Capture of Particles on Spheres by Inertial and Electrical Forces

Proudman, I., Pearson, J. FI. A., J. FluidMech., 2, 237 (1957). Smirnov, L. P., Deryagin. B. V., Kolloid. Zh., 29, 400 (1967). Smvthe. W. R.. "Static and [Nnamic Electricitv." 2nd ed. McGrawHill. New Yak. N.Y., 1950. Whipple, F. J. W.. Chalmers, J. A., Quart. J. Roy. Meteorol. SOC., 70, 103 (1944). Zebel, G.. J. Colloid Interface Sci., 27, 293 (1968).

Received for review February 25,1975 Accepted February 19,1976

This work was supported by the Engineering Research Institute, Iowa stateuniversity,A ~ I ~~ K, ~A. ~ ~, ~ i. was supported ~ l by a~ NDEA (Title IV) Fellowship.

Capture of Particles on Spheres by Inertial and Electrical Forces Kenneth A. Nielsen and James C. Hill'

Downloaded by RUTGERS UNIV on August 25, 2015 | http://pubs.acs.org Publication Date: August 1, 1976 | doi: 10.1021/i160059a002

Department of Chemical Engineering and Nuclear Engineering and Engineering Research Institute, lo wa State University, Ames, lowa 500 7 I

Target efficiencies for the collection of fine particles by a single spherical collector in a gaseous flow field are calculated from the equations of particle motion for the combined influences of particle inertia and electrical forces. The electrical forces considered are coulombic forces, electrical image forces, and external electric field forces. Potential flow and Stokes flow are used to model the gas flow around the collector. Under favorable conditions all of the electrical forces considered can significantly enhance particle collection. Although maximum collection is favored by large particle inertia when electrical forces are absent, collection in the presence of strong electrical forces often reaches a maximum for negligible particle inertia. In general, the introduction of particle inertia to a given electrical force situation produces a greater change in collection efficiency for potential flow than for Stokes flow.

Introduction Analysis of the collection efficiency of particles in wet scrubbers has been inade much easier with the recognition that in most situations inertial impaction of particles onto spherical droplets is the primary mechanism for particle collection, and the many computations and experimental studies of this mechanism rnay be used. With the realization that other mechanisms must be employed to capture very fine particles in wet scrubbers, computations of particle collection under the influence of forces in addition to inertial forces have begun to appear in the literature. For example, Sparks and Pilat (1970) have estimated the combined influences of diffusiophoresis and inertial impaction on particle collection in wet scrubbers. In this paper we examine the combined influence of electrical forces and inertial impaction on particle collection. Inertial Impaction. Herne (1960) has reviewed the work of earlier authors on the computation of particle collection efficiencies for inertial impaction on spheres without electrical forces. Most of the works reviewed involved attempts to solve the equations of motilon before the advent of high speed digital computers made more accurate numerical solutions practical. The most significant early attempts were made by Langmuir and Blodgett (1945) fix potential flow and by Langmuir (1948) for Stokes flow around a sphere. Herne also presented the results of his own numerical integration calculations for potential flow and Stokes flow. Langmuir found a critical Stokes number of impaction of 1.213 for Stokes flow and Michael (1968) derived analytically the critical Stokes number for potential flow as ylz. Michael and Norey (1969) more recently have calculated nunierically the collection efficiencies for inertial impaction in potential flow. Using a perturbation analysis, they derived an expression for the collection efficiency for Stokes numbers greater than five, 7 = 1 - 1/K 1/K2 + O(l/K3) where higher order terms are negligible. Attempts a t experimental verification of the collection efficiencies of particles on spheres by inertial impaction have

