Toward the Development of an Integrated Multiscale Model for

Article pubs.acs.org/IECR

Toward the Development of an Integrated Multiscale Model for Electrostatic Precipitation Bao-Yu Guo,† Si-Yuan Yang,† Mao Xing,† Ke-Jun Dong,† Ai-Bing Yu,*,† and Jun Guo‡ †

Laboratory for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia ‡ Experimental and Research Centre, Fujian Longking Co., Ltd., Longyan, 364000, China ABSTRACT: This paper reports our attempts in developing an integrated multiscale mathematical model to describe the wireplate-type electrostatic precipitator (ESP), aiming to understand the underlying physics and to develop a computer tool for process design and control. In the model, various phenomena in a wide range of length/time scales related to the electric field, gas-particle flow, dust deposition, cake formation, and their interactions are resolved. We apply different numerical methods for different fields in different local regions, leading to various submodels, including a continuum-based electric field model, Euler−Lagrange gas-particle flow model, and discrete-based cake formation model. These submodels are eventually integrated to form an ESP process model, which can generate results useful for better understanding the phenomena and assessing the ESP performance under different conditions.

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resistivity particles create a negative effect on the electric field, which reduces the collection efficiency.6 Although a large number of studies have been reported in this area in the past decades, most of the theoretical studies focused on the local corona region or specific phenomena.7−10 A full scale ESP prediction tool with all these key phenomena considered together, like in the current paper, is not available. As a consequence, the design and optimization of ESP remain at an empirical level due to insufficient knowledge of the effect of complicated structures and multiscale dynamics in a full scale ESP. To date, in the ESP design, engineers still rely on the traditional Deutsch equation and its derivatives,11,12 which are theoretical formula for particle collection in a local corona region based on many oversimplified assumptions. Therefore, there is an urgent need to develop a comprehensive numerical model to describe the coupling among complicated wall boundaries, turbulence, E-field, particle trajectories, and cake layer, which may be significant at different time and length scales. Following an earlier work on the gas-particle flow modeling without electric field,13 the current paper reports our recent attempts in developing an integrated multiscale mathematical model to describe the so-called wire-plate-type ESP, aiming to understand the underlying physics and to develop a computer tool for process design and control. The needs for further studies on some aspects are also discussed.

INTRODUCTION

The electrostatic precipitator (ESP) is widely used in coal-fired power generation plants for removing fine particulate matter.1 The working principle can be described from the following steps. When a high voltage of, say, 20−100 KV (DC) is applied on a wire electrode, an electric corona forms where air is broken down into ions and electrons. The ions stick to the surfaces of passing particles. The charged particles are then driven to a grounded collection plate by the electric force. The deposited particles form a cake or packed bed. The collection plate is rapped periodically to cause the cake to fall down to the hopper. ESP proves to have a high mass efficiency and the advantages of large handling capacity with low pressure loss. However, major problems arise from the increasing demand for collection efficiency of fine particles (e.g., PM2.5) and further requirement on device optimization. These micrometer or submicrometer size particles are particularly harmful to human health. An industrial ESP, in terms of its geometric structure, typically consists of a diffuser, perforated plates, collection chamber, collector plates, hoppers, electrical system, and so on. As shown in Figure 1, it comprises many internal components of different length scales and involves complicated transport phenomena, including multiphase flow (i.e., turbulent gas-polydispersed particles), and so-called multifield (e.g., potential field, ion charge field apart from the gasparticle flow field) and strong interactions among them. This gives us a very complicated multifield and multiscale problem. As an example, we can examine the phenomena associated with the electric field, described as follows. First the corona is generated at the small needle tip of the emitter electrode, which controls the distribution of the electric field intensity and ion charge in the entire flow passage.2 Then nonuniform electric current induces ionic wind that affects the global turbulence level and particle diffusion and re-entrainment.3,4 Particles, after being electrically charged, are then transported to the collector under the electric force and collide with wall and/or other particles. Finally the deposited particles form a cake on the wall surface with a certain pattern,5 and the accumulated charges carried by the high© 2013 American Chemical Society

1. MODEL CONDITIONS AND MULTISCALE APPROACH A two-stage pilot-scale wire-plate type ESP in Fujian Longking Co Ltd. is chosen as a case for model development (Figure 1). Special Issue: Multiscale Structures and Systems in Process Engineering Received: Revised: Accepted: Published: 11282

December 18, 2012 March 4, 2013 March 5, 2013 March 5, 2013 dx.doi.org/10.1021/ie303466g | Ind. Eng. Chem. Res. 2013, 52, 11282−11293

Industrial & Engineering Chemistry Research

Article

Figure 1. ESP model geometry: (a) photo of a pilot ESP; (b) model geometry showing computational domain and internal components considered; and (c) computational meshes on the wall surfaces for overall ESP (left) and barbed wire (right).

