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Ind. Eng. Chem. Fundarn., Vol. 18, No. 4, 1979
ARTICLES
Effect of Packing Density on Collection Efficiencies of Charged Fibers Kuo-chung Fan and James W. Gentry’ Department of Chemical Engineering and Institute of Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Calculations were used to estimate the influence of adjacent fibers on the collection efficiency and deposition pattern of charged and neutral fibers. Significant qualitative differences in collection efficiency were predicted numerically as a result of increased packing density.
Introduction since the development of the H~~~~~gas mask in the 1930’~,there has been considerable discussion of the use of electrostatic forces to increase the effectiveness of fiber filters. Several investigators (Lundgren, 1965; Muhr, 1976a,b; Loffler, 1974) have reported increased efficiency for charged particles. In contrast, other experiments (Fan, 1977, 1978) have been reported in which little difference was observed between charged and neutralized particles. Experimental conditions between the experiments of Muhr and those of Fan differ in several important respects. These include the following: (1) The aerosols in Fan’s experiments were smaller (0.4-1.0 pm) compared to 1.5-10 pm for the Muhr experiments. (2) The particles in the Muhr experiments contained approximately two orders of magnitude more charge. (3) The fibers in the Fan experiments consisted of stainless steel cloth with a packing density of 0.13 compared to the polyamide fibers used by Muhr which had a packing density of 0.07. This paper differs from previous work in three important respects. First, the effect of neighboring fibers on both the electrostatic and velocity fields was considered. In some respects, the simulations presented here may be regarded as a synthesis of the calculations of Ariman (1976) which accounted for the effect of interacting cylinders on the electrical field but assumed potential flow, and the trajectory calculations of Nielsen (1976a,b) which considered a number of combinations of charged and uncharged particles and collectors. Secondly, a parametric study of the effect of packing density or the separation distance between fibers is presented. Thirdly, the pattern of particle deposition on the fiber is studied. This method is based on a procedure developed by Fan and Gentry (Fan, 1976). It suggests a method of comparison of theory and experiment based on electron microscopic analysis of the particle deposition pattern. The principal objectives of the study can be summarized as follows: (1)to test the feasibility of using deposition patterns to discriminate between collection mechanisms; (2) to determine the magnitude of electrostatic parameters necessary to show significant changes in the collection 00 19-7874/79/1018-0306$01.0010
efficiency; and (3) to quantify the effect of packing density on the collection efficiency including the competitive electrostatic interactions of neighboring fibers and their around the influence Of the Model The basic approach used in this model is trajectory calculations in which the momentum equation for the particle is numerically integrated. In order to apply this method it is necessary to describe the flow field around the cylinders, the physical alignment of the cylinders, and the electrostatic forces between particles and cylinder. I t is assumed that the particle is not affected by diffusion or by turbulence. The three-cylinder model applied below has the following properties. 1. The geometry of the three-cylinder model is indicated in Figure 1. The cylinders are arranged with their centers perpendicular to the principal direction of flow. The distance between the centers of adjacent cylinders is 2R,, the radius of the cylinders is RF, and the packing density p and R, are related by
p = - n- RF2 4 R,2 2. The flow field around the cylinder is described by the Kuwabara flow field. The Kuwabara flow field is a “cell” model (Kuwabara, 1959) which assumes that the flow near the cylinder surface is creeping flow, and the flow outside the cell boundary is the free stream velocity. The boundary condition for the Kuwabara model is that the vorticity vanishes at the cell boundary. With this model, the components of radial (UR)and angular (V,) velocity are given by
and 0 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979 307 THREE CYLINDER MODEL
