Cladding Corrosion Model
Zirconium alloy cladding can have an exothermic reaction with coolant water which converts metal to oxide at the cladding outer surface: (1) Such an oxidation process, which is referred to as water-side corrosion, is a fundamental aspect of LWR fuel performance. The resultant oxide film on the outer surface of cladding can affect both the thermal and mechanical properties of cladding. Because of the lower thermal conductivity of zirconium oxide in comparison with zirconium alloys, the oxidation of the cladding adds to thermal resistance to heat transfer from the fuel to the coolant.
Zirconium oxide is a brittle material and can be easily cracked. Thus it is expected that the mechanical strength of cladding is mainly determined by the metallic wall, which is thinned after corrosion. Concurrent to the oxidation process, a fraction of hydrogen can be absorbed into the metal and can diffuse under the influences of both temperature and stress.
Due to the low solubility of hydrogen in zirconium and its alloys, hydrogen can precipitate as -phase hydrides (ZrH), which are known to further reduce the ductility of irradiated cladding material. In fact, the hydrogen content in the zircaloy cladding has become a limiting parameter for burnup extension of LWR fuel. An oxidation model which can predict the growth of oxide layer as a function of operation conditions and metallurgical variables of cladding materials is essential to the study of LWR fuel performance. In addition, it is also important to account for the effects of the oxide layer on the thermal and mechanical properties of cladding.
Zirconium Alloy at Normal Operating Temperatures
Low temperature (250 C/ 523 K to 400 C/ 673 K) oxidation is calculated considering that cladding oxidation under normal LWR conditions occurs in two stages: a pre-transition oxidation process that follows a cubic time dependence up to a transition oxide thickness, and a post-transition process that follows a linear time dependence. The transition between the two stages typically occurs at 2 microns.
For the pre-transition period, the corrosion rate is given by an Arrhenius equation (Ritchie, 1998): (2) For the post-transition period, the corrosion rate is given by Ritchie (1998): (3) where is the oxide thickness, is the metal-oxide interface temperature, is the rate constant for pre-transition oxidation, is the activation energy for pre-transition oxidation, is the rate constant for post-transition oxidation, is the activation energy for post-transition oxidation, is the universal gas constant, and is the transition oxide thickness.
The metal-oxide interface temperature, , is calculated assuming steady-state heat conduction across the oxide thickness as: (4) where is the outer surface (waterside) oxide temperature and is thermal conductivity of zirconium oxide.
In most Bison simulations, the oxide layer is not meshed independently. Instead, the oxide layer is modeled as a virtual layer within the clad, and the code keeps track of the thickness , as shown in Figure 1. Since the oxide causes a larger temperature jump than would be caused by the same thickness of metal, calculated by Bison does not correspond to the true temperature at the coolant-clad interface. Therefore, we must modify the heat transfer coefficient so that the driving force ( is the bulk coolant temperature) is correct.
In this approach, zircaloy material is used in the thermal solution while an effective heat transfer coefficient is used to compute a "fictitious" boundary condition to match the true temperature at the metal and oxide interface. We begin with two equivalent statements for heat flux into the coolant (5) where is the true heat transfer coefficient. The starred values and are the simulated (fictitious) temperature of the oxide surface (waterside) and corresponding effective heat transfer coefficient, respectively. The temperature at the interface between the oxide and metal must also match: (6) where is the temperature at the interface of the oxide and metal, is the thermal conductivity of zirconium alloy, is the thermal conductivity of zirconium oxide, and is the Pilling-Bedworth ratio.
These equations can be combined to eliminate and (7) The oxide growth calculation requires , which can be calculated directly from by (8)
\!include /Materials/ZryOxidation.md start=EPRI KWU CE Model end=Example Input Syntax
Zirconium Oxide Thermal Conductivity
Thermal conductivity of zirconia in the model is a constant value of 1.5 W/m-K for PWR applications (Gilmore et al., December 1995). However the reported value of zirconium oxide thermal conductivity varies greatly from different sources.
