Solution de référence
=====================


Méthode de calcul utilisée pour la solution de référence
--------------------------------------------------------

On dispose d'une solution semi-analytique faisant intervenir les fonctions d'erreurs :

:math:`\mathit{erf}(x)=\frac{2}{\sqrt{\pi}}{\int}_{0}^{x}{e}^{-{t}^{2}}\mathit{dt}` et :math:`\mathrm{erfc}(x)=\frac{2}{\sqrt{\pi}}{\int}_{x}^{+\infty }{e}^{-{t}^{2}}\mathrm{dt}`

Cette solution est valide pour un milieu semi-infini, elle ne pourra donc être utilisée que dans un domaine de variation limité de la variable de temps.

Soit

.. image:: images/Object_5.svg
    :width: 17
    :height: 22


.. _RefImage_Object_5.svg:

la position de l'interface solide/liquide. Soient :math:`{s}_{t}=\frac{L}{\sqrt{{t}_{\mathrm{total}}}}` et :math:`\lambda =\frac{{s}_{t}}{2\sqrt{{d}_{s}}}`  où :math:`{d}_{s}` et :math:`{d}_{l}` désignent la diffusivité des milieux solide et liquide :math:`({d}_{s}=\frac{{k}_{s}}{{c}_{s}},{d}_{l}=\frac{{k}_{l}}{{c}_{l}})` . La solution de l'équation de la chaleur est de la forme :

:math:`{T}_{s}(x,t)={T}_{0}+\frac{{T}_{m}-{T}_{0}}{\mathrm{erf}(\lambda )}\mathrm{erf}(\frac{x}{2\sqrt{{d}_{s}t}})` si :math:`x\le {x}_{t}`

:math:`{T}_{l}(x,t)={T}_{i}+\frac{{T}_{m}-{T}_{i}}{\mathrm{erfc}(\lambda \sqrt{\frac{{d}_{s}}{{d}_{l}}})}\mathrm{erfc}(\frac{x}{2\sqrt{{d}_{l}t}})` si :math:`x\ge {x}_{t}`

La donnée de :math:`{t}_{\mathrm{total}}` suffit à définir la solution, on fixe donc :math:`{t}_{\mathrm{total}}=420.`


Résultats de référence
----------------------


.. csv-table::

    "**TEMPS :**
    **Abscisse**", "**0.5**", "**1.0**", "**1.5**", "**2.0**", "**2.5**", "**3.0**"
    ".000", "580.", "580.", "580.", "580.", "580.", "580."
    ".005", "682.43", "661.33", "647.50", "638.74", "632.69", "628.20"
    ".010", "726.05", "705.75", "692.06", "682.43", "675.24", "669.63"
    ".015", "738.11", "728.70", "718.44", "709.60", "702.23", "696.06"
    ".020", "739.86", "737.22", "731.99", "726.05", "720.27", "714.94"
    ".025", "740.", "739.50", "737.56", "734.47", "730.81", "727.00"
    ".030", "740.", "739.93", "739.39", "738.11", "736.20", "733.88"
    ".035", "740.", "739.99", "739.88", "739.45", "738.61", "737.40"
    ".040", "740.", "740.", "739.98", "739.86", "739.55", "739.00"
    ".045", "740.", "740.", "740.", "739.97", "739.87", "739.65"
    ".050", "740.", "740.", "740.", "740.", "739.97", "739.89"
    ".055", "740.", "740.", "740.", "740.", "740.", "739.97"
    ".060", "740.", "740.", "740.", "740.", "740.", "740."
    ".065", "740.", "740.", "740.", "740.", "740.", "740."
    ".070", "740.", "740.", "740.", "740.", "740.", "740."
    ".075", "740.", "740.", "740.", "740.", "740.", "740."
    ".080", "740.", "740.", "740.", "740.", "740.", "740."
    ".085", "740.", "740.", "740.", "740.", "740.", "740."
    ".090", "740.", "740.", "740.", "740.", "740.", "740."
    ".095", "740.", "740.", "740.", "740.", "740.", "740."
    ".100", "740.", "740.", "740.", "740.", "740.", "740."



.. csv-table::

    "**TEMPS :**
    **Abscisse**", "**3.5**", "**4.0**", "**4.5**", "**5.0**", "**5.5**", "**6.0**"
    ".000", "580.", "580.", "580.", "580.", "580.", "580."
    ".005", "624.68", "621.84", "619.48", "617.48", "615.25", "614.25"
    ".010", "665.09", "661.33", "657.43", "653.65", "650.37", "647.49"
    ".015", "690.83", "686.33", "682.43", "678.99", "675.95", "673.22"
    ".020", "710.11", "705.75", "701.81", "698.25", "709.92", "692.06"
    ".025", "723.23", "719.60", "716.17", "712.95", "720.89", "707.09"
    ".030", "731.34", "728.70", "726.05", "723.43", "728.48", "718.44"
    ".035", "735.89", "734.18", "732.34", "730.43", "733.42", "726.53"
    ".040", "738.21", "737.22", "736.07", "734.79", "736.44", "731.99"
    ".045", "739.29", "738.77", "738.11", "737.33", "738.18", "735.47"
    ".050", "739.74", "739.50", "739.15", "738.71", "739.12", "737.56"
    ".055", "739.91", "739.81", "739.65", "739.42", "739.60", "738.75"
    ".060", "739.97", "739.93", "739.86", "739.75", "739.83", "739.39"
    ".065", "739.99", "739.98", "739.95", "739.90", "739.93", "739.72"
    ".070", "740.", "739.99", "739.98", "739.96", "739.97", "739.88"
    ".075", "740.", "740.", "740.", "739.99", "739.99", "739.95"
    ".080", "740.", "740.", "740.", "740.", "740.", "739.98"
    ".085", "740.", "740.", "740.", "740.", "740.", "739.99"
    ".090", "740.", "740.", "740.", "740.", "740.", "740."
    ".095", "740.", "740.", "740.", "740.", "740.", "740."
    ".100", "740.", "740.", "740.", "740.", "740.", "740."


(En :math:`°C` , en fonction de l'abscisse en mètre et du temps en secondes).

**Remarque :**

*On se limite aux variations pendant les 6 premières secondes, au-delà de 10 secondes la condition au limite à l'extrémité* :math:`x=1` *n'est plus assurée.*


Incertitude sur la solution
---------------------------

Inconnue, due à l'évaluation des fonctions d'erreur.


Références bibliographiques
---------------------------


*   M. Necati Özisik - Heat Conduction - Chapter 10 : Phase-change problems example 10-3 - John Wiley & Sons.