# Differentially weighted direct simulation Monte Carlo method for particle collision in gas–solid flows

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<ul><li><p>P</p><p>Dp</p><p>YS</p><p>a</p><p>ARRA</p><p>KDDGPF</p><p>I</p><p>seTindaHsw</p><p>PP</p><p>h1</p><p>ARTICLE IN PRESSG ModelARTIC-730; No. of Pages 11Particuology xxx (2014) xxxxxx</p><p>Contents lists available at ScienceDirect</p><p>Particuology</p><p>jo ur nal home page: www.elsev ier .com/ locate /par t ic</p><p>ifferentially weighted direct simulation Monte Carlo method forarticle collision in gassolid flows</p><p>ongxiang He, Haibo Zhao , Haoming Wang, Chuguang Zhengtate Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, China</p><p> r t i c l e i n f o</p><p>rticle history:eceived 13 December 2013eceived in revised form 28 April 2014ccepted 12 May 2014</p><p>eywords:irect simulation Monte Carloifferentially weighted methodassolid flowarticleparticle collisionour-way coupling</p><p>a b s t r a c t</p><p>In gassolid flows, particleparticle interaction (typical, particle collision) is highly significant, despitethe small particles fractional volume. Widely distributed polydisperse particle population is a typicalcharacteristic during dynamic evolution of particles (e.g., agglomeration and fragmentation) in spite oftheir initial monodisperse particle distribution. The conventional direct simulation Monte Carlo (DSMC)method for particle collision tracks equally weighted simulation particles, which results in high statis-tical noise for particle fields if there are insufficient simulation particles in less-populated regions. Inthis study, a new differentially weighted DSMC (DW-DSMC) method for collisions of particles with dif-ferent number weight is proposed within the framework of the general EulerianLagrangian models forhydrodynamics. Three schemes (mass, momentum and energy conservation) were developed to restorethe numbers of simulation particle while keeping total mass, momentum or energy of the whole systemunchanged respectively. A limiting case of high-inertia particle flow was numerically simulated to validatethe DW-DSMC method in terms of computational precision and efficiency. The momentum conservation</p><p>scheme which leads to little fluctuation around the mass and energy of the whole system performedbest. Improved resolution in particle fields and dynamic behavior could be attained simultaneously usingDW-DSMC, compared with the equally weighted DSMC. Meanwhile, computational cost can be largelyreduced in contrast with direct numerical simulation.</p><p> 2014 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by Elsevier B.V. All rights reserved.</p><p>svpobtdwts</p><p>ntroduction</p><p>Gassolid flows are frequently found in industrial processesuch as fluidized beds, pneumatic conveying, boilers and furnaces,lectrostatic precipitators, and particle separation in cyclones.hese systems often involve complicated flow dynamics andnteraction between flow constituents and their surroundings. Aumber of numerical studies of gassolid flow have been con-ucted focusing on one-way coupling (Deutsch & Simonin, 1991)nd two-way coupling (Squires & Eaton, 1990; Wang, Zhao, Guo,Please cite this article in press as: He, Y., et al. Differentially weightegassolid flows. Particuology (2014), http://dx.doi.org/10.1016/j.partic</p><p>e, & Zheng, 2013). One-way coupling that the influence of theolid particle on the continuous phase is neglected, is reasonablehen the particle fractional volume v and mass loading m are</p><p> This paper is adapted from the presentation at the 4th UK-China Internationalarticle Technology Forum, October 1519, Shanghai, China, as recommended byrof. Xiaoshu Cai and Dr. Jerry Heng, the co-chairs of the scientific committee. Corresponding author. Tel.: +86 27 87545526; fax: +86 27 87545526.</p><p>E-mail addresses: hzhao@mail.hust.edu.cn, klinsmannzhb@163.com (H. Zhao).</p><p>pwpttaWuOc</p><p>ttp://dx.doi.org/10.1016/j.partic.2014.05.013674-2001/ 2014 Chinese Society of Particuology and Institute of Process Engineering, Cmall (e.g., v < 106). However, with an increase in fractionalolume v (e.g., 106 < v < 103), the effect of the continuoushase on the dynamics of the dispersed phase and the feedbackf the dispersed phase on the continuous phase dynamics shoulde considered simultaneously. The conservation equations of con-inuous phase include appropriate source terms resulting from theispersed phase. The interaction between two phases is called two-ay coupling. While the volume fraction of particles v continues</p><p>o increase (e.g., v > 103), flows are referred as dense suspen-ion. The particleparticle collision plays an important role on therofiles of continuous phase and dispersed phase, the term four-ay coupling effect emerges. In fact, it is essential to considerarticleparticle collision even though the fractional volume ofhe particles v is small (e.g., v < 103), because the turbulentransport effect and the preferential concentration effect lead ton increase in inter-particle collision rates by a factor of 30 (Wang,d direct simulation Monte Carlo method for particle collision in.2014.05.013</p><p>exler, & Zhou, 2000). Indeed, the average particle fractional vol-me is not the only measure of the importance of particle collision.esterle and Petitjean (1993) performed a developed horizontalhannel flow calculation in a non-dilute gassolid suspension flow.