Numerical models in simulating wire-plate electrostatic precipitators: A review

Journal of Electrostatics 71 (2013) 673e680

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Review

Numerical models in simulating wire-plate electrostatic precipitators: A review K. Adamiak Department of Electrical Engineering, University of Western Ontario, London, Ontario, Canada

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 November 2012 Accepted 3 March 2013 Available online 19 March 2013

This paper attempts to review the most important works on numerical simulation of processes in electrostatic precipitators published so far. Only the wire-plate configuration is considered, although the discharge electrode may have different geometries: smooth cylinder, barbed wire of different shape or helical electrode. Different mathematical models and numerical algorithms for gas flow, electric field, corona discharge and particle transport have been compared. The discussion is focused on coupling between different phenomena. A continuous progress has been shown from early works published about 30 years ago, which dealt with much idealized models of the problem, to recent publications, where the numerical predictions show close agreement with the experimental data. Ó 2013 Elsevier B.V. All rights reserved.

Keywords: Electrostatic precipitation Electrohydrodynamics Numerical simulation Particle transport and deposition Ionized electric fields Turbulent flows

1. Introduction It is difficult to believe that the first idea related to electrostatic precipitation (ESP) was discovered more than 400 years ago e in early 1600’s Gilbert has noticed smoke attraction to electrified bodies [1]. The first electrohydrodynamic experiment, demonstrating the so-called ionic wind, was performed 303 years ago by Hauksbee [2]. At that time, this was just a curiosity and for long time nobody even believed it will ever have any practical significance. The principle of particle removal by their electrical charging and exposing to an external electric field was demonstrated 189 years ago [3] and the first commercial invention was patented in 1884 [1]. In 1925 Deutch connected both ideas and showed the influence of ionic wind on electrostatic precipitation [4]. Considering this long history, it is surprising that ESP still attracts interests of engineers and researchers. The fact is that it is relatively easy to collect large particles, for example, larger than 1 mm, if they are not too resistive. The transport of such particles is dominated by electric forces, their collection efficiency is very high and details of the flow pattern and ionic wind are not really so critical. Collection of very small, submicron, particles is much more difficult and many of them usually escape from the conventional devices. The new environmental regulations pay more and more attention to these particles, as they can be potentially hazardous for

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human health. Therefore, many different research teams are looking for new ways to improve this process. However, designing a new kind of precipitator is not possible without detailed understanding of all phenomena affecting the process. It is well known that there are three basic phenomena taking place in any ESP: gas flow, ionized electric field and particle transport (Fig. 1). All of them are mutually coupled, although some couplings are weaker than others and can be neglected. It is commonly accepted that while the EHD flow generated due to corona discharge may have a relatively strong influence on the flow pattern, the reverse coupling (charge convection) is much weaker. This is obviously true in the area close to the corona wire: electric field is strong there, so the ion velocity is much higher than the gas velocity and the charge convection can be neglected. However, in the precipitation channel there are also some areas relatively remote from the corona electrode and close to the ground plate, where the electric field is much weaker and the ion drift velocity can be comparable with the gas velocity. Nobody has yet verified if this has any effect on the particle collection. Another coupling, which is usually neglected in the precipitation model, is the particle-gas coupling. Mechanical coupling between particles and gas for small particle concentrations can be neglected without causing substantial errors. However, there is an indirect coupling: particles are charged, so they form a space charge. This charge interacts with the electric field and body force is generated, which affects the flow pattern. Experimental investigation of such complicated systems is very expensive, so there is a natural idea to simulate the process

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K. Adamiak / Journal of Electrostatics 71 (2013) 673e680

EHD flow

Gas flow

Ion convection

Electric field

Coulomb force

Particle-gas coupling

Particle charging

Air drag

Space charge

Particle transport Fig. 1. Mutual coupling between different phenomena in electrostatic precipitation.

