A TECHNIQUE FOR CALCULATING
OVERALL EFFICIENCIES
OF PARTICULATE CONTROL DEVICES
by
William M. Vatavuk
Monitoring and Data Analysis Division
ENVIRONMENTAL PROTECTION AGENCY
Office of Air and Water Programs
Office of Air Quality Planning and Standards
Research Triangle Park, North Carolina 27711
August 1973
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This report is issued by the Environmental Protection Agency to report technical
data of interest to a limited number of readers. Copies are available free of
charge to Federal employees, current contractors and grantees, and nonprofit
organizations - as supplies permit - from the Air Pollution Technical Information
Center, Environmental Protection Agency, Research Triangle Park, North Carolina
27711, or from the National Technical Information Service, 5285 Port Royal Road,
Springfield, Virginia 22151.
Publication No. EPA-450/2-73-002
11
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CONTENTS
LIST OF ABBREVIATIONS i.v
ABSTRACT v
BASIC DEVELOPMENT 1
APPLICATIONS TO SINGLE CONTROL DEVICES 3
Gravity Settling Chambers 3
Cyclones 6
Venturi Scrubbers 7
Electrostatic Precipitators 11
APPLICATIONS TO CONTROL DEVICES OPERATED IN SERIES 14
REFERENCES 17
LIST OF FIGURES
Figure Page
1 Cumulative Distribution Versus Particle Diameter 5
2 Reciprocal Efficiency Versus Ratio of Cut Size to Mean Diameter for Cyclone 8
3 Reciprocal Efficiency Versus e/y for Venturi Scrubber 10
4 Reciprocal Efficiency versus B/CT for Electrostatic Precipitator 12
5 n Control Devices Operated in Series 14
m
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LIST OF ABBREVIATIONS
A Electrostatic precipitator collecting surface, ft2
Bc Width of gravity settling chamber, ft
C Gravity settling chamber empirical factor, dimensionless
C' Venturi scrubber correlation coefficient, dimensionless
D Particle diameter, microns
Dc Cut size, microns
Dm Mean particle diameter, microns
esu Electrostatic units
Ej Overall collection efficiency of a single control device
Ejn Overall collection efficiency of n control devices operated in series
F Electrostatic precipitator migration velocity constant, microns"!
f(D) Frequency size distribution of particles
g Gravitational acceleration, ft/sec2
L Venturi scrubber: liquid injection rate, ft3/1000 ft3
Lc Length of gravity settling chamber, ft
MCi Total mass of particles collected by a control device, Ib
M-J Total mass of particles entering a control device, Ib
NC Number of parallel chambers in gravity settling chamber
Q(D) Size collection efficiency of a control device
V Gas volumetric flow rate, ft3/sec
Vj. Venturi scrubber: gas stream throat velocity, ft/sec
W(D) Migration velocity of a particle of diameter D in an electrostatic precipita-
tor, ft/sec
Y(D) Cumulative size distribution of particles
a Cyclone empirical size efficiency parameter, micron~l
3 Slope of a semi logarithmic particle cumulative distribution, micron-1
y Gas viscosity, Ib/ft-sec
p Gas density, lb/ft3
Pp Particle density, lb/ft3
W-j Width of the cyclone gas inlet, ft
N£ Number of revolutions the gas stream makes in the cyclone
Vj Inlet gas velocity to cyclone, ft/sec
En Charging field in electrostatic precipitator, kV/cm
E0 Collecting field in electrostatic precipitator, kV/cm
K Dielectric constant of particles collected in electrostatic precipitator
i v
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ABSTRACT
A generalized mathematical technique is developed to calculate the overall col-
lection efficiency of particulate control devices. Equipment operating parameters
and the size distribution of the particles in the inlet gas stream are used in the
calculation. The technique is successively applied to efficiency calculations for
settling chambers, cyclones, venturi scrubbers, and electrostatic precipitators.
Extension of this mathematical method is made to encompass control devices operated
in series. Specific examples are also included to illustrate the technique when it
is applied to single and multiple devices.
