Advanced PHWR Safety Technology: PHWR Challenging Issues for Safe Operation and LongTerm Sustainability
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Seung Yeol Yoo, Hyung Jin Shim, Chang Hyo Kim, "Monte Carlo FewGroup Constant Generation for CANDU 6 Core Analysis", Science and Technology of Nuclear Installations, vol. 2015, Article ID 284642, 11 pages, 2015. https://doi.org/10.1155/2015/284642
Monte Carlo FewGroup Constant Generation for CANDU 6 Core Analysis
Abstract
The current neutronics design methodology of CANDUPHWRs based on the twostep calculations requires determining not only homogenized twogroup constants for ordinary fuel bundle lattice cells by the WIMSAECL lattice cell code but also incremental twogroup constants arising from the penetration of control devices into the fuel bundle cells by a supercell analysis code like MULTICELL or DRAGON. As an alternative way to generate the twogroup constants necessary for the CANDUPHWR core analysis, this paper proposes utilizing a B_{1} theory augmented Monte Carlo (MC) fewgroup constant generation method (B_{1} MC method) which has been devised for the PWR fuel assembly analysis method. To examine the applicability of the B_{1} MC method for the CANDU 6 core analysis, the fuel bundle cell and supercell calculations are performed using it to obtain the twogroup constants. By showing that the twogroup constants from the B_{1} MC method agree well with those from WIMSAECL and that core neutronics calculations for hypothetical CANDU 6 cores by a deterministic diffusion theory code SCAN with B_{1} MC method generated twogroup constants also agree well with whole core MC analyses, it is concluded that the B_{1} MC method is well qualified for both fuel bundle cell and supercell analyses.
1. Introduction
A B_{1} theory augmented Monte Carlo (MC) homogenized fewgroup constant generation method [1, 2] (B_{1} MC method hereafter) has been proposed as an alternative way to generate homogenized fewgroup constants of nuclear fuel systems like fuel pins or fuel assemblies or bundles by deterministic fuel assembly spectrum codes like CASMO [3], HELIOS [4], WIMSAECL [5], and so forth. The applicability of the B_{1} MC method to PWR core analyses has been demonstrated by showing that fewgroup constants from the method implemented in a Seoul National University (SNU) MC code, McCARD [6] lead to nodal core neutronics calculations in a good agreement with whole PWR core reference MC calculations [2]. The purpose of this paper is to demonstrate that the B_{1} MC method is also applicable to the neutronics analysis of CANDUPHWRs by showing that homogenized twogroup constants of fuel bundles of CANDU 6 from it can result in core neutronics calculations that agree very well with a reference CANDU 6 whole core analysis.
As described in detail in [2], the B_{1} MC method generates the homogenized fewgroup constants of fuel assemblies or fuel bundles in much the same way as its deterministic counterparts [3–5]. Like the latter, the former consists of conducting the infinite medium spectrum (IMS) calculations to determine IMSweighted homogenized multigroup reaction cross sections of fuel assemblies or fuel bundles, solving multigroup B_{1} equations to determine the critical spectrum (CS) and the critical buckling, and group condensing to obtain the fewgroup constants including fewgroup diffusion constants. Unlike the latter, however, the former utilizes continuous energy cross section data available in the evaluated nuclear data libraries and models the geometry of fuel assemblies or bundles exactly as they are instead of using builtin multigroup cross section libraries and approximate modelling of the complex geometry of the fuel assemblies or bundles. Because of these characteristics of treating the nuclear cross section and geometrical data input exactly, the B_{1} MC method may perform inherently more exact IMS calculations, which in turn makes subsequent CS calculations conducted on more precise multigroup B_{1} equations, than its deterministic counterparts.
