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


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:


Geometry
---------

**Figure 1.1-a: Problem Geometry**

It is a cylinder with radius :math:`0.05m` and height :math:`0.05m`.

The square in bold corresponds to the axisymmetric modeling used in [:ref:`§3 <§3>`].


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

Material properties are described by the following parameters:

(Zirconium)

:math:`\rho {C}_{P}=2000000{\mathrm{J.m}}^{-3}\mathrm{.}°{C}^{-1}`

:math:`\lambda =9999.9{\mathrm{W.m}}^{-1}\mathrm{.}°{C}^{-1}`

Coefficient for metallurgy:

:math:`\mathrm{teqd}=809°C`, :math:`K=1.135E-2`, :math:`n=2.187`

:math:`\mathrm{t1c}=831°C`, :math:`\mathrm{t2C}=0.`, :math:`\mathrm{qsr}=14614`, :math:`\mathrm{Ac}=1.58E-4`

:math:`m=4.7`, :math:`\mathrm{t1r}=\mathrm{949,1}°C`, :math:`\mathrm{t2r}=0`, :math:`\mathrm{Ar}=-5.725`, :math:`\mathrm{Br}=0.05`


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

The temperature is imposed on the entire cylinder at times :math:`t=\mathrm{0s},\mathrm{120s}` and :math:`\mathrm{240s}`.

:math:`T(x,y,t=0)=20°C`

:math:`T(x,y,t=120)=1200°C`

:math:`T(x,y,t=240)=20°C`


Initial conditions
--------------------

The following variables are initialized to the following values:

:math:`\mathrm{V1}(x,y,t=0)=1.0`

:math:`\mathrm{V2}(x,y,t=0)=0.0`

:math:`\mathrm{V3}(x,y,t=0)=0.0`

:math:`\mathrm{V4}(x,y,t=0)=0.0`

:math:`\mathrm{V1}` *: proportion of the cold phase* :math:`\alpha`

:math:`\mathrm{V2}` *: proportion of the cold phase* :math:`\alpha` *, mixed with the phase* :math:`\beta`

:math:`\mathrm{V3}` *: node temperatures*

:math:`\mathrm{V4}` *: time corresponding to the temperature at which the start or end of the equilibrium transformation*