Comparison of absorption XS and reaction rate for 238U in Group 27 (Mosteller benchmark 5 wt% UO2 pin-cell). Comparison of absorption XS and reaction rate for 239Pu in group 29 (burned UO2 pin cell).
Introduction
Second, the equivalence theory cannot address the spatial distribution of the effective XS within the fuel pellet. The new method has been verified on several light water reactor (LWR) problems and shows significant improvements in the accuracy of the effective XS and multiplication factor.
Overestimation of 238 U Cross Sections
- Equivalence Theory
- Numerical Test with Equivalence Theory
- Pointwise Energy Approach
- Numerical Test with Pointwise Energy Approach
- Numerical Test without Resonance Scattering Cross Section
- Contemporary Spatially Dependent Self-shielding Method
One of them is inconsistency between the derivation of the effective multi-group XS and its use. The PW 0-D equations were solved without considering distributions of the scattering sources and the fluxes in the fuel subregions.
Pin-based Pointwise Energy Slowing-down Method
- General Derivation
- Collision Probability Calculation: First step – Isolated Fuel Pellet
- Collision Probability Calculation: Second step – Fuel Pin in Lattice
- Resonance Upscattering Treatment
- Techniques to Achieve High Performance
- Calculation Flow
When creating the ˆPijiso table, only geometric information (ie, the radii of the fuel pellet sub-regions) is needed because the grids of total XSs are fixed. 9 the shading correction factor is expressed as a function of the total XS of the fuel pellet.
Numerical Result
Sensitivity Test for Calculation Option in PSM
There are two parameters to test, namely the number of energy points in the PW XS libraries, the number of sub-regions of the fuel pellet. Some techniques (i.e. the efficient scattering source integration algorithm, the nuclide grouping) introduced in section 3.5 and the ˆPijiso table introduced in section 3.2 were tested. The number of sub-areas of the pellet is 5. Due to the spatial self-protection effect in the fuel pellet, the material compositions in the pellet are not uniform.
The ˆPijiso table is generated as a function of the total XS of the fuel pellet. The pin-base cell problem described in Section 2.2 was solved by PSM with the different number of fuel pellet sub-regions. In fact, the multi-group calculation does not require many sub-regions of the fuel pellet.
The maximum value of the macroscopic multigroup XS is approx. 0.6 cm-1 in the resonance energy range.
Sensitivity Test for Ratio of Fuel Diameter to Pin-pitch
If the multigroup XS is constant for all subregions of the pellet, only one radial subregion gives a nearly convergent solution in terms of grid divisions. In fact, the energy limit of the multi-group XS has a great influence on the size of the multi-group XS. In order to reduce the total calculation time, a different number of flat source regions can be used for multi-group calculation and resonant self-shielding calculation (eg 3 radial sub-regions for multi-group calculation and 15 radial sub-regions for PSM).
The ratio of the realistic design is between 0.6 and 0.7, where k-inf has the highest value. For the pin-cell design of the cold zero-power critical experiment, the ratio is greater than 0.7 because the moderator density is high. The ratio can be less than 0.6 for the pin cell located at the edge of the nucleus.
The dilution condition can be changed depending on the following factors: moderator density, fuel pellet radius, amount of propellant in the fuel, etc.
Base Pin-cell Problem
On the other hand, PSM shows good agreement in absorption XS with the MCNP6 result. The resonance treatment of the unresolved resonance region is not important for the LWR application in terms of reactivity and current distribution, therefore PSM can still calculate accurate solutions. The relative radius is defined as the ratio of the outer radius of annular subregions to that of the fuel pellet.
The SDDM also underestimates the absorption XS in the inner region of the fuel in group 21. The underestimated absorption XS in the inner region of the fuel leads to a ∼1.5% difference in the regional average XSs for group 21 in Fig. In the comparison of the average absorption XSs, the SDDM tends to overestimate the XSs in groups 25 and 26.
As discussed in Section 2.6, SDDM is limited in modeling the spatially distributed resonance XS because the distribution of the resonance scattering source is not properly considered during the resonance treatment.
Pin-cell with Uniform Material Composition and Temperature Profile
The differences in the reaction rates with EQ are very significant along the radial direction. The difference in reaction rate with MCXS and PSM is of the order of 10 pcm in the inner region. Due to the overestimated XSs, SDDM underestimates the absorption reaction rate in the inner regions.
In the resonance energy range, there are significant differences in the response speed of 235U and 238U with EQ, DRI and SDDM. In Group 29, the production rate contributes more to the difference in the reaction rate than the absorption rate. EQ, DRI and SDDM show differences of 50 ~ 80 pcm, while PSM shows a difference of 75 pcm in response speed.