The current neutronics design methodology of CANDUPHWRs is based on the twostep neutronics calculation method represented by lattice cell spectrum codes WIMSAECL [5] and DRAGON [7] and the twogroup diffusion theory core analysis code RFSP [8]. The homogenized twogroup constants are obtained by performing the two types of lattice cell computations: the standard unit lattice cell and the supercell calculations. The former is designed to obtain the homogenized twogroup constants of a unit lattice cell comprising a fuel bundle, coolant, pressure tube, and associated moderator as a function of reactor state variables, such as temperatures of fuel, moderator, and coolant and fuel depletion, and is done by the WIMSAECL code. The latter is designed to obtain the incremental cross sections of various control devices such as liquid zone controllers (LZCs), adjuster rods (ADJs), and mechanical control absorbers. The supercell means the standard cell penetrated horizontally or vertically by various control devices. The supercell computations used to be performed by the MULTICELL code [9]. They are currently done by a 3dimensional collision probability transport theory code, DRAGON, which is developed to tackle the strong heterogeneity posed by the supercell configuration more satisfactorily than the MULTICELL code.
As mentioned previously, the B_{1} MC method is inherently advantageous in handling the geometrical heterogeneity and evaluated cross section data without approximation. Therefore, it can serve as an alternative to the twogroup constant generation codes for CANDUPHWR neutronics design, WIMSAECL and DRAGON. To warrant this qualification, this paper will show how well the twogroup constants generated from McCARD [6] by the B_{1} MC method compare with those from WIMSAECL. Needless to mention, the comparison of the twogroup constants from the B_{1} MC and deterministic methods alone is not sufficient to guarantee the qualification of the B_{1} MC method or its applicability to neutronics design computations for CANDUPHWRs. It is prerequisite to show how well the twogroup constants from it can predict core neutronics design parameters including the effective multiplication factor and normalized channel power distribution of CANDUPHWRs. In order to do so, this paper will perform McCARD whole core transport calculations for three states of a reference CANDU 6 core with all control devices out, a uniform level of LZCs at 50% fill and all ADJs in. The McCARD whole core analyses for ’s and normalized channel power distributions of the three core states will be compared with the corresponding deterministic neutronics analysis by a SNU diffusion theory code SCAN [10] using the B_{1} MC methodgenerated twogroup constants as inputs. The qualification of the B_{1} MC method as the twogroup constant generator for twostep neutronics design of CANDUPHWRs is then demonstrated by showing that deterministic SCAN and the reference McCARD analyses agree well with one another.
Followed by this introduction, the B_{1} MC method for both the standard unit lattice cell and the supercell is briefly described in Section 2 to make this paper selfcontained. In Section 3, the three hypothetical CANDU 6 core analysis problems are specified against which the qualification of the B_{1} MC method as an alternative to the current CANDUPHWR fuel lattice spectrum codes is examined. The B_{1} MC method calculations and WIMSAECL calculations for the twogroup constants of the CANDU 6 lattice cells as well as the CANDU 6 core analyses by SCAN and McCARD are compared in Section 4.
2. The B_{1} Monte Carlo Method
A detailed description of the B_{1} MC method is available in [1, 2]. In order to make this paper selfcontained, it is briefly described here. The essential step of the B_{1} MC method involves infinite medium spectrum (IMS) calculations for angular flux by the MC neutron transport calculations, which are performed to determine finegroup cross sections of a nuclear fuel system like a fuel assembly or a fuel bundle defined as ( = scattering, absorption, fission) is the type IMSweighted group reaction cross section of the nuclear system. () is the IMSweighted th coefficient of Legendre expansion of group transfer scattering cross section. It must be noted that (2) for derives from an approximation that the energy dependence of the component of Legendre expansion of is proportional to component [1, 2]. Once the IMSweighted finegroup cross sections in (1) and (2) are obtained through the MC calculations, they are used to specify B_{1} equations:
Like its deterministic counterparts, the B_{1} MC method makes use of the solution to the B_{1} equations above, namely, the critical spectrum , the critical current spectrum , and the critical buckling , to determine the CSweighted fewgroup cross sections byand the fewgroup diffusion constants by
The B_{1} MC method has been implemented into the fewgroup generation module of the SNU MC code McCARD [6]. The qualification of the method as a twogroup constant generator for the standard unit lattice cell and the supercell will be examined in terms of the CANDU 6 core analysis problems described below.