Figure 2. A schematic of mechanical structures and major local transport phenomena simulated in ESP.

ones. Two perforated plates with porosity f = 0.5 and 0.4, respectively, are used to even out the gas flow before entering the first electric field. Ten vertical guide vanes are attached to the

The rectangular chamber (6 m long by 1.25 m wide by 2.2 m high) consists of two stage electric fields in series. There are four hoppers at the bottom with partition plates in the first and the last 11283

dx.doi.org/10.1021/ie303466g | Ind. Eng. Chem. Res. 2013, 52, 11282−11293


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perforated plates on the back side. Several baffles are installed to reduce the bypassing flow (interstage baffle, inserted baffle). There are totally 64 wire electrodes of 8 mm in diameter. The barb on the wire is 10 mm long and 2 mm in diameter with a sharp conical tip. Wire-to-wire distance is 0.24 m, and wireto-plate distance is 0.2 m. Because of geometric symmetry, only one-half of the ESP is considered. Figure 1c shows typical finite element grid patterns on the wall surfaces. The ESP model comprises 492 086 grid points and 927 258 elements of mixed tetrahedra, pyramids, and hexahedra. A separate grid is used to solve the local electric field, where significantly refined grid size is imposed at the barb surface. As noted above, the current ESP has complicated internal mechanical structures of wide range of dimensions. The major internal structural components and some important small scale phenomena in local regions are schematically shown in Figure 2. By means of multiscale numerical simulations, we resolve various phenomena in a wide range of length/time scales, respectively, in the following three aspects: (1) electric field, from an electrode discharge needle of 1 mm, to a single wire-plate cell of length scale of 0.1 m, up to several meters of electric field with hundreds of wire electrodes; (2) for gas flow, eddies of different length scales are resolved, that is, recirculation eddies behind the perforated plate with a single orifice diameter scale of several mm and electro-hydrodynamic (EHD) secondary flow with length scale of 0.2 m-wide collector passage, up to the large scale flow pattern in the ESP unit of several meters; (3) for particles flow, we handle particles as small as 0.1 μm, both in the region of dense cake layer and the wide space of dilute particle system elsewhere in the device. Different numerical methods are applied for different fields in different local regions, leading to various submodels. The submodels are combined to form an integral ESP process model. The methods, governing equations, and modeling results are detailed respectively in the subsequent sections.

where v=

qqe 2πε0d pk 0T

,

w=

κp

Ed pqe

κ p + 2 2k 0T

,

τq =

bρion t ε0 (6)

⎧(w + 0.475)−0.575 , w ≥ 0.525 f (w) = ⎨ w < 0.525 ⎩1, ⎪

⎪

(7)

The specific multiscale approach used here for the electric field is explained as follows. The solution of electric field is started with a small scale corona-emitting barb where the boundary condition for ion density is determined. Then the distribution of local E-field and ion density in a single wire-plate unit is solved. The distribution is assumed to be independent of location and can hence be directly used in the large-scale ESP model. In this way, the electric fields at different scales can be inter-related, and the E-field in the whole ESP can be established. The approach is further discussed in the subsections below. 2.2. Local E-Field Simulation. The local E-field can be solved in a representative unit cell with properly defined symmetry planes and periodic boundaries, as detailed by Xing et al.15 Here, two types of wire electrodes are considered: simple smooth cylindrical rod (Type-A) and barbed rod with alternately arranged needles pointing to the plate (Type-B). Assuming static gas flow for simplicity (U = 0), the predicted current density on the plate as a function of the applied voltage (V−I characteristics) is shown in Figure 3, together with the literature values for

2. ELECTRIC FIELD 2.1. Governing Equations. The distributions of the electric potential and space charge density are controlled by a Poisson equation and current continuity equation, respectively, given by ∇2 V = −ρion /ε0

(1)

∇·(ρion (b E + U) − D∇ρion ) = 0

(2)

E = −∇V

(3)

Figure 3. Current density as a function of applied voltage.

