Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional

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1 Journal of Nuclear Science and Technology ISSN: (Print) (Online) Journal homepage: Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional Transport Code PARAGON Hideki MATSUMOTO, Mohamed OUISLOUMEN & Toshikazu TAKEDA To cite this article: Hideki MATSUMOTO, Mohamed OUISLOUMEN & Toshikazu TAKEDA (2006) Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional Transport Code PARAGON, Journal of Nuclear Science and Technology, 43:11, To link to this article: Published online: 05 Jan Submit your article to this journal Article views: 108 View related articles Citing articles: 4 View citing articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 26 November 2017, At: 08:37

2 Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 43, No. 11, p (2006) ORIGINAL PAPER Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional Transport Code PARAGON Hideki MATSUMOTO 1;, Mohamed OUISLOUMEN 2 and Toshikazu TAKEDA 3 1 Mitsubishi Heavy Industries, Ltd., Minatomirai, Nishi-ku, Yokohama , Japan 2 Westinghouse Electric Company, Pittsburgh, PA, , USA 3 Osaka University, Yamadaoka, Suita-shi, Osaka , Japan (Received April 7, 2006 and accepted in revised form July 13, 2006) The Spatially Dependent Dancoff Method (SDDM) was recently developed to evaluate the power distribution within a fuel rod that has spatial variation of isotopic contents. The method was validated and verified by comparison to Monte Carlo calculations and measurements. However, those evaluations and comparisons were based on the assumption that the temperature distribution within a rod is flat. In this study, an equation used in the SDDM is enhanced in order to more accurately treat the temperature distribution. The enhancement was carried out with the knowledge that a Monte Carlo calculation shows no effect of temperature distribution on spatial flux within a rod. This leads to the cancellation of reaction rate changes due to temperature distribution between inner and outer regions within a rod. The knowledge gained from these evaluations was then applied to the equation used in the SDDM with the temperature distribution. The improved SDDM was validated and verified by comparison to MCNP4C calculations. PARAGON with the improved SDDM is now capable of performing micro-nuclear physics calculations with greater accuracy. KEYWORDS: resonance self-shielding, temperature distribution, SDDM I. Introduction The spatially dependent Dancoff method (SDDM) 1,2) was recently developed in order to determine the radial power distribution for fuel integrity evaluations. The SDDM module, the micro-depletion module and the multi-region flux solver of the PARAGON 3,4) code have been validated and verified by comparison to Monte Carlo calculations and measurements. However, these evaluations were based on the flat temperature model that assumes a constant temperature within the fuel rod. In this paper, we will present the newly enhanced SDDM model that better accounts for the radial temperature distribution within a fuel rod. In a routine core design calculation, the effective temperature that preserves the total isotopic absorption reaction based on detailed modeling is usually used. A more accurate method is to use a radial temperature distribution within the rod along with a neutron flux distribution. The results from a Monte Carlo calculation showed that the volume average temperature was very close to the effective average temperature. 5) For typical PWR operating conditions, the earlier version of the SDDM can lead to an effective temperature that is 60 K lower than the volume average temperature. 6) This inconsistency with Monte Carlo is too large and therefore makes the SDDM impractical as a calculation model with temperature distribution capability. In this paper, we will present the newly enhanced SDDM model that better accounts for the radial temperature distribution within a fuel pin. This performance of fuel and core designs are more accurately evaluated by incorporating the fuel temperature distribution into determination of the pellet power distribution. Corresponding author, hideki matsumoto@mhi.co.jp ÓAtomic Energy Society of Japan II. Enhancement of Spatially Dependent Dancoff Method 1. Derivation of the SDDM Equation with Temperature Distribution The detailed derivation of the SDDM has already been published in Ref. 1). In this paper, the concept of treating temperature distribution will be explored more closely. In the SDDM, the blackness ( i ) of a concentric fuel ring (i) is expressed by the following equation: i ¼ ð i Þ ð i 1 Þ¼ð i C i A Þ ði 1 C i 1 A Þ: The meanings of the notations are explained in Fig. 1. If the pellet has a temperature distribution, it becomes very difficult to show the blackness for each region. However, whatever temperature distribution the pellet has, the regions other than the target ring i approximately cancel each other out as shown in Fig. 2. If one wants to obtain the blackness of ring region i, its blackness can be expressed as, i ¼fC i ðt iþ A i ðt iþg fc i 1 ðt iþ A i 1 ðt iþg: ð2þ One can assume that the temperature is flat in the target ring. ased on this assumption, the SDDM equation can be easily derived as, g x;i ðt iþ¼ nmk b X 4 m¼1 1 X4 m¼1 F m X 2 n¼1 F m X 2 n¼1 n Ix;g k ðnmk b ; T i Þ Ia;g k ðnmk b ; T i Þ n b nmk ð1þ ð3þ ¼ b þ n = m N k ; ð4þ where b is the background cross-section of the fuel lump and N denotes number density, and k, g, a and i denote nuclide, energy-group, absorption and ring-number, respective- 1311

