Application of the finite element method to the three. Kanpur, india numerical solution of two group uncertain neutron diffusion equation for multi region reactor sukanta nayak and s. Iterative schemes for the neutron diffusion equation upv. Application of the finite element method to the threedimensional neutron diffusion equation paper submitted to the neacrp 21th meeting tokai, november 1978 takeharu ise, toshio yamazaki and yasuaki nauwra japan atomic energy research institute sumitomo heavy industries, ltd. The solution of coupled fractional neutron diffusion equations with delayed neutrons. The multigroup neutron diffusion to equations1 space. This is the energydependent neutron diffusion equation. Approximation of the neutron dif usion equation on. Pdf the solution of coupled fractional neutron diffusion. The steadystate diffusion equation 3 substituting the source term from eq. Diffusion equation laboratory for reactor physics and systems behaviour neutronics comments 1 domain of application of the diffusion equation, very wide describes behaviour of the scalar flux not just the attenuation of a beam equation mathematically similar to those for other physics phenomena, e. Ambrosini mainly on the basis of the material adopted by prof.
When the diffusion equation is linear, sums of solutions are also solutions. The neutron transport equation can be expressed in a form identical to that of the diffusion equation in the elementary theory of diffusion. Numerical techniques for the neutron di usion equations in. View notes lecture 5 neutron diffusion equation3 from mie 407 at university of toronto.
Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. Hence, for its integration, it is convenient to use an implicit backward difference formula bdf 7. Because diffusion is composed by a sum of different terms, and if your temrs assorbtion and generation that describe the nuclear reaction are 0, there still a term that describe the neutron collision and this term involves the flick law so there still neutron diffusion also without nuclear reacion. Everyone breathes a sigh of relief as it is shown to be very solvable, and a criticality relation a balance between neutrons created and destroyed links the geometry of a reactor to its material of construction. However, by combining all mesh points lying on a line.
Many methods of solving the diffusion equation have been used in the past. Development of a three dimensional neutron diffusion code. Introduction we have seen that the transport equation is exact, but difficult to solve. Neutron diffusion 90 if we insert the diffusion approximation 23 into our balance equation 4, we obtain. Lecture notes for the course on numerical models for nuclear. First, the stationary neutron di usion equation is studied. Chapter 7 the diffusion equation the diffusionequation is a partial differentialequationwhich describes density. Numerical solution of two group uncertain neutron diffusion. During the diffusion process some neutrons ar e absorbed and any which eventually cross the reactor boundary or surface are lost by leakag e. Introduction the diffusion theory model of neutron transport plays a crucial role in reactor theory since it is simple enough to allow scientific insight, and it is sufficiently realistic to study many important design problems. This leads to the steady state diffusion equation which, in simple terms, states that neutron production is equal to neutron absorption plus neutron leakage for stable conditions to be maintained. Solution of the twodimensional multigroup neutron diffusion.
Walter ambrosini university of pisa, italy unit 1 eigenvalue problems with neutron diffusion and solution strategies. From the point of view of image segmentation, these. This work deals with this model for nuclear reactors with hexagonal geometries. Solutions of the neutron diffusion equation in nonmultiplying media plane isotropic source in an infinite homogeneous medium plane isotropic source in a finite homogeneous medium line source in an infinite homogeneous medium homogeneous cylinder of infinite axial extent with axial line source point source in an. A quadratic boundary element formulation for neutron di. To evaluate the constant a we need to impose a suitable. Apr 06, 2008 because diffusion is composed by a sum of different terms, and if your temrs assorbtion and generation that describe the nuclear reaction are 0, there still a term that describe the neutron collision and this term involves the flick law so there still neutron diffusion also without nuclear reacion.
Chapter 2 derivation of the neutron diffusion equation from transport theory. Pdf numerical techniques for the neutron diffusion equations in. So we will have to use some average flux and cross section that have been averaged over the property in the group energy range in question. There may be no flow of neutrons, yet many interactions may occur i. The derivation of the diffusion equation depends on ficks law, which states that solute diffuses from high concentration to low. Zweifel, linear transport theory, addisonwesley, 1967 neutron motion boltzmann transport equation. The nuclear group constants for the diffusion equation are expressed as functions of the thermal hydraulic condition at each block in the core. The gaussian function is the greens function of the linear diffusion equation. Transport crosssection the effect of the scattering angular distribution on. Keywordsiterative methods, neutron diffusion equation, second degree methods. Solution of the multigroup neutron diffusion equations by. Multidimensional neutron diffusion research online uow. The particular solution was obtained by assuming some source distribution, and solving the neutron diffusion equation for the particular solution based upon the assumed source distribution. It turns out that this set can be created by convolving the image with gaussian functions of dif ferent scales.
The technique employs onedimensional trial solutions to reduce the twodimensional transport equation to the form of. Lecture notes for the course on numerical models for. A description is given of a program for the ferranti mercury computer which solves the onedimensional multigroup diffusion equations in plane, cylindrical or spherical geometry, and also approximates automatically a twodimensional solution by separating the space. Diffusion equation and neutron diffusion theory physics forums. It consists of a set of secondorder partial differential equations over the spatial coordinates that are, both in the academia and in the industry, usually solved by discretizing the neutron leakage term using a structured grid.
