2. Benchmark solution#
2.1. Calculation method used for the reference solution#
\(T(r)=\frac{{T}_{e}-{T}_{i}}{\frac{1}{{R}_{e}}-\frac{1}{{R}_{i}}}\frac{1}{r}+\frac{\frac{{T}_{i}}{{R}_{e}}-\frac{{T}_{e}}{{R}_{i}}}{\frac{1}{{R}_{e}}-\frac{1}{{R}_{i}}}\)
\({\varphi }_{e}={h}_{e}({T}_{e}-{T}_{e}^{e})\) , \(\phi =4\pi {R}_{e}^{2}{h}_{e}({T}_{e}-{T}_{e}^{e})\) eq 2.1-1
\(\begin{array}{}{\varphi }_{i}=\sigma \varepsilon \left[{({T}_{i}^{e}+273.15)}^{4}-{({T}_{i}+273.15)}^{4}\right]\\ \phi =4\pi {R}_{i}^{2}\sigma \varepsilon \left[{({T}_{i}^{e}+273.15)}^{4}-{({T}_{i}+273.15)}^{4}\right]\end{array}\) eq 2.1-2
\(\phi =4\pi {r}^{2}\varphi =\mathrm{constante}\) \(\phi =4\pi \lambda \frac{{T}_{e}-{T}_{i}}{1/{R}_{e}-1/{R}_{i}}\) eq 2.1-3
\(\sigma =5.73\mathrm{.}{10}^{-8}W/{m}^{2}{K}^{4}\) (Stefan’s constant) with \(T\) in \(°C\)
The reference temperatures are obtained by solving numerically by the Newton method an equation of the 4th degree in \({T}_{i}\) obtained from the equations [éq 2.1-1] [éq 2.1-2] and [éq 2.1-3].
2.2. Benchmark results#
in \(A\): |
in \(B\): |
|
Temperatures |
\({T}_{i}=91.77°C\) |
|
Flow densities |
\({\phi }_{i}=11675.W/{m}^{2}\) |
|
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
Guide for the validation of structural calculation software packages. French Society of Mechanics, AFNOR 1990 ISBN 2-12-486611-7