3. Hypothetical CANDU 6 Core Analysis Problems
The hypothetical CANDU 6 reactor has the same geometry and components as the typical CANDU 6 reactor. As shown in Figure 1, it is composed of the reactor core, the D_{2}O reflector, and the stainless steel Calandria reactor vessel. The Calandria is a horizontal cylindrical vessel which envelopes 380 fuel channels comprising the core and contains heavy water moderator and reflector. Eighty out of the 380 fuel channels are depleted ones while the rest are fresh ones. Each fuel channel consists of 12 fuel bundles aligned horizontally inside the pressure tube. Depleted fuel channels contain 2 depleted fuel bundles each, which are positioned at the 3rd and the 4th sites of the 12 fuel bundle sites in the order from the front or end of the fuel channel while all the fresh fuel channels comprise the fresh fuel bundles. The depleted fuel channels are arranged bidirectionally in the center region of the core so that none of two adjacent depleted fuel channels are aligned in the same direction. All the fresh fuel channels are put outside the center region of depleted fuel channels. The core has 4400 fresh fuel bundles and 160 depleted fuel bundles loaded into 380 fuel channels. Figure 2 shows a cross sectional view of the core indicating the positions of the depleted fuel channels.
In order to validate the effectiveness of the B_{1} MC method for the incremental twogroup constant generation, three different states of the hypothetical CANDU such as (i) allthecontroldevicesfree core, (ii) a core with all the LZCs filled with 50% water, and (iii) a core with all the ADJs positioned inside the designed locations in the core. Figure 3 shows locations of 14 LZCs which are lumped into 3 types: type 1 representing 6 LZCs in regions 1, 4, 6, 8, 11, and 13, type 2 6 LZCs in 2, 5, 7, 9, 12, and 14, and type 3 the remaining 2 LZCs in regions 3 and 10. In the LZC core model, the water level of each LZC is set to 50%. Figure 4 shows the core configuration in which all the six types of ADJs are inserted.
The hypothetical cores are presumed to be at hot full power condition with fresh fuels, no xenon and no poisoning material in moderator. Regional temperatures are set to 960.2 K at fuel, 561.2 K at coolant and cladding, and 342.2 K at moderator and structure material. These problems call for determining the and the normalized channel power distribution.
4. Numerical Results and Discussions
The twostep deterministic solutions to the hypothetical CANDU 6 core analysis problems require specifying the twogroup constants for every 3dimensional node. The required twogroup constants are produced through the standard unit lattice cell and the supercell calculations by the fewgroup generation module of the McCARD code based on the B_{1} MC method. Figure 5 shows the standard CANDU 6 lattice cell which comprises a 37element fuel bundle, pressurized heavywater coolant in a pressure tube, and the associated unpressurized heavy water moderator. Two sets of twogroup constants are produced from the standard unit cell calculations with one set with the natural uranium fuel bundle and another with the depleted bundle. Table 1 shows a comparison of the twogroup constants from the McCARD calculations and those from the WIMSAECL Release 2–5d calculations for the natural uranium fuel bundle unit cell. Note that the B_{1} MC and deterministic WIMSAECL calculations produce very similar twogroup constants in magnitude. The thermal and fast diffusion constants from the two methods are very close to each other with much less than 1% relative differences. The twogroup reaction cross sections from the two methods are also very similar with about 1~2% relative differences. The largest difference between the two is observed in the thermal absorption cross section with the relative difference of about 2.2%. Table 2 shows similar comparison of the twogroup constants from the B_{1} MC and WIMSAECL calculations for the depleted uranium fuel bundle cell.
 
Group 1 (>0.625 10^{−06} MeV), group 2 (<0.625 10^{−06} MeV). ^{ 2}Rel. Diff: relative difference between McCARD and WIMSAECL. 
 
Group 1 (>0.625 10^{−06} MeV), group 2 (<0.625 10^{−06} MeV). ^{ 2}Rel. Diff: relative difference between McCARD and WIMSAECL. 
