A plane wall 15 cm thick has a thermal conductivity given by the relation k = 2.0 + 0.0005 T W/(m K) where T is in degrees Kelvin. If one surface of this wall is maintained at 150 °C and the other at 50 °C, determine the rate of heat transfer per square meter. Sketch the temperature distribution through the wall.
GIVEN
A plane wall
Thickness (L) = 15 cm = 0.15 m
Thermal conductivity (k) = 2.0 + 0.0005 T W/(m K) (with T in Kelvin)
Surface temperatures: Th = 150 °C Tc = 50 °C
FIND
(a) The rate of heat transfer per square meter (q/A)
(b) The temperature distribution through the wall
ASSUMPTIONS
The wall has reached steady state
Conduction occurs in one dimension
SKETCH
Simplifying for steady state conduction with no internal heat generation but allowing for the
variation of thermal conductivity with temperature yields
Integrating
The constant can be evaluated using the first boundary condition
(a) The rate of heat transfer can be evaluated using the second boundary condition:
(b) Therefore, the temperature distribution is
0.00025 T 2 + 2.0 T = 890.7 – 1458 x
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