Time-dependent Transport Equation Research Articles

Initiation and structure of axisymmetric eddies in a rotating stream

Axisymmetric flow of a rotating stream is examined numerically to determine conditions under which an isolated eddy will form on the axis of rotation. An explicit finite difference procedure is used to integrate the time dependent transport equations. Solutions provide details of the flow structure and are presented for a range of Reynolds numbers and swirl ratios. Calculated results are interpreted in terms of recent physical experiments insofar as is possible.

On nonlinear neutron transport equations

The purpose of this paper is to investigate a class of time-dependent neutron transport equations in which the total and differential scattering cross sections are nonlinear functions of neutron density function. Sufficient conditions on the nonlinear cross sections are given to insure the existence, uniqueness and asymptotic stability of a solution in one, two, or three-dimensional space domains under various boundary and initial conditions. The approach to the problem is based on abstract analysis on nonlinear evolution equations which are closely related to nonlinear semigroup theory.

Solutions of Monoenergetic Time Dependent Neutron Transport Equation in Slab Geometry

Approximate solutions to the one-velocity time dependent neutron transport equation are derived by means of the eigenfunction expansion method of Case for a slab geometry. A finite set of discrete time eigenvalues is determined by a simple formula. This formula, which is derived for the case of large slab thickness, is found to be valid also for fairly thin slabs. Each time eigenvalue generates a pair of spatial modes. The expansion coefficients to these modes are examined by numerical evaluation for various slab thicknesses. Higher spatial modes are found to play an important role in the transient time evolution of neutrons.

Solutions of Monoenergetic Time Dependent Neutron Transport Equation in Slab Geometry

Approximate solutions to the one-velocity time dependent neutron transport equation are derived by means of the eigenfunction expansion method of Case for a slab geometry. A finite set of discrete time eigenvalues is determined by a simple formula. This formula, which is derived for the case of large slab thickness, is found to be valid also for fairly thin slabs. Each time eigenvalue generates a pair of spatial modes. The expansion coefficients to these modes are examined by numerical evaluation for various slab thicknesses. Higher spatial modes are found to play an important role in the transient time evolution of neutrons.

Solution of Neutron Transport Equation by Superposition of Beams

The analytical solution of the isotropic, one-dimensional and time-dependent transport equation is expressed by the superposition of a newly defined Green function of the virgin neutron flow. As a result, the solution of the neutron transport equation is given by the superposition of beams possessing wave fronts. The present study has introduced the possibility of clearly explaining neutron transport behavior by beam processes, thus providing a further insight into the interpretation of the transport equation. This method should prove useful when the continuous mode is superior to discrete modes. The solution for the stationary case is also derived.

The RCI model—A general model for semiconductor devices

The RCI model is an electric network consisting of lumped resistors, capacitors, and controlled current sources. It simulates the three-dimensional time-dependent transport and continuity equations and the Poisson equation for nondegenerated semiconductors without additional restriction except the spatial discretization error. Therefore, the model is suitable to describe the steady-state solution as well as the small- and large-signal transient behavior of semiconductor devices. The disadvantages of the Linvill model and the Ohtsuki-Kani model have been avoided by introducing excess variables and new model potentials. For constant mobility, the passive components of the network become linear. All nonlinearities are focused on the voltage-controlled current sources that have simple transfer functions.

Multigroup Neutron Transport

We investigate the nature of the approximations involved in the multigroup treatment of the time-dependent neutron transport equation by using the method of approximating sequences of Banach spaces. We prove that solutions of the multigroup system converge, in a suitable sense, to the corresponding solutions of the exact transport equation. Moreover, we indicate the order of magnitude of the rate of convergence.

Solution to the Linear Time-Dependent Neutron Transport Equation with Time-Dependent Source and Cross Sections

Systems that are below prompt critical are considered, and the linear time-dependent neutron transport equation in a quite general setting is studied. Both source and cross sections are allowed to depend on space, energy, and time. The method of matched asymptotic expansions is used to find an asymptotic solution uniformly valid in time. This solution is written in the form of a sum of solutions to simpler problems and for most practical problems is essentially exact. After a short initial time period, the transport equation (with delayed neutrons neglected) may be solved at a given time by a single inversion of the steady-state transport operator; i.e., with a steady-state code.

