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Development of analysis method for sodium void reactivity of step type FBR cores by using group-wise Monte Carlo code
Annals of Nuclear Energy 38 (2011) 1382–1388
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Development of analysis method for sodium void reactivity of step type FBR cores by using group-wise Monte Carlo code Tsugio Yokoyama a,⇑, Hiroshi Endo b, Hisashi Ninokata c a
Toshiba Nuclear Engineering Services Corporation, 8 Shin-sugita-cho, Isogo-ku, Yokohama-shi, Kanagawa 235-8523, Japan Safety Analysis and Evaluation Div., Japan Nuclear Energy Safety Organization (JNES), Kamiya-cho MT Bldg., 4-3-20, Toranomon, Minato-ku, Tokyo 105-0001, Japan c Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan b
a r t i c l e
i n f o
Article history: Received 30 September 2010 Received in revised form 24 January 2011 Accepted 27 January 2011 Available online 23 February 2011 Keywords: FBR Step type core Sodium void reactivity Perturbation theory Monte Carlo method GMVP
a b s t r a c t A new method is proposed to separate the sodium void reactivity of step type FBR cores to components including non-leakage terms and a leakage term by using a newly developed perturbation code MCPERT where fluxes and adjoint fluxes are derived from a group-wise Monte Carlo code. The step type FBR core is a core where the height of the inner core is smaller than that of the outer core and a large sodium plenum region is located above the core so as to decrease the sodium void reactivity. The conventional diffusion perturbation method cannot treat such a large void region due to the diffusion approximation, while the Monte Carlo code can treat it exactly. In this study, a group-wise Monte Carlo code GMVP with a 70-group constant set JFS-3-J3.3 is employed to evaluate the neutron fluxes and adjoint fluxes which are used as inputs to the MCPERT code to evaluate the non-leakage terms. The leakage term is derived from the difference of the total sodium void reactivity evaluated by the direct calculation of GMVP and the summation of the non-leakage terms. It is found that the proposed method can provide the result approximately consistent to the ratio of the reactivity components derived from the conventional method. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In typical large FBR cores, the loss of coolant sodium can result in a large positive reactivity effect. This problem, not existed in thermal reactors, will present an important safety issue for FBRs. The sodium loss reactivity effect is made of two components. One is non-leakage term, which expresses the effect of neutron reaction change when the system is infinite and the neutron leakage does not exist, and the other is leakage term, which is the reactivity effect due to the increase of neutron leakage though the space of the sodium voided. The former is positive and the latter negative in typical large FBRs because the former is reactivity mainly due to the hardening of neutron spectrum where g, the number of emitted neutrons per absorption to fissile nuclides increases along with neutron energy increase. The summation of two components is usually positive in typical cores, but various efforts have been dedicated to decrease the reactivity. The step core concept is one of them. The step core is a combined core of a lower height of the inner region and a higher height of the outer core region. The typical way to decrease sodium void reactivity in large
⇑ Corresponding author. Tel.: +81 45 770 2419; fax: +81 45 770 2336. E-mail address: [email protected] (T. Yokoyama). 0306-4549/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2011.01.033
size cores is to decrease core heights, but it leads higher enrichment which sometimes ends up with poor core performances. Then we applied this way only to the inner region of the core, where sodium void reactivity is large positive, and the height of the outer region is remained higher, where it usually negative due to large neutron leakage. In addition we employed a combination of large fuel volume fraction fuel core to decrease the enrichment. Another effective way to decrease the sodium void reactivity is to elide the upper axial blanket. The increase in neutron leakage through the voided upper region of the core can enhance the negative void reactivity effect of the core. Thus we have adopted this type of step cores with a negative void reactivity as the target core to which our new method apply. In our previous studies, the continuous energy Monte Carlo method code, MVP (Yokoyama et al., 2006), was employed to analyze the core characteristics as it can treat complicated configurations with various neutron spectrum. But it sometimes requires long time to achieve statistically accurate results due to Monte Carlo method and cannot analyze the components of sodium void reactivity because it cannot calculate adjoint fluxes. In this paper, the separation of the components, non-leakage term and leakage term, was tried for the step core by using a multi-group version of MVP code, which can calculate real fluxes in a short time in addition to be able to calculate adjoint flux. The separation to the
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Nomenclature FBR ULOF JAEA
fast breeder reactor unprotected loss of flow event Japan atomic energy agency
components is a common way to identify physical features of the sodium void reactivity and useful to fit the calculation data to various experimental data of sodium void reactivity of critical experiments.
2. Step core concept and specifications For the commercialization of large FBRs, increase in safety characteristics and improve in economy are the most important aspects. Recent concerns on the safety design are to prevent the release of large kinetic energy at Unprotected Loss of Flow (ULOF) events (Yokoyama et al., 2006). The core with a negative void reactivity and an excess reactivity less than 1 $ is the most desirable core to provide a comprehensive persuasion under a simple developing scenario of accidents. If the sodium void reactivity is negative, there is no positive reactivity factor at loss of flow events. The step core where the core height of the inner core is lower than that of the outer core can resolve the above problems at the same time when metal fuel is employed by adjusting the core height. The typical way to decrease sodium void reactivity in large size cores is to decrease the core height, but it leads higher enrichment which sometimes ends up with large burn-up reactivity. Then we employed the combination of large fuel volume fraction metal fuel core to decrease the enrichment. Another effective way to decrease the sodium void reactivity is to elide the upper axial blanket. In our study of the no upper axial core, it was found that the sodium void reactivity in outer core is usually negative thus the flattening of the inner core only is effective to decrease the sodium void reactivity when the core diameter is limited. Table 1 lists the specifications of the typical step core when MOX fuel is employed. Fig. 1 illustrates the core layout and core profile. Table 1 Specifications of MOX step core. Specifications
