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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:
At the interface between the surface of a solid object and air, the convective boundary condition must be enforced:
<math>-k \frac{\partial T}{\partial n} = -h \left[ T(\mathbf{r}) - T_{\infinfty} \right] </math>
where T<sub>∞</sub> is the ambient temperature, and h is the coefficient of convective heat transfer having units of W/(m<sup>2</sup>.K).
<math> T(i,j,k) = \frac{1}{6} \big[ T(i+1,j,k) + T(i-1,j,k) + T(i,j+1,k) + T(i,j-1,k) + T(i,j,k+1) + T(i,j,k-1) \big] </math>
The standard types of domain boundary conditions take the following forms:
*Dirichlet boundary condition: T = T<sub>0</sub> =const.
*Neumann boundary condition: ∂T/∂n = -q<sub>s0</sub>/k = const.
*Adiabatic boundary condition: ∂T/∂n = 0.
*Convective boundary condition: ∂T/∂n = h(T-T<sub>∞</sub>)/k.
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