In PSM results, there is no noticeable difference in absorption XSs and reaction rates.
Pin-cell with Non-uniform Material Composition and Uniform Temperature Profile
When comparing reaction rates relative to nuclides, different actinoids and fission products cause a difference in reaction rates. Different fission products cause significant differences in reaction rates for EQ, DRI and SDDM. 150Sm causes differences in reaction rates of 90 pcm for the three methods.
There are significant differences in the reaction rate of 239Pu with EQ, DRI and SDDM. There are 10 ~ 30 pcm differences in the absorption reaction rates in the outer region from MCXS, PSM and PSM-CPM. In group 27, more than 90 pcm differences occurred in the reaction rates with EQ, DRI and SDDM.
EQ, DRI, SDDM show significant differences in XS and reaction rates of actinides and fission products.
Pin-cell with Non-uniform Material Composition and Temperature Profile
SNU Non-uniform Temperature Pin-cell Benchmark
PSM and PSM-CPM have less than 30 pcm differences in the reaction rates in the groups. The differences in the XSs are less than 1% in all subregions of the grain. PSM and PSM-CPM show relatively accurate results in the reaction rates of the resonance energy groups.
Finally, the response rates of the resonance are overestimated, therefore the multi-group XSs are overestimated in the inner regions.
VERA 17x17 Fuel Assembly Problem
The RMS difference and the maximum difference in the pin current distribution are approximately 0.10% and 0.25%, respectively. The two types of FAs are used in the 2x2 multi-assembly problem, as shown in Figure. The temperature and density of the moderator in the FA-A are 580 K and 0.7067 g/cc, respectively.
Although the quite different fuel pins are used in the problem, PSMs still calculate the exact k-inf and leg current distribution. With PSM and PSM-CPM, the differences in reaction rates for each nuclide are less than 53 pcm. It is difficult to say how much of the difference in k-inf is due to the difference in the depletion libraries.
In order to achieve high accuracy in the depletion calculation, it is important to calculate the exact reaction rates of each nuclide.
Test for Computing Time
This is why PSM needs about half the computation time in fixed source computation (MOC FSP in Table 22) compared to EQ. Of course, the time spent in the fixed-source MOC calculation depends on the MOC beam conditions. The same problem was solved with the different number of radial sub-regions in the fuel pellet.
84 shows the calculation time as a function of the number of subareas in the fuel pellet. As the number of subregions increases, the elapsed time used in the XS generation increases significantly with PSM-CPM. On the other hand, the elapsed time in the XS generation with PSM is not much compared to the total simulation time.
PSM-CPM takes a very long time to calculate multi-cluster XSs with the large number of sub-regions in the fuel pellet.
Discussion
First, PSMs eliminate errors from the resonance scattering source, which cannot be addressed in existing equivalence theories. As discussed in Section II, the resonance scattering source causes significant error in the effective XSs. The derivation of the effective XS with the multi-term rational equation and the distribution of the scattering source in the fuel pellet causes error when the fuel has a significant XS resonance distribution.
Third, PSMs can reduce the number of fixed-source MOC transport calculations in the resonance treatment. The lattice physics codes based on equivalence theory use the multi-group RI library generated in the homogeneous system using the NJOY code. On the other hand, PSMs use the multi-group XS library and the pointwise energy XS library.
It should be noted that the STREAM code uses equivalence theory to calculate the effective multigroup XS of the structural materials such as the cladding.
Conclusions
먼저, 학부에서 대학원까지, 그리고 박사과정까지 지도해주신 이덕정 교수님께 감사의 말씀을 전하고 싶습니다. 제가 졸업하는 동안 여러분께서 저를 격려해 주시고, 감사해 주시고, 믿어 주신 사실은 저에게 큰 동기부여가 되었고, 여기까지 올 수 있었던 원동력이 되었습니다. 많이 혼났지만 지금 생각해보면 좋은 경험이자 더 발전할 수 있는 기회로 남아있습니다.
교수님의 전문분야인 비물리학 분야에서 연구를 계속할 수 있다는 것도 저에게 큰 축복이자 특권이었습니다. 박사님께 특별한 감사의 말씀을 전하고 싶습니다. 아르곤 연구소에서 반년 동안 인턴으로 연구할 수 있는 기회를 주신 이창호 선생님. 저에게 이런 좋은 기회와 지도를 주셔서 감사드립니다.
덕분에 많은 것을 배우고 여러모로 즐거웠습니다.
