2. Benchmark solution#
2.1. Calculation method used for the reference solution#
\(\frac{T(x,t)\mathrm{-}{T}_{p}}{{T}_{0}\mathrm{-}{T}_{p}}\mathrm{=}\mathrm{\sum }_{n\mathrm{=}1}^{\mathrm{\infty }}{A}_{n}\mathrm{exp}(\mathrm{-}{\xi }_{n}^{2}\frac{\lambda }{\rho {C}_{p}{L}^{2}}t)\mathrm{cos}({\xi }_{n}\frac{x}{L})\)
\(x=\) |
abscissa |
\(t=\) |
Time |
\({T}_{0}=\) |
Initial temperature |
\({T}_{p}=\) |
Imposed temperature |
\(n=\) |
|
With \({\xi }_{n}\) positive roots from \({\xi }_{n}\mathrm{tan}{\xi }_{n}\mathrm{=}\mathit{hL}\mathrm{/}\lambda \mathrm{=}10.\)
and \({A}_{n}=\frac{4\mathrm{sin}{\xi }_{n}}{2{\xi }_{n}+\mathrm{sin}(2{\xi }_{n})}\)
2.2. Benchmark results#
Temperatures at points \(\mathrm{M1}\) (\(x=0.02\)) and \(\mathrm{M2}\) (\(x=0.08\)),
and at different times (\(t=0.1,0.5,2.0\) and \(10.0\)).
The reference values are obtained by calculating the first 30 terms of the series (Mathematica).
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
INCROPERA F.P., DE WITT D.P., Fundamentals of heat and mass transfer. Third edition. 1990.