Analysis of First-Harmonic Eigenvalue Separation Experiments on KUCA Coupled-Core

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1 Journal of Nuclear Science and Technology ISSN: (Print) (Online) Journal homepage: Analysis of First-Harmonic Eigenvalue Separation Experiments on KUCA Coupled-Core Yoshiki KATO, Toshihisa YAMAMOTO, Takanori KITADA, Toshikazu TAKEDA, Kengo HASHIMOTO, Seiji SHIROYA, Hironobu UNESAKI & Otohiko AIZAWA To cite this article: Yoshiki KATO, Toshihisa YAMAMOTO, Takanori KITADA, Toshikazu TAKEDA, Kengo HASHIMOTO, Seiji SHIROYA, Hironobu UNESAKI & Otohiko AIZAWA (1998) Analysis of First-Harmonic Eigenvalue Separation Experiments on KUCA Coupled-Core, Journal of Nuclear Science and Technology, 35:3, , DOI: / To link to this article: Published online: 15 Mar Submit your article to this journal Article views: 144 Citing articles: 7 View citing articles Full Terms & Conditions of access and use can be found at

2 Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 35, No. 3, p (March 1998) TECHNICAL REPORT Analysis of First-Harmonic Eigenvalue Separation Experiments on KUCA Coupled-Core Yoshiki KATO*'.t, Toshihisa YAMAMOTO", Takanori KITADA*l;tt, Toshikazu TAKEDA", Kengo HASHIR/IOT0*2, Seiji SHIROYA*3, Hironobu UNESAKI*3 and Otohiko AIZAWA*4 *'Department of Nuclear Engineering, Graduate School of Engineering, Osaka University *2 Atomic Energy Research Institute, Kinki University +3 Research Reactor Institute, Kyoto University *4 Atomic Energy Research Institute, Musashi Institute of Technology (Received March 21, 1997), (Revised October 30, 1997) The first-harmonic eigenvalue separation, the difference between the fundamental and the first order eigenvalues of the higher harmonic neutron transport equations, which were measured at the Kyoto University Critical Assembly (KUCA) has been analyzed. A method was proposed to calculate the first order eigenvalue based on the discrete ordinate method. The 3-D effect, energy group effect, mesh size effect, and transport effect were investigated. Among these effects, the transport effect was significant and when it was taken into account, the calculated eigenvalue separation approached the measured value on the KUCA coupled-core. KEYWORDS: first-harmonic eigenvalue separation, fundamental eigenvalue, first-order eigenvalue, discrete ordinate method, measurement, KUCA reactor, coupled-core, analysis, three-dimensional effect, energy group effect, mesh site effects, transport effects I. Introduction The eigenvalue separation is an important parameter to evaluate the safety feature of reactors. It has been investigated by many authors and it has been pointed out that the parameter can be employed as an indication of the stability against the out-of-phase (regional) power oscillations in BWR So, whether the eigenvalue separation can be evaluated with sufficient accuracy is a great concern when we refer to the reliability of the time dependent safety analyses. For the purpose of validating numerical methods for the eigenvalue separation, the first-harmonic eigenvalue separation was measured at the C-core tank of the Kyoto University Critical Assembly (KUCA) with a wide range of deco~pling(~). The core is a water-moderated and -reflected coupled-core system which is arranged in a rectangular parallelepiped geometry. In this core, the gap between the two halves of the core is modifiable and a wide range of decoupling can be covered. For example, *' Yamadaoka, Suita-shi *2 Kowakae, Higashi-Osaka-shi r3 Kumatori-cho, Sennan-gun, Osaka-fu Ozenji, Aso-ku, Kawasaki Present address: Shikoku Electric Power Co., Ikata-machi, Nishiuwa-gun, Ehime-ken Corresponding author, Tel. $ , Fa. $ , kitada@nucl.eng.osaka-u.ac.jp a core gap of 20cm gives approximately the same degree of decoupling as a large BWR. So, the experiments with various core gaps are regarded as good benchmarks to validate the methodology of evaluating the eigenvalue separation. Many investigations have been performed for the evaluation of eigenvalue separation, however, the following points are still remained to be considered: (1) As for the measurement of the eigenvalue separation in a loosely coupled-core system, several different methods have been proposed. However, their advantages, drawbacks and reliability are not yet known. In order to establish a reliable benchmark, comparison among the methods is necessary. (2) Numerical method to obtain the eigenvalue separation based on transport theory has not been established. The transport effect will be possibly important especially for small cores such as the KUCA. (3) In the analysis of a critical assembly, various correction factors should be taken into account to obtain the final solution. This may be same for the analysis of the eigenvalue separation. The information on this point will be useful when a standard method to analyze the eigenvalue separation is determined in the future. The main goal of this study is to establish a calculational method to analyze the eigenvalue separation of a coupled-core system based on transport theory, and then to validate the method through the analysis of the experiments on the KUCA coupled-core. 216