numerically. However, due to enormous complexity of the problem, a full numerical model is still impossible and many simplifications are necessary in order to keep computational costs at a reasonable level. Different authors have used various idealizations of the process trying to simulate the process using available computing equipment in reasonable time. An impressive progress in the speed and capacity of the computing equipment can be observed during the last 20 years or so. In this situation, there is also a dramatic improvement in the size of considered models and accuracy of calculations. The paper attempts to review the most important works on numerical simulation of ESP from earlier works in this area until the most recent publications. 2. Models of ESP Only single-stage wire-plate configurations will be considered in this review. This is by far the most common configuration used in commercial applications, even though there is a large number of others [5]. This includes variety of different geometries [6e8], twostage precipitation [9], wet processes [1] or electrostatic scrubbing [10]. The new ideas developed recently try, for example, to use dielectric barrier discharge [11] or integrate particle collection with removal of gaseous pollutants [12]. The most important papers published in this area and their short characteristics are presented in Table 1. 2.1. Geometry A few authors attempted to simulate the entire multichannel structure of ESP [23,28,51,52], but in this case a detailed discussion of processes in one channel is not possible. This can be done if only one channel, or one section of the channel, is investigated assuming that inlet velocity of gas is known. Out of two essential elements of the considered ESP, the grounded plate is pretty standard; even though it is not always perfectly flat, the details of its shape are not really critical and can be easily incorporated in any model. The most common geometry for the corona wire is a smooth cylinder. Due to its simplicity for long time it was the only shape considered; in 2D models this is practically the only choice (for example [13]). Phenomena occurring near the corona wire are the major source of computational difficulties and this forced many authors to consider just one corona electrode [13,14,17,32,41,43,45,46,48], although there were also attempts to assume a larger number: two [15,33],

three [16,20,21,34,36,37,44], five [47] or more [19,30,38,39]. Surprisingly, the case of infinite number of electrodes is even simpler than the single one, because the problem becomes periodic and just one section can be analyzed with symmetry boundary conditions [18,24,27,29,42]. Helical wires have also been considered for some special supplications [53]. Even if a smooth cylinder is used as a corona electrode, application of negative voltage (used in most cases due to a better discharge stability) leads to non-uniform discharge: some number of tufts is created along the wire, which automatically adds third dimension to the model. For an ideally smooth cylinder the distance between tufts varies, controlled for example by the supply voltage. In order to have a regular set of corona injection points, the spiked corona wires can be used. They can be cylindrical [26,31,35,40] or cut from a flat tape [49,50], but more complicated shapes have also been proposed [54]. 2.2. Gas flow Predicting the gas flow in the ESP channel is probably the most computationally intensive component of the simulation. Some earlier works, which attempted to grasp the general characteristics of the problem only, were based on the assumed flow patterns, without solving the flow equations [15,18,24,27]. More complete analysis was possible using the laminar model, which was done first in 2D [13,26,31,43] then in 3D cases [33]. Electric body force was considered in majority of these papers, but sometimes this effect was considered unimportant and neglected [17,24]. It is well-known that even without corona discharge the gas flow in ESP is turbulent. However, more than 20 years ago Atten proved theoretically and experimentally that interactions between gas, ions and particles intensify the turbulence level [55,56]. It is natural that the turbulent flow model was incorporated in many models to simulate ESP; from many different options, the k-3 approach was the most popular [16,17,19e23,25,28e30,32,36e50]. The most accurate approach used so far seemed to be direct NaviereStokes solution suggested by Soldati [34]. However, majority of these models do not include the presence of the corona wire. Often it was assumed that the size of the corona wire is so small, that it distorts the flow pattern only locally, even though this distortion can be interesting and quite complicated (swirling von Karman’s vortices). These effect was investigated in the ESP geometry for the first time by Skodras et al. [39] and then by other authors [41,43,45,46,48e50].

Table 1 Summary of papers on numerical simulation of wire-plate ESP. Authors

Geometry

Yamamoto, Velkoff [13] Single cylinder Yamamoto, Sparks [14] Yamamoto [15]

Single cylinder Two cylinders

Kallio, Stock [16] Soldati et al. [17]

Three cylinders Single cylinder

Fluid flow

Electric field

Corona discharge

Particles

Results

2D, FDM, laminar, wire not included 3D, FDM Predefined flow pattern

2D, FDM, wire not included

2D, FDM, experimental BC, no upwinding 3D, FDM, tufts, experimental BC 2D, FDM, experimental BC, no particle charge First-order FDM Not included

Not included

EHD flow pattern, experimental validation Vortex rings, electrical sneakage Particle concentration, EHD effect on turbulence EHD flow pattern Particle trajectories

3D, FDM 2D, FDM

2D, conformal mapping 2D, FDM, no particle charge

2D, current continuity 2D, FDM, analytical prediction of total current

Choi, Fletcher [20]

