General Diffusion Equation

Describes the fundamental conservation of heat in a material in a perfected case of no heat absorption or generation

$$\frac{\partial T}{\partial t}=D{\nabla }^{2}T$$

Derivation

$$-\frac{\partial q}{\partial t}=\nabla \cdot h$$

With $q$ being heat per unit volume of the material and $\nabla \cdot h$ the outward normal of the “heat flow density” that is represented by $h$. We use the definition of heat conduction $h=-\kappa \nabla T$ to arrive at

$$\frac{\partial q}{\partial t}=\kappa {\nabla }^{2}T$$

With $T$ being the total temperature across and kappa a constant. Then, we find that the overall change in temperature is proportional to the change in heat per unit volume with the proportionality being governed by a yet unknown proportionality constant.

$$\frac{\partial q}{\partial t}=c\_v\frac{\partial T}{\partial t}$$

Then we rewrite our heat change equation as

$$\frac{\partial T}{\partial t}=\frac{\kappa }{c\_v}{\nabla }^{2}T$$

or simplified as

$$\frac{\partial T}{\partial t}=D{\nabla }^{2}T$$

which frankly is the heat-diffusion equation.