Numerical Study of Particle Collection by Single Water Droplets

Ind. Eng. Chem. Res. 1999, 38, 4433-4442

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Numerical Study of Particle Collection by Single Water Droplets Shekar Viswanathan Department of Chemical and Environmental Engineering, 10 Kent Ridge Crescent, National University of Singapore, Singapore, 119260

Particle collection efficiencies of water droplets were calculated by considering thermophoresis, diffusiophoresis, inertial impaction, wake capture, and effects of drop deformation. The computations were carried out for a droplet temperature of 10 °C with saturated gas temperatures of 20, 35, 65, and 95 °C. The temperature, water vapor, and velocity distributions around the collector were determined by direct numerical integration of the appropriate governing partial differential equations using a convenient orthogonal grid generation technique. The investigation shows the following: deposition of fine particles can be significantly enhanced by phoretic forces; flux deposition of fine particles can be related to Reynolds number through the proportionality, Eflux ∝ Re-0.78; the flux deposition of fine particles on the rear of the collector is significant for low Reynolds numbers; wake capture is relatively insignificant; drop deformation can improve the collection of larger particles by inertial impaction and of smaller particles as a result of increased hydrodynamic effects on such particles. Introduction One of the most effective and economical methods of cleaning dirty gas streams involves contact with water sprays, such as Venturi scrubbers and countercurrent spray towers. A fundamental parameter that determines the effectiveness of a typical unit is the single drop collection efficiency. Reliable experimental single drop collection data are difficult to obtain. Therefore, researchers turn to theoretical approaches to study mechanisms responsible for the collection of particles by water droplets. Large particles (rp g 5 µm) are captured primarily by inertial impaction and interception, while small particles are collected as a result of phoretic forces, wake capture, and Brownian motion.1,2 The particle sizes varying from 0.1 to 2.0 µm in radius, commonly known as the Greenfield gap,1 are difficult to capture as no single mechanism responsible for their capture in this range is known. In addition, evaporation from sprays of liquid droplets into a hot gas stream occurs in scrubbers. This evaporation (or condensation) is complicated by the wide range of droplet sizes normally present, each with a constantly changing velocity, temperature, and evaporation rate. This requires the development of a realistic model that takes into account heat and mass transfer. This work focuses on the collection of these particles taking into account inertial impaction, thermophoresis, and diffusiophoresis. Brownian motion is neglected in this investigation3 because it has been shown to be unimportant for particles with radii greater than 0.05 µm. The effect of wake formation in the capture of smaller particles is also investigated. In addition, the effects of drop deformation on the collection efficiency are studied since this phenomenon occurs on drop size ranges that are encountered in countercurrent scrubber operations. Literature Review There is a considerable body of literature describing particle deposition on single spherical targets assuming

viscous and inviscid flow conditions.4-9 Langmuir and Blodgett10 were perhaps the earliest researchers who computed collection efficiencies using a model assuming potential flow (high Reynolds numbers) or Stokes flow (low Reynolds numbers) with the dominant collection mechanism being inertial impaction. To evaluate collection efficiencies at intermediate Reynolds numbers, Langmuir proposed a scheme of interpolation between the two limiting flow conditions. Unfortunately this approach appears to be based on intuition as it has never been subjected to rigorous analysis. Other authors, such as Johnstone and Roberts,12 Ranz and Wong,14 Pemberton,13 Fonda and Herne,11 and Michael and Norey,7 made refinements to the basic calculations of Langmuir. Some experimental verification was also provided. All of these investigators restricted themselves to the single mechanism of inertial impaction under ideal flow conditions. Wu,15 one of the earliest to consider the effects of flows at intermediate Reynolds numbers, used boundary layer flow approximations to model the flow of the fluid around the collector. Wu reported that his calculations agreed reasonably well with the experimental data. Beard and Grover4 made improvements on the effects of intermediate Reynolds numbers by solving the steadystate Navier-Stokes equations in spherical coordinates. Tardos et al.9 and Ellwood et al.16 also considered the effects of intermediate Reynolds numbers by using the flow field proposed by Hamielec et al.17 Pilat and Prem19 and Mehta18 developed models for micron and submicron particle collection by phoretic forces, inertial impaction, and Brownian diffusion. Although these models assumed potential flow conditions and temperature, vapor, and particulate distributions linear in thin boundary layers, there appears to be an inconsistency in the results. It has been pointed out by many that the potential flow model does not account for standing eddies on the downstream side of the drop. Jacober and Matteson20 and Matteson and Prince21 experimentally performed target efficiencies for single and multiple targets resulting only from Cou-