+

been made by Ranz and Wong (1952), Walton and Woolcock (1960), Jarman (1959), and Goldshmid and Calvert (1963). The results generally confirm that below a critical Stokes number collection by inertial impaction does not occur. The collection efficiencies obtained agree approximately with those predicted for potential flow. Electrical Forces. In the previous paper (Nielsen and Hill, 1976), hereafter referred to as paper I, theory is developed and previous work is reviewed on the collection of particles having negligible inertia, using various electrical forces. Analytical results show that the collection efficiency of inertialess, point particles under the influence of coulombic and/or external electric field forces is the same for a family of flow velocity profiles which includes potential flow and Stokes flow. Combined Inertial and Electrical Forces. Recently, two studies were reported in which the combined influences of particle inertia and coulombic forces were used to calculate particle collection efficiencies for potential flow around a spherical collector. The first study, by George and Poehlein (1974), presented results different from those presented here. Examination of those results by the present authors (Nielsen and Hill, 1974) revealed inadequacies in the calculations by George (1972), whose thesis was the basis of the paper by George and Poehlein. The second study, by Pilat et al. (1974), presented a calculation made by Sparks (1971) which included Brownian diffusion as well as electrical attraction, but considered collection only on the upstream surface of the target sphere. There have also been a number of studies (e.g., Sartor, 1960; Plumlee and Semonin, 1965; Semonin and Plumlee, 1966) of the collection of free-falling water droplets, considering both inertial and electrical forces. The results of those studies are difficult to compare to the present one, since they usually involve the collision of droplets whose sizes are the same order of magnitude (and for which hydrodynamic interactions should be considered) and also consider the influence of gravity. Ind. Eng. Chem., Fundam., Vol. 15,No. 3, 1976

157

~

Table I. Dimensionless Fluid Velocities Potential Flow

u, = 1 + (xz- 2 2 2 ) / 2 ~ 5 Ux = -3ZX/2R5 Stokes Flow

+

U, = 1 - ( 3 + 1/R2)/4R 3Z2(1/R2- 1)/4R3


Ux = 3 Z X ( 1 / R 2- 1)/4R3

+

This paper presents collection efficiencies of fine particles on a single spherical collector in gaseous flow fields, in Stokes and potential flows, under the combined influences of particle inertia and an electrical force. The electrical forces considered are (1)the coulombic force between a charged particle and a charged collector, (2) the electrical image force between a charged particle and a neutral collector, (3) the electrical image force between a neutral particle and a charged collector, (4) the force on a charged particle in the presence of a neutral collector by a uniform external electric field directed parallel to the flow field, and (5) the electric dipole interaction force between a neutral particle and a neutral collector, both bodies being polarized by a uniform external electric field directed parallel to the flow field. Combinations of the electrical forces are not considered in this paper. Mathematical Model Equations of Motion. The equation of motion for a small spherical particle in a gas is given in paper I (for a single electrical force) as

where u is the local mean velocity of the fluid, vis the velocity of the particle, and Fe is the electrical force on the particle. The equation of motion may be written in dimensionless form as dV K - V=U dT

+

eters K , are defined in the Nomenclature and the rectangular Cartesian vector components off, are listed in Table 11.The nature of these forces and the approximations involved in obtaining the tabulated forms of fi, and fip are described in paper I. The equation of particle motion, eq 2, consists of two coupled second order nonlinear differential equations, one for each vector component of V. Solution of eq 2 gives the particle trajectory for a given Stokes number, electrical parameter, and initial conditions of particle position and velocity. The initial conditions used in the mathematical model are V, = 1 K,, and V, = 0 a t the far upstream location 2 = -03 and X = X,. For calculating collection efficiencies it is necessary to know the initial condition X, of the limiting trajectory for particle impaction, Xi. Particles having trajectories closer to the centerline than the limiting trajectory will be collected, while particles outside of the limiting trajectory will not be collected. We have found no exceptions to this working definition of limiting trajectory. The total collection efficiency 7,defined in paper I as the fraction of particles removed from the projected area swept out by the collector, is 7 = Xi2. The interception parameter .%? does not appear in the equation of motion but enters the problem as a boundary condition when calculating the limiting trajectory for impaction. Method of Solution. In general, no analytical solution exists for the particle trajectory and, hence, collection efficiency when the Stokes number is nonzero. Therefore, particle trajectories were found by numerical integration of the equations of motion. In order to avoid having to start the integration at a very large distance upstream from the collector, an asymptotic series solution was obtained from eq 2 for each component of the particle velocity, V, and V,. In the series expansion in 1/Z of the components of the flow velocity and electrical force, the X-coordinate dependence was approxi. dependence of Ux and f e x mated by setting X2 = X O 2 Linear on X , which results in V, being proportional to X, was retained, however. An analytical integration of the particle trajectory equation

