A Preliminary Study of an Improved Area Method, Adapted to Short Time Transients in Sub-Critical Systems.

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(1)PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future The Westin Miyako, Kyoto, Japan, September 28 - October 3, 2014, on CD-ROM (2014). A PRELIMINARY STUDY OF AN IMPROVED AREA METHOD, ADAPTED TO SHORT TIME TRANSIENTS IN SUB-CRITICAL SYSTEMS. P. Saracco, R. Marotta, G. Lomonaco, D. Chersola I.N.F.N. National Institute for Nuclear Physics - Sez. Genova Via Dodecaneso, 33 16146 Genova, Italy and GeNERG-DIME/TEC, University of Genoa Via all’Opera Pia, 15/a 16145 Genova, Italy paolo.saracco@ge.infn.it riccardo.marotta@ge.infn.it guglielmo.lomonaco@unige.it davide.chersola@edu.unige.it L. Mansani Ansaldo Nucleare S.p.A. Corso F.M. Perrone 25 16161 Genova, Italy Luigi.Mansani@ann.ansaldo.it. ABSTRACT Since the ’90s interest has grown in the characterization of sub-critical systems. Because they should not have control/shutdown devices, it is necessary to prevent by design neutron flux divergence in all conditions. Most of the available theoretical models on neutron kinetic behavior in multiplying systems are tailored on the description of critical systems; then they have decreasing validity as well as the system is far and far from criticality; as a consequence many of the available monitoring methods are of questionable use for sub-critical systems and give contradictory informations: this is true, among the others, for the Area Method. One of the difficulties in understanding the kinetic behavior of a sub-critical system is the scarcity of experimental measurements available. Then we decided to develop a full time-dependent MCNP6 simulation of a sub-critical core to obtain some ”phenomenological-like” data whose analysis leads to conclude that geometrical inhomogeneity of the system plays a key role in determining its time behavior over short time scales. Starting from an original observation by Weinberg and Wigner, we conclude that, for the short time scale considered (∼ µs), we cannot use the ”classical” diffusion approximation but a little bit more complicated formulation, similar to what is known as telegrapher’s equation: along this line we find a reasonable justification for the idea that, before using any analysis method grounded on the assumption that the system has relaxed into its fundamental mode, we could have to wait for a longer time than the prompt neutrons lifetime. We conclude that from an experimentalist’s point of view it should be at lest necessary to cut from the analysis the lowest part of the time interval before using the Area Method, but more refined analyses are needed..

(2) P. Saracco et al.. Key Words: Neutron transport, Sub-critical systems, Area method, Telegrapher’s equation, Non homogeneous systems. 1. INTRODUCTION. Since the ’90s interest has grown in the characterization of sub-critical systems: some sub-critical cores have been studied experimentally in the past, such as Yalina [1] in Belarus or MUSE [2] in France, others are currently running an experimental program, such as FREYA [3] at SCK-CEN in Belgium or are foreseen, like the MYRRHA project [4]. The efficiency of subcritical facilities for the purpose of transmutation can be questionable at the industrial scale if the accelerator parameters are not properly set [8, 9]. From the point of view of safety, it is mandatory that the nuclear design ensures that criticality conditions are not attained, with adequate margin, under any foreseeable occurrence pertaining either to Design Basis Conditions (DBC) or Design Extension Conditions (DEC). The above can be achieved, in principle, with or without reliance on neutron absorbers. A margin of 0.016 ∆k was proposed [5] for DBCs (0.016 ∆k is derived assuming ∼ 3$ of margin and ∼ 2$ of allowance for measurements errors). Hence keff shall not exceed the limiting value of 0.984 at any time during DBC. As starting point for the design a keff = 0.97 was assumed for EFIT [7] (ADS cooled by Lead and developed in the FP6 EU IP-EUROTRANS project [6]) as the maximum value for the nominal full power conditions (upper limit of the green region). Figure 1 below illustrates safety criteria reported in, showing that at no time keff shall move into the red regions, which represent the margins to criticality allocated to the DBC, DEC and Refueling conditions. The green region covers the range of keff allowed for normal operation. The yellow regions cover the excursion ranges of keff predicted by the transient analyses for DBC and DEC respectively. Since the margin to sub-criticality depends on the specific and on the results of the transient analyses, both regions are depicted, but not figured out. In an ADS the keff shall be monitored both during operation at power and during refueling with qualified methods. For nominal operation, the accelerator beam will work in Continuous Wave (CW) mode which means a continuous uninterrupted beam without any holes. In this operation mode, the current-to-power indicator will provide the on-line measurement of the reactivity. Since the proportionality constant in the current-to-power indicator has to be verified regularly, interim reactivity monitoring techniques have to be applied. These dynamic reactivity measurements require specific beam time structures for the accelerator. There are advantages and disadvantages in sub-critical systems: they should be simpler both in project phase and during operation because they are expected to be inherently safe, but, on the other hand, they are very expensive machines (one Nuclear Reactor plus one Accelerator). So the use of this system for electricity production are not envisage due to its high cost, but they can be effectively used to burn the waste produced in the current fleet of thermal reactors (its efficiency is. 2/14. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014.