10.1021/ie990199s CCC: $18.00 © 1999 American Chemical Society Published on Web 10/14/1999

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Ind. Eng. Chem. Res., Vol. 38, No. 11, 1999

Cf ) 1 + Kn (1.257 + 0.40 exp[-1.1 Kn-1])

(3)

Equations 1 and 2 can be combined and rearranged into a dimensionless form by making the following substitutions:

U∞2 U∞2mp A h ) a j; F he ) hf e; U h ) U∞ u j; V h p ) U∞vj p; a0 a0 a0 t T) U∞ Figure 1. Collection efficiency definition.

lombic attraction with inertial effects. However, this work has had limited implications for wet-scrubbing conditions. Theoretical Model A single drop collection efficiency can be defined in many different ways. In this work, it has been taken as the ratio of the number of particles predicted to strike the collector to the number of particles flowing through the projected area of the collector. Particle collection will occur from within an area defined by the trajectories that would be traced by point masses that just graze the collector as shown in Figure 1. If Y0 is the maximum vertical displacement from the collector centerline that will allow the particle to collide with the collector, the droplet collection efficiency can be defined as E ) Y0/ a0)2, where a0 is the volume radius of the collector. To model the capture of aerosol particles, it is necessary to understand the influence of several important physical variables including: particle motion that affects the flow of the fluid over the body of the collector; collector shape that is other than spherical;22,23 presence of vapor and temperature gradients around the collector that influence the path of a moving particle. Equation of Particle Motion. The equation of particle motion is derived from Newton’s law of motion with the assumption that the particle size is so small in comparison to the collector that it will not influence the collector flow field, particle-particle interactions are negligible, heat and mass transfer between the particle and fluid are negligible, the particle is spherical in shape. A simple force balance on a particle provides

m pA h )F h drag + F he

(1)

The drag force on an object can be expressed by

F h drag )

1 Cd F (U h -V h p)|U h -V h p|Ap 2 f Cf

(2)

The value of F h e consists of thermophoretic and diffusiophoretic forces, called collectively flux forces, and is calculated by the procedure illustrated by Pilat and Prem.19 The Cunningham correction factor, Cf, is introduced to correct the reduction of the drag force from the theoretical value when the particle size is of the order of the mean free path length, λ, of the gas. This correction is done in terms of a dimensionless Knudsen number, Kn ) (λ/rp):

Kn )

()

λ , rp

Combining eqs 1 and 2 and making the appropriate substitution yields

a j)

γ (u j - vj p) + hf e 2K

(4)

where Stokes number K ) CfFpU∞rp2/9µfa0 and Fp and µf are particle density and fluid viscosity, respectively. The deviation of the particle from Stokesian behavior is given in terms of particle Reynolds number, Rep, as24

γ)

Rep CdRep )1+ + 0.15Rep0.6 24 109

(5)

Equations 3 and 4 can be solved simultaneously given any external force, hfe, to produce particle trajectories. The particle acceleration vector, a j , is given by25

a j ) a1ej1 + a2ej2

(6)

Fluid Flow Equations. To accurately predict the flow of a Newtonian fluid at intermediate Reynolds numbers, the Navier-Stokes equations must be solved. The equations that relate to the conservation of linear momentum, with constant density and viscosity in the dimensionless form, are

1 2 ∂u j j x(∇xju j ) + ∇2 u j ) -∇(hp* + u2 + u ∂t 2 Re

(7)

∇ h ‚u j)0

(8)

Du j ∂u j ∂u j 1 2 j x(∇ h xju j) ) + (u j ‚∇ h )u j) + ∇ hu j -u Dt ∂t dt 2

(9)

where p* ) p/FfU∞2 and fluid Reynolds number Re ) 2FfU∞a0/µf. The form of the momentum equation used to solve for the fluid flow involves the velocity/stream function formulation. This equation is obtained by taking the curl of eq 7 to remove the pressure term, and introducing the stream function. Ellwood26 provides all the necessary differential operators in general orthogonal coordinates to permit the formulation of eq 8 as

