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Topics in Chemical Engineering A series edited by R. Hughes, University of Salford, UK

Volume J

HEAT AND MASS TRANSFER IN PACKED BEDS by N. Wakao and S. Kaguci

Volume 2

THREE-PHASE CATALYTIC REACTORS by P. A. Ramachandran and R. V. Chaudhari

Volume 3

DRYING: Principles, Applications and Design by Cz. Strumillo and T. Kudra

Volume 4

T HE ANALYSIS OF CHEMICALLY REACTING SYSTEMS: A Stochastic Approach by L. K. Doraiswamy and B. D. Kulkarni

VolumeS

CONTROL OF LIQUID-LIQUID EXTRACTION COLUMNS by K. Najim

Volume 6

CHEMICAL ENGINEERING DESIGN PROJECT: A Case Study Approach by M. S. Ray and D. W. Johnston

Volume 7

MODELLING, SIMULATION AND OPTIMIZATION OF INDUSTRIAL FIXED BED CATALYTIC REACTORS by S. S. E. H. Elnashaie and S. S. Elshishini

Modelling, Simulation and Optimization of Industrial Fixed Bed Catalytic Reactors S. S. E. H. Elnashaie and S. S. Elshishini King Saud University, R iyadh, Saudi Arabia, and Cairo University, Egypt

This book is part of a series. The publisher wi ll accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

GORDON AND BREACH SCIENCE PUBUSHERS Switzerland • Australia • Belgium • France • Germany • Gt B1111 India • Japan • Malaysia • Netherlands • Russia • Singapon· • \

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1993 by OPA (Amsterdam) B.V. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A.

Contents

Gordon and Breach Science Publishers

Introduction to the Series Preface Notation Acknowledgements for Figures and Tables

Y-Parc Chemin de Ia Sallaz CH-1400 Yverdon, Switzerland

Post Office Box 90 Reading, Berkshire RG 1 8JL Great Britain

Private Bag 8 Camberwell , Victoria 3124 Australia

3-14-9, Okubo Shinjuku-ku, Tokyo 169 Japan

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Christlowger Strasse 11 10405 Berlin Germany

820 Town Center Drive Langhorne, Pennsylvania 19047 United States of America

XV

xvii xix XXV

INTRODUCTION

Library of Congress Cataloging-in-Publication Data Modelling, simulation, and optimization of industrial fixed bed catalytic reactors I S.S.E. H. Elnashaie and S.S. Elsbisbini. p. em. -- (Topics in chemical engineering ; v. 7) Includes bibliographical references and index. ISBN 2-88 124-883-7 I. Chemical reactors. 2. Catalysis. I. Elnashaie, S. S. E. H., 1947- . II. Elshishini, S. S., 1948- . III. Series. TP157.M53 1993 660' .2995--dc20 92-29469 CIP

No J)nrt or th1s book may be reproduced or utilized in any form or by any mcons, clcct&onic 0 1 rncehanicnl, including photocopying and recordi ng, I) I hy uny 111to rmat ion ~ torngc or rctricvnl system, without

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CHAPTER 1 SYSTEMS THEORY AND PRINCIPLES FOR DEVELOPING MATHEMATICAL MODELS OF lNDUSTRIAL FIXED BED CATALYTIC REACfORS 1.1 Systems and Mathemat ical Models 1.1. 1 A Brief Background 1.1.2 Mathematical Model Building: General Concepts 1.1.3 Outline of the Procedure for Model Building 1.2 Basic Principles of Mathematical Modelling for Industrial Fixed Bed Catalytic Reactors

7 8 8 11 12 13

CHAPTER 2 CHEMISORPTION AND CATALYSIS 2.1 Adsorption and Catalysis 2.2 Physical and Chemical Adsorption 2.3 Heats of Adsorption and Desorptjon 2.4 The Kinetics of Adsorption and Desorption 2.5 Activated and Non-activated Adsorption 2.5.1 Activated Adsorption 2.5.2 Non-activated Adsorption 2.6 Equilibrium and Non-equilibrium Adsorption-Desorption 2.6.1 Equilibrium Adsorption-Desorption 2.6.2 Steady State Non-equilibrium Adsorption-Desorption 2.7 Adsorption Isotherms 2.8 The Effect of Surface Coverage 2.8. 1 The Role of the Surface in Heterogeneous Catalysis 2.8.2 Heat of Adsorption and Surface Coverage

31

CHAPTER 3 INTRINSIC KINETICS OF GAS-SOLID CATALYTIC REACTIONS 3. 1 Kinetic Models for Gas-Solid Cat alytit: Reactions

35 38

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3.1.1 Power Law (PL) Kinetic Models 38 3.1.2 Chemisorption-Surface Reaction-Desorption (CSD) Kinetic Models for Unimolecular Reactions 39 3.1.2. 1 The equilibrium adsorption-desorption case with negligible product inhibition 40 3.1.2.2 Effect of product inhibition 42 3. 1.2.3 The steady state assumption (SSA) case with negligible product inhibition 42 3.1.3 Chemisorption-Surface Reaction-Desorption (CSD) Kinetic Models for Bimolecular Reactions 43 3.1.4 The Number of Kinetic Parameters to be Estimated for the Bimolecular Case 48 3.2 Chemisorption-Surface Reaction-Desorption (CSD) Kinetic Models for Some Industrially Important Gas-Solid Catalytic Reactions 50 3.2.1 Steam Reforming of Methane 51 3.2.1.1 Kinetics of steam reforming reactions 51 3.2.1.2 A more general chemisorption-surface reaction-desorption (CSD) kinetic model for methane steam reforming 60 3.2.1.3 Rate dependence on steam partial pressure 62 3.2.1.4 Reaction rate dependence observed by earlier investigators 63 3.2. 1.5 Comparison with Bodrov kinetics 65 3.2.1.6 Comparison with De Deken kinetics 67 3.2.1. 7 Implications of non-monotonic kinetics on fixed bed reactors 69 3.2.1.8 The implications of non-monotonic kinetics on the selection of feed conditions for steam reformers 73 3.2.1.9 Concluding remarks 77 3.2.2 High and Low Temperature Water-Gas Shift Reactions 79 3.2.2.1 Kinetic models for the high temperature shift catalysts 79 3.2.2.2 Comments on reaction mechanism 85 3.2.2.3 Actual industrial kinetic models for high and low temperature shift catalysts 86 3.2.3 Gas Phase Catalytic Hydrogenation Reactions 87 3.2.3. 1 Hydrogenation of aromatics 87 3.2.3.2 Hydrogenation of olefins 91 3.2.4 Catalytic Ammonia Synthesis 94 3.2.4.1 Mechanism of the ammonia reaction 94 3.2.4.2 Rate equations 95 3.2.4.3 Widely used kinetic models for industrial ammonia synthesis 97 3.2.5 Partial Oxidation Reactions 100

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Reaction network for the partial oxidation of o-Xylene to phthalic anhydride 3.2.5.2 Kinetic models of the o-Xylene partial oxidation reactions 3.2.5.3 Kinetic models and parameters for the partial oxidation of o-Xylene to phthalic anhydride

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CHAPTER 4 PRACllCAL RELEVANCE OF BIFURCATION, INSTABILITY AND CHAOS IN CATALYTIC REACTORS 4.1 Sources of Multiplicity 4.1.1 Isothermal Multiplicity (or Concentration Multiplicity) 4.1.2 Thermal Multiplicity 4.1.3 Multiplicity Due to the Reactor Configuration 4.2 Simple Quantitative Discussion of the Multiplicity Phenomenon 4.3 Bifurcation and Stability 4.3.1 Steady State Analysis 4.3.2 Dynamic Analysis CHAPTERS EFFECT OF DIFFUSIONAL RESISTANCES. THE SINGLE PELLET PROBLEM 5.1 Non-porous Catalyst Pellets 5.1.1 Isothermal Catalyst Pellets with Linear Kinetics 5.1.2 Isothermal Catalyst Pellets with Non-linear Kinetics 5.1.3 The Isothermal Effectiveness Factor for Single Reactions (Irreversible and Unimolecular) 5. 1.4 The Non-isothermal Catalyst Pellets with Linear Kinetics (Non-porous Catalyst PeUet with Unimolecular Reaction) 5.1.5 Non-isothermal Catalyst Pellets with Non-monotonic Kinetics 5.1.6 Preliminary Remarks on Complex Reaction Networks 5.1.7 Preliminary Comments on the Effectiveness Factor Concept for Gas-Solid Catalytic Reactions. Reaction versus Component Effectiveness Factor 5.1.8 Results and Discussion of the Steady State for Non-isothermal, Non-porous Catalyst Pellet for a Single Unimolecular Irreversible Reaction 5.1.8.1 Practical applications and range of parameters for industrial catalysts

101 105 108

112 114 114 115 1 16 116 II 7 118 125

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5.1.8.2 General model with finite pellet thermal conductivity: The symmetrical case 5.1.8.3 Numerical solution of the equations 5.1.8.4 Simplified models 5.1.8.5 Steady state model equations 5.1.9 Results and Discussion for Complex Reaction Network on Non-porous Catalyst Pellets. The Steady State Analysis for the Catalytic Partial Oxidation of o-Xylene to Phthalic Anhydride 5.1.9.1 The mathematical models for the partial oxidation of o-Xylene to phthalic anhydride on non-porous catalyst pellets for two different kinetic models 5.1.9.2 Mathematical expression for the effectiveness factors 5.1.9.3 Simulation results for the partial oxidation of o-Xylene to phthalic anhydride on non-porous pellets 5.2 Porous Catalyst Pellets 5.2.1 Lumped Models 5.2.1.1 The importance of surface processes in the dynamic behaviour of porous catalyst pellets 5.2.1.2 Chemisorption and catalysis 5.2. 1.3 Rates of chemisorption 5.2.1.4 Heats of adsorption 5.2.1.5 Practical example for the rates of activated adsorption 5.2. 1.6 Rates of desorption 5.2. 1.7 Equilibrium adsorption desorption 5.2.1.8 The lumped model for porous catalyst pellets 5.2.2 Distributed Models 5.2.2.1 Fickian-type models 5.2.2.2 Dusty gas models 5.2.2.3 The use of the dusty gas model for multiple reversible reaction networks. Steam reforming of methane CHAPTER 6 THE OVERALL REACTOR MODELS

6.1 General Classification of Reactor Models 6.1.1 The Continuum Models 6.1.1.1 Pseudo-homogeneous models 6.1.1.2 Heterogeneous models 6.1.1.3 One-dimensional models 6.1.1.4 Two-dimensional models 6.1.2 The Cell Models

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Systems Theory and Principles for Developing Mathematical Models of Industrial Fixed Bed Catalytic Reactors PerhapJ maJhematir.s 1s effective in organizing phrsicaf exi,lt(•nce because It Is msptred by physical existence. The pragmatic reality is that mathematics 1s rlw most effect/l'e and mmworthy method that we know for rmderstanding what we see around liS. Ian Srewan (1989)

Mathematical modelling of diffusion and reaction in petrochemical and petroleum refining systems is a very strong tool for design and research. It leads to a more rational approach for the design of these systems in addition to elucidating many important phenomena associated with the coupling between diffusion and reaction. Rigorous highly sophisticated mathematical models of varying degrees of complexity are being used in industrial design as well as academic and industrial research (e.g. Eigenberger, 1981 a,b~ Muller and Hofmann, 1986; Salmi, 1988). An important point to be noticed with regard to the state of the art in these fields, is that steady state modelling is more advanced than unsteady state modelling due to the additional complexities associated with unsteady state behaviour and the additional physico-chemical information necessary (Elnashaie, 1977; Eigenberger, 1981 a,b; Baiker and Bergougnou, 1985). Based on these advancements, continuous processing has progressed in these two fields over the years and became the dominant processing mode. Computerized design packages have been developed (Einashaie and Alhabdan, 1989a,b) and computer control of units and whole plants has been introduced widely (e.g. Jutan eta/., 1977; Schnelle and Richards, 1986; Richards and Schnelle, 1988). This has been coupled with a considerable increase in the productivity of industrial units and plants with a reduction in manpower and tight control over the product quality. On the industrial and academic level, these advancements have led to innovative designs and configurations that open new and exciting avenues for developing compact units with very high productivity

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(Boreskov and Martos, 1984; Itoh et al., 1985; Itoh, 1987; Elnashaie and Adris, 1989). Despite these advances there is stiU a considerable gap between academic progress and industrial applications regarding the use of rigorous models for the design, simulation and optimization of catalytic reactors. Amundson ( 1984) stresses this by saying: "The theoretical side of chemical reaction engineering is well in hand or will be shortly. Unfortunately the experimental and practical implementation of these results lags. Universities are not equipped to operate chemical reactors on a scale necessary to validate models appropriate to the industrial scene. The chemical and petroleum companies are loath to present proprietary results and for good reason. This is regrettable, and one cannot fail to notice that there is not one industrial paper presented in this volume". Amundson wrote this almost 8 years ago, the situation to date is to a considerable extent different with regard to laboratory experimental results, however, on the industrial front the situation remains almost the same. It is hoped that this book with its emphasis on gearing the accumulated fundamental knowledge towards the modelling, simulation and optimization of industrial fixed bed reactors will be one step in the direction of bridging this gap between academia and industry. A system approach will be adopted which treats the fixed bed reactor as a system consisting of subsystems with their properties and interactions giving the overall system (the reactor) its characteristics. Before describing details, it is important to give a brief discussion of system theory and the principles of mathematical models with emphasis on fixed bed catalytic reactors.

1.1 SYSTEMS AND MATHEMATICAL MODElS Here the basic concepts of system theory and principles for mathematical modelling are presented for the development of diffusion reaction models for fixed bed catalytic reactors.

1.1.1

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A Brief Backgound

A system is a whole consisting of elements or subsystems. The concept of system-subsystem-element is relative and depends upon the level of analysis. The system has a boundary that distinguishes it from the environment. It may exchange matter and/or energy with the environment depending upon the type of system from a thermodynamical point of view. A system (or subsystem) is described by its elements (or subsystems), the interaction between the elements

and its relation with the environment. The elements of the system can be material elements distributed topologically within the boundaries of the system and giving the configuration of the system, or they can be processes taking place within the boundaries of the system and defining its function. They can also be both, together with their complex interactions. An important property of the system, wholeness, is related to the principle of the irreducibility of the complex to the simple, or of the whole to its elements, i.e. the whole system will possess properties and qualities not found in its constituent elements. This does not mean that certain information about the behaviour of the system cannot be deduced from the properties of its elements, but it rather adds to it (Biauberg et a/. , 1977). Systems can be classified on different basis. The most fundamental of whkb is that based on thermodynamic principles and on this basis they can be classified into (Prigogine el a/., 1973; Nicolis and Prigogine, 1977):

1. Isolated systems: These are systems that exchange neither energy nor matter with the environment. The simplest chemical reaction engineering example is an adiabatic batch reactor. These systems tend towards their thermodynamic equilibrium which is characterized by maximum entropy (highest degree of disorder).

2. Closed systems: These are systems that exchange energy with the environment through their boundaries but do not exchange matter. The simplest chemical reaction engineering example is a non-adiabatic batch reactor. These systems tend towards their thermodynamic equilibrium which is characterized by maximum entropy (highest degree of disorder), or more precisely they tend toward a state of minimum free energy.

