Monte Carlo calculation of the effective neutron generation time

Transcription

1 annals of NUCLEAR ENERGY Annals of Nuclear Energy 33 (26) Monte Carlo calculation of the effective neutron generation time B. Verboomen a, *, W. Haeck a,b, P. Baeten a a Belgian Nuclear Research Centre (SCKÆCEN), Reactor Physics & MYRRHA Department, Boeretang 2, B-24 Mol, Belgium b Ghent University (UGent), Subatomic and Radiation Physics, Proeftuinstraat 86, B-9 Gent, Belgium Received 13 March 26; accepted 5 May 26 Available online 27 June 26 Abstract The Monte Carlo calculation of the effective (i.e. adjoint weighted) neutron generation time K eff, especially for continuous energy simulations, is not straightforward nor standard in Monte Carlo codes. The use of the non-adjoint weighted neutron generation time K (standard in most Monte Carlo codes) as an approximation of the effective one can lead to a serious bias. We show here that the difference between K eff and K can be more than 2% for a thermal fission system (VENUS reactor, SCKÆCEN, Belgium) and more than 6% for a fast fission system (MASURCA reactor, Cadarache, France). For this, we have implemented, for the first time, in the Monte Carlo code MCNP(X), an asymptotically exact method for the calculation of K eff. Our method was benchmarked against measurements in MASURCA: the difference between calculated and experimental values is less than 1%. Ó 26 Elsevier Ltd. All rights reserved. 1. Introduction The knowledge (and thus calculation) of kinetic parameters such as the effective neutron generation time K eff and the effective delayed neutron fraction b eff are important in nuclear power plant operation since they determine different kinds of safety procedures. The calculation of these parameters is usually performed with deterministic codes. There are three general methods available for this: (i) the direct integration method or adjoint weighted method which requires the calculation of the adjoint flux (e.g. Keepin, 1965); (ii) the eigenvalue method or perturbation method which requires the calculation of the reactivity of the system for a well chosen perturbation (Henry, 1958); and (iii) the iterated fission probability method (e.g. Bohl and Margolis, 1964) where the neutron importance is evaluated by counting the total number of fission generated per neutron history. In advanced nuclear systems, the need for reliable kinetics parameters is of considerable importance because of the lower value for b eff (due to the large amount of transuranic * Corresponding author. Tel.: ; fax: address: bverboom@sckcen.be (B. Verboomen). elements loaded in the core of those systems). For such systems, the Monte Carlo method is the preferred calculation tool since (i) Monte Carlo has the ability to handle nuclear data not only in its most basic but also most complex form: continuous energy cross-sections, complex interaction laws, detailed energy angle correlations, multi-particle physics, S(a, b) tables for thermal neutron scattering by molecules and crystalline solids, unresolved resonance probability tables; (ii) Monte Carlo can handle geometries from simple 1D to very complex 3D. As a result, normal critical systems as well as sub-critical systems with an external source can all be calculated with a single code, practically without making any approximation. Due to this growing use of Monte Carlo codes one now also desires the calculation of the kinetic parameters by such codes. If the calculation of b eff by the iterated fission probability method is now straightforward 1 with MCNP(X) (Meulekamp and Van Der Marck, 26), it is not yet the case for the calculation of K eff, especially for continuous energy simulations with the direct integration method. 1 In fact, Meulekamp and Van Der Marck (26) calculate the next fission probability as an approximation for the iterated fission probability. They show that this approximation work very well for all type of systems /$ - see front matter Ó 26 Elsevier Ltd. All rights reserved. doi:1.116/j.anucene