Unit
Step core
Fuel composition Coolant Thermal power Cycle length Inlet temperature Outlet temp. Peak clad. temp. Max. linear powera Core press. drop Core diameter Core height (innerb/outer) Axial blanket thickness (top/bottom) Gas plenum length (upper/lower) Fuel pin diameter Fuel pin p/d No. of pins Assembly pitch Fuel vol. fraction Fuel smear density Pu enrichment (inner/outer)
– – MWt Year °C °C °C W/cm MPa m m m m mm – /Ass. mm % % %
MOX Sodium 3000 1 355 510 650 400 0.5 4.4 0.7/1.0 0.0/0.4 0.3/1.0 7.8 1.16 271 162.8 45 85 18.7/25.5
3. Calculation method of GMVP The sodium void reactivity of the core has been evaluated with group-wise Monte Carlo code GMVP (Nagaya et al., 2004) and the JFS-3-J3.3 70 energy group constant set (Nakagawa et al., 1995) as shown in Table 2. Each fuel assembly has been modeled as a hexagonal homogeneous cell fore each axial region. The history numbers of the Monte Carlo calculations has been selected to provide enough accuracy where the k-eff error is less than 0.01%. The ANISN-type cross sections or the double-differential cross sections can be used in the multi group code GMVP, while the special cross section libraries are used in the continuous-energy code MVP. The libraries are generated from the evaluated nuclear data (JENDL-3.1, -3.2, -3.3, ENDF/B-IV, -V, -VI, etc.) by using the LICEM code (Mori et al., 1996). The cross sections in the unresolved resonance region are described by the probability table method. In this study, we employed JENDL-3.3 for MVP as the cross section library and JFS-3-J3.3 70 group set (NEA-0796, 2001) for GMVP. This group constant set has been derived from JENDL-3.3 by using a fast neutron spectrum for the use of FBR core design, where the energy group structure is composed of 70 neutron groups over 10 MeV to 0.00001 eV. The effective cross sections of the core for ANISN-type cross section have been generated with SLAROM (Nakagawa and Tsuchihashi, 1984) and SLTOCCC code that are developed by JAEA. The JFS-3-J3.3 70 group set is collapsed by using the typical FBR core spectrum at flooded condition (NEA-0796, 2001). In order to evaluate the collapsing effect of the spectrum used, the difference of the cross section of Pu-239 between the flooded and voided condition has been calculated for the inner core of the step core using the continuous energy MVP code with 10 million neutron histories. Table 3 presents the comparison of mrf of Pu-239 between the flooded spectrum and voided spectrum and statistical errors. The error increases among lower energy groups where reaction rates decrease. In particular, no reactions has taken place in the lowest three groups even such a large number of histories are employed. The ratio of the cross section is about 0.02% in average over 1 keV in neutron energy, where about 80% of fission reactions occur. Even among the lower energy groups, the cross sections between the two conditions agree within the statistical errors. Thus we have concluded the collapsing effect within each group of the JFS-3J3.3 set is too small to affect the evaluation of sodium void reactivity. For the adjoint flux calculation of GMVP, ordinary adjoint matrix of the 70 groups cross section set similar to the diffusion calculation of adjoint fluxes are employed, while the sampling of fission source in Monte Carlo calculation is derived from the fission spectrum, not from mrf, which is used for the fission source of real flux calculation.
4. Component analysis method using quasi perturbation theory for sodium void reactivity
a
Maximum linear power of the core with 0.7 m of inner core height. The inner core height is varied from 0.5 m to 0.9 m for the parametric survey of void reactivity. b
Based on multi group exact perturbation theory, a reactivity effect q is expressed as follows:
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Fig. 1. Reference MOX fuel step core (inner core height: 70 cm).
hR P
q¼
g /g
vg
P
mRf Þg0 /g0 dV þ
g 0 dð
RP
g
r/g dDg r/g dV RP
g /g
vg
P
g0 ð
RP
g /g d
Rrg /g dV þ
mRf Þg0 /g0 dV
Table 2 Calculation method of k-eff. Item
Condition
Cross section library Code Method
JFS-3-J3.3 GMVP 70 group Three dimensional Monte Carlo Three dimensional Hex-Z 10,200,000 10,000 20
Core Model Number of neutron histories Number of neutrons per batch Number of skipped batches for neutron source convergence
where vg is the fission spectrum; /g, the neutron flux of group at perturbed condition; /0g the adjoint flux of group at unperturbed
RP
g /g
vg
P
g 0
Rg0 !g /g0 dV
i ð1Þ
condition; mRf, the macroscopic neutron generation cross section; Dg, the diffusion coefficient; Rr, the macroscopic removal cross section; Rg?g0 , the macroscopic scattering matrix; d, the difference of cross sections between perturbed and unperturbed conditions; and dV is the differential region volume. The difference of cross sections are just derived from the difference between the cross sections of the regions voided to those of the regions flooded, where these cross sections are evaluated by the SLAROM code for generating effective 70 group cross sections for GMVP. GMVP calculate real fluxes and adjoint fluxes while MVP can calculate only real fluxes. The fluxes and adjoint fluxes of GMVP are calculated as the total length of neutron tracks crossing the region concerned (Nagaya et al., 2004). Ordinary defined fluxes can be derived from the total length of tracks divided by the volume of the region. Thus the non-leakage terms have been calculated
Table 3 Comparison of Pu-239 cross sections collapsed with voided core spectrum. Upper energy (eV)
mrf at
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
1.00E+07 7.79E+06 6.07E+06 4.72E+06 3.68E+06 2.87E+06 2.23E+06 1.74E+06 1.35E+06 1.05E+06 8.21E+05 6.39E+05 4.98E+05 3.88E+05 3.02E+05 2.35E+05 1.83E+05 1.43E+05 1.11E+05 8.65E+04 6.74E+04 5.25E+04 4.09E+04 3.18E+04 2.48E+04 1.93E+04 1.50E+04 1.17E+04 9.12E+03 7.10E+03 5.53E+03 4.31E+03 3.35E+03 2.61E+03 2.03E+03
9.525 7.842 6.21 6.143 6.175 6.247 6.265 6.019 5.498 5.152 4.849 4.684 4.572 4.446 4.347 4.294 4.315 4.357 4.414 4.461 4.482 4.506 4.555 4.774 5.003 5.219 5.332 5.549 6.497 6.369 6.926 8.328 10.248 8.365 10.193
flooded core spectrum (barn)
Statistical error of mrf at flooded (%)
mrf at voided core spectrum (barn)
Statistical error of mrf at voided (%)
Voided/ flooded ratio
Statistical error of the ratio (%)
Group no.
