Skip to main content

Kinetics of Nuclear System Solution Methods for the Space-Time Dependent Neutron Diffusion Equation

  • Chapter
Advances in Nuclear Science and Technology

Part of the book series: Advances in Nuclear Science and Technology ((ACRE,volume 10))

Abstract

Within the large variety of methods of analysis for transient phenomena in nuclear reactors, this article is limited to the discussion of computational methods for the analysis of operational transients, off-normal transients, and hypothetical accidents in power reactors. Though any realistic analysis of such transients requires a coupled treatment of thermodynamics, fluid dynamics and neutronic phenomena, all details of the modeling and computations of thermodynamic and fluid dynamic quantities are omitted in the following presentation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Yasinsky, J. B and Henry, A. F., “Some Numerical Experiments Concerning Space-Time Kinetics Behavior,” Nucl. Sci. Eng., 22, pp. 171–181, 1965.

    Google Scholar 

  2. Yasinsky, J. B., “On the Use of Point Kinetics for the Analysis of Rod-Ejection Accidents,” Nucl. Sci. Eng., 39, pp. 241–256, 1970.

    Google Scholar 

  3. Kessler, G., “Space-Dependent Dynamic Behavior of the Fast Reactors Using the Time-Discontinuous Synthesis Method,” Nucl. Sci. Eng., 41, pp. 115–148, 1970.

    Google Scholar 

  4. Jackson, J. F., and Kastenberg, W. E., “Space-Time Effects in Fast Reactor Dynamics,” Nucl. Sci. Eng., 42, pp. 278–294, 1970.

    Google Scholar 

  5. Salah, S., Rossi, G. E., and Geets, J. M., “Consequences of Asymmetric Cold Water Addition to a PWR Core from an Inactive Loop,” Trans. Am. Nucl. Soc., 14, Page 756, 1971.

    Google Scholar 

  6. Salah, S., Rossi, C. E., and Geets, J. M., “Three-Dimensional Kinetic Analysis of an Asymmetric Boron Dilution in a PWR Core,” Trans. Am. Nucl. Soc., 15, Page 831, 1972.

    Google Scholar 

  7. Lamarsh, J. R., Introduction to Nuclear Reactor Theory, Addison-Wesley, Reading, Mass., 1966.

    Google Scholar 

  8. Meghreblian, R. V. and Holmes, D. K., Reactor Analysis, McGraw-Hill, New York, 1960.

    Google Scholar 

  9. Henry, A. F., Nuclear Reactor Analysis, MIT Press, Cambridge, Mass., 1975.

    Google Scholar 

  10. Mikhlin, S. G., and Smolitsky, K. L., Approximate Methods for Solution of Differential and Integral Equations, Elsevier, New York, 1967.

    MATH  Google Scholar 

  11. Aziz, A. K., and Babuska, I., The Mathematical Foundation of the Finite Element Method with Application to Partial Differential Equations, Academic Press, New York, 1972.

    Google Scholar 

  12. Strang, G., and Fix, G. J., An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.

    Google Scholar 

  13. Kang, C. M., and Hansen, K. F., “Finite Element Methods for Reactor Analysis,” Nucl. Sci. Eng., 51, pp. 456–495, 1973.

    Google Scholar 

  14. Babuska, I., “Error-Bounds for Finite Element Methods,” Numer. Math, 16, pp. 322–333, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  15. Schultz, M. H., “L2 Error Bounds for the Rayleigh-Ritz-Galerkin Method,” SIAM J. Numer. Anal., 8, pp. 737–748, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  16. Nitsche, J. A., “Convergence of Nonconforming Methods,” in Mathematical Aspects of Finite Elements in Partial Differential Equations,” C. deBoor, Ed., Academic Press, New York, 1974.

    Google Scholar 

  17. Birkhofer, A., and Werner, W., “Efficiency of Various Methods for the Analysis of Space-Time Kinetics,” Proc. Conf. Mathematical Models and Computational Techniques for Analysis of Nuclear Systems, CONF-730414, Vol. 2, pp. IX-31–41, 1973.

    Google Scholar 

  18. Birkhofer, A., Langenbuch, S., and Werner, W., “Coarse-Mesh Method for Space-Time Kinetics,” Trans. Am. Nucl. Soc., 18, Page 153, 1974.

