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


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

The reference solution is the one given in sheet TPLA04 /89 of guide VPCS.


* Temperature according to :math:`r`:

:math:`T(r)={T}_{i}+\frac{Q}{4\lambda }\left[({R}_{e}^{2}-{R}_{i}^{2})\frac{\mathrm{ln}(\frac{r}{{R}_{i}})}{\mathrm{ln}(\frac{{R}_{e}}{{R}_{i}})}-({r}^{2}-{R}_{i}^{2})\right]`


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

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

The cylinder is assumed to be infinitely long (:math:`l\gg {R}_{e}`)


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

Temperature and flow density for :math:`r=1.0,1.2\text{et}1.5`


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