+ Kef,

V, d X = V, dZ

(3)

up to a convenient starting point for the numerical integration gives a solution of the general form

where K is the Stokes number, K E 2CppUJ?p2/9~Rc, K , is a dimensionless characteristic particle mobility corresponding to the force F,, and f, is the dimensionless spatial dependence of the force. The coordinate system is the same as shown in paper I. Coordinates and velocities are made dimensionless with respect to the collector radius R, and the free-stream velocity U,. In this study, potential flow and Stokes flow are used to model the velocity field around the collector. The dimensionless flow velocities are expressed in component form as U = Uxax UZa, and are given in Table I. The electrical forces considered in this study are the same as those used in paper I. The various electrical force param-

+

X = X, exp( - 5 Di/Zi> i=l

(4)

where D;= Di(X,, K , Ke),The initial conditions for the numerical integration then lay on the trajectory X = X(X,, Z ) given by eq 4. The numerical integration was done in the rectangular coordinate system using a fourth-order Runge-Kutta scheme for the set of equations for dX/dT, dV,ldT, dZldT, and dV,/dT derived from eq 2. Only two equations are required for the case K 0. To avoid a significant accumulation of round-off errors, the calculations were performed in double precision on an IBM 360165 digital computer. Whenever a

Table 11. Electrical Force Expressions Force F,

e C

ic iP ex icp

158

Coulombic force C harged-particle image force C harged-collector image force External electric field force Electric dipole interaction force

fex

X/R - [ R / ( R 2- 1)' -X/R6

- 1/R3]X/R

fez

Z/R - [ R / ( R 2- 1)'-

1/R3]2/R

-Z/R6

+ yc(2Z2- X 2 ) / R 5 -[5Z2/R2- 1 + ~ , (+l4 Z 2 / R 2 ) / R 3 ] X / R 5 -[(3Z2 - 2 X 2 ) / R 2+ yc(3Z2- X 2 ) / R 5+ yJR3

3y c X Z / R5

Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976

1

- 1]Z/Rj

Table 111. Collection Efficiencies for Inertial Impaction in Potential Flow, Tabulated as Xi = $/2, with no Electrical Forces

Limiting trajectory values, X i Stokes no., K

This work (Nielsen,1974)

Michael and Norey (1969)

0.1

0.0049 0.1848 0.3257 0.4914 0.5872 0.6754 0.7586 0.8071 0.8618

0.011 0.186 0.327

0.2 0.3 0.5 0.7

1.0 1.5 2.0 3.0

3.5 4.0 5.0 6.0 7.0


a

K-I

0.8920 0.9115 0.9249 0.9347

Asymptotic large Stokes numbera

-

0.588 0.807 0.86 0.88 0.89 0.906

ficiencies for inertial impaction alone agree well with the results of earlier investigators (and, in fact, are probably more accurate), as shown in Table 111, for example; (2) results for collection of inertialess particles agree almost exactly with exact theory, as shown in Table IV; (3) for finite inertial and electrical forces, computed trajectories far upstream of the collector agree precisely with the asymptotic theory used to begin the calculations closer to the collector (Nielsen, 1974, includes extensive tabulations of the agreement for various K , K,, and X,); and (4) computed efficiencies for combined 0 electrical and inertial forces extrapolate smoothly as K to the known theoretical results for K p 0 or the previous computations of Kraemer and Johnstone (1955).

