Will the stability of this system be any different than that of the standard feedback system? Why?

Chemical Engineering
Process_5_Modelling Assignment_2021 Page 1 of 4
SHC4032 Process Control Coursework: Modelling and Systems Analysis Assignment

Present all equations, substitutions and place a box around the final answer, where
applicable.
1. A model for a batch reactor has been derived as follows:
𝑑𝑦
𝑑𝑡 = (𝐼𝐷5)𝑦 − (𝐼𝐷6)𝑦2
𝑑𝑥
𝑑𝑡 = (𝐼𝐷7)𝑦
where the initial values of x and y are 0 and 0.03, respectively.
Using MS Excel, determine y(1) using the following:
a) Euler’s method
b) Fourth Order Runge Kutta method
2. The following chemical reaction takes place in a CSTR:
𝐴
𝑘1
⇌
𝑘2
𝐵
𝑘3
⇌
𝑘4
𝐶
where the rate constants are as follows:
k1 = (ID4) min–1 k2 = (ID5) min–1
k3 = (ID6) min–1 k4 = (ID7) min–1
Determine the following:
a) Rate expressions for components A, B and C.
b) Final steady state concentrations of Component B and C using the Euler
method.
Process_5_Modelling Assignment_2021 Page 2 of 4
3. Determine 𝑦(2) from the following 2nd order differential equation:
𝑑2𝑦
𝑑𝑡2 + 𝑑𝑦
𝑑𝑡 + 𝑦 = (𝐼𝐷7)
where 𝑦(0) = 𝑦′(0) = 0. Use:
a) Euler method
b) Fourth Order Runge Kutta method
4. (Bequette, 2003) Use the initial and final value theorems to determine the initial and
final values of the process output for a unit step input change for the following
transfer functions:
a) 5𝑠+12
7𝑠+4
b) (7𝑠2+4𝑠+2)(6𝑠+4)
(4𝑠2+4𝑠+1)(16𝑠2+4𝑠+1)
c) 4𝑠2+2𝑠+1
8𝑠2+4𝑠+0.5
5. (Bequette, 2003) Derive the closed loop transfer function between L(s)and Y(s) for
the following control block diagram (this is known as a feed forward / feedback
controller).
Figure 0-1 Feedforward / Feedback Control Loop
a) Describe how you would be able to check for stability for this closed loop system.
b) Will the stability of this system be any different than that of the standard feedback
system? Why?
Process_5_Modelling Assignment_2021 Page 3 of 4
6. (Bequette, 2003) A process has the following transfer function:
𝐺𝑃(𝑠) = 2(−3𝑠 + 1)
(5𝑠 + 1)
a) Using a P-controller, find the range of the controller gain that will yield a stable
closed loop system.
b) Simulate the process with the P controller to confirm the range of stability as
determined in part (a).
7. Consider the open-loop unstable process transfer function:
𝐺𝑃(𝑠) = 1
(𝑠 + 2)(𝑠 − 1)
a) Find the range of KC for a P-only controller that will stabilize this process.
b) As it turns out, 𝐾𝐶 = 4 will yield a stable closed-loop (does this match with your
answer in part (a)?). Typically, there is a measurement lag in the feedback loop.
Assuming a first-order lag on the measurement, find the maximum measurement
time constant which is allowed before the system (with 𝐾𝐶 = 4) is destabilized.
c) Confirm all of your calculations with the system reproduced using
MATLAB/Simulink. Print out your results.
8. A PI controller is used on the following second order process:
𝐺𝑃(𝑠) = 𝐾𝑃
𝜏2𝑠2 + 2𝜁𝜏𝑠 + 1
The process parameters are:
𝐾𝑃 = 1, 𝜏 = 2, 𝜁 = 0.7
The tuning parameters are:
𝐾𝐶 = 5, 𝜏𝐼 = 0.2
a) Determine if the process is closed-loop stable.
b) Reproduce your results using MATLAB/Simulink and print out your results.
Process_5_Modelling Assignment_2021 Page 4 of 4
9. (Marlin, 2000) The process shown below consists of a mixing tank, mixing pipe, and
continuously stirred tank reactor. The following assumptions are applied to the
system:
I. Both tanks are well mixed and have constant volume and temperature.
II. All pipes are short with negligible transportation delay.
III. All flows and densities are constant.
IV. The first tank is a mixing tank.
V. The mixing pipe has no accumulation, and the concentration, CA3, is
constant.
VI. The second tank, CSTR, with 𝐴 → 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠, and 𝑟𝐴 = −𝑘𝐴𝐶𝐴
3 2⁄ .Mixing
Point
q1
CA0
q2
CA2
q3
CA3
q4
CA4
q5
CA5
V1
V2
Mixing
Tank
CSTR
Figure 0-2 Mixing tank, mixing pipe, and CSTR Process
a. Derive a linearized model relating 𝐶𝐴2
′ (𝑡) to 𝐶𝐴0
′ (𝑡).
b. Derive a linearized model relating 𝐶𝐴4
′ (𝑡) to 𝐶𝐴2
′ (𝑡).
c. Derive a linearized model relating 𝐶𝐴5
′ (𝑡) to 𝐶𝐴4
′ (𝑡).
d. Combine the models in (a) to (c) into one equation 𝐶𝐴5
′ (𝑡) to 𝐶𝐴0
′ (𝑡) using
Laplace transforms.