Page 221 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Distributed Processes 195

1. Each tank is completely mixed in the sense that the exit
temperature is the same as the temperature throughout the
lump or tank. These lumps are often called continuous stirred
tanks (CSTs).
2. Each tank is jacketed and is exposed to the jacket temperature
~· Although this need not be case in general, each tank sees
the same jacket temperature. That is, the jacket temperature
does not vary from tank to tank.
3. The temperature leaving the kth tank is the inlet temperature
for the k+ 1 th tank.
4. Parameterwise, each tank is identical. This is not necessary
but it does make the mathematics more manageable.

The mixing in each lump, mentioned in condition 1, will change
the propagation of step changes significantly. The relationship
between the number of lumps and the degree of axial mixing (or axial
diffusion) will be examined later.
A dynamic energy balance over the kth tank yields the following:

In from (k- 1)th tank: vAvCppTk-t
Out from kth tank: vAvCppTk
Into the kth tank from steam jacket: UAr(Ts- Tk)
Accumulation in kth tank: VC,p dt
dT.

Where ~ is the cross-sectional area for flow into and out of the
tank, Ar is the transfer area between the jacket and the tank, and V is
the volume of the tank.
The balance becomes

Equation (7-16) can be simplified slightly with the introduction of
a time constant and two gains, as in

(7-17)

_ vAvCpp
K; - UAr + vAvCpp

Note that both gains, independent of the parameterization, have
to be less than unity and, what's more, they add up to unity, as in

K; + Ks = 1 (7-18)