This chapter explores advanced and specialized aspects of heat transfer beyond the fundamentals. Topics covered include heat transfer in nanomaterials, phase change phenomena, renewable energy systems, thermal management in electronic devices, and heat transfer in biological systems. It also discus...
Heat Generation in a Plane Wall
The wall, with temperature boundary conditions
1
,The appropriate heat conduction equation is
T
k + qgen = 0
x x
With conductivity and internal heat generation
uniform and constant,
d 2T qgen
2
+ = 0
dx k
Integrating this twice, we get
dT −qgen
= x + C1
dx k
−qgen 2
T (x) = x + C1 x + C 2
2k
Boundary conditions are
At x = 0, T = T1 and At x = L, T = T2
−qgen
T1 = (0)2 + C1 (0) + C2 C2 = T1
2k
−qgen T2 − T1 qgenL
T2 = L2 + C1L + T1 C1 = +
2k L 2k
qgen T2 − T1
T (x) = T1 + (Lx − x 2 ) + x
2k L
2
, The maximum temperature in the wall is where
dT
= 0. The location is found to be
dx
L k T2 − T1
xmax = +
2 qgen L
Back to the T ( x) equation:
qgen T2 − T1
T (x) = T1 + (Lx − x 2 ) +
x
2k L
If both wall surfaces are the same temperature
T1 = T2 = To , then
qgen
T (x) = T0 + (Lx − x 2 )
2k
3
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