Critical Experiment Analyses by CHAPLET-3D Code in Two- and Three-Dimensional Core Models

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1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 42, No. 1, p (January 2005) TECHNICAL REPORT Critical Experiment Analyses by CHAPLET-3D Code in Two- and Three-Dimensional Core Models Shinya KOSAKA 1; and Toshikazu TAKEDA 2 1 TEPCO Systems Corporation, Eitai, Koutou-ku, Tokyo Osaka University, 2-1 Yamadaoka, Suita-shi, Osaka (Received July 30, 2004 and accepted October 4, 2004) A series of critical experiments has been analyzed by the deterministic method code CHAPLET-3D in two- and three-dimensional core configurations in which explicit core structures are represented. The results show that the three-dimensional core calculation model employed in CHAPLET-3D code is valid and useful to obtain fine resolution results by the deterministic method. Moreover, the conventional two-dimensional axial buckling calculation for critical experiment analysis has also been validated, through the comparison between the results of two- and threedimensional experimental core analyses by CHAPLET-3D code. KEYWORDS: critical experiment analysis, three-dimensional calculation, axial buckling calculation, deterministic method, CHAPLET-3D I. Introduction Corresponding author, Tel , Fax , kosaka-shinya@tepsys.co.jp In order to obtain reference measurement data for the validation of lattice physics codes, many critical experiments and related measurements have been performed from various viewpoints of reactor physics and nuclear data. Almost all of the lattice physics codes, however, treat two-dimensional problems and cannot model the whole three-dimensional core exactly. Therefore, the axial buckling theory is usually introduced to two-dimensional core analysis in order to consider the core axial profile, though buckling values and diffusion coefficients introduced to simulate axial leakage might affect to the result due to their uncertainties. Recently, as computer performance improves, Monte Carlo codes become very useful tools to analyze critical experiments without any geometrical approximations. Due to its statistical methodology, the Monte Carlo method is good at evaluating cumulative values such as a multiplication factor, but there are some difficulties to obtain accurate pointwise values such as detail flux distribution in a large core calculation. Therefore, deterministic method codes are still required to obtain fine resolution results which cannot be traced accurately by the statistical way. Under such a situation, some studies to realize the threedimensional heterogeneous core calculation by a deterministic method have been reported. 1 3) In one of those studies, a three-dimensional heterogeneous core analysis code CHAP- LET-3D has been developed. 3) In order to achieve threedimensional core calculation in the explicit heterogeneous geometry, CHAPLET-3D code employs an idea in which the solutions of radial two-dimensional method of characteristics (MOC) analyses are linked axially by utilizing the non-linear iteration technique. 4) In the previous study, 3) CHAPLET-3D code has been extensively validated by the numerical benchmarks using Monte Carlo solutions. As a result, it was showed that an explicit geometry three-dimensional core analysis by the deterministic method can be performed with the accuracy of two-dimensional MOC solutions by CHAPLET-3D code. In this study, a series of critical experiments has been analyzed by CHAPLET-3D code in the explicit three-dimensional core configuration and also in the two-dimensional core configuration considering axial buckling. Firstly, in order to investigate the applicability of the three-dimensional deterministic analysis method used in CHAPLET-3D code to the critical experiment analyses, the accuracy of the three-dimensional analysis has been verified by comparing calculated results with measured data. Then, the validity of the conventional two-dimensional buckling calculation for critical experiment analysis has been confirmed, through the comparison between the results of two- and three-dimensional experimental core analyses by CHAPLET-3D code. In this paper, the critical experiments to be analyzed are introduced in Chap. II. In the following chapter, the analysis methods of two- and three-dimensional core calculations by CHAPLET-3D code are described briefly. Finally, the calculation results and conclusions are shown in Chaps. IV and V, respectively. II. Experiments The experiments 5) to be analyzed in this study were performed using NCA criticality facility owned by Toshiba Corporation for the purpose of investigating so-called neutron spectrum mismatch effect caused by uranium enrichment difference between BWR fuel assemblies. Figure 1 shows the experimental core configurations. In the test region, aluminum tubes were inserted at the corner of fueled cells to simulate moderator-to-fuel ratio in hot operating condition at 40% void. Under this condition, two types of fuel arrangement were constructed. In the first case, 2.0 wt% enriched UO 2 fuels were uniformly placed over the test region (uniform core case). In the other case, 1.0 wt% 101