∂ ∂ (u h h ) + (u h h ) ) 0 ∂a1 1 2 3 ∂a2 2 1 3

(10)

assuming that the velocity component in the R3 direction is zero. Equation 8 is satisfied everywhere if the stream function, ψ, is introduced as

u1 )

1 ∂ψ 1 ∂ψ ; u2 ) h1h3 ∂R2 h2h3 ∂a1

(11)

Ind. Eng. Chem. Res., Vol. 38, No. 11, 1999 4435

By removing the pressure term from eq 7 and using eq 11, the final form of the vorticity transport equation becomes

[

]

h3 ∂ψ ∂f ∂ψ ∂f 2 4 ∂ 2 + (D ψ) ) Dψ ∂t h1h2 ∂a1 ∂a2 ∂a2 ∂a1 Re

(12)

with

1 2 D2ψ D ψ; f) 2 ; ξ)h h3 3 h3 ∂ h2 ∂ h1 ∂ ∂ D2 ) + h1h2 ∂R1 h1h3 ∂R1 ∂R2 h2h3 ∂R2

[ {

} {

Figure 2. Viscous flow around a sphere for a Reynolds number of 200.

}]

Equation 12 is a convenient scalar relationship that can be used to solve for incompressible, planar, or axisymmetric flow of a Newtonian fluid over an arbitrarily shaped body. Heat and Mass Transfer Equations. Similarly by applying the principle of conservation of total energy (kinetic and internal) to an ideal fluid and introducing the stream function, the steady-state energy equation takes the form26

[ {

} {

∂ψ ∂η ∂ψ ∂η 2 ∂ h2h3 ∂η ) + ∂R2 ∂R1 ∂R1 ∂R2 Pe ∂R1 h1 ∂R1 ∂ h1h3 ∂η ∂R2 h2 ∂R2

}]

(13)

The convective diffusion of a component of a gas mixture can be expressed with the assumption that the diffusing component is dilute as:

DC1 ) Ω12∇2C1 Dt

(14)

The heat and mass transfer equations are very similar and have identical dimensionless forms with

η)

µf C1 - C1∞ ; Pe ) Re Sc; Sc ) C1c - C1∞ FfΩ12

Results and Discussion During the investigation, several separate and distinct problems are solved before particle trajectory calculations are performed. The preliminary calculations involve the generation of a grid to represent the computational domain, the establishment of the fluid velocity profile around the collector, and the determination of the temperature and vapor distributions around the collector. To ensure the integrity of the model as a whole, each step of the calculation procedure is validated before the final trajectories are computed. Orthogonal Grids Generation and Numerical Solution. Orthogonal grids are generated for droplets of various sizes ranging between 50 and 4000 µm in radius. Droplets with radii smaller than 620 µm are essentially spherical in shape. Significant deformation does not occur until the droplet radius is larger than 1800 µm.22,23 The grid generation procedure provides a convenient method of effectively distributing grid lines around these droplets for better resolution of the flow field and temperature/vapor distributions. The shapes of the deformed droplets are determined by the semi-

empirical equation provided by Pruppacher and Pitter.23 Details pertinent to the generation of grids and the numerical solution employed are provided elsewhere.26,27 The important objective in the grid generation is to account for the large deviations in the fluid and particle trajectories in the region close to the collector. In applications where there will be large deviations in velocity, temperature, and concentration in the flow field, it would be desirable to “pack” the grid lines in the regions where steep gradients will occur to achieve better resolution. Very fine grids are required to model variables that change quickly. Hence variable grid size selection as proposed by Mobley and Stewart27 was used in this work. They proposed four general packing functions to accomplish various grid line density distributions. The size of the grids varied from 1/10000 to 1/100 of a drop size chosen. The procedure was exactly the same as that explained by Mobley and Stewart. The iterative procedure used to solve the equations required an initial guess for the entire solution. The initial guess was based on the multidirectional trans-finite interpolation technique provided by Thompson et al.28 The procedure was repeated until the convergence criterion n+1 n - φi,j | e 10-4, between successive iterations, max|φi,j was satisfied. The gas was assumed to be fully saturated during the simulation. Collector Flow Field. The governing equations for viscous flow around a rigid body are the Navier-Stokes equations described earlier. These are used to determine the flow fields around collectors for Reynolds numbers between 1.54 and 400. This range in Reynolds numbers corresponds to collector diameters varying between 100 and 1240 µm assuming that they are falling at their terminal velocities.23 For Reynolds numbers above 400, potential flow conditions have been shown to provide good approximations to the flow fields when determining particle collection.4 Viscous Flow Model. The method used to solve the equations governing viscous flow is the hybrid firstsecond-order algorithm, in which the first-order algorithm is used to produce an initial guess for the secondorder algorithm. This process is accomplished by assuming the potential flow solution for an initial guess and solving the equations until the convergence criterion is reached. For the Reynolds number range considered for the viscous flow model, it is accepted that water droplets at their terminal velocities are still spherical.22, Figure 2 illustrates the solution for the stream function, ψ, for a 433 µm radius droplet at its terminal Reynolds number of 200. The flow model can be validated by comparison with rigorous studies of various flows around spheres, using several important parameters such as the angle of flow separation (θs), the vortex length (L), and the surface vorticity distribution (ξsurface). Tables 1 and 2 and Figure 3 summarize these comparisons. The good agreement between the data validates the assumptions of the present model and

