Thermal Analysis Equations

The heat transfer equation is a parabolic partial differential equation that describes the distribution of temperature in a particular region over given time:

$$\rho c\frac{\partial T}{\partial t}-\nabla \cdot \left(k\nabla T\right)=Q$$

Parameters of the heat transfer equation are as follows:

• ρ is the density of the material.

• c is the specific heat of the material.

• k is the thermal conductivity of the material.

• Q is the heat source.

Boundary conditions include temperatures on the boundaries or heat fluxes through the boundaries.

• For convective heat flux through the boundary $$htc\left(T-T_{\infty }\right)$$, specify the ambient temperature $$T_{\infty }$$ and the convective heat transfer coefficient htc.

• For radiative heat flux $$\epsilon \sigma \left(T^4-T_{\infty }{}^4\right)$$, specify the ambient temperature $$T_{\infty }$$, emissivity ε, and Stefan-Boltzmann constant σ.

By default, the toolbox uses the zero Neumann boundary condition and assumes that the boundary is insulated, so heat flux through the boundary is 0.