Eine mehrskalige Näherungslösung für die zeitabhängige Boltzmann-Transportgleichung

Abstract

The basis of all transient simulations for nuclear reactor cores is the reliable calculation of the power production. The local power distribution is generally calculated by solving the space, time, energy and angle dependent neutron transport equation known as Boltzmann equation. The computation of exact solutions of the Boltzmann equation is very time consuming. For practical numerical simulations approximated solutions are usually unavoidable. The objective of this work is development of an effective multi scale approximation solution for the Boltzmann equation. Most of the existing methods are based on separation of space and time. The new suggested method is performed without space-time separation. This effective approximation solution is developed on the basis of an expansion for the time derivative of different approximations to the Boltzmann equation. The method of multiple scale expansion is used for the expansion of the time derivative, because the problem of the stiff time behaviour can't be expressed by standard expansion methods. This multiple scale expansion is used in this work to develop approximation solutions for different approximations of the Boltzmann equation, starting from the expansion of the point kinetics equations. The resulting analytic functions are used for testing the applicability and accuracy of the multiple scale expansion method for an approximation solution with 2 delayed neutron groups. The results are tested versus the exact analytical results for the point kinetics equations. Very good agreement between both solutions is obtained. The validity of the solution with 2 delayed neutron groups to approximate the behaviour of the system with 6 delayed neutron groups is demonstrated in an additional analysis. A strategy for a solution with 4 delayed neutron groups is described. A multiple scale expansion is performed for the space-time dependent diffusion equation for one homogenized cell with 2 delayed neutron groups. The result is once more compared with the exact analytical solution obtaining good agreement. In the next steps multiple scale expansion solutions are developed for the space-time dependent P 1 and P 3 transport equations for the homogenized cell and 2 delayed neutron groups. These results are analysed versus the solution for the diffusion equation emphasizing the differences in the space-time structure between the time dependent diffusion- and transport solutions. The effect of the additional derivation terms in the transport equations can be observed during the analytical expansion process and in the graphical analysis of the differences between the solutions. The developed solution is tested for direct calculation of the time behaviour of single nodes in the framework of a nodal code and the results are compared. It is evident that the nature of the inserted perturbation has major impact on the discrepancy of the results compared to the reference nodal method. Finally a solution strategy usable for big time steps is given for the space-time dependent neutron flux based on superposition of perturbations. This newly developed strategy is verified in a series of test calculations using a nodal code. A new efficient way for the simulation of the space-time dependent neutron flux distribution in nuclear reactor cores is given.