Type-A.16 As can be seen, Type-B electrode offers a higher current density and a lower onset voltage, in addition to the known advantage in corona stability. Regarding the field distribution, Figure 4 shows the maximum E-field intensity (or potential gradient) on the plate as a function of voltage, together with the equation given in ref 17. The peak field intensity on the plate may be used to determine the operation point, that is, the spark-over voltage. According to the experiment of Sekar and Stomberg,17 the E-field intensity at the plate just before sparkover is 7.2−7.9 × 105 [V/m]. Therefore, the effective voltage applied should not exceed, approximately, 100 000 [V] for a direct current (DC) or 70 000 [V] for a pulsating DC, if sparks are to be controlled. This estimated voltage is close to the average on-site readings for similar configurations. In the future, the truly dynamic E-field will be simulated for the case of pulsating DC voltage. Meanwhile, the effects of other factors, such as EHD-related ionic wind, temperature, and chemical composition, will be considered. The distribution of electric field for Type-A as shown in Figure 5 is symmetric in both horizontal and vertical directions. It can be

In addition, the Peek equation is used to determine the corona onset field intensity on the wire electrode, ⎛ 0.0308[m0.5] ⎞ Eo = 3.1 × 106[V /m]δ ⎜1 + ⎟ ⎠ ⎝ δr

(4)

The particle charge is calculated in the Lagrangian framework by integrating the charging rate of Lawless,14 ⎧ v − 3w , v > 3w ⎪ f (w) − 3w) − 1 exp( v ⎪ ⎪ 2 ⎪ 3w ⎛ dv v ⎞ ⎟ + f (w), = ⎨ ⎜1 − − 3w ≤ v ≤ 3w dτq 3w ⎠ ⎪ 4 ⎝ ⎪ −v − 3w ⎪−v + f (w) , v < −3w ⎪ exp( −v − 3w) − 1 ⎩ (5) 11284



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Figure 4. Maximum E-intensity as a function of the applied voltage.

seen that, in the plane passing the wires, the E-field intensity is highly nonuniform (the highest on the wire surface and lowest at the middle point of the neighboring wires). On the wire surface, stronger corona is expected at the points exactly facing or closest to the plate. The ion charge density decreases with the distance from the wire and still far from uniform on the plate. Peanutshaped contours in the proximity of the wire are surrounded by a wide range of ovals. The distribution of electric field for Type-B configuration is depicted in Figure 6. It is found that the tiny barb interferes with the E streamlines only in its close region, whereas those on the plate are barely affected. However, the distributed pattern of the ion charge and current density for this case are significantly different from those in Type-A. Since the corona is generated only on the conical tip of the barb, the charge density and current density appear to be discrete with a regular pattern corresponding to the barb−barb spacing. In some regions, for example, near the rod surface, particularly on the opposite side of the barb, ion charges may not exist at all. 2.3. Implementation in ESP Model. The modular modeling procedure by Varonos et al.1 is very useful. The above results on the detailed local electric field (E-intensity and charge density) are used directly as a database in the ESP model,

Figure 6. Electric field for Type-B configuration: (a) streamlines of potential gradient and (b) contours of space charge density with logarithmic scale.

where there are 15 barbs on each of the 32 Type-B electrodes. E-streamlines and iso-surfaces in Figure 7 show a discrete pattern corresponding to that of Type-B wire electrodes. Therefore, the electric field in the ESP model keeps similar local distribution. The global E-field looks much like that of smooth wireplate (Type-A), whereas the electrical current is close to that of

Figure 5. Electric field for Type-A configuration: (a) streamlines of potential gradient; (b) contours of potential gradient; and (c) contours of space charge density. 11285



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turbulence quantities. The pressure drop coefficient depends on the Reynolds number, plate open porosity, thicknessto-diameter ratio, flow impact angle, and wall roughness. For a smooth face and right angle condition, the pressure drop coefficient can be expressed as ζ=

⎞ ⎛ C3 + C2 exp⎜ − ⎟ ⎝ max(500, Re) ⎠ f Re C1 2

(13)

where the value of the inertial loss coefficient C2, that is, the limit of ζ for infinitely large Re, is tabulated in the literature,20 or can be obtained from microscopic simulations as demonstrated by Guo et al.19 The pressure drop coefficient generally decreases as plate porosity increases and as thickness-to-diameter ratio increases up to two. The particle behavior through the perforated plate will be studied in the future. In the present model, it is treated following the flow of gas in a perforated plate. 3.3. EHD Flow. The electric force on the ion charge generates the so-called EHD secondary flow with organized structures in the gas flow around each wire, which affects the pressure drop and modulates the turbulence level.21 For Type-A configuration, Figure 8 shows a fairly regular EHD flow pattern of a wavy main

Figure 7. Visualization of the electric field in the ESP model.

point-plate configuration. Note that this approach inevitably introduces errors near the boundaries of the electric field, where the symmetry or periodicity assumptions no longer hold. However, it can be used as the first approximation of the E-field in the ESP considered. How to improve the predicted E-field will be studied in the future.

3. GAS FLOW 3.1. Governing Equations. The gas flow in the ESP is simulated using the Reynolds average Navier−Stokes momentum equation (RANS) with a turbulence model. The steady state continuity equation and momentum equation are respectively ∇·(ρ U) = 0

(8)

and

Figure 8. Streamlines of gas flow showing the EHD secondary flow for the case of four Type-A electrodes.

