United States • ' "Industrial Environmental Research
Environments. Protection •—-tafeeratory
Agency Research Triangle Park NC 27711
EPA-600/7-78-111a
June 1978
Research and Development
A Mathematical
Model of
Electrostatic
Precipitation
(Revision 1):
Volume I.
Modeling and
Programming
Interagency
Energy/Environment
R&D Program Report
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine 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 (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the INTERAGENCY ENERGY-ENVIRONMENT
RESEARCH AND DEVELOPMENT series. Reports in this series result from the
effort funded under the 17-agency Federal Energy/Environment Research and
Development Program. These studies relate to EPA's mission to protect the public
health and welfare from adverse effects of pollutants associated with energy sys-
tems. The goal of the Program is to assure the rapid development of domestic
energy supplies in an environmentally-compatible manner by providing the nec-
essary environmental data and control technology. Investigations include analy-
ses of the transport of energy-related pollutants and their health and ecological
effects; assessments of, and development of, control technologies for energy
systems; and integrated assessments of a wide range of energy-related environ-
mental issues.
REVIEW NOTICE
This report has been reviewed by the participating Federal Agencies, and approved
for publication. Approval does not signify that the contents necessarily reflect the
views and policies of the Government, nor does mention of trade names or commercial
products constitute endorsement or recommendation for use.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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DISCLAIMER
This report was prepared as an account of work sponsored by
the United States Government. The report has been reviewed by the
Industrial Environmental Research Laboratory, U.S. Environmental
Protection Agency, and approved for publication. Approval does
not signify that the contents necessarily reflect the views and
policies of the U.S. Environmental Protection Agency, nor does
mention of trade names or commercial products constitute endorse-
ment or recommendation for use. Neither the United States nor
the U.S. Environmental Protection Agency, nor any of their employees,
nor any of their contractors, subcontractors, or their employees,
makes any warranty, express or implied, or assumes any legal
liability or responsibility for the accuracy, completeness or
usefulness of any information, apparatus, product, process or
computer program disclosed, or represents that its use would not
infringe privately owned rights.
11
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ABSTRACT
The objectives of this research program were to upgrade the
fundamental basis of the existing model of electrostatic precipi-
tation developed under the sponsorship of the Environmental Pro-
tection Agency, to make the computer program which performs the
calculations required by the model more user oriented, and to
fully document those subroutines in the computer program that
perform fundamental calculations or utilize numerical techniques.
In this report, the fundamental mechanisms and limiting
factors involved in the electrostatic precipitation process are
described briefly. The theories and procedures used in the
model to describe the physical mechanisms are discussed. A
general description of the major operations which are performed
in the computer program is given. A listing of the entire com-
puter program and the definitions of all the variables used in
the program are provided.
Major improvements to the fundamental basis of the model
include the capability of generating theoretical voltage-current
characteristics for wire-plate geometries, a new method for
describing the effects of rapping reentrainment, and a new pro-
cedure for predicting the effects of particles on the electrical
conditions.
The computer program has been made more user oriented by
making the input data less cumbersome, by making the output data
more complete, by making modifications which save computer time,
and by providing for the construction of log-normal particle size
distributions.
Those subroutines in the computer program that perform funda-
mental calculations or utilize numerical techniques are described
in sufficient detail to provide an understanding of their content
and usage. A detailed flow chart is provided for each of these
subroutines. Input and output variables are described and any
limitations on these variables are noted.
A complete description of the input data to the computer pro-
gram is provided so that the program can be utilized. Modifications
which can be made to the computer program to adapt it to different
computers and to extend its capabilities are discussed.
111
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This report was submitted in partial fulfillment of Task VI
of Contract No. 68-02-2114 by Southern Research Institute under
the sponsorship of the U.S. Environmental Protection Agency. This
report covers a contract period from June 30, 1975 to February 28,
1978, and work was completed as of February 15, 1978.
IV
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CONTENTS
Disclaimer ii
Abstract iii
Figures vii
Tables x
Nomenclature xi
Metric Conversion Factors xx
1. Introduction 1
2 . Conclusions 4
3. Recommendations 5
4. Fundamental Steps in the Electrostatic Precipitation
Process 7
Creation of an electric field and corona
current 7
Particle charging 9
Particle collection 14
Removal of collected material 16
5. Limiting Factors Affecting Precipitator Performance... 17
Allowable voltage and current density 17
Methods for predicting fly ash electrical re-
sistivity 19
Nonideal effects 19
6. Description of the Mathematical Model 22
Ideal calculation of particle collection
efficiency 22
Methods for estimating nonideal effects 29
Empirical corrections to no-rap migration
velocities 36
Estimation procedure for calculating particle
collection efficiencies 39
7. Computer Programing of the Mathematical Model 42
Description of the computer program 42
Descriptions of the subroutines 45
8. Description of Input Data 141
General description 141
Construction of the basic data set 141
Construction of shortened data sets 160
9. Machine-Dependent Aspects of the Computer Program 165
References 170
v
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Appendices
A. Development of New Procedure for Determining Space
Charge Effects 174
B. Definitions of Variables Used in the Main Program
and Subroutines 190
C. Complete Listing of the Computer Program 262
VI
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FIGURES
Number Page
la Region near small-radius electrode 8
b Electric field configuration for wire-plate
geometry 8
2 Electric field configuration during field charging.. -11
3a Electric field configuration and ion distribution
for particle charging with no applied field 12
b Electric field configuration and ion distribution
for particle charging in an applied electric
field after saturation charge is reached 12
4 Measured rapping emissions versus calculated par-
ticulate removal by last electrical section. These
curves are a result of work sponsored by the Electric
Power Research Institute 34
5 Apparent rapping puff size distribution for six
full-scale precipitators. These data are a result
of work sponsored by the Electric Power Research
Institute. 37
6 Average rapping puff size distribution for six full-
scale precipitators. These data are a result of
work sponsored by the Electric Power Research
Institute 38
7 Empirical correction factors for the "no-rap"
migration velocities calculated from the mathe-
matical model. This work was sponsored by the
Electric Power Research Institute 40
8 Simplified flow chart for the entire program 46-49
9 Flow chart for subroutine SPCHG1 50
10 Flow chart for subroutine SPCHG2 53-54
11 Flow chart for subroutine CMAN 58-59
vii
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12 Nomenclature used in the numerical analysis of the
electrical conditions in wire-plate precipitators... 61
13 Partial grid showing nomenclature used in the
numerical analysis of the electrical conditions 62
14 Flow chart for subroutine EFLDl 67-69
15 Flow chart for subroutine EFLD2 73-80
16 Flow chart for subroutine CHARGN 85
17 Flow chart for statement function RATE 88-92
18 Flow chart for subroutine ARCCOS 95
19 Flow chart for subroutine ZERO .97
20 Flow chart for subroutine CHGSUM 99-101
21 Flow chart for subroutine ADJUST 106-117
22 Flow chart for subroutine WADJST 123
23 Flow chart for subroutine LNDIST 127-129
24 Flow chart for subroutine QTFE 131
25 Flow chart for subroutine LNFIT 135
26 Flow chart for subroutine CFIT 139
27 Flow chart for the input data logic 163-164
28 Nomenclature used in the procedure which determines
particulate space charge effects 176
29 Theoretical variation of average current density
at the plate with precipitator length for dif-
ferent specific collection areas and inlet mass
loadings at 33 kV 181
30 Theoretical variation of average current density
at the plate with precipitator length for dif-
ferent specific collection areas and inlet mass
loadings at 35 kV 182
31 Theoretical variation of average current density
at the plate with precipitator length for dif-
ferent specific collection areas and inlet mass
loadings at 40 kV 183
viii
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32 Theoretical voltage-current curves for a specific
collection area of 19.7 m2/(m3/sec) 185
33 Theoretical voltage-current curves for a specific
collection area of 59.1 m2/(m3/sec) 186
34 Theoretical voltage-current curves for a specific
collection area of 98.4 m2/(m3/sec) 187
35 Comparison of theoretical voltage-current curves
for different specific collection areas 188
36 Comparison of model predictions using the dif-
ferent space charge schemes with field test data
from a full-scale precipitator. Model predic-
tions are for unadjusted, no-rap efficiencies
where a =0.25 and S = 0 189
g
IX
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TABLES
Number Page
1 Particle Sizes and Correction Factors for No-Rap
Migration Velocities Tabulated in Subroutine
WADJST 121
2 Reduced Effective Negative Ion Mobilities for
Various Gas Compositions 150
3 Values of Viscosity for Air at Various Temperatures
and Water Contents 158
4 Core Requirements for Various Segments of the
Computer Program 166
x
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NOMENCLATURE
w Migration velocity near the collection electrode of a
P particle of radius a, m/sec
q Charge on a particle, coul
E Electric field near the collection electrode, V/m
P
a Particle radius, m
y Gas viscosity, kg/m-sec
C Cunningham correction factor (or slip correction factor)
X Mean free path of gas molecules, m
A" Quantity defined by [1.257 + 0.4 exp (-1.1 a/A)]
ri Collection fraction for a monodisperse aerosol
A Collection electrode area, m2
Q Gas volume flow rate, m3/sec
VT Voltage drop across the collected particulate layer, V
_LJ
j Current density in the collected particulate layer,
A/cm2
p' Resistivity of the collected particulate layer, ohm-cm
t Thickness of the collected particulate layer, cm
E Average electric field in the collected particulate
layer, V/cm
n. . Ideal collection fraction for the i-th particle size
"""'^ in the j-th increment of length of the precipitator
w. . Migration velocity of the i-th particle size in the
1' -' j-th increment of length of the precipitator, m/sec
A. Collection electrode area in the j-th increment of
-1 length of the precipitator, m2
xi
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n. Ideal collection fraction for a given particle size
1 over the entire length of the precipitator
N. . Number of particles of the i-th particle size per
1' "J cubic meter of gas entering the j-th increment, #/m
n Ideal overall mass collection efficiency for the
entire polydisperse aerosol, %
p. Percentage by mass of the i-th particle size in the
1 inlet size distribution, %
V Electric potential at a given point in a precipitator,
V
p Total space charge density at a given point in the
gas in a precipitator, coul/m3
b Effective charge carrier mobility, m2/V-sec
ti
y Coordinate parallel to the gas flow from wire-to-wire,
m
x Coordinate perpendicular to the gas flow from wire-to-
plate, m
e0 Permittivity of free space, coul/N-m2
J Average current density at the collection plate, A/m2
p . Space charge densities for various points on the
" collection plate, coul/m3
E . Electric field strengths for various points on the
P1 collection plate, V/m
N Number of grid points in the direction of gas flow in
the electric field calculations
q Instantaneous particle charge, coul
q Saturation charge due to field charging, coul
6 Azimuthal angle in a spherical coordinate system with
origin at the center of the particle, radians
80 Maximum azimuthal angle for which electric field lines
enter a charged particle, radians
NO Free ion density, #/m3
e Electronic charge, coul
xii
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Eo Average electric field between the discharge elec-
trodes, V/m
t> Ion mobility, m2/V-sec
a/
v Mean thermal speed of ions, m/sec
k Boltzmann's constant, J/°K
T Absolute temperature of the gas, °K
t Time, sec
K Dielectric constant of the particle
ro Radial distance along 6 at which the radial component
of the total electric field is zero, m
qD Charge predicted from classical diffusion charging
theory, coul
qp Charge predicted from classical field charging theory,
coul
t. Initial time for charging under a fixed set of con-
ditions , sec
tf Final time for charging under a fixed set of con-
ditions, sec
q. Charge on the particle at time t., coul
b"* Effective ion mobility, m2/V-sec
j Total current density at the collection plate due to
ions and particles, A/m2
j Particulate current density at the collection plate,
p A/m2
w Migration velocity for a given particle diameter which
is calculated from fundamental principles and applies
only to a given length increment as used in the model,
cm/sec
w Effective migration velocity for a given particle
e diameter which is calculated from fundamental prin-
ciples and applies to the entire length of the pre-
cipitator, cm/sec
w- Apparent effective migration velocity for a given par-
e ticle diameter which is obtained by making an empirical
correction (or corrections) to w , cm/sec
xiii
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w precipitation rate parameter which provides a measure
Pr of how well the entire mass which enters the pre-
cipitator will be collected, cm/sec
F. Particle diameter-dependent correction factors for
1 nonuniform gas velocity distribution
a Normalized standard deviation of the gas velocity
9 distribution
PN Penetration of a given particle size from the last
baffled section which is corrected for gas sneakage
S Fractional amount of gas sneakage per baffled section
N Number of baffled sections
s
B. Particle diameter-dependent correction factors for
1 gas sneakage and/or nonrapping reentrainment
w . Effective migration velocity for the i-th particle
' diameter, cm/sec
w" . Apparent effective migration velocity for the i-th
e'X particle diameter, cm/sec
p
N Penetration of a given particle size which is corrected
for nonrapping reentrainment
R Fraction of collected material reentrained per section
NR Number of sections over which reentrainment is assumed
to occur
n'/section Total mass collection fraction per linear electrical
section under normal operating conditions
no Overall mass collection fraction determined from mass
train measurements under normal operating conditions
X" Quantity which is equal to -In (1-rio)
NE Number of electrical sections in series
X Calculated mass removal by the last electrical section,
mg/DSCM
Yi Measured rapping emissions from cold-side precipitators,
mg/DSCM
Yz Measured rapping emissions from hot-side precipitators,
mg/DSCM
xiv
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dso(MMD) Mass median diameter of a log-normal particle size
distribution, um
ap Geometric standard deviation of a log-normal particle
size distribution
Vw Applied voltage, V
S Wire-to-plate spacing, m
X
S One half wire-to-wire spacing, m
r Radius of corona wire, m
a Increment size in the x-direction used in calculating
electrical conditions, m
a Increment size in the y-direction used in calculating
^ electrical conditions, m
E x-component of the electric field, V/m
X
E y-component of the electric field, V/m
VQ Electric potential at an arbitrary point in a numerical
grid, V
po Space charge density at an arbitrary point in a numerical
grid, coul/m3
a Parameter in the equation for p0, coul/m3
3 Parameter in the equation for po, cou!2/m6
a Values of a along a line from wire to wire, coul/m3
.TT..D
a. Values of a along a line midway between wires from
the plane of the wires to the plate, coul/m3
a Values of a along a line from the wire to the plate,
AD coul/m3
a Values of a along the plate, coul/m3
3 Values of 3 along a line from wire to wire, cou!2/m6
3 Values of 3 along a line midway between wires from
the plane of the wires to the plate, cou!2/m6
3 Values of 3 along a line from the wire to the plate,
AD
XV
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3 Values of 3 along the plate, coul2/m6
p Space charge density at the outer boundary of the
s ionized corona sheath, coul/m3
b Effective charge carrier mobility at the outer boundary
S of the ionized corona sheath, m2/V-sec
f Roughness factor of the corona wires
<5 Relative density of the gas
P.
Space charge density near the corona wire, coul/m3
r Radius of the ionized corona sheath, m
s
E Electric field at the outer boundary of the ionized
s corona sheath, V/m
E Corona starting electric field, V/m
Derivative of the variable y with respect to the
variable x where x and y represent unspecified
quantities
f(x,y) Arbitrary function of x and y
k Weighting factors in a Runge-Kutta integration scheme
n (n - 1,2,3,4)
Ay Increment for advancing the dependent variable in the
Runge-Kutta integration scheme
h Increment size for the independent variable in the
Runge-Kutta, Simpson's Rule, and trapezoidal rule
integration schemes
xn(xi) Values of the independent variable in the Runge-Kutta
and Simpson's Rule integration schemes
yn(yi) Values of the dependent variable in the Runge-Kutta,
Simpson's Rule, and trapezoidal rule integration
schemes
n Number of charges on a particle
GI Coefficient of X in the cubic equation (53)
£2 Factor in the constant term in the cubic equation (53)
fL_N(z) Log-normal distribution function
xv i
-------
d Particle diameter, ym
z Independent variable for the log-normal distribution
z Mean value of z
tf Standard deviation of z
£*
M Total mass contained in a log-normal particle size
distribution, kg/m3
F^ Mass fractions for the different size bands in a log-
normal particle size distribution
S. Cumulative mass fractions in a log-normal particle
size distribution
Z . Cumulative integrals obtained in a trapezoidal rule
integration
S(X) Cumulative fraction up to a given particle size in a
log-normal distribution
t Transformation variable for a log-normal distribution
t" Lower limit for the integration of a Gaussian integral
over the variable t
Q(t) Cumulative fraction greater than a given particle size
cj) Variable in terms of which t can be expressed
ao,a1,a2 Coefficients in an approximate expression for t"
bi,b2fb3 Coefficients in an approximate expression for t"
z' Natural logarithm of a known or measured particle
diameter corresponding to a known or measured cumula-
tive mass fraction
~z" Mean value of z"
A Quantity which is equal to -z"/a
&
B Quantity which is equal to I/a
£4
m Number of data points in a least squares fit to a
straight line
A Parameter obtained in a least squares fit to a
straight line
xvii
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r Linear-correlation coefficient
J Average current density in the 1-th subincremental
length of a given length increment, A/m2
if Average electric field in the 1-th subincremental
^ length of a given length increment, V/m
p" Average total particulate charge density in the 1-th
subincremental length of a given length increment,
coul/m3
p". Average charge density for the i-th particle size at
x' the end of the 1-th subincremental length of a given
length increment, coul/m3
X. „ Number of particles per unit volume of gas of the
1' i-th particle size entering the 1-th subincremental
length of a given length increment, #/m3
q. . Charge on the i-th particle size at the end of the
lf 1-th subincremental length of a given length incre-
ment, coul
b,. Weighted particulate mobility due to all particles
in the 1-th subincremental length of a given length
increment, m2/V-sec
C. Cunningham correction factor (or slip correction
factor) for the i-th particle size
a. Radius of the i-th particle size, m
Xp Total number of particles per unit volume of gas
entering the 1-th subincremental length of a given
length increment, #/m3
P£ Average ionic charge density with a particulate mass
loading in the 1-th subincremental length of a given
length increment, coul/m3
b" Molecular ion "effective mobility", m2/V-sec
P£ Average ionic charge density without a particulate
mass loading in the 1-th subincremental length of a
given length increment, coul/m3
APj£ Average charge density shifted from molecular ions
to particles in the 1-th subincremental length of a
given length increment, coul/m3
XVlll
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k>£ Effective mobility due to both ions and particles in
the 1-th subincremental length of a given length in-
crement, m2/V-sec
w^ ^ Migration velocity of the i-th particle size in the
' 1-th subincremental length of a given length increment,
m/sec
E£ Average electric field at the collection plate in the
1-th subincremental length of a given length increment,
V/m
r\. „ Ideal collection fraction for the i-th particle size
' in the 1-th subincremental length of a given length
increment
Q
b Average effective mobility for ions and particles over
a length equal to one wire-to-wire spacing, m2/V-sec
j Average current density near the wire without particles,
W A/m2
j "* Average current density near the wire with particles,
W A/m2
A"* Collection plate area receiving current from a single
P wire, m2
A Surface area of a single wire, m2
w 3
j Average current density at the collection plate for an
P area receiving current without particles from a single
wire, A/m2
Jx Average current density at the collection plate for an
p area receiving current with particles from a single
wire, A/m2
xix
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To Convert From
grams/ft3
ft
ft2
in
ft3/min
ft/sec
METRIC CONVERSION FACTORS
To
kg/m3
m
m2
m
m3/sec
m/sec
°K
Multiply by
0.00229
0.3048
0.0929
0.0254
0.000472
0.3048
(°F+459) x
xx
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SECTION 1
INTRODUCTION
The electrostatic precipitation process involves several com-
plicated and interrelated physical mechanisms: the creation of a
nonuniform electric field and ionic current in a corona discharge;
the ionic and electronic charging of particles moving in combined
electro- and hydro-dynamic fields; and the turbulent transport of
charged particles to a collection surface. The removal of the
collected particulate layer from the collection surface presents
a serious problem in many practical applications since the removal
procedures introduce collected material back into the gas stream
and cause a reduction in collection efficiency. Other practical
considerations which reduce the collection efficiency are non-
uniform gas velocity distribution, bypassage of the electrified
regions by particle-laden gas, and particle reentrainment during
periods when no attempt is being made to remove the collected
material.
In recent years, increasing emphasis has been placed on de-
veloping theoretical relationships which accurately describe the
individual physical mechanisms involved in the precipitation
process and on incorporating these relationships into a complete
mathematical model for electrostatic precipitation. From a prac-
tical standpoint, a reliable theoretical model for electrostatic
precipitation would offer several valuable applications:
(1) precipitator design could be easily and completely per-
formed by calculation from fundamental principles;
(2) a theoretical model could be used in conjunction with a
pilot-plant study in order to design a full-scale pre-
cipitator;
(3) precipitator bids submitted by various manufacturers
could be evaluated by a purchaser with respect to meeting
the design efficiency and the costs necessary to obtain
the design efficiency;
(4) the optimum operating efficiency of an existing precipi-
tator could be established and the capability to meet
particulate emissions standards could be ascertained; and
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(5) an existing precipitator performing below its optimum
efficiency could be analyzed with respect to the differ-
ent operating variables in a procedure to troubleshoot
and diagnose problem areas.
