Study on the Impact of Thermal Agitation on Doppler Coefficient in Epithermal Range for Gd-Bearing Fuel
The impact of thermal agitation on Doppler coefficient for Gd-bearing fuel was analyzed. It was found through the analysis that the impact increases when a small amount of Gd2O3 is added to pure UO2 fuel although the impact decreases for a large amount of Gd2O3. This tendency was discussed with the usage of simplified expression for the difference of Doppler coefficient. The simplified expression was used to consider the tendency, and it was revealed that the tendency mainly comes from the rapid decrement of multiplication factor and the relatively slow decrement of the magnitude of sensitivity coefficient of U-238 capture cross section at low Gd2O3 concentration. Similar tendency which shows a maximum impact on Doppler coefficient at interior concentration is expected for other UO2 fuel with a slight content of strong absorber. This indicates that Doppler coefficient of UO2 fuel system with low content of strong absorber should be analyzed carefully by considering thermal agitation in epithermal range.
Scattering kernel used in epithermal range is usually treated as the asymptotic model in solving the slowing down equation, which does not take thermal agitation into account. If the targets are scatterings of light isotopes, the asymptotic model is appropriate since the change of scattering kernel by thermal agitation is trivial. But for heavy isotopes, the asymptotic model is not appropriate for scattering kernel especially in epithermal range  where Doppler effect is remarkable in thermal reactor. Therefore, the impact of thermal agitation on Doppler coefficient has been investigated by many researchers [2–5]. And a simplified treatment of the exact scattering model for deterministic slowing down equation was also developed to reduce the huge computational cost. This simplified treatment called Upscattering Approximation (UA) method  was applied to GROUPR module of the NJOY  system and was verified through the comparison of impact on Doppler coefficient with other reported calculation results .
Many evaluations were performed by many researchers but the target composition to evaluate Doppler coefficient is mainly UO2 without any poison. As easily expected, the addition of poison such as Gd2O3 to UO2 fuel causes the change in multiplication factor and also causes the change in Doppler coefficient by the increment of capture rate of the poison. Thus, it is expected that the impact of thermal agitation on Doppler coefficient varies as a function of poison content in UO2 fuel.
The purpose of this paper is to evaluate the impact of thermal agitation on the Doppler coefficient for Gd-bearing fuel and to reveal the mechanism of the impact of thermal agitation on Doppler coefficient in Gd-bearing fuel. Section 2 deals with the evaluation of the impact for Gd-bearing fuel for various Gd2O3 concentrations. Section 3 shows detailed calculation results to describe the dependence of the impact on Gd2O3 concentration. Discussions and conclusions are summarized in Sections 4 and 5, respectively.
2. Calculation of Impact of Thermal Agitation on Doppler Coefficient
The calculation procedure and the calculation results are described in this chapter.
2.1. Calculation Procedure
The calculation geometry is modeled from conventional PWR cell for 4.8 wt% enriched UO2 fuel with Gd2O3 as shown in Figure 1 and Table 1. Fuel temperature is changed from 600 to 900 K and cladding and coolant temperature are unchanged to evaluate Doppler coefficient.
The Doppler reactivity is calculated from two multiplication factors of different fuel temperatures aswhere and are multiplication factors calculated at low temperature and high temperature, respectively, and shows the difference between low and high temperatures. The impact of Doppler coefficient is obtained as the relative difference of Doppler coefficient between asymptotic model and exact scattering model.
2.2. Calculation Results
The calculation is performed by continuous energy Monte Carlo code MVP which can handle thermal agitation in epithermal range  with JENDL-4.0 library . The impact calculated by 109 histories is summarized in Table 2. Table 2 shows that the impact increases by adding small amount of Gd2O3 to pure UO2 fuel and the impact decreases by adding large amount of Gd2O3 (10~20 wt%). The smaller impact at high Gd2O3 concentration can be explained by smaller amount of U-238 which is the main isotope to cause Doppler reactivity. On the other hand, the impact at low Gd2O3 concentration may be enhanced compared to that at pure UO2 case. Further evaluation is necessary to reveal the mechanism of the enhanced impact at low Gd2O3 concentration, but detailed evaluation was performed by deterministic method to reveal the mechanism because the statistical error by stochastic method will bring the ambiguity to consider the mechanism.
