.. _section-3.16:

Thermal Properties of Fuel and Cladding
---------------------------------------

All material property data for fuel and cladding are cast as functions
or subroutines to allow for modularization and the ease of making
changes. This also allows for the incorporation of different materials
data in a straightforward manner. In a number of the correlations used,
the units are inconsistent with the SI unit system adopted by
|SAS|. The routines that use these correlations carry out the
appropriate units conversions internally.

The thermal properties for fuel and cladding are described in this
section. Sodium properties are described in :numref:`Chapter %s<section-12>`.

.. _section-3.16.1:

Fuel Density
~~~~~~~~~~~~

The fuel density can be obtained either from a user-supplied table of
density vs temperature or from a correlation with

.. math::
    :label: 3.16-1

	\rho_{\text{f}} = \frac{\rho_{\text{o}}}{1 + C_{1}\left( T - 273 \right) + C_{2}\left( T - 273 \right)^{2}}

where

:math:`\rho_{\text{o}}` = The theoretical density at 273 K, kg/m\ :sup:`3`

:math:`C_1`, :math:`C_2` = Input coefficients

:math:`T` = Temperature, K

This applies between 273 K and the solidus temperature.

The liquid fuel density is given by

.. math::
    :label: 3.16-2

	\rho_{\text{l}} = \frac{\rho_{\text{o}}}{1 + C_{3}\left( T - 273 \right)}

where

:math:`C_3` = Input coefficient

This applies to temperatures above the liquidus. For the range between
the solidus and liquidus temperatures, a linear interpolation is
performed.

These equations are found in the function RHOF. Suggested values of
coefficients are from the Nuclear Systems Materials Handbook [3-13].

:math:`\rho_{\text{o}}` = :sasinp:`COEFDS`\(1) = :math:`11.05 \times 10^{3} \text{kg/m}^{3}` (mixed
oxide)

:math:`C_1` = :sasinp:`COEFDS`\(2) = :math:`2.04 \times 10^{-5} ~\text{K}^{-1}`

:math:`C_2` = :sasinp:`COEFDS`\(3) = :math:`8.70 \times 10^{-9} ~\text{K}^{-2}`

:math:`C_3` = :sasinp:`COEFDL`\(2) = :math:`9.30 \times 10^{-5} ~\text{K}^{-1}`

.. _section-3.16.2:

Fuel Thermal Conductivity
~~~~~~~~~~~~~~~~~~~~~~~~~

Four different options exist for the fuel thermal conductivity. These
are controlled through the input parameter :sasinp:`IRHOK`.

:sasinp:`IRHOK` = 0

The thermal conductivity as function of temperature is input in table
form through the variable arrays XKTAB and XKTEM.

:sasinp:`IRHOK` = 1

For this option, the conductivity equations are given by:

.. math::
    :label: 3.16-3

    k_{1}\left( T \right) = 1.1 + \frac{1 \times 10^{2}}{T\left( .4888 - .4465f_{\text{D}} \right)}

for :math:`800^{\circ}C  \leq T \leq 2000^{\circ}` C

.. math::
    :label: 3.16-4

    k_{2}\left( T \right) = k_{1}\left( 800 \right)\frac{168.844}{12.044 + \left( 0.196 \right) T}

for :math:`T \leq 800^{\circ}` C

.. math::
    :label: 3.16-5

    k_{3}\left( T \right) = k_{1}\left( 2000 \right)

for :math:`T > 2000^{\circ}` C

where

:math:`k_1`, :math:`k_2`, :math:`k_3` = Fuel thermal
conductivity, W/m-k

:math:`T` = Temperature, °C

:math:`f_{\text{D}}` = Fuel fraction of theoretical density

:sasinp:`IRHOK` = 2

This form of the conductivity is given by

.. math::
    :label: 3.16-6

    k_{1}\left( T \right) = \left\lbrack \left( C_{1} - f_{\text{D}} \right) f_{\text{D}} - 1 \right\rbrack\left\lbrack \frac{1}{\left( C_{2} + C_{3}T \right)} + C_{4}T^{3} \right\rbrack

for :math:`0.75  \leq f_{\text{D}} \leq 0.95`

.. math::
    :label: 3.16-7

    k_{2}\left( T \right) = \left\lbrack 3.0 f_{\text{D}} - 1 \right\rbrack\left\lbrack \frac{1}{\left( C_{5} + C_{6}T \right)} + C_{7}T^{3} \right\rbrack

for :math:`f_{\text{D}} > 0.95`

where

:math:`C_1`, :math:`C_2`, :math:`C_3`, :math:`C_4`, :math:`C_5`, :math:`C_6`, :math:`C_7`
= Input variables

:math:`k_1`, :math:`k_2` = Fuel conductivity W/m-k

T = Temperature, K

If :math:`T` is greater than the melting temperature, it is set to the melting
temperature.

Suggested values:

:math:`C_1` = :sasinp:`COEFK`\(1) = :math:`2.1`

:math:`C_2` = :sasinp:`COEFK`\(2) = :math:`2.88 \times 10^{-3}`

:math:`C_3` = :sasinp:`COEFK`\(3) = :math:`2.52 \times 10^{-5}`

:math:`C_4` = :sasinp:`COEFK`\(4) = :math:`2.83 \times 10^{-10}`

:math:`C_5` = :sasinp:`COEFK`\(5) = :math:`5.75 \times 10^{-2}`

:math:`C_6` = :sasinp:`COEFK`\(6) = :math:`5.03 \times 10^{-4}`

:math:`C_7` = :sasinp:`COEFK`\(7) = :math:`2.91 \times 10^{-11}`

:sasinp:`IRHOK` = 3

This conductivity form is [3-14]

.. math::
    :label: 3.16-8

    k_{1}\left( T \right) = \frac{4.005 \times 10^{3}}{\left( T - 273 \right) + 402.4} + 0.6416 \times 10^{- 10}T^{3}

where

:math:`T` = Temperature, K

:math:`k_1` = Conductivity in W/m-k

This is the correlation for :math:`\text{UO}_2` and is converted to mixed oxide
by subtracting 0.2.

.. math::
    :label: 3.16-9

    k_{2}\left( T \right) = k_{1}\left( T \right) - 0.2

The porosity correction term was derived for use in the COMETHE-IIIJ
[3-15] code and is given by

.. math::
    :label: 3.16-10

    f_{\text{p}} = 1 - 1.029 \varepsilon  - 3.2 \varepsilon^{2} - 40.1 \varepsilon^{3} +  158 \varepsilon^{4}

where

:math:`f_{\text{p}}` = Porosity multiplier

:math:`\epsilon = 1 - \rho_{\text{f}}` = Fractional porosity

:math:`\rho_{\text{f}}` = fractional fuel density = actual density/theoretical
density

The conductivity is therefore given by

.. math::
    :label: 3.16-11

    k\left( T \right) = f_{\text{p}} k_{2}\left( T \right)

Two different routines contain the above correlations, FK and KFUEL. The
function FK returns a single value of the conductivity for a single
invocation and is used in the steady-state calculation. The subroutine
KFUEL returns the conductivity values for each radial node in the
current axial segment. It is used in the transient calculational
procedure.