Three cylinders

2D, FVM, turbulent

2D, FVM

2D, FVM, Kaptzov’s BC

Choi, Fletcher [21]

Three cylinders

2D, FVM, turbulent

2D, FVM

2D, FVM, Kaptzov’s BC

Soldati [22] Gallimberti [23]

Multiple cylinders Entire structure

2D, FDM FDM, includes particle charges

2D, FDM FDM/FCT, glow, streamer and back corona

2D, FDM 2D, FDM

2D, FDM, experimental BC 2D, FDM, experimental BC

2D, FDM

MoC, experimental BC

Kim et al. [27]

3D, turbulent EHD effects Hybrid 3D/2D, FDM stationary, turbulent Periodic cylinders Predefined, no EHD Multiple cylinders 2D, turbulent, time dependent Multiple barbed2D, laminar, FDM vorticitycylinders stream function Multiple cylinders Uniform velocity

2D, FDM, no particle charge

2D, FDM, experimental BC

Varonos et al. [28]

Entire structure

3D, FVM, turbulent

2D, FDM, no particle charge

MoC Kaptzov’s BC

Schmid et al. [29] Schmid [30]

Multiple cylinders Fifteen cylinders

2D, turbulent 2D, FVM, turbulent

2D, FEM/FVM, particle charge 2D, FEM/FVM Kaptzov’s BC 2D, FEM, no particle charge 2D, FVM, Kaptzov’s BC

Dumitran et al. [31]

Multiple barbedcylinders Single cylinder Two cylinders Three cylinders

2D, FDM, vorticitystream function 3D, FVM, turbulent 3D, FDM, laminar 2D, FDM, turbulent (direct NaviereStokes) 3D, FDM, steady-state

3D, FDM, no particle charge

3D, MoC, experimental BC

3D, FVM, no particle charge 3D, FDM 2D, FDM

FVM Kaptzov’s BC 3D, FDM, tufts, experimental BC 2D, FDM

3D, FDM, particle charge

3D, FDM, experimental BC

3D, turbulent

3D

MoC, Kaptzov’s BC

2D, upwind FVM, experimental BC Lagrangian, dynamic charging, deposition model based on particle energy 2D, FDM, empirical BC Lagrangian, dynamic charging, polydispersed particles 2D, FEM Kaptzov’s BC Lagrangian, dynamic charging (field)

Lu, Huang [24] Soldati [25] Atten et al. [26]

Bӧttner [32] Yamamoto et al. [33] Soldati [34] Fujishima et al. [35] Nikas et al. [36]

Multiple spikedcylinders Three cylinders

Zhang et al. [37]

Three cylinders

2D, FVM, turbulent

2D, FVM

Talaie [38]

Ten cylinders

2D, FDM, turbulent

2D, FDM

Skodras et al. [39]

Twelve cylinders

2D, turbulent, FEM (wires included)

2D, FEM, particle charge

Fujishima et al. [40]

Multiple spiked cylinders

3D, FDM, steady state, turbulent (wires ignored)

3D, FDM, periodic

3D, FDM, experimental BC

Collection efficiency Flow streamlines, turbulent kinetic energy, particle trajectories Particle trajectories and concentration Particle trajectories Drag reduction, EHD flow pattern Flow velocity, particle concentration, collection efficiency

Eulerian, polydispersed particles Lagrangian, polydispersed particles, precharged Not included

Collection efficiency Particle trajectories

Eulerian, poly-dispersed particles, constant charge and mobility Lagrangian, turbulent dispersion, dynamic charging, re-entrainment Not included Lagrangian, dynamic charging, random walk method Lagrangian, dynamic charging (field/diffusion) Lagrangian, field-diffusion charging Not included Eulerian, precharged particles

Collection efficiency

Lagrangian, dynamic charging (field-diffusion) Lagrangian, dynamic charging

Not included

Flow streamlines

Flow distribution, collection pattern Flow streamlines EHD flow pattern, particle concentration Helical particle trajectories, particle charging dynamics Flow streamlines EHD flow pattern Cost-efficiency analysis

K. Adamiak / Journal of Electrostatics 71 (2013) 673e680

2D, FEM Image charges

Khare, Sinha [18] Goo, Lee [19]

