D.C. Sahni

An application of reactor noise techniques to neutron transport problems in a random medium

Abstract Neutron transport problems in a random medium are considered by defining a joint Markov process describing the fluctuations of neutron population and the random changes in the medium. Backward Chapman-Kolmogorov equations are derived which yield an adjoint transport equation for the average neutron density. It is shown that this average density also satisfies the direct transport equation as given by the phenomenological model.

Spectrum of one-speed Neutron transport operator with reflective boundary conditions in slab geometry

Abstract The time eigenvalue spectrum of the one-speed neutron transport equation has been investigated in the case of an infinite slab with reflective boundary conditions and isotropic scattering. The region Re ( ) v Σ t , too, is mainly contained in the resolvent set of the operator, the continuous spectrum consisting of only a set of discrete lines. Further it is shown that a finite number of real decay constants also exist in this region. We also give an upper limit for the magnitude of these eigenvalues. The analysis is supported by numerical calculations.

Non-monotonic variation of the criticality factor with the degree of anisotropy in one-speed neutron transport

Abstract The criticality factor c has been calculated for one—speed neutrons in spheres, infinite slabs and cylinders. The scattering is assumed to be a combination of linearly an isotropic scattering and strongly forward (Inonu) scattering. The earlier observed non—monotonic variation of the criticality factor with the degree of forward scattering has been shown to exist in this case, too, for slabs and cylinders. However, for such systems below a certain size the variation is monotonic. To facilitate these studies a simple criterion has been derived.

One-speed neutron transport in reflected spheres

Abstract The transport of one-speed neutrons has been studied in isotropically scattering spheres with reflective boundaries. An integral equation for the total flux has been derived and solved numerically for various radii and reflexion coefficients. Calculations have also been done for spheres in vacuum with varying degree of strongly peaked forward and backward scattering, so called Inonu scattering. Some special cases have been studied in detail, such as that when the forward-backward scattering dominates over the isotropic scattering.

Modelling of a vibrating reactor boundary and calculation of the induced neutron noise

Abstract Three different models of a moving (vibrating) reactor boundary in time-dependent diffusion theory are investigated. The models are: (a) a localized absorber of variable strength at the boundary (equivalent to a perturbational treatment); (b) a time-varying extrapolation length; (c) explicit treatment of the moving boundary with a new transformation technique. The induced neutron noise was calculated in first order of the perturbation parameter both exactly and in the adiabatic approximation. All three models lead to equivalent results, confirming the applicability of perturbation techniques in treating moving perturbations (e.g. vibrating control rods). Application of the adiabatic approximation in model (c) required the extension of the Henry formalism, i.e. the use of orthogonality relations expressed as integrals over the system, to cases with non-constant system volume. The incentives for investigating a time-varying boundary arose from problems related to vibrating control rods; however, the results have some general relevance for systems with a varying volume such as gaseous core or liquid fuel reactors.

Time eigenvalue spectrum for one-speed neutron transport in spheres with strong forward-backward scattering

Abstract The spectrum of time eigenvalues has been studied for one-speed neutrons in spheres with strong forward and backward scattering together with isotropic scattering. It is shown that in the presence of strong backward scattering there are both discrete eigenvalues and a continuum of eigenvalues. The lower limit of the continuum has been determined. The results are supported by numerical calculations.

Green's function approach to the solution of the time dependent Fokker-Planck equation with an absorbing boundary

The solution of the time dependent multivelocity Fokker-Planck equation in plane geometry with an absorbing boundary is formulated in terms of the infinite medium Greens function and the boundary distribution. The boundary distribution is to be determined by solving an integral equation. It is shown that the velocity moments such as density, current and energy density can be expressed in terms of three reduced distributions which depend on the longitudinal velocity component only. Numerical results for monovelocity pulsed and steady sources are presented.

Evaluation of higher K-rmeigenvalues of the neutron transport equation by Sn-method

Abstract Many semi-analytical methods are being developed to evaluate higher eigenvalues of mono-energetic neutron transport equation. Here, a numerical approach is presented where the well-known S n -method is used to generate higher K -eigenvalues of neutron transport equation via the generation of a fission matrix. Although less accurate than the semi-analytical methods, the inclusion of spatial inhomogeneities, scattering anisotropies and even more energy groups would be straightforward in this method owing to the versatility of the S n -codes. It is also shown that the K -eigenvalues are always real in the mono-energetic case even if scattering is anisotropic. The inadequacy of the S n -method to reproduce the flux-source reciprocity relation in spherical geometry, noticed during the course of this work, is also briefly discussed.

Computation of higher spherical harmonics moments of the angular flux for neutron transport problems in spherical geometry

Abstract The integral form of one-speed, spherically symmetric neutron transport equation with isotropic scattering is considered. Two standard problems are solved using normal mode expansion technique. The expansion coefficients are obtained by solving their singular integral equations. It is shown that these expansion coefficients provide a representation of all spherical harmonics moments of the angular flux as a superposition of Bessel functions. It is seen that large errors occur in the computation of higher moments unless we take certain precautions. The reasons for this phenomenon are explained. They throw some light on the failure of spherical harmonics method in treating spherical geometry problems as observed by Aronsson.

Real criticality eigenvalues of the one-speed linear transport operator

Abstract The criticality eigenvalue problem has been studied for the one-speed neutron transport equation. Convex bodies of arbitrary shape and with vacuum boundary conditions have been considered. The cross sections may be space dependent, and the scattering is assumed to be anisotropic. Several new conditions have been derived which ensure that a point spectrum of eigenvalues exists and that all the eigenvalues are real. The most general such condition is that the even order coefficients in the development of the scattering function have a different sign than the odd order ones. As a consequence, for linearly anisotropic scattering the eigenvalues are all real if the average cosine of the scattering angle is negative. Numerical results computed for homogeneous bodies in the form of spheres and infinite slabs and cylinders confirm the theoretical considerations.