EPA-650/2-75-037
April 1975 Environmental Protection Technology Series
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EPA-650/2-75-037
A MATHEMATICAL MODEL
OF ELECTROSTATIC PRECIPITATION
by
John P. Gooch, Jack R. McDonald,
and Sabert Oglesby Jr.
Southern, Research Institute
2000 Ninth Avenue South
Birmingham, Alabama 35205
Contract No. 68-02-0265
ROAP No. 21ADJ-026
Program Element No. 1AB012
EPA Project Officer: Leslie E. Sparks
Control Systems Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Prepared for
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
WASHINGTON, D.C. 20460
April 1975
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EPA REVIEW NOTICE
This report has been reviewed by the National Environmental Research
Center - Research Triangle Park, Office of Research and Development,
EPA. and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the Environmental
Protection Agency, nor does mention of trade names or commercial
products constitute endorsement or recommendation for use.
RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environ-
mental Protection Agency, have been grouped into series. These broad
categories were established to facilitate further development and applica-
tion of environmental technology. Elimination of traditional grouping was
consciously planned to foster technology transfer and maximum interface
in related fields. These series are:
1. ENVIRONMENTAL HEALTH EFFECTS RESEARCH
2. ENVIRONMENTAL PROTECTION TECHNOLOGY
3. ECOLOGICAL RESEARCH
4. ENVIRONMENTAL MONITORING
5. SOCIOECONOMIC ENVIRONMENTAL STUDIES
6. SCIENTIFIC AND TECHNICAL ASSESSMENT REPORTS
9. MISCELLANEOUS
This report has been assigned to the ENVIRONMENTAL PROTECTION
TECHNOLOGY series. This series describes research performed to
develop and demonstrate instrumentation, equipment and methodology
to repair or prevent environmental degradation from point and non-
point sources of pollution. This work provides the new or improved
technology required for the control and treatment of pollution sources
to meet environmental quality standards.
This document is available to the public for sale through the National
Technical Information Service, Springfield, Virginia 22161.
Publication No. EPA-650/2-75-037
11
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ABSTRACT
This report describes a mathematical model which relates
collection efficiency to electrostatic precipitator (ESP)
size and operating parameters. It gives procedures for
calculating particle charging rates, electric field as a
function of position in wire-plate geometry, and the theo-
retically expected collection efficiencies for various
particle sizes and ESP operating conditions. It proposes
methods for empirically representing collection efficiency
losses caused by non-uniform gas velocity distributions,
gas bypassing the electrified regions, and particle reentrain-
ment due to rapping of the collection electrodes. Incorporating
these proposed techniques into a mathematical model of ESP
performance reduces the theoretically calculated overall
collection efficiencies. It compares the reduced efficiencies
with those obtained from measurements on ESPs treating flue
gas from coal-fired generating stations. It also presents the
effects of changes in particle size distributions on calcu-
lated collection efficiencies obtained from the mathematical
model. A procedure for estimating the program output by hand
calculation is given, and a complete listing of the FORTRAN
computer program is contained in an Appendix.
111
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TABLE OF CONTENTS
Page
Abstract iii
List of Figures v
List of Tables vii
Conclusions viii
Sections
I Introduction 1
II Description of Calculations 8
III Input Data and Program Output 63
IV Results 88
V Collection of High Resistivity Dust 105
VI Estimation of Program Output 108
VII Acknowledgements 115
VIII References 116
IX Appendices 119
IV
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FIGURES
No. Page
1 Simplified flow diagram of precipitator model
computer program • 6
2 Partial grid showing nomenclature used in the
numerical analysis 13
3 Nomenclature used in the numerical analysis 15
4 Potential profiles in a wire plate precipitator 18
5 Current density and electric field at the plates
as a function of displacement 19
6 Two dimensional physical model for developing a
charging theory 26
7 Model for mathematical treatment of charging rate 28
8 Comparison of charge values for 0.18 ym diameter
particle 38
9 Comparison of charge values for 0.28 ym diameter
particle 39
10 Comparison of charge values for 0.56 ym diameter
particle 40
11 Comparison of charge values for 0.92 ym diameter
particle 41
12 "F" as a function of ideal efficiency and gas flow
standard deviation 52
13 Degradation from 99.9% efficiency with sneakage 56
14 Correction factor for sneakage when
Ns=5 57
15 Effect of reentrainment on the efficiency of a
four-section precipitator designed for a no
reentrainment efficiency as indicated for a mono-
disperse particulate 60
16 Inlet size distribution obtained from modified
Brink impactor from power station burning an
Eastern coal 68
v
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FIGURES
(Continued)
17 Size data from Brink impactor measurements at inlet
of precipitator collecting a low sulfur Western coal 69
18 Current density as a function of resistivity 74
19 Comparison between the voltage vs current character-
istics for cold side and hot side precipitators 78
20 Voltage vs current characteristic for second field
clean electrode and 1 cm layer of IxlO^ohm-cm dust 79
21 Typical output information from computer model 85
22 Effective migration velocity as a function of current
densitv and particle size 91
23 Effective migration velocities for a full-scale
precipitator on a coal-fired boiler 93
24 Fractional collection efficiencies for a full-scale
precipitator on a coal-fired power boiler 95
25 Computed performance curves at 5 nA/cm2 96
26 Computed performance curves at 20 nA/cm2 97
27 Computed performance curves at 40 nA/cm2 98
28 Computed performance curves for "hot" precipitator 99
29 Effect of mass median diameter on computed
performance 102
30 Effect of particle size distribution standard
deviation on computed performance 103
VI
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TABLES
No. Page
1 Input data and data card format 64
2 Size fractionating points of some commercial
cascade impactors for unit density spheres 67
3 Input data for Figures 22 through 28 89
4 Theoretical effective migration velocities as a
function of current density, temperature, and
particle diameter 111
5 Example of manual calculation of overall
collection efficiency 112
VII
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CONCLUSIONS
Calculation of overall collection efficiency of polydisperse
particulate in an electrostatic precipitator from theoretical
relationships gives results higher than those obtained from
performance measurements on coal-fired power boilers. Correc-
tions to the idealized or theoretical collection efficiency
to estimate the effects of non-uniform gas flow, rapping
reentrainment, and gas by-passing the electrified sections
reduce the overall values of calculated efficiency to the
range of values obtained from field measurements. These
calculations suggest that the theoretical model may be used
as a basis for quantifying performance under field conditions
if sufficient data on the major non-idealities become
available. The computer model in its present state of
development is useful for qualitatively indicating performance
trends caused by changes in specific collecting area, electrical
conditions, and particle size distributions, provided that
back corona does not exist. Current density, applied voltage,
and the particle size distribution are the most important
variables in the calculation of overall mass collection
efficiency for a given specific collection area.
Vlll
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A MATHEMATICAL MODEL OF ELECTROSTATIC PRECIPITATION
SECTION I
INTRODUCTION
Because of the complexity of the factors which influence collec-
tion efficiency in electrostatic precipitators, it is necessary
to use a high speed computer to predict precipitator performance
from the applicable theoretical relationships. A computer model
of electrostatic precipitation has been developed by Southern
Research Institute as part of a research program sponsored by
the Environmental Protection Agency with the objective of
improving the understanding of the factors which influence
precipitator performance. The computer model predicts particulate
collection efficiency under ideal conditions as a function of the
dust properties and the operating parameters and includes
relationships for estimating the effect of gas velocity distribu-
tion, particle reentrainment, and gas sneakage.
In general, a comparison of the overall mass efficiency predicted
from the theoretical model with measured efficiencies obtained
under field conditions indicates that the theoretical projections
are higher than the field measurements. Corrections to the
idealized or theoretical collection efficiency to include
estimated effects of the previously mentioned non-idealities
reduce the theoretical values to the range obtained from field
measurements. These calculations and comparisons suggest that
the theoretical model may be used as a basis for quantifying
performance under field conditions, if sufficient data on the
major non-idealities become available.
The Southern Research Institute mathematical model uses the
Deutsch equation to predict the collection fraction rii-i for the
i-th particle size in the j-th incremental length of the
precipitator. Thus, the Deutsch equation is applied in the
form
-WJL .: A-i/Q
nif j = 1 - e 1':I D (1)
-------
where
w. . = migration velocity of the i-th particle size
1 ' J
in the j-th increment
Aj = collection plate area in the j-th increment
Q = volumetric flow rate.
Since the Deutsch equation is based on the assumption that the
migration velocity is constant over the collection area of the
precipitator , it is necessary to make the incremental lengths
sufficiently small so that the electric field at the plate and
the charge accumulated by a given particle size remain essentially
constant over the increment.
The collection fraction (fractional efficiency) r\^ for a given
particle size over the entire length of the precipitator is
determined from
~W °
= j _ = _ _ ! _ , (2)
where Nj. j is the number of particles of the i-th particle size
per cubic meter of gas entering the j-th increment. The quantity
N can be written in the form
-wi,j-l AJ-1/Q
"'"'-I-'- •> \
(3)
where N. -, = N^ Q / the number of particles of the i-th particle
size per cubic meter of gas in the inlet size distribution.
-------
The overall collection efficiency n for the entire polydisperse
aerosol is obtained from
= X,
n = LJ niPi , (4)
i
where PI is the percentage by mass of the i-th particle
size in the inlet size distribution.
The following list gives the major operations which are performed
by the computer program in evaluating equations 2 and 4:
1) Read input data, which include the particle size
distribution in the form of a histogram, applied voltage, total
current, total plate area, plate to plate and wire to wire
spacing, gas volume flow, precipitator length, gas temperature
and pressure, average density of dust particles, corona wire
radius, the standard deviation of the gas velocity distribution,
the percentage reentrainment and gas sneakage per stage, the
number of stages over which reentrainment and sneakage are
assumed to occur, and an estimated efficiency. Those data
which are dependent on a given electrical section of the pre-
cipitator are inputed in the sectionalized form.
2) Compute the number of particles in each size band
of the input mass histogram.
3) Divide the precipitator into .305 meter (1 foot) length
increments and compute the amount of material removed per incre-
ment from the estimated efficiency.
4) Calculate space charge due to particulate in each
increment based on the estimated efficiency per increment, and
calculate the reduced free ion density in each increment for use
in the calculation of particle charge.
-------
5) Compute the electric field at the plate and calcu-
late the average charging field from the electrode spacing
and applied voltage.
6) Calculate the charge on each size particle at the
end of each increment of length.
7) Calculate a migration velocity for each size particle
at the end of each length increment.
8) Compute the number of particles removed in each size
band after each length increment of travel from the Deutsch
equation.
9) Calculate the mass median diameter and the weight of
the collected dust for each increment.
10) After the required calculations have been performed in
all length increments, calculate an overall mass efficiency, and
compare this value with the input estimated efficiency. If the
difference is greater than 0.05%, the program returns to the
first length increment, and repeats all calculations using the
newly computed overall efficiency as the input value of efficiency.
Usually, only one iteration is required.
11) After convergence on overall efficiency has been
obtained, print the collection efficiency and compute the
effective migration velocity for each size range. Calculate
a precipitation rate parameter from the overall efficiency.
The above operations complete the calculation of
theoretical or ideal performance that would be expected under
a given set of input conditions. Corrections to these theo-
retical projections are obtained by operating on the effective
migration velocities for each particle size as follows:
12) For a given value of gas velocity standard deviation,
calculate a correction factor for the effective migration
velocities, using the theoretical efficiency for each particle
size.
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13) Calculate a correction factor for the effective
migration velocities, using assumed values of number of
stages and the percent loss per stage from reentrainment and/
or sneakage.
14) Obtain an "apparent" effective migration velocity
for each particle size by dividing the theoretical values by
the product of the two correction factors described above,
and calculate a corrected efficiency for each particle size
from the Deutsch equation.
15) Calculate a corrected overall efficiency and pre-
cipitation rate parameter.
The fundamental steps in the precipitation process are particle
charging, particle collection, and the removal and disposal of
the collected material. Therefore, in order to calculate the
efficiency of particle collection, it is necessary to mathemati-
cally model the electric field, the particle charging process,
the mechanism by which charged particles are transported to the
collection electrode from the gas stream, and efficiency losses
caused by reentrainment or other non-idealities.
Figure 1 gives a simplified block diagram of the precipitator
model computer program. The program is structured around
three major loops, the outermost of which is a direct itera-
tion that converges on the overall mass efficiency. An
initial estimate of the overall mass efficiency is required
because the space charge on the particulate at any point in
the precipitator is a function of the particle charge and
the number and size of particles remaining in the gas.
The program contains a calculation procedure which estimates
the effect of particulate space charge on the average free ion
density and the electric field near the collecting electrode.
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[ READ DATA
| CALC. NO. OF PART. IN EACH SIZE BAND)
CALC. NO. OF LENGTH INCREMENTS AND n0/INCREMENT FROM Ho ESTIMATE
r
| CALC. SPACE CHARGE DUE TO PARTICIPATE BASED ON n0 ESTIMATE h*-
ICALC. REDUCED FREE ION DENSITY FOR PARTICLE CHARGING CALC.|
PCOMPUTE AVERAGE FIELD FOR CHARGING |
f
| CALL E FIELD, COMPUTE FIELD AT PLATE[
CALL CHARGE, CALC. CHARGE ON EACH SIZE PART.
CALC. n FOR EACH SIZE FROM W FOR EACH SIZE |
| CALC. NO. OF PART. REMOVED IN EACH SIZE~|
[""SUM WEIGHT OF PARTICLES REMOVED |
|CALC. SIZE DISTRIBUTION TO NEXT SECTION!
I REPEAT FOR EACH PART. SIZE
ICALC. MMD AND WEIGHT COLLECTED FOR THIS INCREMENT |
REPEAT FOR EACH INCREMENT
| CHECK OVERALL COMPUTED HO WITH r\0 ESTIMATE, REPEAT IF REQUIRED |
REPEAT TILL CONVERGES t 0.05%
| CALC. EFFECTIVE WeFOR EACH SIZE|
t
| CALC. PRECIPITATION RATE PARAMETER |
»
| CALC. CORRECTION FACTOR FOR GAS VELOCITY}**
,
[CALC. CORRECTION FACTOR FOR REENTRAINMENT-SNEAKAGE |
| CALC. REDUCED EFFECTIVE We j
FCALC. REDUCE'D EFFICIENCY"]
I REPEAT FOR EACH PART. SIZE
I CALC. REDUCED OVERALL EFFICIENCY |
fCALC. REDUCED PRECIPITATION RATE PARAMETER"]
t '
I PRINT RESULTS |
I END |
Figure 1. Simplified flow diagram of precipitator model
computer program
6
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The second major loop includes the calculations which must
be performed in each incremental length, and the inner-most
loop contains the calculations dependent upon particle size.
The following sections present the mathematical relationships
used to calculate particulate space charge, electric fields,
particle charging rates, collection efficiency, and the
degradation of collection efficiency caused by non-ideal
effects. The calculation of particle charging rates is given
in considerable detail because the charging model was developed
under contract with EPA's Control Systems Laboratory and the
detailed mathematical development has not been published prior
to the date of this report. Input data format and a typical
example of the program output are also presented, as well as
a discussion of the determination of the input parameters, and
examples of typical results obtained from the program. The
Appendix contains a listing of the FORTRAN variables used in
the program and a listing of the main program with all of
the subroutines.
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SECTION II
DESCRIPTION OF CALCULATIONS
ELECTRICAL CONDITIONS
Space Charge Calculations
It is well known that the introduction of a significant number
of fine dust particles into an electrostatic precipitator
significantly influences the voltage-current characteristics
of the interelectrode space. Qualitatively, the effect is
seen by an increased voltage for a given current compared to
a dust-free situation. The increased voltage results from
the lowered mobility of the charge carriers which occurs as
the highly mobile gas ions are bound to the relatively slow
dust particles, thus creating a "space charge". It is desir-
able to determine the space charge resulting from dust
particles because this quantity influences the electric field
near the collecting electrode as well as the charging dynamics.
Also, the space charge is a function of location along the
length of a precipitator and must be determined on an
incremental basis along this length.
If we ignore the presence of free electrons, the current
density at the collecting electrode results from charge trans-
ported by both ions and particulate in accordance with the
relationship
JT = E0Pibi + E0ppbp = E0PTbe (5)
-------
where
JT = total current density, amps/m2
E0 = average electric field, volts/m
Pi = charge on ions, coul/m3
t>i = ion mobility, m2/ (volt-sec)
Pp = charge on particles, coul/m3
bp = particle mobility, m2/ (volt-sec)
PT = Pi + Pp
be = effective mobility of ions and particulate,
mV(volt-sec)
Thus, an effective mobility may be defined as
be = P . (6)
PT
The values of be are estimated as follows. First,
JP = E°pb
pP
JT EoPpbp + EoPibi jp + Ji
The quantity jp is estimated by (a) calculating the total
charge on all particulate present in the inner-electrode space
in a given length increment using the saturation charge from
field charging, and (b) multiplying this value of charge by an
estimated removal rate for the length increment under considera-
tion to obtain the coul/sec, or current, transported by the
particulate. If it is assumed that the current density due to
the particulate is only a small fraction of the total current
density (Ji»jp) and that the mobility of the ion charge carriers
is, on the average, 200 times that of the particulate, then
-------
and
. = Ppbp/Pibi ^ Ppbp ^ PpbP _ PP (9)
JT Pb/Pibi + 1 Pibi pi(200bp) 200Pi
Oglesby and Nichols1 have shown that equation 6 can be
re-arranged to yield
bi (10)
be = IPi + Pp - Pp d - bi )] p^ '
Now, since we have assumed that bi = 200 bp,
be = ^i + Pp - Pp (1 - 0.005)] £L * £) bi (ID
The value be can therefore be estimated from a knowledge of
carrier ion mobility and the ratio of ionic space charge to
total space charge. The space charge ratio can be obtained
by manipulation of equation q :
= 200 IB- '
Pi JT
pi PT _ 200 j
Pi Pi PI DT
Substitution of equation (12) into equation (11) gives
JT _
be = bi (200 jp + JT ) • (13)
Although the foregoing procedure provides a basis for esti-
mating the effect of particulate space charge, several of
the assumptions are of questionable accuracy. Specifically,
• The current carried by the particulate may be
significant.
• Particle mobility varies with size, and the
assumption that bi = 200 bp on the average
10
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may be considerably in error.
• For small particles, the saturation field
charge is not an appropriate value of charge
to use for estimation of the particle contri-
bution to current.
As a result of these problem areas, additional work on
developing a more accurate space charge calculation pro-
cedure is planned.
11
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Electric Field Calculations
Since the particle migration velocity is a function of the elec-
tric field at the plate, it is necessary to calculate the electric
field adjacent to the collection electrode. The method employed
for this calculation is a numerical technique introduced
by Leutert and Bohlen.2 The equations which must be solved
are written in discrete form in two dimensions as
A2V . A2V p , .,,»
7 — 5- + T — r = - ±— , and (14)
Ax2 Ay2 e0
2
,AV Ap , AV Ap
where p = space charge, coul/m3
y = distance parallel to gas flow from wire
to wi re, m
x = distance perpendicular to gas flow from
wire to plate, m
e0 = permittivity of free space, coul2/(N-m2).
Figure 2 shows a partial grid illustrating the nomenclature
used in the numerical solution to the above equations. The
initial point for which a solution is obtained is designated
point "0". As the calculation progresses to neighboring points,
each point in the grid becomes point "0". Using the relation-
ships
AVo 1 ,„ „ . A2Vo 1 ,V»-Vo V0-V2v
Ax~ = 21 ' Ax^~ = a (~~I— —} d6)
and
A2V0 !_ ,V i -V o _ y o -y 3.
Ayz ~ a a a '
equation 14 becomes, in terms of the grid points on figure 2,
Vo = T (Vi +V2 + V3 +Vi» + a PO) (17)
* EO
12
-------
^
r" f
f t
a
1 ,
,
f '
, 3 ,
Eox
,
r «
2
0 Eoy
4
,
'
1
Figure 2. Partial grid showing nomenclature used in
the numerical analysis.
13
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From the relationships
AV _ - AV
Ax ~ Eox' Ay
Ap in x direction = po - 02 »
Ap in y direction = po - Pa /
and Ax = Ay = a,
equation 15 may be transformed into a function of the space
charge at the surrounding points and EOX and EOy. Solving the
resulting equation for the space charge at the point "0" gives
Eoy) - 1/2 [{fJL (Eox + Eoy)}2
1
EoyP3))]7. (18)
Figure 3 shows additional nomenclature used in the numerical
analysis. The boundary conditions are:
V = Vo on the wire,
V = 0 on the plate,
AY. = 0 along line AB,
Ax
£Y. = 0 along lines BC, CD, and AD, and
57
p = ^/(~ (b)] near the plate.
14
-------
V = Vo ON WIRES
V = 0 ON PLATES
2345
Xl2
1
•a
A
CO
2
X
D
21
31
/
'22
3?
42
1
y
1
23
Tft
43
-*•
24
a
] __ y AXIS
i
4 _
i
4. _
i
r
\
I
j^
'
s
1
c
AREA OF
INTEGRATION
Sy - ONE HALF WIRE TO WIRE SPACING
Sx = WIRE TO PLATE SPACING
a = INCREMENT SIZE FOR INTEGRATION
V0 = APPLIED VOLTAGE
Ex - COMPONENT OF ELECTRIC FIELD PERPENDICULAR
TO PLATE
Ey = LONGITUDINAL COMPONENT OF ELECTRIC FIELD
j = AVERAGE CURRENT DENSITY
Figure 3 . Nomenclature used in the numerical analysis. The
problem is considered two-dimensional so that edge
effects and variations in E, V, and p parallel to
the wires are ignored.
15
-------
In order to begin the iteration, it is necessary to establish
an initial estimate of the potential grid in the inter-
electrode region and to estimate a space charge for the corona
region. The initial estimate of the potential grid is obtained
from an expression developed by Cooperman3 which describes the
electric field for wire-plate geometry at voltages less than
that required for initiation of the corona. A computer program
was written which obtains a numerical solution to equations 14
and 15 by the following steps:
1. V is computed at every point in the integration
grid using Cooperman's expression.
2. p is computed at every point in the integration
grid from equation 18.
3. V is recomputed at every point in the integration
grid using equation 17.
4. Steps 2) and 3) are repeated alternately until
convergence occurs. Convergence on the potential
grid is obtained when the value of the potential
at each point in the grid is within one volt of
the value calculated at that point in the previous
iteration.
5. The computed current density [obtained using the
relationship j = p (-r—) b] is compared with the
measured current density. If the computed and
measured current densities do not agree within .1%
then the space charge representing the corona
region is adjusted and steps 1) through 5) are repeated
until agreement is obtained.
This procedure iterates on a grid of electric field and space
charge density until convergence is obtained. The major
approximation, and one that is seemingly unavoidable in practice,
is the assumption that the motion of all charge carriers can, on
the average, be described by a single effective mobility. The
space charge introduced by the particulate present in flue gas
would reduce the effective mobility. The program estimates
16
-------
the effect of reduced mobility by using equation 13 of Section
II. However, it is necessary to limit the mobility reduction
in order to prevent nonconvergence of the grid under certain
conditions.
In order to check the accuracy of the calculation procedure,
the computer program has been used to calculate potential
profiles and electric fields based on the geometry and
operating conditions for electric field measurements reported
in the literature. Figure 4 shows calculations based on the
geometry and operating conditions reported by Penney and
Matick "* and their experimental results. Reasonable agreement
is found for the potential profiles from the wire to the
plate and from a point midway between wires to the plate.
Also, excellent agreement is found for the field near the
plate (the slope of the potential curve).
Tassicker5 performed a series of experiments to measure the
field and current density at the plate in precipitators of
different geometry. Figure 5 shows some of his data on a
wire-plate precipitator. Corona wires were adjacent to the
points x = -10 and x = +10 cm. Thus, the positions x = +5,
x = -5 correspond to positions at the plate midway between
the corona wires, and the position x = 0 corresponds to a
position at the plate adjacent to a corona wire. The general
shape and magnitude of the electric field at the plate show
good agreement. Notice that, although the field is fairly
uniform along the plate, there is a maximum opposite the corona
wire. Since the calculated results appear to match the
available data rather well, it may be concluded .that Leutert
and Bohlen's technique provides a basis for computing
electric fields in the region of interest adjacent to the
collecting electrode.
