Dynamic modeling of a non-isothermal CSTR
Raymond B Cyiza
Robotics and Cybernetics
Applied Computer Information
Technology
Oslo, Norway
s326055@oslomet.no
Abstract—A nonlinear model dynamic model of a nonisothermal continuous stirred tank reactor was developed. The
model has two output variables and four input variables. The
nonlinear model was linearized using Taylor series expansion.
Four experiments were designed around the operating point
with step changes in the input variables. The performance of the
linear model was compared to the nonlinear model using the
integral of absolute error. None of the performance from the
linear model to step changes in cooling medium temperature Tc,
inlet temperature Ti, inlet composition cAi and inlet flow rate q
can used future control purposes, since it was suspected that the
composed transfer function matrix inside the simulink process
was perceived to be incorrect. Also each of the performances of
the linear model to all step changes showed wrong direction and
magnitude, which would not be acceptable for further use. A
system identification approach was done in order to develop
linear models with better fitness criteria.
Keywords—first principles modeling, linearization, step
responses, integral error criteria, continuous stirred tank reactor,
systemidentification toolbox
B. Schematic diagram of the process and list of symbols
Figure 1.A non-isothermal continuous stirred-tank reactor.
The constants and variables of the process are given in
Table I List of symbols.
TABLE I.
I. INTRODUCTION
The aim of Assignment 2 is to develop linear transfer
function models based on the non-linear, non-isothermal
continuous stirred tank reactor developed in Assignment1.
First, the data collection experiments are designed. Then, the
data is collected from the nonlinear model implemented in
Simulink for 4 input variables and 2 output variables. The four
data sets are imported to SystemIdentification toolbox transfer
function models are identified between the input variables and
output variables. The best models are exported to Simulink
and compared with the nonlinear model in Simulink.
This article addresses the following questions: Can the
linear model be used for control studies? Is the dynamic
behavior of the linear model similar to the dynamic behavior
of the non-linear model?
II. SYSTEMATIC APPROACH FOR DEVELOPING DYNAMIC
Symbol
𝑇𝑖
𝑐𝐴𝑖
𝑇(0)
Nom.value
Temperature in
K
350
Component A in
mol/L
1
K
350
g/L
1000
mol/l
0.5
Component A
𝑉
Volume
L
100
𝑞
Flow rate
L/min
100
Heat of reaction per mole
J/mol
5*10^4
K
8750
J/min K
5*10^4
min^-1
7.2*10^10
K
300
j/g K
0.239
−∆𝐻𝑅
𝐸/𝑅
𝑈/A
𝑘0
Systematic approach for developing dynamic models,
presented in Chapter 2.2. in (Seborg, Edgar, Mellichamp, &
Doyle, 2017) will be used to develop the models.
𝐶
ACIT4810 Assignment2 2020 ©tiina.komulainen@oslomet.no
Unit
Density
𝑐𝐴 (0)
𝑇𝑐 (0)
The end use is to compare the best data models exported
from SystemIdentification toolbox with the non-linear model
Simulink.
Description
Temperature in tank
𝜌
MODELS
A. Modeling objectives and end use
The model’s objective is to create a dynamic model based
on the non-linear, non-isothermal continuous stirred tank
reactor developed in Assignment1.
LIST OF SYMBOLS
Activation of energy/gas
constant
Overall
heat
transfer
coefficient/heat transfer area
Frequency factor
Cooling medium temperature
Heat capacity
C. Assumptions
1. The CSTR has perfect mixing → 𝑇𝑜𝑢𝑡 = 𝑇𝑡𝑎𝑛𝑘 (𝑇)
2.
Density and heat capacity C of the liquid are constant
in the tank, this means that temperature dependence is
negligible
3.
The liquid hold-up volume 𝑉 is kept constant by an
overflow line → 𝑐𝐴𝑖 = 𝑐𝐴
4.
The thermal capacitances of the coolant and the
cooling coil wall are negligible compared to the
thermal capacitance of the liquid in the tank.
5.
The increase in coolant temperature as the coolant
passes through the coil is neglected
6.
