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== Heat Diffusion Equation ==
<math>q = -k\nabla T(\mathbf{r})</math>
where q is the heat flux density with units of W/m<sup>2</sup>, T (<b>r</b>) is the temperature expressed in °C or °K, ∇ is the gradient operator and k is the thermal conductivity with units of W/(m.K). It can be shown that the distribution of temperature is governed by the heat diffusion equation subject to the appropriate boundary conditions:
<math> \nabla^2 T(\mathbf{r}) - \frac{1} {\alpha}\frac{\partial T}{\partial t} = \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) - \frac{1} {\alpha}\frac{\partial T}{\partial t} = - \frac{w(\mathbf{r})}{k} </math>
where α = k/(ρ<sub>V</sub>c<sub>p</sub>) is the thermal diffusivity with units of m<sup>2</sup>/s, ρ<sub>V</sub> is the volume mass densityhaving units of kg/m<sup>3</sup>, c<sub>p</sub> is the specific heat capacity of the medium having units of J/(kg.K), and w(<b>r</b>) is the volume heat source density with units of W/m<sup>3</sup>.
In the steady-state regime, the time derivative vanishes and the diffusion equation reduces to the Poisson equation: