Calculating the effective delayed neutron fraction using Monte Carlo techniques

Transcription

1 Calculating the effective elaye neutron fraction using Monte Carlo techniques 1. Introuction Steven C. van er Marck * an Robin Klein Meulekamp NRG, Petten, the Netherlans We present a true Monte Carlo estimator of the effective elaye neutron fraction β eff. The avantage of this metho is that by using the physics at the microscopic level, it obviates the nee for ajoint calculations, without making any approximations. We have implemente this estimator into MCNP. In a stanar k eff calculation, the coe now reports a β eff value. The metho oes not slow own the coe by more than.5%. We efine an extensive benchmark set for β eff, which we use to test our metho, an two known approximate methos. Our metho reprouces all experimental values. KEYWORDS: effective elaye neutron fraction, Monte Carlo, MCNP In reactor kinetics, the effective elaye neutron fraction, β eff, plays a key role. It is, in effect, the unit of reactivity, conventionally being referre to as a 'ollar'. Because of the ifficulties in measuring β eff, one has traitionally relie heavily on calculations to etermine its value in a specific situation. Keepin, in 1965, provie a theoretical framework that has been use ever since for such calculations [1]. These involve an ajoint an spectrum weighting of the elaye neutron prouction rate, an hence require a calculation of both a flux an an ajoint function, an on top of that a suitable post-processing to calculate the weighte prouction rate. The amount of work that this entails is consierable. Not many coe packages can calculate β eff in a stanar proceure, base on nuclear ata without making any major assumptions or approximations along the way. Monte Carlo programs are especially har presse to o this, because ajoint calculations are cumbersome in continuous energy Monte Carlo coes [2]. Nevertheless one can argue that there is a nee for reliable β eff calculations for systems other than the familiar light water reactor type (LWR), since there is a strong interest in transmutation of plutonium an minor actinies. These isotopes prouce fewer elaye neutrons, which reuces the istance to prompt criticality, an hence increases the nee for accuracy in the calculation of β eff. Moreover, many propose systems for transmutation are strongly heterogeneous, which troubles some of the currently use methos for calculating β eff. A Monte Carlo base metho woul be ieally suite for such a purpose. Spriggs et al. [3] recently escribe an approximation that allows the computation of β eff using two eigenvalues. In fact, the approximation was introuce earlier [4], but the theoretical framework was provie in Ref. [3]. This is potentially a consierable improvement, because the ajoint function is not neee anymore, paving the way for continuous energy Monte Carlo coes. However, although the metho was shown to work well for a bare, homogeneous uranium sphere (Goiva), it remains unclear how goo the approximation is for LWR systems, or for any new reactor type that is contemplate. In fact, * Corresponing author, Tel. (++31) , FAX (++31) , vanermarck@nrg-nl.com

2 Spriggs et al. mentione that for one, heterogeneous, uranium system the approximation i not work well. At this point, we want to take a step back an think funamentally about what it is that we want to calculate: the effect of elaye neutrons on reactor transient behavior. As reactors are generally operate on the basis of the power that is generate, the effect that neutrons have on reactor behavior is through their ability to generate fissions (an hence power). Therefore the quantity we are after is the elaye neutron fraction insofar as it is effective in leaing to fission. In this paper, we propose an approach to β eff that is entirely in microscopic terms, an, consequently, easy to implement in a Monte Carlo scheme. This approach involves no approximations. We have implemente our metho into MCNP [5], an show that in a stanar MCNP eigenvalue calculation, one can get a β eff value by incluing only some minor bookkeeping in the coe. Given the fact that one calculates k anyway, one gets a value for β eff at no extra (CPU) cost. We have compute β eff for a wie variety of benchmark systems, proving that the metho is vali for all systems. This paper is organize as follows. In Section 2 we outline our approach to the effective elaye neutron fraction, an we iscuss other commonly use approaches. In Section 3 we report numerical results for many experimental benchmark systems. Section also contains a brief iscussion of these results an some conclusions. 2. Theory of Effectiveness As a first step towar assessing the importance of elaye neutrons in a reactor, one nees to etermine how many elaye neutrons are generate. We will restrict ourselves in this paper to a reactor near criticality, an without external source. In general, one writes for the total neutron prouction rate by fission: & & & P = ν ( E) Σ ( r, E, Ω) ϕ( r, E, Ω) EΩr, (1) f where E, Ω, an r & are the energy, soli angle, an position of the neutrons, ϕ is the neutron flux, Σ f the macroscopic fission cross section of the material at position r &, an ν the average neutron multiplicity per fission. For the prouction P of elaye neutrons, one replaces the factor ν(e) by ν (E), the average elaye neutron multiplicity per fission. For the sake of simplicity, we will not istinguish here between the several elaye neutron time groups, which each have their own energy spectrum. The ratio P /P then is the 'funamental' elaye neutron fraction β. So far, this is fairly straightforwar. The problems start when trying to assess how effective this fraction is in terms of reactor kinetics. As state in the introuction, the effect that neutrons have on reactor behavior is through their ability to generate power, i.e. to inuce the 'next' fission. It follows that we shoul compute the number of fissions that are inuce by elaye neutrons, as well as by all neutrons. 2.1 Transport Theory In transport theory one calculates the effectiveness in generating fission by multiplying by the energy spectrum of the generate neutrons, (E'), an by the ajoint function ψ ( r &, E, Ω ), (often referre to as 'ajoint flux'). The ajoint function is efine as the funamental moe eigenfunction of the equation ajoint to the time inepenent transport equation. Here r &, E', an Ω are the position, energy, an soli angle of neutrons generate by the fissions that were inuce by the incient neutrons characterize by r &, E, Ω. The position r & is obviously the same for both. The factor (E') is neee because the energy the neutrons start with has an