0.9014 0.9165 0.9280 0.9368

Michael and Norey (1969) derived the result X i 2 = 1 + K-* + O ( K - 3 ) valid for large K in potential flow.

trajectory intersected the collector, the particle was considered to be collected and the numerical integration was stopped. Different criteria were used for the different cases to decide that a particle would not ultimately be collected (and hence the integration could be stopped) when a trajectory did not intersect the collector: (1)when the limiting trajectories always graze the surface of the collector, integration was stopped when the particle distance from the collector began t o increase; (2) when a velocity node exists downstream from the collector in the inertialess case ( K = 0), integration was stopped when the particle reached a distance of 0.4 collector radii beyond the location of the appropriate velocity node; (3) when a velocity node exists upstream from the collector in the inertialess case, integration was stopped one collector radius beyond the center of the collector. Numerous selected trajectories were integrated further to verify the above criteria. The value of X,was then found by trial and error using an interval halving search procedure based on selecting initial upper and lower bounds of an interval of X,believed to contain X,. In the course of this study, 609 collection efficiencies were calculated (366 with potential flow and 243 with Stokes flow) for the five different electrical force cases with the interception parameter equal to zero. The results are given below in terms from which the collection efficiency can be readily calof X,, The validity and accuracy of the numerical culated as 7 = X,2. integration procedure is attested to by the following special results: (1)for the case of no electrical forces, collection ef-

Coulombic Force Results Particle Trajectories. The effect of particle inertia on particle trajectories with potential flow is shown sequentially in Figure 1for Stokes numbers of 0, 1.0, and 1.5, respectively, and K , = -2.0. Trajectories for Stokes flow are similar but are displaced further from the collector (see Supplementary Material). For comparison, the values of X, used for the complete trajectories in Figure 1 represent the same fractional values of the limiting trajectory X,/Xifor each K.This helps to illustrate changes in the deposition density of collected particles over the surface of the collector with increased particle inertia. Partial near limiting trajectories are also included to illustrate the sensitivity of near limiting particle paths to small changes in X,. Examination of Figure 1 and similar figures reveals several features of inertial collection of particles using attractive coulombic forces. When inertia is present particle trajectories may cross one another, may cross the downstream centerline, may increase in distance from the collector although inside the limiting trajectory, and may graze the collector. These features do not occur in trajectories of inertialess particles. In order to collect particles that have gone past the collector, the attractive coulombic force must overcome particle momentum directed away from the collector as well as the drag force, with the result that the collection efficiency is decreased from that of the inertialess case. The maximum distance downstream from the collector that a particle following the limiting trajectory will reach before being collected decreases with increasing particle inertia, and the particle never goes beyond the inertialess-case velocity node. For large Stokes numbers the effects of particle inertia become dominant over the effects of the coulombic force: particles that go past the collector are no longer collected and the limiting trajectory grazes the surface of the collector. An especially dramatic illustration of the effects of large values of particle inertia and coulombic forces is shown in Figure 2. In this situation X,is a multiple-valued function of the angle 0 a t impaction over a large portion of the collector. Collection Efficiencies. Collection efficiencies are given

Table IV. Comparison of Numerical and Exact Analytical Solutions for Collection of Inertialess Particles ( K = 0) with Coulombic Force in Potential Flow

Limiting trajectory, Xi Method of calculation Exact TheoryD Numerical with K Ob Numerical with K extrapolated to zeroc

Velocity node, R ,

K , = -0.5

K , = -2.0

K , = -0.5

1.414214 1.414215 1.414311

2.828427 2.828424 2.828440

1.1654 1.1650(+0.0008)

K , = -2.0

1.6180 1.6176(+0.0016)

a Xi and R , are taken from Nielsen and Hill (1976). Xi is determined from integration of eq 2 with acceleration terms omitted; R n is obtained by averaging values for computed trajectories closest t o limiting trajectory. c Xi is obtained by linear extrapolation of values from eq 2 for K = 0.01 and 0.02.


