Consider a one-dimensional plane wall of thickness \(2 L\) that experiences uniform volumetric heat generation. The surface temperatures of the wall are maintained at \(T_{s, 1}\) and \(T_{s, 2}\) as shown in the sketch. Verify, by direct substitution, that an expression of the form

\[T(x)=\frac{\dot{q} L^{2}}{2 k}\left(1-\frac{x^{2}}{L^{2}}\right)+\frac{T_{s, 2}-T_{s, 1}}{2} \frac{x}{L}+\frac{T_{s, 1}+T_{s, 2}}{2}\]

satisfies the steady-state form of the heat diffusion equation. Determine an expression for the heat flux distribution, \(q^{\prime \prime}(x)\).

Transcribed Image Text:

Ts.1 -L 7+ Tx2