T s,1
h 1
T ∞,1
q
.
x
h 2
T ∞,2
T s,2
−L
O
L
�
�
18
Analytical Heat Transfer
FIGURE 2.4
Flat plate heat conduction with internal heat generation and asymmetrical boundary conditions.
where k is the conductivity of insulation material, h is outside convection
coefficient from insulation material. If r i is less than (r o ) critical , q is increased
as insulation is added until r o = (r o ) critical . Further increases in r o cause q to
decrease. If, however, r i is greater than (r o ) critical , any addition of insulation
with decrease q (heat loss rate).
2.2 Conduction with Heat Generation
1-D steady-state heat conduction equation with heat generation in the plane
wall is (from Equation 1.17)
∂ 2 T
q ˙
+ = 0
∂x 2
k
(2.19)
dT
q ˙
= − x + c 1
dx
k

The general solution is

q ˙
T = − x
2
+ c 1 x + c 2
2k
with asymmetrical boundary conditions [2] shown in Figure 2.4,
q ˙
at x = L, T = T s,L = − L
2
+ c 1 L + c 2
2k
q ˙
at x = −L, T = T s,1 = − (−L)
2
+ c 1 (−L) + c 2
2k
Solve for c 1 and c 2 , one obtains the temperature distribution
2
qL ˙ 2
x
T s,2 − T s,1 x T s,2 + T s,1
T(x) =
1 −
+
+
2k
L 2
2
L
2