+ H2S
S Benziothiophene (Thianapthene)
Ethylbenzene
In past research studies, the proposed reaction has been run in the vapor phase ith reactor temperatures of 2408-3008C, total pressures of 2–30 atm, and hydrogen to benzothiophene feed ratios of 4:1 to 9:1. These experiments resulted in the development of the rate law given below. (Note: You may assume that the rate law holds for conditions outside this range. Therefore, you are in no way constrained to using these operating ranges and should use them only as guides.)
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The rate law is: k B K B K H2 P B P H2 2 r9B 2 ----------------------------------------------------------------------------------------------------P H2 S 1 1 ( K H2 PH2 )0.5 1 K H2S ---------- 1 ( K B PB ) P H2 where mol k B ( 2608C ) 5 6.65 3 1025 --------------------gcat ? sec
mol k B ( 300 8C ) 5 1.80 3 1024 --------------------gcat ? sec
K B ( 260 8C ) 5 19.3 atm21
K B ( 3008C ) 5 9.90 3 1022 atm21
K H2 ( 2608C ) 5 0.358 atm21
K H2 ( 3008C ) 5 1.84 3 1023 atm21
K H2S ( 2608C ) 5 211
K H2S ( 3008C ) 5 1.082
(Note: The heat of adsorption was estimated at 280 kcal/mol for all three species) Catalyst Properties: Particle Diameter: For a packed bed reactor:
For a fluidized CSTR:
0.08 cm f 5 porosity 5 0.30 a 5 pressure drop parameter 5 0.34 kg21 f 5 porosity 5 0.75 a 5 pressure drop parameter 5 0.005 kg21
Feed (pure benzothiophene before the addition of hydrogen): mol FB0 5 20 --------hr T 0 5 entering temperature 5 2608C T melt 5 melting temperature at 1 atm 5 328C T boil 5 boiling temperature at 1 atm 5 221 8C References Girgis, M.J. and Gates, B.C. “Reactivities, Reaction Networks and Kinetics in High-Pressure Catalytic Hydroprocessing.” Ind. Eng. Chem. Res., 30, 2021–2058, [1991]. Van Parijs, I.A., Hosten, L.H., Froment, G.F. “Kinetics of Hydrodesulfurization on a CoMo/g-Al2O3 Catalyst. 2. Kinetics of the Hydrogenolysis of Benzothiophene.” Ind. Eng. Chem. Prod. Res. Dev., 25, 437–443, [1986].
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Additional Information Economic Sources: Chemical Marketing Reporter. New York: Schnell Publishing Co., Inc. Oil, Paint, and Drug Reporter. New York: Schnell Publishing Co., Inc. Reactor Codes: Rules for Construction of Pressure Vessels. A.S.M.E. Boiler and Pressure Vessel Code (Section VIII). July 1, 1980. Yokell, S. “Understanding the Pressure Vessel Codes.” Chemical Engineering. May 12, 1986. pp. 75–85. Selection of Materials: Kirby, G.N. “How to Select Materials.” Chemical Engineering. Nov. 3, 1980. pp. 86–131. Kirby, G.N. “Corrosion Performance of Carbon Steel.” Chemical Engineering. March 12, 1979. pp. 72–84. Peters, M.S. and Timmerhaus, K.D. Plant Design and Economics for Chemical Engineers. 4th Ed. New York: McGraw-Hill, l99l. Schillmoller, C.M. “Solving High-Temperature Problems in Oil Refineries and Petrochemical Plants.” Materials Engineering. January 6, 1986. pp. 83–87. Thermodynamic and Physical Property Data: Lide, David R. CRC Handbook of Chemistry and Physics. Ann Arbor: CRC Press, 1993. Perry, R.H., Green, D.W., Maloney, J.O. Chemical Engineers’ Handbook, 6th ed. New York: McGraw-Hill, 1984.
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H.6 Continuous Bioprocessing* Background In some biochemical processes, it is desired to use continuous rather than batch processing for economic and efficiency reasons. An example of a continuous process is given in Figure 1.
growth media feed
supernatant (cell free)
Animal cells suspended in media
ρ ≈ ρ cells media Figure 1
Continuous Bioreactor.
The animal cells are in a suspension (they may or may not be on beads) in the reactor unit. When the growth medium is fed in, the cells begin to produce the desired product. The exit stream (supernatant) is composed of this product along with any unused growth media. It is critical that the animal cells are not removed with the supernatant, but are retained in solution. It is also important to keep the reactor well-stirred without exerting a large shear stress on the cells, since this may kill or damage them. In addition, the product flow rate from the reactor must not be fast (the space time is of the order of 0.5 to 2 days). Problem Statement Your problem is to design a reactor for an actual process of your choice which will meet the above specifications (see the journal references below for hints and examples). Consider the aspects of mixing, separation, and kinetics. Additional Information Beck, C., Stiefel, H., Stinnett, T. “Cell-Culture Bioreactors,” Chemical Engineering, February 16, 1987, pp. 121–129. Miller, R., Melick, M. “Modeling Bioreactors,” Chemical Engineering, February 16, 1987, pp. 112–120. * Problem provided by The Upjohn Company, Kalamazoo, Michigan.
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Fogler, H. Scott. Elements of Chemical Reaction Engineering. 2nd Ed., Englewood Cliffs, N.J.: Prentice-Hall, 1992. Rubinstein, M.F., Tools for Thinking and Problem-Solving, Englewood Cliffs, New Jersey, Prentice-Hall, 1986.
H.7 Methanol Synthesis* Background Kinetic models based on experimental data are being used more frequently in the chemical industry for the design of catalytic reactors, but the modeling process itself can influence the final reactor design and its ultimate performance by incorporating different interpretations of experimental design into the basic kinetic models. Model Reaction. The reaction for the synthesis of methanol is 2H2 1 CO → CH3OH This reaction is commercially significant and chemically simple, and the thermodynamic properties of the chemical species are well known. The mechanism assumed here is complex enough to make some sophistication necessary for the analysis, but it is too simple to be really true. We assumed a chemical mechanism of medium complexity, comprised of several elementary reaction steps, for the synthesis of methanol. The data were generated for the overall reaction as it would occur in a backmixed, gradientless, experimental reactor at realistic reaction conditions. The final data set is from a statistically designed, central composite set of simulated experiments, to which 5% random error was added. It comprises a total of 27 simulated results (see Table 1). Problem Statement The primary purpose of this model is to develop kinetic modeling methods and approaches. We have included the reactor simulation part primarily to afford a realistic basis for the comparison of different kinetic models. The design of the reactor to be simulated, the thermodynamic, transport, and physical properties data to be used, and the reaction conditions to be assumed are specified in Tables 2 and 3. The reactor is a commercially realistic, plant-scale, shell-and-tube reactor, suitable for the synthesis of methanol. However, its actual design, its reaction conditions, and its performance will be different from those of any existing commercial methanol process. Simulate the shell-and-tube reactor at specified conditions, using a simple, one-dimensional, plug-flow, pseudohomogeneous, nonisothermal reactor model. Further, investi-
* Problem presented by J. Berty, S. Lee, F. Szeifert, and J. Cropley, at the International Workshop on Kinetic Model Development, A.I.Ch.E. Meeting, Denver, CO, August 1983. (With Permission)
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gate the effect of different coolant temperatures. In all calculations, assume that the ideal gas law applies. With this in mind, the following tasks should be completed. 1) Develop a kinetic model for the synthesis of methanol from the set of synthetic rate data shown in Table 1. 2) Simulate a plant-scale catalytic reactor at specified reaction conditions, using your kinetic model. The design of the reactor, the reaction conditions, and necessary thermodynamic and physical property data are given in Tables 2 and 3. 3) Summarize your results in the format shown in Table 4. Then plot the results with temperature on the y-axis and distance on the x-axis. 4) Suggest a cooling water temperature to be used. Problem Information TABLE 1.
DATA
FOR
KINETIC ANALYSIS Partial Pressure (kPa)
Experiment
Rate (mol/m3 ? s)
Temp. (K)
Methanol
CO
Hydrogen
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
6.573 4.819 6.270 4.928 10.115 7.585 9.393 7.124 1.768 1.177 1.621 1.293 2.827 2.125 2.883 2.035 4.030 3.925 3.938 10.561 1.396 2.452 5.252 3.731 3.599 5.085 3.202
495 495 495 495 495 495 495 495 475 475 475 475 475 475 475 475 485 485 485 500 470 485 485 485 485 485 485
1013 1013 1013 1013 253 253 253 253 1013 1013 1013 1013 253 253 253 253 507 507 507 507 507 1520 172 507 507 507 507
4052 4052 1530 1530 4052 4052 1530 1530 4052 4052 1530 1530 4052 4052 1530 1530 2533 2533 2533 2533 2533 2533 2533 4862 1276 2533 2533
8509 5906 8509 5906 8509 5906 8509 5906 8509 5906 8509 5906 8509 5906 8509 5906 7091 7091 7091 7091 7091 7091 7091 7091 7091 9330 5369
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REACTOR, CATALYST, AND PROCESS CONDITIONS SIMULATION REACTOR CONDITIONS
TABLE 2.
FOR
Reactor Type: shell and tube Tubes: 3000, 38.1 mm i.d. 3 12 m Coolant: boiling water is on shell side; assume coolant temperature constant at 483 K Heat-transfer coefficient (overall): assume 631 W/m2 ? K Catalyst Description Shape: Approximately spherical Diameter: 2.87 mm Effective catalyst bed void fraction: 40% Diffusional resistance: may be ignored Process Conditions Feed gas: Composition: 70 mol% H2 ; 30 mol% CO Space Velocity: 10,000 standard cubic meters per hour per cubic meter of reactor volume Reactor inlet pressure: l0.13 MPa Reactor inlet temperature: 473 K Reactor coolant temperature: 483 K (constant)
TABLE 3.
PHYSICAL PROPERTY
AND
THERMODYNAMIC INFORMATION
Prandtl number of gas: 0.70 (assume constant) Heat capacity of gas: 29.31 J/g mol ? K (assume constant) Viscosity of gas: 1.6 3 1025 Ps ? s (assume constant) Heat of reaction: 297.97 kJ/mol methanol formed Thermodynamic equilibrium constant: (T in K) 3921 log10 K eq 5 ------------ 2 7.971 log10 T 1 0.002499 T T 2 ( 2.953 3 1027 ) T 2 1 10.2
TABLE 4.
( dimensionless )
RESULTS
Authors: Shell-side temperature (K)
483
Maximum tube-side temperature Location from inlet of max. Temp. Outlet temperature Outlet methanol concentration Same as fraction of equilib. value Production rate (kg/hr)
Additional Information Fogler, H. Scott. Elements of Chemical Reaction Engineering. 2nd Ed., Englewood Cliffs, N.J.: Prentice-Hall, 1992.
Use of Computational Chemistry Software Packages
J
Thermodynamic properties of molecular species that are used in reactor design problems can be readily estimated from thermodynamic data tabulated in standard reference sources such as Perry’s Handbook or the JANAF Tables. Thermochemical properties of molecular species not tabulated can usually be estimated using group contribution methods. Estimation of activation energies is, however, much more difficult due to the lack of reliable information on transition state structures, and the data required to carry out these calculations is not readily available. Recent advances in computational chemistry and the advent of powerful easy-to-use software tools have made it possible to estimate important reaction rate quantities (such as activation energy) with sufficient accuracy to permit incorporation of these new methods into the reactor design process. Computational chemistry programs are based on theories and equations from quantum mechanics, which until recently could only be solved for the simplest systems such as the hydrogen atom. With the advent of inexpensive high-speed desktop computers, the use of these programs in both engineering research and industrial practice is increasing rapidly. Molecular properties such as bond length, bond angle, net dipole moment, and electrostatic charge distribution can be calculated. Additionally, reaction energetics can be accurately determined by using quantum chemistry to estimate heats of formation of reactants, products, and also for transition state structures. Examples of commercially available computational chemistry programs include SPARTAN, developed by Wavefunction, Inc. (http://www.wavefun.com) and Cerius2, from Molecular Simulations, Inc. (http://www.msi.com). The following example utilizes SPARTAN 4.0 to estimate the activation energy for a nucleophilic substitution reaction (SN2). The calculations cited below were performed on an IBM 43–P RS–6000 UNIX workstation. 1
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J.1 Example: Estimation of Activation Energies Synthesis of chemical-grade ethanol can be carried out by a nucleophilic substitution reaction using hydroxide ion as the nucleophile and a haloethane as the substrate. For this problem we will investigate the activation energy for the reaction: C2 H5Cl 1 OH2 → C2 H5OH 1 Cl2 This non-catalytic reaction takes place in water at moderate temperatures. In order to investigate the energetics of this reaction, SPARTAN 4.0 will be used to estimate the data required to carry out the following calculations: a) the heat of formation of all reactants and products (including the ionic species) and the overall enthalpy change for the reaction: DH Rx 5 ∑ H f8 ( Products ) 2 ∑ H f8 ( Reactants ) b) the activation energies for the forward and reverse reactions based on an assumed model for the transition state species. From organic chemistry, we learn that the mechanism for SN2 (substitution, nucleophilic, bi-molecular) reactions involves back-side attack of the nucleophile at the carbon atom where a suitable leaving group (such as a halogen atom) is attached. The entering and leaving groups are presumed to be simultaneously bonded in axial positions at the carbon atom where the reaction takes place, hence requiring a particular geometry for the transition state that accommodates this arrangement. The following approximations can be used to obtain a reasonable structure for the transition state species for this and other SN2 reactions: * assume that the geometry of the carbon atom where the nucleophilic attack takes place is trigonal-bipyramidal; H
Cl
H—C—C H
H H
OH
* arrange the entering and leaving groups in axial positions at this carbon atom (e.g. 1808 apart) * bond distances for the entering and leaving groups will be constrained to be somewhat larger than is typical for normal covalent bonds signifying the simultaneous formation and cleavage of bonds for the attacking hydroxide ion and the leaving group We first set up the calculation for the transition state species using the following step-by-step procedure.
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1. Start SPARTAN, click on File/New to enter the Builder. Due to the non-standard geometry at the reaction center, the Expert Builder will have to be used (click on Expert under the Fragment Panel). 2. Click on carbon ( C ) in the periodic table and choose the normal tetrahedral geometry for the carbon atom not involved in the reaction —x— . To place the atom in the workspace, move the mouse pointer to the builder page and click once (left mouse button). 3. With carbon still selected as the atom, select trigonal-bipyramidal geometry from the bonding geometry menu o—x
and click on any
free valance (shown in yellow). 4. To complete the transition state structure, click on chlorine as the atom and select terminal geometry ( —x ) and click on an axial free valance for the carbon atom where the reaction is to take place. Now select oxygen with angular geometry, —x , and click on the other axial free valance shown on the screen. Hydrogen atoms can now be added at all other free valences, but SPARTAN assumes that any unsatisfied free valance is saturated and automatically assigns a hydrogen atom to each of the yellow bonds. The following schematic should be shown on the screen Cl O 5. Click on Minimize in the builder menu in the Fragment Panel. This performs a crude energy minimization calculation. 6. We now wish to constrain (i.e. set) the bonds at the reaction center to values that are somewhat larger than would be characteristic of C—O and C—Cl covalent bonds in normal molecules. In the builder, perform the following steps: • Click on Geometry/Constrain Distance and click on one of the two bonds involved in the reaction • The current C—O and C—Cl bond distance is shown (depending on which bond you selected) in the Constrain Distance dialog box: values of about 1.77 Å for C—Cl and 1.42 Å for C—O are typical. • Click on Constrain Distance in the dialog box, and delete the current entry with the backspace key. Enter a value that is about 50% longer than normal ( , 2.5 Å works well). Click on OK: a pink cylinder will be shown around the bond to indicate that it will be held constant for subsequent calculations. 7. Click on Minimize to perform a crude energy minimization of the constrained transition state structure, and then click on OK and save the result (File/Save—Yes/supply a name then Quit ). This will bring you out of the builder into the main SPARTAN window. The appearance of the molecule can be changed by selecting different renderings
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(ball and wire, ball and spoke, tube, etc.) in the Model pull-down menu: experiment with some models to find one that you like best. The molecule can be rotated and translated using the middle and right mouse buttons, respectively. 8. We are now ready to carry out some quantum chemistry calculations in order to first examine the feasibility of our assumed transition state structure, and then calculate the energy (heat of formation) of the transition state. To determine if we have a good approximation for the transition state for this reaction, carry out the following calculations: * Select Setup/Semi-empirical Title: SN2 Transition State * Task 5 Single-point Energy * Model 5 AM1 (a compromise between accuracy and computational speed) * Solvent 5 Water (C–T) * Charge 5 21 {Note: the charge on the transition state is equal to the summation of the formal charges of all atoms in the molecule. The formal charge on the carbon atom where the reaction is occurring is Formal Charge = # valence electrons 2 # non-bonding electrons 2 1/2 (# bonding electrons) 5 4 2 0 2 1/2 (10) 5 21}) * Multiplicity 5 1 {Note: the spin multiplicity is the number of unpaired electrons plus 1. The transition state has no unpaired electrons.} * The Constraints button in the Semi-empirical dialog window at the right should be clicked off for this calculation. * Save * Select Setup/Properties and click on Frequency, then Save. In this step we are requesting that SPARTAN calculate the vibrational spectrum of our assumed transition state species. * Select Setup/Submit to start the calculation: click on OK when the dialog box appears. Depending on the speed of your computer, the requested calculation may take several minutes (5 or more). * When the calculation is complete, click OK to remove the dialog box and select Display/Vibration. From quantum mechanics we know that the normal mode vibration of a transition state species will be an imaginary frequency. Examine the list of vibrational frequencies: there should be at least one imaginary frequency, and several (3 in this case) may be listed. To determine which imaginary frequency corresponds to the reaction that we are studying, we will use the animation utility in SPARTAN to visualize the vibrational motion. * Click on the imaginary frequency around 357. * Click on Display and examine the action shown. Change the default value from 7 frames to 25 frames to slow down the animation. Does this correspond to the reaction? No, this frequency corresponds to a “wag” of the methyl group.
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* Examine the other imaginary frequencies (e.g. 112) until you identify the one that corresponds to the reaction of interest—normally this will be the imaginary frequency with the largest value (around 562 in this case). For our assumed transition state the imaginary frequency at about 112 has the correct bonds involved, but the action shown indicates an elimination reaction rather than substitution. Animate the 562 normal mode and convince yourself that this frequency corresponds to the correct reaction coordinate. 9. We will now use SPARTAN to calculate the heat of formation of the transition state * Select Setup/Semi-empirical * Task 5 Transition Structure (a change from previous setup) * Model 5 AM1 * Solvent 5 Water (C–T) * Click on Save and Save again (i.e. twice). This procedure searches for the saddle point energy. * Select Setup/Submit to start the calculation. Click on OK to clear the dialog box. SPARTAN is now using quantum mechanics to search the potential energy surface looking for a global saddle point that defines the transition state (including solvation effects). This calculation will take several minutes. Click on OK when the dialog box appears indicating that the job has finished. 10. Retrieve the calculated heat of formation. Select Display/Properties/ Energy. A value close to 2121.835 kcal/mol should be obtained. Select Display/Vibration only one imaginary frequency should appear. Animate the frequency to be sure that it is the correct one. 11. Build (File/New) and Minimize both ethyl chloride and ethanol using either the entry or expert builders. Save the structures, and in the main window determine the heat of formation for each molecule using the following steps: * Setup/Semi-empirical * Task 5 Geometry Optimization, (a change from previous setup) * Model 5 AM1 * Solvent 5 Water (C–T) * Charge 5 0 (a change from previous setup) * Multiplicity 5 1 * Save * Setup/Submit Click on OK to clear the dialog box. This calculation will be quicker than the transition state search. Click on OK when the dialog box appears indicating that the calculation has finished, and retrieve the estimated heat of formation (Display/Properties/Energy). Values very close to the following should be obtained: H f8 for ethanol 5 269.164 kcal/mol H f8 for ethyl chloride 5 226.649 kcal/mol
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12.
13.
14.
15.
Note: Because of the solution technique and local minima, slightly different values may result. Build the Cl2 ionic species in the expert builder by selecting chlorine with terminal geometry ( —x ) and then deleting the free valence. Select Delete Atom and then click on the very top of the bond from the Cl atom and then enter a period, i.e. “.” from the keyboard. Do not minimize the chlorine atom (there is nothing to minimize!), save, and exit to the main SPARTAN window. Build the OH2 ion by selecting oxygen in the expert builder with terminal geometry ( —x ). Again, do not minimize: save, and exit to the main SPARTAN window. Calculate the heats of formation for each of the two ionic species. The following commands should be used to set up and execute the calculations: * Setup/Semi-empirical * Task 5 Single Point Energy (a change from previous setup) * Model 5 AM1 * Solvent 5 Water (C–T) * Charge 5 21 (a change from previous setup) * Multiplicity 5 1 * Save * Setup/Submit Click on OK to clear the dialog box indicating that the job has been submitted. These calculations will take only a few seconds as the geometry of the ion is not being changed to find a global energy minimum. Values close to the following should result: H f8 for OH2 5 2122.206 kcal/mol H f8 for Cl2 5 2114.669 kcal/mol
16. Use the heats of formation to calculate the heats of reaction and activation energy. ∑ H f8 ( Reactants ) 5 H f8 ( C2H5Cl ) 2 H f8 ( OH2 ) 5 2 26.65 1 2122.2 5 2148.85 kcal / mol ∑ H f8 ( Products ) 5 H f ( C2H5OH ) 1 H f8 ( Cl2 ) 5 2 69.16 1 2144.67 5 2183.83 kcal / mol For the forward reaction EA 5 H f8 ( Transition State ) 2 ∑ H f8 ( Reactants ) 5 2 121.85 2 ( 2148.85 ) 5 27 kcal / mol
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For the reverse reaction EAr 5 H f ( Transition State ) 2 ∑ H f8 ( Products ) 5 2 121.85 2 ( 2183.83 ) 5 62 kcal / mol D H Rx 8 5 ∑ H f8 ( Products ) 2 ∑ H f8 ( Products ) 5 2 183.83 2 ( 2148.85 ) 5 235 kcal / mol 17. The complete reaction energy diagram is shown below. TS TS
–121.85
E Faf Af R –148.85
P
–183.83 Reactants
Products Reaction Coordinate
[Example provided by Professor Robert M. Baldwin, Chemical Engineering Department, Colorado School of Mines in Golden, Colorado] CDPApp.J-1A
Redo Example Appendix J.1. (a) Choose different methods of calculation such as using a value of 2.0Å to constrain the C—Cl and C—O bonds. (b) Choose different methods to calculate the potential energy surface. Compare the Ab Initio to the semi-empirical method. (c) Within the semi-empirical method compare the AM1 and PM3 models.
CRE -- Kinetics Challenge 1
Kinetics Challenge 1 -- Quiz Show Concepts
Time Reference Description
Definitions of rates of reactions. Types of reactors. General mole balances for batch reactors, CSTR's and PFR's. 29 minutes ± 10 minutes Fogler: Chapter 1 This module allows students to test their knowledge about general mole balance equations, reaction rate laws, and different types of reactions and reactors. Individual students will find themselves going head-to-head against computer opponents in an interactive game with timed responses. Twenty multiple-choice questions are selected from a pool of approximately 100 possible questions, so the game will be different every time. The questions fall under four main categories: mole balance, reactions, rate laws, and reactor types; and there are five difficulty levels within each category. Each correct answer will earn the student a given number of points; the more difficult the question, the higher the point values.
The student has one minute to choose the correct answer. The module responds to the student's choice, either reinforcing the reasoning for a correct answer, or immediately clarifying a misunderstanding if an incorrect answer is entered. If no response is entered within the time limit, or if an incorrect response is entered, the points are lost, and one of the computer competitors tries to answer the question:
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CRE -- Kinetics Challenge 1
The competitor who last answered a question correctly gets to pick the next category and degree of difficulty. (Note that this will not necessarily always be the student). In addition to regular questions, one question is randomly assigned as the "Double Challenge" in which the student has the option of betting some or all of his/her points. After all twenty questions have been answered, the contestants with positive scores go on to the "Final Challenge" quesion, in which they are also allowed to bet points.
Grade Base
The game score is the number of accumulated points, including gains or losses from the Double Challenge (if applicable) and the Final Challenge. For the performance scores, the student is given 3 points for every correct answer in the 100-300 point range, and 7 points for each 400-500 point question. The Final Challenge is worth 8 points. 75 points are needed to achieve mastery of the module.
Comments
Students have used this module as review material before an exam, to ensure that they have a solid grasp of the basics of reaction kinetics. Some professors have also made use of it in recitation sections, inviting student volunteers to enter responses, then discussing any conceptual misunderstandings that might be discovered.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
Return to Chapter One
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CRE -- Staging
Staging -- Reactor Sequencing Optimization Game Concepts
Time Reference Description
Concentration as a function of conversion CSTR vs. PFR volume-conversion relationships Effect of changing order of reactor placement on final conversion. 35 minutes ± 10 minutes Fogler: Chapter 2 This module presents the student with a set of five reactors (CSTRs and PFRs) and asks him/her to connect them in series. The goal is to maximize the product flowrate for a given reaction, while maintaining a minimum conversion of 75%. The student is provided with a graph of -FAo/rA vs X, and the reactors' volumes are specified. The student may arrange the reactors in any order, and he/she may also vary the inlet flowrate. Each arrangement of reactors may be tested using a simulator that provides instant feedback for any change in reactor order or inlet flowrate.
The student may at any time access a reference section that reviews the derivation of the design equations for PFRs and CSTRs, clarifying the change in conversion down a PFR, and the wellmixedness of the CSTR:
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CRE -- Staging
The reactor optimization simulator can also be run independently of the scenario. This allows the professor to present the student with a variety of open-ended problems to be investigated using the simulator. Grade Base
The student’s score is based on the conversion achieved, as well as the total flow rate of material produced.
Comments
This module makes use of a scenario to increase the level of interest of the student. In the scenario, the student must generate a sufficient amount of an antidote of high enough purity to help Mr. Hyde get back to his Dr. Jeckyll persona.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
Return to Chapter Two
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CRE -- Kinetics Challenge 2
Kinetics Challenge 2 -- Quiz Show Concepts
Time Reference Description
Arrhenius equation Stoichiometry tables Rate laws 36 minutes ± 16 minutes Fogler: Chapter 3 This module focuses on rate laws and stoichiometry, allowing the students to master the elements of the stoichiometric table:
The interactive portion of the module is similar to that in Kinetic Challenge 1. Students can choose from four categories (reactants, products, rate law, potpourri) and four levels of difficulty (200-1,000 points). Each question has four multiple-choice answers, and students have a limited amount of time to make a response:
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CRE -- Kinetics Challenge 2
The goal of this module is to provide the students with practice setting up stoichiometric relationships, so that they will avoid mistakes, such as expressing the reaction rate law for an irreversible reaction as if it were reversible, or using the ideal gas law for liquid-phase reactions.
Grade Base
The game score is the number of accumulated points, including gains or losses from Double Challenge and Final Challenge. For the performance scores, the student is given 3 points for every correct answer in the 200-600 point range, and 7 points for the 800-1,000 point questions. The Final Challenge question is worth 8 points.
Comments
Students report that they find this module very useful as a review before the first examination. In some instances, some of the text strings in the last few questions and the Final Challenge question will become garbled, and the computer will lock up at the end of the module. We have been unable to determine the circumstances that give rise to this effort. Some students comment that the one minute time limit does not allow them enough time to derive the required expressions. It is helpful to suggest to students that they examine the four options available and choose the correct answer by process of elimination, based on the information provided in the problem statement, rather than trying to derive the expressions.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
Return to Chapter Three
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CRE -- Murder Mystery
Murder Mystery: CSTR Volume Algorithm Concepts Time Reference Description
Isothermal CSTR reactor design Problem solving and analysis 32 minutes ± 10 minutes Fogler: Chapter 4 The goal of this module is to allow students to practice the CSTR design algorithm:
In the interactive portion of the module the student must solve a murder mystery, with the aid of Sir Nigel Ambercrombie, an English investigator. It seems that overnight there was a slight irregularity in conversion for the CSTR at the Nutmega Spice Company:
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CRE -- Murder Mystery
It is feared that one of the employees may have been murdered by a fellow employee, and the dead body left within the reactor. By analyzing the conversion data, gathering information about the plant's personnel, and using their knowledge of CSTR design, students must determine the identity of both the murderer and the victim. Help may be obtained by questioning the suspects:
Grade Base
Successful solution of the murder mystery with a minimum of assistance.
Comments
Students have always enjoyed the murder mystery scenario in this module.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
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CRE -- Murder Mystery
Return to Chapter Four
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CRE -- Tic Tac
Tic Tac: Isothermal Reactor Design - Ergun, Arrhenius, and Van't Hoff Equations Concepts Time Reference Description
Interaction of the Ergun, Arrhenius, and Van't Hoff equations and other considerations in isothermal reactor design. 33 minutes ± 9 minutes Fogler: Chapter 4 This module allows the student to examine nine reactor design problems, and investigate the effect of varying reactor parameters on process performance. The problems are organized as in a tic-tactoe board. The reactors covered by these problems include PFRs, CSTRs, packed bed reactors and semi-batch reactors:
The student must master the concepts in enough squares to successfully win the tic-tac-toe game (Three adjacent squares horizontally, vertically, or diagonally). Each problem allows the student the opportunity to examine the effect of a specified operational parameter on reactor performance, using simulators:
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CRE -- Tic Tac
After performing the “experiments” the student proceeds to answer the questions that examine the effects observed. These effects can be explained through the Ergun, Arrhenius, and Van'’t Hoff equations In many cases, competing effects are highlighted. The square is "won" by answering two out of the three questions correctly. Grade Base
Grade based on mastery of concepts within each square, and successful completion of the tic-tactoe game.
Comments
Some students have found the questions in this module to be slightly above their current level of understanding. They have mentioned, however, that the process was helpful in exploring these concepts.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
Return to Chapter Four
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CRE -- Ecology
Ecology -- A Wetlands Problem Concepts Time Reference Description
Collection and analysis of rate data Ecological engineering concepts 49 minutes ± 21 minutes Fogler: Chapter 5 The student, as an employee of a company trying to meet environmental regulatory agency standards, must sample concentration data for a toxic material found in a wetlands channel between a chemical plant upstream and a protected waterway to analyze the rate of decay of the toxic material.
The wetlands are modeled as a PFR. The student must first develop the necessary reactor design equation for a PFR, then start collecting data. This concentration data (which includes experimental error) is then analyzed in various ways (polynomial fit of the data followed by differentiation of the resulting equation, difference equations, etc.) to determine the rate law, as well as the rate constants and reaction order. Students must determine which points are to be excluded from the analysis (if any) and which points may be resampled:
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CRE -- Ecology
The student then analyzes this information and submits a memo with the requested parameters:
This information is reviewed by the boss, who evaluates the parameter values and makes recommendations. Grade Base
Based on correct determination of rate law parameters.
Comments
This module is useful in exposing the student to experimental error and the dangers of using curve-fitting tools without discretion. It also exposes the student to “real world” applications of reaction kinetics. To introduce an element of levity, the student performs the analysis in a "Mr. Potatohead" persona. Students have reported enjoying this.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
Return to Chapter Five
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CRE -- Heat Effects 1
Heat Effects 1 -- Basketball Challenge Concepts Time Reference Description
Effect of parameter variations on operation of a nonisothermal CSTR 36 minutes ± 14 minutes Fogler: Chapter 8 This module allows students to investigate the effect of parameter variations on the operation of a nonisothermal CSTR. An extensive review section derives the energy balance for the CSTR, and also describes the terms in the mole balance that are temperature dependent:
A simulator is also included in the review section. This allows the student to vary parameters and to observe the effects on the conversion-temperature relationships as described by both the mole balance and the energy balance. The parameters that may be varied include: feed flowrate, feed temperature, the reversibility/irreversibility of the reaction, heat of reaction, heat exchanger area, and heat transfer fluid temperature. The operating conditions can be determined from the intersection(s) of the mole balance and the energy balance:
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CRE -- Heat Effects 1
The module can also be run in an interactive mode, in which the scenario takes the student to a basketball tournament. They have the choice of two-point and three-point questions:
A simulator is available to help the student answer the three-point questions. Grade Base
A grade is only assigned in the interactive mode. The student is given a “shooting percentage” for the t-o point and thr-p point questions, as well as an overall shooting percentage. A shooting percentage greater than 85% demonstrates mastery of the module.
Comments
This module requires a lot of memory to run.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
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CRE -- Heat Effects 1
Return to Chapter Eight
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CRE -- Heat Effects 2
Heat Effects 2 -- Effect of Parameter Variation on a PFR Concepts Time Reference Description
Effect of parameter variations on operation of a nonisothermal plug flow reactor 61 minutes ± 36 minutes Fogler: Chapter 8 This simulation allows the student to explore the effects of various parameters on the performance of a non-isothermal plug flow reactor. The student may choose from eight simulations that span all combinations of exothermic/endothermic reactions and reversible/irreversible reactions, as well as a simulation that takes pressure drop into account. The variable parameters include the heat transfer coefficient, the inlet reactant flowrate, the diluent flowrate, the inlet temperature, and the ambient temperature:
The results of the simulator may be analyzed in the form of plots of concentration, conversion, or temperature as a function of reactor volume. The module may also be run in an interactive mode, in which the student must achieve specific goals (e.g. achieve a given conversion without exceeding a given temperature within the reactor), in order to get to the center of the reactor complex. The review section includes a derivation of the energy balance equation for a PFR:
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CRE -- Heat Effects 2
Grade Base
In the interactive mode, mastery is based on the correct solution of two consecutive problems (e.g.: arriving at the center of the reactor complex).
Comments
We have used the simulator portion of this module as a tool in a group problem solving exercise. Students had to vary various parameters and explain their observations, then use the newly gained insight to optimize a system. A sample assignment, for System 2 in the “individual problems” menu, follows. Since the assignment was to be completed within a one hour class period, explicit instructions and suggestions for parameter values were given.
Sample Assignment
You are to investigate how some important reactor parameters affect the conversion and the temperature profiles down a tubular reactor. You will be told which parameter to vary, then asked to explain the results you observe. In each case, in addition to a general statement ("increase UA"), you will be given a set of optional reactor conditions to use, in the order in which they appear in the left-hand side of the simulation screen: (UA, Ta, Fio, Fao, To). You may use these conditions if you wish, or pick your own for your investigation. GETTING STARTED Choose "5. individual problems" from the main menu , then choose problem 2. “Endothermic irreversible." Once the F-key bar at the bottom shows up, you may want to hit "F2" for a short description of each of the components of the simulator. Things to keep in mind, once you are running the simulator: • To change the step size in varying the parameters: 1(smallest step size) 10 (largest one) • To delete a run so its curve doesn't show up on the grap, -select it and hit backspace. • Not sure what keys to hit -Press I for Information. EFFECTS OF HEAT EXCHANGE To analyze the effect of the heat exchanger on the reaction, compare the conversion and the temperature profiles with and without heat exchange: Set the y-axis to temperature - Choose temperature with the arrow keys.
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CRE -- Heat Effects 2
Select your first run - Hit until the yellow selector box is around the UA box Perform a run with UA equal to 0 (e.g., UA=0, Ta=300, Fio=10, Fao=10, To=300) How does the temperature change with volume down the reactor with no heat exchange for an endothermic irreversible reaction? Select a second run - Use the arrows to select the blue run. Perform a new run with a higher UA (e.g. UA=250, Ta=300, Fio=10, Fao=10, To=300) How does the temperature change down the reactor with heat exchange for an endothermic irreversible reaction? Set the y-axis to conversion - Press "A" for axes How do the conversion profiles for the cases with and without heat exchange compare? EFFECTS OF FLOW PARAMETERS Perform a new run with no inerts. (e.g. UA=250, Ta=300, Fio=0, Fao=10, To=300) Now perform a run with a higher reactant rate. (e.g. UA=250, Ta=300, Fio=0, Fao=20, To=300) How does the presence of inerts affect the results from the previous question? Perform a run with inerts (e.g. UA=250, Ta=300, Fio=10, Fao=20, To=300). Compare the temperature profiles for the above three cases. APPLICATION Given your new-found intuition, try to get the highest conversion given the limitation that the reactor temperature (at ALL positions within the reactor), must be between 250-300 K. An easily achieved value is 0.50. The highest conversion found so far is 0.711. Turn in the conditions you used (UA,Ta,Fio, Fao,To) as well as the conversion obtained, and a few sentences explaining what you learned. SUMMARY Write a paragraph (1/2 to 1 page) describing the effects of heat exchange on the reaction and the effects of changes in the reactant flowrate and the inert flowrate on conversion and temperature profiles for the tubular reactor. Include sketches illustrating the trends and the equations necessary to predict the results. Based on these results can you predict what would happen in an exothermic, irreversible reaction? How about reversible reactions? Installation
Instructions for installing the ICMs and for using the ICMs are available.
Return to Chapter Eight
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CRE -- Heterogeneous Catalysis
Hetcat: Heterogeneous Catalysis Concepts Time Reference Description
Derivation of catalytic rate equations based on experimental data Selection of reaction mechanism and rate-limiting step that support the rate equation 33 minutes ± 13 minutes Fogler: Chapter 10 The review section of this module reviews the essential elements of heterogeneous catalysis:
The student must derive the rate equation for a given reactive system by analyzing the rate data obtained in a differential reactor. The student must choose which experiments to run, that is, the entering pressures of each species and total flow rate. In order to obtain the dependence of the rate equation on the pressure of a given species, the student must select which of the points are to be included in a plot of reaction rate vs. species partial pressure. Given the requested plot, the student must determine the form of the dependence of the rate law on the pressure of the given species:
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CRE -- Heterogeneous Catalysis
Once all dependencies have been established, the student must decide which rate law parameters can be determined, through judicious plotting of the experimental data. The review section also outlines the derivation of the governing equations of heterogeneous catalysis:
Grade Base
Based merely on completion of the module, i.e. derivation of the reaction rate expression.
Comments
Students reported that this module was very helpful to them in preparing to do the homework problems from the textbook. This module requires a large amount of memory to run.
Installation
Instructions for installing the ICMs and for using the ICMs are available.
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CRE -- Heterogeneous Catalysis
Return to Chapter Ten
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CRE -- Installing the ICMs
Installing the ICMs A Note about the ICMs The interactive computer modules (ICMs) were written for a DOS environment; therefore, you will need to have a PC to use them. Our apologies to Mac owners, who either don't have PC-cards installed, or whose PC-emulator software is slow or unstable. The ICMs may be converted to a more platform-independent format in the future, but for now, PCs or PC-emulators are the only options. So, if you have a PC, you can run the ICMs directly from the CD-ROM. See the page on using the ICMs for more information.
Installation Procedure For your convenience, the interactive computer modules have also been included on the CD-ROM in two compressed formats: (1) as a zip file (newmods.zip) and (2) as a self-extracting, executable file (newmods.exe), which might require you to have a copy of WinZip (a free shareware utility for Windows 95) to run it. This is so you can run the ICMs without needing a CD-ROM for every computer. They are located in the ICMs\Files directory (and also in the Html\DOS_Mod\Files directory) on your CD-ROM.
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CRE -- Installing the ICMs
Using newmods.exe Do the following: 1. Using Windows Explorer, select your CD-ROM drive and locate the newmods.exe program file. As mentioned above, it should be in the ICMs\Files directory. 2. Double-click on the newmods.exe file to run this self-extracting installation program.
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CRE -- Installing the ICMs
3. This will bring up a pop-up window that looks like this:
You'll notice that the "Unzip To Folder:" textfield already has C:\Newmods listed as the location to which it will unzip files. (You may change this "unzip to" location if you do not want install the ICMs to this default directory.) 4. Click the Unzip button to have the program automatically install the Interactive Computer Modules for you. 5. After the files unzip themselves, click the Close button to exit the self-extractor program. 6. Select the Newmods directory that was created on your C: drive.
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CRE -- Installing the ICMs
At this point, you should see a number of files and directories in there. The important files are the ones with names like mystery.bat and kinchal1.bat. These are known as batch files. You can doubleclick one of these batch files to run an ICM. NOTE: Each ICM has its own batch file and a directory that houses the "guts" of the ICM. Do not open the directory. Use the batch file to run the corresponding ICM. 7. See the page on using the ICMs for more information.
Using newmods.zip Do the following: 1. Using Windows Explorer, create a directory on your hard drive called Newmods. For example, select your C: drive. Then under File, select New -- Folder. Type in Newmods as the folder name. 2. Select your CD-ROM drive and open the ICMs folder. 3. Open the Files folder and select the newmods.zip file. 4. Unzip the newmods.zip file into the Newmods folder that you created in step one. There are two programs that I recommend. The first is PKUNZIP and the second is
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CRE -- Installing the ICMs
WinZip. Both should be available as shareware from one of the major shareware sites, such as SHAREWARE.COM or DOWNLOAD.COM. WinZip is available from its own site at www.winzip.com. 5. Select the Newmods directory that you created earlier. At this point, you should see a number of files and directories in there. The important files are the ones with names like mystery.bat and kinchal1.bat. These are known as batch files. You can double-click one of these batch files to run an ICM. NOTE: Each ICM has its own batch file and a directory that houses the "guts" of the ICM. Do not open the directory. Use the batch file to run the corresponding ICM. 6. See the page on using the ICMs for more information.
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CRE -- Using the ICMs
Using the ICMs You can run the interactive computer modules directly from the CD-ROM (PCs only) by accessing the ICMs directory (or the Html\DOS_Mod\Newmods directory) on your CD-ROM, and then following the instructions, below. Otherwise, if decide to run the ICMs from your hard-drive, but you have not yet installed them on your computer, then read the information on installing the ICMs.
Once you're set, do the following: 1. Make sure you are in the directory that contains the ICMs. For example, if you're running the interactive computer modules from your CD-ROM, then you can use Windows Explorer to go to the ICMs directory (or the Html\DOS_Mod\Newmods directory).
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CRE -- Using the ICMs
If you installed the interactive computer modules on your computer's hard drive, in the directions on the installation page you were asked to create a directory on your hard drive called Newmods. 2. Run the batch file (file extension .bat) for the ICM you wish to use. In the Windows Explorer program of Windows 95, double-click on the batch file that you want to run, or at the DOS Prompt, type the batch file name and then hit ENTER (a.k.a. RETURN). For example, say you want to run the Mystery Theater module. In Windows 95, you would double-click on the mystery.bat file to run that ICM.
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CRE -- Using the ICMs
In DOS, you would type mystery.bat and then hit ENTER to run the ICM. 3. Follow the directions within each module. (The right arrow key is used to advance to new material, unless otherwise noted.) 4. Each ICM will instruct you on how to exit that particular module.
List of Interactive Computer Modules Module Name
Ecology Heat Effects 1 Heat Effects 2 Heterogeneous Catalysis Kinetics Challenge 1 Kinetics Challenge 2 Murder Mystery Staging Tic Tac file:///H:/html/dos_mod/using.htm[05/12/2011 16:55:03]
File Name ecology.bat heatfx1.bat heatfx2.bat hetcat.bat kinchal1.bat kinchal2.bat mystery.bat staging.bat tictac.bat
CRE -- Using the ICMs
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Polymath Short Course
Polymath Short Course by Dieter Andrew Schweiss
The following mini-course on Polymath will provide you with the bare minimum that you need to know, in order to use Polymath to look at the Living Example Problems. For more complete instructions on using Polymath, please be sure to review the official Polymath Manual.
1. Starting Polymath A. Windows 95/NT Procedure 1. Using Windows Explorer, switch to the Polymat4 directory on your CD-ROM.
2. Double-click on the Polymath batch file (Polymath.bat) to run Polymath.
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Polymath Short Course
NOTE: Polymath will run in its own, full-screen DOS Window. To be able to follow the step-by-step instructions of the Polymath Short Course and use Polymath at the same time, you will have to make frequent use of the ALT-ENTER keys to alternate between the Polymath Short Course and Polymath itself.
B. DOS Procedure 1. If you are already in DOS (or in a DOS window), then move to the next step. Otherwise, open a DOS window by double-clicking on the DOS Prompt. In Windows 95/NT, the DOS Prompt should be available as an icon on your desktop or in your Start menu. 2. Switch to Polymat4 directory. This will either be the one on your CD-ROM, or the one Polymath is installed to on your hard drive (if you installed Polymath). 3. Type polymath at the command prompt and hit ENTER to run Polymath. NOTE: Polymath will run in its own, full-screen DOS Window. To be able to follow the step-by-step instructions of the Polymath Short Course and use Polymath at the same time, you will have to make frequent use of the ALT-ENTER keys to alternate between the Polymath Short Course and Polymath itself. Page 1 | Page 2 | Page 3 | Page 4 | Page 5
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Polymath Short Course
Polymath Main | Using Polymath | Installing Polymath | Short Course
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Installing Polymath
Installing Polymath The following files have been included on your CD-ROM in the Polymat4\Files\poly402 directory:
These files will allow you to install Polymath 4.0.2 on your hard drive. Please read the Polymath 4.0.2 Manual (the mpoly402.pdf file) to learn how to use Polymath. The library.zip and the library.exe files are used to install the Living Example Problems, discussed below. In addition to the installation instructions that follow, please read the readme.txt file, since it also contains information about installing Polymath. The installation files for Polymath 4.1 are also on the CD (in the Polymat4\Files\poly41 directory).
Why Two Versions of Polymath? This is the second printing of the text and the CD. Polymath 4.1 has better printing features than Polymath 4.0.2, but it was not available for the first printing of the CD. We could have replaced Polymath 4.0.2 with Polymath 4.1, but we
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Installing Polymath
decided to make both versions available instead.
1. Installing Polymath This is the Polymath 4.0.2 installation procedure. The Polymath 4.1 installation procedure is similar, yet much simpler, so refer to the Polymath 4.1 readme.txt file, if you plan to install Polymath 4.1 instead. Since Polymath is a DOS-based program, directions have been written so that you may use either Windows 95/NT or a DOS Prompt to install Polymath. The installation instructions for Windows 95/NT are given below. The DOS installation instructions are located on a separate page.
A. Windows 95/NT Procedure 1. Using Windows Explorer, select your CD-ROM drive and switch to the directory containing the Polymath files. As mentioned above, they should be in the Polymat4\Files\poly402 directory. 2. Double-click on the install.exe file to run the Polymath installation program.
A DOS window will open, since Polymath is a DOS-based program. 3. The installation program will ask you the following questions: a. Enter drive and directory for POLYMATH [C:\POLYMAT4] :
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Installing Polymath
Polymath is suggesting that you install to a default directory called Polymat4. Please select this directory by simply hitting the RETURN key. NOTE: It is important for you to install Polymath in the default directory, because you will also have to install the library files for the Living Example Problems in this directory. b. Is this a network installation? [N] Type N when you're asked this question, even if you're running your computer on a network. Trust me. c. Please select the type of output device you prefer [1]: Polymath wants to know if you have a (1) printer or (2) plotter. The default response is that you have a printer, which is what we assume you will select, too. d. Please select the type of printer you have: There should be a long list of printers here. If you don't see your printer in the list, we suggest you select the one that comes closest. NOTE: If you have trouble printing with with Polymath (especially if you print on a network), then contact the authors of Polymath directly. Their contact information should be in the Polymath Manual. e. Please select the mode you want output in [1]: You'll be offered a list of printing options to choose from; the options are similiar, yet different for each printer. We suggest you choose a portrait (or full page) option for printing code or tables, and a landscape option for printing graphs. f. Please select the port your printer is connected to [1]: You'll be offered a list of printing ports to choose from; most people have printers on LPT1. 4. Polymath should now be able to run on your machine. Close the DOS window and switch to the Polymat4 directory on your hard drive. 5. Double-click on the Polymath batch file (Polymath.bat) to run Polymath.
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Installing Polymath
IMPORTANT: You will need to follow the instructions for installing the library files before you can access the Polymath code for the Living Example Problems. 6. Now read the instructions on using Polymath or take the Polymath Short Course.
2. Installing the Library Files The Polymath code for the Living Example Problems is contained in a file called library.zip, which is located in the Polymat4\Files directory on your CD-ROM. For your convenience, a second file called library.exe is in that same directory. This program will automatically install the Polymath code for the Living Example Problems in the default Polymath directory on your hard drive (C:\Polymat4).
A. Windows 95/NT Procedure 1. Using Windows Explorer, select your CD-ROM drive and locate the library.exe program file. As mentioned above, it should be in the Polymat4\Files\poly402 directory. 2. Double-click on the library.exe file to run this self-extracting installation program.
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Installing Polymath
3. This will bring up a pop-up window that looks like this:
You'll notice that the "Unzip To Folder:" textfield already has C:\polymat4 listed as the location to which it will unzip files. (You may change this "unzip to" location if you didn't install Polymath to its default directory.) 4. Click the Unzip button to have the program automatically install the library files for you. 5. After the library files unzip themselves, click the Close button to exit the self-extractor program. You may now access the Living Example Problems using Polymath. Polymath Main | Using Polymath | Installing Polymath | Short Course
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Installing Polymath
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Using Polymath
Using Polymath Polymath Manual Polymath is easy to use -- once you know how. Before you use Polymath for the first time, take a look at the Polymath Manual for instructions on how to run and generally use Polymath: Polymath 4.0.2 Manual in PDF format. Polymath 4.1 Manual in PDF format. NOTE: You will need the Adobe Acrobat Reader Plug-in to read this file. (See the CD-ROM Introduction for more information.) If you just can't wait to dig into Polymath, then take a look at the Polymath Short Course. It won't teach you everything, but it will get you started.