The overall heat transfer coefficient 𝑈 and the heat
transfer area 𝐴 are assumed to be constant. The rate of
heat transfer from the reactor contents to the coolant
is given by
𝑄 = 𝑈𝐴(𝑇𝑐 − 𝑇)
8.
Shaft work and heat losses to the ambient can be
neglected.
1) The nonlinear model
The non-isothermal tank reactor is described in Chapter
2.4.6 and Example 2.5 in (Seborg, Edgar, Mellichamp, &
Doyle, 2017).
𝑑𝑇
𝑑𝑡
=
𝑞(𝑇𝑖 −𝑇)
𝑉
=
+
𝑞(𝑐𝐴𝑖 −𝑐𝐴 )
𝑉
𝑈𝐴(𝑇𝑐 −𝑇)
𝑉𝜌𝐶
− 𝑐𝐴 𝑘0 𝑒−𝐸/𝑅𝑇 (2)
+
𝑑𝑐𝐴 ′(𝑡)
𝑑𝑡
′ (𝑡)
= 𝑎11 𝑐𝐴′ (𝑡) + 𝑎12 𝑇 ′ (𝑡) + 𝑎13 𝑞′ (𝑡) + 𝑎14 𝑐𝐴𝑖
(4)
Where:
𝑎11 = (−𝑞̅/𝑉 − 𝑘_0 𝑒^(−𝐸/(𝑅𝑇̅ )) )
The heat mixing is negligible compared to the heat of
reaction.
𝑑𝑡
Component A: The linearized equation for concentration of
component A. Notice the variables denoted with ‘ are
deviation variables from steady state marked with ̅ .
𝑒𝑞(1)
7.
𝑑𝑐𝐴
𝑞 ′ = 𝑞 − 𝑞̅
(−∆𝐻𝑅 )𝑐𝐴 𝑘0 𝑒−𝐸/𝑅𝑇
𝐶𝜌
(3)
𝑎12 = (−
̅̅̅̅𝐸k
cA 0 e
R𝑇̅ 2
E
− ̅
R𝑇
)
𝑐𝐴i − 𝑐𝐴
)
V
q̅
𝑎14 =
v
𝑎13 = (
Energy: The linearized equation for temperature in tank T.
Notice the variables denoted with ‘are deviation variables
from steady state marked with ̅ .
𝑑𝑇′(𝑡)
= 𝑎21 𝑐′𝐴 (𝑡) + 𝑎22 𝑇 ′ (𝑡) + 𝑎25 𝑞′ (𝑡) + 𝑎23 𝑇𝑐′ (𝑡) + 𝑎24 𝑇𝑖′ (𝑡) (5)
𝑑𝑡
Where:
D. Degrees of freedom analysis
(−∆𝐻𝑅 k 0 e
𝑎21 =
Listing all the quantities:
9Constant parameters:
𝑉, 𝜌, 𝐶, 𝑅, 𝐸, 𝑘0 , 𝑈, 𝐴, ∆𝐻𝑅
𝑎22
E
− ̅
R𝑇 )
𝜌𝐶
(−∆𝐻𝑅 )𝑉c̅̅̅𝐸k
1
A
0e
=
[−(𝑞̅𝜌𝐶 + 𝑈𝐴) +
𝑉𝜌𝐶
R𝑇̅ 2
E
− ̅
R𝑇
]
𝑈𝐴
𝑉𝜌𝐶
𝑞̅
=
𝑉
𝑎23 =
6Variables:
𝑇, 𝑐𝐴 , 𝑇𝑐 , 𝑞, 𝑐𝐴𝑖 , 𝑇𝑖
2Equations:𝑒𝑞(2)𝑎𝑛𝑑 𝑒𝑞(3)
Identifying 𝑁𝐸 and 𝑁𝑉 :
Number of equations: 𝑁𝐸 = 2
Number of variables: 𝑁𝑉 = 6
Degrees of freedom 𝑁𝐹 :
𝑁𝐹 = 𝑁𝑉 −𝑁𝐸 = 6 − 2 = 4
Identifying the output variables:
Controlled output variables: 𝑇, temperature in the tank and
𝑐𝐴 ,component A
Identifying the input variables: 𝑇𝑖 , 𝑐𝐴𝑖 , 𝑞
E. Simplify the ODEs and linearize the model
Define deviation variables,
𝑐𝐴′ = 𝑐𝐴 − 𝑐̅𝐴
𝑇 ′ = 𝑇 − 𝑇̅
𝑇𝑐′ = 𝑇𝑐 − 𝑇̅𝑐
′
𝑐𝐴𝑖
