Reference solution
=====================


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

The reference result is of the semi-analytical type. The 1D equation to be solved is as follows:


.. _RefEquation 2.1-1:

:math:`\{\begin{array}{}V\beta {(T)}_{,x}-K{T}_{,\mathrm{xx}}=0\\ \text{avec}{T}_{(x=0)}={T}_{0}\text{et}{T}_{(x=L)}={T}_{L}\end{array}` eq 2.1-1

by integrating the equation [:ref:`éq 2.1-1 <éq 2.1-1>`] we get:


.. _RefEquation 2.1-2:

:math:`\frac{V}{K}\beta (T)-\frac{\mathrm{dT}}{\mathrm{dx}}=A` eq 2.1-2

where :math:`A` is a constant depending on the boundary conditions, the :math:`\frac{V}{K}` ratio, and the enthalpy function :math:`\beta (T)`.

This constant will be determined analytically.

Equation [:ref:`éq 2.1-2 <éq 2.1-2>`] leads to:


.. _RefEquation 2.1-3:

:math:`x={\int }_{{T}_{0}}^{T(x)}\frac{\mathrm{dT}}{A+\frac{V}{K}\beta (T)}` eq 2.1-3

who should check:


.. _RefEquation 2.1-4:

:math:`L={\int }_{{T}_{0}}^{{T}_{L}}\frac{\mathrm{dT}}{A+\frac{V}{K}\beta (T)}` eq 2.1-4

Knowing :math:`{T}_{\mathrm{0,}}{T}_{L},L,V,t` and :math:`\beta (T)`, equation [:ref:`éq 2.1-4 <éq 2.1-4>`] should give the value of the integration constant :math:`A`.

However, it is difficult (if not impossible) to determine this constant analytically, hence the use of numerical resolution of equation [:ref:`éq 2.1-4 <éq 2.1-4>`] to determine :math:`A`.

With the data for problem :math:`({T}_{\mathrm{0,}}{T}_{L},{T}_{\mathrm{1,}}{T}_{\mathrm{2,}}{C}_{S}={C}_{l},{C}_{\mathrm{Sl}}\mathrm{.}\mathrm{.}\mathrm{.})`, we got the (physical) solution of :math:`A` which takes the value :math:`A=-294.9117`.

Based on this constant, the analytical solution of the problem [:ref:`éq 2.1-1 <éq 2.1-1>`] is analytical.


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


.. csv-table::

    "**Abscissa**", "**Temperature**"
    "0.6", "387.98514"
    "0.7", "451.51001"
    "0.725", "469.72232"
    "0.750", "488.97505"
    "0.775", "509.32766"
    "0.80", "530.84296"
    "0.825", "553.58738"
    "0.85", "577.63114"
    "0.9", "683.71269"
    "0.9125", "719.51615"
    "0.925", "756.32221"
    "0.9375", "794.16795"
    "0.95", "833.07971"
    "0.9625", "873.08751"
    "0.9750", "914.22222"
    "0.9875", "956.51557"