implementation of the quasi-static method for neutron transport

International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011) Rio de Janeiro, RJ, Brazil, May 8-12, 2011, on CD-ROM, Latin American Section (LAS) / American Nuclear Society (ANS) ISBN 978-85-63688-00-2

IMPLEMENTATION OF THE QUASI-STATIC METHOD FOR NEUTRON TRANSPORT Fabio Alcaro, Sandra Dulla and Piero Ravetto Dipartimento di Energetica Politecnico di Torino 10129 Torino, Italy C.so Duca degli Abruzzi, 24 [email protected]; [email protected]; [email protected] Romain Le Tellier and Christophe Suteau CEA, DEN, DER/SPRC/LEPh Cadarache, F-13108 Saint Paul-lez-Durance, France [email protected]; [email protected]

ABSTRACT The study of the dynamic behavior of next generation nuclear reactors is a fundamental aspect for safety and reliability assessments. Despite the growing performances of modern computers, the full solution of the neutron Boltzmann equation in the time domain is still an impracticable task, thus several approximate dynamic models have been proposed for the simulation of nuclear reactor transients; the quasi-static method represents the standard tool currently adopted for the space-time solution of neutron transport problems. All the practical applications of this method that have been proposed contain a major limit, consisting in the use of isotropic quantities, such as scalar fluxes and isotropic external neutron sources, being the only data structures available in most deterministic transport codes. The loss of the angular information produces both inaccuracies in the solution of the kinetic model and the inconsistency of the quasi-static method itself. The present paper is devoted to the implementation of a consistent quasi-static method. The computational platform developed by CEA in Cadarache has been used for the creation of a kinetic package to be coupled with the existing SNATCH solver, a discrete-ordinate multi-dimensional neutron transport solver, employed for the solution of the steady-state Boltzmann equation. The work aims at highlighting the effects of the angular treatment of the neutron flux on the transient analysis, comparing the results with those produced by the previous implementations of the quasi-static method. Key Words: Neutron transport, Reactor kinetics, Computer simulation, Quasi-static method

1.

INTRODUCTION

The innovative features introduced by next generation nuclear reactors require as much innovation in the analysis of their performances and safety aspects. The safety assessment of such reactors, in particular, requires the accurate simulation of the behavior of the reactor core during typical operational and accidental conditions, with the need for the solution of the neutron

Fabio Alcaro et al.

transport equation. Being the full time-inversion of the neutron Boltzmann equation an impracticable task, the study of the dynamics of a reactor has to rely on the definition of approximate mathematical models. Hence, several approaches have been proposed during the years; as a result of such efforts the quasi-static method has become a reference procedure to address reactor dynamics and several implementations of the method can be found in the literature [1–8], both in diffusion and in transport. The main drawback for the application of the quasi-static method in neutron transport is the necessity to handle the neutron angular flux, in order to guarantee the consistency of the mathematical model. The deterministic transport codes usually do not provide access to the angular flux, mainly for computer memory reasons; thus the scalar flux is often adopted in the kinetic modeling of the quasi-static method, in conjunction with an isotropy hypothesis, introducing unavoidable inaccuracies of the method itself. The present work is devoted to the design of a computational module that implements two approaches to the quasi-static method, the Improved Quasi-static Method (IQM) and the Predictor-Corrector Quasi-static Method (PCQM), in a fully consistent form, coupled to a wellestablished neutron transport solver [4]. The code has been developed within the framework of a broader computational platform for nuclear reactor applications designed by CEA-Cadarache [9]. The rationale of the use of this code lies in the possibility of handling the angular flux, provided by the transport solver, for the correct implementation of the characteristic equations of the quasi-static method. The comparison of the results between the consistent and the standard quasi-static methods is also carried out, in order to highlight the difference of accuracy in the evaluation of the kinetic parameters and in the reactor power evolution. 2.

