Reference problem
=====================


Geometry
---------

Let's say a bar moving, at speed :math:`V`, in line with the temperature conditions imposed in :math:`X=0` and :math:`X=L` expressed in a fixed frame of reference (with respect to the bar moving).


.. image:: images/100026E0000069D50000330E98E799A372599B9D.svg
    :width: 571
    :height: 276


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Material properties
------------------------


* thermal conductivity is constant: :math:`K=150W/m°C`


* the enthalpy function is such that:

:math:`\beta (T)=\{\begin{array}{cc}{C}_{S}T& \text{;}T\le {T}_{1}\\ {C}_{S}T+{C}_{\mathrm{Sl}}(T-{T}_{1})& \text{;}{T}_{1}\le T\le {T}_{2}\\ {C}_{S}T+{C}_{\mathrm{Sl}}({T}_{2}-{T}_{1})+{C}_{l}(T-{T}_{2})& \text{;}{T}_{1}\le T\le {T}_{2}\end{array}`

with the following values:

:math:`\begin{array}{}{C}_{S}={C}_{l}=1/3\mathrm{.}{10}^{7}J/{m}^{3}°C\\ {C}_{\mathrm{Sl}}=8.333{10}^{7}J/{m}^{3}°C\\ {T}_{1}=585.°C\\ {T}_{2}=615°C\end{array}`


.. image:: images/10001B3C0000284F000021B181ADDA7F90ED407F.svg
    :width: 571
    :height: 276


.. _RefImage_10001B3C0000284F000021B181ADDA7F90ED407F.svg:


Boundary conditions and loads
-------------------------------------

Temperatures imposed at the extremities

:math:`{T}_{0}=200°C` for :math:`x=0`

:math:`{T}_{L}=1000°C` for :math:`x=L=\mathrm{1m}`

Solid movement speed: :math:`V={10}^{-4}m/s`