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Consider one-dimensional steady-state heat conduction without heat generation in a plane wall, with boundary conditions as shown in the figure below. The conductivity of the wall is given by $k = k_0(1 + bT)$, where $k_0$ and $b$ are positive constants, and $T$ is temperature. As $x$ increases, the temperature gradient $(dT/dx)$ will
- remain constant
- be zero
- increase
- decrease
Correct answer: decrease
Solution
In steady one-dimensional conduction without heat generation, the heat flux remains constant through the wall. Since $k = k_0(1+bT)$ increases with temperature, the temperature gradient needed to maintain the same heat flux becomes smaller in magnitude as $T$ increases. Hence, as $x$ increases, $dT/dx$ decreases.
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