Example 2: semi-infinite solid with surface convection

Files

• Comprehensive test file: main.cpp
• Reference results for comparison: time_specialized.csv

Statement of the problem

This test corresponds to a 1D simulation of the evolution of temperature in a semi-infinite solid due to surface convection.

The domain $\Omega$ is a segment $[0,10^{-3}]$

$$\begin{align} \rho C_p \frac{\partial T}{\partial t}=[\nabla \cdot{} k\nabla T] \text{ in }\Omega \end{align}$$

Boundary conditions

Robin boundary conditions are prescribed on $\Gamma_{left}$ and Dirichlet boundary conditions are prescribed on $\Gamma_{right}$:

$$\begin{align} {\bf{n}} \cdot{} k \nabla T + h T &= hT_{\infty} \text{ on }\Gamma_{left} \\[6pt] T(x=L,t) &= T_i \end{align}$$

Initial condition

The temperature is assumed to be constant at $t=0$:

$$\begin{align} T(x,t=0) = T_i \end{align}$$

Parameters used for the test

For this test, the following parameters are considered:

Parameter	Symbol	Value
Density	$\rho$	$10^4$
Heat capacity	$C_p$	$10^4$
Thermal conductivity	$k$	$1$
Convection coefficient	$h$	$1$
Fluid temperature	$T_{\infty}$	$1$
Initial temperature	$T_i$	$0$

Numerical scheme

• Time integration: Euler Implicit over the interval $t\in[0,5]$ with a time-step $\delta t=0.1$
• Spatial discretization: uniform grid with $N=30$ nodes
• Newton solver: absolute tolerance $10^{-10}$

Results

The simulated temperature is compared to the analytical solution ([1]), which can be written as:

$$\begin{align} T(x,t) = T_i + (T_{\infty} - T_i) \left[ \text{erfc}\left(\frac{x}{L_c}\right) - \exp\left(\frac{hx}{k} + \frac{1}{4} \left(\frac{hL_c}{k}\right)^2 \right) \text{erfc}\left( \frac{x}{L_c} + \frac{1}{2} \frac{hL_c}{k} \right)\right] \end{align}$$

with $L_c = \sqrt{ 4 \alpha t } \text{, } \alpha=\frac{ k }{ \rho C_p }$.

The results show good agreement with the analytical solution.

References

[1] F. Incropera and D. DeWitt, “Fundamentals of Heat and Mass Transfer,” 6th Edition, J. Wiley & Sons, New York, 2007.