⎛ 2 ⎞ ∇·(ρ U ⊗ U) = ∇·⎜P + ρk ⎟ ⎝ 3 ⎠

flow with an eddy on each side. A nondimensional number has been used in the literature, representing the ratio between the electrostatic body force and inertial force acting on a fluid parcel,22 i NEHD = bρf U 2 (14)

+ ∇·[(μ + μt )(∇U + (∇U)T )] + FEHD (9)

The widely used two-equation turbulence models (k−ε, SST) provide a good compromise between computational cost and accuracy. The equations for the SST model are μt = ρ

a1k max(a1ω , SF2)

⎡⎛ μ⎞ ⎤ ∇·(ρ Uk) − ∇·⎢⎜μ + t ⎟∇k ⎥ = Pk − β′ρkω + Sk σk ⎠ ⎥⎦ ⎣⎢⎝

where i is the current per unit length of the wire. This number, however, does not seem to account for the uniformity of the electric field for different geometries, which is regarded as the root cause of EHD flow. Figure 9 shows the effect of EHD flow on the pressure drop coefficient as the mean velocity varies. It is indicated that the pressure drop can be overpredicted without considering the EHD effect, particularly for the case of a large EHD number. This is consistent with the previous study of Soldati and Banerjee.23 Accordingly, the local drag in the EHD areas is decreased by adjusting the drag force term in the momentum equations for the global scale ESP model. The gas flow predicted for Type-B at a typical mean velocity of 1 m/s is shown in Figure 10, which shows a more complicated EHD flow. The velocity at the barb tip could be as high as 10 m/s, and a couple of recirculation eddies are induced in the central region between the neighboring plates. A reverse flow is also possible at the wire. Note that since the local velocity near the barb tip is too large to be neglected compared with the ion drift velocity, its effect on the electric field should be an interesting topic for future study. This local flow structure then generates additional shear, so that the turbulence kinetic energy is increased.

(10)

(11)

⎡⎛ μ⎞ ⎤ ∇·(ρ Uω) − ∇·⎢⎜μ + t ⎟∇ω⎥ σω ⎠ ⎦⎥ ⎣⎢⎝ = 2(1 − F1)

ω 1 ∇k∇ω + α3 Pk − β3ρω 2 σω2ω k

(12)

where the model constants and parameters are well documented elsewhere (ANSYS CFX).18 3.2. Perforated Plate. The microscopic gas flow around a single hole was studied by Guo et al.,19 which has provided a base for a perforated plate model. A recirculation zone around each hole behind the plate mainly contributes to the pressure drop. Resolving the hundreds or thousands of holes in the ESP model is impractical. To solve this problem, a thin-wall perforated plate is treated as a “porous jump” boundary in pressure and 11286



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4. PARTICLE FLOW AND CAKE FORMATION 4.1. Governing Equations. The suspended particle flow in most regions is modeled using the Lagrangian approach to track a large number of discrete particles statistically. The equation of motion is mp

du p dt

=

1 1 C Dπd p2ρf |u f − u p|(u f − u p) + πd p3 8 6 × g(ρp − ρf ) + q E + Fdis

(16)

where ⎛ 24 ⎞ C D = max⎜⎜ (1 + 0.15Rep0.687), 0.44⎟⎟ ⎝ CcRep ⎠

Figure 9. Effect of EHD on gas pressure drop for Type-A configuration.

Based on the energy balance, the kinetic energy production rate is related to the electric power input per unit volume,

Sk = cρion bE

2

CC = 1 +

(15)

where the proportion coefficient c is 0.03 for the present configuration. The EHD flow not only affects the particle mixing locally, but also has an implication to the large scale ESP modeling, where detailed resolution of the EHD flow is impractical and its effect should be considered by the so-called EHD submodel in the form of volumetric sources in the momentum equation and turbulence equations, respectively. Figure 11 shows a higher turbulence kinetic energy downstream of the perforated plate and at the electric field, respectively. The bulk flow pattern is changed by the inclusion of EHD, as the large eddy in the first stage electric field is suppressed. In the second stage chamber, a large eddy in the lower part diminishes so that the main flow becomes more uniform. Whether such modifications on the gas flow quantities can account for the correct particle behaviors is still under investigation. But the results seem reasonable and hence useful for the next step in developing an integrated process model.

