Page 205 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Distributed Processes 179

The thickness of the disc is decreased to differential or infinitesi-
mal size as in

From App. A one sees that the above equation contains the defini-
tion of the derivative ofT with respect to z, as in

dT
vApCP Tz = U1rD(T - T(z)] (7-1)
5
This ordinary differential equation describes the steady-state
behavior of the idealized jacketed tube energy exchanger. From Chap. 3
we already know how to solve this equation if we know an inlet tem-
perature, as in T(O) = T •
0
If Eq. (7-1) is rearranged slightly, the reader can see the similarity
to the equation for the liquid tank presented in Chap. 3.
vApCpdT
U(1rD) Tz + T = Ts
dT
ytdz+T=Ts (7-2)

vApCP vDpCP
Yl = U1rD = --ru-

The reader has seen Eq. (7-2) before, at least structurally. By
inspection, the reader can arrive at a solution to Eq. (7-2) as

(7-3)

The parameter yt can be considered as a kind of Space constant,"
11
somewhat analogous to the time constant used in transient analysis.
In fact, yt is the tube length needed for T(z) to reach 63% of the jacket
temperature ~-
The reader should spend a few moments looking at Eq. (7-3) to
see how T(z) changes as various parameters change. First, it shows
that as one travels down the tube axially the temperature T(z)
approaches ~- Second, as the overall heat transfer coefficient U
increases, the parameter yt decreases. Thus, T(z) approaches ~ more