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


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

A semi-analytical solution involving error functions is available:

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

This solution is valid for a semi-infinite medium, so it can only be used in a limited range of variation in the time variable.

Either

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


.. _RefImage_Object_5.svg:

the position of the solid/liquid interface. Let's say :math:`{s}_{t}=\frac{L}{\sqrt{{t}_{\mathrm{total}}}}` and :math:`\lambda =\frac{{s}_{t}}{2\sqrt{{d}_{s}}}` where :math:`{d}_{s}` and :math:`{d}_{l}` designate the diffusivity of solid and liquid media :math:`({d}_{s}=\frac{{k}_{s}}{{c}_{s}},{d}_{l}=\frac{{k}_{l}}{{c}_{l}})`. The solution to the heat equation is of the form:

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

The data of :math:`{t}_{\mathrm{total}}` is enough to define the solution, so we set :math:`{t}_{\mathrm{total}}=420.`


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


.. csv-table::

    "**TEMPS :**
    **Absciss**", "**0.5**", "**1.0**", "**1.0**", "**1.5**", "**2.5**", "**2.5**", "**3.0**"
    ".000", "580. ", "580. ", "580. ", "580. ", "580. ", "580."
    ".005", "682.43", "661.33", "661.33", "647.50", "638.74", "632.69", "628.20"
    ".010", "726.05", "705.75", "705.75", "692.06", "682.43", "675.24", "669.63"
    ".015", "738.11", "728.70", "728.70", "718.44", "709.60", "702.23", "696.06"
    ".020", "739.86", "737.22", "737.22", "731.99", "726.05", "720.27", "714.94"
    ".025", "740. ", "739.50", "737.56", "737.56", "734.47", "730.81", "727.00"
    ".030", "740. ", "739.93", "739.39", "739.39", "738.11", "736.20", "733.88"
    ".035", "740. ", "739.99", "739.88", "739.88", "739.45", "738.61", "737.40"
    ".040", "740. ", "740. ", "739.98", "739.86", "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 :**
    **Absciss**", "**3.5**", "**4.0**", "**4.0**", "**5.0**", "**5.5**", "**5.5**", "**6.0**"
    ".000", "580. ", "580. ", "580. ", "580. ", "580. ", "580."
    ".005", "624.68", "621.84", "621.84", "619.48", "617.48", "615.25", "614.25"
    ".010", "665.09", "661.33", "661.33", "657.43", "653.65", "650.37", "647.49"
    ".015", "690.83", "686.33", "686.33", "682.43", "678.99", "675.95", "673.22"
    ".020", "710.11", "705.75", "705.75", "701.81", "698.25", "709.92", "692.06"
    ".025", "723.23", "719.60", "719.60", "716.17", "712.95", "720.89", "707.09"
    ".030", "731.34", "728.70", "728.70", "726.05", "723.43", "728.48", "718.44"
    ".035", "735.89", "734.18", "734.18", "732.34", "730.43", "733.42", "726.53"
    ".040", "738.21", "737.22", "737.22", "736.07", "734.79", "736.44", "731.99"
    ".045", "739.29", "738.77", "738.77", "738.11", "737.33", "738.18", "735.47"
    ".050", "739.74", "739.50", "739.50", "739.15", "738.71", "739.12", "737.56"
    ".055", "739.91", "739.81", "739.81", "739.65", "739.42", "739.60", "738.75"
    ".060", "739.97", "739.93", "739.93", "739.86", "739.75", "739.83", "739.39"
    ".065", "739.99", "739.98", "739.98", "739.95", "739.90", "739.93", "739.72"
    ".070", "740. ", "739.99", "739.98", "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."


(In :math:`°C`, according to the abscissa in meters and the time in seconds).

**Note:**

*We limit ourselves to variations during the first 6 seconds, beyond 10 seconds the limit condition at the end* :math:`x=1` *is no longer assured.*


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

Unknown, due to evaluation of error functions.


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


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