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Table 1. Comparison of Angle of Flow Separation (θs) (θs deg) Re

present work

Garnera,b

Tanedaa,b

LeClaireb

30 100 200 300 400

23 51 60 66 70

61 65 71

51 -

26 53 63 67 73

a

Experimental data. b From LeClaire.31

Table 2. Comparison of Vortex Length (L) L (m)

a

Re

present

Tanedaa,b

LeClairea

30 100

0.154 0.882

0.125 0.920

0.155 0.950

Experimental data. b From LeClaire.31

Figure 4. Comparison of NuL with data of Woo.

Figure 3. Comparison of ξsurface with data of LeClaire.

demonstrates its ability to predict the complex flow behavior behind a collector. It is interesting to note that the waviness in the surface vorticity, ξsurface, as shown in Figure 3 at higher Reynolds numbers, is not the result of numerical instabilities. Rather, it is indicative of the beginning of a formation of a second vortex behind the collector as observed experimentally. However, an extremely high resolution grid would be needed to document this phenomenon numerically for a sphere.29 Potential Flow Model. The method used to solve the equations governing potential or irrotational flow involves the block successive relaxation algorithm. With straight flow as an initial guess to the velocity field, the solution for the irrotational flow is obtained using iterative procedure. This portion of the model is validated by comparison to the analytical solution for potential flow around a sphere:

ψ)

1 2 1 r sin2(θ) 2 2r

[

]

(15)

The results of the numerical solution agree with the analytical results within 0.1% everywhere in the solution domain. Convective Diffusion Model. The convective diffusion eq 14 is solved using the standard point relaxation method with variable relaxation factors and calculations for Reynolds numbers ranging between 1.54 and 400, with both the Prandtl number and Schmidt number set to 0.7. The solutions to this model provide closely spaced contour plots at the front of the collector

indicating high-temperature gradients. As the fluid flows from the front of the sphere toward the rear along the surface, it is cooled by conduction from the collector. The fluid is found to leave the collector at the point of flow separation. High-temperature gradients are found to occur at the rear of the collector in the presence of a wake because the fluid moves directly toward the rear of the body. Since the center of the wake is well circulated, it produces uniform temperatures. As a result, there are low-temperature gradients at the center of the wake. It should be noted that the discussion pertaining to temperature gradients also applies to vapor gradients surrounding a droplet on which condensation occurs. To ensure that the solutions obtained for this portion of the model are valid, comparisons are made with pertinent data from Woo and Hamielec30 as shown in Figure 4. The good agreement indicates that the model used in this investigation accurately predicts the heat transfer and condensation rate associated with water droplets. Flux Deposition Model. The method used to solve for the flux deposition efficiency of fine particles involves trial and error calculation of particle trajectories. The trajectories are evaluated by integrating the generalized particle equations of motion with the variable-step-size Runge-Kutta-Fehlberg method. The formulation of the inertial terms in the model is tested by inserting the potential flow field and generating well-known solution for the case where flux forces are absent. Figures 5 and 6 compare model calculations with published literature data.11,19 The data in Figure 5 are calculated from a condition without flux forces, whereas the data in Figure 6 are calculated with flux terms. The good agreements in the comparisons made indicate that the inertial and flux terms are correctly formulated in general orthogonal coordinates. Therefore, the application of more accurate velocity, temperature, and vapor distributions would result in valid particulate matter deposition rates. Additionally, the agreement with the data of Pilat and Prem reinforces the fact that Brownian motion need not be included for the particle size range considered in this investigation when flux forces are present. Pilat and Prem19 identified in their investigation that the diffusiophoretic and thermophoretic forces are larger than the Brownian diffusion force by about 1 order of magnitude at 0.01 µm particle diameter and by about 4 orders of magnitude at 10 µm particle diameter.


