Steady-State Thermal Analysis

From Emagtech Wiki
Revision as of 20:49, 21 June 2018 by Kazem Sabet (Talk | contribs)

Jump to: navigation, search
Maxwell1.png
Cube-icon.png Cad-ico.png Fdtd-ico.png Prop-ico.png Static-ico.png Planar-ico.png Metal-ico.png Po-ico.png

Heat Diffusion Equation

The Fourier law of heat conduction relates the heat transfer rate to the temperature variation:

[math]q = -k\nabla T(\mathbf{r})[/math]

where q is the heat flux density with units of W/m2, T(r) 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/(ρVcp) is the thermal diffusivity with units of m2/s, ρV is the volume mass density, cp is the specific heat capacity of the medium having units of J/(kg.K), and w(r) is the volume heat source density with units of W/m3.

In the steady-state regime, the time derivative vanishes and the diffusion equation reduces to the Poisson equation:

[math] \nabla^2 T(\mathbf{r}) = - \frac{w(\mathbf{r})}{k} [/math]

In a source-free region, the Poisson equation reduces further to the Laplace equation:

[math] \nabla^2 T(\mathbf{r}) = 0 [/math]

The steady-state heat diffusion equations are elliptic partial differential equations. These equations can be solved analytically only for a few canonical geometries with very simple boundary conditions. For most practical and realistic problems, you need to utilize a numerical technique and seek a computer solution. The Poisson and Laplace equations can be solved numerically using the finite difference (FD) method.

Thermal Boundary Conditions

The simplest thermal boundary condition at the walls of the computational domain is the Dirichlet boundary condition:

[math] T(\mathbf{r}) = T_0 [/math]

The Neumann boundary condition constrains the rate of heat flow through the domain boundary walls:

[math]-k \frac{\partial T}{\partial n} = -k \mathbf{\hat{n}} . \nabla T(\mathbf{r}) = q_s [/math]

where qs is the heat flux passing through the boundary walls.

If qs = 0, the Neumann boundary condition is also known as the adiabatic boundary condition, which represents a perfectly insulated surface:

[math]\frac{\partial T}{\partial n} = 0 [/math]

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_{\infty} \right] [/math]

where T is the ambient temperature, and h is the coefficient of convective heat transfer having units of W/(m2.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(r) can be an arbitrary function in 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(r) is analogous to the electric scalar potential Φ(r), the volume heat source density w(r) is analogous to the volume charge density ρ(r), and the thermal conductivity k is analogous to the permittivity ε.

Similarly, one can establish an analogy between the heat flux q(r) and the static electric field E(r):

[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:

Thermal Item Corresponding Electrical Item
Temperature Electric Scalar Potential
Heat Flux Density Electric Field
Perfect Thermal Conductor Perfect Electric Conductor
Insulator Material Dielectric Material
Volume Heat Source Volume Charge

The Finite Difference Technique

The general form of Poisson's equation for temperature can be expressed as:

[math] \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = -f(\mathbf{r}) [/math]

When f(r) = 0, one obtains the well-known Laplace equation, which applies to source-free regions.

The second derivative of T with respect to the x coordinate can be approximated by the second-order difference:

[math] \frac{\partial^2 T(\mathbf{r})}{\partial x^2} \approx \frac{T(x+\Delta x,y,z)-2T(x,y,z)+T(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.

In the finite difference method, the computational domain is discretized using a 3D rectangular grid as shown on the figure below. The grid spacing along the three principal coordinate axes is denoted by Δx, Δy and Δz, respectively. In this grid, the coordinates of any point (x,y,z) in the space can be expressed as x = iΔx, y = jΔy and z = kΔz. Therefore, every point can simply be represented by an index triplet (i,j,k).

The 3D rectangular grid used to mesh the computational domain.

The twmperature at the point (x,y,z) can be expressed in terms of the temperatures at its six neighboring grid points along the principal axes. This creates a 7-point computational molecule shown in the figure below:

The 7-point computational molecule used by the finite difference solver.

In the special case of a uniform grid with Δx = Δy = Δz, it can be shown that in a source-free region:

[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 boundary conditions take the following forms:

  • Dirichlet boundary condition: T = T0 =const.
  • Neumann boundary condition: ∂T/∂n = -qs0/k = const.
  • Adiabatic boundary condition: ∂T/∂n = 0.
  • Convective boundary condition: ∂T/∂n = h(T-T)/k.



Top icon.png Back to the Top of the Page

Back icon.png Back to Maxwell's Equations Page

Back icon.png Back to EM.Ferma Manual

Back icon.png Back to EM.Cube Main Page