The following equations describe the computation of orthotropic thermal conductivity. To perform this computation, 3 load cases are needed, each of which applies a temperature gradient in X, Y, and Z direction, respectively.

For a material with orthotropic thermal conductivity, Fourier's law specifies the following relation between the heat flux and the temperature gradient :

(3–44)

where [D] is the conductivity matrix:

(3–45)

If we now apply a fixed temperature gradient in x direction, that is, if has a fixed value and ,, we obtain

(3–46)

We can solve for to obtain:

(3–47)

By integrating and normalizing the heat flux on the boundary face normal to X-axis, we can easily obtain and thus, we get the thermal conductivity in X direction .

In a similar manner, we can also obtain and .

It remains to specify how we enforce the fixed temperature gradients, which we will do in the following sections.

 

3.2.2.4.1. Periodic Boundary Conditions

In each load case one of the quantity of , , and (the components of the temperature gradient) is set to a predefined value and all the other quantities are set to 0.

On the faces normal to the X-axis, enforce

(3–48)

On the faces normal to the Y-axis, enforce

(3–49)

On the faces normal to the Z-axis, enforce

(3–50)

To fully constrain the model, enforce

(3–51)

 

3.2.2.4.2. Non-Periodic Boundary Conditions

For the load case with temperature gradient in X-direction, we enforce only boundary conditions on the faces normal to the X-axis and they are given by

(3–52)

with a non-zero .

For the load case with temperature gradient in Y-direction, we enforce only boundary conditions on the faces normal to the Y-axis and they are given by

(3–53)

with a non-zero .

For the load case with temperature gradient in Z-direction, we enforce only boundary conditions on the faces normal to the Z-axis and they are given by

(3–54)

with a non-zero .