17
-------
y = o
25
20
15
UJ
O
Q.
10
y = Sy
I
12
10
8 6
DISPLACEMENT, x (cm)
Figure 4. Potential profiles in a wire plate precipitator.
(a) x=0 corresponds to the wire, x=12 cm the
plate. Solid line, Penney and Matick
(experimental). Dashed line, SRI (calculated)
(b) x=0 corresponds to a point midway between
wires, x=12 cm the plate. Solid line Penney
and Matick (experimental). Dashed line,
SRI (calculated).
18
-------
0.06
0.04
C\J
£
o
0.02
SRI CALC.
3.0r-
-TASSICKER MEAS.
W/DISCHARGE
E
u
2.0
•x.
UJ
1.0
I
I
-5.0 0 5.0
DISPLACEMENT, x
(a)
Figure 5.
(a) Current density at the plate
as a function of displacement.
Only the shape of the curves is
significant. The peak values
were matched by adjusting
parameters. (Positive corona).
'SRI CALC.
W/DISCHARGE
,TASSICKER MEAS.
W/0 DISCHARGE
SRI CALC.
W/0 DISCHARGE
I
-5.0 0 5.0
DISPLACEMENT, x
(b)
(b) Electric field at the plates
as a function of displacement,
with and without discharge.
Although the shape of the field
agrees fairly well, the magni-
tude is slightly low. A change
in th$ effective mobility
might correct this. The
mobility for oxygen was used.
(Positive corona).
-------
CHARGING RATE CALCULATIONS
Particle charging in industrial electrostatic precipitators
is performed by electrons and negatively charged gas ions
which originate from a negative corona discharge. Although
the gases present in stack gases are ordinarily good elec-
trical insulators, processes such as naturally occurring
radiation and flame ionization continuously produce a small
number of ion-electron pairs in each cubic centimeter of gas.
These sources of ionization provide initiating electrons for
the corona process. If an electric field is applied to
these gases, positive ions and free electrons will be driven
by the electrical force, and the electrons, because of their
greater mobility, will move at a higher velocity than the
relatively sluggish positive ions. When the electric field
reaches some critical value, the free electrons will acquire
sufficient energy to remove a valence electron from some of
the neutral gas molecules through collisions. The newly
freed electrons, together with the ionizing electrons, will
again accelerate and ionize other neutral gas molecules.
This process, termed avalanche multiplication, will continue
as long as the localized electric field exceeds the critical
value for electron avalanche. Peek has studied this phe-
nomenon in detail and has derived a semi-empirical relation-
ship that relates the field strength required for electrical
breakdown to electrode geometry, temperature, and pressure.
For the case of industrial electrostatic precipitators with
negative corona electrodes, the corona process will produce
negative charge carriers in the region between the corona
and the collecting electrodes. The electrons will flow
toward the collecting electrode, and the positive ions will
20
-------
travel toward and strike the corona electrode. The free
electrons flowing from the corona region will travel
toward the collection electrode and attach to electro-
negative gases such as oxygen, water vapor, and sulfur
dioxide. The ionized electronegative gases are the
major carriers of the corona current and thus predominate
in the particle charging process.
When a stable corona current has been established under
typical flue gas conditions, two particle charging
mechanisms are active in moving the ions to the dust
particles: field charging and diffusion charging. For
the purposes of this discussion, it will be assumed that,
with both mechanisms, negatively charged ions are the
exclusive carriers of charge in the space between the
corona region immediately surrounding the discharge
electrode and the collecting electrode.
The mathematical relationships describing field charging and
diffusion charging have been derived in the literature and
are summarized by White.7 The charging rates predicted by
the field charging equations are in good agreement with
experimental data for large particles (r > 2 pm) and moderate
to high external electric fields;8' 9 and the charging rates
predicted by the diffusion charging equations are in good
agreement with experimental data over a fairly broad range of
particle sizes where the external electric is low.8'10 Neither
the field nor the diffusion charging equations are adequate
for predicting charging rates for particles with radii in the
.8 ym to 2 ym region11 when an electric field is applied.
21
-------
The computer program described in this report employs
two different models to account for the charging process.
The program user specifies which model should be employed,
based upon the specific objectives for using the program.
The following is a discussion of the two particle charg-
ing models and their intended uses.
Sum of the Classical Field and Diffusional Charging Rates
The classical field charging rate 7 is given by
where N0 - undisturbed ion concentration, #/m3
e = electronic charge, coulombs
bj^ = ion mobility, m2/ (volt-sec)
e0 = permittivity constant, cou!2/(N-m2)
q = instantaneous charge on the particle, coulombs
qs = saturation charge, coulombs
The conventional saturation charge q does not take into
s
account the persistence of the momentum of the ions due to
the external electric field and the transient behavior of
the field. Thus, a modified saturation charge q ' is used
to account for these factors. It is given by
m
22
-------
where a = particle radius, m
A = an adjustable parameter = mA , where
A = ion mean free path, m
m = number of mean free paths
E0 = average charging field, volts/m
K = relative dielectric constant of the
particle.
The effect of the modified saturation charge is to allow
a greater charge to accumulate on the particle than the
conventional saturation charge before the field charging
mechanism ceases.
The classical diffusional charging rate7 is given by
'b
= N0eTra2v exp (-eV/kT) , (21)
diffusion
where v = ion mean thermal speed, m/sec
V = potential near the surface of the charged
particle, volts
k = Boltzmann's constant, joules/°K
T = absolute temperature, °K .
The potential near the surface of the charged particle is
approximated by
V = TfJL SL (22)
4ire0 a
23
-------
The total charging rate can be estimated by adding the
rates given by equations 19 and 21. Then,
dt
QT: field diffusion
(l - -9_)2+ Noerra'v- exr> (-eV/kT)
Equation 23 reduces to the classical diffusional equation
in the absence of an applied electric field and approaches
the results obtained from the classical field charging
equation for large particles and high fields. The charging
rates predicted are lower than those found experimentally.
The disagreement is largest in the particle size range where
both field and diffusional charging are important. The disagree-
ment is due mainly to the fact that the effects of the external
electric field on the diffusional charging process have been
neglected.
Equation 23 is solved by summing up incremental amounts of
charge Aq acquired in successive time intervals At, where
initially q = 0 and t = 0. This procedure is straightforward
and the computer program runs relatively fast when it utilizes
this charging model. Thus, if the user is interested in an
approximate precipitator performance or wants to examine
trends under certain conditions, employment of this charging
model would save significant amounts of computer time. This
charging model is selected for use by the computer program
when a certain indicator (NCALC) is read into the program as
1 in the input data. This indicator is the third piece of
information of data card set 7 as described in the section
which discusses data card format.
24
-------
Southern Research Institute Model
Figure 6 shows the two dimensional physical model which
is used as the basis for the theoretical development of
the charging process. The particle shown in this sketch,
and its environment, are considered to be representative of
the average of a large number of similar systems which make
up the aerosol under investigation. Because the ion concen-
tration may only be 10 times as large as the particle
concentration, and because of the screening effect of
neighboring charged particles, macroscopic theories based on
diffusion due to ion concentration gradients may not be
applicable. The approach used in the SRI model is to apply
some ideas from kinetic theory in order to calculate the
charging rate in terms of the probability of collisions
between ions and the particle of interest.
In this theory particle charging is largely attributed to the
thermal motion of the ions and the electric field acts as a
perturbation on the thermal charging process. Although in
practical situations the thermal kinetic energy of the ions is
always much greater than the kinetic energy gained from the
external electric field, experiments show that the charging
rate is greatly enhanced by the application of a field.8 The
effect of the applied electric field is to modify the ion dis-
tribution near the particle in such a way that the average
ion concentration is increased. Murphy et al12 estimated
an increase in ion concentration by a factor up to 440
times as large as the normal Maxwell Boltzmann distribu-
tion would predict when the particle field and applied field
are antiparallel, depending on the amplitude of the applied
field. When the two fields are parallel, however, the
decrease was less than a factor of 3. Thus, the changes in
ion concentration do not cancel and there is a large net
increase in ion concentration near the particle.
25
-------
o
o
o.
05
o
o2-
O
o;
o
Figure 6. TWO dimensional physical model for developing
a charging theory
26
-------
Figure 7 is a simplified diagram which is used to define the
nomenclature used in the theoretical development. The
physical description, however, is based on the conceptual
representation shown in Figure 6, where the particle of
interest is surrounded by gas molecules, ions, and other
charged particles. The particle is assumed to be spherical
and only components of electric field due to charge on the
particle and the applied field are considered. The external
electric field is taken to be uniform and directed along the
negative z axis. The dashed line in Figure 7 labeled ro
corresponds to points in space where the radial component of
the total electric field is equal to zero. The angle 9o
corresponds to the azimuthal angle at which ro is equal to
the particle radius, a. The point of intersection between
ro and the particle surface will always lie on the hemisphere
defined by 8 o £ ir/2. As the charge on the particle increases,
60 will go to zero and ro will exceed "a" for all angles.
If the space charge in the region outside the volume of
interest is homogeneous, we can write an expression for the
radial component of the electric field very near the particle
as follows:l3
K—1 a ^ ne
Er = Eo cos 9(1 + 2 -£ p.) - ^p- , (24)
27
-------
2 AXIS
Figure 7. Model for Mathematical Treatment of Charging
Rate. Along the line r = ro(9)/ the radial
component of the electric field is equal to
zero.
28
-------
where
E = radial component of electric field
(V/m),
EQ = external field (V/m),
K = particle dielectric constant,
a = particle radius (m),
r = radial distance to point of interest (m),
n = number of charges on particle,
e = electronic charge (1.6 x 10~19 coul.),
£Q = the permittivity of the gas (^8.85 x 10"12
cou!2/N-m2), and
9 = the azimuthal angle measured from the z axis (radians)
For the purposes of discussion, we will define three areas
of interest on the particle surface. One area, designated
by Region I, is that bounded by 6 = 0 and 0 = 0O; a second
region, Region II, is bounded by 0 = 0O and 0 = ir/2; and the
third region of interest, Region III, is the "dark side" of
the particle where 0 > ir/2 . Our approach to arriving at an
equation for the charging rate, dq/dt, is to quantify the
probability that ions can reach the particle surface in each
of these three regions.
The rate at which ions reach the particle surface is
— = - ^SL = P.N (E a 9)
dt e dt 'so''
P = the probability that a given ion will move towards
the particle. From kinetic theory, P = 1/4 vA, where
v is the mean thermal speed of the ions given by
(26)
29
-------
and A is the surface area of the particle on which the
ions may impinge.
N (E ,a,8) = the ion concentration near the particle
so r
surface. From classical kinetic
theory, N (E ,a,6) can be related to
o U
the equilibrium ion concentration, N ,
by the expression
N (E ,a,6) = N e-AV(Eo/a,G)/kT _
SO O
AV(E0,a,8) = the energy difference between the particle
surface (r=a) and some larger distance (r=r')
where the equilibrium ion distribution is
undisturbed.
Thus, for diffusion to the entire surface of a
spherical particle, we write
& = Me* a2^-AV(Eo,a,6)/kT> (27)
at o
Up to this point, the derivation is similar to that
given by White1"1 for the classical diffusional charging
rate. For diffusional charging, AV is set equal to
ne2/47ieQa, the potential energy at the particle surface,
and the influence of the applied field is not taken
into account. (In this case, r" = <».)
30
-------
Because of collisions with neutral molecules, the
energy of the ions is not conserved, and hence,
there is no potential energy function associated
with the electric field given by equation 24.
However, a minimum amount of work must be done
in moving an ion from some point in space (defined
by r = r1) to the particle surface. This minimum
work is given by
/a
F-dr = /qErc
r1 r'
AV(EQ,a,9) = I F-dr = / qErdr = ^r^— + eEQr cos 9 (28)
K-I *3 r=a
|rJ ,
r -i r=r'
If expression (28) is used for AV, Ns(Eo,a,9) becomes
- |"/nez(r'-a)
I I A 71- r* V T'^ y '
I ^ T II t, JxJ.CH.
[3ar'2-r' 3 (K+2) + a3(K-l)]eE cos
N (E ,a,9) = N exp
o O (j
(29)
kT(r')2(K+2)
If 9 is set equal to zero in equation (29) and the resulting
equation is used for Ns, equation 25 becomes identical to that
given by Liu and Yeh15 as a solution to the diffusional equa-
tion. Our approach to the solution of this differential equa-
tion for the charge as a function of time is quite different,
however, and yields a smaller charging rate.
31
-------
Ideally, we would like to apply equation (29) to
the entire particle surface by averaging over the
angle 6. To do this, it is necessary to choose
some radial distance r' (9) where the ion density
is undisturbed. If we choose r1 = r , the point at
which the radial component of the electric field (Er)
is zero, we can only apply equation 27 in its
present form to Region II, as defined in Figure 5.
In Region I , r < a and the argument of the
exponential becomes positive. This can be inter-
preted from energy considerations to mean that any
ion which is near the particle surface in Region I
and moving toward the particle has a 100% probability
of impacting on the particle surface. In charging
Region III, no finite r exists. Thus,
the charging rate in each region must be
calculated separately and the rates added to
yield the total charging rate :
/dq\
Xdt/jjj
Equation 27 was developed using expressions from
kinetic theory which, in turn, are based on the
assumption that the system is in equilibrium. In
solving equation 27 for the charge as a function
of time, we will assume that the charging dynamics
can be approximated by a series of steady states
so that the expressions in equation (30) may be applied.
Nevertheless, in reaching the ultimate expression for
32
}
'
-------
q(N0t), we will consider the motion of the ions due to
the applied field.
In Region I, the argument of the exponential in equation 27
becomes positive. In this case, equation 27 predicts
charging rates which are too large to be approximated by
steady state solutions. In fact, the charging rate is
limited to the rate at which ions are brought into the
system by the external field. This rate is given by the
product of the current density and the surface area (Aj) of
Region I :
a?) = S-3*! =bieN°
This is identical to the charging equation developed by
Pauthenier16 which we refer to as the classical field
charging equation. We may write this equation in a more
conventional and useful form,
,., = N0biTmse(l - ^-)2 , (32)
*-' T n
1 s
where
ns = (1 + 2 |^-)E0a2/e, (33)
9O = arc cos (n/ns), and (34)
the other symbols have been previously defined.
33
-------
When n > n , r is greater than "a" for all
s o
values of the angle 8, and this charging
mechanism ceases (A •* 0) .
In Region II charge is acquired by the particle due to ion
diffusion which is enhanced by the presence of the applied
electric field. In this case, equations 25 and 29 apply,
and the charging rate is
ii .
N is the magnitude of the ion density at the
particle surface averaged over Region II.
Explicitly,
2TT IT/2
IT/2
JQ Ns(Eo,a,8)sin8ded
sin8d0d
Ns ~~ ~~o w ~ , (36)
2^ ^72 ' Where
Ns(Eo,a,8) is given by equation 29. Using
this expression for N and writing A in terms
•3 ± ^
of "a" and 8, we find
(*3\
\dtl
eN v
s
II
2TT TT/2
*- "a»
/
a2sin6d8d
-------
For each value of the particle charge , a value of 9O
is calculated using equation (34) and the integration of
equation (37) is performed. The integration is complicated
by the dependence of ro on the angle 6. Thus for each
value of 0 , a value of ro must be calculated. The magnitude
of r0 is found from the condition that Er(Eo,a,6) = 0,
where Er(Eo,a,8) is given by equation 24. We find that ro
varies by factors of 20-60, depending on a and Eo, as 0
varies from 0 to ir/2.
In Region III, the particle surface between the angles
6 = iT/2 and 6 = ir , the electric fields due to the particle
charge and the external field are in the same direction and
there is no radial point ro for which the total electric
field is equal to zero. In this case, equations 27 and 29
would predict that no charging could occur on this side of
the particle. This is a result of our application of equili-
brium thermodynamics to a dynamic problem. Physically,
this means that the ions move in the direction of the electric
force and are swept from the system. In reality, additional
ions are swept into the system by the same electric field.
As Murphy e_t a_l pointed out}2 the change in the ion density
(and hence, charging rate) near the particle surface is much
greater for small values of 9 than for the region 9 > ir/2
when an electric field is applied. Our calculations also
indicate that this is true. As an approximation, the effects
of the applied field are neglected in Region III and the
classical diffusional equation is used:
/da\ 0
(dt) = - 2 - exP (-ne2/47re0akT) . (38)
35
-------
In the preceding paragraphs, we have developed equations for
the charging rate for each of three charging regions on the
surface of the particles. The charging rate of the particle
is the sum of these rates:
• (it) + (if ) + (if )
o
[3arQ2-ro3 (K+2) + a3(K-l)J eEQcos 6)~|sinede
kT rQ2(K+2)
I]
?ra vN e
=—— exp(-ne2/4-rre akT) (39)
Equation (39) is integrated numerically using the quartic
Runge-Kutta method17 in the following procedure:
The initial conditions are taken to be n=o at t=0.
ns is calculated using equation (33).
For each increment in the Runge-Kutta scheme, a value
of 00 is calculated from equation (34) .
The integral over 6 in equation (39) is performed using
Simpson's Rule, and for each value of 9 which is
chosen for this integration, ro is calculated.
The three individual charging rates are calculated and
then added to give the total instantaneous charging
rate for a particular value of n.
36
-------
The charging model described by equation (39) reduces to the
classical diffusional equation in the absence of an applied
electric field and approaches the results obtained from the
classical field charging equation for large particles and
high fields. As is discussed below, the charging rates
predicted by equation (39) agree within 25% of Hewitt's
data over all particle sizes, electric field strengths, and
charging times. For practical charging times in electrostatic
precipitators, the agreement is within 15%. The model also
gives a good description of particle charging in the range of
particle sizes where both field and diffusional charging are
important.
Figures 8, 9, 10, and 11 give comparisons of charge values as
a function of N0t for particle sizes of 0.18, 0.28, 0.56, and
0.92 ym diameter. The calculated charge values shown were
obtained with the SRI model and the sum of the field and
diffusion rates model for the indicated multiples of mean
free path. These comparisons indicate that Am has no appre-
ciable effect on the results obtained for m values of 0, 1,
and 2, with the SRI model, but considerable variation in
charge with Am occurs with the sum of the rates model. These
comparisons also demonstrate close agreement obtained between
Hewitt's data and the results obtained with the SRI model.
Since the numerical method of finding the particle charge as a
function of time is extensive, the program runs slowly when it
utilizes this charging model. The user would want to employ this
model when he needs the best charging mechanism available in
order to compare with experimental results or for actual
precipitator sizing where time is not a factor. This charging
model is selected for use by the computer program when the
indicator NCALC is read into the program as 0 in the input
data.
37
-------
100
tn
C
O
M
4J
O
0)
10
O SRI THEORY
• SUM OF FIELD AND
DIFFUSION RATES
X HEWITT EXPERIMENTAL
E = 3.6 kV/cm
O m=0,l,2
Not x 1013
Figure 8. Comparison of charge values for 0.18 pm
diameter particle
38
-------
100
U)
c
o
>-l
-p
o
0)
10
O SRI THEORY
• SUM OF FIELD AND
DIFFUSION RATES
X HEWITT EXPERIMENTAL
E = 3.6 kV/cm
Not x 10
1 3
Figure 9. Comparison of charge values for 0.28 ym diameter
particle
39
-------
1000
c
o
M
4J
u
OJ
M-l
O
0)
100
10
O SRI THEORY
• SUM OF FIELD AND
DIFFUSION RATES
X HEWITT EXPERIMENTAL
E0 = 3.6 kV/cm
0123456
N0t x 1013
Figure 10. Comparison of charge values for 0.56 ym diameter
particle
40
-------
1000
en
c
o
n
4J
u
(1)
>+-(
o
QJ
100
50
O SRI THEORY
• SUM OF FIELD AND
DIFFUSION RATES
X HEWITT EXPERIMENTAL
E0 = 3.6 kV/cm
Figure 11. Comparison of charge values for 0.92 ym
diameter particle
41
-------
PARTICLE COLLECTION
Migration Velocity
Once the particle charge and electric field values have been
computed, the next step in calculating theoretical collection
efficiency is the calculation of the electrical drift velocity,
or migration velocity, resulting from the coulomb and viscous
drag forces acting upon a suspended particle. For particle
sizes and electrical conditions of practical interest, the
time required for the particle to achieve the steady-state
value of velocity is negligible, and the migration velocity
is given by : * 8
w = 2__ , (40)
where w = migration velocity of a particle of radius a,
m/sec
q = charge on particle, coul
Ep = electric field near the collection electrode, volt/m
a = particle radius, m
y = gas viscosity, kg/m-sec
C = Cunningham correction factor, or slip correction
factor1 9
= (1 + AX/a) ,
where A = 1.257 + 0.400 exp (-1.10 a/X)
and X = mean free path of gas molecules .
Particle Collection Fractions
For the idealized case of laminar flow, the collection length
required for 100% collection of a particle with a known migration
velocity is easily calculated. However, laminar flow never occurs
in industrial precipitators , so the calculation is of academic
interest only. Consideration will therefore be limited to
turbulent flow conditions.
42
-------
Gas flow velocities in most cases of practical interest are
between 0.60 and 2.0 m/sec, while theoretical migration
velocities for particles smaller than 6.0 urn are usually less
than 0.30 m/sec. The path of these smaller particles therefore
tends to be dominated by the turbulent motion of the gas
stream in the interelectrode region. Under these conditions,
the path of the particles is random, and the determination
of the collection efficiency of a given particle becomes, in
effect, the problem of determining the probability that a
particle will enter a laminar boundary zone adjacent to the
collection electrode in which capture is assured. The classical
equation for describing collection of monodisperse particles in
electrostatic precipitators under turbulent flow conditions was
derived by Deutsch:20
n = 1.0 - exp (-ApW/Q), (41)
where n = collection fraction of the particle size under
consideration
Ap = collection area, m2
w = migration velocity of particle of radius a, m/sec
Q = gas volume flow rate, m3/sec.
The assumptions on which the derivation of this equation is
based are:
a) Gas turbulence provides sufficient mixing to
establish a uniform particle concentration at any cross
section of the precipitator.
b) The gas velocity through and across the precipi-
tator is uniform except for a boundary layer near the wall.
c) The particle migration velocity near the collecting
surface is constant for all particles and small compared with
the average gas velocity. This implies that the equation is
strictly applicable only to a monodisperse aerosol with particle
diameters less than about 6 to 10 ym, under conditions such
that the migration velocity and hence the electric field at the
plate and the charge accumulated on a particle do not vary over
the length of collection area.
43
-------
d) There are no disturbing effects, such as reentrain-
ment, back corona, etc. 21
In order to use equation (41) in the precipitator model and
approximate assumption c) it is necessary to use the incre-
mental length approach. White22 reports a series of experi-
ments using nearly monodisperse oil fumes under experimental
conditions that were consistent with all of the above
assumptions. The results demonstrated convincingly that
equation (41) adequately describes the collection of mono-
disperse aerosols in an electrostatic precipitator under
idealized conditions.
It has been common practice to correlate data from field
electrostatic precipitators with an equation with the same
functional form as equation 41:
Ho = 1 - exp (-ApWp/Q) , (42)
where no = overall mass collection fraction
w = an empirical parameter (precipitation rate
parameter).
The parameter Wp characterizes the performance of a given
precipitator under a specified set of operating conditions,
and often varies widely, even among installations treating a
similar flue gas. Equation 42 is inadequate for design purposes
because it is a gross over-simplification in which particle
size effects and variations in electrical conditions are
lumped into the quantity Wp.