159


K

Figure 3. Collection efficienciesfor coulombic force in potential flow, plotted as Xi = q'/z vs. K for various K,.

Figure 1. Particle trajectories for attractive coulombic force in potential flowwith K , = -2.0 A, K = 0;B, K = 1.0;C, K = 1.5. Complete trajectories correspond to values of (X,/XJZ from li$ to loh in increments of %.

Figure 4. Collection efficiencies for coulombic force in potential flow, showing greater range of parameters than in Figure 3.

4.

py -

I

i

Figure 2. Particle trajectories for strong coulombic force and high particle inertia in potential flow with K = 5.0 and K , = -10.0. in Figures 3 and 4 for potential flow and Figure 5 for Stokes flow as plots of vl/z = Xi vs. K for different values of the coulombic parameter K,. Although equal when inertia is not present (K = 0), the collection efficiencies for the two flows become significantly different when particle inertia is introduced. For potential flow and -0.6 < K, < 0, as the Stokes number increases from zero, the collection efficiency passes through a minimum and then approaches 1 as the Stokes number becomes very large. For small values of -K, the drop in collection efficiency can be quite large. For example, for K , = -0.1 the efficiency drops more than 7 5 percent as K increases from zero to 0.1. For K, < -0.6 the collection efficiency drops off monotonically with increasing K and approaches a minimum of 1.Figure 4 shows that for small K and large -Kc, Xi is nearly linear in K . For this region the empirical equation 7 = (2(-K,)1'2 160

- 0.80K)'


(5)

Figure 5. Collection efficienciesfor coulombic force in Stokes flow, vs. K for various K,. plotted as Xi = gives the collection efficiency to within three percent for 0
10 as well since the slope dXi/dK at K = 0 for constant K, appears to approach a limit as -K, becomes large.

The introduction of particle inertia influences the collection efficiency to a lesser extent with Stokes flow, as seen in Figure 5 . For small negative values of K, there is a minimum in collection efficiency but the amount of decrease is small. T h e

0.4

c

0.1 t-

I

f

u

I///

'


Figure 6. Percentage of total collection attributed to the downstream half of the collector for coulombic force in potential flow.

regions of rapidly decreasing Xi in Figures 3-5 are characterized by limiting trajectories that go past the collector and return (coulombic force is dominant), whereas regions of increasing or slowly decreasing Xi are characterized by limiting trajectories that graze the collector (inertial forces dominate). The mechanism of particle interception is likely to be more important in the latter case than the former. Comparison of the collection efficiency calculations of George and Poehlein (1974) with ours presented in Figures 3 and 4 has already been made (Nielsen and Hill, 1974); despite computational deficiencies by George, order of magnitude agreement was obtained. The computations presented by Pilat et al. (1974) include image forces between particle and collector and Brownian diffusion, as well as coulombic and inertial forces, but consider collection only on the upstream side of the collector. Their results were presented as collection efficiency vs. particle size for a given set of physical conditions (droplet velocity, charging field strength, etc.). For the physical conditions reported, orders of magnitude higher collection efficiencies than ours were obtained for the smaller particles; their results are now suspect (Pilat, 1975),however, since the electrical force parameters seem to be much too small for significant enhancement of collection. Distribution of Collection. I t is of interest to know how much collection occurs, on the downstream side of the collector because the potential flow description is not realistic for this region. When K = 0 one half of the total collection occurs on the downstream half whereas, when K all collection takes place on the upstream half. For intermediate conditions, values of X, were calculated for trajectories which intersect the collector at 8 = ~ / 2 The . percentage of the total collection attributed to the downstream half of the collector for potential flow is shown in Figure 6. For -K, less than about 1, the fraction collected on the downstream side of the collector decreases from one half to zero as K increases, but for K c < -1.0, the fraction first increases from one half when K = 0 to a maximum value before decreasing to zero for very large values of K . Similar behavior was found for Stokes flow. The form of the deposition density function in potential flow was examined in detail for the case where the coulombic force parameter K , is held constant a t -10.0 and the Stokes number is varied. Fractional efficiencies, in the form of Xo2/v, vs. angular position 8 ,are shown in Figure 7. In the inertialess case X o 2 / v = (1+ cos 8)/2. For K = 1.5 coulombic forces still dominate, but inertia causes particles t o cross the downstream centerline and to be deposited in regions of negative 0 so that the fractional efficiency curve is multivalued, although continuous. With a Stokes number of 5.0 both inertial and coulombic force effects are important. The fractional efficiency curve is discontinuous and particles are collected over a wide range of the negative 8 portion of the collector. In fact, some particles have nearly reached the upstream half of the collector after having passed completely around the downstream

