Steady-State Thermal Analysis - Revision history

Kazem Sabet at 21:20, 23 July 2018

2018-07-23T21:20:30Z

Kazem Sabet: /* Heat Diffusion Equation */

2018-06-21T20:51:10Z

‎Heat Diffusion Equation

Kazem Sabet: /* Heat Diffusion Equation */

2018-06-21T20:49:22Z

‎Heat Diffusion Equation

Kazem Sabet: /* The Finite Difference Technique */

2018-06-17T02:41:43Z

‎The Finite Difference Technique

Kazem Sabet: /* The Finite Difference Technique */

2018-06-17T02:33:26Z

‎The Finite Difference Technique

Kazem Sabet: /* Thermal Boundary Conditions */

2018-06-17T02:30:49Z

‎Thermal Boundary Conditions

Kazem Sabet: /* Thermal Boundary Conditions */

2018-06-17T02:30:20Z

‎Thermal Boundary Conditions

Kazem Sabet: /* Thermal Boundary Conditions */

2018-06-17T02:30:00Z

‎Thermal Boundary Conditions

Kazem Sabet at 02:27, 17 June 2018

2018-06-17T02:27:43Z

Kazem Sabet at 01:20, 17 June 2018

2018-06-17T01:20:23Z

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 <tr>
 </table>
 == Heat Diffusion Equation ==

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 <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 &alpha; = k/(&rho;<sub>V</sub>c<sub>p</sub>) is the thermal diffusivity with units of m<sup>2</sup>/s, &rho;<sub>V</sub> is the volume mass density, c<sub>p</sub> 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/m<sup>3</sup>.
+where &alpha; = k/(&rho;<sub>V</sub>c<sub>p</sub>) is the thermal diffusivity with units of m<sup>2</sup>/s, &rho;<sub>V</sub> is the volume mass density having 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:

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 <math>q = -k\nabla T(\mathbf{r})</math>
-where q is the heat flux density with units of W/m<sup>2</sup>, T is the temperature expressed in &deg;C or &deg;K, &nabla; 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:
+where q is the heat flux density with units of W/m<sup>2</sup>, T(<b>r</b>) is the temperature expressed in &deg;C or &deg;K, &nabla; 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>

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 <hr>
-[[Image:Top_icon.png|30px]] '''[[#Static_Modeling_Methods | Back to the Top of the Page]]'''
+[[Image:Top_icon.png|30px]] '''[[#Heat_Diffusion_Equation | Back to the Top of the Page]]'''
 [[Image:Back_icon.png|30px]] '''[[A_Review_of_Maxwell%27s_Equations_%26_Computational_Electromagnetics_(CEM)  | Back to Maxwell's Equations Page]]'''

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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>
-The standard types of domain boundary conditions take the following forms:
+The standard types of boundary conditions take the following forms:
 *Dirichlet boundary condition: T = T<sub>0</sub> =const.
 *Neumann boundary condition: &part;T/&part;n = -q<sub>s0</sub>/k = const.
 *Adiabatic boundary condition: &part;T/&part;n = 0.
 <br />

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 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_{\infin} \right] </math>
+<math>-k \frac{\partial T}{\partial n} = -h \left[ T(\mathbf{r}) - T_{\infty} \right] </math>
 where T<sub>&infin;</sub> is the ambient temperature, and h is the coefficient of convective heat transfer having units of W/(m<sup>2</sup>.K).

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 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} \right] </math>
+<math>-k \frac{\partial T}{\partial n} = -h \left[ T(\mathbf{r}) - T_{\inf} \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).
+where T<sub>&infin;</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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 <tr>
 <td>
-[[Image:FD 7Point.png|thumb|left|480px| The 7-point computational molecule used by the finite difference solver.]]
+[[Image:FD 7PointA.png|thumb|left|480px| The 7-point computational molecule used by the finite difference solver.]]
 </td>
 </tr>
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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 types of domain boundary conditions can be applied:
+Two standard types of domain boundary conditions take the following forms:
-*Dirichlet boundary condition: &psi; = k =const.
+*Dirichlet boundary condition: T = k =const.
-*Neumann boundary condition: &part;&psi;/&part;n = k = const.
+*Neumann boundary condition: &part;T/&part;n = k = const.
 In the above, &part;&psi;/&part;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 &part;&psi;/&part;x, &part;&psi;/&part;y or &part;&psi;/&part;z. Using the respective finite difference approximations of these derivatives, one arrives at fairly complicated difference equations involving the constitutive parameters &epsilon;, &mu; and &sigma;, which must be solved simultaneously with the primary potential difference equations.