# Orthotropic Heat Conduction

## Model Data

**Type:**
heat transfer

**Physics Modes:**
heat transfer

**Keywords:**
heat conduction
orthotropic
equation editing
validation

This is an example of modeling anisotropic heat conduction in an
orthotropic material where a heated solid bar suddenly is cooled by
submerging it in a cool fluid. The material features different thermal
conductivities in the *x* and *y*-directions which is accounted for by
modifying the diffusion term in the heat transfer equation. Instead of
solving the usual isotropic case

$$ \rho C_p\frac{\partial T}{\partial t} - k(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}) = 0 $$

we will instead solve the following

$$ \rho C_p\frac{\partial T}{\partial t} - (k_x\frac{\partial^2 T}{\partial x^2} + k_y\frac{\partial^2 T}{\partial y^2}) = 0 $$

where $k_x$ = *34.6147* and $k_y$ = *6.23687 W/mK* are the thermal
conductivities in the different directions. Fully anisotropic cases
can technically also be handled by introducting off-diagonal
$k_{x_i,x_j}\frac{\partial^2 T}{\partial x_i \partial x_j}$ diffusion
terms.

The following material and simulation parameters are given; initial
temperature *T _{0}* =

*260 °C*, density ρ =

*6407.4 kg/m*, heat capacity $C_p$ =

^{3}*37.688 J/kgK*. Due to symmetry we only need to model a quarter domain of a 2D cross-section with dimensions

*5.08*by

*2.54*cm. Conductive heat flux boundary conditions are used on the external boundaries with a convection coefficient

*h*=

*1361.7 W/m*and surrounding temperature

^{2}K*T*=

_{inf}*37.7778 °C*. The cooling process is simulated for 3 seconds after which the temperature is measured in the center core and corners, and compared to the reference values [1].

This model is available as an automated tutorial by selecting **Model
Examples and Tutorials…** > **Heat Transfer** > **Orthotropic Heat
Conduction** from the **File** menu. Or alternatively, follow the
linked step-by-step instructions.

## Reference

[1] P.J. Schneider, Conduction Heat Transfer, Addison-Wesley, 2nd Ed., Ex. 10-7, pp. 261, 1957.