Consider the problem of transient heat transfer with a constant heat source in a slab.

Show that the governing equation in dimensionless form is

\[\begin{equation*}\frac{\partial \theta}{\partial t^{*}}=\frac{\partial^{2} \theta}{\partial \xi^{2}}+1 \tag{11.83}\end{equation*}\]

Identify and define the parameters in this problem stated in the above dimensionless form. State the boundary conditions for (i) constant surface temperature of the slab and (ii) convective heat loss from the surface.

Now consider the series solution to the case of constant surface temperature. The initial temperature is \(\theta_{\mathrm{i}}\), which is assumed to be a constant for this problem. For simplicity take this as equal to the surface temperature.

Use the superposition to split the problem into two problems, (ii) a position-dependent steady-state part and (ii) a time-dependent part.

Solve for each of these parts for the case of constant surface temperature and find the composite solution.

Sketch and plot illustrative temperature distributions for different values of the Fourier number \(\left(t^{*}\right)\).