2. Benchmark solution#

2.1. Calculation method used for the reference solution#

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

  • Temperature according to \(r\):

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

  • Flux density as a function of \(r\):

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

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

2.2. Benchmark results#

Temperature and flow density for \(r=1.0,1.2\text{et}1.5\)

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

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