2D, FDM, turbulent 3D, turbulent, no coupling with electric field Multiple cylinders 2D, predefined Eight cylinders 2D, FDM, turbulent

Not included Eulerian, turbulent diffusion, upwind FDM Not included Lagrangian, negligible electric charge Eulerlian, polydispersed Lagrangian-Monte Carlo, dynamic charging (field/diffusion), monodispersed particles Lagrangian, dynamic (field) charging, monodispersed particles Lagrangian, turbulent diffusion, polydispersed particles Not included Polydispersed particles, dynamic charging (field/diffusion)

EHD flow pattern, particle trajectories EHD flow pattern, collection efficiency Particle trajectories

Collection efficiency Gas velocity and streamlines, particle trajectories and 7concentration EHD flow pattern

(continued on next page) 675

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Flow velocity, particle trajectories, collection efficiency Lagrangian, dynamic charging, (field/diffusion), polydispersed particles 3D, FEM/FCT, experimental BC 3D, FEM, particles included

EHD flow pattern 3D, FEM/FCT, experimental BC

Farnoosh et al. [50]

Flat spiked tape

3D, turbulent, FVM, wire included 3D turbulent, FVM, wire included Farnoosh et al. [49]

Flat spiked tape

3D, FEM, particles included

3D, FEM/FCT, Kaptzov’s BC 3D, turbulent, FVM, wire included Farnoosh et al. [48]

Single cylinder

3D, FEM, particles included

2D, FVM, Kaptzov’s BC 2D, FVM, particles included 2D, turbulent, FVM Five cylinders

Single cylinder Farnoosh et al. [46]

Single cylinder Adamiak, Atten [45]

Three cylinders Lei et al. [44]

Single cylinder Zhao et al. [43]

Periodic cylinders Liang, Lin Ref. [42]

Neimarlija et al. [47]

3D, FCT/FEM 3D, FVM, particles included

2D, FEM, (particles included)

2D, FEM/MoC, Kaptzov’s BC

EHD flow pattern, turbulence intensity, particle trajectories Effect of particle charge on EHD flow and trajectories Flow pattern, particle trajectories, collection efficiency Flow velocity, particle trajectories, effect on turbulence, collection efficiency Flow velocity, particle trajectories, collection efficiency

Lagrangian, dynamic charging, monodispersed particles Lagrangian, dynamic charging, (field/diffusion) Lagrangian, dynamic charging, (field/diffusion) Lagrangian, dynamic charging, (field/diffusion), polydispersed particles Lagrangian, dynamic charging (field/diffusion), polydispersed particles Not included 3D, FDM

3D, FDM, experimental BC

EHD flow pattern Not included 2D, FEM/MoC 2D, FEM, no particles

Flow streamlines 2D, FDM, experimental BC

Not included

EHD flow pattern

2D, FDM

Particles

Not included 2D, analytical

Corona discharge

2D, turbulent (wire included) 2D, turbulent, wires neglected 2D, laminar (wire included) 3D, FDM, turbulent, wires neglected 2D, turbulent, FVM (wire included) 3D, turbulent, FVM Single cylinder

Electric field Fluid flow Geometry

Chun et al. [41]

Authors

Table 1 (continued )