3 Analysis of First-Harmonic Eigenvalue Separation Experiments on KUCA Coupled-Core 217 In Chap.11, experimental results are shown and consistency among different methods is discussed. In Chap. 111, a calculational method including the firstharmonic mode flux calculation based on transport t,heory is proposed. In Chap. IV, the results of the analysis are shown together with the evaluation of the various effects in the calculational method. Then the comparison between the analysis and the measurement will be discussed. 11. Experimental Results The experimental core configuration is shown in Fig. 1. In this experiment, the core gap was varied by three steps: 10 cm, 14cm, and 20 cm. The identifications of the configurations are G10, G14 and G20, respectively. The measurements were performed by three methods: the flux tilt method, the rod-drop met,liod and the reactor noise method. Details of the experimental methods are shown in Ref. (4). In Ref. (4), the experimental results are shown only for G20, whereas in this paper the results for G10, G14 and G20 are shown. The critical mass of these cores increases as the core gap becomes wider, because the wide gap brings the weak coupling of the core, and then it needs more fuel plates to maintain the criticality. Only a simple description of the methods is given in the following. 1. Flux Tilt Method In this method, the eigenvalue separation is evaluated from the magnitude of the flux distortion(flux tilt) across the two halves of the core caused by an asymmetric configuration of the control rods (C1 and C2) which are inserted in individual half cores. The eigenvalue separation is given by the following equation(5): Pdiff &=- (E.S.) -k "' where E is the magnitude of the flux tilt, Pdiff is the difference of control rod worths (=pel - pcz), EO is the initial flux tilt, and (E.S.) is the eigenvalue separation. The control rod worths pcl and pcz were measured by the positive period method, and the flux tilt E was evaluated from the thermal flux distribution which was measured using optical fibers #1 and #2 with scintillators@) as shown in Fig. 1. Figure 2 shows the relation between the difference of control rod worths and the flux tilt observed in the measurements with various core gaps. It is clear that the linear relation holds for every configuration and for each counter. Table 1 summarizes the eigenvalue separations of each configuration. 2. Rod-drop Method In this method, the eigenvalue separation of a loosely coupled-core system is given by the following equati~n(~): n where p1 and p2 represent the control rod worths obtained with a counter at the rod-dropped side and with a h 0.2 v CI } 5 G 'c 'CI g o.- CI c % Difference of rod worths (pc1-pc2) ( Q) 614 } G10 Fig. 2 Linear relation between the difference of rod worths and flux tilt Fig. 1 Top view of experimental core VOL. 35, NO. 3, MARCH 1998