2 102 S. KOSAKA and T. TAKEDA Case1: Uniform core Case2: Mismatch core Water reflector Water reflector Test bundle (1wt% UO 2 ) 9x9 array with 7 water holes Test bundle (2wt% UO 2 ) 9x9 array with 7 water holes Test bundle (3wt% UO 2 ) 9x9 array with 7 water holes Driver region (2wt% UO 2 ) Al-tube Water density reduced region by inserting Al-tubes Fig. 1 Core configuration of NCA critical experiment and 3.0 wt% enriched UO 2 fuels were arranged as shown in Fig. 1 to realize enrichment mismatch between the center fuel assemblies (mismatch core case). Figure 2 shows the side view of the experimental core. Critical condition is achieved by adjusting water level. In order to fix fuel rod positions, an aluminum spacer is placed at 69 cm height from the bottom of fuel. In this series of critical experiments, critical core heights (water levels), horizontal pin-wise power distributions at the middle of core height and axial power distributions at some fuel pin positions in the core center region were measured. III. Analyses Fig. 2 Axial profile of experimental core CHAPLET-3D code was utilized to analyze NCA critical experiments in the two- and three-dimensional core configurations. The basic concept of the analysis method utilized in CHAPLET-3D code is described in Fig. 3. In CHAPLET- 3D code, three-dimensional core calculation can be achieved by linking radial 2D MOC solution of the respective axial planes utilizing the non-linear iteration technique. As for the axial solver, finite difference method (FDM), nodal expansion method (NEM) or one-dimensional MOC can be selected as an option. Consequently, it enabled us to obtain very accurate results as good as those of multi-group Monte Carlo calculations. (see Ref. 6) for details). The basic two-dimensional MOC solver of CHAPLET-3D code is the same as that of the original CHAPLET code, 7) which is a two-dimensional MOC code employing the equidistant ray tracing scheme used in CACTUS 8) code and has been well-verified for two-dimensional heterogeneous core problems. JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

3 Critical Experiment Analyses by CHAPLET-3D code in Two- and Three-Dimensional Core Models 103 Explicit core configuration Radial 2D Separate 3D core problem into 2D and 1D calculation model Axial 1D 2D heterogeneous fixed source MOC calculations Axial information Radial information 1D FDM/NEM/MOC homogeneous pin-cell calculations Radial information Axial information Main 3D FDM-form core calculation by non-linear iteration technique Fig. 3 Outline of CHAPLET-3D core calculation model Fig. 4 Fuel Cladding Al-tube (smeared) Flat source region sample of internal MOC calculation 1. Three-Dimensional Analyses by CHAPLET-3D The three-dimensional experiment analyses have been performed by CHAPLET-3D code in heterogeneous core geometries. In these calculations, the exact core structures such as fuel pins, aluminum-tubes and spacers are modeled explicitly. Figure 4 shows a sample of flat source mesh divisions for the internal planar MOC calculations. The macroscopic cross sections for each heterogeneous material in these analyses were supplied by CASMO. 9) Main other analysis conditions are shown in Table Two-Dimensional Analyses by CHAPLET-3D In the two-dimensional core analysis by CHAPLET-3D code, the core axial profile is considered by the axial buckling model as an usual critical experiment analysis by a lattice physics code. When the axial flux distribution in the whole core region can be expressed as cosðb z zþ, the axial leakage can be considered in two-dimensional calculation by introducing the pseudo absorption 10) as ~ a ¼ a þdb 2 z, where a and ~ a are the original and modified macroscopic absorption cross section, D is the diffusion coefficient and B 2 z is the axial buckling corresponding to the core height. The axial buckling values used for the experiment analyses are obtained by the core heights (H: critical water level) Table 1 Conditions of CHAPLET-3D code for 2D and 3D calculations Number of neutron energy groups: 7 Isotropic scattering assumed Quadrature set of planar MOC calculation Number of azimuthal angles: 64/octant Number of polar angles aþ : 2/octant Ray tracing pitch bþ : 1.0 mm Axial solver used for 3D calculation Diffusion calculation by FDM (Finite Differential Method) Axial mesh width: 0:5 cm Convergence criteria Eigenvalue: 1:010 6 Point wise scalar flux: 1:010 5 aþ Leonard s optimal polar angles 11) bþ Equidistant ray tracing scheme of CACTUS and the extrapolation distance () which is normally utilized for NCA core analyses, as B 2 z ¼ð=ðHþ2ÞÞ2. Then the buckling values utilized in this study are 1: and 5: cm 2 for uniform core and mismatch core analyses, respectively. As for the diffusion coefficients to provide the pseudo absorption (DB 2 z ), three types of diffusion coefficients were considered in order to investigate the effects on the results by their definition. (1) Whole core (including reflector) average diffusion coefficient: D all D all is obtained by averaging all of the diffusion coefficients in the whole core including the reflector region as Z DðrÞðrÞdr r2whole-core D all ¼ Z ; ð1þ ðrþdr r2whole-core VOL. 42, NO. 1, JANUARY 2005

