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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_{amb\infty} \right] </math>
where T<sub>amb∞</sub> is the ambient temperature, and h is the coefficient of convective heat transfer having units of W/(m<sup>2</sup>.K).
The convective boundary condition is a special case of Robin boundary condition:
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[[Image:FD 7Point7PointA.png|thumb|left|480px| The 7-point computational molecule used by the finite difference solver.]]
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<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>
Two The standard types of domain boundary conditions can be appliedtake the following forms: *Dirichlet boundary condition: ψ = k =const.*Neumann boundary condition: ∂ψ/∂n = k = const. In the above, ∂ψ/∂n denotes the normal derivative of the potential at the surface of the domain boundary. [[EM.Ferma]]'s default domain boundary condition for both the electrostatic and magnetostatic solvers is Dirichlet. At the interface between different material media, additional boundary conditions must be applied. These boundary conditions involve electric or magnetic field components. The field components can be expressed as partial derivatives of the potential, i.e. in the form of ∂ψ/∂x, ∂ψ/∂y or ∂ψ/∂z. Using the respective finite difference approximations of these derivatives, one arrives at fairly complicated difference equations involving the constitutive parameters ε, μ and σ, which must be solved simultaneously with the primary potential difference equations.
Note that the electrostatic Poisson and Laplace equations are of the scalar type, while the magnetostatic Poisson and Laplace equations are vectorial*Dirichlet boundary condition: T = T<sub>0</sub> =const. As a result, the size of the numerical problem in the latter case is three times as large as the former case for the same mesh size*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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