= 𝑐𝐴𝑖 − ̅̅
𝑐𝐴𝑖
̅̅
𝑇𝑖′ = 𝑇𝑖 − 𝑇̅𝑖
𝑎24
𝑎25 =
𝑇𝑖 − 𝑇
𝑉
Present the linearized models in transfer function format
𝐹11
𝐹12
𝐹13
𝐹14
𝑞𝑖′ (𝑠)
′
′ (𝑠)
(𝑠)
𝐶
𝑠 − 𝑎11 𝑠 − 𝑎11 𝑠 − 𝑎11 𝑠 − 𝑎11 𝑐𝐴𝑖
[ 𝐴 ]=
𝐹22
𝐹23
𝐹24
𝐹25
𝑇′(𝑠)
𝑇𝑐′ (𝑠)
[ 𝐹26
𝐹26
𝐹26
𝐹26 ] [ 𝑇𝑖 ′(𝑠)]
Where:
𝐹11 = 𝑎12 𝐹22 𝐹26 + 𝑎13
𝐹12 = 𝑎12 𝐹23 𝐹26 + 𝑎14
𝐹13 = 𝑎12 𝐹24 𝐹26
𝐹14 = 𝑎12 𝐹25 𝐹26
𝑎13 𝑎21
𝐹22 =
𝑠 − 𝑎11
𝑎14 𝑎25
𝐹23 =
𝑠 − 𝑎11
𝐹24 = 𝑎23
𝐹25 = 𝑎24
1
𝐹26 =
𝑎 𝑎
𝑠 − 𝑎22 − 21 22
𝑠 − 𝑎11
F. Classify the inputs as DVs and MVs
The cooling medium temperature Tc is a manipulated
variable in the CSTR shown in example 2.5 in (Seborg,
Edgar, Mellichamp, & Doyle, 2017). The inlet concentration
is disturbance variable. The inlet flow is a manipulated
variable and the inlet temperate is a disturbance variable.
First we have the manipulated variables: 𝑇𝑐 , 𝑞
Second, we have the disturbance variables: 𝑐𝐴𝑖 , 𝑇𝑖
III. MATERIALS AND METHODS
A. Hardware and Software
The software was run on Acer Nitro AN517-51.
MATLAB software package version R2020a was used for the
simulations. The simulations method was ode15s with
automatic settings for the time step and error tolerance.
The simulation model was built in Simulink according to the
nonlinear model and the linear state space model. The model
parameters and test procedures are implemented in m script,
see Appendix 1.
In the first test cooling medium temperature will be
changed from nominal value of 300K to 290K, then back to
nominal value, then up to 302K and finally back to nominal
value.
TABLE II.
TEST PROCEDURE1 COOLING MEDIUM
Time
[min]
Action
t=0
Initial values
Tc0=300K
Start simulation
t=10
Tc1=290K
t=20
Tc2=300K
t=30
Tc3=302K
t=40
Tc4=300K
t=50
Stop simulation
t=0
Collected
variables
Tc, T, cA
In the second test the flowrate will be changed from
nominal value of 100L/min to 90L/min, then back to nominal
value, then up to 110L/min and finally back to 100L/min.
TABLE III.