2.52λ dp

(17)

(18)

Fdis is a force due to turbulent dispersion and Brownian diffusion. The former is based on an eddy interaction model,24 while the latter is based on the work of Li and Ahmadi.25 These are modeled by a random walk method. The Lagrangian approach does not apply to the region of high particle concentration at the collection plate, where a packed bed or cake layer forms. Discrete element method (DEM) is used here to model the cake formation process considering detailed particle−wall, particle−particle interactions. In DEM, both translational and rotational motion of a particle, for example, particle i and its neighboring particle j, are described by Newton’s second law of motion, given by mi Ii

du i = dt

dωi = dt

∑ (Fijn + Fijs + Fijvdw ) + Fie

(19)

∑ (R i × Fijs + Tijr )

(20)

Figure 10. Planar gas flow showing the EHD secondary flow for Type-B configuration: (a) velocity vectors near a barb; (b) velocity streamlines; and (c) turbulence kinetic energy. 11287



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gas flow. The deposited portions subsequently form a cake layer on the wall surface. Two major submodels are needed to describe these behaviors as given below. 4.2. Particle Collection Criteria. The near-wall behavior of particles on the ESP collection plate surface and the outcome of particle−wall interactions (deposition or re-entrainment) are more complicated than the particle−wall collision process. They also depend on whether a cake bed has formed already. For the simple case of no cake bed, the viscous sublayer of gas flow plays an important role in the collection efficiency. The special gas flow features (e.g., sharp velocity profile and low turbulence) of the sublayer are easily overlooked during the particle tracking in engineering applications, where sufficient grid resolution is impractical and the wall function is used. Here the importance of the drag is related to the Stokes number, Stk =

Up ρpd p2Cc 18μL

(21)

The sublayer thickness L is found to be much larger than the particle size considered and typical Stk for PM2.5 is much less than 1. These suggest that proper care is necessary in modeling the near-wall behaviors of fine particles correctly. In many previous numerical treatments,8,26 particles close to the collection wall are simply assumed to be collected. We propose a three-step deterministic approach, where the fate of a coming particle depends on fluid/solid properties, local flow conditions (shear stress, particle initial velocity), and forces applied: Step 1. Penetration through the sublayer toward the wall. In this process, a particle that arrives at the sublayer border with low momentum may fail to reach the wall surface before its velocity drops to zero (e.g., 1 × 10−20 m/s). This particle does not deposit, but returns to the main flow with identical velocity. Step 2. Particle-wall collision. A penetrated particle in Step-1 may stick to the wall or rebounds, depending on particle/wall properties and impact velocity. Step 3. Back penetration through the sublayer away from the wall. In this process, the rebound particle in step 2 may or may not be able to penetrate the sublayer, depending on its energy retained. A particle that fails to reach the bulk−sublayer interface before its velocity drops to zero will deposit, while a successful particle will be re-entrained into the main flow with a known velocity. Specifically in step 2, the coefficient of restitution for a particle−wall collision is calculated by the adhesive, elastic-plastic particle impaction model in the perpendicular direction.27 In this model, a particle, if slower than the critical sticky velocity, will stick to the wall. Otherwise it will rebound with a damped

Figure 11. Gas flow in the ESP: (a) turbulence kinetic energy; (b) streamlines without EHD; and (c) streamlines with EHD.

where the forces considered are listed in Table 1. Major challenges to simulate the particle phase are (a) submicrometer particles of concern are much smaller than the ESP size by at least 6 orders of magnitude, (b) dilute and dense particle regions coexist, and (c) particles strongly interact with the complicated electric field and gas flow. Figure 12 shows some quite different characteristic behaviors for particle phase. In the local electric field regions, fine particles tend to move along jig-jag paths due to the EHD flow and turbulent fluctuations. When they approach the collection plate, they may deposit permanently under certain conditions or rebound and be re-entrained in the

Table 1. A List of the Equations to Calculate the Force and Torque Used in the DEM Model forces

equations

normal contact force

⎡2 ⎤ Fijn = ⎢ E R̅ ξn3/2 − γnE R̅ ξn (uij·n̂ ij)⎥n̂ ij ⎣3 ⎦

tangential contact force

3/2 ⎤ ⎡ ⎛ min(ξs , ξs ,max ) ⎞ ⎥ ⎟⎟ Fijs = − sgn(ξs)μs |Fijn|⎢1 − ⎜⎜1 − ⎢ ξs ,max ⎝ ⎠ ⎥⎦ ⎣

van der Waals force

Fijvdw = −

electrostatic force

Fie = (qEa + q2 /16πε0h2)n̂E

torque

Ti =

64R i3R j3(h + R i + R j) Ha × 2 n̂ ij 6 (h + 2R ih + 2R jh)2 (h2 + 2R ih + 2R jh + 4R iR j)2

∑ (R i × Fijs + Tijr ) = ∑ (R i × Fijs − μr R i|Fijn|ω̂ i)

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Figure 12. Particle transport behavior in the local region of the electric field: (a), suspended particle trajectories in turbulent flow; (b), particle trajectories near the collection plate; and (c), cake structure of deposited particles.