Upper energy (eV)
mrf at flooded core spectrum (barn)
Statistical error of mrf at flooded (%)
mrf at voided core spectrum(barn)
Statistical error of mrf at voided (%)
Voided/ flooded ratio
Statistical error of the ratio (%)
0.13 0.49 0.12 0.00 0.01 0.01 0.01 0.06 0.07 0.03 0.04 0.02 0.01 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.01 0.25 0.29 0.32 0.35 0.41 0.46 0.51 0.60 0.72 0.88 1.29 1.38
9.51 7.902 6.207 6.143 6.174 6.246 6.265 6.026 5.5 5.149 4.852 4.686 4.572 4.447 4.347 4.294 4.315 4.357 4.414 4.46 4.482 4.506 4.555 4.786 5.001 5.204 5.328 5.571 6.479 6.375 7.02 8.507 10.159 8.749 10.005
0.12 0.49 0.12 0.00 0.01 0.01 0.01 0.07 0.07 0.03 0.04 0.02 0.01 0.02 0.01 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.01 0.27 0.30 0.35 0.38 0.44 0.49 0.54 0.59 0.76 0.81 1.27 1.45
0.998 1.008 1 1 1 1 1 1.001 1 0.999 1.001 1 1 1 1 1 1 1 1 1 1 1 1 1.002 1 0.997 0.999 1.004 0.997 1.001 1.014 1.021 0.991 1.046 0.982
0.25 0.98 0.24 0.00 0.03 0.02 0.01 0.13 0.14 0.07 0.09 0.03 0.03 0.03 0.02 0.00 0.01 0.01 0.02 0.01 0.00 0.01 0.02 0.52 0.59 0.67 0.73 0.85 0.95 1.05 1.19 1.47 1.69 2.56 2.83
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70a
1.58E+03 1.23E+03 9.61E+02 7.49E+02 5.83E+02 4.54E+02 3.54E+02 2.75E+02 2.14E+02 1.67E+02 1.30E+02 1.01E+02 7.89E+01 6.14E+01 4.79E+01 3.73E+01 2.90E+01 2.26E+01 1.76E+01 1.37E+01 1.07E+01 8.32E+00 6.48E+00 5.04E+00 3.93E+00 3.06E+00 2.38E+00 1.86E+00 1.45E+00 1.13E+00 8.76E01 6.83E01 5.32E01 4.14E01 3.22E01
13.532 18.369 15.764 12.032 44.771 21.751 39.033 52.713 52.359 50.124 60.523 113.98 180.37 215.4 59.466 11.59 49.319 205.71 260.68 381.32 39.568 285.41 20.115 21.062 26.742 32.231 40.571 55.091 70.643 109.44 173.86 311.05 NA NA NA
1.30 1.50 1.07 1.33 1.47 1.80 1.83 1.75 3.19 2.67 4.38 3.44 10.67 5.02 11.83 13.19 16.74 15.98 12.87 24.27 27.46 30.64 0.09 0.81 1.94 2.31 1.65 2.25 3.20 4.37 6.59 3.28 NA NA NA
13.192 18.228 16.432 11.823 46.199 21.049 37.743 56.086 58.599 55.049 64.127 115.24 165.7 186.41 56.356 7.823 51.272 326.32 268.03 502.06 61.21 224.36 20.613 21.404 25.531 32.344 42.893 53.816 69.398 111.84 178.14 314.93 NA NA NA
1.34 1.66 1.15 1.38 1.58 1.78 1.72 2.10 3.75 3.24 5.48 4.03 10.34 6.02 11.14 7.95 15.53 13.75 12.13 22.39 48.37 39.31 2.17 1.31 0.66 1.52 2.80 2.89 2.93 3.71 6.45 4.17 NA NA NA
0.975 0.992 1.042 0.983 1.032 0.968 0.967 1.064 1.119 1.098 1.06 1.011 0.919 0.865 0.948 0.675 1.04 1.586 1.028 1.317 1.547 0.786 1.025 1.016 0.955 1.004 1.057 0.977 0.982 1.022 1.025 1.012 NA NA NA
2.63 3.17 2.21 2.71 3.05 3.58 3.55 3.84 6.93 5.91 9.86 7.46 21.01 11.04 22.97 21.13 32.27 29.73 25.00 46.66 75.83 69.95 2.26 2.12 2.60 3.83 4.45 5.14 6.14 8.08 13.05 7.45 NA NA NA
T. Yokoyama et al. / Annals of Nuclear Energy 38 (2011) 1382–1388
a
Group no.
Lower energy boundary of 70th group is 1.00E5 eV.
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by multiplying the two fluxes and the difference of the cross sections of the perturbed regions in the MCPERT code, which has been newly developed in this study. The second term of Eq. (1) is difficult to calculate in Monte Carlo codes because the gradient of fluxes cannot be exactly evaluated due to the increase in statistical error to in a thin zone that is needed to evaluate the gradient of fluxes. In this study, the second term, which is correspond to leakage term in sodium void reactivity, is assumed to be derived from the difference between the total reactivity and the non-leakage term, which is expressed as the summation of the 1st, 3rd and 4th term of Eq. (1), i.e.,
ql ¼ qtotal qnon
ð2Þ
where ql is the leakage term of sodium void reactivity; qtotal, the total sodium void reactivity evaluated with direct calculation of GMVP; and qnon is the Non-leakage term of sodium void reactivity derived by Eq. (1) without the second term of the numerator. All of the non-leakage terms shown in formula (1) are calculated in the MCPERT code to evaluate the non-leakage component. The statistical errors of the total reactivity have been evaluated with the statistical errors of the direct calculation results of GMVP by using the error propagation formula. The statistical errors of the non-leakage term have been evaluated by using the statistical errors of the fluxes and adjoint fluxes, which are calculated by the perturbed and unperturbed GMVP calculations, respectively.
5. Analysis of sodium void reactivity of reference core In the step core without upper axial blanket, the main contributor to decrease sodium void reactivity is the neutron leakage through the upper gas plenum region. Thus we have simulated the upper gas plenum region exactly for GMVP calculations. As shown in Table 1, the upper gas plenum length of the fuel is set at 0.3 m, where springs to constrain fuel pellets are installed. In the conventional model employed for diffusion or transport calculations, each zone was modeled with a homogeneous region as shown in Fig. 2. In this study, however, this plenum region has been simulated with a heterogeneous assembly model as shown in Fig. 3 (horizontal cross section of the fuel) and Fig. 4 (vertical cross section). The results of the sodium void reactivity of the reference step core, which has a inner core height of 70 cm, is shown in Table 4 for this model. The sodium void reactivity for the void of the core and upper plenum, when all of the sodium above the top level of lower axial blanket, is 0.22%Dq, while the void reactivity of the active core region is 1.03%Dq. Table 5 is a result of components of the sodium void reactivity derived from the separation method described above. In this table, (C), the void reactivity of plenum region is calculated with GMVP for the model when only the upper region above the top level of the active core is voided, and the (D), the non-leakage term of the core region is calculated by Eq. (1) without the second term.
Control rod
upper gas plenum of Inner core
Outer core Inner core
gas plenum
Outer core
Lower axial Blanket Fig. 2. Cross section model of the step core.
Fig. 3. Horizontal cross section of pin heterogeneous model at upper gas plenum level.