In addition to its many applications, a mathematical model
can be a valuable tool for analyzing precipitator performance due
to its cost and time savings capability. The approach is cost
effective because it (1) allows for the analysis and projection
of precipitator operation based upon a limited amount of data
(extensive field testing is not necessary), (2) can predict
trends caused by changing certain precipitator parameters and
thus, in many cases, can prevent costly modifications to a pre-
cipitator which will not significantly improve the performance,
(3) can be used as a tool in sizing precipitators and preyent
excessive costs due to undersizing or significant oversizing, and
(4) can be used to obtain large amounts of information without
extensive use of manpower but, instead, with reasonable use of a
computer.
The approach is time effective because (1) large amounts of
information can be generated quickly, (2) it does not necessarily
depend on time-consuming field tests which involve travel, exten-
sive analysis, and plant and precipitator shut-downs, (3) it can
prevent losses in time due to unnecessary or insufficient modifi-
cations to a precipitator, and (4) it can prevent losses in time
due to the construction of an undersized precipitator.
In the present work, a revised model of electrostatic pre-
cipitation developed by Southern Research Institute under the
sponsorship of the Environmental Protection Agency (Industrial
Environmental Research Laboratory, Research Triangle Park) is
discussed. The first version of the model is described in the
publication entitled "A Mathematical Model of Electrostatic
Precipitation."1 The present report is separated into two volumes.
Volume 1 contains a description of the physical mechanisms involved
in the electrostatic precipitation process, the physical and mathe-
matical formulation of the model, and a documentation of a computer
program which implements the model. Volume 2 is a user's manual
which describes how to use the model for various purposes. This
volume includes a description of input and output data and relates
these quantities to the various applications of the model.
The version of the model described in the present text has
the following features:
(1) it predicts collection efficiency as a function of par-
ticle diameter, electrical operating conditions, and gas
properties;
(2) it can calculate clean-plate, clean-air voltage-current
characteristics for wire-plate geometries;
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(3) it determines particle charging by unipolar ions as a
function of particle diameter, electrical conditions,
and residence time;
(4) it can estimate the effects of particles on the elec-
trical conditions under the assumption that effects due
to the particulate layer can be ignored;
(5) it accounts for electrical sectionalization;
(6) it predicts particle capture at the collection electrode
based on the assumptions of completely-random, turbulent
flow, uniform gas velocity, and particle migration ve-
locities which are small compared to the gas velocity;
(7) it employs empirical correction factors which adjust the
particle migration velocities obtained without rapping
losses;
(8) it accounts for the nonideal effects of nonuniform gas
velocity distribution, gas bypassage of electrified
regions, and particle reentrainment from causes other
than rapping by using empirical correction factors to
scale down the ideally-calculated particle migration
velocities; and
(9) it accounts for rapping reentrainment by using empirical
relationships for the quantity and size distribution of
the reentrained mass.
In its present form, the model has the capability of pre-
dicting trends caused by changes in specific collection area,
applied voltage, current density, mass loading, and particle size
distribution„ Comparisions of the predictions of the model with
laboratory-scale precipitators2 and full-scale precipitators col-
lecting fly ash from coal-fired boilers1'3' "* indicate that the
model can be used successfully to predict precipitator performance,
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SECTION 2
CONCLUSIONS
The version of the mathematical model of electrostatic pre-
cipitation presented in this report offers greater predictive
capabilities and is more user oriented than the previous version.
Greater predictive capabilities are provided by allowing for the
calculation of theoretical voltage-current characteristics for
wire-plate geometries, by use of a new method for determining
the effects of rapping reentrainment that is directly related
to full-scale precipitators, by incorporation of a new method for
estimating the effects of particles on the electrical conditions,
and by the use of experimentally-determined, empirical correction
factors for individual particle migration velocities that results
in increased agreement between the theory and field test data.
The computer program which performs the calculations required by
the model is more user oriented than the previous program due to
modifications that make the input data less cumbersome, make the
output data more complete and useful, result in savings of com-
puter time, and allow for the construction of log-normal particle
size distributions. Detailed documentation of those subroutines
which perform fundamental calculations or utilize numerical tech-
niques should provide a firm basis for understanding their content
and usage.
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SECTION 3
RECOMMENDATIONS
Although the mathematical model of electrostatic precipitation
presented in this report represents a significant improvement over
the previous version, more work still needs to be performed in
order to improve the fundamental basis and user oriented aspects
of the model.
With respect to the fundamental basis of the model, it is
recommended that the following research be pursued:
1. Theoretical and experimental studies of the effects of
particles on the electrical conditions should be continued in
order to better describe the effect on the electric field dis-
tribution.
2«, Theoretical and experimental studies of electrical break-
down mechanisms in the collected particulate layer should be given
greater emphasis in an attempt to acquire the capability of theo-
retical prediction of when electrical breakdown will ensue for a
given resistivity.
3. Since the model underpredicts the collection efficiencies
for fine particles without the use of empirical correction factors,
theoretical and experimental studies should be continued in order
to remove the empiricism. These studies should include a reeval-
uation of the theories presently used in the model and an exam-
ination of those effects which are presently neglected such as
particle charging near corona wires and phenomena due to the gas
flow fields
4, The mathematical model should be restructured to take
into account time-dependent effects. The effects due to the
growth of the particulate layer and the rapping schedule should
be included as a function of time. Although the empirical pro-
cedure employed in the present version of the model represents
a useful interim technique for estimating the effects due to
rapping reentrainment in precipitators, it does not describe
the temporal and dynamic aspects of the rapping process. The
inclusion of time-dependent effects is necessary in order to
optimize the electrical operating conditions and the rapping
schedule and intensity.
-------
The above research is needed in order to make the model indepen-
dent of empiricism and of the experience and judgment of the user.
With respect to the user oriented aspects of the model, it
is recommended that the following work be performed:
1. Alternative numerical techniques need to be investigated
and implemented in order to make the computer program run signif-
icantly faster.
2. Procedures which edit the input data should be implemented.
3. Documentation of the computer program needs to be included
in abbreviated form in the computer card deck.
The above work is needed in order to continue the transition in
which the model is transformed from a research tool to a production
tool.
-------
SECTION 4
FUNDAMENTAL STEPS IN THE ELECTROSTATIC
PRECIPITATION PROCESS
CREATION OF AN ELECTRIC FIELD AND CORONA CURRENT
The first step in the precipitation process is the creation
of an electric field and corona current. This is accomplished
by applying a large potential difference between a small-radius
electrode and a much larger radius electrode, where the two elec-
trodes are separated by a region of space containing an insulating
gas. For industrial applications, a large negative potential is
applied at the small-radius electrode and the large-radius elec-
trode is grounded.
At any applied voltage, an electric field exists in the
interelectrode space. For applied voltages less than a value
referred to as the "corona starting voltage", a purely electro-
static field is present. At applied voltages above the corona
starting voltage, the electric field in the vicinity of the small-
radius electrode is large enough to produce ionization by electron
impact. Between collisions with neutral molecules, free electrons
are accelerated to high velocities and, upon collision with a
neutral molecule, their energies are sufficiently high to cause
an electron to be separated from a neutral molecule. Then, as
the increased number of electrons moves out from the vicinity of
the small-radius electrode, further collisions between electrons
and neutral molecules occur,, In a limited high electric field
region near the small-radius electrode, each collision between
an electron and a neutral molecule has a certain probability of
forming a positive molecular ion and another electron, and an
electron avalanche is established. The positive ions migrate
to the small-radius electrode and the electrons migrate into
the lower electric field regions toward the large-radius elec-
trode. These electrons quickly lose much of their energy and,
when one of them collides with a neutral electro-negative molecule,
there is a probability that attachment will occur and a negative
ion will be formed. Thus, negative ions, along with any electrons
which do not attach to a neutral molecule, migrate under the
influence of the electric field to the large-radius electrode
and provide the current necessary for the precipitation process.
Figure la is a schematic diagram showing the region very near
the small-radius electrode where the current-carrying negative
-------
SMALL-RADIUS ELECTRODE AT
HIGH NEGATIVE POTENTIAL
REGION OF ELECTRON AVALANCHE
WHERE POSITIVE IONS AND ELECTRONS
ARE PRODUCED
REGION OF IONIZATION WHERE ELECTRONS
ATTACH TO NEUTRAL MOLECULES TO
FORM NEGATIVE IONS
Figure la. Region near small-radius electrode.
SMALL-RADIUS ELECTRODE AT
HIGH NEGATIVE POTENTIAL
ELECTRIC FIELD
LINES
V
EQUIPOTENTIAL
SURFACES
IONS WHICH CONSTITUTE A CURRENT
AND A SPACE CHARGE FIELD
\
GROUNDED LARGE-
RADIUS ELECTRODE
Figure 1b. Electric field configuration for wire-plate geometry.
-------
ions are formed. As these negative ions migrate to the large-
radius electrode, they constitute a steady-state charge distri-
bution in the interelectrode space which is referred to as an
"ionic space charge". This "ionic space charge" establishes an
electric field which adds to the electrostatic field to give the
total electric field. As the applied voltage is increased, more
ionizing sequences result and the "ionic space charge" increases.
This leads to a higher average electric field and current density
in the interelectrode space.
Figure Ib gives a qualitative representation of the electric
field distribution and equipotential surfaces in a wire-plate
geometry which is most commonly used. Although the electric field
is very nonuniform near the wire, it becomes essentially uniform
near the collection plates. The current density is very nonuni-
form throughout the interelectrode space and is maximum along a
line from the wire to the plate.
In order to maximize the collection efficiency obtainable
from the electrostatic precipitation process, the highest possible
values of applied voltage and current density should be employed.
In practice, the highest useful values of applied voltage and
current density are limited by either electrical breakdown of
the gas throughout the interelectrode space or of the gas in the
collected particulate layer. High values of applied voltage and
current density are desirable because of their beneficial effect
on particle charging and particle transport to the collection
electrode. In general, the voltage-current characteristics of
a precipitator depend on the geometry of the electrodes, the gas
composition, temperature, and pressure, the particulate mass
loading and size distribution, and the resistivity of the col-
lected particulate layer. Thus, maximum values of voltage and
current can vary widely from one precipitator to another and from
one application to another.
PARTICLE CHARGING
Once an electric field and current density are established,
particle charging can take place. Particle charging is essential
to the precipitation process because the electrical force which
causes a particle to migrate toward the collection electrode is
directly proportional to the charge on the particle. The most
significant factors influencing particle charging are particle
diameter, applied electric field, current density, and exposure
time.
The particle charging process can be attributed mainly to
two physical mechanisms, field charging and thermal charging.5'6'7
These two mechanisms are discussed below.
(1) At any instant in time and location in space near a par-
ticle, the total electric field is the sum of the electric field
-------
due to the charge on the particle and the applied electric field.
In the field charging mechanism, molecular ions are visualized as
drifting along electric field lines. Those ions moving toward the
particle along electric field lines which intersect the particle
surface impinge upon the particle surface and place charge on the
particle.
Figure 2 depicts the field charging mechanism during the
time it is effective in charging a particle. In this mechanism,
TT
only a limited portion of the particle surface (0<_0<-) can suffer
an impact with an ion and collisions of ions with other portions
of the particle surface are neglected. Field charging takes place
very rapidly and terminates when sufficient charge (the saturation
charge) is accumulated to repel additional ions. Figure 3b depicts
the electric field configuration once the particle has attained
the saturation charge. In this case, the electric field lines
circumvent the particle and the ions move along them around the
particle.
Theories based on the mechanism of field charging agree rea-
sonably well with experiments whenever particle diameters exceed
about 0.5 ym and the applied electric field is moderate to high.
In these theories, the amount of charge accumulated by a particle
depends on the particle diameter, applied electric field, ion
density, exposure time, ion mobility, and dielectric constant of
the particle.
(2) The thermal charging mechanism depends on collisions be-
tween particles and ions which have random motion due to their
thermal kinetic energy. In this mechanism, the particle charging
rate is determined by the probability of collisions between a
particle and ions. If a supply of ions is available, particle
charging occurs even in the absence of an applied electric field.
Although the charging rate becomes negligible after a long period
of time, it never has a zero value as is the case with the field
charging mechanism. Charging by this mechanism takes place over
the entire surface of the particle and requires a relatively long
time to produce a limiting value of charge.
Figure 3a depicts the thermal charging process in the absence
of an applied electric field. In this case, the ion distribution
is uniform around the surface of the particle and each element of
surface area has an equal probability of experiencing an ion col-
lision. Thermal charging theories which neglect the effect of
the applied electric field adequately describe the charging rate
over a fairly broad range of particle sizes where the applied
electric field is low or equal to zero. In addition, they work
well for particles less than 0.2 ym in diameter regardless of the
magnitude of the applied electric field.
10
-------
X, Z, 6 - SPHERICAL COORDINATE SYSTEM
NEGATIVELY CHARGED PARTICLE
0
ELECTRIC FIELD LINES
Figure 2. Electric field configuration during field charging
11
-------
NEGATIVE IONS
X, Z - COORDINATE AXES
NEGATIVELY CHARGED
PARTICLE
ELECTRIC FIELD LINES
Figure 3a. Electric field configuration and ion distribution for
particle charging with no applied field.
X, Z- COORDINATE AXES
PARTICLE HAS SATURATION CHARGE
0-
Figure 3b. Electric field configuration and ion distribution for
particle charging in an applied field after saturation
charge is reached.
12
-------
Figure 3b depicts the thermal charging process in the pre-
sence of an applied electric field after the particle has attained
the saturation charge determined from field charging theory. The
effect of the applied electric field is to cause a large increase
in ion concentration on one side of the particle while causing
only a relatively small decrease on the other side. Although the
ion concentration near the surface of the particle becomes very
nonuniform, the net effect is to increase the average ion con-
centration, the probability of collisions between ions and the
particle, and the particle charging rate.
In thermal charging theories, the amount of charge accumulated
by a particle depends on the particle diameter, ion density, mean
thermal velocity of the ions, absolute temperature of the gas,
particle dielectric constant, residence time, and the applied elec-
tric field. The effect of the applied electric field on the ther-
mal charging process must be taken into account for fine particles
having diameters between 0.1 and 2.0 ym. Depending most importantly
on the applied electric field and to a lesser extent on certain
other variables, particles in this size range can acquire values
of charge which are 2-3 times larger than that predicted from
either the field or the thermal charging theories. For these par-
ticles, neither field nor thermal charging predominates and both
mechanisms must be taken into account simultaneously.
In most cases, particle charging has a noticeable effect on
the electrical conditions in a precipitator. The introduction of
a significant number of fine particles or a heavy concentration
of large particles into an electrostatic precipitator signifi-
cantly influences the voltage-current characteristic. Qualita-
tively, the effect is seen by an increased voltage for a given
current compared to the particle-free situation. As the particles
acquire charge, they must carry part of the current but they are
much less mobile than the ions. This results in a lower "effec-
tive mobility" for the charge carriers and, in order to obtain
a given particle-free current, higher voltages must be applied to
increase the drift velocities of the charge carriers and the ion
densities.
The charged particles, which move very slowly, establish a
"particulate space charge" in the interelectrode space. The dis-
tribution of the "particulate space charge" results in an electric
field distribution which adds to those due to the electrostatic
field and the ionic field to give the total electric field dis-
tribution. It is desirable to determine the space charge re-
sulting from particles because of its influence on the electric
field distribution, especially near the collection plate where,
for the same current, the electric field is raised above the
particle-free situation. In addition, the "particulate space
charge" is a function of position along the length of the pre-
cipitator since particle charging and collection are a function
of length.
13
-------
PARTICLE COLLECTION
As the particle-laden gas moves through a precipitator, each
charged particle has a component of velocity directed towards the
collection electrode. This component of velocity is called the
electrical drift velocity, or migration velocity, and results
from the electrical and viscous drag forces acting upon a sus-
pended charged particle. For particle sizes of practical interest,
the time required for a particle to achieve a steady state value
of migration velocity is negligible and, near the collection elec-
trode, the magnitude of this quantity is given by8
(1)
where w = migration velocity near the collection electrode of a
particle of radius a (m/sec),
q = charge on particle (coul),
E = electric field near the collection electrode (volt/m),
a = particle radius (m),
\JL = gas viscosity (kg/m-sec) ,
C = Cunningham correction factor, or slip correction
factor9 = (1 + A'X/a),
where A" = 1.257 + 0.400 exp (-1.10 a/A), and
A = mean free path of gas molecules (m).
In industrial precipitators, laminar flow never occurs and
the effect of turbulent gas flow must be considered. The tur-
bulence is due to the complex motion of the gas itself, electric
wind effects of the corona, and transfer of momentum to the gas
by the movement of the particles. Average gas flow velocities
in most cases of practical interest are between 0.6 and 2.0 m/sec.
Due to eddy formation, electric wind, and other possible effects,
the instantaneous velocity of a small volume of gas surrounding
a particle may reach peak values which are much higher than the
average gas velocity. In contrast, migration velocities for
particles smaller than 0.6 ym in diameter are usually less than
0.3 m/sec. Therefore, the motion of these smaller particles
tends to be dominated by the turbulent motion of the gas stream.
Under these conditions, the paths taken by the particles are
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
14
-------
adjacent to the collection electrode in which capture is assured.
Using probability concepts and the statistical nature of the
large number of particles in a precipitator, White10 derived an
expression for the collection efficiency in the form
nm = 1 - exp (-A w^/Q) , (2)
m
where nm = collection fraction for a monodisperse aerosol,
A = collection area (m2),
w = migration velocity near the collection electrode of the
particles in the monodisperse aerosol (m/sec), and
Q = gas volume flow rate (m3/sec).
The simplifying assumptions on which the derivation of equa-
tion (2) is based are:
(1) The gas is flowing in a turbulent pattern at a constant,
mean forward-velocity.
(2) Turbulence is small scale (eddies are small compared to
the dimensions of the duct), fully developed, and completely random.
(3) The particle migration velocity near the collecting sur-
face is constant for all particles and is small compared with the
average gas velocity.
(4) There is an absence of disturbing effects, such as par-
ticle reentrainment, back corona, particle agglomeration, or uneven
corona.
Experimental data11 under conditions which are consistent with
the above assumptions demonstrate that equation (2) adequately
describes the collection of monodisperse aerosols in an electro-
static precipitator under certain idealized conditions.
In industrial precipitators, the above assumptions are never
completely satisfied but they can be approached closely. With
proper design, the ratio of the standard deviation of the gas
velocity distribution to the average gas velocity can be made to
be 0.25 or less so that an essentially uniform, mean forward-
velocity would exist. Although turbulence is not generally a
completely random process, a theoretical determination of the
degree of correlation between successive states of flow and be-
tween adjacent regions of the flow pattern is a difficult problem
and simple descriptive equations do not presently exist for typical
precipitator geometries. At the present, for purposes of modeling,
it appears practical and plausible to assume that the turbulence
15
-------
is highly random. For particles larger than 10 ym diameter, the
turbulence does not dominate the motion of these particles due
to their relatively high migration velocities. Under these con-
ditions, equation (2) would be expected to underpredict collection
efficiencies. The practical effect in modeling precipitator per-
formance will be slight, however, since even equation (2) predicts
collection efficiencies greater than 99.6% for 10 ym diameter par-
ticles at relatively low values of current density and collection
area [i.e., a current density of 10 nA/cm2 and a collection area
to volume flow ratio of 39.4 m2/(m3/sec)].
REMOVAL OF COLLECTED MATERIAL
In dry collection, the removal of the precipitated material
from the collection plates and subsequent conveyance of the mate-
rial away from the precipitator represent fundamental steps in the
collection process. These steps are fundamental because collected
material must be removed from the precipitator and because the
buildup of excessively thick layers on the plates must be pre-
vented in order to ensure optimum electrical operating conditions.
Material which has been precipitated on the collection plates is
usually dislodged by mechanical jarring or vibration of the plates,
a process calling rapping. The dislodged material falls under the
influence of gravity into hoppers located below the plates and is
subsequently removed from the precipitator.
The effect of rapping on the collection process is determined
primarily by the intensity and frequency of the force applied to
the plates. Ideally, the rapping intensity must be large enough
to remove a significant fraction of the collected material but
not so large as to propel material back into the main gas stream.
