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


Geometry
---------


.. image:: images/10000200000007BD000007CAA619283CC1C8280D.png
    :width: 3.178in
    :height: 3.2402in


.. _RefImage_10000200000007BD000007CAA619283CC1C8280D.png:

Cylindrical bar:

Overall height: :math:`\mathrm{2H}=150\mathrm{mm}`

Diameter: :math:`\mathrm{2L}=\mathrm{100mm}`


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

The bar is made of 16 MND5 steel. The thermal properties (conductivity and enthalpy of the material) vary with temperature and are given in [:ref:`bib1 <bib1>`].

Coefficient for metallurgy:

TRC "standard"

:math:`\mathrm{AR3}=830°C`, :math:`\mathrm{alpha}=-0.0249`

:math:`\mathrm{MS0}=415°C`, :math:`\mathrm{AC1}=724°C`, :math:`\mathrm{AC3}=846°C`

:math:`{\tau }_{1}=0.034`, :math:`{\tau }_{3}=0.034`

Microhardness of the various metallurgical phases:

for ferrite :math:`d=184.\mathrm{HV}`

For perlite :math:`d=184.\mathrm{HV}`

For the bainite :math:`d=309.\mathrm{HV}`

for martensite :math:`d=450.\mathrm{HV}`

for austenite :math:`d=100.\mathrm{HV}`


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

Coefficient of heat exchange with the outside for 14 sec. (before tempering) based on temperature measurements by thermocouples:


.. csv-table::

    "Time :math:`(s)` "," :math:`H({\mathrm{W.m}}^{-2}\mathrm{.}°{C}^{-1})`"
    "0. ", "100."
    "13", "100."
    "14. ", "10000."


:math:`{T}_{\infty }=25°C`

Surface heat flow density as a function of temperature :math:`T` on the borders (during tempering) based on temperature measurements:


.. csv-table::

    ":math:`T(°C)` ", "25. ", "50. ", "100. ", "150. ", "200. ", "250", "300."
    "Flow :math:`(\mathrm{W.m}-2)` ", "0. E+06", "-0.03 E+06", "-0.08 E+06", "-0.08 E+06", "-0.71 E+06", "-1.51 E+06", "-1.51 E+06", "-1.99 E+06", "-1.99 E+06"



.. csv-table::

    ":math:`T(°C)` ", "350. ", "40. ", "450. ", "500. ", "50. ", "600."
    "Stream :math:`(\mathrm{W.m}-2)` ", "-2.53 E+06", "-3.34 E+06", "-3.78 E+06", "-6. E+06", "-7. E+06", "-7.9 E+06"



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

:math:`T(r,z)=906°\mathrm{C.}`

Bar entirely austenitized in the initial state i.e. :math:`\mathrm{Zf}(r,z)=0.`, :math:`\mathrm{Zp}(r,z)=0.`, :math:`\mathrm{Zb}(r,z)=0.` and :math:`\mathrm{Zm}(r,z)=0.`