Some New Thoughts on the Multipoint Method for Reactor Physics Applications. Sandra Dulla, Piero Ravetto, Paolo Saracco,

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1 Jeju, Korea, April 16-20, 2017, on USB 2017 Some New Thoughts on the Multipoint Metho for Reactor Physics Applications Sanra Dulla, Piero Ravetto, Paolo Saracco, Politecnico i Torino, Dipartimento Energia, NEMO Group, Corso Duca egli Abruzzi 24, Torino, Italy I.N.F.N., Via Doecaneso, 33, Genova, Italy sanra.ulla@polito.it, piero.ravetto@polito.it, paolo.saracco@ge.infn.it Abstract - The multipoint approach proves to be an efficient metho for the neutron kinetics of nuclear reactors an accelerator-rive systems. In the present work some features of multipoint kinetics are iscusse with reference to stanar iscretization schemes, both in space an in energy. In the secon part of the paper a physically-base approach to construct a multipoint moel is presente. It is shown how the parameters of the moel can be erive through a Monte Carlo simulation evaluating transfer probabilities an characteristic times. Some results are obtaine for an iealize fast system to evience the properties of the multipoint parameters. At last, a stanar point kinetic moel is reconstructe from the multipoint equations, showing the relationship between the classic integral parameters of a multiplying systems an the multipoint parameters. This allows to consier a multipoint moel for inverse applications aiming at the experimental etermination of integral parameters. I. INTRODUCTION In the stuy of nuclear reactors it is of great importance to accurately simulate the spatial an spectral effects uring transient situations. In many applications only the very simple point kinetic moel is use. This moel is particularly avantageous when ealing with inverse problems, e.g. for the interpretation of kinetic experiments to reconstruct integral parameters such as reactivity or effective elaye neutron fractions, from flux measurements [1]. However, point kinetics is not suitable at all for applications to large reactors or uncouple systems characterize by large spatial istortions uring transients. It is also well-known that point-kinetic results may be on the unsafety sie, as the values of the power may turn out to be lower than the exact ones. A full space-energy approach may be computationally too intensive in many applications. Alternatively, the multipoint approach can prove to yiel aequate results in many applications with an acceptable computational effort. The iea of the multipoint metho is base on subiviing the reactor phase space geometrical omain an/or energy range into separate subomains an assuming that the characteristic function amplitue for each subomain evolves accoring to a point-like equation, incluing coupling terms with the other subomains, ue to neutron streaming in space an transfer in energy through collisions. Therefore a system of couple first-orer orinary ifferential equations is obtaine. The multipoint moel was propose by Avery in the theory of couple reactors [2, 3]. Later several formulations were propose in ifferent frameworks [4, 5, 6, 7]. In this paper some basic aspects of the multipoint formulation are consiere, starting from simple physical configurations. The attention is focuse separately on spatial an spectral effects. For spatial problem the criticality problem is analyse, giving some attention also to the full eigenvalue spectrum of the multipoint equations. One attractive feature of the multipoint metho is the possibility of its application to inverse problems for the interpretation of kinetic experiments, in particular for the measurement of integral parameters in source-riven experiments carrie out in subritical assemblies for the assessment of acceleratorriven technology. In many situations point kinetics has prove to yiel unsatisfactory results [8]. In this work the point kinetic moel is reformulate starting from the multipoint one, in orer to relate the stanar integral parameters of point kinetics to the multipoint parameters. A physical-base general approach for the formulation of the multipoint equations is thoroughly iscusse. The parameters are relate to probabilities associate to the transport an collision phenomena within the system. Such parameters are evaluate using a Monte Carlo approach: results for a challenging fast system configuration are presente. The general features of the metho are escribe in the following without irectly introucing the source term. However, the contribution of an external source can be reaily inclue. Therefore, the metho is applicable for the escription an the simulation of an accelerator-riven system. II. SPACE MULTIPOINT THEORY The multipoint metho can be rigorously erive through a stanar separation-projection