The rapping frequency must be adjusted so that a larger thickness
which is easy to remove and does not significantly degrade the
electrical conditions is reached between raps. In practice, the
optimum rapping intensity and frequency must be determined by
experimentation. With perfect rapping, the sheet of collected
material would not reentrain, but would migrate down the col-
lection plate in a stick-slip mode, sticking by the electrical
holding forces and slipping when released by the rapping forces.
16
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SECTION 5
LIMITING FACTORS AFFECTING PRECIPITATOR PERFORMANCE
ALLOWABLE VOLTAGE AND CURRENT DENSITY
The performance of a precipitator which has good mechanical
and structural features will be determined primarily by the elec-
trical operating conditions. Any limitations on applied voltage
and current density will be reflected in the optimum collection
efficiency which can be obtained. A precipitator should be oper-
ated at the highest useful values of applied voltage and current
density for the following reasons:
(1) high applied voltages produce high electric fields;
(2) high electric fields produce high values of the saturation
and limiting charge that a particle may obtain;
(3) high current densities produce high rates at which par-
ticles charge to the saturation or limiting values of
charge;
(4) high current densities produce an increased electric
field near the collection electrode due to the "ionic
space charge" contribution to the field; and
(5) high values of electric field and particle charge produce
high migration velocities and increased transport of par-
ticles to the collection electrode.
Electrical conditions in a precipitator are limited by either
electrical breakdown of the gas in the interelectrode space or by
electrical breakdown of the gas in the collected particulate layer.
In a clean-gas, clean-plate environment, gas breakdown can origi-
nate at the collection electrode due to surface irregularities
and edge effects which result in localized regions of high elec-
tric field. If the electric field in the interelectrode space is
high enough, the gas breakdown will be evidenced by a spark which
propagates across the interelectrode space. The operating applied
voltage and current density will be limited by these sparking
conditions.
If a particulate layer is deposited on the collection elec-
trode, then the corona current must pass through the particulate
17
-------
layer to the grounded, collection electrode. The voltage drop
(V_) across the particulate layer is
j_i
VT = jp't , (3)
j_i
where j = current density (A/cm2),
p" = resistivity of particulate layer (ohm-cm), and
t = thickness of the layer (cm).
The average electric field in the particulate layer (EL) is given
by
EL = jp'. (4)
The average electric field in the particulate layer can be
increased to the point that the gas in the interstitial space
breaks down electrically. This breakdown results from the accel-
eration of free electrons to ionization velocity to produce an
avalanche condition similar to that at the corona electrode. When
this breakdown occurs, one of two possible situations will ensue.
If the electrical resistivity of the particulate layer is moderate
(^0.1-1.0 x 1011 ohm-cm), then the applied voltage may be suffi-
ciently high so that a spark will propagate across the interelec-
trode space. The rate of sparking for a given precipitator geom-
etry will determine the operating electrical conditions in such a
circumstance. If the electrical resistivity of the particulate
layer is high (>1011 ohm-cm), then the applied voltage may not
be high enough to cause a spark to propagate across the inter-
electrode space. In this case, the particulate layer will be
continuously broken down electrically and will discharge positive
ions into the interelectrode space. This condition is called back
corona. The effect of these positive ions is to reduce the amount
of negative charge on a particle due to bipolar charging and re-
duce the electric field associated with the "ionic space charge".
Both the magnitude of particle charge and rate of particle charging
are affected by back corona. Useful precipitator current is there-
fore limited to values which occur prior to electrical breakdown
whether the breakdown occurs as sparkover or back corona.
Field experience shows that current densities for cold side
precipitators are limited to approximately 50-70 nA/cm2 due to
electrical breakdown of the gases in the interelectrode space.
Consequently, this constitutes a current limit under conditions
where breakdown of the particulate layer does not occur.
Electrical breakdown of the particulate layer has been studied
extensively by Penney and Craig12 and Pottinger13 and can be in-
fluenced by many factors. Experimental measurements show that
18
-------
particulate layers experience electrical breakdown at average
electric field strengths across the layers of approximately 5 kV/
cm. Since it takes an electric field strength of approximately
30 kV/cm to cause electrical breakdown of air, the low breakdown
strength of particulate layers suggests that high localized fields
exist in the particulate layer and produce the breakdown of the
gas in the layer. The presence of dielectric or conducting par-
ticles can cause localized regions of high electric field which
constitute a negligible contribution to the average electric field
across the layer. The size distribution of the collected particles
also influences the electrical breakdown strength by changing the
volume of interstices.11* It has also been found that breakdown
strength varies with particulate resistivity with the higher
breakdown strength being associated with the higher resistivity.
METHODS FOR PREDICTING FLY ASH ELECTRICAL RESISTIVITY
Since the electrical resistivity has a pronounced effect on the
electrostatic collectability of fly ash, it is desirable to have
advanced knowledge regarding the magnitude of resistivity one might
expect from a given coal. Obviously the best source of this infor-
mation would be in situ resistivity measurements made during the
burning of the sub~ject coal in a commercial boiler. If the coal
has not been used commercially, one has the option of burning the
coal in a small scale pilot furnace and measuring the resistivity
in situ or in the laboratory, or one can utilize one of the
methods l 5 ' 16 ' l 7 for predicting fly ash resistivity.
These methods for predicting resistivity are based on corre-
lations that have been established between resistivity and fly ash
compositions for specific laboratory test conditions. The tech-
niques leave much to be desired. First, although coal ash analyses
can be used, the predictors are based on fly ash analyses. Second,
the predictors do not take into account the effect of environmental
variations. Presently, research18 is being conducted to develop
a predictive technique that will utilize the chemical composition
of a coal ash and the stoichiometrically calculated flue gas.
NONIDEAL EFFECTS
The nonidealities which exist in full-scale electrostatic
precipitators will reduce the ideal collection efficiency that
may be achieved with a given specific collection area. The non-
ideal effects of major importance are (1) nonuniform gas velocity
distribution, (2) gas sneakage, and (3) particle reentrainment.
These nonideal effects must be minimized by proper design and
optimization of a precipitator in order to avoid serious degra-
dation in performance.
Nonuniform Gas Velocity Distribution
Uniform, low-turbulence gas flow is essential for optimum pre-
cipitator performance. Nonuniform gas flow through a precipitator
19
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lowers performance due to two effects. First, due to the expo-
nential nature of the collection mechanism, it can be shown mathe-
matically that uneven treatment of the gas lowers collection
efficiency in the high velocity zones to an extent not compensated
for in the low velocity zones. Secondly, high velocity regions
near collection plates and in hopper areas can sweep particles
back into the main gas stream.
Although it is known that a poor gas velocity distribution
results in reduced collection efficiency, it is difficult to
formulate a mathematical description for gas flow quality. White19
discusses nonuniform gas flow and suggests corrective actions.
Preszler and Lajos20 assign a figure-of-merit based upon the
relative kinetic energy of the actual velocity distribution com-
pared to the kinetic energy of the average velocity. This figure-
of-merit provides 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. At the inlet
of a precipitator, a value of 0.25 or less for the ratio of the
standard deviation of the gas velocity distribution to the average
gas velocity is generally recommended. However, it must be noted
that the gas velocity distribution can change significantly
throughout the length of a precipitator and, depending upon the
design of the precipitator and the manner in which it is inter-
faced with other plant equipment, the gas velocity distribution
may improve or degrade.
Gas Sneakage
Gas sneakage occurs when gas bypasses the electrified regions
of an electrostatic precipitator by flowing through the hoppers
or through the high voltage insulation space. Gas sneakage can
be reduced by the use of frequent baffles which force the gas to
return to the main gas passages between the collection plates.
If there were no baffles, the percent gas sneakage would establish
the maximum possible collection efficiency because it would be
the percent volume having zero collection efficiency. With baffles,
the sneakage gas remixes with part of the main gas flow and then
another fraction of the main gas flow re-bypasses in the next un-
baffled region. The upper limit on collection efficiency due to
gas sneakage will therefore depend on the amount of sneakage gas
per baffled section, the degree of remixing, and the number of
baffled sections. Gas sneakage becomes increasingly important
for precipitators designed for high collection efficiencies where
only a small amount of gas sneakage per section can result in a
severe limitation on collection efficiency.
Particle Reentrainment
Particle reentrainment occurs when collected material reenters
the main gas stream. This can be caused by several different
effects and, in certain cases, can severely reduce the collection
20
-------
efficiency of a precipitator. Causes of particle reentrainment
include (1) rapping which propels collected material into the
interelectrode space, (2) the action of the flowing gas stream
on the collected particulate layer, (3) sweepage of material from
hoppers due to poor gas flow conditions, air inleakage into the
hoppers, or the boiling effect of rapped material falling into
the hoppers, and (4) excessive sparking which dislodges collected
material by electrical impulses and disruptions in the current
which is necessary to provide the electrical force which holds
the material to the collection plates.
Recent studies'*'21 have been made to determine the effect of
particle reentrainment on precipitator performance. In studies
where the rappers were not employed, real-time measurements of
outlet emissions at some installations showed that significant
reentrainment of mass was occurring due to factors other than
rapping. These same studies also showed that for high-efficiency,
full-scale precipitators approximately 30-85% of the outlet par-
ticulate emissions could be attributed to rapping reentrainment.
The results of these studies show that particle reentrainment,
especially rapping reentrainment, is a significant factor in
limiting precipitator performance.
21
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SECTION 6
DESCRIPTION OF THE MATHEMATICAL MODEL
IDEAL CALCULATION OF PARTICLE COLLECTION EFFICIENCY
The mathematical model of electrostatic precipitation is based
on the exponential-type relationship given in equation (2). This
equation was derived subject to several assumptions which have
been stated earlier. In order to use the equation it is necessary
to structure the mathematical model such that the assumptions are
not violated. As discussed earlier, the assumptions are never
completely satisfied in an industrial precipitator but they can
be approached closely.
The assumption that the particle migration velocity near the
collection surface is constant for all particles has the most sig-
nificant effect on the structure of the model. This assumption
implies two things:
(1) The particles are all of the same diameter.
(2) The electrical conditions are constant.
Because the particles entering a precipitator are not all of
the same diameter, the assumption of uniform particle diameters
creates a problem. This problem is dealt with in the model by
performing all calculations for single diameter particles and
then summing the results to determine the effect of the electro-
static precipitation process on the entire particle size distri-
bution.
Because the electrical conditions change along the length of
a precipitator, the assumption of constant electrical conditions
creates a problem. This problem is dealt with in the model by
dividing the precipitator into small length increments. These
length increments can be made small enough that the electrical
conditions remain essentially constant over the increment. The
number of particles of a given diameter which are collected in
the different length increments are summed to determine the col-
lection efficiency of particles of a single diameter over the
entire length of the precipitator.
In summary, a precipitator is divided into essentially many
small precipitators in series. Equation (2) is valid in each of
22
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these small precipitators for particles of a given diameter. A
large majority of the time used in the computer program which per-
forms the calculations in the model is devoted to calculating the
values of quantities needed to determine the migration velocity
for each particle diameter in each length increment.
The collection fraction, n- ., for the i-th particle size in
i/D
the j-th increment of length of the precipitator is mathematically
represented in the form
n. • = 1 - exp rwifj VQ> , (5)
-1- / J
where w. . (m/sec) is the migration velocity of the i-th particle
size in the j-th increment of length and A. (m ) is the collection
plate area in the j-th increment of length^
The collection fraction (fractional efficiency) T]-J_ for a given
particle size over the entire length of the precipitator is deter-
mined from
where N. .is the number of particles of the i-th particle size
1/3
per cubic meter of gas entering the j-th increment. The quantity
N. . can be written in the form
N. . = N. . exp (~Wi,j-l Aj-l/Q) , (7)
-1- / J -1- / J -1-
where N. , = N. , the number of particles of the i-th particle
size per cubic meter of gas in the inlet size distribution which
is expressed in the form of a histogram.
The overall mass collection efficiency n for the entire poly-
disperse aerosol is obtained from
n = V TUP, / (8)
where P^ is the percentage by mass of the i-th particle size in
the inlet size distribution.
In order to determine the migration velocities for use in
equation (5), the electrical conditions and the particle charging
23
-------
process in a precipitator must be modeled.
Calculation of Electrical Conditions
If the operating voltage and current are known and a voltage-
current curve is not desired, then the electric potential and
electric field distributions are determined by using a relaxation
technique described by Leutert and Bohlen.1'22 In this numerical
technique, the appropriate partial differential equations which
describe the electrodynamic field are solved simultaneously under
boundary conditions existing in a wire-plate geometry. The equa-
tions which must be solved are written in discrete form in two
dimensions as
+ 0 = - £- , and (9)
,AV Ap , AV Ap>
£op ,AV Abe AV
Ax Ay Ay '
where p = space charge density (coul/m3),
b = effective charge carrier mobility (m2/V-sec) ,
y = coordinate parallel to gas flow from wire-to-wire (m) ,
x = coordinate perpendicular to gas flow from wire-to-
plate (m) , and
£o = permittivity of free space (coul/N-m2).
In order to find the solutions for V and p from equations
(9) and (10) , the known boundary conditions on applied voltage
and current are held fixed while the space charge density at the
wire is adjusted until all the boundary conditions are satisfied.
For each choice of space charge density at the wire, the procedure
iterates on a grid of electric potential and space charge density
until convergence is obtained and then checks to see if the bound-
ary condition on the average current density at the plate is met
by using the expression
JP = (be Z Ppi Epi)/N '
1=1
24
-------
where J = average current density at the plate (A/m2),
b£ = effective charge carrier mobility (m2/V-sec),
p • = space charge densities for points on the plate
(coul/m3),
E . = electric field strengths for points on the plate
(V/m), and
N = number of grid points in the direction of gas flow.
If the boundary condition on the average current density at
the plate is not met, then the space charge density at the wire
is adjusted and the iteration procedure is repeated.
If the operating voltage and current are unknown or if a
voltage-current curve is desired, then the voltage-current char-
acteristic for a wire-plate geometry is determined by using the
technique described by McDonald et a_l_.2 3 In this technique, the
electric potential and electric fTeld distributions are deter-
mined for each point on the voltage-current curve„ Equations
(9) and (10) are solved simultaneously using the same mathematical
procedure employed by Leutert and Bohlen but an alternate set of
boundary conditions is imposed. The space charge density in the
region of ionization near the discharge electrode is calculated
from an arbitrarily chosen value of average current density at
the plate. The space charge density near the wire and the aver-
age current density at the plate provide boundary conditions
which are held fixed while the electric potential at the wire is
adjusted until simultaneous solutions are found to equations (9)
and (10) which satisfy all the boundary conditions„
Calculation of Particle Charge
Particle charge is calculated from a unipolar, ionic-charging
theory formulated by Smith and McDonald.24 In this theory, par-
ticle charge is predicted as a function of particle diameter,
exposure time, and electrical conditions. The charging equation
is derived based on concepts from kinetic theory and determines
the charging rate in terms of the probability of collisions be-
tween particles and ions. The theory accounts simultaneously for
the effects of field and thermal charging and accounts for the
effects of the applied electric field on the thermal charging
process. According to this theory, the charging rate is given by
25
-------
dt
Noebq
° s
Tra2vN0e c"/2 r/qe(r0-a)
2 J0 c^ \4TreokTaro
DO L
[3ar02 - r03(K + 2) + a3(K - l)]eE0cos9^
-nrm ' sin 6de
exp (-qe/4Tre0akT) , (12)
where qg = 47re0E0a2 (1 + 2 ^) , (13)
90 = arccos (q/q ) , (14)
o
and q = instantaneous charge on the particle (coul) ,
q - saturation charge due to field charging (coul) ,
S
6 = azimuthal angle in a spherical coordinate system with
origin at the center of the particle (radians) ,
80 = maximum azimuthal angle for which electric field lines
enter the particle (radians) ,
No = free ion density (m~3) ,
e = electronic charge (coul) ,
eo - permittivity of free space (coul/V-m) ,
E0 = average electric field between the electrodes (V/m) ,
b = ion mobility (m2/V-sec) ,
%
v = mean thermal speed of ions (m/sec) ,
a = particle radius (m) ,
k = Boltzmann's constant (J/°K) ,
T = absolute temperature (°K),
t - time (sec) ,
K = dielectric constant of the particle, and
26
-------
r0 = radial distance along 0 at which the radial component
of the total electric field is zero (m).
For large particles and high applied electric fields, the theory
predicts essentially the same charging rate as the classical field
charging equation. For low applied electric fields, the charging
equation reduces to the classical thermal equation.
Equation (12) can be solved on a computer by simple numerical
techniques. The integral on the right-hand side of equation (12)
is evaluated using Simpson's Rule and the charge as a function
of time is determined by using the quartic Runge-Kutta method.
In cases where the use of computer time is an important con-
sideration, the computer model for electrostatic precipitation
allows for considerable savings in computer time by providing the
option of using an analytical expression for charge as a function
of time. This expression is given by
e /
e2avNo
:-t.
+ q<
Nobe \/t-t.\ +
Nobe \/tf~ti
- 1
(15)
where q_ = charge predicted from classical diffusion charging
theory (coul),
q = charge predicted from classical field charging theory
(coul),
initial time for charging under a fixed set of con-
ditions (sec),
final time for charging under a fixed set of con-
ditions (sec) ,
t.
27
-------
q. = charge on the particle at t. (coul) ,
and all other symbols are as defined previously.
This equation represents the sum of the charges from clas-
sical field and diffusion charging theories. In principle, the
sum of the charging rates should be added to be physically meaning-
ful; however, fortuitously, equation (15) yields a reasonable pre-
diction of particle charge for particles in the size range 0.09-
1.4 ym in diameter.25 The forms of q and q used in equation
(15) reflect the fact that N0 and E0 change along the length of
a precipitator and, in the model, are assumed to remain fixed
only over each incremental length.
Calculation of Space Charge Effects
In the calculation of the electrical conditions, it is as--
sumed that the motion of all the charge carriers can, on the
average, be described by a single "effective mobility". The
presence of particles in the flue gas will cause a reduction in
the "effective mobility" because the particles, which acquire
charge from the ions and are much less mobile than ions, must
carry part of the total current.1'26'27 When the electrical con-
ditions are calculated by using measured values of applied voltage
and current density, the "effective mobility" is determined from*'26
200 j +
(16)
where bg = effective mobility for ions and particles (m2/V-sec) ,
b' = effective ion mobility (m2/V-sec) ,
Jt = total current density at the plate (A/m2 ) , and
jp = particulate current density at the plate (A/m2) .
If the electrical conditions are calculated by generating a
voltage-current curve, then the model employs a different method
for determining the effects of space charge. Since this method
has not been published prior to this writing, it is discussed in
Appendix A in more detail than the other calculations which are
presented. In this part of the text, this method will be dis-
cussed only briefly in order to acquaint the reader with the
basic concepts involved in the method.
In this method, the precipitator is divided into successive
length increments which are equal to the wire-to-wire spacing.
Each of these increments is divided into several subincrements .
The first calculation in the procedure involves the determination
28
-------
of a clean-gas, voltage-current curve which terminates at some
specified value of applied voltage. At the specified applied
voltage, the average electric field and ion density are calcu-
lated in each subincrement. This allows for the nonuniformity
of the electric field and current density distributions to be
taken into account.
As initially uncharged particles enter and proceed through
the precipitator, the mechanisms of particle charging and par-
ticle collection are considered in each subincrement„ In each
subincrement, the average ion density, average particulate den-
sity, weighted particulate mobility, and effective mobility due
to both ions and particles are determined. At the end of each
increment, the effective mobilities for the subincrements are
averaged in order to obtain an average effective mobility for
the increment. Then, for the specified value of applied voltage,
the average effective mobility is used to determine the reduced
current for the increment by either calculating a new voltage-
current curve or using an approximation which is discussed in
Appendix A.
In its present state of development, this method provides
good estimates of reduced current due to the presence of par-
ticles. However, it does not have the capability of predicting
the redistribution of the electric field due to the presence of
particles. Work is going on at the present time to improve the
model in this respect. This work involves the use of an iter-
ation procedure over each increment in order to obtain self-
consistency.
METHODS FOR ESTIMATING NONIDEAL EFFECTS
Since the model is structured around an exponential-type
equation for individual particle sizes, it is convenient to rep-
resent the effect of the nonidealities in the model as correction
factors which apply to the exponential argument. These correc-
tion factors are used as divisors for the ideally-calculated
effective migration velocities.
Since four types of migration velocities will be referred
to in the following sections, it is important to define the
terminology which will be used. The migration velocity w is a
quantity which is calculated from fundamental principles and
applies only to a given length increment as used in the model.