In the NFIR experimental program, the ZrO thermal conductivity was estimated using cladding elongation measurements during power ramps as a representation of cladding temperature changes (Ikeda and Kolstad, February, 1996). By comparing the cladding elongation of a fuel rod with an external oxide to a reference rod without an external oxide, the thermal impact of the oxide layer was determined. Experiments were performed at oxide layer thicknesses between 30 and 82 m. In determining the thermal conductivity from the measured data, considerations were made for external crud layers, power increases, power decreases, and oxide layer thickness. The results of the experiments found that the thermal conductivity of ZrO is independent of oxide thickness and temperature in the temperature range between 240C and 300C. An NFIR corrosion model was developed with a constant thermal conductivity value of 2.7 W/m-K (which tends to be on the high side of the data). The NFIR model is based on a series of in-pile experiments performed in the Halden test reactor that were designed to determine the thermal conductivity of external oxide layers on fuel rods (Ikeda and Kolstad, February, 1996).
The MATPRO-11 Rev. 2 model for Zircaloy oxide thermal conductivity is based on several different data sources of thermal conductivity measurements (Allison et al., 1993). These measurements were performed using a variety of oxide morphologies (stabilized oxides, nodular, and black) and oxide formation techniques (steam oxidation and plasma sputtering).
Using thermal diffusivity measurements, the thermal conductivity was determined for the different oxide types as a function of temperature. The MATPRO model used primarily data from tests with black oxide layers to develop the thermal conductivity as a function of temperature (Allison et al., 1993).
The resultant correlation is (9) where is the oxide thermal conductivity (W/m-K) T is the oxide temperature (K). The correlation above is applicable to solid Zircaloy oxide found on fuel rods. These other values are typical of other models found in the literature. Further information on the MATPRO Zircaloy oxide model can be found in Allison et al. (1993).
Nuclear Electric (NE PLC) use a different correlation starting at a value of 1.5 W/m-K. The value then decreases with oxide thickness according to the following relationship (Gilmore et al., December 1995): (10) The CEA Cochise code uses a constant value of 1.6 W/m-K (Gilmore et al., December 1995).
Numerical solution of the oxide thickness growth consists of pre-transition and post-transition period. In the pre-transition period: (11) where is oxide thickness at previous time step (m), is rate constant for pre-transition oxidation (m/day), is activation energy for pre-transition oxidation (cal/mol), is gas constant (here set as 1.987 (cal/mol-K)), is cladding outer surface temperature (K), is time increment (day), and is oxide thickness increment (m).
In the post-transition oxidation period, an approximate integral method is used (Garzarolli et al., 1982) to account for the metal-oxide interface temperature change on the oxygen weight gain: (12)
(13) where is cladding outer surface temperature (K), is thermal conductivity of zirconium oxide (W/cm-K), is weight gain (g/cm ), (=0.6789 cm/g) is a factor that converts weight gain (g/cm) to thickness (cm), is activation energy for post-transition phase, is heat flux (W/cm), is rate constant for post-transition phase (g/cm-day), is time increment (day), is ideal gas constant (that is 1.987 (cal/mol-K)), is weight gain at previous time step (g/cm), and is oxide layer thickness at previous time step (m).
According to Griess et al. (1964) and Griebenow et al. (1971), corrosion of aluminum in ATR follows (14) where is the oxide thickness (m), is time (hours), and is temperature (K). This corrosion thickness may be used in the coolant channel model.
FCCI Interaction Layer Thickness
See the ThicknessLayerFCCI page for information on this capability and model.
FCCI Eutectic Penetration Thickness
See the EutecticThicknessFCCI page for information on this capability and model.
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SCDAP/RELAP5/MOD3.1 code manual, volume IV: MATPROâ€“A library of materials properties for light-water-reactor accident analysis.
Technical Report NUREG/CR-6150, EGG-2720, Idaho National Engineering Laboratory, 1993.[BibTeX]
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Review of PWR fuel rod waterside corrosion behavior.
Technical Report EPRI NP-2789 Project 1250 Final Report, Kraftwerk Union A.G. and Combustion Engineering Inc., 1982.[BibTeX]
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In American Nuclear Society 1971 Winter Meeting Transactions, volume 14(3), 761â€“762. 1971.[BibTeX]
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Effect of heat flux on the corrosion of aluminum by water, part IV, tests relative to the advanced test reactor and correlation with previous results.
Technical Report ORNL-3541, Oak Ridge National Laboratory, February 1964.[BibTeX]
- J. Ikeda and E. Kolstad.
In-pile determination of thermal conductivity of oxide layer on LWR cladding.
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Waterside corrosion of zirconium alloys in nuclear power plants.
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