Particle Transport Equation Research Articles

Discrete-Analytical Difference Scheme for Solving the Nonstationary Particle Transport Equation by the Splitting Method

Discrete-Analytical Difference Scheme for Solving the Nonstationary Particle Transport Equation by the Splitting Method

Particle Acceleration in Relativistic Shearing Flows: Energy Spectrum

We consider the acceleration of charged particles in relativistic shearing flows, with Lorentz factor up to Γ0 ∼ 20. We present numerical solutions to the particle transport equation and compare these with results from analytical calculations. We show that in the highly relativistic limit the particle energy spectrum that results from acceleration approaches a power law, , with a universal value for the slope of this power law, where α parameterizes the power-law momentum dependence of the particle mean free path. At mildly relativistic flow speeds, the energy spectrum becomes softer and sensitive to the underlying flow profile. We explore different flow examples, including Gaussian and power-law-type velocity profiles, showing that the latter yield comparatively harder spectra, producing for Γ0 ≃ 3 and Kolmogorov turbulence. We provide a comparison with a simplified leaky-box approach and derive an approximate relation for estimating the spectral index as a function of the maximum shear flow speed. These results are of relevance for jetted, high-energy astrophysical sources such as active galactic nuclei, since shear acceleration is a promising mechanism for the acceleration of charged particles to relativistic energies and is likely to contribute to the high-energy radiation observed.

An improved spectral approach for solving the nonclassical neutral particle transport equation

An improved modification of the Spectral Approach (SA) used for approximating the nonclassical neutral particle transport equation is described in this work. The term “Spectral” is used to indicate that the nonclassical angular flux is approximated as an expansion in terms of spectral basis functions. In the SA the basis functions are the Laguerre polynomials. The main focus of the modified SA lies on a slight modification of the nonclassical angular flux representation as a truncated Laguerre series. This leads, in some cases, to a considerable decrease of the Laguerre truncation order required to generate accurate solutions, thus improving efficiency. Numerical results are given to illustrate this offered improvement.

Open-source modelling of aerosol dynamics and computational fluid dynamics: Bipolar and unipolar diffusion charging and photoelectric charging

Electric charging is one of the essential aerosol dynamic mechanisms and is harnessed for detection, capture and control of ultrafine aerosol particles in a range of devices. For simplicity, charging and transport mechanisms are commonly modelled with zero spatial dimensions (0-D) and averaged properties such as mean charge or mean particle diameter. These models often neglect localised effects of the flow distribution, diffusion, discrete charge states, and particle polydispersity, often proving inadequate to explain experimental data. This work aims to provide an open-source three-dimensional (3-D) aerosol charging and transport model including bipolar and unipolar diffusion charging, and photoelectric charging algorithms for use in detailed design and analyses of aerosol systems. The computational model consists of more than 200 particle transport equations for discrete charge states and polydisperse sizes coupled with ion conservation equations in the framework of OpenFOAM, an open-source computational fluid dynamics platform. Three test cases are introduced to verify implementation of three charging models by comparison with published literature: bipolar and unipolar diffusion charging, and photoelectric charging. Tutorial cases, which model three distinct aerosol sensors, are described and demonstrate the capabilities of the 3-D aerosol charging and transport models within the predetermined flow field. The aerosolChargingFoam code is available at https://openaerosol.sourceforge.io for widespread use and can be further modified under the GNU general public licence. Program summaryProgram title: aerosolChargingFoamCPC Library link to program files:https://doi.org/10.17632/4pf4n2vg9f.1Developer's repository link:https://openaerosol.sourceforge.io/Licensing provisions: GNU General Public License 3Programming language: C++Nature of problem: aerosolChargingFoam solves generalised aerosol electrical charging and transport equations coupled with computational fluid dynamics using the open-source computational platform, OpenFOAM [1]. The electric charging algorithm including unipolar diffusion charging, bipolar diffusion charging, and photoelectric charging is verified with zero-dimensional test cases [2-4] and generalized to three-dimensional monodisperse or polydisperse particle distributions for more than 50 particle charge states. aerosolChargingFoam can be straightforwardly coupled with other existing solvers, which enables computations of complex multi-physics aerosol charging processes in practical conditions.Solution method: aerosolChargingFoam employs an explicit time-stepping for the time-dependent source terms for generalised aerosol charging. The solution methods and schemes provided by OpenFOAM 9 are used for solving the spatial and time derivatives in the transport equations.

