Abstract
In this paper, a novel model is proposed to investigate the neutron transport in scattering and absorbing medium. This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method (LBM) with the collision and streaming process. The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time. The fully implicit backward difference scheme is used to ensure the numerical stability, and relaxation time and equilibrium particle distribution function are obtained. To validate the new lattice Boltzmann model, the LBM formulation is tested for a homogenous media with different sources, and both transient and steady-state LBM results get a good agreement with the benchmark solutions.
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W.A. Wieselquist, D.Y. Anistratov, J.E. Morel, A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes. J. Comput. Phys. 273, 343–357 (2014). doi:10.1016/j.jcp.2014.05.011
P. Picca, R. Furfaro, A hybrid method for the solution of linear Boltzmann equation. Ann. Nucl. Energy 72(5), 214–236 (2014). doi:10.1016/j.anucene.2014.05.014
G. Orengo, C. de Oliveira Graça, A model of the 14MeV neutrons source term, for numerical solution of the transport equation to be used in BNCT simulation. Ann. Nucl. Energy 42(2), 161–164 (2012). doi:10.1016/j.anucene.2011.12.008
J. Hu, L. Wang, An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit. J. Comput. Phys. 2015(281), 806–824 (2015). doi:10.1016/j.jcp.2014.10.050
Z.J. Xiao, S.Z. Qiu, W.B. Zhuo et al., The development and verification of thermal-hydraulic code on passive residual heat removal system of Chinese advanced PWR. Nucl. Sci. Tech. 17(5), 301–307 (2016). doi:10.1016/S1001-8042(06)60057-2
A.M. Mirza, S. Iqbal, F. Rahman, A spatially adaptive grid-refinement approach for the finite element solution of the even-parity Boltzmann transport equation. Ann. Nucl. Energy 34(7), 600–613 (2007). doi:10.1016/j.anucene.2007.02.015
M.A. Goffin, A.G. Buchan, S. Dargaville et al., Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation. J. Comput. Phys. 281, 1032–1062 (2015). doi:10.1016/j.jcp.2014.10.063
R.T. Ackroyd, Finite element methods for particle transport, applications to reactor and radiation physics (Research Studies Press, New York, 1997), pp. 526–748
K.R. Olson, D.L. Henderson, Numerical benchmark solutions for time-dependent neutral particle transport in one-dimensional homogeneous media using integral transport. Ann. Nucl. Energy 31(13), 1495–1537 (2004). doi:10.1016/j.anucene.2004.04.002
J. Tickner, Arbitrary perturbations in Monte Carlo neutral-particle transport. Comput. Phys. Commun. 185(6), 1628–1638 (2014). doi:10.1016/j.cpc.2014.03.003
D. She, J. Liang, K. Wang et al., 2D full-core Monte Carlo pin-by-pin burnup calculations with the RMC code. Ann. Nucl. Energy 64(64), 201–205 (2014). doi:10.1016/j.anucene.2013.10.008
K. Yue, W.Y. Luo, Y.Z. Zha et al., MC simulation of shielding effects of PE, LiH and graphite fibers under 1 MeV electrons and 20 MeV protons. Nucl. Sci. Tech. 19(6), 329–332 (2008). doi:10.1016/S1001-8042(09)60013-0
Q.L. Ma, S.B. Tang, J.W. Zuo, Numerical simulation of high-energy neutron radiation effect of scintillation fiber. Nucl. Sci. Tech. 19(4), 236–240 (2008). doi:10.1016/S1001-8042(08)60056-1
D. Li, C.S. Wang, W.Y. Luo et al., Energy loss caused by shielding effect of steel cage outside source tube. Nucl. Sci. Tech. 18(2), 86–87 (2007). doi:10.1016/S1001-8042(07)60025-6
B. Askri, A. Trabelsi, B. Baccari, Method for converting in-situ gamma ray spectra of a portable Ge detector to an incident photon flux energy distribution based on Monte Carlo simulation. Nucl. Sci. Tech. 19(6), 358–364 (2008). doi:10.1016/S1001-8042(09)60019-1
R.T. Ackroyd, The why and how of finite elements. Ann. Nucl. Energy 8(81), 539–566 (1981). doi:10.1016/0306-4549(81)90125-0
A. Vidal-Ferrandiz, R. Fayez, D. Ginestar et al., Solution of the Lambda modes problem of a nuclear power reactor using an h-p finite element method. Ann. Nucl. Energy 72(5), 338–349 (2014). doi:10.1016/j.anucene.2014.05.026
J.R. Askew, A characteristics formulation of the neutron transport equation in complicated geometries. UKAEA, Winfrith, AAEW- M 1108, (1972)
Z. Liu, H.C. Wu, L.Z. Cao et al., A new three-dimensional method of characteristics for the neutron transport calculation. Ann. Nucl. Energy 38(2), 447–454 (2011). doi:10.1016/j.anucene.2010.09.021
W. Boyd, S. Shaner, L. Li et al., The OpenMOC method of characteristics neutral particle transport code. Ann. Nucl. Energy 68, 43–52 (2014). doi:10.1016/j.anucene.2013.12.012
D. Priimak, Finite difference numerical method for the superlattice Boltzmann transport equation and case comparison of CPU (C) and GPU (CUDA) implementations. J. Comput. Phys. 278, 182–192 (2014). doi:10.1016/j.jcp.2014.08.028
