.. include:: /images.rst

FLD02: Steady, 1D heat conduction
---------------------------------

.. _description-10:

Description
~~~~~~~~~~~

Steady-state, one-dimensional heat conduction occurs across a rectangular plane-shaped slab of length :math:`L` with constant material properties. As shown in :numref:`fld02fig1`, two opposing slab boundaries are maintained at constant temperatures. All other faces are perfectly insulated such that the heat flux along these boundaries is zero. Without heat generation, heat transfer through the :math:`x = 0` face must equal that through the :math:`x = L` face.

.. _fld02fig1:
.. figure:: ../media/fld02-setup.png
   :align: center

   Plane slab with constant material properties and no internal heat generation is shown with constant temperatures specified on opposing faces. The slab is assumed to be perfectly insulated along all other faces.

For constant thermal conductivity, the energy equation reduces to a second order ODE with Dirichlet boundary conditions as given by :eq:fld02eq1. The analytical solution for temperature distribution within the slab follows a line as given by :eq:fld02eq2.

.. math::
   \frac{d}{\text{dx}}\left( \lambda_{g}\frac{\text{dT}}{\text{dx}} \right) = 0;	T\left( x = 0 \right) = T_{1};	T\left( x = L \right) = T_{2}
   :label: fld02eq1

.. math::
   T\left( x \right) = T_{1} + \frac{\left( T_{2} - T_{1} \right)}{L}
   :label: fld02eq2

.. _setup-10:

Setup
~~~~~

.. literalinclude:: /subprojects/mfix/tests/fluid/FLD02/mfix.dat
      :language:  fortran

.. .. _fld02table1:
.. .. csv-table:: FLD-02 Setup, Initial and Boundary Conditions.
..    :widths: auto
..    :header: "Computational/Physical model", " ", " "
..
..    "1D, Steady-state, incompressible", " ", " "
..    "Single-phase (no solids)", " ", " "
..    "No gravity", " ", " "
..    "Momentum equations are not solved", " ", " "
..    "Thermal energy equation is not solved", " ", " "
..    "Turbulence equations are not solved (Laminar)", " ", " "
..    "Uniform mesh", " ", " "
..    "Superbee discretization scheme", " ", " "
..    " ", " ", " "
..    "**Geometry**", " ", " "
..    "Coordinate system", "Cartesian", " ", "Grid partitions"
..    "x-length", "1.0", "\(m\)", "8, 16, 32, 64"
..    "y-length", "1.0", "\(m\)", "1"
..    " ", " ", " "
..    "**Material** :sup:`†`", " ", " "
..    "Thermal conductivity, :math:`\lambda_{g}`", "1.0", "(W·m\ :sup:`-1`\ ·K\ :sup:`-1`)"
..     " ", " ", " "
..    "**Initial Conditions**", " ", " "
..    "Temperature, :math:`T_{g}`", "250.0", "\(K\)"
..     " ", " ", " "
..    "**Boundary Conditions**", " ", " ", " "
..    "West boundary", "400.0", "\(K\)", "Constant T"
..    "East boundary", "300.0", "\(K\)", "Constant T"
..    "North/South boundaries", "0.0", "(J·s\ :sup:`-1`\ ·m\ :sup:`-2`)", "Constant flux"

.. _results-10:

Results
~~~~~~~

The analytical and numerical solutions for temperature, :math:`T_{g}`, are shown in :numref:`fld02fig2`. Only a subset of the numerical solution data points are plotted causing the appearance of a slight shift in presented data points. The largest observed absolute error is bounded above by :math:`10^{- 12}` and occurs for the finest mesh. This error is attributed to convergence criteria of the linear equation solver.

.. _fld02fig2:
.. figure:: ../media/image35.png
   :align: center

   Steady, 1D heat-conduction. (Left) numerical solution vs analytical solution, and (right) absolute error between the analytical and numerical solutions.