    Google Scholar 

  19. Langenbuch, S., Maurer, W., and Werner, W., “Simulation of Transients with Space-Dependent Feedback by Coarse Mesh Flux Expansion Method,” MRR 145, Proc. of Joint NEACRP/CSNI Specialists’ Meeting on New Development in Three-Dimensional Neutron Kinetics, pp. 173–188, 1975.

    Google Scholar 

  20. Schäfer, A., Über das Konvergenzverhalten eines Galerkin-Petrov-Verfahrens Thesis, TU Munchen, 1976.

    Google Scholar 

  21. Finneraan, H., “A Consistent Nodal Method for the Analysis of Space-Time Effects in Large LWR’s,” MRR 145, Proc. of Joint NEACRP/CSNI Specialists’ Meeting on New Developments in Three-Dimensional Neutron Kinetics, pp. 145–172, 1975.

    Google Scholar 

  22. Selengut, D. S., “Variational Analysis of a Multidimensional System,” Page 89, HW-59126, Hanford Laboratory, 1959

    Google Scholar 

  23. Dougherty, D. E., and Shen, C. N., “The Space-Time Neutron Kinetics Equations Obtained by the Semidirect Variational Method,” Nucl. Sci. Eng., 13, pp. 141–152, 1962.

    Google Scholar 

  24. Kaplan, S., Marlowe, O. J., and Bewick, J., “Application of Synthesis Techniques to Problems Involving Time-Dependence,” Nucl. Sci. Eng., 18, pp. 163–176, 1964.

    Google Scholar 

  25. Yasinsky, J. B., “The Solution of the Space-Time Neutron Group Diffusion Equations by a Time Discontinuous Synthesis Method,” Nucl. Sci. Eng., 29, pp. 381–391, 1967.

    Google Scholar 

  26. Stacey, W. M. Jr., “Variational Functionals for Space-Time Neutronics,” Nucl. Sci. Eng., 30, pp. 448–463, 1967.

    Google Scholar 

  27. Yasinsky, J. B., and Kaplan S., “Synthesis of Three-Dimensional Flux Shapes Using Discontinuous Sets of Trial Functions,” Nucl. Sci. Eng., 28, pp. 426–440, 1967.

    Google Scholar 

  28. Stacey, W. M. Jr., “A Variational Multichannel Space-Time Synthesis Method for Nonseparable Reactor Transients,” Nucl. Sci. Eng., 34, pp. 45–56. 1968.

    Google Scholar 

  29. Stacey, W. M. Jr., Space-Time Nuclear Reactor Kinetics, Academic Press, New York, 1969.

    Google Scholar 

  30. Yasinsky, J. B., and Henry, A. F., “Some Numerical Experiments Concerning Space-Time Reactor Kinetics Behavior,” Nucl, Sci. Eng., 22, pp. 171–181, 1965.

    Google Scholar 

  31. Boresen, S., “A Simplified, Coarse-Mesh, Three-Dimensional Diffusion Scheme for Calculating the Gross Power Distribution in a Boiling Water Reactor,” Nucl. Sci. Eng., 44, pp. 37–43, 1971.

    Google Scholar 

  32. Boresen, S., “Characteristics and Performance of the 3D LWR Simulator PRESTO,” Trans. Am. Nucl. Soc., 15, Page 956, 1972.

    Google Scholar 

  33. Delp, D. L., Fischer, D. L., Harriman, J. M., and Stedwell, M. J., “FLARE — A Three-Dimensional Boiling Water Reactor Simulator,” GEAP-4598, General Electric, 1964.

    Google Scholar 

  34. Goldstein, L., Nakache, F., and Veras, A., “Calculation of Fuel-Cycle Burnup and Power Distribution of Dresden-I Reactor with the TRILUX Fuel Management Program,” Trans. Am. Nucl. Soc., 10, Page 300, 1967.

    Google Scholar 

  35. Deppe, L. O., Hansen, K. F., “Applications of the Finite Element Method to Two-Dimensional Diffusion Problems,” Nucl. Sci. Eng., 54, pp. 456–465, 1974.

    Google Scholar 

  36. Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, Englewood Cliff, New Jersey, 1971.

    MATH  Google Scholar 

  37. Lapidus, L., and Seinfeld, J. H., Numerical Solution of Ordinary Differential Equations, Academic Press, New York, 1971.

