There is no indication that this detection technique, examined over the concentration range of 100-10,000 ppm, should not also be effective a t lower concentrations (-1 PPd. Furthermore, in the event that a catalytic converter is not performing to full potential, yet peak broadening and tailing are negligible, the percent conversion is independent of sample concentration if the residence time in the converter is kept constant.
References Anderson, R. B., “Catalysis IV, Hydrogenation,” p 331, Chap 4, Reinhold, New York, N.Y., 1956. Bond, G. C., “Catalysis by Metals,” Academic Press, New York, N.Y.. 1962. Bufalini, J. J.. Gay, B. W., Brubaker, K. L., Environ. Sci. Technol., 6 (9), 816-21 (1972). Dwer, F. L.. “Hiah TemDerature Oxidation of Carbon Monoxide and Methane 1; a Turiulent Flow Reactor,” AFOSR Scientific Rept.. TR-72-1109, March 1972.
Hightower, F. W., White, A. H., Ind. Eng. Chem., 20, 10-15 (1928). Leighton, P. A., “Photochemistry of Air Pollution,” Academic Press, New York, N.Y., 1961. McKee, D. W., J. Catal., 8,240-9 1967. Medsforth, S.,J. Chem. SOC.,123,1425-69 (1923). Pichler, H., Aduan. Catal., 4,289-92 (1952). Porter, K., Volman, D. H., Anal. Chern., 34 ( 7 ) ,748-9 (1962). Randhava, S. S., Amirali Rehmat, Camara, E. H., Ind. Eng. Chem.. Process Des. DeueloD.. 8, (4). 482-6 (1969). Schwenk, V., Hachenberg, H., Forderreuther, M., Brennst. Chem , 42,295-300 (1961). Williams. F. W.. Woods. F. J., Unstead. M. E., J Chrornat Scz., 10,570-72, (1972). Received for review February 12, 1973. Accepted September 18, 1973. The authors would like t o acknowledge financial support toward completing this study from both the Air Force Office of Scientific Research, Energetics Division of the Directorate of Aeromechanics and Energetics (Grant AF-AFOSR-69-1649) and the Air Pollution Control Office, Enuironmental Protection Agency (Grant R-801194).
Capture of Aerosol Particles by Spherical Collectors Electrostatic, Inertial, Interception, and Viscous Effects Herman F. George and Gary W. Poehlein’ Department of Chemical Engineering, Lehigh University, Bethlehem. Pa. 18015
A mathematical model is presented to predict the effi-
W
ciency of capture of aerosol particles by spherical collectors. This model includes consideration for inertial, viscous, gravity, and electrostatic forces, and the interception phenomena. The model shows that the efficiency of collection can be improved by the presence of electrostatic charges on the particles. The model, based on potential flow around the collector, is written in dimensionless form. Model predictions are plotted in terms of the appropriate dimensionless groups and are compared with previous works that considered electrostatic and viscous forces separately. The collection of aerosol particles smaller than 10 p is an important and difficult area of air pollution control. These smaller particles are difficult to collect because they tend to follow the air streamlines through the collection equipment. Collection may be achieved by one or more of the following six mechanisms (Strauss, 1966): inertial impaction, interception, gravity, electrostatic attraction, diffusion, or temperature gradient motion. Inertial forces are normally most significant for aerosol collection in scrubbers and high-speed filters. Gravity forces and interception can be important for larger particles and slow-speed operations, while the relative magnitudes of the diffusion (Brownian Motion) and the temperature gradient mechanisms become greater for small particles. The temperature gradient mechanism is almost never significant for industrial collection processes. The diffusion mechanism can be important for low-speed filters but is not significant in scrubbers and high-speed fil-
l
46
To whom correspondence should be addressed. Environmental Science 8, Technology
ters. Thus, for these devices, higher collection efficiencies are likely to be achieved only through the inertial and electrostatic mechanisms. The book by Strauss (1966) and the papers of Davies (1952) and Goldshmid and Calvert (1963) discuss the various collection mechanisms and list references to numerous related papers. When a small particle approaches a collector, the balance between particle inertia, electrostatic forces, and fluid friction, which tends to pull the particle past the collector, determines if collision (and thus collection) is to occur. Theoretical consideration of this problem involves the application of the equations of motion to determine the limiting collision trajectory for particle collection. The simultaneous effects of inertial, electrostatic, gravity, and viscous forces are considered in this work. The results are compared with those of previous publications in which the inertia-viscous and electrostatic-viscous interactions were considered separately. Our work serves to connect these earlier results. The relationships can be used to estimate the magnitude of electrostatic charge necessary to improve collection by spherical collectors, such as droplets in spray scrubbers.