A.G. Buchan, C.C. Pain, M.D. Eaton et al., Linear and quadratic octahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation. Ann. Nucl. Energy 32(11), 1224–1273 (2014). doi:10.1016/j.anucene.2005.01.005
A. Pirouzmand, K. Hadad, Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry. Part I, Modeling and verification for time-independent solution. Ann. Nucl. Energy 38(6), 1288–1299 (2011). doi:10.1016/j.anucene.2011.02.012
A. Hussein, M.M. Selim, Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique. Appl. Math. Comput. 213(1), 250–261 (2009). doi:10.1016/j.amc.2009.03.016
J. Mandrekas, GTNEUT, a code for the calculation of neutral particle transport in plasmas based on the Transmission and Escape Probability method. Comput. phys. commun. 161(1), 36–64 (2004). doi:10.1016/j.cpc.2004.04.009
S. Van Criekingen, F. Nataf, P. Havé, Parafish, a parallel FE–P N neutron transport solver based on domain decomposition. Ann. Nucl. Energy 38(1), 145–150 (2011). doi:10.1016/j.anucene.2010.08.002
M.A. Goffin, A.G. Buchan, A.C. Belme et al., Goal-based angular adaptivity applied to the spherical harmonics discretisation of the neutral particle transport equation. Ann. Nucl. Energy 71, 60–80 (2014). doi:10.1016/j.anucene.2014.03.030
W. Sweldens, Lifting scheme, a new philosophy in biorthogonal wavelet constructions, in SPIE’s 1995 International Symposium on Optical Science, Engineering, and Instrumentation, San Diego (1995) (to be published)
A.G. Buchan, C.C. Pain, M.D. Eaton et al., Chebyshev spectral hexahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation. Ann. Nucl. Energy 35(6), 1098–1108 (2008). doi:10.1016/j.anucene.2007.08.021
Y.Q. Zheng, H.C. Wu, L.Z. Cao, An improved three-dimensional wavelet-based method for solving the first-order Boltzmann transport equation. Ann. Nucl. Energy 36(9), 1440–1449 (2009). doi:10.1016/j.anucene.2009.06.006
P. Picca, R. Furfaro, B.D. Ganapol, A hybrid transport point-kinetic method for simulating source transients in subcritical systems. Ann. Nucl. Energy 38(12), 2680–2688 (2011). doi:10.1016/j.anucene.2011.08.005
P. Picca, R. Furfaro, Hybrid-transport point kinetics for initially-critical multiplying systems. Prog. Nucl. Energ. 76, 232–243 (2014). doi:10.1016/j.pnucene.2014.05.013
X.Y. He, L.S. Luo, M. Dembo, Some progress in lattice Boltzmann method. Part I. Nonuniform mesh grids. J. Comput. Phys. 129(2), 357–363 (1996). doi:10.1006/jcph.1996.0255
S.C. Mishra, H.K. Roy, Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method. J. Comput. Phys. 223(1), 89–107 (2007). doi:10.1016/j.jcp.2006.08.021
S.C. Mishra, A. Lankadasu, K.N. Beronov, Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction–radiation problem. Int. J. Heat Mass Transf. 48(17), 3648–3659 (2005). doi:10.1016/j.ijheatmasstransfer.2004.10.041
S. Chen, G.D. Doolen, Lattice Boltzmann method for fluid flows. Annu. Rev. fluid Mech. 30(1), 329–364 (1998). doi:10.1146/annurev.fluid.30.1.329
G. Mayer, J. Páles, G. Házi, Large eddy simulation of subchannels using the lattice Boltzmann method. Ann. Nucl. Energy 34(1), 140–149 (2007). doi:10.1016/j.anucene.2006.10.002
G. Házi, A.R. Imre, G. Mayer et al., Lattice Boltzmann methods for two-phase flow modeling. Ann. Nucl. Energy 29(12), 1421–1453 (2002). doi:10.1016/S0306-4549(01)00115-3
Y. Zhang, H.L. Yi, H.P. Tan, The lattice Boltzmann method for one-dimensional transient radiative transfer in graded index gray medium. J. Quant. Spectrosc. Radiat. Transf. 137, 1–12 (2014). doi:10.1016/j.jqsrt.2014.01.006
S.S. Chikatamarla, I.V. Karlin, Lattices for the lattice Boltzmann method. Phys. Rev. E 79(4), 046701 (2009). doi:10.1103/PhysRevE.79.046701
S. Succi, The Lattice Boltzmann Equation, for Fluid Dynamics and Beyond (Oxford University Press, Oxford, 2001), pp. 25–133
J.W. Negele, K. Yazaki, Mean free path in a nucleus. Phys. Rev. Let. 47(2), 71 (1981). doi:10.1103/PhysRevLett.47.71
G.A. Bird, Definition of mean free path for real gases. Phys. Fluids 26(11), 3222–3223 (1983). doi:10.1063/1.864095
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This work was supported by the Foundation of National Key Laboratory of Reactor System Design Technology (No. HT-LW-02-2014003) and the State Key Program of National Natural Science of China (No. 51436009).
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Wang, YH., Yan, LM., Xia, BY. et al. Lattice Boltzmann method for simulation of time-dependent neutral particle transport. NUCL SCI TECH 28, 36 (2017). https://doi.org/10.1007/s41365-017-0185-z
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DOI: https://doi.org/10.1007/s41365-017-0185-z