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8.5.5 Anisotropic Thermal Conductivity for Solids

The anisotropic conductivity option in FLUENT solves the conduction equation in solids with the thermal conductivity specified as a matrix. The heat flux vector is written as


 q_i = -k_{ij}\frac{\partial T}{\partial x_{j}} (8.5-9)

The following options are available for defining anisotropic thermal conductivity in FLUENT. These are discussed below.

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Note that the anisotropic conductivity options are available only with the pressure-based solver; you cannot use them with the density-based solvers.



Anisotropic Thermal Conductivity


For anisotropic diffusion, the thermal conductivity matrix (Equation  8.5-9) is specified as


 k_{ij} = k \hat{\bf e}_{ij} (8.5-10)

where $k$ is the conductivity and $\hat{\bf e}_{ij}$ is a matrix (2 $\times$ 2 for two dimensions and 3 $\times$ 3 for three-dimensional problems. Note that $\hat{\bf e}_{ij}$ can be a non-symmetric matrix.

To define anisotropic thermal conductivity for a solid material, select anisotropic for Thermal Conductivity in the Materials panel (Figure  8.5.1). This will open the Anisotropic Conductivity panel (Figure  8.5.2).

Figure 8.5.2: The Anisotropic Conductivity Panel
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In the Anisotropic Conductivity panel, enter the Matrix Components of matrix $\hat{\bf e}_{ij}$ and then select the Conductivity ( $k$ in Equation  8.5-10) to be a constant, polynomial function of temperature ( polynomial, piecewise-linear, piecewise-polynomial), or user-defined function. See Sections  8.5.1 and 8.5.2 for details on constants and thermal polynomial functions.

When you select the user-defined option, the User-Defined Functions panel will open allowing you to hook a DEFINE_PROPERTY UDF only if you have previously loaded a compiled UDF library or interpreted the UDF. Otherwise, you will get an error message. Refer to the separate UDF Manual for details on user-defined functions.



Biaxial Thermal Conductivity


Biaxial thermal conductivity is mainly applicable to solid materials used for the wall shell conduction model. To define a biaxial thermal conductivity, select biaxial in the drop-down list for Thermal Conductivity in the Materials panel. This opens the Biaxial Conductivity panel (Figure  8.5.3).

Figure 8.5.3: The Biaxial Conductivity Panel
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In the Biaxial Conductivity panel, both the conductivity normal to the surface of the solid region ( Transverse Conductivity) and the conductivity within the shell or solid region ( Planar Conductivity) can be defined as constant, polynomial, piecewise-linear, or piecewise-polynomial. See Sections  8.5.1 and 8.5.2 for details on these parameters. Within the shell, however, the conductivity is isotropic. See Section  7.13.1 for more information about shell conduction in walls.



Orthotropic Thermal Conductivity


When the orthotropic thermal conductivity is used, the thermal conductivities $(k_\xi,k_\eta,k_\zeta)$ in the principal directions $(\hat{\bf e}_\xi,\hat{\bf e}_\eta,\hat{\bf e}_\zeta)$ are specified. The conductivity matrix is then computed as


 k_{ij} = k_{\xi} e_{\xi i} e_{\xi j} + k_{\eta} e_{\eta i} e_{\eta j}+ k_{\zeta} e_{\zeta i} e_{\zeta j} (8.5-11)

To define an orthotropic thermal conductivity in solids, select orthotropic in the drop-down list for Thermal Conductivity in the Materials panel. This opens the Orthotropic Conductivity panel (Figure  8.5.4).

Figure 8.5.4: The Orthotropic Conductivity Panel
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Since the directions $(\hat{\bf e}_\xi,\hat{\bf e}_\eta,\hat{\bf e}_\zeta)$ are mutually orthogonal, only the first two need to be specified for three-dimensional problems. $\hat{\bf e}_\xi$ is defined using X,Y,Z under Direction 0 Components, and $\hat{\bf e}_\eta$ is defined using X,Y,Z under Direction 1 Components. You can define Conductivity 0 $(k_\xi)$, Conductivity 1 $(k_\eta)$, and Conductivity 2 $(k_\zeta)$ as constant, polynomial, piecewise-linear, piecewise-polynomial functions of temperature, or user-defined. See Sections  8.5.1 for and 8.5.2 for details on constant and temperature profile functions.

When you select the user-defined option, the User-Defined Functions panel will open allowing you to hook a DEFINE_PROPERTY UDF only if you have previously loaded a compiled UDF library or interpreted the UDF. Otherwise, you will get an error message. Refer to the separate UDF Manual for details on user-defined functions.

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For two-dimensional problems, only the functions $(k_\xi,k_\eta)$ and the unit vector $(\hat{\bf e}_\xi)$ need to be specified.



Cylindrical Orthotropic Thermal Conductivity


The orthotropic conductivity of solids can be specified in cylindrical coordinates. To define the orthotropic thermal conductivity in cylindrical coordinates, select cyl-orthotropic in the drop-down list for Thermal Conductivity in the Materials panel. This opens the Cylindrical Orthotropic Conductivity panel (Figure  8.5.5).

Figure 8.5.5: The Cylindrical Orthotropic Conductivity Panel
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In three-dimensional cases, the origin and the direction of the cylindrical coordinate system must be specified along with the radial, tangential, and axial direction conductivities. In two-dimensional cases, the origin of the cylindrical coordinate system must be specified along with the radial and tangential direction conductivities. Note that in two-dimensional cases, the direction is always along the + z axis. FLUENT will automatically compute the anisotropic conductivity matrix at each cell from this input. The calculation is based on the location of the cell in the cylindrical coordinate system specified.

You can define the Radial Conductivity, Tangential Conductivity, and Axial Conductivity as constant, polynomial, piecewise-linear, piecewise-polynomial, or as user-defined functions of temperature. See Sections  8.5.1 and 8.5.2 for details on constant and thermal profile functions.

When you select the user-defined option, the User-Defined Functions panel will open allowing you to hook a DEFINE_PROPERTY UDF only if you have previously loaded a compiled UDF library or interpreted the UDF. Otherwise, you will get an error message. Refer to the separate UDF Manual for details on user-defined functions.

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For conductivity calculations near the wall, the cell next to the wall is chosen for computing the conductivity matrix instead of the wall itself.


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