3 1312 H. MATSUMOTO et al. R Table 1 Definition of the function F m m m F m O A Fig. 1 A C A r ly. The weighting functions F m correspond to the ones defined by Stoker-Weiss as shown in Table 1. The mean chord length of the region OA or OC shown in the Fig. 1 is described as m. The and denote the coefficients of the twoterm approximation where fuel-to fuel collision probability in an assembly is preserved. I x and I a denote resonance integrals per energy group lethargy for reaction type x and absorption, respectively. ρ C i = r R 2R π = 1 ρ + sin ρ ρ π ρ 2 2R = π 1 ρ + sin ρ + ρ π ρ 2 Illustration of chord lengths used in SDDM O A C i C - 1 C ð i Þ S 0 i C ð i Þ 4V i 2 A ð i Þ S 0 i A ð i Þ 4V i 3 C ð i 1 Þ S 0 i 1 C ð i 1 Þ 4V i 4 A ð i 1 Þ S 0 i 1 A ð i 1 Þ 4V i S 0 : fuel pellet surface C and A denote OC and OA as shown in Fig Comparison of the Reactivity Change due to Temperature Profile with MCNP The SDDM equation described in Sec. II-1 was implemented into the PARAGON lattice physics code. PARAGON is a 2-dimensional lattice physics code for LWR assemblies that generates a few group constants for use in a core simulator code such as ANC. 7,8) At the assembly calculation level, the calculation speed is one of the most important requirements. For neutronics core design applications, the mechanical characteristics of the fuel rods are not directly evaluated. Routine PARAGON calculations should be performed with a fuel lump model where flat flux and flat temperature are usually assumed. Therefore the effective temperature that preserves the isotopic reaction rates with detailed modeling of spatially dependent self-shielding and O A C - ' - ' O A A' C' C O A A' C' C = ' A A' C' C Fig. 2 Illustration of the concept expressing the blackness used in SDDM JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

4 Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional Transport Code 1313 temperature distribution was evaluated by using the SDDM equation. Table 2 shows the calculation model for Case 1: temperature distribution model and Case 2: volume average temperature model. The MCNP4C calculation with 100 million histories using the same model in Table 2 was also performed to establish the reference result. The result of PARAGON with SDDM was compared to the MCNP4C result as shown in Table 3. The reactivity differences between Case 1 and Case 2 are 0.142%k and 0:0510:010%k for PARAGON with SDDM and MCNP4C, respectively. The PARAGON result is clearly much larger than the MCNP4C result. In addition, we can conclude that the volume average temperature model cannot represent the detailed temperature distribution model even if a Monte Carlo calculation is used. y using a typical Doppler coefficient of %k=k for a PWR, those reactivity differences are roughly equivalent to about 60 K and 20 K Doppler effects for PARAGON and MCNP4C, respectively. The effective temperature obtained with SDDM becomes 40 K smaller than that of the Monte Carlo calculation. To solve the inconsistency, SDDM was improved using Monte Carlo results. 3. Improvement of SDDM Using Monte Carlo Results The 70 energy group reaction rates and neutron spectrum corresponding to a PARAGON calculation were obtained from the MCNP4C calculation discussed in the previous section. Figures 3 and 4 show the 238 U absorption rate and the neutron flux distributions, respectively. The typical 238 U resonance energy groups around 6.7 ev, 21.0 ev and 36.8 ev are selected. From the figures, it is shown that the peripheral 238 U absorption rate of the temperature distribution model becomes smaller compared to the flat temperature model. At the center, the opposite is true: the 238 U absorption rate Absorption Rate σ<0.13(%) for each Table 2 Calculation model of Monte Carlo Calculation for detailed edit Case 1 Case 2 Cell pitch 1.33 (cm) Pellet outer radius 0.41 (cm) Cladding outer radius 0.48 (cm) 235 U enrichment 4.1 (%) Fuel region (K) 01 (center) 1, , , , , for each region Cladding (K) Moderator (K) Unit: Kelvin The pellet region is divided into 10 equal ring regions for both Cases 1 and 2. Table 3 Comparison of the reactivity change between the volume average flat temperature model and the parabolic distribution model Reactivity change Flat Temperature Model PARAGON with SDDM Temperature profile Model MCNP4C Difference :0510:010 0:0910: Fig U absorption rate distribution in a typical UO 2 (4.1 wt%) rod obtained with the Monte Carlo code MCNP4C calculations for large resonance energy ranges VOL. 43, NO. 11, NOVEMER 2006

5 1314 H. MATSUMOTO et al Flat Temperature Model Temperature profile Model Neutron Flux (n/cm2) σ<0.05(%) for each Fig. 4 Neutron flux distribution in a typical UO 2 (4.1 wt%) rod obtained with the Monte Carlo code MCNP4C calculations for large resonance energy ranges Relative Difference (%) (Profile - Flat)/Flat x Fig. 5 Comparison of 238 U absorption rate distribution in a typical UO 2 (4.1 wt%) rod obtained with the Monte Carlo code MCNP4C calculations for large resonance energy ranges of the temperature distribution model is larger compared to the flat temperature model. Regarding neutron flux, there are no remarkable differences between the profile case and the flat case. Taking a closer look at the discrepancies, the relative differences are shown in Figs. 5 and 6 for the 238 U reaction rate and the neutron flux, respectively. It can be seen that the relative differences between reaction rates are large, about 5%, but the differences between neutron fluxes are very small, within 0.3%. The analysis above can be applied to Eq. (3). The numerator of Eq. (3) denotes the spatially dependent reaction rates. And the denominator denotes the neutron flux. Therefore, to match the Monte Carlo result, the temperature dependency of the neutron flux simply has to be eliminated JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

6 Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional Transport Code Relative Difference (%) (Profile - Flat)/Flat x 100 and the flat temperature profile, such as volume average or chord average temperature, 7) is used instead of the regionwise model: g x;i ðt iþ¼ X 4 m¼1 1 X4-1.0 m¼1 F m X 2 n¼1 F m X 2 n¼1 n Ix;g k ðnmk b ; T i Þ Ia;g k ðnmk b ; T Avg Þ n b nmk : ð5þ The SDDM with Eq. (5) will be hereafter referred to as the improved SDDM or ISDDM. Later in Chap. IV we will discuss the way in which the temperature distribution should be averaged. The ISDDM still has a temperature dependency. The average temperature is applied only to the denominator of the ISDDM equation. The reaction rate change due to temperature distribution can be directly treated in the numerator of Eq. (5). III. Numerical Results Fig. 6 Comparison of neutron flux distribution in a typical UO 2 (4.1 wt%) rod obtained with the Monte Carlo code MCNP4C calculations for large resonance energy ranges 1. Validation of the ISDDM To validate the ISDDM, the effects of temperature distribution on reactivity are compared between a flat and a radial profile. The results of SDDM, ISDDM and MCNP4C are given in Table 4. It can be seen that the ISDDM model reduces the radial temperature distribution effect from 142 to 5 pcm. This means the ISDDM is an adequate model. MCNP4C estimated the effect to be 51 pcm with a standard deviation of 10 pcm. Hence, the ISDDM based on Eq. (5) shows better agreement with MCNP than the current SDDM based on Eq. (3). The chord average temperature instead of the volume average temperature is also applied to the denominator of Eq. (5). The result is in excellent agreement Table 4 Comparison of the temperature distribution effect on the reactivity Reactivity (pcm) PARAGON SDDM ISDDM a) ISDDM b) MCNP4C Statistical error (1) a) The volume average temperature is used in the denominator of Eq. (5). b) The chord average temperature is used in the denominator of Eq. (5). Note: The reactivity denotes the difference between Cases 1 and 2 in Table 2. with the MCNP4C results, also shown in Table 4. To make the ISDDM result consistent with the MCNP result, the chord average temperature should be applied to the denominator of Eq. (5) rather than the volume average temperature. 2. Comparison of Power Distribution and Reaction Rates Figure 7 shows the comparison of power distribution ( 235 U fission rate) of the ISDDM, the SDDM and MCNP4C. The specifications of the Case 1 of Table 2 were used in the computation. It can be seen that the power distribution is identical between the ISDDM and the SDDM. Note also that both results agree very well with the MCNP result. Figure 8 shows the comparison of 238 U absorption rate distribution. The SDDM and the ISDDM also agree very well with MCNP4C. It is found that the new ISDDM model does not affect the spatial distribution of the power or the total reaction rates over energy. VOL. 43, NO. 11, NOVEMER 2006

