Page 208 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

182 Chapter Seven

H the space element 4z and the time element At are both decreased
to an infinitesimally small size, then the following partial differential
equation results.

aT oT
-vApCP az-+U(nD)[T - T(z,t)) = ApCP at (7-4)
5
Dividing all terms by ApCP and remembering that A= nfY/4,
gives
aT 'i1f 4U
a,+vaz-= DpC [T 5 -T(z,t)] (7-5)
p

The quantity DpCP I 4U has units of sec, so Eq. (7-5) could be
written as

aT 'i1f 1
-+v-=-[T -T(z t)]
at dz 'fr 5 I
(7-6)
DpC
P =
'rr = __ VI
4U v
where 'rr has units of time and is a time constant. Equation (7-6) is
a partial differential equation describing the time-space behavior
of the temperature in the tube. It is subject to initial conditions,
such as T(z,O) = T , 0 S z S L, and a boundary condition on the inlet,
0
such as T{O,t) = 1f, t > 0. Since we have added the dependence on
time, this process model can be used in simulations to test control
algorithms.
As an aside, the quantity

oT 'i1f
-+v-
at az
is often called the total derivative or the convective derivative of tem-
perature and is sometimes given the symbol DT I Dt.

7 ·2·1 Transfer by Diffusion
The model in Eq. (7-6) describes the transfer of energy along the tube
by convection. Energy can also be transported axially by molecular
diffusion where the rate is proportional to the axial gradient of tem-
perature, as in -k('i1f I d:z) where k is the thermal conductivity. Hone
modifies the above energy balance on an element of length 4z by
adding the contribution of diffusion, the result is

(7-7)