# Simulation and Modeling of Heat Transfer

FEATool supports modeling heat transfer through both conduction, that is heat transported by a diffusion process, and convection or advection, which is heat transported through a fluid through convection by a velocity field. The heat transfer physics mode supports both these processes, and is defined by the following equation

\[ \rho C_p\frac{\partial T}{\partial t} + \nabla\cdot(-k\nabla T) = Q - \rho C_p\mathbf{u}\cdot\nabla T \]

where $\rho$ is the density, $C_p$ the heat capacity, $k$ is the
thermal conductivity, $Q$ heat source term, and $\mathbf{u}$ a
vector valued convective velocity field. In the *Equation Settings*
dialog box show below the equation coefficients, initial value for the
temperature $T$, and finite element shape function space can be
specified (note that here the convective velocities **u** and **v**
are the dependent variables from a coupled incompressible fluid flow
physics mode, but can also be constants and complex expressions).

The heat transfer physics mode allows for four different boundary conditions types.

**Temperature**, $T=T_0$ Prescribes the temperature on the boundary to $T_0$.

Note that $T_0$ does not have to be a constant either but as all coefficients in FEATool can be complex expression involving space coordinates, dependent variables, and derivatives.

**Convective flux**, $-\mathbf{n}\cdot k\nabla T = 0$

This boundary condition prescribes a zero diffusive flux leaving the convective flux unspecified and free which is appropriate for outflow boundaries in fluids.

**Thermal insulation/symmetry**, $-\mathbf{n}\cdot (k\nabla T + \rho C_p\mathbf{u}c) = 0$

This condition sets the heat flux at the boundary to zero which is appropriate for insulated and symmetry boundaries.

**Heat flux**, $-\mathbf{n}\cdot (k\nabla T + \rho C_p\mathbf{u}T) = q_0$

The heat flux boundary condition allows the heat flux $q_0$ at the
boundary to be prescribed. As with the temperature condition, $q_0$
allows for complex expressions such as the common convective and
radiation conditions to a surrounding medium, in this cases one could
for example set **q_0 = k_ht (T_amb-T) + c_rad(T_amb-T)^4** where

**T_amb**is the ambient temperature of the surrounding fluid, and

**c_rad**a constant for the radiation term.

**T**is the name of dependent variable physics mode and

**k_ht**is the thermal conductivity specified in the physics mode subdomain settings.

A model example that incorporates these heat transfer effects is a
transient cooling for shrink fitting a two part assembly [1]. A
tungsten rod heated to 84 *C* is inserted into a -10 *C* chilled steel
frame part. The time when the maximum temperature has cooled to 70 *C*
should be determined. The assembly is cooled due to convection through
a surrounding medium kept at 17 *C* and a heat transfer coefficient of
750 *W/m^2 K* thus heat flux boundary conditions are prescribed on all
boundaries as **k_ht*(17-T)**. Note that the model involved several
subdomains with different thermal conductivities but using **k_ht** in
the boundary prescription will automatically choose the right value.

The FEATool tutorial for the model can be viewed in the tutorial section of the User’s Guide

**References**

[1] Krysl P. A Pragmatic Introduction to the Finite Element Method for Thermal and Stress Analysis. Pressure Cooker Press, USA, 2005.