Page 207 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Distributed Processes 181

assumed to be much quicker than those of the liquid flowing through
the tube.
As with the steady-state derivation in Sec. 7-1, the reader can skip
to the result in Eq. (7-6) which is boxed. For the adventurous, the bal-
ance proceeds as follows.
Energy rate at z due to convection at time t during the interval At:

Energy rate out at z+LU due to convection at time t during the
interval At:

Energy rate in from jacket at time t during the interval At:

Accumulation of energy in the disc between time t and time t +At
in the volume AAz:

AAzpC T(z + z + 4z t At)- ALUpC T(z + z + 4z t)
p 2 ,+ p 2 I

Entering the various elements into the balance equation

In- out= accumulation
gives

z+z+LU )~
vApCPT(z,t)At+U(nDLU) ~-T ,t ~At-vApCPT(z+LU,t)At
[ ( 2

Dividing by At4z and doing a little rearranging gives