Plane Wall with Uniform Heat Generation (surfaces maintained at different temperatures):

From one dimensional heat conduction equation, for a flat wall we have, \[\frac{d^2T}{dx^2} = -\frac{\dot{g}}{k}\] Integrating, we get \[\frac{dT}{dx} = -\frac{\dot{g}x}{k} + C_1\] Integrating further, we get \[T = -\frac{\dot{g}x^2}{2k} + C_1x + C_2 \tag*{(27)}\] Boundary conditions: \[\begin{aligned} T&= T_1 \qquad & \text{ at } x&=-L \\ T&=T_2 & \text{ at } x&=L\end{aligned}\] From B.C. 1 and 2, \[\begin{align*} T_1 &= -\frac{\dot{g}L^2}{2k} - C_1L + C_2 \tag*{(28)} \\ T_2 &= -\frac{\dot{g}L^2}{2k} + C_1L + C_2 \tag*{(29)}\end{align*}\] Adding Eqns.(28) and (29), we get \[\begin{align*} T_1 + T_2 &=-\frac{\dot{g}L^2}{2k}2 + 2 C_2 \nonumber \\ \Longrightarrow \quad C_2 &= \frac{\dot{g}L^2}{2k} + \frac{T_2+T_1}{2} \tag*{(30)}\end{align*}\] Eqn.(29) \(-\) Eqn.(28) \(\Longrightarrow\) \[C_1 = \frac{T_2-T_1}{2L} \tag*{(31)}\] Using Eqns.(30) and (31) in Eqn.(27), we get \[\begin{aligned} T &= -\frac{\dot{g}x^2}{2k} + \frac{T_2-T_1}{2L}x + \frac{\dot{g}L^2}{2k} + \frac{T_2+T_1}{2} \\ &= \frac{\dot{g}L^2}{2k}-\frac{\dot{g}x^2}{2k} + \frac{T_2-T_1}{2}\frac{x}{L} + \frac{T_2+T_1}{2}\end{aligned}\] i.e., \[T = \frac{\dot{g}L^2}{2k}\left(1-\frac{x^2}{L^2} \right) + \frac{T_2-T_1}{2}\frac{x}{L} + \frac{T_2+T_1}{2}\] For \(T_1=T_2=T_s\), we get \[T = \frac{\dot{g}L^2}{2k}\left(1-\frac{x^2}{L^2} \right) + T_s\] The maximum temperature exists at the mid-plane. \[T_{\text{max}} = T(0) = \frac{\dot{g}L^2}{2k} + T_s\]