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Chemical Process Dynamics and Control /Summer 2021
Instructors: Prof. Ronald Hedden and Dr. Matthew Titus
Student Name:
Exam #1
● The exam is closed book – no calculators/laptops permitted.
● Answer all questions. Show all work for full credit.
● Write legibly, justify your answers, and state your assumptions.
1. The following system of tanks is found in a mixing/blending facility. Volumetric flowrates (m 3/min)
are given the symbol “F” and the concentration of a dissolved salt (mol/m 3) is given the symbol “C.”
Both tanks are well-mixed at all times. The gate valves both have a flow coefficient of
Cv = 2.0 m2.5/min. F2 can flow either to the left or the right, depending on liquid levels.
a. (8 pts.) Formulate the ODEs describing dh1/dt and dh4/dt, assuming Cin is held constant, but
disturbances in Fin are possible.
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b. (8 pts.) Formulate the ODEs describing dC1/dt and dC4/dt, assuming all liquid levels and flow rates
are constant and remain at their steady-state values, but disturbances in C in are possible.
c. (8 pts.) If Fin has a constant value of 1.0 m3/min and Cin has a constant value of 30 mol/m3, find the
steady-state values of h1, F2, F3, and h4.
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d. (10 pts.) Suppose disturbances in both Fin and Cin can occur. Classify each of the variables listed by
circling the correct answer. For any variables that might be considered both states and outputs, please
choose "State." No credit for answers with more than one choice circled!
Variable
Cv
Input
Output
State
Parameter
Fin
Input
Output
State
Parameter
F2
Input
Output
State
Parameter
F3
Input
Output
State
Parameter
A1
Input
Output
State
Parameter
Cin
Input
Output
State
Parameter
C1
Input
Output
State
Parameter
C4
Input
Output
State
Parameter
h1
Input
Output
State
Parameter
h4
Input
Output
State
Parameter
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2. A stirred tank reactor is running the exothermic reaction A  2B in the liquid phase, with equal inlet
and outlet flow rates F. Liquid density is assumed constant, but temperature can vary with time.
Using (system = liquid in the reactor), the following ODEs describe the time dependence of the state
variables, CA and CB.


dC A  F  in
   C A  C A  kC A
dt
 VR 


dC B  F  in
   CB  CB  2kC A
dt
 VR 
Reactor performance also depends on temperature, so the energy balance is important to describe system
dynamics. In order to remove heat from the reactor during operation, a cooling jacket is added, which
removes heat at a rate of 𝑄̇. With the cooling jacket in place, the energy balance becomes:
=
̇
+ 𝐹 𝑇
−𝑇 −
(𝑘𝐶 )
ΔHR is the heat of reaction, VR is the liquid volume in the reactor, “n” is the total number of moles in the
reactor, Cp is molar heat capacity, and ρ is molar density; these quantities are taken to be constants
(parameters). The kinetic rate constant is temperature dependent and has the following form:
𝐸
𝑘 = 𝑘 ∗ exp (− )
𝑅𝑇
Additional parameters: Ea is an activation energy, R is the gas constant, and k 0 is a kinetic pre-factor.
a. (10 pts.) Using the following definitions, convert the energy balance ODE to deviation variables to
obtain dx3/dt = f3(x,u).
in
u1  C A  C A, s
in
in
u2  C B  C B ,s
u3  F  Fs
in
x1  C A  C A,s
x2  CB  C B , s
𝑥 =𝑇−𝑇
y1  x1
y 2  x2
𝑦 =𝑥
𝑢 = 𝑄̇ − 𝑄̇
𝑦 =𝑘
𝑢 =𝑇
−𝑇
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b. (8 pts.) For the state-space model x  Ax  Bu , write the dimensions of the A, B, C, and D matrices
(rows  columns).
A matrix:
(
 )
C matrix:
(
 )
B matrix:
(
 )
D matrix:
(
 )
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c. (8 pts.) Find the component A33 of the A matrix in the state-space model.
d. (8 pts.) Find the component B33 of the B matrix in the state-space model.
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3. (32 pts.) The following MATLAB code describes a system of two tanks, but it contains errors. Once
the script is debugged, it should produce the following plot, exactly as shown:
Please correct the coding errors. Coding errors may include incorrect syntax, missing or extra
characters, misplaced lines, math errors, etc. The number of incorrect lines is not given. Hint: no lines
of code are completely missing.
a. Circle the line number of each line that needs to be corrected or moved. No credit if instructions are
not followed.
b. Next to each incorrect line (or underneath it), write the corrected line of code, or else draw an arrow
to where the line should be moved if a line is misplaced.