-

6 (d0F.M)

Figure 7. Fractional efficiencies for coulombic force in potential flow, plotted as X O 2 / qvs. B for K , = -10.0 and various Stokes numbers. q = 40( = - 4 K J for K = 0,26.11 for K = 1.5,7.434 for K = 5.0,3.527for K = 7.0, and 1.0 for K = m.

o.M

t

im

\

IY)

im

w

u

3

0

e (W) Figure 8. Deposition density function for coulombic force in potential flow, plotted as ( ( 0 ) l q vs. B for K , = -10.0 and various Stokes numbers.

half, and others the downstream centerline two or three times before they are collected, as shown in ~i~~~~ 2. When the Stokes number is 7,0 inertial effects become dominant and the collection resembles inertial impaction. The corresponding curves for the deposition density ((e), with respect to D, were evaluated with the equation

and are displayed in Figure 8. In the inertialess case the normalized deposition density is constant at 1/47r. For K = 1.5 the deposition density is nearly constant over a large portion of the collector but increases rapidly in the vicinity of the downstream centerline where inertial effects are most important. The curve for K = 5.0 is quite discontinuous due to the overlap of "layers" of particles that originate from different portions of the undisturbed flow (and from X, < 0 as well). The radial velocity at impaction was also calculated for the Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976

161

Figure 11. Collection efficiencies for charged-particle image force in potential flow,.plottedas X i = q1l2 vs. K for various Ki,. 2.

1

1

1

L

=o

I

1

1

,

1

/

1

1

I

1

)

I

1

I

l

I

1

l

"

,

Figure 9. Radial component of particle velocity at impaction for coulombic force in potential flow, plotted as V,/V,max vs. 0 for K , = -10.0 and various Stokes numbers. VrmaX= -10.0 ( = K c )for K = 0, -2.83 for K = 1.5, -1.96 for K = 5.0, -1.76 for K = 7.0, and -1.0 for


K=m.

x 0.5 i

0.

pp I

I

0.0

I

I 1 I

0.5

I

1

I

I .o

I

I

I

1

I l I .5

l

I

l

0.0

l

2.0

cases discussed above. Curves of the radial impaction velocity, normalized with respect to the maximum radial impaction velocity Vrmax (corresponding to a particle traveling along the cedterline) us. angular position are shown in Figure 9. In the inertialess case V , equals K , over the surface of the collector but equals cos 0 for infinite K . Etternal Electric Field Force Limiting trajectories for positive values of K,, in the presence of particle inertia look similar to those for purely inertial collection (see Supplementary Material). Because the limiting trajectories graze the surface of the collector for all values of K,, and K , the collection efficiencies would normally depend also on particle interception. Although particles are collected only on the upstream surface when K = 0, a small amount are also deposited in the region 70' < 0 < 90' when both K,, and K are large. An interesting situation arises with potential flow, for when K,, = 1/27, the collection efficiency is not affected by particle inertia. From eq 2 , written as

dZ

d2X

+ (1- 27,Ke,)(X2 - 2Z2)/R5

dX

(7)