Pdf the spacetime neutron diffusion equations with multigroup of. These two routines are combined by a subroutiw crossace. In chapters 5 and 6 respectively, numerical results were presented for the one and two. The neutron transport equation is a balance statement that conserves neutrons. A quick short form for the diffusion equation is ut. Numerical techniques for the neutron di usion equations 651 di usion equations with adiabatic heat up and doppler feedback aboanber and hamada, 20095. H is the full hamiltonian operator, hi is the incident neutron kinetic energy operator and v is the neutronnucleus interaction potential. To do this we must first solve for the spaceenergytime distribution of the neutrons that cause fission. The energy of the neutron does not change as a result of a collision with the nuclei of the medium, 4.
Equation 9 is an eigenproblem, whose solution behaves in a typical way. This notebook is an entirely selfcontained solution to a basic neutron diffision equation for a reactor rx made up of a single fuel rod. These equations are based ontheconceptoflocal neutron balance, which takes int pdf available in advanced studies in theoretical physics 6 january 2012 with 585 reads. This work introduces the alternatives that unstructured grids can provide. Neutron diffusion equation 9 to integrate equation 3, we must take into account that it constitutes a system of stiff differential equations, mainly due to the elements of the diagonal. Solution of the neutron diffusion equation in hexagonal. In previous chapters we introduced two bases for the derivation of the diffusion equation. But first, we have to define a neutron flux and neutron current density. Diffusion equation laboratory for reactor physics and systems behaviour neutronics comments 1 domain of application of the diffusion equation, very wide describes behaviour of the scalar flux not just the attenuation of a beam equation mathematically similar to. Transport crosssection the effect of the scattering angular distribution on the motion of a neutron is taken into. Figure 6 shows an illustrative 5 group approximation.
This video describes the neutron diffusion in nuclear reactors. Feb 15, 2017 this video describes the neutron diffusion in nuclear reactors. A generalized, finitedifferenced, diffusion equation for. Article pdf available in advanced studies in theoretical physics 6. In order to obtain that, we must then use the diffusion equation. One such approach is directional diffusion, a typical example of. Solution of the multigroup neutron diffusion equations by the finite element method. Ei and es are the eigenvalue energies for the incident neutron and for the scattered neutron. The solution of twodimensional neutron diffusion equation with delayed neutrons 341 table 3 dependence of solution on physical properties. The solution of reactor diffusion problems oxford academic journals. Finite difference solution of steadystate diffusion equation. The neutrons are here characterized by a single energy or speed, and. The technique employs onedimensional trial solutions to reduce the twodimensional transport equation to the form of the neutron diffusion equation, thereby al.
The helmholtz equation is derived, and the limitations on diffusion equation as well as the boundary conditions used in its application to realistic engineering and physics problems are discussed. Third international conference on advances in control and optimization of dynamical systems march 15, 2014. The kinetic equation expressed in such a form is defined as the kinetic diffusion equation. Finite element method applied to neutron diffusion problems. Methods for solving the neutron diffusion equation in hexagonal geometries various methods have been developed and are applied for solving the neutron diffusion equation in hexagonal geometries, e. Lecture 5 neutron diffusion equation3 mie 407h1f\1129h. Pdf this paper presents a general theoretical analysis of the neutron motion problem in a nuclear reactor, where large variations on neutron. A generalization of the neutron diffusion equation is derived which improves the results obtainable by conventional diffusion theory for multidimensional systems.
The steady state and the diffusion equation the neutron field basic field quantity in reactor physics is the neutron angular flux density distribution. This is a di erential eigenvalue problem, called lambda modes problem. Multigroup diffusion 7 recall that the cross sections and flux can vary greatly as a function of neutron energy, e. Iterative schemes for the neutron diffusion equation. Chapter 2 the diffusion equation and the steady state. The fullcore calculation consists of solving a simplified transport equation, either. Neutron diffusion theory nuclear reactor physics wiley. The onegroup diffusion equation that we will be stepping through time and space is. The kinetic diffusion equation in a onedimensional planar geometry is derived and physical validation is given for the parameters in the equation. Unstructured grids and the multigroup neutron diffusion equation.
Chapter 2 the diffusion equation and the steady state weshallnowstudy the equations which govern the neutron field in a reactor. The hideous neutron transport equation has been reduced to a simple oneliner neutron diffusion equation. The famous diffusion equation, also known as the heat equation, reads. Kinetic diffusion equation in neutron transport theory. The neutron flux is used to characterize the neutron distribution in the reactor and it is the main output of solutions of diffusion equations. The neutron di usion equation describes the neutron population in a nuclear reactor core. The neutron diffusion equation is often used to perform corelevel neutronic calculations. The neutron flux is used to characterize the neutron distribution in the reactor and it is the main output of solutions of diffusion.
For calculating it, one may combine equations 10 and 14 to obtain. Approximation of the neutron di usion equation on hexagonal. The solution of twodimensional neutron diffusion equation. In this paper we are concerned with the diffusion of neutrons in fissile material where collisions. We give a brief derivation of the transport equation to show that it is nothing more than just a balance. The multigroup integrodi erential equations of the neutron di usion kinetics was presented and solved numerically in multislab. Here is an example that uses superposition of errorfunction solutions. Neutron diffusion introduction into reactor theory. Matkowsky, uniform asymptotic expansions in tranport theory with small mean free paths and the diffustion approximation, j.
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