UNIVERSITY OF THE WEST INDIES

DEPARTMENT OF CHEMICAL ENGINEERING

CH26B – Process Dynamics and Control I

Midterm Examination: Tuesday, March 21/00, 8:00 to 10:00 AM.

Answer Questions 1 and 2, and EITHER Question 3 OR Question 4.

N.B. Calculators may be used in this examination. They must be silent, cordless and non-programmable. Calculators will be examined to ensure that they comply with these requirements.

1. [20 marks]

(a) Find the Laplace transform of

(b) Find the inverse transform y(t) corresponding to

(c) Let

Compute the final value of i.e.

(d) Let

Compute the final value of

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2. [40 marks] Two cylindrical tanks operate in series in a noninteracting mode as shown in Fig. Q2. Figure Q2 - Tanks in series.

The initial steady-state fluid flowrate is producing the steady-state tank levels and The cross-sectional area A of each tank is 20 ft2. It may be assumed that the Tank 1 outlet flowrate is linearly related to the head of liquid, i.e. and similarly that

An operator decides to dump an extra 50 ft3 of fluid into Tank 1 over a very short period of time.

(a) Does Tank 1 overflow?

(b) Does Tank 2 overflow?

(c) If Tank 2 does overflow, when does it stop overflowing?

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q(t)

R 1

R 2

9 ft.

9 ft.

q1(t)

q2(t)

TANK 1

TANK 2

h1(t)

h2(t)

3. [40 marks] Consider the continuous stirred-tank reactor (CSTR) depicted below:

Figure Q3 - Continuous stirred-tank reactor.

The inlet stream concentrations of components A and B expressed in units of kg/m3 are and respectively. Component A reacts irreversibly with a specific rate

constant k to form product, component B:

The rate of reaction is n’th order with respect to reactant A. Hence, the mass rate of disappearance of A in the CSTR is given by It may be assumed that the densities and volumetric flowrates of the inlet and outlet streams are constant at and respectively.

(a) Derive an expression which describes the time domain response of the outlet product concentration to a step change of magnitude M in the inlet concentration The inlet and (hence) outlet concentrations of reactant A are held fixed at their initial steady-state values.

(b) Write a differential equation which describes the behaviour of the outlet reactant concentration in response to changes in the inlet concentration

(c) Linearize the relationship developed in 3(b) via first-order Taylor series approximation.

(d) Solve the linear differential equation derived in 3(c) to obtain an expression for the response of the outlet reactant concentration to a step change of magnitude M in

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)(),(

,,

0

0

tCtC

q

B

A

)(),(

,,

tCtC

q

B

A

)(),(,,

tCtC

V

B

A

4. [40 marks] You have been asked to design a response test for the purpose of modelling the mixing dynamics of a reaction vessel in a chemical plant. The fluid in the vessel is fairly well agitated, has a constant holdup of about 18,000 ft3, and a flowrate through the vessel of 900 ft3/min at normal operating conditions.

Your supervisor has suggested that you approach this problem by adding a dye tracer to the vessel inlet stream and collecting 20 samples of the outlet stream at 2 minute intervals thereafter. The samples are to be taken to the plant laboratory, where the concentration of tracer in each can be measured on a bench colourimeter. The dye costs about $10 per pound and the sensitivity of the colourimeter is such that the lowest detectable concentration of dye is 0.001 lbm/ft3.

(a) What is the minimum mass of dye required if the tracer is to be added in step fashion?

(b) What is the minimum mass of dye required if the tracer is to be injected as an ideal impulse?

(c) Which of the two response test methods ((a) or (b)) is more cost effective?

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