(3) IMPROVED AREA METHOD.... Figure 1. Reactivity criteria for EFIT[7] 42 kg of MA for TWh). The electricity production would be considered only a byproduct. It would be leave to the Fast Neutron Reactors the electricity generation: Fast Neutron Reactors that thanks to the adoption of a so called ”adiabatic fuel cycle” (one possible way to realize a closed fuel cycle [10]) could produce only Fission Products that can be sent to ”historical” repository. Most of the available theoretical models on neutron kinetic behavior in multiplying systems are tailored on the description of critical ones; many well known approximations leading to these models have decreasing validity as well as the system is far and far from criticality [11]: as a consequence many of the available methods to monitor the system behavior - in particular those concerning sub-critical reactivity level - are of questionable use for sub-critical systems and give apparently contradictory informations. This is true, for instance, for the Area Method [12], for the source jerk method and also for noise technique; the robustness of these well known methods with respect to measurement position is not assessed. We decided to develop a full time-dependent MCNP6 simulation of a sub-critical core to obtain some ”phenomenological-like” data whose analysis leads to conclude that geometrical inhomogeneity of the system plays a key role in determining its time behavior over short time scales. This contribution is a preliminary report on a current research activity jointly performed by I.N.F.N., Genova University and Ansaldo Nucleare S.p.A. with the goal of identify and validate robust measurement methodologies of the sub-criticality level of the systems also in view of the MYRRHA project.. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014. 3/14.

(4) P. Saracco et al.. 2. SOME ”PHENOMENOLOGICAL” OBSERVATIONS. One of the major difficulties in understanding the kinetic behavior of a sub-critical system is the scarcity of experimental measurements available to date: this is also a consequence of the restrictions that safety authorities put on the operation of such facilities in absence of assessed reactivity monitoring techniques; so it seems that we are in a loop. Then we decided to follow a somehow unusual path and to develop a full time-dependent Monte Carlo (MC) simulation of a sub-critical core, based on MCNP6 code, allowing only for prompt neutrons (see in the following): we know that this is a questionable approach because discrepancies are known to be present among MC simulation and measurements. This disagreement appears both in static and in kinetic simulations, the most important reason being the low quality of the used nuclear data - and, on a minor extension, because usually one does not simulate the ”exact” experimental configuration, but some more or less coarse approximation to it: however there are no reasons to believe that MC simulation gives a ”wrong” solution to the transport equation, apart from statistical errors. Then by using simulations we can safely compare, for instance, static and kinetic quantities for an assigned configuration, trying to figure out relations between them. Moreover a full time-dependent simulation has an important advantage in giving the possibility to vary freely system parameters and ”operating conditions” - sub-criticality level, duration of beam trips, temperatures, and so on - eventually yielding a wide set of ”phenomenologies” against which it is possible to verify validity of approximations and theoretical models. Even configurations not experimentally accessible - like, e.g. a completely unreflected system - are instead possible configurations for a simulation. We underline that this is a necessity when looking to sub-critical systems primarily in view of the inherent ambiguity in the meaning of the word ”sub-criticality” itself: keff is evidently a good parameter to define what is ”criticality”, that is a system with keff = 1, but it is more difficult to use it as a parameter ”measuring” the distance from criticality, because it is not so clear which is the scale on which we can ”measure” this distance. Then, if it is clearly expected - on the basis of some experience - that a system with, for instance, keff = 0.9 or keff = 0.95 is far from criticality or, to use FREYA terminology, that this is by sure a deep sub-critical configuration, it is not so clear which is the maximum value for which we are allowed to consider a configuration as deeply sub-critical. On the other hand it is qualitatively obvious that the curvature of the neutron flux1 discriminates a system dominated by the external source from a system dominated by internal fission chain, so one could assume that a change in the sign of the curvature of the static neutron flux indicates the transition from one configuration to the other: however within a simple 1 group diffusion model it was observed [13] that this ”critical value” is around keff = 0.995, a seemingly very high value. A second preliminary observation regards the kind of experiments that, for various reasons, are relevant and practicable in sub-critical systems: the time scale is very short with respect to typical 1. We refer here to the mathematical notion of curvature, that is the sign of the 2nd derivative, instead of buckling to avoid confusion. The notion of buckling is properly referred to critical systems.. 4/14. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014.

