.. _section-A9.4:

Appendix 9.4: Fuel Swelling and Fission Gas Release Model Parameters
====================================================================

This section provides material properties and the constitutive models
used in fuel swelling and fission gas release model. :numref:`table-A9.4-1a` and :numref:`table-A9.4-1b`
summarize the parameters. Ref. [9‑10] and Ref. [9‑11] have been
utilized to derive the constitutive models. Fission gas diffusion
constants are the main fitting constants.

*Fission Gas Diffusion Coefficient*

The thermal bulk and surface fission gas diffusion coefficients,
:math:`D_{fg_b}` and :math:`D_{fg_s}`, are dependent on the phase
and irradiation conditions:

.. math::
   :label: eq144

   D_{fg\_ b} = \begin{cases}
   10^{- 4}\exp\left( - \frac{52,000} {RT} \right) + \ 4.9810^{- 39}F_{r}  & T < T_{\gamma} \\
   10^{- 8}\exp\left( - \frac{28,500}{RT} \right) + 4.9810^{- 39}F_{r}  & else \\
   \end{cases}

.. math::
   :label: eq145

   D_{fg\_ s} = \begin{cases}
   10^{- 1}\exp\left( - \frac{52,000} {RT} \right) + {4.9810^{- 36}F}_{r}  & T < T_{\gamma} \\
   10^{- 5}exp\left( - \frac{28,500}{RT} \right) + {4.9810^{- 36}F}_{r}  & else \\
   \end{cases}


where :math:`D_{fg\_ b}` and :math:`D_{fg\_ s}` are the bulk and
surface fission gas diffusion coefficients (m\ :sup:`2`/s),
respectively, R is the gas constant, 1.987 cal/mol/K, T is the
temperature in Kelvin, and :math:`T_{\gamma}` is the phase transition
temperature, and :math:`F_{r}` is the fission density rate
(fission/m\ :sup:`3`/s).

*Bubble Nucleation*

Bubble-1 nucleation rate is given as a function of matrix gas
concentration and bubble-1 gas atom number as follows:

.. math::
   :label: eq146

   J_{b1,nucl} = {k_{b1nuc}C}_{g}\rho_{g1}

Where :math:`k_{b1nuc}` is :math:`10^{- 20}`\ (bub-1/s),
:math:`C_{g}` is the matrix gas atom density (atom/m\ :sup:`3`), and
:math:`\rho_{g1}` is the constant atom number per bubble.

*Gas Diffusion into the Bubbles*

Diffusion of gas atoms into the bubbles can be calculated using the
following equation:

.. math::
   :label: eq147

   J_{gi} = {k_{gi}C}_{g}N_{bi}A_{open}A_{ellipsoidal}

.. math::
   :label: eq148

   k_{gi} = {E_{gbi}4\pi r}_{bi}D_{g}

where :math:`J_{gi}` is the atomic flux into bubble-i by diffusion,
:math:`k_{gi}` is the gas diffusion constant to bubble-i
(m\ :sup:`3`/s), :math:`E_{gbi}` is the empirical bias factor,
currently set to 1, :math:`r_{bi}` is the radius of bubble-i (m),
:math:`D_{g}` is the diffusion coefficient of gas atom
(m\ :sup:`2`/s), :math:`A_{open}` is area correction for the open
porosity network (the value is unity for the diffusion into closed
bubbles), and\ :math:`\ A_{ellipsoidal}` is the area correction for
the ellipsoidal bubbles (the value is unity for spherical bubbles and
pores).

*Integration rate via Bubble Diffusion*

The integration rate of bubble-i into bubble-j due to collision
between bubble-i and j is given as follows:

.. math::
   :label: eq149

   {ab}_{ij} = {k_{ij}N}_{bi}N_{bj}\rho_{gi}

.. math::
   :label: eq150

   k_{ij} = 4\pi  \left( r_{bi} + r_{bj} \right)\left( D_{bi} + D_{bj} \right)

.. math::
   :label: eq151

   D_{bi} = \frac{3a_{0}^{4}}{2\pi r_{bi}^{4}}D_{s}

The integration rate of bubble-i into bubble-j due to collision
between two bubble-i is given as follows:

.. math::
   :label: eq152

   {ab}_{ii} = 2k_{ii}N_{bi}^{2}\rho_{gi}

.. math::
   :label: eq153

   k_{ii} = 16\pi r_{bi}D_{bi}

The transition probability of bubble-i into
bubble-i+1, :math:`\ f_{i,i + 1}`, can be obtained by:

.. math::
   :label: eq154

   f_{i,i + 1} = \frac{2\rho_{gi}}{\rho_{gi + 1}}

where :math:`D_{bi}` is the bubble-i diffusion coefficient
(m\ :sup:`2`/s), :math:`D_{s}` is the surface diffusion coefficient
(m\ :sup:`2`/s),\ :math:`\ r_{bi}` and :math:`r_{bj}` are the radius of
bubble-i and bubble-j (m), respectively, and
:math:`a_{0}^{2} = 9 \times 10^{- 20}`\ m\ :sup:`2`.

*Integration rate via Bubble Growth*

.. math::
   :label: eq155

   {gab}_{ij} = P_{ij}\rho_{gi}N_{bi}

For collision between closed bubbles:

.. math::
   :label: eq156

   P_{ij} = \begin{cases}
    \frac{2{{\Delta}r}_{bi}}{{0.5l}_{j} - r_{bi} - r_{bi}}  & i = j \\
    \frac{{{\Delta}r}_{bi}}{0.5l_{j} - r_{bi} - r_{bj}}  & i \neq j \\
    \end{cases}

For collision between closed bubble and open pores:

.. math::
   :label: eq157

   P_{ij} = \frac{{{\Delta}r}_{bi}}{l_{j}f_{p}}


Where :math:`P_{ij}` is the probability of bubble-I colliding with
bubble-j due to radial growth of bubble-i,
:math:`{{\Delta}r}_{bi}` is the radial growth of bubble-i with
in a time step (m),\ :math:`\ r_{bi}` and :math:`r_{bj}` are the
radius of bubble-i and bubble-j (m), respectively, :math:`l_{j}`
approximate distance between bubble-js, and :math:`f_{p}` is an
empirical fitting factor.

.. _table-A9.4-1a:

.. list-table:: Fission gas atom density :math:`\rho_{g}` (atom)
    :header-rows: 0
    :align: center
    :widths: auto

    * - Small Bubble
      - 0.75E+08
    * - Medium Bubble
      - 0.2E+12
    * - Large Bubble
      - 0.25E+14

.. _table-A9.4-1b:

.. list-table:: Fuel swelling and fission gas release parameters
    :header-rows: 0
    :align: center
    :widths: auto

    * - Y (atoms/fission)
      - 0.25
    * - :math:`k_{b1nuc}` (bubble*atom/s)
      - 1.0E-20
    * - Threshold gas swelling (\%)
      - 10
    * - :math:`A_{open}`
      - 0.1
    * - :math:`A_{ellipsoidal}`
      - 1.4
    * - :math:`f_{p}`
      - 0.7