EX40HC 2020-21 CA Assignment 1

Problem 1

Consider the perfectly mixed blending tank shown below, where two streams with different

concentrations in a soluble component are mixed to obtain a certain composition ππ(π‘π‘) in this

component at the outlet of the tank. Input variables are the concentrations of this soluble

component in the inlet streams ππ1(π‘π‘) and ππ2(π‘π‘) in mol m-3 and the flowrates of the inlet streams

ππ1(π‘π‘) and ππ2(π‘π‘) in m3 min-1

. The volume of the liquid in the tank ππ can be assumed constant and

equal to 40 m3

, while the variation of stream densities with composition can be neglected.

Consider that the tank is initially at steady state and it is mixing a stream with concentration

ππ1π π = 80 mol m-3 with a stream with concentration ππ2π π = 30 mol m-3 to produce 4 m3 min-1 of

concertation πππ π = 50 mol m-3

.

a) Starting from the mathematical model of this process in the time domain, determine the

transfer functions in the Laplace domain that describe this mixing tank and link the four input

variables to the outlet concentration in deviation state. Clearly show all steps taken. (Hint: The

total mass balance needs to be considered, even if the volume in the tank is constant)

b) Determine the gains and time constants for the four transfer functions obtained in a) and clearly

indicate the units of all involved parameters.

c) Based on the obtained transfer functions construct the block diagram of the process and build it

in Simulink. Name all blocks and streams in a descriptive way, so that they are identifiable in

relation to the actual process. Provide a screenshot of the final Simulink diagram.

d) Simulate the cases described below and provide plots of the time profile of ππβ²

(π‘π‘) with time.

Based on the transfer functions determined above use the final value theorem to calculate the

expected new steady state values to verify these agree with the simulated ones. Cases to

simulate:

i. Step change at time π‘π‘ = 0 for inlet flowrate ππ1(π‘π‘) equal to 0.1 m3 min-1

.

ii. Step change at time π‘π‘ = 0 for inlet flowrate ππ2(π‘π‘) equal to 0.1 m3 min-1

.

f1(t)

c1(t)

V

f2(t)

c2(t) c (t)

f (t)

Problem 2

Consider the two perfectly mixed blending tanks in series shown below, where two streams with

different concentrations in a soluble component are mixed to obtain a certain composition ππ(π‘π‘) in

this component at the outlet of tank 2. A recirculation stream exists between tank 2 and tank 1,

while all flows can be assumed constant. Input variables are the concentrations of the soluble

component in the inlet streams ππ1(π‘π‘) and ππ2(π‘π‘) in mol m-3

. The volume of the liquid in the tanks ππ1

and ππ2 can be assumed constant, while the variation of stream densities with composition can be

neglected.

a) Starting from the mathematical model of this process in the time domain, show that the Laplace

domain transfer functions of the outlet concentration to the inlet concentrations in deviation

state are the following:

ππβ²(π π )

ππ1

β² (π π ) = πΎπΎ1

(ππ1π π +1)(ππ2π π +1)βπΎπΎ3

and ππβ²(π π )

ππ2

β² (π π ) = πΎπΎ2

(ππ1π π +1)(ππ2π π +1)βπΎπΎ3

, where:

πΎπΎ1 = ππ1

ππ+πππ
π

, πΎπΎ2 = ππ2

ππ+πππ
π

, πΎπΎ3 = πππ
π

ππ+πππ
π

, ππ1 = ππ1

ππ+πππ
π

and ππ2 = ππ2

ππ+πππ
π

b) Using the steady state values from Problem 1 and clearly indicating all involved units, calculate

the parameters of the above transfer functions, assuming that ππ1 = ππ2 = 0.5 β ππ and that the

recirculation flow, πππ
π
, is equal to:

i. π₯π₯+1

10 ππ

ii. 50βπ₯π₯

5 ππ, π₯π₯ being the last digit of your Student ID number for both cases.

c) Based on the obtained transfer functions construct the block diagram of the process and build it

in Simulink. Name all blocks and streams in a descriptive way, so that they are identifiable in

relation to the actual process. Provide a screenshot of the final diagram for case i of b).

d) Considering that initially the system is at steady state, simulate for both cases of b) a step

change at time π‘π‘ = 0 for the inlet concentration ππ1(π‘π‘) equal to 5 mol m-3

. Compare the obtained

responses of ππβ²

(π‘π‘) with that of the system of Problem 1 for the same input change and discuss

on the observed differences between the three simulated cases and the effect of πππ
π
.

f1

c1(t)

V1

f2

c2(t) ci (t)

fi V2

c (t)

f

fR

Submission

1. Submit the report, including plots, calculations and discussion of the results, as a single word or

pdf file via the Turnitin link that is available in MyAberdeen.

(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_CA1_XXX.pdf/.doc (XXX to be

replaced by your surname).

(c) The length of the report (excluding title page, plagiarism sheet, etc.) should not exceed

10 pages. This page count is a limit and not a target.

2. Not complying with the above specifications can incur penalties or invalidate your submission.

3

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

6. The submitted work is part of the assessment and will contribute 40% to your EX40HC mark.

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