proceure, generalizing the Henry s process to erive the point kinetic equations [7]. We are using here formulations base on a irect approach. If one focuses on space aspects only, a multipoint set of equations can be obtaine by irect iscretization of the spatial variable by means of a stanar iscretization scheme. Of course, this is not what is one in effective multipoint moels, because stanar iscretization schemes require to set the spatial mesh to values that can effectively account for the physical process of neutron motion. Therefore, ue to the values of cross sections in realistic systems, the number of points may turn out to be too large an far from matching the philosophy of the multipoint approach. This is the reason why more sophisticate proceures must be use, such as the alreay mentione separation-projection technique or the physical here propose. However, the use of spatially iscretize equations

2 Jeju, Korea, April 16-20, 2017, on USB 2017 may prove useful to unerstan the numerical features of the multipoint framework. k 1 1. The space iscretize moel Let us refer to a simple one-group problem, let us also efine a n+1-point spatial gri an a vector φ containing the values of the scalar fluxes at each point of the gri. The iscretize balance equations in the absence of a neutron source take the form: 1 v φt = ˆFφt ˆLφ, 1 where the loss matrix ˆL an the fission matrix ˆF are relate to the material properties of the omain, the physical moel aopte an the iscretization scheme. For instance, in a simple homogeneous one-imensional plane meium of wih l, with a uniform gri characterize by the mesh length h an using the iffusion moel while isregaring elaye emissions, the loss matrix ˆL is triiagonal, with l ii = 2 D h 2 + Σ a an l i,i+1 = l i,i 1 = D h 2, while ˆF is a iagonal matrix whose elements are simply equal to νσ f. In this case the matrices have imensions n-1 because bounary conitions impose φ 0 = φ n = 0. A criticality theory may be evelope from the above moel, by stuying the steay-state version of Eq. 1 an introucing a coefficient 1/k in front of the fission term, thus moving to an eigenvalue problem. Matrix ˆL is a very peculiar one, that is a triiagonal Toeplitz matrix, characterize by constant elements on each of the escening iagonals, for which an analytical formula for the eigenvalues is available [9], leaing, with some algebraic manipulations, to an expression for the spectrum of the k eigenvalues k m = νσ f 2Dn 2 mπ, m = 1,..., n cos + Σ l 2 a n for an n interval mesh. In the limit n, using [ ] n qπ lim n n2 1 + cos = π2 q 2 n 2, we obtain the spectrum for the k eigenvalues as νσ f k q =, q = 1,...,, 3 D π2 q 2 + Σ l 2 a where we use q = n m, so that now k 1, the highest eigenvalue can be truly interprete as k. In Fig. 1 we show the spectrum of the eigenvalues as a function of the number of intervals in the mesh: we represent all the eigenvalues for a given mesh imension with the same color; it is interesting to observe that all eigenvalues are real an that the m th eigenvalue, whose first appearance is when the number of intervals is equal to m + 1, is really an approximation to the m th k eigenvalue, to which it converges monotonically as the number of meshes increases. For a non-homogeneous meium the same formulation of the problem applies as well, provie one replaces the k 2 k 3 k 4 k n-1 Fig. 1. The k eigenvalue spectrum for a one-imensional homogeneous slab as a function of the number of points in the spatial mesh. constant values D, Σ a, νσ f with the corresponing values evaluate at the mesh points D j = D x j, Σa, j = Σ a x j, νσ f j = νσ f x j ; in this case, however, the Toeplitz nature of the triiagonal matrix ˆL 1 k ˆF is lost an the analytical form for the spectrum is no more available, but a numerical evaluation of the require eterminant remains feasible. 2. A physical approach to multipoint moel A general approach is possible, groune on a physical basis: we consier a reactor as subivie into a set of non overlapping geometrical parts, whose union covers the full omain, an we try to buil a proper balance moel. Let us start for simplicity with a 2-zone system, an ask how the number of neutrons born in each zone - which at time t are everywhere in the system - can vary over time. The balance equations in the absence of an external source, that can be written almost immeiately, rea: N 1 = ν a 11 N 1 + a 12 N 2 C 1 N 1 4 N 2 = ν a 21 N 1 + a 22 N 2 C 2 N 2. A neutron is generate - in this scheme - only via a fission process. Hence, for instance, a fission can happen in zone 1 generating ν new neutrons, but loosing the original one: this event can be inuce by a neutron born in zone 1 with a probability per unit time a 11 or by a neutron originate in zone 2 with a probability per unit time a 12 ; the same processes can happen in zone 2, but there is also the possibility that a neutron in zone 12 is absorbe or leak out of the system with probabilities per unit time C 12. In orer to be more explicit, for instance, the quantity a 12 N 2 is the contribution of neutrons generate in zone 2 that prouces a fission zone 1 at time t an thence are in zone 1 at time t itself. To write own these equations we consiere solely two physical processes to happen, in both zones: the prouction of ν neutrons by a fission process an the absorption of neutrons in each of the reactor zones; this approach implies that the material properties of the system are assume to be stationary. If, as in many realistic cases,

