.. _section-12.13:

Physical Properties of Sodium
-----------------------------

The expressions used in SASSYS-1 and SAS4A to model the physical
properties of sodium as functions of temperature are derived from data
correlations generated by Padilla :ref:`[12-11] <section-12.references>` and by Fink and Leibowitz
:ref:`[12-12] <section-12.references>`. These correlations were developed using the most recent
measurements available, namely those of Bhise and Bonilla :ref:`[12-13] <section-12.references>` and Da
Gupta and Bonilla :ref:`[12-14] <section-12.references>`. The experiments described in Refs. :ref:`[12-13] <section-12.references>`
and :ref:`[12-14] <section-12.references>` resulted in the first measurements of sodium properties in
the temperature range from about 1250 K up to the critical point (2503.3
K). Because of this new information, sodium properties can be modeled in
SASSYS-1 and SAS4A over a much wider temperature range than was possible
in previous versions of the SAS code.

The correlations given in Refs. :ref:`[12-11] <section-12.references>` and :ref:`[12-12] <section-12.references>` could have been used
directly in SASSYS-1. However, in order to increase computational
efficiency, least-squares approximations to these correlations were
generated and incorporated in the code. These approximations introduce
little additional error into the calculation (they fit the correlations
to 1.5% or better in all cases), and they require less central
processing time than do the more complex expressions given in the
references. In addition, one polynomial fit is made over the temperature
range of interest per sodium property. The references often present two
or more correlations (each valid over different sections of the
temperature range) for a given property. Implementing different
correlations over different temperature ranges for a single property
would require computationally expensive branching logic in the code;
this is avoided by the polynomial fits.

Because most sections of the code do not require properties at
temperatures above about 90% of the critical point (2270 K), the
polynomial fits do not extend past 2270 K in order to avoid the
difficulty of fitting a polynomial to the rapid changes that occur in
most properties near the critical point. On the low end, the temperature
range goes down to 590 K, well below any temperature needed in fast
reactor accident analyses.

The polynomial expressions for each property are listed below, along
with a brief explanation of their origin. In all cases, the temperature
:math:`T` is in Kelvins.


.. rubric:: Heat of Vaporization (:math:`\lambda` in J/kg)

.. math::
    :label: NaVapHeat

    \lambda = A_{1} + A_{2}T + A_{3}T^{2} + A_{4}T^{3}

..

    :math:`A_{1} = 5.3139 \times 10^{6}`

    :math:`A_{2} = -2.0296 \times 10^{3}`

    :math:`A_{3} = 1.0625`

    :math:`A_{4} = -3.3163 \times 10^{-4}`

The expression for :math:`\lambda` is a fit to two equations recommended by Padilla;
the first (valid below 1644 K) is from Golden Tokar's :ref:`[12-15] <section-12.references>`
quasichemical approach, while the second (for T > 1644 K) is the model
of Miller, Cohen, and Dickerman :ref:`[12-16] <section-12.references>`, as adjusted by Padilla. The
polynomial fit approximates both equations to within 1%.


.. rubric:: Saturation Vapor Pressure (:math:`P_{s}` in Pa)

.. math::
    :label: NaVapSaturationPressure


    \ln\left( P_{s} \right) = A_{5} - \frac{A_{6}}{T} - \frac{A_{7}}{T^{2}}

..

    :math:`A_{5} = 2.169 \times 10^{1}`

    :math:`A_{6} = 1.14846 \times 10^{4}`

    :math:`A_{7} = 3.41769 \times 10^{5}`

Fink and Leibowitz present the equation

.. math::
    :label: NaVapSaturationPressureFL

    \ln{\left( P_{s} \right) = 3.0345 \times 10^{1} - \frac{13113}{T} - 1.09481\ln{\left( T \right) + 1.9777 \times 10^{- 4}T}}

However, SASSYS-1 and SAS4A require an expression for temperature as a
function of pressure, which must be obtained by inverting the equation,
giving pressure as a function of temperature. Since this formula cannot
be inverted directly, the least-squares fit shown above was used. The
polynomial approximates the reference equation to within 1.2%.


