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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:
<math> \left[ -k \frac{\partial T}{\partial n} + h T(\mathbf{r}) \right]_\Omega = f(\mathbf{r}) </math>
where Ω is the boundary surface and f(<b>r</b>) is a can be an arbitrary functionin general. == The Analogy between Thermal and Electrostatic Equations == Let us now compare the steady-state thermal Poisson equation to the electrostatic Poisson equation: <math> \nabla^2 T(\mathbf{r}) = - \frac{w(\mathbf{r})}{k} \quad \leftrightarrow \quad \nabla^2 \Phi(\mathbf{r}) = -\frac{\rho(\mathbf{r})}{\epsilon} </math> One can see a one-to-one correspondence between the electrostatic and thermal quantities: Temperature T(<b>r</b>) is analogous to the electric scalar potential Φ(<b>r</b>), the volume heat source density w(<b>r</b>) is analogous to the volume charge density ρ(<b>r</b>), and the thermal conductivity k is analogous to the permittivity ε. Similarly, one can establish an analogy between the heat flux <b>q</b>(<b>r</b>) and the static electric field <b>E</b>(<b>r</b>): <math> \mathbf{q(r)} = -k\nabla T(\mathbf{r}) \quad \leftrightarrow \quad \mathbf{E(r)} = - \nabla \Phi(\mathbf{r})</math> The table below summarizes the analogous thermal and electrical quantities: {| class="wikitable"|-! scope="col"| Thermal Item! scope="col"| Corresponding Electrical Item|-| style="width:200px;" | Temperature| style="width:200px;" | Electric Scalar Potential|-| style="width:200px;" | Heat Flux Density| style="width:200px;" | Electric Field|-| style="width:200px;" | Perfect Thermal Conductor| style="width:200px;" | Perfect Electric Conductor|-| style="width:200px;" | Insulator Material| style="width:200px;" | Dielectric Material|-| style="width:200px;" | Volume Heat Source| style="width:200px;" | Volume Charge|}
== The Finite Difference Technique ==
The general form of Poisson's equation for any potential ψ temperature can be expressed as:
<math> \frac{\partial^2\psiT}{\partial x^2} + \frac{\partial^2\psiT}{\partial y^2} + \frac{\partial^2\psiT}{\partial z^2} = -f(\mathbf{r}) </math>
When f(<b>r</b>) = 0, one obtains the well-known Laplace equation, which applies to source-free regions.
The second derivative of ψ T with respect to the x coordinate can be approximated by the second-order difference:
<math> \frac{\partial^2\psiT(\mathbf{r})}{\partial x^2} \approx \frac{\psiT(x+\Delta x,y,z)-2\psi2T(x,y,z)+\psiT(x-\Delta x,y,z)}{(\Delta x)^2} </math>
Similar expressions can be written for the second derivative with respect to the y and z coordinates.
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The potential twmperature at the point (x,y,z) can be expressed in terms of the potentials temperatures at its six neighboring grid points along the principal axes. This creates a 7-point computational molecule shown in the figure below:
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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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In the special case of a uniform grid with Δx = Δy = Δz, it can be shown that in a source-free region:
<math> \psiT(i,j,k) = \frac{1}{6} \big[ \psiT(i+1,j,k) + \psiT(i-1,j,k) + \psiT(i,j+1,k) + \psiT(i,j-1,k) + \psiT(i,j,k+1) + \psiT(i,j,k-1) \big] </math>Â Two types of domain boundary conditions can be applied:Â *Dirichlet boundary condition: ψ = k =const.*Neumann boundary condition: ∂ψ/∂n = k = const.
In the above, ∂ψ/∂n denotes the normal derivative The standard types 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 take the primary potential difference equations.following forms:
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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