Serway & Jewett — The First Law of Thermodynamics: Problem 46.

A spherical shell has inner radius \(3.00 \text{ cm}\) and outer radius \(7.00 \text{ cm}\). It is made of material with thermal conductivity \(k = 0.800 \text{ W/m} \cdot {}^\circ\text{C}\). The interior is maintained at temperature \(5^\circ\text{C}\) and the exterior at \(40^\circ\text{C}\). After an interval of time, the shell reaches a steady state with the temperature at each point within it remaining constant in time. (a) Explain why the rate of energy transfer \(P\) must be the same through each spherical surface, of radius \(r\), within the shell and must satisfy \(\frac{dT}{dr} = \frac{P}{4\pi kr^2}\). (b) Next, prove that \(\int_{5}^{40} dT = \frac{P}{4\pi k} \int_{0.03}^{0.07} r^{-2} dr\) where \(T\) is in degrees Celsius and \(r\) is in meters. (c) Find the rate of energy transfer through the shell. (d) Prove that \(\int_{5}^{T} dT = 1.84 \int_{0.03}^{r} r^{-2} dr\) where \(T\) is in degrees Celsius and \(r\) is in meters. (e) Find the temperature within the shell as a function of radius. (f) Find the temperature at \(r = 5.00 \text{ cm}\), halfway through the shell.