Program Calculations
The computer program calculates a migration velocity for each
representative particle size at the end of each length increment
using equation 40. Particle charge and electric field values
employed in the calculation are calculated in the electric field
and particle charge subroutines from the residence time and
44
-------
electrical conditions pertaining to the length increment
under consideration. The fraction of particles collected
for each representative particle size in each length incre-
ment is obtained from the Deutsch equation as follows:
Hj = 1.0 - exp (-Ajw/Q) , (43)
where ru = collection fraction for length increment
Aj = collection area for length increment, m2
Q = gas volume flow, m3/sec.
The number of particles removed in each size band is obtained
from
X- - X' V n- (44)
Ajc ~ Ajo x nj i
where Xjc = the number of particles of radius a per m3 of
gas collected in a given length increment
Xjo = the number of particles of radius a per m3 of
gas at the beginning of each length increment.
The total number of particles collected, the mass of particles
collected, and the number of particles entering the next length
increment are obtained from the quantity Xjc and the entering
size distribution. All the calculations are performed on the
basis of a cubic meter of gas. After these operations have
been completed for all representative particle sizes, the
program calculates the mass median diameter and the total
mass of the particulate collected in the length increment
under consideration, and then returns to the beginning of the
length increment loop to repeat the calculations for the
following length increment.
When the program has completed the required calculations for all
length increments, the overall mass efficiency is computed and
compared with the estimated value which was used for the space
charge calculations. If the disagreement is greater than
45
-------
±.05%, the computed overall mass efficiency is used for a
new estimate, and the program returns to the beginning of the
length increment loop and repeats the calculations for the
entire precipitator length. Usually, less than three itera-
tions are required to obtain agreement within the specified
limits.
The next operation performed by the program, after convergence
on the overall mass efficiency has been obtained, is the calcu-
lation of effective or length averaged migration velocities for
the different particle sizes from the Deutsch equation:
1" (I^> , (45)
where we = effective migration velocity of particle of
radius a, m/sec
AT = total collecting area
n = collection fraction for given particle size over
total length.
The values obtained for we in this manner are in effect an
average of the values of w obtained in the incremental length
sections. A precipitation rate parameter, Wp, is computed
by the program from equation 42 after the individual efficiencies
and effective migration velocities have been obtained for all
particle sizes.
The calculation procedure described here in effect consists of
assuming that the Deutsch equation adequately describes the
mechanism by which monodisperse particles are transported to
the collection electrode. For particles larger than 10 ym
diameter, the assumption is invalid because the motion of these
particles is not dominated by turbulence due to their relatively
high migration velocities. Under these conditions, the Deutsch
46
-------
model would be expected to under-predict efficiencies. The
practical effect in modelling precipitator performance will
be slight, however, since even the Deutsch equation predicts
ideal collection efficiencies greater than 99.6% for 10.0 ym
diameter particles at relatively low values of current density
and collection area [i.e., a current density of 10xlO~9 amps/
cm2 and a collection area to volume flow ratio of 39.4 m2/
(m3/sec) or 200 ft2/(1000 ft
A more serious objection to the assumption of uniform turbulent
mixing of the particulate may be found in experimental measure-
ments obtained with a laser obscurometer which suggest that,
under certain conditions, a concentration of the fine fraction
of the particulate occurs in the space adjacent to the collec-
tion electrode.23 The causes for such a gradient and its
effect on predicted collection rates have not been determined
as of the date of this report.
47
-------
METHODS FOR REPRESENTING NON-IDEAL EFFECTS
In the preceding sections, a basis for calculating theoretical
collection efficiencies has been described. This section will
discuss the non-idealities which exist in full-scale electro-
static precipitators and describe calculation procedures for
estimating the effects on predicted collection efficiencies.
The factors of major importance are:
(1) Gas velocity distribution
(2) Gas sneakage
(3) Rapping reentrainment .
These non-idealities will reduce the collection efficiency
that may be achieved for a precipitator operating with a
given specific collecting area. Since the model is structured
around the Deutsch equation for individual particle sizes, it
is convenient to represent the effect of the non-idealities in
the model as correction factors which apply to the exponential
argument of the Deutsch equation. In the subsequent discussions,
these correction factors will be used as divisors for the
theoretical migration velocities. The resulting "apparent"
migration velocities are empirical quantities only and should
not be thought of as an actual reduction in the migration
velocity in the region of space adjacent to the collecting
electrode.
Effect of Gas Velocity Distribution
Although it is widely known that a poor velocity distribution
gives a lower than anticipated efficiency, it is difficult
to apply a numerical description for gas flow quality. White21*
discusses non-uniform gas flow and suggests corrective actions.
Prezler and Lajos25 assign a figure-of-merit based upon the
relative kinetic energy of the actual velocity distribution
compared to the kinetic energy of a uniform velocity. This
48
-------
figure of merit will be a measure of how difficult it may be
to rectify the velocity distribution but not necessarily a
measure of how much the precipitator performance would be
degraded. The following discussion will describe an approach
to the calculation of degradation of performance based upon
the velocity distribution, the theoretical or ideal efficiency,
and the Deutsch equation.
It will be assumed that the Deutsch equation as written applies
to each particle size with a known migration velocity, w, and
that the specific collection area and size of precipitator
are fixed.
_ Apw
Given:
n = 1 - e Q
It can be seen that
1 - n = e _K_ , (46)
and
1 , An w k
In
J.-H AI ua ua
where
Ap = plate area
AI = inlet cross sectional area
Q = inlet volume flow rate
w = migration velocity for a given particle size
ua = average inlet velocity
Al
n = ideal collection fraction .
49
-------
From this form of the Deutsch equation it can be seen that the
logarithm of the inverse of the penetration is proportional to
the inverse of the velocity (and thus the transit time). The
precipitator can now be divided into a number of imaginery
channels corresponding to pitot traverse points. Using the
altered form of the Deutsch equation, the losses for all the
channels can be summed and averaged to obtain the mean loss in
the precipitator using an actual velocity distribution instead
of an assumed uniform distribution. This can be accomplished
as follows:
(1) Calculate constant k from the efficiency
predicted under ideal conditions:
k = u-,ln •=
(2) Calculate the mean penetration:
N
P = JT^- XI Ui (1-ni) , (48)
c Mil *M^ 1 x ' -L ' * x '
or
N
<49)
where
N = number of points for velocity traverse
ui = point values of velocity
Hi = point values of collection fraction for
the particle size under consideration .
Note that the average penetration is a weighted average to
include the effect of higher velocities carrying more
particles per unit time than lower velocities.
50
-------
For any practical velocity distribution and efficiency, the
mean penetration obtained by summation over the velocity
traverse will be higher than the calculated penetration based
on an average velocity. If an apparent migration velocity for
a given particle size is computed based upon the mean penetra-
tion and the Deutsch equation, the result will be a value lower
than the value used for calculation of the single point values
of penetration. The ratio of the original migration velocity
to the reduced "apparent" migration velocity is a numerical
measure of the performance degradation caused by a non-uniform
velocity distribution. An expression for this ratio may be
obtained by setting the penetration based on the average
velocity equal to the corrected penetration obtained from a
summation of the point values of penetration, and solving
for the required correction factor, which will be a divisor
for the migration velocity.
The correction factor "F" may be obtained from:
N
exp (- F- ui> = N^ Z) uiexp (-k/v = p • (50)
i=l
Therefore,
F = £ r . (51)
ua(ln p)
Whether the quantity F correlates reasonably well with statis-
tical measures of velocity non-uniformity is yet to be
established. A limited number of traverse calculations seem
to indicate a correlation between the factor F and the
normalized standard deviation of the velocity traverse.
Figure 12 shows F as a function of the ideal efficiency for
several values of gas velocity standard deviation. These
curves were obtained by computer evaluation of equation 51,
and the data on which the calculations are based were obtained
from Preszler and Lajos.25 The standard deviations have
51
-------
1.58
Figure 12.
CORRECTION FACTOR F
"F" as a function of ideal efficiency and gas
flow standard deviation.
52
-------
been normalized to represent a fraction of the mean. The
overlapping of the curves for standard deviations of 1.01
and 0.98 indicates that the standard deviation alone does
not completely determine the relationship between F and
collection efficiency.
The data in Figure 12 were used to obtain the following
empirical relationship between F, the normalized standard
deviation of the gas velocity distribution, and the ideal
collection predicted for the particle size under considera-
tion:
F = 1 + 0.766 nag1'786 + 0.0755 ag In (^) , (52)
where
vr? '^
ag = = . (53)
"~a
This relationship is based on a pilot plant study, and should
be regarded as an estimating technique only. If it is desirable
to simulate the performance of a particular precipitator, the
preferred procedure would be to obtain the relationship
between F, n and ag for the conditions to be simulated from
a velocity traverse at the entrance to the unit.
Equation 52 is included in the computer program following
the theoretical calculations. The program evaluates F from
the ideal collection predicted for each particle size and
the value of Og chosen for the input data. The quantity F
is used in combination with an empirical representation of
reentrainment and sneakage as described in the following
section.
53
-------
Effect of Gas By-Passage
Gas sneakage occurs when gas by-passes the electrified areas
of an electrostatic precipitator by flowing through the hoppers
or through the high voltage insulation space. Sneakage is
reduced by frequent baffles which force the gas to return to
the main gas passages between the collection plates. If there
were no baffles, the percent sneakage would establish the minimum
possible penetration because it would be the percent volume
having zero collection efficiency. With baffles, the sneakage
re-mixes with part of the main flow and then re-by-passes in
the next unbaffled area. The limiting penetration due to
sneakage will therefore depend on the amount of sneakage gas
per section, the degree of re-mixing, and the number of sections.
If we make the simplifying assumption that perfect mixing occurs
following each baffled section, an expression for the effect of
gas sneakage may be derived as follows:
Let S = fractional amount of gas sneakage per section
n = collection fraction of a given size particle
obtained with no sneakage for total collection
area
ru = collection fraction per section of a given particle
size =!-(!- n)1/Ns
Ns = number of baffled sections
p^ = penetration from section j.
Then the penetration from section one is given by:
pi = S + (1 - rij) (1 ~ S) , (54)
and from section 2
P2 = SPl + (1 - rij) (1 - S)Pl
= P! [S + (1 - n..) (1 - S)]
= [s + (i - n-j) (i - s)]2 , (55)
54
-------
and from section Ns (the last section),
PNs = is + (1 - njMi - S)]NS
= [S + (1 - S)(l - n)VNs]Ns . (56)
Figure 13 shows a plot of the degradation of efficiency from
99.9% design efficiency versus percent sneakage with .number of
baffled sections as a parameter. For high efficiencies, the
number of baffled sections should be at least four and the amount
of sneakage should be held to a low percentage. With a high
percentage of sneakage, even a large number of baffled sections
fails to help significantly. As the next section will indicate,
this graph can also be applied to reentrainment.
We can define a by-pass or sneakage factor, B, analogous to
the gas flow quality factor, in the form of a divisor for
the migration velocity in the exponential argument of the
Deutsch equation:
n in (1 - n)
B = i/M ' (57)
NS In [S + (1-S) (l-n)1/Nsl
Figure 14 shows a plot of the factor versus sneakage for a
family of ideal efficiency curves for five baffled sections.
Similar curves can easily be constructed for different numbers
of sections.
The foregoing estimation of the effects of sneakage is a
simplification in that the sneakage gas passing the baffles will
not necessarily mix perfectly with the main gas flow, and
the flow pattern of the gas in the by-passage zone will not
be uniform and constant. The formula is derived to help in
designing and analyzing precipitators by establishing the
order of magnitude of the problem. Considerable experimental
data will be required to confirm the theory and establish
numerical values of actual sneakage rates.
55
-------
99.9
N=NUMBER OF
BAFFLED SECTIONS
.001
1/10%
S%SNEAKAGE OR R%RE-ENTRAINMENT PER SECTION
Figure 13. Degradation from 99.9% efficiency with sneakage.
56
-------
2.4
ui
CD
cr
o
CD
1.5
1.2
I.I
0 10 20 30 40
S % SNEAKAGE OR R % RE-ENTRAINMENT PER SECTION FOR A MONODISPERSE PARTICULATE
Figure 14. Correction factor for gas sneakage when Ns-5.
-------
Effect of Rapping Reentrainment
Rapping reentrainment is defined as the amount of material
that is recaptured by the gas stream after being knocked from
the collection plates by rapping or vibration. With perfect
rapping, the sheet of collected material would not reentrain,
but would migrate down the collection plate in a stick-slip
mode, sticking by the electrical holding forces and slipping
when released by the rapping forces. However, the rapping
forces are necessarily large to overcome adhesion forces, and
much of the material is released into the gas stream as sheets,
agglomerates, and individual particles. Most of the material
is recharged and recollected at a later stage in the precipitator,
We will make the simplifying assumptions that (1) the fraction of
material reentrained does not vary with particle size or
position, (2) the reentrained material is perfectly mixed in
the gas stream following rapping.
Let R = fraction of mass of a given particle size that
is reentrained
n = collection fraction of a given particle size
obtained with no reentrainment for total
collection area
ru = collection fraction per section of a given
particle size = 1 - (l-n)1/NR
NR = number of stages over which the reentrainment is
assumed to occur
PJ = penetration from section j.
Then the penetration from section 1 is given by
Pl = Rnj + 1-rij , (58)
58
-------
and from section 2,
p2 = RH.P! + (1 - n. )P!
= p [Rn + (1 - n-j) 1
1 D J
= [Rn-j + (1 - Hj)]2 , (59)
and from the last section,
= [R(i - (l - n)1/NR) + (1 - n)1/NR]NR
= [R . R(1 - n)1/NR+ {1 _ n)i/NR]NR
= [R + (i - n)1/NR (i - R)]NR - (60)
This is analogous to the formula for sneakage , so the effect
of reentrainment can be expected to be similar to the effect
of sneakage, provided that a constant fraction of the material
is always reentrained. It is doubtful that such a condition
exists, since precipitators frequently use different rapping
programs on different sections, agglomeration occurs during
collection, and different holding forces exist in different
sections. However, until sufficient data on rapping losses
PER SECTION as a function of particle size can be accumulated,
the relationship may be used to estimate the effect of rapping
reentrainment on precipitator performance.
Figure 15 shows the effect on resultant efficiency for a given
size particle of various degrees of reentrainment for a four-
section precipitator with the indicated values of no-reentrainment
efficiency .
59
-------
99.9
W
D
U
H
EH
O4
W
co
«
W
04
CO
H
Q
O
2
O
a
u
3
W
H
U
PL4
!S
O
H
EH
U
W
O
u
D
CO
0
Figure 15,
% REENTRAINMENT PER SECTION
90
80-
50-
10
20 30 40 50 60 70 80
% OF COLLECTED OUST REACHING HOPPER
100
Effect of reentrainment on the efficiency of a
four-section precipitator designed for a no
reentrainment efficiency as indicated for a
monodisperse particulate.
60
-------
This analysis has considered only reentrainment due to rapping
Other forms of reentrainment are known to occur, however, such
as: a) "saltation", losses which occur when large particles
impact previously deposited smaller particles on the collec-
tion electrode, b) losses due to bouncing of particles
following impaction on the collection surface, c) losses from
erosion of the deposited dust layer caused by excessive gas
velocity. These losses are expected to be relatively insigni-
ficant when compared with rapping losses, and the influence of
such losses on the performance of a precipitator designed for
high collection efficiencies is expected to be minor.
Since reentrainment and sneakage effects are estimated with
identical mathematical expressions, a combined correction
factor, B, is used in the computer model. From input values
of the fraction of material assumed to be lost by reentrain-
ment and sneakage, and the number of stages over which losses
are assumed to occur, the program computes B from the ideal
collection fraction for each particle size.
The correction factors B and F are used to calculate an
"apparent" migration velocity for each particle size as
follows:
61
-------
The program contains an "IF" statement which truncates all
collection fractions greater than 0.999999 at this value.
This procedure holds the product of F and B constant at these
high values of collection fraction for a given set of condi-
tions.
From we ', a reduced collection efficiency is obtained from the
Deutsch equation. A reduced precipitation rate parameter is
obtained from
V = ^ ln 'T^ ' (62)
where n-p' is the overall mass collection fraction obtained
from the individual reduced collection fractions.
62
-------
SECTION III
INPUT DATA AND PROGRAM OUTPUT
INPUT DATA FORMAT
The input data required by the program consist of an estimate
of the overall efficiency, the operating parameters and
geometry of the electrostatic precipitator under considera-
tion, pertinent characteristics of the gas and suspended
particulate, and estimated values of parameters which account
for non-ideal effects. Table 1 gives the input data required
with the data card format. All input data are converted to
MKS units prior to performing the calculations. The provisions
for electrical sectionalization require that the length of
each electrical section be rounded to the nearest integer
number of feet. If there are sections in parallel, these
must be combined by hand into overall sections across the
width of the precipitator. Procedures for measuring or
estimating the important input parameters are discussed below.
PARTICLE SIZE DISTRIBUTION
Since particle charge and electrical migration velocity are
functions of particle diameter, collection efficiency under a
given set of electrical conditions is also a function of
particle size. The overall mass collection efficiency is
therefore influenced by the size distribution of the particu-
late entering the precipitator. In-situ measurements of
particle size distributions in the range of interest for
coal-fired power plants are conducted primarily by inertial
sizing devices known as cascade impactors. The impactors can
be inserted directly into the duct or flue, thus eliminating
any condensation and sample loss problems which occur when
external sampling is used.
63
-------
TABLE 1
INPUT DATA AND DATA CARD FORMAT
Card
Set
Variable
Number of different particle
sizes (maximum of 20)
Number of submicron particle
sizes (maximum of 8)
Units
Format
12
12
Overall Card Format
2
3
4
5
8
9
Particle diameters
microns
percent
Percentages of size distribution
for all particle diameters
Identification information
Gas volume flow rate
Dust load
Precipitator length
Gas velocity
Estimated efficiency
Dust density
Dust resistivity-multiplier
of 10X
Value of x
Gas temperature
Pressure
Fraction of sneakage and/or
reentrainment
Normalized standard deviation
of gas velocity distribution
Number of stages for sneakage
and/or reentrainment
Number of steps for Runge-Kutta
integration for charge sub-
routine 0
Number of points used in numerical none
integration for charge sub-
routine 0
Charge subroutine selector, 0
or 1 (see text, Section II)
Dielectric constant
Ion mobility
Gas viscosity
Mean thermal speed of ions
Overall Card Format
none
none
none
(I2,8X,I2)
10F8.0
10F8.0
40A2
ft3/min
gr/ft3
ft
ft/sec
%
kg/m3
ohm- cm
_
op
atm
none
none
none
10F8.0
10F8.0
10F8.0
10F8.0
10F8.0
10F8.0
10F8.0
10F8.0
10F8.0
10F8.0
10F8.2
10F8.2
10F8.2
12
12
12
none
m2/volt-sec
kg/(m-sec)
cm/sec
Ell. 4
Ell. 4
Ell. 4
Ell. 4
Number of electrical sections
in direction of gas flow
Lengths of electrical sections
ft
(3(12),4(E11.4))
12
40(12)
64
-------
TABLE 1
(Continued)
Card
Set Variable Units Format
10 Area of first electrical section ft2 Ell.4
Applied voltage of first section volts Ell. 4
Current for first section amps Ell.4
Wire length for first section ft Ell.4
Corona wire radius for first section in. Ell.4
Wire to plate spacing for first in. Ell. 4
section
No. of wires per linear section none Ell.4
for first section
1/2 wire to wire spacing for in. Ell.4
first section
Repeat for each electrical section
Overall Format (7(1PE11.4))
65
-------
Table 2 shows some characteristics of several commercially
available cascade impactors..26 Measurements have been
conducted by Southern Research Institute with a modified
Brink impactor at the inlet of several precipitators
collecting ash from coal-fired boilers. Results from two
such measurements are presented in Figures 16 and 17.
Although the principle of operation of inertial impactors is
relatively simple, accurate results are obtained only with
careful attention to the technique employed in the use of the
devices. A detailed discussion of operating procedures is
available elsewhere.27 The size distribution data are entered
into the program in the form of a histogram. Data card sets
1, 2, and 3 in Table 1 are employed to transmit the particle
size information.
MEASUREMENT OF RESISTIVITY
Measurement of dust resistivity is influenced by a number of
factors which cause the values as measured by the various
methods to differ by as much as two decades. Since the
useful current density in a precipitator can be strongly
influenced by resistivity, it is apparent that resistivity
must be known precisely if it is to be used as a basis for
precipitator performance or sizing.
The relationship between dust resistivity and current density
as given in the following section is based on resistivity
values as determined with a point-plane probe. The following
is a brief discussion of factors involved in measurement of
resistivity for the purpose of defining the problem. A more
complete discussion is presented in a report entitled
"Techniques for Measuring Fly Ash Resistivity."28
66
-------
Table 2. SIZE FRACTIONATING POINTS OF SOME COMMERCIAL CASCADE
IMPACTORS FOR UNIT DENSITY SPHERES
Modified Andersen U. of W. ERC
Stage Brink Mark III (Pilat) Tag
0.85 LPM 14 LPM 14 LPM 14 LPM
Cyc
0
1
2
3
4
5
6
7
8
18.0 ym
11.0
6.29
3.74
2.59
1.41
0.93
0.56
14.0 ym
8.71
5.92
4.00
2.58
1.29
0.80
0.51
39.0 ym
15.0
6.5
3.1
1.65
0.80
0.49
11.1 ym
7.7
5.5
4.0
2.8
2.0
1.3
0.9
0.6
67
-------
22.89
10.0
100.0
0.002289
DIAMETER,
Figure 16.
Inlet size distribution obtained from modified
Brink impactor from power station burning an
Eastern coal.
68
-------
22.89
2.289
w
o
.2289
UJ
>
i
o
0.02289
0.001
0.002289
10
100
DIAMETER,
Figure 17. Size data from Brink impactor measurements at
inlet of precipitator collecting a low sulfur
Western coal.
69
-------
Electrical resistivity of a dust layer is measured by deter-
mining the current flow through a defined volume of dust when
a given voltage is impressed across the dust layer. The ratio
of the voltage to current is the resistance of the dust, and
resistivity is calculated from the geometry of the measurement
cell.
Procedures for measurement of electrical resistivity are
classified as laboratory methods if the dust is extracted
from the duct and the measurement is made in an environment
other than the flue gas, and in-situ if measurement is made
either in the duct or in the gaseous atmosphere of the duct.
Laboratory measurements generally agree with those taken
in-situ at temperatures of about 200°C and higher. Above
200°C, resistivity of fly ash is determined primarily by ash
composition and is largely independent of flue gas composition,
Bickelhaupt29 has developed an expression to predict volume
resistivity for fly ashes produced from the efficient burning
of coals as a function of ash chemistry, temperature, and
porosity for the high temperature region at one level of field
strength. At lower temperature, resistivity is influenced by
flue gas composition and laboratory measurements do not,
as a rule, correlate with in-situ values. Research is
continuing in an effort to predict low temperature resistivity
values from ash and flue gas composition.
Among the factors which cause different results to be obtained
from different measuring techniques are the particle size
distribution of the collected sample, the method of depositing
the sample, and the magnitude of the electric field at which
the measurement is made. Although no existing method is
considered to be an ideal procedure for in-situ resistivity
70
-------
determinations, the point-to-plane probe is recommended
because of the following advantages:
• The particulate collection mechanism is the same
as that in an electrostatic precipitator.
• The dust-gas and dust-electrode interfaces are
the same as those in an electrostatic precipitator.
• The measured electric fields and current densities
are comparable to those in an electrostatic precipitator.
• Flue gas conditions are preserved.
• Values obtained for the resistivity are in
general consistent with the electrical behavior
observed in the precipitator.
It should be noted, however, that resistivity data obtained
with this technique usually exhibit considerable scatter.
The scatter may be caused in part by the variations of the
properties of the small collected sample due to composition
changes. A precipitator tends to average short-term compo-
sitional variations because considerable time is required to
collect a dust layer. Resistivity measurements should thus
be averaged over a long period of time to be statistically
valid.
