.. _section-A9.2:

Appendix 9.2: Thermodynamics and Kinetics Parameters of Fuel Constituent Redistribution Model
=============================================================================================

Diffusion constant and activation energy, heat of transport, and solubility limits for the phases present are given in :numref:`table-A9.2-1a`, :numref:`table-A9.2-1b`, :numref:`table-A9.2-2`, :numref:`table-A9.2-3`, respectively.
:eq:`eq139` and :eq:`eq140` are used to predict the phase transition temperatures.
Ref. [9-11] and [9-27] has been utilized to obtain most of the diffusion constants, diffusion activation energies and heat of transport values.
Similar to Ref. [9-11] and [9-27], diffusion constants and heat of transport values are used as fitting parameters.
Some of those parameters differs from the corresponding references.
Phase transition Equations (:eq:`eq139` and :eq:`eq140`) are modeled as given in Ref. [9-11].

.. _table-A9.2-1a:

.. list-table:: Zirconium diffusion constant (m :math:`^2`/s)
    :header-rows: 1
    :align: center
    :widths: auto

    * -  Phase
      -  Pu < 7 at\%
      -  Pu :math:`\geq` 7 at \%
    * - :math:`\alpha`
      - 2.0e-07
      - 4.0e-06
    * - :math:`\delta`
      - 2.0e-07
      - 4.0e-06
    * - :math:`\beta`
      - 5.7e-06
      - 7.0e-04
    * - :math:`\gamma`
      - :math:`0.4 \times 10^{-5.1-8.05C_{Zr}+9.13C_{Zr}^2}`
      - :math:`10^{-5.1-8.05C_{Zr}+9.13C_{Zr}^2}`

.. _table-A9.2-1b:

.. list-table:: Zirconium activation energy :math:`Q_0` (kJ/mol)
    :header-rows: 1
    :align: center
    :widths: auto

    * -  Phase
      -  Pu < 7 at\%
      -  Pu :math:`\geq` 7 at \%
    * - :math:`\alpha`
      - 170.0
      - 170.0
    * - :math:`\delta`
      - 150.0
      - 150.0
    * - :math:`\beta`
      - 180.0
      - 180.0
    * - :math:`\gamma`
      - :math:`128 - 107 C_{Zr} + 174 C_{Zr}^2`
      - :math:`128 - 107 C_{Zr} + 174 C_{Zr}^2`

.. _table-A9.2-2:

.. list-table:: :math:`Q_h` - Heat of Transport (kJ/mol)
    :header-rows: 1
    :align: center
    :widths: auto

    * -  Phase
      -  Pu < 7 at\%
      -  Pu :math:`\geq` 7 at \%
    * - :math:`\alpha`
      - 0
      - 150
    * - :math:`\delta`
      - 0
      - 150
    * - :math:`\beta`
      - 0
      - 650
    * - :math:`\gamma`
      - -80
      - -200

.. _table-A9.2-3:

.. list-table:: Zr Solubility Limit
    :header-rows: 1
    :align: center
    :widths: auto

    * -  Phase
      -  Zr Solubility limit (at \%)
    * - :math:`\alpha`
      - 5
    * - :math:`\delta`
      - 40
    * - :math:`\beta`
      - 5


.. math::
   :label: eq139

   T_{\alpha + \delta \leftarrow \beta +\gamma} = \frac{a_{Pu}}{0.164} \left( 868.15 - 935.15 \right) + 935.15

.. math::
   :label: eq140

   T_{\beta +\gamma \leftarrow \gamma } = \frac{a_{Pu}}{0.164} \left( 923.15 - 965.15 \right) + 965.15