Problem 1: The Uncontrolled Isothermal CSTR

 

Consider a CSTR with no control in which a first-order reaction takes place. Assuming a constant tank volume (i.e., constant flow rate F) then a mole balance on the reactant A gives

 

(1)

 

Given the data below, use MATLAB to solve for the time-dependent concentration leaving the tank following a step change in inlet concentration from 0.5 units to 1.0 units.

 

V = 5.0 units

K = 1 units

CA1(t=0) = 0.25 units

F = 10 units

 

 

Problem 2: The PI Controlled Isothermal CSTR

 

Consider the same reactor, this time with PI (proportional-integral) control added to adjust F so that CA1 is maintained at 0.25 units. If the flow rate is set equal to the controller signal (i.e., the gain on the final control valve is unity) then

 

(2)

 

where KC is the controller’s proportional gain and t I is the integral mode’s reset time. and represent the initial steady-state values (i.e., prior to any disturbance). The transient concentration of A, meanwhile, is given by Eq. (1).

 

We can re-express Eqs. (1) and (2) as a coupled set of o.d.e.s by first differentiating Eq. (2):

 

 

 

(3)

 

Noting from Eq. (1) that

 

(4a)

 

and substituting this into Eq. (3) gives

 

(4b)

 

 

Eqs. (4a) and (4b) describe the controlled CSTR and can be used to determine the time dependent outlet concentration and flow rate.

 

Use these equations and MATLAB to model the response of the CSTR to the same step change in inlet concentration as considered in Problem 1. In this case assume KC equals 1 unit and vary the reset time, t I as follows:

 

Reset time = ¥ (i.e., pure proportional control),

Reset time = 1

Reset time = 0.05

Reset time = 0.005

 

Comment on your findings.