Analytic Solution for 1D Transient Heat Conduction
The problem geometry and boundary conditions are shown below. An initially isothermal (Tinitial) semi-infinite medium is suddenly subject to a surface temperature Th.
The temperature field can be non-dimensionalized as:
$$\theta (x,t)=\frac{T(x,t)-T_{\text{initial}}}{T_h-T_{\text{initial}}}$$
The governing differential equation (with spatially one-dimensional heat flow) is
$$ \frac{\partial \theta (x,t)}{\partial t} = \alpha \frac{\partial^2 \theta (x,t)}{\partial x^2} $$
The solution for all locations x and times t is:
$$\theta(x,t) = 1-\text{erf}\left[\frac{x}{2\sqrt{\alpha t}}\right]$$
where $$\alpha$$ is the material’s thermal diffusivity.
Graphical CDF Tool
The following is an embedded, active Mathematica CDF tool. The units for $$\alpha$$ are cm2/sec, with corresponding units of cm and sec for x and t, respectively.