(5) IMPROVED AREA METHOD.... 0.001 à. à. à. à. à. à. à à. à à. à à. à à. à à. à à. 10-5. 0.1. keff =0.95. Φ @arbitrary unitsD. Φ @arbitrary unitsD. 0.1. à à. à à. à à. à. à à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. 10-7. à à. à à. à à. à à. à. à à. à à. à à. à. à à. à à. à. à à. à. à à. à à. à à. à à. à à. à à. à à. à à. à à. à à. à à. à à. à. à à. à à. à. à à. à à. à à. à. à à. à à. à. à à. à à. à à. à. à à. à. à. à à. à à. à. à à. à. -9. à. 10. à à. à à. à. à. à. à à. à. à. à. à à à. à à à. à. à à. à à. à. à. à à. à à. à. à. 0.001 à. à. à. à. à. à. à. -5. 10. à à. à. à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. 10-7. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à. à à. à à. à à. à à. à à. à à. à à. à à. à à. à à. à à. à à à. à à. à. à à. à. à. à. à. à à. à à à. à. à à. à à. à. à. à. à à. à. à à. à. à. à à. à à. à à. à à. à à. à. à. à. à à. à. à à. à. à. à. à à. à. à. à. à. à à. à à à. à à. à. à. à. à à. à à. à. à. à. à. à à. à. à. à. à à à. à. à à. à à. à à. à. à. à. à à. à à. à. à. à à à. à. à. keff =0.97. à. à. à à. à à. à à à. à. à à. à à. à. à. à. à. à à. à. à à. -11. à. à. à à. 10. à à. 0. 20. 40. 60. 80. t @ ΜsecD. 0. 20. 40. 60. à. à. 80. t @ ΜsecD. Figure 2. The time dependent flux for an homogeneous reflected system. The off-source interval is preceded by a full precursors precharging interval. Tallies are at 1/2 R (blue), 7/10 R (red) and 6/5 R (green), where R is the radius of the active zone. The last tally is then in the reflector. critical reactor times, of the order of tens of µs or so. Main reasons for this are: by one side in a sub-critical system neutron kinetics is not ruled by delayed neutrons [13], but on the other side it is not practical to operate an ADS with long beam trips because of thermo-mechanical stresses. Then the time scale over which experiments are carried on are not so large with respect to typical prompt neutron lifetimes. In Figure 2 we show in logarithmic scale the time behavior of the neutron flux for a homogeneous spherical core of uranium with ∼20% enrichment in 235 U : to make different simulations easily comparable and, on the other side, to be able to vary keff , we leave the radius of the assembly fixed at 50 cm and we vary (slightly) the percentage of enrichment; the homogeneous core is surrounded by a 20 cm thick lead reflector; flux are evaluated at different positions in the core. Apart the first few microseconds these fluxes are manifestly ruled by a single time frequency common to all detector positions; for larger times statistical errors become evident and make data not usable (the system manifestly relaxed into its fundamental mode for lower times, so we do not need these data for the. Figure 3. The x − y scheme of more ”realistic” simulation (see text): in blue the lead zones, in red the fuel, in yellow the position of the tallies. Source is at the center.. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014. 5/14.

