Izvestia Vysshikh Uchebnykh Zawedeniy. Yadernaya Energetika

The peer-reviewed scientific and technology journal. ISSN: 0204-3327

Quadrature formulas for integral equations of kinetics and for digital reactimeters

6/21/2017 2017 - #02 Modelling processes at nuclear facilities

Yuferov A.G.

DOI: https://doi.org/10.26583/npe.2017.2.09

UDC: 621.039.515:621.039.516.2:621.039.514.4

The aim of work is the derivation of quadrature formulas for a kinetic equations nuclear reactor in the form of Volterra integral equations of the second kind and for equation of a reactimeter in the form of integral convolution, the core of which is a function of decay of the delayed neutron precursors (DNP) in the nongroup form. The expediency of the transition to integral equations is caused by the unification of the direct (calculation of power dynamics) and the reverse (calculation of current reactivity) tasks of reactor kinetics. As a result, the solution reduces to the calculation of the delayed neutrons integral. This eliminates the source of discrepancy calculated and experimental evaluations of reactivity due to the difference in computational algorithms direct and inverse problems. The paper describes a general scheme to convert different transport equation approximations to describe the contribution of delayed neutrons by a integral convolution without using dynamics of the DNP concentration. This conversion reduces the dimension of the model, simplifies the software implementation, eliminates the stiffness problem of differential equations of the kinetics, provides stability of calculations. The dimension of the model is preserved in the case of several fissile nuclides. The integral form of the equations admits in quadrature formulas the use the samples of experimental function of the decay, which can be identified in the operating conditions of a nuclear reactor and stored by pointwise in the non-group form – without expansions in sum of the exponents. This eliminates the need to address the nonlinear problem of identification of the group parameters of delayed neutrons and increases the adequacy of modeling. In work is obtained a series of quadrature formulas for the calculation of the delayed neutrons integral and describes the corresponding algorithms of digital reactimeter and the numerical simulation of the reactor kinetics.


