M.Ç. Güleçyüz

The HN method for solving linear transport equation: theory and applications

Abstract The system of singular integral equations which is obtained from the integro-differential form of the linear transport equation using the Placzek lemma is solved. The exit distributions at the boundaries of the various media and the infinite medium Greens function are used. The process is applied to the half-space and finite slab problems. The neutron angular density in terms of singular eigenfunctions of the method of elementary solutions is also used to derive the same analytical expressions.

Solution of the CN equations using singular eigenfunctions and applications

Abstract The third form of Boltzmann equation involves only the angular flux at the boundary while the usual transport equation deals with the angular flux at any point. The kernel of this equation is the infinite medium Greens function and satisfies the lineer transport equation. The method of solution of this equation is known as the CN method and is based on the Placzek lemma and depends on the calculation of the infinite medium Greens function. Here, the well-known form of the Greens function in terms of elementary solutions is used to solve the third form of the transport equation and applications for the half-space albedo problems for both isotropic and extremely anisotropic scatterings are given. Uncollided neutrons are also taken into account.

The slab albedo problem using singular eigenfunctions and the third form of the transport equation

Abstract The albedo and the transmission factor for slabs are obtained using the infinite medium Greens function in terms of the singular eigenfunctions in the third form of the transport equation. Our analytical results are simple as in the F N method and the convergence of the numerical results is as fast as in the C N method. Calculations are also carried out by various incoming angular fluxes and uncollided neutrons are taken into account. Our numerical results are in very good agreement with the results of the C N method and, as shown in the tables, are obtained up to the eighth order of approximation.

A new approach of solving the third form of the transport equation in plane geometry: Half-space albedo-problem

Abstract In a new approach of solving the third form of the transport equation, one should consider and compare the two methods C N and F N . Both of the methods depend on the use of Greens function. In C N method the Greens function obtained by the Fourier-transform technique is used to solve the system of integral equations which is provided at the boundary of the various media. In F N method which is a modified version of C N method. The Greens function in terms of singular eigenfunctions is used to solve the integro-differential form of the Boltzmann equation. In our approach we use the Greens function of F N method to solve C N equations. That is Greens function in terms of singular eigenfunctions is used to solve the system of integral equations of C N method. As in F N method this approach yields simple analytical equations that can be solved numericaly even more efficiently than the C N method and the convergence of the numerical results is faster than the numerical results of F N method.

The solution of the third form transport equation using singular eigenfunctions: the slab and the sphere criticality problems

Abstract A singular eigenfunction approach is used to solve the classical slab and the sphere criticality problems. This approach is based on the use of the infinite medium Greens function which is obtained by the method of elementary solutions in the C N equations, that is in the third form of the transport equation. It is shown that our numerical results, even in the approximations of the lowest order are very accurate.

SLAB ALBEDO PROBLEM FOR ANISOTROPIC SCATTERING USING SINGULAR EIGENFUNCTION SOLUTION OF THE CN EQUATIONS

Abstract The albedo and the transmission factor for slabs for extremely anisotropic scattering are calculated. The solution of the problem is obtained using a new approach; that is the solution of the third form of the transport equation using the singular eigenfunctions of the method of elementary solutions. This approach leads simple equations for solving numerically when compared with C N method and also leads the convergence of the numerical results to be faster than that of the other methods in literature (i.e. F N -method).

The FN method for anisotropic scattering in neutron transport theory: The critical slab problem

Abstract The F N method which has been applied to many physical problems for isotropic and anisotropic scattering in neutron transport theory is extended for problems for extremely anisotropic scattering. This method depends on the Placzek lemma and the use of the infinite medium Greens function. Here the Greens function for extremely anisotropic scattering which was expressed as a combination of the Greens functions for isotropic scattering is used to solve the critical slab problem. It is shown that the criticality condition is in agreement with the one obtained previously by reducing the transport equation for anisotropic scattering to isotropic scattering and solving using the F N method.

The singular eigenfunction analysis of the third form transport equation using half-range orthogonality relations: the half-space problems

Abstract The third form transport equation is solved using the infinite medium Greens functions in terms of the singular eigenfunctions of the method of elementary solutions and their half-range orthogonality relations. For simplicity, we consider only the Fredholm equation of the first kind and obtain the solution of the two problems for isotropic scattering: (i) the albedo problem for a source free half-space and (ii) the extrapolation length for the Milne problem. The convergence of the numerical results is fast as in C N method and the analytical expressions are simple for solving numerically.

The critical slab problem for linearly anisotropic scattering and reflecting boundary conditions with the HN method

Abstract The HN method is used to calculate the critical slab thickness for a parallel media surrounded bay a reflector for linearly anisotropic scattering. It is shown that the method leads to concise equations and accurate numerical results even in the lowest order of approximation.

An application of transport theory for optical oceanography: the isotropic point source

Transport theory methods can be applied to optical oceanography to compute the spatial dependence of irradiance and scalar irradiance of the light field from an isotropic point source deep within a spatially uniform ocean. The infinite medium Greens function for plane geometry is used to obtain scalar and plane irradiances. The isotropic point source expressions can be obtained from the isotropic plane source solutions using plane-to-point transformations. Then the analytical solutions for the diffuse attenuation coefficient and the mean cosine far from the isotropic point source are obtained. The problem is numerically solved for the Henyey–Greenstein phase function.