Resistivity values are entered into the program in data card
number five in Table 1. The program uses the resistivity
value and the input current density to calculate the electric
field in the deposited dust layer. With respect to the use
of the computer program, however, the most important usage
of the dust-resistivity value is the determination of
allowable current density, which is discussed in the
following section.
71
-------
ALLOWABLE CURRENT DENSITY
Corona current influences the electric field and particle
charging rate, and hence is one of the most significant
input parameters in determining precipitator performance.
Current and voltage relationships in a precipitator are
governed by electrode geometry and by the mobility of the
charge carriers. Limitations on current therefore limit the
precipitator operating voltage and hence the electric field.
The current also influences the field due to the space charge
resulting from the presence of ions or charged particulate in
the interelectrode space.
Electrical conditions in a precipitator are limited by either
breakdown of the gas in the interelectrode space or by
breakdown in the collected dust layer. For any given geometry,
an increase in voltage is accompanied by an increase in
electric field. When the field exceeds that required to
initiate a spark and to propagate it across the interelectrode
space, the electrical operating conditions will be limited
by these sparking conditions. Breakdown of the collected
dust layer can occur as a spark or as a back corona depending
upon the resistivity and thickness of the dust layer. In a
single stage precipitator, the corona current must pass
through the dust layer to the grounded or collecting electrode.
The voltage drop across the dust layer is
Vd = JPt , (63)
where
j = current density, amps/cm2
p = dust resistivity, ohm-cm
t = dust layer thickness, cm.
The electric field in the dust layer (E^) is the product of
the current density (j) and the resistivity (p).
72
-------
The electric field in the dust layer can be increased to the
point that the gases in the interstitial space break down
electrically. This breakdown results from the acceleration
of free electrons to ionization velocity to produce an
avalanche condition similar to that at the corona electrode.
When this breakdown occurs, the voltage drop across the dust
layer is added to that between the corona electrode and dust
surface. If this voltage is sufficiently high, a spark will
occur. The rate of sparking for a given precipitator geometry
will determine the operating electrical conditions in such a
circumstance. If the electrical resistivity of the dust is very
high and if the dust layer is sufficiently thin, the voltage
drop across the dust layer can occur at a condition which would
not cause breakdown of the gases in the interelectrode space.
Under these conditions, the dust layer will be continuously
broken down electrically and will discharge positive ions into
the interelectrode space. The effect of these positive ions
is to reduce the effective density of negative ions for charging
and reduce the electric field associated with the space charge.
Both the magnitude and rate of charging are affected by such a
back corona condition. Effective precipitator current is
therefore limited to that corresponding to the electrical
breakdown conditions whether the breakdown occurs as sparkover
or back corona.
The maximum permissible current density as a function of ash
resistivity based on dust breakdown strengths of 10 and 20
kV/cm are shown in Figure 18. Line 3 on this figure is based
on field observations, and is discussed later. Average current
densities in practical precipitators are less than those pre-
dicted on the basis of electrical breakdown because of non-
uniform current densities resulting from electrode geometry
and because the plate area or length of corona wire powered by
a single TR set is large. Consequently, theoretical current
73
-------
10"
RESISTIVITY, ohm cm
Figure 18. Current density as a function of resistivity.
74
I0»
-------
density-resistivity relationships must be modified to
accomodate these conditions. At present there is no
theoretical basis for making the required adjustments.
Field experience shows that current density for cold side
precipitators is limited to around 50 to 70 nA/cm2
(lxlO~9 A = 1 nA) due to electrical breakdown of the gases
in the interelectrode region. Consequently, this consti-
tutes a current limit under conditions where breakdown of
the dust layer does not occur.
Practically, current can be increased in a precipitator some-
what beyond that corresponding to the onset of back corona
since current density in a wire-plate geometry is fairly
non-uniform. If there is severe non-uniformity, breakdown
can occur at localized points corresponding to areas of
highest current density and it is possible for beneficial
effects to be achieved by increasing the current density above
that corresponding to the onset of back corona. However, for
reasons stated, the improved performance would be marginal
and for practical purposes, current density should be con-
sidered as being limited to that at which breakdown of the
dust layer occurs.
Breakdown of the dust layer has been studied extensively by
Penney and Craig, 30 Pottinger, 31 and others and can be influenced
by many factors. The presence of conducting particles can cause
localized areas of high electric field and hence breakdown may
occur at lower than average electric fields. The size distri-
bution of the dust also influences breakdown strength by changing
the volume of interstices. It has also been found that break-
down strength varies with dust resistivity, the higher breakdown
strength being associated with the higher resistivity.
75
-------
All of these conditions contribute to the difficulty of
determining allowable current densities on a purely theo-
retical basis. However, if allowance is made for current
non-uniformity due to electrode geometry, variations in
spacing due to erection and manufacturing tolerance, etc.,
some estimates can be made that agree to a fair approxima-
tion with observed conditions. Curve 3 of Figure 18 was
obtained from the literature,32 and is based on the observa-
tion that critical current densities in full-scale precipi-
tators can be reduced from the theoretical dust breakdown
values by a factor of about 10. The use of this curve should
give a conservative estimate of the allowable current density
as a function of resistivity.
DETERMINATION OF VOLTAGE-CURRENT CHARACTERISTICS
The voltage-current relationships for an electrostatic pre-
cipitator are governed by the mechanical design of the
collector system, the size and concentration of dust
particles in the gas stream, the presence of a dust layer
on the collection electrode, and the temperature and compo-
sition of the gas stream.
With respect to mechanical design characteristics, the corona
electrode may consist of round wires, formed metallic strips,
or specially formed members of barbed wire. Each of these
structures provide different voltage versus current character-
istics. Collection electrode spacing also influences the
voltage-current characteristics. Typical plate spacing
ranges from 22.86 cm to 30.48 cm (9 to 12 inches). Corona
current starts at a lower voltage for close spacing than for
wide spacing. The increased space charge associated with the
wide spacing causes a further decrease in current for a given
applied voltage.
76
-------
As has been discussed previously, the partieulate matter
suspended in the effluent gas stream influences the elec-
trical conditions. Large quantities of fine particles
will acquire an electrical charge with a resultant decrease
in current at a given voltage. As the charged partieulate is
collected, this relatively immobile space charge is removed
with an increase in current for a given applied voltage.
This current suppression with dust load is shown in Figure 19
for a precipitator operating on a power station boiler
utilizing coal with a sulfur content of about two and one
half percent.
The voltage drop across the dust layer results in an increase
in the applied voltage required to achieve a given current
density as shown in Figure 20. The condition illustrated is
for a DC voltage-current curve with a one centimeter thick
dust layer with a resistivity of 1x1011 ohm-centimeter on a
typical second field volt-ampere curve.
The actual electrical conditions that are active for elec-
trical collection exclude the voltage drop in the dust layer.
Therefore, that portion represented by the shift in Figure 20
must be neglected in estimating performance. For estimating
purposes, some average volt-ampere characteristics should be
selected. In the inlet section, the voltage and consequently
the electric field will be high while the outlet section with
the very light dust load will operate at a somewhat reduced
voltage. Therefore, the proper choice for a representative
voltage-current characteristic will lie somewhere in between.
A reasonable approximation to the average electrical conditions
in the precipitator is given in Figure 19 by the curves
labeled "typical".
77
-------
cvj
£
o
X
- 30
V)
z
Ul
o
UJ
(T
cr
3
o
SOLID- 370°C
DASHED- I50°C
OUTLET / TYPICAL
(370°C)
Figure 19
30 40
APPLIED VOLTAGE, kilovolts
Comparison between the voltage vs current
characteristics for cold side and hot side
precipitators. Corona wire radius =
.277 cm (.109"), plate spacing = 22.86 cm
(9").
78
-------
70
60
50
CM
E
o
V)
Z
u
Q
LJ
o:
40
30
20
10
10
SECOND
FIELD
CLEAN
ELECTRODE
*
/
*
/
SECOND FIELD
WITH I CM
LAYER _
/* = ! XIO"J1CM
20 30 40
APPLIED VOLTAGE,kilovolts
50
60
Figure 20 .
Voltage vs current characteristic for second
field clean electrode and 1 cm layer of 1x10ll
ohm-cm dust.
79
-------
The voltage-current characteristics are different for pre-
cipitators operating prior to the air preheater from those
operated subsequent to it for two reasons: the gas temperature
is on the order of 270°C for the former and 150°C for the
latter, and because of the increased gas volume at the higher
temperature, the dust loading is less for the hot side unit.
The increased temperature leads to an increase in current at
each applied voltage because of the reduced gas density, as
well as electrical sparkover occurring at a lower voltage.
The decreased dust loading at the high temperature causes a
reduction in the space charge that suppresses the current.
Figure 19 illustrates the difference in the V-I characteristics
of precipitators in the two temperature regions.
Voltage and current data are entered in the program in
data card set 10 (Table 1).
ELECTRICAL SECTIONALIZATION
Sectionalization of a precipitator can be expressed in terms
of the number of transformer rectifier (TR) sets used per
unit of gas flow, or the plate area served by a TR set.
In terms of performance, a large number of TR sets for a
given gas flow or plate area, will permit each section to
operate at a higher voltage and current and hence at higher
collection efficiency. From a maintenance standpoint, large
power supplies with the large attendent plate area contribute
to excessive electrical failure for the discharge electrodes
and greater precipitator outage time. Another consequence of
poor Sectionalization is that outage of one section disables
a larger portion of the precipitator.
80
-------
Performance requirements can be met by several combinations
of precipitator size and degree of sectionalization. Based
on only performance considerations, the choice might be
determined by economic trade off. However, reliability and
maintenance considerations should be overriding factors that
dictate the degree of sectionalization.
White33 has shown that theoretically expected reductions in
operating voltage occur as identical wire pipe sections are
connected to the same power supply due to the variation in
voltage with spark rate. For a given overall spark rate, each
pipe would operate at a lower effective spark rate and lower
voltage as the number of units is increased.
In full scale precipitators an additional factor influencing
precipitator electrical conditions is variation in mechanical
and erection details resulting in electrode misalignment and
sharp edges or points that cause high localized electric fields
With large plate areas, variations are likely to occur in
localized areas resulting in voltage limitations for an entire
section. A high degree of sectionalization tends to minimize
the effect of these mechanical and constructional variations.
The trend in precipitator design has been toward larger TR sets
and larger electrical sections. Some TR sets of up to 4 amps
secondary current have been installed on larger fly ash precipi-
tators. Such sets can supply up to 37,180 m2 (400,000 ft2) of
plate area. Earlier precipitator TR sets were designed to
supply 1858 to 3716 m2 (20-40,000 ft2) of plate area. During
sparkover, the energy that can be dissipated is determined
both by the energy stored in the precipitator capacitance, and
the energy that the power supply can provide. The energy
stored in the precipitator-distributed capacitance continues
to supply the spark even though the power supply voltage is
81
-------
removed. In the case of large power supplies and large
plate areas, the energy dissipated can cause rapid failure
of the discharge electrodes and severe maintenance problems.
From a design standpoint, precipitators should be engineered
according to one of two philosophies. If sparking is to be
permitted, a reasonably high degree of sectional!zation should
be provided. If large power supplies are to be used, spark
rates should be maintained at very low levels (around 1 per
minute) in order to minimize electrical erosion of the elec-
trode material. In the latter case, a lower operating voltage
and current would result and increased plate area must be
provided.
The computer program is capable of representing electrical
sectionalization in the direction of gas flow. If there are
multiple TR sets across the width of a precipitator longi-
tudinal section, these sets must be combined manually to give
an average voltage and current for the section.
PROGRAM OUTPUT
Figure 21 gives a typical example of the output from the
program using data from a laboratory-scale precipitator.
The following information is given.
(1) Identification of the data set.
(2) Input data which pertain to the entire precipitator.
(3) Input data and derived quantities which pertain
to a particular electrical section.
(4) Results obtained for each incremental length which
indicate the status of the calculations. These
quantities are:
82
-------
(a) the ratio of total space charge to ionic
space charge,
(b) the average electric field between wire
and plate, volt/m,
(c) the electric field adjacent to the collecting
electrode, volt/m,
(d) the average free ion density, no./m3,
(e) the current density, nanoamperes/cm2*
(f) the mass median diameter of the particulate
collected in the length increment, m/
(g) the mass of dust collected in the length
increment, kg/m3, and
(h) the increment number.
(5) Inlet particle size distribution.
(6) Calculated collection efficiency, the Cunningham
correction factor, the effective migration velocity
(cm/sec), the reduced effective migration velocity,
and the reduced collection efficiency for each
particle size.
(7) Estimated and computed values of the overall
mass efficiency.
(8) The mass median diameter of the uncollected dust, m
(from the theoretical calculations).
(9) The precipitation rate parameter, cm/sec.
(10) Input values of normalized gas velocity standard
deviation, the fraction of dust assumed lost per
stage from reentrainment and/or sneakage, and the
number of stages over which the losses are assumed
to occur.
(11) The reduced overall efficiency obtained from the
use of the empirical corrections.
(12) The reduced precipitation rate parameters obtained
from the reduced overall efficiency, cm/sec.
83
-------
(13) The ratio of the charge accumulated to the
modified saturation charge (Section II), based
on conditions in the last increment, for the
submicron particles as a function of increment
number.
(14) The charge accumulated (in coul) on submicron
particles as a function of increment number.
84
-------
"FT WALL CO = »'j
USAT*\,2 EPS=S.l
PHH LENGTH = O
OUST LOAO = U,I037F.-05 KG/Mi
OUST DtNSIfY s O.lOOOt+OU KG/M3
OUST WEIGHT = 0,b899E-05 KG/SEC
VISCOSITY = 0,180UE-0« KG/M-SEC
MEAN THERMAL SPEED a 0,<|UOOE»03 M/SEC
INPUT EFFICIENCY/INCREMENTS 36,03
CALCULATION IS IN SECTION NO, » J
COLLECTION AREA s 0.6975E+00 M2
-IIRF TO PLATE B 0.6350E-01 M
CURRENT/M s 0.6B31E-OU AHP/M
1/2 WIRE TO HIRE B 0.6350E-01 M
'-in, up INCREMENTS = to
GAS VELOCITY = 0,1373K.*Ol M/SEC
OUST HES1STIVTY a 0.1000F>09 OHM-M
TEMPERATURE = 299,UUU K
ION MOBILITY o 0.2200E-03 M2/VOLT-SEC
APPLIED VOLTAGE = 0,2950£*05 VOLTS
CORONA HIRE RADIUS m 0,13«6E-02 M
CURRENT DENSITY a 0,2b88E-03 AMP/M2
GAS Fl'J« HA1E c 0,ofcSr,K-01 r,5/3tC
tST, FKHCIENCY B o*,00 PFRC.ENT
DUST VOLUME s 0,10S7E-Ob M3/Mi
PHESSUSt B 1.000 ATM
REL. DIELECTRIC COnSTANl a 0 ,S100fc.*01
TOTAL CUHHEMT = 0,187b£-03 AMHS
CORONA WIRE LENGTH a 0.27'tSt + ni M
DEPOSIT E FIELD = 0.2b8Bf-.«05 V
00
ROVRI
1,312
1,155
1,076
ERAVG
O.U6U6F+06
O.U6U6E+06
0,«6UbEt06
EPLT
0,2397EtOb
0,235JE*06
0.2331E+06
CALCULATION is IN SECTION NO, » 2
COLLECTION AREA • 0,fcV75E*00 M2
WIRE TO PLATE e 0,6350E«01 M
CURRENT/M s 0,683tE-0« AMP/M
1/2 HIRE TO WIRE = o,6350E»oi M
ROVRI
1,038
1,019
1 ,009
ERAVG
0,U661E«06
O.U661E+06
EPLT
0,2326E*Ob
0,232lEf06
0.2321E406
CALCULATION is IN SECTION NO, = 3
COLLECTION AREA • 0,9300E*00 M2
HIRE TO PLATE • 0.6350E-01 M
CURRENT/M B 0,6831E"0« AMP/M
1/2 HIRE TO HIRE • 0.63SOE-01 M
AFID
0.1253E+14
CMCD
26.•>
26,9
26.9
MMD
0.3223E-05
0.2711E-OS
0.2337E-05
HEIGHT
0,3289E-Oa
0.2329E-OU
0.1506E-OU
INCREMENT NO.
i
2
3
APPLIED VOLTAGE s 0.2960E+05 VOLTS
CORONA WIRE RADIUS * 0.1346E-02 M
CURRENT DENSITY a 0,2688f-03 AMP/M2
TOTAL CURRENT e 0.1875E-03 AMPS
CORONA HIRE LENGTH • 0,27«5E*01 M
DEPOSIT E FIELD s 0.2688F+05 VOLT/M
0.1579E + U
Q.1608E*ia
0.1623E*1«
CMCO
26,9
26.9
26.9
MMO
0.2039E-05
0.1718E-05
0.1541F-OS
WEIGHT
0.9790E-05
INCREMENT NO,
APPLIED VOLTAGE • 0.2900E+05 VOLTS
CORONA WIRE RADIUS = 0,1302 M
CURRENT DENSITY = 0.268BE-03 AMP/M2
TOTAL CURRENT • 0.2500E-03 AMP3
CORCNA HIKE LENGTH • 0,3660E+01 M
DEPOSIT E FIELD s 0,2688E+Ob VOLT/M
ERAVG
EPLT
ROVRI
l.OOS
1,002 0.4567E+06
1,001 0,«567E»06
1,001 0,4S67Et06
INPUT EFFICIENCY/INCREMENT 27,72
CALCULATION IS JN SECTION NO. s 1
COLLECTION AREA B 0,6975E*00 M2
HIRE TO PLATE e o,635oE-oi M
CURRENT/M a 0,6«31E-0« AMP/M
1/2 HiRE TO HIRE = 0.6350E-01 M
AFID
CMCD
MMQ
HtlGHT
0.2321F+06
0,2321Et06
0,2321Et06
0,16feSE+l«
0. 1668E*1«
0.1670E>1«
0,16.71E + 1«
26.9
26.9
26.9
26.9
0.1«2UE-05
0.132«E-05
0.1228E-05
O.U«lE-OS
0,30b3E-05
0.2taaE-05
0. 1387E-05
0.1171E-05
APPLIED VOLTAGE = 0,2950f.»05 VOLTS
CORONA WIRE RADIUS s 0,t3U6E-02 M
CURRFNT DENSITY s 0.2688E-03 AMP/M2
INCREMENT NO.
7
8
9
10
TOTAL CUSRfNT = 0.1B75E-03 AMPS
CORONA HIRE LENGTH = 0.2745E+01 M
DEPOSIT E FIELD * 0,2688E*05 VOLT/,1
Figure 21. Typical Output Information from Computer Model.
-------
HUVK ]
E w t v c;
AM!)
0,E + 06
1.07U 0,464»»E*Oh
C*LCULAlI"N IS J* SPCUON NO.
COLLECTION AKEA s 0,h975EtOO
HIRE fO PI ATE. s 0.63SOE-01 M
CuKKtM/c = o.oMlk -.014 AMP/M
1/2 hl»F TO fjHt e O.fcl^^t-ni
0.2131E+06
s 2
0. 1«50E*14
26,9
26.9
APPLIED vnLTAGt = o.29hOE*os VOLTS
CORONA KIRf. KAiMUS a 0,13«6E-02 f
CURRENT DENSITY = 1.2686E-03 AMP/M?
0,^221t-05
0.2706E-OS
. '< !l fl ! F - .1 n
TOTAL CURRENT i 0,ia75E-03 AMPS
COHONA "I HE LENGTH o 0.2/4SEf01 M
DEPOSIT E FIELD » 0.2b»HE*05 VOLT/I^
ROVR1
EPLT
1,002 0,U66lE*06 0,2326E+06
1.02U 0.«6»i|E + 06 0.2322Et06
1,013 O.UbblE+06 0,2320Et06
CALCULATION I§ IN SECTION NO, a J
COLLECTION AREA o 0,9300EtOO M£
HIRE TO PLATE • 0.6J50E-01 M
CURRENT/M E 0,68JlE-Ofl AMP/M
1/2 WIRE TO WIRE a o,6350E»01 M
AFIO
0,lS73t«l«
CMCO
26,9
26,9
26.9
HMD
0.2035E-OS
Q.1744E-05
0.15UOE-05
HEIGHT
0.9756E.05
0,644SE>05
O.U375E.05
NO,
APPLIED VOLTAGE = 0.2900C+05 VOLTS
CORONA MIRE RADIUS a 0.1346E-02 H
CURRENT DENSITY = 0.2688E-OJ AHP/M2
TOTAL CURRENT • 0.2500E-03 AMPS
CORONA MIRE LENGTH 3 0,JM>OttOl
DEPOSIT E FIELD • 0.2668E+05
ROVRI
ERAVG
EPLT
AFID
00
CTi
1,008 0.4567E+06
1,004 0,4567Et06
1,003 0.4S67E«06
1.001 0,4567E«06
0,2320E«06
0.2320E*06
0.2320E+06
0,2320E»0(>
0.1660E+14
0,1665E«14
0.1668E+14
0,1670E«14
PARTICLE SIZE RANGE STATISTICS
DIAMETER-MtTEHS
0.2500000E-06
0.3500000E-06
0,«500000E»06
0.5500000E-06
0.7000000E-06
0.9000000E.06
0.1 100000E-05
0.1300000E-05
0.1600000E-05
0.2000000E-05
0.2600000E-OS
0.3500000E-OS
0,SOOOOOOE-05
PERCENT OF TOTAL
0,280000
0.550000
1.000000
1.400000
3,700000
4.500000
4.500000
5.300000
10.700000
9,000000
16,000000
13.000000
30,069999
EFFICIENCY
71,860634
73,541920
76,458817
79,001150
43,314363
87,473246
90.606186
92,953346
95,415375
97.412830
98,905010
99.702742
99.967566
EFFICIENCY - STATED = 96.31
HMD 0* EFFLUENTS 0.6h70£-06
PRECIPITATION RATE PARAMETER
COMPUTED = 96.33
9.U52
CMCD
26.9
26.9
26.9
26.9
CCF
1.5942
,4170
.3222
,2629
.2063
.1604
.1312
.1110
.0902
,0722
.0555
.0412
,0289
HMD
O.I423E-05
0,1323E-05
0.1227E-05
0.1140E-OS
H
3,627
3.803
4.137
4.519
5,122
5,942
6,765
7.587
8,017
10,454
12.913
16,643
22.979
CONVERGENCE OBTAINED
HEIGHT
0.3046E-05
0.2172E-05
0.1S76E-05
0.1165E-05
INCREMENT NO.
7
8
9
in
»•
3,291
3.448
3.745
4,084
4,616
5.335
6.050
6.757
7.801
9,162
11.143
13,975
18.218
EFF'
68.3485
70,0421
73.0016
7*.01BO
80,0890
84.5126
87.9377
90,5799
93.4596
95.936J
97,9669
99.2445
99.6266
SIGMAB O.OCH'O MT
CORR, EFF, a 95.12U6
CORRECttn PRECIPITATION «ATt PARAMETEH =
O.OBOOSNEAKAGE OVER 4,0000 STAGF.S
Figure 21 (continued).