This quantity represents the actual drift velocity of a particle
toward the collection electrode in the region near the collection
electrode. The effective or length-averaged migration velocity
w is a quantity which is calculated from fundamental principles
but it applies to the entire length of the precipitator0 This
quantity is obtained by replacing w in equation (2) by w and
determining a single value of w which is necessary to produce
29
-------
the same collection efficiency over the entire length of the pre-
cipitator that is obtained from the values of w. The apparent
effective migration velocity w^ is a quantity which is obtained
from making an empirical correction (or corrections) to the effec-
tive migration velocity w . This quantity bears no relationship
to the actual migration velocities in the region of space adjacent
to the collection electrode and has no physical interpretation.
The quantities w, w , and w" apply to particles of a given diam-
eter. The precipitation rate parameter w is a quantity which
provides a measure of how well the entire mass which enters the
precipitator will be collected. This quantity is obtained by
replacing w in equation (2) by w and determining a single
value of w which is necessary to produce the same overall mass
collection efficiency that is obtained from the collection effi-
ciencies for all particle diameters, as determined by the values
of w or w'', and the inlet particle size distribution.
Ci "
Calculation of Effect of Nonuniform Velocity Distribution
It is possible to develop an approach to estimating the de-
gradation of performance due to a nonuniform velocity distribution
based upon the velocity distribution, the ideal collection effi-
ciencies, and the exponential-type collection equation.1 In
this approach, it is assumed that equation (2) applies to each
particle size with a known effective migration velocity and that
the specific collecting area and size of the precipitator are
fixed.
For any practical velocity distribution and efficiency, the
mean penetration obtained by summation over the point values of
velocity will be higher than the penetration calculated from the
average velocity. If an effective migration velocity for a given
particle size is calculated based upon the mean penetration and
equation (2), the resulting effective migration velocity will
have a value lower than the value necessary to obtain the same
mean penetration from a summation of point values of penetration.
The ratio of the effective migration velocity determined by the
summation of point values of penetration to that determined by
equation (2) is a numerical measure of the performance degradation
caused by a nonuniform 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 effective migration velocity obtained from equation (2).
Whether the correction factor obtained from the above pro-
cedure correlates reasonably well with statistical measures of
30
-------
velocity nonuniformity is yet to be established. A limited
number of traverse calculations which have been performed seem
to indicate a correlation between the correction factor and the
normalized standard deviation of the velocity traverse. Based
upon a pilot plant study,20 the following empirical relationship
between the correction factors F., the normalized standard de-
viation of the velocity distribution a , and the ideal collection
fractions n. for the i-th particle size under consideration has
been obtained:1
F± = 1 + 0.766 n.j_a 1>786 + 0.0755 a In (l/l-r^) • (17)
In simulating the performance of a particular precipitator, the
preferred procedure would be to obtain the relationship [F. =
F. (n • f cr ) ] between F., n . , and a for the conditions to be
simulated from a velocity traverse at the entrance to the pre-
cipitator. If this cannot be done, equation (17) can be used,
but only in the sense of obtaining a rough estimate of the effects
of a given nonuniform velocity distribution.
Calculation of Effect of Gas Sneakage
If the simplifying assumption is made that perfect mixing
occurs following each baffled section, then an expression for the
p
penetration Nc of a given particle size from the last baffled
O
section which is corrected for gas sneakage can be derived1 in
the form
PNS = [s + (i-s) (i-ni)1/Ns]Ns , (18)
where S is the fractional amount of gas sneakage per baffled sec-
tion and N is the number of baffled sections. Estimations based
S
on equation (18) indicate that, 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 per-
centage of sneakage, even a large number of baffled sections
fail to help significantly.
Gas sneakage factors B. can be defined in the form of divi-
sors for the effective, or length-averaged, migration velocities
in the exponential argument of equation (2). The factors B. are
obtained by taking the ratio of the effective migration velocities
w . under ideal conditions to the apparent effective migration
velocities w" . under conditions of gas sneakage so that
e, i
31
-------
in (i-n-) ln d-n-)
= - ^- = • - - - ™— , (19)
In PN Ns in [S + (1-S) (l-ni)VJNs]
o
where the subscript i refers to the different particle diameters.
The foregoing estimation of the effects of gas 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 bypass zones will not be uniform
and constant. Equation (18) has been formulated to help in de-
signing and analyzing precipitators by establishing the order of
magnitude of the problem „ Considerable experimental data will
be required in order to evaluate the method and to establish
numerical values of actual sneakage rates.
Calculation of Effect of Reentrainment Without Rapping
Although it is difficult to quantify the complex mechanisms
associated with particle reentrainment due to (1) the action of
the flowing gas stream on the collected particulate layer, (2)
sweepage of particles from hoppers caused by poor gas flow con-
ditions or air inleakage into the hoppers, and (3) excessive
sparking, the effect of these nonideal conditions on precipitator
performance can be estimated if some simplifying assumptions are
made. If it is assumed that a fixed fraction of the collected
material of a given particle size is reentrained and that the
fraction does not vary with length through the precipitator, an
expression can be derived which is identical in form to that
obtained for gas sneakage:1
PNR = [R + (l-R)l-ni)1/NR] NR , (20)
p
where NR is the penetration of a given particle size corrected
for reentrainment, R is the fraction of material reentrained, and
NR is the number of stages over which reentrainment is assumed to
occur.
Since equations (18) and (20) are of the same form, the ef-
fect of particle reentrainment without rapping can be expected to
be similar to the effect of gas sneakage, provided that a constant
fraction of the collected material is reentrained in each stage.
It is doubtful that such a condition exists, since the gas flow
pattern changes throughout the precipitator and different holding
forces and spark rates exist in different electrical sections.
However, until detailed studies are made to quantify the losses
in collection efficiency as a function of particle size for these
types of reentrainment, equation (20) provides a means of esti-
mating the effect of particle reentrainment without rapping on
precipitator performance.
32
-------
Since the equation which is obtained for calculating the
correction factors for particle reentrainment without rapping is
of the same form as that obtained for calculating the correction
factors for gas sneakage, only equation (19) is used in the model.
Thus, only a value of S is used as input data for the model. How-
ever, the value of S represents the combined effects of the frac-
tional amount of gas sneakage per baffled section and the frac-
tional amount of collected material which is reentrained per
baffled section without rapping.
No-Rap Calculations
The combined nonideal effects of nonuniform gas velocity dis-
tribution, gas sneakage, and particle reentrainment without rapping
are taken into account by reducing the ideally calculated effective
migration velocities w . by the correction factors F. and B. .
G f 1 1 -L
Apparent effective migration velocities w"* . are determined from
e , i
w: . = - , (2D
where the subscript i refers to the different particle diameters.
Corrected fractional collection efficiencies are calculated using
equation (2) and the values of the w** . .
e , i
The apparent effective migration velocities and corresponding
collection efficiencies obtained from equation (21) may be referred
to as "no-rap" migration velocities and collection efficiencies.
These quantities are of practical interest because they can be
measured by turning the rappers off, whereas ideal quantities can
never be truly measured.
Calculation of Effect of Rapping Reentrainment
As part of a program sponsored by the Electric Power Research
Institute, an approach to representing losses in collection effi-
ciency due to rapping reentrainment has been developed based upon
studies performed on six different full-scale precipitators col-
lecting fly ash.1* In these studies, outlet mass loadings and
particle size distributions were measured both with rapping losses
and without rapping losses. Based on these data, outlet mass
loadings and particle size distributions which can be attributed
to rapping were obtained.
The rapping emissions obtained from the measurements on the
six precipitators are graphed in Figure 4 as a function of the
amount of dust calculated to have been removed by the last elec-
trical section. The dust removal in the last electrical section
was approximated by
n "/section = l-exp(-X"/N ) , (22)
£j
33
-------
100 -,
y2 = .618X'894
o
00
Q
"Si
E
CO
00
Q.
O.
cc
10
= 0.155X
-905
0.1
10 100
CALCULATED MASS REMOVAL BY LAST FIELD
mg/DSCM
Figure 4. Measured rapping emissions versus calculated paniculate
removal by last electrical section. These curves are a
result of work sponsored by the Electric Power Research
Institute.
34
-------
where X' = -ln(l-ru), (23)
rio = overall mass collection fraction determined from mass
train measurements under normal operating conditions,
and
N = number of electrical sections in series.
£j
These data suggest a correlation between rapping losses and
particulate collection rate in the last electrical section. Data
for the two hot-side installations (4 and 6) which were tested
show higher rapping losses than for the cold-side units. This
would be expected due to reduced dust adhesivity at higher tem-
peratures. Data 2a and 2b are for a cold-side unit operating at
normal and approximately one-half normal current density, re-
spectively. The decrease in current density at installation 2
resulted in a significant increase in rapping emissions due to
the increased mass collected in the last field and smaller elec-
trical holding force for the same rapping intensity.
The simple exponential relationships
y1 = (Ool55)X°'905 (24)
and
(25)
can be used for interpolation purposes in determining the rapping
emissions (mg/DSCM) for a given calculated mass removed by the
last field (mg/DSCM) for cold- and hot-side precipitators , re-
spectively. In constructing Figure 4, the calculated mass re-
moved in the last field was determined by using the measured
overall mass collection efficiency during normal operation of the
precipitator . This was done because complete traverses were made
by the mass trains during the normal tests whereas this was not
the case for the measurements made during the no-rap tests. In
principle, the no-rap efficiencies should be used to calculate
the mass removed in the last field and this is what is done in
the mathematical model. Obviously, the limited amount of data
obtained thus far is not sufficient to validate in general the
approach presented here. However, this approach gives reason-
able agreement with the existing data and offers a quantitative
method for estimating rapping losses.
The apparent size distribution of emissions attributable to
rapping at each installation was obtained by subtracting the
cumulative distributions during non-rapping periods from those
35
-------
with rappers in operation, and dividing by the total emissions
(based on impactor measurements) resulting from rapping in order
to obtain a cumulative percent distribution. Figure 5 contains
the results of these calculations. Although the data indicate
considerable scatter, an average particle size distribution has
been constructed in Figure 6 for use in modeling rapping puffs.
In the mathematical model, the data in Figure 6 are approximated
by a log-normal distribution with a mass median diameter of 6.0
ym and a geometric standard deviation of 2.5.
In summary, the mathematical model determines a "rapping
puff" by using either equation (24) or (25) to obtain the outlet
mass loading due to rapping and by using a log-normal approxi-
mation to the data in Figure 6 to represent the particle size
distribution of the outlet mass loading due to rapping. This
"rapping puff" is added to the "no-rap" outlet emissions to obtain
the total outlet emissions as a function of mass loading and par-
ticle size distribution. Then, the model generates migration
velocity, collection efficiency, penetration, and AM/AlogD (rate
of change of mass over a given size interval) as a function of
particle diameter for the "no-rap", "rapping puff" and "no-rap"
plus "rapping puff" outlet emissions.
Although rapping is an important part of the electrostatic
precipitation process, the present version of the model does not
take into account the temporal and dynamic nature of the rapping
process. The time-dependent aspects of the rapping process are
of significance because different electrical sections are rapped
at different time intervals and the thickness of the collected
particulate layer changes with time. The dynamic aspects of the
rapping process are of significance because (1) a suitable me-
chanical force must be applied to a collection electrode in order
to remove the collected particulate layer, (2) the force which is
necessary to remove the collected particulate layer from the col-
lection electrode depends on such variables as the electrical
forces in the layer, the cohesiveness and adhesiveness, etc.,
and (3) the reentrained particles are recharged and re-collected
as the gas flow carries them downstream. Although the empirical
procedure employed in the present version of the model represents
a useful interim technique for estimating the effects due to
rapping reentrainment in precipitators, it is important that
models be developed in the future to describe the temporal and
dynamic aspects of the rapping process.
EMPIRICAL CORRECTIONS TO NO-RAP MIGRATION VELOCITIES
Comparisons of measured apparent effective migration velo-
cities for full-scale precipitators under "no-rap" conditions with
those predicted by the model indicate that the field-measured
values exceed the theoretically projected values (in the absence
of back corona, excessive sparking, or severe mechanical problems)
in the smaller size range. Based on these comparisons, a size-
dependent correction factor has been constructed and incorporated
36
-------
E
a.
of
LLI
i—
LLI
S
<
Q
^u
10
9
8
7
5
4
3
2
1
1 1
0 4
• 6
A 2a
"A3
-05
~ • 1
-
••
a
a
i P
0.5 1 2
ii i i i i i i i i i i i i
• 0 ADA
0» DA A
-
—
0 ••DA *
-
• O»AA
-
—
• a o A*
0 • A A
0 «A A
A*^ I I I I I I I I | II I
5 10 20 30 40 50 60 70 80 90 95 98 99 99.89
%LESS THAN
Figure 5. Apparent rapping puff size distribution for six full-scale
precipitators. These data are a result of work sponsored
by the Electric Power Research Institute.
37
-------
20
E
3.
LU
10
9
8
7
6
5
Experimental
• Log-normal approximation
for MMD = 6.0 ;um,
ap = 2.5
I 1 I
1
J
I
10 20 30 40 50 60 70 80
%LESS THAN
90 95
Figure 6. A verage rapping puff size distribution for six full-scale
precipitators. These data are a result of work sponsored
by the Electric Power Research Institute.
-------
into the model. "* This correction factor is shown in Figure 70
The empirical correction factor accounts for those effects
which enhance particle collection efficiency but are not included
in the present model. These effects include particle charging
near corona wires, particle concentration gradients, and flow
field phenomena. In future work which is planned, efforts will
be made to develop theoretical relationships to describe the
above effects and to incorporate them into a more comprehensive
model for electrostatic precipitation.
ESTIMATION PROCEDURE FOR CALCULATING PARTICLE COLLECTION
EFFICIENCIES
The mathematical model for electrostatic precipitation allows
for the use of estimation procedures for calculating particle
collection efficiencies. Use of these procedures results in con-
siderable savings in computer time since involved numerical tech-
niques are not extensively employed. These procedures can be used
to advantage when only gross trends in precipitator performance
are required or when an estimating technique is desired in order
to approximate the specific collection area required for a chosen
overall mass collection efficiency so that a starting point for
the more rigorous calculation can be easily obtained.
In the case where the operating applied voltage and current
are known, particle charge, average electric field at the plate,
and space charge effects are only estimated. Particle charge is
calculated by using equation (15). The average electric field
at the plate is calculated by dividing the applied voltage by
the wire-to-plate spacing and scaling this value down by a factor
of 1.75o This method of determining the average electric field
at the plate is based on the examination of the results from
model simulations of several full-scale precipitators which were
collecting fly ash. Space charge effects are determined by re-
ducing the free ion density and effective charge carrier mobility
in the same procedure which leads to equation (16)0
In the case where the operating applied voltage and current
are not known, a voltage-current curve must be generated up to
some specified operating applied voltage0 The voltage-current
calculation also determines the average electric field at the
plate which is used in the estimation procedure. Particle charge
is calculated by using equation (15). Space charge effects are
determined by applying the new procedure discussed earlier and
in Appendix A.
It must be emphasized that these procedures are not expected
to give results which will always be reasonable estimates. For
any given set of conditions, these procedures may lead to pre-
dictions of precipitator performance which are in considerable
error. However, in most cases, they should yield reasonable
39
-------
_ 3
I—I—I I I II
0.2
0.8 1.0
DIAMETER,
Figure 7. Empirical correction factors for the "no-rap" migration
velocities calculated from the mathematical model. This
work was sponsored by the Electric Power Research
Institute.
40
-------
estimates of precipitator performance and their judicious use
can lead to considerable savings in computer time.
41
-------
SECTION 7
COMPUTER PROGRAMING OF THE MATHEMATICAL MODEL
DESCRIPTION OF THE COMPUTER PROGRAM
A computer program has been written in Fortran IV language
in order to perform the mathematical operations associated with
the model of electrostatic precipitation discussed in Section 60
Although the program has been developed using a Digital Equipment
Corporation, PDF 15/76 computer, efforts have been made to make
the program sufficiently generalized so that it can be easily
implemented on other computers which have a Fortran compiler.
Due to the lack of sufficient storage capacity on the POP 15/76
computer, the program contains some duplication because the use
of arrays for storing the values of certain variables is avoided
and the values of these variables are recalculated each time
they are needed in the program. Appendix B contains a list of
the symbols used in the program along with their definitions and
Appendix C contains a listing of the entire program. The pro-
gram consists of a main program and 20 subroutines„ Excluding
job control language (JCL) cards, the program card deck contains
2,240 cards.
The following is a sequential list of the major operations
which are performed by the computer program in order to determine
fractional collection efficiencies and overall mass collection
efficiency.
1. Data which are necessary to characterize precipitator
performance are read into the main program.
20 If the inlet size distribution is known, it is read into
the main program in the form of a histogram and it is "fit" to a
log-normal distribution in subroutine LNFITo Alternatively,
parameters characteristic of a log-normal distribution can be
read into the main program and a histogram is constructed in sub-
routine LNDIST in order to represent the inlet size distribution.
30 The number of particles in each size band of the inlet
size distribution is calculated.
4. The precipitator is divided into specified incremental
lengths.
42
-------
5. If the operating applied voltage and current are not
known, then subroutine EFLD2 is used to generate a clean-plate,
clean-gas, voltage-current curve up to a specified value of
operating applied voltage, to determine the free ion densities
and charging fields for particle charging, and to calculate the
electric field at the plate. If the operating applied voltage
and current density are known, then subroutine SPCHG1 is used to
calculate the amount of material removed per increment and the
"particulate space charge" in each increment based on an esti-
mated overall mass collection efficiency, to determine an effec-
tive charge carrier mobility, and to establish a reduced free
ion density in each increment for use in determining particle
charge.
6. If the operating applied voltage and current are known,
then the average charging field is calculated using the applied
voltage and the wire-to-plate spacing and the electric field at
the plate is calculated in subroutine EFLDl.
7. The charge on each particle size at the end of each in-
crement or subincrement of length is calculated in either CHARGN
or by using equation (15), whichever is specified. In order to
save computer time, the program contains a procedure which by-
passes the charge calculation for a given particle size whenever
the charge on that size does not change by more than 0.5% in two
successive length increments in the same electrical section. Also,
if the charge calculation for a given particle size has been by-
passed in the last increment of a given electrical section and
the applied voltage in this section is equal to or greater than
that in the next section, then the charge calculation for this
particle size will be bypassed in each increment in the next
section.
8. If the operating applied voltage and current are unknown,
then subroutine SPCHG2 is used to determine the "particulate space
charge", the effective charge carrier mobility, and the operating
current density.
9. A migration velocity for each particle size is calculated
at the end of each length increment using equation (1).
10. The number of particles removed in each size band after
each length increment of travel is calculated using equation (2).
11. After the required calculations have been performed in
all length increments, an overall mass collection efficiency is
calculated. If the operating applied voltage and current are
known, then the calculated overall mass collection efficiency is
compared 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 mass collection efficiency. Usually, only one iteration
43
-------
is required. If the calculation of the overall mass collection
efficiency has been based on a generated voltage-current curve,
then no iteration over incremental lengths is performed.
12. After the overall mass collection efficiency has been ob-
tained, an "effective" migration velocity is calculated for each
size band and a precipitation rate parameter is computed based on
the overall mass collection efficiency and equation (2).
The above operations complete the calculation of ideal per-
formance that would be expected under a given set of input con-
ditions and based on those physical mechanisms which are included
in the model. In the following operations which are performed in
subroutine ADJUST, corrections are made to the ideal projections
by operating on the "effective" migration velocity for each par-
ticle size in order to account for unmodeled and nonideal effects.
13. For a given value of normalized gas velocity standard
deviation, a correction factor is calculated for each "effective"
migration velocity using the ideal efficiency for a given particle
size and equation (17).
14. Using assumed values of number of stages and the percent
loss per stage from reentrainment without rapping and/or gas
sneakage, a correction factor is calculated for each "effective"
migration velocity using the ideal efficiency for a given par-
ticle size and equation (19) .
15. An "apparent" effective, no-rap migration velocity is
obtained for each particle size by dividing the ideal values by
the product of the two correction factors described above and a
no-rap collection efficiency is calculated for each particle size
using equation (2) .
16. Using the correction factors given in Figure 7, sub-
routine WADJST corrects the "apparent" effective, no-rap migration
velocities in order to account for unmodeled effects and "adjusted"
no-rap efficiencies are determined. An "adjusted" no-rap overall
mass collection efficiency and precipitation rate parameter are
calculated.
17. Losses in collection efficiency due to rapping reentrain-
ment are obtained by reducing the mass collected in each size
band under "adjusted" no-rap conditions according to either equa-
tion (24) or (25) and Figure 6. A collection efficiency and
migration velocity with rapping are calculated for each particle
size. An overall mass collection efficiency and precipitation
rate parameter which account for losses due to rapping are calcu-
lated.
18. No-rap and no-rap + rap outlet size distributions are
determined and outlet emissions are characterized by calculation
44
-------
of AM/AlogD for each size band for the "rapping puff" and no-rap
and no-rap + rap conditions.