3. Open systems: T hese are systems that exchange both energy and matter with the environment through their boundaries. The simplest chemical reaction engineering example is the continuous stirred tank reactor. These systems do not tend toward their thermodynamic equilibrium, but rather towards a state called "stationary non-equilibrium state" and is characterized by minimum entropy production. Open systems near equilibrium have unique stationary non-equilibrium state, regardless of the initial conditions. However far from equilibrium these systems may exhibit multiplicity of stationary states and may also exhibit periodic states.

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1. Lumped systems: These are systems where the state variables describing the system are lumped in space (invariant in all space dimensions). The simplest chemical reaction engineering example is th~ perfectly mixed continuous stirred tank reactor. These systems are described at steady state by algebraic equations while in the unsteady state they are described by initial value ordinary differential equations where time is the independent variable.

2. Distributed systems: These are systems where the state variables are varying in one or more directions of the space coordinates. The simplest chemical reaction engineering example is the plug flow reactor. These systems are described at steady state either by an ordinary differential equation (where the variation of the state variables is only in one direction of the space coordinates, i.e. one dimensional models, and the independent variable is this space direction), or partial differential equations (when the variation of the state variables is in more than one direction of the space coordinates, i.e. two dimensional models, and the independent variables are these space directions). The ordinary differential equations of the steady state of the one-dimensional distributed model can be either initial value differential equations (e.g. plug flow models) or two-point boundary value differential equations (e.g. models with superimposed axial dispersion). The equations describing the unsteady state of distributed models are invariably partial differential equations. Another classification of systems which is very important for deciding the algorithm for model solution, is that of linear and non-linear systems. The equations of linear systems can usually be solved analytically, willie the equations of non-linear systems are almost always solved numerically. ln this respect, it is important to recognize the important fact that physical systems are almost always non-linear and linear systems are either an approximation that should be justified, or the equations are intentionally linearized in the neighbourhood of a certain state of the system and are strictly valid only in this neighbourhood.

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It is clear from the above classification that batch processes are usually of the isolated or closed systems type while the continuous processes are usually of the open systems type. For continuous processes a classification from a mathematical point of view is very useful for both model formulation and algorithms for model solution. According to this basis, systems can be classified as (Douglas, 1972):

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A third classification, which is very relevant and important in chemical engineering, is the classification based on the number of phases involved within the boundaries of the system. According to this classification, systems are divided into (Froment and Bischoff, 1979):

1. Homogeneous systems: These are systems where only one phase is involved in the processes taking place within the boundaries of the system. In chemical reactors, the behaviour of these systems is basically governed by the kinetics of the reactions taking place without the interference of any diffusion processes.

2. Heterogeneous systems: These are systems where more than one phase is involved in the processes taking place. In chemical reactors, the behaviour of these systems is not only governed by the kinetics of the reactions taking place but also by the complex interaction between the kinetics and the relevant diffusion processes. The modelling and analysis of these systems is obviously much more complicated than for homogeneous systems. It is clear that the systems for fixed bed catalytic reactors fall into this category of heterogeneous systems, and more specifically into the category of gas-solid systems and therefore the behaviour of the system is dependent upon a complex interaction between kinetics and diffusion. The reader interested in more details about system theory and the general concepts of mathematical modelling will find a large number of good books (Dransfield, 1968; Forrester, 1968; Von Bertalantfy, 1968; Robert et a/., 1978).

1.1.2

Mathematical Model Building: General Concepts

Building a mathematical model for a gas-solid reactive system, depends to a large extent on the knowledge of the physical and chemical laws governing the processes taking place within the boundaries of the system. This includes the diffusion mechanism and rates of diffusion of reacting species to the neighbourhood of active centers (or active sites) of reaction, and the chemisorption of the reacting species on these active sites for non-porous pellets, while for porous pellets it also includes the diffusion of reactants through the pores of the catalyst pellets (intraparticle diffusion), the mechanism and kinetic rates of the reaction of these species, the desorption of products and the diffusion of products away from the reaction

centers. It also includes the thermodynamic limitations that decide the feasibility of the process to start with, and also includes heat production and absorption as well as heat transfer rates. Of course, diffusion and heat transfer rates between the bulk gas and lhe surface of the pellet are both dependent to a great extent on the proper description of the fluid flow phenomena in the system. The ideal case is when all these processes are determined separately and then combined into the system's model in a rigorous manner. However, very often this is quite difficult with regard to experimental measurement, therefore special experiments need to be devised, coupled with the necessary mathematical modelling, in order to decouple the different processes interacting in the measurements. Mathematical models of different degrees of sophistication and rigor were built to take their part in directing design procedure as well as directing scientific research in this field. It is important in this respect to recognize the fact that most mathemtical models are not completely based on rigorous mathematical formulation of the physical and chemical processes taking place in the system. Every mathematical model contains a certain degree of empiricism. The degree of which, of course, limits the generaJity of the model and as our knowledge of the fundam entals of the processes taking place increases, the degree of empiricism decreases and the generality of the model increases. The existing models at any stage, with this stage's appropriate level of empiricism, helps greatly in the advancement of the knowledge of the fundamentals and therefore helps to decrease the degree of empiricism and increase the level of rigor in the mathematical models. In addition any model will contain simplifying assumptions which are believed, by the model builder, not to alfect the predictive nature of the model in any manner that sabotages the purpose of the model. With a given degree of fundamental knowledge at a certain stage of scientific development, one can build different models with different degrees of sophistication depending upon the purpose of the model building and the level of rigor and accuracy required. The choice of the appropriate level of modelling and the degree of sophistication required in the model is an art that needs a high level of experience. Models which are too simplified will not be reliable and will not serve their purpose. While models which are too sophisticated will present unnecessary and sometimes expensive overburden. Models which are too sophisticated can be tolerated in academia and may sometimes prove to be useful in discovering new phenomena. However, over sophistication in modelling can hardly be tolerated or justified in industrial practice.

1.1.3 Outline of the Procedure for Model Building The procedure for the development of a mathematical model can be summarized in the following steps:

(1) Identification of the system configuration, its environment and the

modes of interaction between them. (2) Introduction of the necessary justifiable simplifying assumptions. (3) Identification of the relevant state variables that describe the system. (4) Identification of the processes taking place within the boundaries of the system. (5) Determination of the quantitative laws governing the rates of the processes in terms of the state variables. These quantitative laws can be obtained from the literature and/or through an experimental research program coupled with a mathematical modelling program. (6) Identification of the input variables acting on the system. (7) Formulation of the model equations based on the principles of mass, energy and momentum balances appropriate to the type of system. (8) Development of the necessary algorithms for the solution of the model equations. (9) Validation of the model against experimental results to ensure its reliability and to re-evaluate the simplifying assumptions which may result in imposing new simplifying assumptions or relaxing others. It is clear that these steps are interactive in nature and the results of each step should lead to a reconsideration of the results of previous steps. In many instances steps 2 and 3 are interchanged in the sequence depending on the nature of the system and the degree of knowledge regarding the processes taking place within its boundaries.

1.2 BASIC PRINCIPLES OF MATHEMATICAL MODELLING FOR INDUSTRIAL FIXED BED CATALYTIC REACTORS Fixed bed catalytic reactors have a wide variety of configurations ranging from the single bed adiabatic configuration to the multitubular, non adiabatic configurations with cocurrent or countercurrent cooling or heating. The configurations can sometimes be complex such as the TVA type ammonia converters where there are internal cooling tubes immersed into the catalyst bed as we'll as an external beat exchanger. In other cases, the reactor may be quite simple with an adiabatic single bed or multiple adiabatic beds in series such as the high and low temperature shift converters. Intermediate degrees of complexity of reactor configuration are exemplified by reactors such as the multibed ammonia converters with interstage cooling.

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It is of course not possible, nor desirable, to present in such a book, the development of mathematical models for all (or even an appreciable percentage of) the configurations used industrially. The most suitable is to present the basic modelling principles for what may be considered the heart of all these configurations, that is the catalyst bed itself, followed by some details and examples regarding the special features of each configuration. Different configurations of catalytic reactors differ in the arrangement of catalyst beds, the techniques used for the introduction of the feed and the means by which the beat is removed (for exothermic reactions) or added (for endothermic reactions). In the first chapters ( 2, 3, 5) emphasis is given to the link common for all configurations, the catalyst bed itself. In chapter 6 specific industrial configurations are presented, modelled and analyzed. The catalyst bed consists of the catalyst particles and the bulk gases passing through the voids of the bed. The reactants diffuse from the bulk gas to the surface of the catalyst pellet, then through its pores where it is chemisorbed and reacts forming the products which desorb and diffuse back into the bulk of the fluid. From a systems point of view, the overall reactor can be considered a system, with the different parts of the entire reactor, e.g. the catalyst bed, cooling (or heating) tubes, external beat exchangers, condensers etc., being considered as subsystems. Obviously in this book (specifically in chapters 2, 3, 5), we consider the "subsystem": the catalyst bed, to be our main concern, and in what follows, we will consider it to be "the system". In chapter 6, industrial reactors are discussed and therefore the catalyst bed becomes a subsystem of the overall reactor system. The catalyst bed as a system can be looked upon from a topological point of view and be considered as formed of subsystems: the catalyst pellets which are stationary and d istributed along the length of catalyst tubes (which can be considered as the second subsystem), and the flowing gases through the packed bed. In fixed bed catalytic reactors, the function of the system is to transform a raw material A to a product P. This transformation takes place according to a number of consecutive or parallel processes, these processes can be considered to be the elements (or subsystem) of the function of the catalyst bed system. This is equivalent to considering the function of transforming A to P to be an overall process and then distinguishing the steps which lead to the required transformation.

Steps of catalytic reactions At this stage it is useful to give a simple and relatively detailed verbal description of what is taking place withln the catalyst bed in order to achieve its final function:

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'::>sJnoo JO ·s::>H~J J::>JSUeJl 1eaq se II::lM. se UO!tdJOsqe ptrn aoqo.npoid lC::llf S::>pnJOU! OSJC pue 'l{l!M. lJClS 01 SS::lOOJd ::ll{l JO A1!1TqtSC3J ::ll{l ::lpfO::>p )Cq) SUO!lElfW!J OfWCUApOWJ::ll{l ::>q) S::>pnfOU! OSyB .11. "SJ::l)U::lO ( 1) The reactant molecules transfer from the entrance of the reactor to the neighbourhood of the catalyst pellets. This transfer takes place by convection and/or diffusion. When axial diffusion is negligible and radial diffusion is instantaneous, we get the simplest description for the bulk phase, that is one-dimensional plug;ffow. (2) The reactant molecules diffuse from the bulk of the fl\.iid phase to the surface of the catalyst pellets. This process is usuaily described by mass transfer rate over a hypothetical external mass transfer resistance. The mass transfer coefficient is usually calculated using J-factor correlations (e.g. Hill, 1977). This step is dependent upon the properties of the gas mixture and the flow conditions around the catalyst pellets as well as the size and shape of the catalyst pellets. This step is usually referred to as external mass transfer of reactant molecules. (3) For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. (4) The chemisorbed molecules, whether on the external surface for non-porous pellets or the internal surface for porous catalyst pellets, undergo surface reaction producing cbemisorbed product molecules. Thls surface reaction is the truly intrinsic reaction step. However, in chemical reaction engineering it is usual practice to consider that intrinsic kinetics include this surface reaction step coupled with the chemisorption steps. This is due to the difficulty of separating these steps experimentally and the ease by which they are combined mathematically in the formulation of the kinetic model. (5) The chernisorbed product molecules desorb from the surface into the gas phase adjacent to the catalyst surface. (6) The chernisorbed product molecules undergo intraparticle back diffusion toward the surface of the catalyst pellets (for porous pellets). (7) The product molecules, whether for a porous or non-porous pellet, diffuse through the external mass transfer resistance into the bulk gas phase.

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(8) The product molecules transport away from the pellet by convection and/or diffusion in order to reach the exit of the reactor. In addition to the above sequence of events, heat transfer resistances also play a part in the overall behaviour of the system. For exothermic reactions, the heat produced at the reaction centers dissipates through the pellet by conduction creating temperature gradient along the depth of the pellet. Then heat gets dissipated from the surface of the pellet to the bulk of the fluid creating temperature difference between the bulk fluid phase and the surface of the pellet. For endothermic reactions, the same sequence of events occurs creating temperature gradients which are opposite to those created for exothermic reactions. It is this complex and interactive sequence of events that gives the overall behaviour of the reactor. The modeller must model these events in an accurate and rationally integrated manner in order to simulate the behaviour of the industrial reactor.

Some common physically justified simplifying assumptions in modelling industrial catalytic reactors Fortunately, some of these processes are of such high rates that they can be neglected in the model formulation. A list of commonly encountered situations for straight forward simplifications is given below. However, a word of warning is essential here. There are of course situations where such simplifying assumptions are not valid, therefore the following cases should be considered as "commonly occurring" not " rules"; (1) Usually, dispersion in the axial direction is negligible in industrial reactors. This is because of the high flow rates and long length of the catalyst tubes, resulting in Peclet numbers which are high enough to justify the assumption of plug flow. (2) Although radial dispersion has been investigated extensively, it seems in many industrial cases that it is appropriate to neglect radial concentration and temperature gradients and use a one dimensional model. (3) In many cases, the external mass and beat transfer resistances are negligible because of the high gas flow rate that destroys the external resistances (see for example the ammonia converter, ch. 6, sec. 6.3.3, and the steam reformer, ch. 6, sec. 6.3.4). A counter example is the case of the partial oxidation of o-Xylene to phthalic anhydride, ch. 6, sec. 6.3.6, where external mass and heat transfer resistances must be taken into consideration for precise modelling of these reactors. (4) In many cases the thermal conductivity of the catalyst pellet is

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kinetic model. This task may be as simple as using an intrinsic kinetic rate model together with the kinetic parameters from the literature as in the case of ammonia synthesis, ch. 6, sec. 6.3.3. It may be slightly more difficult necessitating the use of a full reactor model to extract the intrinsic parameters from industrial data as in the case of the water-gas shift reactions, ch. 6, sec. 6.3.1, and the dehydrogenation of ethylbenzene to styrene, ch. 6, sec. 6.3.5. It can also be more difficult, allowing the use of the functional form of the intrinsic kinetic rate model from the literature, while the kinetic parameters have to be estimated for the specific catalyst using laboratory bench scale differential or integral reactors and using an efficient algorithm, such as the Marquardt algorithm (Marquardt, 1963), for parameter estimation. An example for such a case is the partial oxidation of a-Xylene to phthalic anhydride where most of the published data are for specific catalysts prepared in the laboratories of the investigators (ch. 6, sec. 6.3.6.), while the activity as well as the selectivity of such catalysts vary widely from one catalyst to another. The task can be extremely difficult when the chemistry, mechanism and structure of the reaction network are not well established. In this case, an entire experimental research program aiming at establishing these missing links, should be initiated in conjunction with a project for the development of a reactor model. All the above degrees of complexity apply reasonably well to petrochemical reactions where the feedstock and the products are usually well defined. However, in petroleum refining reactors such as catalytic cracking, hydrocracking, catalytic reforming, etc, the situation is much more complicated and necessitates the use of lumping into pseudo-components (Hasten and Froment, 1984). This class of reactions is beyond the scope of this book, especially since the basic questions regarding the kinetic modelling of these reactions are not settled. The second step the modeller is faced with in some reactors, is to decide on the suitable j-factor correlations to be used for estimating the external mass and heat transfer resistances between the bulk gas and the surface of the catalyst pellet (external resistances). In many industrial cases the answer to this task is rather trivial, that is, the external mass and heat transfer resistances are negligible. However, in· other cases the appropriate choice of the j-factor correlation together with the correlations for estimating the change of physical properties along the length of the reactor (and also radially for two-dimensional models) due to the change of temperature and composition accompanying the reaction, are of great importance for the accurate modelling of the reactor. This external mass and heat transfer problem is closely linked to the appropriate description of the conditions in the bulk phase especially with regard to the hydrodynamics and the axial and radial diffusion mechanisms.