</p><p>hinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.</p><p>dx.doi.org/10.1016/j.partic.2014.05.013dx.doi.org/10.1016/j.partic.2014.05.013http://www.sciencedirect.com/science/journal/16742001http://www.elsevier.com/locate/particmailto:hzhao@mail.hust.edu.cnmailto:klinsmannzhb@163.comdx.doi.org/10.1016/j.partic.2014.05.013</p></li><li><p>ARTICLE IN PRESSG ModelPARTIC-730; No. of Pages 112 Y. He et al. / Particuology x</p><p>Nomenclature</p><p>c particle velocity, m/sd diameter, mEts initial total energy of the whole systemFother external and inter-particle forcesg gravity acceleration, m/s2</p><p>l grid length, mMts initial total mass of the whole systemN total particle numbersNc average collision rate per unit volume, m3</p><p>Nf total simulation particlesP the probability of one particle interacting with any</p><p>other particlePts initial total momentum of the whole systemS source termt time scale, st time step, sTe the integral Lagrange time scale, su fluid velocity, m/s; velocityVs volume of the whole system, m3</p><p>w number weight</p><p>Greek letters collision cross-section, m3/s turbulent kinetic energy effective transport tensor turbulent kinetic energy dissipation rate, m2/s3</p><p> relative error time-accumulated relative error fluid density, kg/m3; density, kg/m3</p><p>p relaxation time scale of particle, sk Kolmogorov time scale, sv particle volume fraction, m3</p><p>Subscriptsa analytical valuee discarded particlei, j, k index of simulation particlek coordinate directionnew condition after dynamic eventold condition before dynamic eventp particle phaser property between two simulation particlest detected valuex, y, z coordinate directions1 indication of one of the decomposed particles2 indication of the other decomposed particle the general transport variable</p><p>Superscriptsmax maximum valuemin minimum value ensemble average value* particle velocity after collision event post-collision and pre-restoration; a new coordi-</p><p>nate system</p><p>Iewl</p><p>ehtwbd</p><p>pEidscmtpplauppp</p><p>tastmsmtsoCdlirfoloco1niefcsTgbawdpcr</p><p> post-collision and post-restorationPlease cite this article in press as: He, Y., et al. Differentially weightegassolid flows. Particuology (2014), http://dx.doi.org/10.1016/j.partic</p><p>t was found that particleparticle collision has a strong influ-nce on the profile of the vertical particle concentration comparedith dilute flow simulation, revealing that particleparticle col-</p><p>ision plays a non-negligible role once the particle mass loading</p><p>i</p><p>cC</p><p>xx (2014) xxxxxx</p><p>xceeds unity. In many cases, the particle loading is moderate toigh and hence particleparticle collisions significantly influencehe flow field. Therefore, the particleparticle collision and four-ay coupling between the continuous and discrete phases shoulde processed for the purpose of accurate description and control inispersed systems.Various models have been proposed to consider</p><p>articleparticle collision in the general framework ofulerianLagrangian models, where the continuous flow behaviors studied in Euler coordinates and the motion of dispersed phase isescribed by Lagrangian equation. Wassen and Frank (2001) clas-ify these models in two principally different ways: the trajectoryalculation (TC) method and simultaneous particle tracking (SPT)ethod. The inherent assumption of the TC method is that each</p><p>rajectory represents a constant flow of particles with identicalhysical properties, and macroscopic properties of the dispersedhase can be acquired by averaging all trajectories that cross theocal numerical cell. As a consequence, this method is limited topplications of steady flow. While the SPT method is inherentlynsteady because each particle represents a certain number of realarticles with identical physical properties, and the macroscopicroperties of the particulate phase are achieved by averaging allarticles in the same cell.To take particleparticle collision into account in the frame of</p><p>he TC method, Oesterle and Petitjean (1993) proposed an iter-tive technique with which particleparticle collision is treatedtochastically on the basis of the local concentration and veloci-ies obtained from the previous iteration. In the frame of the SPTethod, collisions can be computed either deterministically ortochastically. The most straightforward deterministic trajectoryethod, in which all particles have to be tracked simultaneously</p><p>hrough the flow field and the occurrence of particleparticle colli-ion can be judged by particle position and relative motion duringne time step, is based on molecular dynamics. Sundaram andollins (1996) used direct numerical simulation (DNS) method toeal with particleparticle collision (hard-sphere collisions, simi-ar to the present study) to