Why Two Versions of Polymath? This is the second printing of the text and the CD. Polymath 4.1 has better printing features than Polymath 4.0.2, but it was not available for the first printing of the CD. We could have replaced Polymath 4.0.2 with Polymath 4.1, but we decided to make both versions available instead.
Accessing the Polymath Examples Before you begin, refer to the Polymath Manual for specific instructions on using Polymath. More-detailed instructions for using Polymath and opening the Living Example Problems are available in the Polymath Short Course. Some of the examples are for use with Polymath's ordinary differential equation (ODE) solver, while others can be used with the non-linear regression solver. If you follow the link to a specific example problem to see the code for it, the code will give you an idea as to which tool to use to open the example problem. You can tell the difference between the two types of code, in the way the data is presented: For instance, Example 5-6 holds Polymath code for a non-linear regression. The code has data in a tabular format, since you enter regression data in a table. By comparison, Example 4-10 holds Polymath code for the ODE solver. The code presents differential equations and constants as a list of equations, since that is how the information would be entered in the ODE solver.
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Using Polymath
To actually load a Living Example Problem: 1. Select the correct option from Polymath's main menu. (You'll probably select either option 1 or option 4 most of the time.) 2. On the next screen, hit F9 for file options. 3. At the TASK MENU, hit F9 again for LIBRARY OPTIONS. 4. Select the example you want to load from the list. (NOTE: The files in the list will depend on the solver you're using.) 5. When the example loads, you will be instructed to hit any key to look at the example problem, so do that. 6. Now you can play around with the example as you see fit. See the Polymath Manual for further instructions. Confused? Please take a look at the Polymath Short Course for a more-detailed explanation of the information given here, as well as a step-by-step example of how to navigate through Polymath. Polymath Main | Using Polymath | Installing Polymath | Short Course
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POLYMATH VERSION 4.02
USER-FRIENDLY NUMERICAL ANALYSIS PROGRAMS - SIMULTANEOUS DIFFERENTIAL EQUATIONS - SIMULTANEOUS ALGEBRAIC EQUATIONS - SIMULTANEOUS LINEAR EQUATIONS - POLYNOMIAL, MULTIPLE LINEAR AND NONLINEAR REGRESSION for IBM and Compatible Personal Computers THIS MANUAL AND SOFTWARE ARE TO ACCOMPANY Elements of Chemical Reaction Engineering - 3rd Edition by H. Scott Fogler and published by Prentice Hall
POLYMATH Internet Site http://www.polymath-software.com Users are encouraged to obtain the latest general information on POLYMATH and its use from the above Internet site. This will include updates on this version and availability of future versions.
Copyright 1998 by M. Shacham and M. B. Cutlip This manual may be reproduced for educational purposes by licensed users. IBM and PC-DOS are trademark of International Business Machines MS-DOS and Windows are trademarks of Microsoft Corporation
i -2 PREFACE
POLYMATH 4.0 PC
POLYMATH LICENSE AGREEMENT The authors of POLYMATH agree to license the POLYMATH materials contained within this disk and the accompanying manual.pdf file to the owner of the Prentice Hall textbook Elements of Chemical Reaction Engineering, Third Edition, by H. Scott Fogler. This license is for noncommercial and educational uses exclusively. Only one copy of this software is to be in use on only one computer or computer terminal at any one time. One copy of the manual may be reproduced in hard copy only for noncommercial educational use of the textbook owner. This individual-use license is for POLYMATH Version 4.02 and applies to the owner of the textbook. Permission to otherwise copy, distribute, modify or otherwise create derivative works of this software is prohibited. Internet distribution is not allowed under any circumstances. This software is provided AS IS, WITHOUT REPRESENTATION AS TO ITS FITNESS FOR AND PURPOSE, AND WITHOUT WARRANTY OF ANY KIND, EITHEREXPRESS OR IMPLIED, including with limitation the implied warranties of merchantability and fitness for a particular purpose. The authors of POLYMATH shall not be liable for any damages, including special, indirect, incidental, or consequential damages, with respect to any claim arising out of or in connection with the use of the software even if users have not been or are hereafter advised of the possibility of such damages. HARDWARE REQUIREMENTS POLYMATH runs on the IBM Personal Computer and most compatibles. A floating-point processor is required. Most graphics boards are automatically supported. The minimum desirable application memory is 450 Kb plus extended memory for large applications. POLYMATH works with PC and MS DOS 3.0 and above. It can also execute as a DOS application under Windows 3.1, Windows 95, and Windows NT. It is important to give POLYMATH as much of the basic 640 Kb memory as possible and up to 1024 Kb of extended memory during installation. A variety of drivers are provided for most printers and plotters. Additionally some standard graphics file formats are supported. POLYMATH 4.0 PC
PREFACE i-3
TABLE OF CONTENTS - POLYMATH PAGE INTRODUCTION POLYMATH OVERVIEW..................................................................... 1 MANUAL OVERVIEW.......................................................................... 1 INTRODUCTION................................................................................ 1 GETTING STARTED.......................................................................... 1 HELP.................................................................................................... 1 UTILITIES........................................................................................... 1 APPENDIX........................................................................................... 1 DISPLAY PRESENTATION................................................................. 1 KEYBOARD INFORMATION............................................................. 1 ENTERING VARIABLE NAMES........................................................ 1 ENTERING NUMBERS......................................................................... 1 MATHEMATICAL SYMBOLS............................................................ 1 MATHEMATICAL FUNCTIONS........................................................ 1 LOGICAL EXPRESSIONS.................................................................... 1 POLYMATH MESSAGES...................................................................... 1 HARD COPY........................................................................................... 1 GRAPHICS.............................................................................................. 1 -
1 2 2 2 2 2 2 3 3 4 4 5 5 6 6 6 6
GETTING STARTED HARDWARE REQUIREMENTS......................................................... POLYMATH SOFTWARE .................................................................... INSTALLATION TO INDIVIDUAL COMPUTERS & NETWORKS. FIRST TIME EXECUTION OF POLYMATH..................................... EXITING POLYMATH PROGRAM.....................................................
2 2 2 2 2
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1 1 1 2 3
HELP MAIN HELP MENU............................................................................... ACCESSING HELP BEFORE PROBLEM ENTRY.......................... ACCESSING HELP DURING PROBLEM ENTRY........................... CALCULATOR HELP........................................................................... UNIT CONVERSION HELP.................................................................
3 3 3 3 3
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1 2 2 3 3
UTILITIES CALCULATOR....................................................................................... 4 CALCULATOR EXPONENTIATION............................................... 4 AVAILABLE FUNCTIONS................................................................ 4 ASSIGNMENT FUNCTIONS............................................................. 4 CALCULATOR EXAMPLES............................................................. 4 UNIT CONVERSION............................................................................ 4 PREFIXES FOR UNITS...................................................................... 4 UNIT CONVERSION EXAMPLE...................................................... 4 i-4 PREFACE POLYMATH 4.0
- 1 - 1 - 1 - 3 - 3 - 5 - 5 - 6 PC
PROBLEM STORAGE........................................................................... FILE OPERATIONS............................................................................ LIBRARY OPERATIONS..................................................................... LIBRARY STORAGE.......................................................................... LIBRARY RETRIEVAL...................................................................... PROBLEM OUTPUT AS PRINTED GRAPHICS............................. SAMPLE SCREEN PLOT.................................................................... OPTIONAL SCREEN PLOT................................................................ PRESENTATION PLOT....................................................................... PROBLEM OUTPUT TO SCREEN AND AS PRINTED TABLES.. PROBLEM OUTPUT AS DOS FILES.................................................. PROBLEM OUTPUT AS GRAPHICS FILES.....................................
4 - 7 4 - 7 4 - 8 4 - 8 4 - 8 4 - 9 4 - 9 4-10 4-10 4- 10 4-11 4-11
DIFFERENTIAL EQUATIONS SOLVER QUICK TOUR.......................................................................................... DIFFERENTIAL EQUATION SOLVER............................................. STARTING POLYMATH.................................................................... SOLVING A SYSTEM OF DIFFERENTIAL EQUATIONS.............. ENTERING THE EQUATIONS.......................................................... ALTERING THE EQUATIONS.......................................................... ENTERING THE BOUNDARY CONDITIONS................................. SOLVING THE PROBLEM................................................................. PLOTTING THE RESULTS................................................................. EXITING OR RESTARTING POLYMATH....................................... INTEGRATION ALGORITHMS.......................................................... TROUBLE SHOOTING......................................................................... SPECIFIC ERROR MESSAGES.......................................................... NONSPECIFIC ERROR MESSAGES.................................................
5 5 5 5 5 5 5 5 5 5 5 5 5 5
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1 1 1 2 3 4 4 5 6 7 8 9 9 9
ALGEBRAIC EQUATIONS SOLVER QUICK TOUR.......................................................................................... NONLINEAR ALGEBRAIC EQUATION SOLVER......................... STARTING POLYMATH.................................................................... SOLVING ONE NONLINEAR EQUATION...................................... SOLVING A SYSTEM OF NONLINEAR EQUATIONS.................. EXITING OR RESTARTING POLYMATH....................................... SELECTION OF INITIAL ESTIMATES FOR THE UNKNOWNS METHOD OF SOLUTION.................................................................... TROUBLE SHOOTING.........................................................................
6 6 6 6 6 6 6 6 6
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1 1 1 3 5 7 7 7 8
POLYMATH 4.0 PC
PREFACE i-5
LINEAR EQUATIONS SOLVER QUICK TOUR.......................................................................................... LINEAR EQUATION SOLVER......................................................... STARTING POLYMATH.................................................................... SOLVING FIVE SIMULTANEOUS EQUATIONS............................ EXITING OR RESTARTING POLYMATH.......................................
7 7 7 7 7
REGRESSION QUICK TOUR......................................................................................... REGRESSION PROGRAM.................................................................. STARTING POLYMATH.................................................................... QUICK TOUR PROBLEM 1................................................................ RECALLING SAMPLE PROBLEM 3.... ............................................ FITTING A POLYNOMIAL................................................................ FITTING A CUBIC SPLINE................................................................ EVALUATION OF AN INTEGRAL WITH THE CUBIC SPLINE... MULTIPLE LINEAR REGRESSION.................................................. RECALLING SAMPLE PROBLEM 4................................................. SOLVING SAMPLE PROBLEM 4...................................................... TRANSFORMATION OF VARIABLES............................................. NONLINEAR REGRESSION.............................................................. EXITING OR RESTARTING POLYMATH....................................... SOLUTION METHODS.........................................................................
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
APPENDIX INSTALLATION AND EXECUTION INSTRUCTIONS................... DOS INSTALLATION......................................................................... DOS EXECUTION............................................................................... WINDOWS 3.1 INSTALLATION....................................................... WINDOWS 3.1 EXECUTION.............................................................. WINDOWS 95 INSTALLATION........................................................ WINDOWS 95 EXECUTION............................................................... INSTALLATION QUESTIONS............................................................ "OUT OF ENVIRONMENT SPACE" MESSAGE............................. CHANGING PRINTER SELECTION.................................................. PRINTING FOR ADVANCED USERS................................................ PRINTING TO STANDARD GRAPHICS FILES...............................
9 9 9 9 9 9 9 9 9 9 9 9
i-6 PREFACE
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POLYMATH 4.0 PC
INTRODUCTION POLYMATH OVERVIEW POLYMATH is an effective yet easy to use computational system which has been specifically created for professional or educational use. The various programs in the POLYMATH series allow the user to apply effective numerical analysis techniques during interactive problem solving on a personal computer. Whether you are student, engineer, mathematician, scientist, or anyone with a need to solve problems, you will appreciate the ease in which POLYMATH allows you to obtain solutions. Chances are very good that you will seldom need to refer to this manual beyond an initial reading because POLYMATH is so easy to use. With POLYMATH, you are able to focus your attention on the problem at hand rather than spending your valuable time in learning how to use or reuse the program. You are encouraged to become familiar with the mathematical concepts being utilized in POLYMATH. These are discussed in most textbooks concerned with numerical analysis. The available programs in POLYMATH include: - SIMULTANEOUS DIFFERENTIAL EQUATION SOLVER - SIMULTANEOUS ALGEBRAIC EQUATION SOLVER - SIMULTANEOUS LINEAR EQUATION SOLVER - POLYNOMIAL, MULTIPLE LINEAR AND NONLINEAR REGRESSION Whether you are a novice computer user or one with considerable computer experience, you will be able to make full use of the programs in POLYMATH which allow numerical problems to be solved conveniently and interactively. If you have limited computer experience, it will be helpful for you to read through this manual and try many of the QUICK TOUR problems. If you have considerable personal computer experience, you may only need to read the chapters at the back of this manual on the individual programs and try some of the QUICK TOUR problems. This manual will be a convenient reference guide when using POLYMATH. POLYMATH 4.0 PC
INTRODUCTION 1-1
MANUAL OVERVIEW This manual first provides general information on features which are common to all of the POLYMATH programs. Particular details of individual programs are then presented. Major chapter topics are outlined below: INTRODUCTION The introduction gives an overview of the POLYMATH computational system and gives general instructions for procedures to follow when using individual POLYMATH programs. GETTING STARTED This chapter prepares you for executing POLYMATH the first time, with information about turning on the computer, loading POLYMATH, and making choices from the various menu and option screens. HELP On-line access to a general help section is discussed. UTILITIES This chapter discusses features that all programs have available. These include a scientific calculator and a convenient conversion for units and dimensions. This chapter discusses saving individual problems, data and/or result files on a floppy or hard disk. It also describes the use of the problem library for storing, retrieving and modifying problems on a disk. Options for the printing and plotting of results are explained. The remaining chapters of the manual present a QUICK TOUR of each individual POLYMATH program and are organized according to the following subsections: 1. PROGRAM OVERVIEW This subsection gives general details of the particular program. 2. QUICK TOUR You can use this subsection to see how easy it is to enter and solve a problem with a particular POLYMATH program. APPENDIX Detailed installation instructions and additional output options are presented for advanced users. 1-2 INTRODUCTION
POLYMATH 4.0 PC
DISPLAY PRESENTATION Throughout this manual, a full screen is indicated by a total enclosure:
An upper part of screen is contained within a partial enclosure:
A lower part of screen is shown by a partial enclosure:
An intermediate part of a screen is given between vertical lines:
The option box is given by:
KEYBOARD INFORMATION When using POLYMATH, it is not necessary to remember a complex series of keystrokes to respond to the menus, options, or prompts. The commands available to you are clearly labeled for easy use on each display. Normally the keystrokes which are available are given on the display as indicated on the PROBLEM OPTIONS display shown below.
POLYMATH 4.0 PC
INTRODUCTION 1-3
USING THE KEY symbol is used to indicate In this manual as in POLYMATH, the the carriage return key which is also called the enter key. Usually when you are responding to a menu option, the enter key is not required. However, when data or mathematical functions are being entered, the enter key is used to indicate that the entry is complete. SHIFTED KEYPRESSES Some options require that several keys be pressed at the same time. This is indicated in POLYMATH and in this manual by a dash between the keys such as a ⇑ F8 which means to press and hold the ⇑ or "shift" key, then press the F8 function key and finally release both keys. THE EDITING KEYS Use the left and right arrow keys to bring the cursor to the desired position, while editing an expression. Use the Del key to delete the character (Back Space) key to delete the first character to above the cursor or the the left of the cursor. Typed in characters will be added to the existing expression in the first position left to the cursor. BACKING UP KEYS Press either the F8 or the Esc key to have POLYMATH back up one program step. ENTERING VARIABLE NAMES A variable may be called by any alphanumeric combination of characters, and the variable name MUST start with a lower or upper case letter. Blanks, punctuation marks and mathematical operators are not allowed in variable names. Note that POLYMATH distinguishes between lower and upper case letters, so the variables 'MyVar2' and 'myVar2' are not the same. ENTERING NUMBERS All numbers should be entered with the upper row on the key board or with the numerical keypad activated. Remember that zero is a number from the top row and not the letter key from the second row. The number 1 is from the top row while letter l is from the third row.
1-4 INTRODUCTION
POLYMATH 4.0 PC
The results of the internal calculations made by POLYMATH have at least a precision of eight digits of significance. Results are presented with at least four significant digits such as xxx.x or x.xxx . All mathematical operations are performed as floating point calculations, so it is not necessary to enter decimal points for real numbers. MATHEMATICAL SYMBOLS You can use familiar notation when indicating standard mathematical operations. Operator +
Meaning addition subtraction multiplication division power of 10
Symbol + * / x.x10a
Entry + x * -: / x.xea x.xEa (x.x is numerical with a decimal and a is an integer) exponentiation rs r**s or r^s MATHEMATICAL FUNCTIONS Useful functions will be recognized by POLYMATH when entered as part of an expression. The arguments must be enclosed in parentheses: ln (base e) abs (absolute value) sin arcsin sinh log (base 10) int (integer part) cos arccos cosh exp frac (fractional part) tan arctan tanh POLYMATH 4.0 PC
exp2(2^x) round (rounds value) sec arcsec arcsinh exp10 (10^x) sign (+1/0/-1) csc arccsc arccosh sqrt (square root) cbrt (cube root) cot arccot arctanh INTRODUCTION 1-5
LOGICAL EXPRESSIONS An "if" function is available during equation entry with the following syntax: if (condition) then (expression) else (expression). The parentheses are required, but spaces are optional. The condition may include the following operators: > greater than < less than >= greater than or equal 0) then(log(x)) else(0) b=if (TmaxT) then (maxT) else (T)) POLYMATH MESSAGES There are many POLYMATH messages which may provide assistance during problem solving. These messages will tell you what is incorrect and how to correct it. All user inputs, equations and data, are checked for format and syntax upon entry, and feedback is immediate. Correct input is required before proceeding to a problem solution. HARD COPY If there is a printer connected to the computer, hard copy of the problem statements, tabular and graphical results etc. can be made by pressing F3 key wherever this option is indicated on the screen. Complete screen copies can also be made if the DOS graphics command is used before entering POLYMATH and your printer accepts this graphics mode. Problem statements and results can be also printed by saving them on a file and printing this file after leaving POLYMATH. GRAPHICS POLYMATH gives convenient displays during problem entry, modification and solution. Your computer will always operate in a graphics mode while you are executing POLYMATH. 1-6 INTRODUCTION
POLYMATH 4.0 PC
GETTING STARTED This chapter provides information on the hardware requirements and discusses the installation of POLYMATH. HARDWARE REQUIREMENTS POLYMATH runs on IBM compatible personal computers from the original IBM PC XT to the latest models with Pentium chips. Most graphics boards are automatically supported. The minimum application memory requirement is 560Kb, and a new feature uses extended memory when it is available. POLYMATH works with PC and MS DOS 3.0 and above. POLYMATH can be a DOS application under Windows 3.1 and Windows 95. Calculations are very fast since the floating point processor is used when it is available. POLYMATH SOFTWARE The complete set of POLYMATH application programs with a general selection menu is available on a single 3-1/2 inch 1.44 Mb floppy in compressed form. It is recommended that a backup disk be made before attempting to install POLYMATH onto a hard disk. Installation is available via an install program which is executed from any drive. INSTALLATION TO INDIVIDUAL COMPUTERS AND NETWORKS POLYMATH executes best when the software is installed on a hard disk or a network. There is a utility on the POLYMATH distribution disk which is called "install". Detailed installation instructions are found in the Appendix of this manual. Experienced users need to simply put the disk in the floppy drive, typically A or B. Type "install" at the prompt of your floppy drive, and press return. Follow the instructions on the screen to install POLYMATH on the particular drive and directory that you desire. Note that the default drive is "C:" and the default directory is called "POLYMAT4". Network installation will require responses to additional questions during installation. Latest detailed information can be found on the INSTALL. TXT and README.TXT files found on the installation disk. The PRINTSET program which can be used to change the printer specifications without completely reinstalling POLYMATH is discussed in the Appendix. POLYMATH 4.0 PC
GETTING STARTED 2-1
FIRST TIME EXECUTION OF POLYMATH The execution of POLYMATH is started by first having your current directory set to the subdirectory of the hard disk where POLYMATH version 4.0 is stored. This is assumed to be C:\POLYMAT4 C:\POLYMAT4 > Execution is started by entering "polymath" at the cursor C:\POLYMAT4 > polymath and then press the Return ( ) key. The Program Selection Menu should then appear:
The desired POLYMATH program is then selected by entering the appropriate letter. You will then taken to the Main Program Menu of that particular program. Individual programs are discussed in later chapters of this manual. GETTING STARTED 2-2
POLYMATH 4.0 PC
EXITING POLYMATH PROGRAM The best way to exit POLYMATH is to follow the instructions on the program display. However, a Shift F10 keypress (⇑F10) will stop the execution of POLYMATH at any point in a program and will return the user to the Polymath Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM so be sure to store your problem as an individual file or in the library before exiting the program in this manner. A query is made to determine if the user really wants to end the program in this manner while losing the current problem. This ⇑ F10 keypress is one of the few POLYMATH commands which is not always indicated in the various Display Menus. It is worth remembering.
POLYMATH 4.0 PC
GETTING STARTED 2-3
HELP MAIN HELP MENU Each individual POLYMATH program has a detailed help section which is available from many points in the program by pressing F6 when indicated. The Help Menu allows the selection of the topic area for specific help as shown below for the Differential Equation Solver:
For example, pressing "a" gives a discussion on entering the equations.
3-1 HELP
POLYMATH 4.0 PC
Once the current topic is completed, the Help Options Menu provides for additional options as shown below:
The ⇑ - F8 option to return to the program will take you to the display where you originally requested HELP ACCESSING HELP BEFORE PROBLEM ENTRY The Main Help Menu is reached during the startup of your POLYMATH program from the Main Menu as shown below and from the Problem Options Menu by pressing F6.
ACCESSING HELP DURING PROBLEM ENTRY When you are entering a problem, the HELP MENU is available from the Problem Options Menu. This will allow you to obtain the necessary help and return to the same point where the HELP MENU was originally requested. As an example, this access point is shown in the Problem Options Menu shown on the next page. POLYMATH 4.0 PC
HELP 3-2
CALCULATOR HELP A detailed discussion of the POLYMATH Calculator is given in Chapter 4 of this manual. The Calculator can be accessed from by pressing F4 from any point in a POLYMATH program.
An F6 keypress brings up the same page help which provides a brief instruction inside the Calculator window. UNIT CONVERSION HELP The Unit Conversion Utility is discussed in Chapter 4 of this manual. There is no on-line help for this utility.
3-3 HELP
POLYMATH 4.0 PC
UTILITIES CALCULATOR A sophisticated calculator is always available for use in a POLYMATH program. This calculator is accessed by pressing the F4 key . At this time a window will be open in the option box area which will give you access to the calculator.
CALCULATOR: Enter an expression and press to evaluate it. Press to leave or press F6 for information
The POLYMATH calculator allows you to enter an expression to be evaluated. After the expression is complete, press to have it calculated. You may then press again to clear the expression, or you may edit your expression using the standard editing functions. When you wish to leave the calculator, just press the F8 or Esc key. CALCULATOR EXPONENTIATION Numbers may also be entered in scientific notation. The calculator will recognize E or e as being equivalent to the notation *10**. Either ** or ^ indicates general exponentiation. For example, the following three expressions are equivalent for a particular value of A: 4.71*10**A = 4.71eA = 4.71*10^A AVAILABLE FUNCTIONS A number of standard functions are available for use in the calculator. The underlined portion of the following functions is all that is required provided that all arguments are enclosed in parentheses. The arguments may themselves be expressions or other functions. The nesting of function is allowed. ln ( ) or alog ( ) = natural logarithm to the base e log ( ) or alog10 ( ) = logarithm to the base 10 exp ( ) = exponential (ex) exp2 ( ) = exponential of 2 (2x) exp10 ( ) = exponential of 10 (10x) sqrt ( ) = square root abs ( ) = absolute value POLYMATH 4.0 PC
UTILITIES 4-1
int ( ) or ip ( ) = integer part frac ( ) = or fp ( ) = fractional part round ( ) = rounded value sign ( ) = returns + 1 or 0 or -1 N! = factorial of integer part of number N (this only operates on a number) sin ( ) = trigonometric sine with argument in radians cos ( ) = trigonometric cosine with argument in radians tan ( ) = trigonometric tangent with argument in radians sec ( ) = trigonometric secant with argument in radians csc ( ) = trigonometric cosecant with argument in radians cot ( ) = trigonometric cotangent with argument in radians arcsin ( ) = trigonometric inverse sine with result in radians, alternates arsin ( ) and asin ( ) arccos ( ) = trigonometric inverse cosine with result in radians, alternates arcos ( ) and acos ( ) arctan ( ) = trigonometric inverse tangent with result in radians, alternate atan ( ) arcsec ( ) = trigonometric inverse secant with result in radians arccsc ( ) = trigonometric inverse cosecant with result in radians arccot ( ) = trigonometric inverse cotangent with result in radians sinh ( ) = hyperbolic sine cosh ( ) = hyperbolic cosine tanh ( ) = hyperbolic tangent arcsinh ( ) = inverse hyperbolic sine arccosh ( ) = inverse hyperbolic cosine arctanh ( ) = inverse hyperbolic tangent You should note that the functions require that their arguments be enclosed in parentheses, but that the arguments do not have to be simple numbers. You may have a complicated expression as the argument for a function, and you may even nest the functions, using one function (or an expression including one or more functions) as the argument for another.
4-2 UTILITIES
POLYMATH 4.0 PC
ASSIGNMENT FUNCTIONS The assignment function is a way of storing your results. You may specify a variable name in which to store the results of a computation by first typing in the variable name, then an equals sign, then the expression you wish to store. For example, if you wish to store the value of sin (4/3) 2 in variable 'a', you would enter: a = sin (4/3)**2 Variable names must start with a letter, and can contain letters and digits. There is no limit on the length of the variable names, or on the number of variables you can use. You can then use the variable 'a' in other calculations. These variables are stored only as long as you remain in the current POLYMATH program. Please note that all stored values are lost when the particular program is exited. Calculator information is not retained during problem storage. CALCULATOR EXAMPLES Example 1. In this example the vapor pressure of water at temperatures of 50, 60 and 70 o C has to be calculated using the equation: log10 P = 8.10765 – 1750.29 235.0 + T For T = 50 the following expression should be typed into the calculator: 10^(8.10675 - 1750.29 / (235+50)) CALCULATOR: Enter an expression and press to evaluate it. Press to leave or press F6 for information.
Pressing brings up the desired answer which is 92.3382371 o mm Hg at 50 C. To change the temperature use the left arrow to bring the cursor just right to the zero of the number 50, use the (BkSp or delete) key to erase this number and type in the new temperature value.
POLYMATH 4.0 PC
UTILITIES 4-3
Example 2. In this example the pressure of carbon dioxide at temperature of T = 400 K and molal volume of V = 0.8 liter is calculated using the following equations: P = RT – a V – b V2
Where
2 2 a = 27 R Tc Pc 64
b = RTc 8 Pc
R = 0.08206, Tc = 304.2 and Pc = 72.9. One way to carry out this calculation is to store the numerical values to store in the named variables. First you can type in Pc = 72.9 and press this value as shown below.
Pc=72.9 CALCULATOR: Enter an expression and press to evaluate it. =72.9.
After that you can type in Tc = 304.2 and R = 0.08206. To calculate b, you must type in the complete expression as follows: b=R*Tc/(8*Pc) CALCULATOR: Enter an expression and press to evaluate it. =0.0428029012
The value of a is calculated in the same manner yielding a value of 3.60609951. Finally P can be calculated as shown:
P=R*400/(0.8-b)-a/(0.8*0.8) CALCULATOR: Enter an expression and press to evaluate it. =37.7148168
4-4 UTILITIES
POLYMATH 4.0 PC
UNIT CONVERSION A utility for unit conversion is always available for use within a POLYMATH program. Unit Conversion is accessed by pressing F5 wherever you desire. This will result in the following window in the option box area: Type the letter of the physical quantity for conversion. a) Energy b) Force c) Length d) Mass e) Power f) Pressure g) Volume h) Temperature F8 or ESC to exit
The above listing indicates the various classes of Unit Conversion which are available in POLYMATH. A listing of the various units in each class is given below: ENERGY UNITS: joule, erg, cal, Btu, hp hr, ft lbf, (liter)(atm), kwh FORCE UNITS: newton, dyne, kg, lb, poundal LENGTH UNITS: meter, inch, foot, mile, angstrom, micron, yard MASS UNITS: kilogram, pound, ton (metric) POWER UNITS: watts, hp (metric), hp (British), cal/sec, Btu/sec, ft lbf /sec PRESSURE UNITS: pascal, atm, bar, mm Hg (torr), in Hg, psi [lbf /sq in] VOLUME UNITS: cu. meter, liter, cu. feet, Imperial gal, gal (U.S.), barrel
(oil), cu. centimeter
TEMPERATURE UNITS: Celsius, Fahrenheit, Kelvin, Rankine
PREFIXES FOR UNITS It is convenient to also specify prefixes for any units involved in a Unit Conversion. This feature provides the following prefixes: deci 10 -1 hecto 10 2
centi 10 -2 kilo 10 3
POLYMATH 4.0 PC
milli 10 -3 mega 10 6
micro 10 -6 giga 10 9
deka 10
UTILITIES 4-5
UNIT CONVERSION EXAMPLE Suppose you want to convert 100 BTU's to kilo-calories. First you should access the Unit Conversion Utility by pressing F5. This will bring up the following options Type the letter of the physical quantity for conversion. a) Energy b) Force c) Length d) Mass e) Power f) Pressure g) Volume h) Temperature F8 or ESC to exit
Press "a" to specify an Energy conversion: From units: Type in a letter (F9 to set a prefix first) a. joule b. erg c. cal d. Btu f. ft lbf g. (liter)(atm) h. kwh
e. hp hr
Type a "d" to specify Btu: From units : Btu To units: a. joule b. erg c. cal f. ft lb, g. (liter)(atm) h. kwm
(F9 for a prefix) d. Btu e. hp hr
Use F9 to indicate a Prefix: Press the number of the needed prefix or F9 for none. 1) deci 10 -1 2) centi 10 -2 3) milli 10 -3 4) micro 10 -6 2 3 5) deka 10 6) hecto 10 7) kilo 10 8) mega 106 9) giga 109
Please indicate kilo by pressing the number 7. From units: Btu a. joule b. erg f. ft lbf g. (liter)(atm)
To units: kilo c. cal d. Btu h. kwh
e. hp hr
Complete the units by pressing "c" for calories. Indicate the numerical value to be 100 and press enter: From units: Btu Numerical value: 100 100.00 Btu = 25.216 kilo-cal
4-6 UTILITIES
To units: kilo-cal
POLYMATH 4.0 PC
PROBLEM STORAGE POLYMATH programs can be stored for future use as either DOS files or in a "Library" of problems. The Library has the advantage that the titles are displayed for only the problems for the particular POLYMATH program which is in use. Both the DOS files and the Library can be placed in any desired subdirectory or floppy disk. In both cases, only the problem and not the solution is stored. The storage options are available from the Task Menu which is available from POLYMATH programs by pressing either F9 from the Main Menu or ⇑ F8 from the Problem Options Menu.
FILE OPERATIONS A current problem can be saved to a DOS file by selecting "S" from the Task Menu. The desired directory and DOS file name can be specified from the window given below:
Note that the path to the desired directory can also be entered along with the file name as in "A:\MYFILE.POL" which would place the DOS file on the Drive A. A previously stored problem in a DOS file can be loaded into POLYMATH from the Task Menu by selecting "L". A window similar to the one above will allow you to load the problem from any subdirectory or floppy disk. An F6 keypress gives the contents of the current directory for help in identifying the file name for the desired problem. POLYMATH 4.0 PC
UTILITIES 4-7
LIBRARY OPERATIONS The Library is highly recommended for storing problems as the titles of the problems are retained and displayed which is a considerable convenience. Also, only the problems for the particular POLYMATH program in current use are displayed. The Library is accessed from the Task Menu by pressing F9 as shown below:
If there is no current Library on the desired subdirectory or floppy disk, then a Library is created. LIBRARY STORAGE The Library Options menu allows the current POLYMATH problem to be stored by simply entering "S". The title as currently defined in the active problem will be displayed. The user must choose a file name for this particular problem; however, it will then be displayed along with the Problem title as shown above. LIBRARY RETRIEVAL The Library Options window allows the current POLYMATH problem to be recalled by first using the cursor keys to direct the arrow to the problem of interest and then entering "L". A window will confirm the library retrieval as shown below:
Problems may be deleted from the Library by using the arrow to identify the problem, and then selecting "D" from the Library Options menu. Users are prompted to verify problem deletion. 4-8 UTILITIES
POLYMATH 4.0 PC
PROBLEM OUTPUT AS PRINTED GRAPHICS One of the most useful features of POLYMATH is the ability to create graphical plots of the results of the numerical calculations. The command to print graphical output is F3. The first step in printing graphical output is to display the desired output variables. The POLYMATH programs allow the user to make plots of up to four variables versus another variable. An example which will be used to demonstrate plotting is the Quick Tour Problem 1 from the next chapter. Here the POLYMATH Differential Equation Solver has produced a numerical solution to three simultaneous ordinary differential equations. The calculations are summarized on a Partial Results display which has the following Display Options Menu:
SIMPLE SCREEN PLOT The selection of "g" from the Display Options Menu allows the user to select desired variables for plotting. A plot of variables A, B, and C versus the independent variable t can be obtain by entering "A,B,C" at the cursor and pressing the Return key ( ).
The resulting graph is automatically scaled and presented on the screen.
POLYMATH 4.0 PC
UTILITIES 4-9
OPTIONAL SCREEN PLOT The selection of "g" from the Display Options Menu with the entry of "B/A" results in B plotted versus A. This demonstrates that dependent variables can be plotted against each other. PRESENTATION PLOT A simple plot can be printed directly or it can be modified before printing by using options from the Graph Option Menu shown below:
This menu allow the user to modify the plot before printing as desired to obtain a final presentation graphic with specified scaling and labels. PROBLEM OUTPUT TO SCREEN AND AS PRINTED TABLES The Display Options Menu also allows the user to select tabular output from the Partial Results Display by pressing "t":
This is shown below for the same entry of "A, B, C" for the Quick Tour Problem 1 from the next chapter on differential equations.
The output shown above gives variable values for the integration interval at selected intervals. The maximum number of points is determined by the numerical integration algorithm. Output variable values for a smaller number of points are determined by interpolation. A Screen Table can be printed by using F3. 4-10 UTILITIES
POLYMATH 4.0 PC
PROBLEM OUTPUT AS DOS FILES The output from many of the POLYMATH programs can also be stored for future use as DOS files for use in taking results to spreadsheets and more sophisticated graphics programs. Typically this is done after the output has been sent to the screen. This is again accomplished with option "d" from the Display Options Menu.
This option take the user to a display where the name and location of the DOS data file is entered:
Please note that the user can change the drive and the directory to an desired location. One the location is indicated and the file name is entered, the desired variable names must be provided and the number of data points to be saved. The file shown below was created as shown for the request of "A,B,C" and 10 data points for Quick Tour Problem 1 from the next chapter: t 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3
A B C 1 0 0 0.74081822 0.19200658 0.067175195 0.54881164 0.24761742 0.20357094 0.40656966 0.24127077 0.35215957 0.30119421 0.21047626 0.48832953 0.22313016 0.17334309 0.60352675 0.16529889 0.13797517 0.69672595 0.12245643 0.10746085 0.77008272 0.090717953 0.082488206 0.82679384 0.067205513 0.062688932 0.87010556 0.049787068 0.047308316 0.90290462
Note that the separate columns of data in this DOS output file are separated by tabs which is suitable format for input to various spreadsheet or graphics programs. POLYMATH 4.0 PC
UTILITIES 4-11
PROBLEM OUTPUT AS GRAPHICS FILES Advanced POLYMATH users can direct their printed output to many standard graphics files for direct use in word processing, desktop publishing, etc. This is accomplished though special polymath.bat files which direct any printer output to specified graphics files with user-defined file names. Details of this option are found in Chapter 9 of this manual. A typical example would be to create the following output in a TIFF file for inclusion in a written reporting using word processing or desktop publishing. The problem is again the for Quick Tour Problem 1 from the next chapter. Note that the figure below has utilized the title and axis definition options.
The modified bat file which lead to the above TIFF image was ECHO OFF SET BGIPATH=C:\POLYMAT4\BGI SET PM_WORKPATH=C:\POLYMAT4 SET PM_PRINTER=_TIF,0,FILE:E:\PMOUT+++.TIF POLYMENU CLS and the first tiff image was stored as PMOUT001.TIF in C:\POLYMAT4 directory. See Chapter 9 for more information. 4-12 UTILITIES
POLYMATH 4.0 PC
DIFFERENTIAL EQUATION SOLVER QUICK TOUR This section is intended to give you a very quick indication of the operation of the POLYMATH Differential Equation Solver Program. DIFFERENTIAL EQUATION SOLVER The program allows the numerical integration of up to 31 simultaneous nonlinear ordinary differential equations and explicit algebraic expressions. All equations are checked for syntax upon entry. Equations are easily modified. Undefined variables are identified. The integration method and stepsize are automatically selected; however, a stiff algorithm may be specified if desired. Graphical output of problem variables is easily obtained with automatic scaling. STARTING POLYMATH To begin, please have POLYMATH loaded into your computer as detailed in Chapter 2. Here it is assumed that your computer is set to the hard disk subdirectory or floppy drive containing the POLYMATH package. At the prompt (assumed C:\POLYMAT4 here), you should enter "polymath" C:\POLYMATH4 > polymath then press the Return ( ) key. The Program Selection Menu should then appear, and you should enter "1" to select the Simultaneous Differential Equation Solver. This should bring up the Main Program Menu:
POLYMATH 4.0 PC
DIFFERENTIAL EQUATIONS 5 -1
Now that POLYMATH is loaded, please press F6 and then the letter "a" to get information on "Entering the equations". The first page of the Help Section should be on your screen as shown here:
Please press F8 to return from the Help to the program, and then press the Enter key ( )to continue this Quick Tour example. SOLVING A SYSTEM OF DIFFERENTIAL EQUATIONS Let us now enter and solve a system of three simultaneous differential equations: d(A) / d(t) = - kA (A) d(B) / d(t) = kA (A) - kB (B) d(C) / d(t) = kB(B) In these equations, the parameter kA is to be constant at a value of 1.0 and the parameter kB is to be constant at the value of 2.0. The initial condition for dependent variable "A" is to be 1.0 when the initial value of the independent variable "t" is zero. The initial conditions for dependent variables "B" and "C" are both zero. The solution for the three differential equations is desired for the independent variable "t" between zero and 3.0. Thus this problem will be entered by using the three differential equations as given above along with two expressions for the values for kA and kB given by: kA = 1.0; kB = 2.0 5-2 DIFFERENTIAL EQUATIONS
POLYMATH 4.0 PC
ENTERING THE EQUATIONS The equations are entered into POLYMATH by first pressing the "a" option from the Problem Options Menu. The following display gives the first equation as it should be entered at the arrow. (Use the Backspace key, to correct entry errors after using arrow keys to position cursor.) Press the ) to indicate that the equation is to be entered. Don't worry Return key ( if you have entered an incorrect equation, as there will soon be an opportunity to make any needed corrections. d(A)/d(t)=-ka*A_
The above differential equation is entered according to required format which is given by: d(x)/d(t)=an expression where the dependent variable name "x" and the independent variable name "t" must begin with an alphabetic character and can contain any number of alphabetic and numerical characters. In this Quick Tour problem, the dependent variables are A, B and C for the differential equations, and the independent variable is t. Note that POLYMATH variables are case sensitive. The constants kA and kB are considered to be variables which can be defined by explicit algebraic equations given by the format: x=an expression In this problem, the variables for kA and kB will have constant values. Note that the subscripts are not available in POLYMATH, and in this problem the variable names of ka and kb will be used. Please continue to enter the equations until your set of equations corresponds to the following: Equations: → d(A)/d(t)=-ka*A d(B)/d(t)=ka*A-kb*B d(C)/d(t)=kb*B ka=1 kb=2
As you enter the equations, note that syntax errors are checked prior to being accepted, and various messages are provided to help to identify input errors. Undefined variables are also identified by name during equation entry. POLYMATH 4.0 PC
DIFFERENTIAL EQUATIONS 5-3
ALTERING THE EQUATIONS with no After you have entered the equations, please press equation at the arrow to go to the Problem Options display which will allow needed corrections:
The Problem Options Menu allows you to make a number of alterations on the equations which have been entered. Please make sure that your equations all have been entered as shown above. Remember to first indicate the equation that needs altering by using the arrow keys. When all equation are correct, press ⇑ F7 (keep pressing shift while pressing F7) to continue with the problem solution. ENTERING THE BOUNDARY CONDITIONS At this point you will be asked to provide the initial values for the independent variable and each of the dependent variables defined by the differential equations. Enter initial value for t _
Please indicate this value to be the number "0" and press Return. The next initial value request is for variable "A". Please this value as the value "1." Enter initial value for A 1_ 5-4 DIFFERENTIAL EQUATIONS
POLYMATH 4.0 PC
The initial values for B and C will be requested if they have not been previously entered. Please enter the number "0" for each of these variables. Next the final value for t, the independent variable, will be requested. Set this parameter at "3": Enter final value for t 3_
As soon as the problem is completely specified, then the solution will be generated. However, if you corrected some of your entries, then you may need to press ⇑ F7 again to request the solution. Note that a title such as "Quick Tour Problem 1" could have been entered from the Problem Option Menu. SOLVING THE PROBLEM The numerical solution is usually very fast. For slower computers, an arrow will indicate the progress in the independent variable during the integration. Usually the solution will be almost instantaneous. The screen display after the solution is given below:
POLYMATH 4.0 PC
DIFFERENTIAL EQUATIONS 5-5
Another Return keypress gives the partial Results Table which summarizes the variables of the problem as shown below:
The Partial Results Table shown above provides a summary of the numerical simulation. To display or store the results you can enter "t" (tabular display), "g" (graphical display), or "d" (storing the results on a DOS file). This Table may be printed with the function key F3. PLOTTING THE RESULTS Let us now plot the variable from this Problem 1 by entering "g" for a graphical presentation. When asked to type in the variable for plotting, please enter the input indicated below at the arrow: Type in the names of up to four (4) variables separated by commas (,) and optionally one 'independent' variable preceded by a slash(/). For example, myvar1, myvar2/timevar
A, B, C __
A Return keypress ( ) will indicate the end of the variables and should generate the graphical plot on the next page of the specified variables A, B, and C versus t, the independent variable, for this example.
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POLYMATH 4.0 PC
Suppose that you want to plot variable B versus variable A. Select the option "g" from the Display Options Menu and enter B/A when asked for the variable names. Type in the names of up to four (4) variables separated by commas (,) and optionally one 'independent' variable preceded by a slash(/). For example, myvar1, myvar2/timevar
B/A
This will results in a scaled plot for variable B versus variable A. This concludes the Quick Tour problem using the Differential Equation Solver. If you wish to stop working on POLYMATH, please follow the exiting instructions given below. EXITING OR RESTARTING POLYMATH A ⇑ - F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. POLYMATH 4.0 PC
DIFFERENTIAL EQUATIONS 5-7
INTEGRATION ALGORITHMS The program will first attempt to integrate the system of differential equations using the Runge-Kutta-Fehlberg (RKF) algorithm. A detailed discussion of this algorithm is given by Forsythe et al.* This algorithm monitors the estimate of the integration error and alters the step size of the integration in order to keep the error below a specified threshold. The default values for both relative and absolute (maximal) errors are less than 10-10. If this cannot be attained, then the absolute and relative errors are set as necessary to 10-7 and then to 10-4. If it is not possible to achieve errors of 10-4, then the integration is stopped, and the user is given a choice to continue or to try an alternate integration algorithm for stiff systems of differential equations. Under these circumstances, the system of equations is likely to be "stiff" where dependent variables may change in widely varying time scales, and the user is able to initiate the solution from the beginning with an alternate "stiff" integration algorithm The algorithm used is the semi-implicit extrapolation method of Bader-Deuflhard**, and the maximal errors are again started at 10-10. When the integration is very slow, the F10 keypress will allow the selection of the stiff algorithm, and the problem will be solved from the beginning.
* Frosythe, B. E., M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computation, Prentice-Hall, Englewood Cliffs, NJ, 1977. ** Press, W. H., P. B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, 2nd Ed., Cambridge University Press, 1992, pp. 735739.
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POLYMATH 4.0 PC
TROUBLE SHOOTING SPECIFIC ERROR MESSAGES Most error messages given by POLYMATH are self-explanatory, and they suggest the type of action which should be taken to correct the difficulty. NONSPECIFIC ERROR MESSAGES "Circular dependency detected." This message appears during the inputting of equations when the equations are not all explicit. For example, an attempt to define y=z/x when z has been previously defined will cause this error message to appear. This version of POLYMATH Differential Equations Solver can only solve variables which can be explicitly expressed as a function of other variables. "The expression ... is undefined at the starting point." This common problem can be solved by starting the integration from t=eps where eps is a very small number and t represents the independent problem variable. "Solution process halted due to a lack of memory." This message may result when the default Runge-Kutta-Fehlberg algorithm is used for a stiff system of differential equations, and thus very small step sizes are taken. Consequently, a large number of data points for possible plotting of the results. Use the F10 to stop the integration and switch to the stiff algorithm. If the message persists, then take the following steps to resolve the difficulty: (1) If you are running under Windows, make sure the PIF for POLYMATH specifies 640K of conventional memory and 1024K or more of XMS(but see item (3) below). (2) Remove other memory-resident programs from your computer. (3) Reduce the number of equations. This is most easily accomplished by introducing the numerical values of the constants into the equations, instead of defining them separately. Due to a limitation of some versions of HIMEM, POLYMATH may not be able to access most of the XMS in the number of equations exceeds approximately 24. (The exact number depends on the computer's configuration.) (4) Reduce the integration interval. POLYMATH 4.0 PC
DIFFERENTIAL EQUATIONS 5-9
"Solution process halted because it was not going anywhere." This message usually appears when the problem is very stiff, and the default RKF algorithm is used for integration. The stiff algorithm should be used, or the interval of integration should be reduced. If the error message persists, there are probably errors in the problem setup or input. Please check for errors in the basic equation set, the POLYMATH equation entry, and the numerical values and the units of the variables.
.
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POLYMATH 4.0 PC
ALGEBRAIC EQUATION SOLVER QUICK TOUR This chapter is intended to give a very brief discussion of the operation of the POLYMATH Nonlinear Algebraic Equation Solver. NONLINEAR ALGEBRAIC EQUATION SOLVER The user can solve up to a combination of 32 simultaneous nonlinear equations and explicit algebraic expressions. Only real (non-complex) roots are found. All equations are checked for correct syntax and other errors upon entry. Equations can be easily be modified, added or deleted. Multiple roots are given for a single equation. STARTING POLYMATH To begin, please have POLYMATH loaded into your computer as detailed in Chapter 2. Here it is assumed that your computer is set to the hard disk subdirectory or floppy drive containing the POLYMATH package. At the prompt (assumed C:\POLYMAT4 here), you should enter "polymath" C:\POLYMAT4>polymath then press the Return ( ) key. The Main Program Menu should then appear: The Program Selection Menu should then appear, and you should enter "2" to select the Simultaneous Algebraic Equation Solver.
POLYMATH 4.0 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-1
Once that POLYMATH is loaded, please utilize the Help Menu by pressing F6 and then the letter "a" to obtain details on in order to learn how to input the equations. The first page of this Help Section is given below:
This Help Section gives detailed information for entering the nonlinear and auxiliary equations. Press F8 to return from the Help Section to the program, and then press the Enter key ( ) to enter an equation for the first Quick Tour example. The Problem Options Menu at the bottom of your display allows entry of equations with the keypress of "a". Now you are ready for the first problem.
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POLYMATH 4.0 PC
SOLVING ONE NONLINEAR EQUATION The first nonlinear equation to be solved as Quick Tour Problem 1 is: x2 -5x + 6 = 0 The solution is to be obtained over the range of x between 1 and 4. This equation is entered into POLYMATH using the equation entry guidelines where the equation is to be zero at the solution. The following display gives the equation as it should be entered at the arrow: (Use the or the delete key to erase entered characters. Standard DOS editing is available at the cursor.)
f(x)=x^2-5*x+6_ The format for the above equation for f(x) is that the left side of the equation will be equal to zero when the solution has been obtained. The variable which is to be determined is set as an argument within the parentheses for the function f( ). Thus in this case, the variable is x and the function to be solved as being zero is x2-5x+6. Also note that in POLYMATH one way of entering x2 is x^2. An alternative entry is x**2. After you have correctly typed the equation at the arrow, please press once to enter it and then again to end equation entry. This should result in the Problem Options Menu at the bottom of the display and the equation at the top:
The Problem Options Menu indicates which options are now available for you to carry out a number of tasks. In this case, the problem should be complete, and these options for the equation at the arrow will not be needed. If an equation needed to be changed, then you would enter a "c" at the above display. (The arrow is moved by the arrow keys on the keyboard.) POLYMATH 4.0 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-3
Once you have the equation entered properly, please press ⇑F7 to solve the problem. You will then be asked to provide the interval over which you wish to find solutions for the equation. This interval is only requested during the solution of a single nonlinear equation.