Figure 5. Potential flow collection efficiencies for a sphere.

Figure 7. Flux deposition for a Reynolds number of 1.54.

Figure 8. Flux deposition for a Reynolds number of 200.

Figure 6. Recalculation of the data of Pilat and Prcm.

General Results. Particle collection efficiencies are calculated for a range of collector radii varying between 50 and 4000 µm. Accurate velocity, temperature, and vapor distributions are determined numerically as discussed earlier. The particle sizes range between 0.05 and 10 µm to include particles in the “Greenfield Gap”. Based on literature review19,22,23 and preliminary simulation results from this work, the mechanisms assumed to be responsible for particle collection are the following: for drop radius e 620 µm, inertial impaction, thermophoresis, diffusiophoresis and wake capture; for drop radius > 620 µm, inertial impaction (with drop deformation). The temperature of the water is assumed to be at 10 °C while the gas temperature is assumed to vary between 20 and 95 °C. The general results of this investigation are obtained for the case where condensation occurs on cool water droplets moving in warm humid gas streams. To perform the calculations, the following simplifying assumptions are made: the water droplet is assumed to be at a constant temperature during the time that a particle is passing the collector; the amount of water condensing on the collector during the time that a particle is passing the collector is small compared to the droplet volume. Figures 7-9 depict the results of particle collection efficiency calculations for a water temperature of 10 °C and a gas temperature ranging between 20 and 95 °C. From the illustrations, it is evident that particle collection efficiencies could be increased significantly by

Figure 9. Flux deposition for a Reynolds number of 400.

actions of phoretic (flux) forces. The increases can be as large as several orders of magnitude, and they are most prominent at lower Reynolds numbers. It is not surprising that efficiencies are greater than unity in some instances. This condition simply implies that some particles are collected from an area greater than the projected area of the collector due to the influence of flux forces. The data presented demonstrate only trends for the various parameters involved in the determination of collection efficiencies. Flux deposition rates are influenced primarily by the complex flow behavior from the formation of a vortex, the effects of changing Reynolds number, the effects of increasing particle mass, and the role of droplet formation. Clearly, several of these

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Figure 10. Rear collision of a 0.3 µm particle. Figure 12. Fraction of deposition occurring on the front of a collector.

Figure 11. Radial velocity diagram for rear collision.

parameters influence one another, as demonstrated by the effects of Reynolds number on vortex formation. However, each parameter has a distinct individual influence that merits further discussion. Droplet deformation does not affect flux deposition because of the extreme Reynolds numbers involved. Collection in the Wake. Wake capture is determined by predicting the complex flow behavior behind the collector. Figure 10 illustrates the wake capture of a 0.3 µm radius particle by a droplet for a Reynolds number of 100 when the gas and droplet temperatures are 95 and 10 °C, respectively. From this representation, a definite mechanism can be visualized for wake capture when flux forces are weak in some regions and strong in others. The changes in the strengths of the flux forces are illustrated in Figure 11 in terms of the dimensionless radial component of the flux velocity and the fluid velocity. Referring to Figures 10 and 11, the flux velocities are small at point A and a particle is moved to the collector predominantly by the hydrodynamic effects. As the particle is carried close to the collector, high flux forces at point B help the particle move toward the collector and into the wake region. Near the point of flow separation, as discussed earlier, the flux forces are reduced. As a result, the particle is carried away from the collector in the wake at point C. The total absence of flux forces in the wake causes the particle to follow the fluid, essentially, until it is pushed back toward the collector at point D. As the particle comes into close proximity with the collector, the high flux forces, resulting from the reverse flow, capture the particle near point E. The presence of the wake has two effects relative to the capture of a particle. The wake