investigate the particle collision rate insotropic turbulent flows. Two different techniques (proactive andetroactive) are used for particle collisions in the DNS method. Theormer (proactive) technique anticipates all the collisions that willccur within one time step and then enacts them in order. While theatter (retroactive) looks for particleparticle overlaps at the end ofne time step, and then resolves these overlaps by enacting particleollisions retroactively in the order in which they occurred until allverlaps have been considered (Chen, Kontomaris, & McLaughlin,999; Sundaram & Collins, 1996). However, the DNS method isot able to handle multiple particle collisions that are of greatmportance for the modeling of quasi-static systems. The distinctlement method (DEM) first proposed by Cundall and Strack (1979)or granular dynamic simulation provides new possibilities for dis-rete particle simulation to calculate the dense phase flows using aoft-sphere collision model to deal with particle collision dynamics.suji, Kawaguchi, and Tanaka (1993) performed a two-dimensionalas-fluidized bed simulation where particle collisions are modeledy DEM and observed the phenomena of the formation of bubblesnd slugs and the process of particle mixing which is consistentith experiment. Note that the inherent computation cost of theeterministic trajectory method while choosing colliding particleairs is proportional to N2, where N is the number of traced parti-les. For this reason, the deterministic calculation is of no practicalelevance in some engineering applications such as high mass load-d direct simulation Monte Carlo method for particle collision in.2014.05.013</p><p>ng impinging streams (Xu, Zhao, & Zheng, 2014).In fact, in the majority of the applications of the SPT method,</p><p>ollisions were treated stochastically. The direct simulation Montearlo (DSMC) method developed by Bird (1976, 1994) is a physically</p><p>dx.doi.org/10.1016/j.partic.2014.05.013</p></li><li><p>ARTICLE IN PRESSG ModelPARTIC-730; No. of Pages 11ology x</p><p>aTiletttctaa(NtaitYgctstmfOtdtsfuoos</p><p>lomwdcsdcisnswtatsa&f</p><p>i(satw</p><p>Tispmt</p><p>M</p><p>S</p><p>voea</p><p>wSoe(fam</p><p>d</p><p>wttpvi</p><p>ftAtadEaputciedt</p><p>T</p><p>Y. He et al. / Particu</p><p>ccurate method for the computation of non-equilibrium gas flows.his technique is most useful in circumstances where there arensufficient numbers of collisions in the flow to maintain the equi-ibrium forms of the distribution functions describing the variousnergy modes of the gas. Several stochastic models were developedo reduce the numerical effort to a linear proportionality of theotal number of simulation particles. Bird (1976, 1989) proposedhe event-driven time-counter method and time-driven no-time-ounter (NTC) method for gas molecular collisions. In the frame ofhe NTC method, the number of collisions in each grid is calculatednd then an acceptancerejection method is used to judge whether</p><p> collision event will occur. In addition, other stochastic modelsIllner & Neunzert, 1987; Ivanov & Rogasinsky, 1988; Koura, 1986;anbu, 1980) for selecting colliding pairs were also developed inhe dilute molecular gas dynamics. These stochastic models takedvantage of the analogy between the motion of discrete particlesn gassolid flows and the motion of molecules in dilute gas flowo consider particleparticle collision in gassolid flows. Tanaka,onemura, and Tsuji (1995) simulated a two-dimensional upwardassolid flow in a rectangular domain with a periodic boundaryondition to study the effect of the particle property on the struc-ure of the particle cluster. Tsuji, Kawaguchi, and Tanaka (1998)imulated the riser of a circulating fluidized bed to investigatehe formation of particle aggregation compared with the two-fluidodel. Furthermore, a stochastic particleparticle collision model</p><p>or particle-laden flow was suggested by Sommerfeld (2001) andesterle and Petitjean (1993), in which a generated simulation par-icle, whose size and velocity are sampled from local probabilityensity functions of particle diameters and velocities, collides withhe selected particle during each time step. The post-velocity of theelected simulation particle is solved by the conservation equationor linear and angular momentum, while the properties of the sim-lation particle generated are of no further interest. The advantagef this model is that it does not require information on the positionf surrounding particles. However, it needs to conserve the particleize and velocity distribution of each grid.</p><p>The studies mentioned above focused on the collision of simu-ation particles with equal number weight. In many applicationsf interest, the traced particles possess only a small portion ofulti-species or polydisperse particle population, and the equallyeighted DSMC faces enormous difficulties or complex particleynamics in terms of either huge computational cost or signifi-ant statistical noise or even both. To circumvent this difficulty, acheme to apply different...</p></li></ul>

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