4 ~ 218 Y. KATO et al. Table 1 Eigenvalue separation obtained by flux tilt measurement G 10 G14 G20 Fiber #1 E.S.t 7.903f f f k f f Fiber #2 E.S.t 9.468k t * f Averaged E.S.t 8.686k f &0.011 tin $ Table 2 Reactivity worthst of safety rods S4 and S5 Rod BF3#1 BF3#2 FC#1 FC#2 FC#3 G10 S , S G14 S S G20 S tin $ counter at the opposite side, respectively. The measured worths of safety rods S4 and S5 are shown in Table 2. Substituting the counter-position dependent worths into Eq. (2), we obtained the eigenvalue separation of each configuration. The results are shown in Table Feynman-a Method In the two-detector Feynman-a method@), t,he eigenvalue separation is given as a1 - a0 (E.S.) = ~, (3) a0 where the prompt-neutron decay constants of the fundamental mode (ao) and the first- harmonic mode (a1) are obtained from the time sequential response of two detectors at different positions. They are obtained from the same time sequential response but with different formula in the data processing(8). Table 4 shows the eigenvalue separations obtained by Feynman-a method. 4. Comparison among the Methods The measured results are summarized in Table 5. As shown in Table 5, flux tilt method gave the smallest experimental error for all the three cores (G10, G14 and G20). Especially for G10, Feynman-a method revealed a large underestimation with large error compared to the other methods. In the case of Feynman-a method, the data are obtained under the critical condition where the fundamental mode is very dominant. Thus it is difficult to detect the information of the higher harmonic mode accurately, especially for GI0 which has t,he deepest sub- Table 3 Eigenvdue separationt obtained by rod-drop method Counter pair G10 G 14 G20 s4 s5 s4 s5 s4 s5 BF3#1 & BF3# FC#1 & FC# FC#1 &FC# Average 8.754f f f0.029 Table 4 Eigenvalue separation obtained by Feynman-a method ff f f f0.94 ff ~ f f3.67 Averaged E.S.t 4.447f f0.027 JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

5 Analysis of First-Harmonic Eigenvdue Separation Experiments on KUCA Coupled-Core 219 Table 5 Eigenvalue separationt obtained by various techniques G10 G14 G20 Flux tilt method 8.686f ~ f0.011 Rod drop method 8.754f f &0.029 Feynman-a method 4.447f f f0.027 criticality among the three cores. On the other hand, for the cases of flux tilt method and rod-drop method, the data are measured by artificially exciting the higher harmonic mode. However rod-drop method excites not only the first-harmonic mode but also the higher harmonic mode, flux t,ilt method can excites only the firstharmonic mode by considering the shape of the firstharmonic mode. So, flux tilt method can give the best results with small experimental error among the three methods. Therefore in this paper, t8he results of only flux tilt method are used as the experimental results to obtain the CIE value Calculational Methods In this chapter, the calculational methods used in the analysis are described. 1. Cross Section Preparation In the analysis, the heterogeneous cell structure was taken into account by using the SRAC95 system(g), which was developed in Japan Atomic Energy Research Institute (JAERI) in The JENDL group nuclear 1ibrary(l0) was used. The energy structure is composed of 74 groups as fast groups, and 33 groups as thermal groups. The PEACO routine(") which can treat the ultra-fine energy structure was used in the cell calculations. Figure 3 describes a simplified core model of the KUCA. The core is composed of multiple fuel frames between which water exists. The fuel frame used in this experiment is loaded with up to 40 fuel plates. Figures 4(a) and (b) describe the structures of a fuel Core Tank Core Gap I Fuel Frame Fig. 3 Concept of loading fuel frames into KUCA core tank 62mm- Fig. 4 (a) Fuel frame 0.5mm 0.5mm Fuel Meat (Uranium-Aluminum) 570mm U g u 9.55 g U-235 enrichment 93.1 % I 1 l l &+- Clad (Aluminum) (b) Fuel plate Structure of a fuel frame and a fuel plate frame and a fuel plate. Water is also filled in the gap between the fuel plates. In order to take into account the existence of water accurately, we have applied two steps of homogenization. In the first step, only the fuel plate and water arrangement in the fuel frame is considered using the primary cell model shown in Fig. 5. Then the averaged cross sections obtained for the primary cell were applied to the secondary cell model shown in Fig. 6, which treats the existence of water gap between two adjacent fuel frames. The 107 group cell averaged cross sections obtained for the fuel frame were collapsed into fewer groups using VOL. 35, NO. 3, MARCH 1998