(6) P. Saracco et al.. à. 5 ´ 10-5. à. à à à à. à. à. keff =0.95. à à à. 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Φ @arbitrary unitsD. à. à. 5 ´ 10-5. à. à à à à à à à. à à à. à. à. à à. à à à à à. à à. à. à à à. à à à à à à. à à à à à à. à à. à à. à à. à à à à à à. à à à à. à à. à à à. à à à à. à à à à à. à. à à à à. à à à à. à à. à à. 2 ´ 10-6. à. à à à. à. à à. à à à. à à à à. à à à à à à. à. à à. à à à à. à à à à à à. à à à à à à à à. à à à à à à à. à à à à. à à à. à à à à. à à à à à à à à à. à à à à à à à à. à à. à à. à à à à à. à. à. à à à à. à à à à à. à à à à à à à à à à. 0. 50. 100. 150. à à à. à. à à à à à à. à. à. à à. à. à. 200. t @ ΜsecD. Figure 4. The time dependent flux for the more ”realistic case” (see text). The off-source interval is preceded by a full precursors precharging interval. Tallies are in fuel cells as in Figure 3; in blue in the first cell on the horizontal yellow line, in red in the ninth cell on the horizontal yellow line, in green in the ninth cell on the diagonal yellow line purpose of the present discussion; then we stopped the (long) simulations to save CPU time.). The different behavior in the first few microseconds has to be ascribed, on the basis of diffusion theory, to the effect of higher modes frequencies. In Figure 4 we show the same result for a more ”realistic” core built from cylindrical fuel assemblies with the same fuel embedded in a square lead lattice; the whole system is contained in a cubic 20 cm thick lead reflector (see Figure 3). Flux presented are mediated over the volume of each fuel pin. Again we can see in the first few microseconds the effect of higher modes, but in the intermediate region (5 < t < 150 µs approximately) flux is no more ruled by a single frequency, as it is clearly evident from Figure 4. From the analysis of this first ”phenomenological” data a striking, and rather surprising to us, evidence arises: the geometrical inhomogeneity of the system plays a key role in determining the time behavior of the system over so short time scales. We then deduce that a net effect of non homogeneity of the system is the unexpected, at least to us, appearance of frequencies other than the fundamental one in the region t > 5 µs. It should be evident that this feature, if confirmed, invalidates the main assumption on which point kinetic is grounded, namely that system has relaxed into its fundamental mode. Then it is highly questionable to make use of any analysis method grounded on this approximation, like e.g. the Area Method, if this effect is not properly assessed. Naively we could expect these methods work well if we cut that part of flux that is not linear on logarithmic scale, because in this way we wait the system relaxes to fundamental mode: however this prescription should be properly justified to make this suggestion something more than a ”cooking recipe”. The first improvement manifestly needed is a more spatially refined tally definition: it could be possible that the features observed depends on spatial extension of the used tallies. Unfortunately this requirements conflicts with CPU requirements: tallies over smaller volumes require higher CPU times to have sufficient statistic; a possible way-out is to perform a simulation for the Green’s function of the system - that is we simulate the prompt response of the system to a instantaneous pulse - and then, if needed, to perform a source integration.. 6/14. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014.

(7) IMPROVED AREA METHOD.... In Figure 5 we present in a logarithmic plot results for such a simulation for a non homogeneous system in absence of the reflector. We show with the same color tallies at different positions taken for the same value of kef f : different tallies have been rescaled to emphasize the common long time behavior. Thickness of the different lines is a consequence of statistical uncertainties. Figure 5 shows expected features with one notable exception: for longer times fluxes have the same exponential behavior, as expected, but the time needed to reach such condition is somehow larger than expected: higher modes seem to contribute in a non negligible way till 10 µs or so. This is a larger time than for a homogeneous system: in Section 3 we give some hints on these features. But, at the end, nothing dramatic: to be safe in using analysis methods grounded on point kinetic one should, at most, cut the first 10 µs or so of the measurements: for longer times only one frequency, strongly kef f dependent, rules the system. However a realistic system always has a reflector outside. So we repeated the simulation with a thick reflector around the system and in Figure 6 we present in a logarithmic plot results for such a simulation. Now things changes in a rather dramatic way: the two time scales of the system, the one characteristic of prompt neutrons - enhanced, as we have seen, by the effect of the spatial inhomogeneity of the system - and the second coming from the reflector, which sends back neutron to the core with some delay, interfere to give a very different space-time behavior of the system. The time needed to reach the fundamental mode enhances to ∼ 2 − 3ms, a relatively very high value, at least for subcritical systems: we were able to obtain such values with some refined variance reduction methods, which enabled our simulation to run up to 10 ms, spanning in a statistically reliable way values for the flux over 4-5 order of magnitude. There could be, in principle, other reasons for the appearance of higher frequencies in the system in this ”intermediate times” region: for instance it was shown [14] that even within a simple. Figure 5. The time dependent flux for the more ”realistic case” (see text). Fluxes at different positions have been (spatially) rescaled to have the same long time behavior: PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014. 7/14.