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Upper gas plenum of outer core
dition of the plenum region does not affect the reactivity components of the core region.
Upper gas plenum of inner core
6. Application to step cores of different inner core heights In this section, an application of the method to step cores with different inner core heights is intended, which will be useful to verify how the method could be adopted to predict the sodium void reactivity of a given core height. The relation of the components of sodium void reactivity to the inner core height of step cores is analyzed using the method described above. In this study, we have changed only the height of the inner core. When we determine the specifications of the core with a negative sodium void reactivity, the change of the inner core height may affect other core characteristics such as average power density or peak linear power, but sodium void reactivity is considered to be most sensitive among the main core characteristics because it strongly depends on neutron leakage. Other core parameters can be easily adjusted if the core height is selected appropriately to satisfy the criteria of sodium void reactivity. Table 6 and Fig. 5 presents the results for the inner core height of 50–90 cm. Fig. 5 indicates the main component dominating the change of void reactivity with inner core heights is the leakage term of the core. The non-leakage component increases with the inner core height because the neutron leakage of the higher core is lower than that of the lower core and the neutron spectrum shift dominates the reactivity change when the core is voided. As shown in the figure, the curves of each component are smooth over the range, which implies the method can be applicable to predict the components of the reactivity of the cores where the existing diffusion perturbation method cannot be applied such as a core that contains a large void region. According to the experiences on the component study with conventional method using diffusion perturbation calculations, the ratios of the non-leakage term to the leakage term for the typical large MOX critical assemblies ZPPR-9 and ZPPR-10A, where the core heights are 101.8 cm, are 0.47 and 0.52, respectively, when the whole core height of 101.8 cm is voided (Chiba, 2006). In Table 6, this ratios are 0.56 and 0.50 when the inner core height is 80 cm and 90 cm, respectively. Thus the result of this method is approximately consistent to that of the conventional method for the component analysis of sodium void reactivity.
Fig. 4. Vertical cross section of pin heterogeneous model at gas plenum.
Table 4 Region-wise sodium void reactivity of reference (70 cm in inner core height) for heterogeneous plenum model. Voided regions
k-eff
Non (intact)
1.0372
Core + upper plenum
1.0348
Core
1.0484
Plenum
1.0242
Void reactivity(%Dq) –
Statistical error (%Dq,1r)
Comments
–
Direct calculation GMVP Direct calculation GMVP Direct calculation GMVP Direct calculation GMVP
0.014
0.22
1.03
0.014
0.014
1.23
with
with
with
with
(E) is the leakage component of the core derived from the formula (B)–(D). (F) is that of the core derived from the formula (A)–(C)–(D). These two results are about the same within the statistical error, which is the summation of the error of the flux and that of the adjoint flux for the perturbed region. This means the void conTable 5 Component-wise sodium void reactivity of the reference core (70 cm inner core height). Voided regions
Components
Void reactivity (%Dq)
Statistical error (%Dq,1r)
Comments
(A) Core + upper plenum (B) Core (C) Plenum (D) Core (E) Core (F) Core
Non-leakage + leakage Non-leakage + leakage Non-leakage + leakage Non-leakage Leakage Leakage
0.22 1.03 1.21 2.81 1.78 1.82
0.014 0.014 0.014 0.11 0.18 0.18
Direct calculation with GMVP Direct calculation with GMVP Direct calculation with GMVP Perturbation calculation with GMVP adjoint flux (B)–(D) (A)–(C)–(D)
Table 6 Component-wise sodium void reactivity for different inner core heights unit: (%Dq).
a
Voided regions
Inner core height (cm)
50
60
70
80
90
Statistical errora (%Dq,1r)
(A) Core + upper plenum (B) Core (C) Plenum (D) Core (E) Core (E)/(C)
Non-leakage + leakage Non-leakage + leakage Non-leakage + leakage Non-leakage Leakage –
1.55 0.15 1.4 2.23 2.38 1.07
0.75 0.55 1.3 2.56 2.01 0.79
0.22 1.03 1.21 2.81 1.78 0.63
0.25 1.35 1.1 3.04 1.69 0.56
0.55 1.6 1.05 3.09 1.56 0.50
0.02 0.02 0.02 0.12 0.2 –
Maximum among different core heights.
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4.0
Sodium void reactivity (%Δρ)
3.0
(A)Core+upper plenum (Non-leakage+leakage)
2.0
(B)Core (Non-leakage+leakage) 1.0
(C)Plenum (Non-leakage+leakage) 0.0
(D)Core (Non-leakage)
-1.0
(E)Core (leakage)
-2.0
-3.0
50
60
70
80
90
100
Inner core height (cm) Fig. 5. Component-wise sodium void reactivity for different inner core heights.
7. Conclusions The reactivity component of sodium void reactivity of the step core, which has a lower inner core height relative to the height of the outer core, and the upper axial blanket is eliminated, has been evaluated with a new method using GMVP, a group-wise Monte Carlo code and the cross section library JFS-3-J3.3 70-group constant set. The ratio of the components of non-leakage to the leakage term of the active core region is similar to that of conventional method, which implies this method proposed will provide reasonable results for complicated core configurations that can be evaluated only by Monte Carlo codes. The analysis of the relation of void reactivity to different core heights using the proposed method shows that the method can provide the result approximately consistent to the ratio of the reactivity components derived from the conventional method for the whole core height void. Acknowledgments The authors express their gratitude to the members of Ninokata Lab. Tokyo Institute of Technology, Mr. K. Hayashi of Toshiba
Nuclear Engineering Services Corporation for the supports and discussions in the development work. References Chiba, G., 2006. Overestimation in parallel component of neutron leakage observed in sodium void reactivity worth calculation for fast critical assemblies. J. Nucl. Sci. Technol. 43, 946. Mori, T., et al., 1996. LICEM, JAERI-Data/Code 96-018 (in Japanese). Nagaya, Y., et al., 2004. MVP/GMVP: general purpose Monte Carlo codes for neutron and photon transport code based on continuous energy and multi-group methods. JAERI-1348. Nakagawa, M., Tsuchihashi, K., 1984. SLAROM: a code for cell homogenization calculation of fast reactor. JAERI 1294. Nakagawa, T. et al., 1995. Japanese evaluated nuclear data library Version 3 revision-2: JENDL-3.2. J. Nucl. Sci. Technol. 32, 1259. NEA-0796, 2001. Cross-sections Library 25-Groups ABBN and 70-Group JFS for Fast Reactor Calculation. Yokoyama, T., Tokiwai, M., Endo, H., Ninokata, H., 2006. Aluminum–metal fueled long life fast reactor cores with inherent safety features. Trans. Am. Nucl. Soc. 95, 163723.
