Particle Transport Equation Research Articles

Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation

A method for applying goal-based adaptive methods to the angular resolution of the neutral particle transport equation is presented. The methods are applied to an octahedral wavelet discretisation of the spherical angular domain which allows for anisotropic resolution. The angular resolution is adapted across both the spatial and energy dimensions. The spatial domain is discretised using an inner-element sub-grid scale finite element method. The goal-based adaptive methods optimise the angular discretisation to minimise the error in a specific functional of the solution. The goal-based error estimators require the solution of an adjoint system to determine the importance to the specified functional. The error estimators and the novel methods to calculate them are described. Several examples are presented to demonstrate the effectiveness of the methods. It is shown that the methods can significantly reduce the number of unknowns and computational time required to obtain a given error. The novelty of the work is the use of goal-based adaptive methods to obtain anisotropic resolution in the angular domain for solving the transport equation.

Even- and odd-parity kinetic equations of particle transport. 1: Algebraic and centered forms of the scattering integral

We consider an equivalent formulation of the linear kinetic transport equation for neutral particles (neutrons, photons) as a system of two equations for even and odd parts of the distribution function. The particle scattering integral of even- and odd-parity transport equations is converted into a non-linear algebraic form and into a centered form. In the algebraic form of the integral we clearly identify the net result of two opposite processes, i.e., particle scattering from a beam and into the beam. In the centered form of the integral the principal terms of scattering processes are canceled out. An iterative method is proposed for the solution of the system of even- and odd-parity equations with these forms of the scattering integral. Convergence of iterations is studied for a one-dimensional plane problem.

A Discontinuous Phase-Space Finite Element Discretization of the Linear Boltzmann-Vlasov Equation for Charged Particle Transport

We examine the modeling of charged-particle transport when both collision processes with background media and electromagnetic effects are important using the Boltzmann-Vlasov equation. We derive and transform the Boltzmann-Vlasov equation into a form very similar to the standard linear Boltzmann equation with additional operators. We apply the discontinuous finite element methods for discretization in the spatial, energy, and angular variables. An implementation of these methods demonstrates correct transport behavior for fixed electric and magnetic fields. We also demonstrate coupling to Maxwell's equations with a simple electromagnetic solver to generate self-consistent fields.

Discontinuous finite element discretizations for the [formula omitted] neutron transport equation in problems with spatially varying cross sections

We examine the accuracy of arbitrary degree discontinuous finite element spatial discretizations of the SN particle transport equations with cross sections that are continuous functions of space. For cell-wise constant cross section problems in purely absorbing materials, it has previously been shown that using arbitrary degree DFEM with quadrature-based mass matrix lumping techniques can result in fully accurate schemes with strictly positive angular flux outflows in 1-D slab geometry. We adapt these quadrature-based, or “self-lumping”, schemes to problems with arbitrarily varying spatial cross sections and incorporate the cross-section spatial dependence into the definition of the mass matrices.We compare two approaches to deal with the cross-section spatial dependence. The first approach approximates the true cross section as a cell-wise constant that preserves the cell-average cross-section value. The second method uses self-lumping quadrature to evaluate the exact cross section at quadrature points. Regardless of DFEM trial space degree, approximating a spatially varying cross section with a cell-wise constant cross section results in schemes that are at most second-order accurate in space. Additionally, we demonstrate that assuming a cell-wise constant cross section generates interaction rate profiles that have highly non-physical, non-monotonic discontinuities. Using the self-lumping technique to account for cross-section spatial variation with Gauss or Lobatto quadrature yields fully accurate DFEM schemes for problems with spatially varying cross section. Self-lumping schemes that evaluate the cross section at quadrature points do not exhibit non-physical interaction rate profiles. Unfortunately, only a self-lumping linear DFEM with Lobatto quadrature is guaranteed to produce strictly positive angular flux outflow in a pure absorber when accounting for cross-section spatial variation.Robustness and convergence order comparisons are first carried out using a pure absorber test problem. The cross-section spatial variation is chosen such that an analytical solution can be obtained. Next, we study the spatial effects of isotopic concentration changes during fuel depletion. The depletion problem confirms the accuracy results of the pure-absorber problem.

A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes

We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes.

Goal-based angular adaptivity applied to the spherical harmonics discretisation of the neutral particle transport equation

A variable order spherical harmonics scheme has been described and employed for the solution of the neutral particle transport equation. The scheme is specifically described with application within the inner-element sub-grid scale finite element spatial discretisation. The angular resolution is variable across both the spatial and energy dimensions. That is, the order of the spherical harmonic expansion may differ at each node of the mesh for each energy group. The variable order scheme has been used to develop adaptive methods for the angular resolution of the particle transport phase-space. Two types of adaptive method have been developed and applied to examples. The first is regular adaptivity, in which the error in the solution over the entire domain is minimised. The second is goal-based adaptivity, in which the error in a specified functional is minimised. The methods were applied to fixed source and eigenvalue examples. Both methods demonstrate an improved accuracy for a given number of degrees of freedom in the angular discretisation.