(8) P. Saracco et al.. Figure 6. The time dependent flux for the more ”realistic” case (see text). Fluxes at different positions have been (spatially) rescaled to have the same long time behavior: multigroup diffusion model a multiplicity of time frequencies must be associated with each of the normal modes of the system (for a NG calculation one ends with NG frequencies associated to the fundamental mode and NG others associated to each of the other normal modes of the system), so in principle the outcome of the ”realistic” simulation could be not so surprising. But few considerations, even without making very detailed calculations, show things are not so simple; Figure 2 explicitly shows that in the range 5 < t < 80 µs only one frequency - by the way the higher - survives: this one is necessarily associated with the fundamental mode of the system and even other frequencies associated with this fundamental mode arising from energy dependence are faster, not only those associated with higher modes; one could then infer that even in diffusion theory geometric inhomogeneity slows the system modifying the time eigenfrequencies associated with fundamental mode. 1.0. 0.5. -80. -60. -40. -20. 0. -0.5. -1.0. Figure 7. Plot of the denominator of Laplace transform solution φ(x, s) for one-group diffusion equation in one-dimension for different non homogeneous systems made by alternate zones of fuel and moderator: black 2 zones, blue 4 zones, red 6 zones, green 8 zones, orange 10 zones. Plots have been rescaled to fit in the same figure, being relevant to the discussion only the position of respective zeros. It is easy to verify that it is not so; following the methodology developed in [15] we determined the position of the poles for the Laplace transform of the solutions of diffusion equation for a non 8/14. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014.

(9) IMPROVED AREA METHOD.... homogeneous one dimensional system: in Figure 7 we show the plot of the denominator of the Laplace transform φ(x, s) of the solution for the one-group time-dependent diffusion equation for a one-dimensional system made by alternate zones of fuel and moderator2 , with the same material compositions we used in the simulations; it is clear that, even if position of the poles depends on the number of zones in the system3 , there is no evident ”slowing” of the system: in this case, to explain simulation result, we should observe zeros denser and right shifted as the number of zones increase. This argument is not conclusive, because it has been carried on in one group approximation, but it seems sufficiently convincing: alternatively one should explain also why energy dependence of the flux is relevant for a non homogeneous system, but not for a homogeneous reflected system. A possible hint on the solution of this problem is to observe that on so short time scales the information about the shutdown of external source arrives at the various fuel pins at different times, so yielding possibly to different shutdown ramps shifted in time, because many different paths can be run by neutrons in the system, before they finally arrive to the detector: obviously we imagine shutdown process as instantaneous. This effect, a true transport effect, is surely included in a time dependent MC simulation and, possibly, discriminates a homogeneous system - featuring a continuous distribution of fission sources - from a non homogenous one. We have not attempted till now a full phenomenological description of this process, because it is first necessary to assess a (simple) mathematical description of the system allowing finite velocities for the transmission of the information - a feature that is not possible in diffusion theory. We simply observe that this effect, if any, is clearly expected to be strongly keff -dependent: in fact by varying keff we change the (mean) number of neutron generations from the source to the detector and then also the number of different paths that must be considered to reconstruct the total flux: only by summing over all possible paths we can in principle reconstruct the form of the time dependent neutron flux at a given position.. 3. A POSSIBLE MOTIVATION: AN ”OLD” SUGGESTION. Many years ago Weinberg and Wigner observed [16] that the correct P1 approximation to monoenergetic neutron transport is not given by time-dependent diffusion equation because the Fick’s law strictly holds only for stationary problems. In fact when carrying on honestly the P1 approximation the equation for the current is # " ~ r, t) ∂ J(~ 3 ~ r, t) + ~ r, t) = −D ∇φ(~ J(~ v ∂t for isotropic sources and only assuming that the rate of time variation of the current density is very much lower than the collision frequency vΣt ∼ 105 s−1 for thermal system[17] (a factor 10-100 more for a fast system) one finally recover Fick’s law; when justifying negligibility of this term, 2. By the way the denominator of φ(x, s) is independent from x and all poles are real: proof of this second feature is a bit complex and it is omitted. 3 For scale reasons this is really appreciable only for the two last zeros on the right of the plot in Figure 7. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014. 9/14.