velocity due to either surface tension or plastic deformation. The tangential coefficient of restitution is given in ref 28. As a result of the collection criteria, different behaviors for different particles are predicted. In general, finer particles take a longer time to settle down onto the plate, but once in contact, will deposit more easily. Large particles tend to bounce many times and may glide along the plate before deposition. Note that the proposed three-step approach for the deposition kinetics is to be corrected after cake formation, since more complicated forces and energy transfers are involved when particle-cake collisions become dominant, which is under investigation currently. The overall particle flow in the ESP is shown in Figure 13. An appreciable proportion of particles travel around the electric field

electric region will redistribute before entering the second one, so there are still significant deposits on the second stage plates. 4.3. Cake Formation. Particle−particle interactions, though not considered for the suspended particle flow and the near-wall behavior of particles in the viscous sublayer, are essential in simulating the cake formation process. They are naturally taken into account in DEM. The simulated cake structures, in terms of macroscopic and microscopic properties, for example, porosity and coordination number, vary with particle size and E-field intensity, as shown in Figure 15.29 Generally, porosity decreases

Figure 15. Cake structure varying with particle size and E-field intensity: (a) porosity and (b) coordination number.

as particle size or E-field intensity increases. The change in porosity is more sensitive to particle size in a weaker E-field, and to E-field intensity for a smaller particle size. A lower limit similar to the random loose packing under gravity30 can be achieved when particle size is increased up to 1 mm. The coordinates of each particle in a cake can be obtained from computer simulation, allowing detailed structural analysis to be conducted.29 For example, the mean coordination number (CN), which is the number of contacts of a particle, is shown to decrease with the increase of porosity in Figure 15b. Here two particles are defined in contact when the distance between them is less than 0.01d. The local structure including a particle and its neighboring particles are shown in the inset of Figure 15b, where spheres represent the center of particle and sticks represent two particles in contact. It can be seen that with the decrease of porosity, the typical local structure changes from 1D chains (CN=2) to 2D triangles (CN=3, 4), and finally to 3D tetrahedral (CN ≥ 6). Such changes in microstructure are similar to those observed for the packings formed under gravity,31 which results from the similar variation of the van der Waals force relative to the gravity or the electrical force.30,31 On the basis of the packing structure, other bulk properties, such as the electric resistivity and particle charge density, can also be calculated. Some of these properties will be useful when the quantification of the effect of cake layer on the electric field or the so-called cake-E-field coupling is of interest, as discussed in section 5.

Figure 13. Particle trajectories in the ESP with color scaled to particle charge.

through the large space at the lower part, and exit the outlet. The local particle collection rate in Figure 14 shows a relatively larger

Figure 14. Distribution of particle deposition rate on the ESP collection plate.

value near the front entry and the upper/lower edges of the electric field. The particles passing the first chamber around the 11289



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model. These submodels are eventually integrated to form an ESP process model according to the multiscale approach and the coupling methods (database, analytical correlations) as discussed above. At present, the effect of the cake on E-field has not yet been incorporated. A steady-state ESP model without considering this effect could already be useful in providing some qualitative and quantitative information. For example, given a list of inputs: • pilot ESP design with total collection plate area of 25 [m2] • gas flow rate, 5500 [m3/h] • particle size distribution • gas temperature, 25 °C • particles: density, 2000 [kg/m3]; dialectic constant, 14 (based on a typical subbituminous fly ash composition; 50%SiO2, 25%Al2O3, 10%Fe2O3, 15%CaO) A list of outputs can be produced by the steady-state simulation, • initial operation voltage for pulsating DC: 70,000 [V] • total current: 0.015 [A] • collection efficiency for different particle size, as shown in Figure 19. In Figure 19, the case without electric field is also given as a reference. Without electric field, there is a sharp cutoff particle size at 25 μm, below which little can be collected. In contrast, however, the electric field greatly improves the collection of smaller particles, so that the collection efficiency predicted is not as much sensitive to the particle size in the range of 0.1−100 μm considered. Particularly, it is nearly constant for particle sizes below 2.5 μm. This can be attributed to two possible factors: (a) the particle mixing by turbulence and EHD plays a dominant role over the sedimentation process, thereby smaller particles always take a longer time to settle; (b) a full collection has been reached in the electric field, so that the maximum efficiency is limited by the bypassing flow. In the latter case, it is possible to optimize the flow condition by further design modification. Interestingly in this device, the electric field does not necessarily promote the collection of the largest particles (100 μm), which can settle down easily by gravity alone. In fact, compared with the gravity, the electric force is of lower order with respect to particle size and becomes negligible for large particles. Given a particle size distribution (PSD) at the inlet, the PSD of the escaped part via the outlet can be calculated (Figure 20). In this example, PM2.5 of interest takes up 15.5% initially, but it rises to 31.3% after the ESP. The estimated collection efficiency for PM2.5 (75%) is lower than the total efficiency (88%). While these results partly demonstrate the model capability, validation by laboratory experiments and plant data will be carried out in the future. It should be pointed out that the results in Figure 20 were obtained without considering the effect of cake formation. Implications arise after a cake layer builds up, as the flow boundary (e.g., surface location, roughness, coefficient of restitution) and importantly, the electric conditions change. The electricity-discharging behaviors of deposited particles are complicated and would affect the electric field by three possible mechanisms: (1) Charge accumulation. Accumulated charges generate their own E-potential and a negative E-field. (2) Current resistance. Particles on the cake surface keep being charged and neutralized dynamically, and a current passing the packed bed (i.e., cake) creates a voltage loss. (3) Back corona: particles release charges by local gas breakdown. Figure 21 gives some preliminary results for Type-A configuration predicted based on the first mechanism. Assuming