!
jKey words: control efficiency, particle size, settling chamber, cyclone, venturi
iscrubber, electrostatic precipitator, mathematical technique
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A TECHNIQUE FOR CALCULATING
OVERALL EFFICIENCIES
OF PARTICULATE CONTROL DEVICES
BASIC DEVELOPMENT
Before the effectiveness of a given participate control device can be deter-
mined, it is necessary that its overall collection efficiency be calculated under
existing operating conditions. Collection efficiency is a function of such operating
conditions as temperature, pressure, and velocity of the gas stream entering the
device; dimensions of the control device; and statistical size distribution of the
incoming particles. For most devices it is possible to determine either the theo-
retical or empirical functional relationship between efficiency and particle size.
That is:
Size collection efficiency = Q(D) (1)
Where: D = diameter of particle, microns
In many cases, the cumulative size distribution of the particles in the incoming gas
stream can be readily determined from a few measurements. The cumulative distribu-
tion i:; the function used to calculate the mass or weight fraction of particles in a
given sample having diameters greater than a stated diameter, D:
Cumulative size distribution = Y(D) (2)
Once known, the cumulative distribution can be used to determine the frequency dis-
tribution, which is used to calculate the masses of particulates that fall within
the various ranges of size (size-fractions).
The following mathematical development shows how the frequency size distribution,
f(D), can be found from the cumulative distribution, Y(D). Consider a particulate
sample of mass, M(lb), with diameters ranging from 0 to <=•. By definition, Y(D)
equals the mass fraction of particles larger than diameter D.
r° /*D
Y(D) =J f(x)dx = -/ f(x)dx (3)
D «•
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This relationship holds because Y(D) is the integral of f(D). Differentiating Y(D)
with respect to D:
/•D
d/dD [Y(D)1 = d/dD [-/ f(x)dx] = -f(D) (4)
oo
Now, consider all the particles in the size interval D to D + dD. The mass of these
particles collected by the device, dMc, would equal the mass of incoming particles in
the interval, Mf(D) dD, multiplied by the collection efficiency at size D, Q(D):
dMc = Mf(D) Q(D) dD (5)
To determine the overall collection efficiency, Ej, one would simply integrate dMc
from 0 to M and divide by the total incoming mass, M:
ET =!/ dMc =/ f(D)Q(D)dD (6)
" 0 0
This is the basic equation for overall collection efficiency. Equation 6 can be
further simplified by the substitution of f(D). For many, if not most, particulate
streams, the weight distribution very closely follows a log-normal curve, which is
a normal distribution skewed toward very small or very large particle sizes. Sub-
stitution of the log-normal distribution function into equation 6 would result in an
expression that probably could not be integrated to yield a closed-form function.
Either numeric or graphic integration would have to be used to calculate the overall
efficiency (see Example 1). A simpler approach can be taken if one recognizes that
for values of Y(D) ranging from about 0.15 to 0.85, the distribution approximates an
exponential of the form:
Y(D) = e-eD (7)
Where: 6 = slope of the semilog distribution, micron"'
The parameter @ can be obtained by plotting the log of the cumulative distribu-
tion of particle size-fractions against the particle diameters. These size-fraction
measurements can be obtained from one or more particle separation procedures such as
sieving (for coarse particles), elutriation, and sedimentation (for fine particles).
(It must be emphasized that equation 7 deviates from the actual weight distribution
for relatively small and relatively large particles.)
Recalling equation 4: f(D) = -d/dD [Y(D)j. Then, substituting equation 7 into
equation 4:
f(D) = -d/dD [Y(D)] = -d/dD [e^D] = 6e-6D (8)
Substitution into equation 6 yields:
dD (9)
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APPLICATIONS TO SINGLE CONTROL DEVICES
To obtain efficiencies, the above equations can be applied to some of the com-
monly used parti cul ate control devices; e.g., gravity settling chambers, cyclones,
venturi scrubbers, and electrostatic precipitators.
GRAVITY SETTLING CHAMBERS
According to R. T. Shigehara, the size efficiency of a settling chamber is
based on Stoke's settling law and such parameters as the spatial dimensions of the
device and the characteristics of the gas stream:
[Cg(pD - P)LCBCNC]D2
Q(D) - — P CCC - [k]D2 (10)
Where :
C = dimensionless empirical factor (use 0.5 if no other information is
available)
g = gravitational acceleration, ft/sec2
Pp = particle density,
p = gas density,
p = gas viscosity, Ib/ft-sec
V = volumetric flow rate, ft3/sec
Bc = chamber width, ft
Lc = chamber length, ft
Nc = number of parallel chambers: 1 for a simple chamber and N trays + 1 for a
Howard setting chamber
For a given set of operating conditions, the terms in brackets are considered to be
constants. Thus, substitution into equation 9 yields the overall collection effi-
ciency: /%
ET = 3/ kD2 e-SD dD
T 0 (11)
This integral must be evaluated in two parts: first, between the limits of D = 0 and
D = /Vk and second, between the limits of D = A/k and D = °°. This is necessary
because the size efficiency remains at a constant value of 100 percent for all par-
ticles having diameters greater than or equal to /Vk. Thus:
ET = fey kD2 e-SD do) + (e/ e-^D dD] = ^-{l - e-eM[l + 0//k] ) (12)
\ o / \ vVk / e2 \ /
Example 1 :
A settling chamber, operated as the primary cleaner on a heating plant spreader
stoker, has the following parameters:
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Empirical factor, C = 0.60
Particle density, pp = 150 lb/ft3
Number of parallel chambers, Nc = 1 (simple chamber)
Gas viscosity, y = 1.68-10'5 lb./ft-sec at 400° Fahrenheit, 1 atmosphere
Gas density, p = 0.0462 lb/ft3
Volumetric flow rate, V = 120 acf/sec
Chamber length, Lc = 20 ft
Chamber width, Bc = 10 ft
Gravitational constant, g = 32.2 ft/sec2
Slope of the semi logarithmic particle distribution, B =0.025micron~'
Substituting these values into equation 10:
k = Cg (pp - p) LCBCNC = ["(0.60)(32.2 ft/sec2)(150 1b/ft3)(20 ft)1
.68-10-5 lb/ft-sec)(120 ft3/sec) J
(10
18yV
[
[(
J(30.48-104 microns/ft)
k = 1.719-10-4 micron-2
/ \
And: ET = ^n - e-3//k" [l + e//k~]j = 0.3131 or 31.31 percent
It is interesting to compare this calculated efficiency with the value obtained
from a numerical integration method. In Figure 1 the cumulative distribution function,
Y(D), is plotted against the particle diameter. As the dotted lines on the graph
indicate, the weight fraction of particles falling between 10 and 20 microns, AY(D),
is 0.1723. The average size collection efficiency for this size interval corresponds
to the average particle size, 15 microns:
Q(15) = kD^v = (1.719-10'4 micron"2) (15 microns)2 =0.0387 or 3.87 percent (13)
From this equation, Table 1 can be constructed.
Table 1. AVERAGE OVERALL COLLECTION EFFICIENCIES
Size range,
microns
0 to 5
5 to 10
10 to 20
20 to 30
30 to 40
40 to 50
50 to 60
60 to 70
70 to 76
>76
DAV»
microns
2.5
7.5
15.0
25.0
35.0
45.0
55.0
65.0
73.0
--
AY(D)
0.1175
0.1037
0.1723
0.1341
0.1045
0.0814
0.0634
0.0493
0.0242
0.1496
Q(DAV)
0.00107
0.00967
0.0387
0.1074
0.2106
0.3481
0.5200
0.7263
0.9161
1.0000
Q(DAV)-AY(D)
0.00013
0.00100
0.00667
0.01440
0.02201
0.02834
0.03297
0.03581
0.02217
0.14960
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100
90
80
70
60
50
40
30
20
\
o
i-
hg
(£
te
f-
.-i
10
9
8
7
6
\
10 20 30 40 50 60 70 80 90 100 110 120 130 140
PARTICLE DIAMETER, microns
Figure 1. Cumulative distribution versus particle diameter.
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The overall efficiency is given by the sum of the products of AY(D) and
Ey =0.3131 or 31.31 percent (14)
Note that this is exactly equal to the calculated value.