    MATH  Google Scholar 

  38. Enright, W. H., “Second Derivative Multistep Methods for Stiff Ordinary Differential Equations,” SIAM J. Anal., 11, pp. 321–331, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  39. Hofer, E., “A Partially Implicit Method for Large Stiff Systems of ODEs with Only Few Equations Introducing Small Time-Constants,” SIAM J. Numer. Anal., 13–5, 1976.

    Google Scholar 

  40. Richtmeyer, R. D., and Morton, K. W., Difference Methods for Initial Value Problems, Interscience Publishers, New York, 1967.

    Google Scholar 

  41. Birkhofer, A., and Werner, W., “Eine Methode zur Berechnung der raum- und zeit-abhängigen Leistungsverteilung in Kernreaktoren,” Atomkernenergie, 15, pp. 97–102, 1970.

    Google Scholar 

  42. Wight, A. L., Hansen, K. F., Ferguson, D. R., “Application of Alternating-Direction Implicit Methods to Space-Dependent Kinetics Equations,” Nucl. Sci. Eng., 44, pp. 239–251, 1971.

    Google Scholar 

  43. Ferguson, D. R., and Hansen, H. F., “Solution of the Space-Dependent Reactor Kinetics Equations in Three Dimensions,” Nucl. Sci. Eng., 51, pp. 189–205, 1973.

    Google Scholar 

  44. MEKIN: MIT-EPRI Nuclear Reactor Core Kinetics Code, 1975.

    Google Scholar 

  45. Gordon, P., “Nonsymmetric Difference Equations,” J. Soc. Indust. Appl. Math., 13, pp. 667–673, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  46. Peaceman, D. W., and Rachford, H. H. Jr., “The Numerical Solution of Parabolic and Elliptic Differential Equations,” J. Soc. Indust. Appl. Math., 3, pp. 42–65, 1955.

    Article  MathSciNet  Google Scholar 

  47. Douglas, J., and Gunn, J. E., “A General Formulation of Alternating Direction Methods,” Num. Math., 6, pp. 428–453, 1965.

    Article  MathSciNet  Google Scholar 

  48. Janenko, N. N., “Die Zwischenschrittmethode zur Lösung mehrdimensionaler Probleme der mathematischen Physik,” Lecture Notes in Mathematics, 109, Springer-Verlag, Heidelberg, 1969.

    Google Scholar 

  49. Birkhofer, A., and Werner, W., “Fully Implicit Matrix Decomposition Method for Space-Time Kinetics,” Trans. Am. Nucl. Soc., 15, pp. 789–790, 1972.

    Google Scholar 

  50. Langenbuch, S., and Werner, W., “Implicit Matrix Decomposition Scheme for Coarse-Mesh Methods”, Trans. Am. Nucl. Soc., 21, Page 224, 1975.

    Google Scholar 

  51. Varga, R. S., Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.

    Google Scholar 

  52. Wachspress, E. L., Iterative Solution of Elliptic Systems, Prentice-Hall, Englewood Cliffs, New Jsersy, 1966.

    MATH  Google Scholar 

  53. Nakamura, S., “A Variational Rebalancing Method for Linear Iterative Convergence Schemes for Neutron Diffusion and Transport Equations,” Nucl. Sci. Eng., 39, pp. 278–283, 1970.

    Google Scholar 

  54. Nakamura, S., “Coarse Mesh Acceleration of Iterative Solution of Neutron Diffusion Equation,” Nucl. Sci. Eng., 43, pp. 116–120, 1971.

    Google Scholar 

  55. Fröhlich, R., “A Theoretical Foundation of Coarse Mesh Variational Techniques,” Report CNM-R-2, Vol, 1, Page 219, (CONF-670501), 1967.

    Google Scholar 

  56. Anderson, M. M., Buckner, M. R., Carswell, J. H., Dodds, H. L., Gregory, M. V., Honeck, H. C., Routt, K. R., and Stewart, J. W., “Three-Dimensional Coupled Neutronic and Engineering Calculations of Savannah River Reactors,” Proc. Conf. Computational Methods in Nuclear Engineering, CONF-750413, Vol. II, VI, pp. 123–141, 1975.

    Google Scholar 

  57. Hestenes, M. R., and Stiefel, E., “Methods of Conjugate Gradients for Solving Linear Systems,” Nat. Bur. Standards, J. of Res., 49, pp. 409–436, 1952.