Theor? The trajectory of an aerosol-particle can be predicted from Newton’s law of motion ( F = mu) if the initial position, velocity, and applied forces can be specified. If the trajectory of such a particle intersects the surface of a collector and sticks, it is removed from the aerosol stream. The theoretical equations for calculating the trajectories of spherical particles approaching a spherical collector are presented in this section. These equations were numerically integrated on a digital computer to determine that trajectory which just intersects the collector surface (the limiting trajectory). The offset distance of this limiting trajectory was then used to compute a collection efficiency
.
assuming a sticking coefficient of 1.0. The applied forces due to electrostatic potentials, gravity, and stream viscosity were equated to the inertial forces for the trajectory computations. Physical System a n d Collection Efficiency. Figure 1 shows the geometry and coordinates of the two-sphere system. Rectangular and spherical coordinate systems are shown with their axes a t the center of the collector. Fluid streamlines and typical particle trajectories are also illustrated. Gravity is acting in the negative z direction. Trajectory calculations for an aerosol particle begin when the particle is 450 collector diameters upstream ( t o = -450 D c ) , At this position the z-velocity of the bulk gas stream toward the collector is u o and the velocity of the aerosol particle toward the collector is assumed to be the difference between the terminal velocities of the collector and the aerosol particle ( u o - u p ) . The > velocities are assumed to be zero. The starting point of t o = -450 Dc for trajectory calculation was arbitrarily selected to ensure that the calculated results are independent of starting position. The object of the numerical calculation was to determine the largest offset distance which will just lead to a collision (the limiting trajectory). The collection efficiency for this simple two-body system is defined as the ratio of the area from which all particles are collected to the projected cross-sectional area of the collector. This efficiency is given by Equation 1, where Yllm is the off-center coordinate of the limiting trajectory.
components of the motion equation in spherical coordinates are given by Equations 3 and 4.
r - component:
F,, -k F,, f F,,
=
[ :;;
m -- r
[$]*I
(3)
- component:
where F,, and Fso are the fluid resistance forces in the r and 0 directions, respectively, and Fgr and F g e are sjmilarly defined for the gravity force. Please note that Fe = F e r because the electrostatic force acts in the radial direction. Viscous Forces. The fluid resistance force (F,) can be represented by the Stokes equation, with the Cunningham correction factor (C) for molecular slip, as follows (Bird et al., 1960; Kraemer and Johnstone, 1955). -
F,
=
3 x p D P V-*
c
where is fluid viscosity, C is the Cunningham correction factor, and L?* is the relative velocity between the particle and the surrounding fluid stream. The quantity U* can be expressed as the difference between the particle velocity ( U p ) and the fluid velocity ( f i r ) as given in Equation 6.
i*
-
-
= UD
-
(6)
L',
The velocity components for the particle are:
dr dt
Aerosol Particle Motion Equations. The equation of motion for an aerosol particle can be expressed, in vector form, as follows:
Upr = -
de =
where F e is the electrostatic force, Fs is the fluid viscous force, Fg is the gravity force, m is the mass of the particle, V is the velocity, and t is time. Although the initial conditions are specified relative to the rectangular coordinate system, because of symmetry, the motion equations are most easil: olved in-the spherical system. The r and 0
2
-o
------O---
Figure 1.
FLUID STREAMLINES ELECTROSTATIC COLLECTION COLLECTION BY INTERCEPTION I N E R T I A L IMPACTION
Collection system and coordinates
"dt
The fluid velocity components depend on the type of flow around the spherical collector. The velocity component equations for potential flow as given by Equation 8 were used in this work.
Grauit)' Forces. The gravity forces (gravity-buoyancy), although not significant in many collection processes (exceptions include such items as large particles and low velocity filters), were included to make the motion equations consistent with the physical system and the initial
1 0
//I 1
0
0 5
Figure 2.