CYCLONES
Unlike gravity settling chambers, no useful theoretical expression has ever been
developed for cyclones that ties operating parameters, equipment dimensions, etc. to
particle size to yield a size efficiency function. However, several attempts have
2
been made at deriving empirical functions. For example, Gallaer determined that the
cyclone size efficiency curve, when plotted on semi logarithmic paper, results in a
straight line, the equation of which is:
Q(D) = 1 - e-<*D (15)
Where: a = a constant for the particular cyclone in question, micron-1
Direct substitution of equation 15 into the basic equation results in:
/""
0 6 a + e (16)
3 4
Grove ' developed a method for replacing the constants a and e in this expression
with two commonly measured particle parameters, the cut size, Dc, expressed in
microns, and the mean particle diameter, Dm, also given in microns. By definition,
the cut size is the diameter corresponding to a size efficiency of 50 percent, or:
Q(D) = 0.50 = e"BDc (17)
The cut size can, in turn, be calculated from the following expression:
DC = -,
Where:
W-j = width of the cyclone gas inlet, ft
Nt = number of revolutions the gas stream makes in the cyclone (5 to 10 is
typical)
V-j = inlet gas velocity, ft/sec
Pp = particle density, Ib/ft3
p = gas density, Ib/ft3
y = gas viscosity, Ib/ft-sec
The mean particle diameter is defined as the size above which 50 percent of the
particles lie or where the cumulative distribution equals 0.50:
Y(D) = 0.50 = e"30"1 (19)
Combining equations 16 and 17, and solving for a yields:
ft)
(20)
-------
Finally, substitution of this expression for a into the Ej relationship yields:
p
The reciprocal efficiency, 1/Ey, is plotted against the ratio Dc/Dm = 6/a irvFigure 2,
Example 2:
A dry cyclone is used as a secondary cleaner following the gravity settling
chamber in Example 1. The device has the following operating characteristics:
Width of gas inlet, Uj = 1 ft
Number of revolutions, Nj- = 10
Inlet gas velocity, V-j = 41.4 ft/sec
Substitution of these parameters into equation 18 yields:
r * • tn \ I 9 (1.68-10-5 Ib/ft-sec) 1 ft „ .. ,..4 . ...
Cut size (Dc) = /— L «m> 4 .1... i. n». i -30.48-10^ microns/ft
V2TT.10 (150 Ib/ft3)(41.4 ft/sec)
- 6 microns
Because the size distribution is semi logarithmic, equation 19 can be rearranged
to yield:
Mean particle diameter (Dj = 0.025" micron'1 = ?'8 m1crons
Hence, the efficiency of the cyclone alone would be:
. _ 1 1 1
= 82-37 percent
VENTURI SCRUBBERS
The collection mechanism of venturi scrubbers is based on the impingement of
the particles on a spray of liquid (usually water) droplets. The efficiency of
collection is a function of water droplet size, velocity and viscosity of the gas
stream, and rate of liquid injection to the scrubber, as well as particle diameter and
5
density. The combined work of Ranz and Wong has resulted in a theoretical size
efficiency function, similar in form to that of the cyclone:
Q(D) = 1 - exp f-C'LD /VtPP 1 (22)
f
Where:
C'= dimensionless correlation coefficient (varies from 1 to 2; 1.5 is a
reasonable estimate)
L = liquid injection rate, ft3/!000 ft3
Vt = gas stream velocity through venturi throat, ft/sec
Pp = particle density, lb/ft3
-------
>-
o
$£ 2.0
o
o
EC
0.
O
0.5
2.5
1.0 1.5 2.0
RATIO OF CUT SIZE TO MEAN DIAMETER
Figure 2. Reciprocal efficiency versus ratio of cut size to mean diameter for cyclone.
-------
DW = average liquid droplet diameter, microns
u = gas viscosity, Ib/ft-sec
D = particle diameter, microns
For a given venturi, all of the terms above become constants, except D, so that
equation 22 can be simplified to:
Q(D) = 1 - e'YD (23)
Where: Y = C'L |VtpP (24)
\ 18DwM
The average liquid droplet diameter can, in turn, be calculated from the fol-
lowing relationship:
D 1^+28.51.1.5
Vt (25)
Substitution into the basic equation results in an expression for overall efficiency
identical in form to that of the cyclone:
r - Y _ 1 __
TTT i + 3/Y)
Refer to Figure 3 for a graphic representation of equation 26.