(10) P. Saracco et al.. Weinberg and Wigner in fact observed that ”one hopes that the neutron density does not change materially in periods of the order of 1/Σt v ∼ 10−5 s”. However one can easily argue that this condition is hardly met when looking at time variation of a sub-critical system over time periods of the same order of magnitude or less, immediately after a beam shutdown (or starting up). If this is the case, the equation to be solved is not the time-dependent diffusion equation but a form similar to ”telegrapher’s equation”4 , namely 3D ∂ 2 φ(~r, t) 1 + 3DΣa (1 − k∞ ) ∂φ(~r, t) = D∇2 φ(~r, t) − Σa (1 − k∞ )φ(~r, t) + S . + v2 ∂t2 v ∂t. (1). We note that a similar observation was done, with some modifications with respect to present approach, by Merk [18] and later by Heizler[19]. There is no detailed study in literature of the properties of the solutions of equation (1) in the specific context of neutrons multiplicative systems, so in the following we give some details on this topic. 35 30. ΦΦmax. 25 20 15 10 5 0 0. 1. 1. 2. xL Figure 8. Time dependent neutron flux after the shutdown of the beam as a function of the distance from source position in one dimension (solid lines: equation (1), dashed: diffusion theory). Stationary solution is obviously the same. Velocity is assumed to be unity, so that time scales must be rescaled correspondingly to proper neutrons velocity. Black: stationary solution in presence of the beam, red: t = 5, green: t = 15, blue: t = 25, brown: t = 35. Time is measured in units such that v = 1. This equation has a peculiar feature with respect to diffusion equation in that it preserves an important (at these time scale) feature: velocity √ 5 that propagates into the system the information about perturbations is not infinite, but is v/ 3 . Telegrapher’s equation is very much like a wave √ equation in this respect: it presents a well definite wave-front at, in one dimension x = vt/ 3; so, for instance after a beam shutdown for x larger than this value, the solution for the time dependent 4. Telegrapher’s equation, to be precise, is the form of the P1 equation we obtain for non multiplicative systems, k∞ = 0; neutron multiplication implies a change of sign with respect to traditional form of telegrapher’s equation, as it clear in (1): any analogy with known solutions of the traditional form of it must be taken with care, because this change of sign would imply a negative transverse resistivity. 5 This precise value is a consequence of the P1 approximation.. 10/14. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014.

(11) IMPROVED AREA METHOD.... neutron flux coincides with the stationary solution, in contrast with diffusion theory, see Figure 8. Oscillations in figure are not physical, but they are a numerical consequence of the calculation √ scheme adopted: solutions before and after the wave front are proportional to θ(±(x − vt/ 3)) respectively. On the other side of the wave-front there is a persistent perturbation: when the wavefront reaches the boundaries of the system, this persistent solution is similar in form to diffusion theory one, but with a different amplitude. Solution of (1) can be found in a standard way both for an homogeneous and for a non homogeneous system: we note, however, that it will not be possible to find a detailed solution for realistic geometries even for this equation, simpler with respect to full transport theory; the purpose of this preliminary study is to find a theoretical justification for the ”multiple neutron paths” model we are looking for to understand the ”observed” different behaviors in Figs. 2 and 4. We expect such a model to be useful both to a better understanding of the physics involved in the description of sub-critical systems and to have hints on how to build up a phenomenologically useful model analogous to point kinetic one - for sub-critical systems. For the moment we assume as initial condition for (1), in the beam shutdown case

(12) ∂φ(x, t)

(13)

