.. _section-A9.6:

Appendix 9.6: Metallic Fuel Thermal Conductivity Model
======================================================

The metallic fuel thermal conductivity equation given in Ref. [9‑32] for
the unirradiated U-Pu-Zr fuel is utilized. It is given as follows:

.. math::
   :label: eq169

   k_{0s} = a + bT + cT^{2} \\
   a = 17.5\left( \frac{1 - 2.23w_{Zr}}{1 + 1.61w_{Zr}} - 2.62w_{Pu} \right) \\
   b = 1.54 \times 10^{- 2}\left( \frac{1 + 0.061w_{Zr}}{1 + 1.61W_{Zr}} + 0.9w_{Pu} \right) \\
   c = 9.38 \times 10^{- 6}\left( 1 - 2.7w_{Pu} \right) \\

where :math:`k_{0s}` is  unirradiated  solid metallic fuel thermal conductivity  (W/m-K), T is temperature (K),
a,b, and c are constants,  :math:`w_{Pu}`, :math:`w_{Zr}` are the Plutonium and Zirconium weight fractions in fuel alloy

**Irradiated Metal Fuel Thermal conductivity**

Irradiated metallic fuel thermal conductivity is developed in this study
in parallel with development and validation of MFUEL metallic fuel
performance models of SAS.

.. math::
   :label: eq170

   k_{irr} = X_{Na}X_{p}k_{0}

where :math:`k_{irr}` effective thermal conductivity of irradiated fuel ( W/m-K),
:math:`X_{Na}` is the thermal conductivity enhancement factor due to sodium
infiltration into the fuel porosity, only applicable in outer
:math:`\left( (\alpha + \zeta) + \delta \right)` phase region Ref.[9‑16],
:math:`X_{p}` is the thermal conductivity correction factor due to fission gas porosity,
:math:`P_{tot} = P_{g} + P_{Na}`,
:math:`P_{tot},P_{g},P_{Na}` are the total, gas filled, and sodium filled porosity.

.. math::
   :label: eq171

   X_{p} = \left( 1 - P_{g} \right)^{1.5}

.. math::
   :label: eq172

   X_{Na} = 1 - 3\frac{1 - \left( \frac{k_{Na}}{0s} \right)}{1.163 + 1.837\left( \frac{ k_{Na}}{k_{0s}} \right)}\frac{P_{Na}}{1 - P_{g}}

where   :math:`k_{Na}` is the sodium thermal conductivity (W/m-K)