9.8.4. Appendix 9.4: Fuel Swelling and Fission Gas Release Model Parameters

This section provides material properties and the constitutive models used in fuel swelling and fission gas release model. Table 9.8.11 and Table 9.8.12 summarize the parameters. Ref. [9‑10] and Ref. [9‑11] have been utilized to derive the constitutive models. Fission gas diffusion constants are the main fitting constants.

Fission Gas Diffusion Coefficient

The thermal bulk and surface fission gas diffusion coefficients, \(D_{fg_b}\) and \(D_{fg_s}\), are dependent on the phase and irradiation conditions:

(9.8-6)\[\begin{split}D_{fg\_ b} = \begin{cases} 10^{- 4}\exp\left( - \frac{52,000} {RT} \right) + \ 4.9810^{- 39}F_{r} & T < T_{\gamma} \\ 10^{- 8}\exp\left( - \frac{28,500}{RT} \right) + 4.9810^{- 39}F_{r} & else \\ \end{cases}\end{split}\]
(9.8-7)\[\begin{split}D_{fg\_ s} = \begin{cases} 10^{- 1}\exp\left( - \frac{52,000} {RT} \right) + {4.9810^{- 36}F}_{r} & T < T_{\gamma} \\ 10^{- 5}exp\left( - \frac{28,500}{RT} \right) + {4.9810^{- 36}F}_{r} & else \\ \end{cases}\end{split}\]

where \(D_{fg\_ b}\) and \(D_{fg\_ s}\) are the bulk and surface fission gas diffusion coefficients (m2/s), respectively, R is the gas constant, 1.987 cal/mol/K, T is the temperature in Kelvin, and \(T_{\gamma}\) is the phase transition temperature, and \(F_{r}\) is the fission density rate (fission/m3/s).

Bubble Nucleation

Bubble-1 nucleation rate is given as a function of matrix gas concentration and bubble-1 gas atom number as follows:

(9.8-8)\[J_{b1,nucl} = {k_{b1nuc}C}_{g}\rho_{g1}\]

Where \(k_{b1nuc}\) is \(10^{- 20}\)(bub-1/s), \(C_{g}\) is the matrix gas atom density (atom/m3), and \(\rho_{g1}\) is the constant atom number per bubble.

Gas Diffusion into the Bubbles

Diffusion of gas atoms into the bubbles can be calculated using the following equation:

(9.8-9)\[J_{gi} = {k_{gi}C}_{g}N_{bi}A_{open}A_{ellipsoidal}\]
(9.8-10)\[k_{gi} = {E_{gbi}4\pi r}_{bi}D_{g}\]

where \(J_{gi}\) is the atomic flux into bubble-i by diffusion, \(k_{gi}\) is the gas diffusion constant to bubble-i (m3/s), \(E_{gbi}\) is the empirical bias factor, currently set to 1, \(r_{bi}\) is the radius of bubble-i (m), \(D_{g}\) is the diffusion coefficient of gas atom (m2/s), \(A_{open}\) is area correction for the open porosity network (the value is unity for the diffusion into closed bubbles), and\(\ A_{ellipsoidal}\) is the area correction for the ellipsoidal bubbles (the value is unity for spherical bubbles and pores).

Integration rate via Bubble Diffusion

The integration rate of bubble-i into bubble-j due to collision between bubble-i and j is given as follows:

(9.8-11)\[{ab}_{ij} = {k_{ij}N}_{bi}N_{bj}\rho_{gi}\]
(9.8-12)\[k_{ij} = 4\pi \left( r_{bi} + r_{bj} \right)\left( D_{bi} + D_{bj} \right)\]
(9.8-13)\[D_{bi} = \frac{3a_{0}^{4}}{2\pi r_{bi}^{4}}D_{s}\]

The integration rate of bubble-i into bubble-j due to collision between two bubble-i is given as follows:

(9.8-14)\[{ab}_{ii} = 2k_{ii}N_{bi}^{2}\rho_{gi}\]
(9.8-15)\[k_{ii} = 16\pi r_{bi}D_{bi}\]

The transition probability of bubble-i into bubble-i+1, \(\ f_{i,i + 1}\), can be obtained by:

(9.8-16)\[f_{i,i + 1} = \frac{2\rho_{gi}}{\rho_{gi + 1}}\]

where \(D_{bi}\) is the bubble-i diffusion coefficient (m2/s), \(D_{s}\) is the surface diffusion coefficient (m2/s),\(\ r_{bi}\) and \(r_{bj}\) are the radius of bubble-i and bubble-j (m), respectively, and \(a_{0}^{2} = 9 \times 10^{- 20}\)m2.

Integration rate via Bubble Growth

(9.8-17)\[{gab}_{ij} = P_{ij}\rho_{gi}N_{bi}\]

For collision between closed bubbles:

(9.8-18)\[\begin{split}P_{ij} = \begin{cases} \frac{2{\mathrm{\Delta}r}_{bi}}{{0.5l}_{j} - r_{bi} - r_{bi}} & i = j \\ \frac{{ \mathrm{\Delta}r}_{bi}}{0.5l_{j} - r_{bi} - r_{bj}} & i \neq j \\ \end{cases}\end{split}\]

For collision between closed bubble and open pores:

(9.8-19)\[P_{ij} = \frac{{\mathrm{\Delta}r}_{bi}}{l_{j}f_{p}}\]

Where \(P_{ij}\) is the probability of bubble-I colliding with bubble-j due to radial growth of bubble-i, \({\mathrm{\Delta}r}_{bi}\) is the radial growth of bubble-i with in a time step (m),\(\ r_{bi}\) and \(r_{bj}\) are the radius of bubble-i and bubble-j (m), respectively, \(l_{j}\) approximate distance between bubble-js, and \(f_{p}\) is an empirical fitting factor.

Table 9.8.11 Fission gas atom density \(\rho_{g}\) (atom)

Small Bubble

0.75E+08

Medium Bubble

0.2E+12

Large Bubble

0.25E+14
Table 9.8.12 Fuel swelling and fission gas release parameters

Y (atoms/fission)

0.25

\(k_{b1nuc}\) (bubble*atom/s)

1.0E-20

Threshold gas swelling (%)

10

\(A_{open}\)

0.1

\(A_{ellipsoidal}\)

1.4

\(f_{p}\)

0.7