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CHEMICAL ENGINEERING 3P04

PROCESS CONTROL TUTORIAL #5

2008

1. The continuous stirred tank chemical reactor shown in the Figure 1 is to be analyzed.Assumptions and data for deriving a dynamic model are given below.

i) the tank is well mixedii) the density is constant, and the heat capacity

(CpCv) is constant

iii) the heat of reaction, Hrxn=0; heat transfer,

Q=0; work, W=0

iv) A B, rate of reaction, rA= - k0e-E/RTCA [mole/

(volume*time)]

v) system is initially at steady-statevi) flow in and out [volume/time] and the volume

are constant

Figure 1. CSTR

Goal: Determine the concentration of A when the inlet temperature changes in a step, T0.

A. Starting from the basic balances, derive the differential equation(s) that describe the dynamic

behavior of component A.

B. Express the equations derived in Part A as linear (or linearized, as needed) equations in

deviation variables. Identify the steady-state gains and the time constants.

C. Using the results in Part B, solve for the temperature and concentration for a step.

1. (10 points) Derive the analytical expression for T'(t).

2. (15 points) Derive the analytical expression for CA'(t).

D. Sketch the shapes of the responses for T'(t) and CA'(t) in response to the input. Be as specific

as possible in defining features of the transient response.

E. Using the results in previous parts, answer the following questions for the relationshipbetween T0 CA. Briefly justify your answers.

1. What is the order?

2. Is the response stable?

3. Is the response over-, critically, or underdamped?

4. Is the steady-state gain proportional to V/F?

5. Does the time constant (or time constants) depend on the value of temperature?

F

CA0VCA

T0

T

A B

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2. In this question, we will consider the most common flow meter, the orifice meter.

a. Sketch a pipe with an orifice meter, describe the measurements taken, and how the flow

rate is determined from the measurements taken (i.e., the principle and the resulting

equation).

b. In the sketch, identify the non-recoverable pressure drop and discuss why this isimportant.

c. Identify two situations (process conditions) when the orifice meter should not be used.

3. The system in Figure 2 is a cylindrical tank that has gas entering and leaving. The roof of

fixed mass (m) floats on the gas, i.e., it moves up and down depending on the amount

of gas in the vessel, and no gas can leak from at roof-wall interface. The gas inlet flow

rate is determined by a compressor and valve (not shown), so that the inlet flow rate is

independent of the pressure in the tank. There is no reaction occurring in this vessel.

The pressure is low, so that the gas behaves as an ideal gas.

Is the position of the roof (L) self-regulatory or non-self-regulatory for changes in the

inlet flow rate? Explain your answer clearly based on modelling and/or physical

reasoning.

Figure 2. Process system for Question 3.

m

Fin Fout

The control valve is

partially opened and fixed

in this position.

Cross sectional area A

L

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3. We recognize the importance of determining whether a process system is self-regulatory

(which tends to reach a steady-state after a disturbance) or non-self-regulatory (which does not

achieve a steady-state after a step input). The non-self-regulatory process variable requires

control by a computer or close supervision by a person.

The question addresses a floating roof gas holder. This question is from the end-of-chapter

questions in Chapter 5 (Q5.14c) of the textbook. It builds on the learning from the liquid levelproblems throughout the course, the definition of self-regulatory in Chapter 5, and the constant-

volume gas holder in question 5.14d that was covered in a tutorial.

We note the following from the physical system.

1. The volume in the holder changes

2. The pressure of the gas in the holder is constant . This conclusion follows from (a) the

roof mass being constant and (b) the cross sectional area being constant.

3. From the ideal gas law, the volume of gas in the vessel is proportional to the moles of gas

at any time.

4. The flow out does not depend on the amount of gas in the holder. This is because the

pressure in the tank is constant.

2)( tan

PPvKF

k

vout

= = constant for this problem statement

with Ptank = constant

P2 = exhaust pressure after valve

v = valve % open (constant)

Kv = constant depending on valve manufacture

= density (constant because pressures are constant)

5. The flow out depends on the tank pressure and the % valve opening, both of which areconstant. Therefore, the flow out of the tank is constant in this scenario in which the

valve position is not changed.

We proceed to formulate a model of the process.

Variable: Volume of gas in the vessel. Since the area is constant, the volume will give the

level of the roof.

System: The volume in the vessel below the roof. In this case, the system dimensions

change with time.

Balance: To determine the total mass (or moles for this non-reacting system), we develop a

total mass balance.

(accumulation) = (in) - (out)

with m= mass and , we can formulate the following total mass balance

}F)t(F)t{(})m()m{( outinttt =+

We can divide by delta time and take the limit as delta time approaches 0 to give

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outin FFdt

dm =

Applying the ideal gas law and the assumption that the temperature is constant,

outin FFdt

dL

ART

MWP

dt

dV

RT

MWP

dt

dm ===

)()(

We have a model for the behaviour of the roof level, L. This model for the level of the roof

demonstrates that the derivative of the level is independent of the level, i.e., the level does not

influence the right-hand side of the equation.

The flow in is independent of the level and pressure in the vessel, by the problem statement. Theflow out depends on the pressure in the vessel and the % valve opening. Neither of these depends

on the level. Also, the density out depends on the vessel pressure, which does not depend on the

level.

Thus, we conclude that the roof level is non-self-regulatory. It is unstable and requires

automatic control using a computer or close supervision by a person.

For a non-self-regulatory variable, the variables derivative is independent of the

variable itself.