Analysis of Corner Balance Nonlinear Diffusion Acceleration and Moment-Based, Spatially Linear Transport Discretizations

The recently developed High-Order, Low-Order scheme for the solution of thermal radiative transfer problems is applied as an acceleration method to the neutral particle transport equation. The resulting Corner Balance Nonlinear Diffusion Acceleration (CB-NDA) is derived, and a stability analysis is performed in conjunction with moment-based, spatially linear discretizations. These spatial discretizations correspond to the lumped Linear Discontinuous (LD) and Linear Characteristic (LC) schemes, which possess the thick diffusion limit. The lumped LD and LC schemes satisfy corner balance equations, which in turn are used to derive the CB-NDA. Two variants of the CB-NDA include the net current and partial current formulations. Numerical results are presented that verify the theoretical predictions and implementation. Theoretical spectral radius from the analysis is verified by comparison to values from the numerical solution of a one-dimensional transport problem. Results indicate similar stability between the CB-NDA–accelerated lumped LD and LC schemes. The net current–based CB-NDA is found to be unstable whereas the partial current formulation remains stable over the range of scattering ratios and optical thicknesses.

Effects of the Nature of Boundaries Conditions and Their Truncation Errors on the Distribution of Minority Carriers in Silicon Solar Cell

In this work, the effects of boundaries conditions and truncation errors in the distribution of minority carriers in the semiconductor are studied. It is a one-dimensional digital study of a polycrystalline silicon solar cell under polychromatic illumination in a dynamic state. Starting from the Boltzmann equation of semiconductors, the author establishes the general equation of particle transport. Assumptions made on the latter allow it to give the equation of distribution of minority carriers in a general way in its case to be studied. This dimensioned distribution equation reveals the parameters of influences on the distribution of carriers. It obtains a partial derivative equation for the carrier distribution function. The boundary conditions are then discretized to order one and then to order two. By considering boundary conditions and the nature of the carriers, the author numerically resolves the discretized general equation by assessing the influence of the nature of the boundary conditions and truncation errors and the influence of the discretization step on the density of the charge carriers by setting certain parameters and varying others. The work ends with a conclusion and logical follow-up to this work.

Research on numerical simulation for partial discharge of epoxy interface excited by high‐frequency sinusoidal voltage

High-frequency sinusoidal voltage excitation is proposed as a feasible method to solve the difficulty of partial discharge detection at the epoxy interface. In order to study the development process of partial discharge under high-frequency stress, a two-dimensional plasma simulation model of partial discharge for a needle-plate electrode structure is established by coupling particle transport equation, Poisson equation and plasma chemical reaction. The model adopts a reaction set to reduce the difficulty of modelling generation, transport and disappearance of charged particles. It realises dynamic simulation of the partial discharge at the epoxy interface. The spatial–temporal distribution characteristics of microscopic parameters such as electron density, electron temperature and surface charge are obtained. Based on this simulation model, the partial discharge characteristics of the epoxy interface at different frequencies are further studied in this work. The results show that with the increase of applied voltage frequency, the electron temperature shows an increasing trend, but its increasing rate gradually slows down; while surface charge accumulation decreases uniformly with the increase of applied voltage frequency. These two factors make the partial discharge severest under the frequency of 10 kHz. The results of the model are verified from two aspects of discharge form and discharge intensity by experimental means.