7 1316 H. MATSUMOTO et al Relative Power MCNP SDDM ISDDM 1σ<0.02(%) for each Fig. 7 Comparison of power distribution for a typical UO 2 (4.1 wt%) rod among SDDM, ISDDM and MCNP4C Reaction Rate (Arbitrary Unit) MCNP4C 1σ<0.01(%) for each ISDDM SDDM Fig. 8 Comparison of 238 U absorption rate distribution for a typical UO 2 (4.1 wt%) rod among SDDM, ISDDM and MCNP4C IV. Discussion It is found that the ISDDM is able to evaluate the reactivity change due to the temperature distribution effect. The ISDDM with the chord average temperature shows excellent agreement with the MCNP4C result. In this section, the reason why the chord average temperature is appropriate for the denominator of Eq. (5) is discussed. 1. Effective Temperature In the MCNP4C calculations mentioned in Sec. III-1, the 5110 pcm difference was observed due to the temperature distribution change. It is certain that the effect comes only from the difference between the volume average flat temperature and the radial temperature profile. This means that the volume average temperature model has to be corrected to be consistent with the exact model. Although the reaction rate JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

8 Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional Transport Code 1317 Table 5 Chord average weighting factors for ten rings with equalvolumes in fuel rod Ring region (Inner to outer) changes due to the temperature distributions of the volume average and parabolic profile are almost canceled out as previously shown in Fig. 5, the total absorption rate of the average temperature case becomes larger than the profile case because the peripheral reaction contributes more than inner reaction as shown in Fig. 3. This usually results in a reactivity loss for the flat model. Therefore, a flat temperature that is lower than the volume average temperature is needed in order to be consistent with the profile model in terms of an effective temperature. The chord average temperature is defined in Ref. 10) as follows: Z T ChordAvg ¼ ds S Z n>0 dgðr s ; Þ Lðr Z s ;Þ 0 d Tð Þ Lðr s ; Þ ; ð6þ where S: Surface of the fuel lump n: Inner surface normal at S : Unit vector at the position r s on the surface gðr s ; Þ: Probability distribution for the chords Lðr s ; Þ: Length of the chord. For the isotropic and uniform incidence of the neutrons, gðr s ; Þ is given by gðr s ; Þ ¼ n S : ð7þ If the pellet is divided into ten rings of equal-volume, the chord average temperature is given by T ChordAvg ¼ X10 W k T k ; ð8þ k¼1 Weighting factors (W k in Eq. (8)) where T k denotes the average temperature of the ring k, and W k denotes the weighting factor given in Table 5. The radial distribution of the weighting factors is similar to the reaction rate distribution. To show the validity of the use of the chord average temperature, numerical comparisons were made based on Monte Carlo calculations. The MVP code was used for the comparison because the temperature distribution can be easily modeled. Figure 9 shows the com parison between the effective temperature obtained with MVP and the chord average temperature for several power levels at the beginning of life and at about 15 GWd/t. It can be seen that the chord average temperature agrees well with the effective temperature obtained by the Monte Carlo code. Although the effective temperatures obtained by the Monte Carlo calculations are close to the volume average temperatures up to about 1,200 K, the chord average temperatures are closer to the Monte Carlo results than the volume average temperatures within the entire temperature range. 2. ISDDM Temperature Here, let us summarize the cases of the SDDM calculations including the ISDDM. Table 6 shows the comparison of k-infinities for all cases with their model specifications. All PARAGON results are in agreement with MCNP results within about 100 pcm for the pin cell calculation. Although a difference of 100 pcm is not small in comparison to the effect of the temperature distribution, the difference does not affect the conclusion of this discussion. This is because the objective of the development of the ISDDM is to evaluate the effect of temperature distribution on reactivity with high accuracy. From the viewpoint of the reactivity change, it is found that the SDDM with the chord average temperature in Eq. (5) agrees well with MCNP4C. This model has been defined as the improved SDDM or ISDDM in this paper. For the temperature distribution cases labeled Parabolic in Table 6, the earlier version of the SDDM shows the best k-infinity agreement with MCNP4C. However, it shows the worst result, that is 142 pcm for the temperature effect. For the conventional one-region model used in the current design calculation, the temperature distribution effect has been considered as the effective flat temperature. The effective flat temperature is equivalent to the chord average temperature. The result is labeled as One Region model in Table 6. It can be seen that the reactivity effect agrees well with the ISDDM and MCNP4C. In other words, the ISDDM gives almost the same reactivity effect for temperature distribution as the conventional effective temperature model that is well established and commonly applied in current design calculations. In order to be consistent with both MCNP and the conventional model, the ISDDM should employ the chord average temperature as T Avg in Eq. (5). V. Conclusion ased on Monte Carlo results, the SDDM (Spatially Dependent Dancoff Method) was improved to treat the spatial self-shielding by accounting for the radial profile temperature distribution. Although the SDDM was able to correctly predict the power distribution and the reaction rate distribution, the method over predicted the reactivity when the temperature distribution was considered. Comparison to MCNP results showed that the inconsistency can be solved by means of the flux distribution used in the SDDM. Considering the fact that the Monte Carlo flux distribution is almost flat, the SDDM equation was modified to force the flux distribution to be flat within the pellet. The ISDDM can now accurately predict reactivity relative to a Monte Carlo VOL. 43, NO. 11, NOVEMER 2006