Please indicate the xmin to be 1 and press ( and press ( ).
). Then indicate xmax to be 4
The entire problem is then display above the Problem Options Menu:
For this single equation, the solution is presented graphically over the search range which you indicated. The solution is where the function f(x) is equal to zero. POLYMATH has the ability to determine multiple solutions to a single equation problem, and the first of two solutions is shown below:
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POLYMATH 4.0 PC
Press enter (
) for the second solution.
A shift-return (⇑
) will return you to the Problem Options Display.
SOLVING A SYSTEM OF NONLINEAR EQUATIONS Next, you will solve two nonlinear equations with two unknowns. To enter this new set of equation press ⇑ F8 for a new problem, then press followed by "y" to enter a new problem. The equations that will be solved are:
v CAf – CA1 v CA1 – CA2 2 and k CA2 = V V where k = 0.075; v = 30; CAf = 1.6 ; CA2 = 0.2CAf . Thus there are two unknowns: CA1 and V. 2 k CA1 =
To solve this system of equations, each nonlinear equation must be rewritten in the form f(x) = (an expression that is to have the value of zero at the solution). The appropriate forms for these equations are: v CAf – CA1 2 f CA1 = kCA1 – V v CA1 – CA2 2 f V = kCA2 – and V All equations can be entered into POLYMATH as shown below. Note that each of the problem unknowns (CA1 and V) should appear once and only once inside the brackets in the left of the equal sign. The unknown variable may not be in that particular equation. POLYMATH just needs to know the variable names that you are using in your problem. The explicit algebraic equations may be entered directly. Please enter the equations as given below. The order of the equation is not important as POLYMATH will order the equations during problem solution. Equations f(Ca1)=k*Ca1^2-v*(Caf-Ca1)/V f(V)=k*Ca2^2-v*(Ca1-Ca2)/V k=0.075 Caf=1.6 v=30 Ca2=0.2*Caf
Press ⇑F7 to solve this system of equations. POLYMATH 4.0 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-5
For two or more nonlinear equations, POLYMATH requires an initial estimate to be specified for each unknown.
While the solution method used is very robust, it often will not be able to find the solution if unreasonable initial estimated are entered. In this example, physical considerations dictate that CA1 must be smaller than CA0 and larger than CA2. So please enter initial estimate of Ca1 as 1.0. As for V, any positive value up to about V=3900 can be a reasonable estimate. Please use the initial value of 300 for V in this Quick Tour example. After entering the initial values, this example problem should be:
Please press ⇑-F7 to solve the problem. The solution process will start and its progress will be indicated on the screen by an arrow moving along a ruler scale. For most computers, the solution is so fast that the display of the iterations in the numerical solution to a converged solution will not be seen. When visible, the arrow indicates how far from zero the function values are at a particular stage of the solution on a logarithmic scale. Details are given in the Help Section by pressing F6. The results are given after any keypress as shown below:
Please note that the values of the various nonlinear equation functions (nearly zero) are given along the with values of all the problem variables. 6-6 NONLINEAR ALGEBRAIC EQUATIONS
POLYMATH 4.0 PC
This concludes the Quick Tour problem using the Simultaneous Nonlinear Algebraic Equation Solver. If you wish to stop working on POLYMATH, please follow the exiting instructions given below. EXITING OR RESTARTING POLYMATH A ⇑ - F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. SELECTION OF INITIAL ESTIMATES FOR THE UNKNOWNS The solution algorithm requires specification of initial estimates for all the unknowns. Generally speaking, closer initial estimates have a better chance of converging to the correct solution. If you wish to solve only a single nonlinear equation, the program will plot the equation so that the location of the roots (if any) can be seen. The program will then show the roots. If no roots exist in the chosen range, the plot will indicate what range should be explored to have the nonlinear function f( ) cross zero. When several equations are to be solved, the selection of the initial values is more complicated. First, the user should try to find the limiting values for the variables using physical considerations. (For example: The mole or mass fraction of a component can neither be negative nor greater that 1; the temperature of cooling water can be neither below freezing nor above boiling; etc.) Typical initial estimates are taken to be mid range. Users should be particularly careful no to select initial estimates where some of the functions may be undefined. (For example, f(xa)=1/(xa-xb)+... is undefined whenever xa=xb; f(xb)=log(1-xb) is undefined whenever xb>=1; etc.) The selection of such initial estimates will stop the POLYMATH solution, and an error message will be displayed. METHOD OF SOLUTION For a single nonlinear equation, the user must specify an interval in which the real root(s) can be found. The program will first attempt to locate points or regions where the function is undefined inside this interval*. If the equations are too complicated for determination of discontinuity points, a warning message is issued. ______________________ *For details of the method, see Shacham, O. and Shacham, M., Acm. Trans. Math. Softw., 16 (3), 258-268 (1990). POLYMATH 4.0 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-7
The function is plotted and smaller intervals in which root(s) are located by a sign change of the function. The Improved Memory Method*, which employs a combination of polynomial interpolation and bisection, is used to converge to the exact solution inside those intervals. Iterations are stopped when the relative error is polymath then press the Return ( ) key. The Main Program Menu should then appear: The Program Selection Menu should then appear, and you should enter "3" on the keyboard to select the Linear Equation Solver.
POLYMATH 4.0 PC
LINEAR ALGEBRAIC EQUATIONS 7-1
Once that POLYMATH is loaded, please utilize the Help Menu by pressing F6 for information regarding the use of the Linear Equation Solver. This Help Section is given below:
This Help Section gives detailed information for entering a system of linear equations. Press any key to return from the Help Section to the program, and your display should be at the Main Menu for the Linear Equation Solver. (An alternate command to reach the Main Menu is the ⇑ F10 keypress.) To begin the first Quick Tour example, please press the Return key ( ) from the Main Menu. This will give the Task Menu as shown below:
7-2 LINEAR ALGEBRAIC EQUATIONS
POLYMATH 4.0 PC
SOLVING FIVE SIMULTANEOUS LINEAR EQUATIONS A typical problem for simultaneous linear equations is given below for the variables x1 through x5: x1 + 0.5 x2 + 0.333333 x3 + 0.25 x4 + 0.2 x5 = 0.0 0.5 x1 + 0.333333 x2 + 0.25 x3 + 0.2 x4 + 0.166667 x5 = 1.0 0.333333 x1 + 0.25 x2 + 0.2 x3 + 0.166667 x4 + 0.142857 x5 = 0.0 0.25 x1 + 0.2 x2 + 0.166667 x3 + 0.142857 x4 + 0.125 x5 = 0.0 0.2 x1 + 0.166667 x2 + 0.142857 x3 + 0.125 x4 + 0.111111 x5 = 0.0 The above problem in stored as a Sample Problem in POLYMATH. To recall the above problem, press F7 from the Task Menu of the Linear Equation Solver. Then select problem number "2" to obtain the Problem Options Menu shown below:
Solve this system of equations by pressing ⇑ F7 which should yield the results and the Display Options Menu on the next page. Remember that this keypress combination is accomplished by pressing and holding the Shift key and then pressing the F7 function key.
POLYMATH 4.0 PC
LINEAR ALGEBRAIC EQUATIONS 7-3
Lets explore making changes to this system of equations. This is accomplished by first pressing ⇑ to "make changes" to the problem. Use the arrow keys to take the highlighted box to the top of the "b" of constants for the equation. Please delete the 0 and enter 1.0 in this box which corresponds to changing the first linear equation to: x1 + 0.5 x2 + 0.333333 x3 + 0.25 x4 + 0.2 x5 = 1.0 This involves using the arrow key and pressing the return key ( highlighted box is in the desired location as shown.
) when the
Then enter the new value at the cursor:
Please solve the problem by pressing ⇑F7. The results are shown below:
This concludes the Quick Tour Problem for Simultaneous Linear Equations. When you are ready to leave this program and return to the Program Selection Menu, use the ⇑ F10 keypress which is discussed below. EXITING OR RESTARTING POLYMATH A ⇑ F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. 7-4 LINEAR ALGEBRAIC EQUATIONS
POLYMATH 4.0 PC
REGRESSION QUICK TOUR This chapter is intended to give a very brief overview of the operation of the POLYMATH Polynomial, Multiple Linear and Nonlinear Regression program. REGRESSION PROGRAM This program allows you to input numerical data into up to 30 columns, with up to 100 data points in each column. The data can be manipulated by defining expressions containing the names of previously defined columns. Relationships between different variables (columns of data) can be found using polynomial, multiple linear and nonlinear regression as well as cubic spline interpolation. Fitted curves can be interpolated, differentiated and integrated. Graphical output of the fitted curves and expressions is presented, and a statistical analysis of the parameters found during the regressions is given. STARTING POLYMATH To begin, please have POLYMATH loaded into your computer as detailed in Chapter 2. Here it is assumed that your computer is set to the hard disk subdirectory or floppy drive containing the POLYMATH package. At the prompt (assumed C:\POLYMAT4 here), you should enter "polymath" C:\POLYMAT4 > polymath ) key. The Program Selection Menu should then then press the Enter ( appear, and you should enter "4" on the keyboard to select the Polynomial, Multiple Linear and Nonlinear Regression program. This should bring up the Main Program Menu as given in the next page. In order to save time in entering data points during this quick tour, we will use sample problems which have been stored in POLYMATH. Press F7 to access the Sample Problems Menu from the Main Program Menu. QUICK TOUR PROBLEM 1 Let us consider a fairly typical application of the Regression Program in which some data are available. When these data are fitted to a polynomial
POLYMATH 4.0 PC
REGRESSION 8 -1
within POLYMATH, the polynomial expression has the form: P(x) = a0 + a1x + a2x2 +... + anxn where y is the dependent variable, x is the independent variable, and the parameters are a0 ...an. Variable "n" here represents the degree of the polynomial. In POLYMATH, the maximum degree which is shown is 5. The above polynomial expression gives a relationship between the dependent variable and the independent variable which is obtained by determining the parameters according to a least squares objective function. Data points are usually available which give x and y values from which the parameters a0... an can be determined.
RECALLING SAMPLE PROBLEM 3 After pressing F7 at the Main Program Menu, the Sample Problems Menu should appear on your screen as shown on the next page.. The sample problem to be discussed should be retrieved by pressing "3" on the keyboard. This will result in the Problem Options Display which includes 10 data points of x and y as shown on the next page.
8 -2 REGRESSION
POLYMATH 4.0 PC
POLYMATH 4.0 PC
REGRESSION 8 -3
FITTING A POLYNOMIAL The Problem Options Menu includes problem editing, library, printing, help and solution options. To fit a polynomial to the data of Y versus X you should select the "⇑ F7 to fit a curve or do regression" option. After pressing ⇑ F7 the following "Solution Options" menu appears:
After pressing "p" (lower case), you should be asked for the name of the independent variable's column, as shown below:
You should enter a capital "X" (upper case) as name of the independent variable and press . The same question regarding the dependent variable will be presented. Please enter a capital "Y" (upper case) at the arrow. The following display should appear:
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POLYMATH 4.0 PC
On this display the coefficients of the polynomial P(x), up to the fifth order are shown together with the value of the variance. One of the polynomials is highlighted by having a box around it. This is the lowest order polynomial, such that higher order polynomial does not give significantly better fit. The same polynomial is also plotted versus the experimental data. Other polynomials can be highlighted and plotted by pressing a number between 1 and 5. There are many additional calculations and other operations that can be carried out using the selected polynomial. Please make sure the highlighted box is on the 4th degree polynomial. Let us find the value of X for Y = 10. To do that you should press "y" and enter after the prompt regarding the value of Y: "10". The following display results:
The resultant X values are shown both graphically and numerically. For Y = 10 there are two X values, X = 1.36962 and X = 5.83496. FITTING A CUBIC SPLINE We will now fit a cubic spline to these data of Sample Problem 3. Please press F8 two times to return to the Problems Options Menu. Then press ⇑ F7 to "fit a curve or do regression". The Solutions Options Menu should appear. POLYMATH 4.0 PC
REGRESSION 8-5
Enter "s" (lower case) for a cubic spline followed by "X" and then "Y". The following display should present the results:
EVALUATION OF AN INTEGRAL WITH THE CUBIC SPLINE Please take options "i" and request the initial value for the integration to be "1" at the arrow:
Press integration.
and then enter "6" at the arrow for the find value of the
Press to have the resulting integration shown on the next display with both graphical and numerical results:
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POLYMATH 4.0 PC
Please press any key to end this Sample Problem 3. MULTIPLE LINEAR REGRESSION It will often be useful to fit a linear function of the form: y(x) = a0 + a1x1 + a2 x2 +... + anxn where x1, x2, ..., xn are n independent variables and y is the dependent variable, to a set of N tabulated values of x1,i, x2,i, ... and y (xi). We will examine this option using Sample Problem 4. RECALLING SAMPLE PROBLEM 4 First exit to the main title page by pressing ⇑ F10. Press F7 to access the Sample Problems Menu, and select problem number 4 by pressing "4". (The problem display is shown on the next page).
POLYMATH 4.0 PC
REGRESSION 8-7
SOLVING SAMPLE PROBLEM 4 After you press ⇑ F7 "to fit a curve or do regression", the following Solution Options Menu should appear:
This time press "l" (lower case letter "l") to do "linear regression". You will be prompted for the first independent variable (column) name.
Please type in "X1" at the arrow and press . You will be prompted for the 2nd independent variable. Enter "X2" as the second independent variable name and press once again. A prompt for the 3rd independent variable will appear. You should press here without typing in anything else, since there are no additional independent variables. 8 -8 REGRESSION
POLYMATH 4.0 PC
At the prompt for the dependent variable (column) name shown below you should type "Y" and press .
Once the calculations are completed, the linear regression (or correlation) is presented in numerical and graphical form.
Please note that the correlation the equation for variable "Y" has the form of the linear expression: Y = a0 + a1X1 + a2X2 where a0 = 9.43974, a1 = -0.1384 and a2 = 3.67961. This graphical display of Sample Problem 4 presents the regression data versus the calculated values from the linear regression. The numerical value of the variance and the number of the positive and negative residuals give an indication regarding the validity of the assumption that Y can be represented as linear function of X1 and X2. The results in this case indicate a good fit between the observed data and the correlation function.
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REGRESSION 8-9
The Display Options Menu allows the user to use an "s" keypress "to save results in a column". This refers to saving the calculated value of Y from the linear regression to the Problems Options Display under a column name provided by the user. The "r" keypress from the Display Options Menu give a statistical residual plot as shown below:
The "F9" keypress from the Display Options Menu give a statistical summary:
The confidence intervals given in the statistical summary are very useful in interpreting the validity of the linear regression of data. This concludes Sample Problem 4 which illustrated multiple linear regression.
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TRANSFORMATION OF VARIABLES A nonlinear correlation equation can be often brought into a linear form by a transformation of the data. For example, the nonlinear equation: Y = a0 X1a 1 X2a 2 can be linearized by taking logarithm of both sides of the equation: ln Y = ln a0 + a1 ln X1 + a2 ln X2. To demonstrate this option please recall Sample Problem 5. To do this, please press ⇑ F10 to get to the Main Program Menu, F7 to access the Sample Problems Menu and select Sample Problem number 5. This should result in the Problem Option Display below:
In this display X1, X2 and Y represent the original data, the variables (columns) lnX1, lnX2 and lnY represent the transformed data. You can see the definition of ln X1 , for example, by moving the cursor (the highlighted box), which located in row number 1 of the first column, into the box containing "lnX1" (using the arrow keys) and press .
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REGRESSION 8-11
The following window is brought up:
Note that the expression in the right hand side of the column definition equation must be a valid algebraic expression, and any function arguments used in the expression should be enclosed within parentheses. Since we do not want to change this expression, please press to close the window. Now press ⇑ F7 to do regression, then "l" to do linear regression. Type in "lnX1" as the name of the first independent variable, "lnX2" as the name of the second independent variable and "lnY" as the name of the dependent variable. The results should be displayed as shown below:
All of the statistical analyses are available for the transformed variable. Please note that the results indicate that the equation for variable "Y" can be written as: Y = a0 X1a 1 X2a 2 where a0 = exp (-0.666796) = 0.5133, a1 = 0.986683 and a2 = -1.95438. This concludes the transformation of variable and the multiple linear regression for Sample Problem 5. 8-12 REGRESSION
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NONLINEAR REGRESSION It is often desirable to fit a general nonlinear function model to the independent variables as indicated below: y(x) = f(x1, x2, ..., xn; a0, a1, ..., am) In the above expression, x1, x2, ..., xn are n independent variables, y is the dependent variable, and a0, a1, ..., am are the model parameters. The data are represented by a set of N tabulated values of x1,i, x2,i, ... and y(xi). The regression adjusts the values of the model parameters to minimize the sum of squares of the deviations between the calculated y(x) and the data y(xi). The nonlinear regression capability of POLYMATH allows a general nonlinear function to be treated directly without any transformation. Lets return to Sample Problem 5 and this time treat the model for Y directly where Y = a0 X1a 1 X2a 2 . Please recall Sample Problem 5. From the Problem Options Display press ⇑ F7 and then enter "R" (upper case R) to "Do nonlinear regression." The user is then prompted to:
The user can then enter the model equation using any of the variables from the columns of the Problem Options Display and any unknown parameters (maximum of five) which are needed. For this example, please enter
Thus in this problem, the unknown parameters are k, alpha, and beta. The next query for the user is to supply initial estimates for each of the unknown parameter in turn:
It is good practice to provide good initial parameter estimates from either reasonable physical/chemical model values or from a linearized treatment of the nonlinear model. In this example however, please set all initial guesses for the parameters as unity, "1.0". Then POLYMATH will provide a summary of the problem on the Regression Option Display as shown on the next page.
POLYMATH 4.0 PC
REGRESSION 8-13
The Regression Options Menu gives several useful options for model changes and alterations to initial parameter guesses; however, please press ⇑F7 to solve this problem. The program search is shown to the user and the converged solution is indicated below:
There are a number of options from the Display Options Menu (not shown here). Perhaps the most useful is the "statistical analysis" which is given on the next page. 8-14 REGRESSION
POLYMATH 4.0 PC
This concludes the Quick Tour section dealing with nonlinear regression and the Chapter on the Polynomial, Multiple Linear and Nonlinear Regression Program. Remember, when you wish to stop POLYMATH, please follow the exiting instructions given below. EXITING OR RESTARTING POLYMATH A ⇑ F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. SOLUTION METHODS When fitting a polynomial of the form P(x) = a0 + a1x + a2x2 +...+anxn to N points of observed data, the minimum sum of square error correlation of the coefficients a0, a1, a2...an can be found by solving the system of linear equation (often called normal equations): POLYMATH 4.0 PC
REGRESSION 8-15
XT XA = XT Y where
Y=
y y1 .2 . . y
A=
N
ao a1 . . . an
x0 X=
1 x0 2 x0 N
x1
1 x1 2 x1 N
. . . . . . . . . . . .
xn
1 xn 2 xn N
and where y1, y2...yN are N observed values of dependent variable, and x1, x2...xN are N observed values of the independent variable. Multiple linear regression can also be expressed in the same form except that the matrix X is redefined as follows:
X=
1
x1,1
x2,1 . . . x n,1
1
x1,2
x2,2 . . . x n,2
1
x1,N x2,N . . . x n,N
where xi,j is the j-th observed value of the i-th independent variable. When polynomial or multiple linear regression are carried out without the free parameter (a0), the first element in vector A and the first column in matrix X must be removed. In POLYMATH the normal equations are solved using the GaussJordan elimination. It is indicated in the literature that direct solution of normal equations is rather susceptible to round off errors. Practical experience has should this method to sufficiently accurate for most practical problems. The nonlinear regression problems in POLYMATH are solved using the Levenberg-Marquardt method. A detailed description of this method can be found, for example, in the book by Press et al.*
*Press, W. H., P. B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, 2nd Ed., Cambridge University Press, 1992, pp. 678683. 8-16 REGRESSION
POLYMATH 4.0 PC
APPENDIX INSTALLATION FROM THE CRE-99 CD-ROM ACCOMPANYING "ELEMENTS OF CHEMICAL REACTION ENGINEERING" AND EXECUTION INSTRUCTIONS FOR VARIOUS OPERATING SYSTEMS This Appendix provides complete instructions for the installation and execution of POLYMATH for the DOS, Windows 3.1 and Windows 95 operating systems. Detailed information is provided for advanced options. Latest updates are on the README.TXT and INSTALL.TXT files. DOS INSTALLATION Place the CD-ROM in an appropriate drive. This is usually drive D. First set your current directory to that of the CD-ROM and indicate the location of the POLYMATH installation software. For example, if the drive to be used is D, then insert “D:” at the cursor and press Enter. Then enter “cd\html\toolbox\polymath\files”. At the cursor enter “dir” to verify that you have the following POLYMATH installation files in that directory: PMUNZIP.EXE BGI.EXE INSTALL.EXE README.TXT INSTALL.TXT POLYMATH installation is initiated by typing “install” at the cursor the drive and subdirectory containing the above files. For a CD-ROM as drive D, this would be: D:\html\toolbox\polymath\files>install The installation program will ask a number of questions. These are listed later on this file. Please refer to the details of these questions as needed. Follow the instructions on the screen. Note that the initial recommendation for the Printer Output setting is option 1. Half Page, Low Resolution. When the installation is complete, press Enter. DOS EXECUTION 1. Change your directory to the POLYMATH directory by entering "C:" and then "cd\polymat4" at the cursor. Substitute your specified location if you did not use the default directory location. 2. Enter "polymath" at the cursor. 3. Always end POLYMATH by exiting from the Program Selection Menu. POLYMATH 4.0 PC
APPENDIX 9-1
WINDOWS 3.1 INSTALLATION 1. Put the CD-ROM in the appropriate drive such as “D:”. 2. Double click on the Main icon in the Program Manager window. 3. Double click on the MS-DOS Prompt icon. 4. Change the directory to the CD-ROM drive to where POLYMATH is stored. For example, if the drive to be used is D, then insert “D:” at the cursor and press Enter. Then enter “cd\html\toolbox\polymath\files”. 5. At the D:\html\toolbox\polymath\files> prompt, enter “dir”. 6. The following five files should be listed: PMUNZIP.EXE, INSTALL.EXE, INSTALL.TXT, BGI.EXE, and README.TXT. 7. At the D:\html\toolbox\polymath\files\poly402> prompt, enter “install” 8. Follow the instructions on the screen. Please note that the initial recommended setting for Output is option 1. Half Page, Low Resolution. 9. At the DOS prompt enter “exit” to return to Windows CONTINUE STEPS BELOW TO CREATE A PIF FILE FOR POLYMATH ONLY IF POLYMATH DOES NOT EXECUTE PROPERLY 10. From the Program Manager Window click on Main. 11. From the Main Window double click on PIF Editor. 12. Please enter the following in the PIF Editor: Program Filename: POLYMATH.BAT Window Title: POLYMATH 4.0 Startup Directory: C:\POLYMAT4 Video Memory: Low Graphics Memory Requirements: -1 -1 EMS Memory: 0 1024 XMS Memory: 1024 1024 Display Usage: Full Screen 13. From File use Save As “POLYMAT4.PIF”. WINDOWS 3.1 EXECUTION* 1. From the File options in the Program Manager Window, select Run. 2. The Command Line for your POLYMATH location should be entered such as “c:\polymat4\polymath”. 3. Click on “OK” 4. If POLYMATH does not run properly, then create a PIF file starting with Step 10 as given above. 5. Always end POLYMATH by exiting from the Program Selection Menu. * A Program Group and a Program Item can be created under Windows to allow POLYMATH to be executed conveniently from the desktop. 9-2 APPENDIX
POLYMATH 4.0 PC
Windows 95 Installation 1. Put the installation disk in your floppy drive. 2. Click on the Start button. 3. Click on the My Computer icon. 4. Double click on the CD-ROM icon indicating cre-99. Then continue double clicking on html, toolbox, polymath, files, and poly402 5. Double click on the Install.exe file 6. Follow the directions on the screen. Please note that the initial recommended setting for Output is option 1. Half Page, Low Resolution. 7. When installation is complete, press Enter. 8. Close the DOS window. Windows 95 Execution* 1. Click on the Start button. 2. Click on the Run icon. 3. Enter “c:\polymat4\polymath” 4. Always end POLYMATH by exiting from the Program Selection Menu. * A Program Group and a Program Item can be created under Windows 95 to allow POLYMATH to be executed conveniently from the desktop. INSTALLATION QUESTIONS 1. Enter drive and directory for POLYMATH [C:\POLYMAT4] : ==> The default response is indicated by the contents of the brackets [...] which is given by pressing Enter key. The full path (drive and directory) where you wish the POLYMATH program files to be stored must be provided here. If the directory does not exist, then the installation procedure will automatically create it. NOTE: Network clients will need read and execute permission for this directory and its subdirectories. This procedure does not provide the needed permissions. 2. Is this a network installation? [N] If you are installing POLYMATH on a stand-alone computer, take the default or enter "N" for no and GO TO 5. on the next page of this manual. If you are installing on any kind of network server, answer "Y" and continue with the installation. POLYMATH 4.0 PC
APPENDIX 9-3
3. What will network clients call [POLYdir]? ===> _ This question will only appear if you answered "Y" to question 2 to indicate a Network installation. Here "POLYdir" is what was provided in question 1. On some networks, the clients "see" server directories under a different name, or as a different disk, than the way the server sees them. This question enables POLYMATH to print by indicating where the printerdriver files are located. They are always placed in subdirectory BGI of the POLYMATH directory by the installation procedure. During run time, they must be accessed by the client machines, thus POLYMATH must know what the client's name is for the directory. 4. Enter drive and directory for temporary print files [C:\TMP]: ===> _ This question will only appear if you answered "Y" to question 2 to indicate a Network installation. Depending on the amount of extended memory available, and the type of printer is use, POLYMATH may need disk workspace in order to print. Since client machines are not normally permitted to write on the server disk, you are requested to enter a directory where files may be written. The temporary print files are automatically deleted when a print is completed or cancelled. 5. The POLYMATH installation program now copies all of the needed files according to your previous instructions. This may take some time as the needed files are compressed on the installation diskette. 6. Please select the type of output device you prefer [1]: _ 1. Printer 2. Plotter ===> _ Your answer here will take you to an appropriate selection menu. 7 A. POLYMATH INSTALLATION - printer selection Please select the type of printer you have: 1. Canon Laser Printer 2. Canon BJ 200 3. Canon BJC 600 4. HP LaserJet II 9-4 APPENDIX
POLYMATH 4.0 PC
5. HP LaserJet III 6. HP LaserJet IV 7. HP LaserJet III or IV (HPGL/2) 8. Epson 9-pin Dot Matrix Printer 9. Epson 24-pin Dot Matrix Printer 10. Epson 9-pin Dot Matrix Printer (color) 11. Epson 24-pin Dot Matrix Printer (color) 12. Epson Stylus 13. Epson Color Stylus 14. IBM Proprinter 15. IBM Proprinter X24 16. IBM Quietwriter 17. Kodak Diconix 18. OkiData Dot Matrix Printer (native mode) 19. PostScript printer 20. Toshiba 24-pin Dot Matrix Printer (native mode) 21. Xerox CP4045/50 Laser Printer (USA) 22. Xerox CP4045/50 Laser Printer (International) 23. HP DesignJet 24. HP DeskJet (Black & White) 25. HP DeskJet 500C (Color) 26. HP DeskJet 550C (Color) 27. HP DeskJet 1200C 28. HP PaintJet 29. HP PaintJet XL300 30. HP ThinkJet ===> _ Type in the number of your printer (or a type of printer that your printer can emulate) and press Enter. 7 B. POLYMATH INSTALLATION - plotter selection Please select the type of plotter you have: 1. HP 7090 Plotter 2. HP 7470 Plotter 3. HP 7475 Plotter 4. HP 7550 Plotter 5. HP 7585 Plotter 6. HP 7595 Plotter 7. Houston Instruments DMP/L Plotters ===> _ POLYMATH 4.0 PC
APPENDIX 9-5
Type in the number of your plotter (or a type of plotter that your plotter can emulate) and press Enter. Once you have selected either a printer or a plotter, you will be asked to select the mode for your output. 8. POLYMATH INSTALLATION - printer/ plotter mode selection Please select the mode you want output in [1]: 1. Half Page, Low Resolution 2. Half Page, Medium Resolution 3. Half Page, High Resolution 4. Landscape, Low Resolution 5. Landscape, Medium Resolution 6. Landscape, High Resolution 7. Portrait, Low Resolution 8. Portrait, Medium Resolution 9. Portrait, High Resolution ===> _ Generally, the fastest printing with adequate resolution is option (1), which is also the default. This choice is recommended. It is easy to change your printer/plotter selection and mode by using the PRINTSET program which is installed in the POLYMATH directory. 9. POLYMATH INSTALLATION - printer/ plotter port selection Please select the port your printer/plotter is connected to [1] : 1. LPT1 2. LPT2 3. COM1 4. COM2 5. LPT3 ===> _ The default port, LPT1, is the first parallel port on most personal computers and usually is connected to the printer. For network installations, these are usually "logical" printer names, but they work just as well. If your network uses other printer logical names, please read the Appendix section entitled "PRINTING FOR ADVANCED USERS" which starts on the next manual page.
9-6 APPENDIX
POLYMATH 4.0 PC
10. POLYMATH Execution The POLYMATH program can now be executed by first changing the directory of your personal computer to the one where the program has been installed. Then you should enter "polymath" at the cursor as shown below and press Enter: C:\POLYMAT4> polymath_ Windows users must use POLYMATH as a DOS program. Advanced users may wish to place the polymath program in the "path" statement of the autoexec.bat file so that POLYMATH is more easily available from any cursor. "OUT OF ENVIRONMENT SPACE" MESSAGE If you receive this message or if you are having difficulty in printing from POLYMATH, then follow ONE of the following instruction set for your particular operating system. Window 95, Windows 98, or Windows NT 4.0 1. Open a DOS prompt window (if yours opens full-screen, hit Alt+Enter to get a window). 2. Click on the "Properties" button at the top. 3. At the end of the Cmd line, add the text "/e:2048" (if you already have this, then change the existing number to a higher number in increments of 1024). Windows 3.x and Windows 95 1. Open a DOS prompt window. 2. At the C:\ prompt type "CD/WINDOWS, and press Enter. 3. Type "EDIT SYSTEM.INI" and press Enter. 4. Locate a line that reads "[NonWindows App]". 5. Make sure that this section contains the following entry: "CommandEnvSize=2048". 6. Save the modified file. 7. Reboot your computer. 8. Adjust the size upwards in increments of 1024 as necessary. DOS or Windows 3.1 1. Open a DOS prompt window. 2. Type "SET" at the prompt. POLYMATH 4.0 PC
APPENDIX 9-7
3. Look for the the line that displays the value of the COMSPEC environment variable. If COMSPEC is set to C:\COMMAND.COM then add the following line to the CONFIG.SYS file: SHELL=C:\COMMAND.COM C:\ /E:2048 /P If COMSPEC is set to C:\DOS\COMMAND.COM then add this line to the CONFIG.SYS file: SHELL=C:\DOS\COMMAND.COM C:\DOS /E:2048 /P 4. Reboot your computer. 5. Adjust the size of E: in Step 3 upwards in increments of 1024 as necessary. CHANGING PRINTER SELECTION The printer selection may be changed without a complete reinstallation of POLYMATH by using the PRINTSET utility program which is stored on the directory containing POLYMATH. This program is executed by entering "printset" while in the POLYMATH directory. If there are several printers to be used with POLYMATH, then different POLYMATH.BAT files can be created for each printer. The batch file for printers is discussed in the following section. PRINTING FOR ADVANCED USERS POLYMATH printing is accomplished using GRAF/DRIVE PLUS which is a trademark for copyrighted software from Fleming Software. The printer setup is controlled by three environmental variables which are set in the POLYMATH.BAT file. This file is created by the INSTALL procedure, and it can be subsequently be modified by the PRINTSET utility or with an editor. CAUTION: This file should only be altered with an editor by advanced users. A typical POLYMATH.BAT file is shown below: ECHO OFF SET BGIPATH=C:\POLYMAT4\BGI SET PM_WORKPATH=C:\POLYMAT4 SET PM_PRINTER=_LJ3R,0,LPT1 POLYMENU CLS 9-8 APPENDIX
POLYMATH 4.0 PC
In this file, the three environmental variables are: BGIPATH - This is the directory in which the printer driver "BGI" files for POLYMATH reside. It is usually the BGI subdirectory of the POLYMATH directory. This variable is not normally changed. PM_WORKPATH - This is the directory in which temporary print files are stored. PM_PRINTER - This variable has the following form: ,, This variable in the previous POLYMATH.BAT file is defined as: PM_PRINTER=_LJ3R,0,LPT1 where _LJ3R indicates the printer_type as HP LaserJet III, 0 (the number zero) indicates the page_format as half page with low density print, and LPT1 indicates the printer_port as LPT1. PM_PRINTER Options for and A detailed listing of the and options is given in Table 1 at the end of this Appendix. PM_PRINTER Options for The may be any of the following: LPT1, LPT2, LPT3, COM1, COM2 or other physical/logical device names. PRINTING TO STANDARD GRAPHICS FILES It is possible to have all output which is "printed" by POLYMATH to be saved as various graphics files for use in word processing, desktop publishing, etc. This involves specialized use of the PM_PRINTER variable which is not available during the INSTALL procedure or the PRINTSET program for printer modification. All printing to graphics files requires the creation of a special batch file for this purpose. POLYMATH 4.0 PC
APPENDIX 9-9
A typical file for this purpose which is arbitrarily called POLYGRAP.BAT is shown below: ECHO OFF SET BGIPATH=C:\POLYMAT4\BGI SET PM_WORKPATH=C:\POLYMAT4 SET PM_PRINTER=_TIF,0,FILE:C:\GRAPHS\PMOUT+++.TIF POLYMENU CLS In the above file, the PM_PRINTER variable _TIF indicates the printer_type as a Tagged Image Format (TIFF), 0 indicates low resolution with two colors, and FILE:C:\GRAPHS\PMOUT+++.TIF indicates the location and name of the resulting graphics file. PM_PRINTER Options for and for Graphics Files A detailed listing of the and options for graphics file output is given in Table 2 at the end of this Appendix. PM_PRINTER Options for for Graphics Files This variable can be defined as FILE: when the output is to be to a file and not a port. (Note that if is a logical device name, such as LPT3, the output will be printed.) may be a particular filename or a general template, including '+' signs where a sequence number is to be written. The above example will create a sequence of files named PMOUT001.TIF, PMOUT002.TIF, etc., stored in directory C:\GRAPHS .
POLYMATH 4.0 PC
APPENDIX 9-10
Table 1.1 - PM_PRINTER OPTIONS
Output Page /Resolution /Color
Output Page /Resolution /Color
Epson-compatible 9-pin Dot Matrix, and IBM Proprinter _FX
0
HalfLo
5
LandHi
1
HalfMed
6
FullLo
2
HalfHi
7
FullMed
3
LandLo
8
FullHi
4
LandMed
Epson-compatible 24-pin Dot Matrix _LQ
0
HalfLo
5
LandHi
1
HalfMed
6
FullLo
2
HalfHi
7
FullMed
3
LandLo
8
FullHi
4
LandMed
Epson-compatible 9-pin Dot Matrix Printer (color) _CFX
0
HalfLoC
3
LandMedC
1
HalfMedC
4
FullLoC
2
LandLoC
5
FullMedC
Epson-compatible 24-pin Dot Matrix Printer (color) _CLQ
9-11 APPENDIX
0
HalfLoC
3
LandMedC
1
HalfMedC
4
FullLoC
2
LandLoC
5
FullMedC
POLYMATH 4.0 PC
Table 1.2 - PM_PRINTER OPTIONS
Output Page /Resolution /Color
Output Page /Resolution /Color
0
HalfLo
5
LandHi
1
HalfMed
6
FullLo
2
HalfHi
7
FullMed
3
LandLo
8
FullHi
4
LandMed
0
HalfLo
5
LandHi
1
HalfMed
6
FullLo
2
HalfHi
7
FullMed
3
LandLo
8
FullHi
4
LandMed
2
Full
2
Full
IBM Proprinter X24 _PP24
IBM Quietwriter _IBMQ
OkiData Dot Matrix Printer (native mode) _OKI92
0
Half
1
Land
Toshiba 24-pin Dot Matrix Printer (native mode) _TSH
0
Half
1
Land
POLYMATH 4.0 PC
APPENDIX 9-12
Table 1.3 - PM_PRINTER OPTIONS
Output Page /Resolution /Color
Output Page /Resolution /Color
LaserJet II, LaserJet III, LaserJet IV, DeskJet (black cartridge), Canon laser _LJ _LJ3R _LJ4 _DJ _Canon
0
HalfLo
5
LandHi
1
HalfMed
6
FullLo
2
HalfHi
7
FullMed
3
LandLo
8
FullHi
4
LandMed
DeskJet 500C (color cartridge), and DeskJet 550C _DJC _DJC550
0
HalfLoC8
5
LandHiC8
1
HalfMedC8
6
FullLoC8
2
HalfHiC8
7
FullMedC8
3
LandLoC8
8
FullHiC8
4
LandMedC8
0
HalfLoC2
8
FullLoC8
1
LandLoC2
9
HalfHiC8
2
FullLoC2
10
LandHiC8
3
HalfHiC2
11
FullHiC8
4
LandHiC2
12
HalfLoC16
5
FullHiC2
13
LandLoC16
6
HalfLoC8
14
FullLoC16
7
LandLoC8
PaintJet _PJ
9-13 APPENDIX
POLYMATH 4.0 PC
Table 1.4 - PM_PRINTER OPTIONS
Output Page /Resolution /Color
Output Page /Resolution /Color
0
Half
6
HalfC16
1
Land
7
LandC16
2
Full
8
FullC16
3
HalfGR
9
HalfC256
4
LandGR
10
LandC256
5
FullGR
11
FullC256
0
DraftPL
2
DraftPLB
1
LQPL
3
LQPLB
_HP7470
0
DraftPL
1
LQPL
_HP7475 _HP7550
0
DraftPL
4
DraftPLr
1
LQPL
5
LQPLr
2
DraftPLB
6
DraftPLBr
3
LQPLB
7
LQPLBr
0
DraftPL
5
LQPLC
1
LQPL
6
DraftPLD
2
DraftPLB
7
LQPLD
3
LQPLB
8
DraftPLE
4
DraftPLC
9
LQPLE
PostScript Printers _PS
Hewlett-Packard Plotters _HP7090
_HP7585 _HP7595
POLYMATH 4.0 PC
APPENDIX 9-14
Table 1.5 - PM_PRINTER OPTIONS
Output Page /Resolution /Color
Output Page /Resolution /Color
0
HalfLo
5
LandHi
1
HalfMed
6
FullLo
2
HalfHi
7
FullMed
3
LandLo
8
FullHi
4
LandMed
Canon BJ 200, and Epson Stylus _BJ200 _ESCP2
Canon BJC600, and Epson Color Stylus _BJC600 _ESCP2C
0
HalfLoC8
5
LandHiC8
1
HalfMedC8
6
FullLoC8
2
HalfHiC8
7
FullMedC8
3
LandLoC8
8
FullHiC8
4
LandMedC8
HPGL/2 (DeskJet 1200C, PaintJet XL300, and DesignJet) _HGL2
9-15 APPENDIX
0
HalfC2
5
FullC8
1
LandC2
6
HalfC256
2
FullC2
7
LandC256
3
HalfC8
8
FullC256
4
LandC8
POLYMATH 4.0 PC
Table 2.1 - PM_PRINTER OPTIONS
Output Page /Resolution /Color
Output Page /Resolution /Color
0
LoResC16
3
LoResC2
1
MedResC16
4
MedResC2
2
HiResC16
5
HiResC2
0
ColorMode
1
MonoMode
0
LoResC2
2
HiResC2
1
MedResC2
Zsoft PCX _PCX
Windows BMP _BMP GEM IMG _IMG
Tagged Image Format (TIFF) compressed and uncompressed _TIF _UTIF
0
LoResC2
1
MedResC2
2
HiResC2
1
MonoMode
1
MonoMode
Computer Graphics Metafile (ANSI) _CGM
0
ColorMode
0
ColorMode
AutoCad DXF _DXF
Video Show (ANSI NAPLPS) _VSHO
0
ColorMode
Word Perfect Graphics _WPG
0
POLYMATH 4.0 PC
ColorMode
APPENDIX 9-16
Table 2.2 - PM_PRINTER OPTIONS
Output Page /Resolution /Color
Output Page /Resolution /Color
0
ColorMode
1
MonoMode
ColorMode
1
MonoMode
ColorMode
1
MonoMode
Windows Metafile _WMF
Adobe Illustrator PostScript _AI
0
Color QuickDraw (PICT) _PCT
9-17 APPENDIX
0
POLYMATH 4.0 PC
POLYMATH VERSION 4.1 Provides System Printing from Windows 3.X, 95, 98 and NT
USER-FRIENDLY NUMERICAL ANALYSIS PROGRAMS - SIMULTANEOUS DIFFERENTIAL EQUATIONS - SIMULTANEOUS ALGEBRAIC EQUATIONS - SIMULTANEOUS LINEAR EQUATIONS - POLYNOMIAL, MULTIPLE LINEAR AND NONLINEAR REGRESSION for IBM and Compatible Personal Computers Internet: http://www.polymath-software.com Users are encouraged to obtain the latest general information on POLYMATH and its use from the above Internet site. This will include updates on this version and availability of future versions. Michael B. Cutlip and Mordechai Shacham, the authors of POLYMATH, have prepared this software to accompany the book entitled Problem Solving in Chemical Engineering with Numerical Methods. This book is a companion book for students and professional engineers who want to utilize the POLYMATH software to effectively and efficiently obtain solutions to realistic and complex problems. Details on this Prentice Hall book, ISBN 0-13-862566-2, can be found at www.polymathsoftware.com or at www.prenhall.com.
Copyright 1998 by M. Shacham and M. B. Cutlip This manual may be reproduced for educational purposes by licensed users. IBM and PC-DOS are trademark of International Business Machines MS-DOS and Windows are trademarks of Microsoft Corporation
i -2 PREFACE
POLYMATH 4.1 PC
POLYMATH LICENSE AGREEMENT The authors of POLYMATH agree to license the POLYMATH materials contained within this disk and the accompanying mpoly41.pdf file to the owner of the Prentice Hall textbook Elements of Chemical Reaction Engineering, by H. Scott Fogler. This license is for noncommercial and educational uses exclusively. Only one copy of this software is to be in use on only one computer or computer terminal at any one time. One copy of the manual may be reproduced in hard copy only for noncommercial educational use of the textbook owner. This individual-use license is for POLYMATH Version 4.1 and applies to the owner of the textbook. Permission to otherwise copy, distribute, modify or otherwise create derivative works of this software is prohibited. Internet distribution is not allowed under any circumstances. This software is provided AS IS, WITHOUT REPRESENTATION AS TO ITS FITNESS FOR AND PURPOSE, AND WITHOUT WARRANTY OF ANY KIND, EITHEREXPRESS OR IMPLIED, including with limitation the implied warranties of merchantability and fitness for a particular purpose. The authors of POLYMATH shall not be liable for any damages, including special, indirect, incidental, or consequential damages, with respect to any claim arising out of or in connection with the use of the software even if users have not been or are hereafter advised of the possibility of such damages. HARDWARE REQUIREMENTS POLYMATH runs on the IBM Personal Computer and most compatibles. A floating-point processor is required. Most graphics boards are automatically supported. The minimum desirable application memory is 450 Kb plus extended memory for large applications. POLYMATH works with PC and MS DOS 3.0 and above. It can also execute as a DOS application under Windows 3.1, Windows 95, Windows 98, and Windows NT. It is important to give POLYMATH as much of the basic 640 Kb memory as possible and up to 2048 Kb of extended memory during installation. This version of POLMATH supports only the Windows printers that are available to your operating system. POLYMATH 4.1 PC
PREFACE i-3
TABLE OF CONTENTS - POLYMATH PAGE INTRODUCTION POLYMATH OVERVIEW..................................................................... 1 MANUAL OVERVIEW.......................................................................... 1 INTRODUCTION................................................................................ 1 GETTING STARTED.......................................................................... 1 HELP.................................................................................................... 1 UTILITIES........................................................................................... 1 APPENDIX........................................................................................... 1 DISPLAY PRESENTATION................................................................. 1 KEYBOARD INFORMATION............................................................. 1 ENTERING VARIABLE NAMES........................................................ 1 ENTERING NUMBERS......................................................................... 1 MATHEMATICAL SYMBOLS............................................................ 1 MATHEMATICAL FUNCTIONS........................................................ 1 LOGICAL EXPRESSIONS.................................................................... 1 POLYMATH MESSAGES...................................................................... 1 HARD COPY........................................................................................... 1 GRAPHICS.............................................................................................. 1 -
1 2 2 2 2 2 2 3 3 4 4 5 5 6 6 6 6
GETTING STARTED HARDWARE REQUIREMENTS......................................................... POLYMATH SOFTWARE .................................................................... INSTALLATION TO INDIVIDUAL COMPUTERS & NETWORKS. FIRST TIME EXECUTION OF POLYMATH..................................... EXITING POLYMATH PROGRAM.....................................................
2 2 2 2 2
-
1 1 1 2 3
HELP MAIN HELP MENU............................................................................... ACCESSING HELP BEFORE PROBLEM ENTRY.......................... ACCESSING HELP DURING PROBLEM ENTRY........................... CALCULATOR HELP........................................................................... UNIT CONVERSION HELP.................................................................
3 3 3 3 3
-
1 2 2 3 3
UTILITIES CALCULATOR....................................................................................... 4 CALCULATOR EXPONENTIATION............................................... 4 AVAILABLE FUNCTIONS................................................................ 4 ASSIGNMENT FUNCTIONS............................................................. 4 CALCULATOR EXAMPLES............................................................. 4 UNIT CONVERSION............................................................................ 4 PREFIXES FOR UNITS...................................................................... 4 UNIT CONVERSION EXAMPLE...................................................... 4 i-4 PREFACE POLYMATH 4.1
- 1 - 1 - 1 - 3 - 3 - 5 - 5 - 6 PC
PROBLEM STORAGE........................................................................... FILE OPERATIONS............................................................................ LIBRARY OPERATIONS..................................................................... LIBRARY STORAGE.......................................................................... LIBRARY RETRIEVAL...................................................................... PROBLEM OUTPUT AS PRINTED GRAPHICS............................. SAMPLE SCREEN PLOT.................................................................... OPTIONAL SCREEN PLOT................................................................ PRESENTATION PLOT....................................................................... PROBLEM OUTPUT TO SCREEN AND AS PRINTED TABLES.. PROBLEM OUTPUT AS DOS FILES.................................................. PROBLEM OUTPUT AS GRAPHICS FILES.....................................
4 - 7 4 - 7 4 - 8 4 - 8 4 - 8 4 - 9 4 - 9 4-10 4-10 4- 10 4-11 4-11
DIFFERENTIAL EQUATIONS SOLVER QUICK TOUR.......................................................................................... DIFFERENTIAL EQUATION SOLVER............................................. STARTING POLYMATH.................................................................... SOLVING A SYSTEM OF DIFFERENTIAL EQUATIONS.............. ENTERING THE EQUATIONS.......................................................... ALTERING THE EQUATIONS.......................................................... ENTERING THE BOUNDARY CONDITIONS................................. SOLVING THE PROBLEM................................................................. PLOTTING THE RESULTS................................................................. EXITING OR RESTARTING POLYMATH....................................... INTEGRATION ALGORITHMS.......................................................... TROUBLE SHOOTING......................................................................... SPECIFIC ERROR MESSAGES.......................................................... NONSPECIFIC ERROR MESSAGES.................................................
5 5 5 5 5 5 5 5 5 5 5 5 5 5
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1 1 1 2 3 4 4 5 6 7 8 9 9 9
ALGEBRAIC EQUATIONS SOLVER QUICK TOUR.......................................................................................... NONLINEAR ALGEBRAIC EQUATION SOLVER......................... STARTING POLYMATH.................................................................... SOLVING ONE NONLINEAR EQUATION...................................... SOLVING A SYSTEM OF NONLINEAR EQUATIONS.................. EXITING OR RESTARTING POLYMATH....................................... SELECTION OF INITIAL ESTIMATES FOR THE UNKNOWNS METHOD OF SOLUTION.................................................................... TROUBLE SHOOTING.........................................................................
6 6 6 6 6 6 6 6 6
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1 1 1 3 5 7 7 7 8
POLYMATH 4.1 PC
PREFACE i-5
LINEAR EQUATIONS SOLVER QUICK TOUR.......................................................................................... LINEAR EQUATION SOLVER......................................................... STARTING POLYMATH.................................................................... SOLVING FIVE SIMULTANEOUS EQUATIONS............................ EXITING OR RESTARTING POLYMATH.......................................