3 Jeju, Korea, April 16-20, 2017, on USB 2017 the material properties are change, either because of control operations, or acciental events or feeback phenomena, the parameters must be upate uring the transient. If this is the case, an approximation is implicit, to neglect the effect of the time elay neee to transport neutrons from the zone where it was generate by a fission process to the zone where finally it will be absorbe, giving origin or not to a new fission process. However the time scale associate to the neutron transport is much faster than any other physical phenomena involve in the changes of the material properties of the system. Naturally the quantities N i are by themselves not irectly measurable. On the other han the quantity a 11 N 1 + a 12 N 2 being the total fission rate in zone is, in principle, an experimental observable. Within this scheme a consistency requirement is implicitly assume: namely, the subivision of the system in spatial zones must be performe in such a way that each one of them contains fissile materials. This is consistent with the assume philosophy unerlying lumpe moels [2]. It must be observe that the choice of the subivision of the spatial omain is not unique: the moel obtaine must be anyway assesse on physical grouns an valiate to assure the quality of the results. For instance, a quite natural suggestion coul be to "equalize" the values of the total content of the neutron importance - that is the integral over a zone of the prouct of the importance an the neutron flux - in each of the zones. The coefficients just introuce efine coupling matrices for the system. This approach recalls, in many aspects, the collision probability metho use in integral neutron transport [10]. Equations 4 can be written in matrix form - so to be easily generalizable to a M zone reactor - as: Nt = νânt ĈNt, 5 where Ĉ is a iagonal matrix with elements Ĉ = C i an obviously  = a i j. An eigenvalue equation can be generate ii i j by consiering an artificial time inepenent system, where only ν/k neutrons are generate by fission. The multiplication eigenvalue k is then etermine by: [ ν ] et k  Ĉ = 0. 6 Again we have M values for k, but in principle they can be real or even complex: however we know by the previous example that in the large M limit they must become real, approximating the true spectrum of k values, at least when aopting the iffusion moel. The inclusion of elaye neutrons oes not present particular ifficulties: N N = ν1 βâ N Ĉ N + λ α C α α=1 7 C α = β α νâ N λ α C α, with the same meaning as before for the matrices Â, Ĉ; here β = N α β α ; we inicate with C k the concentrations of precursors for the k th family of elaye neutrons in all the zones of the reactor, with ecay constant λ α, respectively. This heuristic approach to space multipoint kinetics generates a richer physics than the naïve one obtaine by space iscretization, because, in general, all zones are couple, not only near neighbours. In a realistic situation, the evaluation of the coupling matrices can be carrie out effectively by the Monte Carlo metho. This approach is particularly viable in the case the properties of the system are stationary, such as for subcritical experiments, where the transient is cause by neutron source variations. In the case of transients inuce by material changes, the matrices shoul be evaluate along the evolution to account for such changes, as previously pointe out. III. SOME RESULTS In this Section we present the values of the multipoint matrices obtaine by Monte Carlo evaluation of the require rates. The evaluations have been carrie out using the coe MCNP6 [11]; as a first step we estimate the probabilities that neutrons born uniformly in zone j ultimately fission in zone i or that neutrons from zone j are absorbe or leak anywhere in the system in steay state conitions. The source neutrons are assume spatially uniformly istribute, which of course is an approximation that looks reasonable in a fast reactor system as the one uner consieration. As a result we obtain the probability matrix {p i j } an the