ChE 475

Process Dynamics and Control

Problem Set 2

October 15, 2014

1) A mixing tank is shown below. If mixing occurs isothermally, apply the necessary

conservation principles.

F1 (m3/s), C1(kg/m3) , ρ1 kg/m3

2) For the reaction A + B 2C taking place in a batch reactor, starting from the

conservation of energy equation, develop an equation which will give the rate of

change of temperature with respect to time. Since the reaction is endothermic heat is

supplied to the reactor at a rate Q Joules/s (Watts). What is the condition for an

isothermal operation?

3) Linearize the following ODE’s.

dy

dt y + y + lny , , are constants

dy

dta y - b + cy a, b, c, n are constants

2

n

4) A fluid of constant density (kg/m3) is pumped into a cone-shaped tank of total

volume HR2/3. The flow out of the bottom of the tank is proportional to the square

root of the height h of liquid in the tank. Derive the equations describing the system.

If Fi is a variable, linearize the ODEs

you have obtained. Also obtain the

linearized ODE in terms of the

perturbation variables.

F2 m3/s

C2 kg/m3

ρ2 kg/m3

F (m3/s) , C (kg/m3), ρ

kg/m3

5) The following nonlinear ODE’s are obtained for a variable volume well-mixed

heating tank.

QTTcFdt

dThAc fpfp )(

2/3Chdt

dhA

Where C, Cp, Tf, , A and Q are constants.

i) Identify the nonlinearities and obtain the linearized equations.

ii) Obtain the linearized model in terms of the perturbation variables that you clearly

define.

6) The following ODE's represent the dynamic behaviour of a CSTR.

)(

1

exp 1

2

21

1 XXqX

XX

dt

dXf

uXqX

XX

dt

dX

2

2

21

2 )(

1

exp

Where X1 is the dimensionless concentration, X2 is the dimensionless temperature

(controlled variable), and u is the dimensionless cooling jacket temperature

(manipulated variable). The parameters appearing in the equations are also

dimensionless and have the following numerical values:

= 8.0 = 0.3 = 0.072 = 20 q = 1.0

In addition to this, if the dimensionless inlet concentration X1f = 1.0 and u = 0, then

the process has three steady states one of which is X1ss = 0.5528, X2ss = 2.7517.

Linearize the ODE's around this steady state.

Note: This problem is taken from Sistu and Bequette "Nonlinear Predictive Control of

Uncertain Processes: Application to a CSTR, AIChE Journal, Vol 37, pp. 1711-1723

7) Problems 18, 19, 20 on page 270-271 of Chau