2D, analytical

Results

2.3. Electric field Calculation of the electric field is a relatively minor component of the ESP simulation. Poisson equation, governing the scalar electric potential distribution, has been studied in numerous papers and can be solved with good accuracy for practically any geometry. Despite of this, some simplifications were still adopted in different models in order to reduce required computer memory and computing time: thin wires were neglected and replaced with a point electrode [13] or the method of images used to avoid solving the partial differential equations equation [17]. The space charge formed by precipitated particles was often neglected [18,19,27,28,30e32], but considered in more complete models [23,29,35,39,43,45e50]. 2.4. Corona discharge Corona discharge is responsible for creating the ionic space charge, which is essential for ESP operation, as it provides the mechanism for the particle charging. On the other side this is a very complicated phenomenon and still challenging even for the simplest electrode configurations and without considering other effects. In almost all discussed papers a much simplified corona model was analyzed, which can be called a single-species stationary one. In this model, the ionization layer was completely neglected and steady-state flow of just one ionic species was simulated. This model is valid for both negative and positive polarity of applied voltage and arbitrary gas conditions (composition, temperature, pressure), if the mobility of a dominant ionic species is known. The only exception was the work of Gallimberti [23], who tried to incorporate not only different models for the main discharge (glow, streamer), but also the back corona from the layer of deposited particles. The main problem with such approach is formulation of the boundary conditions, which would specify the amount of space charge injected to the gas zone. The most natural approach is to resort to experimentally measured corona current, which also eliminates many secondary factors affecting the simulation accuracy. A simple iterative process in the numerical algorithm could efficiently determine needed space charge density on the surface of the corona wire [13e15,24e27,31,33,35,37,38,40,42,44]. An analytical prediction of the total corona current was also attempted [19]. A more elegant solution is provided by Kaptzov’s hypothesis that the electric field on the surface of corona electrode remains constant at a value given by Peek’s formula [20,21,28e 30,32,36,39,43,45e48]. Even though semi-empirical Peek’s formula is known only for highly symmetric configurations of electrodes (point-sphere, wire-cylinder), its extension to any configurations of electrodes is pretty straightforward, if the corona electrode radius is replaced with the sum of principal radii of curvature [57]. Unfortunately, this approach is not possible for more complicated electrode configurations (spiked electrodes), where the local electric field cannot be determined with sufficient accuracy [49,50]. 2.5. Particulate transport Particle trajectories and deposition were for long time considered as the simplest element in ESP modeling, so the simulation concentrated on electrohydrodynamic flow and particles were neglected [13,14,16,22,26,29,33,40e43,49]. When particles are included the first basic decision is which one of the two basic formulations should be used: Eulerian or Lagrangian. The former yields the particle concentration by solving just one equation, which can save a lot of computing time [15,18,24,27,34], but doesn’t

K. Adamiak / Journal of Electrostatics 71 (2013) 673e680

allow to include many factors affecting particle motion, for example dynamic particle charging. The latter can include all essential factors into the equation of motion, but requires individual tracing of all trajectories, which can be very time consuming [17,19e 21,25,28,30e32,35e39,44e48,50]. The particle trajectories in the Lagrangian model are traced by numerical integration of Newton’s equation with two major forces: air drag and electric ones. The first one depends on the velocity difference between the particle and airflow velocities. The turbulent models for the gas flow are usually stationary with statistical information about velocity fluctuations. When particle trajectories are traced in time domain there is no detailed information about instant and spatial distribution of the gas velocity, so some methods for turbulent particle dispersion must be implemented [21,28,30]. In practical situations the particles are practically neutral when they are injected to the precipitation channel e some residual charging created by tribocharging is usually negligible, but become charged in the ionized electric field. The particle charge depends on the particle properties (size and permittivity) and the magnitude of electric field, with the charging dynamics also affected by the space charge density. In order to avoid complications with simulating this process numerically pre-charged particles were sometimes assumed [25,27,34]. However, more meaningful results must calculate the dynamics of particle charging; a few slightly different models for the combined field and diffusion charging have been presented in Refs. [19,20,23,28,30e32,35e39,44e48,50]. Conditions for the particle deposition have not been studied in details and it was commonly accepted that if a particle touches the collecting plate it is deposited, as the electric force can reach a strong value due to the induced image charge. Only in Ref. [37] a more general model was analyzed, where the kinetic energy of a particle was compared with the local surface energy. Particle deposition is not always final and they can be eventually reentrained to the gas stream due to local gas turbulence or back corona discharge. This effect cannot be simulated without additional assumptions about the nature of the process, this is why it was discussed rarely [28]. Finally, the precipitated particles are usually characterized by variety of different shapes and sizes. In practically all works spherical particles were assumed. While in some papers a monodispersed particles were analyzed [15,17,46], only the general behavior of differently sized particles can be predicted. As it was experimentally proved by Podlinski et al. [58,79] the space charge of charged particles can significantly contribute to the EHD flow pattern and particle transport. Separate treatment of different particles would not be able to capture this effect, so poly-dispersed particles should lead to much accurate predictions. This approach was used in Refs. [18,21,23e25,27,38,47,48,50]. During the last decade the emphasis in the ESP research has been shifted from the conventional devices for relatively large particles to sub-micron particles. The new configurations of ESP with improved collection efficiency of such particles were investigated in Refs. [48e50]. 2.6. Numerical techniques The oldest numerical technique for the partial differential equations is based on the finite difference approximation (FDM). Because it is simple and can be easily expanded for practically any equation, it was very popular in earlier works on ESP [13e17,19,22e 28,31,33e35,38,40,42,44]. An important disadvantage of this technique is necessity to work with a rectangular discretization, which make it practically impossible to consider exact shapes of all electrodes. The Finite Volume (FVM) [20,21,29,32,37,44,47] and the Finite Element (FEM) [16,39,43] methods are much more