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The third crucial step is that related to intraparticle diffusion of mass and heat. As mentioned earlier, most industrial catalysts will have large enough thermal conductivities to justify the assumption of flat temperature profiles inside the pellet. In the rare cases where this assumption is not valid, Fourier type heat conduction equations can be used with effective thermal conductivity which is best estimated experimentally if not available in the literature for the specific catalyst under consideration. However, intraparticle mass diffusion is quite important and although different types of discrete models are available (Pisarenko and Kafarov, 1991 ), the mosl appropriate and widely used model seems to be the continuum model (Evans et a!., 196la,b, 1962; Mason and Malinauskas, 1964; Malinovskaya et al., 1975; Jackson, 1977; Kaza et a/., 1980). The continuum model when adopted for the catalyst pellets, has two levels of rigor and sophistication. The first is the Fickian type model which is the most widely used and the second is the dusty gas model based on the Stefan-Maxwell equations for the simultaneous diffusion in multicomponent systems. In bothcases effective diffusivities (molecular and Knudsen) must be used which necessitates estimating not only binary diffusivities, but also the porosity and tortuosity for the catalyst pellets. The porosity is quite easy to measure, but the tortuosity is more difficult and should be carefully estimated experimentally if not available in the literature for the specific catalyst. This discussion regarding the level of rigor for the modelling of the intraparticle dJffusion can be related to the industrial cases presented in chapter 6 as follows: ( l) for an ammonia converter, section 6.3.3, where the system is formed of a single reaction, the use of the dusty gas model does not necessarily complicate the model. Also the results obtained are close to those obtained using the Fickian type model, and both are very close to those of the industrial reactor. (2) for the steam reforming and methanation reactions (section 6.3.4.), it is clear that for this multiple reactions system, the model based on the Stefan-Maxwell equations is much more complicated in formulation and in solution compared with the model using Fick's law. The two models give almost identical results in certain regions of parameters, but differ considerably in other regions of parameters. (3) for the high and low temperature shift converters (sections 6.3.1, 6.3.2), it is clear that the Fickian type model is quite sufficient to obtain very accurate results. Also, because it is a single reaction, the use of the Stefan-Maxwell equations will not add additional complexity. (4) for the dehydrogenation of ethylbenzene to styrene (section 6.3.5) only the model based on the Stefan-Maxwell equations can be used because of the uncertainities associated with the kinetics of the reaction. Therefore it will not be wise from an industrial point

J•IJIP/11

of view, to introduce extra uncertainties associated with the degree of rigor used in the diffusion-reaction model. The next step for the modeller is to integrate the catalyst pellet equations with the bulk gas phase equations forming the necessary model equations for the catalyst bed module, the heart of the overall reactor as discussed earlier. The modelling of the bulk gas phase depends upon many factors related to the pellet to reactor tube diameter ratio, the velocity of the gas flow, the uniformity of the catalyst packing, the axial and radial Peclet numbers, the mode of operation (adiabatic, non-adiabatic) etc. It is adequate for most (but not all) industrial reactors to use a one dimensional plug flow model for the bulk gas phase. This is due to the fact that the key physical events taking place are those within the catalyst pellet, and therefore the crucial sides in the reliable modelling of gas phase catalytic reactors are those associated with the catalyst pellets. This does not mean that the gas phase flow conditions are not important, but rather it sets the main emphasis in modelling, for it is of very little use to invariably emphasize that "everything is important". In the few case where more complex description of the bulk gas phase is necessary, the one-dimensional model of the bulk gas phase can be extended to a two dimensional model or the assumption of plug flow in the axial direction can be relaxed and an axial dispersion term superimposed. However, this should only be carried out when there are strong justifications, for although these extensions of the bulk gas phase modelling are quite simple mathematically, they increase the computational effort considerably and also require the determination of a larger number of parameters. In addition to the above, the model for the catalyst bed module should include equations for the pressure drop across the bed. Although these equations are reasonably simple and their solution as a part of the model equation is a straightforward exercise, they are of crucial importance for without appreciation of the pressure drop consideration fast increases in reactor productivity can be theoretically estimated using the model with very fine particles. Obviously, this is not practically possible because of the excessive pressure drop associated with fine particles. In fact, it is the excessive pressure drop associated with small catalyst pellets that necessitates the use of relatively large catalyst particles in fixed beds. The use of these relatively large particles in tum, is the reason behind the existence of diffusional resistances and thus all the complexities associated with reliable modelling of industrial fixed bed catalytic reactors. The integration of the catalyst bed module into the overall reactor model depends largely on the configuration of the reactor system and the mode of its operation. In principle, the adiabatic single bed reactor is the simplest since for this configuration and mode of operation the catalyst bed module represents the overall

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model of the reactor system, while usually the most complicated cases are those associated with non-adiabatic operations with countercurrent cooling or heating. The next step the modeller faces is the determination of all physico-chemical parameters and the suitable correlations for computing their changes with the variations in composition. temperature and pressure at different points in the reactor (in general axially and radially) and also along the depth of the catalyst pellets. These parameters include physical parameters such as specific heats, densities, viscosities etc.; transport parameters such as diffusivities and thermal conductivities; kinetic parameters as discussed earlier as well as thermodynamic parameters such as equilibrium constants and heats of reactions. When the modeller built his model to the chosen degree of sophistication, including the appropriate assumptions and approximations and all physico-chemical parameters and their correlations are determined, he is then faced with solving the model. That is for a certain design configuration with its design parameters and feed conditions, what are the output conditions and the changes in the values of the variables along the reactor length? Tbis is the usual simulation problem. The design problem will consist of identifying the output conditions and determining for a certain configuration and design parameters what feed conditions will produce the desired output. The situation may also be that most of the feed parameters are determined from previous units and it is required to find say the volume of the reactor that gives the desired output. The problem may also be a combination of the above situations. Of course, it is also usually necessary to find the optimum operating conditions and/or design parameters that will give the maximum desired product, or to maximize a certain profit function or minimize energy consumption. ln all cases, it is essential to solve the model equations efficiently and accurately. Some techniques are discussed in this book and in the appendices, for the solution of the highly non-linear algebraic differential and integral equations arising in the modelling of fixed bed catalytic reactors. The most difficult equations to solve are usually the equations for diffusion and reaction in the porous catalyst pellets, especially when diffusional limitations are severe. The orthogonal collocation technique has proved to be very efficient in the solution of this problem in most cases. In cases of extremely steep concentration and temperature profiles inside the pellet, the effective reaction zone method and its more advanced generalization, the spline collocation technique, prove to be very efficient. With todays computers and the state of the art regarding numerical techniques, it does not seem that the numerical solution of the model equations presents any serious problems. With the fast development of computer hardware and software, this problem wiU become almost trivial in the near future.

I l'lt/lmtum oj the developed mvdel

Before the developed model can be reliably put to any useful purpose, it has to be verified against industrial or pilot plant units. The verification process is quite difficult, the first problem being associated with obtaining data from industry which is usually very conservative regarding this matter. Although Amundson (1984) states this problem and indicates some kind of sympathy with industry regarding this attitude, we do not think it right either in the short or long term. ln fact most of the "secrets" that industry is worried about are not publishable. What the model verification requires is the input-output, reactors dimensions and pellet sizes used. The investigator does not need the secret recipe for the catalyst preparation nor the specific secrets of the process. In fact, there are a number of industrially oriented publications where the omission of a few little details keeps the "interests" of the industrial side completely intact. The close co-operation between industry and researchers and the problems of exposing certain process secrets to the researcher can always be solved through "secrecy agreements" similar to that between the company and its employees. Inspection of the literature reveals an extreme shortage of industrial data which indicates that industry is not ready yet for this non-conservative attitude. Industry has a lot to gain in the short and long term in becoming more flexible and pragmatic regarding this issue. When the first obstacle is overcome and the industrial data is in hand, the modeller has work to do. First he has to make sure that the data is consistent. This may be a simple consistency test on the mass and heat balances of the overall system, or it may go deeper into the process and the events taking place according to the modeller's physical visualization of the system in order to verify the consistency of the data, otherwise he will have to go back and discuss the data with the industrial people operating the specific reactor. Suitable physicochemical data must then be found to put into the model. Usually this step does not represent a serious problem except with regard to the intrinsic kinetics of the specific catalyst. This was discussed earlier and may range from using intrinsic kinetic model with its kinetic parameters from the literature to developing a whole experimental kinetic modelling program in order to obtain the necessary kinetic model and its parameters. Using the supposedly efficient numerical algorithm, the model equations with the operating and design parameters of the industrial unit are solved and the output is compared with the output of the industrial unit. It is not unusual even with the utmost care in model formulation, choice of the physico-chemical parameters and the use of an accurate solution algorithm, that the predictions of the model differ from the industrial data. Blind empirical fitting using one or more adjustable parameters will make the model Jose almost all its

prediCtive value. t-u~t lhc moral scientific population as being the last strong hold ofale/rem)•.

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maximum value as the pressure increased. Langmuir attributed this kind of adsorption with saturation to short range attractions between surface and adsorbent giving bonds which are essentially chemical in nature and limited in number by the number of sites available for bonding on the surface. This type of adsorption has become universally known as chemisorption to distinguish it from physical adsorption (Langmuir, 1929; Taylor and Langmuir, 1933). In principle, therefore, the study of catalyzed reactions is intimately bound up with studies of chemisorption. The rate of the reaction may be controlled by the rate of chemisorption of the reactants, or the rate of reaction between chemisorbed molecules or by the rate of desorption of the product.

David Trimm (1980)

2.2 PHYSICAL AND CHEMICAL ADSORYflON Three main subjects are discussed in this chapter: the effect of adsorption on catalysis, the distinction and the differences between physical adsorption and chemisorption with regard to rates of adsorption, and heats of adsorption. As the surface of the catalyst plays an important role in the catalytic reaction the effect of surface coverage especially on the heat of adsorption is discussed in the second part.

2.1 ADSORPTION AND CATALYSIS During the nineteenth century, attempts to explain the action of heterogeneous catalysts were based on one or other of two general theories. The intermediate compound theory (Ashmor, 1963) proposed that the reaction took place betw~~n the bulk solid and the reactant to give an intermediate compound. This intermediate compound decomposed or reacted with any other necessary reactant to give the product of the main reaction and to regenerate the catalyst. As long as the intermediates were considered as bulk compounds, the intermediate compound theory was of limited applicability. The other theory had its origin in ideas put forward in 1834 by Faraday (Ashmor, 1963). He assumed the condition of adhesion between solid and fluid. This condition leads, under favourable circumstances, to the combination of bodies simultaneously subjected to the attraction. The chief evidence that this simple idea is inadequate, is that some substances can decompose to give quite different products in the presence of different catalysts. The nature of this specific interaction became clear after Langmuir's work on adsorption and its application to chemical reactions. Experiments showed, however, that the adsorption often reached a constant 24

Adsorption is due to attraction between the molecules of the surface and those of the fluid. If this attraction is mild (of tbe same nature as that between like molecules) this is called physical adsorption . In other cases, the forces of attraction are more nearly akin to the forces involved in the formation of chemical compounds; this is called chemical adsorption or chemisorption. The adsorbing molecule loses entropy since its motion on the surface is more restricted than in the gas phase. The free energy of the system also decreases as the surface valencies become saturated so it can be concluded that the adsorption process is always exothermic. There is no single criterion which distinguishes between physical adsorption and chemisorption in aU systems, but there are a few which are generally valid: (i) The magnitude of the heat of adsorption: when the bonds are physical, the beat is a little more than the latem beat of condensation of the sorbate, usually about 1.5 times the latent heat. For chemisorption however, the heat may be as large as ten times the latent heat. (ii) The rate of the process: physical adsorption is fast while the chemisorption process bas an appreciable energy of activation which limits the rate at low temperature and leads to a rapid increase in rate with temperature. Physical adsorption is unlikely to occur to any appreciable extent at temperatures above the boiling point of the sorbate. (iii) Since electrons are shared between solid and sorbate molecules, no more chemisorption can occur when the surface of the solid has been covered with enough sorbate to satisfy the residual valency requirement of the surface atoms. Chemisorption cannot result in more than one layer of sorbate molecules although the situation does not preclude additional layers of physically adsorbed molecules forming.

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2.3 HEATS OF ADSORPTION AND DESORPTION In all physical adsorption and in most chemisorptions heat is evolved. Heat release upon spontaneous adsorption would be expected for the following reasons. There must be a descrease in the free energy of the system adsorbent-adsorbate for the spontaneous process at constant temperature and pressure, and as the adsorbate is more localized it loses some of its translational entropy and some of its rotational entropy. Thus !lG and Mare both negative and since ll.H • !lG + TM then 6.H must also be negative and therefore heat is released. The activation energy of desorption Ed is related to the heat ( - 6H)11 and the activation energy of adsorption E 0 by the equation,

For the desorption process, equation (2.2) is read from right to felL. Provided that the adsorbed molecules possess the necessary activation energy for desorption, then the rate of desorption becomes: (2.5)

Where kd and Ed are the velocity constant and the activation energy of desorption respectively and /'(8) is the fraction of sites available for desorption at coverage 8.

2.5 ACTIVATED AND NON-ACTIVATED ADSORPTION 2.5.1

Activated Adsorption

(2.1}

This is characterized by: Since adsorption is always exothermic, Ed is appreciable even when E 11 - 0. That is desorption is always activated.

There are few systems in which there is a genuine activation energy for chemisorption at zero coverage. Much of the activated adsorption which has been recorded is due to the surface not being initially in a clean state. Table 2.1 summarizes some of the available informa-

2.4 THE KINETICS OF ADSORPTION AND DESORPTION The adsorption and desorption process may be expressed by: (2.2)

where S defines an active site. The sticking probability~ which is the fraction of collisions between adsorbate molecules and surface resulting in chemisorption, may be defined as, ~- uf(O) exp

( -EJRT)

(2.3)

In this equation u is the condensation coefficient the probability that a molecule is adsorbed provided it has the necessary activation energy Ea and collides with a vacant surface site. /(8) is the function of surface coverage 0 and represents the probability that a collision will take place at a vacant site. The rate of adsorption is given by: r0

-

op f(O) exp (- EjRT) ~2nmkT

(a) Exponential increase in the rate with increasing temperature. (b) Continuous fall in rate with increasing coverage.

(2.4)

where m is the mass of the molecule, k is the Boltzmann constant, T is the temperature and pis the partial pressure of adsorbate in the gas above the surface.

Table 2.1 Systems showing activatud adsorption E.,

Gas

Catalyst

Adsorption acm·alion energies, kJ/mol.

Temperature, •C

Cu

85.77 61.09 43.51 41.84

200-400 190-280 (- 78)-(- 96) 300-500

Ge

Fe Fe

tion on activated systems. Activation energies are usually in the range of 40-80 kJ/mol. Careful study of nitrogen chemisorption on singly promoted iron has shown that the activation energy increases linearly with coverage.

2.5.2 Non-activated Adsorption This is characterized by: (a) Weak or zero dependence of adsorption rate on temperature. (b) Initial rate independent of coverage.