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Development of analysis method for sodium void reactivity of step type FBR cores by using group-wise Monte Carlo code Tsugio Yokoyama a,⇑, Hiroshi Endo b, Hisashi Ninokata c a
Toshiba Nuclear Engineering Services Corporation, 8 Shin-sugita-cho, Isogo-ku, Yokohama-shi, Kanagawa 235-8523, Japan Safety Analysis and Evaluation Div., Japan Nuclear Energy Safety Organization (JNES), Kamiya-cho MT Bldg., 4-3-20, Toranomon, Minato-ku, Tokyo 105-0001, Japan c Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan b
a r t i c l e
i n f o
Article history: Received 30 September 2010 Received in revised form 24 January 2011 Accepted 27 January 2011 Available online 23 February 2011 Keywords: FBR Step type core Sodium void reactivity Perturbation theory Monte Carlo method GMVP
a b s t r a c t A new method is proposed to separate the sodium void reactivity of step type FBR cores to components including non-leakage terms and a leakage term by using a newly developed perturbation code MCPERT where fluxes and adjoint fluxes are derived from a group-wise Monte Carlo code. The step type FBR core is a core where the height of the inner core is smaller than that of the outer core and a large sodium plenum region is located above the core so as to decrease the sodium void reactivity. The conventional diffusion perturbation method cannot treat such a large void region due to the diffusion approximation, while the Monte Carlo code can treat it exactly. In this study, a group-wise Monte Carlo code GMVP with a 70-group constant set JFS-3-J3.3 is employed to evaluate the neutron fluxes and adjoint fluxes which are used as inputs to the MCPERT code to evaluate the non-leakage terms. The leakage term is derived from the difference of the total sodium void reactivity evaluated by the direct calculation of GMVP and the summation of the non-leakage terms. It is found that the proposed method can provide the result approximately consistent to the ratio of the reactivity components derived from the conventional method. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In typical large FBR cores, the loss of coolant sodium can result in a large positive reactivity effect. This problem, not existed in thermal reactors, will present an important safety issue for FBRs. The sodium loss reactivity effect is made of two components. One is non-leakage term, which expresses the effect of neutron reaction change when the system is infinite and the neutron leakage does not exist, and the other is leakage term, which is the reactivity effect due to the increase of neutron leakage though the space of the sodium voided. The former is positive and the latter negative in typical large FBRs because the former is reactivity mainly due to the hardening of neutron spectrum where g, the number of emitted neutrons per absorption to fissile nuclides increases along with neutron energy increase. The summation of two components is usually positive in typical cores, but various efforts have been dedicated to decrease the reactivity. The step core concept is one of them. The step core is a combined core of a lower height of the inner region and a higher height of the outer core region. The typical way to decrease sodium void reactivity in large
⇑ Corresponding author. Tel.: +81 45 770 2419; fax: +81 45 770 2336. E-mail address: [email protected] (T. Yokoyama). 0306-4549/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2011.01.033
size cores is to decrease core heights, but it leads higher enrichment which sometimes ends up with poor core performances. Then we applied this way only to the inner region of the core, where sodium void reactivity is large positive, and the height of the outer region is remained higher, where it usually negative due to large neutron leakage. In addition we employed a combination of large fuel volume fraction fuel core to decrease the enrichment. Another effective way to decrease the sodium void reactivity is to elide the upper axial blanket. The increase in neutron leakage through the voided upper region of the core can enhance the negative void reactivity effect of the core. Thus we have adopted this type of step cores with a negative void reactivity as the target core to which our new method apply. In our previous studies, the continuous energy Monte Carlo method code, MVP (Yokoyama et al., 2006), was employed to analyze the core characteristics as it can treat complicated configurations with various neutron spectrum. But it sometimes requires long time to achieve statistically accurate results due to Monte Carlo method and cannot analyze the components of sodium void reactivity because it cannot calculate adjoint fluxes. In this paper, the separation of the components, non-leakage term and leakage term, was tried for the step core by using a multi-group version of MVP code, which can calculate real fluxes in a short time in addition to be able to calculate adjoint flux. The separation to the
T. Yokoyama et al. / Annals of Nuclear Energy 38 (2011) 1382–1388
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Nomenclature FBR ULOF JAEA
fast breeder reactor unprotected loss of flow event Japan atomic energy agency
components is a common way to identify physical features of the sodium void reactivity and useful to fit the calculation data to various experimental data of sodium void reactivity of critical experiments.
2. Step core concept and specifications For the commercialization of large FBRs, increase in safety characteristics and improve in economy are the most important aspects. Recent concerns on the safety design are to prevent the release of large kinetic energy at Unprotected Loss of Flow (ULOF) events (Yokoyama et al., 2006). The core with a negative void reactivity and an excess reactivity less than 1 $ is the most desirable core to provide a comprehensive persuasion under a simple developing scenario of accidents. If the sodium void reactivity is negative, there is no positive reactivity factor at loss of flow events. The step core where the core height of the inner core is lower than that of the outer core can resolve the above problems at the same time when metal fuel is employed by adjusting the core height. The typical way to decrease sodium void reactivity in large size cores is to decrease the core height, but it leads higher enrichment which sometimes ends up with large burn-up reactivity. Then we employed the combination of large fuel volume fraction metal fuel core to decrease the enrichment. Another effective way to decrease the sodium void reactivity is to elide the upper axial blanket. In our study of the no upper axial core, it was found that the sodium void reactivity in outer core is usually negative thus the flattening of the inner core only is effective to decrease the sodium void reactivity when the core diameter is limited. Table 1 lists the specifications of the typical step core when MOX fuel is employed. Fig. 1 illustrates the core layout and core profile. Table 1 Specifications of MOX step core. Specifications