probability absorption vector {c j }. Fig. 2. Geometry use for the test case. Fuel regions are shown in re an yellow, lea ones in blue, see text for etails. This plot was obtaine by the moel in MCNP6 an the numbers ientify ifferent MCNP6 cells. For the present approach we ientify zone 1 with the union of cells 101 an 201, zone 2 with the union of cells 102 an 202, an so on. The problem is bi-imensional, having use reflecting surfaces as bouns along the z irection. As a secon step the prompt neutron generation times associate to neutrons generate in zone j, Λ j, are estimate as the average time value from the birth of a neutron to its eath. This was one by tallying the times at which each neutron born in zone j is absorbe anywhere in the system

4 Jeju, Korea, April 16-20, 2017, on USB 2017 an evaluating the mean. As a consequence we set a i j = p i j Λ j, C j = c j Λ j, 8 which are the multipoint parameters neee in moel 7. A comment concerning the Monte Carlo simulations is here in orer. Unlike what has to be one for normal simulations, here every neutron history is terminate whenever the neutron either is absorbe or leaks out of the system, without the nee to simulate the seconary neutrons emitte by fission. In fact, recalling the previous efinitions of the multipoint parameters, only the information on the birth an on the eath of the neutron is neee. As a consequence, the present metho can be applie to any type of system, incluing even supercritical reactors. Furthermore, the computational buren is reuce with respect to stanar simulations an it oes not epen on the value of the multiplication factor. We use a physically challenging non symmetric heterogeneous geometry, shown in Figure 2, to better investigate the implications of zone subivisions. The system is a soli lea parallelepipe of cm 3 volume with cylinrical fuel insertions of 10 cm raius. The problem is mae effectively bi-imensional by using reflective bouning surfaces in the z irection. We use lea in the blue areas, while fuel is a mixture of oxygen an uranium. The 235 U atomic fraction was 25% in cells 103 an 106, while in the other four was chosen so to fix k e f f, ranging from 22.6% for k e f f = 0.94 to 26.6% for k e f f = 1. k e f f ν k 2 k 3 k 4 k 5 k TABLE IV. Higher orer k-eigenvalues. ν values have been obtaine by fixing the highest multiplication eigenvalue to the k e f f value obtaine from simulation. k e f f = 0.99 k e f f = 0.94 zone z Λ z [µs] Λ z,leak [µs] Λ z [µs] Λ z,leak [µs] TABLE V. Values of the characteristic mean absorption times Λ z an mean leakage times Λ z,leak for neutrons generate in all zones an for two values of k e f f. The probability matrices an absorption vectors for two values of k e f f are presente in Table I, II, where also fission an leakage probabilities are reporte. It can be clearly seen that the probability matrix are largely ominate by iagonal elements. Furthermore, one can observe the effect of the subcriticality is apparently not very strong, although such slight ifferences prouce a large reactivity change: as for other reactor physics problems the parameters of the moel have to be accurately evaluate. For the present results the Monte Carlo accuracies obtaine are below 0.1%. Table III collects the values of the mean neutron absorption times corresponing to all the zones of the system for ifferent values of k e f f. When consiering systems characterize by lowabsorption non fuel materials, such as generally is for fast systems an for the case here consiere, the choice of geometrical splitting of the spatial omain into the multipoint structure is marginal because the contribution to the rates from non fuel portions of the zones is consequently low. For instance, moving the bounary between region 2, 5 an 4, 6 by 50 cm to the right, it has been verifie that the probability matrix an absorption vector are slightly affecte. We now present some results for the spectrum at various k e f f : however we o not have any way to know "a priori" the precise value for ν to be use for the problem. Then a first possibility is to fix the highest eigenvalue from 6 to the corresponing k e f f value an use it for the evaluation of ν to be use to extract higher orer k eigenvalues. Results are presente in Table IV. The analysis of higher orer eigenvalues is of interest to preict the potential presence of spatial effects in time-epenent configurations [12]. As it can be immeiately realize from the secon colums of Table IV there is a minimal increment of ν as k e f f ecreases: in fact, to ecrease k e f f we ecrease the 235 U fraction, as it was explaine before; corresponingly the 238 U fraction increases, yieling a very small increment on the effective value of ν ue to fast fission in the incient neutron energy range above 1 MeV, where the fission