3. Detailed Evaluation of Impact
The calculation process and the calculation results for detailed evaluation of the impact are described in this chapter.
3.1. Calculation Process
Calculation geometry and calculation conditions are the same as previous chapter. Deterministic evaluations of Doppler coefficient with considering thermal agitation were performed by the following process. The multiplication factor with considering thermal agitation is evaluated with the usage of sensitivity coefficients for the multiplication factor and the relative difference of cross sections between asymptotic model and exact model as expressed inwhere is a multiplication factor calculated by SRAC2006  based on asymptotic model and is the suffix of energy group and is the sensitivity coefficient for multiplication factor calculated by SRAC2006 and SAINT-II  system and is calculated as the relative difference of cross section between asymptotic model and UA method evaluated by NJOY with ENDF-B/VII.0 . Here the exact model is replaced as UA method which has developed and verified to evaluate the cross section with considering thermal agitation for various fuel compositions . The sensitivity coefficient is defined as the ratio of relative change of multiplication factor to the relative change of cross section. Thereby the sensitivity coefficient is expressed aswhere is group-averaged cross section of energy group . The impact of thermal agitation on Doppler coefficient was evaluated by using the multiplication factors of conventional calculations without considering thermal agitation and those calculated by (2) for considering thermal agitation.
The technique shown in this section is useful to obtain breakdown in isotopes, reaction types, and energy group because each of the parameters is evaluated separately.
3.2. Calculation Results
The impact on Doppler coefficient is summarized in Table 3. Table 3 shows that the impact on Doppler coefficient is small at high Gd2O3 concentration and the impact is enhanced at low Gd2O3 concentration, as also shown in Table 2 where the results were obtained by Monte Carlo calculations. There are some differences in magnitude between Tables 2 and 3, but the tendency of the impact is the same. Therefore further considerations were carried out to reveal the mechanism of the tendency especially to reveal the factor to cause the enhanced impact at low Gd2O3 concentration.
Doppler coefficients calculated by conventional asymptotic kernel model and UA method are shown in Figure 2. Figure 2 shows that the magnitude of Doppler coefficient becomes large monotonically and linearly according to the increment of Gd2O3 concentration.
Table 4 presents the nuclide and reaction breakdown of the impact on Doppler coefficient. Another breakdown by energy range is summarized in Table 5. As shown in Tables 4 and 5, U-238 capture reaction is the dominant cause of the impact of thermal agitation on Doppler coefficient, and the impact is remarkable at the energy around 21, 37, and 66 eV as already reported in the literature [2–5]. In addition to this, the impact of thermal agitation on Doppler coefficient depends on Gd2O3 concentration and the impact increases at low Gd2O3 concentration compared to pure UO2 fuel as shown in Tables 4 and 5.
The impact of thermal agitation on Doppler coefficient increases up to about 12% when Gd2O3 concentration is around 0.2 wt% and 2 wt% and the impact decreases according to the increment of Gd2O3 concentration although Gd isotopes do not have the impact of thermal agitation on Doppler coefficient as shown in Table 4.
In order to discuss this tendency, a simplified expression of the difference in Doppler coefficient is derived in Section 4.1 for detailed consideration of the mechanism. Section 4.2 shows the numerical results by using the simplified expression and the discussion of the mechanism especially to cause the enhanced impact at low Gd2O3 concentration.
4.1. Derivation of Simplified Expression
In this section, the impact on Doppler coefficient () caused by thermal agitation is expressed in simple expression for easy consideration to reveal the mechanism.
The impact of Doppler coefficient by thermal agitation at energy group only is expressed aswhere is a multiplication factor which treats thermal agitation effect at energy group and calculated by as easily expected from (2). Therefore, (4) can be transformed asThe change of by the change of temperature is not remarkable compared to the other parameters as shown in later sections (Figure 4). Thus, the sensitivity coefficient can be approximated as Then, (5) can be transformed as
For further simplification of (7), following approximations are considered.