7 7 7 7 7
REGRESSION QUICK TOUR......................................................................................... REGRESSION PROGRAM.................................................................. STARTING POLYMATH.................................................................... QUICK TOUR PROBLEM 1................................................................ RECALLING SAMPLE PROBLEM 3.... ............................................ FITTING A POLYNOMIAL................................................................ FITTING A CUBIC SPLINE................................................................ EVALUATION OF AN INTEGRAL WITH THE CUBIC SPLINE... MULTIPLE LINEAR REGRESSION.................................................. RECALLING SAMPLE PROBLEM 4................................................. SOLVING SAMPLE PROBLEM 4...................................................... TRANSFORMATION OF VARIABLES............................................. NONLINEAR REGRESSION.............................................................. EXITING OR RESTARTING POLYMATH....................................... SOLUTION METHODS.........................................................................
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
- 1 - 1 - 1 - 1 - 2 - 4 - 5 - 6 - 7 - 7 - 8 -11 -13 -15 -15
9 9 9 9 9 9 9 9
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1 1 2 2 2 3 3 3
9 9 9 9 9
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4 4 6 6 7
APPENDIX INSTALLATION AND EXECUTION INSTRUCTIONS................... WINDOWS 3.X INSTALLATION....................................................... WINDOWS 3.X EXECUTION............................................................. WINDOWS 3.X SHUTDOWN............................................................. USING PRINT METAFILES IN DOCUMENTS IN WINDOWS 3.X. WINDOWS 95, 98, AND NT INSTALLATION.................................. WINDOWS 95, 98, AND NT EXECUTION........................................ WINDOWS 95, 98, AND NT SHUTDOWN........................................ USING PRINT METAFILES IN DOCUMENTS FOR WINDOWS 95, 98, AND NT......................................................... INSTALLATION QUESTIONS............................................................ ADDITIONAL NETWORK INSTALLATION NEEDS..................... TROUBLESHOOTING.......................................................................... "OUT OF ENVIRONMENT SPACE" MESSAGE.............................
i-6 PREFACE
-
1 1 1 3 4
POLYMATH 4.1 PC
INTRODUCTION POLYMATH OVERVIEW POLYMATH is an effective yet easy to use computational system which has been specifically created for professional or educational use. The various programs in the POLYMATH series allow the user to apply effective numerical analysis techniques during interactive problem solving on a personal computer. Whether you are student, engineer, mathematician, scientist, or anyone with a need to solve problems, you will appreciate the ease in which POLYMATH allows you to obtain solutions. Chances are very good that you will seldom need to refer to this manual beyond an initial reading because POLYMATH is so easy to use. With POLYMATH, you are able to focus your attention on the problem at hand rather than spending your valuable time in learning how to use or reuse the program. You are encouraged to become familiar with the mathematical concepts being utilized in POLYMATH. These are discussed in most textbooks concerned with numerical analysis. The available programs in POLYMATH include: - SIMULTANEOUS DIFFERENTIAL EQUATION SOLVER - SIMULTANEOUS ALGEBRAIC EQUATION SOLVER - SIMULTANEOUS LINEAR EQUATION SOLVER - POLYNOMIAL, MULTIPLE LINEAR AND NONLINEAR REGRESSION Whether you are a novice computer user or one with considerable computer experience, you will be able to make full use of the programs in POLYMATH which allow numerical problems to be solved conveniently and interactively. If you have limited computer experience, it will be helpful for you to read through this manual and try many of the QUICK TOUR problems. If you have considerable personal computer experience, you may only need to read the chapters at the back of this manual on the individual programs and try some of the QUICK TOUR problems. This manual will be a convenient reference guide when using POLYMATH. POLYMATH 4.1 PC
INTRODUCTION 1-1
MANUAL OVERVIEW This manual first provides general information on features which are common to all of the POLYMATH programs. Particular details of individual programs are then presented. Major chapter topics are outlined below: INTRODUCTION The introduction gives an overview of the POLYMATH computational system and gives general instructions for procedures to follow when using individual POLYMATH programs. GETTING STARTED This chapter prepares you for executing POLYMATH the first time, with information about turning on the computer, loading POLYMATH, and making choices from the various menu and option screens. HELP On-line access to a general help section is discussed. UTILITIES This chapter discusses features that all programs have available. These include a scientific calculator and a convenient conversion for units and dimensions. This chapter discusses saving individual problems, data and/or result files on a floppy or hard disk. It also describes the use of the problem library for storing, retrieving and modifying problems on a disk. Options for the printing and plotting of results are explained. The remaining chapters of the manual present a QUICK TOUR of each individual POLYMATH program and are organized according to the following subsections: 1. PROGRAM OVERVIEW This subsection gives general details of the particular program. 2. QUICK TOUR You can use this subsection to see how easy it is to enter and solve a problem with a particular POLYMATH program. APPENDIX Detailed installation instructions and additional output options are presented for advanced users. 1-2 INTRODUCTION
POLYMATH 4.1 PC
DISPLAY PRESENTATION Throughout this manual, a full screen is indicated by a total enclosure:
An upper part of screen is contained within a partial enclosure:
A lower part of screen is shown by a partial enclosure:
An intermediate part of a screen is given between vertical lines:
The option box is given by:
KEYBOARD INFORMATION When using POLYMATH, it is not necessary to remember a complex series of keystrokes to respond to the menus, options, or prompts. The commands available to you are clearly labeled for easy use on each display. Normally the keystrokes which are available are given on the display as indicated on the PROBLEM OPTIONS display shown below.
POLYMATH 4.1 PC
INTRODUCTION 1-3
USING THE KEY symbol is used to indicate In this manual as in POLYMATH, the the carriage return key which is also called the enter key. Usually when you are responding to a menu option, the enter key is not required. However, when data or mathematical functions are being entered, the enter key is used to indicate that the entry is complete. SHIFTED KEYPRESSES Some options require that several keys be pressed at the same time. This is indicated in POLYMATH and in this manual by a dash between the keys such as a ⇑ F8 which means to press and hold the ⇑ or "shift" key, then press the F8 function key and finally release both keys. THE EDITING KEYS Use the left and right arrow keys to bring the cursor to the desired position, while editing an expression. Use the Del key to delete the character (Back Space) key to delete the first character to above the cursor or the the left of the cursor. Typed in characters will be added to the existing expression in the first position left to the cursor. BACKING UP KEYS Press either the F8 or the Esc key to have POLYMATH back up one program step. ENTERING VARIABLE NAMES A variable may be called by any alphanumeric combination of characters, and the variable name MUST start with a lower or upper case letter. Blanks, punctuation marks and mathematical operators are not allowed in variable names. Note that POLYMATH distinguishes between lower and upper case letters, so the variables 'MyVar2' and 'myVar2' are not the same. ENTERING NUMBERS All numbers should be entered with the upper row on the key board or with the numerical keypad activated. Remember that zero is a number from the top row and not the letter key from the second row. The number 1 is from the top row while letter l is from the third row.
1-4 INTRODUCTION
POLYMATH 4.1 PC
The results of the internal calculations made by POLYMATH have at least a precision of eight digits of significance. Results are presented with at least four significant digits such as xxx.x or x.xxx . All mathematical operations are performed as floating point calculations, so it is not necessary to enter decimal points for real numbers. MATHEMATICAL SYMBOLS You can use familiar notation when indicating standard mathematical operations. Operator +
Meaning addition subtraction multiplication division power of 10
Symbol + * / x.x10a
Entry + x * -: / x.xea x.xEa (x.x is numerical with a decimal and a is an integer) exponentiation rs r**s or r^s MATHEMATICAL FUNCTIONS Useful functions will be recognized by POLYMATH when entered as part of an expression. The arguments must be enclosed in parentheses: ln (base e) abs (absolute value) sin arcsin sinh log (base 10) int (integer part) cos arccos cosh exp frac (fractional part) tan arctan tanh POLYMATH 4.1 PC
exp2(2^x) round (rounds value) sec arcsec arcsinh exp10 (10^x) sign (+1/0/-1) csc arccsc arccosh sqrt (square root) cbrt (cube root) cot arccot arctanh INTRODUCTION 1-5
LOGICAL EXPRESSIONS An "if" function is available during equation entry with the following syntax: if (condition) then (expression) else (expression). The parentheses are required, but spaces are optional. The condition may include the following operators: > greater than < less than >= greater than or equal 0) then(log(x)) else(0) b=if (TmaxT) then (maxT) else (T)) POLYMATH MESSAGES There are many POLYMATH messages which may provide assistance during problem solving. These messages will tell you what is incorrect and how to correct it. All user inputs, equations and data, are checked for format and syntax upon entry, and feedback is immediate. Correct input is required before proceeding to a problem solution. HARD COPY If there is a printer connected to the computer, hard copy of the problem statements, tabular and graphical results etc. can be made by pressing F3 key wherever this option is indicated on the screen. This version of POLYMATH allows printing from the Windows printers, and the Windows meta files can also be used in various documents. See pages 9-2 and 9-4 for more details. Problem statements and results can be also printed by saving them as a file and printing this file after leaving POLYMATH. GRAPHICS POLYMATH gives convenient displays during problem entry, modification and solution. Your computer will always operate in a graphics mode while you are executing POLYMATH. 1-6 INTRODUCTION
POLYMATH 4.1 PC
GETTING STARTED This chapter provides information on the hardware requirements and discusses the installation of POLYMATH. HARDWARE REQUIREMENTS POLYMATH 4.1 runs on IBM compatible personal computers that support the Windows-based operating systems. Most graphics boards are automatically supported. The minimum application memory requirement is approximately 450Kb, and extended memory is used when it is available. POLYMATH 4.1 runs as a DOS application under Windows 3.X, Windows 95, Windows 98, and Windows NT. Printing is accomplished by a separate program operating under Windows. This allows the printing from POLYMATH to be done by any printer that can be used with the particular Windows operating system. POLYMATH SOFTWARE The complete set of POLYMATH application programs with a general selection menu is available on a single 3-1/2 inch 1.44 Mb floppy in compressed form. It is recommended that a backup disk be made before attempting to install POLYMATH onto a hard disk. Installation is available via an install program which is executed from any drive. INSTALLATION TO INDIVIDUAL COMPUTERS AND NETWORKS POLYMATH executes best when the software is installed on a hard disk or a network. There is a utility on the POLYMATH distribution disk which is called "install". Detailed installation instructions are found in the Appendix of this manual. Experienced users need to simply put the disk in the floppy drive, typically A or B. Type "install" at the prompt of your installation floppy drive, and press return. Follow the instructions on the screen to install POLYMATH on the particular drive and directory that you desire. Note that the default drive is "C:" and the default directory is called "POLYMAT4". Network installation will require responses to additional questions during installation. Latest detailed information can be found on the README.TXT file found on the installation disk.
POLYMATH 4.1 PC
GETTING STARTED 2-1
FIRST TIME EXECUTION OF POLYMATH The execution of POLYMATH is started by first having your current directory set to the subdirectory of the hard disk where POLYMATH version 4.1 is stored. This is assumed to be C:\POLYMAT4 C:\POLYMAT4 > Execution is started by entering "polymath" at the cursor C:\POLYMAT4 > polymath and then press the Return ( ) key. The Program Selection Menu should then appear:
The desired POLYMATH program is then selected by entering the appropriate letter. You will then taken to the Main Program Menu of that particular program. Individual programs are discussed in later chapters of this manual. GETTING STARTED 2-2
POLYMATH 4.1 PC
EXITING POLYMATH PROGRAM The best way to exit POLYMATH is to follow the instructions on the program display. However, a Shift-¡F10 keypress (⇑F10) will stop the execution of POLYMATH at any point in a program and will return the user to the Polymath Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM so be sure to store your problem as an individual file or in the library before exiting the program in this manner. A query is made to determine if the user really wants to end the program in this manner while losing the current problem. This ⇑ F10 keypress is one of the few POLYMATH commands which is not always indicated in the various Display Menus. It is worth remembering.
POLYMATH 4.1 PC
GETTING STARTED 2-3
HELP MAIN HELP MENU Each individual POLYMATH program has a detailed help section which is available from many points in the program by pressing F6 when indicated. The Help Menu allows the selection of the topic area for specific help as shown below for the Differential Equation Solver:
For example, pressing "a" gives a discussion on entering the equations.
3-1 HELP
POLYMATH 4.1 PC
Once the current topic is completed, the Help Options Menu provides for additional options as shown below:
The ⇑ - F8 option to return to the program will take you to the display where you originally requested HELP ACCESSING HELP BEFORE PROBLEM ENTRY The Main Help Menu is reached during the startup of your POLYMATH program from the Main Menu as shown below and from the Problem Options Menu by pressing F6.
ACCESSING HELP DURING PROBLEM ENTRY When you are entering a problem, the HELP MENU is available from the Problem Options Menu. This will allow you to obtain the necessary help and return to the same point where the HELP MENU was originally requested. As an example, this access point is shown in the Problem Options Menu shown on the next page. POLYMATH 4.1 PC
HELP 3-2
CALCULATOR HELP A detailed discussion of the POLYMATH Calculator is given in Chapter 4 of this manual. The Calculator can be accessed from by pressing F4 from any point in a POLYMATH program.
An F6 keypress brings up the same page help which provides a brief instruction inside the Calculator window. UNIT CONVERSION HELP The Unit Conversion Utility is discussed in Chapter 4 of this manual. There is no on-line help for this utility.
3-3 HELP
POLYMATH 4.1 PC
UTILITIES CALCULATOR A sophisticated calculator is always available for use in a POLYMATH program. This calculator is accessed by pressing the F4 key . At this time a window will be open in the option box area which will give you access to the calculator.
CALCULATOR: Enter an expression and press to evaluate it. Press to leave or press F6 for information
The POLYMATH calculator allows you to enter an expression to be evaluated. After the expression is complete, press to have it calculated. You may then press again to clear the expression, or you may edit your expression using the standard editing functions. When you wish to leave the calculator, just press the F8 or Esc key. CALCULATOR EXPONENTIATION Numbers may also be entered in scientific notation. The calculator will recognize E or e as being equivalent to the notation *10**. Either ** or ^ indicates general exponentiation. For example, the following three expressions are equivalent for a particular value of A: 4.71*10**A = 4.71eA = 4.71*10^A AVAILABLE FUNCTIONS A number of standard functions are available for use in the calculator. The underlined portion of the following functions is all that is required provided that all arguments are enclosed in parentheses. The arguments may themselves be expressions or other functions. The nesting of function is allowed. ln ( ) or alog ( ) = natural logarithm to the base e log ( ) or alog10 ( ) = logarithm to the base 10 exp ( ) = exponential (ex) exp2 ( ) = exponential of 2 (2x) exp10 ( ) = exponential of 10 (10x) sqrt ( ) = square root abs ( ) = absolute value POLYMATH 4.1 PC
UTILITIES 4-1
int ( ) or ip ( ) = integer part frac ( ) = or fp ( ) = fractional part round ( ) = rounded value sign ( ) = returns + 1 or 0 or -1 N! = factorial of integer part of number N (this only operates on a number) sin ( ) = trigonometric sine with argument in radians cos ( ) = trigonometric cosine with argument in radians tan ( ) = trigonometric tangent with argument in radians sec ( ) = trigonometric secant with argument in radians csc ( ) = trigonometric cosecant with argument in radians cot ( ) = trigonometric cotangent with argument in radians arcsin ( ) = trigonometric inverse sine with result in radians, alternates arsin ( ) and asin ( ) arccos ( ) = trigonometric inverse cosine with result in radians, alternates arcos ( ) and acos ( ) arctan ( ) = trigonometric inverse tangent with result in radians, alternate atan ( ) arcsec ( ) = trigonometric inverse secant with result in radians arccsc ( ) = trigonometric inverse cosecant with result in radians arccot ( ) = trigonometric inverse cotangent with result in radians sinh ( ) = hyperbolic sine cosh ( ) = hyperbolic cosine tanh ( ) = hyperbolic tangent arcsinh ( ) = inverse hyperbolic sine arccosh ( ) = inverse hyperbolic cosine arctanh ( ) = inverse hyperbolic tangent You should note that the functions require that their arguments be enclosed in parentheses, but that the arguments do not have to be simple numbers. You may have a complicated expression as the argument for a function, and you may even nest the functions, using one function (or an expression including one or more functions) as the argument for another.
4-2 UTILITIES
POLYMATH 4.1 PC
ASSIGNMENT FUNCTIONS The assignment function is a way of storing your results. You may specify a variable name in which to store the results of a computation by first typing in the variable name, then an equals sign, then the expression you wish to store. For example, if you wish to store the value of sin (4/3) 2 in variable 'a', you would enter: a = sin (4/3)**2 Variable names must start with a letter, and can contain letters and digits. There is no limit on the length of the variable names, or on the number of variables you can use. You can then use the variable 'a' in other calculations. These variables are stored only as long as you remain in the current POLYMATH program. Please note that all stored values are lost when the particular program is exited. Calculator information is not retained during problem storage. CALCULATOR EXAMPLES Example 1. In this example the vapor pressure of water at temperatures of 50, 60, and 70 o C has to be calculated using the equation: log10 P = 8.10765 – 1750.29 235.0 + T For T = 50 the following expression should be typed into the calculator: 10^(8.10675 - 1750.29 / (235+50)) CALCULATOR: Enter an expression and press to evaluate it. Press to leave or press F6 for information.
Pressing brings up the desired answer which is 92.3382371 o mm Hg at 50 C. To change the temperature use the left arrow to bring the cursor just right to the zero of the number 50, use the (BkSp or delete) key to erase this number and type in the new temperature value.
POLYMATH 4.1 PC
UTILITIES 4-3
Example 2. In this example the pressure of carbon dioxide at temperature of T = 400 K and molal volume of V = 0.8 liter is calculated using the following equations: P = RT – a V – b V2
Where
2 2 a = 27 R Tc Pc 64
b = RTc 8 Pc
R = 0.08206, Tc = 304.2 and Pc = 72.9. One way to carry out this calculation is to store the numerical values to store in the named variables. First you can type in Pc = 72.9 and press this value as shown below.
Pc=72.9 CALCULATOR: Enter an expression and press to evaluate it. =72.9.
After that you can type in Tc = 304.2 and R = 0.08206. To calculate b, you must type in the complete expression as follows: b=R*Tc/(8*Pc) CALCULATOR: Enter an expression and press to evaluate it. =0.0428029012
The value of a is calculated in the same manner yielding a value of 3.60609951. Finally P can be calculated as shown:
P=R*400/(0.8-b)-a/(0.8*0.8) CALCULATOR: Enter an expression and press to evaluate it. =37.7148168
4-4 UTILITIES
POLYMATH 4.1 PC
UNIT CONVERSION A utility for unit conversion is always available for use within a POLYMATH program. Unit Conversion is accessed by pressing F5 wherever you desire. This will result in the following window in the option box area: Type the letter of the physical quantity for conversion. a) Energy b) Force c) Length d) Mass e) Power f) Pressure g) Volume h) Temperature F8 or ESC to exit
The above listing indicates the various classes of Unit Conversion which are available in POLYMATH. A listing of the various units in each class is given below: ENERGY UNITS: joule, erg, cal, Btu, hp hr, ft lbf, (liter)(atm), kwh FORCE UNITS: newton, dyne, kg, lb, poundal LENGTH UNITS: meter, inch, foot, mile, angstrom, micron, yard MASS UNITS: kilogram, pound, ton (metric) POWER UNITS: watts, hp (metric), hp (British), cal/sec, Btu/sec, ft lbf /sec PRESSURE UNITS: pascal, atm, bar, mm Hg (torr), in Hg, psi [lbf /sq in] VOLUME UNITS: cu. meter, liter, cu. feet, Imperial gal, gal (U.S.), barrel
(oil), cu. centimeter
TEMPERATURE UNITS: Celsius, Fahrenheit, Kelvin, Rankine
PREFIXES FOR UNITS It is convenient to also specify prefixes for any units involved in a Unit Conversion. This feature provides the following prefixes: deci 10 -1 hecto 10 2
centi 10 -2 kilo 10 3
POLYMATH 4.1 PC
milli 10 -3 mega 10 6
micro 10 -6 giga 10 9
deka 10
UTILITIES 4-5
UNIT CONVERSION EXAMPLE Suppose you want to convert 100 BTU's to kilo-calories. First you should access the Unit Conversion Utility by pressing F5. This will bring up the following options Type the letter of the physical quantity for conversion. a) Energy b) Force c) Length d) Mass e) Power f) Pressure g) Volume h) Temperature F8 or ESC to exit
Press "a" to specify an Energy conversion: From units: Type in a letter (F9 to set a prefix first) a. joule b. erg c. cal d. Btu f. ft lbf g. (liter)(atm) h. kwh
e. hp hr
Type a "d" to specify Btu: From units : Btu To units: a. joule b. erg c. cal f. ft lb, g. (liter)(atm) h. kwm
(F9 for a prefix) d. Btu e. hp hr
Use F9 to indicate a Prefix: Press the number of the needed prefix or F9 for none. 1) deci 10 -1 2) centi 10 -2 3) milli 10 -3 4) micro 10 -6 2 3 5) deka 10 6) hecto 10 7) kilo 10 8) mega 106 9) giga 109
Please indicate kilo by pressing the number 7. From units: Btu a. joule b. erg f. ft lbf g. (liter)(atm)
To units: kilo c. cal d. Btu h. kwh
e. hp hr
Complete the units by pressing "c" for calories. Indicate the numerical value to be 100 and press enter: From units: Btu Numerical value: 100 100.00 Btu = 25.216 kilo-cal
4-6 UTILITIES
To units: kilo-cal
POLYMATH 4.1 PC
PROBLEM STORAGE POLYMATH programs can be stored for future use as either DOS files or in a "Library" of problems. The Library has the advantage that the titles are displayed for only the problems for the particular POLYMATH program which is in use. Both the DOS files and the Library can be placed in any desired subdirectory or floppy disk. In both cases, only the problem and not the solution is stored. The storage options are available from the Task Menu which is available from POLYMATH programs by pressing either F9 from the Main Menu or ⇑ F8 from the Problem Options Menu.
FILE OPERATIONS A current problem can be saved to a DOS file by selecting "S" from the Task Menu. The desired directory and DOS file name can be specified from the window given below:
Note that the path to the desired directory can also be entered along with the file name as in "A:\MYFILE.POL" which would place the DOS file on the Drive A. A previously stored problem in a DOS file can be loaded into POLYMATH from the Task Menu by selecting "L". A window similar to the one above will allow you to load the problem from any subdirectory or floppy disk. An F6 keypress gives the contents of the current directory for help in identifying the file name for the desired problem. POLYMATH 4.1 PC
UTILITIES 4-7
LIBRARY OPERATIONS The Library is highly recommended for storing problems as the titles of the problems are retained and displayed which is a considerable convenience. Also, only the problems for the particular POLYMATH program in current use are displayed. The Library is accessed from the Task Menu by pressing F9 as shown below:
If there is no current Library on the desired subdirectory or floppy disk, then a Library is created. LIBRARY STORAGE The Library Options menu allows the current POLYMATH problem to be stored by simply entering "S". The title as currently defined in the active problem will be displayed. The user must choose a file name for this particular problem; however, it will then be displayed along with the Problem title as shown above. LIBRARY RETRIEVAL The Library Options window allows the current POLYMATH problem to be recalled by first using the cursor keys to direct the arrow to the problem of interest and then entering "L". A window will confirm the library retrieval as shown below:
Problems may be deleted from the Library by using the arrow to identify the problem, and then selecting "D" from the Library Options menu. Users are prompted to verify problem deletion. 4-8 UTILITIES
POLYMATH 4.1 PC
PROBLEM OUTPUT AS PRINTED GRAPHICS One of the most useful features of POLYMATH is the ability to create graphical plots of the results of the numerical calculations. The command to print graphical output is F3. The first step in printing graphical output is to display the desired output variables. The POLYMATH programs allow the user to make plots of up to four variables versus another variable. An example which will be used to demonstrate plotting is the Quick Tour Problem 1 from the next chapter. Here the POLYMATH Differential Equation Solver has produced a numerical solution to three simultaneous ordinary differential equations. The calculations are summarized on a Partial Results display which has the following Display Options Menu:
SIMPLE SCREEN PLOT The selection of "g" from the Display Options Menu allows the user to select desired variables for plotting. A plot of variables A, B, and C versus the independent variable t can be obtain by entering "A,B,C" at the cursor and pressing the Return key ( ).
The resulting graph is automatically scaled and presented on the screen.
POLYMATH 4.1 PC
UTILITIES 4-9
OPTIONAL SCREEN PLOT The selection of "g" from the Display Options Menu with the entry of "B/A" results in B plotted versus A. This demonstrates that dependent variables can be plotted against each other. PRESENTATION PLOT A simple plot can be printed directly or it can be modified before printing by using options from the Graph Option Menu shown below:
This menu allow the user to modify the plot before printing as desired to obtain a final presentation graphic with specified scaling and labels. PROBLEM OUTPUT TO SCREEN AND AS PRINTED TABLES The Display Options Menu also allows the user to select tabular output from the Partial Results Display by pressing "t":
This is shown below for the same entry of "A, B, C" for the Quick Tour Problem 1 from the next chapter on differential equations.
The output shown above gives variable values for the integration interval at selected intervals. The maximum number of points is determined by the numerical integration algorithm. Output variable values for a smaller number of points are determined by interpolation. A Screen Table can be printed by using F3. 4-10 UTILITIES
POLYMATH 4.1 PC
PROBLEM OUTPUT AS DOS FILES The output from many of the POLYMATH programs can also be stored for future use as DOS files for use in taking results to spreadsheets and more sophisticated graphics programs. Typically this is done after the output has been sent to the screen. This is again accomplished with option "d" from the Display Options Menu.
This option take the user to a display where the name and location of the DOS data file is entered:
Please note that the user can change the drive and the directory to an desired location. One the location is indicated and the file name is entered, the desired variable names must be provided and the number of data points to be saved. The file shown below was created as shown for the request of "A,B,C" and 10 data points for Quick Tour Problem 1 from the next chapter: t 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3
A B C 1 0 0 0.74081822 0.19200658 0.067175195 0.54881164 0.24761742 0.20357094 0.40656966 0.24127077 0.35215957 0.30119421 0.21047626 0.48832953 0.22313016 0.17334309 0.60352675 0.16529889 0.13797517 0.69672595 0.12245643 0.10746085 0.77008272 0.090717953 0.082488206 0.82679384 0.067205513 0.062688932 0.87010556 0.049787068 0.047308316 0.90290462
Note that the separate columns of data in this DOS output file are separated by tabs which is suitable format for input to various spreadsheet or graphics programs. POLYMATH 4.1 PC
UTILITIES 4-11
PROBLEM OUTPUT AS GRAPHICS FILES This 4.1 version of POLYMATH creates Windows Meta files (WMFs) that can be printed or entered into documents. These files are normally printed directly by special MetaFile Print programs for Windows 3.X (MFP16) or for Windows 95, 98, and NT (MFP32) . If these special print programs are not running, the generated WMFs are found in the SPOOL subdirectory under the POLYMAT4 directory. Thus the user can place these file directly into word processors and desktop publishing software. Details of this option are found in Chapter 9 of this manual. A typical example would be to create the following output in a WMF file for inclusion in a written reporting using word processing or desktop publishing. The problem is again the for Quick Tour Problem 1 from the next chapter. Note that the figure below has utilized the title and axis definition options.
4-12 UTILITIES
POLYMATH 4.1 PC
DIFFERENTIAL EQUATION SOLVER QUICK TOUR This section is intended to give you a very quick indication of the operation of the POLYMATH Differential Equation Solver Program. DIFFERENTIAL EQUATION SOLVER The program allows the numerical integration of up to 31 simultaneous nonlinear ordinary differential equations and explicit algebraic expressions. All equations are checked for syntax upon entry. Equations are easily modified. Undefined variables are identified. The integration method and stepsize are automatically selected; however, a stiff algorithm may be specified if desired. Graphical output of problem variables is easily obtained with automatic scaling. STARTING POLYMATH To begin, please have POLYMATH loaded into your computer as detailed in Chapter 2. Here it is assumed that your computer is set to the hard disk subdirectory or floppy drive containing the POLYMATH package. At the prompt (assumed C:\POLYMAT4 here), you should enter "polymath" C:\POLYMATH4 > polymath then press the Return ( ) key. The Program Selection Menu should then appear, and you should enter "1" to select the Simultaneous Differential Equation Solver. This should bring up the Main Program Menu:
POLYMATH 4.1 PC
DIFFERENTIAL EQUATIONS 5 -1
Now that POLYMATH is loaded, please press F6 and then the letter "a" to get information on "Entering the equations". The first page of the Help Section should be on your screen as shown here:
Please press F8 to return from the Help to the program, and then press the Enter key ( )to continue this Quick Tour example. SOLVING A SYSTEM OF DIFFERENTIAL EQUATIONS Let us now enter and solve a system of three simultaneous differential equations: d(A) / d(t) = - kA (A) d(B) / d(t) = kA (A) - kB (B) d(C) / d(t) = kB(B) In these equations, the parameter kA is to be constant at a value of 1.0 and the parameter kB is to be constant at the value of 2.0. The initial condition for dependent variable "A" is to be 1.0 when the initial value of the independent variable "t" is zero. The initial conditions for dependent variables "B" and "C" are both zero. The solution for the three differential equations is desired for the independent variable "t" between zero and 3.0. Thus this problem will be entered by using the three differential equations as given above along with two expressions for the values for kA and kB given by: kA = 1.0; kB = 2.0 5-2 DIFFERENTIAL EQUATIONS
POLYMATH 4.1 PC
ENTERING THE EQUATIONS The equations are entered into POLYMATH by first pressing the "a" option from the Problem Options Menu. The following display gives the first equation as it should be entered at the arrow. (Use the Backspace key, to correct entry errors after using arrow keys to position cursor.) Press the ) to indicate that the equation is to be entered. Don't be Return key ( concerned if you have entered an incorrect equation, as there will soon be an opportunity to make any needed corrections. d(A)/d(t)=-ka*A_
The above differential equation is entered according to required format which is given by: d(x)/d(t)=an expression where the dependent variable name "x" and the independent variable name "t" must begin with an alphabetic character and can contain any number of alphabetic and numerical characters. In this Quick Tour problem, the dependent variables are A, B and C for the differential equations, and the independent variable is t. Note that POLYMATH variables are case sensitive. The constants kA and kB are considered to be variables which can be defined by explicit algebraic equations given by the format: x=an expression In this problem, the variables for kA and kB will have constant values. Note that the subscripts are not available in POLYMATH, and in this problem the variable names of ka and kb will be used. Please continue to enter the equations until your set of equations corresponds to the following: Equations: → d(A)/d(t)=-ka*A d(B)/d(t)=ka*A-kb*B d(C)/d(t)=kb*B ka=1 kb=2
As you enter the equations, note that syntax errors are checked prior to being accepted, and various messages are provided to help to identify input errors. Undefined variables are also identified by name during equation entry. POLYMATH 4.1 PC
DIFFERENTIAL EQUATIONS 5-3
ALTERING THE EQUATIONS with no After you have entered the equations, please press equation at the arrow to go to the Problem Options display which will allow needed corrections:
The Problem Options Menu allows you to make a number of alterations on the equations which have been entered. Please make sure that your equations all have been entered as shown above. Remember to first indicate the equation that needs altering by using the arrow keys. When all equation are correct, press ⇑ F7 (keep pressing shift while pressing F7) to continue with the problem solution. ENTERING THE BOUNDARY CONDITIONS At this point you will be asked to provide the initial values for the independent variable and each of the dependent variables defined by the differential equations. Enter initial value for t _
Please indicate this value to be the number "0" and press Return. The next initial value request is for variable "A". Please this value as the value "1." Enter initial value for A 1_ 5-4 DIFFERENTIAL EQUATIONS
POLYMATH 4.1 PC
The initial values for B and C will be requested if they have not been previously entered. Please enter the number "0" for each of these variables. Next the final value for t, the independent variable, will be requested. Set this parameter at "3": Enter final value for t 3_
As soon as the problem is completely specified, then the solution will be generated. However, if you corrected some of your entries, then you may need to press ⇑ F7 again to request the solution. Note that a title such as "Quick Tour Problem 1" could have been entered from the Problem Option Menu. SOLVING THE PROBLEM The numerical solution is usually very fast. For slower computers, an arrow will indicate the progress in the independent variable during the integration. Usually the solution will be almost instantaneous. The screen display after the solution is given below:
POLYMATH 4.1 PC
DIFFERENTIAL EQUATIONS 5-5
Another Return keypress gives the partial Results Table which summarizes the variables of the problem as shown below:
The Partial Results Table shown above provides a summary of the numerical simulation. To display or store the results you can enter "t" (tabular display), "g" (graphical display), or "d" (storing the results on a DOS file). This Table may be printed with the function key F3. PLOTTING THE RESULTS Let us now plot the variable from this Problem 1 by entering "g" for a graphical presentation. When asked to type in the variable for plotting, please enter the input indicated below at the arrow: Type in the names of up to four (4) variables separated by commas (,) and optionally one 'independent' variable preceded by a slash(/). For example, myvar1, myvar2/timevar
A, B, C __
A Return keypress ( ) will indicate the end of the variables and should generate the graphical plot on the next page of the specified variables A, B, and C versus t, the independent variable, for this example.
5-6 DIFFERENTIAL EQUATIONS
POLYMATH 4.1 PC
Suppose that you want to plot variable B versus variable A. Select the option "g" from the Display Options Menu and enter B/A when asked for the variable names. Type in the names of up to four (4) variables separated by commas (,) and optionally one 'independent' variable preceded by a slash(/). For example, myvar1, myvar2/timevar
B/A
This will results in a scaled plot for variable B versus variable A. This concludes the Quick Tour problem using the Differential Equation Solver. If you wish to stop working on POLYMATH, please follow the exiting instructions given below. EXITING OR RESTARTING POLYMATH A ⇑ - F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. POLYMATH 4.1 PC
DIFFERENTIAL EQUATIONS 5-7
INTEGRATION ALGORITHMS The program will first attempt to integrate the system of differential equations using the Runge-Kutta-Fehlberg (RKF) algorithm. A detailed discussion of this algorithm is given by Forsythe et al.* This algorithm monitors the estimate of the integration error and alters the step size of the integration in order to keep the error below a specified threshold. The default values for both relative and absolute (maximal) errors are less than 10-10. If this cannot be attained, then the absolute and relative errors are set as necessary to 10-7 and then to 10-4. If it is not possible to achieve errors of 10-4, then the integration is stopped, and the user is given a choice to continue or to try an alternate integration algorithm for stiff systems of differential equations. Under these circumstances, the system of equations is likely to be "stiff" where dependent variables may change in widely varying time scales, and the user is able to initiate the solution from the beginning with an alternate "stiff" integration algorithm The algorithm used is the semi-implicit extrapolation method of Bader-Deuflhard**, and the maximal errors are again started at 10-10. When the integration is very slow, the F10 keypress will allow the selection of the stiff algorithm, and the problem will be solved from the beginning.
* Frosythe, B. E., M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computation, Prentice-Hall, Englewood Cliffs, NJ, 1977. ** Press, W. H., P. B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, 2nd Ed., Cambridge University Press, 1992, pp. 735739.
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POLYMATH 4.1 PC
TROUBLE SHOOTING SPECIFIC ERROR MESSAGES Most error messages given by POLYMATH are self-explanatory, and they suggest the type of action which should be taken to correct the difficulty. NONSPECIFIC ERROR MESSAGES "Circular dependency detected." This message appears during the inputting of equations when the equations are not all explicit. For example, an attempt to define y=z/x when z has been previously defined will cause this error message to appear. This version of POLYMATH Differential Equations Solver can only solve variables which can be explicitly expressed as a function of other variables. "The expression ... is undefined at the starting point." This common problem can be solved by starting the integration from t=eps where eps is a very small number and t represents the independent problem variable. "Solution process halted due to a lack of memory." This message may result when the default Runge-Kutta-Fehlberg algorithm is used for a stiff system of differential equations, and thus very small step sizes are taken. Consequently, a large number of data points for possible plotting of the results. Use the F10 to stop the integration and switch to the stiff algorithm. If the message persists, then take the following steps to resolve the difficulty: (1) If you are running under Windows, make sure the PIF for POLYMATH specifies 640K of conventional memory and 1024K or more of XMS(see the Appendix for more details). (2) Remove other memory-resident programs from your computer. (3) Reduce the number of equations. This is most easily accomplished by introducing the numerical values of the constants into the equations, instead of defining them separately. (4) Reduce the integration interval.
POLYMATH 4.1 PC
DIFFERENTIAL EQUATIONS 5-9
"Solution process halted because it was not going anywhere." This message usually appears when the problem is very stiff, and the default RKF algorithm is used for integration. The stiff algorithm should be used, or the interval of integration should be reduced. If the error message persists, there are probably errors in the problem setup or input. Please check for errors in the basic equation set, the POLYMATH equation entry, and the numerical values and the units of the variables.
.
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POLYMATH 4.1 PC
ALGEBRAIC EQUATION SOLVER QUICK TOUR This chapter is intended to give a very brief discussion of the operation of the POLYMATH Nonlinear Algebraic Equation Solver. NONLINEAR ALGEBRAIC EQUATION SOLVER The user can solve up to a combination of 32 simultaneous nonlinear equations and explicit algebraic expressions. Only real (non-complex) roots are found. All equations are checked for correct syntax and other errors upon entry. Equations can be easily be modified, added or deleted. Multiple roots are given for a single equation. STARTING POLYMATH To begin, please have POLYMATH loaded into your computer as detailed in Chapter 2. Here it is assumed that your computer is set to the hard disk subdirectory or floppy drive containing the POLYMATH package. At the prompt (assumed C:\POLYMAT4 here), you should enter "polymath" C:\POLYMAT4>polymath then press the Return ( ) key. The Main Program Menu should then appear: The Program Selection Menu should then appear, and you should enter "2" to select the Simultaneous Algebraic Equation Solver.
POLYMATH 4.1 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-1
Once that POLYMATH is loaded, please utilize the Help Menu by pressing F6 and then the letter "a" to obtain details on in order to learn how to input the equations. The first page of this Help Section is given below:
This Help Section gives detailed information for entering the nonlinear and auxiliary equations. Press F8 to return from the Help Section to the program, and then press the Enter key ( ) to enter an equation for the first Quick Tour example. The Problem Options Menu at the bottom of your display allows entry of equations with the keypress of "a". Now you are ready for the first problem.
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POLYMATH 4.1 PC
SOLVING ONE NONLINEAR EQUATION The first nonlinear equation to be solved as Quick Tour Problem 1 is: x2 -5x + 6 = 0 The solution is to be obtained over the range of x between 1 and 4. This equation is entered into POLYMATH using the equation entry guidelines where the equation is to be zero at the solution. The following display gives the equation as it should be entered at the arrow: (Use the or the delete key to erase entered characters. Standard DOS editing is available at the cursor.)
f(x)=x^2-5*x+6_ The format for the above equation for f(x) is that the left side of the equation will be equal to zero when the solution has been obtained. The variable which is to be determined is set as an argument within the parentheses for the function f( ). Thus in this case, the variable is x and the function to be solved as being zero is x2-5x+6. Also note that in POLYMATH one way of entering x2 is x^2. An alternative entry is x**2. After you have correctly typed the equation at the arrow, please press once to enter it and then again to end equation entry. This should result in the Problem Options Menu at the bottom of the display and the equation at the top:
The Problem Options Menu indicates which options are now available for you to carry out a number of tasks. In this case, the problem should be complete, and these options for the equation at the arrow will not be needed. If an equation needed to be changed, then you would enter a "c" at the above display. (The arrow is moved by the arrow keys on the keyboard.) POLYMATH 4.1 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-3
Once you have the equation entered properly, please press ⇑F7 to solve the problem. You will then be asked to provide the interval over which you wish to find solutions for the equation. This interval is only requested during the solution of a single nonlinear equation.
Please indicate the xmin to be 1 and press ( and press ( ).
). Then indicate xmax to be 4
The entire problem is then display above the Problem Options Menu:
For this single equation, the solution is presented graphically over the search range which you indicated. The solution is where the function f(x) is equal to zero. POLYMATH has the ability to determine multiple solutions to a single equation problem, and the first of two solutions is shown below:
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POLYMATH 4.1 PC
Press enter (
) for the second solution.
A shift-return (⇑
) will return you to the Problem Options Display.
SOLVING A SYSTEM OF NONLINEAR EQUATIONS Next, you will solve two nonlinear equations with two unknowns. To enter this new set of equation press ⇑ F8 for a new problem, then press followed by "y" to enter a new problem. The equations that will be solved are:
v CAf – CA1 v CA1 – CA2 2 and k CA2 = V V where k = 0.075; v = 30; CAf = 1.6 ; CA2 = 0.2CAf . Thus there are two unknowns: CA1 and V. 2 k CA1 =
To solve this system of equations, each nonlinear equation must be rewritten in the form f(x) = (an expression that is to have the value of zero at the solution). The appropriate forms for these equations are: v CAf – CA1 2 f CA1 = kCA1 – V v CA1 – CA2 2 f V = kCA2 – and V All equations can be entered into POLYMATH as shown below. Note that each of the problem unknowns (CA1 and V) should appear once and only once inside the brackets in the left of the equal sign. The unknown variable may not be in that particular equation. POLYMATH just needs to know the variable names that you are using in your problem. The explicit algebraic equations may be entered directly. Please enter the equations as given below. The order of the equation is not important as POLYMATH will order the equations during problem solution. Equations f(Ca1)=k*Ca1^2-v*(Caf-Ca1)/V f(V)=k*Ca2^2-v*(Ca1-Ca2)/V k=0.075 Caf=1.6 v=30 Ca2=0.2*Caf
Press ⇑F7 to solve this system of equations. POLYMATH 4.1 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-5
For two or more nonlinear equations, POLYMATH requires an initial estimate to be specified for each unknown.
While the solution method used is very robust, it often will not be able to find the solution if unreasonable initial estimated are entered. In this example, physical considerations dictate that CA1 must be smaller than CA0 and larger than CA2. So please enter initial estimate of Ca1 as 1.0. As for V, any positive value up to about V=3900 can be a reasonable estimate. Please use the initial value of 300 for V in this Quick Tour example. After entering the initial values, this example problem should be:
Please press ⇑-F7 to solve the problem. The solution process will start and its progress will be indicated on the screen by an arrow moving along a ruler scale. For most computers, the solution is so fast that the display of the iterations in the numerical solution to a converged solution will not be seen. When visible, the arrow indicates how far from zero the function values are at a particular stage of the solution on a logarithmic scale. Details are given in the Help Section by pressing F6. The results are given after any keypress as shown below:
Please note that the values of the various nonlinear equation functions (nearly zero) are given along the with values of all the problem variables. 6-6 NONLINEAR ALGEBRAIC EQUATIONS
POLYMATH 4.1 PC
This concludes the Quick Tour problem using the Simultaneous Nonlinear Algebraic Equation Solver. If you wish to stop working on POLYMATH, please follow the exiting instructions given below. EXITING OR RESTARTING POLYMATH A ⇑ - F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. SELECTION OF INITIAL ESTIMATES FOR THE UNKNOWNS The solution algorithm requires specification of initial estimates for all the unknowns. Generally speaking, closer initial estimates have a better chance of converging to the correct solution. If you wish to solve only a single nonlinear equation, the program will plot the equation so that the location of the roots (if any) can be seen. The program will then show the roots. If no roots exist in the chosen range, the plot will indicate what range should be explored to have the nonlinear function f( ) cross zero. When several equations are to be solved, the selection of the initial values is more complicated. First, the user should try to find the limiting values for the variables using physical considerations. (For example: The mole or mass fraction of a component can neither be negative nor greater that 1; the temperature of cooling water can be neither below freezing nor above boiling; etc.) Typical initial estimates are taken to be mid range. Users should be particularly careful no to select initial estimates where some of the functions may be undefined. (For example, f(xa)=1/(xa-xb)+... is undefined whenever xa=xb; f(xb)=log(1-xb) is undefined whenever xb>=1; etc.) The selection of such initial estimates will stop the POLYMATH solution, and an error message will be displayed. METHOD OF SOLUTION For a single nonlinear equation, the user must specify an interval in which the real root(s) can be found. The program will first attempt to locate points or regions where the function is undefined inside this interval*. If the equations are too complicated for determination of discontinuity points, a warning message is issued. ______________________ *For details of the method, see Shacham, O. and Shacham, M., Acm. Trans. Math. Softw., 16 (3), 258-268 (1990). POLYMATH 4.1 PC
NONLINEAR ALGEBRAIC EQUATIONS 6-7
The function is plotted and smaller intervals in which root(s) are located by a sign change of the function. The Improved Memory Method*, which employs a combination of polynomial interpolation and bisection, is used to converge to the exact solution inside those intervals. Iterations are stopped when the relative error is polymath then press the Return ( ) key. The Main Program Menu should then appear: The Program Selection Menu should then appear, and you should enter "3" on the keyboard to select the Linear Equation Solver.
POLYMATH 4.1 PC
LINEAR ALGEBRAIC EQUATIONS 7-1
Once that POLYMATH is loaded, please utilize the Help Menu by pressing F6 for information regarding the use of the Linear Equation Solver. This Help Section is given below:
This Help Section gives detailed information for entering a system of linear equations. Press any key to return from the Help Section to the program, and your display should be at the Main Menu for the Linear Equation Solver. (An alternate command to reach the Main Menu is the ⇑ F10 keypress.) To begin the first Quick Tour example, please press the Return key ( ) from the Main Menu. This will give the Task Menu as shown below:
7-2 LINEAR ALGEBRAIC EQUATIONS
POLYMATH 4.1 PC
SOLVING FIVE SIMULTANEOUS LINEAR EQUATIONS A typical problem for simultaneous linear equations is given below for the variables x1 through x5: x1 + 0.5 x2 + 0.333333 x3 + 0.25 x4 + 0.2 x5 = 0.0 0.5 x1 + 0.333333 x2 + 0.25 x3 + 0.2 x4 + 0.166667 x5 = 1.0 0.333333 x1 + 0.25 x2 + 0.2 x3 + 0.166667 x4 + 0.142857 x5 = 0.0 0.25 x1 + 0.2 x2 + 0.166667 x3 + 0.142857 x4 + 0.125 x5 = 0.0 0.2 x1 + 0.166667 x2 + 0.142857 x3 + 0.125 x4 + 0.111111 x5 = 0.0 The above problem in stored as a Sample Problem in POLYMATH. To recall the above problem, press F7 from the Task Menu of the Linear Equation Solver. Then select problem number "2" to obtain the Problem Options Menu shown below:
Solve this system of equations by pressing ⇑ F7 which should yield the results and the Display Options Menu on the next page. Remember that this keypress combination is accomplished by pressing and holding the Shift key and then pressing the F7 function key.
POLYMATH 4.1 PC
LINEAR ALGEBRAIC EQUATIONS 7-3
Lets explore making changes to this system of equations. This is accomplished by first pressing ⇑ to "make changes" to the problem. Use the arrow keys to take the highlighted box to the top of the "b" of constants for the equation. Please delete the 0 and enter 1.0 in this box which corresponds to changing the first linear equation to: x1 + 0.5 x2 + 0.333333 x3 + 0.25 x4 + 0.2 x5 = 1.0 This involves using the arrow key and pressing the return key ( highlighted box is in the desired location as shown.
) when the
Then enter the new value at the cursor:
Please solve the problem by pressing ⇑F7. The results are shown below:
This concludes the Quick Tour Problem for Simultaneous Linear Equations. When you are ready to leave this program and return to the Program Selection Menu, use the ⇑ F10 keypress which is discussed below. EXITING OR RESTARTING POLYMATH A ⇑ F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. 7-4 LINEAR ALGEBRAIC EQUATIONS
POLYMATH 4.1 PC
REGRESSION QUICK TOUR This chapter is intended to give a very brief overview of the operation of the POLYMATH Polynomial, Multiple Linear and Nonlinear Regression program. REGRESSION PROGRAM This program allows you to input numerical data into up to 30 columns, with up to 100 data points in each column. The data can be manipulated by defining expressions containing the names of previously defined columns. Relationships between different variables (columns of data) can be found using polynomial, multiple linear and nonlinear regression as well as cubic spline interpolation. Fitted curves can be interpolated, differentiated and integrated. Graphical output of the fitted curves and expressions is presented, and a statistical analysis of the parameters found during the regressions is given. STARTING POLYMATH To begin, please have POLYMATH loaded into your computer as detailed in Chapter 2. Here it is assumed that your computer is set to the hard disk subdirectory or floppy drive containing the POLYMATH package. At the prompt (assumed C:\POLYMAT4 here), you should enter "polymath" C:\POLYMAT4 > polymath ) key. The Program Selection Menu should then then press the Enter ( appear, and you should enter "4" on the keyboard to select the Polynomial, Multiple Linear and Nonlinear Regression program. This should bring up the Main Program Menu as given in the next page. In order to save time in entering data points during this quick tour, we will use sample problems which have been stored in POLYMATH. Press F7 to access the Sample Problems Menu from the Main Program Menu. QUICK TOUR PROBLEM 1 Let us consider a fairly typical application of the Regression Program in which some data are available. When these data are fitted to a polynomial
POLYMATH 4.1 PC
REGRESSION 8 -1
within POLYMATH, the polynomial expression has the form: P(x) = a0 + a1x + a2x2 +... + anxn where y is the dependent variable, x is the independent variable, and the parameters are a0 ...an. Variable "n" here represents the degree of the polynomial. In POLYMATH, the maximum degree which is shown is 5. The above polynomial expression gives a relationship between the dependent variable and the independent variable which is obtained by determining the parameters according to a least squares objective function. Data points are usually available which give x and y values from which the parameters a0... an can be determined.