3 impact on their effectiveness in inucing fission. The ajoint function ψ ( r &, E, Ω ) is use because it is an importance function, which represents the significance of a neutron with properties r &, E', Ω' for proucing fission. In the wors of Keepin, ψ is proportional to the asymptotic power level resulting from the introuction of a neutron ( ) in a critical system at zero power [1] (p. 163). This leas to the so-calle spectrum an ajoint weighte neutron prouction & & & & Peff = ψ ( r, E, Ω ) ( E ) ν ( E) Σ r E Ω r E Ω EΩE f (,, ) ϕ(,, ) Ω r (2) One can calculate the same quantity for elaye neutrons only (P,eff ), by replacing (E ) by (E ) an ν(e) by ν (E). When one takes the ratio P,eff /P eff, one arrives at the Keepin efinition of β eff. It is instructive to interpret P as the neutron source (the number of neutrons prouce per unit of time), an P eff as the number of fissions prouce by this source per unit of time. 2.2 New Metho Within the context of a Monte Carlo scheme, the transport theory approach is more complicate than nee be. In such a scheme, the neutrons are simulate by generating them with a probability that is proportional to P. Assessing the effectiveness of these neutrons in generating the 'next' fission, is then intuitive an straightforwar in a Monte Carlo scheme. All neutrons are labele either 'prompt' or 'elaye' at birth, an subsequently they are tracke through the reactor until they are 'remove' from it by either an interaction such as fission or capture, or by escape to the surrounings. In those cases where the removal is ue to a fission, one nees to check whether the incient neutron is a elaye one or not. One can then calculate the average number of fissions generate by elaye neutrons, ivie by the average number of fissions generate by all neutrons. This is β eff. Notice that a calculation of this fraction can be one in a Monte Carlo program by means of some minor bookkeeping in the coe, which will give a result for β eff in the same run with which one calculates k. The coe will not even be slowe own significantly. We have checke that for a Goiva run the extra amount of CPU time involve was less than.5% of the total. For thermal systems, this fraction is even smaller. In Section 3 we will present calculations base on this metho for many ifferent neutron multiplication systems. We will compare it to two often use approximate methos of calculating β eff. In the remainer of this section we will outline these two methos. 2.3 Prompt Metho Denoting the integral in Eq. (2) as ν, one can rewrite the expression for β eff as follows, making use of the fact that the integrals are linear. ν ν ν ν ν ν p ( ) p p β eff = = 1 = 1 1, (3) ν ν ν ν where we have use ν p =ν-ν. The approximation in the last step is base on the following arguments. The term ( -)ν is two orers of magnitue smaller than the one with ν p, because ν is two orers of magnitue smaller than ν p. For the same reason, the shape of (E ) is almost equal to that of p (E ). At this point a crucial step is taken. Often it is simply state that p ν p k p k p = β eff = 1. (4) ν k k

