Describe how you would be able to check for stability for this closed loop system. Will the stability of this system be any different than that of the standard feedback system? Why?

Process_5_Modelling Assignment_2021 Page 1 of 4

SHC4032 Process Control Coursework: Modelling and Systems Analysis Assignment

For a change, this assignment is to be handwritten with additional files with results from

MATLAB/Simulink and MS Excel. Ensure that all of your writing is legible .

Ensure no spelling, punctuation or grammatical errors when writing your answers.

For a couple of questions, you will be required to utilize the digits from your student ID number. For example, Student ID – 1234567, when the question requests (ID4), use the 4th digit of your ID number. If the digit is a 0, then utilize the number 1 as a substitute.

Present all equations, substitutions and place a box around the final answer, where applicable.

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

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.

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

(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

(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?

(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).

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.

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.

(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:

Both tanks are well mixed and have constant volume and temperature.

All pipes are short with negligible transportation delay.

III. All flows and densities are constant.

The first tank is a mixing tank.

The mixing pipe has no accumulation, and the concentration, CA3, is constant.

The second tank, CSTR, with 𝐴 → 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠, and 𝑟𝐴 = −𝑘𝐴𝐶𝐴

3 2⁄ .Mixing

Point

CA0

CA2

CA3

CA4

CA5

Mixing

Tank

CSTR

Figure 0-2 Mixing tank, mixing pipe, and CSTR Process

Derive a linearized model relating 𝐶𝐴2

′ (𝑡) to 𝐶𝐴0

′ (𝑡).

Derive a linearized model relating 𝐶𝐴4

′ (𝑡) to 𝐶𝐴2

′ (𝑡).

Derive a linearized model relating 𝐶𝐴5

′ (𝑡) to 𝐶𝐴4

′ (𝑡).