The Soviet Journal of Atomic Energy

, Volume 6, Issue 5, pp 396–403 | Cite as

On an approximate method for the homogenization of a heterogeneous reactor

  • V. V. Smelov


In most cases of calculations on heterogeneous reactors use has been made of a method of homogenization based on the replacement of the heterogeneous reactor by a homogeneous one (with effective values of the parameters). Whereas the calculation of the effective cross sections for capture and fission is a simple problem [1–3], the determination of the effective diffusion coefficient involves great difficulties, and until very recently there has been no adequately grounded method for its calculation. In this connection an important advance has been achieved in a paper by Shevelev [4].

In the present paper it is shown that the problem of homogenization is basically associated with a specific functional, which depends on the statement of the problem. A general principle of homogenization is discussed, which uses as this functional the quantity keff. It is shown that in calculations on heterogeneous reactors,the use of effective capture and fission cross sections obtained by averaging over the spatial neutron spectrum in an infinite lattice gives an error in keff that is proportional to the Laplacian k2 of the reactor. Attempts to increase the degree of precision lead to a situation in which the effective parameters lose their “universality,” that is, come to depend on the geometry of the reactor. It is shown that the use of universal parameters, in principle, excludes the possibility of a unique determination of the effective diffusion coefficient, even if one considers the diffusion of the neutrons in a definite direction. Nevertheless,formulas are obtained that give the most correct determination of the diffusion coefficient that is possible. The diffusion coefficient so obtained depends on the direction in the lattice. The method of homogenization is considered for plane, cylindrical, and spherical geometries of the cells.


Diffusion Coefficient Great Difficulty Approximate Method Effective Parameter Heterogeneous Reactor 
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Literature cited

  1. 1.
    Ya. V. Shevelev, Atomnaya Énergiya2, 224 (1957).Google Scholar
  2. 2.
    W. E. Milne, Numerical Calculus [Russian translation] (IL, 1951).Google Scholar
  3. 3.
    S. Glasstone and M. Edlund, Foundations of the Theory of Nuclear Reactors [ Russian translation] (IL, 1954).Google Scholar
  4. 4.
    G. I. Marchuk, Numerical Methods for Calculations on Nuclear Reactors [in Russian] (Atomizdat, Moscow, 1958).Google Scholar

Copyright information

© Consultants Bureau Enterprises, Inc. 1960

Authors and Affiliations

  • V. V. Smelov

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