19. No-rap and no-rap + rap outlet size distributions are
"fit" to a log-normal distribution.
20o All input data and relevant parameters which have been
calculated are printed.
In Figure 8, a simplified flow chart for the main program
is given0 This flow chart shows the major operations and logic
branches and all subroutine callings0 The input and output data
for the computer program, along with the various uses of the pro-
gram, will be discussed in detail in Volume 2 of this report. In
the following subsection, the subroutines which are called by
the main program will be discussed in detail0
DESCRIPTIONS OF THE SUBROUTINES
Subroutine SPCHG1
This subroutine determines the effect of "particulate space
charge" in each increment of length by using the procedure leading
to equation (16) in order to calculate an "effective" charge
carrier mobility and average reduced ion density for particle
charging. Figure 9 shows a detailed flow chart for this sub-
routine. All information which is transmitted between the main
program and this subroutine is transferred through calling argu-
ments. The following is a sequential list of the calling argu-
ments and their descriptions.
SW - Estimated sum of material removed in successive incre-
ments of the ESP (kg/m3).
ROVRI - Ratio of total charge density to ionic charge density
in a given increment of the ESP. Initialized to 10.0
to start procedure.
OROVRI - Ratio of total charge density to ionic charge density
in previous increment of ESP. Initialized to 2000 to
start procedure.
XS - Computed value of exponential argument in equation (2)
for the estimated overall efficiency.
ETAPF - Computed estimated overall collection fraction per
given length increment.
DW(I) - Computed estimated amount of material removed in a
given length increment (kg/m3).
QSAT(J) - Saturation charge on a given particle size (coul).
45
-------
START LOOP OVER
INCREMENTAL LENGTHS
VISAME = 1
AND NSECT>1
VISAME = 1
AND NDSET>1
(£>»<
START CONVERGENCE LOOP
ON OVERALL EFFICIENCY
[NO
ITER + 1
I
CALL
SPCHG1
YES
CALC. NO. OF PARTICLES
IN EACH SIZE BAND
START LOOP OVER SUB
INCREMENTAL LENGTHS
CALL PRTINP
START LOOP OVER
PARTICLE SIZES
A
YES
NO
PRINT OUT ALL
INPUT DATA (IN PRTINP).
Figure 8. Simplified flow chart for the entire program (Sheet 1 of 4).
46
-------
CALC. IDEAL PARTICLE MIGRATION
VELOCITY AND EFFICIENCY
CALC. NO. OF PARTICLES REMOVED AND
SUM OF WEIGHT REMOVED
CALC. NO. OF PARTICLES ENTERING
NEXT INCREMENT
END OF LOOP OVER
PARTICLE SIZES
CALC. TOTAL WEIGHT
COLLECTED AND MMD
PRINT SECTIONALIZED
DATA ( IN PRTINC
PRINT INCREMENTAL
DATA (IN PRTINC)
END OF LOOP OVER
INCREMENTAL LENGTHS
A
CALC. OVERALL MASS
COLLECTION EFFICIENCY
PRINT OUT RESULTS OF
CHARGE CALCS (IN PRTCHG)
PRINT OUT PARTICLE SIZE RANGE
STATISTICS (IN ADJUST)
PRINT OUT UNADJUSTED MIGRATION
VELOCITIES AND EFFICIENCIES, AND
DISCRETE OUTLET MASS LOADINGS
(IN ADJUST)
CALL PRTSUM
(IN ADJUST)
PRINT OUT SUMMARY
TABLE (IN PRTSUM)
END OF MAIN
PROGRAM
Figure 8. Simp/if fed flow chart for
49
the entire program (Sheet 4 of 4).
-------
START SUBROUTINE )
I
REAL: LINC
DIMENSION: DW, QSAT,
XMO, I.SFCT
CALC. VALUE OF EXPONENT IN
EQ. (2) FOR EST. EFFICIENCY (XS)
CALC. EST. EFFICIENCY PER
LENGTH INCREMENT (ETAPF)
CALC. AMOUNT OF MATERIAL
REMOVED IN INCREMENT (DW(D)
SUM WEIGHT REMOVED (SW)
A
c
START LOOP OVER
PARTICLE SIZES
CALC. SUM OF CHARGE DENSITIES
IN INCREMENT (SUM)
END LOOP OVbH
PARTICLE SIZES
)
CALC. RATIO OF PART. CURRENT TO
TOTAL CURRENT X200 (ZC)
CALC. RATIO OF TOTAL CHARGE
DENSITY TO IONIC CHARGE DENSITY (ROVRI)
CALC. REDUCED FREE
ION DENSITY
-------
XNO(J) - Number of particles per unit volume of gas of a given
particle size entering a given length increment (#/m3).
W - Weight of material per unit time (mass flux) in a given
length increment (kg/sec).
LSECT(K) - Number of length increments in a given electrical sec-
tion.
TC - Total current in a given electrical section (A).
VG - Gas volume flow rate in a given electrical section
(m3/sec).
ETAO - Estimated overall mass efficiency of ESP (%).
FID - Computed free ion density in a given electrical section
(#/m3).
AFID - Computed reduced free ion density for particle charging
in a given electrical section (#/m3).
AVGFID - Reduced free ion density (#/cm3).
XCD - Average current density at the plate in a given elec-
trical section (nA/cm2).
U - Ion mobility in a given electrical section (m2/V-sec).
UEQ - Effective charge carrier mobility in a given length in-
crement (m2/V-sec). Restricted to a lower limit of
1 x 10"4 m2/V-sec in main program to avoid convergence
difficulties when used in subroutine EFLDl.
I - Index specifying the given length increment. Can not
exceed a value of 50.
NSECT - Indicator specifying the given electrical section of
the ESP. Can not exceed a value of 10.
LING - Length of each increment in a given electrical section
(m) .
PL - Total electrical length of ESP (m).
CD - Average current density at the plate in a given elec-
trical section (A/m2).
E - Electronic charge (coul).
ERAVG - Average electric field between the wire and plate (V/m),
51
-------
NS - Number of particle size bands in size distribution
histogram. Can not exceed a value of 20.
XPI - Computed estimated overall collection efficiency per
given length increment (%) .
Of the above variables, the values of the following must be
provided by the main program: QSAT(J), XNO(J), W, LSECT(K), TC,
VG, ETAO, U, I, NSECT, LING, PL, CD, E, ERAVG, and NS. The values
of the following variables are determined in the subroutine: SW,
ROVRI, OROVRI, XS, ETAPF, DW, FID, AFID, AVGFID, XCD, UEQ, and XPI.
In the above arrays, I, J, and K can not exceed 45, 20, and 10,
respectively. The restrictions on I, J, and K limit the number
of length increments, the number of particle size bands, and the
number of electrical sections, respectively.
Subroutine SPCHG2
This subroutine determines the effect of "particulate space
charge" in each subincrement of length by using the new procedure
discussed earlier in this report and in Appendix A to calculate an
"effective" charge carrier mobility. Figure 10 shows a detailed
flow chart for this subroutine. Information which is transmitted
between the main program and this subroutine is transferred through
calling arguments and block common statements. The following is
a sequential list of the calling arguments and their descriptions.
NS - Number of particle size bands in size distribution
histogram. Can not exceed a value of 20.
XNO(J) - Number of particles per unit volume of gas of a given
particle size entering a given length increment (#/m3).
Gas viscosity in a given electrical section (kg/m-sec)
Radius of a given particle size (m).
VIS
RAD(J)
LING - Length of each increment in a given electrical section
E -
U -
ERAVG -
DNSION -
DELTNP -
SUMMOB -
(m) .
Electronic charge (coul).
Ion mobility in a given electrical section (m2/V-sec).
Average electric field between the wire and plate (V/m) ,
Ion density in the absence of particles (#/m3).
Number density of charges transferred from ions to par-
ticles in a given subincrement of length (#/m3).
Weighted summation of particle mobilities (m2/V-sec/m3) ,
52
-------
f START SUBROUTINE J
F
}EAL: LINC, ND
DIMENSION: XNO, RAD,
CCF, OLDQ, Q
' A
CALC. SUM OF PARTICLES PER
VOLUME OF GAS (PNUM)
CALC. CHARGE ACQUIRED BY PARTICLES IN A
GIVEN INCREMENTAL LENGTH (DIFF)
BLOCK COMMON: XDC /
BLOCK COMMON: EAVG, CHFID
BLOCK COMMON: NREAD, NPRNT
YES/
\
NO
~v
CALC. VALUE OF EXPONENT IN EQ. (2)
FOR EST. EFFICIENCY (XS)
CALC. EST. EFFICIENCY PER
LENGTH INCREMENT (ETAPF)
,_
DELTNP 0.
SUMMOB = 0.
PNUM = 0.
RHOP = 0.
/" START LOOP OVER "\
\^ PARTICLE SIZES J
CALC. PARTICULATE CHARGE
DENSITY (TCHRG)
A
CALC. SUM OF PARTICLE
CHARGE DENSITIES (RHOP)
CALC. WEIGHTED SUM OF PARTICLE
MOBILITIES (SUMMOB)
NO
OF\YES
/ 1
RECALCULATE DIFF
, 1
CALC. SUM OF CHARGE DENSITIES
TRANSFERRED TO PARTICLES IN A
GIVEN INCREMENTAL
LENGTH (DELTNP)
(END LOOP OVER A
PARTICLE SIZES )
CALC. WEIGHTED PARTICULATE
MOBILITY (PMOB)
CALC. TOTAL CHARGE NUMBER
DENSITY OF PARTICLES (TDNSP)
ESTABLISH DENSITY OF IONS
WITHOUT PARTICLES PRESENT (DNSION)
CALC. TOTAL CHARGE NUMBER DENSITY
TRANSFERRED TO PARTICLES (DELTNP)
CALC. REDUCED FREE
ION DENSITY (RDNSI)
Figure 10. Flow chart for subroutine SPCHG2 (Sheet 1 of 2).
53
-------
YES
IMO
CALC. RATIO OF CHARGE DENSITY
TRANSFERRED TO THAT AVAILABLE (PIR)
PRINT ION DENSITY NEEDED TO
MEET CHARGING RATE
REDEFINE REDUCED FREE
ION DENSITY (AFID)
CONVERT REDUCED FREE ION
DENSITY TO #/cm3 (AVGFID)
CALC. EFFECTIVE MOBILITY FOR
IONS AND PARTICLES (UEQ)
CALC. RATIO ION CHARGE DENSITY TO
TOTAL CHARGE DENSITY (RIOVR)
CONVERT ETAPF TO
A PERCENT (XPI)
c
END SUBROUTINE
Figure 10. Flow chart for subroutine SPCHG2 (Sheet 2 of 2).
54
-------
PNUM - Total number of particles per unit volume of gas enter-
ing a given subincrement of length (#/m3).
RHOP - Total average particulate charge density in a given sub-
increment of length (coul/m3).
TCHRG - Average particle charge density for a given particle
size in a given subincrement of length (coul/m3).
PMOB - Weighted particulate mobility in a given subincrement
of length (m2/V-sec).
TDNSP - Total average particulate charge number density in a
given subincrement of length (#/m3).
RDNSI, AFID - Average reduced ion density in a given subincrement
of length (#/m3).
UEQ - Effective charge carrier mobility in a given subincre-
ment of length (m2/V-sec).
AVGFID - Average reduced ion density in a given subincrement of
length (#/cm3).
RIOVR - Ratio of ionic charge density to total charge density
in a given subincrement of length.
I - Index specifying the given length increment. Can not
exceed a value of 45.
XS - Computed value of exponential argument in equation (2)
for the design overall efficiency.
ETAO - Design overall mass efficiency of ESP (%).
PL - Total electrical length of ESP (m).
ETAPF - Computed design overall collection fraction per given
length increment.
CCF(J) - Cunningham slip correction factor for a given particle
size.
XPI - Computed design overall collection efficiency per given
length increment (%).
OLDQ(J) - Value of charge for a given particle size acquired
through all subincrements of length up to the subincre-
ment under consideration (coul).
Q(J) - Value of charge for a given particle size acquired
through the subincrement of length under consideration
(coul).
55
-------
II - Index specifying the given subincrement of length. Can
not exceed a value of 30=
NSECT - Indicator specifying the given electrical section of
the ESP. Can not exceed a value of 10.
The following is a list of the necessary variables which are
in common with the main program.
XDC(I,J) - Charge on a given particle size at the end of a given
length increment (coul).
EAVG(K) - Average electric field in a given subincrement of
length (V/m)0
CHRID(K) - Average ion density in the absence of particles in a
given subincrement of length (#/m3).
NPRNT - Indicator which specifies the logical unit number of
the printer.
Of the above variables, the values of the following must be
provided by the main program: NS, XNO, VIS, RAD, LING, E, U,
ERAVG, I, ETAO, PL, CCF, OLDQ, Q, II, ND, NSECT, XDC, EAVG, CHFID,
and NPRNT. The values of the following variables are determined
in the subroutine: DNSION, DELTNP, SUMMOB, PNUM, RHOP, TCHRG,
PMOB, TDNSP, RDNSI, AFID, UEQ, AVGFID, RIOVR, XS, ETAPF, and XPI.
In the above arrays, I, J, and K can not exceed values of 45, 20,
and 30, respectively. The restrictions on I, J, and K limit the
number of length increments, the number of particle size bands,
and the number of subincremental lengths in a given length incre-
ment, respectively. If, in a given subincrement of length, there
are not enough free ions available to meet the charging rates of
all the particle sizes, then the subroutine prints out a message
which states the increase in ion density necessary to meet the
charging rates. In this case, the free ion density is defined
as zero and it is assumed that the charging rate was met.
Subroutine CMAN
This subroutine calculates an initial estimate of the elec-
tric potential at each point in a grid which is established in
either EFLD1 or EFLD2 for the purpose of determining the elec-
trical conditions in a wire-plate precipitator. The calculation
is based on an electrostatic solution for a wire-plate geometry.
Thus, this initial estimate does not include the effects of space
chargeo The equation which is used to calculate the initial
values of electric potential at the grid points is given by28
56
-------
f
L
cosh -rr(y-2mS )/2S -cos(irx/2S
V(x,y) = V m cosh *
-------
START SUBROUTINE
REAL:
NUM,
M,
NWIRE
DIMENSION:
VCOOP
DETERMINE #OF GRID STRIPS
IN EACH DIRECTION (NXI AND NYI)
CALC. INCREMENT SIZE IN
EACH DIRECTION (AX AND AY)
c
START LOOP OVER
X-DIRECTION
/START LOOP OVER
V. Y-DIRECTION
CALC. X AND Y
POSITIONS (X AND Y)
YES
CALC. ARGUMENTS FOR COS
AND COSH FUNCTIONS IN
EQ. (26) [El, Fl, Gl, AND HI]
VCOOP (I, J) = VW
CALC. COS AND COSH FUNCTIONS
IN EQ. (26) [E2, F2, G2, AND H2]
Figure 11. Flowchart for subroutine CM AN (Sheet 1 of 2).
58
-------
CALC. ARGUMENTS FOR LN
FUNCTIONS IN EQ. (26) [TT AND TB]
CALC. LN FUNCTIONS
IN EQ. (26) [F AND G]
CALC. SUM IN NUMERATOR
OF EQ. (26) [NUM]
CALC. SUM IN DENOMINATOR
OR EQ. (26) [DENOM]
CALC. POTENTIAL AT
POINT (X,Y) [V COOP (1, J)]
C
END LOOP OVER
Y-DIRECTION
END LOOP OVER
X-DIRECTION
>
C
END SUBROUTINE
M = M + 1.
}-KD
Figure 11. Flow chart for subroutine CMAN (Sheet 2 of 2).
59
-------
AC - Radius of discharge electrode (m).
NWIRE - Number of wires per gas passage per electrical section.
The following variable is in common with the main program
and subroutine EFLD20
VCOOP(I,J) - Initial estimate of potential at points in the grid
(V). I and J can not exceed a value of 15.
Of the above variables, the values of the following must be
provided by the main program: VW, NX, NY, SX, SY, PI, AC, and
NWIRE. VCOOP is determined in the subroutine. The restrictions
on I and J limit the number of grid points in the x-direction
and the number of grid points in the y-direction, respectively.
Subroutine EFLD1
This subroutine calculates the average electric field at the
plate of a wire-plate precipitator0 Its usage depends upon mea-
sured or known values of applied voltage and current. The electric
field at the plate is found by solving equations (9) and (10) si-
multaneously and subject to the existing boundary conditions using
a numerical relaxation technique.29'30'51'32 In solving the
equations, the corona zone surrounding the discharge wire is con-
sidered only as a source of ions and the high values of electric
field intensity existing in this region are not treated in the
calculation. The region outside the corona zone, where the values
of electric field intensity are relatively low, is referred to as
the space-charge zone and is assumed to contain unipolar ions.
Figure 12 shows the basic area of interest in the precipitator
for which equations (9) and (10) are solvedo Figure 12 also in-
cludes the geometry and nomenclature used in the numerical anal-
ysis. Due to the symmetry of the problem it is necessary to
integrate only over the rectangle shown in Figure 12 and then
assert that the solutions hold for all similar rectangles within
the duct. In this manner the entire two-dimensional area can be
taken into account. In the actual numerical procedure, occasions
occur when values of V, E, and p from positions outside the
rectangle are used0 In this case it is justified to assign to
the functions V, E, and p the same values they had at symmetrical-
ly located points within the area of the rectangle. This is
necessary in order for the integration to include values of V
and p which exist on the boundaries in such a way that they con-
verge toward an exact solution along with the interior points0
In order to employ the numerical technique, the area of in-
terest is divided into a fine rectangular grid as shown in Figure
13o Point "0" is the point of interest for the following dis-
cussiono However, once V0, E0, and p0 are calculated here, the
label "0" is moved to a neighboring point and the calculation
60
-------
V = Vo ON WIRES
V = 0 ON PLATES
A |
CO
5 3
X.I2.
345
13
21
31
/
22
32
42
^
T
/
23
33
43
-»•
24
V
_ >v. —
1 y AXIS
4.
i
4.
1
r
\
\
\
\
s
1
c
AREA OF
INTEGRATION
Sy = ONE HALF WIRE TO WIRE SPACING
Sx= WIRE TO PLATE SPACING
a, av = INCREMENT SIZES FOR INTEGRATION
* y
V0 = APPLIED VOLTAGE
Ex = COMPONENT OF ELECTRIC FIELD PERPENDICULAR
TO PLATE
Ey = LONGITUDINAL COMPONENT OF ELECTRIC FIELD
j = AVERAGE CURRENT DENSITY
Figure 12. Nomenclature used in the numerical analysis of the electrical
conditions in wire-plate precipitators.
61
-------
0 Eoy
Eox
Figure 13. Partial grid showing nomenclature used in the numerical
analysis of the electrical conditions.
62
-------
repeated. Each point in the grid would eventually become point
"0".
From Figure 13,
A2V0 _ V
AlT2 a5 ' (27)
and
A2V0 _ V3+Vi-2V0
Ay? a
Y
2 (28)
Equations (27) and (28) can be used in equation (9) and the re-
sulting equation can be solved for VQ to give
V0 =
2(ax2+ay2)
Again, from Figure 13,
2~ 2Po
ax2(Vi+V3) + y . (29)
J\
App _ PQ-P3
—
'
(3Q)
,-,, .
(31>
_ b0-b2 ,__.
~ ~~" ' (32)
and
Abp _ bp —b3
Ay ~ a
(33)
Equations (30)-(33) can be used in equation (10) and the resulting
equation can be solved for po to given
Po = -« +\a2+B , (34)
63
-------
where
e0(2a b0Eo +2a b0E0 -a b2E0 -a b3E0 )
~ _ _ i _ x x Y _ X _ x x _ Y
2axayb0
and
e0 (a EO p2-fa EO pa)
g =
a a
x y
In subroutine EFLDl, solutions to equations (29) and (34) are
obtained subject to the following boundary conditions:
(a) V = V (applied voltage) at the wire;
(b) J = known or measured average current density at the
plate;
(c) V = 0 on the plate (along line CD);
(d) E = - -r-rr = 0 and E - - -r-rr = 0 at points A and B;
X AX y AY
(e) E = - -rrr = 0 along line AB;
X AX
(f) E = - ^Y = 0 along lines BC, CD, and AD.
jL
The boundary conditions greatly simplify equation (34) along
the lines AB, BC, AD, and CD and, in fact, make the calculation
of simultaneous solutions to equations (29) and (34) possible.