Quasilinear turbulent particle and heat transport modelling with a neural-network-based approach founded on gyrokinetic calculations and experimental data

A novel quasilinear turbulent transport model DeKANIS has been constructed founded on the gyrokinetic analysis of JT-60U plasmas. DeKANIS predicts particle and heat fluxes fast with a neural network (NN) based approach and distinguishes diffusive and non-diffusive transport processes. The original model only considered particle transport, but its capability has been extended to cover multi-channel turbulent transport. To solve a set of particle and heat transport equations stably in integrated codes with DeKANIS, the NN model embedded in DeKANIS has been modified. DeKANIS originally determined turbulent saturation levels semi-empirically based on JT-60U experimental data, but now it can also estimate them using a theory-based saturation rule. The new saturation model is still partly connected to experimental data, but it offers the potential for applying DeKANIS independently of the device.

Large-Scale Heterogeneous Computing for 3D Deterministic Particle Transport on Tianhe-2A Supercomputer

Scalable parallel algorithm for particle transport is one of the main application fields in high-performance computing. Discrete ordinate method (Sn) is one of the most popular deterministic numerical methods for solving particle transport equations. In this paper, we introduce a new method of large-scale heterogeneous computing of one energy group time-independent deterministic discrete ordinates neutron transport in 3D Cartesian geometry (Sweep3D) on Tianhe-2A supercomputer. In heterogeneous programming, we use customized Basic Communication Library (BCL) and Accelerated Computing Library (ACL) to control and communicate between CPU and the Matrix2000 accelerator. We use OpenMP instructions to exploit the parallelism of threads based on Matrix 2000. The test results show that the optimization of applying OpenMP on particle transport algorithm modified by our method can get 11.3 times acceleration at most. On Tianhe-2A supercomputer, the parallel efficiency of 1.01 million cores compared with 170 thousand cores is 52%.

Self-similarity in particle accumulation on the advancing meniscus

When a mixture of viscous oil and non-colloidal particles displaces air between two parallel plates, the shear-induced migration of particles leads to the gradual accumulation of particles on the advancing oil–air interface. This particle accumulation results in the fingering of an otherwise stable fluid–fluid interface. While previous works have focused on the resultant instability, one unexplored yet striking feature of the experiments is the self-similarity in the concentration profile of the accumulating particles. In this paper, we rationalise this self-similar behaviour by deriving a depth-averaged particle transport equation based on the suspension balance model, following the theoretical framework of Ramachandran (J. Fluid Mech., vol. 734, 2013, pp. 219–252). The solutions to the particle transport equation are shown to be self-similar with slight deviations, and in excellent agreement with experimental observations. Our results demonstrate that the combination of the shear-induced migration, the advancing fluid–fluid interface and Taylor dispersion yield the self-similar and gradual accumulation of particles.

On the Test Particle Monte-Carlo method to solve the steady state Boltzmann equation, the congruity of its results with experiments and its potential for shared memory parallelism

The Test Particle Monte Carlo is a known method to solve the steady state Boltzmann particle transport equation in rarefied gas systems. A description of the Test Particle Monte-Carlo procedure is outlined and the accuracy of this method is investigated by analyzing its consistency to experimental data of steady state effusions in isothermal systems. Computational results present deviations from the experiments that are at most 6.7%. After which, this paper investigates the potential of this method when it is parallelized using multi-core CPUs with shared memory for large Knudsen numbers. Scalability is expected since the method relies on a mean particle field that renders particle trajectories nearly independent from one another, therefore reducing data communication between processing cores. The mean particle distribution field helps linearize the collision term of the Boltzmann equation. Shared Memory Parallelism is an interesting feature when combined with distributed memory parallelism for design optimization of vacuum systems.