-------
CHARGING NATtb FUK SUB-iICRi)N PARTICLES fktlM RUbRUUUNt. CHAHGN
INCREMENT NO, U/USATF FCH INDICATED PARTICLE SIZES
A ^ C M n r- At * *»-.«.. .., « ,.»--.i_ - - .— .-
0
1
2
3
4
5
6
7
6
9
10
.2500E-06
0.7880
0.9J2H
0,9702
1,0078
1,0344
1.0562
1 .0744
1 ,0906
1 .1052
1.11BU
0.3500F.-06
O.B01 3
0, V *bb
1.0046
1.0515
1.0871
1.1158
1 .138U
1.1579
1 , 1 750
1,1903
O.USOUE-Ob
0,8090
0.950^
1,0276
1,0791
1,1171
1.1471
1.1698
1 ,1892
1,2061
1.2211
O.S500E-0*
0.81P5
0.9585
1,0393
1,0919
1.129H
1.1591
1,1808
1.1993
1.2153
1,2?94
0.7000E-Ob
0,8134
0,9615
1 .0438
1 .0954
1 .I3lb
1.1593
1,1 791
1 ,^9bO
1,2105
1,2234
0.r»OOPh-06
0,810h
0.957U
1.0383
1 .OH76
1.1213
1 .1467
1.1644
1 .1793
l.!*^
1,2035
oo
CHARGE ACCUMULATED ON SUHMICRON PARTICLES EACH INCREMENT
INCREMENT THARRE FOR INDICATED PARTICLE SIZES
i
2
3
4
5
6
7
8
9
10
STOP
SfND
0.2500E-06
0.321U5E-17
0,37237E-17
0.39577E-17
0.4H12E-17
0.42197E-17
0.43085E-17
0,43826E-17
0.44487E-17
0.45082E-17
0.45623E-I7
011111
0.3500E-06
0.51U57F-) 7
0,fc0093L-17
0.64514E-17
0.67525E-17
0.69808E-17
0.71656E-17
0.73106E-17
0.74359E-17
0.75459E-17
0,76«39t-17
0,45006-06
0./6030F-17
0.89353E-17
0.96577E-17
0.10142E-16
0.10499E-16
0.10781E-16
0.10994E-16
0.11177E-16
0,1 1335F-16
0,1 J47^E-16
n.550nE-06
C. 10577E-16
0.12478E-16
0.13530E-16
O.U215E-16
0.1U707E-16
0.15089E-16
0,lb372t-16
0,15612E-16
0.15821E-16
0,lb005E-16
0, 7000F.-06
0,15992fc-l6
0.18905E-16
0.20S23K-16
0.21538E-16
0.22250E-J6
0,22794t-16
0.23164F-16
0,235lSE-16
0,23«0?E-16
0.24054E-16
O.VOOOfc— 06
0,2«9h7f- -16
0,29«9Ct-16
0.31980E-16
0.33500E-16
0.34539E-16
0,3532 If- 16
0,35864f--16
0.363?4t-i6
0.3672U.-16
0,370 701 -16
Figure 21 (continued).
-------
SECTION IV
RESULTS
COAL-FIRED POWER BOILERS
This section presents results obtained from the computer model
using input data representative of conditions encountered for
coal-fired power boilers. Table 3 describes the source of
the more important input parameters. These input data will
serve to illustrate the trends predicted by the model and will
also provide an indication of the approximate plate area
requirements which may be expected for various control effi-
ciencies as a function of conditions. Coal-fired power
boilers were chosen to illustrate the use of the model
because this is the most extensive application area for
electrostatic precipitators and the only one in which appro-
priate data are available.
Implicit in the use of the current density-resistivity relation-
ship of Figure 18/ curve 3, is the assumption that the electric
field in the deposited dust layer is sufficiently low to
prevent the occurrence of back corona or excessive sparking.
The selection of dielectric constant values of 6.0 and 100 for
the low and high current densities, respectively, may be
justified on the basis of resistivity. For a resistivity of
2x10ll ohm-cm, curve 3 of Figure 18 indicates a current
density of 5xlO~9 amps/cm2 may be used. Particles with such
a resistivity would behave as a dielectric, and a dielectric
constant of 6 may be used to approximate the dielectric
behavior of fly ash under high resistivity conditions.31* For
the current density values of 20 and 40xlO~9 amps/cm2,
resistivity values from Figure 18 are 5x1010 ohm-cm and
2.5xl010 ohm-cm, respectively. White35 indicates that
particles with resistivity in this range may be considered
88
-------
Table 3. Input Data for Figures 22 through 28.
Data Element
Particle size distribution
Current-voltage
Current density-resistivity
relationship
Dust loading
Dust density
Dielectric constant
Gas ion mobility
Precipitator dimensions
Gas viscosity
Charge subroutine selection
Data
Histogram from Figure 16
Typical 150°C curve, Figure 19
Curve 3 of Figure 18
9.16 grams/m3
2.27 grams/cm3
6.0 for current density of
5xlO~9amps/cm2
100 for current density of
20 and 40xlO~9 amps/cm2
2.2 cm2/(volt-sec) at 150°C
3.35 cm2/(volt-sec) at 370°C
Length 11 m
Plate to plate spacing 22.8 cm
Plate area 28930 m2
Inlet cross sectional
area 301 m2
Corona wire radius .135 cm
Corona wire spacing in
direction of gas flow 20 cm
2.2xlO~5 kg/(m-sec)
2.8xlO~5 at 370°C
CHARGN (SRI model)
89
-------
as conductors for purposes of calculating particle charge
values. The use of a dielectric constant value of 100 is
equivalent to assuming that the particle is behaving as a
conductor.
There is considerable uncertainty in the value of gas ion
mobility which should be used in the program. The value
reported in the literature36 for oxygen ions in air at
room temperature is about 2.2 cm2/(volt-sec). If tempera-
ture were the only variable influencing the gas ion mobility,
the value of 2.2 cm2/(volt-sec) would be increased in going
from 20°C to 150°C by the ratio of the absolute temperatures.
However, the presence of fine particulate and small amounts
of highly electronegative gases with low mobilities in flue
gases would be expected to lower the overall effective ion
mobility by a significant amount. Since there is no
rigorous procedure for calculating effective mobility
under these conditions, the value to be used in the program
was estimated by using the room temperature value for
oxygen. The program estimates the effects of particulate
on the effective mobility as described in Section II.
For the calculations at 370°C, the mobility value
of 2.2 cm2/(volt-sec) was increased by the ratio of the
absolute temperature change from 150°C in order to indicate
the trend caused by the temperature change. The selection
of an appropriate value for ion mobility is important because
of the influence the value has on the calculated electric
field at the plate and the value of particle charge.
Figure 22 gives the effective, i.e., length averaged
migration velocities obtained by the program for the indi-
cated current densities and temperatures. These results do
not include the correction factors discussed previously.
90
-------
0
OJ
en
-P
•H
O
o
H
QJ
C
O
-H
4-1
id
•H
E
-H
-P
0
0)
4-1
U-4
w
%40na/cm2(l50°C)P
20 no/cm2 (150 °C
5na/cm2 ( I50°C)
SPECIFIC COLLECTION
AREA = 39.4 m2/(m3/sec)
(200 FT2/1000 cfm)
i.o
0.1
1.0
10.0
100.0
Particle Diameter, ym
Figure 22. Effective Migration Velocity as a Function of Current Density
and Particle Size.
-------
The specific collection area (SCA) influences the effective
migration velocities through the residence time which is
available for particle charging. With larger SCA's, more
residence time is allowed, and the we values will increase.
This variation is most significant for the lower values of
current density (10 nanoamps/cm2 or less). Since low current
densities generally require SCA's higher than 39.4 m2/(m3/sec)
in order to meet design efficiency levels, the values indi-
cated on Figure 22 for 5 nanoamps/cm2 will be conservative
from the standpoint of particle charging dynamics.
Note that high temperature operation results in a significant
decrease in we values for comparable current densities. This
is a consequence of the differing voltage-current characteris-
tics, and the changes in physical properties of the gas with
increasing temperature. The practical implication of this
trend is that for comparable values of resistivity, current
density, and size distribution, higher SCA's will be required
for a given collection efficiency. However, if the dust
resistivity severely limits the allowable current density,
the relative potential advantage for high temperature opera-
tion is indicated by a comparison of the high temperature we curve
with the low temperature curve at a current density of 5 nA/cm2.
Figure 23 gives a comparison of effective migration velocity
values obtained from the computer program with those obtained
from inertially-determined fractional efficiency measurements
on a coal-fired power boiler. The computed we values are
somewhat higher than those given in Figure 22 because of the
change in SCA. The value of 55.7 m2/(m3/sec) from Figure 23
represents conditions during the test period. In addition
to the theoretical predictions, Figure 23 shows the effect of
correcting for a gas velocity, standard deviation (ag) of 0.25 using
the empirical relationship given in Section II. The
92
-------
28.0
U>
o
0)
B
•p
-H
U
O
c
o
-H
-p
-H
S
-P
U
w
CALCULATED FROM INERTIALLY
DETERMINED FRACTIONAL
EFFICIENCY MEASUREMENTS
COMPUTED AT
20 no/cm2 ^
2.0
0.1 0.2
Figure 23.
0.3 0.4 0.5 1.0 1.5 2.0
Particle Diameter, ym
3.0 4.0 5.0
Effective Migration Velocities for a Full-Scale
Precipitator on a Coal-Fired Boiler (SCA =
55.7 m2/(m3/sec).
10.0
-------
indication that S=0 means that no corrections for reentrain-
ment or sneakage losses were made. A comparison of the we
values in Figure 23 indicates that the theoretical predictions
are low for the particle diameters below 2.0 ym diameter and
high for larger particles. Possible causes for this are
(1) the under-prediction of fine particle collection due to
particle concentration gradients and (2) under-prediction of
large particle penetration due to reentrained agglomerates.
Due to limitations in the measurement technique, no data are
available for particle diameters larger than 5.0 ym. Figure 24
shows the fractional efficiency data from which the we values
of Figure 23 were calculated.
Figures 25 through 28 present results obtained from the program
in terms of overall mass efficiency as a function of SCA for
current densities of 5, 20, and 40 nanoamps/cm2 at 150°C/ and
for a current density of 30 nanoamps/cm2 at 370°C- Also
given are test results obtained under conditions approximating
the electrical conditions represented by the given values of
current density. A comparison of the limited amount of
applicable test data with the computed results indicates that
the theoretically-predicted overall mass efficiencies are
higher than those obtained from the field measurements. The
measurements were taken with sampling techniques which
insured that essentially all of the particulate mass larger
than 0.3 ym diameter was captured by the sampling device.
The use of the empirical correction factors discussed in
Section II reduces the computed values of overall mass
collection efficiency to the range of values obtained
from the field measurements.
94
-------
99.99
c
a)
•H
U
w
-H
-P
o
s
'COMPUTED AT
20 no/cm2
10
10.0
Particle Diameter, ym
Figure 24.
Fractional Collection Efficiencies for a
Full-Scale Precipitator on a Coal-Fired
Power Boiler.
95
-------
61
c
Q)
•H
O
99.95
99.90
99.5
99.0
w 98
§
•H
4J
O
0)
95
90
80
70
100
200
300
400
500
600
700
800
Specific Collecting Area,
ft2/(1000 ft3/min)
Figure 25. Computed Performance Curves at 5 nA/cm2.
96
-------
99.99
o
c
o
"m
w
c
.3
«
a)
8
99.95
99.9
99.8
99.5
99
98
95
90
100
200
300
400
500
600
Figure 26
Specific Collecting Area/
ft1/(1000 ft3/min)
Computed Performance Curves at 20 nA/cm2.
97
-------
99.9 [
<*>
c
Q)
•H
O
w
§
-H
-P
u
0)
O
0
99.95
99.9
99.8
99.5 I!
99
98
95
90
100
200
300
400
500
600
Specific Collecting Area,
ft2/(1000 ft'/min)
Figure 27. Computed performance curves at
40 nA/cm2.
98
-------
99.99
c
0)
•H
u
w
§
3
u
0)
99.95
99.9
99.8
99.5
99
98
95
90
100
200 300 400
Specific Collecting Area,
ft3/min)
500
600
Figure 28. Computed Performance Curves for "Hot1
Precipitator.
99
-------
Reentrainment and sneakage effects were combined by assuming
that the indicated fractional losses per stage occurred over
four effective stages. Note that high efficiency (greater
than 99.5%) precipitators operating on the low temperature
side of the air heater fall reasonably close to the computed
line obtained with Og = 0.25 and S=0.1. A gas velocity dis-
tribution with a standard deviation of 0.25 is generally
considered to be a good distribution for a full-scale unit.
The computed results show that a poor distribution (ag = 0.5)
seriously degrades performance. The detrimental effects of
sneakage and reentrainment are also indicated in the decrease
of computed performance resulting from variations in the
parameter S. A more definitive comparison of computed with
measured performance than currently available data permit is
expected to result from future test programs on full scale
precipitators in which the gas velocity distributions will
be determined and efforts will be made to quantify losses
caused by reentrainment and sneakage.
It should be noted that these calculations do not directly
indicate the increasing reentrainment that would occur with
high gas velocities. In general, gas velocities greater than
about 1.5 m/sec (5 ft/sec) will be accompanied by excessive
reentrainment. Therefore, variations in SCA for a given
precipitator installation which are achieved by gas flow
variation may require different correction factors for
reentrainment at each value of SCA.
100
-------
EFFECT OF VARIATIONS IN SIZE DISTRIBUTION
Since the particle size distribution has a major effect on the
results obtained from the computer model, it is of interest to
examine the program output for differing size distributions with
other conditions held constant.
Particle size distributions are characterized by two or more
parameters. One parameter is a measure of the mean particle
diameter, and the other parameter or parameters are measures of
the polydispersity of the size distribution. As is shown below,
both the mean particle diameter and the distribution polydispersity
affect the overall eff iciency-^SCA relationship. Thus, the
common practice of specifying a mean particle diameter is not
adequate for sizing an ESP.
Log normal particle size distributions with mass median diameters
of 25, 10, 5, and 2 ym were used as input data to the computer model,
The geometric standard deviation of the distribution was held con-
stant at 2.8. Other important input data were: a current density of
20 nanoamps/cm2, an applied voltage of 33 kV, ag = 0.25, and S =
0.1 over 4 stages. The results from these computer simulations
are given in Figure 29. As would be expected, the computed
performance is a strong function of the mass median diameter of
the distributions.
Figure 30 presents results obtained from the program using a
log normal particle size distribution with a mass median
diameter of 10.0 ym and geometric standard deviations of 1.0
(a monodisperse distribution), 2.0 and 5.0. Other input data
were identical to those used in generating the curves for
Figure 29. Figure 30 shows that predicted performance decreases
with increasing values of particle size standard deviations.
This decrease results from the influence of the increasing
proportions of fine particulate which are present with the
101
-------
o
c
0)
-H
O
•H
w
-H
-P
U
0)
19.7 39.4
m2/(m3/sec)
59.! 78.7 98.4 118.1 137.8 157.5
99.98
99.95
99.9
99.8
99.5
99.0
98.0
95.0
90.0
80.0
70.0
60.0
100
200
300
400
500
600
700
800
Specific Collecting Area,
ftVdOOO ftVmin)
Figure 29. Effect of Mass Median Diameter on Computed
Performance (Op = 2.8).
102
-------
18. L37.8 157.5
19.7 39.4 59.1
80.0
70.0
100
200
300
400
500
600
700 800
Figure 30,
Specific Collecting Area,
ft2/(1000 ft3/min)
Effect of Particle Size Distribution
Standard Deviation on Computed
Performance.
103
-------
larger values of standard deviation. Note that the use of a
monodisperse distribution with a diameter of 10.0 ym gives
results vastly different from those obtained with realistic
values of standard deviation.
104
-------
SECTION V
COLLECTION OF HIGH RESISTIVITY DUST
The degradation of precipitator performance resulting from the
presence of a layer of high resistivity dust on the collection
electrode has been well documented in the literature and will
not be repeated here. When such a dust is to be collected by
electrostatic precipitation, there are two basic design
philosophies available: a) alter the resistivity of the dust
by introducing chemical conditioning agents or by changing
the operating temperature, or b) enlarge the collection area
so that the design collection efficiency can be obtained under
the electrical constraints imposed by the high resistivity dust.
As the preceding sections have indicated, the required collec-
tion area increases as dust resistivity increases above values
in the 1010 ohm-cm range. The uncertainty in the required
collection area also increases because the sensitivity of
precipitator performance to resistivity changes is greater in
the high resistivity region. Therefore, a direct economic
comparison of the options is often misleading because of the
uncertainty associated with option b) and the economic conse-
quences that may result from undersizing the precipitator. It
is advisable, therefore, to consider methods for lowering
resistivity into the 10 l° ohm-cm range rather than attempt to
design for adverse electrical operating conditions. In the
following paragraphs, a brief description of the methods for
lowering resistivity will be given.
CHANGE IN OPERATING TEMPERATURE
The shape of the resistivity-temperature relationship for most
fly ash from coal-fired boilers suggests that acceptable resis-
tivities may be obtained by operating the precipitator either
below about 110°C or above 345°c- Low temperature operation
105
-------
has been successfully used to solve a high resistivity problem,37
but this method has not found widespread acceptance. The prin-
cipal disadvantages of this approach are that resistivity in
the low temperature range is quite sensitive to changes in fuel
composition, and extensive in-situ resistivity-temperature data
3 3
are required to give a firm basis for design. A more detailed
discussion of the low temperature approach is given elsewhere.
For the coal-fired power boiler application, high temperature
operation of the precipitator ahead of the air preheater is
becoming the most commonly used solution to the high resistivity
problem. The principal advantages of this approach are that,
at temperatures in the 315 to 370°C range, resistivities above
1011 ohm-cm are rarely, if ever, encountered, and the
sensitivity of precipitator operation to fuel composition is
reduced. However, Bickelhaupt39 has shown that in the high
temperature range considerable differences in resistivity occur
due to differences in the chemical composition of the ashes.
Therefore, resistivity measurements in the design temperature
range are desirable prior to selecting a collection area in
order to determine whether the dust resistivity will impose
any constraints on the electrical operating conditions. The
principal disadvantages of high temperature operation are the
increased gas volume which must be treated and the lowered
operating voltages which result from the decreased gas density.
USE OF CHEMICAL CONDITIONING AGENTS
Compounds which have been injected in the flue gas ahead of
the precipitator with the objective of lowering ash resistivity
include sulfuric acid vapor, sulfur trioxide, ammonia, ammonium
sulfate, sulfamic acid, and ammonium bisulfate. Of these
compounds, the most commonly used are sulfuric acid vapor and
106
-------
sulfur trioxide. It has been established that proper injection
of these compounds can effectively "condition" fly ash by
lowering resistivity to an acceptable range .** ° The principal
disadvantages are the possibility of introducing a sulfuric
acid aerosol into the atmosphere, and the difficulties of
maintaining a chemical processing plant to properly inject
the corrosive and toxic compounds. However, the use of
conditioning agents can be economically attractive when
compared with the cost of the other methods of lowering dust
resistivity. Detailed discussions of the use of conditioning
agents are given by Dismukes.1*1'lt2
An additional conditioning agent is currently under investiga-
tion. Bickelhaupt29 has shown that the resistivity of the fly
ash is related to the alkali metal content, principally sodium,
of the ash. Preliminary tests show that the addition of trace
quantities of sodium oxide to the coal feed also results in a
reduction of the resistivity. However, the limited experience
with this material precludes a definitive analysis of the long-
term effect on the behavior of a precipitator.
In general, the use of conditioning agents has not been widely
accepted. The primary usage has been for overcoming high
resistivity problems in existing precipitators.
107
-------
SECTION VI
ESTIMATION OF PROGRAM OUTPUT
It is recognized that the general usefulness of the computer
model would be enhanced by presenting the results in a form
which would not require access to a computer. In principle,
it is feasible to prepare tables or plots of effective migra-
tion velocities for a range of particle sizes, precipitator
operating conditions, and electrode geometries. With this
information, and assumed values of the parameters which estimate
the effect of non-ideal conditions, the computed overall mass
efficiency for a given specific collecting area may be obtained
by integrating over the size distribution of the particulate to
be collected. However, since the procedures used for estimating
the effects of reentrainment, gas sneakage, and gas velocity
distribution have not been evaluated by comparison with field
data specifically relating to these factors, the state of
development of the model is such that the preparation of
generalized nomographs is not justified at this time.
Therefore, this section presents a procedure which will enable
the reader to estimate the program output for electrical condi-
tions and an electrode geometry which pertain primarily to the
largest application area for electrostatic precipitators -
the collection of fly ash from coal-fired power boilers. If
the actual conditions in the precipitator to be simulated
differ from those on which the following estimation procedure
is based, the computer program should be utilized to generate
the appropriate relationships between collection efficiency and
specific collection area.
The estimation procedure involves the solution of the
equation:
no = 1 - J [exp-(we'A/Q)] f(a)da
(64)
108
-------
where no is the overall mass collection fraction, we' is the
apparent migration velocity of particles of radius a corrected
for reentrainment, sneakage, and gas velocity distribution,
A/Q is the specific collecting area, f(a) is the fraction of
the particles between a and a +da, and ai and 32 are the lower
and upper limits on the size distribution. Since we' is a func-
tion of both a and the ideal collection fraction, numerical
techniques are required to evaluate equation 64 even if the
functional form of f(a) is known.
Since the particle size distribution is changing as a function
of length within the computer model, it is more convenient
to represent the size distribution in the program as a histo-
gram. Equation 64 then becomes
N
Ho = 1 - ]C texP (-we'A/Q)]Zi (65)
i=l
where Z^ is the fraction of the particles in the ith radius
interval. The following is an outline of the procedure to be
followed in estimating the program output, and Table 5 gives
an example of the manual calculation using a histogram repre-
sentation of the size distribution.
If equation 65 is to be used:
(1) Construct a histogram representation of the
particle size distribution to be collected.
The distribution should be obtained from in-situ
techniques. The size distribution histogram
illustrated in Table 5 was obtained from Figure 16.
(2) Determine the resistivity of the dust to be
collected at the operating temperature from
direct measurements at the source or from measure-
ments at the same temperature on a similar source.
109
-------
(3) Estimate the allowable current density from
curve 3 of Figure 18. If the allowable current
density is less than 5.0 nanoamps/cm2 (i.e.,
a resistivity above 2x101! ohm-cm) it is recom-
mended that consideration be given to the proce-
dures for lowering dust resistivity to an
acceptable range. Estimating precipitator plate
area requirements for dust resistivities in the
10!1-1012 ohm-cm range is difficult because
relatively small variations in resistivity can
significantly alter performance through sparking
or back corona.
(4) Determine the effective migration velocities from
Table 4, which is a tabulation of the data plotted
in Figure 22. For the illustration in Table 5,
a current density of 20 nanoamps/cm2 is used.
(5) Select values of gas velocity standard deviation
and reentrainment/sneakage. For the illustration
in Table 5, ag = 0.25, S = 0.1, N = 4.
(6) Estimate the specific collection area required for
the desired overall mass efficiency from Figures
25 through 28. For the example in Table 5, an SCA
value of 39.4 m2/(m3/sec)is obtained from Figure 26
for a reduced efficiency of 98.9.
(7) Calculate the collection efficiency for each
particle size from the relationship:
E = 100 n = 100 (1.0 - exp (-Ap we/Q)),
where E = collection efficiency of each particle size
n = collection fraction of each particle size
110
-------
Table 4. THEORETICAL EFFECTIVE MIGRATION VELOCITIES AS A
FUNCTION OF CURRENT DENSITY, TEMPERATURE, AND
PARTICLE DIAMETER
Effective Migration Velocity, cm/sec
Temperature, °C
Current Density, nA/cm2
Particle Diameter,
ym
0.2
0.3
0.55
0.85
1.25
1.75
2.50
3.50
4.50
6.00
8.50
12.50
20.00
27.50
150
5
1.335
1.249
1.377
1.620
1.957
2.375
2.987
3.784
4.569
5.739
7.699
10.903
17.070
23.316
150
20
3.312
3.136
3.500
4.167
5.109
6.297
8.080
10.469
12.885
16.572
22.868
33.582
53.298
73.065
150
40
5.023
4.758
5.307
6.313
7.739
9.542
12.250
15.874
19.532
25.100
34.357
49.610
78.785
108.005
370
30
4.195
3.550
3.478
3.839
4.448
5.251
6.468
8.085
9.697
12.117
16.186
22.831
34.726
47.414
111
-------
TABLE 5. EXAMPLE OF MANUAL CALCULATION OF OVERALL COLLECTION EFFICIENCY
Size Distribution
Histogram
Theoretical
Reduced
Diameter %
ym
0.15
0.20
0.30
0.55
0.85
1.25
1.75
2.5
3.5
4.5
6.0
8.5
12.5
20.0
27.5 and
larger
Conditions:
SCA = 39.37
of total
0.077
0.0729
0.1875
0.3625
0.375
0.725
0.950
2.25
2.50
2.50
4.75
6.75
9.75
13.75
55.0
m2/(m3/sec),
Current Density = 20x10"
ag = 0.25
we
m/sec
0.0380
0.0312
0.0314
0.0350
0.0417
0.0511
0.0630
0.0808
0.105
0.129
0.166
0.229
0.336
0.533
0.730
E
or 200 ft
9 amps/cm2
i^oj. .Lection
Efficiency
77.596
72.813
70.861
74.752
80.575
86.587
91.592
95.830
98.370
99.370
99.852
99.987
99.9998
99.9999
99.9999
o = 99.41
2/(1000 ft3/min)
F
1.078
1.071
1.069
1.074
1.083
1.094
1.106
1.122
1.141
1.160
1.187
1.234
1.312
1.326
1.326
B
1.134
1.131
1.130
1.132
1.137
1.144
1.153
1.168
1.191
1.218
1.267
1.373
1.631
1.683
1.683
we'
m/sec
0.0311
0.0273
0.0260
0.0288
0.0338
0.0408
0.0494
0.0616
0.0770
0.0912
0.110
0.135
0.157
0.239
0.328
collection
Efficiency
70.561
65.855
63.972
67.747
73.574
79.925
85.656
91.146
95.158
97.230
98.685
99.500
99.793
99.992
99.9997
Eo1 = 98.88
S = 0.1
-------
Ap/Q = specific collecting area, [39.4 m2/m3/sec) for
the illustrated example]
we = effective migration velocity of the particle
size under consideration.