Along the lines AB, BC, AD, and CD, a and 3 reduce to the fol-
lowing expressions:
e (2a b E - a b E )
a = _° x ° °Y x 3 Oy n,-N
AB 2a a b ' (35)
X« -T
JL V
e (2a b E - a b E )
a = n - n = ° Y o ox y 2 ox i-,c\
aBC aAD aCD 2a a b— ' (36)
x y o
e E p
3AB = aY ' (37)
and
(38)
64
-------
In order to initiate the numerical procedure which finds the
simultaneous solutions to equations (29) and (34), it is necessary
to make initial estimates of the values of the electric potential
at all grid points and the space charge density for the corona
region. The initial values of electric potential are estimated
using equation (26) which is evaluated in subroutine CMAN. An
initial estimate of the space charge density near the discharge
electrode is obtained by assuming that the corona region is
cylindrical and using continuity of current. This initial esti-
mate is calculated from the expression
2 S J
P_ = ^ , x 1CT3 , (39)
TTb r f (306 + 9-J 6/r )
s w > ' w
where p = space charge density at the outer boundary of the
s
ionized sheath (coul/m3),
S = one-half wire-to-wire spacing (m),
J = average current density at the plate (A/m2),
b = effective charge carrier mobility at the outer boundary
O
of the ionized sheath (m2/V-sec),
r = radius of the wire (cm),
w
f = roughness factor, and
6 = relative density of the gas.
The numerical procedure for finding the solutions to equa-
tions (29) and (34) consists of the following steps.
1. V is computed at every point in the integration grid
using equation (26).
2. p is computed at every point in the integration grid from
equations (34) and (39).
3. V is recomputed at every point in the integration grid
using equation (29) .
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 it-
eration,,
65
-------
5. The computed current density (obtained using the re-
lationship J = -p (-T—) b) is compared with the measured current
ZAX
densityo 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 agree-
ment is obtained.
This procedure iterates on a grid of electric field and space
charge density until convergence is obtained. The major approx-
imation, 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 procedure uses a reduced
mobility which is calculated from equation (16) and is evaluated
in subroutine SPCHG1. The reduction in mobility is limited to
a value of 1 x 10" ^ m2/V-sec in order to prevent nonconvergence
of the grid under certain conditions.
Figure 14 shows a detailed flow chart for this subroutine.
Information which is transmitted between the main program and
this subroutine is transferred through calling arguments and block
common statements. The following is a sequential list of the
calling arguments and their descriptions.
UEQ - Effective charge carrier mobility (m2/V-sec). Limited
to a lower value of 1 x lO"1* m2/V-sec.
CD - Average current density at the plate (A/m2).
AC - Radius of discharge electrode (m).
VO - Electric potential at the wire (V).
SX - Wire-to-plate spacing (m).
SY - One-half wire-to-wire spacing (m).
NX - Number of grid points in the x-direction. Can not exceed
a value of 15.
NY - Number of grid points in the y-direction0 Can not exceed
a value of 15.
TDK - Temperature of the gas (°K).
P - Pressure of the gas (atm).
AEPLT - Average electric field at the plate (V/m).
66
-------
( START SUBROUTINE )
RE
1
AL: MAXJ, MINJ, MOBILT
1
DIMENSION: RHO, VCOOP, EX, OLDRO,
OLDV, CDNSTY, V, EY
B
1
LOCK COMMON: VCOOP
1
BLOCK COMMON: NREAD, NPRINT
1
INITIALIZE TO ZERO: RHO, V, EX, EY,
OLDRO, OLDV, CDNSTY, MOBILT
1
vo =-vo
1
DEFINE CONSTANTS IN
CALCULATIONS
1
CALC. RELATIVE AIR
DENSITY (RELD)
1
CALC. PRODUCT (EORO) OF ELECTRIC
FIELD AND RADIAL DISTANCE
AT IONIZATION BOUNDARY
C
S
T
1
ALC. INITIAL ESTIMATE OF
PACE CHARGE DENSITY AT
HE WIRE (VERGE)
1
DEFINE MIN. AND MAX. LIMITS
ON CURRENT DENSITY FOR CONVERGENCE
1
DEFINE CONSTANTS
IN CALCULATION
1
Z = 0
1
INITIALIZE VALUES IN POTENTIAL
GRID TO THOSE OF COOPERMAN
1
NO /""^^ YES
1
/PRINT: CONVERGENCE CAN NOT BE /
OBTAINED IN 25 ITERATIONS /
ON CURRENT DENSITY /
1
v J
\
\
SET pQ QZERO, Ex Ey = 0
AT POINT (1,1)
I
f START LOOP ALONG ,"\
I LINE AD )
A 1
QZERO = VERGE
1
DEFINE GRID OF
MOBILITY VALUES
1
ESTABLISH GRID
SPACINGS
1
CALCULATE EX, E RHO
ALONG LINE AD
I
/^ END LOOP ALONG "\
I LINE AD J
I
(START LOOP ALONG "\
LINE AB )
L «ft C"\\
Figure 14. Flow chart for subroutine EFLD1 (Sheet 1 of 3).
67
-------
1 1
CALCULATE EX, Ey RHO
ALONG LINE AB
1
1
(^ END LOOP ALONG A
1 LINE AB J I-
\
r START LOOP ALONG A
1 LINE BC )
\
CALCULATE EX, Ey, RHO
ALONG LINE BC L
\
\
f~ END LOOP ALONG A
V LINE BC J
\
s \
f START LOOP OVER \
I INTERIOR POINTS )
\
CALCULATE EX, E RHO
^ AT INTERIOR POINTS
1
( END LOOP OVER ^
^ INTERIOR POINTS J
^
( START LOOP OVER ^
^ l^ GRID POINTS ^J
\
STORE PREVIOUS VALUES
OF V AND p
1
JL
NO/^ ^\YES^
END LOOP OVER GRID POINTS
YES
^ <. LL 2000
r PRINT: CONVERGENCE CAN NOT BE
OBTAINED IN 2000 ITERATIONS ON
THE POTENTIAL GRID
Figure 14. Flow chart for subroutine EFLD1 (Sheet 2 of 3).
68
-------
c
START LOOP OVER
GRID POINTS
YES
END LOOP OVER
GRID POINTS
CALCULATE
CDNSTY (NX, 1)
START LOOP ALONG
LINE DC
CALCULATE CDNSTY
ALONG LINE DC
END LOOP ALONG
LINE DC
CALCULATE AVERAGE CURRENT
DENSITY AT PLATE (ACDNTY)
A
CALCULATE AVERAGE ELECTRIC
FIELD AT PLATE (AEPLT)
CVERGE =QZERO
ADJUST QZERO
DOWNWARD
ADJUST QZERO
UPWARD
Figure 14. Flow chart for subroutine EFLD1 (Sheet 3 of 3).
69
-------
VERGE - Initial estimate of space charge density at the wire
(coul/m3)o
CVERGE - Final value of space charge density at the wire for con-
vergence (coul/m ).
The following is a list of the necessary variables which are
in common with the main program.
VCOOP(I,J) - Initial estimate of potential at points in the grid
(V). I and J can not exceed a value of 15.
NPRNT - Indicator which specifies the logical unit number of
the printer.
Of the above variables, the values of the following must be
provided by the main program: UEQ, CD, AC, VO, SX, SY, NX, NY,.
TDK, P, VCOOP, and NPRNT. AEPLT, VERGE, and CVERGE are determined
in the subroutine. The restrictions on I and J limit the number
of grid points in the x-direction and the number of grid points
in the y-direction, respectively. If convergence on the electric
potential grid can not be obtained in 2000 iterations, a message
stating that convergence can not be obtained is printed and the
subroutine returns to the main program with those values which
were calculated in the last iteration. If convergence on the
average current density at the plate can not be obtained in 25
iterations, a message stating that convergence can not be obtained
is printed and the subroutine returns to the main program with
those values which were calculated in the last iteration.
Subroutine EFLD2
This subroutine calculates a voltage-current curve up to a
specified value of operating applied voltage and calculates the
average electric field at the plate for the operating applied
voltage. The voltage-current curve is generated by (1) speci-
fying a starting value of average current density at the plate,
(2) incrementing upward on the average current density, and (3)
determining the applied voltage at each value of current density.
Once a value of current density results in an applied voltage
which exceeds the specified operating applied voltage, an inter-
polation is performed in order to obtain the operating applied
voltage and current density. At the operating applied voltage,
calculations can also be made to give the average current density,
average electric field, and average electric field at the plate
in subincremental lengths„
The equations which are solved and the mathematical technique
which is used to solve these equations are the same as discussed
for subroutine EFLD1. The major differences in the two subroutines
are the use of different boundary conditions in solving equations
70
-------
(29) and (34) and an added loop in EFLD2 which runs over values
of average current density at the plate0
The boundary conditions imposed on the solutions to equations
(29) and (34) are:
(1) J = given average current density at the plate;
(2) p = p , space charge density near the wire and hence,
o
P = Ps = Pw at point A for calculations outside of r ;
(3) E r = E r = constant;
s s c w
(4) E = - ~ = 0 and E = - -^ = 0 at points A and B;
x AX y i\y
(5) V = 0 on the plate or along line CD;
(6) EX = - ££ = 0 along line AB;
(7) E = - -j— = 0 along lines BC, CD, and AD;
where r = radius of the ionized sheath (m),
o
E = electric field at the outer radius of the ionized
O
sheath (V/m),
E = corona starting electric field (V/m), and
r = radius of the corona wire (m).
w
By using the above boundary conditions, solutions to equations
(29) and (34) can be obtained without measured or known data.
The steps in the numerical procedure are outlined as follows:
1. Choose an average current density at the plate which
corresponds to the lowest value desired on a current-voltage curve.
2. Estimate the potential at the wire that would produce the
chosen value of average current density at the plate and calcu-
late V at every point in the grid using equation (26).
3. Calculate p at every point in the grid using equation
(34), where the space charge density at the wire is given by
equation (39).
71
-------
4. Recalculate V at every point in the grid using equation
(29) c
5. Repeat steps (3) and (4) alternately until each value
of V in the potential grid changes negligibly from its previous
value.
6. Check to see if the computed average current density at
the plate equals the chosen value. If they agree, then the so-
lution has been obtained. If they do not agree, adjust the
potential at the wire and start the calculation over at step
(3) above.
7. Choose a larger value of average current density at the
plate and obtain a new solution by starting at step (3) with the
existing potential grid used to estimate the actual potential
grid.
8. Repeat steps (3)-(7) until the desired current-voltage
curve is obtained.
In the above procedure, the electric potential at the wire is ad-
justed until solutions are found which satisfy equations (29) and
(34) and the boundary conditions, whereas, in EFLDl, the space
charge density at the wire is adjusted.
Figure 15 shows a detailed flow chart for this subroutine.
Information which is transmitted between the main program and
this subroutine is transferred through calling arguments and
block common statements. The following is a sequential list of
the calling arguments and their descriptions.
UEQ - Effective charge carrier mobility (m2/V-sec).
AC - Radius of discharge electrode (m).
VO - Chosen operating applied voltage (V) .
SX - Wire-to-plate spacing (m).
SY - One-half wire-to-wire spacing (m).
NX - Number of grid points in the x-direction0 Can not
exceed a value of 15.
NY - Number of grid points in the y-direction0 Can not
exceed a value of 15.
AEPLT - Average electric field at the plate (V/m).
TDK - Temperature of the gas (°K).
72
-------
START SUBROUTINE
REAL: MAXJ, MINJ, MOBILT,
NWIRE, MAXS
DIMENSION: RHO, VCOOP, EX, OLDRO,
OLDV, CDNSTY, V, EY, EAVGS, CHFIDS, ECOLLS
BLOCK COMMON: EAVG, CHFID
BLOCK COMMON: ECOLL
BLOCK COMMON: VCOOP
BLOCK COMMON: NREAD, NPRNT
INITIALIZE TO ZERO: RHO, V, EX, EY,
OLDRO, OLDV, CDNSTY, MOBILT
vw
= -VSTART •
CALC. RELATIVE
AIR DENSITY (RELD)
CALC. PRODUCT (EORO) OF ELECTRIC FIELD AND
RADIAL DISTANCE AT IONIZATION BOUNDARY
DEFINE GRID OF
MOBILITY VALUES
SSTART = START
A
A
ESTABLISH GRID
SPACINGS
DEFINE CONSTANTS
IN CALCULATIONS
c
START LOOP OVER
CURRENT DENSITIES
ESTABLISH DESIRED
CURRENT DENSITY
DEFINE MIN. AND MAX. LIMITS ON
CURRENT DENSITY FOR CONVERGENCE
CALCULATE SPACE CHARGE DENSITY
AT THE WIRE (QZERO)
CALL
CMAN
INITIALIZE VALUES IN POTENTIAL
GRID TO THOSE OF COOPERMAN
Figure 15. Flow chart for subroutine EFLD2 (Sheet 1 of 8).
73
-------
1
/\YES
NO|
LL = 0
1
1
SET p0 = QZERO, Ex Ey = 0
AT POINT (1,1)
I
/" START LOOP ALONG , ^\
I LINE AD }
|
CALCULATE EX, Ey, RHO
ALONG LINE AD
I
/" END LOOP ALONG ^\
1 LINE AD )
I
(START LOOP ALONG A
LINE AB )
I
CALCULATE EX, E RHO
ALONG LINE AB
I
/" END LOOP ALONG A
1 LINE AB J
|
/" START LOOP ALONG "\
1 LINE BC J
I
CALCULATE EX, E RHO
ALONG LINE BC
I
/PRINT: CONVERGENCE CAN /
NOT BE OBTAINED IN 25 /
INTERATIONS ON CURRENT /
DENSITY /
C END LOOP ALONG LINE BC "}
( START LOOP OVER A
1 INTERIOR POINTS J
/\ 1 '
'-* CALCULATE EX, Ey, RHO
AT INTERIOR POINTS
f END LOOP OVER \
1 INTERIOR POINTS J
(START LOOP OVER \
GRID POINTS J
STORE PREVIOUS VALUES
OF V AND p
A
^ (
Figure 15. Flow chart for subroutine EFLD2 (Sheet 2 of 8).
74
-------
I—. . . I
END LOOP OVER
GRID POINTS
c
YE<3
NO
START LOOP OVER
GRID POINTS
PRINT: CONVERGENCE CAN NOT
BE OBTAINED IN 2000 ITERATIONS
ON THE POTENTIAL GRID
Figure 15. Flow chart for subroutine EFLD2 (Sheet 3 of 8).
75
-------
YES
END LOOP OVER
GRID POINTS
A
J
CALCULATE
CDNSTY (NX, 1)
START LOOP ALONG
LINE DC
)
CALCULATE CDNSTY
ALONG LINE DC
I
(
END LOOP ALONG
LINE DC
A
J
CALCULATE AVERAGE CURRENT
DENSITY AT PLATE (ACDNTY)
TEST = ACDNTY - (MAXJ + MINJl/2
Figure 15. Flow chart for subroutine EFLD2 (Sheet 4 of 8).
76
-------
CALCULATE AVERAGE ELECTRIC
FIELD AT PLATE (AEPLT)
PRINT VW, ACDNTY
AEPLT /
YES
YES
NO
OLDVW = VW
OLDCD = ACDNTY
END LOOP OVER
CURRENT DENSITIES
'PRINT: BREAKDOWN FIELD
IS EXCEEDED, VW, ACDNTY /
INTERPOLATE TO FIND
CURRENT DENSITY AT VO
Figure 15. Flow chart for subroutine EFLD2 (Sheet 5 of 8).
77
-------
YES
START LOOP OVER SUB
INCREMENTAL LENGTHS
START LOOP OVER GRID VALUES
IN SUB INCREMENTAL LENGTHS
YES
CALC. CONTRIBUTION TO
AVG. ELECTRIC FIELD
IN SUB INCREMENT
CALC. CONTRIBUTION TO
AVG. ELECTRIC FIELD IN
SUB INCREMENT
CALC. AVG. ELECTRIC FIELD
IN SUB INCREMENT
CALC. CONTRIBUTION TO
AVG. ION DENSITY IN
SUB INCREMENT
END LOOP OVER GRID VALUES
IN SUB INCREMENTAL LENGTHS
Figure 15. Flow chart for subroutine EFLD2 (Sheet 6 of 8).
-------
STORE VALUES OF AVG. ELECTRIC
FIELD AND ION DENSITY FOR
SUB INCREMENT
END LOOP OVER SUB
INCREMENTAL LENGTHS
NYY
= NY1
START FIRST LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
CALC. EAVG(L)
AND CHFID (L)
NYY
= NYY
- 1
END FIRST LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
A
END SECOND LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
LL = 1
START LOOP OVER SUB
INCREMENTAL LENGTHS
CALC. AVG. ELECTRIC
FIELD AT PLATE
LL =
LL + 1
END LOOP OVER SUB
INCREMENTAL LENGTHS
START FIRST LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
KK 1
M1 NY1 + 1
M2 2(NY1)
CALC.
ECOLL(L)
START SECOND LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
CALC. EAVG(M)
AND CHFID(M)
KK KK + 1
A
END FIRST LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
Figure 15. Flow chart for subroutine EFLD2 (Sheet 7 of 8).
79
-------
START SECOND LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
CALC.
ECOLL(I)
L2
L2 + 1
END SECOND LOOP TO PUT SUB
INCREMENTAL QUANTITIES
IN CORRECT ORDER
START =
SSTART
END SUBROUTINE
Figure 15. Flow chart for subroutine EFLD2 (Sheet 8 of 8).
80
-------
P - Pressure of the gas (atm).
RF - Roughness factor for the discharge wire (0.5 <_ RF <_ 1.0).
START - Chosen initial current density at which the voltage-
current curve calculation starts (A/m2). Current
densities increment in values of START until a change
is specified.
DSTART - Chosen increment in current density which is used in
place of START when specified (A/in2) .
CSTART - Chosen increment in current density which is used in
place of DSTART when specified (A/m2).
IFINAL - Indicator which terminates the loop over average current
densities at the plate after IFINAL times.
VSTART - Initial estimate of applied voltage corresponding to
first value of average current density at the plate on
the voltage-current curve (V).
VW - Operating applied voltage corresponding to a given
current density (V).
ACDNTY - Average current density at the plate (A/m2).
NWIRE - Number of wires per gas passage per electrical section.
NEC - Indicator which governs the calculations of average
current density, average electric field, and average
electric field at the plate in subincremental lengths.
The calculations are performed when NEC = 0 and are
not performed when NEC = 1.
EBD - Electrical breakdown strength of the gas (V/m).
JI1 - Indicator which governs the change in the increment on
average current density at the plate from START TO
DSTART. The change occurs on the Jll-th value of
current density.
JI2 - Indicator which governs the change in the increment on
average current density at the plate from DSTART to
CSTART. The change occurs on the JI2-th value of
current•density.
The following is a list of the variables which are in common
with the main program.
EAVG(M) - Average electric field in a given subincrement of length
(V/m). M can not exceed the value of 30.
81
-------
CHFID(M) - Average ion density in the absence of particles in a
given subincrement of length (#/m3)0 M can not exceed
a value of 30.
ECOLL(M) - Average electric field at the plate in a given sub-
increment of length (V/m). M can not exceed the value
of 300
VCOOP(I,J) - Initial estimate of potential at points in the grid
(V). I and J can not exceed a value of 15.
NPRNT - Indicator which specifies the logical unit number of
the printer.
Of the above variables, the values of the following must be
provided by the main program: UEQ, AC, VO, SX, SY, NX, NY, TDK,
P, RF, START, DSTART, CSTART, IFINAL, VSTART, NWIRE, NEC, EBD,
JI1, JI2, and NPRNT. AEPLT, VW, ACDNTY, EAVG, CHFID, ECOLL, and
VCOOP are determined in the subroutine. The restrictions on I,
J, and M limit the number of grid points in the x-direction, the
number of grid points in the y-direction, and the number of sub-
incremental lengths in a given length increment, respectively.
The subroutine calls subroutine CMAN in order to determine VCOOP.
If, for a given current density, convergence on the electric
potential grid can not be obtained in 2000 iterations, a message
stating that convergence can not be obtained is printed and those
values which were calculated in the last iteration are used for
that particular point on the voltage-current curve. If conver-
gence on a given average current density at the plate can not be
obtained in 25 iterations, a message stating that convergence
can not be obtained is printed and those values which were calcu-
lated in the last iteration are used for that particular point on
the voltage-current curve.
There are three possible conditions which will result in
termination of the voltage-current curve at a particular voltage
and current. The curve is terminated if (1) the specified oper-
ating applied voltage is reached, or (2) the number of points on
the curve is equal to value of IFINAL, or (3) the specified value
of electrical breakdown strength near the collection electrode is
exceededo If the breakdown strength is exceeded, a message
stating that this is the case is printed.
Subroutine CHARGN
This subroutine calculates particle charge as a function of
residence time, electrical conditions, gas conditions, and par-
ticle characteristicso In order to use this subroutine, statement
function RATE and subroutines ARCCOS and ZERO are required.