Evaluation of Single-Node Performance of Parallel Algorithms for Multigroup Monte Carlo Particle Transport Methods

Monte Carlo (MC) methods have been widely used to solve the particle transport equation due to their high accuracy and capability of processing complex geometries. History-based and event-based algorithms that are applicable to different architectures are two methods for parallelizing the MC code. There is a large work on evaluating and optimizing parallel algorithms with continuous-energy schemes. In this work, we evaluate the single-node performance of history-based and event-based algorithms for multigroup MC methods on both CPUs and GPUs with Quicksilver, a multigroup MC transport code that has already implemented the history-based algorithms. We first implement and optimize the event-based algorithm based on Quicksilver and then perform the evaluation work extensively on the Coral2 benchmark. Numerical results indicate that contrary to continuous-energy schemes, the history-based approach with multigroup schemes outperforms the event-based algorithm on both architectures in all cases. We summarize that the performance loss of the event-based algorithm is mainly due to: 1) extra operations to reorganize particles, 2) batched atomic operations, and 3) poor particle data locality. Despite the poor performance, the event-based algorithm achieves higher memory bandwidth utilization. We further discuss the impact of memory access patterns and calculation of cross sections (xs) on the performance of the GPU. Built on the analytics, and shed light on the algorithm choice and optimizations for paralleling the MC transport code on different architectures.

Density Transform Method for Particle Transport Problems in Spherical Geometry with Linearly Anisotropic Scattering

We develop the density transform method for treating particle transport problems in spherical geometry with linearly anisotropic scattering. We consider both, the interior and exterior problems of a homogeneous sphere and show that the transform satisfies an equation that resembles particle transport equation in slab geometry. The boundary conditions for these two problems are different. We also work out the density transform method for linearly anisotropic scattering for a region of arbitrary shape.

Electro-Chemical Modelling of Laser Structured Electrodes

A simulation study performed in the scope of the project RealLi! is presented. One of the project’s main goals is to improve NMC811 and graphite electrode cycling capacities at high C-rates.The rapid charging and discharging capability of batteries is improved using laser ablation to introduce structures into the surface of the electrode composite layers. Due to improved transport kinetics, this not only improves the electrochemical properties in the high-current range, but also homogenizes and accelerates the electrolyte wetting during production as a side effect. This is particularly advantageous in thick-film electrodes for providing high energy densities.This study supports the laser structuring process of battery electrodes [1][2] via a virtual optimisation, based on electro-chemical battery models. The electrodes are structured by ultrafast laser ablation, with parallel channels being introduced along the electrode surface. This modification enables an easier electrolyte penetration, a reduced charge transfer resistance, and shortened lithium-ion transport pathways which finally leads to a reduced diffusion overpotential at high C-rates. The geometrical parameters of this process (pitch distance, width, and cross-sectional shape of laser-generated micro-channels) and their impact on cell performance are virtually optimised by simulations.The simulations are based on a homogenised multi-scale model, applied in 2D/3D macroscopic cuts, coupled with 1D microscopic particle cuts. The 2D/3D macroscopic electrolyte transport equations are common concentrated electrolyte equations. The microscopic particle transport equations are either a set of non-linear Fick’s Diffusion equations [3] that are used to describe spherical symmetric NMC811 materials or a set of Cahn-Hilliard equations [4] that consistently describe the phase separating nature of graphite anodes in cylindrically symmetric particles.The underlying numerical method is an implicit-multi-scale finite-element-method [3] that allows for a flexible implementation of such models. The first results of this ongoing project will be presented along with the overall structure of the method and its implementation. The results include geometrical as well as electro-chemical parameter variations and their respective sensitivity analysis.Furthermore, in the discussed electrode geometry the possible anisotropic structure of an electrode (due to particle shape and distribution) has a bigger impact than in unstructured electrodes. The improved transport pathways along the channels, therefore, imply the necessity of a more thorough homogenisation than it is usually done, for example in a Newman-Model approach. A long-term goal of this work is to enable a significant increase in areal energy density, i.e., the use of thicker electrode films and the use of advanced high energy materials in battery electrodes.[1]3D silicon/graphite composite electrodes for high-energy lithium-ion batteries, W. Pfleging et.al., Electrochimica Acta, Volume 317, 2019, Pages 502-508, J Power Sources 145 (5), 2345-2356[2]Recent progress in laser texturing of battery materials: a review of tuning electrochemical performances, related material development, and prospects for large-scale manufacturing,W. Pfleging,International Journal of Extreme Manufacturing, Vol 3, 2020[3]Derivation of a multi-scale battery model and its high-performance computing implementation, F. Pichler, Doctoral Thesis, Graz, 2018[4]Phase Transformation Dynamics in Porous Battery Electrodes, R. Ferguson, M. Z. Bazant, Electrochimica Acta, Volume 146, Pages 89-97, 2014 Figure 1