where 1 3 P2 Ku = - - In p - - p - (4) 2 4 4 In eq 2 and 3, p is the packed density defined in eq 1, RF is the cylinder diameter, and Uois the upstream velocity. 3. The electrostatic forces acting on any particle within the Kuwabara cell are given by the vector sum of the electrostatic forces of the three closest cylinders. This is shown schematically in Figure 1 where Pi are the electrostatic forces associated with the j t h Cylinder. The X (the direction parallel to the flow) and Y (the direction perpendicular to the flow and perpendicular to the axes of the cylinder) components for the four cases discussed here are presented in Table I. These four cases are: (a) charged particles and uncharged cylinder which have an attractive image force; (b) oppositely charged cylinder and particles, which is designated below as Coulombic attraction; (c) an external electric field with dielectric particles and cylinders; and (d) an external dielectric field for charged particles and dielectric cylinders. 4. Outside the hypothetical cylinder R,, the velocity is uniform and the particles are not affected by electrostatic forces. 5. The cylinders have the same number and polarity of charge. With these assumptions,the equation for particle motion
+
(5) can be solved. In eq 5, U , P, and F are the dimensionless gas velocity, particle velocity, and time defined by u/u, (6)
o=
P = VIVO
(7)
and
F = t Uo/RF The electrostatic parameter K, is defined in the first column of Table I for each of the four types of electrostatic interactions considered, and Stk is the Stokes number defined by
P are the dimensionless electrostatic forces whose components in Cartesian coordinates are given in columns 3 and 4 of Table I. The calculational procedure is to assume a displacement 5’. Equation 5 is then solved numerically. If the center of the particle passes within a distance R, + RF of the center of the cylinder, it is captured. The value of displacement Y, for which trajectories beginning at Y > Y, will never pass within a distance R, + RF of the cylinder center and for which trajectories beginning at Y < Y, will always pass within R, + RF of the cylinder center is called the critical displacement. The assumption is made that particle trajectories do not cross. The single fiber efficiency is defined by E = Y*/RF (10)
3
Figure 1. Diagram of three cylinder model.
It should be noted that with this definition it is possible to obtain single fiber efficiencies greater than one. The overall efficiency is defined by
Eo = Y * / R ,
(11)
and is always less than or equal to one. In most calculations, the initial location was the boundary of the hypothetical cell. Several criticisms can be made regarding the model. Physically, the closest analogue is the layers of parallel cylinders. Model filters of this type with packing densities of 0.08 and 0.055 have been fabricated from polyamide. The central criticism that can be made of the “cell model” is that except for the outer layers, it seems unlikely the velocity outside the cell would be U,. Nevertheless, these models are widely used (Davies, 1973). Of the cell models, Davies recommends the Kuwabara model based on two sets of experiments (Kirsch, 1967a,b) at low Reynolds numbers. [Re C 0.051. A second criticism that can be made is that the ordered parallel uniformly sized cylinders do not correspond to actual fabric filters. However, they are not a bad representation of model grid filters which form the bridge between isolated single fibers and fabric filters. Finally, the model can be criticized on the nature of charges assigned to the cylinders. It is possible to envision a model filter consisting of conducting, grounded cylinders which would approximate the case of charged particles and uncharged filter, and it is possible to construct a model filter of parallel wires which have the cylinders maintained at a constant potential. But for nonconducting cylinders such as the polyamide fibers used by Loffler, it is improbable that one would have equally charged fibers of the same polarity. In answering these criticisms, the objectives of the study should be reiterated. These objectives include: (1)the use of numerical simulations to test the feasibility of using deposition patterns to discriminate between collection mechanisms; (2) a determination of the magnitude of electrostatic parameters needed to show significant changes in the collection efficiency. Specifically, this study was designed to examine the effects of packing density (R), inertial effects (indicated by the Stokes number (Stk),and electrostatic effects (indicated by the electrostatic parameters (Kc),Consequently,only one particle size RP and one cylinder size RF were used. (3) Although a more realistic model of the fiber filter would include particles
308
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
+
INAGE FORCE
h W
4:
5 E
0I3
-+*
I
KC= 05
X
2 5
Y p 4
-
w
Kc* 01
I -
STOKES
NUMBER
Figure 2. Single fiber efficiency as a function of Stokes number.