To complete the specification of the twogroup constants for all nodes of the CANDU 6 core problems, it is necessary to estimate the incremental cross sections representing the effects of the presence of the reactivity devices as well as their guide tubes inside the fuel bundle unit cell on the twogroup constants. The B_{1} MC method can estimate them in the same way as the standard unit lattice cell through socalled the supercell model. Figure 6 shows a supercell geometry model adopted in this study for estimating the incremental cross sections arising from the penetration of LZCs or ADJs into the fuel bundle lattice cell. As noted in Figure 6, the dimension of the supercell is 1 lattice pitch × 1 lattice pitch × 1 bundle length, which represents the normal supercell size. The incremental cross sections are computed by the difference of twogroup constants of the supercell with and without the guide tube and its control device. Tables 3 and 4 show the B_{1} MC method estimates for the incremental cross sections of LZCs and ADJs. The second columns in Tables 3 and 4 list the base twogroup constants, , of the fuel bundle cell without any control devices or guide tubes, for example, LZC and its guide tube in Table 3. The airn () columns in Table 3 list the incremental twogroup constants of type LZCs with their zone control units filled with air, . The H_{2}On () columns list the incremental twogroup constants of type LZCs with their zone control units filled with H_{2}O.


As shown above, the B_{1} MC method can generate the CSweighted twogroup constants of the ordinary fuel bundle cells free from control devices and the incremental twogroup constants of the supercell. In addition, the IMSweighted twogroup constants are generated to investigate the effect of the critical spectrum on the twogroup constants. To validate their qualification for the CANDUPHWR neutronics design calculations, they are used for the core neutronics analysis of the CANDU 6 core problems by the finite difference method option of the SCAN code [10]. The SCAN calculations are conducted in the fine mesh model with reflector cross sections given with RFSPIST version REL 304 and zero flux conditions at extrapolated boundaries in the axial and radial directions. Table 5 shows comparisons of ’s of the three different core states of the hypothetical CANDU 6 core calculated by SCAN with reference solutions obtained by the continuous energy McCARD whole core calculations with 1200 cycles including 200 inactive cycles with 1,000,000 histories per cycle. One can see that from the SCAN calculations with the CSweighted twostep constants agrees very well with the reference McCARD whole core predictions. In this conjunction, it is noted that the differences in ’s between SCAN with the CSweighted twogroup constants and the reference McCARD are −54 pcm for the nocontroldevice core, −62 pcm for the LZC50% fill core, and −43 pcm for the allADJin core while those from SCAN with the IMSweighted twogroup constants 104 pcm, 99 pcm, and 117 pcm, respectively. Tables 6, 7, and 8 show comparisons of the channel power distributions obtained by folding the full core results in the 1/4 core model for the three different core states with no control devices, LZCs at 50% fill, and all ADJs in, respectively. From these figures, one can again see that the channel power distributions from McCARD/SCAN twostep calculations agree very well with those from the reference McCARD calculations. Note that the root mean square (RMS) errors of the SCAN predictions with the CSweighted twogroup cross sections to the reference McCARD calculations are 0.39%, 0.83%, and 1.36% and those with the IMSweighted twogroup cross sections 0.36%, 0.80%, and 1.46% for (i) the nocontroldevice core, (ii) the LZC50% fill core, and (iii) the allADJin core, respectively. Note also that the maximum channel power errors of the SCAN predictions to the reference McCARD calculations are 2.33%, 2.18%, and 3.25% for the core (i), (ii), and (iii), respectively.
 
Continuous energy McCARD whole core transport calculation. ^{ 2}Fewgroup constant. ^{ 3}McCARD/SCAN − Ref. . 
 
= McCARD power, Rel. S.D. (%) = relative standard deviation of . = McCARD/SCAN power by infinite medium spectrum. = McCARD/SCAN power by critical spectrum. (%) = ( − )/ 100, and (%) = ( − )/ 100. 
 
= McCARD power, Rel. S.D. (%) = relative standard deviation of . = McCARD/SCAN power by infinite medium spectrum. = McCARD/SCAN power by critical spectrum. Diff._{INF} (%) = ( − )/ 100, and Diff._{CRI} (%) = ( − )/ 100. 