    Article  MathSciNet  MATH  Google Scholar 

  58. Langenbuch, S., and Werner, W., “Eine Methode zur Verbesserung der Zeitintegration in 3d Neutronenkinetik-Rechnungen durch eine Form der Periodenfaktorisierung,” Proc. Reaktortagung, 1976.

    Google Scholar 

  59. Garland, W. J., Vlachopoulos, J., Harms, A. A., “A Summation-Exponent Analysis for Space-Dependent Reactor Transients,” Trans. Am. Nucl. Soc., 18, Page 322, 1974.

    Google Scholar 

  60. Devought, J, and Mund, E., “A-Stable Algorithms for Neutron Kinetics,” MRR 145, Proc. of the Joint NEACRP/CSNI Specialists 1 Meeting on New Developments in Three-Dimensional Neutron Kinetics, pp. 21–71, 1975.

    Google Scholar 

  61. Yasinsky, J. B., “Combined Space-Time Synthesis with Axially Discontinuous Trial Functions,” USAEC Report, WAPD-TM-736, Westinghouse Electric Corp., Bettis Atomic Power Laboratory, 1967.

    Google Scholar 

  62. Yasinsky, J. B., “Numerical Studies of Combined Space-Time Synthesis,” Nucl. Sci. Eng., 34, pp. 158–168, 1968.

    Google Scholar 

  63. Henry, A. F., “Review of Computational Methods for Space-Dependent Kinetics,” Dynamics of Nuclear Systems, University of Arizona Press, Tuscon, Ariz., 1972.

    Google Scholar 

  64. Ott, K., and Madell, J. T., “Quasistatic Treatment of Spatial Phenomena in Reactor Dynamics,” Nucl. Sci. Eng., 26, pp. 563–565, 1966.

    Google Scholar 

  65. Ott, K., and Meneley, D. A., “Accuracy of the Quasistatic Treatment of Spatial Reactor Kinetics,” Nucl. Sci. Eng., 36, pp. 402–411, 1969.

    Google Scholar 

  66. Wagner, M. R., Finnemann, H., Lee, R. R., Meneley, D. A., Michelsen, B., Misfeldt, I., Vondy, D. R., Werner, W., “Multidimensional LWR Benchmark Problems,” Trans. Am. Nucl. Soc., 23, 1976.

    Google Scholar 

  67. Werner, W., Finnemann, H., Langenbuch, S., “Two- and Three-Dimensional BWR Kinetics Benchmark Problem,” Trans. Am. Nucl. Soc., 23, 1976.

    Google Scholar 

  68. “Nuclear Reactor Core Analysis Code: VENTURE,” Oak Ridge National Lab., 1976.

    Google Scholar 

  69. Cadwell, W. R., PDQ-7 Reference Manual, WAPD-TM-678, January 1967.

    Google Scholar 

  70. Misfeldt, I., “Solution of the Multigroup Neutron Diffusion Equations by the Finite Element Method,” RIS-M-1809, Danish AEC, Research Establishment RIS, Denmark, July 1975.

    Google Scholar 

  71. Buckel, G., “Vorschlag für ein Benchmark Problem in xyz- und Dreiecks-z-Geometrie,” INR Notiz, 335, 1975.

    Google Scholar 

  72. Buckel, G., Approximation der stationären, dreidimensionalen Mehrgruppen-Neutronen-Diffusionsgleichung durch ein Syntheseverfahren mit dem Karlsruher Synthese-Programm KASY, KfK-1349, 1971.

    Google Scholar 

  73. Dodds, H. L. Jr., Honeck, H. C, Hostetier, D. E., “Coarse-Mesh-Method for Two-Diraensional Mixed-Lattice Diffusion Theory Calculations,” Trans. Am. Nucl. Soc., 21, Page 223, 1975.

    Google Scholar 

  74. Schmidt, F. A. R., IKE Stuttgart, FRG, Private Communication.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1977 Plenum Press, New York

About this chapter

Cite this chapter

Werner, W. (1977). Kinetics of Nuclear System Solution Methods for the Space-Time Dependent Neutron Diffusion Equation. In: Henley, E.J., Lewins, J., Becker, M. (eds) Advances in Nuclear Science and Technology. Advances in Nuclear Science and Technology, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-9913-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-9913-1_4

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-9915-5

  • Online ISBN: 978-1-4613-9913-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics