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

.. image:: images/1000000000000143000001E568884D96E37996C6.png
    :width: 1.339in
    :height: 2.1661in


.. _RefImage_1000000000000143000001E568884D96E37996C6.png:


Geometry
---------

The structure consists of a tube in which a fluid carrying the thermal shock circulates.


* Straight tube section


* Tube length: :math:`L=0.1m`


* Outer tube diameter: :math:`{\varphi }_{\mathit{ext}}=0.048m`


* Tube Thickness: :math:`e=0.006m`


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


* :math:`\lambda =15W/m°C` Thermal Conductivity Coefficient


* :math:`{C}_{P}=500J/\mathit{Kg}°C` Specific heat


* :math:`a=3.8x{10}^{-6}{m}^{2}/s` Thermal diffusivity


* :math:`E=200000\mathit{MPa}` Young's module


* :math:`\nu =0.3` Poisson's Ratio


* :math:`\alpha =15x{10}^{-6}/°C` Coefficient of thermal expansion


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

The thermal shock is transmitted to the tube by the fluid via convective exchange. Two thermal shocks are studied:

.. csv-table::

    "Thermal shock", "Intrados", "Extrados"
    ":math:`\Delta T=1°C` "," :math:`{T}_{\mathit{fluide}}=21°C` :math:`h=20000W/{m}^{2}/°C` "," :math:`{T}_{\mathit{imposée}}=20°C`"
    ":math:`\Delta T=50°C` "," :math:`{T}_{\mathit{fluide}}=70°C` :math:`h=20000W/{m}^{2}/°C` "," :math:`{T}_{\mathit{imposée}}=20°C`"


Thermal shock corresponds to the difference between the initial temperature of the tube and that of the fluid.

The flow is zero in the axial direction of the tube (:math:`\frac{\partial T(x,t)}{\partial z}=0`).

For mechanical analysis, the tube is locked at its base in the axial direction.


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


* :math:`T(x,t=0)=20°C` for the :math:`\mathrm{\Delta }T=50°C` shock


* :math:`T(x,t=0)=20°C` for the unity shock :math:`\Delta {T}_{U}=1°C`


Details concerning the models
---------------------------------------

To define time discretization, we use the time constant :math:`\tau =\frac{{e}^{2}}{(a{\pi }^{2})}=1s`, which gives us, in the case of a thermal shock, the instant beyond which the thermal behavior becomes almost stationary in the tube. The duration of the thermal shock is set to :math:`{10.}^{-1}s`. We chose discretization in the following time:


.. csv-table::

    "5", "not for", ":math:`[0.0,0.1]` ", "either", ":math:`\Delta t=0.01s`"
    "4", "not for", ":math:`[0.1,0.5]` ", "either", ":math:`\Delta t=0.04s`"
    "4", "not for", ":math:`[0.5,3.0]` ", "either", ":math:`\Delta t=0.25s`"
    "1", "not for", ":math:`[3.0\mathrm{,5}.0]` ", "either", ":math:`\Delta t=0.4s`"
    "1", "not for", ":math:`[5.0\mathrm{,10}\mathrm{.}]` ", "either", ":math:`\Delta t=2.5s`"