can influence the particle motion directly through the hydrodynamic forces, and it can influence particle motion through its effects on the temperature/vapor distribution. In regions where the wake produces low temperature/vapor gradients, the flux forces will be reduced. Resolution of the question of whether the presence of the wake actually enhances flux deposition of particles is very complicated. For higher Reynolds numbers, the low temperature/vapor gradients in the wake region are more pronounced. If the wake was not present, the resulting higher and more uniform gradients could produce higher collection efficiencies than possible through the combined action of the wake and flux forces. Another type of wake capture that can occur has also been studied. This involves a small (nearly massless) particle being forced into a wake and circulating forever in the absence of other forces. The decaying spiral phenomenon observed appears to be as a result of the complex interaction of a weak oscillating flux force and the fluid drag as shown in Figure 11. The magnitude of the oscillating flux velocity is generally higher when the particle is moving away from the collector. As a result, there is a larger reduction in the particle velocity when it is being dragged away by the fluid than the increase when it is moving toward the collector. The reduction in particle velocity moves it closer to the center of the vortex where the fluid velocities are lower. Imagining that the particle has now moved into a “ring” of slower moving fluid, it appears that as the particle moves toward the collector its velocity is enhanced by the flux velocity which would otherwise have pulled the particle closer to the collector or onto the “ring” of higher fluid velocity. However, since the velocity enhancement is less when the particle moves toward the collector, the flux force is not strong enough to move the particle as far out of the slower “ring” of fluid as before. The net result of the cycling is a constant reduction of fluid velocity and particle velocity. In addition, the flux force decays as the particle moves toward the well-mixed center of the vortex where there are no forces acting on it. Figure 12 illustrates the ratio of the number of particles captured on the front of the collector to the total collected, for particles of 0.1 and 0.5 µm radius on a droplet at 10 °C from a gas at 65 °C. The small particle sizes and relatively large temperature difference are chosen to promote rear capture. When the Reynolds number is low, the fraction captured on the rear of the collector is as high as 50%. For higher Reynolds


Figure 13. Effects of Reynolds number on flux deposition.

numbers most of the deposition occurs on the front of the collector. Although it has been shown that wakes can capture particles, its relative importance to flux deposition is low because most of the particles do not appear to reach the wake. Effects of Reynolds Number. The effects of Reynolds number on flux deposition rates is not as straightforward because of the complex interaction of the wake formation, increased inertial forces on the particle, and boundary layer thickness. Figure 13 depicts the dependence of collection efficiency of small particles on Reynolds number for various temperatures. When considering an explanation for this behavior, two factors must be considered: 1. An increase in Reynolds number would reduce thermal/diffusive boundary layer. Consequently, flux deposition is retarded because the region in which the particle can be affected by the flux forces will be smaller. 2. The same reduction in boundary layer thickness would increase the temperature/vapor gradients. Consequently, flux deposition should increase because of the increased flux forces in this boundary layer. Also, there is a greater inertial force moving the particle to the sphere at higher Reynolds numbers. The data in Figure 13 appear to support the first of these considerations as the controlling factor that governs flux deposition. An unexpected result that is present in Figure 13 is the remarkably linear relationship between the collection efficiency of small submicron particles and the Reynolds number. This linear relationship on a log-log plot suggests a correlation of the form E ) a‚Reb , where a and b are, generally, functions of the droplet and gas temperatures. It is clearly evident from Figure 13 that the slopes of the lines for different temperatures are relatively constant. This trend suggests that b is a constant that is independent of the temperatures of the gas and the collector. Therefore, a generalized expression for the flux deposition of small particles can be written as a product of two functions according to E ) F(Td,Tf) G(Re), where G ) Re-0.78. The function F reflects the change in flux deposition due to an increase in the temperature difference between the gas and the collector. The function G represents the change in deposition rate as a result of changes in Reynolds number. Since the amount of collection at the rear of the collector is small when the wake is most prominent, the simple expression for the effect of Reynolds number is the result of the reduction of the uniform portion of the thermal/diffusive boundary layer.

Figure 14. Illustrated minimum collection for Reynolds number of 30.

