TUTORIAL QUESTIONS

CHAPTER 1

INTRODUCTION TO CHEMICAL PROCESS DYNAMICS AND CONTROL

1.1 Consider a tank heater system shown in Figure 1.1. A liquid enters the tank with a

flow rate Fi (m3/min) and a temperature Ti(0C), where it is heated with steam

(having a flow rate Fst kg/min). Let F and T be the flow rate and temperature of

the stream leaving the tank. The tank is considered to be well stirred.

a. Identify the controlled variables, manipulated variables, and

disturbance variables.

b. Propose feedback control methods to control the liquid level and the

outlet temperature and sketch the schematic diagram

c. Suggest a feed forward control method to control the outlet

temperature

2.1 Consider a home heating system consisting of a natural gas-fired furnaces and a

thermostat. In this case the process consists of the interior space to be heated. The

thermostat contains both the measuring element and the controller. The furnace is

either on (heating) or off. Draw a schematic diagram for this control system. On

your diagram, identify the controlled variable, manipulated variables, and

disturbance variable that can affect the room temperature?

Steam

Fst

Condensate

F i, Ti

F , T

ASSIGNMENT 1

The distillation column shown in the drawing is used to distill a binary mixture.

Symbols x, y and z denote the mole fractions of the more volatile component, while

B, D, R and F represent molar flow rates. It is desired to control distillate composition

y despite disturbances in feed flow rate F. All flow rates can be measured and

manipulated with the exception of F, which can only be measured. A composition

analyzer provides measurements of y.

a. Propose a feedback control method and sketch the schematic diagram

b. Suggest a feed forward control method and sketch the schematic

diagram

COLUMN

D , y R

F, z

B , x

CHAPTER 2

MATHEMATICAL MODELING OF CHEMICAL PROCESSES

2.1 A perfectly stirred, constant-volume tank has two input streams, both consisting

of the same liquid. The temperature and flow rate of each of the streams can vary

with time.

(a) Derive a dynamic model that will describe transient operation. Make a degree of

freedom analysis assuming that both Streams 1 and 2 come from upstream units

(i.e., their flow rates and temperature are known functions of time).

(b) Simplify your model, if possible, to one or more differential equations by

eliminating any algebraic equations. Also, simplify any derivatives of products

of variables

2.2 Two surge tanks are used two dampen pressure fluctuations caused by erratic

operations of a large air compressor (see Figure 2.2).

If the discharge pressure of the compressor is Pd(t) and the operating pressure of the

furnace is Pf (constant), develop a dynamic model for the pressure in the two surge

tanks as well as for the air mass flow at points a, b, and c. You may assume that the

valve resistance is constant, that the valve characteristics are linear, that the surge

process isothermally, and the ideal gas law works.

T1

w1

T2

w2

T3

w3

Stream 1 Stream 2

Stream 3

Figure 2.1

Compressor Pd

wa wb wc

Air Supply

Process furnace Pf

Ra Rb Rc

Figure 2.2

ASSIGNEMENT 2

Irreversible consecutive reactions A B C occur in a jacketed, stirred tank reactor

as shown in Figure 2.3. Derive a dynamic model based on the following assumptions:

(i) The contents of the tank and cooling jacket are well mixed. The volumes

of material in the jacket and in the tank do not vary with time.

(ii) The reaction rates are given by

(iii) The thermal capacitance of the tank contents the jacket contents are

significant relative to the thermal capacitance of the jacket and tank walls,

which can be neglected.

(iv) Consider physical properties and heat transfer coefficients can be

assumed.

Note:

All flow rates are volumetric flow rates in L/h. The concentrations have units of

mol/L. The heats of reaction are H1 and H2 .

Product cA, cB, cC,, T, q

Coolant outqc, Tc

Figure 2.3

Coolant inqc i , Tc i

Jacketed reactor

Feed cAi, cBi qi , Ti,

CHAPTER 3

LAPLACE TRANSFORM

3.1 The mathematical model of a process is given by (3.1). Where y(t) and f(t) are the

input and output variables respectively. It is given that y(0)=y’(0)=0.

(3.1)

Calculate and also plot y(t) for the following change in the input f(t)

(b)

3.2 Using partial fraction expansion where required, find x(t)

(a)

(b)

(c)

(d)

ASSIGNMENT 3

1. Using partial fraction expansion where required, find x(t)

(a)

(b)

2. The mathematical model of a process is given by (3.1). Where y(t) and f(t) are the

input and output variables respectively. It is given that y(0)=y’(0)=0.