NONTHERMAL RADIATION OF YOUNG SUPERNOVA REMNANTS: THE CASE OF CAS A

The processes responsible for the broad-band radiation of the young supernova remnant Cas A are explored using a new code which is designed for a detailed treatment of the diffusive shock acceleration of particles in nonlinear regime. The model is based on spherically symmetric hydrodynamic equations complemented with transport equations for relativistic particles. Electrons, protons and the oxygen ions accelerated by forward and reverse shocks are included in the numerical calculations. We show that the available multi-wavelength observations in the radio, X-ray and gamma-ray bands can be best explained by invoking particle acceleration by both forward and reversed shocks. Although the TeV gamma-ray observations can be interpreted by interactions of both accelerated electrons and protons/ions, the measurements by Fermi LAT at energies below 1 GeV give a tentative preference to the hadronic origin of gamma-rays. Then, the acceleration efficiency in this source, despite the previous claims, should be very high; 25% of the explosion energy (or approximately $3\cdot 10^{50}$ erg) should already be converted to cosmic rays, mainly by the forward shock. At the same time, the model calculations do not provide extension of the maximum energy of accelerated protons beyond 100 TeV. In this model, the acceleration of electrons is dominated by the reverse shock; the required $10^{48}$ erg can be achieved under the assumption that the injection of electrons (positrons) is supported by the radioactive decay of $^{44}$Ti.

Extended gyrokinetic field theory for time-dependent magnetic confinement fields

A gyrokinetic system of equations for turbulent toroidal plasmas in time-dependent axisymmetric background magnetic fields is derived from the variational principle. Besides governing equations for gyrocenter distribution functions and turbulent electromagnetic fields, the conditions which self-consistently determine the background magnetic fields varying on a transport time scale are obtained by using the Lagrangian, which includes the constraint on the background fields. Conservation laws for energy and toroidal angular momentum of the whole system in the time-dependent background magnetic fields are naturally derived by applying Noether's theorem. It is shown that the ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work agree with the results from the conventional recursive formulation with the WKB representation except that collisional effects are disregarded here.

Characteristic interactions and evaluation of the parameters of the transport equation and formation, including porosity, from data of measurements by some nuclear geophysical methods

An important but problematic component of nuclear geophysical technologies is computer inversion of measurement data based on the equation of particle transport. In this paper, we propose an approach and iterative methods for this inversion that are applicable to many problems of evaluation formation characteristics, including elemental composition, from data of corresponding logging methods. This approach is based on defining characteristic elements, interactions, and trajectories and using the superposition principle for transport processes, and linear a priori constraints on unknowns are used. In comparison with the previous publications of the author, this paper deals with a broader type of measurement data, including the reading ratio of detectors and normalized measurements. The iterative methods are specified for problems of determining the porosity and density from data of neutron-neutron and gamma-gamma logs, respectively. For the first of them, the results of numerical experiments are presented.

An Eulerian model for particles nonisothermally carried by a compressible fluid

An Eulerian model describing the transport of particles that nonisothermally interact with their carrying flow is developed in the general time-dependent, three-dimensional frame. The Eulerian model derives from the Lagrangian kinematic, momentum and energy particle equations based on a combination of the filtered Liouville equation and the method of moments. The filtered Liouville equation governs the filtered density function of the particle velocities and temperature in phase space. Moments of the Liouville equation result in a coupled system of Eulerian governing equations for the particle number density, velocity and temperature and their second moments. The equations are closed by expressing third order moments in terms of lower order moments based on the assumption that third-order correlations are negligible. The particle transport equations are two-way coupled with the inviscid, Euler equations governing compressible flows. Consistent with the hyperbolic conservation form of the Eulerian–Eulerian model, a flux-vector splitting and characteristics-based Roe averaging method discretizes the equations. Computations of a cloud of particles initially at rest in a uniform, subsonic flow and in an accelerated flow behind a normal shock in one dimension, show very good comparison with a recent higher order WENO based Eulerian–Lagrangian model.

Predicting transient particle transport in enclosed environments with the combined computational fluid dynamics and Markov chain method

To quickly obtain information about airborne infectious disease transmission in enclosed environments is critical in reducing the infection risk to the occupants. This study developed a combined computational fluid dynamics (CFD) and Markov chain method for quickly predicting transient particle transport in enclosed environments. The method first calculated a transition probability matrix using CFD simulations. Next, the Markov chain technique was applied to calculate the transient particle concentration distributions. This investigation used three cases, particle transport in an isothermal clean room, an office with an underfloor air distribution system, and the first-class cabin of an MD-82 airliner, to validate the combined CFD and Markov chain method. The general trends of the particle concentrations vs. time predicted by the Markov chain method agreed with the CFD simulations for these cases. The proposed Markov chain method can provide faster-than-real-time information about particle transport in enclosed environments. Furthermore, for a fixed airflow field, when the source location is changed, the Markov chain method can be used to avoid recalculation of the particle transport equation and thus reduce computing costs.

