.. ############################################################################
.. manual/packages/vertex-cfd/closure_model_conduction.rst
.. ############################################################################

.. _sec_closure_model_conduction:

***********************************************
Conduction equation
***********************************************
      
.. literalinclude:: ../xml-blocks/closure_model.xml
   :language: xml
   :lines: 1-7
   :caption: `Closure Model` block in XML input file.

.. py:data:: Closure Factory Type

   :type: string
   :value: Conduction
   :usage: Required
   :description: Closure factory type to set source terms and material properties to solve the conduction equation.

.. literalinclude:: ../../../../examples/inputs/conduction/conduction_inverse_proportional_2d.xml
   :language: xml
   :linenos:
   :lines: 105

.. dropdown:: Material Properties
   :icon: multi-select
   :color: primary
   :animate: fade-in-slide-down

   .. py:data:: Type

      :type: string
      :value: ConstantMaterialProperties
      :usage: Required
      :description: Material property type to set density, heat capacity and thermal conductivity.

   .. py:data:: Density

      :type: double
      :value: :math:`[0,\infty]`
      :usage: Required
      :description: Constant density.

   .. py:data:: Specific Heat Capacity Value

      :type: double
      :value: :math:`[0,\infty]`
      :usage: Required
      :description: Constant specific heat capacity.

   .. py:data:: Thermal Conductivity Type

      :type: string
      :value: constant, inverse_proportional
      :usage: Optional
      :default: constant
      :description: Model to use for the thermal conductivity.

   .. py:data:: Thermal Conductivity Value

      :type: double
      :value: :math:`[0,\infty]`
      :usage: Required if `Thermal Conductivity Type` is set to `constant`.
      :description: Constant thermal conductivity value: :math:`k(T) = k_0`.

   .. py:data:: Thermal Conductivity Coefficient

      :type: double
      :value: :math:`[0,\infty]`
      :usage: Required if `Thermal Conductivity Type` is set to `inverse_proportional`.
      :description: Coefficient :math:`k_0` for the thermal conductivity :math:`k(T) = \frac{k_0}{T}`.

   .. literalinclude:: ../../../../examples/inputs/conduction/conduction_inverse_proportional_2d.xml
      :language: xml
      :linenos:
      :lines: 122-128

.. dropdown:: <Error Norm>
   :icon: multi-select
   :color: primary
   :animate: fade-in-slide-down

   .. py:data:: Type

      :type: string
      :value: ConductionErrorNorms
      :usage: Required if computing error norms against an exact solution.
      :description: Computes the :math:`L_1` and :math:`L_2` error norms between the numerical solution and an exact solution.

   .. literalinclude:: ../../../../examples/inputs/conduction/conduction_inverse_proportional_2d.xml
      :language: xml
      :linenos:
      :lines: 106-108

.. dropdown:: <Heat Source>
   :icon: multi-select
   :color: primary
   :animate: fade-in-slide-down

   .. py:data:: Type

      :type: string
      :value: ConductionVolumetricSource
      :usagge: Optional.
      :description: Add a volumetric source term to the conduction equation.

   .. py:data:: Heat Source Type

      :type: string
      :value: constant, xlinear
      :usage: Optional
      :default: constant
      :description: Model to use for the volumetric source term.

   .. py:data:: Volumetric Heat Source Value

      :type: double
      :value: :math:`[-\infty,\infty]`
      :usage: Required.
      :description: Constant volumetric heat source value :math:`q_0`.

   .. py:data:: X-value of the left boundary

      :type: double
      :value: :math:`[-\infty,\infty]`
      :usage: Required if `Heat Source Type`  is set to `xlinear`.
      :description: x-coordinate of the left boundary :math:`x_{left}`.

   .. py:data:: X-value of the right boundary

      :type: double
      :value: :math:`[-\infty,\infty]`
      :usage: Required if `Heat Source Type`  is set to `xlinear`.
      :description: x-coordinate of the left boundary :math:`x_{right}`.

   .. note:: Volumetric heat source definition

      A `constant` volumetric heat source is defined as :math:`S = q_0`. A `xlinear` volumeteric heat source
      is defined as :math:`S = q_0 \frac{x - x_{left}}{x_{right} - x_{left}}`. The later is used for a
      verification case.

   .. literalinclude:: ../../../../examples/inputs/conduction/conduction_inverse_proportional_2d.xml
      :language: xml
      :linenos:
      :lines: 115-121

.. dropdown:: <Exact Solution>
   :icon: multi-select
   :color: primary
   :animate: fade-in-slide-down

   .. py:data:: Type

      :type: string
      :value: ConductionExactSolution
      :usage: Required if computing error norms for the verification case.
      :description: Computes the :math:`L_1` and :math:`L_2` error norms between the numerical solution and an exact solution.

   .. py:data:: Volumetric Heat Source Value

      :type: double
      :value: :math:`[-\infty,\infty]`
      :usage: Required if `Thermal Conductivity Type` is set to `inverse_proportional`.
      :description: Constant volumetric heat source value :math:`q_0`.

   .. py:data:: Thermal Conductivity Coefficient

      :type: double
      :value: :math:`[0,\infty]`
      :usage: Required if `Thermal Conductivity Type` is set to `inverse_proportional`.
      :description: Coefficient :math:`k_0` for the thermal conductivity :math:`k(T) = \frac{k_0}{T}`.

   .. py:data:: Right Temperature Boundary Value

      :type: double
      :value: :math:`[0,\infty]`
      :usage: Required if `Thermal Conductivity Type` is set to `inverse_proportional`.
      :description: Value of the temperature at the right boundary.

   .. literalinclude:: ../../../../examples/inputs/conduction/conduction_inverse_proportional_2d.xml
      :language: xml
      :linenos:
      :lines: 109-114