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UNIVERSITY OF LAGOS Chemical Engineering Department 2015/2016 SESSION, 1st SEMESTER

CHG 501: Process Dynamics and Control Tutorial I –Introductory Concepts and Review of Mathematical tools Date Given: Date Due: Test Date:

February 01, 2016 February 09, 2016, 10:00 am February 10, 2016 5:00 – 6:00 pm

Question 1 A process furnace heats a process stream from near ambient temperature to a desired temperature of 300oC. The process stream outlet temperature is regulated, by manipulating the flow of fuel gas to the furnace, as shown below.

a. b. c. d. e. f.

What are the objectives of this control strategy? What is the measured output? What is the manipulated input? what are the possible disturbances? Is this a feed-forward or feedback control system? Discuss safety, environmental and economic issues.

1

Question 2

The process in the figure below vapourises liquid butane and mixes the vapour with compressed air. The mixture flows to a packed bed reactor after it has been cooled in a heat exchanger. In order to meet our control objectives of safety and quality control, the following needs to be implemented: The pressure in the vapouriser (P1), which is important for safety, should be controlled by adjusting the vapour release from the vapouriser (v3); Liquid level in the vapouriser (L1), should not overflow the vessel or drain empty, and this can be controlled by adjusting the flow of liquid butane from storage to the vaporizer (v1); the percentage of butane in the mixed stream (A1), which is important in order to avoid an explosive concentration, can be controlled by adjusting the flow of air through the valve in the compressor suction, v4. a.

b.

List the output (controlled) and input (manipulated) variables. Show the implementation of the above control schemes on a simple P&ID.

2

Question 3 Draw the process instrument diagram for the Control of the Sugar-Refining Process discussed below. Use either AutoCAD, Microsoft Office Visio, the draw toolbar of Microsoft Word or PowerPoint or any other flow-sheeting Software. The process units, shown below, form part of the process to refine sugar. Raw sugar is fed to the process through a screw conveyor. Water is sprayed over it to form sugar syrup. The syrup is heated in the dilution tank. From the dilution tank the syrup flows to the preparation tank where more heating and mixing is accomplished. From the preparation tank the syrup flows to the blending tank. Phosphoric acid is added to the syrup as it flows to the blending tank. In the blending tank lime is added. This treatment with acid, lime, and heat serves two purposes. The first is that of clarification, that is, the treatment causes the coagulation and precipitation of the nosugar organics. The second purpose is to eliminate the coloration of the raw sugar. From the blending tank the syrup continues to other processes in the refinery plant.

Sugar refining process The following variables are thought to be important to control. i. ii. iii. iv. v.

Temperature in the dilution tank. Temperature in the preparation tank. Density of the syrup leaving the preparation tank. Level in preparation tank. Level in 50% acid tank. The level in the 75% acid tank can be assumed constant.

vii. viii. ix. x.

The flow of syrup and 50% acid to the blending tank The pH of the solution in the blending tank. Temperature in the blending tank. The blending tank requires only a high-level alarm.

vi.

The strength of the 50% acid. The strength of the 75% acid tank can be assumed constant.

Question 4

3

Consider the conical water tank shown below. Obtain the model of the process if the flowrate out of the tank is a function of the square root of height of water in the tank ( F0   h ). List state variables, input variables and parameters. (Hint: Use height as a state variable.)

Question 5 Model a mixing tank with two feed streams, as shown below. Assume that there are two components, A and B. C represents the concentration of A. (C1 is the mass concentration of A in stream 1 and C2 is the mass concentration of A in stream 2): Model the following cases: a. Constant volume, constant density b. Constant volume, density varies linearly with concentration c. Variable volume, density varies linearly with concentration

Question 6 Consider two tanks in series where the flow out of the first tank enters the second tank. Our objective is to develop a model to describe how the height of liquid in tank 2 changes with time, given the input flowrate Fo (t) . Assume that the flow out of each tank is a linear function of the height of liquid in the tank ( F1 sectional area.