Example 3:
Because the spreader stoker described in Example 1 emitted fairly large quanti
ties of small-diameter particulates, it became necessary to install a venturi
water scrubber to follow the settling chamber and cyclone. The operating para
meters of this device were:
Correlation coefficient, C'=1.40
L-quid injection rate, L = 1.50 ft3/!000 ft3
Throat velocity, Vt = 250 ft/sec
Average liquid droplet diameter, Dyy = 100 microns
From equation 24:
y - (1.40)0.50) J (250 ft/sec) (150 1b/ft3)
18-100 microns (30.48-104 microns/ft)(l.68.10"5 Ib/ft-sec))
Y = 1.40 microns"^
The overall efficiency for the venturi is calculated from equation 26:
ET = 1 + (S/Y) = 1 + (0.025/1.40)= 98'25 percent
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3.5
3.0
2.5
o
z
UJ
o
o
o
a:
o_
2.0
1.5
1.0
0.5
0.5
1.0
1.5
2.0
2.5
ft/y
Figure 3. Reciprocal efficiency versus /?/y for venturi scrubber.
10
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ELECTROSTATIC PRECIPITATORS
According to Engelbrecht, the size collection efficiency of an electrostatic
precipitator also follows the familiar exponential function:
Q(D) = 1 - exp
Where:
A = collecting surface of the precipitator, ft2
V = volumetric flow rate of the gas stream through the device, ft^/sec
W(D) = migration velocity of a particle of diameter D in the precipitator,
ft/sec
This migration velocity is, in turn, dependent on several operating parameters, as
well as on the particle size:
W(D) =
12irw
-
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(3/5
Figure 4. Reciprocal efficiency versus /3/8 for electrostatic precipitator.
-------
Average effective collecting field, EQ ~ 1.15 kV/cm
Dielectric constant of particle, K = 7
Gas stream viscosity, y = 1.68-10"5 Ib/ft. - sec
Precipitator collecting surface, A = 2,000 ft2
Gas stream volumetric flow rate, V = 120 ft-Vsec
Therefore:
F - (1.92 kV/cm)(1.15 kV/on)(3.33 esu-cm/kV)2 L + 2J7-l( ] g/cm_sec2_esu2
(12ir)(l. 68-10-5 lb/ft-sec)(14.88 g-ft/lb-cm) |_ (7+2) J
F = 0.269 sec"1
According to equation 29:
x AF (2,000 ft2)(0.269 sec-T) . .-,
6 " 7" = (120 ft3/sec)(30. 48-104 microns/ft) = °'333 microns
Thus, the overall efficiency is:
1^(0.025/0. 333)
13
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APPLICATIONS TO CONTROL DEVICES OPERATED IN SERIES
Because it is necessary in certain operations to remove a higher percentage of
the incoming particulates than is possible with a single control device, two or more
o
devices are often operated in series. Van Der Kolk evaluated the overall efficiency
of series-connected cyclones. The following development extends his argument to
cases where different devices are operated in series. First, consider n devices
arranged as illustrated in Figure 5:
Figure 5. n control devices operated in series.
In this arrangement, the outlet stream of device 1 becomes the inlet stream of
device 2, and so forth.