 
= McCARD power, Rel. S.D. (%) = relative standard deviation of . = McCARD/SCAN power by infinite medium spectrum. = McCARD/SCAN power by critical spectrum. Diff._{INF} (%) = ( − )/ × 100, and Diff._{CRI} (%) = ( − )/ × 100. 
5. Conclusion
The above results show that not only do the CSweighted twogroup constants generated by the B_{1} MC method compare well with those by WIMSAECL but also core neutronics analysis for the hypothetical CANDU 6 core problems by the SCAN calculation with the B_{1} MC method generated twogroup constants agrees well with that by the whole core reference McCARD calculation. Therefore, it is safely concluded that the B_{1} MC method is well qualified as a twogroup constant generator for the standard unit lattice cell and the supercell and therefore it can serve a valuable alternative to its deterministic counterparts for the neutronics analysis of CANDU 6 reactors. The striking advantage of the B_{1} MC method as a twogroup constant generator for the neutronics analysis of CANDUPHWR is its inherent capability to utilize the continuousenergy cross section library data and to model the geometrical heterogeneity of the fuel bundle cells—particularly those with control devices—exactly as they are. In this study, the CANDUPHWR analysis utilized for the qualification test of the B_{1} MC method has been confined to the hypothetic core problems. Further tests for the qualification of the B_{1} MC method will be made in terms of realistic CANDU 6 reactor core problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Research Foundation of Korea Grant funded by the Government of Republic of Korea (Ministry of Science, ICT, and Future Planning) (no. NRF2012M2A8A4011779).
References
 H. J. Shim, J. Y. Cho, J. S. Song, and C. H. Kim, “Generation of few group diffusion theory constants by Monte Carlo code,” Transactions of the American Nuclear Society, vol. 99, pp. 343–345, 2008. View at: Google Scholar
 H. J. Park, H. J. Shim, H. G. Joo, and C. H. Kim, “Generation of fewgroup diffusion theory constants by Monte Carlo code McCARD,” Nuclear Science and Engineering, vol. 172, no. 1, pp. 66–77, 2012. View at: Google Scholar
 M. Edenius and B. H. Forrsen, “CASMO3 a fuel assembly burnup program user's manual,” Tech. Rep. Studsvik/NFA893, Rev. 2, Studsvik AB, 1992. View at: Google Scholar
 R. J. J. Stammler, HELIOS Methods, Studsvik Scandpower, 2002.
 J. D. Irish and S. R. Douglas, “Validation of WIMSIST,” in Proceedings of the 23rd Annual Conference of Canadian Nuclear Society, Toronto, Canada, June 2002. View at: Google Scholar
 H. J. Shim, B. S. Han, S. J. Jong, H. J. Park, and C. H. Kim, “McCARD: Monte Carlo code for advanced reactor design and analysis,” Nuclear Engineering and Technology, vol. 44, no. 2, pp. 161–176, 2012. View at: Publisher Site  Google Scholar
 G. Marleau, A. Hebert, and R. Roy, “A user's guide for DRAGON version DRAGON_980911 release 3.03,” Tech. Rep. IGE174, Revision 4, Ecole Polytechnique de Montreal, 1998. View at: Google Scholar
 M. Ovanes, D. A. Jenkins, F. Ardeshiri et al., “Validation of the RFSPIST code against powerreactor measurements,” in Proceedings of 22nd Annual Conference of Canadian Nuclear Society, Toronto, Canada, June 2001. View at: Google Scholar
 A. R. Dastur and D. Buss, “MULTICELL—a 3D program for the simulation of reactivity devices in CANDU reactors,” AECL 7544, Atomic Energy of Canada Limited, 1983. View at: Google Scholar
 I. S. Hong, C. H. Kim, B. J. Min, and H. C. Suk, “Development of CANDUPHWR neutronics code SCAN,” in Proceedings of the 7th International CANDU Fuel Conference, vol. 1, pp. 77–86, Canadian Nuclear Society, Kingston, Canada, September 2001. View at: Google Scholar
Copyright
Copyright © 2015 Seung Yeol Yoo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.