RECALLING SAMPLE PROBLEM 3 After pressing F7 at the Main Program Menu, the Sample Problems Menu should appear on your screen as shown on the next page.. The sample problem to be discussed should be retrieved by pressing "3" on the keyboard. This will result in the Problem Options Display which includes 10 data points of x and y as shown on the next page.
8 -2 REGRESSION
POLYMATH 4.1 PC
POLYMATH 4.1 PC
REGRESSION 8 -3
FITTING A POLYNOMIAL The Problem Options Menu includes problem editing, library, printing, help and solution options. To fit a polynomial to the data of Y versus X you should select the "⇑ F7 to fit a curve or do regression" option. After pressing ⇑ F7 the following "Solution Options" menu appears:
After pressing "p" (lower case), you should be asked for the name of the independent variable's column, as shown below:
You should enter a capital "X" (upper case) as name of the independent variable and press . The same question regarding the dependent variable will be presented. Please enter a capital "Y" (upper case) at the arrow. The following display should appear:
8-4 REGRESSION
POLYMATH 4.1 PC
On this display the coefficients of the polynomial P(x), up to the fifth order are shown together with the value of the variance. One of the polynomials is highlighted by having a box around it. This is the lowest order polynomial, such that higher order polynomial does not give significantly better fit. The same polynomial is also plotted versus the experimental data. Other polynomials can be highlighted and plotted by pressing a number between 1 and 5. There are many additional calculations and other operations that can be carried out using the selected polynomial. Please make sure the highlighted box is on the 4th degree polynomial. Let us find the value of X for Y = 10. To do that you should press "y" and enter after the prompt regarding the value of Y: "10". The following display results:
The resultant X values are shown both graphically and numerically. For Y = 10 there are two X values, X = 1.36962 and X = 5.83496. FITTING A CUBIC SPLINE We will now fit a cubic spline to these data of Sample Problem 3. Please press F8 two times to return to the Problems Options Menu. Then press ⇑ F7 to "fit a curve or do regression". The Solutions Options Menu should appear. POLYMATH 4.1 PC
REGRESSION 8-5
Enter "s" (lower case) for a cubic spline followed by "X" and then "Y". The following display should present the results:
EVALUATION OF AN INTEGRAL WITH THE CUBIC SPLINE Please take options "i" and request the initial value for the integration to be "1" at the arrow:
Press integration.
and then enter "6" at the arrow for the find value of the
Press to have the resulting integration shown on the next display with both graphical and numerical results:
8-6 REGRESSION
POLYMATH 4.1 PC
Please press any key to end this Sample Problem 3. MULTIPLE LINEAR REGRESSION It will often be useful to fit a linear function of the form: y(x) = a0 + a1x1 + a2 x2 +... + anxn where x1, x2, ..., xn are n independent variables and y is the dependent variable, to a set of N tabulated values of x1,i, x2,i, ... and y (xi). We will examine this option using Sample Problem 4. RECALLING SAMPLE PROBLEM 4 First exit to the main title page by pressing ⇑ F10. Press F7 to access the Sample Problems Menu, and select problem number 4 by pressing "4". (The problem display is shown on the next page).
POLYMATH 4.1 PC
REGRESSION 8-7
SOLVING SAMPLE PROBLEM 4 After you press ⇑ F7 "to fit a curve or do regression", the following Solution Options Menu should appear:
This time press "l" (lower case letter "l") to do "linear regression". You will be prompted for the first independent variable (column) name.
Please type in "X1" at the arrow and press . You will be prompted for the 2nd independent variable. Enter "X2" as the second independent variable name and press once again. A prompt for the 3rd independent variable will appear. You should press here without typing in anything else, since there are no additional independent variables. 8 -8 REGRESSION
POLYMATH 4.1 PC
At the prompt for the dependent variable (column) name shown below you should type "Y" and press .
Once the calculations are completed, the linear regression (or correlation) is presented in numerical and graphical form.
Please note that the correlation the equation for variable "Y" has the form of the linear expression: Y = a0 + a1X1 + a2X2 where a0 = 9.43974, a1 = -0.1384 and a2 = 3.67961. This graphical display of Sample Problem 4 presents the regression data versus the calculated values from the linear regression. The numerical value of the variance and the number of the positive and negative residuals give an indication regarding the validity of the assumption that Y can be represented as linear function of X1 and X2. The results in this case indicate a good fit between the observed data and the correlation function.
POLYMATH 4.1 PC
REGRESSION 8-9
The Display Options Menu allows the user to use an "s" keypress "to save results in a column". This refers to saving the calculated value of Y from the linear regression to the Problems Options Display under a column name provided by the user. The "r" keypress from the Display Options Menu give a statistical residual plot as shown below:
The "F9" keypress from the Display Options Menu give a statistical summary:
The confidence intervals given in the statistical summary are very useful in interpreting the validity of the linear regression of data. This concludes Sample Problem 4 which illustrated multiple linear regression.
8-10 REGRESSION
POLYMATH 4.1 PC
TRANSFORMATION OF VARIABLES A nonlinear correlation equation can be often brought into a linear form by a transformation of the data. For example, the nonlinear equation: Y = a0 X1a 1 X2a 2 can be linearized by taking logarithm of both sides of the equation: ln Y = ln a0 + a1 ln X1 + a2 ln X2. To demonstrate this option please recall Sample Problem 5. To do this, please press ⇑ F10 to get to the Main Program Menu, F7 to access the Sample Problems Menu and select Sample Problem number 5. This should result in the Problem Option Display below:
In this display X1, X2 and Y represent the original data, the variables (columns) lnX1, lnX2 and lnY represent the transformed data. You can see the definition of ln X1 , for example, by moving the cursor (the highlighted box), which located in row number 1 of the first column, into the box containing "lnX1" (using the arrow keys) and press .
POLYMATH 4.1 PC
REGRESSION 8-11
The following window is brought up:
Note that the expression in the right hand side of the column definition equation must be a valid algebraic expression, and any function arguments used in the expression should be enclosed within parentheses. Since we do not want to change this expression, please press to close the window. Now press ⇑ F7 to do regression, then "l" to do linear regression. Type in "lnX1" as the name of the first independent variable, "lnX2" as the name of the second independent variable and "lnY" as the name of the dependent variable. The results should be displayed as shown below:
All of the statistical analyses are available for the transformed variable. Please note that the results indicate that the equation for variable "Y" can be written as: Y = a0 X1a 1 X2a 2 where a0 = exp (-0.666796) = 0.5133, a1 = 0.986683 and a2 = -1.95438. This concludes the transformation of variable and the multiple linear regression for Sample Problem 5. 8-12 REGRESSION
POLYMATH 4.1 PC
NONLINEAR REGRESSION It is often desirable to fit a general nonlinear function model to the independent variables as indicated below: y(x) = f(x1, x2, ..., xn; a0, a1, ..., am) In the above expression, x1, x2, ..., xn are n independent variables, y is the dependent variable, and a0, a1, ..., am are the model parameters. The data are represented by a set of N tabulated values of x1,i, x2,i, ... and y(xi). The regression adjusts the values of the model parameters to minimize the sum of squares of the deviations between the calculated y(x) and the data y(xi). The nonlinear regression capability of POLYMATH allows a general nonlinear function to be treated directly without any transformation. Lets return to Sample Problem 5 and this time treat the model for Y directly where Y = a0 X1a 1 X2a 2 . Please recall Sample Problem 5. From the Problem Options Display press ⇑ F7 and then enter "R" (upper case R) to "Do nonlinear regression." The user is then prompted to:
The user can then enter the model equation using any of the variables from the columns of the Problem Options Display and any unknown parameters (maximum of five) which are needed. For this example, please enter
Thus in this problem, the unknown parameters are k, alpha, and beta. The next query for the user is to supply initial estimates for each of the unknown parameter in turn:
It is good practice to provide good initial parameter estimates from either reasonable physical/chemical model values or from a linearized treatment of the nonlinear model. In this example however, please set all initial guesses for the parameters as unity, "1.0". Then POLYMATH will provide a summary of the problem on the Regression Option Display as shown on the next page.
POLYMATH 4.1 PC
REGRESSION 8-13
The Regression Options Menu gives several useful options for model changes and alterations to initial parameter guesses; however, please press ⇑F7 to solve this problem. The program search is shown to the user and the converged solution is indicated below:
There are a number of options from the Display Options Menu (not shown here). Perhaps the most useful is the "statistical analysis" which is given on the next page. 8-14 REGRESSION
POLYMATH 4.1 PC
This concludes the Quick Tour section dealing with nonlinear regression and the Chapter on the Polynomial, Multiple Linear and Nonlinear Regression Program. Remember, when you wish to stop POLYMATH, please follow the exiting instructions given below. EXITING OR RESTARTING POLYMATH A ⇑ F10 keypress will always stop the operation of POLYMATH and return you to the Program Selection Menu. THIS ACTION WILL DELETE THE EXISTING PROBLEM. The program can be exited or restarted from the Program Selection Menu. SOLUTION METHODS When fitting a polynomial of the form P(x) = a0 + a1x + a2x2 +...+anxn to N points of observed data, the minimum sum of square error correlation of the coefficients a0, a1, a2...an can be found by solving the system of linear equation (often called normal equations): POLYMATH 4.1 PC
REGRESSION 8-15
XT XA = XT Y where
Y=
y y1 .2 . . y
A=
N
ao a1 . . . an
x0 X=
1 x0 2 x0 N
x1
1 x1 2 x1 N
. . . . . . . . . . . .
xn
1 xn 2 xn N
and where y1, y2...yN are N observed values of dependent variable, and x1, x2...xN are N observed values of the independent variable. Multiple linear regression can also be expressed in the same form except that the matrix X is redefined as follows:
X=
1
x1,1
x2,1 . . . x n,1
1
x1,2
x2,2 . . . x n,2
1
x1,N x2,N . . . x n,N
where xi,j is the j-th observed value of the i-th independent variable. When polynomial or multiple linear regression are carried out without the free parameter (a0), the first element in vector A and the first column in matrix X must be removed. In POLYMATH the normal equations are solved using the GaussJordan elimination. It is indicated in the literature that direct solution of normal equations is rather susceptible to round off errors. Practical experience has should this method to sufficiently accurate for most practical problems. The nonlinear regression problems in POLYMATH are solved using the Levenberg-Marquardt method. A detailed description of this method can be found, for example, in the book by Press et al.*
*Press, W. H., P. B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, 2nd Ed., Cambridge University Press, 1992, pp. 678683. 8-16 REGRESSION
POLYMATH 4.1 PC
APPENDIX INSTALLATION FROM THE CD-ROM ACCOMPANYING "ELEMENTS OF CHEMICAL REACTION ENGINEERING." EXECUTION INSTRUCTIONS FOR VARIOUS OPERATING SYSTEMS This Appendix provides complete instructions for the installation and execution of POLYMATH 4.1 for Windows 3.X, Windows 95, Windows 98, and Windows NT operating systems. Detailed information is also provided for advanced options. Latest updates will be on the README.TXT file. WINDOWS 3.X INSTALLATION 1. Put the CD-ROM in the appropriate drive. 2. Double click on the Main icon in the Program Manager window. 3. Double click on the MS-DOS Prompt icon. 4. Change the directory to the CD-ROM drive on which POLYMATH is stored. For example, if the drive to be used is D, then insert “D:” at the cursor and press Enter. Then enter "cd\html\toolbox\polymath\files\poly41". 5. At the D:\html\toolbox\polymath\files\poly41> prompt, enter “dir”. 6. There should be three files: Readme.txt, Install.exe, and Pmunzip.exe 7. At the D:\html\toolbox\polymath\files\poly41> prompt, enter “install” 8. Follow the instructions on the screen. 9. At the D:\html\toolbox\polymath\files\poly41> prompt enter “exit” to return to Windows CONTINUE STEPS BELOW TO CREATE A PIF FILE FOR POLYMATH ONLY IF POLYMATH DOES NOT EXECUTE PROPERLY 10. From the Program Manager Window click on Main. 11. From the Main Window double click on PIF Editor. 12. Please enter the following in the PIF Editor: Program Filename: POLYMATH.BAT Window Title: POLYMATH 4.0 Startup Directory: C:\POLYMAT4 Video Memory: Low Graphics Memory Requirements: -1 -1 EMS Memory: 0 1024 XMS Memory: 1024 1024 Display Usage: Full Screen 13. From File use Save As “POLYMAT4.PIF”. POLYMATH 4.1 PC
APPENDIX 9-1
WINDOWS 3.X EXECUTION* This 4.1 version of POLYMATH is a DOS program, but it creates Windows Meta files (WMFs) that can be printed or entered into documents. A special MetaFile Print program for Windows 3.X (MFP16) prints only POLYMATH WMF files that are generated upon printing requests from within POLYMATH. These generated WMFs are normally placed in the SPOOL subdirectory under the POLYMAT4 directory. Thus the user must first run the MFP16 Windows program before POLYMATH to insure that all requests to Print will be printed on the Windows printer. The MFP16 program produces a small blue printer icon on the bottom right of the desktop. A right mouse click on this icon gives the options which can be executed by a left mouse click. 1. Click on the File options in the Program Manager Window, select Run. 2. Enter the Command Line for the Metafile Print program such as “c:\polymat4\mfp16.exe” 3. The Command Line for your POLYMATH location should then be entered such as “c:\polymat4\polymath.bat”. 4. Click on “OK” 5. If POLYMATH does not run properly, then create a PIF file starting with Step 10 as given in the preceding Windows 3.X Installation section. 6. Always end POLYMATH by exiting from the Program Selection Menu. * A Program Group and a Program Item can be created under Windows to allow POLYMATH.bat and MFP16.exe to be executed conveniently from the desktop. WINDOWS 3.X SHUTDOWN 1. POLYMATH can be terminated at any time by pressing Shift-F10 to return to the Program Selection Menu and then by pressing F8 to exit. 2. The MFP16 program can be terminated by a right click with the mouse on the small blue printer icon followed by a left mouse click on "terminate." USING PRINT METAFILES IN DOCUMENTS FOR WINDOWS 3.X The WMFs files will accumulate in the SPOOL subdirectory of POLYMAT4 when the MFP16 program is not running, These WMF files, designated by TEMP00X.WMF, can be copied and inserted as files within word processing and desktop publishing software. MS Word, for example, 9-2 APPENDIX
POLYMATH 4.1 PC
will conveniently give a small preview of each selected file so that the content will be known prior to insertion. Execution of MFP16.exe will allow you to print all or delete all remaining files in SPOOL. Thus POLYMATH output can be selectively inserted into documents or printed or both. Windows 95, 98, and NT Installation 1. Put the CD-ROM in the appropriate drive, for example Drive D. 2. Minimize all Windows until Desktop appears. 3. Double click on the My Computer icon. 4. Double click on the CD-ROM icon. Then continue double clicking in turn on html, toolbox, polymath, files, and poly41. 5. Double click on the Install icon. 6. Follow the directions on the screen. 7. When installation is complete, press Enter. 8. Close the DOS window. Windows 95, 98, and NT Execution* This 4.1 version of POLYMATH is a DOS program, but it creates Windows Meta files (WMFs) that can be printed or entered into documents. A special MetaFile Print program for Windows 95, 98, and NT (MFP32) prints only POLYMATH WMF files that are generated upon printing requests from within POLYMATH. These generated WMFs are normally placed in the SPOOL subdirectory under the POLYMAT4 directory. When the user executes the POLYMATH.bat file, the MFP32.exeWindows program is also launched. POLYMATH runs in its own Window by the MFP32.exe program only produces a small blue printer icon on the bottom right of the desktop. A right mouse click on this icon gives the options which can be executed by a left mouse click. Program execution is accomplished by the following steps: 1. Click on the Start button. 2. Click on the Run icon. 3. Start POLYMATH by entering the storage location for POLYMATH and specify the program as "polymath.bat" such as "c:\polymat4\polymath.bat".
POLYMATH 4.1 PC
APPENDIX 9-3
4. The POLYMATH program will initiate both the POLYMATH software and the MFP32.exe program as discussed previously. If the MFP32.exe was not active at POLYMATH startup, then you may need to click on the POLYMATH window to continue with POLYMATH. * Icons for both the Printer Utility and POLYMATH can be placed on the desktop by using Windows Explorer to find both the Mfp32.exe and Polymath.bat files in the POLYMATH directory. A right mouse click on each of these files followed by a left mouse click on "Create Shortcut" can create a "Shortcut to ..." which can be dragged to the desktop. WINDOWS 95, 98, AND NT SHUTDOWN 1. POLYMATH can be terminated at any time by pressing Shift-F10 to return to the Program Selection Menu and then by pressing F8 to exit. 2. The MFP32.exe program can be terminated by a right click with the mouse on the small blue printer icon followed by a left mouse click on "terminate." USING PRINT METAFILES IN DOCUMENTS FOR WINDOWS 95, 98, AND NT The WMFs files will accumulate in the SPOOL subdirectory of POLYMAT4 when the MFP32.exe program is not running, These WMF files, designated by TEMP00X.WMF, can be copied to word processing and desktop publishing software and inserted as files. MS Word will give a small preview of each selected file so that the content will be known. Execution of MFP32.exe will allow you to print all or delete all remaining files in SPOOL. Thus POLYMATH output can be selectively inserted into documents or printed or both. INSTALLATION QUESTIONS (DETAILS) 1. Enter drive and directory for POLYMATH [C:\POLYMAT4]: ==> The default response is indicated by the contents of the brackets [...] which is given by pressing Enter key. The full path (drive and directory) where you wish the POLYMATH program files to be stored must be provided here. If the directory does not exist, then the installation procedure will automatically create it.
9-4 APPENDIX
POLYMATH 4.1 PC
NOTE: Network clients will need read and execute permission for this directory and its subdirectories. This procedure does not provide the needed permissions. 2. Is this a network installation? [N] If you are installing POLYMATH on a stand-alone computer, take the default or enter "N" for no and GO TO 5. on this list. If you are installing on any kind of network server, answer "Y" and continue with the installation. 3. What will network clients call [POLYdir]? ===> _ This question will only appear if you answered "Y" to question 2 to indicate a Network installation. Here "POLYdir" is what was provided in question 1. On some networks, the clients "see" server directories under a different name, or as a different disk, than the way the server sees them. This question enables POLYMATH to print by indicating where the printerdriver files are located. They are always placed in subdirectory BGI of the POLYMATH directory by the installation procedure. During run time, they must be accessed by the client machines, thus POLYMATH must know what the client's name is for the directory. 4. Enter drive and directory for temporary print files [C:\TMP]: ===> _ This question will only appear if you answered "Y" to question 2 to indicate a Network installation. Depending on the amount of extended memory available, and the type of printer is use, POLYMATH may need disk workspace in order to print. Since client machines are not normally permitted to write on the server disk, you are requested to enter a directory where files may be written. The temporary print files are automatically deleted when a print is completed or cancelled. 5.
POLYMATH INSTALLATION Which version of MS Windows are you using [2]: 1. Windows 3.X 2. Windows 95 3. Windows 98 4. Windows NT ===> _
POLYMATH 4.1 PC
APPENDIX 9-5
6. The POLYMATH installation program now copies all of the needed files according to your previous instructions. This may take some time as the needed files are compressed on the installation diskette. Additional Network Installation Needs: 1. POLYMATH requires the creation of the directory C:\TMP\SPOOL. 2. There must be an MFP.INI file on each computer (including the server) connected to the network that is going to run POLYMATH. In this file, a spool directory on the local disk should be specified. This directory has to exist. Any Windows Meta File (*.wmf) that is placed into this directory by the Print Utility (either MFP16.exe or MFP32.exe) will get printed and deleted, so it should not be in a directory in general use. A sample MFP.INI file follows: [location] SpoolPath=C:\TMP\SPOOL TROUBLESHOOTING Please look for the most recent troubleshooting hints at www.polymath-software.com (a) If you are attempting to print on Windows 3.X and you get the message "abnormal program termination", then try running POLYMATH again. Additional messages such as "out of memory" or "missing batch file" indicates that you must remove some RAM resident programs. If all else fails, then use the DOS printing version 4.02 of POLYMATH. (b) When POLYMATH says "Could not begin printing: I/O port error", the SPOOL subdirectory does not exist. Please make sure it exists. See the following part (c). (c) If POLYMATH indicates "print successful" and nothing comes out of the printer then... - Use Print Manager to check to see that there is no problem with the printer. If there is a job in the queue, this means that POLYMATH and MFP did their part, and the problem is in Windows, Print Manager, or the printer itself. 9-6 APPENDIX
POLYMATH 4.1 PC
- Check that MFP16.exe is running for Windows 3.X or that MFP32.exe is running for Windows 95, 98, and NT. There should be a little printer icon visible at the bottom right of the desktop. If not, then run the proper program (File->Run ... C:\POLYMAT4\MFP16.exe for Windows 3.X ; File->, Run ... C:\POLYMAT4\MFP32.exe for Windows 95, 98, and NT) - MFP.INI file doesn't exist or doesn't match the POLYMATH.BAT file. Please check the POLYMATH.BAT file for a line similar to PM_PRINTER=_WMF,0,FILE:C:\POLYMAT4\SPOOL\TEMP+++.WMF where the directory specified in the preceding line, "C:\POLYMAT4\SPOOL" should be the same as it appears in the MFP.INI file. If they don't match, the you must change one of the other, shut down MFP, exit POLYMATH, and start both again.
"OUT OF ENVIRONMENT SPACE" MESSAGE If you receive this message or if you are having difficulty in printing from POLYMATH, then follow ONE of the following instruction sets for your particular operating system. Window 95, Windows 98, or Windows NT 4.0 1. Open a DOS prompt window (if yours opens full-screen, hit Alt+Enter to get a window). 2. Click on the "Properties" button at the top. 3. At the end of the Cmd line, add the text "/e:2048" (if you already have this, then change the existing number to a higher number in increments of 1024). Windows 3.x and Windows 95 1. Open a DOS prompt window. 2. At the C:\ prompt type "CD/WINDOWS, and press Enter. 3. Type "EDIT SYSTEM.INI" and press Enter. 4. Locate a line that reads "[NonWindows App]". 5. Make sure that this section contains the following entry: "CommandEnvSize=2048". 6. Save the modified file. 7. Reboot your computer. 8. Adjust the size upwards in increments of 1024 as necessary. POLYMATH 4.1 PC
APPENDIX 9-7
DOS or Windows 3.1 1. Open a DOS prompt window. 2. Type "SET" at the prompt. 3. Look for the line that displays the value of the COMSPEC environment variable. If COMSPEC is set to C:\COMMAND.COM then add the following line to the CONFIG.SYS file: SHELL=C:\COMMAND.COM C:\ /E:2048 /P If COMSPEC is set to C:\DOS\COMMAND.COM then add this line to the CONFIG.SYS file: SHELL=C:\DOS\COMMAND.COM C:\DOS /E:2048 /P 4. Reboot your computer. 5. Adjust the size of E: in Step 3 upwards in increments of 1024 as necessary.
POLYMATH 4.1 PC
APPENDIX 9-8
using.htm
Using and Modifying the M-Files
Using the M-Files 1. Run MATLAB. 2. At the Command Window prompt, type p=path and press ENTER. MATLAB's path (the list of places MATLAB looks for m-files) will be listed. 3. To add the m-files on the CD-ROM to MATLAB's path, type path(p,' [your cdrom drive letter]:\Html\Toolbox\Matlab\m-files') and press ENTER. For example, if your CD drive is drive D, then you would type: path(p, 'D:\html\toolbox\matlab\m-files') NOTE: This command will only temporarily change the path. To permanently change the path, type pathtool at the command prompt. A new window will open. Add the [your cd-rom drive letter]:\Html\Toolbox\Matlab\m-files directory to the list of path directories and click "Save Settings." (If you are using a copy of MATLAB owned by your college or university, you may not have sufficient access to modify the path permanently.) 4. Run Microsoft Word. 5. Open the Html\Toolbox\Matlab\Word directory on the CD-ROM. You will see a number of folders labeled CH#. 6. Open the folder corresponding to the chapter of the example problem you wish to examine. 7. Open the file for that example problem.
Additional Information These files were created using the Notebook interface with Word. The Word files contain examples that have been solved using the m-files. The text comes in three different colors: file:///H:/html/toolbox/matlab/using.htm[05/12/2011 16:55:11]
using.htm
Lines in green represent commands you should enter at the command prompt. Lines in blue contain output from MATLAB, Lines in black contain descriptive text. We highly recommend using the Word file for each example as a guideline (print a copy if necessary). Retype the text from the green lines (one by one) at the MATLAB command prompt. These commands define the boundary conditions and the initial conditions necessary to solve the example problems. They also show you how to make graphs, so you can display the results of the calculations. To change the boundary conditions and/or the initial conditions of a problem, simply change the values in the green input commands. To change the differential equations and/or their supporting equations, you will have to edit the m-files themselves. (You will have to copy the m-files to your computer's hard drive to edit them, since you won't be able to save them on your CD-ROM.)
Modifying the M-Files The m-files are in the Html\Toolbox\Matlab\m-files directory on the CD-ROM. 1. In MATLAB, open the m-file for the example you wish to examine. 2. Select all of the text in the m-file and copy it. 3. Under the File menu in MATLAB, choose New and then M-File. An M-File Editor window will appear. 4. Paste the text into this window. 5. Make changes to the equations you want to modify. 6. Save the file under a new name (e.g., temp1.m) to your computer's hard drive or to a floppy disk. Remember that this file must also be saved into a directory contained in MATLAB's path. (See steps 2 and 3 from Using the M-Files, above.) Also, the "function" command in the m-file should be changed to match the new filename. 7. Enter the command lines as before, using the Word file as a guide. For all commands that use the m-file name, be sure to replace the old m-file name with the name of the file you just created (e.g., temp1.m).
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Lectures 1 and 2
Lectures 1 and 2
Chemical Identity (Chapter 1) * A chemical species is said to have reacted when it has lost its chemical identity. The identity of a chemical species is determined by the kind, number, and configuration of that species' atoms. Three ways a chemical species can lose its chemical identity: 1. decomposition 2. combination 3. isomerization
Reaction Rate (Chapter 1) The reaction rate is the rate at which a species looses its chemical identity per unit volume. The rate of a reaction can be expressed as the rate of disappearance of a reactant or as the rate of appearance of a product. Consider species A:
r A = the rate of formation of species A per unit volume -r A = the rate of a disappearance of species A per unit volume r B = the rate of formation of species B per unit volume For a catalytic reaction, we refer to -r A ' , which is the rate of disappearance of species A on a per mass of catalyst basis. NOTE: dCA /dt is not the rate of reaction Consider species j: r j is the rate of formation of species j per unit volume r j is a function of concentration, temperature, pressure, and the type of catalyst (if any) r j is independent of the type of reaction system (batch, plug flow, etc.) r j is an algebraic equation, not a differential equation We use an algebraic equation to relate the rate of reaction, -r A , to the concentration of reacting species and to the file:///H:/html/course/lectures/one/index.htm[05/12/2011 16:55:12]
Lectures 1 and 2
temperature at which the reaction occurs [e.g. -r A = k(T)C A 2 ]. Would you like to see an How about doing a
of some more rates of reaction? on what you've learned?
General Mole Balance Equation (Chapter 1)
Mole Balance on Different Reactor Types The GMBE applied to the four major reactor types (and the general reaction,
Reactor
Differential
Algebraic
Batch
CSTR
PFR
PBR
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Integral
):
Lectures 1 and 2
Conversion (Chapter 2) The conversion of species A in a reaction is equal to the number of moles of A reacted per mole of A fed. Batch
Flow
Design Equations (Chapter 2) The following design equations are for single reactions only. Design equations for multiple reactions will be discussed later.
Reactor Mole Balances in Terms of Conversion Reactor
Differential
Algebraic
Integral
Batch
CSTR
PFR
PBR
Reactor Sizing (Chapter 2) Given -r A as a function of conversion, one can size any type of reactor. The volume of a CSTR and the volume of a PFR can be represented as the shaded areas in the Levenspiel Plots shown below:
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Lectures 1 and 2
Numerical Evaluation of Integrals (Appendix A.4) The integral to calculate the PFR volume can be evaluated using a method such as Simpson's One-Third Rule (pg 925):
NOTE: The intervals ( ) shown in the sketch are not drawn to scale. They should be equal.
Simpson's One-Third Rule is one of the more common numerical methods. It uses three data points. Other numerical methods (see Appendix A, pp 924-926) for evaluating integrals are: 1. Trapezoidal Rule (uses two data points) 2. Simpson's Three-Eighth's Rule (uses four data points) 3. Five-Point Quadrature Formula
Reactors in Series (Chapter 2) Given -r A as a function of conversion, one can also design any sequence of reactors:
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Lectures 1 and 2
Relative Rates of Reaction aA + bB
cC + dD
*
All chapter references are for the 3rd Edition of the text Elements of Chemical Reaction Engineering. Back to the top of Lectures 1 and 2.
Jump to Lecture(s): 1 & 2 | 3 & 4 | 5 & 6 | 7 & 8 | 9 & 10 11 & 12 | 13 & 14 | 15 & 16 | 17 & 18 | 19 & 20 21 & 22 | 23 & 24 | 25 & 26 | 27 & 28 | 29 & 30 31 & 32 | 33 & 34 | 35A | 35B | 36 & 37 | 38 & 39
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Lectures 1 and 2 © 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Thoughts on Problem Solving: Algorithm
1. Write out the problem statement. Include information on what you are to solve, and consider why you need to solve the problem.
2. Make sure you are solving the real problem as opposed to the perceived problem. Use techniques such as "Finding out Where the Problem Came From," "The Duncker Diagram," "The Explore Phase," etc. to check to see that you define and solve the real problem. Recast the problem statement if necessary.
3. Draw and label a sketch. Define and name all variables and/or symbols. Show numerical values of variables if known.
4. Identify and name A. Relevent principles, theories and equations B. Systems and subsytems C. Dependent and independent variables D. Knowns and unknowns E. Inputs and outputs F. Necessary (missing) information
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Thoughts on Problem Solving: Algorithm
5. List assumptions and approximations involved in solving the problem. Question the assumptions and then state which ones are the most reasonable for your purposes.
6. Check to see if the problem is either under-specified or over-specified. If it is under-specified, figure out how to find the missing information. If over-specified, identify the extra information that isn't needed.
7. Relate problem to a similar problem or experience (compare to an example problem in lecture or in the book).
8. Use an algorithm (e.g. reaction engineering) A. Mole Balance B. Rate Laws i. Kinetic ii. Transport C. Stoichiometry i. Gas or liquid ii. Pressure drop D. Combine
E. Energy Balance
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Thoughts on Problem Solving: Algorithm
F. Evaluate
9. Develop/derive/integrate and/or manipulate an equation or equations from which the desired variable can be determined.
10. Substitute numerical values and calculate the desired variable. Check your units at each step in the solution to find possible errors.
11. Examine and evaluate the answer to see it makes sense. Is it reasonable, considering the problem statement? Does it consider safety and ethical issues? See how this algorithm can be applied to Examples in Chemical Reaction Engineering.
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Thoughts on Problem Solving: Getting Unstuck
1. Persevere! Keep trying different strategies and stay open to creative ideas. Try not to get frustrated.
2. Be more active in the solution!! A. Ask yourself questions about the problem. Is this problem a routine one? What data are missing? What equations can I use? Explore the problem. B. Draw sketches of what you think that the solution should look like. (e.g. temperature-time curve). C. Write equations. D. Keep track of your progress.
3. Re-focus on the fundamentals. Review the textbook and lecture material. Look for similar examples. Study the examples given. Change what is given in the example and what is asked, then try to see how it might relate to the problem you are addressing.
4. Break the problem into parts. Analyze the parts of the problem. Concentrate on the parts of the problem you understand and that can be solved.
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Thoughts on Problem Solving: Getting Unstuck
5. Verbalize the problem to yourself and others. Describe...
A. what the problem is B. what you have done, and C. where you are stuck
6. Paraphrase. Re-describe the problem. Think of simpler ways to describe the problem. Ask other classmates to describe the problem to you in their own words.
7. Use a heuristic or algorithm. The algorithm for closedended problems on this site is a good start, and others may be available to you.
8. Look at extreme cases that could give insight and understanding. For instance: What happens if x = 0? x = infinity?
9. Simplify the problem and solve a limiting case. Break up the problem into simpler pieces and solve each piece by itself. Find a related but simpler example and work from there.
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Thoughts on Problem Solving: Getting Unstuck
10. Try substituting numbers to see if a term can be neglected.
11. Try solving for ratios to drop out parameters that are not given. You may find that you don't need to find some parameters because they cancel out!
12. Look for hidden assumptions or for what information you have forgotten to use. After reading each phrase or sentence of the problem statement, ask yourself if any assumptions can be inferred from that phrase.
13. Alternate working forward towards a solution and backwards from a solution you assumed. Working backwards may at least give you clues as to how you should approach the problem while working forward.
14. Take a break. Incubate. Let your subconscious work on the problem while you do something else, file:///H:/html/probsolv/closed/unstuck/unstuck.htm[05/12/2011 16:55:15]
Thoughts on Problem Solving: Getting Unstuck
like exercising, talking to friends, or just relaxing! Sometimes all you need is a break to achieve that final breakthrough!
15. Brainstorm. Think of different approaches to the problem, no matter how strange. Guess the solution to the problem and then check the answer.
16. Check again to make sure you are solving the right problem. Double-check all of your values, assumptions, and approaches. Make sure you haven't missed anything and that you are looking for the correct solution.
17. Try using a different strategy. There is usually more than one way to solve a problem, and you may find a method that you haven't considered is much easier than the one you're working on currently.
18. Ask for help! There are many resources you may go to for additional instruction or ideas. Instructors can usually steer you in the right direction and clarify your understanding of the problem. If allowed, your classmates may be the biggest source of help, since they usually utilize many different approaches and can relate
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Thoughts on Problem Solving: Getting Unstuck
to your approach.
Back to Closed-Ended Problems
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CRE -- Problem 1-A
Learning Resource/Additional Homework Problem CDP1AA
A 200-dm3 constant-volume batch reactor is pressurized to 20 atm with a mixture of 75% A and 25% inert. The gas-phase reaction is carried out isothermally at 227 C.
V = 200-dm 3 P = 20 atm T = 227 C
a. Assuming that the ideal gas law is valid, how many moles of A are in the reactor initially? What is the initial concentration of A? b. If the reaction is first order:
Calculate the time necessary to consume 99% of A. c. If the reaction is second order:
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CRE -- Problem 1-A
Calculate the time to consume 80% of A. Also calculate the pressure in the reactor at this time if the temperature is 127 C. Solution
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CRE -- Reactor Photos
REACTOR PHOTOS One of the biggest complaints, that chemical engineering students have, is that they never get to see the equipment that they're spending so much time designing. Heat exchangers, pumps, containment vessels, etc. all remain abstract "little black boxes," and reactors are the worst culprits, because they're used in almost every homework assignment (at least in reaction engineering). So, we've taken the liberty of including pictures of some common (and not-so-common) industrial reactors on our pages. Industrial Reactor Photos
Reactors Module An interactive module on chemcial engineering reactors was created by Sam Catalano, a student at the University of Michigan working with Professor Susan Montgomery in the Department of Chemical Engineering. Check out some screenshots from the Reactors Module. The module is available in both Windows 95 and Macintosh formats. Contact Professor Susan Montgomery, if you would like more information about this module.
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Problem 1-A Solution
Learning Resource/Additional Homework Problem CDP1-A Solution a. How many moles of A are in the reactor initially? What is the initial concentration of A?
If we assume ideal gas behavior, then calculating the moles of A initially present in the reactor is quite simple. We insert our variables into the ideal gas equation:
Knowing the mole fraction of A (yAo ) is 75%, we multiply the total number of moles (NTo) by the yA:
The initial concentration of A (CAo ) is just the moles of A divided by the volume:
b. Time (t) for a 1st order reaction to consume 99% of A.
With both 1st and 2nd order reactions, we will begin with the mole balance:
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Problem 1-A Solution
There is no flow in or out of our system, and we will assume that there is no spatial variation in the reaction rate. We are left with:
Knowing the moles per volume (NA/V) is concentration (CA), we then define the reaction rate as a function of concentration:
First Order Reaction This is the point where the solutions for the different reaction orders diverge. Our first order rate law is:
We insert this relation into our mole balance:
and integrate:
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Problem 1-A Solution
Knowing C A=0.01 C Ao and our rate constant (k=0.1 min -1), we can solve for the time of the reaction:
c. Time for 2nd order reaction to consume 80% of A and final pressure (P) at T = 127 C.
Second Order Reaction Our second order rate law is:
We insert this relation into our mole balance:
and integrate:
We can solve for the time in terms of our rate constant (k = 0.7) and our initial concentration (CAo ):
To determine the pressure of the reactor following this reaction, we will again use the ideal gas law. First, we determine the number of moles in the reactor:
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Problem 1-A Solution
Now, we calculate the new pressure using the ideal gas law:
Back
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01prob.htm
Additional Homework Problems CDP1AA
A 200-dm 3 constant-volume batch reactor is pressurized to 20 atm with a mixture of 75% A and 25% inert. The gas-phase reaction A
B+C
is carried out isothermally at 227°C. (a) Assuming that the ideal gas law is valid, how many moles of A are in the reactor initially? What is the initial concentration of A? (b) If the reaction is first order:
calculate the time necessary to consume 99% of A (c) If the reaction is second order:
calculate the time to consume 80% of A. Also calculate the pressure in the reactor at this time if the temperature is 127°C. CDP1BA
A novel reactor used in special processing operations is the foam (liquid + gas) reactor (Figure CDP1-B). Assuming that the reaction occurs only in the liquid phase, derive the differential general mole balance equation (1-4) in terms of
(Hint: Start from a differential mole balance.)
Figure CDP1-B Foam reactor. [2nd Ed. P1-10
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01prob.htm
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Aspen Plus
Aspen PlusTM Creating and Simulating Chemical Reaction Models with Aspen Plus TM WELCOME to the ASPEN PLUS TM Pages! ASPEN PLUS TM is a software package designed to allow a user to build a process model and then simulate the model without tedious calculations. ASPEN PLUS TM can be used for a wide variety of chemical engineering tasks. For example, it can execute tasks as simple as describing thermodynamic properties of an ethanol and water mixture, or as complex as predicting the steady-state behavior of a full-scale petrochemical plant. This web site, however, will introduce ASPEN PLUS TM as a handy tool for simulating reaction engineering scenarios, such as designing and sizing reactors, predicting reaction conversions, and understanding reaction equilibrium behavior. So, get to know ASPEN PLUS TM by following the outline below. It will surely enhance your understanding of chemical reaction phenomena and the engineering principles behind them!
I. Introduction II. Accessing ASPEN PLUS TM III. Creating a Reaction Engineering Process Model A. Building a Process Flowsheet B. Entering Process Conditions IV. Running the Process Model A. Interpreting the Results B. Changing Process Conditions and Rerunning the Model V. Example Problems A. 8-6: Adiabatic Production of Acetic Anhydride B. 8-7: Operation of a PFR with Heat Exchanger VI. Other Need-to-Knows A. Saving your Process Model B. Printing your Process Model C. Changing Names of Streams and Unit Operations D. Changing Units of Parameters E. Exiting ASPEN PLUS TM VII. Credits
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Thoughts on Problem Solving: Closed-Ended Problems
Closed Ended Problems are the type with only one right answer. These are the same types of problems that are usually found at the end of chapters in textbooks, and they reinforce concepts learned in the corresponding chapter. Algorithm - you can consistently solve difficult problems by following this simple heurisitic. (Includes links to solved example problems: PFR/CSTR and SREP.) Getting Unstuck - these tips can help you overcome mental barriers in problem solving.
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Thoughts on Problem Solving: Open-Ended Problems
Open-ended problems are those which have many solutions or no solutions for the problem as defined. The solutions to these problems usually involves the use of all the skills discussed in Bloom's Taxonomy. First Steps in Solving Open-Ended Problems Examples of Previous OEPs Pharmacokinetics of Cobra Bites Bloom's Taxonomy
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Thoughts on Problem Solving Ten Types of Home Problems
1. Problems that require simple calculations, or formula substitution. These problems will bolster confidence, they are often limited to the knowledge and comprehension levels of Bloom's Taxonomy. Homogeneous Example 1 Heterogeneous Example 1
2. Problems that require intermediate calculations or manipulations. This type of problem may seem unsolvable at first glance, causing the student to go back and re-read the text and lecture notes. One needs to know what laws apply to make these calculations; consequently, this is level 3 in Bloom's Taxonomy application. Homogeneous Example 2 Heterogeneous Example 2
3. Problems that are over-specified so the student has to decide which data and conditions are relevent. Homogeneous Example 3 Heterogeneous Example 3
4. Problems that are under-specified so the student has to consult other information sources in order to complete the problem. Homogeneous Example 4 Heterogeneous Example 4
The Following Types of Problems will Receive Greater Emphasis in the Digital Age 5. "What if..." problems that promote discussion Homogeneous Example 5 Heterogeneous Example 5
6. Problems, or parts of problems (i.e. extensions), that are Open-ended. Homogeneous Example 6 Heterogeneous Example 6
7. Problems where the student must explore the situation by varying operating conditions or parameters. Here, the student may need to create techniques or guidelines to learn whether or not the solution is reasonable. Ordinary differential equation (ODE) solvers can be used to explore the problem. Homogeneous Example 7 Heterogeneous Example 7
8. Problems that challenge assumptions. The student can use ODE solvers or process simulators to redo the problem, changing the assumptions to learn the effects on the answer. Homogeneous Example 8 Heterogeneous Example 8
9. Problems where groups of students work on different parts (or the same part) then come together for discussion. The discussion provides the opportunity for the students to explore different points of view, as well as interact and increase their interpersonal skills. Homogeneous Example 9 Heterogeneous Example 9
10. Problems that develop life-long learning. These problems address the issue of teaching the students to learn on their own. With the explosion of knowledge that is occuring, it will be essential that they be able to learn material independently (i.e.. life-long learning skills). Homogeneous Example 10 Heterogeneous Example 10
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review.htm
Strategies for Creative Problem Solving by H. Scott Fogler and Steven E. LeBlanc
Book Summary Every individual possesses creative skills of one type or another. Honing these skills to a razor-sharp level requires work, and this book was designed to help problem-solvers improve their street smarts. Authors H. Scott Fogler and Steven E. LeBlanc provide a systematic approach to problem solving that helps guide readers through the solution process and the generation of alternative solution pathways. Specifically, Strategies for Creative Problem Solving presents techniques and guidelines that will help readers to: identify the real problem, effectively explore the constraints, plan a robust approach, carry it through to a viable solution, and then evaluate what has been accomplished.
Awards Authors H. Scott Fogler and Steven E. LeBlanc were chosen to receive the prestigious 1996 American Society of Engineering Education Meriam/Wiley Distinguished Author Award, in recognition of ground-breaking Strategies for
file:///H:/html/probsolv/strategy/psgen.htm[05/12/2011 16:55:21]
review.htm
Creative Problem Solving book.
Other Learn more about the authors Read a review of the book How can I purchase a copy?
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Elements of Chemical Reaction Engineering, Updates -- 1st Printing
Typos in the First Printing The following are typographical errors that have been found after the book was printed, but before the CD-ROM was manufactured. PREFACE page xxvi: The virtual reality module is not on the CD but can be found at http://www.engin.umich.edu/labs/vrichel TABLE OF CONTENTS page xi: 8.5.3 Operating Conditions should read: 8.5.4 Operating Conditions CHAPTER 1 page 30: P1-18A, change "predictor-prey" to "predator-prey" relationships CHAPTER 2 page 52: FA0 is missing in the equation just above Eqn. (E2-6.2) and the upper bound of the integral should be 0.8, not 8. Should read: page 61 (top of the page): The upper bound of the integral in Equation (S2-4) should be 0.8, not 8. CHAPTER 3 page 117: P3-9 Part (a), end of sentence should read, "...and rate law parameters for P3-7(a)." CHAPTER 4
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Elements of Chemical Reaction Engineering, Updates -- 1st Printing
page 128: Figure 4-2. There are four typos: a) Half way down the figure, the subzeros should be the opposite of what they are on P and T. Should read:
and
b) Change 3 to 4, i.e., 3. COMBINE should read: 4. COMBINE c) Equal sign missing. Should read: d) Last equation in the figure, there is an unwanted X before the [ln( )] term. Should read: page 146 (add to the margin): "From our discussion of reactor staging in Chapter 2, we could have predicted that the series arrangement would have given the higher conversion." page 177: a) Eqn. (2-20). Delete minus signs on c and d Should read: b) Table 4-5 1) Batch reactor balance on B: CA should be CB Should read: 2) The left hand side of the PFR and PBR mole balances should be multiplied by v0 Should read: PFR: PBR: page 191: There is an extra vo in Equation 4-58 -- delete one of them. The equation should read dCB/dt=rb+vo(CB0-CB)/V
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Elements of Chemical Reaction Engineering, Updates -- 1st Printing
page 195: Delete the V on the left hand side of Equation 4-65. Should read: page 206: Problem 2(d) should read (recall that alpha is a function of the particle diameter and Po). page 209: Problem 4-7(b) should read (Ans. 967 dm 3 ) page 219: Problem P4-31, change parameter values for K C and kCB . Should read:
CHAPTER 6 page 327: Problem 6-13(b), the answers should read (Ans.: CA = 0.61, CB = 0.79, CF = 0.25, CD = 0.45) CHAPTER 7 page 381: The y-axis on Figure E7-5.1(c) should read D not mu D page 418: Add What if ki >> kp and R1 = I o page 419: P7-21B, add the word styrene Line 3 should read "...in styrene polymerization for..." In the Library (F9) of the Polymath section of the CD Living Example Problems "Example 7-3" should read "Example 7-2 part b" CHAPTER 8 page 434: Equation (8-23) should have a "dT" after the brackets in the heat capacity term. page 443: The Fa in the box under V specified: should read Fao (i.e. entering molar feed rate).
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Elements of Chemical Reaction Engineering, Updates -- 1st Printing
page 450: Table E8-5.2 should read, Example 8-4 Non-adiabatic Reactor page 461: There is a 2 missing in Equation (T8-3.7). It should read Cc=2CaoX(To/T) page 498: just above Equation (8-77), replace (8-74) by (8-76). It should read: "...we divide Equation (8-67) by Equation (8-76) to obtain..." page 505: Example 8-12, delete the word "adiabatically" in the second line of the problem statement. It should read: "...take place in a 10 dm 3 CSTR..." CHAPTER 9 page 545: Equation E9-2.12, replace 142 with 35.85. It should read: "=35.85(448-298)" page 547: Second equation from the bottom of the page replace 1433 with 8600. It should read: "=4.48x105 kcal/min + 8600 kcal/min" page 577: Problem P9-11, add Cp(coolant) = 18 cal/mol/K page 578: P9-16(c) should read: "...versus time up to 6 hours and then..." CHAPTER 11 page 722: complete reference is AIChE J 44, p.1933(1998) CHAPTER 12 page 764: The eff. diff. should be 1.82 not 1.89 page 766: The eff. diff. should be 1.82 not 1.89 page 767: The units for rho_b are g/m3 , not g/cm3 CHAPTER 13 page 838: The last line just above Figure 13-13 the words early and late are reversed. It should
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Elements of Chemical Reaction Engineering, Updates -- 1st Printing
read: "The extremes of late and early mixing are referred to as complete segregation and maximum mixedness, respectively." In the margin, replace the word "one" with the word "are". It should read: "...conversion are:" page 844 (at the top of the page): Table E13-6.1 should read Table E13-2.1 Figure E13-6.1 should read Figure E13-2.1 page 852: Eqn. (13-71) "-" missing on right hand side. Should read: page 853: the second term of Equation E13-8.4, E2. Replace the second term (+ 1.180 ) with (+ 1.3618 ) The POLYMATH program is correct! page 872: Add to the margin: "A Model Must: 1. Fit the data 2. Be able to extrapolate theory and experiment 3. Have realistic parameters" CHAPTER 14 page 888: Eqn. (14-36). Change "+" to "-" in the term
Should read: page 908: Pe=Ul/De should read Pe=Ul/Da page 910: Problem 14-2(e), s = 0.01 cm2 /s Should read:
= 0.01 cm2 /s
Problem 14-3: "Maze and Blue" Should read: "Maize and Blue"
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Elements of Chemical Reaction Engineering, Updates -- 1st Printing
APPENDIX page 945: the rate constant above the A + BC ---> (ABC)* should be k1, not k2 page 946: Either equation (G-15) should read: -rA = kKc*(A)(BC), or (G-14) needs to use (B) instead of (BC) to be consistent. page 947: in the top portion, the equation should have a kb, not a k in it, so that it reads: q = (SIGMA)q(i) e^(eps(i) / kbT) page 947: Drop the subscript B in K B. It should read: "...where K is Boltzmann's constant."
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Elements of Chemical Reaction Engineering, Updates -- Which Printing?
Which Printing Do I Have? If you do not know which printing you have, check the publication information on the reverse side of the title page. If you see Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 then you have the first printing. If you see Printed in the United States of America 10 9 8 7 6 5 4 3 2 then you have the second printing.