K 7+ -= -(1 - 2y,Kex)3ZX/2R5 (8) dT dT it can be seen that if K,, = 1/27,, the viscous and electrical forces will cancel exactly for all values of the Stokes number, and that the particle velocity remains constant, yielding a 162

Ind. Eng. Chem.,Fundam., Vol. 15, No. 3, 1976

1

I .O

2.0

1

1

I

3.0

Figure 12. Collection efficiencies for charged-collectorimage force in potential flow; plotted as X i = q1/2 vs. K for various Kip.

Figure 10. Collection efficiencies for external electric field force in potential flow, plotted as X i = ql/* vs. K for various Kex.

d2Z

I

K

K

K -d T 2 + = 1 + K,, dT

0.0

collection efficiency of unity. Because the maximum 7 for a perfectly conducting spherical collector is 3 (with particles following the external field force lines to the collector), one would expect that 7 < 1when K,, < 1hand 1 < < 3 when K,, > Y2, for all values of K . These predictions are verified in Figure 10 (for y c = 1.0), which shows that particle inertia has little effect on the collection efficiency for values of K,, close to 0.5. For small K,, a minimum similar to that in Figure 3 occurs and, as one would expect, 7 does not change much with K,, for large values of K,,. The results for Stokes flow (see Supplementary Material) look similar but with no pronounced minima for small K . Charged-Particle, Charged-Collector, and Electric Dipole Interaction Forces The effect of particle inertia on trajectories for the charged-particle and charged-collector image forces (both are radial forces) is quite similar t o that shown in Figure 1 for coulombic forces, with possible overshooting of the central axis and with great sensitivity of the particle path near the velocity node to changes in X , (see Supplementary Material). The maximum distance downstream from the collector that a particle following the limiting trajectory will reach before being collected decreases with increasing particle inertia, and the particle does not go beyond the inertialess-case velocity node. The charged-particle image force is a short range force that becomes infinite a t the collector surface, and so for W = 0, limiting trajectories never really graze the collector surface itself, even for large K . Limiting trajectories for particles attracted by the charged-collector image force or the electric dipole interaction force do graze the collector (and are subject to capture by interception) for large enough K . For the electric dipole interaction force essentially all collection occurs on the upstream half of the collector. Collection efficiencies for the image forces in potential flow are plotted as $I2 = X i vs. K for selected values of the electrical force parameters in Figures 11-13. For the electric dipole

U = u/U, = dimensionless local mean fluid velocity U , = free-stream velocity = v/U, = dimensionless particle velocity X = x / R , = dimensionless distance from centerline xi = x J R C = value of X,for limiting trajectory X, = x , / R , = value of X a t 2 = --m 2 = z / R , = dimensionless distance along centerline Greek Letters y c = ( 6 , - tf)/(tc + 2 4 = collector polarization coefficient y, = ( t - q ) / ( t p 2 4 = particle polarization coefficient

V

~4-L-41

' ' , l o ' I:

I

ll!llI

'

'1 2.

Figure 13. Collection efficiencies for electric dipole interaction force in potential flow, plotted as Xi = v1/2 vs. K for various Kicp.