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% Consider the following dynamic model for a 2-tank system
%
%
h1dot = (1/A1)*Fi - (Cv/(2*A1*sqrt(h1bar))*h1
%
h2dot = (Cv/2*A2*sqrt(h1bar))*h1 - (1/A2)*F2
%
% The inputs to the system are volumetric flowrates Fi and F2
% The outputs are equal to the states, which are the tank levels h1 and h2
% We wish to plot the dynamic response of the system to step
% changes in the inputs
%
%% Housekeeping
1
clc;
2
clear all;
3
close all;
%% Parameter definitions
4
A1 = 1; % Cross-sectional area of tank 1 [m^2]
5
A2 = 1; % Cross-sectional area of tank 2 [m^2]
6
Cv = 1; % Valve coefficient [m^2.5/hr]
%% Steady-state values of states and inputs
7
Fibar = 1; % Flowrate into tank 1 at steady-state [m^3/hr]
8
F2bar = 1; % Flowrate out of tank 2 at steady-state [m^3/hr]
9
h2bar = 1; % Level of liquid in tank 2 at steady-state [m]
10
F1bar = Fibar; % Flowrate out of tank 1 at steady-state [m^3/hr]
11
h1bar = (Fibar/Cv)^2; % Level of liquid in tank 1 at steady-state [m]
%% State space model definition
12
A = [???];
13
B = [];
14
C = [];
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15
D = [];
%% Set-up simulation
16
tbeg = 0;
17
tend = 35;
18
tsim = tbeg:tend;
19
x0 = [0 0];
20
tstepFi = 5;
21
tstepF2 = 10;
22
Fistep = 0.1;
23
F2step = 0.1;
24
Fi = zeros(length(tsim),1);
25
F2 = zeros(length(tsim),1);
26
for i=1:length(tsim)-1
27
if tsim(i) < tstepFi
28
29
Fi(i) = 0;
else
30
31
Fi(i) = Fistep;
end
32
end
33
for i=1:length(tsim)-1
34
if tsim(i) < tstepF2
35
36
F2(i) = 0;
else
37
38
39
F2(i) = F2step;
end
end
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%% Run simulation
40
tsol = [ ];
41
xsol = [ ];
42 for i=1:length(tsim)
43
u = [Fi(i);F2(i)];
44
if i=1
45
46
47
(t_part,x_part) = ode45(@ssmodeleqn,[tbeg tend],x0,[],u);
else
(t_part,x_part) = ode45(@ssmodeleqn,[tsim(i) tsim(i+1)],x0,[],u);
48
end
49
tsol = [tsol;t_part];
50
xsol = [xsol;x_part];
51
end
%% Convert deviation variables back to physical variables
52 hsol = xsol + [h1bar ; h2bar];
53
Fi = Fi - Fibar;
54
F2 = F2 - F2bar;
%% Plot simulation result
% Give plot a 10% padding to avoid clipping by figure borders
55
min1X = min(tsol);
56
max1X = max(tsol);
57
min1Y = min(hsol);
58
max1Y = max(hsol);
59
pad1Y = 0.10*(max1Y-min1Y);
60
min2X = min(tsim);
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61
max2X = max(tsim);
62
min2Y = min([Fi;F2]);
63
max2Y = max([Fi;F2]);
64
pad2Y = 0.10*(max2Y-min2Y);
65
subplot(1,2,1)
66
plot(tsim,hsol)
67
axis([min1X max1X min1Y-pad1Y max1Y+pad1Y])
68
title('Two-Tank System (Input Step Response)')
69
xlabel('Time [h]')
70
ylabel('Tank Level [m]')
71
legend('Tank 1','Tank 2','location','southeast')
72
subplot(1,2,2)
73
plot(tsim,Fi,'bo-')
74
hold on;
75
plot(tsim,F2,'r-')
76
axis([min2X max2X min2Y-pad2Y max2Y+pad2Y])
77
legend('F_i','F_2','location','southeast')
78
xlabel('Time [h]')
79
ylabel('Flowrate [m^3/h]')
80
hold off;
%% Function block
81
function dxdt = ssmodeleqn(t,x,u,A,B)
82 dxdt = Ax + Bu;
83 end
GOOD LUCK!