A plane wall, 7.5 cm thick, generates heat internally at the rate of 105 W/m3 . One side of the wall is insulated, and the other side is exposed to an environment at 90°C. The convection heat transfer coefficient between the wall and the environment is 500 W/(m2 K). If the thermal conductivity of the wall is 12 W/(m K), calculate the maximum temperature in the wall.
GIVEN
FIND
- The maximum temperature in the wall (Tmax)
ASSUMPTIONS
- The heat loss through the insulation is negligible
- The system has reached steady state
- One dimensional conduction through the wall
SKETCH
The one dimensional conduction equation, given is
This is subject to the following boundary conditions
No heat loss through the insulation
Convection at the other surface
Integrating the conduction equation once
C1 can be evaluated using the first boundary condition
The expression for T and its first derivative can be substituted into the second boundary condition to evaluate the constant C2
Substituting this into the expression for T yields the temperature distribution in the wall
Examination of this expression reveals that the maximum temperature occurs at x = 0
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