Reference problem
=====================


Geometry
---------


.. image:: images/100007B20000181500001AC5F6817B68E575AC1B.svg
    :width: 310
    :height: 345


.. _RefImage_100007B20000181500001AC5F6817B68E575AC1B.svg:

The hollow cylinder is assumed to be infinitely long.


Material properties
-----------------------

Only the conductivity coefficient is involved. The*Code_Aster* makes it mandatory to provide a function representing the volume enthalpy determined from the density heat coefficient.

Volume heat :math:`\rho {C}_{P}=1.00J/{m}^{3}°C`

thermal conductivity :math:`k=40W/{m}^{3}°C`


Boundary conditions and loads
-------------------------------------

Radiation-type condition on the inner surface of the cylinder, convection-type condition (exchange with the external environment) on the outer surface.

No boundary conditions on the ends of the cylinder (which amounts to imposing a zero flow).

Inner surface :math:`k\frac{\partial T}{\partial n}=\varepsilon \sigma [{(T+273.15)}^{4}-{({T}_{\mathrm{ext}}^{i}+273.15)}^{4}]`

 with :math:`\varepsilon =0.6`, :math:`\sigma =5.73{10}^{-8}W/{m}^{2}{K}^{4}` and :math:`{T}_{\mathrm{ext}}^{i}=500.°C`, :math:`T` in Celsius

External surface :math:`k\frac{\partial T}{\partial n}={h}_{e}[{T}_{\mathrm{ext}}^{e}-T]`

 with :math:`{h}_{e}=142.W/{m}^{2}°C` and :math:`{T}_{\mathrm{ext}}^{e}=20.°C`