Nuclear Reactor Theory - Neutron Flux Analysis & Diffusion Approx

Nuclear Reactor Theory - Nuclear Reactor Analysis -

The Neutron Flux ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Neutron Flux ,[object Object],[object Object],[object Object],[object Object],[object Object]

The Neutron Flux ,[object Object],Ref.: http://www.tpub.com/content/doe/h1019v1/css/h1019v1_138.htm

The Diffusion Approximation: Fick´s Law ,[object Object],[object Object],[object Object],[object Object],[object Object],Neutron Current Density Vector x  (x) J x Gradient of Flux

The Equation of Continuity ,[object Object],[object Object],[object Object],V r

The Equation of Continuity ,[object Object],[object Object],[object Object],Surface J x J z J y

The Diffusion Equation ,[object Object],[object Object],Fick´s Law Diffusion Length The integrands must satisfy The integration is over the same volume V

The Diffusion Equation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],A B n d  x)  d)=0

The Diffusion Equation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Low A High A Reflectors have LOW A

One-group Reactor Equation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

One-group Reactor Equation ,[object Object],[object Object],[object Object],[object Object],[object Object]

One-group Reactor Equation ,[object Object],[object Object],[object Object],[object Object],Material Buckling Geometric Buckling

Criticality of a Bare Reactor ,[object Object],[object Object],[object Object],[object Object],[object Object],k eff accounts also for the leakage Material Properties

One-group Critical Reactor Equation ,[object Object],Power Energy per fission The Solution for the Flux is: Buckling Spherical Critical Reactor The flux is a function only of the radius r Homogeneous Reactor The power is given by the integral There are many possible values of B that will satisfy the boundary conditions , but the geometrical buckling is the FIRST eigenvalue B 1 r R

One-group Critical Reactor Equation ,[object Object],Finite Cylindrical Critical Reactor The flux is a function of the radius r and z Two Functions: The solution: Homogeneous Reactor z r H/2 H/2 R

One-group Critical Reactor Equation ,[object Object],[object Object],[object Object],[object Object],[object Object],Too large for a real reactor. Real reactors have FLATTER Flux distributions by using reflectors and distributing the fuel.

Multi-group Reactor Equation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],g 1 g 2 g N …… .. Discretization of the Neutron Energy for Multi-group Calculations

Multi-group Reactor Equation ,[object Object],[object Object],[object Object],[object Object],[object Object],Group 1 ( g 1 ) Group 2 ( g 1 ) Group n-1 ( g n-1 ) Group N ( g N ) … . Energy Increasing Energy Groups for a N-Group Diffusion Calculation Transfer out of g Transfer into g

Multigroup Diffusion Core Analysis Codes ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

State-of-the-Art Nodal Methods Ref.: www.fz-juelich.de/ ief/ief-6/2/htr2-flu.html Node i,j,k Ref.: http://www.polymtl.ca/nucleaire/en/GAN/GAN.php The Multi-group equations are solved for each node i,j,k in which the reactor is divided. The nodes “homogenize” the heterogeneous reactor.

Neutron Transport ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Neutron Transport ,[object Object],Fission Source Scattering External Neutron Source Time variation and removal of neutrons Angular Flux and Neutron Density z x y dV dA

Nuclear Reactor Theory - Neutron Flux Analysis & Diffusion Approx

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Nuclear Reactor Theory - Neutron Flux Analysis & Diffusion Approx