.. rubric:: Saturation Temperature (:math:`T_{s}` in K)

.. math::
    :label: NaLiqSaturationTemperature

    T_{s} = \frac{A_{8}}{A_{9} + \sqrt{A_{10} + A_{11}\ln{(P_{s})}}}

..

    :math:`A_{8} = 2 A_{7}`

    :math:`A_{9} = - A_{6}`

    :math:`A_{10} = A_{6}^{2} + 4 A_{5} A_{7}`

    :math:`A_{11} = -4 A_{7}`

    :math:`3.5 \leq P_{s} \leq 1.6 \times 10^{7}` Pa


.. rubric:: Liquid Density (:math:`\rho_l` in kg/m\ :sup:`3`)

.. math::
    :label: NaLiqDensity

    \rho_{l} = A_{12} + A_{13}T + A_{14}T^{2}

..

    :math:`A_{12} = 1.00423 \times 10^{3}`

    :math:`A_{13} = -0.21390`

    :math:`A_{14} = -1.1046 \times 10^{-5}`

This equation fits two empirical equations recommended by Fink and
Leibowitz to within 0.5%. Below 1644 K, they suggest the equation of
Stone, et al. :ref:`[12-17] <section-12.references>`, while above 1644 K, they use a model of their own.


.. rubric:: Vapor Density (:math:`\rho_v` in kg/m\ :sup:`3`)

.. math::
    :label: NaVapDensity

    \rho_{v} = P_{s}(\frac{A_{15}}{T} + A_{16} + A_{17}T + A_{18}T^{2} + A_{19}T^{3} + A_{20}T^{4})

..

    :math:`A_{15} = 4.1444 \times 10^{-3}`

    :math:`A_{16} = -7.4461 \times 10^{-6}`

    :math:`A_{17} = 1.3768 \times 10^{-8}`

    :math:`A_{18} = -1.0834 \times 10^{-11}`

    :math:`A_{19} = 3.8903 \times 10^{-15}`

    :math:`A_{20} = -4.922 \times 10^{-19}`

with :math:`P_{s}` in pascals. This equation substitutes for the two
correlations presented by Padilla, one generated using the quasichemical
approach (below 1644 K), the other (above 1644 K) being the Clapeyron
equation. The polynomial is accurate to 1.5%.


.. rubric:: Liquid Heat Capacity (:math:`C_l` in J/kg-K)

.. math::
    :label: NaLiqHeatCapacity

    C_{l} = \frac{A_{28}}{\left( T_{c} - T \right)^{2}} + \frac{A_{29}}{T_{c} - T} + A_{30} + A_{31}\left( T_{c} - T \right) + A_{32}\left( T_{c} - T \right)^{2}

..

    :math:`A_{28} = 7.3898 \times 10^{5}`

    :math:`A_{29} = 3.1514 \times 10^{5}`

    :math:`A_{30} = 1.1340 \times 10^{3}`

    :math:`A_{31} = -2.2153 \times 10^{-1}`

    :math:`A_{32} = 1.1156 \times 10^{-4}`

    :math:`T_{c} = 2503.3` K = the critical temperature

This equation fits to within 1.5% the data generated by Padilla using a
thermodynamic relation from Rowlinson :ref:`[12-18] <section-12.references>`.


.. rubric:: Vapor Heat Capacity (:math:`C_{g}` in J/kg-K)

.. math::
    :label: NaVapHeatCapacity

    C_{g} = A_{33} + A_{34}T + A_{35}T^{2} + A_{36}T^{3} + A_{37}T^{4} + A_{38}T^{5} + A_{39}T^{6}

..

    :math:`A_{33} = 2.1409 \times 10^{3}`

    :math:`A_{34} = -2.2401 \times 10^{1}`

    :math:`A_{35} = 7.9787 \times 10^{-2}`

    :math:`A_{36} = -1.0618 \times 10^{-4}`

    :math:`A_{37} = 6.7874 \times 10^{-8}`

    :math:`A_{38} = -2.1127 \times 10^{-11}`

    :math:`A_{39} = 2.5834 \times 10^{-15}`

Padilla recommends a quasichemical approach below 1644 K and a relation
of Rowlinson's above that point. This polynomial approximates both
correlations to better than 1%.


