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 SAS4A/SASSYS‑1. 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 Chapter 12.

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

(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

\(\rho_{\text{o}}\) = The theoretical density at 273 K, kg/m³

\(C_1\), \(C_2\) = Input coefficients

\(T\) = Temperature, K

This applies between 273 K and the solidus temperature.

The liquid fuel density is given by

(3.16‑2)

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

where

\(C_3\) = 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].

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

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

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

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

3.16.2. Fuel Thermal Conductivity

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

IRHOK = 0

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

IRHOK = 1

For this option, the conductivity equations are given by:

(3.16‑3)

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

for \(800^{\circ}C \leq T \leq 2000^{\circ}\) C

(3.16‑4)

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

for \(T \leq 800^{\circ}\) C

(3.16‑5)

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

for \(T > 2000^{\circ}\) C

where

\(k_1\), \(k_2\), \(k_3\) = Fuel thermal conductivity, W/m-k

\(T\) = Temperature, °C

\(f_{\text{D}}\) = Fuel fraction of theoretical density

IRHOK = 2

This form of the conductivity is given by

(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 \(0.75 \leq f_{\text{D}} \leq 0.95\)

(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 \(f_{\text{D}} > 0.95\)

where

\(C_1\), \(C_2\), \(C_3\), \(C_4\), \(C_5\), \(C_6\), \(C_7\) = Input variables

\(k_1\), \(k_2\) = Fuel conductivity W/m-k

T = Temperature, K

If \(T\) is greater than the melting temperature, it is set to the melting temperature.

Suggested values:

\(C_1\) = COEFK(1) = \(2.1\)

\(C_2\) = COEFK(2) = \(2.88 \times 10^{-3}\)

\(C_3\) = COEFK(3) = \(2.52 \times 10^{-5}\)

\(C_4\) = COEFK(4) = \(2.83 \times 10^{-10}\)

\(C_5\) = COEFK(5) = \(5.75 \times 10^{-2}\)

\(C_6\) = COEFK(6) = \(5.03 \times 10^{-4}\)

\(C_7\) = COEFK(7) = \(2.91 \times 10^{-11}\)

IRHOK = 3

This conductivity form is [3-14]

(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

\(T\) = Temperature, K

\(k_1\) = Conductivity in W/m-k

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

(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

(3.16‑10)

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

where

\(f_{\text{p}}\) = Porosity multiplier

\(\epsilon = 1 - \rho_{\text{f}}\) = Fractional porosity

\(\rho_{\text{f}}\) = fractional fuel density = actual density/theoretical density

The conductivity is therefore given by

(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.