Unit
Step core
Fuel composition Coolant Thermal power Cycle length Inlet temperature Outlet temp. Peak clad. temp. Max. linear powera Core press. drop Core diameter Core height (innerb/outer) Axial blanket thickness (top/bottom) Gas plenum length (upper/lower) Fuel pin diameter Fuel pin p/d No. of pins Assembly pitch Fuel vol. fraction Fuel smear density Pu enrichment (inner/outer)
– – MWt Year °C °C °C W/cm MPa m m m m mm – /Ass. mm % % %
MOX Sodium 3000 1 355 510 650 400 0.5 4.4 0.7/1.0 0.0/0.4 0.3/1.0 7.8 1.16 271 162.8 45 85 18.7/25.5
3. Calculation method of GMVP The sodium void reactivity of the core has been evaluated with group-wise Monte Carlo code GMVP (Nagaya et al., 2004) and the JFS-3-J3.3 70 energy group constant set (Nakagawa et al., 1995) as shown in Table 2. Each fuel assembly has been modeled as a hexagonal homogeneous cell fore each axial region. The history numbers of the Monte Carlo calculations has been selected to provide enough accuracy where the k-eff error is less than 0.01%. The ANISN-type cross sections or the double-differential cross sections can be used in the multi group code GMVP, while the special cross section libraries are used in the continuous-energy code MVP. The libraries are generated from the evaluated nuclear data (JENDL-3.1, -3.2, -3.3, ENDF/B-IV, -V, -VI, etc.) by using the LICEM code (Mori et al., 1996). The cross sections in the unresolved resonance region are described by the probability table method. In this study, we employed JENDL-3.3 for MVP as the cross section library and JFS-3-J3.3 70 group set (NEA-0796, 2001) for GMVP. This group constant set has been derived from JENDL-3.3 by using a fast neutron spectrum for the use of FBR core design, where the energy group structure is composed of 70 neutron groups over 10 MeV to 0.00001 eV. The effective cross sections of the core for ANISN-type cross section have been generated with SLAROM (Nakagawa and Tsuchihashi, 1984) and SLTOCCC code that are developed by JAEA. The JFS-3-J3.3 70 group set is collapsed by using the typical FBR core spectrum at flooded condition (NEA-0796, 2001). In order to evaluate the collapsing effect of the spectrum used, the difference of the cross section of Pu-239 between the flooded and voided condition has been calculated for the inner core of the step core using the continuous energy MVP code with 10 million neutron histories. Table 3 presents the comparison of mrf of Pu-239 between the flooded spectrum and voided spectrum and statistical errors. The error increases among lower energy groups where reaction rates decrease. In particular, no reactions has taken place in the lowest three groups even such a large number of histories are employed. The ratio of the cross section is about 0.02% in average over 1 keV in neutron energy, where about 80% of fission reactions occur. Even among the lower energy groups, the cross sections between the two conditions agree within the statistical errors. Thus we have concluded the collapsing effect within each group of the JFS-3J3.3 set is too small to affect the evaluation of sodium void reactivity. For the adjoint flux calculation of GMVP, ordinary adjoint matrix of the 70 groups cross section set similar to the diffusion calculation of adjoint fluxes are employed, while the sampling of fission source in Monte Carlo calculation is derived from the fission spectrum, not from mrf, which is used for the fission source of real flux calculation.
4. Component analysis method using quasi perturbation theory for sodium void reactivity
a
Maximum linear power of the core with 0.7 m of inner core height. The inner core height is varied from 0.5 m to 0.9 m for the parametric survey of void reactivity. b
Based on multi group exact perturbation theory, a reactivity effect q is expressed as follows:
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Fig. 1. Reference MOX fuel step core (inner core height: 70 cm).
hR P
q¼
g /g
vg
P
mRf Þg0 /g0 dV þ
g 0 dð
RP
g
r/g dDg r/g dV RP
g /g
vg
P
g0 ð
RP
g /g d
Rrg /g dV þ
mRf Þg0 /g0 dV
Table 2 Calculation method of k-eff. Item
Condition
Cross section library Code Method
JFS-3-J3.3 GMVP 70 group Three dimensional Monte Carlo Three dimensional Hex-Z 10,200,000 10,000 20
Core Model Number of neutron histories Number of neutrons per batch Number of skipped batches for neutron source convergence
where vg is the fission spectrum; /g, the neutron flux of group at perturbed condition; /0g the adjoint flux of group at unperturbed
RP
g /g
vg
P
g 0
Rg0 !g /g0 dV
i ð1Þ
condition; mRf, the macroscopic neutron generation cross section; Dg, the diffusion coefficient; Rr, the macroscopic removal cross section; Rg?g0 , the macroscopic scattering matrix; d, the difference of cross sections between perturbed and unperturbed conditions; and dV is the differential region volume. The difference of cross sections are just derived from the difference between the cross sections of the regions voided to those of the regions flooded, where these cross sections are evaluated by the SLAROM code for generating effective 70 group cross sections for GMVP. GMVP calculate real fluxes and adjoint fluxes while MVP can calculate only real fluxes. The fluxes and adjoint fluxes of GMVP are calculated as the total length of neutron tracks crossing the region concerned (Nagaya et al., 2004). Ordinary defined fluxes can be derived from the total length of tracks divided by the volume of the region. Thus the non-leakage terms have been calculated
Table 3 Comparison of Pu-239 cross sections collapsed with voided core spectrum. Upper energy (eV)
mrf at
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
1.00E+07 7.79E+06 6.07E+06 4.72E+06 3.68E+06 2.87E+06 2.23E+06 1.74E+06 1.35E+06 1.05E+06 8.21E+05 6.39E+05 4.98E+05 3.88E+05 3.02E+05 2.35E+05 1.83E+05 1.43E+05 1.11E+05 8.65E+04 6.74E+04 5.25E+04 4.09E+04 3.18E+04 2.48E+04 1.93E+04 1.50E+04 1.17E+04 9.12E+03 7.10E+03 5.53E+03 4.31E+03 3.35E+03 2.61E+03 2.03E+03
9.525 7.842 6.21 6.143 6.175 6.247 6.265 6.019 5.498 5.152 4.849 4.684 4.572 4.446 4.347 4.294 4.315 4.357 4.414 4.461 4.482 4.506 4.555 4.774 5.003 5.219 5.332 5.549 6.497 6.369 6.926 8.328 10.248 8.365 10.193
flooded core spectrum (barn)
Statistical error of mrf at flooded (%)
mrf at voided core spectrum (barn)
Statistical error of mrf at voided (%)
Voided/ flooded ratio
Statistical error of the ratio (%)
Group no.