4 104 S. KOSAKA and T. TAKEDA Table 2 Eigenvalue results of 2D and 3D core calculations by CHAPLET-3D Analysis conditions Uniform core Error Mismatch core 3D analysis Explicit geometry (ref.) (ref.) 2D buckling analysis Core average (D-all) (þ0:14%dk) (þ0:17%dk) Core average (D-core) (þ0:07%dk) (þ0:13%dk) Local pin-cell (D-pin) ( 0:16%dk) ( 0:02%dk) Zero buckling (þ4:66%dk) (þ2:71%dk) Error where fluxes to homogenize diffusion coefficients are obtained by zero buckling two-dimensional core calculation. In the two-dimensional core calculation with this axial buckling model, the pseudo absorption (D all B 2 z ) was added to all of the macroscopic absorption cross sections in the core. (2) Active core average diffusion coefficient: D core D core is obtained by averaging all of the diffusion coefficients in the whole core excluding the reflector region as Z DðrÞðrÞdr r2active-core D core ¼ Z ; ð2þ ðrþdr r2active-core where fluxes to homogenize diffusion coefficients are obtained by zero buckling two-dimensional core calculation. In the two-dimensional core calculation with this axial buckling model, the pseudo absorption (D core B 2 z ) was added to all of the macroscopic absorption cross sections in the core. (3) Local pin-cell average diffusion coefficient: D pin D pin is obtained by averaging all of the diffusion coefficients in the pin-cell for each pin type as Z DðrÞðrÞdr r2pin-cell D pin ¼ Z ; ð3þ ðrþdr r2pin-cell where fluxes to homogenize diffusion coefficients are obtained by zero buckling two-dimensional pin-cell calculation. In the two-dimensional core calculation with this axial buckling model, the pseudo absorption (D pin B 2 z ) was added to all of the macroscopic absorption cross sections in the corresponding pin-cell type. In addition to these three cases, zero buckling two-dimensional calculation was also performed in order to know how much the total amount of axial leakage is. The analysis conditions of two-dimensional CHAPLET- 3D calculations, such as a quadrature set of planar MOC calculation, flat source mesh divisions, convergence criteria and macroscopic cross sections are identical to those of the threedimensional core calculations. IV. Results 1. Results of Three-Dimensional Core Analyses Table 2 shows the eigenvalue results of two- and threedimensional core analyses by CHAPLET-3D code. The eigenvalues of three-dimensional core analyses agree among both core configurations. Figures 5 and 6 show measured and calculated axial power distributions at typical fuel pin positions in the uniform core case, and Figs. 7 and 8 show those of the mismatch core case. CHAPLET-3D code can reproduce very detail axial power shapes such as a small power dip observed at the position of the aluminum spacer inserted. Figure 9 shows the relative errors of horizontal pin-power distributions. The RMS errors of axial and horizontal power distributions are within 1.2%, which is almost the same level as the measurement errors. It should be noted that the diffusion theory was applied to the axial solver in these CHAPLET-3D three-dimensional analyses, because the transport theory axial solver failed to converge. However, the results of the three-dimensional core analyses show that the application of the diffusion theory to the CHAPLET-3D axial solver is quite reasonable to be utilized for this kind of LWR core analysis. 2. Results of Two-Dimensional Core Analyses The impact of different definition of the diffusion coefficient for buckling calculation on eigenvalue has been investigated. The results are shown in Table 2. The eigenvalues shift within 0:2%dk from the reference results of three-dimensional analyses. In each definition case, however, the eigenvalue level is very stable independently of the core configuration like as the three-dimensional analyses results. In addition to the definition of the diffusion coefficient, the buckling value itself also affects its eigenvalue result directly. Table 3 shows the axial buckling values of measurements, calculated by three-dimensional analyses and used for two-dimensional analyses. The measured and calculated axial buckling values are evaluated by cosine fitting of the axial pin power distributions which are obtained by measurement and calculation respectively. The cosine fitting processes are applied for the data points at the range of the middle of core height 20 cm excluding around the spacer position. These buckling values agree within 2.0%. This fact leads that expected eigenvalue shifts caused by these buckling value differences will be less than 0:1%dk. Because the total amount of the reactivity loss induced by axial JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

5 Critical Experiment Analyses by CHAPLET-3D code in Two- and Three-Dimensional Core Models Measured data ( σ~1.0%) CHAPLET-3D Relative power distribution Pin position Al-spacer (cm) Axial position Fig. 5 Axial pin-power evaluation by three-dimensional analysis in uniform core case (1) Measured data ( σ~1.0%) CHAPLET-3D Relative power distribution Pin position Al-spacer Axial position (cm) Fig. 6 Axial pin-power evaluation by three-dimensional analysis in uniform core case (2) Table 3 Comparison of measured, calculated and used axial buckling (cm 2 ) Uniform core Error Mismatch core Measured buckling 1: (ref.) 5: (ref.) Calculated buckling 1: ( 1:0%) 5: ( 0:5%) Buckling used for 2D analysis 1: (þ0:8%) 5: (þ2:0%) Error VOL. 42, NO. 1, JANUARY 2005