(3.1)

Calculate and also plot y(t) for the change in input f(t) given by

CHAPTER 4

TRANSFER FUNCTION

4.1 For the process modeled by

Find the four transfer functions relating the outputs (y1, y2) to the inputs (u1, u2). The u and y are deviation variables.

4.2 A simple surge tank with fixed valve on the outflow line is illustrated in Figure 4.1. If the outflow is proportional to the square root of the liquid height, an unstedy state model for the level in the tank is given by

As usual you can assume that the process is initially at steady state

(a) Find the transfer function H’(s)/Q’i (s). Put the transfer function in standard gain /time constant form.

(b) Find the transfer function Q’(s)/Q’i (s) and put it in standard form.(c) If the algebraic function for the outflow rate is linear instead of square root, the

level transfer function can be put into a first order form,

With is the initial steady-state volume.

When written in this way,* is easily interpreted as the liquid residence time at the nominal operating conditions. What are equivalent expressions for K and in the part (a) level transfer function, that is, for the square root out flow relation?

ASSIGNMENT 4

Consider the following transfer function:

qi,

q h

Figure 4.1

(a) What is the steady state gain?

(b) What is the time constant?

(c) If U(s)=2/s, what is the value of the output y(t) when t ?

(d) For the same U(s), what is the value of the output when t=10?

What is the output when expressed as a fraction of the new steady

state value?

(e) If U(s)=(1-e-s)/s, that is the unit rectangular pulse, what is the

output when t ?

(f) If u(t)=(t), that is a unit impulse at t=0, what is the output when

t ?

(g) If u(t)=2 sin 3t, what is the value of the output when t ?

CHAPTER 5

DYNAMIC BEHAVIOR OF FIRST ORDER AND SECOND ORDER PROCESS

5.1 Consider a process that is represented by the mathematical model of which is

given by the following ODE.

y’(0)=y(0)=0;

Find out the standard second order transfer function and answer the following

questions for a step input of size 5.

a. Show that the response of the process is under damped

b. Calculate the following values

i. Rise time

ii. Time to first peak

iii. Percent overshoot

iv. Decay ratio

v. Period of oscillation

vi. Ultimate Value of y(t)

vii. Maximum value of y(t)

5.2 The overall transfer function of a process is given by

Where 1 = 5, 2 = 3

(a) What is the overall gain?

(b) Show that for a step change in input it takes longer time for the

overall response to reach 63.2% of the steady state than the sum of

time constants of the two processes.

(c) Can y(t) show oscillation for a step input? Explain.

ASSIGNEMENT 5

The transfer function of a process is given by

Answer the following questions for a step input of size 3.5.

a. Show that the response of the process is under damped

b. Calculate the following values

i. Rise time

ii. Time to first peak

iii. Percent overshoot

iv. Decay ratio

v. Period of oscillation

vi. Ultimate Value of y(t)

vii. Maximum value of y(t)

CHAPTER 6

HIGHER ORDER SYSTEMS

6.1 The following transfer function is not written in a standard form

(a) Put it in a standard gain/time constant form.

(b) Determine the gain, poles, and dead time

(c) Determine the response for a unit step change in input.

(d) If the time-delay term is replaced by 1/1 Padé approximation, repeat part

(b) and (c).

6.2 A process consists of an integrating element operating in parallel with a first-

order element (Figure 6.1).

(a) What is the overall transfer function, G(s) = Y(s) / U(s)?

(b) What is the gain of G(s)? Under what condition (s) is the process gain negative?

(c) What are the poles of G(s)? Where are they located in the complex s-plane?

(d) What are the zeros of G(s)? Where are they located? Under what condition(s) will

one or more of the zeros be located in the right-half s-plane?

(e) Under what conditions, if any, can this process exhibit both a negative gain and a

right-half plane zero?

(f) For any input change, what functions of time (response modes) will be included in

the response, y(t)?

(g) Is the output bounded for any bounded input change, for example, u(t)=M?

ASSIGNMENT 6

For each of the systems with transfer functions given below, (a) draw the

corresponding block diagram, (b) identify the gain, poles, zeros of the transfer

functions, and (c) sketch the response to a unit step change in input.

s

K1

12

sK

U(s) + +

Figure 6.1

(i)

(ii)

CHAPTER 7

FEED BACK CONTROLLERS

7.1 The parallel form of the PID controller has the transfer function given by

Equation 7.1 Many commercial analogue controllers can be described by the

series form given by Equation 7.2.

(7.1)

(7.2)

(a) For the simplest case, 0, find relations between the settings for

the parallel form (K*c,*I, *

D) and the settings for the series form

(Kc,I, D).