Truncate all collection fractions greater than 0.999999
at this value.
(8) If the theoretical value of overall collection
efficiency is desired, it may be obtained from:
n
Eo = 2_j zi ni •
where Eo = overall mass efficiency, %
n = number of particle sizes
Zj_ = % of mass represented by particle size i
Hi = collection fraction of particle size i.
This is an optional step.
(9) Calculate the corrected migration velocities, we",
for each particle size as follows:
a) Evaluate "F", the empirical correction
factor for gas velocity standard deviation
from
F = 1.0 + 0.766 nog1'786 + 0.0755 ag In (-—)
For the example, aa = 0.25.
113
-------
b) Evaluate B from
In (1-n)
B ~ Ns In [S + (1-S)(1-n:
For the example, 3 =0.1 and Ns = 4.0.
c) Calculate we' from
,, > - we
we - fr-g •
(10) Calculate a reduced collection efficiency for
each particle size from
E1 = lOOn1 = 100 (1.0 - exp (-
(11) Obtain the reduced overall efficiency from
n
E0'=
where Eo ' = reduced overall mass efficiency
f|'j_ = reduced collection fraction of particle
size i .
The variation in theoretical and reduced overall collection
efficiency with variation in specific collection area may be
obtained by repeating the above procedure from steps 6 through
11. Note that the overall efficiencies computed in Table 4
agree with the computed values shown in Figure 26. For
higher values of SCA, the manual estimating procedure will
produce somewhat lower overall efficiencies than would the
computer program, since the effect of increased particle
charging time is not accounted for with the manual calcula-
tions.
114
-------
SECTION VII
ACKNOWLEDGEMENTS
The subroutine for calculating electric fields in wire plate
geometry was written and evaluated by Dr. Wallace B. Smith,
Head of the Physics Section. Procedures for representing
non-ideal effects were developed by Mr. Norman L. Francis,
Research Engineer. The assistance of Dr. Leslie Sparks,
EPA Project Officer, is also gratefully acknowledged.
115
-------
SECTION VIII
REFERENCES
1. Oglesby, Sabert, Jr. and Grady B. Nichols. A Manual of
Electrostatic Precipitator Technology, Vol. I (August 1970).
Southern Research Institute, Birmingham, Alabama.
2. Leutert, G. and B. Bohlen. The Spatial Trend of Electric
Field Strength and Space Charge Density in Plate Type
Electrostatic Precipitators. Staub. 32(7), July 1972.
3. Cooperman, P. A Theory for Space Charge Limited Currents
with Application to Electrostatic Precipitators. Trans.
AIEE, Vol. 79. March 1960.
4. Penney, G. W. and R. E. Matick. A Probe Method for
Measuring Potential in DC Corona. AIEE Preprint. 1957.
5. Tassicker, 0. J. Aspects of Forces on Charged Particles
in Electrostatic Precipitators. Ph.D. Dissertation.
Dept. of Elec. Eng., Wallongong Univ. College, Univ. of
S. Wales. July 1972.
6. Peek, F. W. Dielectric Phenomena in High Voltage Engineering,
3rd Edition. New York, McGraw-Hill, 1929.
7. White, H. J. Particle Charging in Electrostatic Precipi-
tation. AIEE., 1186-1191, May 1951.
8. Hewitt, G. H. The Charging of Small Particles for Electro-
static Precipitation. AIEE. 76:300-306, July 1957.
9. Fuchs, N., I. Petrjanoff, and B. Rotzeig. On the Rate of
Charging of Droplets by an Ionic Current. Trans. Faraday
Society, 1131-1138, February 1936.
10. Liu, Benjamin Y. H., Kenneth T. Whitby, and Henry H. S. Yu.
Diffusion Charging of Aerosol Particles at Low Pressures.
J. Appl. Phys. 38(4):1592-1597, March 1967.
11. Mirzabekyan, G. Z. Aerosol Charging in a Corona-Discharge
Field. Strong Electric Fields in Technological Processes
(Ion Technology). V. I. Popkov, Ed. Energy Publishing
House, Moscow, 20-38, 1969.
12. Murphy, A. T., F. T. Adler, and G. W. Penney. A Theoretical
Analysis of the Effects of an Electric Field on Charging
of Fine Particles. AIEE Trans., 318-326, September 1959.
13. Stratton, J. A. Electromagnetic Theory. McGraw Hill Book
Co., Inc., New York, p. 205, 1941.
116
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14. White, H. J. Industrial Electrostatic Precipitation.
Addison-Wesley Publishing Co., Inc., Reading, Mass.
1963. p. 138.
15. Liu, B. Y. H. and Hsu-Chi Yeh. On the Theory of Charging
of Aerosol Particles in an Electric Field. J. Appl. Phys.,
39(3):1396-1402, February 1968.
16. Pauthenier, M. M. and M, Moreau-Hanot. Charging of Spherical
Particles in an Ionizing Field, Journal de Physique et
Le Radium (Paris)., 3(7):590-613, 1932.
17. Hildebrand, F. B. Introduction to Numerical Analysis.
McGraw-Hill, 1956.
18. White, op.cit., p. 157.
19. Fuchs, N. A. The Mechanics of Aerosols. New York,
The Macmillan Co., 1964. Chap. 2.
20. Deutsch, W. Ann. der Physik. 68:335 (1922).
21. White, op.cit., p. 165.
22. White, op.cit., p. 185.
23. Nichols, G. B. and J. P. Gooch. An Electrostatic Precipi-
tator Performance Model. Southern Research Institute,
Birmingham, Alabama. Contract CPA 70-166. The Environ-
mental Protection Agency, Research Triangle Park, N. C.,
July 1972.
24. White, op.cit., p. 238-293.
25. Preszler, L. and T. Lajos. Uniformity of the Velocity
Distribution Upon Entry into an Electrostatic Precipitator
of a Flowing Gas. Staub. 32(11):1-7, November 1972.
26. Smith. W. B., K. M. Gushing, and J. D. McCain. Particulate
Sizing Techniques for Control Device Evaluation. Southern
Research Institute, Birmingham. Alabama. Contract 68-02-0273
The Environmental Protection Agency, Research Triangle Park,
N. C. July 12, 1974, p. 10.
27. Ibid. Appendix A.
28. Nichols, G. B. Techniques for Measuring Fly Ash Resistivity.
Final Report to Environmental Protection Agency. Contract
No. 68-02-1303. Southern Research Institute, August 5, 1974.
117
-------
29. Bickelhaupt, Roy E. Volume Resistivity-Fly Ash Composition
Relationship. Environ. Science and Technology. 9(4),
April 1975.
30. Penney, G. W. and S. Craig. Pulsed Discharges Preceding
Sparkover at Low Voltage Gradients. AIEE Winter General
Meeting. New York. January 29-February 3, 1961.
31. Pottinger. J. F. The Collection of Difficult Material by
Electrostatic Precipitation. Australian Chem. Process Eng.
20(2):17-23, February 1967.
32. Hall, Herbert J. Trends in Electrical Precipitation of
Electrostatic Precipitators. Proceedings of the Electrostatic
Precipitator Symposium. Birmingham, Alabama. Paper I-C.
February 23-25, 1971.
33. White, op.cit., p. 219-230.
34. Tassicker, 0. J. The Temperature and Frequency Dependence
of the Dielectric Constant of Power Station Fly Ash.
Staub-Reinhalt. Luft. 31(8), August 1971.
35. White, op.cit., p. 146.
36. White, op.cit., p. 135.
37. Berube, D. T. Low Gas Temperature Solution to High
Resistivity Ash Problem. Proceedings of the Electrostatic
Precipitator Symposium, Paper II-E, Birmingham, Alabama,
February 23-25, 1971.
38. Nichols, G. B. and J. P. Gooch, op.cit., p. 103.
39. Bickelhaupt, Roy E., op.cit.
40. Dismukes, Edward B. A Study of Resistivity and Conditioning
of Fly Ash. Final Report to Environmental Protection Agency,
Contract No. CPA 70-149. Southern Research Institute.
February 1972.
41. Dismukes, op.cit.
42. Dismukes, Edward B. Conditioning of Fly Ash with Sulfur
Trioxide and Ammonia. Report to Tennessee Valley Authority
and Environmental Protection Agency (EPA Contract No.
68-02-1303). Southern Research Institute. January 1975.
118
-------
SECTION IX
APPENDICES
119
-------
APPENDIX 1
Listing of FORTRAN Variables
120
-------
001
002
003
00«
005
006
007
008
009
010
Oil
01?
013
014
015
016
017
Olfl
019
020
021
022
023
02"
025
026
027
028
02<>
030
031
032
033
030
035
036
037
038
039
QUO
OU1
042
043
04U
045
046
047
046
049
050
051
052
053
054
055
056
057
05fl
099
060
PI
E
U
BC
EPS
EPSO
VIS
MS
JLY
DIAM(J)
C RAO(J)
C PCNT(J)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C AS(I)
C
c vos(i)
C TCS
-------
061
062
063
06U
065
066
067
068
069
070
071
073
073
07U
075
076
077
07fl
079
080
081
OB2
083
080
085
086
087
088
089
090
091
092
093
09U
095
096
097
098
099
100
101
102
103
104
105
106
107
108
109
110
111
112
113
11"
115
116
117
118
119
120
C
c
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
r
c
c
c
c
c
c
c
c
c
c
c
c
BSH)
NWS(I)
SYS(I)
NF
DV
sx
VERGE
VOL( J)
w
CO
ET
CL
TDK
ZMFP
NPRINT
X
DXS(J)
XNO(J)
ONO(J)
ETAPF
SW
Dw(I)
ZWT
ROVRI
ERAVG
KK
tCHARG
BOLTZ
RATIO
G
VC
FACTRC
COEFFC
TINC
INDEX
NSECT
NCOOP
NWIRE
EZERO
(INCHES TO METERS)
"IRE TO PLATE SPACING IN THE DIFFERENT LINEAR SECTIONS
CINCHES TO METERS)
NUMBER OF CORONA WIRES IN THE DIFFERENT LINEAR SECTIONS
1/2 THE WIRE TO WIRE SPACING IN THE OIFFFRENT LINEAR SECTIONS
(INCHES TO METFRS)
NUMBER OF INCREMFNTS INTO WHICH THE PRECIPITATOR IS DIVIDED
DUST VOLUME (METFR**3(DIRT)/M£TEH**S (GAS))
WIRE TO PLATE SPACING (METERSJ
INJTUL ESTIMATE OF THE SPACE CHARGE DENSITY AT THE CORONA
TO START THE CALCULATION OF THE ELECTRIC FIF.LD AT THE PLATE
(COULOMBS/METER**3)
TOTAL VOLUME FOR EACH PARTICLE SIZE PER CUBIC METER OF GAS
(METER«*3/METER**3(GAS))
WEIGHT OF DUST PER SECOND PASSING INTO THE PRECIPITATOR
(KILOGRAMS/SECOND)
CURRENT DENSITV AT THE PLATE ( AMPERE5/METER**2 )
ELECTRIC FIELD IN DEPOSIT ( VOL TS/M£TER)
CURRENT PER METER OF CORONA WIHE (AMPERES/METER)
TEMPERATURE (DEGREES KELVIN)
MEAN FREE PATH (METERS)
INDICATOR WHICH DESIGNATES WHEN TO PRINT CERTAIN HESUI TS
EXPONENT IN THE DEUTSCH EQUATION FOR THE STATRp EFFICIENCY
TOTAL NUMBER OF PARTICLES REMOVED PER CUHIC METFR OF GAS IN
EACH SIZE RANGE (NUMBER/METER»»3)
NUMBER OF PARTICLES PER CUBIC METER OF GAS IN EACH SIZE RANGE
AT THE START OF EACH INCREMENT (NUMBER/METER**!)
INITIAL NUMBER OF PARTICLES PER CUBIC METER OF GAS IN EACH
RANGE (NUMBER/METFR**3)
EFFICIENCY PER 0,3 METER INCREMENT
TOTAL AMOUNT OF MATERIAL REMOVED (KILOGRAMS)
AMOUNT OF MATERIAL REMOVED PER INCREMENT ON A TOTAL WEIGHT
BASIS (KILOGRAMS)
TOTAL WEIGHT OF OUST PER CUBIC METER OF GAS REMOVED UP TO A
GIVEN 1NCREMFNT (KILOGRAMS/METER**3)
RATIO OF TOTAL SPACE CHARGE DENSITY TO IONIC SPACE CHARGE
DENSITY IN THE I-TH INCREMENT
AVERAGE CHARGING FIELD ( VOLTS/METER)
DO LOOP INDEX USED IN THE RUNGE»KUTTA SCHEME FOR SOLVING THE
DIFFERENTIAL FttlUTlON DESCRIBING THE SRI CHARGING THF.OHY
ELEMENTARY CHARGE UNIT (ST ATCOULOMflS )
BOLTZMANN'S CONSTANT (ERGS/DEGREES KELVIN)
CONSTANT FACTORS USED IN THE SRI CHARGING THEORY
TIME INTERVAL FOR THE GAS TO TRAVEL ONE FOOT (SECONDS)
INDICATOR WHICH KEEPS TRACK OF HOW MANY FEET THE CALCULATION IS
INTO A GIVEN SECTION
INDICATOR WHICH KEEPS TRACK OF WHICH SECTION THE CALCULATION is
IN
INDICATOR WHICH TELLS THE PROGRAM TO MAKE CERTAIN CALCULATIONS
AT THE START OF EACH SECTION
NUMBER OF WIRES
AVERAGE CHARGING FIELD (8TAT-VOLTS/CENTIMETER)
122
-------
121
122
123
12«
12S
126
127
128
129
150
131
132
133
13«
i3s
136
137
13S
13
-------
181
182
183
iea
185
186
1B7
IBS
189
190
191
192
195
194
195
196
197
198
199
200
201
202
203
20U
205
206
207
208
209
210
211
212
213
215
216
217
218
219
220
221
222
223
22a
225
226
227
228
229
230
231
232
233
23"
235
236
237
23«
239
240
CZB
TLl
ZMO
ETC
OIFF
C DXNO
c nxs(j)
c
c
C ZTM
c
c
C CZA
c
c
c
c
c
C
C
c
c
C TL2
C
C
C
C
C
C ATOTAL
C
C SCOREF
C
C X
C
C DIA
C EFESR
C
C
C
C
C XY
C
C XEP
C PI
C F2
C ZNLFF
C
C WY
C
C
C
C
C COREFF
C
C SL
C
C
C
c
c
c
c
EACH SIZE
TOTAL WEIGHT
wSL
WTL
ZTM
NUMBER OF PA&TICLFS PtR CUflIC METER OF GAS REMOVED FOR A
PARTICULAR SIZF RANGE (NUMBER/MfTER**3)
TOTAL NUHBF.H of PARTICLES PER CUBIC METER OF GAS REMOVED IN
AFTER I INCREMENTS (NUMBER/METER* *3)
OUST PER CUBIC METER OF GAS REMOVED IN EACH
SIZE RANGE IN A GIVEN INCREMENT (KILOGRAMS/METER**3)
PARTIAL SUM OF THE WEIGHT OF DUST REMOVED PER CUBIC METER OF
GAS UP TO THE J-TH PARTICLE SIZE IN A GIVEN INCREMENT
(KlLOGRAMS/MfTFR**3)
RATIO OF THE PARTIAL SUM OF THE wE'IGHT OF OUST REMOVED PER
CUBIC METER OF GAS UP TO THE J-TH PARTICLE SIZE IN A GIVEN
INCREMENT TO THE TOTAL WEIGHT OF OUST REMOVED PE« CUBIC MF.TEH
OF GAS IN A GIVEN INCREMENT
RATIO OF THE PARTIAL SUM OF THE WEIGHT OF OUST REMOVED PER
CUBIC METER OF GAS UP TO THE CJ-D-TH PARTICLE SI/E IN * GIVEN
INCREMENT TO THE TOTAL WEIGHT OF DUST REMOVED PF« CUHIC METER
OF GAS IN A GIVEN INCREMENT
DIFFERENCE BETWFEN CZA AND CZB FOH USE IN INTERPOLATING TO
FIND THE MASS MFDIAN DIAMETER OF THF COLLECTED DUST
DIFFERENCE BETWEFN 0,50 AND CZB FOR USE IN INTERPOLATING TO
FIND THE MASS MEDIAN DIAMETER OF THE COLLECTED OUST
INTERPOLATED MASS MEDIAN DIAMETER OF COLLECTED DUST (METERS)
EFFICIENCY FOR THE ENTIRE PREC1PITATQR LENGTH
DIFFERENCE BETWEEN THE CALCULATED EFFICIENCY AND STATED
EFFICIENCY
TOTAL COLLECTION PLATE AREA FOR THE PRECIPITATOR
(METERS ** 2)
TOTAL EFFICIENCY CORRECTED FOR GAS VELOCITY DISTRIBUTION, GAS
SNEAKAGE, OR REFNTRAINMENT
EFFICIENCY COMPUTED WITHOUT ACCOUNTING FOR GAS VELOCITY
DISTRIBUTION, GAS SNEAKAGE, OR REENTRAINMENT
DIAMETER OF A PARTICULAR PARTICLE SIZE (METERS)
RATIO OF THE TOTAL NUMBER OF PARTICLES REMOVED PER CUBIC METF.R
OF GAS IN A CERTAIN SIZE RANGE TO THE INITIAL NUMBER OF
PARTICLES PER CUBIC METER OF GAS IN THAT SIZE RANGE -
COLLECTION FRACTION WITH NO ACCOUNT FOR GAS VELOCITY
DISTRIBUTION, GAS SNEAKAGE, OR REENTRAINMENT
PERCENTAGE OF PARTICLES PER CUBIC METER OF GAS OF EACH SIZE
EXPRESSED AS A PFRCENT
COLLECTION FRACTION EFESR EXPRESSED AS A PFHCENT
AIR QUALITY FACTOR TO ACCOUNT FOR GAS VELOCITY DISTRIBUTION
FACTOR TO ACCOUNT FOR GAS SNEAKAGE AND RFENTRAINMENT
FACTOR TO ACCOUNT FOR GAS VELOCITY DISTRIBUTION, GAS SNEAKAGF,
AND REENTRAINMENT
EFFECTIVE MIGRATION VELOCITY OF A CERTAIN SIZE RANGE WITHOUT
ACCOUNT FOR GAS VELOCITY DISTRIBUTION, GAS SNEAKAGE, OR
REENTRAINMENT (METERS/SECOND)
EFFECTIVE MIGRATION VELOCITY OF A CERTAIN SIZE RANGE ACCOUNTING
FOR GAS VELOCITY DISTRIBUTION, GAS SNEAKAGE, AND HF.ENTRAINMENT
(METERS/SECOND)
COLLECTION EFFICIENCY FOR A CERTAIN SIZE RANGE ACCOUNTING FOR
GAS VELOCITY DISTRIBUTION, GAS SNEAKAGE. AND REENTHAINMENT
NUMBER OF EFFLUENT PARTICLES PER CUBIC M£TFR OF GAS IN F.ACH
SIZE RANGE (NUMBER/METER**3)
WEIGHT OF EFFLUENT PARTICLES PER CUBIC METER OF GAS IN FACH
SIZE RANGE (KILOGPAMS/METER^S)
TOTAL WEIGHT OF EFFLUENT PARTICLES PER CUBIC METFR UK GAS
(KILOGRAMS/ME T£R**3)
PARTIAL SUM OF THE WEIGHT OF EFFLUENT OUST PER CUBIC METER
OF GAS UP TO SOME J-.TH PARTICLE SIZE (K ILOGRAMS/MET£***3)
124
-------
2U1 C CZA RATIO OF THF PARTIAL SUM OF THE WEIGHT OF EFFLUENT OUST P£R
2«2 c CUBIC METER OF GAS UP TO SOME J-TH PARTICLE SIZE TO THE.
2uj c TOTAL WEIGHT OF EFFLUENT OUST PER CUBIC METER OF GAS
2
-------
001
002
003
00«
005
006
007
OOfl
009
010
OH
01?
