Page 214 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

188 Chapter Seven

is so small, the attenuation factor e-tolrT = 4.5 x 10-s snuffs out
the delayed component leaving only the conventional first-order
component.
This small-diameter tube exchanger might be more amenable to
the idea of adjusting the jacket temperature to control the liquid out-
let temperature.

7-5 Studying the Tubular Energy Exchanger
In the Frequency Domain
We wish to analyze the effect of a sinusoidal variation in the jacket
temperature on the liquid outlet temperature. Start with the transfer
function between the process output, which is the liquid temperature
as it emerges from the tube at z = L, and the process input/ output
control which in this case is the jacket temperature, given in
Eq. (7-12). Make the usual substitution of s ~ jro:

_!!!
T{L,jro) 1- e rT e-it»to
T (jro) -r jro+ 1
5 1

Appendix F shows that this transfer function can be reformed in
terms of magnitude and phase as

~ -2!e.
2
T(L,jro)l= 1-2e rT cos (rot )+e rT
0
I T {jro) (-rro) + 1
2
5
(7-15)

Figure 7-7 shows a Bode plot for this process model for the large-
diameter tube exchanger where L = 1, 'rr = 1, and v = 1.
First, note that the magnitude and phase curves start to decrease
near the comer frequency which is

1
(J) =-=1
a" "'r
(J)
fcor = ; = 0.159 Hz
2