The decay of neutron pulse in a multiplying assembly using a simple kernel

Using the Laplace transform technique, we have solved the time dependent neutron transport equation in a multiplying assembly using a simple scattering kernel. The complete discrete and continuum solutions have been obtained. Numerical calculations have been performed for water moderator which give only one discrete mode. The results have been compared with the exact calculations performed using free gas kernel.

The phase space time evolution method

Abstract A new computer method for solving problems of time dependent particle transport is presented and applied to the solution of several one-speed neutron transport problems having planar symmetry. The method employs the basic concept of conservation of points in phase space, enabling the computer to be used simply as a bookkeeping device to directly keep track of the locations of small phase space elements as they move in time. It is felt that this approach embodies a more direct and efficient use of the computer, as it eliminates the difficult, tedious and essentially intermediate step of writing down the time dependent transport equation and programming the computer to effect its solution, The actual program has been refined by the development of a digitized Green's function, which allows the time evolution of the neutron distributions to be generated by simple matrix multiplication.

The time-dependent rectilinear transport equation

Abstract The rectilinear or rod model of neutron transport consists of a segment which is assumed to be capable of transporting particles. Such particles (neutrons) can move only to the right or to the left on the segment at a constant speed, and suffer scattering, capture and fission. We have shown that the time-eigenvalue spectrum of this model is made of a finite, non-void set of real eigenvalues, plus infinitely many complex eigenvalues, distributed in pairs of complex conjugates. A simple asymptotic formula for the high-order complex eigenvalues has been obtained. As an application, we have considered the following initial value problem: find the outgoing density of neutrons, at one end of the rod, at any time after the injection of a neutron pulse at the same end. A complete series expansion of the outgoing density has been given. The successive time-derivatives of this expansion show certain characteristic discontinuities which are due to the motion, from one end of the rod to the other, of discont...

Time-Dependent Neutron Transport Theory for Fast Pulsed Assemblies

The time-dependent transport equation in plane geometry has been solved numerically using the double spherical harmonics angular approximation and first-order finite differences. The monoenergetic case has been shown to meet both necessary and sufficient conditions for stability for reasonable values of the time step. The convergence and wave front propagation characteristics of the difference scheme have also been checked in special cases and found to be satisfactory. A computer program has been written to solve the difference equations of the multienergy, multiregion problem. Monoenergetic and multigroup calculations have been made which compare qualitatively with experimental results.

Time-dependent transport process

A compact and unified derivation of transport theory in the time-dependent case is presented. The development is based on the “star-product” which is an extension of that used by Redheffer ( 2). A “state form” and a corresponding “linearized system” are derived. The former is a system of non-linear integral equations which correspond to the invariant imbedding formulations of Bellman and Kalaba ( 1). The latter shows that such a system can be linearized. We also show that the right and left linearized equations can be derived from each other. This result is a generalization of a theorem given by Reid ( 9). The boundary effects on time-dependent transport equations are discussed. The general results are applicable to problems of transmission-line theory, radiative transfer, and neutron transfer in slab geometry. A specific example of the time-dependent problem for particles moving in a rod is given.

An Analysis of the Time-Dependent Neutron Transport Equation with Delayed Neutrons by the Method of Matched Asymptotic Expansions

The time-dependent neutron transport equation is treated as a problem in singular perturbation theory. The method of matched asymptotic expansions is used to find equations yielding approximate sol...

Quantum Theory of Electrical Transport in a Magnetic Field. I

The relationship between the quantum theory of electrical transport in the presence of a magnetic field and the corresponding Boltzmann transport equation is established for a simple system. The model consists of noninteracting free electrons being elastically scattered by an arbitrary potential in the presence of uniform electric and magnetic fields. Without the aid of a representation, the exact gauge-dependent Liouville equation for the density operator of this system is transformed into a completely gauge-independent equation satisfied by a new density operator. The new density operator is shown to give the correct current density, using the ordinary gauge-independent free-particle velocity operator. No approximations are made in performing the transformations, and the physical content of the new gauge-independent transport formalism is identical in all respects with that contained in the initial gauge-dependent equations. A new density matrix is defined which is essentially the Fourier sum of the matrix of the gauge-independent density operator. By considering the scattering potential resulting from a set of fixed impurity centers, it is shown that the diagonal elements of this new density matrix satisfy the ordinary time-dependent Boltzmann transport equation in which the spatial gradient term appears explicitly. The possible existence of a spatial variation in the average density of scatterers is also taken into account. The final equation for the quantum-mechanical distribution function represents the result obtained by treating the electric field and the effective scattering potential to the lowest possible order in which they contribute.