013
010
015
016
017
OJfl
019
020
021
022
023
024
025
026
027
026
029
030
031
032
033
034
055
036
037
038
039
oao
041
042
0«3
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
r
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
vo
NX
NY
AX
sx
SY
.PI
AC
vcoopn.j)
X
V
M
El
Fl
Gl
HI
E2
F2
G2
H2
TT
TB
F
G
MUM
OENOM
LIST OF VARIABLK NAMES AND UNITS FOR SUBROUTINE CHAN USED IN
THE ELECTROSTATIC PRECIPITATOR PERFORMANCE MODEL
APPLIED VOLTAGE (NEGATIVE OF APPLIED VOLTAGE USED IN
CALCULATIONS) fVnlTS)
NUMBER OF POINTS IN THE X-DIRECTION OF THE GRID
NUMBER HP POINTS IN THE V-OIRECTION OF THE GRID
INCREMENT USED IN THE X-DIRECTION OF THE GRID
*IRE TO PLATE SPACING (METERS)
1/2 WIRE TO WIRE SPACING (METERS)
VALUE OF THE CONSTANT PI
CORONA WIRE RADIUS (METERS)
ARRAY CONTAINING COOPERMAN'S VALUE FOR THE POTENTIAL AT
DIFFERENT POINTS IN THE GRID (VOLTS)
VALUE OF X USED IN COOPERMAN'S EXPRESSION V(X,Y) FOR THE
POTENTIAL (MtTERS)
VALUE OF Y USED IN COOPERMAN'S EXPRESSION V(X,v) FOR THfc
POTENTIAL (METERS)
SERIES SUM IN COOPERMAN'S EXPRESSION IS TAKEN FROM M TO «M
WHERE Ms -1
ARGUMENTS FOR THE HYPERBOLIC COSINE FUNCTIONS IN THE NUMERATOH
OF COOPERMAN'S EXPRESSION
ARGUMENTS FOR THE COSINE FUNCTIONS IN THE NUMERATOR OF
COOPERMAN'S EXPRESSION
ARGUMENTS FOR THF HYPERBOLIC COSINE FUNCTIONS IN THE
DENOMINATOR OF COOPEP-MAN'S EXPRESSION
ARGUMENTS FOR THF COSINE FUNCTIONS IN THE DENOMINATOR OF
COOPERMAN'S EXPRESSION
HYPERBOLIC COSINE FUNCTIONS IN THE DENOMINATOR OF COOPKHMAM'S
EXPRESSION
COSINE FUNCTIONS IN THE NUMERATOR OF COOPERMAN'S EXPRESSION
HYPERBOLIC COSINE FUNCTIONS IN THE DENOMINATOR OF COOPEWMAN'S
EXPRESSION
COSINE FUNCTIONS IN THE DENOMINATOR OF COOPERMAN'S EXPRESSION
ARGUMENT FOR THE LOGARITHMIC FUNCTION IN THE NUMERATOR OF
COOPERMAN'S EXPRESSION
ARGUMENT FOR THE LOGARITHMIC FUNCTION IN THE DENOMINATOR OF
COOPERMAN'S EXPRESSION
LOGARITHMIC FUNCTION IN THE NUMERATOR OF COOPERMAN'S EXPRESSION
LOGARITHMIC FUNCTION IN THE DENOMINATOR OF COOPERMAN'S
EXPRESSION
SUM IN THE NUMERATOR OF COOPERMAN'S EXPRESSION
SUM IN THE DENOMINATOR OF COOPERMAN'S EXPRESSION
126
-------
001
002
003
OOU
OOS
006
007
008
009
010
Oil
012
013
Oil
015
Olb
017
018
019
020
021
022
023
02«
025
026
027
028
029
030
031
032
033
03U
035
036
037
03B
039
040
041
012
043
014
045
046
047
048
049
050
051
052
053
054
055
056
057
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
r
c
c
c
c
c
c
c
c
UEQ
CD
AC
vo
sx
SY
AEPLT
VCOOP(I,J)
VERGE
C.VERGE
(JZERO
PI
EPSO
NX
MAXJ
MINJ
NX1
NY1
z
V(I,J)
RHO(I.J)
EY(I,Ji
E X ( I , J )
LL
OLDV(I,J)
OLflRO(I.J)
CDNSTYfNX, J)
ACONTY
EPLT
LIST OF VARIABLES AND UNITS FOH THE SUBROUTINE EFItLD USED
IN THE ELECTROSTATIC PRECIPITATOR PERFORMANCE MODEL
EFFECTIVE MOBILITY (METERS**2/VOLT-S£COND>
CURRENT DENSITY AT THE PLATE (AHPERES/METER**2)
CORONA WIRE RADIUS (METERS)
APPLIED VOLTAGE (NEGATIVE OF APPLIED VOLTAGE USED IN
CALCULATIONS) (VOLTS)
WIRE TO PLATE SPACING (METERS)
1/2 "IRE TO WIRE SPACING (METERS)
AVERAGE ELECTRIC FIELD AT THE PLATE (VOLTS/METER)
ARRAY CONTAINING COOPERMAN'S VALUES FOR THE POTENTIAL AT THE
DIFFERENT POINTS IN THE GRID (VOLTS)
INITIAL ESTIMATE OF THE SPACE CHARGE DENSITY AT THE -IRE TO
START THE CALCULATION OF THE ELECTRIC FIELD AT THE PLATF
(COULOMBS/METER**3)
CONVERGED VALUE OF THE SPACE CHARGE DENSITY AT THE WIRE IN
CALCULATING THE ELECTRIC FIELD AT THE PLATE (COULOMBS/METER**3)
INITIAL ESTIMATE OF THE SPACE CHARGE DENSITY AT THE "-IHE TO
START EACH ITERATION IN THE CALCULATION OF THE ELFCTkIC
FIELD AT THE PLATES (COULOMBS/METER**J)
VALUE OF THE CONSTANT PI
PERMITTIVITY OF FREE SPACE (couLOMB**2/NEWTON-METER*«>2)
NUMBER OF POINTS IN THE Y-DIRECUON OF THE GRID
UPPER LIMIT THAT THE CALCULATED CURRENT DENSITY CAN NOT EXCEED
(AMPERES/METER**2)
LOWER LIMIT THAT THE CALCULATED CURRENT DENSITY CAN MOT FALL
BELOW (AMPERFS/MET£R**2)
ONE LESS THAN THE TOTAL NUMBER OF POINTS IN THE X-DIRECTION
OF THE GRID
ONE LESS THAN THE TOTAL NUMBER OF POINTS IN THE v-DIRECTION
OF THE GRID
INDICATOR WHICH COUNTS THE NUMBER OF TIMES THE CALCULATION
ITERATES DUE TO LACK OF CONVERGENCE IN THE CURRENT DENSITY
ARRAY CONTAINING THE VALUE OF THE POTENTIAL AT EACH POINT IN
THE GRID DURING AN ITERATION AND INITIALLY CONTAINING
COOPERMAN'S VALUES (VOLTS)
ARRAY CONTAINING THE VALUE OF THE SPACE CHARGE DENSITY AT EACH
POINT IN THE GRID DURING AN ITERATION (COULOMflS/METER**3)
ARRAY CONTAINING THE VALUE OF THE FLECTRIC FIELD PARALLEL TO
THE PLATES AT EACH POINT IN THE GRID DURING AN ITERATION
(VOLTS/METER)
ARRAY CONTAINING THE VALUE OF THE ELECTRIC FIELD PERPENDICULAR
TO THE PLATES AT EACH POINT IN THE GRID DURING AN ITERATION
(VOLTS/METER)
INDICATOR WHICH COUNTS THE NUMBER OF TIMES THE CALCULATION
ITERATES DUE TO LACK OF CONVERGENCE IN THE POTENTIAL AT EACH
POINT IN THE GRID
ARRAY CONTAINING THE VALUE OF THE POTENTIAL AT EACH POINT
IN THE GRID DURING THE PREVIOUS ITERATION (VOLTS)
ARRAY CONTAINING THE VALUE OF SPACE CHARGE DENSITY AT EACH
POINT IN THE GRID DURING THE PREVIOUS ITERATION (COULOMBS/
M£T£R**3)
ARRAY CONTAINING THE VALUE OF THE CURRENT DENSITY AT EACH GRID
AVERAGE CURRENT DENSITY AT THE PLATE (AMPER£S/METER**2)
POINT ON THE PLATES (AMPER£S/METER**2)
TOTAL ELECTRIC FIELD AT THE PLATE (VOLTS/METER)
127
-------
001
002
003
oou
oos
006
007
OOfl
009
010
Oil
012
013
OJU
015
016
017
016
01<»
020
021
022
023
02«
025
026
027
028
02"
030
031
032
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
Nin
NII2
AFID
01
J
VRMS
DELTT
DELT01
I
II
QSAT(J)
FPSO
F
TEMO
RAD(J)
PI
BC
ARC
DELT02
DELTO
LIST Of VARIABLE NAMES AND UNITS FOR SUBROUTINE CHARGE HSFD
IN THE FLECTROSTiTIC PRF.C IP I TATOW PERFORMANCE MODEL
DO LOOP INDEX WHICH BEGINS A TIME PERIOD FOR CHARGING IN AN
INCREMENT
DO LOOP INDEX WHICH TERMINATES A TIME PFRIOD FOR CHAHGIMG IN AN
INCREMENT
AVF"AGE FREE ION DENSITY (NUMBER/M£TER**3)
INITIAL CHARGE ON EACH PARTICLE AT THE BEGINNING OF AN
INCREMENT (COULOMBS)
INDEX REPRESENTING A CERTAIN PARTICLE SIZE
ROOT MEAN SQUARE VELOCITY OF THE IONS CMETERS/SECOND)
DIFFERENTIAL HMF INTERVAL OVER WHICH CHARGING TAKES PLACE
(SECONDS)
DIFFERENTIAL AMOUNT OF CHARGE ACQUIRED BY A CERTAIN PARTICLE
SIZE DURING THE DIFFERENTIAL TIME INTERVAL DELTT DUE TO
CLASSICAL FIELD CHARGING (COULOMBS)
ELEMENTARY CHARGE UNIT (COULOMBS)
ION MOBILITY (METFRS**2/VOLT-SECONO)
SATURATION CHARGE FOR A PARTICULAR PARTICLE SIZE (COULOMBS)
PERMITTIVITY OF FREE SPACE (COULOMB**2/NEWTON.METER*«2)
TEMPERATURE IN DEGREES RANKINE
TEMPERATURE IN DEGREES KELVIN
RADIUS OF A PARTICULAR PARTICLE SIZE
VALUE OF THE CONSTANT PI
VALUE OF BOLTZMANN'S CONSTANT (JOULES/DEGREE KELVIN)
ARGUMENT OF THE EXPONENTIAL IN CLASSICAL DIFFUSION CHARGING
DIFFERENTIAL AMOUNT OF CHARGE ACQUIRED BY A CERTAIN PARTICLE
SIZE DURING THE DIFFERENTIAL TIME INTERVAL DELTT DUE TO
CLASSICAL DIFFUSION CHARGING (COULOMBS)
TOTAL DIFFERENTIAL AMOUNT OF CHARGE ACQUIRED BY A CERTAIN
PARTICLE SIZE DURING THE DIFFERENTIAL TIME INTERVAL DELTT
128
-------
001
002
003
oou
005
006
007
008
00<»
010
Oil
012
01 J
Old
015
016
017
018
019
020
021
022
025
02«
025
026
027
028
029
030
051
052
055
05«
015
036
057
05R
059
QUO
out
C
C
C
C
C
C
C
C
C
C
C
r
c
c
c
c
c
c
c
c
c
c
c
c
c
c
r.
c
c
c
c
c
c
c
c
c
c
c
c
c
ECHARG
SCHARG
NUMlNC
EZERO
KSIZE
AFID
RATE
H
XI
VI
KK
NN
X
y
CONST
V
tCONST
RH
FCONST
FACTOR
COEFF
H2
Tl
T2
T5
T«
LIST OF VARIABLES AND UNITS FOR THE SUBROUTINE CHARGN OF
THE ELECTROSTATIC PRECIPITATOR PERFORMANCE MODEL
ELEMENTARY CHANGE UNIT(STATCOULOMBS)
NUMBER OF CHARGES AT SATURATION FOR A GIVEN PARTICLE SIZE
NUMBER OF POINTS SPECIFIED FOR USE IN THE SIMPSON'S RULE
INTEGRATION OVER THETA IN REGION n op THE SRI CHARGING THEORY
AVERAGE CHARGING F IELO (STAT-VOLTS/CM)
RADIUS OF A GIVEN PARTICLE SIZE
AVERAGE FREE ION DENSITY(NUMBER/CENTIMETER**3)
SUPPLIED STATEMENT FUNCTION WHICH CALCULATES THE CHARGING RATES
AT THE DIFFERENT POINTS IN THE RUNGE«KUTTA SCHEME
INCREMENT SIZE TAKEN IN THE RUNGE-KUTTA SCHEME(SECONDS)
TIME AT THE START OF EACH INCREMENT(SECOND)
NUMBER OF CHARGES ON A PARTICLE AT THE START OF EACH INCREMENT
DO LOOP INDEX USFD IN THE RUNGE-KUTTA SCHEME FOR SOLVING THE
DIFFERENTIAL EQUATION DESCRIBING THE SRI CHARGING THEORY(KKai)
NUMBER OF POINTS SPECIFIED FOR USE IN THE RUNGE-KUTTA SCHEME
SOLVING THE DIFFERENTIAL EQUATION DESCRIBING THE SRI CHARGING
THEORY
TIME AT THE END OF EACH INCREMENT(SECONDS)
NUMBER OF CHARGES ON A PARTICLE AT THE END OF EACH INCREMENT
FACTORS USED IN THE SRI CHARGING THEORY
ONE-HALF THE INCREMENT SIZE CHOSEN FOR THE RUNGE-KUTTA
SCHEME(SECONDS)
NUMBER OF CHARGES ACQUIRED AT A PARTICULAR POINT IN THE
RUNGE-KUTTA SCHEME
NUMBER OF CHARGES ACQUIRED AT A PARTICULAR POINT IN THE
RUNGE-KUTTA SCHEME
NUMBER OF CHARGES ACQUIRED AT A PARTICULAR POINT IN THE
RUNGE-KUTTA SCHEME
NUMBER OF CHARGES ACQUIRED AT A PARTICULAR POINT IN THE
RUNGE-KUTTA SCHEME
129
-------
001
00?
005
004
005
OOb
007
ooe
009
010
Oil
012
013
014
015
016
017
Olfl
019
020
021
022
023
02U
025
026
027
028
029
030
031
032
033
030
035
036
037
038
039
0«0
Out
OU2
043
04U
045
046
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
NTIME
NUMBER
NE
ARCCOS
THZERO
DELTA*
THETA
SUMODD
RSTART
TCONST
ECOS
ZERO
RZFRO
ARGI
YVAL
SUMEVN
ARR2
ZVAL
INTGRL
RATEl
ARG3
RATE2
RATEJ
RATE
LIST OF VARIAHIES AND UNITS FOR THE STATEMENT FUNCTION RATE OF
THE ELECTROSTATIC PRECIPITATOR PERFORMANCE MODEL
INSTANTANEOUS CHARGING TI ME(SECONDS)
INSTANTANEOUS NUMBER OF CHARGES ON A PARTICLE
NEGATIVE OF THE INSTANTANEOUS CHARGE ON A PARTICLE
(STATCOULOMBS)
SUBROUTINE WHICH CALCULATES THE ARCCOSINE OF A NUMBER
INCREMENT SIZE TAKEN FOR THE INTEGRATION OVER THE ANGLE THETA
IN REGION II OF THE SRJ CHARGING THEORY(RADIANS)
VALUES OF THE ANGLE THETA TAKEN FOR THE INTEGRATION OVER THETA
IN REGION n or THE. SRI CHARGING THEORYCRADIANS)
SUM OF THE ODD TERMS CONTRIBUTING TO THE INTFGRAL IN THE
SIMPSON'S RULF INTEGRATION SCHEME
SOME RADIAL DISTANCE FROM WHICH A SEARCH is INITIATED TO FIND
THE RADIAL DISTANCE ALONG A GIVEN ANGLE AT WHICH THE TOTAL
ELECTRIC FIELD IS ZERO FOR USE IN REGION II OF THE SRI
CHARGING THEOHY(cENTIMETERS)
FACTOR APPEARING IN THE EQUATION GIVING THE CONDITION THAT TH£
TOTAL ELECTRIC FIELD BE ZERO FOR A PARTICULAR VALUE OF THETA
FACTOR APPEARING IN THE EQUATION GIVING THE CONDITION THAT THE
TOTAL ELECTRIC FIELD BE ZERO FOR A PARTICULAR VALUE OF THETA
SUBROUTINE WHICH SEARCHES FOR THE RADIAL DISTANCE AT WHICH THF
TOTAL ELECTRIC FIELD IS ZERO
RADIAL DISTANCE AT WHICH THE TOTAL ELECTRIC FIELD is ZERO
FOR USE IN REGION 11 OF THE SRI CHARGING THEORY
ARGUMENT OF THE EXPONENTIAL IN THE CHARGING RATE EXPRESSION
FOR REGION n OF THE SRI CHARGING THEORY
THE PART OF THE CHARGING RATE EXPRESSION IN REGION II WHICH
DEPENDS ON THE ANGLE THETA, USED AS THE INTEGRAND IN THE
INTEGRATION SCHEME
SUM OF THE EVEN TERMS CONTRIBUTING To THE INTEGRAL IN THE
SIMPSON'S RULE INTEGRATION SCHEME
ARGUMENT OF THE EXPONENTIAL IN THE CHARGING RATE EXPRESSION
FOR THF ANGLE THZERO IN REGION n OF THE SRI CHARGING THEORY
THE PART OF THE CHARGING RATE EXPRESSION IN REGION n WHICH
DEPENDS ON THE ANGLE THETA, EVALUATED AT THZERO
VALUE OF THE INTEGRAL APPEARING IN THE CHARGING HATE FOK
REGION ii
CHARGING RATE FOR REGION IKNUMHER/SECOND)
ARGUMENT OF THE EXPONENTIAL IN THE CHARGING RATE EXPRESSION
FOR REGION III
CHARGING RATE FOR REGION III (NUMBER/SECOND)
CHARGING RATE FOR REGION i (NUMBER/SECOND)
TOTAL CHARGING RATE TO THE ENTIRE PARTICLE SURFACE
{NUMBER/SECOND)
047
130
-------
001 C LIST OF VARIABLES »NO UNITS FOR THF SUBROUTINE ARCCOS OF
002 C THF ELECTROSTATIC PRECIPITATOR PERFORMANCE MODEL
003 C RATIO ARGUMENT OF THE ARCCOSINE FUNCTION
OOU C T INDEX WHICH RIVES VALUES THAT GENE-RATE THE DIFFERENT
005 C COEFFICIENTS IN THE SERIES REPRESENTATION FOR THF AHCCOSINE
008 C FUNCTION
007 C FACTORS IN THE COEFFICIENTS OF THE DIFFERENT TERMS OF THE
008 c SERIES
009 C U
010 C V
Oil C w
012 C TERM A PARTICULAR TERM IN THE SERIFS
013 C SUM SUM OF SUCCESSIVE TERMS IN THE SERIES
oiu c ACOS VALUE OF THE ARCCOSINE FUNCTION
ois
131
-------
001 C LIST OF VARIABLES AND UNITS FOH THE SUBROUTINE 7FHO OF THE
002 C ELECTROSTATIC PRECIPITATOR PERFORMANCE MODFL
005 C COEFFICIENTS OF A THIRD DEGREE POLYNOMIAL
00« C A
005 C B
006 C C
007 C 0
00* C Y I-TH POINT IN THE SEARCH FOR THE ZERO OF THE POLYNOMIAL
00<» C Z INCREMENT SIZE USED IN THE SEARCH FOR THE ZERO OF THE
010 C POLYNOMIAL
Oil C W (1 + D-TH POINT IN THE SEARCH FOR THE ZERO OF THE POLYNOMIAL
012 C PI VALUF OF THE POLYNOMIAL AT THE I-TH POINT
013 c P2 VALUE OF THE POLYNOMIAL AT THE (I*D«TH POINT
01« C SUMA SUM OF THE ABSOLUTE VALUES OF THE POLYNOMIAL VALUES AT THE
015 C I-TH AND (I*1)-TH POINTS
016 C ASUM ABSOLUTE VALUE OF THE SUM OP THE POLYNOMIAL VALUES AT THE
017 C I-TH AND (Itl)-TH POINTS
018 c x INTERPOLATED APPROXIMATE VALUE FOR THE ZERO OF THE POLYNOMIAL
010 C E VALUE OF THE CUBIC TF.HM IN THE POLYNOMIAL
020 C f VALUE OF THE QUADRATIC TERM IN THE POLYNOMIAL
021 C G VALUE OF THE LINEAR TERM IN THE POLYNOMIAL
02? C H SUM OF THE VALUES OF THE QUADRATIC TERM AND THE CONSTANT
025 c TERM OF THE POLYNOMIAL
020 C HI LOGARITHM TO THE BASE 10 OF H
025 C H2 VALUE OF HI INCREASED BY 1
026 C N H2 EXPRESSED AS AN INTEGER
027 C P VALUE OF THE POLYNOMIAL SCALED BY 10 RAISED TO THE N POKER
02*
132
-------
APPENDIX 2
Program Listing
133
-------
001 OOUBLF PRECISION EFESR,DLOG
002 RE*l NZtRO,INTGRL,NE, NUMBER, NTIME,NHIRF
003 WFAL NWS(20)
oou 01 MEN si ON DIAM(20), VOl(20),XNO(20),Q(20),ONO(20),DXS(20),PCNT(20),
ins l«S(20),wSl(20),CCF(20),ITL(«0),XMV(20),Ow(aO)
OOb DIMFNSrON XDC(«0,?0),VY(10),VCOOP(il»9)
007 DIMENSION LSEC T (20 ) , AS ( 20 ) , VOS (20 ) , TC8 ( 20 ) , WLS (20 ) , ACS (20 ) , BS (20 ) ,
OOB 1SYSC2U)
009 COMMON H*0(20)«OSAT(20),U,E,EPSO,PI,E»AVG,BC*TEMPIEPS
010 EXTFWNAL HATE
Oil C
01? C CONSTANTS
013 C
01" PI = 3,1
032 17 FORMAT('l')
033 C
03« REAO(2,U) (PCNT(I),I=l,NS)
035 C
03t> DO 7«J2 I a 1,NS
037 PCNT(I) = PCNT(I) * l,E"2
038 7«12 CONTINUE
039 C
0«HICH ARE INDEPENDENT OF SECTION
C
wtAD(2,«) VG,DL ,PL,VGAS,ETAO,DD,RHO,X,TEMP,P
0 FOHMAT(10F8,0)
R£AO(?,500«)SNUCK,ZIGGY,ZNUMS
OU9 500a KORMAT(10F8.2)
OSO RtAD(?fa66il} NN,NUMINC,NCALC,EPS,U,V1S,VAVG
OS1 U6b« FORMAT ( 3 ( I 2 ) , « ( E 1 1 . « ) )
0^2 r
053 c READ IN THoSt PARAMETERS WHICH DEPEND ON SECTION
OSu c
CbS WEAO(2,770) NUMSEC
OS6 770 FORMATd?)
OS/ PEAD(2,771) (LSECT(I),lBl, NUMSEC)
ObH 771 FORMAT («0(I2>)
OS9 00 n<43 NSFCTsl, NUMSEC
06n REAC>t2,7b2) AS(NSECT),VOS(NSECT),TCS(NSECT),«LS(NSEcr),ACS(N8ECT),
134
-------
061 lBS(NSECT),NHS(NSECT),SYS(NSEi:T)
062 762 FORMAT(7(1PE11,93 J3l»NS
096 CCF(J)sU(ZNFP/RAD(J))*(1.257+t
-------
121 700 CONTINUE
i?? r
123 c PKINT THOSE PARAMETERS THAT ARE INDEPENDENT OF SECTION
12" C
)?^> iF(iK)in,iit, jbo
126 r
127 111 *l»ITf (3,in)PL,NF,VG
I2tt 10 FORMAT (' PPP- LENGTH « • , E 11 , « , 1 X , 'ME TER8 • , Ttt 1 , »NO, OF INCREMENTS »'
129 S,I3, 19X,'GAS FLOW RATE s»,Ell,a7 FORMATC MtAN THERMAL SPEED s»,Ell,«,' H/SEC')
i«y c
i«s c
X3tTAPF*100,
1«f 161 FORMATf/' INPUT EFF 1C IENC Y/1NCREMENT= * , F6, 2)
1U9
ISO C
ibi c START INCREMENTAL ANALYSIS OF PRECIPITATOR
152 C
is^ r
Ibo C
IbS
Ibb
) 60 KK= i
Jhl tCHAHGca,8F-10
162 MTiOoftPSwl ,)/(EPS*2,)
GsfPS+2.