DESCRIPTION OF THE ALGORITHM

The starting point for the derivation of the model is given by the dynamic equations:  I   1 ∂ϕ(r , E, Ω, t) 1 X   = L (t)ϕ(r , E, Ω, t) + χi (E)λi Ci (r , t) + S(r , E, Ω, t),  v ∂t 4π i=1   χi (E)  χi (E) ∂Ci (r , t)   =− λi Ci (r , t) + Fid (t)ϕ(r , E, Ω, t), i = 1, 2, . . . , I. 4π ∂t 4π

(1)

The operators appearing in Eqs. (1) are the standard operators of the neutron transport equation and S represents the external neutron source, which is present if a subcritical reactor is driven by an external neutron supply. The rationale of the quasi-static approach relies in the separation of the time scales of the kinetic problem that is obtained through the factorization of the neutron flux in the form given by: ϕ(r , E, Ω, t) = A(t)ψ(r , E, Ω; t),

(2)

where the amplitude function A depends only on time on the fast time scale, being proportional to the reactor power changes, while the shape function ψ depends on all the phase space variables and on time on a slower time scale, being dominated by the spatial and spectral modifications of the neutron population in the reactor. The introduction of the flux factorization into Eqs. (1) allows to transform the mathematical structure of the kinetic equations from a single set of PDEs, depending on the phase space 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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variables and on time, to a coupled system of ODEs on time and a so-called ‘shape equation’ depending only on the phase space variables. The system of ODEs is obtained from Eqs. (1) through a weighting-projection technique: in order to simplify the model, the weighting function is chosen to be the adjoint solution of a reference problem, usually assumed to be the initial state of the system. The system of ODEs resulting from these calculations is the so-called Point Kinetics (PK) problem that describes the evolution of the amplitude function on the fast time scale:  I X ˜  ρ(t) − β(t) dA   ˜  A(t) + λi c˜i (t) + S(t)  dt = Λ(t) i=1  ˜  β(t) d˜ ci   = −λi c˜i (t) + A(t), i = 1, 2, . . . , I. (3)  dt Λ(t) The parameters appearing in the PK equations depend on the shape function, hence the PK problem is solved assuming the shape function is known [4]. The shape function must be calculated from Eqs. (1), once the factorization of the neutron flux has been introduced: " # 1 1 dA(t) L (t) − − ψ(r , E, Ω, t) + Q(r , E, Ω, t) = 0, (4) v∆t vA(t) dt It is evident that the shape equation (4) is solved provided the amplitude function is known. An iterative procedure is therefore established in order to obtain a shape function which is consistent with the PK model results. In an alternative approach, adopted by PCQM [4], the time-dependent neutron balance equation is written in the following form: " # 1 ∆t X d L (t) − + λi Fi (t) ϕ(r , E, Ω; t) + Q(r , E, Ω, t) = 0, (5) v∆t 2 i

and then the shape is obtained by re-normalization of the flux and then used to compute the kinetic parameters, thus leading to a linear formulation of the quasi-static algorithm. The formulation of the balance problems reported in Eqs. (4) and (5) enlightens the pseudostationary characteristics of both models, allowing the use of a steady-state flux solver for the solution of these problems. In the next sections it is explained in more details how this objective is accomplished through the computational platform. 3.

3.1

THE COMPUTATIONAL PLATFORM AND THE IMPLEMENTATION OF THE QUASI-STATIC SCHEME The Computational Platform

Within the framework of the ERANOS code system, a new solver, named SNATCH, has been developed with a specific focus on the modeling of core geometries based on hexagonal assemblies for fast reactor calculations [9]. The multigroup neutron transport equation is discretized angularly using a discrete ordinate approach and the resulting set of coupled first-order hyperbolic 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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transport equations are discretized using a discontinuous Galerkin method with adaptive mesh refinement capabilities [10]. This solver comes with consistent perturbation tools [11] and the capability to retrieve the angular flux and deal with angular dependent sources. These features are needed for the development of a consistent quasi-static method. The analysis of transients through the quasi-static approach requires the coupling of the PK problem with the solution of a ‘pseudo-stationary’ balance equation, which is a general steadystate flux equation unavoidably with an external neutron source. The main task of the implementation of the quasi-static algorithm is the accomplishment of the consistent formulation of such model. The platform has been used as a computational environment in which both the IQM and the PCQM have been implemented in the framework of the neutron transport theory. Equations (4) and (5) represent the transport problems to be solved by the IQM and PCQM, respectively. The transport operators and the source are here modified by the presence of terms which appear as a consequence of the time discretization of the shape/flux equation, the presence of delayed neutrons and the chosen quasi-static algorithm. Such modifications amount to manipulations of the nuclear data necessary to provide the correct formulation of the pseudostationary equation to the transport solver.