Nonthermal processes in hot accretion flows onto supermassive black holes: An inhomogeneous model

Context.Many low-redshift active galactic nuclei harbor a supermassive black hole accreting matter at low or medium rates. At such rates, the accretion flow usually consists of a cold optically thick disk, plus a hot, low density, collisionless corona. In the latter component, charged particles can be accelerated to high energies by various mechanisms.Aims.We aim to investigate, in detail, nonthermal processes in hot accretion flows onto supermassive black holes, covering a wide range of accretion rates and luminosities.Methods.We developed a model consisting of a thin Shakura-Sunyaev disk plus an inner hot accretion flow or corona, modeled as a radiatively inefficient accretion flow, where nonthermal processes take place. We solved the transport equations for relativistic particles and estimated the spectral energy distributions resulting from nonthermal interactions between the various particle species and the fields in the source.Results.We covered a variety of scenarios, from low accretion rates up to 10% of the Eddington limit, and identified the relevant cooling mechanisms in each case. The presence of hadrons in the hot flow is decisive for the spectral shape, giving rise to secondary particles and gamma-ray cascades. We applied our model to the source IC 4329A, confirming earlier results which showed evidence of nonthermal particles in the corona.

Radial Profiles of Non-thermal Emission from Supernova Remnant RX J1713.7-3946

Abstract Supernova remnant RX J1713.7-3946 has exhibited the largest surface brightness and a detailed spectral and shell-type morphology, and is one of the brightest TeV sources. The recent H.E.S.S. observation of RX J1713.7-3946 revealed a broken power-law GeV–TeV gamma-ray and a more extended gamma-ray spatial radial profile than in the X-ray band. Based on the diffusion shock acceleration model, we solve spherically symmetric hydrodynamic equations and particle transport equations, and investigate the multi-band non-thermal emission of RX J1713.7-3946 and radial profiles of its surface brightness for two selected zones in the leptonic scenario for gamma-ray emission. We found (1) the diffusion coefficient has a weak energy dependence, and the Kolmogorov type is favored; (2) the magnetic field strength can vary linearly or nonlinearly with radius for different surrounding environments because of possible turbulence in the shock downstream region, and compressional amplification is likely to exist at the shock front; (3) the non-thermal photons from radio to X-ray bands are dominated by synchrotron emission from relativistic electrons if the GeV–TeV gamma-rays are produced by inverse Compton scattering from these electrons interacting with the background photons; then the X-ray and gamma-ray radial profiles can be reproduced except for the more extended gamma-ray emission.

Sand production by hydraulic erosion during multicycle steam stimulation: An analytical study

The production of sand particles associated with the reservoir hydrocarbons becomes one of the most common problems a well of multi cycle steam stimulation may experience during reservoir lifetime. Therefore, it is essential to the investigation of sand particles migration and sand production mechanisms coupled with fluid flow for optimization design of injection and profile control parameters. This paper presents a model based on hydraulic erosion with heat transfer that can analysis release of sand fines under high temperature. The model describes the continuous hydraulic erosion of rock core by the mass conservation equation and particle transport equation, moreover, the sand production and heat transfer are connected through it is coupled with steam injection model. Then, a coupled nonlinear governing equation is derived by using porosity and liquefied sand concentration to evaluate pore enlargement behavior of flow channel and predict sand production behavior under different steam breakthrough channel. Finally, in terms of numerical processing for model, an effectively difference solution method is adopted based on PDE module in COMSOL Multiphysics. The simulation results show that the sand production model can effectively couple heat and sand production, and the relationship between sand production and channeling flow can be studied through seepage process. The degree of steam channeling affects sand production, that is, sand production increases with the increase of the number of steam breakout channels. In addition, it reveals why the sand production can be effectively slowed down by adjusting reasonable injection parameters for multicycle steam stimulation.