82
-------
The subroutine determines particle charge by solving equa-
tion (12) . Equation (12) is a first-order differential equation
of the form
f(*,y) (40)
with initial values x0 and y0 and is solved numerically using a
quartic Runge-Kutta method. 3 This is a single-step method in
which the value of y at x = x is used to compute y = y(x ,)
and earlier values y _,, y _„, etc. are not usedo
The increment for advancing the dependent variable is given
by
Ay = | (^ + 2k2 + 2k3 + k4) (41)
where, for a given stepsize h,
k1 = hf (xn, yn) , (42)
k2 = hf (xn + |h, yn + | k^ , (43)
and
= hf (xn + h, yn + k2) , (44)
= hf(xn + h, yn + k3) o (45)
The values at (x ,, y .) are given by
xn+1 = xn + h (46)
and
- y + Ay . (47)
The subroutine calls the statement function RATE to calculate the
right hand side of equation (12) at the function values specified
in equations (42)-(45).
The numerical procedure for finding solutions to equation
(12) consists of the following steps.
83
-------
1. The initial conditions are taken to be q = 0 at t = 0.
2. q is calculated in the main program using equation (13)
s
and is supplied to subroutine CHARGN and statement function RATE.
3. For each value of q required in the Runge-Kutta scheme,
a value of 0o is calculated in statement function RATE using
equation (14) .
4. The integration over 0 on the right hand side of equation
(12) is performed in statement function RATE using Simpson's Rule.
For each value of 6 which is chosen for this integration, the
radial distance (r0) from the center of the particle and along 6
for which the total electric field is zero is calculated using
subroutine ZERO.
50 The three individual charging rates are calculated and
then added in statement function RATE to give the total instan-
taneous charging rate for a particular value of q.
6. The total charging rates necessary for use in equations
(41)-(45) are obtained by subroutine CHARGN and q and t are ob-
tained from equations (46) and (47).
Figure 16 shows a detailed flow chart for this subroutine.
All information which is transmitted between the main program and
this subroutine is transferred through calling arguments. The
following is a sequential list of the calling arguments and their
descriptions.
ECHARG - Value of an electronic charge unit (coul).
SCHARG - Value of saturation charge number from the field charging
equation [see eq0 (13)].
NUMINC - Number of increments in the Simpson's Rule integration
over 0. A value of 20 is normally sufficient.
CONST - Value of the quantity [2 -ig^- a3E0] found in equation
(12) [V-m2]. (K+2)
EZERO - Applied electric field strength for particle charging
(V/m).
V - Value of the quantity [^—e___] found in equation (12) .
RSIZE - Radius of the particle (m).
ECONST - Value of the quantity [ff] found in equation (12).
84
-------
START SUBROUTINE
CALC. HALF STEPSIZE (H2)
INITIALIZE X AND Y
TO XI AND Yl
START LOOP OVER
NN STEPSIZES
CALCULATE
T1 = H * RATE
( X, Y)
CALCULATE
T2 = H * RATE
,., X + H2, Y + T1/2.)
CALCULATE
T3 H * RATE
., X + H2, Y + T2/2.!
CALCULATE
T4 = H * RATE ( X + H, Y + T3)
CALC. CUMULATIVE
INTEGRAL (Y)
INCREASE X BY STEPSIZE
END LOOP OVER
NN STEPSIZES
c
END SUBROUTINE
Figure 16. Flow chart for subroutine CHARGN.
85
-------
CMKS - Value of the quantity [4Treo] found in equation (12)
(cou!2/nt-m2).
RR - Value of the quantity
found in equation (12)
FCONST - Value of the quantity
(12) [m2].
found in equation
a.
Value of the quantity [? ] found in equation (12)
[m3/sec] .
bq
Value of the quantity [-JTS-] found in equation (12)
4E°
FACTOR -
COEFF -
AFID - Free ion density for particle charging (#/m3).
RATE - Statement function which must be supplied to subroutine
CHARGN .
H - Increment size for Runge-Kutta integration (sec) .
XI - Initial value of time (sec) .
YI - Initial value of charge number.
NN - Number of increments in the Runge-Kutta integration.
A value of 10 is normally sufficient.
X - Final value of time (sec) .
Y - Final value of charge number.
Of the above variables, the values of the following must be
provided by the main program: ECHARG, SCHARG, NUMINC, CONST,
EZERO, V, RSIZE, ECONST, RR, FCONST, FACTOR, COEFF, AFID, RATE,
H, XI, YI, and NN. X and Y are determined in the subroutine.
For length increments along the precipitator of approximately
0.305 meter or less, the use of 10 increments in the Runge-Kutta
integration and 20 increments in the Simpson's Rule integration
yields solutions to equation (12) which are changed negligibly
by increasing the number of points used. In cases where the use
of computer time is a significant consideration, the use of 5
increments in the Runge-Kutta integration and 10 increments in
the Simpson's Rule integration will result in charge values which
are not severely changed. These values of NN and NUMINC should
be regarded as yielding a lower limit for which reliability can
be expected and they should not be reduced further.
86
-------
Statement Function RATE
This statement function calculates the right hand side of
equation (12) for use in subroutine CHARGN. In order to use this
statement function, subroutines ARCCOS and ZERO must be supplied.
The first and third terms on the right hand side of equation
(12) are calculated in a straightforward manner. However, the
third term involves an integration over the angle 6 which must
be performed numerically. The integration is performed by using
Simpson's Rule34 which is given by
X
r
=- (y0 + 4y i + 2y2 + 4y
X° (48)
where
x. = x0 + ih (i = 0,1,2, •••• n) , (49)
n is even, and h is the increment size. In the application of
this technique, there must be an odd number of points.
The subroutine performs the operations indicated in equation
(48) by first calculating the odd-numbered function values and
summing them. Next, the even-numbered function values are calcu-
lated and those between y0 and y are summed. Thus, equation (48)
is applied in the form
x
I y(x)dx—| (y + 4 \^ vn + 2 T^ ^n + Yn) ' (50)
n even n odd
The lower integration limit 0o in the second term of equation
(12) is determined by calling subroutine ARCCOS. If 00 is less
than or equal to 0.00001 radian, it is set equal to zero. For
each value of 0 in the integration, the radial distance (ro)
from the center of the particle and along 0 for which the total
electric field is zero is determined by calling subroutine ZERO.
If the charge on the particle is equal to or greater than
the saturation charge, the first term on the right hand side of
equation (12) is set equal to zero. Once the three terms on the
right hand side of equation (12) are calculated, then they are
added to give the total charging rate.
Figure 17 shows a detailed flow chart for this subroutine.
All information which is transmitted between subroutine CHARGN
87
-------
START STATEMENT
FUNCTION
REAL: INTGRL, NE,
NUMBER, NTIME
CALC. CHARGE ON
PARTICLE (NE)
ESTABLISH STEPSIZE (DELTAX)
CALC. STARTING POINT ON d FROM
WHICH ODD NUMBERED FUNCTION
VALUES ARE CALCULATED (THETA)
INITIALIZE SUM OF ODD NUMBERED
FUNCTION VALUES TO ZERO (SUMODD)
Figure 17. Flow chart for statement function RATE (Sheet 1 of 5).
88
-------
START LOOP OVER ODD
NUMBERED FUNCTION VALUES
CALC. VALUE OF
8 (THETA)
CALC. PARAMETERS DEPENDENT ON
6 (CTHETA, TCONST, ECOS)
CALC. COEFFICIENTS OF POLYNOMIAL IN
WHICH THE RADIAL COMPONENT OF ELECTRIC
FIELD IS ZERO (Cl AND CO)
CALL ZERO
CALC. ARGUMENT OF EXPONENTIAL FUNCTION
IN CHARGING RATE FOR REGION II (ARG1)
SUM ODD NUMBERED
FUNCTION VALUES (SUMODD)
END LOOP OVER ODD
NUMBERED FUNCTION VALUES
Figure 17. Flow chart for statement function RATE (Sheet 2 of 5).
89
-------
CALC. STARTING POINT ON 0 FROM
WHICH EVEN NUMBERED FUNCTION
VALUES ARE CALCULATED (THETA)
INITIALIZE SUM OF EVEN NUMBERED
FUNCTION VALUES TO ZERO (SUMEVN)
START LOOP OVER EVEN
NUMBERED FUNCTION VALUES
CALC. VALUE OF
9 (THETA)
1
CALC. PARAMETERS DEPENDENT ON
6 (CTHETA.TCONST, ECOS)
1
CALC. COEFFICIENTS OF POLYNOMIAL 1
WHICH THE RADIAL COMPONENT OF
ELECTRIC FIELD IS ZERO (C1 AND CO)
I
CALL ZERO
I
N
CALC. ARGUMENT OF EXPONENTIAL FUNCTION
IN CHARGING RATE FOR REGION II (ARG1)
TNO
UNCTION
(YVAL)
YVAL = 0.
I
Figure 17. Flow chart for statement function RATE (Sheet 3 of 5).
90
-------
SUM EVEN NUMBERED FUNCTION
VALUES EXCLUDING FIRST AND LAST (SUMEVN)
END LOOP OVER EVEN
NUMBERED FUNCTION VALUES
YES
RZERO = RSIZE
CALC. PARAMETERS DEPENDENT ON
d AT 0 0 (CT2ERO, TCONST, ECOS)
CALC. COEFFICIENTS OF POLYNOMIAL IN
WHICH THE RADIAL COMPONENT OF ELECTRIC
FIELD IS ZERO AT d = 0 (C1 AND CO)
CALL ZERO
CALC. ARGUMENT OF EXPONENTIAL FUNCTION
IN CHARGING RATE FOR REGION II (ARG2)
YES
-------
CALC. INTEGRAL OVER
REGION II (INTGRL)
CALC. CHARGING RATE FOR
REGION II (RATED
RATE1 = 0.
CALC. ARGUMENT OF EXPONENTIAL FUNCTION
IN CHARGING RATE FOR REGION III (ARG3)
YES
CALC. CHARGING RATE IN
REGION III (RATE2)
RATE2 = 0.
NUMBER -SCHARG > 0
CALC. CHARGING RATE IN
REGION III ( RATE 3 )
CALC. TOTAL CHARGING
RATE (RATE)
END STATEMENT FUNCTION
Figure 17. Flow chart for statement function RATE (Sheet 5 of 5).
92
-------
and this statement function is transferred through calling argu-
ments. The following is a sequential list of the calling argu-
ments and their descriptions.
ECHARG - Value of an electronic charge unit (coul).
SCHARG - Value of saturation charge number from the field charging
equation [see eq. (13)].
NUMINC - Number of increments in the Simpson's Rule integration
over 6. NUMINC must be even and 20 is normally suffi-
cient.
CONST - Value of the quantity [2 ;^~^( a3E0] found in equation
/TO\rTT.m2l \ KT / )
(±z; LV—m J.
EZERO - Applied electric field strength for particle charging
(V/m) .
e2
V - Value of the quantity h r~] found in equation (12) .
4 7T £ o cl-K J.
RSIZE - Radius of the particle (m).
ECONST - Value of the quantity [>fp /£, 0\ ] found in equation (12) .
CMKS - Value of the quantity [4TT£ol found in equation (12)
(cou!2/nt-m2).
RR - Value of the quantity [^7^] found in equation (12) [m"1].
(TC— 1 ^ c*^? ?\ ^
FCONST - Value of the quantity [,„,~N, " ] found in equation
(12) [m2]. (K+2)kT
'^ 2
FACTOR - Value of the quantity [-^f—] found in equation (12)
[m3/sec]. Z
bqs
COEFF - Value of the quantity [-rp— ] found in equation (12)
[m3/sec]. 4E°
AFID - Free ion density for particle charging (#/m3).
NTIME - Residence time for particle charging (sec).
NUMBER - Particle charge number (coul).
Of the above variables, the values of the following must be
provided by subroutine CHARGN: ECHARG, SCHARG, NUMINC, CONST,
EZERO, V, RSIZE, ECONST, RR, FCONST, FACTOR, COEFF, AFID, NTIME,
and NUMBER. The total charging rate given on the right hand side
of equation (12) is RATE and is determined in the statement
function.
93
-------
Subroutine ARCCOS
This subroutine calculates the inverse cosine of a number.
The calculation is performed by using the series expansion given
by35
= - (x + + j x5 + ,:-, x7 + .-..) ,
£ 2.6 2.4.5 2.4.6.7 '
where x2 a3 = 0 , (52)
where the symbols are defined the same as for equation (12)
94
-------
START SUBROUTINE
CALC. ARGUMENT OF
INVERSE COS FUNCTION (RATIO)
INITIALIZE VARIABLES:
T = 1.
SUM = 0.
TERM = RATIO
SET UP VARIABLES WHICH COMBINE
TO GIVE PROPER COEFFICIENTS IN
SERIES EXPANSION OF COS'1 d (U, V, AND W)
CALC. TERM IN SERIES EXPANSION
OF COS'1 6 (TERM)
SUM TERMS IN SERIES EXPANSION
OF COS'1 9 (SUM)
T = T + 1.
YES
CALC. COS'1 (RATIO) FROM
SERIES EXPANSION (ACOS)
END SUBROUTINE
Figure 18. Flow chart for subroutine ARCCOS.
95
-------
For a given angle 9, this is a cubic equation in r0 of the form
x3 - cix + 2c2 = 0 . (53)
This type of cubic equation has roots given by
x = _2W Si cos J V^ + ^ n } (54)
where n = 0,1,2. For particle charging, the physically meaningful
solution is given for n=l
-> I ne
r° ~ " "V 127Te0E0cose
-i 10-7 /K~l\2 /4Tr£0a2E0cos9
V27 K+2 Si
(55)
The subroutine determines r0 from equation (55).
Figure 19 shows a detailed flow chart for this subroutine.
This subroutine is called by statement function RATE and all in-
formation which is transmitted between these subprograms is
transferred through calling arguments. The following is a se-
quential list of the calling arguments and their descriptions.
Cl - Value of the coefficient [. ^ 5-] of r0 in equation
(52) [m]. 4ue0E0cos9
CO - Value of one-half the constant term [ (|^-) a3] in equa-
tion (52) [mVv] . K+2
RZERO - Radial distance from the center of a charged particle
and along a given angle 9 for which the total electric
field is zero.
The variables Cl and CO are supplied by statement function RATE
and RZERO is determined in the subroutine.
Subroutine CHGSUM
This subroutine calculates particle charge as a function of
residence time, electrical conditions, gas conditions, and particle
characteristics by using equation (15). Although the subroutine
96
-------
START SUBROUTINE
CALC. ARGUMENT OF COS'1 FUNCTION IN
EQ. (55) [B]
CALL ARCCOS AND CALC.
COS"1B (C)
CALC. FACTOR IN FRONT OF
COS IN EQ. (55) [D]
CALC. RADIAL DISTANCE WHERE
ELECTRIC FIELD IS ZERO FROM
EQ. (55) [R2-ERO]
C
END SUBROUTINE
Figure 19. Flow chart for subroutine ZERO.
97
-------
utilizes equation (15) in a straightforward manner, the programing
is involved in that the values of charge at the end of each length
increment due to field and diffusion charging must be kept track
of independent of one another and the charging process must be
incorporated into the incremental or incremental plus subincre-
mental schemes that may be utilized in the main program0 Since
the free ion density and electric field change along the length
of a precipitator, the values of charge due to field and diffusion
charging must be saved at the end of each increment or subincre-
ment so that they can be used as initial values for the next
increment or subincrement. Also, provisions must be made to
ensure that the charge acquired due to the field charging term
in equation (15) does not exceed the saturation charge in any
given increment or subincrement.
Figure 20 shows a detailed flow chart for this subroutine.
This subroutine is called by the main program and all information
which is transmitted between the main program and this subroutine
is transferred through block common statements. The following is
a list with descriptions of those variables which must be trans-
mitted between the main program and this subroutine:
TDK - Temperature of the gas stream (°K).
U - Gas ion mobility (m2/V-sec).
E - Electronic charge unit (coul).
EPSO - Permittivity of free space (coul/V-m).
BC - Boltzmann's constant (J/°K).
VAVC - Mean thermal speed of gas ions (m/sec).
NVI - Indicator which can have the values of 1 and 2. If
NVI = 1, then known or measured operating voltages
and currents are used in the main program and only
incremental lengths are taken. If NVI = 2, then the
operating voltages and currents are calculated in the
main program and incremental plus subincremental
lengths are taken.
I -
SCHARG -
CHRFID -
TIMEI -
TIMEF -
Index which runs over incremental lengths.
Saturation number from field charging theory.
Average free ion density (#/m3).
Time at the start of a given incremental or subincre-
mental length (sec).
Time at the end of a given incremental or subincre-
mental length (sec) .
98
-------
c
START SUBROUTINE
A
REAL:
LING,
LTHICK,
JPART,
JION
BLOCK COMMON: ZMMDI, SIGMI, NONID,
NRAPD, TDK, NUMSEC, NEFF, NTEMP, GFIT
YES
BLOCK COMMON: VOL, XNO, Q, WS, ITL, DW, AS,
VOS, TCS, WLS, ACS, BS, SYS, VGS, VGASS, TEMPS,
VISS, QSAT, U, E, EPSO, PI, ERAVG, BC, TEMP,
EPS, VAVC, OLDQ, OLDXNO, RFS, START1,
START2, STARTS, VSTAR
BLOCK COMMON:
TMFP, NVI
BLOCK COMMON: NPRINT, NSECT, SLNGTH, A, VO,
TC, B, AC, WL, CL, CD, ET, SY, VGAS, P, VIS,
W, LINC, XPI, RIOVR, EPLT, AFID, XCD, ZMD,
WT, LTHICK, JPART, JION, I, ROVRI
SAVE THE VALUES OF FIELD AND
DIFFUSION CHARGES AT THE START
OF A NEW INCREMENT (SOLDQF AND
SOLDQT)
YES
RESET VALUES OF FIELD AND
DIFFUSION CHARGES EQUAL
TO VALUES AT THE START OF
AN INCREMENT (OLDQF AND
OLDQT)
BLOCK COMMON: SCHARG, CHRFID, TIMEI,
TIMEF, V, FRACTRE, RSIZE, CIMUMBER, J,
II, ITER, OLDQF, OLDQT, SOLDQF, SOLDQT
YES
INITIALIZE FIELD AND DIFFUSION
CHARGES TO 0 AT T = 0: OLDQF(J) = 0.,
OLDQT(J) = 0.
Figure 20. Flow chart for subroutine CHGSUM (Sheet 1 of 3).
99
-------
INITIALIZE FIELD AND DIFFUSION
CHARGES TO 0 AT T = 0: OLDQF(J) = 0,
OLDQT(J) = 0
CALC. SATURATION
CHARGE (SATCHG)
OLDQF(J)>
SATCHG
NO
BYPASS
IF SATU
HAS BEE
QF = OL
.
CALC. QUANTITIES APPEARING IN
EQ. (15) (CF1 AND CF2)
CALC. CHARGE DUE TO
FIELD CHARGING (QF)
PREVENT QF FROM EXCEEDING
SATURATION CHARGE:
QF = SATCHG
Figure 20. Flow chart for subroutine CHGSUM (Sheet 2 of 3).
100
-------
SAVE VALUE OF QF AT END
OF INCREMENT OR SUB
INCREMENT (OLDQF(J))
CALC. ARGUMENT FOR EXPONENTIAL
FUNCTION IN EQ.(15) (ARC)
BYPASS CALC. OF QT
AND SET EQUAL TO
VALUE IN PREVIOUS
INCREMENT OR SUB-
INCREMENT: QT=OLDQT(J)
CALC. CHARGE DUE TO DIFFUSION
CHARGING (QT)
SAVE VALUE OF QT AT END
OF INCREMENT OR SUB-
INCREMENT (OLDQT(J))
CALC. TOTAL CHARGE
NUMBER (CNUMBR)
END OF SUBROUTINE
Figure 20. Flow chart for subroutine CHGSUM (Sheet 3 of 3).
101
-------
V - Value of the quantity
RSIZE - Radius of a given particle size (m).
CNUMBR - Total charge number due to the sum of field and
diffusional charges.
J - Index which runs over particle sizes.
II - Index which runs over subincremental lengths.
ITER - Counter that indicates which iteration is being per-
formed over subincremental lengths in a given incre-
ment (necessary when NVI = 2).
OLDQF(J) - Value of field charge at the end of a given increment
or subincrement (coul).
OLDQT(J) - Value of diffusion charge at the end of a given in-
crement or subincrement (coul).
SOLDQF(J) - Value of field charge at the start of an increment
which must be saved when NVI = 2 for the iteration
procedure over subincrements in a given increment
(coul) .
