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


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

The reference solution is the one given in sheet TPLV06 /89 of guide VPCS.


* Temperature according to :math:`r`:

:math:`T={T}_{i}+\frac{Q}{6\lambda }\left[\frac{({R}_{e}^{2}-{R}_{i}^{2})\left[\frac{1}{{R}_{i}}-\frac{1}{r}\right]}{\left[\frac{1}{{R}_{i}}-\frac{1}{{R}_{e}}\right]}-({r}^{2}-{R}_{i}^{2})\right]`


* Flux density as a function of :math:`r`:

:math:`\phi =-4\pi {r}^{2}\lambda \frac{\mathrm{dT}}{\mathrm{dr}}=-\frac{2\pi Q}{3}\left[({R}_{e}^{2}-{R}_{i}^{2})\left[\frac{1}{{R}_{i}}-\frac{1}{{R}_{e}}\right]-2{r}^{3}\right]`


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

Temperature in :math:`r=1.25`; :math:`1.5` and :math:`1.75m`


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

Analytical solution.


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


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