B. Nonlinear simulation model in Simulink
The nominal operating point was chosen to cA =0.8771 and
Time
[min]
T = 324.5K as this is the global minimum of the ordinary differential
functions.
t=0
TEST PROCEDURE2 FLOW RATE
Action
t=0
Initial values
qi0=100L/min
Start simulation
t=10
qi1 =90L/min
t=20
qi2=100L/min
t=30
qi3=110L/min
t=40
qi4=100L/min
t=50
Stop simulation
Collected
variables
q, T, cA
In the third test the inlet concentration will be changed
from nominal value 1 mol/L to 0.9 mol/L, then back to
nominal value 1mol/L, then up to 1.1 mol/L and finally back
to nominal value.
Figure 2 Nonlinear model in Simulink
TABLE IV.
The m-script is given in Appendix1. (Includes the linearmodel-function)
C. Experimental plan – input signal design
The input signal design must have high amount of
variation in order for the estimation method to give enough
information about the dynamics of the system. We will
implement a periodic part with slow variation and large
amplitude. We will then compare the sinus wave to the step
changes in the cooling medium temperature, inlet flow rate,
reactant component A and inlet temperature to see which
model shows the best input-output dynamics.
D. Experimental plan - Test procedure
Time
[min]
TEST PROCEDURE3 INPUT CONCENTRATION
Action
t=0
Initial values
cAi0=1 mol/L
Start simulation
t=10
cAi1=0.9 mol/L
t=20
cAi2=1 mol/L
t=30
cAi3=1.1 mol/L
t=40
cAi4=1.0 mok/L
t=50
Stop simulation
t=0
Collected
variables
cAi, T, cA
In the fourth test the inlet temperature will be changed
from nominal value 350K to 340K, then back to nominal
value, then up o 355K and then finally back to nominal value
again.
TABLE V.
Time
[min]
TEST PROCEDURE4 INLET TEMPERATURE
Action
t=0
Initial values
Ti0=350K
Start simulation
t=10
Ti1=340K
t=20
Ti2=350K
t=30
Ti3=355K
t=40
Ti4=350K
t=50
Stop simulation
t=0
Collected
variables
Ti, T, cA
E. Experimental plan – Modeling error indices
The models are compared using the integral of absolute
error(IAE) index between the nonlinear model and linear
𝑡5
model: 𝐼𝐴𝐸 = ∫0 |𝑒(𝑡)|𝑑𝑡 (6)
Figure 3 Reactor temperature T variation with changes in cooling
water temperature TC (Test procedure1).
F. System identification procedure
In order to identify the transfer functions in system
identification toolbox, input data and output data will be
measured from the CSTRsys2 (the transfer function in
MATLAB, (Appendix1)) using a “to workspace” source from
Simulink, then the data of inputs and outputs will be collected
inside a .txt file so that we can implement it in our .m code
while we prepare for the system identification process.
G. Experimental plan – Criteria for model fitness
In system identification toolbox the identified transfer
function models are compared to the validation data using:
𝑁𝑂𝑅𝑀(𝑌−𝑌̅)
𝑓𝑖𝑡 = [1 −
] ∗ 100 , 𝑤ℎ𝑒𝑟𝑒 𝑌 is the measured output
𝑁𝑂𝑅𝑀(𝑌−|𝑌|)
and 𝑌̅ is the simulated(predicted) model output.
Figure 4 Reactant A concentration variation with changes in
cooling water temperature TC (Test procedure1).
IV. RESULTS
A. Data collection results
Here we have the collected data from the test procedures
of the change in cooling water temperature, inlet flow change,
component A in change and the inlet temperature change.
The simulations were carried out according to the test
procedure.
The data collected is presented in Figures 3 - 14
Figure 8 Reactant A concentration variation with changes in inlet
concentration cAi (Test procedure3).
Figure 5 Reactor temperature T variation with changes in inlet
flow rate qin (Test procedure2).
Figure 6 Reactant A concentration variation with changes in inlet
flow rate qin (Test procedure2).
Figure 7 Reactor temperature T variation with changes in inlet
concentration cAi (Test procedure3).
Figure 9 Reactor temperature T variation with changes in inlet
temperature Ti (Test procedure4).
Figure 10 Reactant A concentration variation with changes in inlet
temperature Ti (Test procedure4).
