Process Control (Assignment)

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 𝑐𝑐(𝑑𝑑) in this
component at the outlet of the tank. Input variables are the concentrations of this soluble
component in the inlet streams 𝑐𝑐1(𝑑𝑑) and 𝑐𝑐2(𝑑𝑑) in mol m-3 and the flowrates of the inlet streams
𝑓𝑓1(𝑑𝑑) and 𝑓𝑓2(𝑑𝑑) in m3 min-1
. The volume of the liquid in the tank 𝑉𝑉 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
𝑐𝑐1𝑠𝑠 = 80 mol m-3 with a stream with concentration 𝑐𝑐2𝑠𝑠 = 30 mol m-3 to produce 4 m3 min-1 of
concertation 𝑐𝑐𝑠𝑠 = 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 𝑐𝑐′
(𝑑𝑑) 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 𝑑𝑑 = 0 for inlet flowrate 𝑓𝑓1(𝑑𝑑) equal to 0.1 m3 min-1
.
ii. Step change at time 𝑑𝑑 = 0 for inlet flowrate 𝑓𝑓2(𝑑𝑑) 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 𝑐𝑐(𝑑𝑑) 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 𝑐𝑐1(𝑑𝑑) and 𝑐𝑐2(𝑑𝑑) in mol m-3
. The volume of the liquid in the tanks 𝑉𝑉1
and 𝑉𝑉2 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:
𝑐𝑐′(𝑠𝑠)
𝑐𝑐1
β€² (𝑠𝑠) = 𝐾𝐾1
(𝜏𝜏1𝑠𝑠+1)(𝜏𝜏2𝑠𝑠+1)βˆ’πΎπΎ3
and 𝑐𝑐′(𝑠𝑠)
𝑐𝑐2
β€² (𝑠𝑠) = 𝐾𝐾2
(𝜏𝜏1𝑠𝑠+1)(𝜏𝜏2𝑠𝑠+1)βˆ’πΎπΎ3
, where:
𝐾𝐾1 = 𝑓𝑓1
𝑓𝑓+𝑓𝑓𝑅𝑅
, 𝐾𝐾2 = 𝑓𝑓2
𝑓𝑓+𝑓𝑓𝑅𝑅
, 𝐾𝐾3 = 𝑓𝑓𝑅𝑅
𝑓𝑓+𝑓𝑓𝑅𝑅
, 𝜏𝜏1 = 𝑉𝑉1
𝑓𝑓+𝑓𝑓𝑅𝑅
and 𝜏𝜏2 = 𝑉𝑉2
𝑓𝑓+𝑓𝑓𝑅𝑅
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 𝑉𝑉1 = 𝑉𝑉2 = 0.5 βˆ™ 𝑉𝑉 and that the
recirculation flow, 𝑓𝑓𝑅𝑅, is equal to:
i. π‘₯π‘₯+1
10 𝑓𝑓
ii. 50βˆ’π‘₯π‘₯
5 𝑓𝑓, 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 𝑑𝑑 = 0 for the inlet concentration 𝑐𝑐1(𝑑𝑑) equal to 5 mol m-3
. Compare the obtained
responses of 𝑐𝑐′
(𝑑𝑑) 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 𝑓𝑓𝑅𝑅.
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 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
6. The submitted work is part of the assessment and will contribute 40% to your EX40HC mark.

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