COULOMBIC
0
;’]
-’%(/
-
1
2
3
4
5 STOKES
6 7 NUMBER
K:20
7
.--
UNIFORM FIELD STRENGTH, Eo Dleleclnc port(cIcs Polarized cylinders
n n
H
+
+ +
8
9
1
0
Figure 3. Fiber efficiency as a function of Stokes number with Coulombic force.
;/
8 ,
X I %
I
FORCE
n
n
a
> 5 2
K
4 -
Y LL
KO :,
3 K
II
>(“
-
.. . .r
n
n
H
+ h n
h
. I
c?
0
2 II I
H
.,.
. n
n h
a
+ n
%
Y
II
d
K ;, I 0
-
/
5
I
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979 309 UNIFORM FIELD
Table 11. Comparison of Numerical Simulations (NoElectrostatic Effects) ( p = 0.11 and Stk = 0 . 5 )
STRENGTH, E.
Charged ~ o r l ~ c l e s Dielectric cyimders
Cp=0131 7
I
I
KC= 3
single fiber efficiency
intercept ion parameter, RPlRf
HarropStenhouse
FanGentry
0.030 0.072 0.340 0.510 0.700
0.006 0.10 0.25 0.45 0.63
0.005 0.11 0.26 0.43 0.62
Table 111. Comparison of Simulations of Single Fiber Collection Efficiency 0
I
2
3
5 STOKES
4
6 7 NUMBER
E
9
I O
Figure 5. Fiber efficiency as a function of Stokes number under uniform electric strength field (charged particles and dielectric fibers).
and the Stokes number S t k . In each of the simulations reported here, the packing density was 0.13 and the particle diameter was 0.79 pm. These values were chosen to correspond to the experimental parameters for the model filters used by Fan. The importance of the simulations presented in this section lies in that they suggest the magnitudes of electrostatic parameter necessary to show significant changes in collection efficiency. In Figure 2, the case of charged particles and a neutral conducting cylinder is given. The values of K, correspond to a heavily charged particle. For example, the NaCl aerosols used by Loffler would have a value of K, = 0.01 at 1 cm/s (a very slow flow rate) and K, N 0.0001 at 100 cm/s. For these larger values of K, there is a significant increase in the velocity. It is interesting to observe the minimum of efficiency with Stokes number which occurs for K , 0.01. For very large values of K, the collection efficiency decreases owing to inertia being less important than electrostatic attraction. In Figure 3, the case of oppositely charged particles and cylinders is considered. The effect on the collection efficiency is similar with a minimum being observed for K, 0.1. It is interesting to note that the collection efficiency approaches a limiting value of approximately -3.5KC as the Stk approaches zero. This is 10% higher than the value of -IIK, of Natanson (1957). The value of K, = 0.1 is approximately the same as used by Loffler. It corresponds to a fiber charge of lo7charges/cm and approximately 10oO elementary charges. In Figures 4 and 5, the case of an external electrical field is considered. The calculations are based on assumptions that the radius of the particle is small compared to the radius of the filter and that the influence of the field of aerosol particles and neighboring fibers on the polarization of the field can be neglected. Because of the novel features in this simulation, direct comparisons with literature values could not be made. However, the calculations could be compared with the simulations of Stenhouse (Harrop, 1969) for cases when electrostatic forces could be neglected and with the calculations of Yoshioka (Yoshioka, 1968) where electrostatic forces were considered for an isolated cylinder. Comparisons of our simulations with those of Harrop and Stenhouse are shown in Table 11. The simulations are compared for different values of the interception parameterthe ratio of particle to fiber radius at a Stokes number of Stk = 0.5 and a packing density ( B )of 0.11. It is apparent that the simulations yield the same results. In order to make a comparison with the Harrop calculation, it was necessary to use a Happel (Happel, 1959) flow field
-
-
electrostatic parameter, -K, 4 2 2 2
x 10-3 x 10-3
x 10-4 x 10-5
single fiber efficiency Yoshiokan
Fan-Gentry
0.08 0.064 0.054 0.054
0.08 0.065 0.055 0.055
a In the Yoshioka calculation, t h e flow field was based o n a Lamb flow field with R e = 0.2 which was modeled in this simulation with a Kuwabara flow field with a packing density of 0.0001630. F o r such a small packing density, t h e calculation can be regarded as a simulation of an isolated fiber.