,.

.......... uv•e:. u1 Htpto, l!ssenually non-activated chemisorption has been observed in the chem isorption of oxygen and nitrogen on tungsten wire and hydrogen on nickel wire. The chemisorption in these cases occurs with activation energies of 2.51 , 1.67 kJ/mol, respectively, and with high beats of adsorption. In the case of oxygen adsorption on tungsten, Langmuir (Langmuir, 1929) visualized that molecules of oxygen are physically adsorbed on top of chemisorbed oxygen atoms. Later work of Taylor and Langmuir (1933) on the adsorption of cesium on tungsten supported this suggestion.

o o t

t to

t

I

o

"' (2.9)

r- k,. k" . pi( l + k" . p)

(2.10)

where

which has the same form as equation (2.7) to which it reduces when kd>> k,.

2.6 EQUILffiRIUM AND NON-EQUILffiRIUM ADSORPTION-DESORPTION 2.7 If the reaction is taking place on a surface the following adsorption conditions are possible.

2.6.1

Equilibrium Adsorption-Desorption

Equilibrium adsorption-desorption is established if the rate of surface reaction is very slow compared to rates of adsorption and desorption of reactants. This case can be simply illustrated for a unimolelcular reaction in which the Langmuir adsorption equilibrium for reactant is given by: ()- K. p/(1 + K. p)

(2.6)

where K = kofkd is the equilibrium constant for adsorption. The rate of reaction is then, r -k,. B- k,. K.pl(l + K.p)

(2.7)

where k, is the surface reaction velocity constant.

2.6.2 Steady State Non-equilibrium Adsorption-Desorption If the rate of surface reaction is not much smaller than the rate of adsorption, then a steady non-equilibrium treatment may be employed. For unimolecular reaction, this steady state relation is given by,

ADSORPTION ISOTHERMS

It is important to be able to derive theoretical isotherms based on various physical models and to compare them with isotherms observed experimentally in order to establish the particular model which is valid under given experimental conditions. There are three possible methods for deriving theoretical isotherms. ( 1) The kinetic method, where the equilibrium conditions are deter-

mined by equality of the rates of adsorbtion and desorption. (2) The thermodynamic method, where the equilibrium conditions

are determined in the usual manner of thermodynamics. (3) The method of statistical thermodynamics, where the equilibrium

conditions involve a knowledge of the partition function of the adsorbed and gas-phase molecules and the sites. The three most important isotherms are those associated with the names of Langmuir, Freundlich and Tempkin. Each is based on a different theoretical model, the principal difference being the way the beat of adsorption varies with surface coverage. The Langmuir isotherm is the most widely used in the derivation of kinetic models for gas-solid catalytic reactions. The derivation of a Langmuir isotherm depends on the kinetic method where the equilibrium conditions are determined by the equality of the rates of adsorption and desorption. At equilibrium the following relation holds, (2.1 1)

Accordingly, the isotherm is expressed by the relation,

(2.8)

where it is assumed that there is no build up of product adsorbed on the surface. In this case the reaction rate is given by,

(2.12)

\U I C.J

(6'(;)

(d . 1/>t + l);cl . ,,)f . '>t ... .I 0

(d' >f +''>f +P>f)jd · ":>f ' J>f =(J f •J • '

lit

• . ••

•

......... '

•

A Langmuir ISOtherm is obtained if it is assumed that the expression (kdla) ,J2nmkT exp [( + I!:Jl)AIRTJ is independent of 0. This coverage independent function is expressed as the reciprocal of the parameter

a, (2.13)

When K is dependent on temperature alone, this isotherm becomes,

(2.14)

For a molecule adsorbed on a single site. /(8)- I - 0

(2.15)

/'(0)- 8

(2.16)

and the isotherm becomes,

0 p .. K(l-8)

(2.17)

8- K.p l +K.p

(2.18)

(i) Porous catalysts may have surface areas as high as 10 7cm 2/g.catal. (ii) It is clear that the activation energy of the heterogeneous reaction must be considerably lower than that of the homogeneous reaction in order that a significant increase in the rate of reaction may be attained through the use of catalyst. (iii) The surface activates the reactants by dissociating the molecules into atoms, which is more difficult in the absence of the catalyst. (iv) The surface can influence the reaction rates by bringing the reactants together in a way which renders the formation of the transition state most probable. The number of possible directions of approach in surface reactions is much restricted. The geometric properties of the surface play an important role since the ease of dissociation depends strongly on the distance of separation of adjacent sites. 2.8.2 Heat of Adsorption and Surface Coverage The heat of adsorption usually decreases markedly with the fraction of the surface already covered by the gas. This decrease often follows

or

Where K is the equilibrium adsorption constant defined as follows. K-

(]

·exp [(- I!:Jl)AIRGTJ

(2. I 9)

kd,./2nmkT

Three important conditions are implied in the derivation of a Langmuir isotherm: (I) Adsorption is localized and takes place only througl1 collision of gas molecules with vacant sites. (2) Each site can only accommodate one adsorbed molecule. (3) The energy of an adsorbed molecule is the same at any site on the surface, and is independent of the presence or absence of nearby adsorbed molecules. 2.8 2.8.1

THE EFFECf OF SURFACE COVERAGE The Role of the Surface in Heterogeneous Catalysis

The role of the solid surface in heterogeneous catalysis can be summarized in the following four points:

0

a

FIGURE 2.1

0

a

a

Effect of surface coverage on the heat of chemisorption.

one of three basic forms, shown in Figure 2.1, although combination of these are also encountered. The effect of surface coverage on the activation energies E 0 , Ed is shown in Figure (2.2), that indicates the dependence of the heat of adsorption and the activation energies on the coverage of 0, 0.5 and 1. There is a corresponding fall in the heat of adsorption. The activation energy for chemisorption which is initially zero, becomes positive and progressively larger as the coverage increases. Ea can never of course exceed Ed otherwise adsorption would become endothermic. The study of adsorption on a solid catalyst has shown that heterogeneous catalytic reactions are complex and consist of at least three single steps adsorption, surface reaction and desorption.

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'P;)!Idde ~q lOUUeo 8u!punoJJns

·~!O!lJed ~q1

building of catalysLs using a molecular surface science approach. Gales discusses the organo metallic chemistry as a basis for the design of supported catalysts. Bell presents discussion on the effect of supports and metal-support interactions on catalyst design. Boudart discusses the use of kinetic information for the design of catalytic cycles and Haag and Chen, the design of zeolite catalysts. Aris gives an elegant presentation for the use of mathematical modelling in catalyst design and Finally Wei gives some useful ideas regarding the design of hydrometallation catalysts. The interested reader should also read the discussion between Kiperman et al. (1989) from one side and Boudart (1989) from the other side, which was published under the title: Classical catalytic kinetics: what is the point of matter?. The discussion brings up a list of important references from both the Russian and Bulgarian (K.iperman et a/., 1989) and Western literature (Boudart, 1989).

CHAPTER 3

Intrinsic Kinetics of Gas-Solid Catalytic Reactions We now understand why it was impossible for the first synthesis produced by science, the Newtonian synthesis, to be complete; the forces of interaction described by dynamics cannot explain the complex and irreversible behavior of mauer. Ignis mutatres. According to this ancient saying, chemical stn1ctures are the creatures of fire, the results of irreversible processes. Ilya Prigogine and Isabelle Stengers (1984)

Among the many processes taking place in industrial fixed bed catalytic reactors discussed earlier, the surface reaction is certainly the most important. Intrinsic kinetics refers to the laws governing the rate of reaction when all other processes, specifically diffusional processes, are much faster than the rate of reaction so that the effect of diffusional limitations on the actual (or apparent) rates of reactions is negligible. The intrinsic rate of reaction includes the chemisorption-desorption rates as well as the rates of the surface reactions, but does not and should not include any diffusional rates since diffusional rates are included separately in the overall reactor model as will be discussed later. The effective reaction rates which include diffusional rates are sometimes used and this reduces the generality, reliability and rigor of the reactor model considerably. When such kinetics are the only ones available they should, for the sake of consistency be used in a pseudo-homogeneous model. Otherwise, diffusional limitation will be effectively included into the model twice. Even when such an obvious inconsistency is avoided and the effective rates are used with a pseudo-homogeneous model, the model will still be neither reliable nor general. This is due to the fact that diffusional resistances usually depend upon the bulk conditions, flow rate and particle size. Thus the results will be, strictly speaking, valid only for the bulk conditions and catalyst particle size which were used to obtain such effective rates. This non-rigorous practice sometimes gives adequate results when the effectiveness factor (17) is not varying appreciably along the length of the reactor and when its sensitivity to flow conditions and catalyst particle size is not very high for the region of interest (Elnashaie et al., 1987a,b; Elnashaie and Alhabdan, 1989a). In general, for the development of reasonably rigorous models for industrial fixed bed reactors, reliable intrinsic kinetics are essential. 35

ll!lljl llll hl

' "II

IIJ " '""

I '"'( "''" Jllll l ·lltllh·l'lll

I'll'

IIJI

011111lf

"II

"'"'""'I'

llllf1111lllll .lj ,,, •• '1111' Ill •II 1'1•111 'I 1111 'I 11111\\ I

Difficulties associated with the establishment of reliable kinetic models for the intrinsic kinetics of gas-solid catalytic reactions Despite the great importance of intrinsic kinetic models and the extensive research effort spent on developing sophisticated kinetic models for different important catalytic reactions it remains the most difficult problem facing the development of'rigorous mathematical models (design equations) for industrial fixed bed catalytic reactors (Agnew, 1985). This is due to many factors: (1) The complexity of the chemisorption - surface reaction desorption processes taking place on the catalytic surfaces. (2) The effect of different factors involved in the preparation of the catalyst on its kinetic characteristics. Such factors include the support material, the dispersion of the active material the interaction of the active material with the support, the pron{oters added to the catalyst, in addition to many other factors related to the detailed catalyst preparation recipe. (3) The tendency of many investigators to develop rate equations for in-house prepared catalysts which makes much of the results related to the specific catalyst prepared and thus considerably decreases the general validity of these rate equations. In addition many investigators publish their kinetic work without giving enough details about the catalyst used which makes it even more difficult to relate their results to other catalysts. It would be much more fruitful practicalJy, and without any loss to the fundamental objectives of the kinetic investigation, if investigators working in the field of the kinetics of gas-solid cataytic reactions concentrated on commercial catalysts. Of course the situation is different when the research work is oriented towards the catalysis side in this case views of certainly the investigators will have to focus on their catalyst preparation. (4) The use of power law kinetics to describe the rates of catalytic reactions. This is an empirically oriented approach which considerably limits the generality of the rate equations obtained and makes it valid only in the region of parameters in which the empirical power law kinetics was obtained. (5) The inconsistency of some of the results in the literature which is due to either: infection by diffusional limitations or the lack of true isothennality in the experiments while isothennality ic; assumed in the model used to extract the kinetic parameters. (6) ~adequacy of the parameter estimation technique used (especially With the large number of parameters involved in a relatively rigorous kinetic model), in addition to the different experimental errors involved especially in the rigorous calibration of the gas chromatographs used.

own

111111'11

"""'II I "I IIII"

II

"11'"1 1'1"""

I' 111~ 1111111 'Ill I" ft11111JI ""~Ill

•P'

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(7) The diversity of the important catalytic reactions used in industry and the difficulty of ensuring rigorous and detailed investigations for each reaction. In fact some important catalytic reactions have no published intrinsic kinetics (e.g. non-oxidative dehydrogenation of ethylbenzene to styrene) and others for which the literature offers only oversimplified power law kinetics.

Petrochemical vs. Petroleum refining catalytic reactions The situation with regard to petrochemical reactions is much better than with petroleum refining reactions since in most petrochemical catalytic reactions the feedstock is usually formed of well defined components, and the products are also well defined, whereas in petroleum refining reactions the feedstock and the products are highly complex and the rigorous reaction network is extremely complex. The usual technique applied for petroleum refining reactions is the lumping technique using pseudo components (e.g. Jacob et a/. 1976; Weekman, 1979; Lee et a/., 1989) which adds to the complexity and loss of rigor of the kinetic model used. Petroleum refining catalytic reactions are beyond the scope of this book. The designer (or modeller) faced with these difficulties may need in some cases to carry out a fuJJ kinetic investigation for the catalytic reaction before moving into developing a model for the reactor. However, this very expensive and time consuming step should not be undertaken except when all other routes are exhausted. The need to link the development of the model with an experimental program for the kinetics of the reaction is usually essential for new processes or a new catalyst for a known process. There are situations where a limited amount of experimental work on kinetics is needed, such as when the structure of the rate equation is known with a reasonable degree of confidence and only the kinetic constants are sought for a new catalyst or a partially deactivated catalyst. In such cases a few runs in a simple differential reactor will be sufficient to obtain the new kinetic constants necessary for accurately modelling the catalytic reactor. In other cases, when the kinetic model is developed on the basis of industrial data and the kinetic constants are thus not intrinsic constants, but effective constants containing diffusional limitations, it is possible for practical purposes to extract the intrinsic kinetics from this data using techniques of different degrees of sophistication depending on the situation. For example, in the relatively simple case of shift converters where pore diffusion is not severe, the diffusional limitations affect the preexponential factors and it is relatively simple to separate the intrinsic kinetics from the diffusional limitations (expressed as effectiveness factor rJ) (Elnashaie eta/., 1987a,b; Elnashaie and Alhabdan, 1989a). In other cases where the pore diffusion limitations are quite severe, the activation energies are also affected and the procedure is more

1

(t llltpltt.. tl c.*d IIIVItlvtttH lht hlltll"' u l lht trtdu\ltt.tl 11''11111 hl•l tt)'lllttll •

heterogeneous model for the reactor.

I

3.1

KINETIC MODELS FOR GAS-SOLID CATALYfiC REACTIONS

(3.3)

Kinetic models for catalytic reactions vary widely in sophistication, generality and accuracy. The simplest, least general and easiest to obtain are the power law kinetic models which are usually strictly empirical and contain a limited number of parameters. Consequently they are valid only in a narrow region of parameters and cannot be safely extrapolated outside this region. Kinetic models which take into consideration the interaction between the gas phase and the solid surface (whether through a reduction-oxidation mechanism or an adsorption-desorption mechanism) are more difficult to formulate and usually contain a large number of parameters. However, they are certainly more reliable, general and accurate than power law kinetics.

3.1.1

r .. k. c~. c~.

-

The procedure based on (3.3) is certainly better than that based on (3.1) and (3.2) at least with regard to the degree of distortion of the determined activation energy based on such an empirical rate equation.

3.1.2 Chemisorption-Surface Reaction-Desorption (CSD) Kinetic Models for UnimoJecular Reactions Most gas-solid catalytic reactions follow some form of CSD kinetic model, except for partial oxidation reactions (and similar reactions) where CSD model and Redox models are still competing. In this first case these types of CSD kinetic models are illustrated using an extremely simple reaction, the unimolecular irreversible reaction,

Power Law (PL) .Kinetic Models

(3.4)

Power law kinetic models depend upon fitting a set of experimental results to a rate equation having the form:

~

\VIlli''''""' h lljllti lh tit ol llllllllltlll• 1111 ·IIIII I Ill' " ·II· I I " ut c..ldlctcnl lcntpctutuacs unc..l lhllll k,, 6, u, /J, y!}Snf (3.14)

Jn uJLic• lu Jcvclop u 1cluliou l>clwccu the gu~ phase cunccnllallun

CA and the surface concentration CAS• the SSA is used and gives: (3.17)

3.1.2.2 Effect of product inhibition

Using the relation,

If the assumption of n egligible C8 x is relaxed, then the rate equation obtained will have the form,

(k. KA . Cm) . CA r1 +KA. CA+Ks. Cn

c,- Cm- CAs

(3.18)

the following relation is obtained, (3.15)

It is clear that B bas a poisoning effect (it acts as an inhibitor). Using power law kinetics instead of equation (3.15) will give negative order for component B, and the power law kinetic expression will be empirically valid only over a narrow range of concentrations of components A, B.