A Thermophoretic Sherwood Number for Characterizing Submicron-Particle Mass Transfer in Laminar Wall-Bounded Flows

New solutions to the Eulerian particle-transport equations are presented which describe concentration profiles in wall-bounded, submicron-particle-laden, one-way coupled flows of gases undergoing advective transport and thermophoresis. These solutions have been deduced for the cases of steady, fully developed, laminar flow of hot gas within pipes and channels with a cold surface at a uniform temperature, when the velocity field and the temperature and particle concentration profiles can be described by their constant-property forms. They are used to show how the effectiveness with which temperature difference drives particulate mass transport can be characterized by a dimensionless mass-transport coefficient or thermophoretic Sherwood number—the product of a particle-concentration ratio and the heat-transfer Nusselt number—that is useful in making engineering predictions of and comparisons between particulate mass transfer rates in different flows. The solutions also reveal how the concentration profiles ...

STOCHASTIC ACCELERATION OF SUPRATHERMAL PARTICLES UNDER PRESSURE BALANCE CONDITIONS

The acceleration of suprathermal charged particles in the heliosphere under pressure balance conditions including for the first time the radial spatial particle diffusion and convection in the solar wind is investigated. The physical conditions are derived for which the stationary phase space distribution of suprathermal particles approaches the power-law distribution f∝p –5, which is often seen in spacecraft observations. For separable source distributions in momentum and position we analytically solve the stationary particle transport equation for a radially constant solar wind speed V 0 and a momentum-independent radial spatial diffusion coefficient. The resulting stationary solution at any position within the finite heliosphere is the superposition of an infinite sum of power laws in momentum below and above the (assumed mono-momentum) injection momentum pI . The smallest spatial eigenvalue determines the flattest power law, to which the full stationary solution approaches at large and small enough momenta. Only for the case of a reflecting inner and a free-escape outer spatial boundary, does one small eigenvalue exist, yielding the power-law distribution f∝p –5 at sufficiently large momentum values. The other three spatial boundary conditions imply steeper momentum spectra. Momentum spectra and radial profiles of suprathermal particles are calculated by adopting a uniform outer ring spatial source distribution.

Algorithms of NCG geometrical module

The methods and algorithms of the versatile NCG geometrical module used in the MCU code system are described. The NCG geometrical module is based on the Monte Carlo method and intended for solving equations of particle transport. The versatile combinatorial body method, the grid method, and methods of equalized cross sections and grain structures are used for description of the system geometry and calculation of trajectories.

Fluid modelling of capacitively coupled radio-frequency discharges: a review

This paper reviews the basis and the successes with the fluid modelling of capacitively coupled radio-frequency discharges, produced within a parallel-plate cylindrical setup at (single) 13.56–80 MHz frequencies, 6 × 10−2–6 Torr pressures and 50–1000 V rf-applied voltages, in SiH4-H2, H2 and N2. The two-dimensional, time-dependent model used in the simulations accounts for the production, transport and destruction of the charged particles (via the electron and ion continuity and momentum-transfer equations, and the electron mean energy transport equations), and of the excited species and/or radicals (via their rate balance equations, including very complete kinetic descriptions with several collisional–radiative production/destruction mechanisms, coupled to the two-term electron Boltzmann equation), accounting also for the self-consistent development of the rf field (via the solution to Poisson's equation). The charged particle transport equations are solved with and without corrective flux terms (due to inertia and friction effects), whose influence on results is discussed. In the case of silane–hydrogen mixtures, the model further includes a phenomenological description of the plasma–substrate interaction to calculate the deposition rate of a-Si : H thin films. In general, the model gives good predictions for the self-bias voltage, the coupled power and the intensities of radiative emission transitions (both average and spatially resolved), underestimating the electron density by a factor of 3–4.

Analytical Solutions for the Pencil-Beam Equation with Energy Loss and Straggling

In this article, we derive equations approximating the Boltzmann equation for charged particle transport under the continuous slowing down assumption. The objective is to obtain analytical expressions that approximate the solution to the Boltzmann equation. The analytical expressions found are based on the Fermi-Eyges solution, but include correction factors to account for energy loss and spread. Numerical tests are also performed to investigate the validity of the approximations.