4 In fact, this is an approximation. It is true that the k-eigenvalue is the ratio of prouction P an loss L, an that this also hols for the ratio of P,eff an L,eff. But the ifficulty lies in the efinition of k p. Since this parameter is suppose to be calculate by means of a transport theory coe, it shoul be efine as the eigenvalue pertaining to a reactor with = p, an ν=ν p. The ifference is that in such a calculation, the shapes of ϕ an ψ will not be the same as for the original system with an ν, for which the eigenvalue is k. This subtlety is generally ignore in papers ealing with the 'prompt' metho of calculating β eff base on Eq. (4), see e.g. Ref. [6]. 2.4 Spriggs Metho Alternatively, Spriggs et al. [3] rewrite β eff as ν ν ν ν β eff = = = β, (5) ν ν ν ν where we have introuce, after Spriggs et al., yet another elaye neutron fraction β. For the present purposes we restrict ourselves to the approximation that β β, because we still nee to perform ajoint weighting to calculate β. By approximating β β we can simplify the calculation to something that can easily be implemente in a Monte Carlo coe. As remarke by Spriggs et al., the approximation β β works well for homogeneous cases. Also for this metho we subsequently introuce a ratio of k-values. ν k = k β eff = β, (6) ν k k As in the case of the 'prompt' metho, this is an approximation. Here the problem lies in the efinition of k. Since this parameter is suppose to be calculate by means of a transport theory coe, it shoul be efine as the eigenvalue pertaining to a reactor with = an ν=ν. Again, the shapes of ϕ an ψ will not be the same as for the original system with an ν, for which the eigenvalue is k. This subtlety is explaine by Spriggs et al. for their metho of calculating β eff. 2.5 Comparison of Monte Carlo Methos Given the fact that Eqs. (4) an (6) are approximations, one woners how goo these approximations are in a practical situation. In the next section we will investigate the accuracy of these approximations by presenting results for experiments one in the past, but base on the above erivation one can expect that the 'prompt' metho is the better approximation. This is because in the erivation, the shape of the flux an of the ajoint function for the k p eigenvalue calculation are approximate by the respective shapes for the k eigenvalue calculation. As the prompt neutrons constitute roughly 99% of the neutron population, this is as goo an approximation as one can get. On the other han, for the Spriggs metho, one approximates ϕ an ψ by ϕ an ψ. The elaye neutrons constitute only 1% of the total neutron populations, so in this case the approximation cannot be expecte to perform equally well. Still, in section 3 we will show that this metho yiels goo results for most systems consiere. Only the results for very heterogeneous systems are unfavorable. There is also the computational aspect of Eqs. (4) an (6). In the first one, the ratio k p /k, which is very close to unity, is subtracte from unity, to obtain a value typically below.1. This implies that, when one uses a Monte Carlo coe to calculate k p an k, one nees very goo statistics. Otherwise the statistical uncertainty in β eff will be larger than the result for β eff