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complicated, but allow much more flexible discretization. In some cases, the last two techniques were combined, with the fluid part solved in the FLUENT commercial software base on FVM and other parts coded as the User-Defined-Functions based on FVM [45,46,48e50]. While the numerical techniques for the gas flow and electric field are relatively well-known, the weakest part of the simulation algorithm is usually the equation for the charge transport. Due to very small physical diffusion of ions, this equation is dominated by the first derivative and a conventional treatment in any numerical technique may introduce an artificial numerical diffusion. Because of difficulties associated with numerical simulation of the corona discharge most of the earlier papers were focused on electrical conditions in the precipitation channel only. Many techniques were suggested for the charge transport and its coupling to the electric field. Experimental data of Penney and Matick [59] were used in most cases for validation of the numerical algorithm. The most important works are summarized in Table 2. An elegant way to predict the ionic charge transport is the Method of Characteristics (MoC) [26,28,31,36,43,45], which is completely immune to numerical diffusion, but it requires two different meshes: one for the electric field and another for space charge density. This not only increases the memory requirements, but also the algorithm complexity, as field and charge distributions must be interpolated from one mesh to another. This is possible in 2D systems, but much more difficult in 3D ones [69]. 3D problems can be solved using some kind of upwinding techniques [37,77] or the Flux Corrected Transport (FCT) algorithm; its implementation with FDM is relatively simple [23], but much more complicated for FEM [46,48e50]. In all cases, it is still impossible to completely avoid some residual diffusion and irregularities in the space charge density. 3. Results 3.1. Flow patterns From early works of Yamamoto and others it was evident that for smooth cylindrical corona wires the secondary EHD flow forms arrays of large-scale, span-wise, counter-rotating circulatory rolls [13,16,22]. They were clearly observed experimentally and confirmed numerically. At zero primary flow velocity four symmetrical rolls were observed and no mean flow was generated. With an increased flow velocity the secondary rolls are shifted: one pair downstream and one toward the collecting plates. At sufficiently high velocity of the primary flow all rolls practically disappear [35,47]. The number of rolls increases with the number of the discharge electrodes [13,29,36]. An interesting insight into the structure of secondary EHD flows can be provided by plotting the streamlines of velocity difference for the cases with and without electric volume forces [29]. Positive corona discharge is usually uniform along the discharge wire and as a consequence the flow pattern is 2D. When negative discharge is used, tufts are usually formed and more complicated 3D flow patterns are observed [14,33]. Each individual tuft creates a vortex ring elongated in the direction of the primary flow. Tufts are not very stable, so these rings are very erratic. With a small distance between tufts the flow can change its pattern to a 2D one [33]. The corona discharge generates wakes behind wires. The EHD flow modifies the main airflow and makes it so complex that at least eight different types of flow patterns can be identified [43]. This also contributes to the increased turbulence intensity [39,47]. The spiked electrodes generate much more complicated flow patterns. The electric body force has very strong values close to the electrode spikes, so the gas is accelerated from the spikes toward

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Table 2 Summary of papers on numerical simulation of corona discharge in ESP. Authors

Geometry

Dimensions

Electric field

Corona discharge

Boundary conditions

Sekar, Stomberg [60] Davies, Hoburg [61] Kallio, Stock [62] Elmoursi, Castle [63] Butler et al. [64] Levin Ref. [65] Cristina et al. [66] Adamiak [67] Lami et al. [68] Davidson et al. [69] Bucella [70] Lami et al. [71] Medlin et al. [72] Talaie et al. [73] Rajanikanth, Sarma [74] Al-Hamouz [75] Anagnostopoulos, Bergeles [76] Long et al. [77] Nouri, Zebboudj A [78]

Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Point, barb-plate Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders Periodic cylinders

