Simulate with complex geometries and complex physics
%BND_ROBIN%
Robin boundary condition
In general, a Robin (third-type) boundary condition is a linear combination of a Dirichlet boundary condition (%BND_DIRICH%)
and a Neumann boundary condition (%BND_NEUMANN%) of the form
\begin{align}
Af + B \frac{\partial f}{\partial n} = g
\end{align}
For the temperature T, this can be used to describe how the convective heat flux across the
boundary/interface depends on the difference between the temperature of the material at the
boundary/interface and the temperature on the opposite side
\begin{align}
H\cdot A\cdot \rho \cdot c_{v}\cdot \frac{T^{n+1}-T^{n}}{\Delta t}=-\alpha \cdot A\cdot \left( T^{n+1}-T_{opp}^{n+1} \right)+\lambda \cdot A\cdot \frac{\partial T^{n+1}}{\partial n} + H \cdot A \cdot q
\end{align} --->>> which leads to --->>>
\begin{align}
& \left( \frac{H \cdot \rho \cdot c_{v}}{\Delta t}+\alpha \right)\cdot T^{n+1}-\lambda \cdot \frac{\partial T^{n+1}}{\partial n}=\alpha T_{opp}^{n+1}+\frac{H\cdot \rho \cdot c_{v}}{\Delta t}\cdot T^{n} + H \cdot q
\end{align}
where
\( H\) , \( A\) = the thickness and the area of the control volume element represented by the boundary point,
\(\lambda\) = heat conductivity of the material,
\(\alpha\) = heat transfer coefficient,
\( c_v\) = specific heat,
\( \rho\) = density,
\( \Delta t\) = current time step size,
\( T^{n+1}\) = yet unknown temperature at new time level,
\( T^{n}\) = temperature of previous time level at the boundary point,
\( T_{opp}^{n+1}\) = current environmental temperature (or temperature of the opposite, adjacent material).
This model rule the heat balance around the given boundary point by the equilibrium of energy contained in the control element, heat flux going to the enviropnment, and heat flux coming from the interior of the domain.
Syntax:
BC_T($xyz$) = (%BND_ROBIN%, \(\alpha\), \( T_\text{opp}\), OPTIONAL: NondimensionalControlThickness, OPTIONAL: %HEAT_EQ_1D_BC%, OPTIONAL: LAMBDA , OPTIONAL: HeatSource )
- \( \alpha\) :: the heat trandfer coefficient in \( \frac{W}{m^2 K}\)
- \( T_\text{opp}\) :: environmental / opposite temperature in \( K\) or \( C\).
- NondimensionalControlThickness :: the control thickness is correlated by this parameter to the smoothing length: \( H = \text{NondimensionalControlThickness} \cdot h\), with \( h\) the local smoothing length. Recommended value: 0.3 , Default: 0.0 .
- use information from detailed resolution of temperature boundary layers, i.e. Result of 1D heat equation can be used by keyword %HEAT_EQ_1D_BC% (see HeatEquation1D) .
- LAMBDA :: defines the \(\lambda\) as stated above, if another LAMBDA than the one given in Y%ind_LAM% has to be used
- HeatSource :: re-defines the heat source \( q\) as stated above, the default is the standard Y%ind_diss%. In order to reduce effect discontinuities at the boundary, use for example projY(%ind_diss%) Example: This replaces the temperature on the opposite side \( T_\text{opp}\) by the result %ind_T1D(1)% of the 1D heat equation. So the value &Topp& is ignored in this case!