VCeECHARG**2/(BOLTZ*TOK)
Ih6 FACTRCe(PI*VAVG)/2,
167 COEFFCsPl*U*ECHARG*3,E*06
168 TINC=0,305/VGA3
I/O NSECTsj
171 NCOOPeO
172 DO 3000 1=1, NF
173 INDEXelNDEXtl
17" IFfI.EQ.l) GO TO 761
US IFCINpfX-LSECTCNSECT)) 760,760,761
I7e« 761 CONTlNUk
177 NCUOPsj
17H NPHJNTsl
IFfl.EO.l) GO TO 760
136
-------
IM iNOEXsl
1H2 7f>0 CUNTJM,'K
83 A = AS(NSe'CT)*9,3E-02
(?y VusvOS(NSECT)
165 TCsTCSCNSECn
Mb *L=WLS(NSECT)*0,305
1H/ AC = ACS(NSECM*2,SUF-02
«fc 3sHS(NSECT)*2t5UE-02
H9 SY=SYS(NSECT)*2,5UF-02
l"l SXcB
192 AxeSx/10.
9i NX=11
19S IFtNfOOP.NE.l) GO TO
196 EKAVfirVO/B
197 EZEROcER
198 DO 6989
200
•?0t b989 CONTINUE
\?- C
213 C COMPUTE CURRENT PfH M, OF CORONA WJRE
*1« r
21S CL = K / WL
218 C
21 / NCOOPsO
21H /6U CONTINUE
219 FIOaCD/(E»U»ERAVG)
2?0 SUMsOP0
2*1 DO 1300 L=1,NS
222 1300 SUM3SUM+USAT(L)*XNO(L)
223 ZC = 200,*(D«(I)/tN)*(FLOATCLSECT(NSECT))/TC)*VG
2rj« ROVRIsZC*SUM-H.O
22S AFIDeFTD/HOVRl
2?h AVGFlpsAF 10*1 ,E-Ofe
227 XCDeCDMoOOOQ,
22H C
229 C COMPUTE EFFECTIVE MOBILITY
230 UEQ3U/ROVRI
231 IFCUEQ,LT.l,OE-
-------
5KIP = | I'LT
.175
245 C
246
c STAKT PAKTICLE SIZE LOOP
r
2S1
2S2
2 S .< f T s 0
DO 2900 J * 1, N3
C
C COMPUTE CHAHGE ON fr.ACH PAHTICLE AFTER ONE INCREMENT OF TRAVEL
IFCNCAlC.tQ.lJ GO TO 9323
IP(l.NE.,1) GO TO 426
2S9 IFM,fiT,l ) GO TO 428
2bf. 1IMF1=0,
261 TlMtfsMNC
262
26i
264 (,0 Id
426 JF(J,G1.1) GU TO 429
TIME Jsf-LOAT f 1-1 )*TINC
267
H»T:
429 XIHCBXDC(I-l,J)/lf6E"l«
«28 CONTINUE
271 RSIZtcRAO(J)*l,E+02
272 SCMARG=QSAT(J)/1.6E-19
CONSTe2,*f)CGNST*EZFRO
S=^,*RSIZt
VBVC/RSIZF
FACTNE=FACT»C*RSIZE**2
280 COFFFsCOEFFC*8CMARG
281 CALL CHARGN(ECHARG,SCHARG,NUMINC,CONST,EZERO,V,RSIZE.ECONST,RR,
262 1FCONST,FAC1"RE,COEFF, AVGFIO,RATE,H,TIMEI|KIPC,KK,NN,CTIME,CNUMBR)
xocn ,j)=o(J)'
9323 CONTINUE
287 cTlMfc=FLOAT(l)*0,305/VGAS
288 "lllsr.TIME/,001
289 IF(I,EQ,1) tO TO 4290
290 IFd.GT.l ) GO TO 4270
!*T12 = N1 I
XIPC=0.0
GU TO 4280
295 4270 IF(J.GT.i) GO TO 4260
296 NI IisNI12
297 MI^eNT 1
298 «26« XTPC=XOC(1-1.J)
299 4280 CONTTNiJF
300 fALL C''ARGE(NII1,NII2,AFID,XIPC,J)
138
-------
301
50? XJ>CU,J)sXIPC
503 932« CONTINUE
ion r. COMPUTE mr;BAiiON VELOCITY FOR EACH SIZE RANGE
ins c
JOO FMV=uJ(J)*EPLT)/(6t*PI*RAD(J)*VIS)
S07 ENVeLCF (J)*fcMV
308 XMV(.1)=E*V
109 C
31C C COMPUTE EFFICIENCY FOR EACH SIZE RANGE
111 C
112 Xa(-A*EMV)/(VG*Fl.OAT(LSECT(NSECT)))
me
il« EfF 3 l. - EXP( X )
MS c
ile r COMPUTE NUMBER OF PARTICLES REMOVED IN EACH SIZE RANGE
Jl 7 c
M tt C •
WS(J)=OXNO*(l,33333*PI*RAD
XNn(J)sX
wl=wit«S
2900 CONTINUE
i^fi C
327 c CALCULATE ^"^0 AND HEIGHT COLLECTED FOR EACH INCREMENT
J=1,N3
CZAsZTM/wT
IFCCZA-0. 5)2901, 2901, 2902
2901 t
2902
136 TL2»0,50-C7b
J37 Kjsj-l
538 IP(KJ)29iO,29JO,29ll
GO TO 29J2
29U ZMDeOlAM(KJ) + (TL2/TLl)*(DIAM(J)-OlAM(KJ))
2912 CONTIMUE
!F(NPkINT,NE,n GO TO 8«39
iuo NPRlNTeo
WPITFt i/7820) NSECT
7620 FOPMAT(» CALCULATION IS IN SECTION NO, o'
WRiTF(3,77tS) A,VO,TC
7715 FORMATC COLLECTION AREA s»fEll.«,» M2 ' / T« 1 , * APPL IED VOLTAGE s',El
11.",' VOLTS', 7X, 'TOTAL CURRENT »',E11.«,' AMPS')
wPlTEt3,7716) B,AC,WL
7716 FORMATC WIRE TO PLATE B',Ell.U,» M» , T«l , 'CORONA WIRE RADIUS «',E1
!!•«,' M»,8X,*CORONA WIRE LF.NGTH o',Ell.«,' M')
WRITE(3,7717) CL,CD,ET
77 1'/ FORMATC .CURRENT/M 8'yfcn((|f» AMP/M», Tttl , 'CURRENT DENSITY =',E11,«
l,' AHP/M2»,6X, 'DEPOSIT E FIELD a',Ell,«,' VOLT/M*)
^RITE(3,7718) SY
771B FORM.1TC 1/2 WIHE TO WIRE «',EU,a,' M')
(1322 FORM4T(//Tb,'ROVRI»,Tl6,'ERAVG',T31,»EPLT',T«8,'AFID',T63,'rMcD',T
l7h,'MMD',nX, 'WEIGHT', 7X, 'INCREMENT NO.'/)
139
-------
tfcl HU39 CONTIM.if
36 jZNUMS)))
ao7 ^'0-S CONTINUE
JF(EFESK,GT.Ot99999)«Y«XMV(I)*100,
«10 IF (EFESR.LT, 0.99999) HYe(VG/ATOTAL)*l 00, *ALOG(1 00. /(100.-XEP))
COREFFaioOi*Cl."EXP(»ATOTAL*l|*YP/(100r*VG)))
SCOREF * SCUKEF * COREFF*PCNT(I)
HRITE(3,?291)OIAM(I),XY,XEPICCF(I),HY,HYP,COREFF
FORMAT(2X,F.17,7,T29,F10,6,T«5,F10,6,T6l,F7,a,T73,F7.J,
«X ,F7,3,6X ,F7,«)
140
-------
a21 x = x * ion,
«22 *KIU (3,?292) E1AO, X
423 ? H)R,v/M v'o',')X, 'EFFICIENCY - STATED s ' ,F5>. 2, 5X, 'COMPUTED »' ,F6,2
U24 SrSx, •OM'F.'WGENCF OBTAINED*)
C CALCOLATt
U27 C
a29 ZfM=o.
on 2V9S i=i,NS
CZA=7TM/WTL
«35 IF (C7A-O.SJ2995, 2995, 2996
TLJ=C7A-CZB
TL2=O.SO«CZ8
KJ=I-)
lF(KJ)29ao, 2^60, 2981
ZMCL=OlAMfl)
GO TO 2
-------
U H ^
MIKM* i ( I 3, Ic', ) 6, 10(L13.S, IX) )
C'.'N': ' "-lit'.
G ( • 1 i 'J '•> 0 (i
S TCH- 1 i i | l
IN:''
142
-------
(101 SUBROUTINE EFHUD(UEQ,CO,AC,VO,SX,SY,AfcPLT,VCOOP,VtHGE,CVERGt)
00? C EVALUATION ()F HEIOS, SPACE CHARGE DENSITY, POTENTIAL , AND
003 c OHKENT ufNSjTY FoK A MIRE-PLATE PRECIPITATOR
oou REAL MAXJ,MJNJ
005 DIMENSION RHO(ilf9),VCOOP(ll»9),EX{li,9),OLDROCU,9),OlOV(ll,9),
007 DATA RHO/09*0,/,V/99io./',EX/99*Oi/,EY/99»0./,OLDRO/99*0./,OLOV/99*
008 10,/,CnKSTY/99*0,/
009 VO=-J,*VO
010 (J^ROsVERGF
Oil Hlsj,i«i6
Olc1 E.PSOs8,8S«E-12
on
0
0
AaSX/10,0
NyeSY/A
MAXJcCD*t,001
0 7 MINJ3CO*.9999
0
0
0?0 2*0
0^1 DO «615 1=1, NX
n?? CO «M5 JB>»NY
OMS V(J, JJaVcOOPd, J3
IZsZ
IF(/.EO.?S) GO TO 700
02.' MHOd, DSOZEHO
O?P on snj is?, .-ix
0L>C) tV( 1,11=0.
030 F*(T, 1 )={V(I-1,1).V(I,1))/A
031 ti = FPSn/A*tX(l,l)*RHOfI-l.l)
0.^ RHCi(I. l )s(-b.)«EPSO/A*EX(I,n-.5»SQFiTr (EPSO/A*EX(j,m**a*
1
-------
061 CO TO 30J
06r? 303 CONTIM.JC
06< IFd.FU.i.i^O.J.NE.n GO TO 30«
Oh"J lFU,".t,l.AND,J,EQ,n GO TO 305
fib5. IF(j.tn.NY) GO TO 600
06b r,0 Tn 306
067 bon V[I,NY)sot2b*(V(I-l,NY)+V(Ul.NY)+2,*lV(I,NY«nHA**2*RHOtI,'JY)/EP
06* ISO)
Oh'' (.0 TU 301
OH1 30" IF(I,f Q,1.AND,J,EQ,NY) GO TO 350
071 vt i,lMao.25* r,(j TO 301
07* 3--.0 V(i,NY)30.b*(V(l,NY«n*V(2,NY)*0.5*A**2*RHO(lfNV)/EP80)
ft7'i t;0 TO 301
07'j 3 os V(I,i)so.2'i«(V{I-l,l)+v(I*l , l)*2.0*V(lf2)+A**2*RhO(I,l)/EPSO)
07p (,() TO 301
07 V toh V(l,J)B0.25«(V(I,Jtl)*V(I.l,J)*V(I,J-n*V(I+l,Jj*A**2*RHO(I,J)/
07(.' itPS'i)
07'' ^01 CONTlMUt
OPO 00 -07 Ial,WX
0^1 DO 3C7 J=1,NY
OR? 1F( r,t0.1,ANR,J,EQ.l) GO
0«3 GO TO ti()3
Oi-iil /I Of! HHC( I , J) s
o « :> ». n . ,j j - " o
OVi: If Cl.tQ.l .AMD.J.NF.i) GO TO 310
091 (,o ro 3.1 1
it 0 f X( 1 , J)sfl,()
tY(1,J )=(-!. )*(V(I,J),V(1,J-1)) /A
C-ifi T(i 3ia
.Ml IF ( f.NE.l.Ai-'D.J.EQ.l) UO TO 312
GO Tf] 313
312 £ v(T. J) = 0,0
t"x(T, l) = f-l . )*CV(I.J)-V(l-l,J))/A
o "> ** t; 1 1 T n 3 1 <;
* FXf I,.l)s(-i ,)*(V(I,J)-V(I-1,J))/A
Fvf I,J) = r-l . )*(VC1,J)-V(I,.
IFM.H-'.NY) EY(I,J)*0,0
)*(RHOl![-l,J
J)+EY(1»J)}
0 la(l>/2. ) **2
,j)=t-l ,)*(D/2.5»SQRT(ABS(D1)+ABS{C))
307 CO
Tf- (Ll..fu,2000) l>0 TO 700
"1 f'f' 3^0 Jrl.MYl
11 IFcAHSfV(I,J)-nLDV(l,j)).LT.l.) RO TO 320
1 '• Cn ifi 300
l 3 3^o CUM I "UK
•'' CONST Y ( NX, 1) SEX (NX, 1)*UEQ*RHO(NX, 1)
^ ACON! Y = (".nN3TY (NX, 1)
DC)
-------
\<>\ IF (4CDNTY,GT.MAXJ) GO TO 910
\?<> IF(ACDVTY.l.T,MINJ> GO TO 9?0
12.S f,n TO 980
\?u 910 07
Icb (.0 TO
1 fPI TsfXJNX, 1)
129 HO 1000 J=2,NY
I JO tPLTcF.PL.TfEXfNX,J)
\*\ 1000 CONl
li'' AFPL
rs^ 700 CONTIMUF
RETURN
145
-------
ooi SUBROUTINE C«AN(VO,NX,NY,AX,SX,SY,PI,AC,NWIHE,VCOOP)
002 c
003 c ***********************************************************************
000 C CUOPtWMA* SERIES DETERMINATION FOR VOLTAGE WIRE TO PLATt
005 r ffiw SilfiROun^c EFIELO
006 RFAL Nllh,M,NWlK£
007 D1MFMSION VCuOP(ll,9)
00» V(1»-1 ,*VO
009 00 MC; 1st, NX
010 00 «30 J=1»NY
on xs( t-i )*AX
Olr- Y«rj»ll*AX
r.o TO a 30
01 r (ibn CONTINUE
019
020
0?1 "90
0??
023 G)sPI»M*SY/SX
EXf (El)*EXP(-f.l))/2.
0?6
0?7
Oil F«ALOG(TT)
Oi? G«ALOG(TB)
O.\u OF NUMsOt NOM*G
OiS IF (n.LT.NWIRF) GO TO U08
05ft Gd Tfi
03P Gn Tf
Oi" '11" VcOOf't I , J)sVO*NlJM/OENOM
uo?
VOe.j f *VO
PfclUKM
146
-------
001 SnHRfUjT INE C H ARGN { ECH ARC,, SC H ARG,NUM INC, CONST, EZERO,V, RSI Zl, ECONST
002 *, HP,mi> ST.FACTDR.COEFf-, AF JD,RATE,H,XI,Y1,KK,NN,X,Y)
pn.< H?Srl/;>
006 I'H ? T =1,NN
On7 I'O 1 J=!,KK
OOH T | =>i*PA!F (EC HARG,SCHARG,NUMt NC, CONST, t Z HRO, V, RSI ZF!, ECONST, KR,
009 if-'JUljST,FACTOH,COtFF,AF IO,X,Y)
Old T? = H*KATF(ECHARG,SCHAHG,NllMjNC,CONST,EZF.KO,V,HSlZF.,Er:ONST,RR,
oil .Fr;driST,f AC TOW , CUF.FF , AF 11), X-t H2 , Y*T 112 . )
01 ? T } = >.*&AU ( FC H A«G. SCHARG,NU1 INC, CONS! , I ZERO , V , KSI ZF , ECONST , RW ,
CM *FC'i'- Sr ,F 4CTUR,COfFF,AFJO,X + H?,r+T3/2. )
A! y 1 (J = H * k 4TK(ECHARG,SCHARG,NUM INC,CONST.tZERO,V,RS!ZE,t CONST,PR,
0 1 S *F(!!.',S
01 * v=y + (
017 1 X = )UM
OIH ? rOCTT
010 H
O/'O F
147
-------
001
002
003
000
OOS
006
007
OOH
009
010
Oil
012
on
010
01S
Olto
017
O/'S
•MO
031
052
03S
Oi6
OW
038
039
0«1
««2
003
007
FUNCTION RATE(ECHARG,SCHARG,NUMINC,CONSr,EZERO,V,RSIZE,ECONST,
PR,FCriNST,FACTOR,COEFF,AFIP,NTIME,NUMBER)
RFAL INTGHL,NE,NUMBER,NTIME
Mi =.Ni|MUF«*ECHARG
*PR
IF (MiMBKW-SCHARG) 7005, 7006, 7006
/ooS Cr.LI ftRCr.OS(NUMBER,SCHARG,THZERO)
1F(TH7F»O.LE.1,E-05) GO TO 7006
JFM .S7-1H7ERO) 7011,7011,7015
701S
7006
7Q07
t;n TO /007
THZEROeO.
|>£LTAX=(1,57-THZERO)/FLOAT(NUMINC)
THFTA31H7ERO-DELTAX
WSTARTcRSlZF
00 7CO>: .I=1,MUMINC,2
Tilt TA=rHfcTA+0£lTAX*2,
KCf1SsE2tRO*COS(7HETA)
TMI Z^H^(Ef:OS,0,,NE.TCON•ST,RSTART,RZERO)
4K(;is-(Mi)MBtM*V*(R2ERO-RSI/E)/RZERO*(tCONST
RR*RZERO*FCONST/RZERO«
IKf ABS(A«(;i),GT,100t) GO TO 70?5
YvAL = f-XP(ARGl)*SIN(THETA)
GO TO
CONTINUE
00«j
050
/OOO CONTINUE
THFTA=THZFHO
SOMEVNOO,
RSTARTeRSIZE
00 7001 J = 2
THFTArTHETA*OEl-TAX»2t
TCONSTeCONST*C08(THETA)
tCOSsEZERO«COS(THETA)
CALL Z£RO(ECOS,0.,NE,TCONST,R3TART,RZF.RO)
APR lc.( NUMBER* V*(RZERO-R3IZE)/RZERO+(ECONST-RR*RZERO+FCOMST/RZERO*
1*2)*C03(THETA))
IFf ABscARGn.GT.lOO,) GO TO 7027
Y\,ALeEXP(ARGl)*SIN(THETA)
GO TO 7028
7027 YVALeo,
/028 r.flNTTNUt
GO TO 7051
OS?
05S
os*
057
RSTART=RZ£RO
7001 CONTINUE
IP(THZfcMn,EQ,0.)
7050 KZEROeRSlZE
GO TO 7052
7obl CONTINlie
TCONST=CONST*COS(THZERO)
ECOSsEZEf?0*COS(THZERO)
CALL ZKRO(ECOS,0.,NE,TCONST,RSIZE,RZEHO)
7o52 CONTINUE
AKG2»"( DUMBER* V*fRZERU-RSIZE)/RZERO+fECONST-«R*RZERO+FcnNST/RZEMO*
1*2)*COS(THZERO>)
IF(ABS(ARG2),GT.100f) GO TO 7029
ZVAL=EXP(ARG2)*SIN(THZ£RO)
148
-------
0*1 GO TO 7030
062 702<> ZVALCQ,
06.1 7030 CONTINUE
06S HArEl
066 <~,n TO 7012
0«7 7011 WATElsQ,
06rt 7012 CONTINUE
070 IF (ABS(AHG3),CT,100,i GO TO 7031
071 RATF2=KACTO»«EXP(AHGJ)*AFID
072 (in TO 70^52
07J 7031 HME2 = P,
07S if CNUMOEH-SCHARG)7008,7009,7009
rt7h /008 f(ATE^ = t:OEF
n77 Mi TO 7010
078 7009 hATEJcft,
07V 7010 CHNTlNl.lt
080
OR1
f.t>\)
149
-------
001 SUBROUTINE ARCCOS(A,B,ACOS)
00? RATIOsA/B
SUMsO,
COS TERNeHATjO
006 1 H*2t*T-l.
007
OOP TtPHETrRM/V*U**2/W*RATIO**2
010 SUM3SHM+TERM
OJ1 TsTfl.
01* 3 ACOSs|.5707963-SUM.RATIO
oi<4 RETURN
o i s f MO
150
-------
001 SUBROUTINE ZERO ( A,B, C,D,R, X)
002 Vett
003 ZsO.UR
004 «=r»Z
005 P1=A*Y**3»B»Y**2+C*Y+D
006 GO TO 1
007 3 Y = w
008 PisP?
009
010 1
Oil SUMA=AH3(P1)*AHS(P2)
012 ASUMSA*S(HI*P2)
013 IF(SUMA,NE.ASUM) GO TO 2
oi« GO TO 3
015 i" Xs(w*(H2-»M )wP2*(«.Y) )/(P2-Pl
016 GO TO b
017 6 X=(WMP2-P)-P2*(W»X))/(P2»P)
018 5 CONTINUK
019 E
020 F
021 G=C*X
022 E
023 FJ=ABS(B*X**2)
024 GI=ABS(C*X)
025 DtsABSCO)
026 IF(tl,Gfc,Fl) GO TO 10
027 HsFl
02fl GO TO 11
029 JO CONTlNl^F
030 HsEl
031 11 CONTINUE
032 IF(H.GF .Gl) GO TO 12
033 H=G1
03« 12 CONTINUE
035 IF(H.GE.Dl) GO 10 13
036 H=D1
037 1 1 C.ONTINUF
038 HlsALOr,10(H)
039 H2=ABs(Hl)+lt
0«0 N=IFIX(H2)
012 1=P*10**N
0«3 IFf ABS(T ),LE,1 .E-OU) GO TO a
OUU GO TO 6
0«5 a RETURN
END
151
-------
001 SUBROUTINE CHARGE(NI11,NII?,AFID,01,J)
002 COMMON RAD<20).«8AT(20>,U,E,EPSO,PI,eRAVG,BC,TEMP,EPS
003 VRMS««.4E2
004 DEITTB.001
005 F»TENP-45<».
006 TEMQ«5./9,*(F»J2.)
007 TEMQoTKMQ*27S.
OOfl DO 401 KoNm,NI12
009 DELTOlB((AFI0*S*U*03AT(J)/(«.*EP30))*(l.-81/03ATfJ>)**2)*DELTT
010 ARC=-01*F/(«,*PI*EP80*R*0(J)*BC*TEMQ)
Oil IF(ABS(ARG),OT,100.)GO TO «02
012 OELTQ2sPI*RAO(J)**?*E*VRM8«AFIO*EXP(ARG)*DELTT
013 402 CONTINUE
Oiu IF(Ol«QSAT(J))aiO«4U,4U
015 «10 OELTO»OFLTQUOELT02
016 GO TO 412
017 «11 DFI.TO«DELT02
018 «12 CONTINUE
01V Q1«OUOE|. TQ
020 aoi CONTINUE
152
-------
APPENDIX 3
Conversion Factors
To Convert' From
Ibs
grains/cf
cfm
lbs/in.2
OF
ftVlOOO cfm
inches w. g.
gallon
ft
inches
To
Kg
grams/m3
m3/sec
Kg/m2
°C
m2/(m3/sec)
mm Hg
liter
m
m
Multiply By
0.454
2.29
0.000472
703
(°F - 32) x 5/9
0.197
1.868
3.785
0.3048
0.0254
153
-------
TECHNICAL REPORT DATA
(Please read Inziruetiuns on the reverse before completing)
1. REPORT NO.
EPA-650/2-75-037
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
A Mathematical Model of Electrostatic Precipitation
5. REPORT DATE
April 1975
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
J.P. Gooch, J.R. McDonald, and S. Oglesby Jr.
8. PERFORMING ORGANIZATION REPORT NO
SORI-EAS-75-171
2887-XXVI
9. PERFORMING ORSANIZATION NAME AND ADDRESS
Southern Research Institute
2000 Ninth Avenue, South
Birmingham, Alabama 35205
10. PROGRAM ELEMENT NO.
1AB012; ROAP 21ADJ-026
11. CONTRACT/GRANT NO.
68-02-0265
2. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
EPA, Office of Research and Development
NERC-RTP, Control Systems Laboratory
Research Triangle Park, NC 27711
Final
14. SPONSORING AGENCY CODE
5. SUPPLEMENTARY NOTES
. ABSTRACT!^ report describes a mathematical model which relates collection effi-
ciency to electrostatic precipitator (ESP) size and operating parameters. It gives
procedures for calculating particle charging rates, electric field as a function of
josition in wire-plate geometry, and the theoretically expected collection efficiencies
'or various particle sizes and ESP operating conditions. It proposes methods for
mpirically representing collection efficiency losses caused by non-uniform gas
velocity distributions, gas bypassing the electrified regions, and particle reentrain-
ment due to rapping of the collection electrodes. Incorporating these proposed tech-
niques into a mathematical model of ESP performance reduces the theoretically
calculated overall collection efficiencies. It compares the reduced efficiencies with
those obtained from measurements on ESPs treating flue gas from coal-fired gener-
ating stations. It also presents the effects of changes in particle size distributions
on calculated collection efficiencies obtained from the mathematical model.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Air Pollution
lectrostatic Precipitation
Mathematical Models
ollection
fficiency
Air Pollution Control
Stationary Sources
13B
13H
14B
. DISTRIBUTION STATEMENT
Unlimited
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES
162
20. SECURITY CLASS (This page/
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
154
------- |