Mathematical Model

Consider the continuous stirred tank reactor (CSTR) shown below, where a simple, liquid-phase,
irreversible, exothermic reaction 𝐴𝐴
π‘˜π‘˜
β†’ 𝐡𝐡 takes place. A coolant is flowing in the jacket of the vessel in
order to regulate the temperature of the fluid in the reactor, 𝑇𝑇(𝑑𝑑), and control in this way the
concentration of component A, 𝐢𝐢𝐴𝐴(𝑑𝑑), in the outlet stream at the desired value. There are two
potential disturbances in the process, the temperature, 𝑇𝑇𝑖𝑖
(𝑑𝑑), and the concentration of component
A 𝐢𝐢𝐴𝐴𝑖𝑖(𝑑𝑑), in the inlet stream, while the manipulated variable is the inlet temperature of the cooling
stream in the jacket, 𝑇𝑇𝑐𝑐𝑖𝑖(𝑑𝑑). The cooling jacket should be considered to be at a uniform temperature
𝑇𝑇𝑐𝑐
(𝑑𝑑). The sensor used to measure the concentration of component A, 𝐢𝐢𝐴𝐴(𝑑𝑑), is an on-line infrared
spectrometer providing real-time quantification of the concentration.
For the purposes of the assignment, the simplification assumption can be made that, in the expected
operating temperature range, the reaction rate is a lot more affected by changes in the temperature,
𝑇𝑇(𝑑𝑑), in the tank in comparison to the concentration of A, 𝐢𝐢𝐴𝐴(𝑑𝑑), so that the reaction rate can be
expressed with a pseudo-zero order expression where:
π‘Ÿπ‘Ÿπ΄π΄(𝑑𝑑) = π‘˜π‘˜οΏ½π‘‡π‘‡(𝑑𝑑)�𝐢𝐢𝐴𝐴(𝑑𝑑) β‰… π‘˜π‘˜οΏ½π‘‡π‘‡(𝑑𝑑)�𝐢𝐢𝐴𝐴𝐴𝐴
Numerical values (assumed constant):
 Overall heat transfer coefficient between the coolant and the fluid in the tank: π‘ˆπ‘ˆ = 425 W/m2
-K
 Heat transfer area between the coolant and the fluid in the tank: 𝐴𝐴 = 3.5 m2
 Inlet/Outlet flow rate of the fluid in the tank: 𝐹𝐹 = 6.30 Γ— 10βˆ’4 m3
/s
 The volume of fluid in the tank: 𝑉𝑉 = 0.375 m3
 The density of the fluid in the inlet feed stream and in the vessel: 𝜌𝜌 = 880 kg/m3
 Specific heat of the fluid in the inlet feed stream and in the vessel: 𝑐𝑐𝑝𝑝 = 3680 J/kg-K
 Reaction frequency factor: π‘˜π‘˜π‘œπ‘œ = 2.87 Γ— 105
s-1
 Reaction activation energy: 𝐸𝐸𝐴𝐴 = 6.47 Γ— 104
J/mol
 The heat of reaction: π›₯π›₯π›¨π›¨π‘Ÿπ‘Ÿ = βˆ’2.79 Γ— 104
J/mol
 Density of the coolant: πœŒπœŒπ‘π‘ = 1650 kg/m3
 Specific heat of the coolant: 𝑐𝑐𝑝𝑝𝑐𝑐 = 4190 J/kg-K
 The volume of coolant in the jacket: 𝑉𝑉𝑐𝑐 = 0.045 m3
F, Cai(t), Ti(t)
F, CA(t), T(t)
M
AC
k
A B →
Tci(t)
Tc(t)
CAm(t)
CA(t)
T(t)
 Inlet/Outlet flow rate of coolant in the jacket: 𝐹𝐹𝑐𝑐 = 4.15 Γ— 10βˆ’4 m3
/s
 Steady-state outlet concentration of A: 𝐢𝐢𝐴𝐴𝐴𝐴 = 3300 mol/m3
 Steady-state outlet temperature: 𝑇𝑇𝐴𝐴 = 105 oC
 The transfer function for the final control element: 𝐺𝐺𝑓𝑓
(𝑠𝑠) = 𝐾𝐾𝑓𝑓, where 𝐾𝐾𝑓𝑓 = βˆ’10 C/atm
 The transfer function for the online analyzer: πΊπΊπ‘šπ‘š(𝑠𝑠) = πΎπΎπ‘šπ‘š, where πΎπΎπ‘šπ‘š = 1 (mol/m3
)/(mol/m3
)
a) Starting from the mathematical model of the system, determine all transfer functions that
describe this process in a deviation state in the open-loop. Determine initially any first-order
dependencies between variables and calculate their respective gains and time constants.