The capture cross section of Gd nuclide is remarkably large at thermal range, but the magnitude is not so large compared to that of U-238 at the energy group where the agitation effect is sensitive to Doppler coefficient. Thus, of U-238 does not strongly depend on Gd2O3 concentration at the energy group and the difference of is nearly independent of Gd2O3 concentration as shown in Figure 3. Therefore, new parameter can be set as independent parameter of Gd2O3 concentration asBy using (8), (7) is transformed asPractically, is quite small compared to in the evaluation of the difference in Doppler coefficient. Therefore, (9) can be simplified by neglecting to obtainEquation (10) shows that the difference in Doppler coefficient as a function of Gd2O3 concentration can be simply discussed as the product of the sensitivity coefficient and the inverse of multiplication factor.
4.2. Impact on Doppler Coefficient
In this section, the impact of thermal agitation on Doppler coefficient in epithermal range is discussed by using (10). The sensitivity coefficient of U-238 capture cross section to multiplication factor is shown in Figure 4. Figure 4 shows that the magnitude of decreases according to the increment of Gd2O3 concentration since the ratio of the capture reaction of U-238 to total capture rate becomes small according to the increment of Gd2O3 concentration. Figure 5 shows the multiplication factor and the inverse of the multiplication factor as a function of Gd2O3 concentration. Figure 5 shows that the inverse of multiplication factor increases monotonically by adding Gd2O3 because of its huge capture cross section. Figure 6 shows the difference in Doppler coefficient evaluated by exact solution and the simplified expression expressed as (10). Two evaluations shown in Figure 6 are almost the same, which shows the correctness of the simplified expression. Particular behavior of the impact as a function of Gd2O3 concentration can be interpreted by using the above mentioned results as follows.
As already described, there is a maximum impact of thermal agitation on Doppler coefficient at low Gd2O3 concentration and this tendency comes from the rapid increase of and slow increase of the magnitude of Doppler coefficient at low Gd2O3 concentration as shown in Figures 2 and 5. In other words, the sensitivity coefficient in Figure 4 shows the decrement in magnitude according to the increment of Gd2O3 concentration because of the increment of Gd2O3 capture rate to total capture rate; on the other hand, the decrement of multiplication factor is remarkable especially at low Gd2O3 concentration as shown in Figure 5. And at high Gd2O3 concentration, the value of becomes small and is mainly caused by the fact that the magnitude of Doppler coefficient becomes huge although is almost constant as shown in Figures 2 and 6. This is the reason why there is a maximum point of the impact on Doppler coefficient for Gd-bearing fuel.
Similar tendency which shows a maximum impact is expected for the fuel with the slight content of strong absorber. Therefore, Doppler coefficient of the system which contains low content of strong absorber should be analyzed carefully with the consideration of thermal agitation.
The impact of thermal agitation on Doppler coefficient of Gd-bearing fuel was analyzed. The results show that the impact increases by adding a small amount of Gd2O3 (~2 wt%) to UO2 fuel, although the impact decreases by adding a large amount of Gd2O3 (~10 wt%) to UO2 fuel. This tendency was analyzed with the usage of simplified expression of the difference in Doppler coefficient. The difference in Doppler coefficient varies as a function of Gd2O3 concentration, and this is mainly caused by the two factors: sensitivity coefficient of U-238 capture cross section to multiplication factor and multiplication factor. The magnitude of both factors decreases according to the increment of Gd2O3 concentration, but the decrement rate of multiplication factor is remarkable compared to that of the sensitivity coefficient at low Gd2O3 concentration, where the self-shielding of Gd isotopes is not so important and the content of U-238 is roughly the same to the fuel without Gd2O3. The decrement rates of both factors are almost the same at high Gd2O3 concentration, which brings almost the same difference in Doppler coefficient between with and without considering thermal agitation. On the other hand, the magnitude of Doppler coefficient increases monotonically and linearly according to the increment of Gd2O3 mainly caused by the decrement of multiplication factor. Therefore, there is a maximum point of the relative difference in Doppler coefficient caused by thermal agitation.
Similar tendency is expected for other strong-absorber-bearing fuel. Thus, it should be noted that the Doppler coefficient of the fuel with a slight amount of strong absorber should be analyzed carefully with the consideration of thermal agitation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank Dr. Nagaya for his kind support for the evaluation by continuous energy Monte Carlo code MVP.
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