Page 364 - Modelling in Transport Phenomena A Conceptual Approach

344 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

is given by
STT k(T) dT = - 1' [I' W(u) d] dz
+(r W(u) du] (9.2-10)

k(T) dT + 6" [I' dz}

Note that when % = 0, Eq. (9.2-10) reduces to Eq. (G) in Table 8.1. Equation
(9.2-10) may be further simplified depending on whether the thermal conductivity
and/or energy generation per unit volume are constant.

Case (i) k= constant

In this case Eq. (9.2-10) reduces to

k (T - To) - 1" [1' %(u) du] dz
=

When W = 0, Eq. (9.2-11) reduces to Q. (H) in Table 8.1.

Case (ii) k = constant; W = constant

In this case Eq. (9.2-10) simplifies to
T=T0+x [z- -(To-TL)z
%L2
z
z
(;)2]
(9.2-12)
The location of the maximum temperature can be obtained from dT/dz = 0 as

(9.2-13)

Substitution of Eq. (9.2-13) into Eq. (9.2-12) gives the value of the maximum
temperature as
To + TL ?R L2 k (To - TL)~
Tma=-+- (9.2-14)
2 8k + 2RL2
The representative temperature profiles depending on the values of To and TL are
shown in Figure 9.6.