6 106 S. KOSAKA and T. TAKEDA Measured data ( σ~1.0%) CHAPLET-3D Relative power distribution Pin position Al-spacer Axial position (cm) Fig. 7 Axial pin-power evaluation by three-dimensional analysis in mismatch core case (1) Measured data ( σ~1.0%) CHAPLET-3D Relative power distribution Pin position Al-spacer Axial position (cm) Fig. 8 Axial pin-power evaluation by three-dimensional analysis in mismatch core case (2) leakage is 4:66%dk (see Table 2) in the case of the uniform core analysis, the eigenvalue shift can be estimated as 4:66%dk2:0%¼0:0932%dk when the linearity of the reactivity loss due to the pseudo absorption of axial buckling can be assumed. Figures show the relative errors of the horizontal pin-power distributions at each definition type of the diffusion coefficient. The accuracy of pin-power evaluation by two-dimensional analysis becomes slightly worse than that of the three dimensional analysis. However, the RMS errors are still within 1.4%. Especially, when applying the pin-cell averaged diffusion coefficient to the buckling calculation, almost the same accuracy as three-dimensional analysis seems to be expected for pin-power evaluation. As a result, it has been confirmed that the experiment analysis by two-dimensional buckling calculation is rather reasonable to be applied for the verification of a lattice physics code. JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

7 Critical Experiment Analyses by CHAPLET-3D code in Two- and Three-Dimensional Core Models 107 Fig. 9 Radial pin-power evaluation by three-dimensional analyses Fig. 10 Radial pin-power evaluation by two-dimensional analyses (D all case) Fig. 11 Radial pin-power evaluation by two-dimensional analyses (D core case) VOL. 42, NO. 1, JANUARY 2005

8 108 S. KOSAKA and T. TAKEDA Fig. 12 Radial pin-power evaluation by two-dimensional analyses (D pin case) V. Conclusions Critical experiment analyses have been performed by CHAPLET-3D code in three- and two-dimensional core configurations in which explicit core structures are represented. The results show that the three-dimensional core calculation model employed in CHAPLET-3D code is valid and useful to obtain fine resolution results by the deterministic method. On the other hand, the conventional two-dimensional buckling calculation for a critical experiment analysis has also been validated, through the comparison between the results of two- and three-dimensional experimental core analyses by CHAPLET-3D code. Consequently, it was found that the effects by the different definition of the diffusion coefficient for pseudo absorption are not so significant on the eigenvalue estimation and very minor on pin-power evaluation for this kind of critical experiments presented here. References 1) N. Z. Cho, et al., Fusion of method of characteristics and nodal method for 3-D whole-core transport calculation, Trans. Am. Nucl. Soc., 86, 322 (2002). 2) H. G. Joo, et al., Dynamic implementation of the equivalence theory in the heterogeneous whole core transport calculation, Proc. Int. Conf. on New Front. of Nucl. Technol. Reac. Phys., Safety and High-Perform. Comp., Seoul, Korea, Oct. 7 10, 2002, on CD-ROM, (2002). 3) S. Kosaka, T. Takeda, Diffusion-like 3-D heterogeneous core calculation with 2-D characteristics transport correction by non-linear iteration technique, Proc. Int. Conf. Nucl. Math. and Comp. Sci., Gatlinburg, U.S.A., Apr. 6 11, 2003, on CD-ROM, (2003). 4) K. S. Smith, Nodal method storage reduction by nonlinear iteration, Trans. Am. Nucl. Soc., 44, 265 (1984). 5) S. Kosaka, E. Saji, Critical experiment analyses to study the neutron spectral mismatch effect between BWR fuel assemblies, Trans. Am. Nucl. Soc., 77, 374 (1997). 6) S. Kosaka, T. Takeda, Verification of 3D heterogeneous core transport calculation utilizing non-linear iteration technique, J. Nucl. Sci. Technol., 41[6], 645 (2004). 7) S. Kosaka, E. Saji, Transport theory calculation for a heterogeneous multi-assembly problem by characteristics method with direct neutron path linking technique, J. Nucl. Sci. Technol., 37[12], 1015 (2000). 8) M. J. Halsall, CACTUS, A Characteristics Solution to the Neutron Transport Equations in Complicated Geometries, AEEW-R1291, UKAEA, Winfrith, (1980). 9) D. G. Knott, M. Edenius, Validation of the CASMO-4 transport solution, Proc. Math. Methods and Supercomputing in Nuclear Application, Karlsruhe, Germany, Vol. 2, 547 (1993). 10) S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory, D. Van Nostrand, New York, (1952). 11) A. Leonard, C. T. McDaniel, Optimal polar angles and weights for the characteristics method, Trans. Am. Nucl. Soc., 73, 172 (1995). JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY