Zirconium Redistribution in U-Pu-Zr
Zirconium redistribution in U-Zr and U-Pu-Zr based fuels is important for both fuel integrity and thermal limits. As zirconium redistributes, the uranium moves in the opposite direction and what was once a uniform isotopic concentration across the fuel rod becomes concentrated in zirconium, primarily at rod center and rod edge. This preferential accumulation at rod edge and rod center, with a corresponding decrease in the central region can affect local thermal limits (margin to melting temperature for example), radial power peaking, with potential enhances to lanthanide and actinide migration, all due to a significantly varying radial isotopic composition. A strong understanding of this process along with a robust method for predicting zirconium concentrations is vital to advanced metallic fuel designs, particularly those designs with minor actinide loadings, lanthanide loadings and new proposed metallic alloys.
The prediction of zirconium redistribution is difficult for several reasons. First, fuel behavior under irradiation is a multi-physics, multi-scale problem that requires the coupling of neutronic solutions due to uranium migration which induces radial power peaking. Correspondingly, burnup effects in the fuel cause changes in both power and the thermo-mechanical-diffusion fuel behavior. Additionally there are thermal considerations primarily related to thermal conductivity; as zirconium migrates, the thermal conductivity changes inversely with zirconium concentration - an increase in zirconium concentration causing a decrease in thermal conductivity and vice-versa. Fuel burnup increases porosity in the fuel, which degrades thermal conductivity. However, near the rod edge liquid sodium infiltration due to fuel cracking can occur, causing some recovery of the degraded conductivity. This porosity itself appears to be phase-dependent, particularly in the beta phase region of the fuel where porosity appears to be at a minimum and thermal conductivity degradation is expected to be lower. Along with neutronic and thermal- mechanical concerns, the phase properties of the fuel and our fundamental understanding of fuel properties and fuel behavior, particularly U-Pu-Zr based fuel, is not well known.
The microstructure of irradiated U-Pu-Zr fuel exhibits three distinct concentric zones where the effective heat of transport drives the direction of zirconium migration atoms in different phases: a zirconium-enriched central zone, a zirconium-depleted and uranium- enriched intermediate zone, and a zirconium-enriched zone on the outer periphery. Phase- dependent diffusion coefficients and heats of transport are not well known, with early evidence indicating the phase transition temperatures, experimentally derived from fresh fuel, may not be consistent with irradiated fuel. Final considerations are related to the phase diagram itself. Figure 1 below shows the experimentally derived phase diagram for U-19Pu-10Zr fuel from Kim et al. (2004), Kim et al. (2006), and Ishida et al. (1993). Noting the complexity and difficulty associated with programming logic for such a diagram, along with the lack of any material property information for several of the phases, a simplified version based on those used in several other models (Karahan, 2009; Kim et al., 2004; Kim et al., 2006) was used in practice, shown here in Figure 2.
There have been several published papers that analyze constituent distribution in metallic fuels mechanistically for U-Zr and U-Pu-Zr fuels (Karahan, 2009; Kim et al., 2004; Kim et al., 2006; Hofman et al., 1996; Ishida et al., 1993). In the most recent paper published by Kim et al. (2006), new computational model was developed to solve the diffusion equations using a simplified pseudo-binary phase diagram (similar to Figure 2). The model was in one dimension and the diffusion equation was not solved simultaneously with the thermal conduction equation due to the difficulties encountered in predicting the experimental data. In order to predict the location of the phase transitions correctly an artificial temperature gradient was imposed as a boundary condition and phasic diffusion coefficients were evaluated to match the data. However, the imposed temperature profile resulted in an unphysical heat flux that is not consistent with the operator-declared linear heating rate. The use of reported enthalpies of solution and the simplified phase diagram have inconsistencies. Karahan (2009) repeated aspects of earlier work by Kim et al. (2004) and Kim et al. (2006), aiming to develop an improved capability but used the same artificial temperature boundary condition in their validation of FEAST metallic fuel code (Karahan, 2009).