Upper energy (eV)
mrf at flooded core spectrum (barn)
Statistical error of mrf at flooded (%)
mrf at voided core spectrum(barn)
Statistical error of mrf at voided (%)
Voided/ flooded ratio
Statistical error of the ratio (%)
0.13 0.49 0.12 0.00 0.01 0.01 0.01 0.06 0.07 0.03 0.04 0.02 0.01 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.01 0.25 0.29 0.32 0.35 0.41 0.46 0.51 0.60 0.72 0.88 1.29 1.38
9.51 7.902 6.207 6.143 6.174 6.246 6.265 6.026 5.5 5.149 4.852 4.686 4.572 4.447 4.347 4.294 4.315 4.357 4.414 4.46 4.482 4.506 4.555 4.786 5.001 5.204 5.328 5.571 6.479 6.375 7.02 8.507 10.159 8.749 10.005
0.12 0.49 0.12 0.00 0.01 0.01 0.01 0.07 0.07 0.03 0.04 0.02 0.01 0.02 0.01 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.01 0.27 0.30 0.35 0.38 0.44 0.49 0.54 0.59 0.76 0.81 1.27 1.45
0.998 1.008 1 1 1 1 1 1.001 1 0.999 1.001 1 1 1 1 1 1 1 1 1 1 1 1 1.002 1 0.997 0.999 1.004 0.997 1.001 1.014 1.021 0.991 1.046 0.982
0.25 0.98 0.24 0.00 0.03 0.02 0.01 0.13 0.14 0.07 0.09 0.03 0.03 0.03 0.02 0.00 0.01 0.01 0.02 0.01 0.00 0.01 0.02 0.52 0.59 0.67 0.73 0.85 0.95 1.05 1.19 1.47 1.69 2.56 2.83
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70a
1.58E+03 1.23E+03 9.61E+02 7.49E+02 5.83E+02 4.54E+02 3.54E+02 2.75E+02 2.14E+02 1.67E+02 1.30E+02 1.01E+02 7.89E+01 6.14E+01 4.79E+01 3.73E+01 2.90E+01 2.26E+01 1.76E+01 1.37E+01 1.07E+01 8.32E+00 6.48E+00 5.04E+00 3.93E+00 3.06E+00 2.38E+00 1.86E+00 1.45E+00 1.13E+00 8.76E01 6.83E01 5.32E01 4.14E01 3.22E01
13.532 18.369 15.764 12.032 44.771 21.751 39.033 52.713 52.359 50.124 60.523 113.98 180.37 215.4 59.466 11.59 49.319 205.71 260.68 381.32 39.568 285.41 20.115 21.062 26.742 32.231 40.571 55.091 70.643 109.44 173.86 311.05 NA NA NA
1.30 1.50 1.07 1.33 1.47 1.80 1.83 1.75 3.19 2.67 4.38 3.44 10.67 5.02 11.83 13.19 16.74 15.98 12.87 24.27 27.46 30.64 0.09 0.81 1.94 2.31 1.65 2.25 3.20 4.37 6.59 3.28 NA NA NA
13.192 18.228 16.432 11.823 46.199 21.049 37.743 56.086 58.599 55.049 64.127 115.24 165.7 186.41 56.356 7.823 51.272 326.32 268.03 502.06 61.21 224.36 20.613 21.404 25.531 32.344 42.893 53.816 69.398 111.84 178.14 314.93 NA NA NA
1.34 1.66 1.15 1.38 1.58 1.78 1.72 2.10 3.75 3.24 5.48 4.03 10.34 6.02 11.14 7.95 15.53 13.75 12.13 22.39 48.37 39.31 2.17 1.31 0.66 1.52 2.80 2.89 2.93 3.71 6.45 4.17 NA NA NA
0.975 0.992 1.042 0.983 1.032 0.968 0.967 1.064 1.119 1.098 1.06 1.011 0.919 0.865 0.948 0.675 1.04 1.586 1.028 1.317 1.547 0.786 1.025 1.016 0.955 1.004 1.057 0.977 0.982 1.022 1.025 1.012 NA NA NA
2.63 3.17 2.21 2.71 3.05 3.58 3.55 3.84 6.93 5.91 9.86 7.46 21.01 11.04 22.97 21.13 32.27 29.73 25.00 46.66 75.83 69.95 2.26 2.12 2.60 3.83 4.45 5.14 6.14 8.08 13.05 7.45 NA NA NA
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a
Group no.
Lower energy boundary of 70th group is 1.00E5 eV.
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by multiplying the two fluxes and the difference of the cross sections of the perturbed regions in the MCPERT code, which has been newly developed in this study. The second term of Eq. (1) is difficult to calculate in Monte Carlo codes because the gradient of fluxes cannot be exactly evaluated due to the increase in statistical error to in a thin zone that is needed to evaluate the gradient of fluxes. In this study, the second term, which is correspond to leakage term in sodium void reactivity, is assumed to be derived from the difference between the total reactivity and the non-leakage term, which is expressed as the summation of the 1st, 3rd and 4th term of Eq. (1), i.e.,
ql ¼ qtotal qnon
ð2Þ
where ql is the leakage term of sodium void reactivity; qtotal, the total sodium void reactivity evaluated with direct calculation of GMVP; and qnon is the Non-leakage term of sodium void reactivity derived by Eq. (1) without the second term of the numerator. All of the non-leakage terms shown in formula (1) are calculated in the MCPERT code to evaluate the non-leakage component. The statistical errors of the total reactivity have been evaluated with the statistical errors of the direct calculation results of GMVP by using the error propagation formula. The statistical errors of the non-leakage term have been evaluated by using the statistical errors of the fluxes and adjoint fluxes, which are calculated by the perturbed and unperturbed GMVP calculations, respectively.
5. Analysis of sodium void reactivity of reference core In the step core without upper axial blanket, the main contributor to decrease sodium void reactivity is the neutron leakage through the upper gas plenum region. Thus we have simulated the upper gas plenum region exactly for GMVP calculations. As shown in Table 1, the upper gas plenum length of the fuel is set at 0.3 m, where springs to constrain fuel pellets are installed. In the conventional model employed for diffusion or transport calculations, each zone was modeled with a homogeneous region as shown in Fig. 2. In this study, however, this plenum region has been simulated with a heterogeneous assembly model as shown in Fig. 3 (horizontal cross section of the fuel) and Fig. 4 (vertical cross section). The results of the sodium void reactivity of the reference step core, which has a inner core height of 70 cm, is shown in Table 4 for this model. The sodium void reactivity for the void of the core and upper plenum, when all of the sodium above the top level of lower axial blanket, is 0.22%Dq, while the void reactivity of the active core region is 1.03%Dq. Table 5 is a result of components of the sodium void reactivity derived from the separation method described above. In this table, (C), the void reactivity of plenum region is calculated with GMVP for the model when only the upper region above the top level of the active core is voided, and the (D), the non-leakage term of the core region is calculated by Eq. (1) without the second term.
Control rod
upper gas plenum of Inner core
Outer core Inner core
gas plenum
Outer core
Lower axial Blanket Fig. 2. Cross section model of the step core.
Fig. 3. Horizontal cross section of pin heterogeneous model at upper gas plenum level.
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Upper gas plenum of outer core
dition of the plenum region does not affect the reactivity components of the core region.