3.1.2.3 The steady state assumption (SSA) case with negligible product inhibition If the equilibrium adsorption-desorption assumption (EADA) is relaxed, then a more complicated kinetic model is obtained. The simplest way to relax this assumption is to replace it by a steady state assumption (SSA), that is assuming that the catalyst surface is at steady state. Both assumptions are strictly valid only for steady state conditions and cannot be used rigorously in the dynamic modelling of catalytic reactors. However, because of the complexity of the system and the lack of sufficient knowledge on the dynamics of the CSD processes, steady state kinetic models are usually used for d ynamic modelling of catalytic reactors. In fact the EADA is the limiting case of the SSA when the rates of adsorption and desorption are much faster than the rate of surface reaction. The use of SSA for the simple case of unimolecular irreversible reaction does not cause significant added complexity, however for more complex reactions the SSA causes considerable complexity and most CSD kinetic models are based on the EADA. For the present case the CSD kinetic model is derived based on the SSA and assuming, for simplicity, negligible Css· In this case equation (3.8) is replaced by: (3.16) Where r0 is the net rate of adsorption in kmol/kg catalyst.hr and k 0 , kd are the adsorption and desorption constants, respectively.

(KA · Cm). CA CAs·-----1 + KA . CA + klkd

(3.19)

The rate of reaction is thus given by: (3.20) When the rate of adsorption desorption is much faster than the rate of surface reaction, i.e. kd » k then the term kdlk is negljgible and the kinetic model based on the EADA (eq. 3.13) is obtained. In this book the EADA will be used unless otherwise specified.

3.1.3 Chemisorption-Surface Reaction-Desorption {CSD) Kinetic Models for Bimolecular Reactions In order to move o ne step forward in the direction of actual industrial reactions, the following relatively simple bimolecular gas-solid catalytic reaction is considered: '

A+ B- Products

(3.21)

The EADA is used in the derivation and furthermore, negligible surface concentration of the product molecules is assumed. There are three possible CSD mechanisms for the above reaction:

1. The first mechanism A is cbemisorbed on the catalytic surface forming the surface complex AS, then the surface complex reacts with B in the gas phase forming the product surface complex PS which is instantaneously desorbed to give P in the gas phase leaving the active site S. This simple sequence of events can be expressed as follows:

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0~

(WI I )

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c

41 41

...41

0 tl

..

5

.

vc1allct.l succcsslully ugu m :,t a hu g~.: uuulbca ol uHJustnal steam reformers. (5) The optimum feed partial pressure of steam that gives maximum output conversion is higher than that giving maximum rate of reaction at the entrance. (6) The non-isothermal case gives optimum steam feed partial pressure which is lower than the isothermal case; this optimum feed partial pressure of steam does exist for industrial steam reformers.

0.70

-e

lv•-• ••••••o •._._.. . . _ ·---

existing in the literature. (2) The comparison with Bodrov et a/. ( 1964) and De Deken et a/. (1982) kinetics clearly reveals the limited nature of their kinetic

Ill

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c

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Q, Q,

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3.2.2 High and U>w Temperature Water-Gas Shift Reactions Shift converters are important par ts of the ammonia and hydrogen plants for the purification of the gases by converting CO to C0 2 and enriching the gas mixture by more hydrogen. The main reaction taking p lace is

.H

"ii

0 .55

c:

(3. 124)

0

s:.

;

E

..

0

0

a:

20 0

0.50 10

20

which is a weakly exothermic reversible reaction. The kinetics of this important reaction have been the subject of a large number of studies, some of them based on simple empirical power law kinetics and the others on CSD kinetic models.

30

Steam fud partial pressure ( MPa x 10 )

FIGURE 3.11 Profile of rate of methane disappearance dependence upon steam feed partial pressure and the corresponding exit methane conversion predicted by a one-dimensional heterogeneou~ model, operating conditions are given as follows: tube length/O.D./1.0.- 11.95/ 0.10210.0795 m; feed composition for point a (mol%): CB 4 - 12.98, C1H 6 - 2.14, C~8 • 0.42, C4B 10 ~ 0.05, H 2 0 - 83.73, N2 - 0.34, H 1 - 0.34; total molar feed rate - 4085.7 kmol/h; inlet conditions: feed temperature - 853K, feed pressure = 2.44 MPa; number of catalyst tubes c 176. (Elnashaie et al., 1990). (3) The kinetic rate equations of Xu and Froment (1989a) predict accurately both positive and negative effective reaction orders with respect to steam. (4) The non-monotonic nature of the kinetics gives rise to an interesting optimization problem with respect to the feed partial pressure of steam. This optimization problem is demonstrated using a pseudo-homogeneous model with constant effectiveness factor for both isothermal and non-isothermal cases. ll ic; ul ~o demonstrated using a detailed heterogeneous modrl whn It '' '

3.2.2.1

Kinetic models for the high temperature shift catalysts

Even though the water-gas shift reaction has been the subject of many investigations and has been used commercially for many years, surprisingly little was known about the influence of the kinetic variables on the rate of the reaction (Laupichler, 1938). Industrial practice has required the development of a usable equation so that workable reaction systems can be designed. Rate equations used commercially have been developed with little consideration being given to the theoretical aspects of the reaction kinetics. Thus information is developed about the action of a carbon monoxide conversion catalyst with little understanding of why it performs in the manner that it does. Most rate equations for the water-gas shift reaction have been developed using an empirical approach. Using this method , one fi rst obtains sufficient information from lhe laboratory data over a range of variables thnl cncompnsst.'ll most indu'il t Hll anntic111 ions. A mathcmatit ·ul l 'lflll'~ IIIII II\ lht' ll d t \'1'111)1\'d 111111 piVl' 'l lhr "hcsl fll" f'or lhc data. " ' "I\ t "'"' Ill lht 11 1111 111p11111l t d 1111 n tlu• t•qu:tl iou lo compensate '"' 11 I I\ It \ "' l111!l tt tul t n tlltttiiiiH o 1 um·L•tlntnltcs 111 the basic data.

'(1961) '[11 fiJ OJO. Ott) ' IUIIIIIIIIf\ illlll 1'1!'1111\l.\11 lllj\1 UCljpll~ l fiUM\IOf Jllf UOflllllhil 0 111}1 {j'f I'IJII'I, f, As pointed out by Bohlboro ( 1962) there are many existing kinetic expressions for the shift reaction over iron-based catalysts, some of which are widely different and even conflicting. Bohlboro believes that much of the confusion is due to impurities in the reacting gases such as H 2S. Both Bohlboro (1969) and Ruthven (1969) also believe that much of the confusion is due to some kinetic data being taken under conditions of diffusional limitation. Kul'kova and Temkin (1949) devised the following rate expression for the water-gas shift reaction over an iron-based catalyst:

(3.125) where K1 , Kb are the forward and backward rate constants, P1 is the partial pressure of species i and r is the rate of reaction. lvanovskii et al. ( 1964) found that this expression adequately describes experimental rate data for the shift reaction from 10 to 30 atm and 35o•c. The activation energy for the reaction was found to be 36 kcal/mol indicating no diffusional limitations. Sen eta/. (1964) used the rate equation of Kul'kova and Temkin (eq. 3.125) to correlate kinetic data. Their activation energies of 25.1 to 46.0 kJ /mol (6 to 11 kcallmol) appear to indicate diffusion control in the shift reaction. Shchibrya et al. ( 1965) used a similar theoretical approach to that of Kul'kova and Temkin, and devised the following kinetic expression for the shift reaction: 1

Pf/20Pco- K - PC0 PH

2 2 r=k--------------

ApHlo+Pcol

(3.126)

where r is the reaction rate, P; is the partial pressure of species i, k is the rate constant, K is the equilibrium constant and A is an empirical constant. Shchibrya et a!. (1965) measured the kinetics of the shift reaction over a 93% Fe2 0 3 -7% Cr2 0 3 catalyst at several temperatures and found that the kinetics of the reaction are described adequately by equation (3.126) with an activation energy of 142 kJ /mol (34 kcal/mol). Similar rate expressions were derived earlier by other workers (Kodama et a/., 1952; Stelling and Krusenstierena, 1958). Moe ( 1962) proposed the following empirical rate expression for the water-gas shift reaction:

r=k(ab-cd/K)

(3.127)

where a, b are the concentrations of reactants, c,d the concentrations of product, K the equilibrium constant and k the rate constant. The integrated form of this equation correlated with experimental data very well, with an activation energy of 40.7 kJ/mol (9.74 kcal /mol), which inclicates diffusion control. This low activation energy may also be due to the form of the rate equation which gives

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rise to an apparent activation energy resulting from substracting. the heats of adsorption from the intrinsic activation energy as descnbed earlier. A form of empirical rate expression which appears to be extremely successful in correlating water-gas shift rate data over ferrochrome catalysts is the power type expression. This equation is of the following form: (3.128) where k is the rate constant, [ ] denote the species concentrations, l,m,n,q are empirical exponents, P= [COJ[H2 ]/K[CO][H2 0], and K is the equilibrium constant. The power law types of rate equations are someti~es good approximations of more fundamental and complex express10ns (Boudart, 1956). A series of papers and a book have been published by Bohlboro concerning experimental water-gas shift data on iron-based catalysts, and the treatment of this data using the power law kinetic expressions (Borgars and Campbell, 1974; Bohlboro, 1961 , 1962, 1963, 1964, 1969). Bohlboro collected experimental data for the shift reaction in the following regions: ( 1) (2) (3) (4)

In the kinetic regime and the diffusion-controlled regime. At atmospheric and elevated pressures. With H 2S addition to reactants. With alkali addition to the catalysts.

Bohlboro ( 1961, 1969) studied the shift reaction at atmospheric pressure over a commercial ferrochrome catalyst of 0.8 to 1.2 mm. Fitting of the experimental data with the power type expression was accomplished by determining each individual exponent by varying the· concentration of one of the species while keeping the concentrations of all other species constant. The values of the rate constants were determined by choosing the value of k which gave the closest agreement between the experimental and calculated conversions. Tables 3. 7, 3.8 and 3.9 illustrate the method where the concentrations of CO, H 2 0, and C02 are varied and nitrogen is used as an inert diluent. The agreement between the calculated and experimental conversion is good and the value of k is fairly constant as expected. Bohlboro ( 1961, 1969) found that the following empirical rate expression provided fairly good accuracy for the shift reaction on an iron-chromium catalyst over the temperature range of 330-500"C and atmospheric pressure: (3.129) The activation energy was found to be 114.6 kJ/mol (27.4 kcal/mol). Bohlboro noted that equation (3.129) is significantly different from

..

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.,..

............ t ; t ' . ;

Je4l saAa!raq oJoqrqog ·~u!lomuoo uaAa pue lU3l:llJ!P AJ3P!M all? l{O!l{M JO awos 'SlSAJ'Ble:> pasuq-uru! J3AO uopouaJ ij!t(S atp JOJ suo,ssaJdxa O!l3U!"{3U!1Spca AU'BW 3J'B :lJ:lQl (Z96 I) OJOQIQOQ .\q 1no P3lU!od sy

f

lnl.tlc .1. 7 k11h: cctullllon lor lorward rcuclluo wllh uvcrngc rule constant. (330 ' C, variation of [C0] 0 , amount of catalyst 3g). Data from Bohlboro tl al. (1961}.

'table 3.9 Rate equation for forwurd reucdon with average rate coostunt. (330'C, variation of [COz]0 , amount of catalyst 12 g). Data from Bohlboro et al. (1961).

( C0)0.8(H20)0.2S

( C0)0.8(H20)0.2S

't• 0.0050 Total Feed

rr0.0054

(C02)06S

Feed gas composition, 96

Conv. calc.

Conv.

Rate

exp.

COliS/.

Total Feed

(C02)0.6S

Feed gas composition, 91>

Conv. calc.

exp.

Conv.

Rate const.

mollh

co

C02

Hl

N2

H2 0

96

96

k

mollh

co

C02

H2

N2

H20

%

96

k

2.388 2.383 2.379 2.363

74.8 37.7 12.0 5.6

0 0 0 0

0.56 0.28 0.09 0.04

0.0 37.5 63. 1 69.9

24.6 24.5 24.8 24.5

5.67 8.05 14.31 20.71

5.8 7.3 13.6 22.0

0.0052 0.0042 0.0046 0.0055

2.329 2.288 2.283 2.392

24.7 25.5 24.8 24.0

0 25.5 13.2 51.0

0.18 0.15 0.13 0.08

50.1 23.6 36.4 0.9

25.0 25.3 25.5 24.0

23.1 6.05 8.92 3.79

22.8 6.3 7.9 4.0

0.0052 0.0056 0.0047 0.0057

Table 3.8 Rate equation for forward reaction with average rate constant. (330'C, variation of [H2 0)0 , amount of catalyst 3g). Data from Bohlboro eta/. (1961). [CO]o.s[H20]o.2s TJ•

Total Feed

0.0048

O [C02) ·65

Feed gas composition. 96

Conv. calc.

Conv.

Rate

exp.

COIUl.

mol/h

co

C02

Hl

N2

H20

%

%

k

2.279 2.254 2.317 2.296

50.7 48.4 47.6 50.2

0 0 0 0

0.38 0.37 0.36 0.38

32.9 4.4 18.4 42.7

26.0 46.8 33.6 6.7

7.00 7.91 7.45 5.54

7.0 8.0 7.6 5.4

0.0048 0.0049 0.0050 0.0046

the rate equation of Kul'kova and Temkin (eq. 3.125), but is in qualitative agreement with the rate equatjon of Shchibrya et a/. (1965), (eq. 3.126). To investigate the effect of H 2S (Bohlboro and Jorgensen, 1970; Bohlboro, 1969), the same type of commercial Fe-Cr catalyst was used as in previous experiments. Two H 2S levels, based on wet feed, were used: (1) 50-100 ppm. (2) A few thousands ppm. Bohlboro stated that the H 2S concentration in the second case was sufficient to convert iron oxide to iron sulfide, while in the first case the catalyst remained Fe3 0 4 • In this investigation, it was determined that the activity of the catalyst was considerably lowered by the

addition of 2000 ppm H 2S. The exponents obtained for the rate equation (eq. 3.129) at different H 2S levels are shown in Table (3.1 ~). The differences in the exponents are very small for the two cases w1th H 2S in the feed. Hence the following average rate expression can be used where H2 S is present. (3.130) Bohlboro found that the rate constant was inversely proportional to the H 2 S concentration in the range of 70-1250 ppm H 2 S and 380-SOo·c, in tbe following manner: (3.131) The activation energy at 75 ppm H 2S was 92.9 kJ/mol (22.2 kcal/mol) and at 2000 ppm H 2S, 94.6 kJ /mol (22.6 kcal/mol). Bohlboro ( 1964, 1969) also studied the effects of alkali on laboratory prepared iron catalysts. The behaviour of the catalyst depended greatly on the catalyst pretreatment and !ldde S!ql P~IJ!lSnf (LL6 I) ~SB1J pu~ (6961) u~Al[lD'M ·1unome p~l!nb~J .\lfB:>H~Jo~ql ,I fer'\: 0

S 1:.