Our model, as we will summarize in the next section and published previously (Galloway et al., 2015) is similar to the models from Kim et al. (2004), Kim et al. (2006), and Ishida et al. (1993), but several significant differences are listed here. Our model corrects an inconsistency between the enthalpies of solution and the solubility limit curves of the phase diagram. It also adds an artificial diffusion term when in the 2-phase regime that stabilizes the standard Galerkin FE method used by Bison (Williamson et al., 2012). Another improvement made is in the formulation of the zirconium flux. The Soret diffusion term used in the previous metallic codes includes a zirconium concentration multiplied by the sum of heat of reaction and enthalpies of solution. To ensure a physical zirconium mole fraction, in the presence of a fixed plutonium mole fraction, it is necessary that the zirconium flux tend to 0 as the concentration of zirconium tends to either limiting value (0 or 1, less the plutonium mole fraction). This is true for the lower limit (0), but not the upper limit (1). To address this issue we introduced an additional factor into the Soret term coefficient. With these new modifications we first reanalyze the data taken from rod T179 (Kim et al., 2004; Kim et al., 2006) of EBR-II to revaluate the previously recommended diffusion coefficients. This is important in two aspects. First, we have an experimental program measuring diffusion coefficients from diffusion-coupled experiments. The predictions reported in this manual may be validated with the diffusion-coupled data to provide a better predictive capability. Secondly, the use of a better-validated coupled solution will provide greater confidence in the design of advanced fuels. Therefore, we believe the formulation and framework presented in this manual are credible and represent a significant improvement relative to previous work. The model we presented below is primarily sourced from the paper by Galloway et al. (2015).
While the constituent redistribution model we discussed below is given for U-Pu-Zr fuels, it is equally applicable to U-Zr fuels, albeit with different material properties. This is because of the EBR data which showed no significant redistribution of Pu in ternary U-Pu- Zr fuels. Thus, the model for U-Pu-Zr fuels assumes Zr moves in U only while Pu is not mobile in U or Zr. Therefore, the mathematical model and numerical implementation of it are valid for U-Zr fuels. The only difference between these two fuel types is the use of different phase diagrams and property models. Thermal conductivity of fresh U-Zr is already described in the property description for ThermalUPuZr for both U-Zr and U-Pu-Zr fuels.
Constituent Redistribution Model
In this section we present a model (Galloway et al., 2015) of constituent migration in U -Pu-Zr fuel. It is a refinement of an earlier model Carlson (2009) that was based on the model developed by Hofman et al. (1997) for U-Zr nuclear fuel and later used in Kim et al. (2004) and Kim et al. (2006) in a pseudo-binary study of U-Pu-Zr fuel. Similar models are considered in Karahan (2009) and Ishida et al. (1993). These models adapt the original work of Shewmon, P. C. (1958) in the approach to 2-phase regions.
Binary constituent redistribution model
We begin by considering a binary substitutional alloy, U-Zr for definiteness. Let be the density of lattice sites, which for simplicity we will assume is independent of composition and temperature, and let , , and be the atomic fractions of zirconium and uranium. In the presence of thermal gradients, the Zr atom flux in a single-phase region is, (1) where is the interdiffusion coefficient of Zr in the phase. The term involving is the so-called Soret effect or thermodiffusion. The coefficient may be positive or negative. This term contributes an advective component to the Zr flux directed toward lower temperatures when positive and higher temperatures when negative. To ensure physical compositions, i.e. , the flux must tend to 0 as tends to either 0 or 1. This leads us to take as a leading order approximation, (2) where is the heat of transport of Zr in the phase and is the gas constant. In the dilute limit , we recover the usual form used in previous models. The additional factor ensures that . Thus in a single- phase region the Zr flux is: (3) where and are coefficients associated with the phase. It is important to recognize that the usual form of proportional to alone is not a fundamental law, but is itself merely a leading order approximation that is only valid for small . This can be seen in two ways. First, as just noted, it gives a non-zero flux when that leads to unphysical solutions with . Second, and its approximations should exhibit a certain symmetry. The problem could just as well be posed in terms of with the U flux given by an expression analogous to Equation Eq. 1. On the other hand . Both should lead to the same flux, and reconciling the two leads to the symmetry. Our approximation Equation Eq. 2 exhibits this symmetry, and the necessary asymptotic behavior, while the usual form does not.
Next consider the flux in a 2-phase region very near a solubility curve defined by . For suppose the alloy is single- phase, but for a second Zr-rich precipitant phase appears. Following Shewmon, P. C. (1958) we will assume that a local equilibrium between the two phases is maintained at each point in a temperature gradient through rapid adjustment of the phase fractions via local diffusion processes. This means the composition of the major continuous phase is fixed, and as a consequence, (4)
Assuming that the flux of Zr (or counter flux of U) occurs only through the continuous phase, we then have, (5) where and are the coefficients associated with the continuous phase. For simplicity we consider solubility curves satisfying, (6) where , , and are constant model parameters. Note that this equation uniquely determines the solubility curve. With this choice Equation Eq. 5 becomes, (7) This approach mimics Shewmon, P. C. (1958) who took, in the dilute limit , (8) The models developed in Karahan (2009), Kim et al. (2004), and Kim et al. (2006) also adopted this approach, but the expressions they use for are inconsistent with their phase diagrams, and have the wrong sign in several cases. If one desires to use arbitrary solubility curves, then equation Equation Eq. 5 must be used for the flux.
