Example 3: solid with surface convection and radiation

Files

• Comprehensive test file: main.cpp
• JSON file to define the Robin coefficient: coefficient.json
• Python file for verification: verify.py

Statement of the problem

This test corresponds to a 2D steady simulation of the temperature in a solid with surface convection and radiation.

The domain $\Omega$ is a rectangle $0.02\times0.01$

$$\begin{align} 0=[\nabla \cdot{} k\nabla T]\text{ in }\Omega \end{align}$$

Boundary conditions

Robin boundary conditions are prescribed on $\Gamma_{up}$, Dirichlet boundary conditions are prescribed on $\Gamma_{left}$ and $\Gamma_{right}$ and Homogeneous Neumann boundary conditions are prescribed on $\Gamma_{low}$:

$$\begin{align} {\bf{n}} \cdot{} k \nabla T + h T + \epsilon \sigma T^4 &= hT_{\infty} + \epsilon \sigma T_{\infty}^4 \text{ on }\Gamma_{up} \\[6pt] T(x,y) &= T_d \text{ on } \Gamma_{left} \text{ and } \Gamma_{right} \\[6pt] {\bf{n}} \cdot{} k \nabla T &= 0 \text{ on }\Gamma_{low} \end{align}$$

Parameters used for the test

For this test, the following parameters are considered:

Parameter	Symbol	Value
Thermal conductivity	$k$	$3$
Convection coefficient	$h$	$50$
Fluid temperature	$T_{\infty}$	$323$
Dirichlet temperature	$T_d$	$1173$

Numerical scheme

• Spatial discretization: uniform grid with $Nx=100$ and $Ny=50$ nodes
• Newton solver: relative tolerance $10^{-10}$, absolute tolerance $10^{-12}$

Results

Following [1] and [2], we expect, with a relative error of less than 0.01:

$$\begin{align} T(0.005,0.005) &= 1092.37 K \\[6pt] T(0.01,0.005) &= 1064.21 K \\[6pt] T(0.005,0) &= 1111.38 K \end{align}$$

The agreement with the expected solution is verified with the script verify.py.

Figure 1 : Steady temperature field.

References

[1] J. P. Holman, Heat Transfer Tenth Edition. McGraw-Hill, pp. 111, Example 3-10 (2008). $\\[6pt]$ [2] https://reference.wolfram.com/language/PDEModels/tutorial/HeatTransfer/HeatTransferVerificationTests.html, HeatTransfer-FEM-Stationary-2D-Single-HeatTransfer-0002