1. Reference problem

1.1. Geometry

The hollow cylinder is assumed to be infinitely long.

1.2. 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 \(\rho {C}_{P}=1.00J/{m}^{3}°C\)

thermal conductivity \(k=40W/{m}^{3}°C\)

1.3. 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 \(k\frac{\partial T}{\partial n}=\varepsilon \sigma [{(T+273.15)}^{4}-{({T}_{\mathrm{ext}}^{i}+273.15)}^{4}]\)

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

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

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