The mirror situation where the alloy is single phase for and two-phase for is precisely the same. The Zr flux in the two- phase region very near the solubility curve is also given by Equation Eq. 7 when is of the form Equation Eq. 6, with the coefficients and associated with the continuous phase.
Constituent migration model for U-Pu-Zr
For U-Pu-Zr fuel it has been argued that a pseudo-binary treatment, in which the Pu fraction is assumed fixed, is justified both theoretically and on the basis of experimental data that show the Pu is largely immobile (Kim et al., 2004; Kim et al., 2006). Thus we consider a ternary U-Pu-Zr alloy where the Pu fraction is a fixed, spatially uniform value . As in a true binary alloy, the remaining U and Zr constituents flow counter to each other, and if we set we may apply the preceding binary model to the relative Zr atom fraction . Expressing the result in terms of the original ternary atom fraction we obtain, (9) for the Zr flux in a single-phase region, where the coefficients and are associated with the phase, and, (10) for the flux in a 2-phase region very near the solubility curve , where the coefficients and are those associated with the continuous phase. The solubility curves are chosen to satisfy, (11) for some choice of parameters , , and , which has the solution: (12)
We use the simple pseudo-binary U-19Pu-Zr phase diagram shown in Figure 2 which is based on the phase diagrams used in Karahan (2009), Kim et al. (2004), and Kim et al. (2006). There are two 2-phase regions, and , depending on the temperature range. The nominal transition temperatures are and , but these two values were varied in the simulations presented in this manual. The parameters defining the solubility curves are given in Table 1.
|Phase/Point||X (solubility intersection)||T ( C) (upper transition)||(kJ/mol)|
The 2-phase flux Equation Eq. 10 only applies very near a solubility curve and supposes that all the flux occurs through the major continuous phase, ignoring the precipitant phase. This is a reasonable assumption near a solubility limit, but we need an expression for the flux that spans the entire 2-phase region from the Zr-poor phase solubility limit to the Zr-rich phase solubility limit. For this we use a simple weighted average of the two fluxes that apply near the solubility limits. For the region we take (13) The weight factor is set equal to the phase fraction . The flux in the region is defined analogously. Some computational experiments were done using the more sophisticated weighting for parameters and , however in the end we opted for the simple linear weighting (, ).
The preceding pseudo-binary model for U-19Pu-10Zr fuel was implemented into Bison with two essential numerical modifications: addition of an artificial diffusion term in the 2-phase region to stabilize the algorithm, and smoothing of the model coefficients near phase diagram curves. The full model with these modifications is described in ZirconiumDiffusion.
There are many inputs needed for the fuel performance code Bison and all are not listed here. However a subset of inputs was determined to be either important for the simulations and worth mentioning, or newly added inputs for the constituent redistribution kernel and are given here.
Linear power for a specific height for single level 2-D simulations,
Rod average linear power scaled with axial power profile for full length 2D-RZ simulations,
Phase transition temperatures,
Alpha-delta to beta-gamma transition temperature quoted to be 595 C,
Beta-gamma to gamma transition temperature 650 C,
Rod edge convective and axial t-infinity boundary condition for full length 2D-RZ simulations,
Fuel surface temperature boundary condition for 2-D simulations,
Thermal conductivity and associated porosity modifiers,
Phase-dependent (alpha, delta, beta and gamma) diffusion coefficients and heats of transport, with values and units given in Table 2, based on values from Ishida et al. (1993), Kim et al. (2004), and Kim et al. (2006).
A sensitivity study was performed on these modeling parameters, with the exception of thermal conductivity and heats of transport, to assess the importance of each parameter. It was found that the parameters that control the radial location of phase transformation, those that directly affect temperature (system power, phase transition temperatures and the temperature boundary conditions) have the most significant effect, whereas parameters that deal with the rate of migration (diffusion coefficients) have a much less significant impact on the location of phase transitions. Not all cases are shown here, however two representative cases are shown that illustrate this impact. In the two cases shown, Figure 4 and Figure 3, the alpha phase diffusion coefficient and power, respectively, both were adjusted by +/-20% in order to see the impact on the simulation. Clearly observed is a minimal impact by the diffusion coefficient adjustment but a significant impact in the power adjustment, greatly adjusting the Zirconium profile shape depending on higher or lower powers. The same trends as Figure 4 were observed for all phase diffusion coefficients, while the same trends as Figure 3 were observed for transition temperatures adjustments and boundary condition adjustments.
The redistribution relies on two material models, PhaseUPuZr and ZrDiffusivityUPuZr, to calculate the necessary parameters for diffusion, and a single kernel, ZirconiumDiffusion, to actually redistribute the zirconium within the mesh.
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