(b) Does the series form make each controller setting (Kc,I, D) larger

or smaller than would be expected for the parallel form?

(c) What are the magnitudes of these interaction effects for Kc=4,

I=10min, D=2min?

7.2 If the input Ym to a PI controller changes stepwise (Ym(s)=2/s) and the controller

output changes initially as in Figure 7.1, what are the values of the controller gain

and integral time?

ASSIGNMENT 7

The physically realizable form of the PD transfer function is given below.

6

Slope=1.2 min-1

Figure 7.1

(a) Show how to obtain this transfer function with a parallel arrangement of

two much simpler functions in Figure 7. 1.

(b) Find expressions for K1, K2 and that can be used to obtain desired values

of Kc, D , and .

(c) Verify the relations for Kc=3, D =2, and =0.1.

CHAPTER 8

DYNAMIC BEHAVIOR OF CLOSED LOOP SYSTEMS

8.1 A black diagram for internal model control is shown in Figure 8.1. Transfer

function denotes the process model, while Gp denotes the actual process

transfer function. It has been assumed that Gv=Gm=1 for simplicity. Drive closed-

loop transfer functions for both the servo and regulator problems.

The block diagram of a feedback control system is shown in Figure 8.2. Find the

expression for the closed loop transfer functions Y/D and Y/Ysp.

Consider the closed loop block diagram of the feedback system shown in Figure 8.2.

For a set point change of magnitude 2, do the following

Gd

Gc Gp+_ ++

D

YYm

Figure 8.1

- +pG

~

+ + Y

Figure 8.2

Gv

Gp

Gd

Kc2

Gm

Kc1

Ysp+ _

D

+ _ME’CE

Ym

(a) Derive an expression for the closed-loop response in the Laplace

domain.

(b) Find how the closed-loop output responds with time to the set point

change in (a).

(c) Compute the maximum value of y(t) and state when it occurs

(d) Compute the offset of the final steady state

(e) Give a qualitative sketch of the closed loop response

ASSIGNMENT 8

A block diagram of a closed-loop system is shown in Figure 8.1

+ + Y

Figure 8.3

)13)(1(

1

ss

Km= 1

Ysp Gc = 5+ _

D

Gv = 1

)13)(1(

2

ssG p

(a) Derive a closed loop transfer functions Y/D, and Y/Ysp .

(b) For a unit step change in load (disturbance) and kc=5 determine y(t) and

the offset at steady state.

(c) For a unit step change in set point and kc=5 determine the offset at steady

state

CHAPTER 9

STABILITY OF CLOSED LOOP CONTROL SYSTEM

Km Kc

G2

G1 G3

Km

+_ ++

++

D

E P Y

Ym

Ysp

Figure 8.1

spY~

9.1 For the standard feedback controlled system with transfer functions given below

determine the value of control parameters for which the response will be stable.

a) , PI-controlle

b) P: controller

9.2 Consider a feedback controlled process indicated in Figure 9.1. If PI controller is

used in the system.

(a) Determine the values of Kc for which the system will be stable for I=0.5.

(b) Determine the general condition of stability.

ASSIGNMENT 9

The block diagram of a feed back control system is shown in Figure 9.1. Determine

the response the values of Kc that result in a stable closed loop system.

+ + Y

Figure 9.1

22

12 ss

1

2

s

Km= 1

Ysp Gc+ _

D

CHAPTER 10

FREQUENCY RESPONSE ANALYSIS

+ + YYsp

Figure 9.1

3

4

s s

1

2

14

6

s

0.5+ _

10

1

s

Kc+ _

D

10.1 For each of the following transfer functions, develop both the amplitude ratio and

phase angle. Find AR and for each transfer function at values of =0.1,1, and

10.

(a) (b)

(c) (d)

10.2 Solve problem (a) and (b)

(a) Plot the bode diagram (0.1 100) of the third order transfer function

with a dead time of 2 using MATLAB.

Find both the value of that yields a -1800 phase angle and the value of

AR at that frequency.

(b) Solve problem (a) with a dead time of 2min.

ASSIGNMENT 10

10.1 For each of the following transfer functions, develop both the amplitude ratio and

phase angle. Find AR and for each transfer function at values of =0.1,1, and

10.

(a)

(b)

10.2 The open loop transfer function of a feedback controlled system is ,

Draw, using MATLAB, the bode diagram for Kc=0.5 , 5 and find the value of

the frequency that yield a phase angle of -1800 (crossover frequency) phase

angle and determine the value of the ARN at that frequency.