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Implementation of the pseudo-stationary equations for IQM and PCQM

Both IQM and PCQM algorithms require the definition of virtual total cross-sections and external sources, including the terms arising from the discretization of the time derivative and the presence of the amplitude function for IQM. In the IQM case, such modifications are: ˜ t,g = Σt,g + 1 + 1 dA(t) , Σ v∆t vA(t) dt " 1 X S(r , E, Ω, t) ψ(r , E, Ω; t − ∆t) χi + + Ci,0 (r )e−λi ∆t Q(r , E, Ω, t) = λi A(t) v∆t A(t) 4π i Z t
# −λ t−τ + Fid (t − ∆t)ψ(r , E, Ω; t − ∆t) dτ A(τ )e i .

(6)

(7)

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Inside the platform, this task is performed creating temporary data structures available only in the calculation step, keeping the original nuclear data unmodified. The definition of the virtual data for the PCQM is slightly more difficult due to the algorithm itself: the preliminary step involving the prediction of the flux and precursor concentrations requires the modification of the steady-state fission operator to include the contribution of delayed precursors: ∆t X λi Fid (t), (8) F˜p (t) = Fp (t) + 2 i

while the virtual total cross section and external source appear as: ˜ t,g = Σt,g + 1 , Σ v∆t 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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ϕ(r , E, Ω, t − ∆t) Q(r , E, Ω, t) = S(r , E, Ω, t) + v∆t " # X χi ∆t + λi Ci,0 (r )e−λi ∆t + e−λi ∆t Fid (t − ∆t)ϕ(r , E, Ω, t − ∆t) . 4π 2

(10)

i

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Consistent Formulation of the Quasi-Static Method

The approximations introduced in the quasi-static method by the standard kinetic codes are mainly related with the adoption of the scalar flux, that affects the calculation process at two different stages: the definition of the pseudo-stationary problem that must be solved in the quasi-static algorithm at first, and the definition of the kinetic parameters appearing in the PK equations. As previously pointed out, Eqs. (4) and (5) contain a virtual external neutron source, formulae (7) and (10), respectively. The terms arise from the discretization of the time derivative of the shape/flux and contains the shape/flux computed at the previous time interval. These quantities are by their nature angle-dependent and, if they are not available, the corresponding isotropic distribution associated to the scalar quantities are adopted as: I 1 ψ(r , E, Ω; t − ∆t) ≈ Ψ(r , E; t − ∆t), where Ψ(r , E; t) = dΩ ψ(r , E, Ω; t), (11) 4π not allowing to exactly formulate the pseudo-stationary transport problem. The same issue arises in the definition of the kinetic parameters, thus affecting the solution of the PK problem. The kinetic parameters are generated by the projection of the terms appearing in the equation on the adjoint angular flux for the initial state of the system, computed with the same transport code, consistently with the direct calculations. It is therefore evident that they are affected by an intrinsic error in the computation of the corresponding integrals, if scalar quantities are adopted. The terms which are particularly affected are: • the weighted integral of the neutron density appearing in the definition of the neutron mean generation time and in the shape convergence process for IQM: Z Z I Z Z 1 1 1 † dr dE dΩ ψ0 (r , E, Ω) ψ(r , E, Ω; t) ≈ dr dE Ψ†0 (r , E) Ψ(r , E; t); (12) v 4π v • the reactivity, when the total cross section is perturbed: Z Z I dr dE dΩ ψ0† (r , E, Ω)δΣt (r , E, t)ψ(r , E, Ω; t) Z Z 1 ≈ dr dE Ψ†0 (r , E)δΣt (r , E, t)Ψ(r , E; t), (13) 4π and when the scattering is modified, since the higher order angular terms of the scattering cross section have to be neglected: Z Z I Z I L X 2l + 1 † 0 0 dr dE dΩ ψ0 (r , E, Ω) dE dΩ δΣs,l (r , E 0 → E, t)Pl (Ω0 · Ω) 2 l=0 Z Z Z 1 † 0 0 dr dE Ψ0 (r , E) dE 0 δΣs,0 (r , E 0 → E, t)Ψ(r , E 0 ; t). (14) × ψ(r , E , Ω ; t) ≈ 4π 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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The SNATCH SN transport solver implemented in the platform can handle the angular neutron flux and the angular-dependent external neutron source distributions are defined, thus allowing a consistent formulation of the method. 4.