Effect of Particle Mass. Figure 14 illustrates the dependence of collection efficiency on particle size for a droplet with a Reynolds number of 30 in a gas stream at various temperatures. The minimum collection efficiency at each temperature can be clearly identified. It is evident that there is a shift to smaller particle sizes as the temperature decreases. The minimum collection efficiency occurs as a result of increasing masses of particles with increasing sizes. These trends can be explained by computing the trajectories for these various sizes. The smallest particle (≈0.1 µm) is influenced by the flow of the gas stream as the fluid initially draws the particle away from the collector. However, as the particle approaches the collector, its low mass offers little resistance to the strong radial flux forces. Consequently, it collides with the droplet. The net result of the particle having a low mass is relatively high collection efficiency. A particle with a slightly larger mass (≈5 µm) is not as affected as the smaller particle (≈0.1 µm) is by the sweeping tangential component of the gas velocity that flows around the collector. The slightly larger mass offers enough resistance to acceleration in the radial direction by flux forces. This resistance allows the particle to be swept past the collector. As a result, there is a decrease in collection efficiency for an increase in mass. However, a much larger particle (≈10 µm) is even less affected by the sweeping tangential component of the gas flow. The increased inertia allows for the large particle to be captured on the front of the collector. Consequently the collection efficiency increases for larger particles. Effect of Drop Deformation. Droplet deformation has been shown to be insignificant until a droplet radius is approximately greater than 1800 µm.23 Strictly speaking, for such large drops, the flow around the collector can no longer be considered steady because of the occurrence of vortex shedding.29 A time-dependent three-dimensional model would be needed to account for fluctuations that would occur in the collection efficiency as a result of the vortex shedding. However, simplification is achieved by assuming that potential flow conditions exist around the droplet. The primary collection mechanism for flows at high Reynolds numbers is inertial impaction due to the extremely small thermal/diffusive boundary layer. Figure 15 illustrates efficiencies calculated for water droplets whose radii are 3000 (Re ≈ 3600) and 4000 µm

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Figure 16. Rigorous recalculation of the data of Pilat and Prem. Figure 15. Collection efficiency for deformed drops.

(Re ≈ 4900). Since the collection efficiency due to inertial impaction is a function of the Stokes number (K) only, these calculations are compared to the potential flow solution for a spherical collector with (K) as the independent parameter. The data in Figure 15 show that the deformation of water droplets is responsible for a significant increase in collection efficiency for large particles over potential flow conditions. However, the greatest relative increase in collection efficiency is for particles smaller than 10 µm in radius (K < 2). For example, the collection efficiency of a 20 µm particle colliding with a 4000 µm collector is 1.38 times higher than the potential flow value for a sphere of the same size. A 5 µm particle is collected 1.68 times more efficiently by deformed collectors, which is a relatively larger increase in collection efficiency. The results can be explained by analyzing trajectories of two particles around a deformed droplet. Larger particles are found collected at the waist of the collector, whereas the smaller particles are collected on the protrusion at the front of the collector. It is known that the fluid deviates insignificantly from a straight path until a point very close to the collector. This flow behavior indicates that there is a larger radial component of velocity (relative to the velocity around an undeformed droplet) created as the fluid flows into the depressed frontal area. This increased radial velocity reduces the time during which the tangential velocity can carry a particle past the collector. The net result is the increase in collection efficiency. Comparison with Existing Models. The results of this work, shown in Figure 16, are compared with the only available data in the literature.19 Their model assumed that potential flow conditions described the flow field around the collector; inertial impaction, thermophoresis, diffusiophoresis, Brownian diffusion, and interception were responsible for the collection of particles; and the temperature, vapor, and particle distributions were linear in a thin boundary layer. It is evident that Pilat and Prem generally overestimated collection efficiencies by as much as 70% for a water temperature of 10 °C. There are two distinct reasons for the difference between the data of Pilat and Prem and this investigation. The first relates to the assumption of potential flow. They considered a collector diameter of 100 µm. This size corresponds to a terminal Reynolds number of 1.54, which is much too low for potential flow conditions. Rigorous numerical determi-

Figure 17. Low Reynolds number comparison of the present temperature profile with the simplified profile of Pilat and Prem.