 1h1

and

F2   2 h2 ) and each tank has a constant cross-

Question 7 Two liquid surge tanks (with constant cross-sectional area) are placed in series. Write the modeling equations for the height of liquid in the tanks assuming that the flowrate from the first tank is a function of the difference in levels of the tanks and the flowrate from the second tank is a function of the level in the second tank. Consider two cases: (i) the function is linear and (ii) the function is a square root relationship. State all other assumptions.

Question 8 Liquid surge tanks, containing hydrocarbons, usually have a gas “blanket” of nitrogen or carbon

4

dioxide to prevent the accumulation of explosive vapours forming above the liquid, as shown below

Develop the modeling equation with gas pressure and liquid volume as the state variables. Let and

qf

q represent the inlet and outlet gas molar flowrates, Ff and F the liquid volumetric flowrates,

V the constant (total) volume, V1 the liquid volume, and P the gas pressure. Assume the ideal gas law. Show that the modeling equations are:

dV  Ff  F dt dP P RT  ( Ff  F )  (q f  q) dt V  V1 V  V1 and state any other assumptions made.

Question 9 Most chemical process plants have a natural gas header that circulates through the process plant. A simplified version of such a header is shown below.

The natural gas enters the process plant from a source (the natural gas pipeline) through a control valve. It flows through the plant piping, which for simplicity has been represented as a perfectly mixed drum. Another valve connects the plant piping to the gas drum for a boilerhouse unit, from where it passes through another valve to the furnaces. Write modeling equations assuming that the pressures in drums 1 and 2 are the state variables. Let the input variables be h1 (valve position 1), h2 (valve position 2), and Pi (source pressure).

Question 10 What is the Laplace transform of the function f(t) shown in the figure below

5

f(t) 1

Question 11

a+b

a

0

t

u (s) of the following input function?

a.

What is the Laplace transform

b.

Evaluate u(s) for A =1, σ =1, b = 1, and c = 2

Question 12 Obtain the partial-fraction expansion of the following function using MATLAB:

F ( s) 

10( s  2)( s  4) ( s  1)( s  3)( s  5) 2

Then, obtain the inverse the inverse Laplace transform of F(s) [Hint: you can also use the syntax conv or poly for the numerator and denominator] ……..

Question 13 A process input has the following Laplace transform:

u(s) 

2 6 3s  e s3 s2

What is the time domain input, u(t)? Sketch the time domain input.

Question 14 For the given transfer function

y ( s) 2  u ( s ) ( s  2)( s 2  9) Derive

y (t )

with respect to a unit impulse input and sketch this response.

Question 15 Use the initial and final value theorems of Laplace transforms to sketch the response of the process output for a unit step input change to the following transfer function

6

g ( s) 

5s  12 7s  4

Question 16 For the given ordinary differential equation 2

d

y

dt2

4

dy  5 y  u (t ), dt

with y (0)  y (0)  0

N ,,,m,

Using any plotting software, sketch the time-domain response for a step input of magnitude 3.

Question 17 a) b)

A car tyre has a slow leak. The flowrate of air out of the tyre is proportional to the pressure of air in the tyre. The initial pressure is 30 psig, and after five days the pressure is down to 20 psig. How long will it take to reach 10 psig? Repeat a) but this time assuming the flowrate of air out of the tyre is proportional to the square root of the pressure of air in the tyre Compare your results.

Question 18 Write, in terms of deviation variables, the linear approximation to the nonlinear function

f (x, y, z)  2x2  xy2  3 at the point

xs  1, ys  2, z s  3,

y z

Question 19 The density of ideal gas is given by the following formula:



MP RT

where M is the molecular weight and R is the ideal gas constant. Find the linear approximation, in terms of deviation variables, of the density as a function of T and P and evaluate the coefficients for air (M = 29) at 300K and atmospheric pressure (101,300 N/m 2). Ideal gas constant = 8,314 N-m/kgmole-K

Question 20 In Summary, what knowledge have you gained in class under the topic “Introductory Concepts and Review of mathematical tools” This tutorial is to help your learning process and understanding of what was taught in class, and note that this understanding will be tested as soon as possible. Therefore, be wise (wisdom is the application of understood knowledge)

All the very best in your learning/understanding process. 7