RESULTS

In this section some test calculations are presented in order to compare the power profiles computed with the classical and the consistent formulation of the quasistatic algorithm. In Fig. 1, a picture of the systems analyzed is presented and the nuclear data concerning the precursor families of delayed neutrons are given. A monoenergetic model has been used for the energy dependence for the sake of simplicity. In the present work, the transport solver tolerances both on the error on the eigenvalue and on the local error on the flux has been set to 10−6 . The choice of ∆t and δt time steps is provided by the user. In the following calculations δt has been chosen to be half the generation time, while parametric studies on the value of ∆t are performed.

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Figure 1: Representation of one quarter of the reactor domain. The shaded area identifies the location of the cross-section perturbation. The neutron transport calculations are performed by imposing reflection boundary conditions on the dashed edges, while vacuum boundary conditions are imposed on the solid edges. The first test calculation performed consists in a reactivity injection due to a step-wise reduction of the total cross-section. The reactivity introduced in the system computed through a first order approximation is 0.35$. A preliminary calculation adopting 100 macro time-steps is carried out in order to assess a reference situation. In Fig. 2 the results are shown for the IQM; a comparison of the power profiles between the consistent formulation (Fig. 2.a) and the classical one (Fig. 2.b) evidences some differences: in particular it can be pointed out that the classical formulation of the quasi-statics generally underestimates the evolution of the power, despite the overall results can be considered acceptable. The same transient test is studied through the PCQM: a calculation adopting 100 macro time2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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steps has been also performed in order to provide a reference case. Figures 3.a and 3.b represent the results for the consistent and classical formulations of the PCQM, respectively. The better accuracy on the power prediction obtained is related to the algorithm itself, while it can be highlighted that also in this case the classical formulation of the quasistatic method underestimates the power.

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Figure 3: Transient induced by a reduction of the total cross-section (ρ = 0.35$) simulated with PCQM. The time interval is subdivided into uniform macro time-steps.

The graphs presented in Fig. 4 show the error evolution on the shape function for the IQM in the reference case. Such error is defined as: εγ =

|γ − γ0 | , γ0

(15)

where the parameter γ is defined as the projection of the shape function on the initial adjoint neutron density function. As described in Eq. (12), the classical formulation of the quasistatic 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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method gives only an approximation of such error. Fig. 4 shows that, despite the expected difference of the values, the behavior is the same: the error rapidly saturates after a few recalculations, and no further improvement on the shape function can be obtained.

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(a) Shape error trend resulting from the use of the consistent formulation of the IQM.

(b) Shape error trend resulting from the use of the classical formulation of the IQM.

Figure 4: Transient induced by a reduction of the total cross-section (ρ = 0.35$) simulated with IQM. The plots refer to the reference calculation.

