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


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

:math:`\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})`


.. csv-table::

    ":math:`x=` ", "abscissa"
    ":math:`t=` ", "Time"
    ":math:`{T}_{0}=` ", "Initial temperature"
    ":math:`{T}_{p}=` ", "Imposed temperature"
    ":math:`n=` "," :math:`\mathrm{1,2}\mathrm{,3},\mathrm{...}`"


With :math:`{\xi }_{n}` positive roots from :math:`{\xi }_{n}\mathrm{tan}{\xi }_{n}\mathrm{=}\mathit{hL}\mathrm{/}\lambda \mathrm{=}10.`

and :math:`{A}_{n}=\frac{4\mathrm{sin}{\xi }_{n}}{2{\xi }_{n}+\mathrm{sin}(2{\xi }_{n})}`


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

Temperatures at points :math:`\mathrm{M1}` (:math:`x=0.02`) and :math:`\mathrm{M2}` (:math:`x=0.08`),

and at different times (:math:`t=0.1,0.5,2.0` and :math:`10.0`).

The reference values are obtained by calculating the first 30 terms of the series (Mathematica).


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


* INCROPERA F.P., DE WITT D.P., Fundamentals of heat and mass transfer. Third edition. 1990.