9.8.3. Appendix 9.3: Thermodynamics and Kinetics Parameters of Fuel Clad Chemical Interaction Model

Lanthanide Migration

The solubility limit of lanthanides in metallic fuel is assumed to be zero [9‑4]. The lanthanide diffusion coefficient is given in Eq. (9.8-4). Below 800 K, the temperature is set to 800 K in order to account for the irradiation-induced diffusivity. Ref. [9‑5] is used to assess the solubility limit of lanthanides. Thermodynamic and kinetics parameters related the lanthanide diffusion and precipitation model are given in Table 9.8.5. Note that solubility limit of D9 cladding is reduced significantly compared to the HT9 cladding in order to account for excessive clad wastage behavior observed in experiments. Diffusion constant and diffusion activation energy is used as fitting factors.

(9.8-3)\[\begin{split} D_{la} = \begin{cases} D_0 exp\left( - \frac{Q_0}{RT} \right) & T \ge 800 \text{ K} \\ D_0 exp\left( - \frac{Q_0}{800R} \right) & T \leq 800 \text{ K} \\ \end{cases}\end{split}\]

Table 9.8.5 Diffusion coefficient, heat of transport, and Lanthanide yield for Lanthanide migration

\(D_{0}\) (m \(^2\)/s)	5.0
\(Q_{0}\) (kJ/mol)	250.0
\(Y_{la}\) (#/J)	\(15.29E+09\)
Solubility in HT9 (atom/ \(m^{3}\))	\(6.4E+27\)
Solubility in D9 (atom/ \(m^{3}\))	\(0.5E+27\)

Iron and Actinide Migration

The iron and actinide diffusion coefficients are calculated using the relationships given in Eq. (9.8-4) and Eq. (9.8-5), respectively. Table 9.8.6 and Table 9.8.8 gives the diffusion constant and activation energy values. Table 9.8.9 and Table 9.8.10 gives the thermodynamics and kinetics parameters used in iron and actinide diffusion and eutectic formation. The experimental observartions in Ref. [9‑6] is used to extract thermodynamics and kinetics parameters.

(9.8-4)\[D_{Fe} = D_{0Fe} exp \left( - \frac{Q_{0Fe}}{RT} \right)\]

(9.8-5)\[ D_{Ac} = D_{0Ac} exp \left( - \frac{Q_{0Ac}}{RT} \right)\]

Table 9.8.6 Diffusion constant and activation energy for iron diffusion in metal fuel

Parameter	T < \(T_{\beta + \gamma }\)	T \(\geq T_{\beta + \gamma }\)
\(D_{0Fe}\) (m \(^2\)/s)	0.225E-03	1.0E-03
\(Q_{0Fe}\) (kJ/mol)	220	220

Table 9.8.7 Diffusion constant and activation energy for iron diffusion in cladding

\(D_{0Fe}\) (m \(^2\)/s)	8.5E-03
\(Q_{0Fe}\) (kJ/mol)	251

Table 9.8.8 Diffusion constant and activation energy for Actinides and eutectic

Parameter	T < 1074.15 K	1074.15 K \(\leq\) T \(\leq\) 1094.15 K	T \(\geq\) 1094.15 K
\(D_{0Ac}\) (m \(^2\)/s)	2.86E-08	\(2.86E-08 + 4.29E-08 \frac{T-1075.15}{20}\)	7.15E-08
\(Q_{0Ac}\) (kJ/mol)	120	120	120

Table 9.8.9 Thermodynamics and kinetics parameters for iron, actinide diffusion, and eutectic formation

Phase Structure	Interaction Parameter (J/mol)	Iron Solubility Limit (%)	Phase growth activation energy (kJ/mol)
(U,Pu)Fe2	-3000	66.7	220
Fe2Zr + (U,Pu)6Fe	-1200	40.5	220
(U,Pu)6Fe + U-32Zr-50Fe	-750	28.6	220
(U,Pu)6Fe + U-42Zr-33Fe	-650	21.8	220
U-Zr matrix + U-42Zr-33Fe	-500	13.2	220
U-Zr matrix + U-23Zr-6Fe	-800	2.5	220
(U,Pu)Zr	-800	0.0	220
Clad	-600	0.05	220
Eutectic	-1600	0.5	220

Table 9.8.10 Phase growth constansts (1/s) for iron, actinide diffusion, and eutectic formation

Phase Structure	T \(\le T_{\beta + \gamma}\)	T \(\geq T_{\beta + \gamma}\)
(U,Pu)Fe2	1.69E+05	7.5E+05
Fe2Zr + (U,Pu)6Fe	3.38E+05	1.5E+56
(U,Pu)6Fe + U-32Zr-50Fe	1.7E+06	7.5E+06
(U,Pu)6Fe + U-42Zr-33Fe	2.81E+06	1.25E+07
U-Zr matrix + U-42Zr-33Fe	3.94E+06	1.75E+07
U-Zr matrix + U-23Zr-6Fe	5.63E+07	1.5E+08
(U,Pu)Zr	0.0	0.0
Clad	0.0	0.0
Eutectic	2.81E+09	1.25E+10