The second test calculation consists in a negative reactivity injection consequent to a step-wise increase of the total cross-section. The reactivity introduced in the system is −2.20$. Figures 5 and 6 show the results for the IQM and PCQM, respectively. The same observations on the characteristics of the standard and consistent formulations of quasistatics can be applied for this case. In addition, coherently with previous works [4], it can be noted that PCQM results with few recalculations provide a more accurate power prediction with respect to the corresponding IQM case. The different quality of the time-dependent simulation is related to the behavior of the reactivity, as can be seen in Fig. 7. The reactivity value in PCQM is quickly converging to the correct value, while in IQM the transition happens with a certain delay, depending on the shape recalculation being performed at the end of the macro time-step.

5.

CONCLUSIONS

A kinetic package capable to perform dynamic calculations through quasistatic algorithms is implemented within a computational platform developed at the CEA in Cadarache. The SNATCH solver is used as shape generator for the computation of the kinetic parameters adopted in the amplitude model to predict the reactor power evolution. For both the IQM and PCQM, the test calculations show the accuracy of the results and evidence some differences between the classical and the consistent formulation of the quasistatic method: the latter leads to a more accurate prediction of the power evolution along the prompt part of the transient. This is mainly due to a correct evaluation of the kinetic parameter, in particular of the reactivity. 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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(a) Power profile resulting from the use of the consistent formulation of the IQM.

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Figure 5: Transient induced by an increase of the total cross-section (ρ = −2.20$) simulated with IQM. The time interval is subdivided into uniform macro time-steps.

6.

ACKNOWLEDGEMENTS

One of the Authors (F. A.) is grateful to CEA for the hospitality in the Cadarache research centre.

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Figure 6: Transient induced by an increase of the total cross-section (ρ = −2.20$) simulated with PCQM. The time interval is subdivided into uniform macro time-steps.

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Figure 7: Reactivity profile resulting from the use of the consistent formulation of the quasi-statc scheme. The time variable spans from 0 to 120 mean generation times.

REFERENCES 1. K. O. Ott, D. A. Meneley, Accuracy of the Quasistatic Treatment of Spatial Reactor Kinetics, Nuclear Science and Engineering, 36, pp. 402-411 (1969). 2. J. Devooght, Quasistatic Solutions of Reactor Kinetics, Annals of Nuclear Energy, 7, pp. 47-58 (1980). 3. J. Devooght, E. H. Mund, Generalized Quasistatic Method for Space-Time Kinetics, Nuclear Science and Engineering, 76, pp. 10-17 (1980). 4. S. Dulla, E. H. Mund, P. Ravetto, The quasi-static method revisited, Progress in Nuclear Energy, 50, pp. 908-920 (2008). 5. P. Picca, S. Dulla, E. H. Mund, P. Ravetto, G. Marleau, Quasi-Static Time-Dependent 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

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Computational Tool Using the DRAGON Transport Code, PHYSOR 2008, Interlaken, September 14-19 (2008). 6. F. Alcaro, S. Dulla, G. Marleau, E. H. Mund, P. Ravetto, Development of Dynamic models for Neutron Transport Calculations, Il Nuovo Cimento C, 33, pp. 13-20 (2010). 7. S. Goluoglu, H. L. Dodds, A Time-Dependent, Three-Dimensional Neutron Transport Methodology, Nuclear Science and Engineering, 139, pp. 248-261 (2001). 8. S. Yun, J. W. Kim, N. Z. Cho, Monte Carlo space-time reactor kinetics method and its verification with time-dependent SN method, PHYSOR 2008, Interlaken, September 14-19 (2008). 9. R. Le Tellier, C. Suteau, D. Fournier, J. M. Ruggieri, High-Order Discrete Ordinate Transport in Hexagonal Geometry: a New Capability in ERANOS, Il Nuovo Cimento C, 33, pp. 121-128 (2010). 10. D. Fournier, P. Archier, R. Le Tellier, C. Suteau, Improvement of Neutronic Calculations on a MASURCA Core using Adaptive Mesh Refinement Capabilities, M&C 2011, Rio de Janeiro, May 8-12 (2011). 11. R. Le Tellier, D. Fournier, C. Suteau, Reactivity Perturbation Formulation for a Discontinuous Galerkin based Transport Solver and its Use with Adaptive Mesh Refinement, Nuclear Science and Engineering, accepted.

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