A simulated cake in a nonuniform E-field similar to the E-field in the ESP is depicted in Figure 16, with color scaled to the height

Figure 16. Simulated cake pattern under nonuniform electric field.

from the plate. The surface particles are clearly higher at the edge than those in the center. This pattern corresponds to a nonuniform porosity distribution, which provides a possible explanation for the cake pattern observed experimentally.5 In fact, it is apparent from Figure 17 that the observed cake

Figure 17. Various phenomena related to the cake pattern.

pattern32 is closely correlated with the barbed electrode geometry, distributed electric field and localized EHD flow, although the dominating mechanism is yet to be explored.

5. INTEGRATED ESP MODEL A flow sheet shown in Figure 18 outlines a step-by-step procedure for the simulations of different fields and the dominant interactions considered for model integration. After the model is set up with sufficient inputs in the form of geometry, operational conditions and material properties, the distributed electric field is solved first. Second the detailed turbulent gas flow considering EHD is simulated, followed by the Lagrangian tracking of the suspended particles. Then the outcomes of the particle-wall collisions are decided on the collection plate surface and the deposition rate can be calculated. Next, the deposited particles cumulate to form a dense cake layer, and the microscopic structures and bulk properties are determined. Finally the formation of a cake layer will change the electric boundary conditions on the collection plate, thus the electric field will be updated. Such a cycle is repeated until convergence is achieved. Several key submodels are defined, which refer to either the detailed simulation limited to local regions or linkages among different submodels or different scales, namely, perforated plate model, local electric field model, EHD model, particle collection criteria, cake formation model, and cake-E-field interaction 11290



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Figure 18. Computational flowsheet for model integration.

Figure 19. Particle collection fraction of the ESP as a function of particle size: (a) without electric field and (b) with electric field.

Figure 21. On-plate electric field as a function of cake structure: (a) effect of particle size (cake layer of 1 mm); and (b) effect of cake layer thickness (dp = 2.5 μm).

collection plate significantly weakens the applied E-field.6 The adverse effect becomes more significant as the cake thickness increases and particle size decreases, until the current vanishes at one point. In addition to the overall V−I characteristics, the distribution pattern of the ion charge is changed (Figure 22), compared with the case of no cake (Figure 5c). The oval contours disappear, giving way to the expanding peanut-shaped contours. This large change is mainly for the following reason. For the case of Type-A electrode, ion charges are generated on the part of the wire surface where the local E-strength exceeds a critical value (as determined by the Peek equation). This region is getting wider and covers the entire wire surface before cake formation. Therefore the ion density appears to be uniform around the wire. When a cake is formed, the overall E-field becomes weaker due to the negative E-field created by the particle charges in the cake. As a result, a part of the wire surface

Figure 20. Particle size distribution before and after ESP.

a uniform cake layer and fixed voltage, both the E-field intensity and current density on the plate (normalized by those for the case of no cake) are generally reduced by the cake layer. This agrees with the observation that the cake formed on the 11291



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collection criteria on a cake layer; (ii) implementation of the cake−E-field coupling and subsequently transient simulation as cake is formed; and (iii) validation of the model through comparison with laboratory and plant data.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Australian Research Council and Fujian Longking Co. Ltd for their support of this project.

Figure 22. Contours of ion charge density after cake formation for Type-A (1 mm thick layer of dp = 2.5 μm).

may reduce or even stop ion emission (e.g., on the left and right sides of Figure 22), so that the ion density becomes narrowly distributed about the vertical centerline. These preliminary results demonstrate the significance of the cake−E-field interaction, thus highlighting the need for further study, specifically in several respects: (a) the electric phenomena inside and at the cake surface; (b) spark frequency control versus the fixed voltage; (c) Type-A versus Type-B configuration; and (d) the effect of the nonuniform cake layer. Note that, since the cake grows with time, its effect on the process performance should be transient and also related to the rapping procedure. As a first step, we will develop a quasi-steady state procedure to simulate the time-averaged behavior within a rapping cycle. Then a rapping time period is divided into several intervals to improve the prediction. Finally the procedure will be extended to the large scale ESP, where it would pose a challenge in handling the distributed E-field coupled with a random cake layer with temporal and spatial variations.