5 itself. The Spriggs metho oes not have this problem, which gives it a clear avantage over the prompt metho, computationally. In a practical situation, the 'prompt' metho is the only metho that can be use with, for instance, a stanar MCNP executable. The Spriggs metho, an the one propose in this paper, both nee a bit of extra programming. Finally, the 'prompt' metho an the Spriggs metho both nee two eigenvalue calculations, whereas the new metho propose here requires only one. 3. Results an Discussion We have searche in the literature for measurements of the effective elaye neutron fraction, the result of which is liste below. We will use these experiments as benchmarks for our calculation of β eff. For some systems we have foun experimental values for the parameter α, which is linke to β eff through α=[k(1-β eff )-1]/l, where l is the prompt neutron life time. All systems escribe below are at elaye criticality, so that the parameter we can compare with is the value α c =α(k=1)=-β eff /l. For a escription of the systems use, incluing literature references for the experimental values for β eff an α c, we refer to Ref. [7]. We have calculate β eff for all experiments escribe in the previous section, base on three ifferent nuclear ata evaluations, viz. JEFF-3., ENDF-B/VI.8, an JENDL-3.3. The reason why we have liste the results for all three ata evaluations in Tables 1 3 is that the ifferences between these ata evaluations are sometimes larger than the ifferences between the three methos of calculating β eff. We have calculate the results for the prompt metho only for a selection of cases, because the run times for the other systems are prohibitively long if we are to get statistically useful results. The results are presente in Tables 1 3 an in Figure 1. The results are roughly orere with respect to the average energy at which fission takes place. We conclue the following. For most cases, in particular for LWR type applications, all methos yiel goo results. The results for the new metho agree with the experimental ata, even for the most heterogeneous systems. The prompt metho also yiels goo results for all systems. However, this metho requires at least 4 more CPU time. The Spriggs metho performs well in most cases, but for the heterogeneous systems there are clear eviations from the experimental values. The reason for the iscrepancy most probably is the aitional approximation of β by β, so we cannot conclue that the Spriggs metho is inaequate for these systems, base on these results alone. On the other han, Spriggs et al. [3] mention that for Topsy (a high enriche uranium sphere with a natural uranium reflector) the results of their metho were 'not very goo'. Moreover, if we have to calculate β instea of β, we nee to apply ajoint weighting, which we cannot easily o in a Monte Carlo program. For homogeneous systems, the ifferences between the results base on the various nuclear ata libraries are often at least as large as the ifferences between the three methos. This allows us to raw conclusions about the nuclear ata. Juging by the results for TCA, Stacy an Winco, the JENDL-3.3 nuclear ata library gives the best α an β eff results for LWR type applications. In summary, we have introuce a new metho to calculate the effective elaye neutron fraction using Monte Carlo techniques. We have implemente this metho in a version of MCNP-4C3, an we calculate the effective elaye neutron fraction for a variety of systems.

6 For all these systems, the new metho reprouces the experimental values for β eff with satisfactory accuracy. The 'prompt' metho gives similarly goo results, although one nees at least 4 longer run times. The metho escribe by Spriggs et al. [3] also yiels goo results, except for heterogeneous systems. Therefore we consier the new metho to be the preferre one in all cases, because it involves no approximations, it reprouces all experimental values consiere, it nees no extra Monte Carlo runs, an it has a low stanar eviation. Table 1 C/E results for the new metho of calculating β eff Benchmark Experiment JEFF-3. ENDF/B-VI.8 JENDL-3.3 Proteus 3.6±.2 s ±.2 1.5± ±.2 SHE ±.34 s ± ±.51.93±.5 Stacy ±4.1 s ± ±.4.985±.39 Stacy ±3.9 s ± ±.4.966±.38 Stacy ±3.7 s ± ± ±.4 Stacy ±2.9 s ± ± ±.31 Stacy ±2.6 s ± ± ±.28 Stacy ±1.8 s ± ± ±.27 Winco 119.3±.3 s ± ± ±.12 TCA 771±17 pcm 1.6± ± ±.25 Masurca R2 721±11 pcm 1.19± ± ±.18 Masurca Z2 349±6 pcm 1.26± ± ±.22 FCA XIX-1 742±24 pcm 1.34± ± ±.34 FCA XIX-2 364±9 pcm 1.63±.3 1.3± ±.29 FCA XIX-3 251±4 pcm 1.16± ± ±.22 Sneak-7A 395±12 pcm.965± ± ±.31 Sneak-7B 429±13 pcm 1.42± ± ±.32 Sneak-9C2 426±19 pcm.927± ± ±.42 Sneak-9C1 758±24 pcm.999± ± ±.32 Zpr-Pu 222±5 pcm 1.95± ± ±.32 Zpr-Mox 381±9 pcm.979± ± ±.27 Zpr-Heu 667±15 pcm 1.61± ± ±.27 Zpr-U9 725±17 pcm 1.43± ± ±.26 BigTen 72±7 pcm 1.62± ± ±.13 Goiva 659±1 pcm 1.27± ±.2 1.6±.2 Topsy 665±13 pcm 1.2± ± ±.22 Jezebel 194±1 pcm 1.31± ± ±.61 Popsy 276±7 pcm 1.25± ± ±.32 Skioo 29±1 pcm.979± ± ± Flattop 36±9 pcm.953± ± ±.3