Hll OXygOII UlOIII Oil

I lie

OXH.I~ s urlu~c

ttlld S IS U VUCU IH

site on the surface caused by the removal of an oxygen atom. According to this scheme, the water-gas shlft reaction proceeds via alternate oxidation and reduction of the partially reduced surface of the oxide (redox mechanism). Several kinetic expressions have been derived from this mechanism. Equation (3.125) was derived by Kul'kova and Temkin ( 1949) by assuming that step (3.134) was the rate determining step.

(b) Mechanism

based on stoichiometric number studies

Fundamental and exhaustive mechanistic studies on the ironcatalyzed water-gas shift reaction were carried out by Old and his colleagues (e.g. Oki and Mezaki, 1973a,b) who employed the stoichiometric number method. This method will not be discussed here as it is not widely used in developing kinetic rate models for catalytic reactions.

3.2.2.3 Actual industrial kinetic models for high and low temperature shift catalysts Like the ammonia synthesis reaction (discussed in section 3.2.4), the shift reaction although known to take place according to a CSD mechanism, power law kinetics are adequate for accurate design and simulation of industrial shift converters. The most successful rate equation is that of Rase (1977) obtained from industrial data (and therefore includes diffusional limitations). For the high temperature shift reaction which can be stoichiometrically written as: (3.1 36) The practical kinetic model given by Rase ( 1977) contains diffusional resistances. This practical apparent rate equation for the Girdler (G3-b) catalyst of particle size 0.62 em equivalent diameter (1/4" x 114" pellets) is given by: (X;~Xn- XcXvlK) r = k. 1/1. P& lbmol. CO reacted/lb catalyst.hr (3.137) 379 k= exp(15.95- 4900/T) Tin K

1/1 = 1.53 + 0.123P for 11.8 < Ps. 20

1/1=4.0

for P> 20

1/1 is the activity factor for the specific type of catalyst. In equation (3.137) Xi is the dimensionless concentration for component i (C;!Cref )•

., ---•.,.

lJ- - ---~,~

~-a···

\V;IVJ/

.... .., . .. ..,l .. u Q " ·"'l

-un.•--'u

'"

~•v'I

...... IJ..:/...6

(Vh..l..a. c ".U\.UVV

U! p~AJ~sqo letn pue SJope~J A.lolruoq~l· U! · p~AJ~sqo inorA eqdq ~!l~U!){ ~l{1 U~dh\l~q d!t{SUOPBl~l ~q1 JO UO!SSD:>S!P lU~U~:>Xd uy

~l-A hwany (lY~6) developed a simple tcchnJquc Jor extracting intrinsic kinetics from this effective rate equation, which was used successfully to model a large number of industrial shift converters (see section 6.3.1). For the low temperature shift catalyst the rate equation is similar to the rate equations for the high temperature shift except for the kinetic parameters which are given as follows:

k = exp(l2.88- 1855.5/T) Tin K 1/1 = 0.86 + O. l 4P for Ps. 24.8 1/1= 4.33

for P> 24.8

3.2.3 Gas Phase Catalytic Hydrogenation Reactions Hydrogenation reactions for aromatics and olefins are industrially important (Wheeler, 1955; Bond, 1962; Cunningham et al., 1965; Irving and Butt, 1967; Satterfield et a/., 1969). Their kinetic modelling is discussed in this section.

3.2.3.1 Hydrogenation of aromatics The hydrogenation of aromatics takes place on most group VIJI metals at moderate temperatures. The most common metal catalyst is nickel supported on a wide range of materials. Although much work has been done on the hydrogenation of aromatics (especially benzene and its substituted compounds) there is still no consensus of opinion on the kinetic behaviour and mechanisms. Table 3.12 from Zrncevic and Rusic ( 1988) shows the wide range of suggested rate equations in the literature for benzene hydrogenation on nickel catalyst. Competition on the surface of the catalyst between benzene and hydrogen is . assumed by some authors (Germain et a/. , 1963a,b; Motard eta!., 1957) and denied by others (Snagovskii et al., 1966; Van Meeten et a/., 1976, 1977; Van Meerten and Coenen, 1977). Canjar and Manning (1962) postulated that benzene reacted from the gas phase while hydrogen is chemisorbed on the surface, while several other authors (Jirasek et a/., 1968; Kehoe and Butt, 1972a,b,c,d; Merangozis et a/., 1979) used a mechanism in which hydrogen reacted from the gaseous phase while the hydrocarbon is chemisorbed. rn general the mechanisms suggested for the hydrogenation of benzene and its substituted compounds (Mono-, Di-, and Trimethylbenzenes) over nickel supported catalyst"can be roughly divided into three groups:

' t\lllt.jU ~ WI.I)

C2H4 +S- C2H4- S

(3.157)

C2H4- S+HS- C2H5 - S+S

(3.158)

C2Hs - S + H- S- C2H6 + 2S

(3.159)

which shows competition between the hydrogen and ethylene molecules for the active sites on the surface of the catalyst. This mechanism, thus, may give rise to non-monotonic kinetics. Therefore while hydrogenation of ethylene over a nickel supported catalyst does not show the tendency to give non-monotonic kinetics, the same reaction over Pt supported on alumina catalyst may give non-monotonic kinetics. A qualitative difference and not only a quantitative one.

alloweu a l>cllcr unclcrslB JO SWJ;)l U! 1UB1SUO:> Wn!JQ![!nb;) ;)Q1 S! 0 )[ pue JUBlSUOO Uln!lq!J!Ob;) UO!ldJOSpe ;)l(l S! ()/ pue lUBlSUOO ;)lBJ ;)l(l S! t'f 'DJlS JO ;)SOe:>;)q

JO S;)nfei\ 1U3J;)jJ!P 'J3i\0;)JOW 'S:>flS!J;)l:>tm~q:> (Tr.>!S.< >(

-. -.

0cHtOH 15

>
(

>< "' >
< )(

16

~H-,----------~----------------~

>


(

>()(K

K

>< >
i\[nJI:;'I sno.JOU-UOll [l!lUJ:l41S! ;)l[l .JO S:>p:>Upf JH:)U![ :'>Jdll'!S II

I I

Ill

(If

:np

JOJ

111111 It~ I I I I I I

)IJIIII, III l f

Si mply when CA is close to CAn then the system is close to the kinetic-control regime, and when CA is close to zero the system is close to the mass-transfer-control regime.

I

ti N

II

II \f II

For this reaction to replace equation (5.3) for the non-porous pellet will mean that the equation relating CA to CAB will have the form, (5.21)

5.1.2 Isothermal Catalyst Pellets with Non-linear Kinetics When one of the processes is not linear, the additive principle for the resistances in series is not valid (it may sometimes be used as an approximation). To illustrate this, consider a second order reaction instead of the first order one discussed above. In this case, equation (5. 3) becomes:

r=k. c~

(5.16)

The mass transfer equation (5.2) remains the same and equation (5.4) becomes:

Trying to solve the cubic equation (5.21) analytically will not give much insight into the problem. Let us try to solve it graphically to obtain CA for given values of CAo and other physico-chemical parameters (k~, k). Call the left hand side S( C11) ; which is the reactant supply function; this is a function of CA only with k~ and CAB as parameters. Similarly call the right hand side C(CA), which is the reactant consumption function; this is a function of CA only with k and KA as parameters. By plotting the two functions S(CA) and C(CA) versus CA, it is obvious that the point(s) of intersection is (are) the solution(s) of equation (5.21). Such a plot is shown in Figure 5.2.

(5.17) C (CA)

The relation between CA and CAB can only be obtained now by solving the quadratic equation (5.17) to give:

5

(CA)

(

C (CA)

(5.18) Obviously the positive is to be taken in order not to obtain a negative concentration, thus the rate of reaction in terms of bulk gas concentration CAB is given by:

t (5.19) Obviously the additive principle for the two resistances in series is not valid. It is very interesting and practically important to show that there are situations where kinetic-control and mass-transfer-control are simultaneously possible for the same values of k~ and k! Let us consider the following rate of reaction equation which shows non-monotonic dependence upon the concentration, CA:

(5.20)

0

FIGURE 5.2 Supply and consumption functions for the non-porous isothermal pellet with non-monotonic kinetics.

For the non-monotonic kinetics under consideration, C(CA) has the non-monotonic shape shown in Figure 5.2 while S(C.-~) is a straight line of slope - k~ and intersects the horizontal line at CAB · For constant values of - k~, when CAB- CADI, we get a value CA- CA 1 which is very close to zero indicating closeness to mass transfer control. For CAB= CA 82 we obtain CA = CA 2 which is quite close to

I I 111

Ill\ II AM

o

I I M ll!iiiiNI

I I I I I I I II II II I I I olo I o I I Ill

CA 82 indicating kinetic control, while for CAB= CA 83 we obtain three values of CA these are CA 31 , CA 32 , CA 33 . CA 31 is close to zero indicating closeness to mass transfer control, CA 33 is close to CAIJJ indicating closeness to kinetic control, while CA 32 which is somewhere between CA 83 and zero is not close to either regions of mass transfer or kinetic control. Thus for the same values of all physico-chemical parameters CAB• k, k~ and KA, states with different controlling mechanisms are obtained. This is a very simple illustration of the fact that in many actual gas-solid catalytic processes the problems and concepts involved are much more complex than the simple additive principles for the resistances in series and their limiting cases which appear in linear processes.

I o

k. CA CA k~ 11- k. CAB - CAB - - -k+k~ -which obviously approaches I for very small k (kinetic control) and zero for very small k~ (mass transfer control). However for the case of non-monotonic kinetics described by equations (5.20, 5.2 1) and shown in Figure 5.2. the situation is much more complicated:

(not much larger than l)

Although for simple cases of isothermal non-porous pellets with linear kinetics the concept of the effectiveness factor is not practically important it is quite useful to present it since the definition and the principles involved are the same as for the more complex cases which will be discussed later. The simplest literal definition for the effectiveness factor 11 is:

actual (or apparent) rare ofreaction • rate ofreaction ifdiffusional resistances do not exist

I~~ 1

For the simple linear kinetics of the isothermal non-porous catalyst pellet described by equations (5.3, 5.4), the effectiveness factor is simply gjven by:

5.1.3 The Isothennal Effectiveness Factor for Single Reactions (Irreversible and Unimolecular)

11

I I

(5.22)

The actual rate of reaction includes the effect of reaction and diffusional resistances, while the rate of reaction without the existence of diffusional resistances is the intrinsic rate of reaction (small catalyst pellets with negligible intraparticle diffusional resistances and high enougl1 flow rate to destroy external diffusional resistances). For the simple problem in hand, the actual rate of reaction is the rate of reaction when CA is used in the evaluation of the rate r. and the intrinsic rate is the rate when CAn is used in the evaluation of r . Obviously when the system is kinetic controlled, i.e. CA _.. CAn Lhen 17- 1.0, while when the reaction is mass transfer controlled, i.e. CA - 0, then 17- 0.0. These limits are only valid for isothermal, monotonic systems. For non-monotonic or nonisothermal systems r7 may be greater than unity but it cannot go below zero for these si mple reaction networks. This limit of '1 > 0 is again valid only for single reactions, for multiple reaction networks (which are quite common industrially) 11 of some intermediate components can be less than zero. These complexities of the effectiveness factor will be discussed sequentially in this chapter, as well as the advantages and disadvantages of the use of the concepl.

(not much smaller than I)

In this case three possible values of 11 are possible: 17 1 at C,.., 31 , 112 at CA 32 and '13 at CA 33 : C(Cm)

173- C(CABJ) > 1.0 C(C;~J2)

112 • C(CAB3) > 1.0 C(CAJI)

17t - C(CABJ) > 1.0

(not much larger than 1) (well above I )

(much larger than I)

It is also important to notice that 11 > l is not only characteristic of the cases with multiplicity (existence of more than one solution for equation (5.21 )), 11 > 1.0 is characteristic of the non-monotonic kinetics. For example if we take in Figure 5.2, the case with CA 84 giving CA 4 we notice that in this case 174 » 1.0 although only one steady state exists. The simple physical meaning behind this 11 > 1.0, is the fact that for 1 non-monotonic kinetic.c; there

;~re

reczionc; where the cfron in

re::tct~nl

1'111

ttl

111•11111

ljl•"l

~:urH..cnuatruu

Ill II I

/,

"')/ ' "1)

'll l:l j:llllllii'CI (0 NIIIHfl:ll ,.{ JI :'IIJJUOlOUOW tfl!M P~leposse ~se;}J:>U! liE U! Sl(OS;)J ~OU'elSJS;)J .l;}jSU'BJ1 SS'elll 01 ;}Op 3:>ejJOS 3l(l Ol )f(Oq 3tfl UIOJJ UO!lEJlU30UOO

Modified Thiele modulus='-

~

WP. ku· e ~ ap.

k

(5.28)

reactions, ll IS poss1ble to have rJ > 1.0 in both uniqueness and multiplicity regions of parameters.

g

and

5.1.5

Thermicity factor~ fJ =

( -M/). kg.

h T.

c,,

• rl!/

(5.29)

Thus the mass and heat balance equations are finally written in a dimensionless form as: (5.30)

and (5.31)

Non-isothermal Catalyst Pellets with Non-monotonic Kinetics

When the kinetics of the reaction are non-monotonic and at the same time the system is non-isothermal, the situation may become complex especially for exothermic reactions. For non-monotonic kinetics multiplicity of the steady states may arise for isothermal as weU as for mildly endothermic reactions fJ ~ 0, fJ < 0. For exothermic reactions both concentration multiplicity (resulting from the nonmonotonic kinetics) as well as thermal multiplicity (resulting from the exothermicity of the reaction) are combined to give a slightly more complicated multiplicity phenomenon than discussed previously. In this case, as discussed earlier the rate of reaction is given by:

The effectiveness factor is defined as:

ko . e YJ• ko. e

E I RcT . CA

(5.35) e7(1-l/y). XA

£ /ReT• . CAB

eY(l-1 /ya)

.XAo

(5.32)

In order to evaluate YJ, equations (5.30) and (5.31) must be solved for XA, y. This task can be simplified considerably by combining the two non-linear equations into one non-linear equation in one variable (y). This can be achieved by multiplying the mass balance equation by - p and adding the two equations to give: XA"'

Ys+ P. Xa - Y

p

(5 .33)

The mass and heat balance equations are given in dimensionless form directly as: (5.36)

and '2

.p. erB (2)

The effectiveness factor for the appearance of A or disappearance of B is different than the effectiveness factors for the forward reaction ( 1) or the backward reaction (2). The effectiveness factor that satisfies the requirement for expressing the effect of diffusion through one number is the effectiveness factor for the components and is therefore the relevant effectiveness factor for this system. For more complex reaction networks, these component effectiveness factors are the correct numbers to indicate the effect of diffusion on the yield and selectivity for the different components. For this reason it is the components effectiveness factors formulation which should be used in connection with complex reaction networks. It wilJ be shown that in these cases not only t'J > I are possible but also 11 < 0 are possible for some intermediate components. This phenomenon will be discussed in sections 5.1.9 and 5.2.2, in connection with the highly endothermic steam reforming reaction, and the highly exothermic partial oxidation of a-Xylene to phthalic anhydride. The physical meaning of 17 < 0 means that diffusional resistances may not only retard the reaction (17 < 1) or enhance the rate of reaction (17 > 1) but may also reverse the direction of the reaction (17 < 0). The reversing of the direction of the reaction due to diffusion has been observed by Kaza et a/. (1980), De Deken et al. ( 1982) and Xu and Froment (1989b). This interesting phenomenon is discussed in some details in sections 5.1.9. and 5.2.2.