6 220 Y. KATO et al. Fuel Meat (Uranium- Aluminum) Fuel Frame \d Fuel Plate I+/ k cm -4 I Clad (Aluminum) Reflective Reflective Boundary Boundary + Light eee 2.0mm 0.5mm 2.0mm Fig. 5 One-dimensional cell model for the primary cell calculation Boundary + e.. Reflective Boundary 0.. Fuel Frame Fig. 6 One-dimensional cell model for the secondary cell calculation the neutron spectrum of each region calculated by diffusion theory with the 3-D core model shown in Fig. 7. Then the same 3-D core model and the collapsed cross sections were used in order to calculate the axial buckling of each region so that the axial leakage from each region is conserved in the 2-D X Y model. The same axial bucklings were used for both diffusion and transport, calculations. Table 6 shows the energy group structures of the collapsed cross sections. 2. Fundamental and First-harmonic Mode Calculations The CITATION code(12) was used for diffusion calculations. For transport calculations, the TWOTRAN code(13) was used. Because of the symmetrical configuration, the calculations were done only for a half part of the core. The refledive boundary condition was set Fig. 7 Three-dimensional core model for eigenvalue calculations (G20) between the two halves of the core in the fundamental mode calculations. In a special configuration where the shape of the first-harmonic mode flux can be predicted easily and the location where the mode flux becomes zero (zero point) is clear, the first-harmonic mode flux can be calculated using the conventional diffusion code with a modified boundary condition along the zero points(14). Therefore in the diffusion calculation, the boundary condition X is set to be infinitive, where X is defined by current J and surface flux ds, as X = J/4s. However, the method has been limited to diffusion theory so far, and the transport effect has not been treated. In the transport calculation, however, the boundary condition is not satisfied with ordinary treatment of the angular fluxes on the boundary surface. Therefore, a new boundary condition was introduced so that the scalar flux at the boundary is exactly zero by using the negative incoming angular flux: $; = -$Z,"", (4) where Grn is the angular flux of direction m, and in/out signify t,he incoming/out,going direction from the boundary surface. Obviously, integration of the Eq. (4) over whole direction leads to +s = 0 and the boundary condition required for the first-harmonic mode is precisely satisfied. In the 2-D XY calculations, the axial buckling was used as a pseudo absorption. It is worth mentioning that the axial buckling can be obtained from the axial shape of the fundamental mode flux even though it is applied to the higher mode calculation. The higher mode JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

7 Analysis of First-Harmonic Eigenvalue Separation Experiments on KUCA Coupled-Core 221 Table 6 Energy group structures of 11, 26 and 41 groups llg 26g 41g Upper energy Lower energy Lethargy width (h4ev) (kev) (kev) (ev) (ev) ~ x lov x 3.34 x lop ~ 1.47~ low x 3.52 x lop x 1.00 x lo groups treated in this study is limited to the direction along the common axis of the two halves of the core. Therefore, the axial shape of the higher mode flux can be approximated as that of the fundamental mode. This will be verified through the numerical results described later. Mesh size was almost uniform and set to 0.5cm. The SN order was 6. The anisotropic scattering was treated for PI. The correction for mesh size effect and SN order will be considered later. Other effects, such as 3-D effect, energy group effect and transport effect are also investigated in the analysis. These data are necessary for estimation of error-free value of the fundamental and the first-harmonic eigenvalues obtained from the calculations. 3. Effective Delayed Neutron Fraction (pee) Forty-one-group XY 2 diffusion calculations were performed to obtain the forward and the adjoint fluxes of each configuration. The fission yields were taken from JNDC Nuclear Data Library of Fission Products: Second Version(l5) and the delayed neutron data were based on the Keepin s evaluation(16). Table 7 lists the calculated Peff for each configuration. The Peff tends to decrease as the gap narrows. A measured value for the same core size with a different core gap size, 21.5cm1 is available as an experimental result( ), which is with the 4-5% experimental error. The calculated Peff became , which is 3% larger than the measurement but within the experimental error. IV. Results and Discussions Table 8 summarizes the eigenvalues of the fundamental mode (ko) and the first-harmonic mode (kl) with different calculational conditions. In Table 9, various effects on criticality are summarized as the difference of the values shown in Table 8. There are several tendencies observed in the results, of which details are discussed in the following D Effect This effect is evaluated as the difference between the 41-group 3-D and 2-D diffusion results. The use of the axial buckling mentioned in Sec produced errors about %Ak/k in the fundamental mode. The magnitude of errors are almost the same for the firstharmonic mode. Therefore, it can be concluded that the application of the axial buckling based on the fundamental mode flux to the 2-D calculation of the first-harmonic mode calculation is validated. Table 7 Calculated effective delayed neutron fraction Peff G10 G14 G20 (221.5 Peff x x x x VOL. 35, NO. 3, MARCH 1998