7 Table 2 C/E results for the Spriggs metho of calculating β eff Benchmark Experiment JEFF-3. ENDF/B-VI.8 JENDL-3.3 Proteus 3.6±.2 s ± ± ±.15 SHE ±.34 s ± ± ±.49 Stacy ±4.1 s ± ± ±.35 Stacy ±3.9 s ± ± ±.35 Stacy ±3.7 s ± ±.39.99±.37 Stacy ±2.9 s ± ± ±.26 Stacy ±2.6 s ± ± ±.22 Stacy ±1.8 s ± ± ±.21 Winco 119.3±.3 s ± ±.7.996±.7 TCA 771±17 pcm 1.49± ± ±.23 Masurca R2 721±11 pcm 1.67± ± ±.17 Masurca Z2 349±6 pcm 1.166± ± ±.21 FCA XIX-1 742±24 pcm 1.5± ±.32.96±.32 FCA XIX-2 364±9 pcm 1.294± ± ±.33 FCA XIX-3 251±4 pcm 1.239± ± ±.22 Zpr-Pu 222±5 pcm 1.77± ± ±.26 Zpr-Mox 381±9 pcm.937± ± ±.24 Zpr-Heu 667±15 pcm 1.43± ± ±.24 Zpr-U9 725±17 pcm 1.41± ± ±.24 Sneak-7A 395±12 pcm 1.218± ± ±.37 Sneak-7B 429±13 pcm 1.124± ± ±.34 Sneak-9C2 426±19 pcm 1.146± ± ±.5 Sneak-9C1 758±24 pcm 1.73± ± ±.34 BigTen 72±7 pcm 1.74± ± ±.11 Goiva 659±1 pcm.983± ± ±.16 Topsy 665±13 pcm 1.248± ± ±.25 Jezebel 194±1 pcm.964± ±.5.938±.51 Popsy 276±7 pcm 1.913± ± ±.5 Skioo 29±1 pcm.921± ± ± Flattop 36±9 pcm 1.661± ± ±.45

8 Table 3 C/E results for the prompt metho of calculating β eff Benchmark Experiment JEFF-3. ENDF/B-VI.8 JENDL-3.3 Stacy ±4.1 s ± ±.43.97±.41 Stacy ±3.9 s ± ± ±.48 Stacy ±3.7 s ± ± ±.5 Stacy ±2.9 s ± ±.4 1.8±.47 Stacy ±2.6 s ± ± ±.42 Stacy ±1.8 s ± ±.4.989±.4 Winco 119.3±.3 s ± ± ±.36 TCA 771±17 pcm 1.54± ± ±.26 BigTen 72±7 pcm 1.46± ± ±.18 Goiva 659±1 pcm.991±.2.97±.2.998±.2 Topsy 665±13 pcm 1.41± ± ±.24 Jezebel 194±1 pcm.995± ±.67.97±.66 Popsy 276±7 pcm 1.47±.45 1.± ±.45 Skioo 29±1 pcm.997± ±.48.99± Flattop 36±9 pcm 1.56± ± ±.39 References [1] G.R. Keepin, Physics of nuclear kinetics, Aison-Wesley, Reaing, USA, 1965 [2] J.E. Hoogenboom, Methoology of continuous-energy ajoint Monte Carlo for neutron, photon an couple neutron-photon transport, Nucl. Sci. Eng. 143 (23) [3] G.D. Spriggs, R.D. Busch, an J.M. Campbell, Calculation of the elaye neutron effectiveness factor using ratios of k-eigenvalues, Annals of Nuclear Energy 28 (21) [4] C. Lee, T.J. Downar, an D.B. Jones, Implementation of the effective elaye neutron fractions with ajoint spectrum weighting into CPM-3, Trans. Am. Nucl. Soc. 8 (1999) 278 [5] J.F. Briesmeister (E.), MCNP - a general Monte Carlo n-particle transport coe, version 4C, Technical Report LA-1379-M, LANL, USA (2), J.S. Henricks, MCNP4C3, Report X-5:RN(U)-JSH-1-17, LANL, USA (21) [6] M.M. Bretscher, Evaluation of reactor kinetic parameters without the nee for perturbation coes, Proc. Int. Meeting on Reuce Enrichment for Research an Test Reactors, Oct. 1997, Wyoming, USA [7] R. Klein Meulekamp an S.C. van er Marck, Calculating the effective elaye neutron fraction with Monte Carlo, subm. to Nucl. Sci. Eng.

9 Figuur 1 C/E results for β eff for the various methos, base on JENDL-3.3