nation of the flow field during this investigation indicates that it would be more realistic to assume viscous flow around the sphere. As a result of viscous flow conditions, lower collection efficiencies than determined by Pilat and Prem are to be expected.4 This decrease is due mainly to the reduction in the radial component of velocity. Since the particle is not driven to the surface as quickly, it experiences a longer contact with the tangential velocity that sweeps it around the collector. The second reason for the difference in the two collection efficiencies is related to the assumed temperature/vapor distributions. Figure 17 compares the linear, symmetrical profile assumed by Pilat and Prem with a numerically generated profile at the waist of the collector. The validity of the Pilat and Prem profile is very restricted because the flux gradients are overestimated in the thin boundary layer and the flux gradients do not exist outside the thin boundary layer. Overestimation of the flux gradients leads to an overestimation of particle collection. The exaggerated flux forces in the thin boundary layer would exert an overestimated flux force on a particle in this region. Consequently, this particle would be more easily captured than one influenced by the actual temperature/vapor distribution. On the other hand, the absence of flux gradients outside the boundary layer would suggest that the model of Pilat and Prem would underestimate flux deposition. As shown in Figure 17, the more accurate temperature/vapor distribution of the present model would predict flux forces outside the boundary layer assumed by Pilat and Prem. As a result,


a particle could be collected from outside the boundary layer if there was a high enough temperature difference between the droplet and the gas. Due to the absence of flux forces, it is impossible for collection to occur from the region outside the boundary layer in the Pilat and Prem study. Consequently, for the collector size studied by Pilat and Prem, the maximum possible collection efficiency, under any circumstance, by flux forces is essentially E ) 1.72 × 100 ) 296%. This value of 1.7 is nothing more than the dimensionless distance from the center of the collector to the outer edge of the boundary layer as illustrated in Figure 17. In short, the model of Pilat and Prem is applicable only at high Reynolds number region where fine particle collection becomes relatively unimportant. For low Reynolds numbers more realistic determinations of velocity and temperature/vapor distributions are required as illustrated by this work. Acknowledgment The author thanks Dr. Kevin Ellwood all help provided in association with this study. Nomenclature a j ) particle acceleration (dimensionless) A h ) particle acceleration (m/s2) a0 ) volume radius of the collector (m) Cf ) Cunningham correction factor (dimensionless) Cp ) heat capacity of the fluid (J/kg‚°C) C1 ) molar concentration of water vapor (mol/m3) C1∞ ) molar concentration of water vapor in the undisturbed gas stream (mol/m3) C1c ) molar concentration of water vapor on the surface of the collector (mol/m3) eji ) unit vector in the direction of Ri increasing E ) droplet collection efficiency (dimensionless) F h d ) diffusiophoretic force (kg‚m/s2) F h drag ) drag force (kg‚m/s2) hfe ) external force (dimensionless) F h e ) external force (kg‚m/s2) F h t ) thermophoretic force (kg‚m/s2) gi,j ) transformation of metric tensor element hi ) grid scale factor K ) Stokes number (dimensionless) Kn ) Knudsen number (dimensionless) kf ) thermal conductivity of the fluid (W/m‚°C) kp ) thermal conductivity of the particle (W/m‚°C) L ) vortex length (m) M1 ) molecular weight of water (kg/mol) M2 ) molecular weight of air (kg/mol) Nr ) interception parameter (dimensionless) NuL ) Local Nusselt Number (dimensionless) Peh ) Peclet number for heat transfer (dimensionless) Pem ) Peclet number for mass transfer (dimensionless) Pr) Prandtl number (dimensionless) Re ) Reynolds number (dimensionless) rj ) position vector (dimensionless) rp ) particle radius (m) Sc ) Schmidt number (dimensionless) T ) dimensionless time t ) time (s) Tf ) fluid temperature (K) u j ) fluid velocity (dimensionless) U h ) fluid velocity (m/s) vj ) particle velocity (dimensionless) V h ) particle velocity (m/s) V h d ) particle velocity due to diffusiophoresis (m/s) V h t ) particle velocity due to thermophoresis (m/s)

Y0 ) maximum vertical displacement from collector centerline Greek Symbols Ri ) transformed boundary conforming coordinates β ) dampening factor (dimensionless) η ) temperature or vapor distribution (dimensionless) Ff ) fluid density (kg/m3) Fp ) particle density (kg/m3) µf ) fluid viscosity (kg/m‚s) Ω12 ) diffusivity of water in air (m2/s) ψ ) stream function (dimensionless) φ ) discrete variable ξ ) vorticity function (dimensionless) θs ) angle of flow separation (degrees)

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Received for review March 22, 1999 Revised manuscript received July 20, 1999 Accepted August 4, 1999 IE990199S

Numerical Study of Particle Collection by Single Water Droplets

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