cross section becomes relevant an ν is sharply increasing. It must be observe that the use of an effective value of ν, etermine in orer to preserve the reference value of k e f f from the Monte Carlo simulation, takes into account an compensates in an overall fashion for all the effects associate to the ifferent effective fission, absorption an leakage proper times for each zone. Formula 8 assumes equal characteristic times for all phenomena, while, of course, physically they may be quite ifferent. In particular, the leakage times may be significantly ifferent, epening on the localization of the zone uner analysis, as can be clearly seen in Table V, where the values aopte in formula 8 are compare to the mean leakage times. As a last observation, the average fission times are expecte to be shorter than absorption times, because most of the fission events happen in the same fuel region as the one of the neutron birth. An alternative more consistent proceure to avoi the intermeiate estimate of the number of seconary neutrons emitte by fission, woul consist in irectly estimating the rate of fission-emitte neutrons, thus obtaining the values of the probabilities p i j, so allowing to irectly evaluate the elements of νâ as: νa i j = p i j Λ j, 9

5 Jeju, Korea, April 16-20, 2017, on USB 2017 Zone z P leak P capt P f iss p z1 p z2 p z3 p z4 p z5 p z TABLE I. Probabilities for leakage P leak, capture P capt, fission P f iss an zone-to-zone transmission for k e f f = Zone z P leak P capt P f iss p z1 p z2 p z3 p z4 p z5 p z TABLE II. Probabilities for leakage P leak, capture P capt, fission P f iss an zone-to-zone transmission for k e f f = where Λ j is the characteristic mean fission time. IV. ENERGY MULTIPOINT THEORY The multigroup equations can alreay be consiere as a multipoint moel on the energy variable: there are however some moifications to be introuce, coming from the fission or elaye spectrum: of the ν fission neutrons prouce only a fraction χ j is really prouce in the j th energy group: N i C α = ν1 βχ i a i j N j j N C i N i + χ α i λ α C k α=1 = β α ν a i j N j λ α C α, i j 10 where i = 1,..., N G, α = 1,..., N an χ i χ α i represents the probability that a prompt elaye from α th family fission neutron has its energy in the i th energy group. It is worthwhile to remark that in the space multipoint approach both prompt neutron numbers an elaye neutron precursor numbers N, C are efine in the zone space, while in energy multipoint the latter are clearly scalars. This fact, as we shall see, requires some attention when builing the general multipoint kinetic equations. In ref. [13] we stuie in etail the multiplication eigenvalue spectrum within a multigroup iffusion theory. In particular we prove a general expression for the eigenvalue k n which is associate with the n th spatial moe of the system which turns out to be necessarily real: the proof erive essentially from the peculiar form of the fission matrix - in that case of rank 1. This feature is in principle lost in the general approach outline in the static version of Eq. 10, because no general assumptions can be mae on the structure of the matrix Â, so allowing again a richer physics: this in strict analogy to what we notice in the iscussion on Eq. 6 an 7. In the previous Section we briefly iscusse the epenence of the spectrum of k as a function of the number of space intervals see Fig. 1. Analogously, it is possible to stuy the epenence of k on the number of energy groups in a naïve multigroup approach for an infinite system: however it can be shown that if one collapses macroscopic cross sections preserving reaction rates, the value of k remains unchange. V. REACTOR INTEGRAL PARAMETERS FROM THE MULTIPOINT MODEL An important issue is the etermination of integral parameters from local flux measurements. When using the simple point kinetic moel, spatial effects may cause large errors in the preiction of integral parameters. Correction factors or flux-signal combination techniques shoul be aopte to obtain reliable values. The use of a multipoint moel may prove to be more effective. It can be assume that the system is represente by a multipoint set of equations, where each point is associate to a local flux etector. The response of a source-riven system is escribe by a set of equations that takes the general form as in Eq. 7, that inclues also the contribution of elaye neutrons, with an aitional source term a vector in the space inexe by zones. The application of an inverse approach to an experimental measurement allows to estimate the elements of the characteristic matrices of the moel. The solution of an ajoint problem is then use to project the above equations in orer to retrieve a point-like set of equations, in the frame of the well-known Henry s philosophy.