Turbulent kinetic energy in 2D isothermal interchange-dominated scrape-off layer E × B drift turbulence: Governing equation and relation to particle transport

This paper studies the turbulent kinetic energy (k⊥) in 2D isothermal electrostatic interchange-dominated E × B drift turbulence in the scrape-off layer and its relation to particle transport. An evolution equation for the former is analytically derived from the underlying turbulence equations. Evaluating this equation shows that the dominant source for the turbulent kinetic energy is due to interchange drive, while the parallel current loss to the sheath constitutes the main sink. Perpendicular transport of the turbulent kinetic energy seems to play a minor role in the balance equation. Reynolds stress energy transfer also seems to be negligible, presumably because no significant shear flow develops under the given assumptions of isothermal sheath-limited conditions in the open field line region. The interchange source of the turbulence is analytically related to the average turbulent E × B energy flux, while a regression analysis of TOKAM2D data suggests a model that is linear in the turbulent kinetic energy for the sheath loss. A similar regression analysis yields a diffusive model for the average radial particle flux, in which the anomalous diffusion coefficient scales with the square root of the turbulent kinetic energy. Combining these three components, a closed set of equations for the mean-field particle transport is obtained, in which the source of the turbulence depends on mean flow gradients and k⊥ through the particle flux, while the turbulence is saturated by parallel losses to the sheath. Implementation of this new model in a 1D mean-field code shows good agreement with the original TOKAM2D data over a range of model parameters.

A new periodic cell model of aerosol diffusion deposition in a fibrous filter

A new periodic cell model for the convective-diffusive transport of small aerosol particles in a fibrous filter is developed. The proposed periodic model takes into account the influence of the fiber diffusion wake on the particle deposition. This allows us to predict the efficiency of a filter more accurately compared to the single-fiber approximations. The fluid flow is described in the framework of Stokes flow in a rectangular periodic cell. New form of periodic boundary conditions for the particle transport equation in the cell is suggested. The equations of fluid flow and particle transport are solved using the Boundary Element Method (BEM) and Boundary Domain Integral Method (BDIM).The diffusive deposition of nanoaerosols in the filter, modeled as a row of the large number of circular cylinders located perpendicular to the flow direction, is studied numerically using the many fibers model. It is shown, that predictions of the new periodic model are in the better agreement with the many fibers model for the filters of greater length. It is also shown, that the widely used approximation for single fiber deposition efficiency (Stechkina, 1967) overestimates the efficiency of multi-fiber filter compared to periodic model. The overestimation increases with increase of Peclet number.The approximate formula for the single fiber particle deposition efficiency in a periodic cell for a square and staggered cylinder arrays is derived. The formula accounts for the effect of the Peclet number and the hydrodynamic factor. The predictions of the derived formula are compared to the numerical data from the work of Hosseini and Tafreshi (2010) for the fibrous filter comprised of many randomly arranged cylinders. The predictions of the diffusive deposition efficiency obtained using the derived formula for the staggered fibers array are shown to be in a good agreement with the numerical model predictions.

On fractional approximations of the Fokker–Planck equation for energetic particle transport

To determine the main effects of anomalous diffusion upon the propagation of energetic particles, the fractional modified telegraph equation is solved using the Laplace–Fourier technique and given in terms of the Fox H and M-Wright functions. Examples are used to illustrate the qualitative differences between the fractional telegraph, fractional advection–diffusion and fractional diffusion equations. The profiles of the particles densities are discussed in each case for different values of fractional orders.