6. CONCLUSIONS An ESP process model has been developed by considering various transport phenomena at different time and length scales. In this model, detailed electric field, gas−particle flow, and cake bed structure in a complex ESP geometry can be simulated, and submodels at different scales and fields can be developed individually. Integration of these submodels based on different numerical approaches leads to an innovative framework to describe the ESP process. This integrated multiscale model is shown to be able to generate meaningful results that may assist the ESP process design and control. However, future studies are needed in order to produce a more accurate and robust model for solving practical problems including, for example, (i) modification of some submodels based on fundamental and/or parametric studies, including the EHD submodel, electric phenomena in a cake layer, and

NOMENCLATURE b = ion mobility d = diameter Cc = Cunningham correction factor CD = drag coefficient D = ionic diffusivity E, E = E-field intensity F = force f = porosity of the perforated plates g = gravitational acceleration h = particle−particle or particle-wall surfaces separation I = moment of inertia of a particle i = current per unit length wire J = current density k = turbulent kinetic energy k0 = Boltzmann constant m = mass NEHD = EHD number P = pressure Pk = turbulence production q = particle charge qe = unit electron charge R̅ = harmonic mean radius Re = Reynolds number Rij = a vector running from the center of the particle i to the contact point with particle j r = radius of curvature for corona wire SEHD = source of turbulence due to electro-hydrodynamics Stk = Stokes number T = temperature t = time U,u = velocity V = electric potential uij = the relative velocity between particles i and j Y = Young’s modulus

Greek Letters

ρ = density μ = gas dynamic viscosity μr = rolling friction coefficient μs = sliding friction coefficient μt = eddy viscosity λ = mean free pass of molecules ζ = pressure drop coefficient δ = relative density of gas with respect to the normal conditions ω = turbulent dissipation frequency ε0 = electric permittivity

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ωi = angular velocity of a particle ρion = ion space charge density κp = dialectic constant of particles γn = normal damping constant ξn = total normal displacement of particle during contact ξs = total tangential displacement of particle during contact

(21) Xing, M.; Guo, B. Y.; Yu, A. B.; Guo, J. Numerical analysis of electrohydrodynamic flow in a wire-plate ESP (manuscript in preparation). (22) Leonard, G. L.; Mitchner, M.; Self, S. A. An experimental study of the electrohydrodynamic flow in electrostatic precipitators. J. Fluid Mech. 1983, 127, 123−140. (23) Soldati, A.; Banerjee, S. Turbulence modification by large-scale organized electrohydrodynamic flows. Phys. Fluids 1998, 10, 1742−56. (24) Gosman, A. D.; Ioannides, E. Aspects of computer simulation of liquid-fuelled combustors. J. Energy 1983, 7 (6), 482−490. (25) Li, A.; Ahmadi, G. Dispersion and deposition of spherical particles from point sources in a turbulent channel flow. Aerosol Sci. Technol. 1992, 16, 209−226. (26) Choi, B. S.; Fletcher, C. A. J. Turbulent particle dispersion in an electrostatic precipitator. Appl. Math. Model. 1998, 22, 1009−1021. (27) Thornton, C.; Ning, Z. A theoretical model for the stick/bounce behaviour of adhesive, elastic-plastic spheres. Powder Technol. 1998, 99, 154−162. (28) Wu, C. Y.; Thornton, C.; Li, L. Y. A semi-analytical model for oblique impacts of elastoplastic spheres. Proc. R. Soc. London 2009, A465, 937−960. (29) Yang, S. Y.; Dong, K. J.; Zou, R. P.; Yu, A. B.; Guo, J. Settling of fine particles in an electrostatic field. Granul. Matter, DOI 10.1007/ s10035-013-0410-3. (30) Dong, K. J.; Yang, R. Y.; Zou, R. P.; Yu, A. B. Role of interparticle forces in the formation of random loose packing. Phy. Rev. Lett. 2006, 96, 145505. (31) Dong, K. J.; Yang, R. Y.; Zou, R. P.; An, X. Z.; Yu, A. B. Critical states and phase diagram in the packing of uniform spheres. Eur. Phys. Lett. 2009, 86, 46003−46008. (32) Yuen, A. W. Collector Current Density and Dust Collection in WirePlate Electrostatic Precipitators. Ph.D. Thesis, University of New South Wales, Australia, 2007.

Subscript

f = fluid p = particle

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REFERENCES

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