Skip to main content

Lattice Boltzmann method for simulation of time-dependent neutral particle transport

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  1. 1.

    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

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    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

    Article  Google Scholar 

  3. 3.

    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

    Article  Google Scholar 

  4. 4.

    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

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    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

    Article  Google Scholar 

  6. 6.

    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

    Article  Google Scholar 

  7. 7.

    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

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    R.T. Ackroyd, Finite element methods for particle transport, applications to reactor and radiation physics (Research Studies Press, New York, 1997), pp. 526–748

    Google Scholar 

  9. 9.

    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

    Article  Google Scholar 

  10. 10.

    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

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    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

    Article  Google Scholar 

  12. 12.

    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

    Google Scholar 

  13. 13.

    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

    Article  Google Scholar 

  14. 14.

    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

    Article  Google Scholar 

  15. 15.

    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

    Google Scholar 

  16. 16.

    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

    Article  Google Scholar 

  17. 17.

    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

    Article  Google Scholar 

  18. 18.

    J.R. Askew, A characteristics formulation of the neutron transport equation in complicated geometries. UKAEA, Winfrith, AAEW- M 1108, (1972)

  19. 19.

    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

    Article  Google Scholar 

  20. 20.

    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

    Article  Google Scholar 

  21. 21.

    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

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    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

    Article  Google Scholar 

  23. 23.

    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

    Article  Google Scholar 

  24. 24.

    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

    MathSciNet  MATH  Google Scholar 

  25. 25.

    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

    Article  Google Scholar 

  26. 26.

    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

    Article  Google Scholar 

  27. 27.

    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

    Article  Google Scholar 

  28. 28.

    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)

  29. 29.

    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

    Article  Google Scholar 

  30. 30.

    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

    Article  Google Scholar 

  31. 31.

    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

    Article  Google Scholar 

  32. 32.

    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

    Article  Google Scholar 

  33. 33.

    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

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    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

    Article  MATH  Google Scholar 

  35. 35.

    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

    Article  MATH  Google Scholar 

  36. 36.

    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

    MathSciNet  Article  Google Scholar 

  37. 37.

    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

    Article  Google Scholar 

  38. 38.

    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

    Article  Google Scholar 

  39. 39.

    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

    Article  Google Scholar 

  40. 40.

    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

    MathSciNet  Article  Google Scholar 

  41. 41.

    S. Succi, The Lattice Boltzmann Equation, for Fluid Dynamics and Beyond (Oxford University Press, Oxford, 2001), pp. 25–133

    MATH  Google Scholar 

  42. 42.

    J.W. Negele, K. Yazaki, Mean free path in a nucleus. Phys. Rev. Let. 47(2), 71 (1981). doi:10.1103/PhysRevLett.47.71

    Article  Google Scholar 

  43. 43.

    G.A. Bird, Definition of mean free path for real gases. Phys. Fluids 26(11), 3222–3223 (1983). doi:10.1063/1.864095

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yu Ma.

Additional information

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).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Transient neutron transport
  • Lattice Boltzmann method
  • Linear Boltzmann equation