interaction force yc = 1.0 was assumed. All three figures show the occurrence of a minimum efficiency with variation of K a t small values of the electrical parameters. Corresponding plots for Stokes flow (13ee Supplementary Material) show less dependence on K and with the minima considerably repressed, just as in Figure 5 for the coulombic force. Conclusions The calculations presented here indicate that under favorable conditions all of the electrical forces considered can significantly enhance particle collection over that obtainable by inertial impaction alone. Although when no electrical forces are present collection is favored by increased particle inertia, in many cases involving strong attractive electrical forces the collection efficiency is greatest for negligible particle inertia and decreases with increasing particle inertia. In light of this, electrified wet scrubbers should operate with low relative droplet-to-gas velocities; since K c: U , and any K, a U,-l, smaller relative velocities decrease the effect of particle inertia and increase the influence of the electrical force with respect to viscous drag. In general, the introduction of particle inertia to a given electrical force situation produces a greater change in the collection efficiency 7 for potential flow than for Stokes flow. With weak electrical forces 7 often passes through a minimum as K increases from zero. When a strong attractive radial force is present, a substantial portion of the total collection may occur on the downstream half of the collector. Nomenclature C = Cunningham-Millikan slip correction factor Eo = uniform external electric field strength f, = dimensionless spatial dependence of the electrical force F, = net electrical force on particle K = Stokes number, 2Cp,UJlP2/9fiR, K, = coulombic force parameter, CQcQp/24?r2tfRpRc2wUo K, = dimensionless particle mobility corresponding to F, K,, = external electric field force parameter, CQ&,/ 6nRnuUn Ki, = charged-particle image force parameter, ycCQp2/ 24~~tfRJ?~~uU, Kicp = -eiectric -dipole interaction force parameter, 2~c~ptfCRp~Eo~IRc~Uo Kip = charged-collector image force parameter, y,CQc2Rp2/121r2t~h1,5~Uo Q c = collector charge Qp = particle charge R = r/Rc = dimensionless spherical radial coordinate R = R p / R , = interception parameter R , = spherical collector radius R , = radial coordinate of velocity node R , = spherical particle radius T = t U , / R c = dimensionless time r.

~

+

= cofiector dielectric constant = fluid dielectric constant t, = particle dielectric constant { = deposition density function, defined by eq 6 7 = Xi2 = total collection efficiency 6' = polar angle of particle position from the downstream tC

centerline = polar angle at which particles on the limiting trajectory impact the surface of the collector fi = viscosity pp = particle density 6'i

Subscripts c = coulombicforce e = any of the electrical forces, e = c, ex, ic, ip, icp ex = external electric field force f = fluid i = limiting trajectory ic = charged-particle image force icp = electric dipole interaction force ip = charged-collector image force n = velocitynode o = condition a t Z = -0)

Literature Cited George, H. F., M.S. thesis, Lehigh University, Bethlehem, Pa., 1972. George, H. F., Poehlein, G. W.. Environ. Sci. Techno/.,8,46 (1974). Goldshmid, Y., Calvert, S.. A.i.Ch.E. J., 9, 352 (1963). Herne. H., int. J. Air Poliut., 3, 26 (1960). Jarman, R. T., J. Agr. Eng. Res., 4, 139 (1959). Kraemer, H. F.,Johnstone, H. F., ind. fng. Chem.,47, 2426 (1955). Langmuir, I., J. Meteoroi.,5, 175 (1948). Langmuir, I., Blodgett, K., General Electric Research Laboratory Rept. No.

RL-225, Schenectady,N.Y.. 1945. Michael, D. H.. J. FiuidMech., 31, 175 (1968). Michael, D. H., Norey, P. W., J. FluidMech., 37, 565 (1969). Nielsen, K. A,, Engineering Research Institute Technical Report 74127, Iowa

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Received for review February 25,1975 Accepted February 29,1976

This work was supported by the Engineering Research Institute, Iowa State University. K. A. Nielsen was supported by a NDEA (Title IV) Fellowship. Supplementary Material Available. Further plots of collection efficiencies and particle trajectories, particularly for the Stokes flow and image-force cases, as well as some tabulated results (including interception) and typical parameter values (18 pages), will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D.C. 20036. Remit check or money order for $5.00 for photocopy or $2.50 for microfiche, referring to code number FUND-76-157.

Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976 163

Capture of Particles on Spheres by Inertial and Electrical Forces

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