2 912 B. Verboomen et al. / Annals of Nuclear Energy 33 (26) Indeed, continuous energy adjoint Monte Carlo is not as straightforward as direct Monte Carlo because it requires extensive preprocessing of cross-section data (Lux and Koblinger, 1991; Hoogenboom, 23). For this reason, only the multi-group 2 adjoint calculation is implemented in most Monte Carlo codes (e.g. Wagner et al., 1994). In what follows, we will focus on the calculation of K eff with Monte Carlo using the perturbation method. This method is easier to implement because, contrary to the iterated fission probability method, it does not require major changes in the Monte Carlo code. The methodology described here has been applied to MCNP(X) (Briesmeister, 2; Pelowitz, 25) for two different systems: the critical MUSE-4 configuration at the MASURCA facility of CEA Cadarache (5th EURATOM FP, 24) and the critical UO 2-4.% configuration at the VENUS reactor (Verboomen, 21) of SCKÆCEN. 2. Theory 2.1. Adjoint and non-adjoint weighted neutron generation time 2 In MCNP(X), only infinite diluted (i.e. non-self-shielded) 3 groups cross-sections (without upscatter groups) are available for forward and adjoint calculations. Only fixed source calculations are possible in the adjoint mode. 3 For near critical systems, it is well known (see for example Ott and Neuhold, 1985, p. 57; Dorning and Spiga, 1978) that the best weighting function is the initial critical adjoint flux and the shape function is simply the initial critical flux. For sub-critical systems, the choice of the best weighting and shape functions is still an open problem. Some authors (e.g. Ott and Neuhold, 1985, p. 69) recommend the use of the initial critical adjoint flux as weighting function and the initial solution of the stationary fixed source problem as shape function, others (e.g. Dorning, 198) recommend the use of the instantaneous fundamental x-adjoint mode as weighting function and the instantaneous fundamental x-mode as shape function. We do not intend to solve this problem here. Neutron generation time 3.5E 6 3.E 6 2.5E 6 2.E 6 1.5E 6 1.E 6 5.E 7.E+ The effective (i.e. adjoint weighted 3 ) neutron generation time, K eff is defined as: K eff ¼ Uþ ; 1 U v U þ ; F ð1þ U where U þ is the critical adjoint flux, v the neutron speed, U the critical flux and F the total (i.e. prompt and delayed) fission operator of the reference critical configuration. K eff can therefore be seen as the reciprocal of the average (i.e. adjoint weighted) neutron production probability or the mean time before one neutron produces another (Lewins, 1965). The evaluation of the non-adjoint weighted neutron generation time K (where the adjoint flux U þ is assumed to be 1 everywhere) is standard in MCNP(X): 1 v K ¼ U hf U i ¼ l p ð2þ k eff where l p is the prompt removal lifetime (the average time from the emission of a prompt neutron in fission to the removal of the neutron by some physical process such as escape, capture, or fission) and k eff is the effective multiplication factor in the system. 4 The use of K as an approximation of K eff can lead to a considerable bias. This is illustrated in Fig. 1 for the MAS- URCA reactor (MUSE-4 critical configuration): the difference between experimental values (cases f and g) and non-adjoint weighted calculated values (cases a and b, see also 5th EURATOM FP, 24) is more than 6%. We observe the same trend in VENUS where the difference between experimental values and non-adjoint weighted calculated values lie between 2% and 3% (Baeten et al., 21; Leenders et al., 1968). We observe the same behaviour (see Table 1) between adjoint and non-adjoint weighted calculated values. For the thermal fission reactor VENUS, K is more than two times higher than K eff. For system having a fast neutron spectrum, the situation is even worse. For MASURCA for example, K is more than six times higher than K eff! The order of magnitude for the effective neutron generation time is not even correct in this case. Such differences do not appear for the