8 222 Y. KATO et al. Table 8 Comparison of eigenvalues with different calculational conditions and energy groupst llgtt 26g 41g 3-D diffusion calculation G20 KO K G14 KO K G10 KO K D diffusion calculation G20 KO K G14 KO K G10 KO K D transport calculation G20 KO Ki G14 KO Ki G10 KO Ki t Convergence criteria for eigenvalue and relative flux error is ttll groups Table 9 Comparison of various effects on criticality G10 G14 G20 3-D effect (41g) KO Ki Energy group effect (41-26g) KO Ki (41-llg) KO K Transport effect (41g) KO Ki Energy Group Effect The 41-group calculation was assumed as the reference and 3-D diffusion results were used for the evaluation. The use of the 11-group suffers considerable discrepancy for each configuration especially for the first-harmonic mode. On the other hand, the use of the 26-group model can diminish the error within o.l%ak/k for the fundamental mode and 0.2%Ak/k for the first-harmonic mode. 3. Transport Effect This effect is evaluated as the difference between the 41-group 2-D transport and diffusion results. The transport effect is as large as 3-4%Ak/k which is the main factor among the effects which should be considered in the analysis. It, is worth mentioning that the tendency is opposite between the modes: As the core gap becomes narrower, the effect on the fundamental mode decreases and that on the first-harmonic mode increases by contrast. In order to investigate this tendency, flux dist,ribution was compared among the configurations as shown in Fig. 8. The flux level of each configuration is normalized with the thermal flux at the peak appeared in the core region (around 20-25cm from the core center). As shown in Fig. 8, the thermal flux distribution in the inner reflector region between the cores calculated by diffusion theory is smaller than that calculated by transport theory. This difference can be attributed to the larger fast neutron leakage caused by diffusion approximation and can be regarded as the cause of underestimation of the fundamental mode eigenvalue, which results in the positive transport effect. The error in fast neutron leakage becomes less significant when the core gap becomes narrower, since the neutrons leaked from one core have more chance to penetrate the core gap region and to reach the other core. Thus we consider that the transport effect for the fundamental mode decreases as the core gap becomes narrower. For the first-harmonic mode, the difference of the flux distribution at the core gap becomes smaller compared with that of the fundamental mode because of the zero boundary at core center. Thus, the difference at the outer reflector region becomes more important,, and the peak of the thermal flux distribution at this region becomes higher as the core gap becomes narrower. So the transport effect for the first-harmonic mode becomes larger as the core gap becomes narrower. Also, we considered that this tendency may be caused by the fact that the crit,ical mass becomes smaller as the core gap becomes smaller as mentioned in Chap. 11. This observation explains the difference in the transport effect between the modes. The tendency of the 26-group is similar to the 41-group. So, the 26-group seems to be enough for transport calculations. In order to take into account the mesh size and SN order effects in the transport calculations, the eigenvalues were compared for various mesh size and SN order. Tables 10 and 11 summarize the results together with the estimat,ion of the error-free value which was extrapolated using the following relations of error in eigenvalue JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

9 Analysis of First-Harmonic Eigenvalue Separation Experiments on KUCA Coupled-Core 223 I I I I e ; -2 3 E distance tom Uw ccm center (an) (a) GI0 fundamental mode die& from me om canter (an) (b) GI4 fundamental mode did& tom me CQB canter (an) (c) G20 fundamental mode - Transparl - Diffusion distance from Uw CQB cenw (cm) (d) G10 first-harmonic mode &tam fmm Uw om center (an) (e) GI4 first-harmonic mode distance from Uw are center (om) (f) G20 first-harmonic mode Fig. 8 Comparison of fast and thermal flux distributions between diffusion and transport calculations Table 10 Space mesh size effect on fundamental and first-harmonic eigenvalues Mesh size (h) GI0 G14 G20 KO Ki KO Ki KO Ki Diffusion 2.0 cm calculation 1.0 cm cm h Transport 2.0 cm calculationt 1.0 cm cm h t.9~ order is 6. Table 11 SN order dependence of fundamental and firstharmonic eigenvaluest SN G10 G14 G20 order KO K1 KO K1 KO K1 S, Ss SS S, 'Mesh size is 0.5cm, 2-D calculation. VOL. 35, NO. 3, MARCH 1998

