Chapter 12 Special Topics in Heat Transfer

Forsberg Heat Transfer
Chapter 12
Special Topics




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