5.1.8 Results and Discussion of the Steady State for Non-isothermal, Non-porous Catalyst Pellet for a Single Unimolecular Irreversible Reaction In this section results are presented to demonstrate the basic behaviour of non-porous catalyst pellets with an exothermic unimolecular irreversible reaction.

p![O!>-l>'U3 U! u op:l'e;}J (W!W:'Il,P pUU UO!SOJHP JO AJO~lJl ~t{l JO ~SOV'J

'P;)SSO:lS!P U:'l~q ~AlHJ ~~!l;)d lSApne:l SnOJOd-UOU 1". .JO ;):ll?]JOS JP.UJ;)lX;) ;)In UO ~:lt>.Jd ~Up{t>.l ('ID nS)

s---v (t) _'t./.8. J

l'HII 'fl

'

5

,.,n It• ditfer•nCII .sotutt.on

1.02

t.oo...t---.---.---....---.--,----,---...---r---r---1 0

(b) Orthogonal collocation technique A more efficient method, the orthogonal collocation, can be used to deal with the conduction equation inside the particle together with its non-linear boundary condition. This method is based on the choice of a suitable trial series to represent the solution. The coefficients of the trial series are determined by making the equation residual vanish at a set of points, ca~ed collocation points, in the solution domain. The orthogonal collocation method (Villadsen and Stewart, 1967; Finlayson, 1972) provides a systematic basis for choosing the collocation points. Figures 5.4 and 5.5 show the results using both the finjte-difference and the orthogonal collocation methods, the Latter of which is more efficient.

0.2

0.4

0.6

0.8

1.0

Various simplified models can be used with varying degrees of accuracy for the simulation of the transient behaviour of non-poro~s catalyst pellets. The most suitable unsteady state model for th1s problem is that with infinite thermal conductivity. This simplified model is quite accurate for metal and metal oxide catalysts. In .this model equation (5.45) disappears and the model becomes stnctly lumped parameter described only by ordinary initial value differential equations.

1.4

1.6

1.8

2.0

FIGURE 5.4 Surface temperature-time response curves. Comparison between finite-difference and collocation methods. A case of moderate thermal conductivity. Initial conditions Yo(cu) = 1, Xso = 0

5.1.8.5 Steady stale model equations The steady state equations are obtained by setting the time derivatives equal to zero, and the system is thus described by the two algebraic equations: ( 1 - Xs) = ii eY< I -

5.1.8.4 Simplified models

1.2

t (time) minutes

l ly,) •

Xs

(5.48)

and (5.49)

Equations (5.48) and (5.49) are similar to those of the adiabatic CSTR, but with different physical meaning of the parameters. Those two equations can be decoupled into the single equation:

1 l 1 ·•

1.18

-r-----------------------, ex 1

1.16

p

...::>

1.14

-;;

G; a. 1.12

1.9

1.8

.,o-'

0:

4

a:k

•o'

CXm

50

1.7

e

"' a

1.6 1. 10

1.5

M~lhod Co11ocollon (L~gtnd~ Polynomials)

t

j

1.08

Curvt

No. of poonts

1.4

)

1.3

c 0

.e ..

~

I

I

02

~ ~

'

2.0

.,o•

2 10

Nu1 50

..

I l fl

1.06

4

;:; 1.04

1. 2

5

...

P1111\e 4ift•nnc•

>

tolu1\o"

1.1

X:

il u

1.02

CD

....

.....; J: 1.00+--...,....---r--"T----.,r--T"""--r---,----r---,r-~

o

ru

~

u

u

~

u

~

~

~

w

s:.

4

1.0 0.9

I II

t (time) minutes

·~

FIGURE 5.5 Surface temperature-time response curves. Comparison between finite-difference and collocation methods. A case of moderate thermal conductivity. Initial conditions Yq{w} .. 1, X10 • 1

.. il

0.8

J/

0. 7 0.6 0.5

(5.50)

0.4

Equation (5.50) has been analyzed extensively for uniqueness and multiplicity by a number of authors (Aris, 1969b; Balakotaiah and Luss, 1984). The following necessary conditions for the existence of multiple steady states are known:

0.3 0.2

(1) ,li(y-4)> 4 (2) a· > a>

0.0

cr

-

-

0::21110

5

0.1 10

where

6

CX:2~t10

I

I

14

18

22

26

30

y: ....!..._ RT 8

eY

a· - - - --

-

0~

Eo

:.c::u

J

~

......

~

1E

1.10

0

_f t.OS

.."

2

u

3

10

Ill

c .~ 1.00

ct 41

c:

Cl

:

•

>. X

jjj

I

0

1'/,IFud

«>oc

0.95

I

1.0 ..__ _ _ _ _ _........_ _ _ _ _ _ _

0.01

0.1

I _ _ _ _ _.,.:u

0.90-f---,---,---.--.-~--,---,----,

..~-__;,l

450

10

1.0

500

550

600

650

700

750

800

850

Bulk Temperature ( K)

4

0- Xyl4!ne conc4!ntration x10

FIGURE 5.10 bank, 1974).

Oxidation of o-Xylene with air in a CSTR (Calder-

rate of o-Xylene was first order up to nearly I mol % of o-Xylene concentration, but at higher o-Xylene concentration the catalyst loses its activity. The kinetic data used here is only for the CO combustion products and the normal conditions (shown in Table 5.2) at which Chandrasekharan and Calderbank (1979) ran their experiments. Figure 5.11 shows the effectiveness factor of the reactions of o-Xylene to o-tolualdehyde, reaction number 1, and o-Xylcne to phthalic anhydride, reaction number 2, and a-Xylene to combustion products, reaction number 3. The figure shows effectiveness factor values slightly above unity with different maximum value for each. In Figures 5. 11-5.14 the reaction rates at the catalyst surface and at the bulk conditions are shown which indicate that the difference between the reaction rate at the bulk and catalyst surface are not very high. The maximum values of the effectiveness factors for the above mentioned reactions result from the difference between the bulk and the catalyst surface reaction rate where the difference is higher for the middle temperature steady states than it is for the low or high temperature steady states. Figure 5.15 shows the effectiveness factors for the reaction of o-tolualdehyde to phthalide, reaction number 4, and phthalide to phthalic anhydride, reaction number 5. The effectiveness factors are extremely high, reaching values as high as rf'" 1000.

FIGURE 5.11

Effectiveness factors for reactions l, 2, 3.

Reaction number= 1

0.004

0 ~ 0.003

E9

X: 0 u

0'>

X:

0.002

" & c

.2

~ ex:

0.001

0.000 +---;::::=-..---r--r---r-~---r---. 450 500 550 600 650 700 750 800 850 Bulk Temperature ( K)

FIGURE 5.12 Reaction rate at the surface and bulk conditions for reaction (1).

Lsn

I

~------~~~--------~----------~oot

R~tlon

l he reactiOn rates al the catalyst surface and the bulk conditions arc shown in Figures 5.16, 5. 17 for reactions 4 and 5, respectively, which shows a big difference between the two rates and shows that the reaction rate at the catalyst surface increases much faster than at the bulk, as the bulk temperature increases. The component net reaction rates can be defined as the rate of consumption minus the rate of formation, or as the rare of appearance or disappearance of any reacting component. The component effectiveness factor can be defined as the ratio of the net rate of appearance or disappearance of any component in the reaction network at the catalyst surface to that in the bulk conditions. It can be mathematically expressed as,

numbrr : '2

0003

...

-· s:.

0

,..

E jj 0.002

"' 0 01

"

.,"'

& c:

:71

~

0.001

·~

(5.65)

~

0:

~~+---~--~--~--~~r-~r-~----

450

500

550

&00

&SO

700

750

800

850

From equation (5.65) the effectiveness fa ctor for the network components can be written as:

Bulk Temperature ( K)

FIGURE 5.13 reaction (2).

r 1 + r 2 + r3 'Ylox- ----------rl , 8 + ~'2,8 + r3, B

(5.66)

Reaction rate at the surface and bulk conditions for (5.67)

Reaction number: 3

00015

4

-i

o_ ~ ~0.0010

5

u

01

.,"'

~ .~

0.0005

i

0: 0.~0 4---~---r--~--~--~--~--~--~

450

500

550

&00

650

700

750

800

650

Bulk Temperature ( K)

450

500

550

600

650

700

750

800

850

Bulk Trmperalure ( K)

FIGURE 5.14 Reaction rate at the surface and bulk conditions for reaction (3).

FIGURE 5.15 Effectiveness factors for reactions 4 and 5.

~u~.E-004 ID

-3-0&006

+-----.----.-----,.---....------, 650

660

670

680

690

700

Bulk Temperature ( K )

FIGURE 5.33 Enlargement of muJtiplicity region for phthalide effectiveness factor.

-4.0E-004 +-~--.....---,.--.....----..--...-----, 450 500 550 600 650 700 750 &00 Bulk Temperature ( K)

FIGURE 5.35 Phthalide net reaction rate at bulk conditions.

U t:J dOUt:l.jJ !>llUllU! jUUJ!> !f-' dWt::> dlf~ f-'Ut: t::>.ldl\

"'"!"

pur>

u

~

J; 1.0 0.5

640

650

660

670

680

690

700

710

SJ :1.t:'llill:'>P.J UJ •tllfU flJ11LJ fl:1q,tOSJ11l 1111 IIIIJI p1111 lll n tl I Ill 1\i llfll 1111 1 111' II

The importance of surface processes in the dynamic behaviour of porous catalyst pellets

Although the surface phenomena associated with gas solid catalytic reactions and their implications on the behaviour of the system was briefly introduced earlier (chapter 2), it is discussed here with more details because of the importance of these phenomena for porous catalyst pellets. Also. more emphasis will be given to the surface phenomena and their effect on both steady state and dynamic characteristics of the porous catalyst pellet. The very early work of Hougen and Watson ( 1947) and the work of Kabel and Johanson ( 1962), Hayward and Trapnell (1964), Lebr eta/. (1968) and Denis and Kabel (1970a,b) show the importance of the adsorption-desorption step on the dynamic response of a tubular heterogeneous catalytic system. In this section beside describing the effect of chemisorption on the dynamics of the system, the effect of these processes on the steady state behaviour of the system is also discussed in some detail.

Bulk Temperature ( K)

5.2.1.2 Chemisorption and catalysis FIGURE 5.42 Phthalic anhydride selectivity at the multiplicity region.

One of the oldest theories relating to catalysis by solid surfaces was proposed by Faraday in 1825. This states that adsorption of

P:llU:lS~Jd~.• S! tU:llSAS ~4UO JnO!Any;:,q :>!WlmAp ;:,ql 'AHB:>pntua tpnw

'::>J:>qJBd :l4l JO ::>:>BjJOS ::>41 lE .. WHJ, lJ!lll B OlU! p;}dtunJ ~U!;}Q J::ljSUUJl SSUtll •~·••l•llth ' " ' " '

111.1 VI.I.U I .111J Lh.tt the rcu~..:11011 p t occc~h 111 I he

adsorbed fluid film. In fact there is much evidence against this simple view. For instance, the more effective adsorbents are not always the more effective catalysts, and catalytic action is highly specific i.e. certain reactions are influenced only by certain catalysts. The modem view, therefore, regards adsorption as a necessary but not sufficient condition for ensuring reaction under the influence of a solid surface. Adsorption is due to attraction between the molecules of the surface, called adsorbent, and those of the fluid, called adsorbate. In some cases the attraction is mild, of the same nature as that between like molecules, and is called physical adsorption. In other cases the force of attraction is more nearly akin to the forces involved in the formation of chemicaJ bonds, so the process is called chemical adsorption or chemisorption .

(a) exponential increase in rate with increasing temperature. (b) continuous fall in rate with increasing coverage.

o,

For activated adsorption, the sticking probability defined as the fraction of collisions between adsorbate molecules and surface resulting in chemisorption, may be defined as,

o• u/(8) exp

In this equation a is the condensation coefficient. It is the probability that a molecule is adsorbed provided it has the necessary activation energy Ead and coUides with a vacant surface site. The rate of adsorption is given by,

5.2. 1.3 Rates of chemisorption

ap

I'a •

The adsorptio n process is generally very fast on the surface of many clean metal films. However, in many cases of adsorption by a bulk adsorbent the rate of adsorption is quite slow, with equilibrium being reached in a few hours. An excellent review of rates of adsorption is given by Hayward and Trapnell (1964) as well as Low ( 1960). Tables 5.5 and 5.6 give some examples of fast and slow chemisorption, respectively. The kinetics of adsorption can be classified roughly into two categories:

1 hL dt.IIUI..LCIJ::Olh.:::. ul llu::. Ly!)c ul uu~utp

h III'U/t•d tlil'IJII.\'IJI'J)IW/1

Lion are:

/(0) exp.t ~lf!P ~tp 01 P;)1l3nb~ q UP-:> 1C>1f;)d lSAJRl~O C>I.Jl JO e>:>epns 1111 1111 Ct.; y) lln l l'lli tl p1rpoli< J'11 111 110111 11111111P 11t I'' .1 11'1 111 1

!POJd J:>pow pay!rdw!s lump111g lccluttquc can be u:;ctl to lump the llllt'upatllc lc t'c::.•::.-

1. 9

tances, adding them to the external resistances, and thus ending up with a lumped model with modified external resistances that include (in an approximate fashion) the intraparticle resistances. One of the most convenient methods for achieving this is through the use of the orthogonal collocation technique with one interior collocation point.

1.8 1.7

a: :3.66xt03 Ya: -2

1. 6 1.5 1.4 1.3

~

1. 2 1.1

CD ® @

1.0 0.9

@

C>X11 dw

(5. 139)

0

and at z-RP

For a sphericaJ pellet, fo r example, this will be: (5.140) (5.137) where r1 is the rate of surface reaction expressed in kmol/kg catalyst . h. The formulation of the equation in the dimensionless form depends upon the functional form of the rate of react ion function rs as shown later in this chapter.

Numerical methods of solution: Hansen ( 1971) used the orthogonal collocation method to solve both the steady state and transient equations of six different models of the porous particle of increasing complexity. He found that only 8 collocation points were necessary to obtain accurate results. This leads to a considerable saving in computing time compared to the conventional finite dift'erencc methods such as the Crank-Nicolson method. Ferguson and Finlayson ( I 970) used the collocation method to study a sim ilar problem. They proved that the method converges to the exact solution and demonst rated its superiority compared with the more conventional finite-difference methods.

The isothermal and non-isothermal effectiveness factors for the single unimolecular irreversible reaction. Equilibrium adsorptiondesorption model with linear isotherm: For simplicity of presentation and without any loss of generality we consider here the case where the bulk temperature and concentration a re taken as the reference temperature and concentration. In this case the boundary conditions (5.127) become: at w • 1.0

_1_

(!L (w2X~~>) ..

w2 dw

t/>2 • e>'< l - l ly>XA

(5.141)

which upon rearrangement and integration can be written as: -dx~~~

dw

w- 1.0

•t/>2

(5. 142)

•

Therefore, 17 can be written as:

3 11- -

dx~~ ~

t/>2

dw

w- 1.0

(5.143)

also fro m the boundary condition (in the case when Sh does not tend to infinity) we can write:

Sh

'1 - 3 . 2 ( 1.0 - XA) t/>

(5. 144)

Any of these formulas can be used. They can also be written in terms of dyldw, (1.0- y), and for any other geometry. The formulae in terms of ( 1.0 - XA) or (1.0- Yb) cannot be used when Sh- co or Nu- co.