Now, let the total mass of particulates entering each device be designated as
MI for device 1, M2 for device 2, etc., and the respective masses collected be
Again, consider the differential size interval D to D + dD. The mass of par-
ticulate entering device 1 in this size interval is:
= Mif(D)dD (31)
The amount collected by device 1, dMc, , simply equals its size efficiency, Q-|(D),
times the entering quantity, or:
dMq = M1Qi(D)f(D)dD (32)
The amount leaving device 1 and entering device 2 is:
dM2 = dMi - dMq = MI [1 - Q!(D)]f(D)dD (33)
Extending this argument to the nth device:
dMn = MT[I - Qn-i(D)] [1 - Qn-2(D)] ... [1 - Q](D)] f(D)dD (34)
14
-------
dMCn = M1Qn(D)[l - Qn-i(D)] [1 - Qn.2(D)] ... [1 - Q1(D)]f(D)dD (35)
The total mass collected by n devices would be:
dMCj = dMCi = dMq + dMC2 + . . . + dMCn,
i ~ I
or, by equation 35:
dMCT = Mif(D)dD {QT(D) + q2(D) [i - Q^D)] + . . . + qn(D) [i - qn_i(D)]
[1 - Qn-2(D)] ...Cl- Ql(D)]} (37)
Integration of equation 36 over all particle sizes (0 to °°) and division by the
inlet mass, M] , yields thj overall collection efficiency for the n devices:
1
'n MI^
J>C1
or
/""
ETn =J (Ql(D) + Q?(D)n - Ql(D)] + . . . + Qn(D) [1 - Qn ,
n 0 N-I
... [1 - Q1(D)]}{f(D)dD} (38)
Application of equation 38 to a specific case illustrates its properties. For
example, consider the series arrangement of a cyclone, followed by a venturi. In
this case, the respective size efficiency functions are:
Ql(D) = 1 - e-«D (cyclone) (39)
Q2(D) = 1 - e-YD (venturi) (40)
Substituting these into equation 38, and assuming that the frequency distribution is
represented by equation 8:
/oo
{1 _ e-«D + e-«D + [1 - e-YD]} 66-^° dD (41)
u
Simplifying, and substituting for a:
F _ a + Y _ BDm + YDC (,9}
T2 ' a + 6 + Y " 6(Dm + Dc) + vD, ^'
Example 5:
Again consider the spreader stoker and its various control devices described in
Examples 1 through 4. Suppose it were necessary to know the overall collection
efficiency of the cyclone and venturi operated in series. This can be done by
merely substituting the previously calculated values for a, e, and Y into
equation 42:
- (0.025 micron"')(28 microns) + (1.40 microns"^)(6 microns) _„ ,_
(0.025 micron-l)(28 + 6 microns) + (1.40 microns ^(6 microns) ' Percen
15
-------
As this calculation shows, the operation of these two control devices results
in an overall efficiency of 98.38 percent—only 0.13 percent higher than that
obtained when the venturi is used alone.
As equation 40 indicates, the overall efficiency for the devices is a function
of their operating parameters, k, a, and y, and the frequency distribution parameter,
3.
In conclusion, one important point should be emphasized. That is, there is no
assurance that the operating parameters (a, Y> <$» and k) for control devices operated
singly would remain constant for the same devices arranged in a series. For instance,
experience has shown that the cyclone parameter a is directly dependent on, among
other things, the inlet dust concentration. This concentration, in turn, varies with
the number of devices used and the operating configuration. Therefore, the equations
developed for series operations cannot be used to accurately calculate the control
efficiency unless the respective equipment parameters remain constant. In any case,
actual field testing at operating conditions should be performed to adequately
evaluate the system.
16
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REFERENCES
1. Shigehara, R.T. Settling Chambers. In: Control of Particulate Emissions.
U.S. Environmental Protection Agency, Office of Air Programs, Air Pollution
Training Institute, Research Triangle Park, N.C. January 1970.
2. Gallaer, C.A. and J.W. Schindler. Mechanical Dust Collectors. J. Air Poll.
Control Assn. 1:3:574-580, December 1963.
3. Grove, D.J. Cyclones. In: Control of Particulate Emissions. U.S. Environ-
mental Protection Agency, Office of Air Programs, Air Pollution Training Insti-
tute, Research Triangle Park, N.C. 1971.
4. Personal communication with W. Smith and D.J. Grove. U.S. Environmental Pro-
tection Agency, Office of Air Programs, Air Pollution Training Institute,
Research Triangle Park, N.C. April 1972.
5. Ranz, W.E. and. J.B. Wong. Industrial Engineering Chemistry. 44:1371ff. 1952.
6. Englebrecht, H.L. Electrostatic Precipitators: Control and Industrial Appli-
cation. In: Control of Particulate Emissions. U.S. Environmental Protection
Agency, Office of Air Programs, Air Pollution Training Institute, Research
Triangle Park, N.C. 1971.
7. White, H.J. Industrial Electrostatic Precipitation. Addison-Wesley Publishing
Co., Inc. Reading, Mass. 1963.
8. Van Der Kolk, H. Linking Cyclones in Series and Its Effect on Total Separation,
Communication to Second Symposium on Cyclones. Utrecht, The Netherlands.
December 10, 1968.
17
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