Figure 12 Inlet flow rate qin steps in Simulink. (Test procedure 2)
Step changes in cooling medium temperature, inlet flow rate,
reactant A concentration through inlet and inlet temperature
are:
Figure 13 Reactant A cAin in Simulink. (Test procedure 3)
Figure 11 Cooling medium temperature Tc steps in Simulink. (Test
procedure 1)
Tp1,
Tp2,…
P1
-27.781
P01
-0.665
P0D1
0.0076
P1D
-27.781
1e-06
P2
0.9013
605.55,
ay
Td
Tz1,
Tz2,.
.
1e-06
8703%
-3608%
0
12.74%
0
-8703%
-1387%
8.1099
P2D
310.09
10000,
0
-160.04%
0.36507
P2DZ
-30.543
601.3,
1.846
419.47
-2765%
P1I
0.0066
945
0.044734
3.499%
P3I
0.3139
7
0.0045064,
6.82%
6.398
5391.3,
7159.4
Figure 14 Inlet temperature Tin steps in Simulink. (Test procedure
4)
The IAE index was calculated for each output-input variable
pairs between the non-linear and the linear model given in
Table 1.
Table 1:Model error indices.
Output
variables
cA
T
IAE_Tc
IAE_qin
IAE_Ti
IAE_cAin
2.62
301.1
351.2
7014
2.053
2.359e+04
2.665
585.5
B. System identification results
′
The deviation variables 𝑇𝑐 ′ , 𝑞 ′ , 𝑐𝐴𝑖
𝑎𝑛𝑑 𝑇𝑖 ′ are collected
as the input values, imported with respect sample time, and the
output values are first, the non-linear temperature out of tank
and component A out of tank utilized as the estimation data.
And for the validation data, we use the linear temperature out
of the tank and linear component A out of then tank, calculated
with the transfer function in the Simulink process. This is seen
in Appendix1. The models chosen for the processes are first
order, integrator, integrator with delay, first order with delay,
second order, second order with delay, second order with
delay and zero, first order integrator and finally third order
integrator.
Table 2: Transfer functions between reactor temperature T and
cooling water temperature TC.
Mod
el
nam
e
Proc
ess
gain
Kp
Time
constan
t(s)
Ti
me
del
Zero
(s)
Fit to
estimat
ion
data
Fit to
validat
ion
data
Figure 15 Model fit between reactor temperature T and cooling
water temperature TC (Data from test procedure1).
Table 3: Transfer functions between reactant A concentration cA
and cooling water temperature TC.
Mo
del
nam
e
Proce
ss
gain
Kp
Time
constan
t(s)
Tp1,
Tp2,…
P1
0.35534
5058.7
P01
0.00192
06
P0D1
6.0052e
-05
Ti
me
del
ay
Td
Zero
(s)
Tz1,
Tz2,.
.
Fit to
estimat
ion
data
18.31%
-1486%
40
15.18%
Fit to
validat
ion
data
P1D
P2
0.49277
6842.4
-4.628
1e-06,
0
15.54%
P2DZ
-2.3433
2670.8,
0
10000
-1483%
1e-06,
0
-1501%
4965.4
-470.8%
P1I
0.00028
011
0.0036001
5.886%
P3I
38.592
8.348e+08
,
-49.86%
2074.2
P2D
-257.68
1040.5
0.0023049,
P2DZ
63667.1
1e-06,
P1I
6.6304e
-05
0.3796
17.96%
P3I
0.00011
807
0.057211,
11.86%
10905
0.23
6
42.515
1.001+06
%
3223.6
12748,
2.1887
Figure 17 Model fit between reactor temperature T and inlet flow
rate q (Data from test procedure2).
Table 5: Transfer functions between reactant A concentration cA
and inlet flow rate q.
Figure 16 Model fit between reactant A concentration cA and
cooling water temperature TC (Data from test procedure1)
Table 4: Transfer functions between reactor temperature T and
inlet flow rate q.
Mo
del
nam
e
P1
Proce
ss
gain
Kp
0.19728
Time
constan
t(s)
Tp1,
Tp2,…
Ti
me
del
ay
Td
1e-06
Zero
(s)
Tz1,
Tz2,.
.
Fit to
estimat
ion
data
4.016e+04
%
Fit to
validat
ion
data
Mo
del
nam
e
Proce
ss
gain
Kp
Time
constan
t(s)
Tp1,
Tp2,…
P1
0.00067
31
1e-06
P01
4.1337e
-05
P0D1
2.1974e
-05
P1D
0.00085
955
P2
-31.035
Ti
me
del
ay
Td
Zero
(s)
Tz1,
Tz2,.
.
Fit to
estimat
ion
data
2.144e+04
%
-55.22%
1e-06
0
19.89%
3
2.144e+04
10000,
-7075%
8.5826
P01
0.00031
049
5.844%
P2D
P0D1
0.00082
567
P1D
349.01
4.1523e+0
5
P2
2857.6
2.3519e+0
8,
P2D
-235.88
0
5.641%
0
-2.335%
0.055843
4.6714e+0
5,
0.34929
40
-0.5191%
1e-06,
0
1.659e+04
%
10000
0.00097
257
1e-06,
P1I
0.04122
4431.9
-2283%
P3I
3.4245e
-05
0.0067825,
13.95%
P2DZ
0.01316%
-65.558
7618.9
8635.4,
0.72611
0
313.25
2.144e+04
%
Fit to
validat
ion
data
Figure 18 Model fit between reactant A concentration cA and inlet
flow rate q (Data from test procedure2).
Table 6: Transfer functions between reactor temperature T and
inlet concentration cAi.
Mod
el
nam
e
Proc
ess
gain
Kp
Time
constan
t(s)
Tp1,
Tp2,…
P1
-914.27
1e-06
P01
-46.266
P0D1
0.6273
5
P1D
3453.1
84499
P2
-7976.7
66.997,
Ti
me
del
ay
Td
Zero
(s)
Tz1,
Tz2,.
.
Fit to
estimat
ion
data
-15570
25949,
-15570
99400,
Mod
el
nam
e
Proc
ess
gain
Kp
-2.0436
P01
0.1185
1
1.347e+04
%
P0D1
0.0057
377
0
P1D
27347
5.7999e+0
6
P2
59362
1.7725e+0
9,
40
32.94%
0
-7.085%
P2D
43889
25.04%
-136.9
36820
61.56%
P1I
0.0041
557
0.85501
0.1053%
P3I
1.6894
0.15938,
-110.9%
19.079
Tz1,
Tz2,.
.
1e-06
Fit to
estimat
ion
data
-2673%
-1171%
5.467%
0
1.15%
0.02747%
7.3254e+0
8,
0
0.04537%
5.1133e+0
8,
1.354
1692.9
17.14%
0.24615
8.4024
8.4911,
Zero
(s)
0.2004
P2DZ
5.045
Ti
me
del
ay
Td
0.11714
-3.956%
37.2
Time
constan
t(s)
Tp1,
Tp2,…
P1
8.0487
P2DZ
Table 7: Transfer functions between reactant A concentration cA
and inlet concentration cAi.
2.934e+04
%
13172
P2D
Fit to
validat
ion
data
Figure 19 Model fit between reactor temperature T and inlet
concentration cAi (Data from test procedure3).
P1I
0.0058
578
0.014738
5.474%
P3I
0.0337
78
0.0047054,
-250.3%
924403,
0.24808
Fit to
validat
ion
data
Figure 20 Model fit between reactant A concentration cA and inlet
concentration cAi (Data from test procedure3).
Table 8: Transfer functions between reactor temperature T and
inlet temperature Ti.
Mod
el
nam
e
P1
Proc
ess
gain
Kp
-20.224
Time
constan
t(s)
Tp1,
Tp2,…
Ti
me
del
ay
Td
Zero
(s)
Tz1,
Tz2,.
.
