Consider the continuous stirred tank reactor (CSTR) shown below, where a simple, liquid-phase,

irreversible, exothermic reaction π΄π΄

ππ

β π΅π΅ takes place. A coolant is flowing in the jacket of the vessel in

order to regulate the temperature of the fluid in the reactor, ππ(π‘π‘), and control in this way the

concentration of component A, πΆπΆπ΄π΄(π‘π‘), in the outlet stream at the desired value. There are two

potential disturbances in the process, the temperature, ππππ

(π‘π‘), and the concentration of component

A πΆπΆπ΄π΄ππ(π‘π‘), in the inlet stream, while the manipulated variable is the inlet temperature of the cooling

stream in the jacket, ππππππ(π‘π‘). The cooling jacket should be considered to be at a uniform temperature

ππππ

(π‘π‘). The sensor used to measure the concentration of component A, πΆπΆπ΄π΄(π‘π‘), 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,

ππ(π‘π‘), in the tank in comparison to the concentration of A, πΆπΆπ΄π΄(π‘π‘), so that the reaction rate can be

expressed with a pseudo-zero order expression where:

πππ΄π΄(π‘π‘) = πποΏ½ππ(π‘π‘)οΏ½πΆπΆπ΄π΄(π‘π‘) β
πποΏ½ππ(π‘π‘)οΏ½πΆπΆπ΄π΄π΄π΄

Numerical values (assumed constant):

ο§ Overall heat transfer coefficient between the coolant and the fluid in the tank: ππ = 425 W/m2

-K

ο§ Heat transfer area between the coolant and the fluid in the tank: π΄π΄ = 3.5 m2

ο§ Inlet/Outlet flow rate of the fluid in the tank: πΉπΉ = 6.30 Γ 10β4 m3

/s

ο§ The volume of fluid in the tank: ππ = 0.375 m3

ο§ The density of the fluid in the inlet feed stream and in the vessel: ππ = 880 kg/m3

ο§ Specific heat of the fluid in the inlet feed stream and in the vessel: ππππ = 3680 J/kg-K

ο§ Reaction frequency factor: ππππ = 2.87 Γ 105

s-1

ο§ Reaction activation energy: πΈπΈπ΄π΄ = 6.47 Γ 104

J/mol

ο§ The heat of reaction: π₯π₯π¨π¨ππ = β2.79 Γ 104

J/mol

ο§ Density of the coolant: ππππ = 1650 kg/m3

ο§ Specific heat of the coolant: ππππππ = 4190 J/kg-K

ο§ The volume of coolant in the jacket: ππππ = 0.045 m3

F, Cai(t), Ti(t)

F, CA(t), T(t)

M

AC

k

A B ο£§ο£§β

Tci(t)

Tc(t)

CAm(t)

CA(t)

T(t)

ο§ Inlet/Outlet flow rate of coolant in the jacket: πΉπΉππ = 4.15 Γ 10β4 m3

/s

ο§ Steady-state outlet concentration of A: πΆπΆπ΄π΄π΄π΄ = 3300 mol/m3

ο§ Steady-state outlet temperature: πππ΄π΄ = 105 oC

ο§ The transfer function for the final control element: πΊπΊππ

(π π ) = πΎπΎππ, where πΎπΎππ = β10 C/atm

ο§ The transfer function for the online analyzer: πΊπΊππ(π π ) = πΎπΎππ, where πΎπΎππ = 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 πΆπΆπ΄π΄ππ

β²

(π‘π‘), ππππ

β²

(π‘π‘) and ππππππ

β²

(π‘π‘) on πΆπΆπ΄π΄

β²

(π‘π‘), ππ

β²

(π‘π‘) and ππππ

β²

(π‘π‘) 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.

ο If your Student ID number is even: At time π‘π‘ = 0 there is a step-change in the inlet

the concentration of component A, πΆπΆπ΄π΄ππ(π‘π‘), equal to 200 mol/m3

.

ο If your Student ID number is odd: At time π‘π‘ = 0 there is a step-change in the temperature, ππππ

(π‘π‘),

equal to -10 oC.

i. Using a P controller, simulate in Simulink the process response, under the above

described step-change, for the following πΎπΎππ 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

πΆπΆπ΄π΄

β²

(π‘π‘), ππ

β²

(π‘π‘) and ππππ

β²

(π‘π‘) with time (one plot per variable including the responses for all πΎπΎππ

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 π‘π‘ = 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 πΎπΎππ value of 0.015 atm/(mol/m3

) and for the

following πππΌπΌ values: 1000 and 170 s. Consider an adequate simulation time to reach a

new steady state. Plot the responses of πΆπΆπ΄π΄

β²

(π‘π‘), ππ

β²

(π‘π‘) and ππππ

β²

(π‘π‘) with time (one plot per

variable including the responses for both πππΌπΌ 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 πππΌπΌ 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 Universityβs 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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