Chemical Engineering Department
University of Florida
ECH 4323 Process Control Theory
HOMEWORK No. 9
Special instructions for exercise 5.21 of the textbook
Using the problem statement 5.21 of the textbook, address the following issues:
a. Write a total mass balance (assume that the density is constant, and that the
density of the mixture is equal to that of the individual species) and a species
(component A) mass balance.
b. Write an energy balance for the system.
c. Linearize the equations obtained in (a)-(b) and take the Laplace transform to
obtain two equations in the Laplace variable s. Note that there are two outputs
CA(s) and T(s), and two inputs Ti(s) and CAi(s). Make sure that you work
using deviation variables.
d. The problem statement assumes that the flow rate q is constant. Since in point
(e) below we will be interested in calculating the transfer function T(s)/CAi(s),
assume that the inlet temperature Ti is also constant. Explain why the latter
assumption is useful.
e. Eliminate CA(s) from the equations and obtain the required transfer function
T(s)/CAi(s).
f. Obtain numerical coefficients for the transfer-function (get C
— A from the steady
state conditions)
g. Obtain an expression for the response of the transfer function found in (f) to a
step of unit height.
h. Approximate the transfer function obtained in (f) by a first-order transfer
function (i.e., neglect the fast time constants and any integrators). As an
example of an approximation, the transfer function G(s)= 3
20 1 3 1()()ss++
can be
approximated by G(s) = 3
20 1()s+
after neglecting and the fast time constant of 3
sec.
i. Obtain an expression for the step-response of the approximate transfer function
obtained in (h) and compare with the answer to (g).