FEATool supports modeling heat transfer through both conduction, that is heat transported by a diffusion process, and convection, 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 coeffcients 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 Userâ€™s guide

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

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