Page 209 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Distributed Processes 183

where the added mechanism of transport is described by the term
2
(k I pCP) I (d T I ()z ). The presence of this term makes the solution
2
procedure significantly more difficult and we will not refer to Eq. (7-7)
until later in this chapter when lumping is discussed.

7-3 Solution of the Tubular Heat Exchanger Equation
There are a variety of approaches to solve Eq. (7-6) but we will pick the
one using the tools already developed in this book and the one that will
lend itself to using the frequency domain to gain insight. This means
transforming the time dependence out of Eq. (7-6) using the Laplace
transform. This will leave us with a first-order ordinary differential
equation in the spatial dimension z which we can solve using stand-
ard techniques. The details are given in App. F.
The result of applying the Laplace transform to Eq. (7-6) is
- dT 1 - -
sT+v-=-(T -T) (7-8)
dz 'Cr s

You should convince yourself that Eq. (7-8) is indeed the result of
multiplying Eq. (7-6) by exp(-st) and integrating over [O,oo) with
respect to time. In any case, after the dust has settled, Eq. (7-8) is a
first-order ordinary differential equation of the form

1 )- f
dt (
v-+ s+- T=......!... (7-9)
dz 'Cr 'rr
where the Laplace variable s is just a parameter. Remember that T is
5
the Laplace transform of the jacket temperature which we specified
could be a function of time but not of axial position, that is, ~or T is
5
not a function of z.
Now, how do we solve Eq. (7-9)? We could apply the Laplace
transform again with a different variable, say p, instead of s and
remove the spatial dimension or we could solve the ordinary differ-
ential equation by trying a solution of the form cecz. Both of these
approaches have been used elsewhere in this book. The details of the
solution are presented in App. F and the result is

(7-10)

where