Time Behavior of a Reactor and Ergodic Theory of Semigroups

The solution of the time-dependent neutron transport equation is a semigroup of linear transformations acting on a Banach space. There are some ergodic theorems that can be used to describe the asymptotic behavior of the solution under very general conditions on the semigroup. The results are compared with Wing's famous approach.

On the Initial-Value Problem of the Transport Equation for a Moderator Slab

A mathematical analysis of the time-dependent neutron transport equation is presented for the case of a moderator slab. The scattering of neutrons is treated on the basis of the Van Hove theory and is assumed to be isotropic. First, the spectrum of the corresponding transport operator is investigated and it is shown that the complex λ-plane is decomposed by the operator into a band spectrum Re(λ) ≤ — (v∑t)min, a discrete spectrum on the real axis λ≤— (v∑t), min a resolvent set Re(λ)> — (v∑t)min deleted by the discrete spectrum. The discrete spectrum is carefully studied and is found to consist of isolated eigenvalues or otherwise empty. For all categories of moderator, there exists an upper limit to the thickness of a slab having empty discrete spectrum. A slab exceeding this limit has a finite number of eigenvalues if the moderator material is gas or solid. This is also true for a certain class of liquid moderators. For other liquids, there is another critical thickness for the slab thickness such that i...

On the Initial-Value Problem of the Transport Equation for a Moderator Slab

A mathematical analysis of the time-dependent neutron transport equation is presented for the case of a moderator slab. The scattering of neutrons is treated on the basis of the Van Hove theory and is assumed to be isotropic. First, the spectrum of the corresponding transport operator is investigated and it is shown that the complex λ-plane is decomposed by the operator into a band spectrum Re(λ) ≤ — (v∑t)min, a discrete spectrum on the real axis λ≤— (v∑t), min a resolvent set Re(λ)> — (v∑t)min deleted by the discrete spectrum. The discrete spectrum is carefully studied and is found to consist of isolated eigenvalues or otherwise empty. For all categories of moderator, there exists an upper limit to the thickness of a slab having empty discrete spectrum. A slab exceeding this limit has a finite number of eigenvalues if the moderator material is gas or solid. This is also true for a certain class of liquid moderators. For other liquids, there is another critical thickness for the slab thickness such that if it is exceeded, the set of eigenvalues turns out to be denumerable with an accumulation point at — (v∑t)min. Each eigenvalue possesses a finite multiplicity, and the index is one. Finally the related initial value problem is considered, and it proves to have a unique solution upon application of the Hille-Yosida theorem. The contribution of the band spectrum to the solution is evaluated and is shown to decay faster than exp{ — (v∑t)mint}.

Time-Dependent Flux in Small Assemblies

The one-dimensional (slab), one-velocity time-dependent transport equation has been investigated using a variational method employing flat spatial trial functions. A simple approximation is found for the variation of the asymptotic decay rate (α) with slab size for small slabs. As expected, little difference is found between the use of a single flat spatial flux trial function and a double stepped flux trial function for thin slabs. The method is then extended to the case of a convex body of arbitrary shape. It is shown that an estimate for α is given by the relationwhere Pc = first collision probability.For the slab case, an effective spatial buckling and an effective extrapolation distance consistent with the exact asymptotic decay constant were obtained. This extrapolation distance is approximately equal to the Milne problem value down to a scattering thickness of about 1.0 mean free path after which it rises to λs for the limiting case of zero thickness. Finally, asymptotic time decay rates based upon low-order PL and DPL approximations in slab geometry are determined either numerically or from the exact analytical solutions; a real eigenvalue may or may not exist depending on the boundary conditions. It is shown further that these low-order approximations yield erroneous time-dependent characteristics in the thin slab limit.

Frequency-Dependent Relaxation Time of Electron-Phonon Systems in a Homogeneous Magnetic Field

The problem of electromagnetic-wave absorption in a simple model of metal, consisting of electrons and phonons in the presence of a uniform static magnetic field, is investigated by a quantum-kinetic description. The treatment, which stems from a proper time-dependent transport equation, does not have the time-scale restriction of the usual transition-probability approach and does give a proper description of the time-dependent collective effects in a self-consistent way.