calculation of the effective delayed fractions (the difference between adjoint and non-adjoint weighted delayed fraction is lower than 2% in most cases). A forthcoming paper will be devoted entirely to the investigation of the physical cause for this bias The perturbation method a b c a : MCNPX/non adjoint/jef 2.2 b : MCNPX/non adjoint/jeff 3.1 c : MCNPX/adjoint/JEF 2.2 d : MCNPX/adjoint/JEFF 3.1 e : ERANOS/adjoint/JEF 2.2 f : Experiment 1 g : Experiment 2 2σ 2σ experimental range Fig. 1. MASURCA reactor, MUSE-4 critical configuration, comparison between calculated and experimental values. The stationary homogeneous transport equation for a given system can be written in compact form as (e.g. Ott and Neuhold, 1985): KU ¼ kf U ð3þ where K is the neutron destruction operator, F is the neutron creation (prompt and delayed fission) operator and 4 In fact, MCNP(X) calculates the prompt removal lifetime l p and the effective multiplication factor k eff from which the non-adjoint weighted neutron generation time K can be derived. d e f g

3 B. Verboomen et al. / Annals of Nuclear Energy 33 (26) Table 1 Non-adjoint and adjoint weighted neutron generation time in VENUS and MASURCA Reactor library MASURCA VENUS JEF-2.2 JEFF-3.1 JEF-2.2 JEFF-3.1 a k a k Dq (pcm) c l b p (s) K b (s) K eff (s) r[k eff ]/K eff (%) K/K eff a The relative error on k and k 1 is.1%. The bias is lower than.1%. b The relative error on the prompt removal lifetime and on the neutron generation time is.2% for MASURCA and.1% for VENUS. with k = k 1 with k being the largest eigenvalue of Eq. (3) (i.e. the effective multiplication factor). The neutron destruction operator K can always be written as: K ¼ L þ R t S ð4þ where L is the streaming operator, R t is the total macroscopic cross-section and S is the scattering source operator. 5 By introducing a perturbation in this system, we can now consider the following equations: K 1 U 1 ¼ k 1 F 1 U 1 ð5þ K þ Uþ ¼ k F þ Uþ ð6þ where Eq. (5) is the direct equation for the stationary perturbed configuration (with index 1) and where Eq. (6) is the adjoint equation for the stationary non-perturbed configuration (with index ). The adjoint operator A + of the operator A is defined by the following relation: ha þ f ; gi ¼ hf ; Agi ð7þ The Æ,æ symbol (scalar product) means an integration over the whole range of the spatial, angular and energy variables. By multiplying Eq. (5) with the adjoint flux U þ and by multiplying Eq. (6) with the flux U 1 of the perturbed system and after performing the integration over all spatial, angular and energy variables, we obtain the following: U þ ; K 1U 1 ¼ k1 U þ ; F 1U 1 ð8þ U 1 ; K þ Uþ ¼ k U 1 ; F þ Uþ ð9þ By using definition (7) for the adjoint operator in Eq. (9) we obtain: U þ ; K U 1 ¼ k U þ ; F 1U 1 k U þ ; DF U 1 ð1þ where DF = F 1 F. 5 To simplify the notations, we ignore other reactions such as (n,2n), (n,3n),... Subtracting Eqs. (8) and (1) and solving for Dk yields the exact perturbation formula for reactivity increments (Ott and Neuhold, 1985): Dq ¼ Uþ ; k ½ DF DKŠU 1 U þ ; F ð11þ 1U 1 where DK = K 1 K and where Dq is the reactivity increment due to the perturbation: Dq ¼ q 1 q ¼ 1 k 1 k 1 ¼ Dk ð12þ If we now consider the special case of a uniform c/v poison (where c is a constant and v is the neutron speed), then the perturbation operators in Eq. (11) will be given by: DK ¼ c v DF ¼ ð13þ ð14þ so that the reactivity increment for this perturbation will be given by: Dq ¼ q c q ¼ Uþ ; c U v c U þ ; F ð15þ U c In this equation, q c is the reactivity of the system submitted to a homogeneous 1/v poisoning as function of the amplitude 6 c. The derivative of this reactivity in c = will be exactly equal to minus the adjoint neutron generation time: oq q ¼ lim c q 1 U þ ¼ lim ; c U v c oc c! c¼ c c! c U þ ; F U c ¼ Uþ ; 1 U v U þ ; F ð16þ U This simply means that K eff is the negative value of the slope of the reactivity curve (or the curve of the reactivity difference Dq) in the origin. oq ¼ K eff ð17þ oc c¼ 6 The dimension of c is (s 1 ).