(14) =0 φ(x, 0) = φ0 (x) , ∂t

(15) t=0 on the basis of ”causality principle” [18], but we note that a more detailed derivation of initial conditions should be carried on, starting from transport equation and applying coherently the P1 approximation, because in our opinion some doubts survive on the second of these conditions. Boundary conditions are instead standard: in absence of the source, both for an homogeneous and for a non homogeneous system we must require that at any times the neutron flux vanishes at the extrapolated boundaries of the system. Then in Laplace transform we have to solve d2 φ(x, s) Σa (1 − k∞ ) 1 3 − φ(x, s) − (1 + 3DΣa (1 − k∞ )) sφ(x, s) − 2 s2 φ(x, s) = 2 D Dv dx v 1 3 (1 + 3DΣa (1 − k∞ )) φ0 (x) . = − 2 (s) + Dv v. (2). For a non homogeneous system macroscopic cross section and diffusion coefficient are position dependent: for a multiple homogeneous slabs system, like one we are here considering, we have simply to use this equation in different zones with different coefficients and to impose well known interface conditions identical to those used in diffusion theory. The most striking difference with respect to diffusion theory is about position of the poles in the complex s-plane: in this case they are not real (and negatives for a sub-critical system), but they lie on a line parallel to the imaginary axis (with negative real part). This difference reflects in the very different behavior of the solutions of telegrapher’s equation (see Figure 9) with respect to diffusion theory. As it can be clearly seen, the net result is that space-time behavior of diffusion theory is smoother: as a consequence, diffusion theory is not capable to describe a phenomenon like the one we are looking for (but this is the same as to say that diffusion theory propagates perturbations in the system with infinite velocity, as it is well known).. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014. 11/14.

(16) P. Saracco et al.. Figure 9. Time dependent neutron flux after the shutdown of the beam as a function of the distance from source position in one dimension and of time: the wave-front is clearly evident 4. CONCLUSIONS AND PERSPECTIVES. On the basis of this discussion, and even without really solving equation (1) for a non homogeneous system, we found a reasonable justification for the originally naive idea we explained in Section 2: we (could) have to wait for a longer time than the prompt neutrons lifetime lpr in a non homogeneous system, before prompt frequencies higher than fundamental one become negligible. This prescription should be followed before using any analysis method grounded on the idea that the system has reached this condition: the amount of this effect is expected to be strongly dependent both on keff and on the geometry of the system. However a careful look to our ”phenomenological” results hints to some more detailed even if preliminary conclusions. There are clearly three (and not simply two) different time regions for the space-time dependent flux: (i) a first one, t < 40 µs, where multiple prompt frequencies are manifestly active; (ii) a second ”intermediate” region, ∼ 40 µs < t < 2 ms, where the log-time scale plot shows that fluxes are not ruled by a single frequency, they are not straight lines, but they however are connected one to the other by a purely space dependent factor (indicating a space-time factorization of the flux); (iii) a higher times region where fluxes are characterized by a single frequency. We must note however that this last region is not really experimentally accessible because here flux is essentially due to delayed neutrons, that are not present in our simulation. Moreover, even if this region would be experimentally accessible in some unclear way, fluxes at different keff values seem to be characterized by the same frequency, that then we must ascribe to reflector properties. Then what remains experimentally accessible and possibly dependent on keff in a reasonably simple way is the behavior in the intermediate region (ii): there time-dependence of the flux is probably due to an interference among reflector and prompt time behaviors. Explanation of what really happens in this region appears possible to us on the basis of some equation simpler than full transport, but able to keep track of the finiteness of the velocity of perturbation transmission in the system (that is Telegrapher’s equation). On these basis, and with the help of some purely space-dependent correction [20] it remains possible that some informations about keff remains experimentally accessible. 12/14. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014.