rather than the Kuwabara flow field used elsewhere in this paper. The two flow fields are similar, differing only by a constant. A direct comparison could also be made with Yoshioka for collection efficiencies with an image electrical force. Calculations were carried out with an interception parameter (Rp/Rf) of 0.5 and a Reynolds number of 0.2. The comparisons as indicated in Table I11 show that our simulations agreed with this limiting case. Effect of Packing Density The packing density influences the trajectory calculation in two ways. First the flow field, as can be seen from the fact that the Kuwabara flow field eq 2 and 3 are explicit functions of P, is perturbed by the packing density. Secondly, as the fibers (or cylinders) are grouped closer together, the electrostatic force field from neighboring cylinders influences the trajectories of particles near the central cylinder. In this section, comparisons between simulations for values of the packing density ranging between 0.05 and 0.43 are made. The packing density ( P ) is defined in eq 1. For these simulations, the Stokes number was kept constant at 0.2 and the particle diameter was 0.79 pm. The Stokes number of 0.2 is sufficiently low that in the absence of electrical forces the collection efficiency is negligible. In the simulations presented here only the two cases of (i) charged cylinders and particles of opposite polarity and (ii) uncharged cylinders and charged particles are considered. The result for the case with both cylinder and particles charged is shown in Figure 6. The charging parameter KC
and the packing density P are the two parameters in the simulation. The interesting point is that the efficiency decreases with increasing packing density. This occurs because of the competitive interactions between neighboring cylinders. It is interesting that the overall efficiency defined by the ratio Y*/Rchas a relatively flat dependence on packing density, and these results are presented in Table IV. The maximum value of the overall efficiency
310
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979 COULOMBIC
>
FORCE
STK: 0 2
8 ,
I
7
-
6
-
5 -
z : 4 iL w
-K,:O01
> 5
z
y .4 W Y
3 W U
m
G 2 I
n 0
2 DENSITY
I PACKING
3
Figure 7. Fiber efficiency BS a function of packing density with image force. Table IV. Total Efficiency as a Function of Charging Parameter and Packing Density
demonstrating enhanced orientation of amosite fibers with nuclepore filters (Gentry, 1978),collection by diffusion for very fine aerosols, and the increase of bounce-off or reentrainment with increasing velocity (Fan, 1976). Quantitatively, the deposition pattern showed good agreement for the re-entrainment from 25-pm gold wires (Dahneke, 1977) but only fair agreement with the deposition pattern on model grid filters. In this section, a description of the procedure used to determine the deposition pattern, illustrative calculations demonstrating its use, and a brief discussion of its applicability are presented. The calculational procedure, consisting of repeated trajectory calculations, is based on the assumptions that the aerosol is uniformly distributed downstream from the fiber, that trajectory curves for different starting displacements do not cross, and that Brownian diffusion and gravitational forces can be neglected. The steps in the calculations are as follows. 1. The initial displacement Y, corresponding to the critical trajectory is found. Because of the assumption that trajectory curves do not intersect, particles beginning a t displacements less than Y, will be captured. Dahneke (1977) has demonstrated that when significant "bounceoff' occurs, particles can escape from the cylinder at displacements less than Y,. 2. The critical trajectory is subdivided into 10 or 100 equal intervals. The trajectories are computed until contact with the cylinder surface is made. The angle of contact 0 is determined from 0 = arcsin ( Y / R f )= arctan ( Y / X ) 3. The frequency or probability of displacement corresponding to an angle 0; was calculated from the relationship
-Kc P
0.3
0.1
0.05
0.01
0.05 0.10 0.15 0.20 0.25
0.167 0.204 0.219 0.230 0.231