  1. Recommendations for comparing the calculated and measured reactivity for substantiation of nuclear safety of reactor VVER-type plants. Мoscow. NTC YaRB Publ., 2010. 21 p. (in Russian).
  2. Ionov V.S. Raspredelennaya neytronnaya dinamika aktivnyh zon VVER [Distributed neutron dynamics of the active zones of the VVER]. Мoscow. IzdAT Publ., 2005. 311 p. (in Russian).
  3. Seleznev E.F. Kinetika reaktorov na bystryh neytronah [Fast Breeder Reactor Kinetics]. Мoscow. Nauka Publ., 2013. 239 p. (in Russian).
  4. Zizin M.N. Metody raschyota neytronno+fizicheskih harakteristik bystryh reaktorov [Methods of calculating neutron-physical characteristics of fast reactors]. Мoscow. NIC «Kurchatovskiy institut» Publ., 2014. 178 p. (in Russian).
  5. Zizin M.N. The presentation of delayed neutron parameters in the calculation test model BN600_IAEA. Available at: http://www.atominfo.runewsjq0464.htm. (in Russian).
  6. Computing Methods in Reactor Physics. Ed.: H. Greenspan, C.N. Kelber and D. Okrent. N.Y. Gordon and Breach, 1968 . 589 p.
  7. Hetrick D. L. Dynamics of Nuclear Reactors. University of Chicago Press, 1971. 542 p.
  8. Kolesov V.F. Aperiodicheskie impulsnye reaktory [Aperiodic pulse reactors]. Sarov. RFNC-VNIIEF Publ., 1999. 1032 p. (in Russian).
  9. Mogilner A.I., Fokin G.N., Chayka Yu.V., Kuznetsov F.M. The use of small computers for measuring reactivity. Atomnaya energiya. 1974, v. 36, iss. 5, pp. 358-361 (in Russian).
  10. Verlan A.F., Sizikov V.S. Integral’nye uravneniya: Metody, algoritmy, programmy [Integral equations: Methods, algorithms, programs]. Kiev. Naukova dumka Publ., 1986. 544 p. (in Russian).
  11. Numerical solution of integral equations. N.Y.: Plenum Press, 1990. 430p.
  12. Yuferov A.G. Bibliografiya po razrabotkam reaktimetrov i metodam izmereniya reaktivnosti v FEI [Bibliography on development of the reactivity meters and methods of the reactivity measurements in IPPE]. Obzor FEI-295. Мoscow. CNIIAtominform Publ., 2003. 39 p. (in Russian).
  13. Schmid P. A basic integral equation of reactor kinetics. Proc. 2nd Intern. Conf. Peaceful Uses Atomic Energy. Geneva, 1958, iss. 11, 277 p.
  14. Lewins J. On the General Solution of the Reactor Kinetic Equations. Nucl. Sci. and Eng., v. 11, no. 1, p. 97.
  15. Adler F.T. Reactor kinetics: integral equation formulation. Journal of Nuclear Energy. Parts A/B. 1961, v. 15, no. 2-3, pp. 81-85.
  16. Maslennikov M. V. On one numerical method of solution of the equations of reactor kinetics. Zhurnal vychislitel’noj matematiki i matematicheskoj fiziki. 1961, v. 1, no. 3, pp. 470-480 (in Russian).
  17. Keepin R.G. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co, 1965. 435 p.
  18. Ash M. Nuclear Reactor Kinetics. McGraw Hill, 1965. 415 p.
  19. Vakhromeeva V.V. The method of solving the equations of reactor kinetics. Otchet FEI TR-741, № 4325, 1965. 16 p. (in Russian).
  20. Rogozhin Yu.A. An integral method for solving the equations of point kinetics of nuclear reactors. Otchet FEI TR-837, № 5365, 1969. 29 p. (in Russian).
  21. Pettus W. G., Snioow N. L. A direct integral formulation of point reactor kinetics. Journal of Nuclear Energy. 1972, v. 26, pp. 489-490.
  22. Gulevich A.V., Kukharchuk O.F. The use of an integrated model of neutron kinetics to the calculation of multiplying multizone systems. Preprint FEI-2129, 1990 (in Russian).
  23. Nechepurenko Yu.M., Shishkov L.K. Determination of reactivity based on the inverse point kinetics equation. Zhurnal vychislitel’noj matematiki i matematicheskoj fiziki. 2002, v. 42, no. 9, pp. 1394-1398 (in Russian).
  24. Quintero-Leyva B. On the numerical solution of the integro-differential equation of the point kinetics of nuclear reactors as an ODE. Annals of Nuclear Energy 2009, v. 36, no. 8, pp. 1280-1284.
  25. Li H., Chen W., Luo L., Zhu Q. A new integral method for solving the point reactor neutron kinetics equations. Annals of Nuclear Energy, 2009, v. 36, no. 4, pp. 427-432.
  26. Davidenko V.D., Zinchenko A.S., Kharchenko I.K. Integral non-stationary neutron transport equations for calculation of the kinetics of nuclear reactors by Monte Carlo method. VANT. Ser. Fizika yadernykh reaktorov. 2015, iss. 1, рр. 11-16 (in Russian).
  27. Yuferov A.G., Ibragimov R.L. Interval estimation of reactivity. Izvestiya vuzov. Yadernaya energetika. 2010, no. 3, pp. 3-9 (in Russian).
  28. Zemelman M.A., Tronova I.M. Metodicheskiy material po primeneniyu GOST 8.009-84 «GSI. Normiruemye metrologicheskie kharakteristiki sredstv izmereniy». Normirovanie i ispolzovanie metrologicheskikh kharakteristik sredstv izmereniy [Methodological material for use GOST 8.009-84 «GSI. Standardized metrological characteristics of measuring instruments». Standardization and use of metrological characteristics of measuring instruments]. Мoscow. Izdatelstvo Standartov Publ., 1985, pp. 43-132 (in Russian).
  29. Yuferov A.G. Reactimeter dispersion equation. Nuclear Energy and Technology (2016). Available at: http://dx.doi.org/10.1016/j.nucet (2016.11.04).
  30. Bell G.I., Glasstone S. Nuclear reactor theory. N.Y.: Van Nostrand Reinhold Co., 1970, 619 p.
  31. Borisenko V. I. What must be determined: the period or the reactivity of the reactor? Problemy bezpeki atomnykh elektrostantsyj і Chornobylya. 2010, iss. 13, pp. 8-18 (in Russian).
  32. Brikker I.N. The inverted decision of a nuclear reactor kinetics equations. Atomnaya energiya. 1966, v. 21, iss. 1, pp. 9-13 (in Russian).
  33. Yuferov A.G. On the problem of identification of integral equations of the kinetics. Izvestiya vuzov. Yadernaya energetika. 2005, no. 4, pp. 25-34 (in Russian).
  34. Kuo B.C. Digital Control Systems. N.Y. Holt, Rinehart and Winston Inc., 1980. 730 p.
  35. Kuzin L.T. Raschet i proektirovanie diskretnyh sistem upravleniya [Calculation and design of discrete control systems]. Мoscow. Mashgiz Publ., 1962. 684 p. (in Russian).
  36. Bykov V.V. Cifrovoe modelirovanie v statisticheskoy radiotehnike [Numerical simulation in statistical radio engineering]. Мoscow. Sovetskoe radio Publ., 1971. 328 p. (in Russian).
  37. Yuferov A.G. Quadrature formulas of the sliding integration. Trudy Mezhdunarodnogo simpoziuma «Voprosy optimizatsii vychisleniy (VOV-XXXIII)» [Proceedings of the International Symposium «Problems of optimization calculations (POC-XXXIII)»]. Kiev. Institut kibernetiki im. V.M. Glushkova NAN Ukrainy Publ., 2007, pp. 319-320 (in Russian).

dynamics of nuclear reactor point kinetics reactivity reactimeter integral equations quadrature formulas