Furthermore, combine these first-order transfer functions so that the transfer functions
describing the effect of 𝐢𝐢𝐴𝐴𝑖𝑖
β€²
(𝑑𝑑), 𝑇𝑇𝑖𝑖
β€²
(𝑑𝑑) and 𝑇𝑇𝑐𝑐𝑖𝑖
β€²
(𝑑𝑑) on 𝐢𝐢𝐴𝐴
β€²
(𝑑𝑑), 𝑇𝑇
β€²
(𝑑𝑑) and 𝑇𝑇𝑐𝑐
β€²
(𝑑𝑑) are derived. Clearly
show all steps taken and clearly indicate the units of all involved variables.
[Marks: 10/22]
b) Based on the obtained transfer functions in a) and using a generic PI controller transfer function
block construct in Simulink the block diagram of the closed-loop process. There are two
alternative but equivalent methods that the open-loop process can be drawn in this block
diagram. Include in the assignment report a screenshot of the Simulink diagram used.
[Marks: 2/22]
c) Consider that initially, the system is at steady-state and the following cases.
οƒ˜ If your Student ID number is even: At time 𝑑𝑑 = 0 there is a step-change in the inlet
the concentration of component A, 𝐢𝐢𝐴𝐴𝑖𝑖(𝑑𝑑), equal to 200 mol/m3
.
οƒ˜ If your Student ID number is odd: At time 𝑑𝑑 = 0 there is a step-change in the temperature, 𝑇𝑇𝑖𝑖
(𝑑𝑑),
equal to -10 oC.
i. Using a P controller, simulate in Simulink the process response, under the above
described step-change, for the following 𝐾𝐾𝑐𝑐 values: 0.001, 0.01, 0.1 and 1 atm/(mol/m3
).
Consider an adequate simulation time to reach a new steady state. Plot the responses of
𝐢𝐢𝐴𝐴
β€²
(𝑑𝑑), 𝑇𝑇
β€²
(𝑑𝑑) and 𝑇𝑇𝑐𝑐
β€²
(𝑑𝑑) with time (one plot per variable including the responses for all 𝐾𝐾𝑐𝑐
values). Discuss the following: Presence or not and trend, if any, of an offset, presence or
not, and trend, if any, of oscillatory behavior, stability or not of the response.
[Marks: 3/22]
ii. Starting from the general equation describing the closed-loop response of the feedback-controlledΒ system and using the Final Value Theorem, calculate the offset for the various
Kc values considered in part i. Compare the calculated values with those predicted by
Simulink and discuss trends in relation to theory.
[Marks: 2/22]
d) Consider again that the system is at a steady state. There is no change in the disturbances this
time. At time 𝑑𝑑 = 0 there is a step change of 200 mol/m3 in the set-point of the process.
i. Using a PI controller, simulate in Simulink the process response, under the above-described step-change, for a constant 𝐾𝐾𝑐𝑐 value of 0.015 atm/(mol/m3
) and for the
following 𝜏𝜏𝐼𝐼 values: 1000 and 170 s. Consider an adequate simulation time to reach a
new steady state. Plot the responses of 𝐢𝐢𝐴𝐴
β€²
(𝑑𝑑), 𝑇𝑇
β€²
(𝑑𝑑) and 𝑇𝑇𝑐𝑐
β€²
(𝑑𝑑) with time (one plot per
variable including the responses for both 𝜏𝜏𝐼𝐼 values). Discuss the following: Presence or
not and trend, if any, of an offset, presence or not and trend, if any, of an oscillatory
behavior, stability, or not of the response.
[Marks: 3/22]
ii. Starting from the characteristic equation of this closed-loop system, investigate its
stability for the 𝜏𝜏𝐼𝐼 values considered in part i. Compare the findings with those predicted
by Simulink and discuss trends in relation to theory.
[Marks: 2/22]
Submission
1. Submit the report, including plots, calculations, and discussion of the results, as a single word or
pdf file via the Turnitin link available in Aberdeen.
(a) All calculations, equations, etc. should be typed within the file using an appropriate
equation editor. No scanned pictures of hand-written files should be provided.
(b) The filename of the report file should be EX40HC_CA2_XXX.pdf/.doc (XXX to be replaced
by your surname).
2. Not complying with the above specifications can incur penalties or invalidate your submission.
3. Deadline for the submission of all files is 12:00 noon Friday, December 18th, 2020.
4. Penalties for late or non-submission are as follows:
(a) up to one week late, 2 CGS points will be deducted
(b) up to two weeks late, 3 CGS points will be deducted
(c) up to three weeks late, 4 CGS points will be deducted, with a maximum achievable grade
of CGS D3.
(d) up to four weeks late, 5 CGS points will be deducted, with a maximum achievable grade
of CGS D3.
(e) more than four weeks late, NP will be recorded.
5. A completed plagiarism cover sheet should be included in your electronic submission. Advice
about avoiding plagiarism, the University’s Definition of Plagiarism, a Checklist for Students, and
instructions for TurnitinUK can be found in the following area of the Student Learning Service
website www.cabin.ac.Uk/sys/online-resources/avoiding-plagiarism. For more information on Mathematical Model check on this: https://en.wikipedia.org/wiki/Mathematical_model

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