Keywords

1 Introduction

MOX fuel contains plutonium oxide grains (Pu spots) within the pellet. Pu spots burn locally and cause a problem with local power distribution in the fuel. Currently, the size of Pu spots is measured and controlled to ensure that local power distribution does not occur. The previous studies showed that the accuracy of criticality predictions can be improved by considering the heterogeneity of Pu spots [1]. The analysis of considering the heterogeneity of Pu spots is performed by Monte Carlo calculations to consider the exact core geometry and self-shielding effects. However, the results of Monte Carlo calculations are obtained with statistical errors, thus the small reactivity worth is difficult to be analyzed.

The deterministic calculation is suitable for the analysis of small reactivity worth. One deterministic approach to treating the self-shielding effect is introducing the reduced order-modeling [3,4,5,6]. Another approach is introducing the heterogeneous model by using a method devised by R. Sanchez [7,8,9,10]. In the previous study, the heterogeneous model was introduced into the MOC code OpenMOC [11]. Although there are numerous sizes of Pu spots in MOX fuel, the heterogeneous model in the modified OpenMOC could consider only a single grain size. Therefore, in this study, the modified OpenMOC code was additionally modified to consider the grain size distribution of Pu spots, and the effect of grain size distribution on criticality was investigated.

2 Calculation Method and Condition

2.1 Calculation Flow

The OpenMOC code was modified so that numerous grain sizes can be calculated to treat the grain size distribution. The calculation flow of the modified OpenMOC code is shown in Fig. 1. In Fig. 1, the input module was modified to read the numerous sizes of the grain, and the angular flux calculation function was modified to calculate the flux in each grain size. At the beginning of the calculation, the escape probabilities and transmission probabilities required to introduce the heterogeneous model are calculated.

Fig. 1.
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Calculation flow of modified code

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2.2 Calculation Condition

This section shows the calculation condition of the modified OpenMOC and additionally modified OpenMOC. The effective cross section is obtained by the collision probability method (PIJ) of SRAC2006 [12]. 107 energy group structure are used. The fast neutron energy region is divided into 61 groups, and the thermal neutron energy region is divided into 46 groups. The modified OpenMOC calculations was performed in the pin-cell model. The geometry of pin-cell model is shown in Fig. 2. The fuel temperature is 900K and moderator temperature is 600K. JENDL-4.0 is used [13]. U-235 enrichment of matrix is 0.2wt%. Pu enrichment of each region is shown in Table 1. Isotopic ratio of Pu is shown in Table 2. The calculations of the modified OpenMOC are performed under the conditions shown in Table 3.

Fig. 2.
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Geometry of Pin-cell model

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Table 1. Pu enrichment of each region
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Table 2. Isotopic Ratio of Pu (wt%)
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Table 3. Calculation Conditions on the modified OpenMOC
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In this study, the calculations were performed for both the homogeneous model, in which uranium and plutonium are homogeneously mixed, and the heterogeneous model, in which Pu spots are included. The effect of Pu spots is expressed as a heterogeneity reactivity effect. The heterogeneity reactivity effect is defined as the reactivity which is obtained by changing from the homogeneous model to heterogeneous model. The effect is defined as:

$$ heterogeneity\,reactivity\,effect = \frac{1}{k(homogenenous\,model)} = \frac{1}{{k^{\prime}(homogenenous\,model)}} $$
(1)

2.3 Estimation of Grain Size Distribution

This section shows the conditions related to Pu spots used in the calculations. In this study, the conditions related to Pu spots are determined based on previously investigated grain size distribution [14]. The volume fraction and particle size distribution of Pu spots vary with plutonium enrichment, thus three analyses (analyses for high, middle, and low enrichment) were performed. In Table 4, the volume fractions of Pu spots at each enrichment are shown. In Tables 5, 6, and 7, the grain size distribution is expressed in terms of four grain sizes and the volume fraction of Pu spots at each grain size are shown.

Table 4. Volume fraction of Pu spots at each enrichment
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Table 5. Volume fraction of each grain size in high enrichment fuel
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Table 6. Volume fraction of each grain size in middle enrichment fuel
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Table 7. Volume fraction of each grain size in low enrichment fuel
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In addition, calculations that simulate single grain size are also carried out to investigate the effect of considering the grain size distribution. In the high, middle, and low enrichment fuels, most of Pu spots are 125, 75, and 65 µm in diameter respectively. Therefore, these grain sizes are used in the calculation of the single grain size. The single grain sizes used in the calculations are shown in Table 8.

Table 8. The single grain sizes used in the calculations at each enrichment
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3 Calculation Result

The calculation results of single grain size and four grain sizes are shown in Fig. 3.

In all conditions, the results of the heterogeneity reactivity effect are negative because the ratio of Pu-239 is high and the ratio of Pu-240 is low in the Pu isotope ratio. Pu 239 has a fission resonance at 0.3 eV and the fission reaction is reduced by the self-shielding effect of Pu spots. The reason for the negative heterogeneity reactivity effect is the high proportion of Pu-239. On the other hand, the isotope ratio with the high proportion of Pu-240 is used, the heterogeneity reactivity effect shifts to the positive side. The reason is that Pu-240 has a neutron absorption resonance at 1.0eV and neutrons escaping this resonance can cause fission.

Fig. 3.
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Comparison of calculation results between the case of single grain size and four grain sizes

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At all enrichments, the heterogeneity reactivity effect of the calculation of single grain size is larger than four grain sizes. Because the larger the grain size, the larger the self-shielding effect. In this study, the calculations of single grain size are carried out by replacing smaller grain sizes with the largest one. Therefore, the calculations of single grain size have a large proportion of Pu spots with a larger grain size than the calculations of four grain sizes. As a result, the heterogeneity reactivity effect is larger in the calculation of single grain size.

At the high enrichment fuel, the difference between the calculation results of single grain size and four grain sizes was the largest. The difference is 0.035%k/kk’. When calculating a slight difference in reactivity effect by Monte Carlo calculation, a huge calculation cost is required to suppress statistical error. This study makes it possible to carry out calculations that consider the grain size distribution of Pu spots in MOX fuel, using a deterministic calculation method that does not have statistical errors.

4 Conclusions

The heterogeneous model introduced in previous studies could only consider a single grain size. In practice, there is a distribution of Pu spots sizes in MOX fuel. Therefore, the purpose of this study is to investigate the effect of the Pu spots size distribution on criticality. Initially, the modified OpenMOC was additionally modified to consider the grain size distribution, thus the effect of the grain size distribution on criticality can be evaluated without statistical errors. At high enrichment fuel, the difference between the calculation results of single grain size and four grain sizes was found to be 0.035Δk/kk’. The additionally modified OpenMOC makes it possible to study the effect of considering the grain size distribution of Pu spots in MOX fuel. Further improvement of the criticality prediction accuracy in MOX fuel is expected by the additionally modified OpenMOC.