IO
I5
20
2 5
30
35
40
Inertial, gravity, and interception effects Volume 8, Number 1, January 1974 47
-
IO
-
z 06
-
Dimensionless Equations. When equations which describe a physical phenomena are written in dimensionless form, the coefficients are dimensionless groups. These groups are useful for concise correlation or presentation of computational and experimental results. The following dimensionless variables were used in this work:
>
r
l
0 5
0
l IO
l
l
l
I 5
i
l
A=
,
I
25
20 CP,VO
l
30
(11)
,
The dimensionless motion Equations 12 and 13 are derived by combining Equations 3 and 4 with the appropriate force components and introducing the variables and t. The results are:
I 35
Ob
{-ziiK}Dp
Figure 3.
I
I
2 r/D,
-
0.-JARMAN'S EXPERIMENTAL DATA
I
=
Comparison of theories and experiment
1
I l l 1
I
I
I / I 1 1 1
I
I
1
1
I l l 1
-
-
100 /,,I17
.z 0
2 -
E+
[I
+
&] @] sin
(13)
where the dimensionless coefficients $, G, and E S are given by Equations 14, 15, and 16:
0 01
01
IO
IO 0
ES
Figure 4.
Collection efficiency-electrostatic and inertial effects
conditions. The spherical coordinate components are given by Equation 9.
F,,
=
-mg
[1 -
F , = ~ mg[l -
I;
- cos 8
g]
(9)
sin
e
where, p p is the density of the aerosol particle, m is its mass, and p is the density of the fluid. Electrostatic Forces. Electrostatic attractive forces between two oppositely charged bodies (or the induced force when only one body is charged) can lead to collection efficiencies greater than 1.0. The force between two point charges can be calculated by Coulomb's law:
where r is the distance between the charges, Q1 and Qz are the point charges, and t o is the permittivity constant of the fluid. The point charge equation is valid for large separations but becomes inaccurate as the particle trajectory approaches the collector because the charge on the collector and particle is located near the surface rather than a t the center point, and it is not uniformly distributed on these surfaces. The problem of computing electrostatic forces for such systems has been examined by others and a technique called, "The Method of Images" (Kurrelmeyer and Mais, 1967) was used in this work. This procedure, which assumes that particle and collector are conductors, is ideally suited for iterative calculation with a digital computer. 48
Environmental Science & Technology
The solutions (experimental or theoretical) t o a differential equation depend on the boundary conditions as well as on the equation coefficients. Thus, additional dimensionless groups may be required to account for the boundary conditions. The boundary conditions for the particle motion equations are established by the limiting trajectory. If the aerosol particle is small compared t o the collector, the limiting trajectory is the path which passes within one collector radius ( D c ,2) of the collector center, thus just touching. Actually the limiting trajectory is the path which passes within (Dc + D p ) / 2 of the collector center. This consideration accounts for the interception mechanism mentioned earlier. The interception factor must be considered as D p /D, increases and a'nother dimensionless group (usually defined as R = D p / D , ) must be used to correlate results.