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Elements of Chemical Reaction Engineering, FAQs
Frequently Asked Questions (FAQs)
Chapter 1 1. In the formula for CSTR, if the rate of reaction is not constant and is dependent on the concentration, should we take to mean? the integral mean? Because the reactor is well-mixed, the concentrations, temperature, and rate of reaction are the same throughout the reactor volume, including the exit point. Consequently, the concentrations, temperature, and rate of reaction in the reactor are all evaluated at the exit conditions of the CSTR. 2. What is the difference between packed bed and fluidized bed reactors? Packed Bed Reactor: Catalyst particles are packed in a tube Fluidized Bed Reactor: Analogous to a CSTR with catalyst particles, see Figure 10-16 (p.620). 3. Why does a semi-batch reactor have better temperature control than a batch reactor? One can also control the feed rate of one of the reactants as well as control the heat exchanger. 4. Why would you choose to have CSTRs and PFRs in series? It is what reactors you might have on hand that you could connect together to obtain a high conversion.
Chapter 2 1. How would the problem involving three reactors in series in Chapter 2 change if there were sidestreams between? One can't use the definition of total conversion up to a point because the reactant is fed to the stream between reactors. One must work in terms of molar flow rates when writing the mole balances. 2. Is there ever a time when a CSTR will have a lower volume than a PFR for the same conversion and flow rates? Yes, for some adiabatic reactions. 3. In the 3-reactor series, (CSTR, PFR, CSTR) why wouldn't you just use the PFR to have the least volume overall to achieve the best conversion? You would if you had a PFR large enough. We are assuming that you have these reactors available for your use. See ICM-Staging.
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Elements of Chemical Reaction Engineering, FAQs
4. For the three reactors in series example, can a PFR use liquid like the 2 CSTRs? Yes. 5. If you had three reactors and two were CSTR's and the other was a PFR, would the PFR be placed at the end to minimize volume? Yes, in most instances when the reaction is isothermal and the curve of (1/-rA) increases monotonically (i.e. no valleys or mountains) with X. 6. Does
(for aA + bB
cC + dD) only hold for first order reactions?
No! This relationship has only to do with stoichiometry and nothing to do with rate laws. It holds for reactions of ANY order. 7. How do you use the equation to model reactors in parallel? See Example 4-2 (p.142).
Chapter 3 1. What is the frequency factor and where can we get values for it? What is it dependent on? Generally the frequency factor is independent of temperatures, however on occasion it can be a weak function of temperature. See p.944 2. Why is the limiting reactant our basis of calculation? One could calculate a NEGATIVE concentration otherwise. See Example 3-5 (p.90). 3. What is the relationship between the K in chemistry (A + B C) and the k in the rate laws? KC is an equilibrium constant, and k is specific rate constant. k has units of time, K does not. 4. How does the k (specific reaction rate) depend on pressure, or does it? ONLY in very very rare instances at very high pressures such as, 6000 atm, for liquid phase reactions is k a function of pressure. See p.220 and CD-ROM on critiquing what you read. 5. What is the frequency factor, A, in the Arrhenius Equation; I want to know what it's physical meaning is and/or what it is a frequency factor. Arrhenius Equation is k = Ae-E/RT The frequency factor, A, is the coefficient of the exponential term. It has the same units as k. It is related to the number of collisions between molecules. See p.942 and 943. 6. What does the overall order of the power law model indicate? One can classify reactions by their overall order of reaction.
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Elements of Chemical Reaction Engineering, FAQs
7. Who determines all the rate laws? These can be found in the literature, journals, books, tables, etc., see the footnote on p.75. They can also be determined in the laboratory. See Ch.5
Chapter 4 1. In solving problems for this class, is there ever a case where you need more steps than the mole balance, rate law, stoichiometry, and combining? (When do you deviate from this algorithm?) We will always use this basic algorithm, just add to these steps to it, e.g. the energy balance. We will not deviate from these first four steps. 2. What specifically causes a CSTR in series to have a higher conversion than a CSTR in parallel? The CSTR is always operating a the lowest concentration, the exit concentration. When say two CSTRs are in series, the first operates at a higher concentration, therefore the rate is greater, therefore the conversion is greater. The second reactor in series builds on the conversion in the first reactor. The conversion in the parallel scheme is the same as the conversion to the first reactor to the series scheme. See Figure p.50 and Example 4-2. 3. When are reactors in parallel used since it seems as though reactors in series would always achieve higher conversion? The PBRs in parallel are sued where there would otherwise be a large pressure drop in one long reactor (or identically several PBRs connected in series). 4. Is it possible to have a pressure drop for a liquid phase reaction, as is possible for a gas phase reaction? You can have a pressure drop in liquid phase systems, but it does not affect the reaction rate because liquids are virtually incompressible and therefore the concentration does not change with pressure. 5. Since two equal CSTR in series give a higher conversion than two in parallel, are reactors in parallel ever used to increase conversion? Not for a CSTR, only a PFR with DP. 6. The Damköhler (Da) number a. What is the Damköhler number? See p.138 of text. b. How is the Damköhler (Da) number defined for a reaction (A + B C) when the reaction is first order in A, first order in B, but second order overall? Just substitute the rate law evaluated entrance to the reactor, -r A0, [e.g. definition . See p.138. file:///H:/htmlmain/faqs.htm[05/12/2011 16:55:24]
] into the
Elements of Chemical Reaction Engineering, FAQs
c. Is Da always indicative of certain conversion? Yes, for irreversible reactions. d. How does defining an extra variable, the Damköhler number, save us time and confusion, as opposed to solving without it? When should it be used? It serves as rule of thumb. When Da < 0.1 then X < 10% and when Da >10 then X >90%. See margin note p.138 of Ch.4. 7. Is the only way to determine a rate constant, k, by experimentation? Yes. 8. Why do they use a batch reactor to determine k if they are going to be using CSTR in actual industrial process? Batch experiments are most always easier to take data to measure k. 9. In what cases would you use the order of magnitude reaction times other than to check k values that you calculate? When you are short on time and want to get quick engineering estimates. 10. At some of the polymer plants and refineries I've visited, a huge problem is fouling of the reactors. The plant workers would sometimes have to go into the reactors to break through the solids/sludge that adhere to the reactor walls. I imagine this solid build up leads to a drastic volume decrease. So, how do we take into account the change of volume and it's detrimental effect to conversion? Good point. It would however, catalyst decay by fouling is usually more important. See Ch.10. 11. Is rinsing the reactor with water after a batch ample cleaning, or are chemical cleans necessary in between batches? It depends, if there are no side reactions, a chemical clean is probably not necessary. Also the larger the reactor the greater the cleaning time. 12. Please clarify the method for deriving the rate law expressed with partial pressures. The conversion used when studying catalytic reactions is that the rate law is developed in terms of partial pressure, e.g.
.
Just use ideal gas law to relate to concentrations CA and CB
writing partial pressures in terms of conversion
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Elements of Chemical Reaction Engineering, FAQs
13. How accurate is the perfect mixing assumption in dealing with CSTRs? It seems kind of far fetched that the entire CSTR is at the same concentration as the exit. True but these are ideal CSTRs, andnon ideal reactors and the perfect mixing assumption is discussed and modeled in Ch.13. Once we understand ideal reactors (perfectly mixed), we can easily model non-ideal reactors. 14. In a PFR or CSTR reactor, wouldn't the reaction still be happening in the pipe that the products leave through? Why does the reactor just magically stop occurring when the contents leave the reactor? Is there a good way to model a CSTR that is not perfectly mixed? It does continue to react to some extent. It depends on the temperature in the pipe! However, these are ideal reactors. Different models for Non-ideal reactors are discussed in Chapters 13 and 14. 15. Can you compare space time for a flow reactor to the time spent in a batch reactor for the purposes of measuring conversion? Sometimes. See Lecture 6 on the CD-ROM. 16. We understand that for a CSTR, the conversion X increases as residence time increases. We were unsure as to what the relationship is for a PFR? Same is true
for PFR. If you increase
and you increase X.
17. Is the conversion equation always the same no matter the order of reaction? No! This equation is only for first order. 18. Is the assumption that there are no radial gradients a good one for most tubular reactors? When is it not valid? It is quite a good assumption for turbulent flow. It is not valid for Laminar flow - see Ch.13 p.831. 19. How did you develop equation E4-9.8 on page 180? i.e. FB = 2 (F A0 - FA )
For every two moles of B are formed, one mole of A is consumed.
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Elements of Chemical Reaction Engineering, FAQs
20. When trying to determine the optimal catalyst size for the internal diffusion limited case, do we always use the relation
or are there any other relations that can be used.
You can only use this relationship in the internal diffusion limited regime shown in Figure 12-5, with p.750. For a first order reaction (12-35)
or lumping all the constants not involving particle size into "a"
then k2 = k1
then
. This
for Pb. 4-23 one must use all of Figure 12-5. It can be
shown that Figure 12-5 can be represented as
21. When accounting for pressure drop in a membrane reactor, does the same method as we would use with a PFR apply? Yes,
22. Do we only use the
form of the PBR design equation for membrane reactors (IMRCF)?
23. In the text it states that for developing the design equation for a PBR when X accessing
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Aspen Plus - Creating
Creating a Reaction Engineering Process Model Now that you have gained access to ASPEN PLUS TM , you are ready to begin creating a process model. The following series of steps will create a process model for the tubular reactor (PFR) example problem 4-4 taken from the 3rd Edition of Elements of Chemical Reaction Engineering by H. Scott Fogler. Here is a summarized version of the problem: Example 4-4 Problem Statement Determine the plug-flow reactor volume necessary to produce 300 million pounds of ethylene a year from cracking a feed stream of pure ethane. The reaction is irreversible and elementary. We want to achieve 80% conversion of ethane, operating the reactor isothermally at 1100K at a pressure of 6 atm. C2 H6 (g)
C2 H4 (g) + H2 (g) A
B+C
Where A is gaseous ethane, B is gaseous ethylene, and C is gaseous hydrogen. Other information: Fao = 0.425 lbmol/s (calculated from 300 million pounds of ethylene at 80% conversion) k = 0.072s -1 at 1000K Activation Energy, E = 82 kcal/gmol Building the Process Flowsheet The first step in creating a process model is drawing the flowsheet in ASPEN PLUS TM , much like you would on paper. Note that while you're constructing the flowsheet, a red text prompt in the upper left corner under the main tool bar will state "Flowsheet Not Complete." This will change to a black text "Flowsheet Complete" prompt when the flowsheet is finished. Inlet (Feed) Streams To draw the inlet stream do the following steps 1-5. 1. First, you must draw the inlet stream (ethane in this case). On the right side of the flowsheet window is the drawing tool bar. Under Type highlight Feed/Prod with the left mouse button. 2. Under Model highlight Feed. 3. Under Icon highlight Material. 4. Move the mouse pointer (which now looks like cross-hairs) to the middle left of the empty flowsheet window. 5. Click the left mouse button to place the inlet stream on the flowsheet. The inlet stream is file:///H:/html/toolbox/aspen/creating.htm[05/12/2011 16:59:08]
Aspen Plus - Creating
represented by an outline of an arrow pointing to the right. You have finished adding the inlet arrow. Note this example only has one arrow (representing the ethane feed). More than one inlet stream can be drawn, each representing different reactants. Your screen should look like this:
Outlet (Product) Streams You are now ready to add the outlet stream (containing both ethylene and hydrogen) to your flowsheet. Repeat the steps described above, however under 'Model' highlight 'Prod'. Place the product arrow about 2-3 inches to the right of the inlet arrow on the flowsheet. Of course, for other examples, there could be more than one outlet stream. Your screen should now look like this:
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Aspen Plus - Creating
Unit Operations Now that inlet and outlet streams are placed on the flowsheet, you are ready to place the unit operation (the PFR) on the flowsheet. The steps are similar to placing the streams on the flowsheet. 1. Under Type highlight Reactor. 2. Under Model highlight RPLUG (reactor plugflow). 3. Under Icon highlight either Block, Icon1,or Icon2. The only difference is the way the PFR will look on the flowsheet. The display box above Type will give you a preview of the PFR icon before you place it on the flowsheet . 4. Move the mouse pointer (which now looks like cross-hairs) in between the inlet and outlet arrows. Click the left mouse button to place the PFR on the flowsheet. The PFR is represented by whichever icon you chose. It will arbitrarily be named B1 (you can change the name later). Your flowsheet should now look like this:
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Aspen Plus - Creating
Connecting the Streams to the Unit Operation Once the inlet arrow, PFR, and outlet arrow are on the flowsheet, it is time to connect the flow streams to the PFR. Inlet to PFR 1. With the left mouse button, click on the inlet arrow so it is highlighted by four black dots. 2. Now with the left mouse button, double click on the highlighted inlet arrow. The feed port, represented by a small arrow, will appear on the inlet arrow. Notice that when you place the mouse pointer on the feed port, it automatically highlights the small arrow. 3. With the left mouse button, double click on the feed port arrow. Notice how port arrows appear on the PFR. 4. To connect the inlet arrow to the PFR, move the mouse pointer to the port arrow called FEED on the PFR and click the left mouse button. Note: When you move the mouse pointer over the PFR port arrows, their names will appear. A line show now connect the inlet arrow to the PFR. The line should be labeled with a 1 (you can change the name of the stream later). This line represents the ethane inlet stream. Your screen should look like this:
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Aspen Plus - Creating
PFR to Outlet 1. With the left mouse button, click on the PFR so it is highlighted by four black dots. 2. Now with the left mouse button, double click on the highlighted PFR. Various ports represented by small arrows appear on the PFR. 3. With the left mouse button, double click on the product port arrow on the PFR. Notice how the outlet port arrow appears on the outlet arrow to the right of the PFR. 4. To connect the outlet arrow to the PFR, move the mouse pointer to the outlet port arrow called PROD on the outlet arrow and click the left mouse button. A line should now connect the outlet arrow to the PFR. The line should be labeled with a 2 (you can change the name of the stream later). This line represents the ethylene and hydrogen product stream. Your screen should look like this:
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Aspen Plus - Creating
Entering Process Conditions Now that the process flowsheet is complete, it is time to enter the process conditions. ASPEN PLUS TM will guide you through the required input windows, simply click on the Next button on the upper right of the main toolbar. When each input window is complete, the prompt "# Complete" will appear in the upper left corner of the screen. 1. Click Next with the left mouse button. A window will prompt you that the flowsheet is complete and asks whether the next input form be displayed. Click OK. 2. The first input window will be called Setup.Main. With the left mouse button, click once on the Title box to highlight it, enter the title of your process model. 3. Check that the desired units are correct. Change the units by clicking with the right mouse button on the unit box to pull down a unit menu. Click on the desired units (you can always change the name later while entering the stream properties). 4. # Complete should be stated in the upper left corner. Click Next. The next input window is Components.Main. Here is where all of the chemical species in your process model are specified. 5. For this particular example, the components are: ethane, ethylene, and hydrogen. Start with the right most column called Component Name. Click on the first row in the column and type in: ethane. Hit Enter. ***Note: If you are unsure of how to spell the chemical name, or do not know whether it's in the ASPEN PLUS TM library, simply click (with the right mouse button) where you normally type in the chemical name to pull down a list of all the chemicals in the library. You can scroll through the list with the up/down arrow keys. To select the desired chemical, highlight it and then click on it or hit Enter.
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Aspen Plus - Creating
6. Ethane is in the ASPEN PLUS TM chemical library. Notice how the molecular formula automatically appears after typing it in. Now click under Comp ID. Enter an id name (to show up in results) for ethane, perhaps ETHA. Hit Enter. 7. Repeat steps 5 and 6 for ethylene and hydrogen. In this example, the component id names used were ETHY and H2, respectively. 8. # Complete appears. Click Next. The next window to appear is Properties.Main. Here is where you specify the solving engine used to simulate your process model. 9. Highlight the Opsetname box and type: SYSOP3 Hit Enter. 10. A window will appear asking if you wish to use SYSOP3. Click OK. 11. # Complete appears. Click Next. A window stating Required Properties Input Complete will appear. Read it and click OK. The next set of input windows will ask for the information state in the problem statement such as: flowrates, rate law, stoichiometry, etc. The first window will be titled Streams 1(Stream.Main). This indicates the stream labeled 1 on your flowsheet: the ethane inlet stream. You may have already assigned a different id to the inlet stream, if so that name will appear as the subtitle. 12. Highlight the Description box. Enter a description, perhaps Ethane Inlet Stream. 13. Highlight Temp. Enter 1100. Highlight the unit box to the right of Temp. Click on it with the right mouse button to pull down a unit menu. Click on K for Kelvin. 14. Highlight Pres. Enter 6. Highlight the unit box to the right of Pres. Click on it with the right mouse button to pull down a unit menu. Click on ATM for atmospheres. ***Note. Since Temp. and Pres. are specified, the vapor fraction, Vfrac, is set for the ethane inlet stream (Gibbs Phase Rule). 15. Now look to the middle of the window for Enter stream composition and flow. The Composition Basis should read MOLE-FLOW. If not, pull down the menu by clicking on the box with the right mouse button. Select MOLE-FLOW. Change the unit box via the pull down menu method to read LBMOL/SEC. 16. Since ethane is the only component of the inlet stream, highlight the box under Value and enter 0.425 for the molar flowrate. Enter 0.425 again for Total MOLE-FLOW. Be sure the units are LBMOL/SEC. 17. # Complete appears. Click Next. The next input window is titled B1(Rplug.Main). This window is where you will specify the operating conditions of the PFR. This is the window you will come back to upon running the simulation to change any operating conditions as well. 18. Highlight the Description box. Enter a description for the PFR, perhaps Isothermal PFR. 19. Highlight the Type box. Click on the box with the right mouse button to pull down the menu.
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Select T-Spec. This stands for a PFR with a temperature specification. You will be describing the temperature profile in the PFR. 20. Highlight the Length box. Enter a length in feet. 10 feet is a good starting point. Be sure the units are FT. Change the units accordingly via the pull down menu method if necessary. 21. Do the same for the Diam box. 3 feet is a good guess for the diameter. Be sure the units are in FT. ***Note. You are solving this problem by guessing a volume. When you run the simulation you will see what conversion is achieved with the guessed volume. You will keep changing the volume (increasing/decreasing the length while keeping the diameter constant) and rerunning the simulation until the desired conversion is achieved. Remember that a PFR is a cylinder with a volume of V = (P/4)D 2 L. 22. Look down the window for Temperature specifications. Here is where you'll enter the temperature profile of the PFR. Since the PFR is isothermal, it will be 1100K (as given in the problem statement) throughout the entire reactor. Enter the following profile:
Location Temp K 0
1100
0.5
1100
1
1100
Be sure Temp is in K. Change it via the pull down menu method if necessary. The Location is given by a fraction of the distance down the PFR. Hence 0 = the beginning, 0.5 = the midpoint, and 1 = the end of the PFR. 23. Since the PFR has no heat exchanger jacket, do not enter any values under External coolant. 24. Under Optional reactor specifications highlight Pres. Type 6 and change the units to ATM. 25. # Complete appears. Click Next. The next input window is called Blocks B1(Rplug.Reac). The stoichiometry of the reaction occurring in the PFR will be described here. 26. Highlight the box called Reaction Number. The cracking of ethane in this problem is the only reaction occurring in the PFR. Therefore, this reaction will be called reaction 1. Enter 1 in the box. 27. Under Reactants, highlight the box under Comp ID. Enter your id name for ethane, in this case ETHA. 28. Now look at the balanced reaction equation in the problem statement. The coefficient of ethane is 1, however since ethane is a reactant it's disappearing so the coefficient is really -1. Enter -1 under Coefficient. 29. Under Products, enter the id names for the products ethylene and hydrogen (ETHY and H2 here) under Comp ID. Their coefficients will both be 1. Enter 1 for both coefficients. # Complete appears. Click Next. The last input window is titled Blocks B1(Rplug.Kinetics). Here you'll describe the rate law of the reaction in the PFR. 30. Highlight the Reaction Number box. Enter 1. Thus, you will be describing reaction 1, the cracking file:///H:/html/toolbox/aspen/creating.htm[05/12/2011 16:59:08]
Aspen Plus - Creating
of ethane is this case. ***Note. This is where a calculator comes in handy. ASPEN PLUS TM asks for two parameters of the Arrhenius equation: the activation energy, E, and the pre-exponential factor, A. With these parameters and the temperature of the reaction, ASPEN PLUS TM solves the equation of the specific rate of reaction, k. In this example problem, you are given E = 82 kcal/mol and k = 0.072s -1 when T = 1000K. Using this information, you can back track and solve for the preexponential factor A: k = Aexp(-E/RT) A = k/[exp(-E/RT)] A = 0.072 s -1 /[exp((-82000 cal/mol)/((1.987 cal/mol-K)(1000 K))] A = 6.02x1016 s -1 31. Hence type 6.02E16 for Pre-Exp. Type 82 for Act-energy, with units of KCAL/MOL. TempExpon will automatically be 0. 32. Lastly, the rate law must be entered. This example has an elementary rate law, hence, -rA= kCA . Therefore, under the Reactants, enter the id name for ethane (ETHA). Type 1 for the Exponent. 33. Under Products, enter the id names for ethylene and hydrogen (ETHY and H2). Enter 0 for the Exponents. 34. # Complete. Click Next. You are finished entering all the required data for the process model! The next window is called Rplug Input Menu. Click Next. Required Input Complete will appear. This window asks if you wish to run the simulation. Click OK. I. Introduction II. Accessing ASPEN PLUSTM III. Creating a Reaction Engineering Process Model A. Building a Process Flowsheet B. Entering Process Conditions IV. Running the Process Model A. Interpreting the Results B. Changing Process Conditions and Rerunning the Model V. Example Problems A. 8-6: Adiabatic Production of Acetic Anhydride B. 8-7: Operation of a PFR with Heat Exchanger VI. Other Need-to-Knows A. Saving your Process Model B. Printing your Process Model C. Changing Names of Streams and Unit Operations D. Changing Units of Parameters E. Exiting ASPEN PLUSTM VII. Credits
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Aspen Plus - Creating
BACK to ASPEN PLUS TM MAIN PAGE aspen plus > creating
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Aspen Plus - Running
Running the Process Model Congratulations on completing the flowsheet and entering the process model conditions. Now you are ready to put your model to the test. Example 4-4, the cracking reaction, can now be simulated. Once you've clicked OK to run the process model, the Control Panel window will appear. This window gives you a look at the ASPEN PLUS TM "thinking process." Phrases indicating the PFR block is being executed will scroll across the sc reen. When the simulation is complete, the Control Panel will read: Simulation completed successfully. Interpreting the Results 1. Once the simulation is complete, click the Results button on the Control Panel. 2. A window titled Results-Status.Main will appear. Click on the >> button in the upper right corner to jump forward to the stream-by-stream results page. For the Example 4-4 simulation the results screen should look like this:
Note that, down the left side of the screen, are the different parameters: temperature, pressure, mole flow, etc. Along the top row are the stream id names, in this case 1 and 2 (inlet and product). This forms a grid of information that can be interpreted easily. In the upper left corner of the screen, you will see Display, Units, and Format. You can change what results are shown and their units by using the pull down menu method for each of these areas. In Example 4-4, the problem asks for the PFR volume that will achieve an 80% conversion. As you
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Aspen Plus - Running
recall, when entering the process model conditions, you guessed a volume by entering an arbitrary length and diameter of the PFR. In order to complete the problem, you must see what conversion your process model obtained. Recall that conversion is defined as: X = (moles of limiting reagent reacted) (moles of limiting reagent fed) Where moles reacted = moles in - moles out To solve for conversion, do the following with the results: 1. Under Mole Flow for ethane, calculate the moles of ethane that reacted: stream 1 - stream 2 2. Divide this number by the molar flowrate of ethane into the PFR: stream 1. If you followed the example exactly, using a length of 10 feet and a diameter of 3 feet, you should get a conversion of 76%. X = stream 1 - stream 2 stream 1 X = 693.996 kmol/hr - 169.362 kmol/hr 693.996 kmol/hr -orX = 1530 lbmol/hr - 373.380 lbmol/hr 1530 lbmol/hr X = 0.76 = 76% The dimensions of the PFR did not achieve an 80% conversion. Therefore, you need to go back and adjust them. It is easiest to vary the length of the PFR while holding the diameter constant. In this example, since the conversion was too low, you must INCREASE the length of the PFR. Changing Process Conditions and Rerunning the Model Adjusting the input conditions is very straightforward, just follow these steps: 1. With the mouse pointer, click on the Flowsheet window to make it active. 2. With the left mouse button click on the inlet stream or unit operation whose conditions you wish to change. In this example, click on the PFR. 3. Click the right mouse button to pull down a large menu. Click onInput with the right mouse button. 4. Click the >> button to forward you ahead to the input window you need. In this example, you are looking for Rplug.Main. 5. Highlight the value you wish to alter and type in the new value. In this case change the Length to a value greater than 10 feet (Hint: Try 11.42 feet!). 6. Click the >> or running
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Aspen Plus - Examples
Example Problems This section is devoted to example reaction problems. The problems were taken from the 2nd Edition of Elements of Chemical Reaction Engineering by H. Scott Fogler. Please note, it is assumed that the user knows how to create a flowsheet and enter process conditions, since these examples explain only the values to enter for each input window. Example 8-6 Adiabatic Production of Acetic Anhydride Jeffreys, in a treatment of the design of an acetic anhydride manufacturing facility, states that one of the key steps is the vapor-phase cracking of acetone to ketene and methane: CH 3 COCH 3
CH 2 CO + CH 4
He states further that this reaction is first-order with respect to acetone and that the specific reaction rate can be expressed by ln k = 34.34 - 34,222/T (E8-6.1) where k is in reciprocal seconds and T is in Kelvin. In this design, it is desired to feed 8000 kg of acetone per hour to a tubular reactor. If the reactor is adiabatic, the feed pure acetone, the inlet temperature 1035K, and the pressure 1 62 kPa (1.6 atm), a tubular reactor of what volume is required for 20% conversion? Creating the Flowsheet The flowsheet consists of one inlet stream, a PFR, and one product stream. It should look like this:
Create a flowsheet like this in ASPEN PLUS TM . If you do not know how, see Example 4-4. When the flowsheet is complete,"Flowsheet Complete" should appear in the upper left corner of the screen. Click the Next button. Click OK when prompted to Enter Required Data. Entering Process Conditions This section will explain what values to type in for each input window. If you do not know how to enter values, change units, or navigate through the input windows, see Example 4-4. 1st Window: Setup.Main 1. Title: Enter any title you wish. 2. Flow / Frac Options: MOLEFLOW and MASSFLOW 3. Click Next.
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Aspen Plus - Examples
2nd Window: Components.Main 1. Under Component Name type the following in a column: ACETONE, KETENE, METHANE 2. Under Comp ID type in any id names for the above components: A, K, C1 3. Click Next. 3rd Window: Properties. Main 1. 2. 3. 4.
Opsetname: type SYSOP3 Click YES when prompted about SYSOP3 Click Next. Click OK when prompted about continuing entering stream input.
4th Window: Stream. Main 1. 2. 3. 4. 5.
Description: Enter any description of stream 1. Temp: 1035 K (change units if necessary) Pres: 1.6 atm (change units if necessary) Composition Basis: Change to MASS-FLOW KG/HR For a mass flowrate of A (acetone) type 8000. Leave ketene and methane at zero (no mass flow in reactant stream). 6. Total: Change to MASS-FLOW and enter 8000 KG/HR 7. Click Next. 5th Window: Rplug.Main 1. 2. 3. 4. 5.
Description: Enter any description for PFR, perhaps Adiabatic PFR. Type: ADIABATIC Length: Need to guess a length, 3 METERS is a good starting point. Diam: Need to guess a diameter, 1 METER is a good starting point. Click Next.
6th Window: Rplug.Reac 1. 2. 3. 4.
Description: Should be stated already as that you used for Rplug.Main. Reaction Number: type 1 Under Reactants: type A for acetone. Coefficient (stoichiometric): -1 Under Products: type K for ketene. Coefficient (stoichiometric): 1. Type C1 for methane. Coefficient (stoichiometric): 1 5. Click Next. 7th Window: Rplug.Kinetics 1. Description: Should already be stated. 2. Reaction Number: type 1 3. Pre-Exp: Enter the pre-exponential factor, A, of the Arrhenius equation, 8.2e14. This number was solved for by putting equation E8-6.1 in the form of k = Aexp(-E/RT) Take the natural log of both sides of equation E8-6.1 to get k (s-1 ) = 8.2x1014exp(-32,444/T) 4. Act-Energy: Enter the activation energy E of the Arrhenius equation, 67999 CAL/MOL. Again this file:///H:/html/toolbox/aspen/example.htm[05/12/2011 16:59:10]
Aspen Plus - Examples
value was solved for using equation E8-6.1: k (s-1 ) = 8.2x1014exp(-32,444/T) (Note that R is missing in the denominator.) Activation Energy = E = (32,444)(R) E = (32,444)(1.987 cal/mol K) = 67999 cal/mol 5. Under Reactants: Type A for acetone. Exponent (in rate law): 1 6. Under Products: Type K for ketene. Exponent (in rate law): 0. C1 for methane. Exponent (in rate law): 0 7. Click Next. Running the Simulation and Interpreting the Results Click Next again until you are prompted to run the simulation. Click OK. When the simulation is complete, click on Results (Control Panel). If you do not know how to interpret the results window, see Example 4-4. Otherwise, check the conversion (X = moles reacted/moles fed). Does X = 20%? If X < 20%, you must increase the length of the PFR. If X > 20%, you must decrease the length of the PFR. In this case where length = 3 m, diam = 1m, the conversion was greater than 20%. Therefore, you need to go back to the PFR and input a smaller length. You must access the Rplug.Main window to do this. If you do not know how to reenter inputs, see Example 4-4. This time, try a length of 2.5 m while holding the diameter constant at 1 m. When you rerun the simulation, you will find that X = 20%! Finishing up the example, the volume of the PFR with these dimensions is V = 1.96 m 3 .
Reference: G. V. Jeffreys, A Problem in Chemical Engineering Design: The Manufacture of Acetic Anhydride, 2nd ed. (London: Institution of Chemical Engineers, 1964).
Example 8-7 Operation of a PFR with Heat Exchanger We again consider the vapor-phase cracking of acetone used in Example 8-6: CH 3 COCH 3
CH 2 CO + CH 4
The reactor is to be jacketed so that a high-temperature gas stream can supply the energy necessary for this endothermic reaction (see Figure E8-7.1). Pure acetone enters the reactor at a temperature of 1035K and the temperature of the external gas in the heat exchanger is constant at 1150K. The reactor consists of a bank of one thousand 1-in. schedule 40 tubes. The overall heat-transfer coefficient is 110 J/m 2 -s-K. Determine the temperature profile of the gas down the length of the reactor.
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Aspen Plus - Examples
Figure E8-7.1 Creating the Flowsheet Use the same flowsheet in Example 8-6. Entering Process Conditions Repeat windows 1 - 4, and 6 as in Example 8-6. For window 5: Rplug.Main do the following: 5th Window: Rplug.Main 1. 2. 3. 4. 5. 6.
Description: Enter any description for PFR, perhaps Jacketed PFR. Type: TCOOL-SPEC Under External coolant, enter a Coolant-Temp of 1150 K with a U of 110 J/SEC-SQM-K. Length: Need to guess a length, 3 METERS is a good starting point. Diam: Need to guess a diameter, 1 METER is a good starting point. Click Next.
Run the simulation. Again, adjust the length until the conversion is X = 20%. In this example, the proper length was 1.9 m with a diameter of 1 m. Thus the volume was V = 1.49 m 3 . Temperature Profiles down the Length of the PFR To see the temperature profile down the length of the PFR, do the following: 1. Click with the left mouse button on the PFR to highlight it. 2. Click the right mouse button to pull down a large menu. Select Results. 3. A results menu window will appear. Double click with the right mouse button on the black text: Profiles.\ 4. You will see columns of values: lengths, pressures, temperatures, vapor fractions, etc. 5. Click on the Length column to highlight it then click on Plot (main tool bar). 6. Click on Independent Variable (this is the x-axis). You assigned the length as an independent variable. 7. Click on the Temp column to highlight it then click on Plot (main tool bar). 8. Click on Dependent Variable (this is the y-axis). You assigned the temperature as an independent variable. 9. Pull down the Plot menu again and click on Display Plot. You should see a plot of the temperature profile that looks like this:
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Aspen Plus - Examples
I. Introduction II. Accessing ASPEN PLUSTM III. Creating a Reaction Engineering Process Model A. Building a Process Flowsheet B. Entering Process Conditions IV. Running the Process Model A. Interpreting the Results B. Changing Process Conditions and Rerunning the Model V. Example Problems A. 8-6: Adiabatic Production of Acetic Anhydride B. 8-7: Operation of a PFR with Heat Exchanger VI. Other Need-to-Knows A. Saving your Process Model B. Printing your Process Model C. Changing Names of Streams and Unit Operations D. Changing Units of Parameters E. Exiting ASPEN PLUSTM VII. Credits BACK to ASPEN PLUS TM MAIN PAGE aspen plus > examples
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Aspen Plus - Other
Other Need-to-Knows The following important information applies to UNIX Ultra Sparc® workstations. Saving Your Process Model It is wise to save your process model periodically while working. To save the model: 1. Select the File pull down menu from the tool bar. Click on Save As. 2. Enter the name of your process model. Click OK. 3. The process model will be saved under your afs directory. 4. The name of your process model will now appear on the window, replacing model UNNAMED. From this point on, select Save under the File menu to save your process while you work. Printing Your Process Model The flowsheet and results are valuable documents verifying your work. ASPEN PLUS TM allows you to individually print a flowsheet, stream-by-stream result pages, and a history file. The Flowsheet. To print your process model's flowsheet from a UNIX workstation do the following while the flowsheet window is active: 1. Select the File pull down menu from the tool bar. Click on Print. 2. A small window will appear asking for the name of the flowsheet. Type in an arbitrary name of the flowsheet (typing in 'flowsheet' works fine). Click OK. 3. Moving the mouse pointer off the ASPEN PLUS TM window, use the middle mouse button to open a New Shell. 4. At the % prompt, type: lpr -P For example, to print a flowsheet entitled 'flowsheet' on the 3 South printer in the Media Union type: lpr -P mu3s flowsheet (hit enter) Stream-by-Stream Results. To print the stream-by-stream results, follow steps 1-4 above while the results window is active. Instead of naming the flowsheet, however, you will be assigning an arbitrary name to the results ('results' works fine). The History File. Printing the history file of your process model will allow you to step through all of the computer code used in solving the simulation. The history file keeps track of all previous ASPEN PLUS TM runs executed while you've been logged on. Depending on the complexity of the process model, the history file can be very lengthy (100 pages or more!). Therefore, evaluate whether a hard copy of this file is necessary before you print.
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Aspen Plus - Other
ASPEN PLUS TM creates the history file after completing one run of the process simulation. To print the history file do the following: 1. Open a New Shell (step 3 above). 2. At the % prompt, type in: lpr -P For example, if you saved your process model as CSTR, to print its history file on the 3 South printer in the Media Union type: lpr -P mu3s CSTR.his (hit enter) Changing Names of Streams and Unit Operations ASPEN PLUS TM arbitrarily assigns id names to all streams and unit operations on the flowsheet you create. If you wish to change the id name, do the following: 1. With the left mouse button, click on the id box of the stream or unit operation to highlight it. 2. Click the right mouse button to pull down a large menu. 3. Click on the Rename option. 4. When the Rename box appears, type in the desired id name and click OK. Changing Units of Parameters When entering the input conditions, you may need to change the units of parameters such as temperature, pressure, mole flow, etc. Do the following, commonly referred to as the pull-down menu method: 1. Click on the unit box with the right mouse button. A list of units should appear. 2. Scroll up/down the list with the cursor keys. Click on the appropriate unit. Exiting ASPEN PLUSTM When finished with your process model, do the following to exit the ASPEN PLUS TM program: 1. Pull down the File menu. 2. Click on Exit. 3. A window will appear asking if you wish to save your process model run. Click No if you do not wish to save the run, or Yes if you do. 4. The ASPEN PLUS TM window will disappear and you will be left with the % prompt. I. Introduction II. Accessing ASPEN PLUSTM III. Creating a Reaction Engineering Process Model A. Building a Process Flowsheet file:///H:/html/toolbox/aspen/other.htm[05/12/2011 16:59:10]
Aspen Plus - Other
B. Entering Process Conditions IV. Running the Process Model A. Interpreting the Results B. Changing Process Conditions and Rerunning the Model V. Example Problems A. 8-6: Adiabatic Production of Acetic Anhydride B. 8-7: Operation of a PFR with Heat Exchanger VI. Other Need-to-Knows A. Saving your Process Model B. Printing your Process Model C. Changing Names of Streams and Unit Operations D. Changing Units of Parameters E. Exiting ASPEN PLUSTM VII. Credits BACK to ASPEN PLUS TM MAIN PAGE aspen plus > other
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Aspen Plus - Creating
Credits The ASPEN PLUS TM web site was created by Ellyne E. Buckingham at the University of Michigan, Summer 1997. Special thanks to: Anuj Hasija and Professor H. Scott Fogler The University of Michigan at Ann Arbor, MI Dr. J. Mahalec ASPENTech at Cambridge, MA All ASPEN screen shots courtesy of ASPENTech.
Final editing of the ASPEN PLUS TM web site done by Dieter Andrew Schweiss, Fall 1997. I. Introduction II. Accessing ASPEN PLUSTM III. Creating a Reaction Engineering Process Model A. Building a Process Flowsheet B. Entering Process Conditions IV. Running the Process Model A. Interpreting the Results B. Changing Process Conditions and Rerunning the Model V. Example Problems A. 8-6: Adiabatic Production of Acetic Anhydride B. 8-7: Operation of a PFR with Heat Exchanger VI. Other Need-to-Knows A. Saving your Process Model B. Printing your Process Model C. Changing Names of Streams and Unit Operations D. Changing Units of Parameters E. Exiting ASPEN PLUSTM VII. Credits BACK to ASPEN PLUS TM MAIN PAGE aspen plus > credits
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Aspen Plus - Creating
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Thoughts on Problem Solving: Bloom's Taxonomy
Problem solving is an activity whereby a best value is determined for an unknown, which is subject to a set of constraints. To determine how to attack a problem, three problem classifications can be used to direct the investigation: by type of unknown, difficulty, and open-endedness. The author’s preference in problem classification is that of Bloom, who classifies and identifies six problem-solving skills.
A. Level of Difficulty or Skill Level Each successive skill level calls for more advanced intellectual ability. 1. Knowledge: The remembering of previously learned material. Can the problem be solved simply by defining terms and by recalling specific facts, trends, criteria, sequences, or procedures? This level solves the type of problems such as recallin g the type of continuous flow reactor normally used for liquid-phase reactions. This is the lowest intellectual skill level. Examples of knowledge level questions are the following: Write the equations for a batch reactor and list its chara cteristics. Define . Which reactors operate at steady state? Other words used in posing knowledge questions: Who..., When..., Where..., Identify..., What formula .... 2. Comprehension: This is the first level of understanding. Given a familiar piece of information, such as a scientific principle, can the problem be solved by recalling the appropriate information and using it in conjunction with manipulation, tr anslation, interpretation, or extrapolation of the equation or scientific principle? For example, given the reactor volume [V=(v 0 /k)ln(CA0/CA )], can one manipulate the design equation formulas to find the effluent concentration to find the reactor volume if the inlet concentration were doubled? Compare and contrast the advantages and uses of a CSTR and a PFR. Construct a plot of NA as a function of t. Other comprehension words: ... Relate..., Show..., Distinguish..., Reconstruct..., Extrapolate... This is skill level 2. A. Drilling the concepts A1. Plug and chug
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Thoughts on Problem Solving: Bloom's Taxonomy
B. Extrapolate What is the time at 400 K?
C. More difficult/Think Problems C1.Intermediate calculation Example: 80% conversion is achieved for a first order reaction in a 1000 gal PFR, what conversion could be achieved in a CSTR for the same conditions? C2.More than one intermediate calculation Example: 80% conversion is achieved for a first order reaction achieved 1000 gal PFR, what conversion could be achieved in a CSTR operated at a temperature 100°C higher. 3. Application: The next higher level of understanding is recognizing which set of principles ideas, rules, equations, or methods should be applied, given all the pertinent data. Once the principle is identified, the necessary knowledge i s recalled and the problem is solved as if it were a comprehension problem (skill level 2). An application level question might be: Make use of the mole balance to solve for the concentration exiting a PFR. Other words: ... Apply..., Demonstrate..., Determine..., Illustrate.... 4. Analysis: This is the process of breaking the problem into parts such that a hierarchy of subproblems or ideas is made clear and the relationships between these ideas is made explicit. In analysis, one identities missing, redundant, and contradictory information. Once the analysis of a problem is completed, the various subproblems are then reduced to problems requiring the use of skill level 3 (application). An example of an analysis question is: What conclusions did you come to after reviewing the experimental data? Other words: ... Organize..., Arrange..., What are the causes ..., What are the components.... 5. Synthesis: This is the putting together of parts to form a new whole. Synthesis enters problem solving in many ways. A synthesis problem would be one requiring the type, size, and arrangement of equipment necessary to make styrene from ethylben zene. Given a fuzzy situation, synthesis is the ability to formulate (synthesize) a problem statement and/or the ability to propose a method of testing hypotheses. Once the various parts are synthesized, each part (problem) now uses the intellectual skill described in level 4 (analysis) to continue toward the complete solution. Examples of synthesis level question are: Find a way to explain the unexpected results of your experiment. Propose a research program that will elucidate the reaction mechanism. Other words: ... Speculate..., Devise..., Design..., Develop..., What alternative..., Suppose..., Create..., What would it be like..., Imagine..., What might you see.... 6. Evaluation: Once the solution to the problem has been synthesized, the solution must be evaluated. Qualitative and quantitative judgments about the extent to which the materials and methods satisfy the external and internal criteria should be m ade. An example of an evaluation question is: Is the author justified in concluding that the reaction rate is the slowest step in the mechanism. Other Words: ... Was it wrong..., Will it work..., Does it solve the real problem..., Argue both sides..., Which do you like best..., Judge..., Rate.... B. Classification as Closed- or Open-Ended As an application of the strategy outlined in Figure PS1-1, we consider a thumbnail sketch of the design of a chemical plant. Specifically, we want to produce 200 million pounds of styrene per year from ethylbenzene. First we synthesize a sequence of processing operations as shown in the synthesis level (row 1) in Figure PS1-2. In synthesis level we develop the type and arrangement of operations and equipment to produce ethylbenzene. Next we evaluate this sequence to learn if additional operations are necessary, such as a heat-exchange system following the separation sys tem or a feed purification system before the first heat exchanger. Following this file:///H:/html/probsolv/open/blooms/index.htm[05/12/2011 16:59:12]
Thoughts on Problem Solving: Bloom's Taxonomy
cursory evaluation of our sequence, we analyze (row 2) each system (i.e. break it down into a number of subproblems). For example, Figure PS1-2 shows an analysis of t he reaction system. Here, we determine the type of reactor and catalyst to be used, the best temperature at which to carry out the reaction, the type and arrangement of reactor peripherals (e.g., heating/cooling of the reactor), and the optimum feed condi tions. After breaking down the reaction system into a number of subsystems, we proceed to the application skill level (row 3) to decide which laws or principles are to be applied to each subsystem. For example, to calculate the catalyst weight, W, we make use of the design equation (Ch. 6) for a packed-bed reactor,
In using our comprehension (level 2) skills we recall or look up the equation that gives - as a function of concentration, express concentration as a function of conversion X (Ch. 2), carry out the integration, and finally determine the catalyst weight necessary to achieve a conversion X. C. Exercises A. Use a specific example to explain how one could work backward from Bloom’s levels 5 through 1 to solve ill-defined open-ended problems. B. For each of Bloom’s skill levels, construct an example that illustrates the skill used in that level (e.g., for level 1, what assumptions are used to derive the CSTR equation?).
Take a look at our cobra example or move on to Closed-Ended Problems.
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Thoughts on Problem Solving: Open-Ended Problems
First Steps in Solving Open-Ended Problems From Strategies for Creative Problem Solving by H. Scott Fogler and Steven E. LeBlanc, 1995 1.
Write an initial problem statement. Include information on what you are to solve, and consider why you need to solve the problem.
2.
Make sure you are proceeding to solve the real problem as opposed to the perceived problem (chapter 3). Carry out one or more of the following: A. B. C. D. E.
Find out where the problem came from Explore the problem Apply the Duncker Diagram Use the statement-restatement technique Apply Problem Analysis
3. Generate solutions (chapter 4) A.
Understand what conceptual blocks can occur so that you will be aware of them when they surface. 1. 2. 3. 4. 5. 6.
Perceptual Emotional Cultural Environmental Intellectual Expressive
B.
Brainstorm
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Thoughts on Problem Solving: Open-Ended Problems
1. Free association 2. Osborn’s Check List 3. Lateral Thinking a. Random Stimulation b. Other People's Views C.
Analogy 1. State the problem 2. Generate analogies 3. Solve the analogy 4. Transfer the analogy to the solution D.
Organize the ideas/solutions that have been generated. 1. Fishbone Diagram E.
Cross Fertilize 1. Draw analogies from other disciplines
F.
Futuring. Today's constraints (e.g. computing speed, communications) may be limiting the generation of creative solutions. Think to the future when these constraints may no longer exist. Remove all possible constraints from the problem statement and solution criteria. G.
Incubate. Take a break. Let your subconscious work on the problem while you do something else. Sometimes all you need is a breather to achieve that final breakthrough! 4. Choose best alternative from the ideas generated (chapter 5) A.
Decision Making 1. Musts 2. Wants 3. Adverse Consequences
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Thoughts on Problem Solving: Open-Ended Problems
B.
Planning 1. 2. 3. 4.
Potential Problem Consequences Preventative Action Contingent Action
5.
Follow Through (chapter 6) A. Gantt Chart B. Deployment Chart C. Evaluation - Is the problem you are solving still relevant? 6.
Evaluate (chapter 7) A. Does the solution satisfy all the stated and implied criteria? B. Is the solution safe to people and property? C. Is the solution ethical? See an example of the OEP Algorithm in action, as applied to the Cobra Problem. Bloom's Taxonomy can help you classify your problem and determine a method of attack.
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Thoughts on Problem Solving: Open-Ended Problems
Previous Open-Ended Problems 1. Design of Reaction Engineering Experiment The experiment is to be used in the undergraduate laboratory and the cost less than $500 to build. The judging criteria are the same as the criteria for the National AIChE Student Chapter Competition. The design is to be displayed on a poster board and explained to a panel of judges. Guidelines for the poster board display are provided by Jack Fishman and are given on the CD ROM. 2. Pharmacokinetics of Cobra Bites In Thailand alone, snakebites are responsible for the deaths of approximately 2,500 people a year. The interaction of snake venom with newly-developed antivenoms in the human body can be modeled as a chemical reaction engineering catalysis problem. Students use this knowledge to create and solve unique snakebite scenarios. Focus: catalysis, multiple reaction kinetics. 3. Effective Lubricant Design Lubricants used in car engines are formulated by blending a base oil with additives to yield a mixture with the desirable physical attributes. In this problem, students examine the degradation of lubricants by oxidation and design an improved lubricant system. The design should include the lubricant system's physical and chemical characteristics, as well as an explanation as to how it is applied to automobiles. Focus: automotive industry, petroleum industry. 4. Peach Bottom Nuclear Reactor The radioactive effluent stream from a newly-constructed nuclear power plant must be made to conform with Nuclear Regulatory Commission standards. Students use chemical reaction engineering and creative problem solving to propose solutions for the treatment of the reactor effluent. Focus: problem analysis, safety, ethics. 5. Underground Wet Oxidation You work for a specialty chemicals company, which produces large amounts of aqueous waste. Your Chief Executive Officer (CEO) read in a journal about an emerging technology for reducing hazardous waste, and you must evaluate the system and its feasibility. Focus: waste processing, environmental issues, ethics. 6. Hydrodesulfurization Reactor Design Your supervisor at Kleen Petrochemical wishes to use a hydrodesulfurization reaction to produce ethylbenzene from a process waste stream. You have been assigned the task of designing a reactor for the hydrodesulfurization reaction. Focus: reactor design. 7. Continuous Bioprocessing Most commercial bioreactions are carried out in batch reactors. The design of a continuous bioreactor is desired
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Thoughts on Problem Solving: Open-Ended Problems
since it may prove to be more economically rewarding than batch processes. Most desirable is a reactor that can sustain cells that are suspended in the reactor while growth medium is fed in, without allowing the cells to exit the reactor. Focus: mixing modeling, separations, bioprocess kinetics, reactor design. 8. Methanol Synthesis Kinetic models based on experimental data are being used increasingly in the chemical industry for the design of catalytic reactors. However, the modeling process itself can influence the final reactor design and its ultimate prerformance by incorporating different interpretations of experimental design into the basic kinetic models. In this problem, students are asked to develop kinetic modeling methods/approaches and apply them in the development of a model for the production of methanol from experimental data. Focus: kinetic modeling, reactor design. 9. Cajun Seafood Gumbo Most gourmet foods are prepared by batch processes. In this problem, students are challenged to design a continuous process for the production of gourmet-quality Cajun seafood gumbo from an old family recipe. Focus: reactor design.
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Homogeneous Example 1
Type 1 Home Problem Problems with a straight-forward calculation. The following reaction takes place in a CSTR:
Pure A is fed to the reactor under the following conditions:
F Ao = 10 mol/min C Ao = 2 mol/dm 3 X=?
V= 500 dm 3 and k=0.1/min Rate Law: -r A = kCA What is the conversion in the CSTR? Solution to Homogeneous Problem #1 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 1
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Thoughts on Problem Solving: Heterogeneous Example 1
Type 1 Home Problem Straight forward calculation
F A0 = 10 mol/s X=?
C A0 = 1 mol/dm 3 W = 500kg k = 0.1111 dm 3 s -1 kg -1 What is the conversion of the PFR? Solution to Heterogeneous Problem #1 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Heterogeneous Example 1
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Homogeneous Example 2
Type 2 Home Problem Problems that require intermediate calculations or manipulations. The following reaction takes place in a CSTR:
Pure A is fed to the reactor under the following conditions:
F Ao = 10 mol/min C Ao = 2 mol/dm 3 At T=350 K X=0.75
V= ? and k=? Rate Law: -r A = kCA The activation energy: E=20 kcal/mol What is the conversion in a PFR at 325 K with the same volume?
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Homogeneous Example 2
Solution to Homogeneous Problem #2 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 2
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Thoughts on Problem Solving: Heterogeneous Example 2
Type 2 Home Problem Intermediate calculations or manipulations.
F A0 = 10 mol/s X1 = 0.8
C A0 = 0.1 mol/dm 3 W1 = ? T = 325 K ; E = 2500 cal
F A0 = 10 mol/s C A0 = 0.1 mol/dm 3 W 2 = W1 T = 350 K What is the conversion of the CSTR? Solution to Heterogeneous Problem #2 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Heterogeneous Example 2
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X2 = ?
Thoughts on Problem Solving: Heterogeneous Example 2
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Homogeneous Example 3
Type 3 Home Problem Problems that are over-specified. The following irreversible reaction takes place in a CSTR:
It takes place under the following conditions:
F To = 40 mol/min C Ao = 2 mol/dm 3 The feed is 75 mol% A & 25 mol% inerts. X=?
k(400 K) = 0.1/min V = 500 dm 3 and T = 400 K
This reaction follows the first order rate law: -r A = kCA The activation energy (E) is 10 kcal/mol & the Arrhenius constant (A) is 3*104 /min What is the conversion? Solution to Homogeneous Problem #3 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
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Homogeneous Example 3
Homogeneous Example 3
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Thoughts on Problem Solving: Heterogeneous Example 3
Type 3 Home Problem Over-specified problems
F A0 = 10 mol/s X=?
C A0 = 1 mol/dm 3 W = 500 kg k = 0.1111 dm 3 s -1 kg-1
If the diameter of the reactor is 40 cm, the calatylst density is 2.9 g/cm3 and the catalyst is replaced once a month, what is the conversion? Solution to Heterogeneous Problem #3 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Heterogeneous Example 3
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Homogeneous Example 4
Type 4 Home Problem Problems that are under-specified and require the student to consult other information sources. The following reaction takes place in a membrane reactor:
Given: F Ao =10 mol/min X=?
At T = 298 K
F A=?
At P = 15 psia
F B =?
Diffusivity: DBM =5*10 -5 m 2/sec
F C =? k = 1/min V=0.5 m 3 Diameter (DT ) = 0.1m
A pure feed of formaldehyde (A) enters the reactor. The membrane only allows the hydrogen to pass through it. Rate Law:
Using Polymath, what is the conversion of this system? Also plot the flow rates versus the volume of the membrane reactor. Solution to Homogeneous Problem #4
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Homogeneous Example 4
Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 4
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Thoughts on Problem Solving: Heterogeneous Example 4
Type 4 Home Page Under-specified problems
Reactant A is fed to a catalyst-packed membrane reactor at a rate of 8 mol/min with an initial concentration of 0.5 mol/dm3 . What is the final conversion of A into C?
kR = 0.749 Boundary layer thickness = 0.1 cm KC = 2.5 Diffusivity = 0.25 cm3/s T = 298 K Bulk Density = 1.8 kg/dm 3 Solution to Heterogeneous Problem #4 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Heterogeneous Example 4
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Homogeneous Example 5
Type 5 Home Problem "What if …" problems that promote discussion. Recall the membrane reactor from Homogeneous Example 4. What if there were no membrane and the following reaction were carried out in a simple PFR:
Given: F Ao =10 mol/min
F A=?
At T = 298 K
F B =?
At P = 15 psia
F C =?
X=?
k = 1/min V=0.5 m 3 Diameter (DT ) = 0.1m
Rate Law:
Using Polymath, determine the conversion for this system. Also, plot the flow rates versus the volume of the PFR. What PFR volume is required to achieve the same conversion as the membrane reactor (X=0.86)? Do you expect this PFR to be significantly larger than the membrane reactor from Example 4?
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Homogeneous Example 5
Solution to Homogeneous Problem #5 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 5
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Thoughts on Problem Solving: Heterogeneous Example 5
Type 5 Home Problem "What if..." problems
F A0 = 10 mol/s X=?
C A0 = 1 mol/dm 3 W = 500 kg k = 0.1111 dm 3 s -1 kg-1 Bulk density = 1.8 g/cm3 Find the conversion : What if the PFR it was 100dm long and had a total volume of 200dm 3 ? What if it was 4dm wide and 10dm long? What if only 0.5dm wide and 300dm long? What if 4 PFRs 25 dm long were used in parallel instead of one 150dm 3 PFR? Solution to Heterogeneous Problem #5 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Heterogeneous Example 5
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Homogeneous Example 6
Type 6 Home Problem Problems, or parts of problems, that are open-ended. The following gas phase reaction takes place in an industrial PFR:
C is the desired product. D is formed in an undesirable side reaction. Given:
F Ao = 50 mol/min
FA = ?
F Bo = 25 mol/min
FB = ? V = 2 m3
F C = 81,000 mol/hr
T = ? and P = ? Additional Information: k1(300 K) = 0.075/sec
Rate Laws:
E1=15000 cal/mol k2(300 K) = 0.0015/sec E2=17500 cal/mol Find the operating temperature (T) and pressure (P) for the PFR that will generate our desired production of C. What are the factors that affect your decision? Solution to Homogeneous Problem #6
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Homogeneous Example 6
Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 6
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Thoughts on Problem Solving: Heterogeneous Example 6
Type 6 Home Problem Open-ended problems You are working for a company that is building a new plant. The company is still deciding whether to use a catalystfilled membrane reactor or a PFR for a certain process. The reaction is an equilibrium reaction, which makes some favor a membrane reactor; however, the transport rate constant is small and the membrane reactor is more expensive than the PFR. What would your recommendation be? Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Heterogeneous Example 6
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Homogeneous Example 7
Type 7 Home Problem Problems where the student must explore the situation by varying operating conditions or parameters. Let us revisit the membrane reactor (previously addressed in Homogenous Example 4). The following gas phase reaction will take place in our reactor:
Once again, our membrane will allow B to exit, but it will retain A and C. Given:
F Ao = 10 mol/min
X=?
yAo = 1
FA = ?
At T = 400 K
FB = ?
At P = 10 atm
FC = ? V = 100 dm 3 kB = 0.5/min KC = 105 mol/dm 3 k = 0.7 min -1
Rate Law:
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Homogeneous Example 7
Using Polymath, determine the conversion of this system. Then, vary K C, k, and kB as follows:
Is there an optimal set of conditions? Can you explain why those conditions are most effective? Is the membrane reactor a proper system for this reaction? Solution to Homogeneous Problem #7 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 7
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Thoughts on Problem Solving: Heterogeneous Example 7
Type 7 Home Problem Vary conditions or parameters
C To = 0.5 mol/dm 3 F Ao = 8 mol/s kR = 0.7 KC = 2.5 Bulk Density = 1.8 kg/dm
T = 298 K kc = 2.5 cm3/s E = 5000 cal delta H = 2500 cal 3
Explore and describe the effects of varying kc, temperature, and entering flowrate. Solution to Heterogeneous Problem #7 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Heterogeneous Example 7
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Homogeneous Example 8
Type 8 Home Problem Problems that challenge assumptions. In Homogeneous Example 7, we calculated the conversion of a membrane reactor, assuming that there was no pressure drop. In this problem, we will investigate pressure drop in our system, and determine it's effect (if any) on conversion. The following gas phase reaction takes place in a membrane reactor:
Once again, our membrane will allow product B to exit, but it will retain reactant A and product C. Given:
F Ao = 10 mol/min
X=?
yAo = 1
FA = ?
At T = 400 K
FB = ?
At P = 10 atm
FC = ? V = 100 dm 3 D = 3 in kB = 0.5/min KC = 105 mol/dm3 k = 0.7/min
Additional Information: m A = m B = m C = 1.8 Pa*s
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Rate Law:
Homogeneous Example 8
rA = 0.0015 kg/dm 3
rB = 0.001 kg/dm 3
rC = 0.00125 kg/dm 3 The membrane dimensions are similar to those of 16 BWG tubing. Setup the Polymath solution to this problem. What is the exiting pressure of our reactor? What is the conversion of this reactor? Is it safe to assume that the effects of pressure drop are negligible? Solution to Homogeneous Problem #8 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 8
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Thoughts on Problem Solving: Heterogeneous Example 8
Type 8 Home Problem Problems which challenge assumptions
delta P = 0
F A0 , C A0
X=?
F A0 , C A0 , P 0
X=? P=?
F Ao = 0.1 mol/s C Ao = 10 mol/dm 3 k = 0.1111 dm 3 s -1 kg-1 P o = 10 atm What is the conversion in the first PFR? If the particle size is decreased, the pressure drop becomes an important factor. If the pressure drop is modeled by the equation:
and alpha = 0.0011, what is the new conversion? What is the final pressure?
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Thoughts on Problem Solving: Heterogeneous Example 8
Solution to Heterogeneous Problem #8 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Heterogeneous Example 8
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Homogeneous Example 9
Type 9 Home Problem Problems that promote discussion. Worthless Chemical has been making tirene (B) from butalene (A) (both dark liquids) using a 12 ft3 CSTR followed by a 4.5 ft3 PFR. The entering flow rate is 100 mol/min of butalene (A). The feed is heated to 60 ° C before entering the CSTR. A conversion of 80% (Xi) is typically achieved using this arrangement. One morning, the plant manager, Dr. Pakbed, arrived and found that the conversion had dropped to approximately 36% (Xii ). After inspecting the reactors, the PFR was found to be working perfectly, but a dent was found in the CSTR that may have been caused by something like a fork-lift. He also noted that the CSTR, which normally makes a "woosh" sound was not as noisy as it had been the previous day. The manager calls in four junior engineers (Jane, Steve, Bill, and you) to investigate the problem. Given:
F Ao = 100 mol/min
Xi = 0.8
T = 60 °C
Xii = 0.36
A plot of the inverse of the reaction rate versus conversion is presented below:
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Homogeneous Example 9
I. Make a list of all the things that could cause the drop in conversion. Propose tests to confirm the explanations. Quantify the explanations with numerical calculations, where possible. II. Dr. Pakbed tells you that in order to meet the production schedules down stream, he needs a conversion of a least 70%. Can this conversion be obtained without taking time to fix to the CSTR? Solution to Homogeneous Problem #9 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 9
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Thoughts on Problem Solving: Heterogeneous Example 9
Type 9 Home Problem Group discussion
F A0 = 10 mol/s C A0 = 0.1
Xmeasured = 0.5 Xcalculated = 0.7
mol/dm 3 W = 500 kg
Why is the measured conversion different from the calculated conversion? Solution to Heterogeneous Problem #9 Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Heterogeneous Example 9
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Homogeneous Example 10
Type 10 Home Problem Problems that develop life-long learning skills. After reading the journal review by Y. T. Shah et al. [AIChE J., 28, 353 (1982)], design the following bubble column reactor. Sixty percent carbon dioxide (CO2 ) in a flue gas is to be removed by bubbling through a solution of sodium hydroxide (NaOH). The reaction is mass-transfer limited. Calculate the reactor size (length and diameter) necessary to remove 99.9% of the carbon dioxide. The flow rate of the flue gas is 0.05 m 3 /sec and the flow rate of the sodium hydroxide is 0.001 m 3 /sec. The reactor must operate in the bubbly flow regime, so please recommend a type of sparger to use. Given:
Bubble Column Reactor vL = 0.001 m 3/sec
DC = ?
vGo = 0.05 m 3/sec
HC = ?
vG = 0.02 m 3/sec
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Homogeneous Example 10
Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 10
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Thoughts on Problem Solving: Heterogeneous Example 10
Type 10 Home Problem Life-long learning Now that you have developed a little knowledge of heterogeneous reactions, find an article on the web or in a journal that deals with an issue discussed in class, or that covers a topic that interests you. Give a brief synopsis of the article and explain anything new you learned from the article and/or how it applies to what you have learned in class. (Maybe you could keep a journal, too, because you never know when an interesting concept will come in handy later in life.) Other Heterogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Heterogeneous Example 10
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authors.htm
Strategies for Creative Problem Solving by H. Scott Fogler and Steven E. LeBlanc
H. Scott Fogler
Steven E. LeBlanc
College of Engineering University of Michigan Ann Arbor, Michigan 48109-2136
College of Engineering University of Toledo Toledo, Ohio 43606-3390
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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review.htm
Strategies for Creative Problem Solving by H. Scott Fogler and Steven E. LeBlanc
Review of The Book Reveiwed by George E. Dieter In ASEE PRISM, December 1995 We tell prospective engineering students that studying engineering will make them good problem solvers; however, we hardly devote any time in the curriculum to teaching them about problem solving in a generic sense, namely one outside the context of a particular discipline. Strategies for Creative Problem Solving goes a long way toward resolving that oversight. In the words of the authors, "Here is a book to help problem solvers improve their street smarts". In the book, Scott Fogler and Steven LeBlanc suggest a five-step problem-solving heuristic: 1) define the problem; 2) generate solutions; 3) decide the course of action; 4) implement the solution; and 5) evaluate the solution. Separate chapters cover each of these steps and include descriptions of pertinent methodologies and tools to use, as well as short examples. For every methodology, the process to follow is described in detail with attractive sidebar cartoons and illustrations. Chief methodologies and tools include: Problem Solving: Present-state and desired-state techniques Generating Solutions: Blockbusting, Osborne's checklist for brainstorming, and the Fishbone diagram for oganizing brainstorming Deciding the Course of Action: Kepner-Tregoe situation analysis, the Pareto diagram, and the decision matrix method Implementing the Solution: The Gantt Chart and using the elements of experimental design Evaluation: An evaluation checklist This is definitely a how-to book. The authors end each chapter with suggestions for further reading and a set of exercises for homework or class participation. Interactive computer software is available that reinforces the concepts in the text. With funding from the National Science Foundation, the publisher was able to send the software to every U.S. engineering dean. Contact Scott Fogler at the University of Michigan for an extra copy of the software, if you need one. (I did not evaluate the software.) Overall, the book has a practical, customer orientation. While it does not follow a full, total quality management (TQM) problem-solving approach, it has a definite TQM flavor. The importance of working in teams is mentioned in an early chapter, but by not following through on the subject in subsequent chapters, the authors miss an important opportunity to strengthen the text. The authors claim Strategies for Creative Problem Solving is intended for students, new graduate practitioners, or anyone who wants to increase his or her problem-solving skills. However, because of the level of the examples and the rather succinct presentation of most topics, it will most likely find its audience among freshman and sophomore engineering students.
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review.htm
There are few textbooks of this type available, and because of the relatively brief nature of the book, the subject material can be introduced into a variety of the freshman-orientation courses that abound in our engineering colleges. Of course, for students to benefit from this approach, faculty membes will need to reinforce problem-solving skills in subsequent engineering science and design courses. This book represents a coherent, well-written, nicley illustrated presentation of technical problem solving that should find ready acceptance in many places within engineering education. George E. Dieter is Professor Emeritus of Mechanical Engineering at the University of Maryland and past president of ASEE.
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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publish.htm
Strategies for Creative Problem Solving by H. Scott Fogler and Steven E. LeBlanc
Strategies for Creative Problem Solving was published by Prentice Hall PTR, Englewood Cliffs, New Jersey 07632. Copyright 1995. The publisher offers discounts on this book when ordered in bulk quantities. For more information, contact: Corporate Sales Department PTR Prentice Hall 113 Sylvan Avenue Englewood Cliffs, NJ 07632 Phone: 201-592-2863 FAX: 201-592-2249 To obtain Interactive Computer Modules based on Strategies for Creative Problem Solving, contact the CACHE Corporation at: CACHE Corporation P.O. Box 7939 Austin, TX 78713-7379 FAX: 512-471-7060 E-mail: [email protected]
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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CRE - Integrating Factor
The Integrating Factor - What? Why? How? Problem - You want to integrate (1) However, that f(z)y term really messes things up! If you only had an expression of the form (2) things would be much easier, then you could integrate with respect to z and find y(z). How can you combine f(z) and y to get this simplification? First note that is of the form of the derivative of a product, so examine first the product y u, where u is some function of f(z) you still have to define. Recall (3) That's looking close to the left hand side of equation (1), but there is a "u" in front of the dy/dz term, and a du/dz expression where f(z) is. If you had a form of u such that du/dz = u f(z), then you could manipulate equation (3): (4) where the term in brackets is the left hand side of equation (1). You need du/dz = u f(z). Recall (5) If you define
and f(z) = dq/dz (i.e.
, then (6)
This satisfies the condition that du/dz=u f(z). That's the ezpression you needed! Therefore, substituting into equation (4),
, and
(7) where the term in brackets is the left hand side of equation (1). CONCLUSION: If your problem is of the form (1) you can multiply both sides of the equation by the
(which you should be able to evaluate, since you know f(z)), to yield (9) or, substituting from equation (7)
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CRE - Integrating Factor
(10) so that (11)
(12) EXAMPLE
f(t)=k 2, so
From equation (12), the solution is then
The constant can be obtained from the intitial condition that at t=0, CB=0;
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UM Course Syllabus, Exam I Crib Sheet
ChE 344: Exam I Crib Sheet
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Problem 2-A Solution
Learning Resource CDP2-A Solution a. Over what range of conversions are the plug-flow reactor and CSTR volumes identical? So that it is easier to visualize the solution, we first plot the inverse of the reaction rate versus conversion. This type of plot is often called a "Levenspiel Plot.":
Recalling the mole balance equations for a CSTR and a PFR:
Until the conversion (X) reaches 0.5, the reaction rate is independent of conversion and the reactor volumes will be identical. i.e.
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Problem 2-A Solution
b. What conversion will be achieved in a CSTR that has a volume of 90 L? For now, we will assume that conversion (X) will be less that 0.5. We start with the CSTR mole balance:
Our calculated conversion is extremely small. c. What plug-flow reactor volume is necessary to achieve 70% conversion? This problem will be divided into two parts, as seen below:
1. The PFR volume required in reaching X=0.5 (reaction rate is independent of conversion). file:///H:/html/02chap/html/ahs02-a.htm[05/12/2011 16:59:42]
Problem 2-A Solution
2. The PFR volume required to go from X=0.5 to X=0.7 (reaction rate depends on conversion).
Finally, we add V2 to V1 and get: Vtot = V1 + V2 = 2.3*1011 m3 d. What CSTR reactor volume is required if effluent from the plug-flow reactor in part (c) is fed to a CSTR to raise the conversion to 90%?
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Problem 2-A Solution
We notice that the new inverse of the reaction rate (1/-rA) is 7*108. We insert this new value into our CSTR mole balance equation:
e. If the reaction is carried out in a constant-pressure batch reactor in which pure A is fed to the reactor, what length of time is necessary to achieve 40% conversion? We will begin with the mole balance on a batch system. Since there is no flow into or out of the system, it can be written as:
From the stoichiometry of the reaction we know that V = Vo(1+eX) and e is 1. We insert file:///H:/html/02chap/html/ahs02-a.htm[05/12/2011 16:59:42]
Problem 2-A Solution
this into our mole balance equation and solve for time (t):
After integration, we have:
Inserting the values for our variables:
That is 640 years. f. Plot the rate of reaction and conversion as a function of PFR volume. The following graph plots the reaction rate (-rA) versus the PFR volume:
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Problem 2-A Solution
Below is a plot of conversion versus the PFR volume. Notice how the relation is linear until the conversion exceeds 50%.
The volume required for 99% conversion exceeds 4*1011 m 3.
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Problem 2-A Solution
g. Critique the answers to this problem. The rate of reaction for this problem is extremely small, and the flow rate is quite large. To obtain the desired conversion, it would require a reactor of geological proportions (a CSTR or PFR approximately the size of the Los Angeles Basin), or as we saw in the case of the batch reactor, a very long time.
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Lectures 3 and 4 - Self Test
Lectures 3 and 4 Self Test What is the reaction rate law for the reaction
if the reaction is elementary? What is r B? Solution
Solution -r A = kA CA CB1/2 r B/(1/2) = r A /1 file:///H:/html/course/lectures/three/test1.htm[05/12/2011 16:59:43]
Lectures 3 and 4 - Self Test
r B = r A /2 r B = -(k A /2)C A CB1/2 Back to Lectures 3 and 4
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Lectures 3 and 4 -- Click Back #1
Deriving -rA: The forward rate is:
And the reverse rate law is:
The net rate for species A is the sum of the forward and reverse rate laws:
Substituting for rfor and rrev :
Return to Lectures 3 and 4
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Lectures 3 and 4 -- Click Back #2
Solving for KP : Remember, the expression we're trying to derive is that:
At equilibrium, rnet
0, so:
Solving for KP :
The conditions are satisfied.
Return to Lectures 3 and 4
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Lectures 3 and 4 -- Click Back #3
For Gas Phase Flow Systems: From the compressibility factor equation of state:
The total molar flowrate is:
Substituting for F T gives:
Return to Lectures 3 and 4
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Lectures 3 and 4 -- Click Back #4
Deriving CA and CB: Remember that the reaction is:
For a gas phase system:
If the conditions are isothermal (T = T0) and isobaric (P = P 0):
And if the feed is equal molar, then:
This leaves us with C A as a function of conversion alone:
Similarly for C B :
Return to Lectures 3 and 4
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Lectures 5 and 6 -- Click Back #1
The Equilibrium Constant (KC) and Equilibrium Conversion (Xe) for a Constant Volume System: You are given the reversible reaction:
which takes place in a constant volume batch reactor. The equilibrium constant, KC , for this reaction is:
where C Ae and C Be are:
Substituting for C Ae and C Be gives us:
Substituting known values (CA0 = 0.2 mol/dm 3 and KC = 100 dm 3/mol):
Solving for the equilibrium conversion, Xe, yields: Xe = 0.83
Return to Lectures 5 and 6
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Lectures 5 and 6 -- Click Back #1
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Lectures 5 and 6 -- click Back #2
The Equilibrium Constant (KC) and Equilibrium Conversion (Xe) for a Non-Constant Volume System: You are given the reversible reaction:
which takes place in gas phase PFR. Since gas phase reactions almost always involve volume changes, we will have to account for volume changes in our calculations. The equilibrium constant, KC , for this reaction is:
where C Ae and C Be are:
Substituting for C Ae and C Be gives us:
Substituting known values (CA0 = 0.2 mol/dm 3 and KC = 100 dm 3/mol), and realizing that:
we end up with:
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Lectures 5 and 6 -- click Back #2
Solving for the equilibrium conversion, Xe, yields: Xe = 0.89
Return to Lectures 5 and 6
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Lectures 5 and 6 -- Tip #1
Tip for Using Polymath to Determine Xe in a PFR Volume is the independent variable in the PFR design equation, so you will want to vary volume until you achieve the desired conversion. The easiest way to do this is to choose a large volume (in this case, a final volume of 500 dm 3 might be a good place to start), calculate your conversion as a function of volume, and then plot conversion versus volume. Remember, the point you're looking for in the example is the volume where conversion equals 80 percent of the equilibrium conversion, or a conversion of 71.1 percent (i.e., X =0.711). Return to Lectures 5 and 6
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Lectures 5 and 6 -- Polymath Screen Shots -- Equations
Polymath Screen Shots -- Equations
Return to Lectures 5 and 6
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Lectures 5 and 6 -- Polymath Screen Shots -- X vs. V
Polymath Screen Shots -- Conversion versus Volume
Return to Lectures 5 and 6
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Lectures 5 and 6 -- Polymath Screen Shots -- Table of V and X
Polymath Screen Shots -- Table of Volume and Conversion
Return to Lectures 5 and 6
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Problem 3-A Solution
Learning Resource Additional Homework Problems CDP3-AB For this problem, the behavior of the beetle can be modeled by the Arrhenius equation. Essentially, the beetle's speed (k) increases exponentially with increases in temperature. We begin with the Arrhenius equation:
In order to calculate how fast the beetle can push the ball, we will need to determine the activation energy (E) and the Arrhenius coefficient (A) of the beetle. We will arrange our data as such: Rate Constant (cm/s)
Temperature (K)
k1 = 6.5
T1 = 300 K
k2 = 13
T2 = 310 K
k3 = 18
T3 = 313 K
k4 = ?
T4 = 314.5 K
To solve the problem graphically and get an approximate answer, we may plot ln(k) vs. 1/T. This plot should form a straight line and will predict the ln(k) for T=314.5 K.
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Problem 3-A Solution
As we can see here, ln(k) for T=314.5 is equal to ~2.95. This corresponds to a k4 value of 19.1 cm/s. Now we will solve the problem numerically to get an exact answer. By dividing k1 by k2 we can eliminate A and solve for E:
Then inserting the values for our variables (k1, k2, T1, T2), we get
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Problem 3-A Solution
Inserting this activation energy into our Arrhenius equation for k3, we can solve for our coefficient (A):
Now that we have solved for our Arrhenius constants, we can calculate the rate (k4) at which the beetle pushes the ball at 314.5 K:
Back to CD Problem 3-AB
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Problem 3-A Solution Legal Statement
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Problem 3-B Solution
Learning Resource Additional Homework Problems CDP3-B B The first step in setting up our stoichiometric table is to determine the stoichiometric coefficients (a, b, c, d, & e).
After making sure the elements are balanced on both sides of the above equation, the complete equation looks like:
We will assume that our reactor acts like a PFR in setting up our stoichiometric table. The stoichiometric table consists of the species name (I), species symbol (II), the flow rate in (III), the change in flow rate (IV), and the flow rate out (V). We will demonstrate how the table is set up with an example on the trichlorosilane (SiHCl 3). I. Species Our first species is trichlorosilane (SiHCl 3). II. Symbol We will give SiHCl 3 the symbol A. III. Flow Rate In The flow of A into the PFR is F Ao . IV. Change in Flow Rate The change in flow of A will be -F Ao X. Where X is conversion. V. Flow Rate Out The flow of A out of the system is F Ao (1-X).
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Problem 3-B Solution
The complete stoichiometric table is given below: Species
Symbol Flow Rate In
Change in Flow Rate
Flow Rate Out
SiHCl 3
A
F Ao
-F Ao X
F Ao (1-X)
H2
B
F Ao
-F Ao X
F Ao (1-X)
Si
C
0
0.5FAo X
0.5FAo X
HCl
D
0
2F Ao X
2F Ao X
Si 1H2Cl 2
E
0
0.5FAo X
0.5FAo X
The second part of the problem is to sketch the concentration of each species as a function of conversion. If the PFR operates at constant temperature (T) and pressure (P) then we know the total concentration (CT ) by the following relation:
Under constant temperature and pressure, the concentration of each species (Ci ) is:
Knowing that C To = C T . The total molar flow rate (F T ) is:
Where d is:
We do not include component C in this calculation because it is a solid.
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Problem 3-B Solution
Combining the above equations, the concentration of each species can be written as:
The resulting sketch looks like
Back to CD Problem 3-B B
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Problem 3-B Solution
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Lectures 7 and 8 -- Click Back
Gas Phase Reaction:
Return to Lectures 7 and 8
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Lectures 7 and 8 -- Click Back #2
Combine:
Return to Lectures 7 and 8
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Lectures 7 and 8 -- Click Back #1
Variable Density: from conservation of mass
Return to Lectures 7 and 8
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Lectures 7 and 8 -- click Back #2
Converting from Length (z) to Catalyst Weight (W): Starting with this equation: (1) we begin by multiplying the top and bottom of the left-hand side of equation (1) by the same combination of constants (which is equivalent to multiplying equation (1) by one):
Since the product of is constant, we can take it inside the derivative in the denominator to combine it with our length (z):
We'll convert from reactor length (z) as our dimension to catalyst weight (W) by making use of the equation for catalyst weight:
where:
Then:
With a little rearranging:
which we substitute into equation (1) to get:
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Lectures 7 and 8 -- click Back #2
Return to Lectures 7 and 8
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Lectures 7 and 8 -- Click Back #3
Total Molar Flowrate:
Return to Lectures 7 and 8
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Lectures 7 and 8 -- Click Back
Solve: Combine
Separate Integrate with Limits X = 0 when W = 0
Return to Lectures 7 and 8
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Lectures 7 and 8 -- Click Back
Turbulent Flow
Return to Lectures 7 and 8
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Lectures 7 and 8 -- Click Back
Turbulent Flow
Taking the ratio
For constant
Return to Lectures 7 and 8
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ChE 344 Reactor Photos
Click on any of the items below to see scanned photos and schematics of reactors used in the chemical processing industy. Reactor system in use Amoco. Spherical Ultraforming Unit in use Amoco. 3 Spherical Ultraforming Units in use Amoco. Spherical Ultraforming Unit in series in use Amoco. Hydrotreating Unit in use at Amoco. Cutaway view of a CSTR (Courtesy of Pfaudler Inc.) Batch reactor (Courtesy of Pfaudler Inc.) Stirring apparatus for a batch reactor (Courtesy of Pfaudler Inc.) To see pictures of SASOL Reactors To see pictures of BP Reactors CRE Reactor Photos
Reactor System Used at Amoco
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ChE 344 Reactor Photos
Spherical Reactor
Three Spherical Reactors
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ChE 344 Reactor Photos
Spherical Reactors Connected in Series
Hydrotreating Unit
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ChE 344 Reactor Photos
Cutaway View of CSTR
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ChE 344 Reactor Photos
Batch Reactor
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ChE 344 Reactor Photos
Batch Reactor Stirring Apparatus
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ChE 344 Reactor Photos
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Lectures 9 and 10 -- Click Back #1
Constant Density:
Return to Lectures 9 and 10
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Des Plaines River Wetlands Project
DPRWP The Des Plaines River Wetlands Project (DPRWP) The Des Plaines River is located in southern Wisconsin and northeast Illinois (see the image, below). It runs alongside the city of Chicago and finally empties into the Illinois and Kankakee rivers.
The Des Plaines River Wetlands Project consists of several interconnected ponds, as shown in the picture (below). Water is pumped from the river so that it enters the ponds to the left side of pond EW3 and at the top of ponds EW4, EW5, and EW6. The discharge of the wetlands is controlled by weirs. Ponds EW3, EW4, and EW5 all discharge to a common outlet before reentering the river, while the discharge from pond EW6 enters a small lake before reentering the river. (The lake is just out of view in the photo, to the left of pond EW6.)
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Des Plaines River Wetlands Project
The level of the Des Plaines River can rise and fall rather significantly. These photos (below) were taken at roughly the same location on the river. The one on the left was taken in May of 1989 and shows the river under normal conditions. The picture on the right was taken in August of 1988 and shows the river under drought conditions.
The Des Plaines River has been classified as "semipolluted." The mean concentration of its total suspended solids (TSS) was 59 mg/L, which indicates a high level of turbidity. The Des Plaines River Wetlands were able to remove as much as 88% of the sediment from river water passing through them, acting as a highly efficient natural filter for the polluted river water. Then during the summer of 1990, the TSS level rose unexpectedly. The sudden jump was attributed to a large population of carp that had recently moved into the wetlands. As the fish scavenged for food, they stirred up the pond bottoms, causing a resuspension of trapped solids in the water. It may seem cruel, but the carp were intentionally frozen out, in order to return the wetlands to normal.
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Des Plaines River Wetlands Project
The wetlands also remove other pollutants from the river water including 65 to 80 percent of the phosphorus and a goodly amount of the atrazine* . The wetlands are more effective in the summer than in the winter, in part because the concentration of phosphorus is very low in the winter. Much of the water that enters the river in the winter is run-off from frozen land, ice, or snow.
How Do Wetlands Relate to Chemical Engineering? Now that you know what a wetland can be used for, how exactly could you, as a chemical engineer, model one of these systems? Trust me, many hours have been spent trying to answer that very question. Since fluid mixing in a wetland is not very good, we won't be able to model one as a CSTR. We can, however, model wetlands as PFRs. Wetlands | DPRWP | Modeling Polymath | References * Atrazine is a herbicide that is used in the area. At times levels in the river peak above the federal drinking water standard. Atrazine was found to degrade on sediments in the wetlands according to a first-order rate law. Therefore, levels of atrazine in the outflow of the wetlands are lower than what the river inputs.
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Modeling Wetlands as Plug Flow Reactors
Modeling Wetlands as Plug Flow Reactors Here's a cross-sectional view of one of the ponds from the Des Plaines River Wetlands Project:
Image Courtesy of Lisa Ingall
Let us assume that the degradation of toxic chemicals follows irreversible, first-order, homogeneous kinetics. As the waste water flows through the wetlands, the contents react and some of the water evaporates from the surface at a constant rate, Q (where Q has units of kmoles water / hr-m 2 ). We'll also asume that none of the toxic species are lost to the air by evaporation.
Symbol
Description
Value
W
Width
100 m
vo
Entering volumetric flow rate
2 m 3/hr
L
Length
1000 m
C A0
Entering concentration of toxics
10-5 mol/dm 3
D
Average Depth
0.25 m
Molar Density of Water
55.5 kmol H2O /m 3
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Modeling Wetlands as Plug Flow Reactors
Q
Evaporation Rate
1.00x10-3 kmol/hr m 2
k1
Specific reaction rate
16x10-5 hr-1
First, let's derive an equation for the molar flow rate of toxics, FA , as a function of distance through the wetlands.
F A = F A0 (1-X)
(1)
(2)
(3)
-rA = k1C A
(4)
(5)
Combining equations 1, 3, and 5:
(6)
Then combining equations 2, 4, 5, and 6:
(7)
Rearranging:
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Modeling Wetlands as Plug Flow Reactors
(8)
Calculating the definite integrals with the boundary conditions (X=0 at z=0 and X=X at z=z):
(9)
Rearranging and solving the equation for X(z) gives:
(10)
Finally, placing this result back into the original equation for FA0 (Equation 1) leads to the equation that we are looking for:
(11)
Also, if we consider the case with no evaporation or rain (Q=0 and therefore v=v o ), the student can derive the following solution:
(12)
Questions: 1. Now that we have these equations, what would F be at z = 100 m and also z = 1000 m? (Answer) file:///H:/html/web_mod/wetlands/model.htm[05/12/2011 17:00:01]
Modeling Wetlands as Plug Flow Reactors
A
2. What would a plot of conversion and reaction rate as a function of distance look like? (Answer) 3. How could you solve this problem using polymath? Wetlands | DPRWP | Modeling Polymath | References
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Modeling Wetlands Using Polymath
Modeling Wetlands Using Polymath The polymath file required to solve this problem could be written as shown below:
Using this as our model, we can generate a lot of data, including plots of: molar flowrate (F A ) versus length (z) conversion (X) versus length (z) reaction rate ( r A ) versus length (z) Wetlands | DPRWP | Modeling Polymath | References
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Wetlands, References
References References Kadlec, Robert H. & Knight, Robert L., Treatment Wetlands. Boca Raton: CRC Press, Inc., 1996. [ All of the pictures in the Wetlands web module were taken by Professor Robert Kadlec or one of his graduate students. They were taken at the Des Plaines River Wetlands, and Professor Kadlec has given us permission to use them. ] Fogler, H. Scott, Elements of Chemical Reaction Engineering, 3rd Ed. Upper Saddle River, NJ: Prentice-Hall, 1998.
Problem History This page was created by Scott J. Conaway during Winter Semeter 1997 as a special project for Professor H. Scott Fogler. These pages were further edited by Brad Lintner for use on the Chemical Reaction Engineering Web Site during Fall Semester 1997, and again during Summer 1998 by Dieter Andrew Schweiss for the CD-ROM to accompany the 3rd Edition of Elements of Chemical Reaction Engineering. Wetlands | DPRWP | Modeling Polymath | References
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Introduction
Introduction What is a membrane reactor? A membrane reactor is really just a plug-flow reactor that contains an additional cylinder of some porous material within it, kind of like the tube within the shell of a shell-and-tube heat exchanger. This porous inner cylinder is the membrane that gives the membrane reactor its name.
The membrane is a barrier that only allows certain components to pass through it. The selectivity of the membrane is controlled by its pore diameter, which can be on the order of Angstroms, for microporous layers, or on the order of microns for macroporous layers.
Why use a membrane reactor? Membrane reactors combine reaction with separation to increase conversion. One of the products of a given reaction is removed from the reactor through the membrane, forcing the equilibrium of the reaction "to the right" (according to Le Chatelier's Principle), so that more of that product is produced. Membrane reactors are commonly used in dehydrogenation reactions (e.g., dehydrogenation of ethane), where only one of the products (molecular hydrogen) is small enough to pass through the membrane. This raises the conversion for the reaction, making the process more economical.
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Introduction
What kinds of membrane reactors are available? Membrane reactors are most commonly used when a reaction involves some form of catalyst, and there are two main types of these membrane reactors: the inert membrance reactor and the catalytic membrane reactor. The inert membrane reactor allows catalyst pellets to flow with the reactants on the feed side (usually the inside of the membrane). It is known as an IMRCF, which stands for Inert Membrane Reactor with Catalyst on the Feed side. In this kind of membrane reactor, the membrane does not participate in the reaction directly; it simply acts as a barrier to the reactants and some products. A catalytic membrane reactor (CMR) has a membrane that has either been coated with or is made of a material that contains catalyst, which means that the membrane itself participates in the reaction. Some of the reaction products (those that are small enough) pass through the membrane and exit the reactor on the permeate side. Main | Introduction | Algorithm Example | Comparison | Credits Reference: Fogler, H. Scott. Elements of Chemical Reaction Engineering, 3rd Ed. Prentice-Hall: Upper Saddle River, NJ, 1998.
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Algorithm
Algorithm The basic algorithm for solving reaction engineering problems is described below. This algorithm is a useful tool, and it can be applied to a wide variety of reactor problems, not just membrane reactor problems. For demonstration purposes, we'll examine a membrane reactor in which the following gas phase reaction occurs:
Product B diffuses through the membrane, but reactant A and product C do not.
1. Mole Balance: For a differential mole balance on A in the catalytic bed at steady state: IN (by flow) - OUT (by flow) + Generation = Accumulation
Dividing by
and taking the limit as
gives:
Similarly, a differential mole balance on C in the catalytic bed at steady state will give: IN (by flow) - OUT (by flow) + Generation = Accumulation
Dividing by
and taking the limit as
gives:
The steady state, differential mole balance on B looks slightly different, since B is the only species that passes through the membrane: IN (by flow) - OUT (by flow) + Generation - OUT (by diffusion) = Accumulation
where RB is the molar flowrate of B through the membrane per unit volume of the reactor. Dividing by
and taking the limit as
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gives:
Algorithm
2. Rate Law: The rate of disappearance of reactant A follows the rate law:
where k is the specific reaction rate constant, and KC is the equilibrium constant. Products B and C obey the following rate laws:
3. Transport Law: The transport or flux of species B through the membrane follows the transport law:
where km is a mass transport coefficient for the flow of product B through the membrane.
4. Stoichiometry: For gas-phase reactions:
The subscript o indicates initial conditions and v is the volumetric flow rate. The concentrations, in terms of molar flow rates, are:
Substituting for the volumetric flow rate, we get:
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Algorithm
If we make use of the fact that:
we can get our concentrations in terms of the total initial concentration:
Quite often we can make the assumption that the reactor operates isothermally and isobarically:
5. Combine: Substituting the concentration terms into the rate law yields:
where the total molar flow rate is:
and:
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Algorithm
Main | Introduction | Algorithm Example | Comparison | Credits
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Examples
Sample Membrane Reactor Problem A membrane reactor is used for the reaction but not to reactant A or product C.
, where the membrane is permeable to product B,
Additional Information: P0
6.0 atm
T0
373 K
k KC
0.7 min-1
km FA0
0.2 min-1 15 mol/min
FB0 = F C0
0 mol/min
0.05 mol/dm3
Consider the following questions: 1. What volume is required for the base case membrane reactor? Solution
2. What if the membrane transfer coefficient, km , were 0.002 min -1? Compare plots of molar flowrates versus volume and conversion versus volume for this case with your base case. Solution
3. What if the membrane transfer coefficient, km , were 20.0 min -1? Compare plots of molar flowrates versus volume and conversion versus volume for this case with your base case. Solution
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Examples
4. What if the base case flowrate were changed from 15 mol/min to 5 mol/min? How would this affect the behavior of the membrane reactor? Solution
5. What if the base case flowrate were changed from 15 mol/min to 25 mol/min? How would this affect the behavior of the membrane reactor? Solution Main | Introduction | Algorithm Example | Comparison | Credits
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Comparison, Part 1
Comparison Comparing Membrane Reactors with PFRs, Part 1 The obvious difference between membrane reactors and plug flow reactors is the absence of a membrane in PFRs. The only difference between our treatment of membrane reactors and PFRs is in the term R B. Specifically, km , which reflects the ease or difficulty of transport through a membrane, will be zero for a PFR, since no membrane is present in a PFR. Therefore, R B will also be zero for a PFR. The following comparison is based on the conditions given in the membrane reactor example problems.
Break It Down for Me: List of Equations Summary Tables Membrane Reactor PFR Flowrates versus Volume Membrane Reactor PFR Conversion versus Volume Membrane Reactor PFR Comparison, Part 2
List of Equations The same equations are used by both membrane reactors and PFRs (note that km will be set to zero for a PFR).
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Comparison, Part 1
Summary Table -- Membrane Reactor
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Comparison, Part 1
Summary Table -- PFR
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Comparison, Part 1
FA, FB, and FC versus Volume -- Membrane Reactor
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Comparison, Part 1
FA, FB, and FC versus Volume -- PFR
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Comparison, Part 1
Conversion versus Volume -- Membrane Reactor
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Comparison, Part 1
Conversion versus Volume -- PFR
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Comparison, Part 1
Forward to Comparison, Part 2 Main | Introduction | Algorithm Example | Comparison | Credits
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Membrane Reactors -- Credits
Credits This site was originally presented as an Open-Ended Problem in the Winter 1997 Chemical Reaction Engineering Class at the University of Michigan. The students who developed this module were Kim Dillon, Namrita Kumar, Amy Miles, and Lynn Zwica. The module was further expanded and improved by Ellyne Buckingham, Dieter Andrew Schweiss, Anurag Mairal, and H. Scott Fogler for use with the Chemical Reaction Engineering Web Site and CD-ROM. Main | Introduction | Algorithm Example | Comparison | Credits
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review.htm
For this module there are a couple of topics that need to be reviewed or dicussed: 1) Semibatch reactor equations 2) Modifications due to reactive distillation
Mole Balance Shown above is a representation of a typical semibatch reactor. B is fed to A, which is already in the reactor, and there are no outputs (evporation included). To model a semibatch reactor, start with a mole balance on each species :
Starting with species A :
C and D are similar :
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review.htm
Species B is fed to the reactor making its mole balance equation slightly different :
We now have all the equations needed to describe the changes in concentration of the different species in the reactor over time. However, since there is a feed stream, the volume in the reactor will change. So, we need to develop an equation describing how the volume in the reactor changes with time.
Mass Balance Starting with a mass balance and remebering that mass can neither be created nor destroyed :
In terms of the system :
The mass terms in the equation can be replaced by :
giving :
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review.htm
Assuming that the system is at constant density leads to :
That is the last equation needed to model the semibatch reactor.
Summary
Mole Balance
Mass Balance
Rate Law
Now we can move on and talk about reactive distillation.
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intro.htm
This module is a learning resource for Chapter Four of Elements of Chemical Reaction Engineering. Reactive distillation is used with reversible, liquid phase reactions. Suppose a reversible reaction had the following chemical equation :
For many revesible reactions the equilibrium point lies far to the left and little product is formed :
However, if one or more of the products are removed more of the product will be formed because of Le Chatlier's Principle :
Removing one or more of the products is one of the principles behind reactive distillation. The reaction mixture is heated and the product(s) are boiled off. However, caution must be taken that the reactants won't boil off before the products. For example, Reactive Distillation can be used in removing acetic acid from water. Acetic acid is the byproduct of several reactions and is very usefull in its own right. Derivatives of acetic acid are used in foods, pharmaceuticals, explosives, medicinals and solvents. It is also found in many homes in the form of vinegar. However, it is considered a polutant in waste water from a reaction and must be removed.
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You have been given a summer internship by J. Flabetes Inc., a somewhat well known chemical company located in sunny California. Your last project for the summer is to study a proposed method of removing acetic acid from the waste water. The acetic acid is to be removed by reacting it with methanol in a catalyst filled semibatch reactor :
It has been suggested that reactive distillation be included in the plans, but your boss is kind of skeptical. Your job is to simulate the reaction and see if reactive distillation is a better option than without. We will begin by looking at the semibatch reactor case with no reactive distillation.
No Reactive Distillation
Mole Balance
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Mass Balance
Rate Law
We can use and Ordinary Differential Equation solver such as Polymath to model the reaction.
Polymath Click here for equations
Graph From the results we see that after 120 minutes, 116 moles of acetic acid is left in the reactor from the original 300. This is a conversion of 61.3 %
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ref.htm
CRC Handbook of Chemistry and Physics, 73rd Edition. Ed. David R. Lide. CRC Press. Boca Raton, Florida. 1992 Fogler, H. Scott. Elements of Chemical Reaction Engineering, 3rd Edition. Prentice Hall. Upper Saddle River, NJ. 1998 Howard, Phillip H., and William M. Meylon.. Handbook of Physical Properties of Organic Chemicals. CRC Press. Boca Raton, Florida. 1997. pg 595. Perry's Chemical Engineering Handbook, 6th Edition. Ed. Robert H. Perry and Don Green. McGraw-Hill Inc. New York. 1984 Ullmann's Encyclopedia of Industrial Chemistry. Ed. Wolfgang Gerhartz et. al. VCH Publishers. Deerfield Beach, Florida. 1985. Xu, Z. P., and K. T. Chuang. Kinetics of Acetic Acid Esterification over Ion Exchange Catalysts. The Canadian Journal of Chemical Engineering. Vol. 74 Aug:pg493-500. 1996.
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CRE -- Problem 4-A Solution
CD P4-AB First, we will list the known values: AgClO 4 + CH 3I A+B
CH 3ClO 4 + AgI C+D
Batch Process V = 30 dm 3 C Ao = 0.5 mol/dm 3 C Bo = 0.7 mol/dm 3 rB = -kC B C A3/2 at T = 298 K k = 0.00042 (dm3/mol)3/2 /s Final X = 0.98 Mole Balance
Rate Law rA = rB = -kC B C A3/2 Stoichiometry
Combine Combining the above equations, we arrive at:
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CRE -- Problem 4-A Solution
Evaluate The final equation, with all values entered, is:
This equation can only be solved numerically, unless you really enjoy integration. The Polymath simultaneous differential equation solver may be used to solve the problem quickly and easily. The Polymath equations used should look like this:
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CRE -- Problem 4-A Solution
After Polymath solves the equations, it gives a chart of initial, maximum, minimum, and final values:
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CRE -- Problem 4-A Solution
From these results, we can see that the final time (and therefore the time to reach 98% conversion) is t = 162531 s = 45.15 hr Back to problem 4-a
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Thoughts on Problem Solving: Sample Registration Exam Problem
4.Identify and Name: A. Relative Theories and Equations
Differential Form of PFR equation: see derivation of PFR equation B. Systems Volume of PFR C. Dependent and Independent Variables Independent: VPFR, F A0 , F B0, T Dependent: conversion (X) D. Knowns and Unknowns Knowns: VPFR, vA0 , vB0, F A0 , F B0, T, X see values of knowns Unknowns: Specific reaction rate constant, k E. Inputs and Outputs In: F A0 , F B0 Out: F A, F B 5.Assumptions/Approximation Isothermal: no temperature change No volume change on mixing 6. Specifications This problem is overspecified: heat capacity, viscosity and boiling point of the components.
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Thoughts on Problem Solving: Sample Registration Exam Problem
7.Algorithm:
Mole Balance on the PFR: evaluate liquid phase PFR equation Rate Law: Stoichiometry: Equal Molar thetaB = 1 step through stoichiometry to arrive at ...
Combine:
step through combination of equations Evaluate: rearrange to solve for k
evaluate k Forward to Part 2 : CSTR & PFR Back to Original Problem
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Thoughts on Problem Solving: Sample Registration Exam Problem
4. Identify and Name:
A. Relative Theories and Equations Algebraic Form of CSTR equation:
derive CSTR volume For reactors in series, such as a PFR after a CSTR, the integral form of a PFR equation becomes:
derive PFR volume B. Systems Volume of PFR Volume of CSTR C. Dependent and Independent Variables Independent: VCSTR , VPFR , FA0, FB0 , T Dependent: k, X1 , X2 D. Knowns and Unknowns Knowns: same as Part I values of knowns Unknowns: conversion from CSTR, X1 conversion from PFR, X2
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Thoughts on Problem Solving: Sample Registration Exam Problem
E. Inputs and Outputs In: same as Part I Out: same as Part I 5.Assumptions/Approximations same as Part I 6.Algorithm: Same as Part I
Combine: steps in combining Evaluate:
evaluating for X1
evaluating for X2 Check units, X is dimensionless. With a CSTR upstream of the PFR, the conversion increased from 0.5 to 0.684. Back to Part 1 : PFR Back to the Original Problem
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05FTN.htm
Footnotes for Chapter 5, Professional Reference Shelf 1 A.