0.058 0.074 0.086 0.089 0.101
0.031 0.039 0.045 0.052 0.059
0.006 0.008 0.010 0.012 0.013
would correspond to the minimum thickness of a filter for a given separation. The relatively weak dependence with packing density suggests an explanation for the successful application of highly porous filters. In Figure 7, the efficiency is presented as a function of the packing density for charged particles and uncharged cylinders. The Stokes number and particle diameter have the values of 0.2 and 0.79 pm. The electrostatic parameter K , is defined by
K, =
q2
24lI7R,U$~~
The results are significantly different. Particle Deposition Pattern In comparing experiment with theory, most investigators have measured the penetration and pressure drop through a test filter as a function of filter properties (thickness, porosity, etc.) and operational variables (flow rate, pressure, gas composition, particle size, etc.). An alternative to this approach is the use of electron microscopy to compare the particle deposition pattern with theory. Qualitatively, this procedure has proven successful
where N is the number of increments in which the critical displacement was subdivided. 4. The trajectory calculations were solved using a fourth-order Runge-Kutta procedure (Carnahan, 1969) with a variable step size. The beginning location for the calculation was the cell boundary. The computational procedure differs from that previously described only in the escape criterion. Since the flow field around the cylinder is symmetrical, the deposition pattern from 0 to 180' completely describes the deposition pattern. For some particles with initial displacements in the upper half plane of the cylinder, the trajectories intersect the cylinder in the lower half plane on the backside of the cylinder. That is, particles are deposited at angles between 180 and 270'. From symmetry, there is an equal deposition between 180 and 90' resulting from particles with initial deposits in the bottom plane. Consequently, when the trajectory calculation indicates deposition at 0 > l_SOo, this contribution is added to the deposition at 360 - 0. In Figure 8, a sample calculation is presented for the case of uncharged cylinder with charged particles. The packing density was 0.13 (corresponding to the values for grid filters) and the charging parameter [ K , ( 2 ) = q 2 / (24lI~$t,U&~~)] was 0.05. As indicated in a previous section, this value corresponds to a heavily charged particle and a low flow rate. Deposition patterns are displayed for three velocities. In Figure 8, 0' corresponds to the stagnation point. Little difference in the shape of the curves is indicated.
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979 311
4OOO CM/SEC 1000 C M / S E C --
500 CMISEC
---
K,= 05 p'013
50
0
I00
150
200
CONTACT ANGLE (DEGREES1
Figure 8. Deposition pattern as a function of contact angle with image force (Stk = 0.6, 1.2, and 4.8).
c
I6
- 500
Dimensionless Groups /3 = packing density (defined by eq 1) K, = electrostatic parameters (defined in Table I) Ku = Kuwabara number (defined by eq 4) R e = Reynolds number Stk = Stokes number (defined by eq 9) NR= interception parameter (Rp/RF)
CM/SEC
- - - 200 CM/SEC - K,= 0,3
Y
,81013
10
5 -1
Nomenclature
COULOMBIC FORCE
H
.>
(0 'C) and the backside of the filter (180'). For uncharged cylinders, the deposition pattern is much closer to the simulation when electrostatic forces are neglected. Quantitatively, it is doubtful if measurements would be sufficiently precise to obtain values of the electrostatic charging parameters from deposition patterns alone. It should be stressed that the simulations presented in this section were for very highly charged particles and cylinders. Acknowledgment K.C.F. and J.W.G. wish to acknowledge financial support under the National Science Foundation Grant No. ENG 78 00738. In addition, the computer time for this project was supported in full through the facilities of the Computer Science Center of the University of Maryland.
'\
\
Roman Letters
0
50
00 150 200 CONTACT ANGLE (DEGREES)
250
Figure 9. Deposition pattern as a function of contact angle with Coulombic force (Stk = 0.24 and 0.6).