4 914 B. Verboomen et al. / Annals of Nuclear Energy 33 (26) The calculation scheme for K eff will therefore be based on the first-order perturbation theory: we choose c small enough such as c/v K and U c U = DU U. Discarding second and higher order terms in Eq. (15) gives the well known (e.g. Lewis and Miller, 1993) first-order perturbation formula for reactivity increments (in the particular case of a 1/v perturbation): Dq ¼ q c q ¼ Uþ ; c U v U þ ; F ð18þ U We finally obtain the first-order perturbation formula for K eff : K eff ¼ Dq ð19þ c 2.3. Implementation in MCNP(X) The simplest way to perform the uniform c/v perturbation is to add, in each point of the reactor, an artificial material which possesses only a 1/v capture cross-section. In this way we ensure that: DK ¼ DR t ¼ R p ðcþ ¼ c ð2þ v where R p (c) is the macroscopic capture cross-section of the artificial material: R p ðcþ ¼NðcÞr p ¼ NðcÞ A ð21þ v where N(c) is the atom density and r p is the microscopic cross-section of the artificial material so that: c ¼ ANðcÞ ð22þ In this equation, A is a constant used to scale the microscopic cross-section so that the number N(c) of atoms of the artificial material does not become too high. In our case, we took A equal to 1 9 cm 3 s 1 so that r p (22 m s 1 )= 4545 barn to obtain the same order of magnitude as the value of 1 B (which is 3843 barn). We have built this artificial 1/v cross-section material in ENDF-6 format to produce, with the aid of NJOY (MacFarlane and Muir, 1994), our own ACE file (linear linear interpolated continuous capture cross-sections) for use in MCNP(X). 7 In accordance with Eq. (19), the adjoint weighted neutron generation time will then be given by: K eff ¼ 1 A ðk c k Þ NðcÞ ¼ 1 A ðq q c Þ NðcÞ ð23þ 7 The fractional reconstruction tolerance used was.5% with integral thinning of barn. These values were chosen so that using the MCNP built-in 1/v integral yields the same result as using a reaction multiplier with the artificial material. for sufficiently small values of N(c). Another possibility to determine the value of K eff is to perform the perturbation for many different values of c in order to construct the q c -curve. By simply taking the negative value of the slope of this curve in the origin, we will obtain the effective neutron generation time. 3. Applications 3.1. Code and data The Monte Carlo calculation of the reactivity due to a small 1/v perturbation can be performed essentially in three ways: by the direct approach, by the differential operator perturbation technique (Hess et al., 1998) and by the correlated sampling technique (e.g. Lux and Koblinger, 1991). Only the two first techniques are available in MCNP(X). We have adopted here 8 the direct approach (calculation of the two eigenvalues in KCODE mode). For these calculations, we used MCNPX 2.5., the latest official release of MCNPX. For the nuclear data, we considered the case of JEF-2.2 and the new JEFF-3.1 library at 3 K (to assess the differences in values for K eff due to nuclear data). Both these libraries are part of ALEPH- LIB (Haeck and Verboomen, 26), the multi-temperature standard calculation library available at SCKÆCEN. To facilitate the calculation of K eff, we also created an auxiliary C++ code called LAMBDA which wraps itself around any version of MCNP(X) to automate the calculation of K eff. The input for LAMBDA is simply an MCNP(X) input file along with a set of values for the perturbation amplitude c and the method to be used (either the direct calculations or the standard MCNP(X) perturbation module). The code then prepares separate input files for every value of c (and also for the reference case if the direct method is employed) and runs them using the MCNP(X) executable indicated by the user. LAMBDA then reads the k eff values from the output files and calculates the reactivity, the reactivity loss and the value of K eff for every value of c. LAMBDA calculates also the relative error on K eff. For the method used in this study (direct method), we use the following relation (linear estimate of the variance for small errors): " 2 2 rk ½ eff Š 1 r½k Š ¼ k 2 K eff ðk 1 k Þ 2 1 k þ r ½ k # 2 1Š k 2 ð24þ k Calculations The perturbation was chosen sufficiently small (3 pcm or.3%) to be in the range of applicability of the first-order perturbation theory (see Figs. 2 and 3) but large enough to keep a good statistical precision (< 5%). 9 Note that the corresponding value of c is much larger for fast systems. This 8 The use of the differential operator perturbation technique seems to give unreliable answers. We will investigate this question in the future. 9 We have used Eq. (23) for the estimation of K eff. Indeed, the use of several perturbation points to calculate K eff by linear or higher order fit was considered too sensitive to the number of perturbations points considered and to the order of the polynomial fit.