9 1318 H. MATSUMOTO et al Effective Temperature obtained with Monte Carlo code (1σ<10(k)) Chord Average Temperature Temperature(K) MWd/t Volume Average Fuel Temperature(K) Temperature(K) 1500 Effective Temperature obtained with Monte Carlo code (1σ<10(k)) Chord Average Temperature GWd/t Volume Average Fuel Temperature(K) Fig. 9 Comparison between the effective temperature obtained with MVP and the chord average method Table 6 Comparison of k-infinity for temperature distribution effect Model Temperature k-inf k=k (pcm) Remarks One region Volume average Chord average Conventional Dancoff Method Volume average Parabolic The earlier version of the SDDM 10 region Volume average Volume average temperature for 5 Parabolic the T Avg in Eq. (5) Volume average Chord average temperature for 58 Parabolic the T Avg in Eq. (5): The ISDDM 10 region Volume average MCNP4C 51 Parabolic : for both cases simulation. The power distribution and the reaction rate distribution obtained with the ISDDM are also in good agreement with MCNP. Moreover, the reactivity predicted with the ISDDM agrees very well with the conventional flat flux and effective temperature models. This consistency with the conventional method makes PARAGON very useful to accurately perform fuel integrity evaluations for various core designs. JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

10 Spatially Dependent Self-Shielding Method with Temperature Distribution for the Two-Dimensional Transport Code 1319 Acknowledgments Authors would like to thank Mr. M. Nakano and Mr. Y. Komano of Mitsubishi Heavy Industries, Ltd. (MHI) for their valuable advice and help. They also wish to acknowledge Ms. T. Shiraki, Mr. D. Sato and Mr. K. Yamaji of MHI for their assistance in the execution of the Monte Carlo calculations and other support. References 1) H. Matsumoto, et al., Development of spatially dependent resonance shielding method, J. Nucl. Sci. Technol., 42[8], (2005). 2) H. Matsumoto, et al., Depletion calculation for PWR assemblies including burnable absorbers with lattice code PARAGON, J. Nucl. Sci. Technol., 43[2], (2006). 3) M. Ouisloumen, et al., PARAGON: The New Westinghouse Lattice Code, Proc. ANS Int. Meeting on Mathematical Methods for Nuclear Applications, Sept. 2001, Salt Lake City, Utah, USA, (2001). 4) M. Ouisloumen, et al., The new lattice code PARAGON and its qualification for PWR core applications, Proc. Int. Conf. on Supercomputing in Nuclear Applications, SNA 2003, Sept. 2003, Paris, France, (2003). 5) T. Kitada, et al., Investigation of effective fuel temperature by continuous energy Monte Carlo calculation, Preprints 2003 Fall Meeting of At. Energy Soc. Jpn., Shizuoka, Sep , 2003, I35, (2003), [in Japanese]. 6) H. Matsumoto, et al., Spatially and temperature dependent Dancoff method for LWR lattice physics code, Proc. Int. Conf. on The Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments, PHYSOR2004, Chicago, IL, USA, (2004). 7) Y. S. Liu, et al., ANC A Westinghouse Advanced Nodal Computer Code, WCAP-10966, Westinghouse Electric Corp., (1985). 8) T. Q. Nguyen, et al., Qualification of the PHOENIX-P/ANC Nuclear Design System for Pressurized Water Reactor Cores, WCAP A, (1988). 9) Los Alamos National Laboratory, MCNP4C: Monte Carlo N-Particle Transport Code System, CCC-700, (2000). 10) W. J. M. de Kruijf, et al., The effective temperature to be used for calculating resonance absorption in a 238 UO 2 lump with nonuniform temperature profile, Nucl. Sci. Eng., 123, (1996). 11) T. Mori, et al., MVP/GMVP: General Purpose Monte Carlo Codes for Neutron and Photon Transport Calculations based on Continuous Energy and Multigroup Methods, JAERI- Data/Code , (1994). VOL. 43, NO. 11, NOVEMER 2006