.. rubric:: Liquid Adiabatic Compressibility (:math:`\beta_s` in Pa\ :sup:`-1`)

.. math::
    :label: NaLiqCompressibility

    \beta_{s} = A_{40} + \frac{A_{41}}{T_{c} - T}

..

    :math:`A_{40} = -5.4415 \times 10^{-11}`

    :math:`A_{41} = 4.7663 \times 10^{-7}`

Padilla computed values for this quantity from experimental measurements
of sonic velocity below 1773 K and from an empirical correlation due to
Grosse :ref:`[12-19] <section-12.references>` above that temperature. The polynomial fits these data to
better than 0.1%.


.. rubric:: Liquid Thermal Expansion Coefficient (:math:`\alpha_p` in K\ :sup:`-1`)

.. math::
    :label: NaLiqThermalExpansion

    \alpha_{p} = A_{42} + \frac{A_{43}}{T_{c} - T} + \frac{A_{44}}{\left( T_{c} - T \right)^{2}} + \frac{A_{45}}{\left( T_{c} - T \right)^{3}} + \frac{A_{46}}{\left( T_{c} - T \right)^{4}} + \frac{A_{47}}{\left( T_{c} - T \right)^{5}}

..

    :math:`A_{42} = 2.5156 \times 10^{-6}`

    :math:`A_{43} = 0.79919`

    :math:`A_{44} = -6.9716 \times 10^{2}`

    :math:`A_{45} = 3.3140 \times 10^{5}`

    :math:`A_{46} = -7.0502 \times 10^{7}`

    :math:`A_{47} = 5.4920 \times 10^{9}`

Fink and Leibowitz express :math:`\alpha_{p}` by applying a thermodynamic
relation to the data of Das Gupta and Bonilla. The above expression fits
their correlation to better than 0.7%.


.. rubric:: Liquid Thermal Conductivity (:math:`k_l` in W/m-K)

.. math::
    :label: NaLiqThermalConductivity

    k_{l} = A_{48} + A_{49}T + A_{50}T^{2} + A_{51}T^{3}

..

    :math:`A_{48} = 1.1045 \times 10^{2}`

    :math:`A_{49} = -6.5112 \times 10^{-2}`

    :math:`A_{50} = 1.5430 \times 10^{-5}`

    :math:`A_{51} = -2.4617 \times 10^{-9}`

The expression approximates the data given by Fink and Leibowitz to
within 0.5%. This information comes from experimental data below 1500 K
and extrapolated values above 1500 K generated by a method described by
Grosse :ref:`[12-20] <section-12.references>`.


.. rubric:: Liquid Viscosity (:math:`\mu_l` in Pa-s)

.. math::
    :label: NaLiqViscosity

    \mu_{l} = A_{52} + \frac{A_{53}}{T} + \frac{A_{54}}{T^{2}} + \frac{A_{55}}{T^{3}}

..

    :math:`A_{52} = 3.6522 \times 10^{-5}`

    :math:`A_{53} = 0.16626`

    :math:`A_{54} = -4.56877 \times 10^{1}`

    :math:`A_{55} = 2.8733 \times 10^{4}`

Fink and Leibowitz present this variable as a combination of
experimental data below 1200 K and extrapolated values based on a method
suggested by Grosse :ref:`[12-21] <section-12.references>`. Their results are
approximated by the above polynomial to within 0.5%.

.. rubric:: Enthalpy of Saturated Liquid (:math:`H` in J/kg)

.. math::
    :label: NaLiqSatEnth

    H = A_{56} + A_{57} T + A_{58} T^2 + A_{59} T^3

..

    :math:`A_{56} = -111136.04`
    
    :math:`A_{57} = 1722.2578`
    
    :math:`A_{58} = -0.45544483`
    
    :math:`A_{59} = 1.4692883 \times 10^{-4}`

The enthalpy of saturated sodium is only used in the fuel failure models :ref:`PLUTO <section-14>` and :ref:`LEVITATE <section-16>`.