12.13. Physical Properties of Sodium

The expressions used in SAS4A/SASSYS‑1 to model the physical properties of sodium as functions of temperature are derived from data correlations generated by Padilla [12-11] and by Fink and Leibowitz [12-12]. These correlations were developed using the most recent measurements available, namely those of Bhise and Bonilla [12-13] and Da Gupta and Bonilla [12-14]. The experiments described in Refs. [12-13] and [12-14] 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 SAS4A/SASSYS‑1 over a much wider temperature range than was possible in previous versions of the SAS code.

The correlations given in Refs. [12-11] and [12-12] 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 $T$ is in Kelvins.

Heat of Vaporization ($\lambda$ in J/kg)

(12.13-1) $$\lambda = A_{1} + A_{2}T + A_{3}T^{2} + A_{4}T^{3}$$

$A_{1} = 5.3139 \times 10^{6}$

$A_{2} = -2.0296 \times 10^{3}$

$A_{3} = 1.0625$

$A_{4} = -3.3163 \times 10^{-4}$

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

Saturation Vapor Pressure ($P_{s}$ in Pa)

(12.13-2) $$\ln\left( P_{s} \right) = A_{5} - \frac{A_{6}}{T} - \frac{A_{7}}{T^{2}}$$

$A_{5} = 2.169 \times 10^{1}$

$A_{6} = 1.14846 \times 10^{4}$

$A_{7} = 3.41769 \times 10^{5}$

Fink and Leibowitz present the equation

(12.13-3) $$\ln{\left( P_{s} \right) = 3.0345 \times 10^{1} - \frac{13113}{T} - 1.09481\ln{\left( T \right) + 1.9777 \times 10^{- 4}T}}$$

However, SAS4A/SASSYS‑1 requires 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%.

Saturation Temperature ($T_{s}$ in K)

(12.13-4) $$T_{s} = \frac{A_{8}}{A_{9} + \sqrt{A_{10} + A_{11}\ln{(P_{s})}}}$$

$A_{8} = 2 A_{7}$

$A_{9} = - A_{6}$

$A_{10} = A_{6}^{2} + 4 A_{5} A_{7}$

$A_{11} = -4 A_{7}$

$3.5 \leq P_{s} \leq 1.6 \times 10^{7}$ Pa

Liquid Density ($\rho_l$ in kg/m³)

(12.13-5) $$\rho_{l} = A_{12} + A_{13}T + A_{14}T^{2}$$

$A_{12} = 1.00423 \times 10^{3}$

$A_{13} = -0.21390$

$A_{14} = -1.1046 \times 10^{-5}$

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. [12-17], while above 1644 K, they use a model of their own.

Vapor Density ($\rho_v$ in kg/m³)

(12.13-6) $$\rho_{v} = P_{s}(\frac{A_{15}}{T} + A_{16} + A_{17}T + A_{18}T^{2} + A_{19}T^{3} + A_{20}T^{4})$$

$A_{15} = 4.1444 \times 10^{-3}$

$A_{16} = -7.4461 \times 10^{-6}$

$A_{17} = 1.3768 \times 10^{-8}$

$A_{18} = -1.0834 \times 10^{-11}$

$A_{19} = 3.8903 \times 10^{-15}$

$A_{20} = -4.922 \times 10^{-19}$

with $P_{s}$ 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%.

Liquid Heat Capacity ($C_l$ in J/kg-K)

(12.13-7) $$C_{l} = \frac{A_{28}}{\left( T_{c} - T \right)^{2}} + \frac{A_{29}}{T_{c} - T} + A_{30} + A_{31}\left( T_{c} - T \right) + A_{32}\left( T_{c} - T \right)^{2}$$

$A_{28} = 7.3898 \times 10^{5}$

$A_{29} = 3.1514 \times 10^{5}$

$A_{30} = 1.1340 \times 10^{3}$

$A_{31} = -2.2153 \times 10^{-1}$

$A_{32} = 1.1156 \times 10^{-4}$

$T_{c} = 2503.3$ K = the critical temperature

This equation fits to within 1.5% the data generated by Padilla using a thermodynamic relation from Rowlinson [12-18].

Vapor Heat Capacity ($C_{g}$ in J/kg-K)

(12.13-8) $$C_{g} = A_{33} + A_{34}T + A_{35}T^{2} + A_{36}T^{3} + A_{37}T^{4} + A_{38}T^{5} + A_{39}T^{6}$$

$A_{33} = 2.1409 \times 10^{3}$

$A_{34} = -2.2401 \times 10^{1}$

$A_{35} = 7.9787 \times 10^{-2}$

$A_{36} = -1.0618 \times 10^{-4}$

$A_{37} = 6.7874 \times 10^{-8}$

$A_{38} = -2.1127 \times 10^{-11}$

$A_{39} = 2.5834 \times 10^{-15}$

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%.

Liquid Adiabatic Compressibility ($\beta_s$ in Pa^-1)

(12.13-9) $$\beta_{s} = A_{40} + \frac{A_{41}}{T_{c} - T}$$

$A_{40} = -5.4415 \times 10^{-11}$

$A_{41} = 4.7663 \times 10^{-7}$

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

Liquid Thermal Expansion Coefficient ($\alpha_p$ in K^-1)

(12.13-10) $$\alpha_{p} = A_{42} + \frac{A_{43}}{T_{c} - T} + \frac{A_{44}}{\left( T_{c} - T \right)^{2}} + \frac{A_{45}}{\left( T_{c} - T \right)^{3}} + \frac{A_{46}}{\left( T_{c} - T \right)^{4}} + \frac{A_{47}}{\left( T_{c} - T \right)^{5}}$$

$A_{42} = 2.5156 \times 10^{-6}$

$A_{43} = 0.79919$

$A_{44} = -6.9716 \times 10^{2}$

$A_{45} = 3.3140 \times 10^{5}$

$A_{46} = -7.0502 \times 10^{7}$

$A_{47} = 5.4920 \times 10^{9}$

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

Liquid Thermal Conductivity ($k_l$ in W/m-K)

(12.13-11) $$k_{l} = A_{48} + A_{49}T + A_{50}T^{2} + A_{51}T^{3}$$

$A_{48} = 1.1045 \times 10^{2}$

$A_{49} = -6.5112 \times 10^{-2}$

$A_{50} = 1.5430 \times 10^{-5}$

$A_{51} = -2.4617 \times 10^{-9}$

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 [12-20].

Liquid Viscosity ($\mu_l$ in Pa-s)

(12.13-12) $$\mu_{l} = A_{52} + \frac{A_{53}}{T} + \frac{A_{54}}{T^{2}} + \frac{A_{55}}{T^{3}}$$

$A_{52} = 3.6522 \times 10^{-5}$

$A_{53} = 0.16626$

$A_{54} = -4.56877 \times 10^{1}$

$A_{55} = 2.8733 \times 10^{4}$

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 [12-21]. Their results are approximated by the above polynomial to within 0.5%.

Enthalpy of Saturated Liquid ($H$ in J/kg)

(12.13-13) $$H = A_{56} + A_{57} T + A_{58} T^2 + A_{59} T^3$$

$A_{56} = -111136.04$

$A_{57} = 1722.2578$

$A_{58} = -0.45544483$

$A_{59} = 1.4692883 \times 10^{-4}$

The enthalpy of saturated sodium is only used in the fuel failure models PLUTO and LEVITATE.