5 B. Verboomen et al. / Annals of Nuclear Energy 33 (26) Antireactivity ( ρ) [pcm] 12 1 MCNPX 2.5. calculation σ 8 ρ = c Amplitude c of the c/v perturbation [s 1 ] Fig. 2. MASURCA reactor, MUSE-4 critical configuration, antireactivity in function of the amplitude of the 1/v perturbation. Antireactivity ( ρ) [pcm] MCNPX 2.5. calculation σ ρ = c Amplitude c of the c/v perturbation [s 1 ] Fig. 3. VENUS reactor, UO 2 /4.% critical configuration, antireactivity in function of the amplitude of the 1/v perturbation. is due to the 1/v character of the perturbation: indeed, the perturbation acts primarily on the thermal neutrons. The number of particles per cycle and the number of cycles was chosen to limit the bias on the k eff to.1%, which is negligible compared to the error on the k eff itself (.1%). The calculations were performed with JEF-2.2 and JEFF-3.1. The results are given in Table 1. We see clearly the influence of the adjoint weighting: the difference between adjoint weighted and non-adjoint weighted generation time is 227% for VENUS and 64% for MASURCA. JEFF-3.1 calculations seems to give slightly higher values than JEF-2.2 calculations. We also note that the MAS- URCA JEF-2.2 MCNPX K eff calculation is in perfect agreement (see Fig. 1) with the corresponding ERANOS (Doriath et al., 1993) JEF-2.2 deterministic calculation Comparison with experimental values For MASURCA (5th EURATOM FP, 24), two experimental values of K eff are available: (5.5 ±.2) 1 7 s and (5.9 ±.1) 1 7 s giving a 2r experimental domain of [ s, s]. JEFF-3.1 estimation gives a nearly identical 2r calculation domain of [ s, s]. JEF-2.2 estimation gives a 2r calculation domain of [ s, s]. JEFF- 3.1 calculations are therefore in slightly better agreement than JEF-2.2 calculations. Contrary to the calculated values of K, the calculated values for K eff by the 1/v method are in good agreement with the experimental values (see Fig. 1). For VENUS, no experimental results are available for the calculated VENUS core configuration. However, based on the measured values for previous core configuration (Baeten et al., 21; Leenders et al., 1968), K eff for this configuration lies between sand3 1 7 s which is compatible with the calculated value of s. 4. Conclusions We have implemented, for the first time, in the Monte Carlo code MCNP(X), an asymptotically exact method for the calculation of the effective neutron generation time. Our method was applied to the thermal fission reactor VENUS (SCKÆCEN, Mol, Belgium) and to the fast fission reactor MASURCA (CEA, Cadarache, France). The difference between calculated values and experimental values is less than 1%. A comparison between JEF-2.2 and JEFF-3.1 libraries indicate that JEFF-3.1 calculations are in slightly better agreement than JEF-2.2 calculations. We have also shown, that the difference between adjoint and non-adjoint weighted neutron generation time is larger, by far, than the difference between adjoint and nonadjoint weighted delayed neutron fraction. This difference is the most important for fast fission systems with large reflector (6%) but the difference is also significant for thermal fission systems (2%). A forthcoming paper will be devoted entirely to this problem. This way, we have solved an important issue in the current neutron transport methodology. Acknowledgements We thank Dr. Nadia Messaoudi (SCKÆCEN) for having provided us the MCNPX model of the MASURCA reactor. References Baeten, P., Paepen, J., van der Meer, K., Ait Abderrahim, H., 21. Absolute measurement of b eff and l on weapon-grade MOX fuel at the VENUS critical facility by means of the RAPJA technique. Ann. Nucl. Energy 28, Bohl, L., Margolis, S.G., Computation of parameters in the kinetics equations. In: Radkowsky, A. (Ed.), Naval Reactors Physics Handbook, vol. I, Selected Basic Techniques, USAEC Division of Reactor Development, Naval Reactors Branch, Washington, DC (US), NRB- HGR

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