R.S. Modak

Sub-space iteration scheme for the evaluation of λ-modes of finite-differenced multi-group neutron diffusion equations

Abstract The recently known technique of sub-space iteration for nonsymmetric matrices is used to evaluate the dominant λ-modes of finite-differenced multi-group neutron diffusion equations over a nuclear reactor. The multi-group problem is first cast in the form of a “reduced problem” in terms of the net fission sources in each mesh. This makes the implementation of sub-space iteration scheme easy since the external source option available in well-established static finite difference codes can be utilised to solve the problem. The scheme is applied to a test case which represents a typical CANDU reactor. The exploitation of reactor core symmetry to simplify the problem is also discussed.

A simple scheme for the direct evaluation of time-eigenvalues of neutron transport equation

Abstract A simple numerical scheme is presented for the evaluation of fundamental and higher discrete time-eigenvalues of neutron transport equation. It is based on the direct solution of a matrix equation obtained by discretisation of integro-differential form of the transport equation by using the Sn-method with Diamond Difference scheme. The scheme is applied to mono-energetic homogeneous slab cases with isotropic scattering and verified against published accurate semi-analytical results. Although less accurate than the semi-analytical method, the scheme is more versatile. It is shown to be applicable to spatially heterogeneous cases also.

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.

On the use of the conjugate gradient method for the solution of the neutron transport equation

Abstract The possibility of using the standard conjugate gradient (CG) method to directly solve the Sn-equations based on the diamond difference scheme is studied for mono-energetic neutron transport problems with isotropic scattering. It is shown that such a direct use is possible for practical heterogeneous problems with a significant speed-up over the conventional source iteration (SI) method except for the problems that are prone to unphysical negative fluxes. Some recipes are suggested to make use of the CG-method even in those cases which need negative flux fix-up in the SI-method. The transport synthetic acceleration scheme, recently developed by Ramone [Nucl. Sci. Eng. 125 (1997) 257] and others, is shown to be useful in such cases. A symmetrisation scheme for the coefficient matrix has also been presented to enable the use of the CG-method. This scheme is compared with another approach of using weighted inner products.

Convergence of the iteration scheme in the nodal expansion method for the solution of the diffusion equation

Abstract Convergence of the iteration scheme in the nodal expansion method for the solution of the diffusion equation has been established. The proof is applicable to 1-D, 2-D and 3-D problems with commonly occurring boundary conditions. It is restricted to square and cubic nodes and parabolic expansion of the flux over a node.

Computational Schemes for Online Flux Mapping System in a Large-Sized Pressurized Heavy Water Reactor

Abstract Large-sized pressurized heavy water reactors (PHWRs) are neutronically loosely coupled and hence are prone to significant changes in flux shape during operation. As a result, they need a sophisticated regulation procedure based on an online flux mapping system (OFMS). During the reactor operation, neutron flux is continuously measured at certain predetermined in-core locations. The purpose of OFMS is to compute a detailed flux map at all points in the reactor, after every 2 min, by making use of the measured fluxes. The knowledge of detailed flux distribution is then used for an appropriate regulating action. The choice of computational method used by OFMS is of crucial importance because the method is expected to be both efficient and accurate and should work for a range of reactor configurations occurring during the operation. In this paper, three different methods, namely, flux synthesis, internal boundary condition, and combined least squares (CLSQ), are analyzed for their prospective use in the forthcoming 700-MW(electric) Indian PHWR. The CLSQ method is found to be most accurate, although it needs significant computation. A hybrid method that combines certain features of other methods is also studied and seems to give good accuracy with moderate computational effort.

A scheme for the evaluation of γ-modes of a neutron diffusion equation

Abstract The static multiplication eigenvalue problem over a nuclear reactor is considered in multi-group diffusion theory with finite difference approximation. An alternative scheme is suggested for the evaluation of higher harmonics where one works in terms of fission sources rather than the group fluxes leading to a reduction in problem size. The scheme is tested for simple cases. The situations arising in more complex problems are briefly discussed.

Use of a New Boundary Condition in Computational Neutron Transport

Abstract This paper deals with the numerical evaluation of fundamental- and higher-mode solutions of the well-known K-eigenvalue problem in nuclear reactor physics using neutron transport theory. If the spatial domain has a plane of reflective symmetry, it is customary to find the fundamental K-mode by considering only the half-domain on one side of the plane for the calculation and applying a reflective boundary condition (RBC) on the plane of symmetry. Here, it is shown that the higher antisymmetric K-mode can also be evaluated in a similar way by applying what is called here the anti-RBC (ARBC) on the plane of symmetry. ARBC was implemented in computer codes based on the discrete ordinates method in Cartesian geometry for some sample problems and was found to work well. The implementation of ARBC in existing codes, although very easy, does not seem to be widely used or reported in the literature. For a one-dimensional homogeneous slab with isotropic scattering, the first antisymmetric K-mode found using ARBC is equivalent to the fundamental mode of a sphere, apart from a scaling factor for the total flux. An interesting result is that the fundamental mode of a sphere computed in this way does not contain the unphysical flux dip near the center, commonly obtained by the discrete ordinates codes in spherical geometry. Although not shown here, it appears that ARBC can be implemented in Monte Carlo codes also to find antisymmetric modes.

Some reciprocity-like relations in multi-group neutron diffusion and transport theory over bare homogeneous regions

Abstract Some simple reciprocity-like relations that exist in multi-group neutron diffusion and transport theory over bare homogeneous regions are presented. These relations do not involve the adjoint solutions and are directly related to numerical schemes based on an explicit evaluation of the fission matrix.

On the reciprocity-like relations in linear neutron transport theory

Abstract The existence of certain reciprocity-like relations in neutron transport theory was shown earlier under some quite restrictive conditions. Here, these relations are shown to be valid in more general situations by using a different approach based on individual neutron trajectories.