Page 369 - Modelling in Transport Phenomena A Conceptual Approach

9.2. ENERGY TRANSPORT WITHOUT CONVECTION 349

W Case (ii) k = constant; R = constant

In this case Eq. (9.2-30) reduces to
T=Tz+S b-(&)2]

4k

1n(r'R2)
[1- ( $!-)2]} ln(R1IR2) (9.232)

The location of maximum temperature can be obtained from dT/dr = 0 as

Type I1 boundary condition

The solution of Eq. (9.2-28) subject to the boundary conditions
dT
at r=R1 -k-= 41
dz (9.2-34)
at r=R2 T=T2

is given by
STT k(T)dT = I"' [iF?J?(u)udu] dr

+ [Jo"; W(u) udu - q~Rl] In (e) (9.2-35)

Note that when R = 0, Eq. (9.2-35) reduces to Eq. (C) in Table 8.4.

W Case (i) k = constant

In this case Eq. (9.2-35) reduces to
k (T - T2) = 1"' f [1'R(u)udu] dr + [iR' WU)~ - qiRi] (&)
du
(9.2-36)
When $2 = 0, Eq. (9.236) simplifies to Eq. (D) in Table 8.4.