2D 2D 2D 2D 2D 2D 2D 2D 2D 3D 2D 2D 2D 2D 2D 2D 2D 3D 2D

Conformal mapping FEM FEM CSM FEM FEM FEM BEM FDM FEM FDTD, pulse energization FTDT FVM FDM Variational FEM FDM Cell-centered FVM FEM

Conformal mapping MoC FDM Current continuity MoC Donor-cell FEM MoC Unipolar drift formula MoC FTDT FTDT FVM, pseudo-transient FDM FDM Current continuity FDM Upwind cell-centered FVM CSM

Experimental current Peek’s formula Experimental current Peek’s formula Peek’s formula Peek’s formula Peek’s formula Peek’s formula Peek’s formula Experimental current Peek’s formula Experimental current Peek’s formula Empirical equation Peek’s formula Peek’s formula Peek’s formula Experimental current Space charge on the surface of ionization layer

the ground plane and then bounced back toward the wires [35]. Interaction between primary and secondary EHD flows can generate rather complicated structures: depending on the geometry and discharge intensity it can change from a regular 3D spiral pattern to a well mixed flow [40]. The particle re-entrainment can be minimized by optimization of the velocity profile in the collection chamber. This can be done by proper shaping of the ESP structure [28]. Bӧttner has noticed that the turbulence created by corona discharge is the major source of error in the numerical simulation of the EHD flow: while the agreement between numerical and experimental values of the mean velocity is satisfactory, the differences in velocity fluctuations are much larger [32]. 3.2. Particle trajectories The effect of EHD flow on the particle transport was discussed by many authors, but no definite conclusion has been reached so far [81]. The motion of all particles is affected by the gas flow with the fine particles practically following the gas streamlines. In the configurations with tuft corona discharge, the particles tend to move between tufts and sneak through the charging zone, which results in the reduced charging level [14]. Soldati performed the Lagrangian simulation of the particle transport, assuming some number of precharged particles distributed in the precipitation channel. Comparing two cases: one with electric forces affecting the particle trajectories and flow and the second with electric forces affecting the particles only, he concluded that the EHD flow has a negligible influence on the particle collection efficiency for particles with diameters varying from 4 to 32 mm [25]. The electric force rapidly decreases when the particles move from the corona wire toward the collecting plates. Therefore the particles can be re-entrained by the turbulent diffusion mechanism of the flow [25]. Smaller particles are much more sensitive to this phenomenon. In multi-wire ESPs, the particle collection is the most intensive in the first section. As it has been predicted numerically and validated experimentally, all largest particles are practically collected there, but submicron particles can be also collected in subsequent sections [28]. Particles in the precipitation channel follow different trajectories: they are injected at different points, so they are charged to different levels. The electrical force deflects them to areas with

different gas velocity values, which affects the drag force, and electric field magnitudes, which affects continuing charging. The statistical study of the charging level for submicron particles of different diameter (between 0.3 mm and 1.5 mm) was performed in Ref. [31]. It has been discovered that the location of the injection point is a major factor affecting the deposition efficiency. Increasing the particle concentration significantly affects the flow pattern, which becomes more and more non-uniform and exhibits stronger and stronger agitation. This also increases the space charge due to particles resulting in a reduction of the total corona current [45,47,79]. The results of numerical simulation of this problems presented in Ref. [45] qualitatively agree with the experimental data [82]. The collection efficiency of submicron particles in a spike electrode-plate laboratory-scale ESP for three geometries of discharge electrodes (two-sided spiked electrode and one-side spiked electrode with the tips directed upstream or downstream of the airflow) was investigated in Refs. [49,50]. It was shown that the ESP with spikes on two sides is the best discharge electrode design for collecting particles in the range of 0.25e1.5 mm, assuming the same voltage applied to the corona electrode. The ESP with spikes in the upstream direction showed slightly better collection efficiency for 0.25 mm particles, especially for lower excitation voltages. The numerical results were in fairly good agreement with the experimental data published in Ref. [80]. 4. Conclusions The most important works on the numerical simulation of wireplate electrostatic precipitators have been reviewed in this paper. A continuous progress in this can be observed, which matches improvements in the computational power of modern computers. While at the earlier stage the considered models were very simplistic, 2D and roughly discretized, the most recent are usually 3D, use very fine mesh and included all essential phenomena affecting the process. This is why the results of the numerical simulation are much more accurate and agree much closely with the experimental data. Acknowledgment This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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