### Physics Model

The heat flux $\vec{P}$, and the $Q$ heat change per volume element has the following relation, the conservation of the energy. The $Q_{ex}$ term is the heat pumped in the volume element. $\nabla\vec{P} + \frac{\partial Q_{in}}{\partial t} = \frac{\partial Q_{ex}}{\partial t}$ From the Newtonian heat conduction theory the heat flux is related to the gradient of the temperature:

$\vec{P} = k \vec{\nabla}T$

Where $k$ is the heat conduction coefficient, which is in an an-isotropic material a 3×3 matrix. In the modelling of the heat transfer the heat change can be expressed: $\frac{\partial Q_{in}}{\partial t} = c\frac{\partial T}{\partial t}$

Where $c$ stands for the specific heat of the material.

The equations are summarized as the following:

$\nabla k \vec{\nabla} T + c \frac{\partial T}{\partial t} =\frac{\partial Q_{ex}}{\partial t}$

In Stationary Study, the $\frac{\partial T}{\partial t} = 0$, the simpler equation to be solved: $\nabla k \vec{\nabla} T =\frac{\partial Q_{ex}}{\partial t};$

### Internal Variables

Name Description Dimension
T_INTERNAL Temperature in the sample $K$
c_INTERNAL Specific heat $\frac{J}{kg K}$
k_11_INTERNAL Heat Conduction coefficient $\frac{W}{m \cdot K}$
k_33_INTERNAL Heat Conduction coefficient $\frac{W}{m \cdot K}$

### Examples

#### Time dependent heat transfer

• physicswiki/heat_transfer.txt