Results and Discussion The equations of motion for the aerosol particle were solved by numerical methods on a CDC 6400 digital computer. The programming used a Runge-Kutta-type integration algorithm to calculate the trajectory of the particle and a convergence scheme was used to obtain the limiting trajectory. Computations were begun a t 450 collector diameters upstream, a point where steady velocities of both particles can be specified precisely. Complete ?letails of programming logic are given by George (1972). Interception, Gravity, and Inertial Collection. Preliminary computations were carried out without consideration for electrostatic effects. Figure 2 shows the influence of inertia $, gravity G, and interception R on the collection efficiency, E The square root of the inertial impac-
tion parameter is plotted so that the abscissa is directly proportional to particle diameter. The center curve shows the relationship between efficiency and inertial forces when gravity and interception are unimportant. The efficiency is low for small particles and approaches a maximum of 1.0 a s $ increases. If the gravity force becomes more important (see bottom curve where G = 8.8 x lo-* $1. the efficiency is less. This effect is directly related to the system we chose to examine. Both particles were assumed to be falling in a stagnant stream. The gravity effect would be different in horizontal and downflow systems. The top curve ( R = 0.1) indicates the importance of interception for larger particles. The maximum efficiency, when interception is considered, is (1 + R ) 2 which is greater than 1.0. This effect is small for the troublesome particles below 10 j~ and will not be considered any further in this paper. Figure 3 shows a comparison of our results with the theoretical work of Langmuir and Blodgett (1945) and the experimental data of Jarman (1959). The dimensionless group, 4, as given in Equation 15 was used by Langmuir and Blodgett t o correct for the case where inertial drag becomes significant:
When and G are zero, as in the top curve, the two results are the same. For nonzero values of and G the curves differ as would be expected. The agreement between theory and experiment is good except for moderate values of $ . Electrostatic Collection. When electrostatic forces become important, small particles can be collected, even if they start a t offset distances greater than the collector radius (Figure 1). If the offset distance is greater than the collector radius, the electrostatic force must overcome inertial and viscous counterforces in order for collection to occur. Particle inertia, which aids collection when the initial offset is less t h a t L),$’2, tends to carry the particle past the collector when the offset is greater than D,:’2. Kraemer and Johnstone (1955) calculated efficiencies for spherical collectors as a function of electrostatic charges. They excluded. however, consideration for inertial and gravity forces; i.e.. $ = G = 0.0.Kraemer and Johnstone’s results are shown as the solid ($ = G = 0.0) line in Figure 4 where efficiency. E is plotted against the dimensionless electrostatic parameter. E S . The variation of the parameter. E.9. with system variables can be better understood by expressing the charges in Equation 16 in terms of surface charge gradient as follows: $J
where c p s is the surface gradient of charge. Equation 16 then becomes
If the surface gradients are constant and tion 19 reduces to:
D, >> D,, Equa-
100
, , , ,
0001
,,[
0 01
, , , , , ,,
, , , , , ,,
01
10
Figure 5. Collection efficiency-inertial
ES
, , , , , ,,
, ,, ,
10 0
100
and electrostatic effects
D
= kP
L.0
where k is the combined constants shown in the brackets of Equation 19. Values of E S can vary from near zero to above 1.0. The solid horizontal line at E = 1.0 represents collection efficiency when inertial forces are large (I) = m ) . The curves between these two extremes are the results of this work for inertial parameters, I), of 0.117, 1.17, and 11.7, respectively. When $ and E S are small, the collection efficiency is low, as would be expected. If E S is small, collection efficiency depends on the value of $, increasing to a maximum of 1.0 as $ increases. When electrostatic forces are large, efficiencies are greater than 1.0 but they decrease toward 1.0 as $ increases. This effect is shown clearly on Figure 5 where E is plotted against $. All curves converge toward E = 1.0 as $ increases. Conclusions The theoretical results presented cover some of the gaps between previous works on aerosol collection. Experimental data are now required to determine if significant improvements can be achieved by generating electrostatic forces in commercial scrubbing equipment.
Literature Cited Bird, R. B.. Stewart. FV. E.. Lightfoot, E. N., “Transport Phenomena,” Riley. New York, S . Y . , 1960. Davies. C. N., Inst. Mech. Eng. (London), Proc., 8, (1B). 185 (1952). George, Herman F., M S Thesis, Department of Chemical Engineering. Lehigh Lniversity, 1972. Goldshmid. Y..Calvert. S.,A . I . C h . E .J . , 9 ( 3 ) ,352 (1963). Jarman. R. T., J . Apr. Eng. Res., 4, 139 (1959). Kraemer, H. F.. Johnstone, H. F., Ind. Eng. Chem., 47 (12). 2426 (1955). Kurrelmeyer B.. Mais. R . H., “Electricity and Magnetism.” Van Nostrand Co., Princeton. N . J . , 1967. Langmuir. I.. Blodgett. K., “A Mathematical Investigation of Water Droplet Trajectories.” Army Air Forces Tech. Report No. 5418, 1945. Strauss. W.. “Industrial Gas Cleaning,” Pergamon Press, London, 1966. Received for reiieu, August 7, 197.3. Accepted .September 18, 1973. This project has been financed in part u.ith federal f u n d s from the Encironmentai Protection Agency under Grant N o . 1 ROl AP0127t501. The support f r o m Lehigh I‘nicersiti. is also ack nouledged.
Volume 8, Number 1, January 1974
49