1e-06
P01
-04833
P0D1
0.0044
191
P1D
1189.1
6.9335e+0
5
P2
17569
6.9411e+0
8,
P2D
0.2817
6
Fit to
estimat
ion
data
Fit to
validat
ion
data
Figure 21 Model fit between reactor temperature T and inlet
temperature Ti (Data from test procedure4).
Table 9: Transfer functions between reactant A concentration and
inlet temperature Ti
Mod
el
nam
e
Proc
ess
gain
Kp
1.254e+04
%
Time
constan
t(s)
Tp1,
Tp2,…
P1
-0.2762
4686.2
-5577%
P01
0.0013
965
P0D1
3.0219
e-09
P1D
005890
1
1e-06
P2
-19.453
1e-06,
0
18.04%
0
-2.842%
0.01324%
0.125
38022,
0
-17.83%
-4.1824
6.3098
11655,
1.279
1155.2
Tz1,
Tz2,.
.
Fit to
estimat
ion
data
22.78%
-1827%
0
20.09%
0
-4114%
-1878%
0.0040
786
0.07424
5.679%
P3I
15.651
6.9411e+0
8,
15.26%
1e-06,
0
-1874%
6749.7
P2DZ
P1I
2768.1
-12.797
-126.8%
6.4453
5.7581e05,
Zero
(s)
10000
P2D
P2DZ
Ti
me
del
ay
Td
0.1664
2
1e-06,
9
17.844
-3561%
P1I
5.7152
e-05
0.21804
22.53%
P3I
4.8471
e-05
0.019638,
12,47%
265.36
10156,
1.1336
Fit to
validat
ion
data
cases, namely the component A, cA and temperature out of
tank T.
From the system identification results the third model
integrator fits the best when steps responses are changed in
the cooling medium temperature Tc from procedure 1, model
fit between reactor temperature T and inlet flow rate q from
procedure 2, and inlet temperature Ti and model fit between
reactant A and inlet temperature Ti from procedure 4.
The first order integrator is best fit for the model fit
between reactant A and inlet flow rate q from procedure 2.
The second order model with zero and delay shows to be
best fit for the model fit between reactor T and inlet
concentration cAi from procedure 3.
Figure 22Model fit between reactant A concentration and inlet
temperature Ti (Data from test procedure4).
Finally, the second order model is best fit to the model fit
between reactant A concentration cA and inlet concentration
cAi from test procedure 3.
None of the gain parameters of Tc, Ti, qi or cAin should
be used for control purposes since the difference between the
nonlinear model and linear model are by a significant amount.
VI. CONCLUSION AND FURTHER WORK
V. ANALYSIS AND DISCUSSION
The data collected from each experiment has large
variation when it comes to the magnitude difference between
the nonlinear and linear model.
Step responses due to change in the cooling medium
temperature Tc: the linear model follows the non-linear
model trail with a minimum variance in temperature during
the 30 to 90 second mark, but fails to follow during the initial
state, which is 0 to 30 seconds mark.
Step responses due to change in inlet flow rate qin: the
linear model fails to follow the nonlinear model both when
we observe the component A and the temperature T.
Step responses due to change in component A in, cAin: in
the temperature experiment the nonlinear model has the same
pattern, but differs with a significant amount when comparing
the temperatures, while observing the component A results,
the linear model is seen to be a constant, while the nonlinear
is not.
Step responses due to change in inlet temperature Ti: the
linear model completely fails to follow the nonlinear in both
The performance observed from each experiment shows us
that the nonlinear model doesn´t correlate with the linear
model obtained through the transfer function, this could be an
issue from the wrong implementation done inside the .m file,
or, the setup of the transfer function in matrix form in
simulink, this would mean that further study needs to be
conducted.
The linearized model in this experiment cannot be used to
describe the dynamic behavior of the CSTR system due to the
possibility of error in transfer function implementation inside
simulink.
VII. REFERENCES
Seborg, D. E., Edgar, T. F., Mellichamp, D. A., & Doyle, F.
J. (2017). Process Dynamics and Control (4th
EMEA ed.). Hoboken NJ: Wiley.
APPENDIX1 M-SCRIPT&SIMULINK