Starting with an energy balance on a spherical shell volume element, derive the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heat generation.

Consider a small spherical shell of thickness \(\delta r\) and radius \(r\). The volume of the shell is given by, \(V = 4 \pi r^2 \delta r\). Let the heat transferred into the shell be \(q_{in}\), and the heat transferred out of the shell be \(q_{out}\). The difference between these two will lead to a change in the internal energy of the shell. Therefore, the energy balance can be written as: \(\frac{\partial E}{\partial t} = q_{in} - q_{out}\).

We will assume that the thermal conductivity of the sphere, \(k\), is constant and there is no heat generation within the sphere. This means that there are no additional heat sources or sinks within the sphere, and heat only flows due to temperature gradients.

According to Fourier's law of heat conduction, the heat flux in the radial direction is given by \(q_r = -k \frac{dT}{d r}\), where \(T\) is the temperature and \(r\) is the radial coordinate. The heat entering the shell is \(q_{in} = -k \frac{dT}{dr} \Big|_{r}4 \pi r^2\) and the heat exiting the shell is \(q_{out} = -k \frac{dT}{dr} \Big|_{r+\delta r}4 \pi (r + \delta r)^2\). Now we can rewrite the energy balance equation: \begin{align*} \frac{\partial E}{\partial t} &= q_{in} - q_{out} \\ \frac{\partial (V \rho c_p T)}{\partial t} &= -k \frac{dT}{dr} \Big|_{r}4 \pi r^2 - (-k \frac{dT}{dr} \Big|_{r+\delta r}4 \pi (r + \delta r)^2) \end{align*} Here, \(\rho\) is the density of the material and \(c_p\) is the specific heat capacity. Dividing both sides by \(V= 4\pi r^2\delta r\), we get: \begin{align*} \frac{1}{r^2}\frac{\partial (r^2 \rho c_p T)}{\partial t} &= -\frac{k}{\delta r}\left( \frac{dT}{dr} \Big|_{r+\delta r} - \frac{dT}{dr} \Big|_{r} \right) \\ \end{align*} Now we take the limit as \(\delta r \to 0\): \begin{align*} \frac{1}{r^2}\frac{\partial (r^2 \rho c_p T)}{\partial t} &= -k \frac{d^2T}{d r^2} \end{align*} This is the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heat generation.