Upper gas plenum of inner core
6. Application to step cores of different inner core heights In this section, an application of the method to step cores with different inner core heights is intended, which will be useful to verify how the method could be adopted to predict the sodium void reactivity of a given core height. The relation of the components of sodium void reactivity to the inner core height of step cores is analyzed using the method described above. In this study, we have changed only the height of the inner core. When we determine the specifications of the core with a negative sodium void reactivity, the change of the inner core height may affect other core characteristics such as average power density or peak linear power, but sodium void reactivity is considered to be most sensitive among the main core characteristics because it strongly depends on neutron leakage. Other core parameters can be easily adjusted if the core height is selected appropriately to satisfy the criteria of sodium void reactivity. Table 6 and Fig. 5 presents the results for the inner core height of 50–90 cm. Fig. 5 indicates the main component dominating the change of void reactivity with inner core heights is the leakage term of the core. The non-leakage component increases with the inner core height because the neutron leakage of the higher core is lower than that of the lower core and the neutron spectrum shift dominates the reactivity change when the core is voided. As shown in the figure, the curves of each component are smooth over the range, which implies the method can be applicable to predict the components of the reactivity of the cores where the existing diffusion perturbation method cannot be applied such as a core that contains a large void region. According to the experiences on the component study with conventional method using diffusion perturbation calculations, the ratios of the non-leakage term to the leakage term for the typical large MOX critical assemblies ZPPR-9 and ZPPR-10A, where the core heights are 101.8 cm, are 0.47 and 0.52, respectively, when the whole core height of 101.8 cm is voided (Chiba, 2006). In Table 6, this ratios are 0.56 and 0.50 when the inner core height is 80 cm and 90 cm, respectively. Thus the result of this method is approximately consistent to that of the conventional method for the component analysis of sodium void reactivity.
Fig. 4. Vertical cross section of pin heterogeneous model at gas plenum.
Table 4 Region-wise sodium void reactivity of reference (70 cm in inner core height) for heterogeneous plenum model. Voided regions
k-eff
Non (intact)
1.0372
Core + upper plenum
1.0348
Core
1.0484
Plenum
1.0242
Void reactivity(%Dq) –
Statistical error (%Dq,1r)
Comments
–
Direct calculation GMVP Direct calculation GMVP Direct calculation GMVP Direct calculation GMVP
0.014
0.22
1.03
0.014
0.014
1.23
with
with
with
with
(E) is the leakage component of the core derived from the formula (B)–(D). (F) is that of the core derived from the formula (A)–(C)–(D). These two results are about the same within the statistical error, which is the summation of the error of the flux and that of the adjoint flux for the perturbed region. This means the void conTable 5 Component-wise sodium void reactivity of the reference core (70 cm inner core height). Voided regions
Components
Void reactivity (%Dq)
Statistical error (%Dq,1r)
Comments
(A) Core + upper plenum (B) Core (C) Plenum (D) Core (E) Core (F) Core
Non-leakage + leakage Non-leakage + leakage Non-leakage + leakage Non-leakage Leakage Leakage
0.22 1.03 1.21 2.81 1.78 1.82
0.014 0.014 0.014 0.11 0.18 0.18
Direct calculation with GMVP Direct calculation with GMVP Direct calculation with GMVP Perturbation calculation with GMVP adjoint flux (B)–(D) (A)–(C)–(D)
Table 6 Component-wise sodium void reactivity for different inner core heights unit: (%Dq).
a
Voided regions
Inner core height (cm)
50
60
70
80
90
Statistical errora (%Dq,1r)
(A) Core + upper plenum (B) Core (C) Plenum (D) Core (E) Core (E)/(C)
Non-leakage + leakage Non-leakage + leakage Non-leakage + leakage Non-leakage Leakage –
1.55 0.15 1.4 2.23 2.38 1.07
0.75 0.55 1.3 2.56 2.01 0.79
0.22 1.03 1.21 2.81 1.78 0.63
0.25 1.35 1.1 3.04 1.69 0.56
0.55 1.6 1.05 3.09 1.56 0.50
0.02 0.02 0.02 0.12 0.2 –
Maximum among different core heights.
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4.0
Sodium void reactivity (%Δρ)
3.0
(A)Core+upper plenum (Non-leakage+leakage)
2.0
(B)Core (Non-leakage+leakage) 1.0
(C)Plenum (Non-leakage+leakage) 0.0
(D)Core (Non-leakage)
-1.0
(E)Core (leakage)
-2.0
-3.0
50
60
70
80
90
100
Inner core height (cm) Fig. 5. Component-wise sodium void reactivity for different inner core heights.
7. Conclusions The reactivity component of sodium void reactivity of the step core, which has a lower inner core height relative to the height of the outer core, and the upper axial blanket is eliminated, has been evaluated with a new method using GMVP, a group-wise Monte Carlo code and the cross section library JFS-3-J3.3 70-group constant set. The ratio of the components of non-leakage to the leakage term of the active core region is similar to that of conventional method, which implies this method proposed will provide reasonable results for complicated core configurations that can be evaluated only by Monte Carlo codes. The analysis of the relation of void reactivity to different core heights using the proposed method shows that the method can provide the result approximately consistent to the ratio of the reactivity components derived from the conventional method for the whole core height void. Acknowledgments The authors express their gratitude to the members of Ninokata Lab. Tokyo Institute of Technology, Mr. K. Hayashi of Toshiba
Nuclear Engineering Services Corporation for the supports and discussions in the development work. References Chiba, G., 2006. Overestimation in parallel component of neutron leakage observed in sodium void reactivity worth calculation for fast critical assemblies. J. Nucl. Sci. Technol. 43, 946. Mori, T., et al., 1996. LICEM, JAERI-Data/Code 96-018 (in Japanese). Nagaya, Y., et al., 2004. MVP/GMVP: general purpose Monte Carlo codes for neutron and photon transport code based on continuous energy and multi-group methods. JAERI-1348. Nakagawa, M., Tsuchihashi, K., 1984. SLAROM: a code for cell homogenization calculation of fast reactor. JAERI 1294. Nakagawa, T. et al., 1995. Japanese evaluated nuclear data library Version 3 revision-2: JENDL-3.2. J. Nucl. Sci. Technol. 32, 1259. NEA-0796, 2001. Cross-sections Library 25-Groups ABBN and 70-Group JFS for Fast Reactor Calculation. Yokoyama, T., Tokiwai, M., Endo, H., Ninokata, H., 2006. Aluminum–metal fueled long life fast reactor cores with inherent safety features. Trans. Am. Nucl. Soc. 95, 163723.