(17) IMPROVED AREA METHOD.... A final comment is due: here we presented results for a fixed source energy and a given angular distribution. Further studies are needed to assess energy and angular dependence of the Green’s function.. ACKNOWLEDGMENTS. We acknowledge Prof. P. Ravetto and S. Dulla (Politecnico di Torino) and Dr. M. Carta (ENEA) for many helpful discussions. This work is supported in part by Regione Liguria in the frame of PO CRO Fondo Sociale Europeo Regione Liguria 2007-2013 Asse IV ”Capitale Umano” ob. specifico l/6.. REFERENCES [1] C. M. Persson et al., ”Analysis of reactivity determination methods in the subcritical experiment Yalina.” NIM in Physics Research, 554A: pp. 374 - 383 (2005). [2] M. Salvatores et al. ”Muse-1: A first experiment at Masurca to validate the physics of subcritical multiplying systems relevant to ADS.” In: Proc 2nd Int Conf on Accelerator-Driven Transmutation Technologies and Applications, OECD Nuclear Energy Agency, Kalmar, Sweden, Vol. 1: p. 513 (1996). [3] ”The FREYA project.”, http://freya.sckcen.be/ (2014). [4] ”The MYRRHA project.”, http://myrrha.sckcen.be/ (2014). [5] L. Mansani, R. Monti, P Neuhold, ”Proposed Subcriticality Level for an 80 MWth Lead Bismuth Cooled ADS, Actinide and Fission Product Partitioning & Transmutation”, Seventh Information Exchange Meeting, Jeju, Republic of Korea, 14-16 October (2002). [6] J. Knebel, et al., ”European Research Programme for the Transmutation of High Level Nuclear Waste in an Accelerator Driven System”, EUROTRANS, IX International Exchange Meeting on Actinides and Fission Products Partitioning and Transmutation (IEMPT9), Nimes, France, 25-29 September (2006). [7] L. Mansani, et al., ”The European Lead-Cooled EFIT Plant: An Industrial-Scale AcceleratorDriven System for Minor Actinide Transmutation - I”, Journal of Nuclear Technology, 180:pp. 241-263 (2012). [8] G. Lomonaco, O. Frasciello, M. Osipenko, G. Ricco, M. Ripani, ”Focus point on: an intrinsically safe facility for forefront research and training on nuclear technologies - Burnup and transmutation”, European Physical Journal - Plus, 129: pag 74, (2014). [9] M. Ripani et al., ”Study of an intrinsically safe infrastructure for training and research on nuclear technologies”, submitted to European Physical Journal, (2014).. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014. 13/14.

(18) P. Saracco et al.. [10] C. Artioli, G. Grasso, C. Petrovich, ”A new paradigm for core design aimed at the sustainability of nuclear energy: The solution of the extended equilibrium state” , Annals of Nuclear Energy, 37:pp. 915 - 922 (2010). [11] D.E. Cullen, J. Clouse, R. Procassini and R.C. Little. ”Static ans Dynamic Criticality: Are they different?”, UCRL-TR-201506 (2003). [12] N.G. Sj¨ostrand, ”Measurement on a subcritical reactor using a pulsed neutron source”, Archiv f¨or Fysik, 11: pp. 233-246 (1956). [13] P. Saracco and G. Ricco. ”Various Operating Regimes of a Subcritical System as a Function of Subcriticality in One Group Theory”, Nucl. Science and Engineering, 162:pp. 167-177 (2009). [14] P. Saracco, S. Dulla and P. Ravetto. ”On the Spectrum of the Multigroup Diffusion Equations”, Progress in Nuclear Energy, 59: pp. 86-95 (2012). [15] S.E. Corno, S. Dulla, P. Picca and P. Ravetto. ”Analytical Approach to the Neutron Kinetics of the Non Homogeneous Reactor”, Progress in Nuclear Energy, 50: pp. 847-865 (2008). [16] A.M. Weinberg and E.P. Wigner, The Physical Theory of Neutron Chain Reactions, Chapt. IX, University of Chicago Press, Chicago, Il, USA (1958). [17] J.J. Duderstadt and L.J. Hamilton, Nuclear Reactor Analysis, Chapt. 4, John Wiley & sons, Lexington, KY, USA (1976). [18] B. Merk, ”An Analytical Solution for a One-Dimensional Time-Dependent Problem with External Source”, Transport Theory and Statistical Physics, 37:pp 535-549 (2008). [19] S.I.Heizler, ”Asymptotic Telegrapher Equation (P1) Approximation for the Transport Equation”, Nucl. Science and Engineering, 166:pp. 17-35 (2010). [20] N. Marie et al., ”REACTIVITY MONITORING USING THE AREA METHOD FOR THE SUBCRITICAL VENUS-F CORE WITHIN THE FRAMEWORK OF THE FREYA PROJECT”, presented at the TCADS conference (2013).. 14/14. PHYSOR 2014 - The Role of Reactor Physics toward a Sustainable Future Kyoto, Japan, September 28 - October 3, 2014.

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