6. Maximum Temperature for Systems with Uniform Heat Generation (both the sides at the same temperature)

Using the Fourier’s law, we can derive the relationship for the center (maximum) temperature of a long cylinder of radius \(r_o\).

\[-kA_r \frac{dT}{dr} = \dot{g}V_r\] Here, \[A_r = 2\pi rL \qquad \text{ and } \qquad V_r = \pi r^2 L\] Integrating with the limits of \(T=T_o\) at \(r=0\); and \(T=T_s\) at \(r=r_o\); we get \[T_o-T_s = \frac{\dot{g}r_o^2}{4k}\] For sphere of radius \(r_o\), \[T_o-T_s = \frac{\dot{g}r_o^2}{6k}\]

For flat plate, \(r=x\), and \(V_r/A_r = x\). Hence, we get \[T_o-T_s = \frac{\dot{g}L^2}{2k}\]