Were there no charge on the particles, the deposition frequency between 90 and 270' would be zero. Although more heavily charged (the particles at 4000 cm/s have 2.83 times as many charges as those at 500 cm/s), the greater backside deposit (deposition at an angle greater than 90') results at slower velocities. The bimodal distribution occurring at low flow rates (indicated in Figure 8 by the maximum at 10') is due to the competition of neighboring fibers. For charged particles and filters, the deposition patterns as presented in Figure 9 are significantly different. The simulations were carried out for velocities of 200 and 500 cm/s. In this case, there were two peaks in the deposition pattern, one occurring a t 9' and the other a t 180'. The charging parameter
has the value 0.3. The essential difference between the two cases shown in Figures 8 and 9 is that the Coulumbic forces (Figure 9) have a longer range than the image forces (Figure 8). The effect of adjacent cylinders is at a minimum at 0" and 180'. For the short range image forces, only a slight effect (the bimodal distribution about 0 = 0) is observed and then only for the lower flow rates. In carrying out these simulations, very highly charged particles and fibers were used. The values of KJ1)and KJ2) used in the simulations were chosen sufficiently large that large changes in the collection efficiency occurred. The importance is that they demonstrate how particle deposition patterns can qualitatively discriminate between collection mechanisms. For charged cylinders, the deposition pattern is broad with the maxima occurring at the forward stagnation point
R = radial distance from center of central cylinder R, = radius of Kuwabara cell RF = radius of cylinder Rp-= radius of particle t , t = time, dimensionless time Uo= upstream gas velocity = dimensionless gas velocity V = dimensionless particle velocity X,Y = Cartesian coordinates F = electrostatic force vector Y* = critical initial displacement f = particle deposition distribution function Greek Letters B = angular coordinate qg = gas viscosity pp = pg
density of particle
= density of gas
Literature Cited Ariman, T., Tang, L., Atmos. Environ., 10, 205 (1976). Carnahan, B., Luther, H. A., Wilkes, J. O., "Applied Numerical Methods", Wiley, New York, N.Y., 1969. Dahneke, B., unpublished data. Unkrerstty of Rochester School of Medicine and Dentistry, 1977. Davies, C. N., "Air Filtration", pp 53-59, 84-87, Academic Press, New York, N.Y., 1973. Fan, K. C., Gentry, J. W., Powder Techno/.. 14, 253 (1976). Fan, K. C., Ph.D. Dissertation, University of Maryland, 1977. Fan, K. C., Leaseburge, C., Hyun, Y., Gentry, J., A f m s . Environ., 12(8), 1797 119781. a n t r y , J., SPurnY, K. R., J . Colloid Interface Sci., 65(1), 174-180 (1978). Hamel. J.. 5. 174-177 (1959). . , . J..~ AIChE . ~ - - ,
.
Harrop. J. A., Stenhouse, J. I. T., Chem. Eng. Sci., 24, 1475 (1969). Kirsch, A. A.. Fuchs, N. A., J . Phys. SOC. Jpn.. 22, 1251-1255 (1967). Kirsch, A. A., Fuchs, N. A., Ann. Occup. Hvg., 10, 23-30 (1967). Kuwabara, S.. J . Phys. SOC. Jpn., 14, 527-532 (1959). Loffler, F., Muhr, W., "A Study of the Influence of the Electrostatic Charge and of the Inertia of Particles on Their Deposltion in Fibrous Filters", GVCAIChE Meeting, Sept. 1974. Lundgren, D. A., Whitby, K. T., Ind. Eng. Chem. Process Des. Dev., 4, 345 (1965). Muhr, W., Loffler, F., Machinenmarkt, Wurzburg, 82, 669-672 (1976). Muhr, W., Ph.D. Dissertation, Karlsruhe, 1976. Natanson, G. L., Proc. Acad. Sci. USSR, Phys. Chem. Sect., 112, 95 (1957). Nielsen, K. A., Hill, J. C., Ind. Eng. Chem. Fundam., 15, 149 (1976). Nielsen. K. A., Hili, J. C., Ind. Eng. Chem. Fundam., 15, 157 (1976). Yoshioka. N., Emi, H., Hattori, M., Tamori, I., Kagaku a g a k u Chem. Eng. Jpn.. 32, 815 (1968).
Received for review September 6, 1977 Resubmitted March 5, 1979 Accepted July 2, 1979
I