On the homotopy analysis method for solving a particle transport equation

The purpose of this paper consists in the finding of the solution for a stationary neutron transport equation that is accompanied by the homogeneous boundary conditions, using the techniques of homotopy analysis method (HAM) and a numerical integration formula. Also, algorithm presented can be used for solving the integral–differential equations in which the unknown function depends on two variables, such as a radiative transfer equation. Results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.

Multiphysics modelling of the separation of suspended particles via frequency ramping of ultrasonic standing waves

A model was developed to determine the local changes of concentration of particles and the formations of bands induced by a standing acoustic wave field subjected to a sawtooth frequency ramping pattern. The mass transport equation was modified to incorporate the effect of acoustic forces on the concentration of particles. This was achieved by balancing the forces acting on particles. The frequency ramping was implemented as a parametric sweep for the time harmonic frequency response in time steps of 0.1s. The physics phenomena of piezoelectricity, acoustic fields and diffusion of particles were coupled and solved in COMSOL Multiphysics™ (COMSOL AB, Stockholm, Sweden) following a three step approach. The first step solves the governing partial differential equations describing the acoustic field by assuming that the pressure field achieves a pseudo steady state. In the second step, the acoustic radiation force is calculated from the pressure field. The final step allows calculating the locally changing concentration of particles as a function of time by solving the modified equation of particle transport. The diffusivity was calculated as function of concentration following the Garg and Ruthven [1] equation which describes the steep increase of diffusivity when the concentration approaches saturation. However, it was found that this steep increase creates numerical instabilities at high voltages (in the piezoelectricity equations) and high initial particle concentration. The model was simplified to a pseudo one-dimensional case due to computation power limitations. The predicted particle distribution calculated with the model is in good agreement with the experimental data as it follows accurately the movement of the bands in the centre of the chamber.

The "step feature" of suprathermal ion distributions: a discriminator between acceleration processes?

Abstract. The discussion of exactly which process is causing the preferred build-up of v−5-power law tails of the velocity distribution of suprathermal particles in the solar wind is still ongoing. Criteria allowing one to discriminate between the various suggestions that have been made would be useful in order to clarify the physics behind these tails. With this study, we draw the attention to the so-called "step feature" of the velocity distributions and offer a criterion that allows one to distinguish between those scenarios that employ velocity diffusion, i.e. second-order Fermi processes, which are prime candidates in the present debate. With an analytical approximation to the self-consistently obtained velocity diffusion coefficient, we solve the transport equation for suprathermal particles. The numerical simulation reveals that this form of the diffusion coefficient naturally leads to the step feature of the velocity distributions. This finding favours – at least in regions of the appearance of the step feature (i.e. for heliocentric distances up to about 11 AU and at lower energies) – the standard velocity diffusion as a consequence of the particle's interactions with the plasma wave turbulence as opposed to that caused by velocity fluctuation-induced compressions and rarefactions.

Modeling Performance of a Tangential Flow Cyclone: Effects of Secondary Outlet Geometry and Boundary Conditions

Numerical calculation of the performance of cyclones for gas-particle separation is becoming a popular tool for cyclone analysis and design. A traditionally accepted way to treat these problems is by solving the flow field and particle transport equations in the spatial domain (C1) limited by the inlet and overflow outlet cross-sections, where the flow is usually assumed fully developed, and the dust outlet cross-section. Formulation of the boundary conditions for the flow and particles’ transport at the dust outlet cross-section remains an open question. We address this problem numerically by comparing results of calculations performed by the above traditional approach, i.e., treating the problem in configuration C1, as well as in extended domains, namely those including a dust collecting bin (configuration C2) and a downcomer tubular outlet (configuration C3). Two boundary conditions at the downflow outlet were checked for each configuration, namely, a blocked outlet (zero velocity) and an open outlet (zero gas flow rate). Toward this goal, an efficient computational fluid dynamic (CFD) model of turbulent flow based on the Reynolds Stress turbulent closure scheme was developed, validated, and used to calculate velocity field and the pressure drop as well as particle cut size and separation efficiency in several Stairmand-type tangential flow cyclones. The results were tested and corroborated against the experimental data reported in the literature, including careful measurements performed by Gottschalk and Bohnet (1995, 1998) for a 225 mm tangential flow cyclone, with downflow outlet connected to a tubular extension. Particle separation in the traditional C1 cyclone configuration was shown to be most sensitive with respect to the choice of the boundary conditions at the downflow outlet, whereas with the dust outlet connected to a bin (C3 configuration) this condition has a much weaker effect on the fractional efficiency. Solutions for the C1 and C2 configurations, both combined with the blocked downflow outlet boundary condition yielded the cyclone's pressure drop in agreement with the Gottschalk and Bohnet's data. However, C2 configuration which contained a downcomer tubular extension was found preferable for correlating the measured fractional efficiency.