6 Jeju, Korea, April 16-20, 2017, on USB 2017 Zone z k e f f = 0.99 k e f f = 0.98 k e f f = 0.97 k e f f = 0.96 k e f f = 0.95 k e f f = TABLE III. Mean neutron absoprtion times Λ j [µs] for neutrons generate in the 6 zones of the system, for ifferent values of k e f f. In the case of a space multipoint theory we introuce the irect eigenvector of the matrix ˆM = ν 11 βâ Ĉ as ˆMN 0 = ω N 0, together with its ajoint ˆM ψ 0 = ω ψ 0, an we write N in the usual factorize form Nt = PtN 0 ; after projecting on ψ 0, we obtain the set of equations ψ0, N 0 P = ω N ψ, N 0 0 Pt + λ α ψ0, C α + ψ 0, S α=1 ψ0, C α = ψ 0, ˆ α N 0 Pt λα ψ0, C α, 11 where ˆ α = β α νâ an also a source term is introuce, as it is essential for applications to source-riven systems. The case of energy multipoint theory takes a slightly ifferent form, because, as one can realize from equation 10 an previously note, the concentrations of elaye neutrons in each family are now scalars, the vector character eriving by their probabilities χ α to be prouce in the given energy group. Then, to be able to project elaye equations on ψ 0, we are require to multiply each of them by χ α, which is equivalent to introucing elaye emissivities: ψ0, N 0 P = ω ψ 0, N 0 Pt+ N + λ α ψ0, C α χ α + ψ, S 0 α=1 ψ0, C α χ α = ψ 0, χ α α N 0 Pt λ k ψ0, C k χ k ; 12 where the matrix ˆM is ˆM i j = ν 1β 1χ ia i j C i δ i j an α j = β αν i a i j. Equations 11 an 12 manifest a point-like structure. Henceforth, the values of the kinetic parameters are inclue in their coefficients, an can thus be easily retrieve. In an inverse kinetic approach, each flux etector can be simulate with the multipoint equations an the integral parameters may be reconstructe by the projection formulae that are efining the coefficients of Eqs. 11 an 12. VI. CONCLUSIONS The paper revisits the basic theory on which the multipoint kinetic metho is base. The features of the metho are investigate by the analysis of simple multipoint equations base on stanar space-iscretize an energy multigroup equations. This exercise allows to construct a consistent criticality theory an to gain some physical insight through the observation of the eigenvalue spectrum of the operator. For realistic applications it is necessary to evelop multipoint moels in the full space-energy omain: this is particularly important in source-riven problems where neutrons are injecte at very high energies in a small volume of the system. This can be naturally one within the propose approach. A general physically-base formulation of the multipoint equations is then presente, together with a physical interpretation of the coefficients in terms of probabilities that allows their evaluation, for instance, by irect Monte Carlo simulations. A collapsing proceure, involving a separation-projection proceure, leas to a point-like set of equations. In such a way, by using an inverse approach base on the multipoint moel to interpret flux measurements from a multiple etector experiment, the stanar integral parameters can be reconstructe. This proceure can be use fruitfully, for instance, in the interpretation of flux measurements for the reactivity preiction an monitoring of accelerator-riven systems, in cases where relevant spatial an spectral effects come into play. VII. ACKNOWLEDGMENTS One of the Authors P. S. is grateful to I.N.F.N. for supporting a long stay at Politecnico i Torino, where this work is being carrie out. This work is also inclue in the IAEA Coorinate Research Project on Accelerator-Driven Systems. REFERENCES 1. I. BELL an S. GLASSTONE, Nuclear Reactor Theory, Van Nostran Reinhol, New York R. AVERY, Theory of couple reactors, in Physics of Nuclear Reactors. Proceeings of the Secon International Conference on the Peaceful uses of Atomic Energy, UN Press, Geneva 1959, pp M. KOMATA, On the erivation of Avery s couple reactor kinetics equations, Nuclear Science an Engineering,

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