10 224 Y. KATO et al. ( Ak)(18). A B C Ak=-+-+- N, N, N, D Ak=---, (5b) Nin where N,, Ny, N, is the number of space meshes along the x, y, z axis, Nsn is SN order, and A, B, C, D are constants. In Table 10, the eigenvalue tends to increase as the mesh size becomes smaller: This tendency is common to the fundamental and the first-harmonic eigenvalues, and also to the diffusion and transport calculations. Figure 9 describes the SN order dependence of the eigenvalue of each mode. It is observed that Ey. (5b) holds not only for the fundamental eigenvalue, but also for the first-harmonic eigenvalue. In Table 11, the eigenvalue decreases as the SN order becomes larger. Judging from these results, the combinat,ion of the mesh size of 0.5 cm and SG is almost sufficient to estimate eigenvalue separations. Table 12 summarizes the calculated eigenvalues after all the effects were taken into account. Table 13 shows the eigenvalue separations, which are defined as l/kl-l/ko, calculated from Table 12 and the comparison with the measured value. The discrepancy in diffusion calculations seems to be much larger than the error in Pen estimation shown in the previous chapter. It is obvious that the use of transport theory has greatly improved the consistency between the calcula- 3 Fig. 9 1.ooo I I I I I I I sn order ( 1 N* ) SN order dependence of fundamental and firstharmonic eigenvalues (G20) Table 12 Fundamental and first-harmonic eigenvalues after corrections G10 G14 G20 ~~ Diffusion KO calculation K Transport KO calculation K Table 13 Comparison of eigenvalue separation between calculation and experiment Calculational Experimental resultt resultt tt CIE Diffusion G (*2.8%) 1.24 calculation G (f2.8%) 1.33 G (f1.3%) 1.49 Transport G (f2.8%) 1.00 calculation G (*2.8%) 1.05 G (&1.3%) 1.27 %dk/kk ttrelative error is shown in parentheses. tionhl results and the experimental results. However, large discrepancy which exceeds the experiment,al error still exists especially for G20. V. Conclusions The first-harmonic eigenvalue separation measured at the KUCA has been analyzed. The effects of 3-D models, energy groups, mesh size, SN order, and transport or diffusion theory were investigated. The use of 2-D models with the axial buckling reproduced almost the same eigenvalues of 3-D models for both the fundamental and the first-harmonic mode. The energy group effect on t,he first-harmonic eigenvalue was more significant t,han on the fundamental mode by diffusion theory. For the mesh size and SN order effects, the combination of the mesh size of 0.5 cm and SG is almost sufficient to estimate the eigenvalue separations. It is worth mentioning that the transport effect is the largest of all the effects. Diffusion theory shows a considerable discrepancy from the measurement. On the other hand, transport theory gave the eigenvalue separations closer to the measured values. In conclusion, the present method based on transport theory was proved to be useful to est,imate the eigenvalue separation of such a coupled-core system with large transport effect. -REFERENCES- (1) hilarch-leuba, J., et al.: Nucl. Sci. Eng., 107: 173 (1991). (2) Takeuchi, Y., et al.: Study on mechanism for BWR regional oscillations, Proc. 5th Int. Topical Meeting on Nuclear Thermal Hydraulics (NURETH-S), Salt Lake City, USA, (1992). ( 3 ) Hashimoto, K.: Ann,. Nucl. Energy, 20, 789 (1993). ( 4 ) Hashimoto, K., et al.: Measurements of first-harmonic eigenvalue separation in loosely coupled-core reactor, Proc. Int. Conf. on the Physics of Reactors (PHYSOR 96), Mito, Japan, Vol. 11, E-171 (1996). (5) Hashimoto, K., et al.: 4nn,. Nucl. Energy, 18, 131 (1991). JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

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