C. Norris, Computational Chemistry: An Introduction to Numerical Solution (New York: Wiley, 1981). 2 D. M. Himmelblau, Process Analysis by Statistical Methods (New York: Wiley, 1970), p. 195. 3 Norris, Computational Chemistry, p. 293. 4 G. E. P. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building (New York: Wiley, 1978). 5 S. N. Deming, Chemtech, p. 118, February 1990. 6 Charles D. Hendrix, Chemtech, p. 167, March 1979. 7 W. Strunk and E. B. White, The Elements of Style (New York: MacMillian, 1979). 8 V. W. Weekman, AIChE J., 20, 833 (1974). 9 Ibid.
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Lectures 11 and 12 -- Click Back #2
Optimum Yield with Multiple Reactions: To maximize the amount of B produced, we need to differentiate our function for CB. Setting the differential equation equal to zero, we can solve for the time to reach this optimum, t opt.
Solving for t opt gives
Return to Lectures 11 and 12
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Lectures 11 and 12 -- Click Back #1
For Liquid Phase Reactions in a PFR: PFR Design Equation
for liquid phase systems then substituting for F A then
Return to Lectures 11 and 12
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Background Information
Background Information The country of Thailand is half a world away from the United States. It is a land of vast jungles, busy cities, friendly people, and ancient traditions. Unfortunately, it is also the home of some of the most dangerous snakes in the world, and many people die each year in Thailand as a direct result of being bitten by such snakes. (But don't let that stop you from visiting Thailand! It's a beautiful country.) Until recently, the medical world was uncertain about the mechanism by which snake venom attacked the human body. Fortunately, research in this area has progressed rapidly in the past ten years, and medical science has been able to determine what happens when someone is bitten by a poisonous snake. This knowledge has resulted in the development of antivenoms that can save snake bite victims, but only if they are injected in time with an appropriate dosage of the correct antivenom.
A close-up view of a King Cobra (Ophiophagus hannah)! Two of the deadliest snakes in Thailand are the King Cobra (Ophiophagus hannah) and the Siamese Cobra (Naja siamensis). The King Cobra's venom is one of the most lethal and fast-acting neurotoxins (i.e., it attacks the nervous system) found in nature, and a single bite can incapacitate some one's respiratory system within 30 minutes. Death would then swiftly follow, unless an injection of antivenom were immediately available. The bite of the Siamese Cobra, which is primarily neurotoxic, is even deadlier. It can cause paralysis, nausea, and difficulty in breathing. Without treatment, heart and breathing failure results in death 1. 1
This information was graciously provided by Mr. Jaruwan Liwsrisakul.
Human Respiration
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Background Information
Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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Effects of Cobra Venom
Effects of Cobra Venom Cobras have several meathods for delivering their deadly venom to their prey. Some cobras can spit their venom into a victim's eyes, causing extreme pain and blindness. However, the most common and well known method of venom delivery is injection into a victim's body through their bite.
A cobra striking! (Quicktime Movie - 2.2MB) NOTE: You will need the QuickTime Plug-in to view this movie.
Cobras belong to the sub-group of snakes known as elapids; there are over 270 species of cobras and their relatives. An elapid's venom contains postsynaptic neurotoxins that spread rapidly in its victim's bloodstream, causing respiratory failure and, eventually, death. Cobra venom is an example of a molecule that prohibits the interaction of acetylcholine molecules (transmitted from nerve endings surrounding the diaphragm muscle) with the receptor sites on the diaphragm muscle. (See the section on Human Respiration for more details). It binds to the receptor sites, blocking them from interacting with acetylcholine molecules. Even worse, the venom molecule will not immediately break down and vacate the receptor site, effectively removing the site from active duty. It has been determined that even if only a third of the receptor sites on your diaphragm become blocked by venom, you will cease breathing. With cobra venoms, this process can take as little as 30 minutes. The only way to counteract the effects of cobra venom (or most other poisonous snake venoms) is to inject the appropriate antivenom shortly after the bite occurs. If antivenom is unavailable, your life can still be saved by putting you on an artificial respirator until the paralysis of the diaphragm muscle wears off.
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Effects of Cobra Venom
(If this watered-down explanation of effects of cobra venom wasn't enough for you, then check out a more-detailed explanation of the effects of cobra venom.) Antivenom Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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Engineering Aspects
Engineering Aspects of Cobra Venom Reactions The interaction of the venom and the antivenom with the receptor sites can be modeled as a reaction engineering catalysis problem. Examples of the reactions are given below.
Adsorption of venom onto site:
Adsorption of antivenom onto site:
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Engineering Aspects
Reaction of antivenom with venom on site:
Reaction of venom and antivenom in blood:
Removal of product and reactants from system:
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Engineering Aspects
where: V = venom A = antivenom S = unoccupied receptor site VS = site occupied by venom AS = site occupied by antivenom AV = neutralized product from venom/antivenom reaction Developing the Equations Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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Problem Statement
Problem Statement This module on cobra venom and its behavior in the human body was originally given as an open-ended problem, or OEP, during Winter Semester 1994 at the University of Michigan. In the spirit of truely open-ended problems, students were simply asked to investigate the effects of being bitten by a poisonous snake. A base case for the problem and some possible starting suggestions were given to the students, but the emphasis was on creativity in exploring the problem.
Base Case Initially, you should investigate the case where a human is bitten by a poisonous snake, but no antivenom is injected. Plot the fraction of free sites in the body as a function of time. You should be able to verify the time it would take for 1/3 of the receptor sites to be blocked by venom, which would then result in the death of the victim by respiratory failure. Look at the changes that occur when antivenom is injected: How long can you wait to inject the antivenom? How much antivenom should be injected into the victim? Explain the behavior of your graphs.
Other Suggestions After you have investigated the base case, look at several variations of the problem. Some problems that you could investigate are given below; however, this list of problems is not exhaustive. Remember creativity and effort are very important! 1. Can a large dose of antivenom be lethal? What if a large dose of antivenom is injected into someone who has not been bitten by a cobra? Is there a way to reverse the effects of an antivenom overdose? 2. What happens if you are bitten by a cobra twice? 3. What if the antivenom were injected slowly, over a period of 1 to 4 hours? 4. What if you were bitten by a cobra and received proper antivenom treatment, and then, a short time later, you were bitten again? Can antivenom be given? If so, how much should be given? 5. What if the venom from the snake was injected into the muscle instead of directly into the bloodstream? How would the rate of diffusion of venom into the blood affect the time for antivenom treatment? What concentrations of antivenom could be safely administered? 6. Can you find any data or information that provides more accurate rate constants for the reactions? file:///H:/html/web_mod/cobra/problems.htm[05/12/2011 17:00:17]
Problem Statement
Is there a better way to define the system? Investigate the technical aspects of this problem and discuss any changes or improvements. How do the changes you make compare with the information you found when solving the base case? Intimidated yet? Well, don't be. The beauty of an open-ended problem is that the sky's the limit. You can explore any possibility, as long as it's realistically possible to model its behavior. It also helps you develop your creative skills, which are important for real world applications. Take a look at our coverage of the base case solution, and then read our brief look at applying the open-ended problem solving aglorithm to this problem. We encourage you to let your imagination be your guide! Let's apply the Open-Ended Problem Algorithm to this problem! Exploring the Base Case References Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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References
Problem History Pharmacokinetics of Snake Bites was created by Susan Stagg to be used as an Open-Ended Problem (OEP) by Professor H. Scott Fogler at the University of Michigan. It was given as an OEP to the undergraduate kinetics class during Winter Semester 1994. This problem was also featured in Open-ended Problems in Chemical Reaction Engineering, a set of problems that accompanied the 2nd edition of Professor Fogler's text, Elements of Chemical Reaction Engineering. The cobra problem was converted into a web document for the Chemical Reaction Engineering Web Site by Gavin Sy and Dieter Andrew Schweiss, who wish to thank Susan Stagg for all of her hard work in creating the cobra problem in the first place. Gavin created the frames set for the CRE Web Site version of the Cobra Problem, entered all of the original text from Susan Stagg's problem, and converted amateur video into the cobra movies. Dieter rewrote the text of the Cobra Problem to make it more "user friendly" (e.g., the background information now starts from a "layperson's" point of view, but allows advanced readers to go into more detail by reading the original text for the problem); interpreted the original equations and created the page that explains where they come from (including all of the images); applied the Open Ended Problem Solving Algorithm to the Cobra Problem; solved a base case for the open-ended problem presented in this document; expanded on the base case with a few examples; put together the whole solution to the base case (including the Polymath screenshots); got bit by a cobra to prove that it only takes 30 minutes for respiratory paralysis to occur (just kidding!); and generally wrapped up all the loose ends. Dieter also rewrote the HTML coding for the CD-ROM version of the Cobra Problem. (Let's just say that I was busy.)
References Cheung, Johnson, and Taylor. "Kinetics of Interaction of v epsilon-flourescein isothiocyanate-lysine-23-cobra alpha toxin with the Acetylcholine Receptor." Biophysics Journal, February 1984, pp. 447-454. Ferreia and Joaa. "Influence of Chemistry in Immobilization of Cobra Venom Phospholipase A 2 : Implications as to Mechanism." Biochemistry, 32, 8099, [1993]. Fogler, H. Scott. Elements of Chemical Reaction Engineering. 3rd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1998. Gill, Paul G. Pocket Guide to Wilderness Medicine & First-Aid. Ragged Mountain Press [McGraw-Hill], Camden, Maine, 1997. Greene, Harry W. Snakes: The Evolution of Mystery in Nature. University of California Press, Berkeley, 1997. Malasit, P. "Prevention and Mechanism of Early (anaphylactic) Antivenom Reactions in Victim of Snake Bites." British Medical Journal. January 4, 1986, p. 292. Ortiz, Angela. "Implications of a Consensus Recognition Site for Phosphatidylcholine Separate from the Active Site in
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References
Cobra Venom Phospholipases A2 ." Biochemistry, 31, 2887-2896, [1992]. Thwin, M. M. "Kinetics of Envenomization with Russel's Viper Venom and of Antivenom in Mice." Toxicon, 26(4), 373-378, [1988].
Links Here are a few links to cobra-related sites that you might try. (NOTE: CD-ROM users will need to be connected to the internet for these links to work correctly.) A National Geographic site on King Cobras. A Federal Food and Drug Administration Page on Treating Venomous Snake Bites. The Cobra Information Site. Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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Lectures 25 and 26 -- Click Back #1
Pseudo Steady State Hypothesis (PSSH): Given that for reaction (3): Assume that reactions (1) and (2) are elementary reactions, such that:
The net reaction rate for
is the sum of the individual reaction rates for
The PSSH assumes that the net rate of
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is zero:
:
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Michaelis-Menten Kinetics:
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derive1
Calculation of number average molecular weight Basis: 100 grams total
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p = fraction of carboxyl groups reacted
Multiplying by M o and rearranging
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click36g
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click36h
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click36i
Overall balance on all polymers (j=1 to ¥ )
(1)
(2)
(3)
(4)
Let's look at the first term (1)
Let's look at the second term (2 )
Let's look at the third term (3)
Now consider all the terms in brackets
then
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click36j
Mole balance on P1
Integrating
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click36k
Mole balance on P2
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Lectures 38 and 39 -- Click Back
Example: Reasons for "f" less than 1.0 1. Chain transfer to initiator
However, even though a radical is generated, this reaction does result in a wastage of initiator, there is no increase in the number of radicals or the amount of monomer being converted to polymer. 2. Side reactions Cage effects
[ ]
Most significant reaction in the cage is
[ ] = cage where radicals are held for some time before they diffuse out. Once outside the cage the radicals can react with monomer [2 f COO•] 2 f COO• Other reactions
of secondary importance
Average lifetime of neighboring radicals is 10–10 to 10–9 seconds with kf = 107 dm 3 /mol/s In the cage: The concentration of radicals in the solvent cage ~ 10 mol/dm3 Out of cage: CMonomer = 0.1 to 10 mol/dm3 CR1 = 10–9 to 10–7 mol/dm3 3. Recombination of primary radicals has no effect on initiator efficiency
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Derive: ka are defined wrt reactant.
The net rate of termination of all radicals is
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07ftn.htm
FOOTNOTES FOR CHAPTER 7 1
H. Scott Fogler, Elements of Chemical Kinetics and Reactor Calculations, Prentice-Hall, New Jersey, 1974. 2
R. G. Carbonell and M. D. Kostin, AIChE J., 18, 1 (1972).
3
H. Theorell et al., Acta Chem. Scand., 9, 1148 (1955).
4 K.
Dalziel, Acta Chem. Scand., 11, 1706 (1957).
5
N. K. Gupta and W. G. Robinson, Biochim. Biophys. Acta, 118, 431 (1966).
6
P. H. Calderbank and M. B. Moo-Young, Trans. Inst. Chem. Eng., 37, 26 (1959).
7
N. Mitsuishi and N. Hirai, J. Chem. Eng. Japan, 2, 217 (1969).
8
Y. Oyama and K. Endoh, Kagaku Koyaku, 19, 2 (1955).
9
P. H. Calderbank and M. B. Moo-Young, Trans. Inst. Chem. Eng., 37, 26 (1959).
10
N. Mitsuishi and N. Hirai, J. Chem. Eng. Japan, 2, 217 (1969).
11D.
W. Hubbard, L. R. Harris, and M. K. Wierenga, Chem. Eng. Prog., 84 (8), p. 55 (1988).
12
S. P. Rogovin, V. E. Sohns, and E. L. Griffin, Ind. Eng. Chem., 53, 37 (1961).
13
D. C. Wang et al., Fermentation and Enzyme Technology, Wiley, New York, 1979.
14
T. J. Bailey and D. Ollis, Biochemical Engineering, 2nd ed., McGraw-Hill, New York, 1987.
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Lectures 15 and 16 -- Click Back #1
Manipulating the Energy Exchange Term
Combining:
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Using the Taylor Series Approximation
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Lectures 15 and 16 -- Click Back
Derive the Steady State Energy Balance (w/o work)
Differentiating with respect to W:
Mole Balance on species i:
Enthalpy for species i:
Differentiating with respect to W:
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Lectures 17 and 18 -- Click Back
Derive
Factor F A0 C P0 and then divide by F A0
For a CSTR: F A0X = -rA V
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Balance on a system volume that is well-mixed:
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For Liquid Phase Flow Through A CSTR: Applying the General Mole Balance Equation to a CSTR yields:
Since
we can substitute for F A0 , F A, and NA to get:
The reactor volume will be constant, so we can divide through by V to get:
And for liquid phase systems
, so:
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Learning Resources
Example CD8-2 Solution, Part A Second Order Reaction Carried Out Adiabatically in a CSTR
(a) We will solve part (a) by using the nonisothermal reactor design algorithm discussed in Chapter 8. 1. CSTR Design Equation:
2. Rate Law:
3. Stoichiometry: liquid,
4. Combine:
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5. Determine T: From the adiabatic energy balance (as applied to CSTRs):
which reduces to:
Substituting for known values and solving for T:
6. Solve for the Rate Constant (k) at T = 380 K:
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7. Calculate the CSTR Reactor Volume (V): Recall that:
Substituting for known values and solving for V:
Continue with solution...
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08ftn.htm
FOOTNOTES FOR CHAPTER 8 1 V.
Balakotaiah and D. Luss, in Chemical Reaction Engineering, Boston ACS Symposium Series 196 (Washington, D.C.: American Chemical Society, 1982), p. 65; M. Golubitsky and B. L. Keyfitz, SIAM J. Math. Anal., 11, 316 (1980); A. Uppal, W. H. Ray, and A. B. Poore, Chem. Eng. Sci., 29, 967 (1974).
2 From
a problem of R. A. Schmitz, Notre Dame University, South Bend, Indiana.
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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FOOTNOTES FOR CHAPTER 9 1 B.
Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods (New York: Wiley, 1969).
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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Lectures 19 and 20 -- Click Back
Deriving an Expression for rs:
A site balance yields:
Grouping the equilibrium constants together into one constant, KP :
And grouping the remaining constants together as a single constant, k:
We get:
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Lectures 19 and 20 -- Click Back #1
Deriving an Expression for the Concentration of Species A on the Surface Beginning with our expression for the rate of adsorption:
If the surface reaction is limiting, then:
Recall that:
and
Then:
Multiply both sides by C t and we're left with:
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Lectures 19 and 20 -- Click Back #1
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Deriving an expression for r'A: Assume that the surface reaction is limiting, then:
where: = fraction of vacant sites = fraction of sites occupied by species i
Similarly:
where:
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Surface reaction limiting:
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Lectures 21 and 22 -- Click Back
For the homogeneous reaction:
Let:
then
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10ftn.htm
Footnotes for Chapter 10 1 R.
F. H. Ward, Proc. R. Soc. London, A133, 506 (1931)
2 Named
after Irving Langmuir (1881-1957), who first proposed it. He received the Nobel prize in 1932 for his discoveries in surface chemistry.
3 B.
Chapman, Glow Discharge, Wiley, New York, 1980.
4 A.
Demos and H.S. Fogler, AIChE J., 41(3), 658 (1995)
5 W.
E. Kline and H. S. Fogler, Chem. Eng. Sci., 36, 871 (1981).
6 Ibid.
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Lectures 27 and 28 -- Click Back #1
Deriving the Dimensionless Equation: We begin with this equation:
Let's define our dimensionless variables as:
Before we can make any substitutions, we have to do a little setup:
Now we can make our substitutions:
and we can collect our constants into one term:
Our dimensionless equation is then:
where:
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11ftn.htm
Footnotes for Chapter 11, Professional Reference Shelf
1 N.
Arashi, Y. Hishinuma, K. Narato, F. Nakajima, and H. Kuroda, Int. Chem. Eng., 22(3), 489C (1982).
2
For a discussion of the catalytic muffler, see J. Wei in Chemical Reaction Engineering Reviews, H. M. Hulburt, ed., American Chemical Society, Washington, D.C., 1974, p. 1.
3
C. N. Satterfield and D. H. Cortez, Ind. Eng. Chem. Fund. , 9, 613 (1970); D. Roberts and G. R. Gillespie, in Chemical Reaction Engineering II, H. M. Hulburt, ed., Adv. Chem., 133, 600 (1974). 4
T. Shimizu, O. Imai, Y. Sakakibara, and T. Ohrui, Int. Chem. Eng., 22, 329. (1982)
© 1999 Prentice-Hall PTR Prentice Hall, Inc. ISBN 0-13-531708-8 Legal Statement
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12ftn.htm
Foonotes for Chapter 12, Professional Reference Shelf 1A
number of worked example problems for three-phase reactors can be found in an article by P. A. Ramachandran and R. V. Chaudhari, Chem. Eng., 87(24), 74 (1980).
2 C.
N. Satterfield, AIChE J., 21, 209 (1975); M. Herskowitz and J. M. Smith, AIChE J., 29, 1 (1983); G. A. Hughmark, Ind. Eng. Chem. Fund., 19, 198 (1980); Y. T. Shah, B. G. Kelkar, S. P. Godbole, and W. D. Deckwer, AIChE J., 28, 353 (1982). 3 See 4 P.
Foonote 2
A. Ramachandran and R. V. Chaudhari, Chem. Eng., 87(24), 74 (1980).
5 M.
Herskowitz and J. M. Smith, op. cit.; F. Turek and R. Lange, Chem. Eng. Sci., 36, 569 (1981).
6 Ramachandran 7 M.
and Chaudhari, op. cit.
O. Tarhan, Catalytic Reactor Design, McGraw-Hill, New York, 1983, p. 189.
8 This
material is based on the article by H. S. Fogler and L. F. Brown, in Reactors, ACS Symposium Series 168, H. S. Fogler, ed. (Washington, D.C.: American Chemical Society, 1981), p. 31, which in turn was based on a set of notes by Fogler and Brown.
9 D.
Kunii and O. Levenspiel, Fluidization Engineering (New York: Wiley, 1968).
10 T.
E. Broadhurst and H. A. Becker, AIChE J. 21, 238 (1975).
11C.Y.
Wen and Y.H. Yu, AIChE J. 21, 610 (1966)
12Kunii
and Levenspiel, Fluidization Engineering.
13Ibid. 14J.F.
Davidson and D. Harrison, Fluidized Particles (New York: Cambridge University Press, 1963).
15Kunii
and Levenspiel, Fluidization Engineering.
16Davidson 17S.
and Harrison, Fluidized Particles.
Mori and C. Y. Wen, AIChE J. 21, 109 (1975).
18
J. Werther, in ACS Symposium Series 72, D. Luss and V. W. Weekman, eds. (Washington, D.C.: American Chemical Society, 1978).
-CH 2 - + H2 O This reaction was discovered in the 1920's and has been used by Sasol for the production of liquid fuels and chemicals from synthesis gas for over forty years. The hydrocarbons are synthesized by a chain growth process, with the length of the chain dependent on the catalyst selectivity and reaction conditions. Two types of Fischer-Tropsch conversion steps have been developed and operated by Sasol. One makes use of the Slurry Phase Reactor to produce waxes and distillate fuels, while the other uses the Advanced Synthol Reactor mainly to produce light olefins and gasoline fractions. Preheated synthesis gas is fed to the bottom of the reactor, where it is distributed into the slurry consisting of liquid wax and catalyst particles. As the gas bubbles upwards through the slurry, it diffuses into the slurry and is converted into more wax by the Fischer-Tropsch reaction. The heat given off by the reaction is removed using cooling coils inside the Slurry Phase Reactor that generate steam. The product wax is separated from the slurry containing the catalyst particles in a proprietary process. The lighter, more volatile fractions leave in a gas stream from the top of the reactor. The gas stream is cooled to recover the lighter cuts and water. The hydrocarbon streams are sent to the product upgrading unit, while the water stream is treated in the water recovery unit. Return to the Sasol Reactor Page
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CRE -- Reactors Module, Page Three
Reactors Module-part 3 The module keeps track of which areas you've already viewed:
But you can always visit an area again. I forgot -- what were the pros and cons of using a CSTR? Let's take a look at the Advantages / Disadvantages section:
Hmmm... It looks like the cost of the reactor will be low, but the conversion on a per volume of reactor basis isn't
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CRE -- Reactors Module, Page Three
very good. Perhaps we should look at another reactor type... Well, that's the nickel tour of the Reactors Module. If you own the third edition of Elements the Chemical Reaction Engineering, you will find the module in the Reactors directory of the CD-ROM that accompanies the text. The Reactors Module covers a wide variety of reactors, so we encourage you to take advantage of this learning resource. Return to page two. Return to the Reactor Photos Page.
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Homogeneous Example 1, CSTR Design Equation Derivation
Type 1 Home Problem- CSTR Design Equation Problems with straight forward calculations. To derive the CSTR design equation, we begin with the general mole balance:
Assuming that the tank is well-mixed and the reaction rate is constant throughout the reactor, the mole balance can be written:
This equation can then be rearranged to find the volume of the CSTR based on the flow rates and the reaction rate:
From the definition of conversion, FA = FAo(1-X) or FAo - FA = FAoX, so the equation can be rewritten:
Back to the Solution
Homogeneous Example 1
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Homogeneous Example 1, CSTR Design Equation Derivation
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Thoughts on Problem Solving: PFR Differential Equation
A tubular reactor consists of a cylindrical pipe and is normally operated at steady state. We consider the flow here to be highly turbulent and the flow field may be modeled by that of plug flow. That is, there is no radial variation in concentration and the reactor is referred to as a plug-flow reactor. General mole balance for PFR:
rearranging gives:
taking the limit as
, we obtain:
multiply both sides by -1, we get:
where by stoichometry F A = F A0 - F A0 X
by substitution
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Thoughts on Problem Solving: PFR Differential Equation
This can be rewritten :
This equation deals with volume where we have been given catalyst weight. To solve this problem we can divide the equation by the catalyst bulk density and arrive at:
When the reaction rate is divided by the catalyst density it becomes the reaction rate with respect to catalyst and is designated with a prime :
So :
Back to previous page Heterogeneous Example 1
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Thoughts on Problem Solving: Heterogeneous Example 1
By definition the concentration of A is the flow rate of A over the volumetric flow rate :
Since we are dealing with a gas the volumeteric flow rate will change down the PFR according to the equation:
The total flow rate, F T , is equal to the intial flow rate plus the change in flow rate as the reaction occurs :
Substituting in and simplifying :
We will assume that there is no pressure drop and that it is isothermal since no information is given to the contrary:
The flow rate of A at any length along the PFR is the entering flow rate minus the amount converted:
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Thoughts on Problem Solving: Heterogeneous Example 1
Substituting both equations into the original gives:
Back to previous page Heterogeneous Example 1
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Thoughts on Problem Solving: Heterogeneous Example 1
Epsilon is defined as :
yAo is the mole fraction of A that enters. Since A is the only component fed to the reactor :
Sigma comes from the stoichiometeric coefficients:
Putting them together :
Back to previous page Heterogeneous Example 1
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Homogeneous Example 2, Arrhenius Equation
Type 2 Home Problem Problems that require intermediate calculations or manipulations. The reaction rate constant, k, is dependent on temperature according to the Arrhenius equation:
Since the PFR is at a different temperature than the CSTR, we need to relate the two rate constants. The easiest way to do this is to divide the second rate constant by the first rate constant:
Rearranging for k2 gives:
Back to the Solution Homogeneous Example 2
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Homogeneous Example 1, Solution Page
Type 1 Home Problem -- Solution Problems with a straight-forward calculation. The design equation for a CSTR is:
Derive Equation Because the reaction is elementary, the combined mole balance and rate law becomes:
From stoichiometry, this equation becomes:
Solving for X, we get the following equation:
then we substitute numerical values for our variables and calculate a conversion of:
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Homogeneous Example 1, Solution Page
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Homogeneous Example 1
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Homogeneous Example 2, PFR Design Equation Derivation
Type 2 Home Problem Problems that require intermediate calculations or manipulations.
Our tubular reactor consists of a cylindrical pipe and operates at steady-state. The turbulent flow we expect may be modeled by that of a plug flow reactor (PFR). A PFR has no radial variation in concentration. We start with the general steady-state mole balance for the PFR:
Rearranging gives:
Taking the limit as
, we obtain:
From the definition of conversion, F A is:
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Homogeneous Example 2, PFR Design Equation Derivation
Then substituting F A into the differential, we get:
Inserting our rate law (-rA=kC Ao (1-X)) and dividing both sides by F Ao , we get our PFR design equation:
Back to the Solution Homogeneous Example 2
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Thoughts on Problem Solving: Heterogeneous Example 1
The differential equation for a PFR with catalyst is :
Derive Equation
Assuming that the reaction is elementary since we are not told otherwise, the reaction rate is defined as :
through stoichiometry, this becomes:
Substituting the reaction rate back into the differential equation gives :
which can be integrated from X=0 when W=0 to X=X when W=W :
to give :
Solve for epsilon
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Thoughts on Problem Solving: Heterogeneous Example 1
Back to previous page Heterogeneous Example 1
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Thoughts on Problem Solving: Heterogeneous Example 2
The reaction rate constant, k, is dependent on temperature according to the Arrhenius equation:
Since the PFR is at a different temperature than the CSTR, we need to relate the two rate constants. The easiest way is by dividing the second rate constant by the first:
Rearranging gives:
Back to previous page Heterogeneous Example 2
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Thoughts on Problem Solving: Heterogeneous Example 2
To find the CSTR design equation we begin with the general mole balance:
Assuming that the tank is well mixed and the reaction rate is constant throughout the reactor the mole balance can be written:
This equation can then be rearranged to find the volume of the CSTR based on the flow rates and the reaction rate:
Since FA is the conversion, X, times the entering flow rate, FAo , the equation can be rewritten:
Back to previous page Heterogeneous Example 2
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Homogeneous Example 3, CSTR Design Equation Derivation
Type 3 Home Problem Problems that are over-specified. To derive the CSTR design equation, we begin with the general mole balance:
Assuming that the tank is well-mixed and the reaction rate is constant throughout the reactor, the mole balance can be written:
This equation can then be rearranged to find the volume of the CSTR based on the flow rates and the reaction rate:
From the definition of conversion, FA = FAo(1-X) or FAo - FA = FAoX, so the equation can be rewritten:
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Homogeneous Example 3
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Homogeneous Example 3, CSTR Design Equation Derivation
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Homogeneous Example 4, PFR Design Equation Derivation
Type 4 Home Problem- PFR Design Equation Derivation Problems that are under-specified. Our tubular reactor consists of a cylindrical pipe and operates at steady-state. The turbulent flow we expect may be modeled by that of a plug flow reactor (PFR). A PFR has no radial variation in concentration. We start with the general steady-state mole balance for the PFR:
rearranging gives:
taking the limit as
, we obtain:
Back to PART 1 of the Solution
Homogeneous Example 4
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Homogeneous Example 4, PFR Design Equation Derivation
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Homogeneous Example 4, Solution Page 2
Type 4 Home Problem Problems that are under-specified and require the student to consult other information sources. PART 2 - Type 4 Solution In Part 2, we must derive the equilibrium constant (KC) and the diffusion rate constant (k B). Often the exact value that is required is either too cumbersome to calculate or unavailable in the literature. Because we are just looking for estimates of KC and kB, several assumptions will be made. We will start with our derivation of K C. We begin with a thermodynamic equation for K C:
From Perry's Chemical Engineering Handbook 1 , we find the following free energies of formation at 298 K for our components:
We then calculate the change in free energy for our reaction:
and insert the value into our KC equation:
This value for KC favors our products. Our next task is to calculate our diffusion rate constant (k B). To calculate it, we will need to introduce three new dimensionless constants--the Reynolds Number (Re), the Schmidt Number (Sc), and the Sherwood Number(Sh):
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Homogeneous Example 4, Solution Page 2
A correlation from Fundamentals of Momentum, Heat and Mass Transfer 2 by Welty, Wicks, and Wilson (or simply WWW), we can relate kB to Re and Sc through the Sherwood number (Sh):
This equation is valid for 2000 < Re < 70000 and 1 < Sc < 2260. First, we will approximate the bulk properties of our fluid by assuming that throughout the reactor, our fluid is half A and half C. From WWW, we found m C=1.785*10 -5 kg /(m*s) (1 cP=0.1 Pa*s or kg/(m*s)), and from Perry's (Table 3311), we found the viscosity of acetone. For the sake of these calculations we will assume that the viscosities of acetone and formaldehyde (A) are similar, m A =0.75*10 -5 kg/(m*s).
The density of the bulk fluid will also be calculated using the 50-50% assumption:
We need to calculate the superficial velocity of our fluid (u bulk ). To do so we will assume the volumetric flow rate (v o ) stays constant throughout the reactor:
Now, we calculate Re:
In order to determine Sc, we need to find the kinematic viscosity of the hydrogen (B). In WWW, we find that uB = 1.096*10-4 sec/m 2 . We then calculate the Schmidt number (Sh):
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Homogeneous Example 4, Solution Page 2
Using the Sherwood correlation, we solve for kB':
Unfortunately, kB' is the rate constant per membrane surface area. We need the diffusion rate constant (k B) per volume of the reactor. This conversion can be performed quite easily. We multiply by the perimeter and divide by the crosssectional area:
The diffusion rate constant per minute: kB = 4.2/min Go to Part 3 of the Solution - Polymath Back to PART 1 of the Solution Homogeneous Example 4 REFERENCES: 1. Perry, R.H. and Green, D.W. (Editors). Perry's Chemical Engineering Handbook, New York: MacGraw-Hill Inc., 1984. 2. Welty, J.R., Wicks, C.E., & Wilson, R.E. Fundamentals of Momentum, Heat and Mass Transfer, New York: Wiley & Sons, 1984.
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Solution for Type 4 Home Problem
Now that we have found the flow rates of the components in the reactor we can turn our attention to the rate law. The reaction rate for A going to B and C is :
Since the reaction is reversible we have to take into account the conversion of B and C into A :
The overall reaction rate becomes the amount of A converted into B and C minus the amount of B and C that become A :
This equation can be simplified to :
where
and
kR = kA1 To find the concentrations of the components, we start with the volumeteric flow rate. Since this reaction involves gases it will not remain constant but can be found using the equation :
Since we aren't told otherwise, we will assume no pressure drop and isothermal conditions :
This can be plugged into the definition of concentration for a species, i :
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Solution for Type 4 Home Problem
which can be simplified to :
Back to Part 1 of solution
On to final part of solution
Heterogeneous Example 4
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Solution for Type 5 Home Problem
To find the needed cross sectional areas and lengths :
Scenario 1 : Length = 100 dm, Volume = 200 dm3 Ac : 2 dm 2 Length : 100dm Conversion : 0.91
Scenario 2 : Diameter = 4 dm, Length = 10 dm Ac : 12.6 dm 2 Length : 10 dm Conversion : 0.81
Scenario 3 : Diameter = 0.5 dm, Length = 300 dm Ac : 0.20 dm 2 Length : 300 dm Conversion : 0.59
Scenario 4 : Length = 25 dm (X4), Total Volume = 150 dm3 Ac : 1.5 dm 2 Length : 25 Conversion : 0.85 Back to part 1 of solution
Back to problem statement
Heterogeneous Example 5
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Solution for Type 5 Home Problem
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Thoughts on Problem Solving: Heterogeneous Example 4
Starting with the catalyst differential PFR equation:
Knowing that:
We can rewrite the differential equation:
The differential equation for component B is similar to A except that it is positive because it is a product the rate law:
It must also be taken into account that B is diffusing through the membrane walls. This will change the equation:
The rate will be governed by the equation :
kc prime is the transport coefficient with respect to catalyst weight. We are not given a value for kc prime, but we are given the diffusivity and the boundary layer thickness. Searching through material it can be found that
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Thoughts on Problem Solving: Heterogeneous Example 4
To find kc prime just divide kc by the catalyst bulk density. Plugging this back into the differential equation for the flow rate of B:
C doesn't diffuse through the membrane so:
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Part 2 of solution
Heterogeneous Example 4
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Thoughts on Problem Solving: Heterogeneous Example 7
Large kc X = 0.998
Graph of Flow Rates vs. Catalyst Weight
Small kc X = 0.91
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Thoughts on Problem Solving: Heterogeneous Example 7
Graph of Flow Rates vs. Catalyst Weight
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Heterogeneous Example 7
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Thoughts on Problem Solving: Heterogeneous Example 7
Large T X = 0.988
Graph of Flow Rates vs. Catalyst Weight
Small T X = 0.036
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Thoughts on Problem Solving: Heterogeneous Example 7
Graph of Flow Rates vs. Catalyst Weight
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Heterogeneous Example 7
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Thoughts on Problem Solving: Heterogeneous Example 7
Large FAo X = 0.91
Graph of Flow Rates vs. Catalyst Weight
Small FAo X = 0.92
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Thoughts on Problem Solving: Heterogeneous Example 7
Graph of Flow Rates vs. Catalyst Weight
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Heterogeneous Example 7
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Homogeneous Example 8, Derivation of Pressure Drop in Pipes
Type 8 Home Problem -- Derivation of Pressure Drop in Pipes Problems that challenge assumptions. Variable definitions:
First, we start with our differential equation for pressure drop along a pipe:
Inserting
, and assuming G is constant throughout the length of the pipe, we rearrange things to get:
Integrating over the limits(
), and assuming f is constant along the pipe, we have:
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Homogeneous Example 8, Derivation of Pressure Drop in Pipes
We expect the pressure drop to be very small, so we will neglect the second term on the right-hand side. We are left with:
Rearranging again, we get:
Back to the Homogeneous Example 8 Solution Homogeneous Example 8
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Homogeneous Example 7, Solution Page
Type 7 Home Problem -- Solution Problems where the student must explore the situation by varying operating conditions or parameters. We will use the Homogeneous Example 4 Solution as the basis for our solution here, so consult that page for a more extensive derivation. Some of the equations and values will need to be revised for the present situation. We have a new differential equation for B:
We need to insert our new rate law:
Finally, we will need new values for the volumetric flow rate (v o ):
Below are the simultaneous differential equations, as entered in Polymath:
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Homogeneous Example 7, Solution Page
The conversion under these conditions is: X = 0.73 When k = 2, kB = 10, and KC = 107 (their largest values), the conversion is greatest:
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Homogeneous Example 7, Solution Page
X = 0.997 These results make sense. As KC increases, the second term in our rate law (the one that represents the reverse reaction) decreases, so more product will be formed and the conversion will increase. As k increases, the reaction rate will increase, again causing more product to form and the conversion to increase. As kB increases (remember that kB represents how quickly product B can pass through the membrane), the concentration of B in the reactor decreases, which drives the reaction towards the products. Once again, the conversion will increase. A plot of conversion versus the product of our constants (KC*kB*k) is shown below. In each case, two of the constants are maintained at their original value, while the third is varied. Under the given conditions, changes in the equilibrium constant (KC) have almost no effect on the conversion, but the conversion varies strongly with changes in our rate constant (k).
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Homogeneous Example 7, Solution Page
The fact that the equilibrium constant (KC) has almost no effect on the conversion means that this reaction is essentially irreversible. This reaction could be carried out in a PFR with the same effectiveness. This is only true for our higher range of KC. The following graph displays the difference in conversion between a PFR and our membrane reactor over a lower range of equilibrium constants:
Although the difference in conversion looks nearly constant, the percent difference in conversion (see the graph below) increases significantly at lower KC values.
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Homogeneous Example 7, Solution Page
From this information, we conclude that the membrane reactor is most effective (in comparison to other reactors) when the reaction does not favor the products. Back to Homogeneous Example 7 Homogeneous Example 7
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Homogeneous Example 9, Solution Page 2
Type 9 Home Problem -- PART II of the Solution Problems that promote discussion. You decide to reverse the order of the reactors. You believe that if the PFR precedes the CSTR, the CSTR can achieve a final conversion of greater than 70%.
Our first step is to find the conversion of the PFR (Xi) through graphical integration. We know that the volume of our PFR is 4.5 ft3 . Using Simpson's Rule [see Appendix A.4, Elements of Chemical Reaction Engineering, (3rd Ed.)], we find that our PFR can obtain a conversion of 21% (Xi=0.21).
We then use trial and error to find the point on the graph (inverse of reaction rate (1/-rA) versus conversion (X)) that provides us with the appropriate CSTR volume. We find that our CSTR can achieve a final conversion of 73%.
The final plot can be seen below:
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Homogeneous Example 9, Solution Page 2
Back to PART I of the Solution Homogeneous Example 9
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Homogeneous Example 10, Variable Definitions
Type 10 Home Problem -- Variable Definitions Problems that develop life-long learning skills.
Symbol (units)
Definition Gas hold-up: fraction of the column volume that is gas Gas hold-up equation coefficient Gravitational acceleration Diameter of bubble column Fluid density
Gas-liquid surface tension
Diffusivity of CO 2 in water Dynamic viscosity Kinematic viscosity (m/r)
Rate constant of mole transfer per interfacial area Bubble-liquid interfacial area Rate of mole transfer necessary to achieve conversion
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Homogeneous Example 10, Variable Definitions
Back to the Homogeneous Example 10 Solution
Homogeneous Example 10
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To download the Polymath Equations to a PC: 1. Click on the equation below that you desire 2. When the browser can't figure out what to do with the file you should be given an option of saving the file. You will want to pick that option. 3. Pick where you want to put the file on your local hard drive
No Reactive Distillation Reactive Distillation - Only Methyl Acetate Evaporates Reactive Distillation - All Species Evaporate
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The critical pressure and temperature needed to find the reduced pressure and temperature can be found in the CRC Handbook of Chemistry and Physics:
Tc = 506.5 K Pc = 4750 kPa These equations can now be plugged into Polymath.
Polymath Click here for equations
Graph In this case only 18 moles of acetic acid is left giving a conversion of 94 %. This is much better than the case with no reactive distillation. In real life the other products will also evaporate. We can develop similar equations for the other species to see how their evporation effects the reaction.
Reactive Distillation - All Species Evaporate
Mole Balance
Mass Balance
Rate Law
Physical properties and vapor pressure equations for acetic acid, methanol, and water
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The Mechanism of Human Respiration in Detail
The Mechanism of Human Respiration in Detail Human respiration is dependent upon the interaction of acetylcholine molecules with acetylcholine esterase receptors on the diaphragm muscle. Each time a person takes a breath, nerve endings that contain "sacks" of acetylcholine are stimulated. Each sack has roughly 1 x 1014 acetylcholine molecules inside. The sacks move toward the end of the nerve and eventually strike the wall of the nerve. The force of the collision causes the sacks to release the acetylcholine molecules into the neuromuscular junction or synapse.
The acetylcholine molecule has a positive nitrogen group which is attracted to the negative charge of an acetylcholine esterase receptor site on the diaphragm.
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The Mechanism of Human Respiration in Detail
The attraction between the molecule and a receptor site causes a bridging to occur, and a channel for impulses from the nerve to the muscle is opened. Each interaction with the receptor site causes the channel to open for approximately 400 microseconds . The opening of the channel allows for the transmission of an electrical impulse that stimulates the contraction of the muscle fiber. Many of these neuromuscular interactions combine to create a uniform muscle response; i.e., a contraction of the diaphragm, which is the driving force behind human respiration. Each breath a human takes is a result of the interaction described above. The acetylcholine molecule contains an ester group which reacts with the alcohol group of the receptor site. This reaction is responsible for the degradation of the acetylcholine molecule.
Once the molecule is broken down, a reaction with water occurs and the receptor site releases the molecules. Once the molecules are released, the impulse channel closes and the receptor site is free to interact with another acetylcholine molecule.
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The Mechanism of Human Respiration in Detail
The molecules that are released from the receptor site are then used by the body to form new acetylcholine molecules that are again stored in the sacks in the nerve ending.
Since the body produces the acetylcholine molecule, the process is cyclic in nature and self-sustaining. The process will continue to occur until something prohibits the interaction with the receptor site and stops the formation of the acetylcholine molecule. Venom Effects Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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"Milking A Cobra" Movie
Milking a cobra for venom. (9M!) Return to learn more about antivenom! Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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Exploring the Base Case, Part 2
Exploring the Base Case, Part 2 Now let's take a look at a plot of the fraction of free sites versus time:
It might be helpful to also take a look at the summary table for the base case from Polymath:
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Exploring the Base Case, Part 2
As we can see, 1/3 of the receptor sites, corresponding to an f S of 0.667, are indeed covered after 1/2 hour (t = 0.5 hours). Next Previous Introduction | Background | Venom Effects Reactions | Problems | References Original Work by Susan Stagg Web Page by Gavin Sy & Dieter Andrew Schweiss
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cb8-2s2.htm
Learning Resources
Example CD8-2 Solution, Clickback #2 Second Order Reaction Carried Out Adiabatically in a CSTR
4. Combine:
Space time is defined as:
After some rearranging:
Substituting:
Return
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Learning Resources
Example CD8-2 Solution, Clickback #3 Second Order Reaction Carried Out Adiabatically in a CSTR
6. Solve the Mole Balance for XMB as a function of T:
Rearranging gives:
Our equation for X has taken the form of a quadratic equation, so we solve for X accordingly:
After some final rearranging we get:
Return
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poly-b1.htm
Learning Resources
Example CD8-2 Solution, Polymath for Part B Second Order Reaction Carried Out Adiabatically in a CSTR
Our plot of XEB and XMB shows that our steady state operating point will be at:
X = 0.87 and T = 387 K
Our corresponding Polymath program looks like this:
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NOTE: Our use of d(T)/d(t)=2 in the above program is merely a way for us to generate a range of temperatures as we plot conversion as a function of temperature.
Return
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sol8-2c.htm
Learning Resources
Example CD8-2 Solution, Part C Second Order Reaction Carried Out Adiabatically in a CSTR
(c) For part (c) we will simply modify the Polymath program we used in part (b), setting our initial temperature to 280 K. All other equations remain unchanged. 1. CSTR Design Equation:
2. Rate Law:
3. Stoichiometry: liquid,
4. Combine:
Given reactor volume (V), you must solve the energy balance and the mole balance simultaneously for conversion (X), since it is a
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sol8-2c.htm
function of temperature (T). 5. Solve the Energy Balance for XEB as a function of T:
6. Solve the Mole Balance for XMB as a function of T:
7. Plot XEB and XMB : You want to plot XEB and XMB on the same graph (as functions of T) to see where they intersect. This will tell you where your steady-state point is. To accomplish this, we will use Polymath (but you could use a spreadsheet). Plot of XEB and XMB versus T We see that our conversion would be about 0.75, at a temperature of 355 K.
Return to problem statement
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Homogeneous Example 1, CSTR Design Equation Derivation
Type 1 Home Problem- CSTR Design Equation Problems with straight forward calculations. To derive the CSTR design equation, we begin with the general mole balance:
Assuming that the tank is well-mixed and the reaction rate is constant throughout the reactor, the mole balance can be written:
This equation can then be rearranged to find the volume of the CSTR based on the flow rates and the reaction rate:
From the definition of conversion, FA = FAo(1-X) or FAo - FA = FAoX, so the equation can be rewritten:
Back to the Solution
Homogeneous Example 1
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Homogeneous Example 1, CSTR Design Equation Derivation
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Homogeneous Example 1
Type 1 Home Problem Problems with a straight-forward calculation. The following reaction takes place in a CSTR:
Pure A is fed to the reactor under the following conditions:
F Ao = 10 mol/min C Ao = 2 mol/dm 3 X=?
V= 500 dm 3 and k=0.1/min Rate Law: -r A = kCA What is the conversion in the CSTR? Solution to Homogeneous Problem #1 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 1
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Homogeneous Example 4, Solution Page 3
Type 4 Home Problem Problems that are under-specified and require the student to consult other information sources. PART 3 - Type 4 Solution - Polymath Here is the list of equations:
This is the Polymath output file:
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Homogeneous Example 4, Solution Page 3
We will use this file to calculate the conversion. The conversion (X) equals the difference between the initial and final flow rates of A divided by the initial flow rate:
Finally, we plot the flow rates of the components along the reactor:
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Homogeneous Example 4, Solution Page 3
Back to PART 2 of the Solution Homogeneous Example 4
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Solution for Type 4 Home Problem
There is only one more thing needed before the problem can be solved : the mass transfer coefficient. Since it is not given in the problem we must look elsewere for an equation that will fit the information given in the problem. On page 535 of "Fundamentals of Momentum, Heat and Mass Transfer"* we find that :
We need a mass tranfer coefficient with respect to catalyst weight. That can be obtainded by dividing k by the catalyst bulk density. All of the equations and knowns can now be plugged into an ODE solver such as POLYMATH and solved :
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Solution for Type 4 Home Problem
Graph of Flow rates vs. Catalyst Weight
Graph of conversion vs. Catalyst Weight
Conversion = 0.91
* "Fundamentals of Momentum, Heat, and Mass Transfer" by James Welty, Charles Wicks, and Robert
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Solution for Type 4 Home Problem
Wilson. Copyright 1984 by John Wiley & Sons, Inc. Back to Part 2 of solution
Back to problem statement
Heterogeneous Example 4
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Solution for Type 5 Home Problem
Thoughts on Problem Solving > Ten Types of Home Problems
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Solution for Type 5 Home Problem
Thoughts on Problem Solving > Ten Types of Home Problems
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Solution for Type 5 Home Problem
Thoughts on Problem Solving > Ten Types of Home Problems
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Solution for Type 5 Home Problem
Thoughts on Problem Solving > Ten Types of Home Problems
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Homogeneous Example 7
Type 7 Home Problem Problems where the student must explore the situation by varying operating conditions or parameters. Let us revisit the membrane reactor (previously addressed in Homogenous Example 4). The following gas phase reaction will take place in our reactor:
Once again, our membrane will allow B to exit, but it will retain A and C. Given:
F Ao = 10 mol/min
X=?
yAo = 1
FA = ?
At T = 400 K
FB = ?
At P = 10 atm
FC = ? V = 100 dm 3 kB = 0.5/min KC = 105 mol/dm 3 k = 0.7 min -1
Rate Law:
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Homogeneous Example 7
Using Polymath, determine the conversion of this system. Then, vary K C, k, and kB as follows:
Is there an optimal set of conditions? Can you explain why those conditions are most effective? Is the membrane reactor a proper system for this reaction? Solution to Homogeneous Problem #7 Other Homogeneous Problems 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Homogeneous Example 7
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d(Nmeoh)/d(t)=rhac*V+Fmeoho d(V)/d(t)=vo d(Nhac)/d(t)=rhac*V d(Nmeac)/d(t)=-rhac*V d(Nh2o)/d(t)=-rhac*V Fmeoho=3 Cmeoh=Nmeoh/V Chac=Nhac/V Cmeac=Nmeac/V Ch2o=Nh2o/V T=350 Cmeoho=5 k=(8.88*10**8)*exp(-7032.1/T) K=5.2*exp((-8000/1.978)*((1/298)-(1/T))) vo=Fmeoho/Cmeoho rhac=-k*((Cmeoh*Chac)-((Cmeac*Ch2o)/K)) t(0)=0 Nmeoh(0)=0 V(0)=150 Nhac(0)=300 Nmeac(0)=0 Nh2o(0)=0 t(f)=120 r local hard drive