The isothermal effecliveness factor When the heat effects are negligible or the external a nd intraparticle heat transfer coefficients are very large, the particle is isothermal and hence y = 1.0 and therefore the two equations reduce to one which in this case is linear:

dXA dw - Sh( 1.0 - X11) ; dy dw • Nu( 1.0- y)

There are a number of other formulae for '7 which can be easily derived from the above expression and the difterentiaJ equations of the system. For example, for the spherical catalyst pellet, the mass balance differential eq uation can be written as:

(5. 138)

(5.145)

1.. .. I

I

1

Wllh lhc UUUIIUUI y CUilUII IUil!>.

at w-0

at w; 1.0 (5.146) This case is easily solved analytically for both the spherical and slab geometries. However, for cylindrical particles, the analytical solution is not as simple as for the cases of slab and sphere. However, this does not present any real difficulty since Aris ( 1957) bas proved that all shapes can be transferred into the equivalent slab with the same specific surface as the original shape. Figure 5.46 shows the effectiveness factor versus Thiele modulus profiles for the cases of slab, cylinder and sphere geometries.

The behaviour at the difrerent regions is obviously more complex than the isothermal case. The corresponding '7- if> curve comprises four regions in general as shown in Figure 5.47. Region 1 corresponds to kinetic control and '7 -+ 1; the reactant concentration is uniform throughout the pellet which is at the same temperature as the gas. For an increased reaction rate, the effectiveness factor falls below unity as a result of " pore" diffusion resistance (region 2). The heat generated is removed rapidly enough to keep the pellet at a similar temperature to the gas. For higher rates of reaction (region 3) the particle becomes progressively hotter than the surrounding gas, although reasonably uniform in temperature. " Pore" diffusion

1-

Non-isothermal effectiveness factor The non-isothermal effective-

'0

ness factor can be obtained numericaJiy only by integrating the two points boundary value differential equations using different numerical techniques, the most efficient of these techniques is the orthogonal collocation method.

u

c: Cl

1.0

•

o.a 0.6

t

0.4

7l

$::'

0 .2

0 . 1~~~~--~~~~~-w~~--~--~~~~~~

0 .1

0.2

0.4

0.6 0.8 1

2

4

6

8 10

,tmponoq aq1 Ql!M

100

so 100

so

/J: Cref (-AH)Oeff

0

v

'-e · Tref

s

..

s

..

10 I::"

s:::: ... 10 ~

s

.~

~

-

~

u 0

O.S

UJ

11'1 11'1

~

....

41

>

0.1

-

o.os

u

UJ

0.1

0.5 1

5 10

so

100

500 1000

Tl

J-l:JTY

UOf7:JV().J .IV/n:JJ,OW!q .JOf (If 'X)":f• (If 'X)f /I()/J.111.lt tn/IIMfOIIIII/11 tO((tf

' \' )1{ (J ' \·).f

1000r--------------------------------,

intraparticle void space can be described by the mass balance equation, (5.147)

I

'\\

\

'\\

where De is the "effective" diffusion coefficient of component A within the porous structure and rs is the rate of surface reaction expressed in "kmol/kg catalyst.b". Similarly, a differential heat balance gives, \

2

\

A. ( d-T+ 2-dT) - = - ( - f:lH)rP r e dzl z dz s s

\ \

(5.148)

\

,'

where A.e is the "effective" thermal conductivity of the porous particle. For the special case of negligible external mass and heat transfer resistances between the surface of the particle and the surrounding fluid (Sh, Nu-+ oo), the boundary conditions are given by, at z = 0

I

I

/

/

I

I

I

0.1

10

:lc~ 8

dCA

=

dz

dT =O dz

FIGURE 5.50 Interphase and intraphase non-isothermal effectiveness factor for LHHW kinetics for the reaction C- P, y • 40. mass and beat transfer resistances. The reaction considered was a first order irreversible reaction catalyzed by the internal surface of the cata lyst pellet. Here the analysis is extended using the more realistic distributed model that takes into consideration the intraparticle mass and heat transfer resistances. The influence of reactant adsorption is displayed on 17 - 4> diagrams. In addition the case of bimolecular catalytic reaction will be presented. This reaction is known to have multiple steady states even under isothermal conditions, (i.e. ( - AH)r • 0). This type of multiplicity is referred to as "concentration multiplicity'' (Einashaie and Yates, 1973) in contrast to the well known "thermal multiplicity" phenomenon (Van Heerden, 1953; Aris, 1969a; Weisz and Hicks, 1962; Ray, 1972; Elnashaie and Cresswell, 1973a,b).

(5.149)

The rate of reaction kinetic model: (a) Unimolecular reaction. Following the derivation given in chapter

3, the rate of reaction for the unimolecular case wit11 non-equilibrium adsorption, is given by, (5.150) The rate constant for adsorption ka, surface reaction rate constant k and equilibrium constant for adsorption KA are all temperature dependent according to the Arrhenius expressions,

ka = kao exp (- EaiRGT) 1. The steady state model For a spherical catalyst pellet of radius R catalysing an irreversible gas phase reaction, the steady state d istribution of reactant A in the

k • k 0 exp ( - EIRGT) KA .. KAo exp ((- AH)AIRGT)

(5. 151)

•uoq~nba

~----------------------------~0001

;}OUBJ-eq ssew ~til ,\q paqpos~p ;)q u1>.0 ;}O~ds P!O" ;:,p!l.J~d~.JlU!

(b) Bimolecular reaction. Consider the reaclion,

/(X. y ) - jj(X, y)for unimolecular reaction

/(X, y) - h(X, y) for bimolecular reaction

A+B - C

It is assumed that B exists in large excess and that the rates of adsorption of A and B are much faster than the rate of surface reaction, i.e. equilibrium adsorption-desorption between the intraparticle gas concentration and surface concentration is established for both components A, B. For simplicity, it is further assumed that component B is weakly adsorbed relative to A, i.e. K8 « KA. Following the derivation given in chapter 3, the rate of reaction under the above assumptions is given by,

and where (5. J 57)

and q}eY(I-IIy)X

f2(X, y) = (

a+ace

(5.152)

where K 8 . C8 is assumed to tend to zero. The equilibrium and rate constants are temperature dependent following the expressions given by equations (5.151) and (5.153).

and Although a decade ago, non-monotonic kinetics of catalytic reactions were considered to be exceptional cases, nowadays it is becoming clear that non-monotonic kinetics in catalytic reactions are much more widespread than previously thought (e.g. Nicholas and Shah, 1976; Peloso eta/., 1979; Satterfield, 1980; Cordova and Gau. 1983; Yue and Olaofe. 1984; Yue and Birk, 1984; Das and Biswas, 1986; Takahashi et a/., 1986; Elnashaie et a/. 1990). The dimensionless equations

After inserting equations (5.150-5.153) into eqns. (5.147) and (5 .1 48) we obtain the follow ing dimensionless equations, where XA, the dimensionless concentration of component A is expressed as X, without the subscript A, V 2X=f(X,y)

(5.154)

and V 2y

= -

Pr/(X, y)

(5.155)

n:IY'X)2

(5.158)

All the dimensionless parameters and variables are defined in the nomenclature. The dimensionless boundary conditions are given by, at w=O dX = dy ~ O dw

(5.153)

(5.156)

dw

and at w = 1.0 (5.159)

From equations (5.154), (5.155) and (5.159) it can be shown that, I X(w)=

+Pr- y(w) Pr

(5. f 60)

and therefore equations (5.154) and (5.155) can be combined into one equation using equation (5 .1 60). The effectiveness factor

The effectiveness factor is defined as, 3(dX/dw)w - I.O rJ = /( 1, 1)

- 3(dyldw)w. 1.0

Prf(i, 1)

(5.161)

The effectiveness factor can also be expressed in the following integral form which is more suitable fo r computational purposes, I.O

where

J 0

1'J =

w 2f(X, y) dw

/(I , I )

(5.162)

.UI:>I1diJ.1nH

- ·-1 o·oz

le·o=09 _o·oz= .t

Method oj solution The non-linear two-point boundary value differential equation describing the system must be solved numerically. For the equilibrium adsorption cases, the r,- t1> diagram can be constructed using the mapping method (Weisz and Hicks, 1962). Non-equilibrium cases cannot be solved using the mapping method but instead we use the collocation technique (Villadsen and Stewart, 1967: Villadsen and Michelsen, 1978) described in Appendix B. The technique is more efficient than the Shooting method. For the kinetic control region (i.e. low values oft/>) 2 to 3 collocation points were sufficient to obtain accurate results. For the multiplicity regjon and the mass transfer control region (high values oft/>), 6-8 collocation points were necessary.

·· ~::::::-·

IOOl

15.0 "(: 20 -

./3 :0.4, "(~: 8

~ Cl')

~

LU

u

I&J

\i I

6.0 5.0 4.0

\H.. \ \\. \\\

~\ \\1 ..I.

3.0

, I

~

Results for the effect of chemisorption on the effectiveness factor (a) Unimolecular reaction

n

~

.

I&J

J:'

2.0

(a.J) Equilibrium adsorption-desorption and negligible surface coverage (t/> 0 - oo, Ot£- 0) This case has been investigated by Weisz and Hicks ( 1962) in great detail (some results are shown in Figures 5.48) and their results have been used to check our programs for the mapping and collocation methods. Although the above authors found only multiplicity of three steady states, Figure 5.48, Hlavacek and Marek (1968) and Copelwitz and Aris ( 1970) found multiplicity regions having as many as twenty-one steady states.

(a.2) Non-equilibrium adsorption-desorption and negligible surface coverage (a£- 0, ¢(1 is finite)

1.0 0.2

2

{;~ . /. .I.I

8.0

>

j:

3

·-MULTIPLICITY

10.0 « 0 ..... u c(

4

J

0.3 0.4

0.6

1.0

2.0

4>·THIELE MODULUS

FIGURE 5.51 Effect of rate of adsorption on the effectiveness factor for a unimolecular reaction. Activated adsorption: (J) 6., = 17.4, oo; (2)

¢.,= 17.4xl0- 4 ; (3) ¢.,- 34.8 x J0 - 5 ; (4) ¢, = 5.8 x to - 5; (6) ¢.,- 17.4 x 10 - 6 ; P• Pr·

6 = 17.4 x l0- 5 ; 11

{5)

The adsorption process is usually fast on evaporated films. However, on bulk solids, e.g. porous catalyst carriers, adsorption rates are usually slow and activated with activation energies typically in the range l 0-40 kcal /g mole {Hayward and Trapnell, 1964). Some activation energies for typical activated chemisorption process are given in Table 5.9. In their investigation of the catalytic dehydration of methylcyclohexane, Sinfelt et a/. ( 1960) found that to obtain a suitable kinetic expression finite rates of the adsorption-desorption process must be taken into consideration. ln this section allowance is made for finite rates of adsorption and both activated and nonactivated adsorption are considered:

Mahfouz, 1978). As the rate of adsorption decreases the effectiveness factor decreases and the multiplicity region shrinks. For very low rates of adsorption the multiplicity disappears altogether. An interesting feature of activated adsorption is that it gives multiplicity even for zero apparent activation energy of the reaction, i.e. y = 0. This type of multiplicity was also observed for the lumped case discussed earlier and was termed "adsorption multiplicity". This phenomenon is shown in Figure 5.52 for high values of¢ wh ich under equilibrium adsorption-desorption conditions will correspond to the "mass transfer control region" (Elnashaie and Mahfouz, 1978).

Activated adsorption

Non-activated adsorption

In the case of activated adsorption the rate of adsorption increases strongly with temperature, i.e. Ya > 0. Figure 5.51 shows the effect of the rate of adsorption on the effectiveness factor (Elnashaie and

For non-activated adsorption the rate of adsorption is insensiti ve to temperature Ya s 0. Figure 5.53 shows the effect of the rate of

t ., s =cJ.. · ., ·o =5t · oz =J.. 1 L----------------'- O'SL ~

200

a: ~

LU

z

fJ

Y:o.o

y : 20

a: 0

u 10.0

.

~

u

~LU

.

r::

2

Ill Ill &&I

z

&&I

~

\

\

10

5 .0

1-

~~.

~

:-

~

\\

20

Y0

:: 0.4

&.1.

\.

&&I

1.2

~

.\

&.1.

-·-MULTIPLICIT Y

20.0

, :().I

' ·'-,.

\.

~

Ill Ill

·~

r\! \

50

0

Y0 :20.o

ll~:=::::::::~ r·-... ~-

100

uopnJOS jo pOlfliJN

u

LU

ItLU

. s:::-

- --RANGE Of' \ . MULTIPLICITY

3-0 2.0

5 1.0

___...J...._ .a 9 1.0 2.0

.,_c:::;~._,__.__.._._..~....~.

.3

.4

2

.s

-6 .7

4> . THIELE

1

0.1

0.2

0.5

1.0

2-0

4>0 · THIELE MODULUS FOR ADSORPTDI FIGURE 5.52 Effect of T hiele modulus on adsorption multiplicity. Unimolecular reaction P101 Pr·

adsorption on the effectiveness factor for this case. "Adsorption multiplicity" is not possible for the non-activated case (Elnashaie and Mahfouz, 1978).

_._----1

3.0

MODULUS

FIGURE 5.53 Effect of rate of adsorption on tbe effectiveness factor for a unimolecular reaction. Non-acth•ated adsorption: (1) ¢a.= oo ; (2) 'BOJddU JJSUlUl W01UdlU0W ql JO lUWl'BT

""

:)"IIIII

1;:)"1'11

(9) (' N 1 1 ~1) ' (

IIOI 'li1JJI(J ( \ )

CqUallOII!>

N 1 total

l01 lhe

'"'' ._" ""a '7 -

• '\

I (

JIU!I. n.:IUllOII!> . J

he

I

UH.IIIHII Y OIIIU!>IUII lCIIIl !> Ill

the

Stefan- Maxwell equation (5.172) arc approximated by:

r-------...&..,!"'!2------. 3

(5.179)

FREE MOLECULE N j dlff

Ntv isc

N j surf

where D ,~ is the effective molecular diffusivity of component i in a multicomponent mixture. Equation (5.177) with negligible viscous flow becomes a simplified equation of the form: N, • - D{ grad C,

CO NTINUUM

(5.180)

where: 1

-a-+-Df D~, Dfm

Nj total

FIGURE 5.58 Electrical analogue circuit for a mnemonic device for combining different transport mechanisms.

and Dfm is obtained from the Wilke ( 1950) equation for diffusion of component i through other stagnant components of the mixture: _1 _ _

D;m

(a) Simplified model I

Equat.ion (_5.177) resulting. from the rigorous model (dusty gas model) for dtffusJOn and flow tS not easy to implement for a multicomponent system. Some investigators (Soliman el a/., I988; Xu and Froment, 1989b; Elnashaie and Abashar, 1992) have used simplified

t_IY' n

I - Y; 1"' 1 D/J

Simplified models

The Fickian diffusion models with constant effective diffusivities pres.ented earlier and the rigorous dusty gas model presented in this sectt