Benchmark solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------

:math:`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}}}`

:math:`{\varphi }_{e}={h}_{e}({T}_{e}-{T}_{e}^{e})` **,** :math:`\phi =4\pi {R}_{e}^{2}{h}_{e}({T}_{e}-{T}_{e}^{e})` **eq** 2.1-1

:math:`\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

:math:`\phi =4\pi {r}^{2}\varphi =\mathrm{constante}` :math:`\phi =4\pi \lambda \frac{{T}_{e}-{T}_{i}}{1/{R}_{e}-1/{R}_{i}}` **eq** 2.1-3

:math:`\sigma =5.73\mathrm{.}{10}^{-8}W/{m}^{2}{K}^{4}` (Stefan's constant) with :math:`T` in :math:`°C`

The reference temperatures are obtained by solving numerically by the Newton method an equation of the 4th degree in :math:`{T}_{i}` obtained from the equations [:ref:`éq 2.1-1 <éq 2.1-1>`] [:ref:`éq 2.1-2 <éq 2.1-2>`] and [:ref:`éq 2.1-3 <éq 2.1-3>`].


Benchmark results
----------------------

.. csv-table::

    "", "in :math:`A`:", "in :math:`B`:"
    "Temperatures", ":math:`{T}_{i}=91.77°C` "," :math:`{T}_{e}=71.22°C`"
    "Flow densities", ":math:`{\phi }_{i}=11675.W/{m}^{2}` "," :math:`{\phi }_{e}=6838.W/{m}^{2}`"



Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


* Guide for the validation of structural calculation software packages. French Society of Mechanics, AFNOR 1990 ISBN 2-12-486611-7