SOLDQT(J) - Value of diffusion charge at the start of an incre-
ment which must be saved when NVI = 2 for the
iteration procedure over subincrements in a given
increment (coul).
Of the above variables, the values of the following must be
provided by the main program: TDK, U, E, EPSO, BC, VAVC, NVI, I,
SCHARG, CHRFID, TIME I, TIMEF, V, RSIZE, J, II, and ITER. The values
of the following variables are determined in the subroutine: CNUMBR,
OLDQF, OLDQT, SOLDQF, and SOLDQT. In the above arrays, I, J, and
II can not exceed values of 45, 20, and 30, respectively. The
restrictions on I, J, and II limit the number of length increments,
the number of particle size bands, and the number of subincremental
lengths in a given length increment, respectively.
Subroutine ADJUST
This subroutine performs the following operations: (1) it
takes the ideally-calculated effective migration velocities and
adjusts them in order to account for unmodeled and nonideal effects;
(2) it determines the discrete outlet mass loadings, AM/AlogioD,
for each particle size band for no-rap and no-rap + rap conditions
and for the rapping puff; (3) it prints out detailed information
of interest concerning precipitator operating conditions and per-
formance; and (4) it prints out a table which summarizes precipi-
tator operating conditions and performance. In order to use this
102
-------
subroutine, subroutines WADJST, LNDIST, LNFIT, and PRTSUM must be
supplied.
The first calculation of significance which is performed is
the determination of the unadjusted, ideal overall mass collection
fraction (X). This quantity is determined by using the expression
V" ---i
(ONO).
1 '
(PCNT).
i
(PCNT), , (56)
where (DXS). = the number of particles per cubic meter of gas
in the i-th size band collected over the entire
length of the precipitator under unadjusted,
ideal conditions (#/m3),
(ONO). = the number of particles per cubic meter of gas
in the i-th size band in the inlet size distri-
bution (#/m3),
(PCNT). = the fraction by mass of the i-th particle size
band in the inlet size distribution, and
(EFESR). = the unadjusted, ideal collection fraction for
the i-th particle size band over the entire
length of the precipitator.
The rest of the subroutine is structured around two major
loops. The outside loop runs over different "rapping puff" size
distributions. The variable NRAPDC is a counter for this loop
and it runs over a number of different "rapping puff" size dis-
tributions which is equal to the specified value of the variable
NRAPD. The inside loop runs over different sets of nonideal con-
ditions of gas velocity nonuniformity, gas sneakage, and particle
reentrainment without rapping. The variable NONCK is a counter for
this loop and it runs over a number of sets of nonideal conditions
which is equal to the specified value of the variable NONID.
103
-------
The initial rapping puff size distribution is fixed to be a
log-normal distribution with a MMD = 6.0 ym and a a = 2.5. These
values correspond to the field data discussed previously and were
obtained from Figure 6. Other distributions can be analyzed in
the procedure discussed previously by specifying different sets
of values of the MMD and a in the input data to the main program.
For each specified set of MMD and a , the subroutine constructs a
log-normal size distribution by calling subroutine LNDIST. The
percentage by mass of each particle size band in the rapping puff
is stored in the array RPCNT(I).
After the rapping puff size distribution is established in
the outer loop, the nonideal conditions of gas velocity nonuni-
formity, gas sneakage, and particle reentrainment without rapping
are established in the inner loop. At this point corrections are
made to the unadjusted, ideal migration velocity for each particle
size band in order to account for unmodeled and nonideal effects.
The unadjusted, ideal migration velocity (WY) for each size band
is calculated from the expression
WY = (VG/ATOTAL)•100'ln (100/(100-XEP)) , (57)
where VG = total gas volume flow rate (m3/sec),
ATOTAL = total collection plate area (m2), and
XEP = 100'EFESR (%).
However, if EFESR >_ 0.99999, WY is set equal to the value of the
unadjusted, ideal migration velocity in the last increment of
the precipitator.
The ideal effective migration velocities are corrected first
for gas velocity nonuniformity using equation (17) and then for
gas sneakage and/or particle reentrainment without rapping using
equation (19) 0 The resulting migration velocities are repre-
sentative of unadjusted no-rap conditions. These unadjusted no-
rap migration velocities are then corrected for unmodeled effects
by using subroutine WADJST which applies the size-dependent cor-
rection factor shown in Figure 7 to each particle size band. The
resulting migration velocities (WY) will be referred to as the
no-rap migration velocities. No-rap collection fractions (EFESR)
are determined from the no-rap migration velocities using equation
(2) • Again, if EFESR >, 0.99999, then WY is set equal to the value
of the unadjusted, ideal migration velocity in the last increment
of the precipitator. The no-rap collection fractions for the
different size bands are used to calculate a no-rap overall mass
collection fraction using the same format as in equation (56).
No-rap penetrations are also calculated.
104
-------
The next set of calculations which are performed reduces the
no-rap migration velocities in order to account for the effects
of rapping reentrainment by using the procedure discussed pre-
viously. The total mass which is reentrained due to rapping is
determined by using either equation (24) or (25). The mass
collected in the last section is calculated using the no-rap
overall mass collection efficiency and equation (22). The total
mass which is reentrained due to rapping and the rapping puff
size distribution are used to determine the number of particles
in each size band which is reentrained. The number of particles
reentrained is subtracted from the total number of particles
collected under no-rap conditions to give the number of particles
collected under no-rap + rap conditions. If the number of par-
ticles collected under no-rap + rap conditions is calculated to
be a negative number, the number collected under no-rap con-
ditions is used in its place. The number of particles in each
size band collected under no-rap + rap conditions and the number
of particles in each size band in the inlet size distribution
are used to calculate no-rap + rap collection efficiencies,
penetrations, migration velocities, and overall mass collection
efficiency.
Next, several calculations are made to describe the outlet
emissions under no-rap and no-rap + rap conditions and for the
rapping puff. In each case, the size band penetrations are
normalized and the outlet size distribution is obtained. These
size distributions are then fitted to a log-normal distribution
by calling subroutine LNFIT. Also, in each case, the discrete
outlet mass loadings are determined by calculating AM/AlogioD
for each size band.
The results of the calculations discussed above are printed
out in three different sections: (1) Particle Size Range
Statistics; (2) Unadjusted Migration Velocities and Efficiencies,
and Discrete Outlet Mass Loadings; and (3) Summary Table of ESP
Operating Parameters and Performance. The third section of
printout is obtained by calling subroutine PRTSUM. The output
data from the program is discussed in detail in Volume II and
will not be discussed further here.
Figure 21 shows a detailed flow chart for this subroutine.
This subroutine is called by the main program and all information
which is transmitted between the main program, subroutine PRTSUM,
and this subroutine is transferred through block common state-
ments. The following is a list with descriptions of those vari-
ables which must be transmitted between the main program and
this subroutine.
DIAM(I) - Midpoint of a given particle size band (m).
ONO(I) - Number of particles per unit volume of gas for a
given particle size band entering the precipitator
(#/m3).
105
-------
START SUBROUTINE
DOUBLE PRECISION:
EFESR, DLOG
REAL:
LINCS
DIMENSION: RPCNT, DMDLD, WUNCOR, RDMDLD,
CDMDLD, PCTOT, CPCTOT, WSL, PXS, PRCUNR,
RPRCU,PRCUC, EUNCOR
BLOCK COMMON: DIAM, ONO, DXS,
XMV, PCNT, RAD, CCF, PRCU
BLOCK COMMON: LSECT, LINCS, PS
BLOCK COMMON: VG, ATOTAL, DD,
ETAO, DL, PL, RHO
BLOCK COMMON:
NS
BLOCK COMMON: ZMMDI, SIGMI, IMONID,
NRAPD, TDK, NUMSEC, NEFF, NTEMP, GFIT
BLOCK COMMON: VOL, XNO, Q, WS, ITL, DW, AS, VOS,
TCS, WLS, ACS, BS, SYS, VGS, VGASS, TEMPS, VISS, QSAT,
U, E, EPSO, PI, ERAVG, BC, TEMP, EPS, VAVC, OLDQ,
OLDXNO, RFS, STARTI, START2, START3, VSTAR
BLOCK COMMON: ENDPT, NENDPT
BLOCK COMMON: ARD50, ARSIGM,
ASNUCK, AZNUMS, AZIGGY
BLOCK
COMMON:
NREAD,
NPRNT
BLOCK COMMON: LK, DV, NN, NUMINC, NX, NY, NDATA, NEST,
NDIST, NITER, IFINAL, JI1, JI2, VISKIP, VISAME, US,
FPATH, EBD, NDSET, NWS, D50, SIGMAP
Figure 21. Flow chart for subroutine ADJUST (Sheet 1 of 12).
106
-------
SET DATA SET INDICATOR EQUAL TO 0 (NRUN)
ESTABLISH NUMBER OF ENDPOINTS IN
PARTICLE SIZE HISTOGRAM (NSI)
ESTABLISH NUMBER OF ELECTRICAL
SECTIONS PRIOR TO LAST ONE (NUMSI)
CALC. FACTOR FOR CONVERTING
FROM gr/acf TO mg/DNCM (CONVF)
START LOOP OVER
PARTICLE SIZES
CALC. IDEAL COLLECTION
FRACTION (EFESR)
YES
NO
SUM TO CALC. OVERALL IDEAL
COLLECTION FRACTION (X)
c
END LOOP OVER
PARTICLE SIZES
NRAPDC
= NRAPDC + 1
EFESR
= 0.999999
ARD50(1) = 6.0
ARSIGMd) 2.5
RMMD = 6.0
RSIGMA 2.5
RMMD = ARD50(NRAPDC)
RSIGMA - ARSIGM(NRAPDC)
J.
Figure 21. Flow chart for subroutine ADJUST (Sheet 2 of 12).
107
-------
w
CALL LNDIST: DETERMINE LOG-NORMA
DISTRIBUTION FOR RAPPING PUFF
( START LOOP OVER ^
1 PARTICLE SIZES J
L
Y = 0.0
A r START LOOP OVER"\
V^PARTICLE SIZE J
RPCNT(I) RPCNTID-100
( END LOOP OVER A
\^ PARTICLE SIZES J
NONCK = 0
NONCK NONCK + 1
SNUCK = ASNUCK(NONCK)
ZIGGY AZIGGY(NONCK)
ZNUMS - AZNUMS(NONCK)
/PRINT OUT HEADING FOR DATA /
/ OUTPUT SECTION /
/PRINT OUT WHICH SET OF NONIDEAL /
CONDITIONS IS BEING CONSIDERED /
/PRINT OUT HEADINGS /
/ FOR TABLE /
CALC. IDEAL COLLECTION
EFFICIENCY (EFESR)
'
NO EFESR ='0.999999
CONVERT EFFICIENCY
TO PERCENT (XEP)
SALI ._ _
<^p^gp ^>
\r
I\1O
XEP = 99.9999
^\ YES
NO
SET IDEAL MIGRATION VELOCITY
EQUAL TO THAT AT END OF LAST
INCREMENT OF ESP (WY)
CALC. IDEAL MIGRATION ' '
VELOCITY (WY)
Figure 21. Flow chart for subroutine ADJUST (Sheet 3 of 12).
108
-------
YES
SET CORRECTION FACTOR FOR
VEL. DIST. EQUAL TO 1. (F1>
CALC. CORRECTION FACTOR
FOR VEL. DIST. (Fl)
YES
SET CORRECTION FACTOR FOR
SNEAKAGE EQUAL TO 1. (F2)
CALC. CORRECTION FACTOR
FOR SNEAKAGE (F2)
CALC. MIGRATION VELOCITY
CORRECTED FOR SNEAKAGE (WYS)
CALC. MIGRATION VELOCITY
CORRECTED FOR VEL. DIST. (WYV)
CALC. COMBINED
CORRECTION FACTOR (ZNLFF)
CALC. MIGRATION VELOCITY CORRECTED
FOR SNEAKAGE AND VEL. DIST. (WYSV)
STORE UNCORRECTED IDEAL
MIGRATION VELOCITIES (WUNCOR)
STORE UNCORRECTED IDEAL
EFFICIENCIES (EUNCOR)
CALL WADJST: ADJUST NO-RAP MIGRATION
VELOCITY AND EFFICIENCY (WYSV AND EFESR)
Figure 21. Flow chart for subroutine ADJUST (Sheet 4 of 12).
109
-------
CONVERT EFFICIENCY
TO PERCENT (XEP)
YES
NO
XEP = 99.9999
1
NO
CALC. ADJUSTED NO-RAP
MIGRATION VELOCITY (WY)
CALC. NO. OF PARTICLES COLLECTED UNDER
ADJUSTED NO-RAP CONDITIONS (PXS)
SUM TO CALC. ADJUSTED NO-RAP
OVERALL COLLECTION FRACTION (Y)
END LOOP OVER
PARTICLE SIZE
SET ADJUSTED NO-RAP MIGRATION
VELOCITY EQUAL TO UNADJUSTED
IDEAL VALUE AT END OF LAST
INCREMENT OF ESP (WY)
Figure 21. Flow chart for subroutine ADJUST (Sheet 5 of 12).
110
-------
START LOOP OVER
PARTICLE SIZES
CALC. ADJUSTED NO-RAP
EFFICIENCY (EFESR)
CONVERT EFFICIENCY
TO PERCENT (XEP)
/ . YES
EFESR > 0.99999 > ^
NO
CALC. PARTICLE
PENETRATION (PENTR)
CALC. FRACTION PENETRATING
ESP (PCTOT)
SET ADJUSTED NO-RAP MIGRATION
VELOCITY EQUAL TO UNADJUSTED
IDEAL VALUE AT END OF LAST
INCREMENT OF ESP (WY)
CALC. ADJUSTED NO-RAP
MIGRATION VELOCITY (WY)
A
EXPRESS INLET SIZE
FRACTIONS AS PERCENTAGES (XY)
YES
NO
CLPTLS
= 0.
START LOOP
OVER NUMSI SECTIONS
SUM LENGTH OF ESP UP
TO LAST SECTION (CLPTLS)
END LOOP
OVER NUMSI SECTIONS
)
NYX = r>
JYX + 1
Figure 21. Flow chart for subroutine ADJUST (Sheet 6 of 12).
Ill
-------
CALC. VALUE OF EXPONENT IN EQ.(2)
FOR CHOSEN EFFICIENCY (EXPONT)
CALC. MASS ENTERING
LAST SECTION (XMELS)
CALC. MASS COLLECTED IN
LAST SECTION (XMCLS)
CALC. MASS LEAVING
LAST SECTION (XMLLS)
CONVERT XMCLS TO
mg/DNCM
YES
\
NTEIV
/ - 1
NO CALC. RAPPING LOSS
FOR COLD ESP (RAPLOS)
- 1
r\YES fc
I'T.Q CALC. RAPPING LOSS
FOR HOT ESP (RAPLOS)
1
^ i
Figure 21. Flow chart for subroutine ADJUST (Sheet 7 of 12).
112
-------
CALC. VALUE OF EXPONENT IN EQ.(2)
FOR NO-RAP EFFICIENCY (EXPONT)
CALC. MASS ENTERING
LAST SECTION (YMELS)
CALC. MASS COLLECTED IN
LAST SECTION (YMCLS)
CALC. MASS LEAVING
LAST SECTION (YMLLS)
CONVERT YMCLS TO
mg/DNCM
YES
CALC. NUMBER OF PARTICLES
FOR A GIVEN SIZE BAND IN
THE RAPPING PUFF (RNS)
WYSV
= WY
CALC. PARTICLE COLLECTION EFFICIENCY
WITH RAPPING LOSS (EFFWR)
CALC. NO. OF PARTICLES
COLLECTED AFTER
RAPPING (CRNP)
EFFWR
EFESR
CALC. MIGRATION VELOCITY CORRECTED
FOR VEL. DIST., SNEAKAGE, AND
RAPPING (WYP)
SET NO-RAP MIGRATION VELOCITY
CORRECTED FOR RAPPING LOSS
EQUAL TO NO-RAP VALUE (WYP)
Figure 21. Flow chart for subroutine ADJUST (Sheet 8 of 12).
113
-------
CALC. CORRECTED PERCENT
EFFICIENCY (COREFF)
SUM TO CALC. CORRECTED OVERALL
COLLECTION EFFICIENCY (SCOREF)
CALC. CORRECTED PERCENT
PENETRATION (CPENTR)
CALC. CORRECTED FRACTION
PENETRATING ESP (CPCTOT)
YES
SUM NO-RAP PARTICLE SIZE FRACTIONS
PENETRATING ESP (SPO)
SUM NO-RAP + RAP PARTICLE SIZE
FRACTIONS PENETRATING ESP (SCPO)
CALC. NUMBER OF PARTICLES
PENETRATING ESP (SL)
CALC. MASS PENETRATING
ESP (WSL)
NO
CALC. NO-RAP PERCENT IN
OUTLET SIZE DISTRIBUTION (PCTOT)
CALC. NO-RAP + RAP PERCENT IN
OUTLET SIZE DISTRIBUTION (CPCTOT)
•^.^
Figure 21. Flow chart for subroutine ADJUST (Sheet 9 of 12)
114
-------
CALC. d(LOGD) FOR
SIZE BAND (OLD)
CALC. NO-RAP dM/d( LOGO) FOR
SIZE BAND (DMDLD)
'PRINT ADJUSTED NO-RAP EFFICIENCY AND FITTED
MMD AND Op OF INLET SIZE DIST.
(Y, ZMMDI, AND SIGMI)
YES
CALC. RAP dlWd(LOGD) FOR
SIZE BAND (RDMDLD)
CALC. NO-RAP + RAP dM/d(LOGD)
FOR SIZE BAND (CDMDLD)
PRINT GOODNESS OF FIT FOR
INLET SIZE DIST. (GFIT)
'PRINT DIAM, CCF, XY, PCTOT, CPCTOT,
XEP, WY, PENTR, COREFF, WYP, AND
CPENTR IN TABLE
INITIALIZE CUMULATIVE % FOR THE
SMALLEST PARTICLE SIZE BAND IN THE
OUTLET NO-RAP SIZE DIST. (PRCUNR(D)
INITIALIZE CUMULATIVE
% SUM (SUMNR)
END LOOP OVER
PARTICLE SIZE
YES
START LOOP OVER
PARTICLE SIZE BANDS
CONVERT OVERALL NO-RAP COLLECTION
FRACTION TO PERCENT (Y)
SUM CUMULATIVE PERCENTS FOR THE
OUTLET NO-RAP SIZE DIST. (SUMNR)
CONVERT OVERALL IDEAL COLLECTION
FRACTION TO PERCENT (X)
ESTABLISH CUMULATIVE PERCENTS UP TO
A GIVEN PARTICLE SIZE (PRCUNR(l)l
END LOOP OVER
PARTICLE SIZE BANDS
PRINT STATED OR DESIGN EFFICIENCY AND
IDEAL EFFICIENCY (ETAO AND X)
Figure 21, Flow chart for subroutine ADJUST (Sheet 10 of 12).
115
-------
FIND FITTED MMD AND Op OF
CALL LNFIT:
NO-RAP EFFLUENT AND GOODNESS OF FIT
PRINT FITTED NO-RAP OUTLET SIZE DIST. MMD
AND Op AND GOODNESS OF FIT (ZMDL, SIGMO,
AND ZGFIT)
END LOOP OVER
PARTICLE SIZE BANDS
CALL LNFIT:
NO-RAP + RAP EFFLUENT AND GOODNESS OF FIT
FIND FITTED MMD AND 0fl of
CALC. OVERALL NO-RAP + RAP
MIGRATION VELOCITY (COREFW)
A
PRINT NONIDEAL PARAMETERS
(ZIGGY, SNUCK, ZNUMS)
CALC. OVERALL ADJUSTED NO-RAP
MIGRATION VELOCITY (WZ)
PRINT TEMPERATURE INDICATOR, AND RAPPING
PUFF SIZE DIST. MMD AND ap (NTEMP, RMMD,
AND RSIGMA)
PRINT NO-RAP PRECIPITATION
RATE PARAMETER (WZ)
PRINT NO-RAP + RAP EFFICIENCY, FITTED MMD
AND ap, AND GOODNESS OF FIT (SCOREF, CZMDL,,
(CSIGMO, AND CGFIT)
INITIALIZE CUMULATIVE % FOR THE
SMALLEST PARTICLE SIZE BAND IN THE
OUTLET NO-RAP + RAP SIZE DIST. (PRCUC(D)
PRINT NO-RAP + RAP PRECIPITATION
RATE PARAMETER (COREFW)
INITIALIZE CUMULATIVE % SUM (SUMO
SET UP TABLE HEADINGS
START LOOP OVER
PARTICLE SIZE BANDS
START LOOP OVER
PARTICLE SIZE
A
SUM CUMULATIVE PERCENTS FOR THE
OUTLET NO-RAP + RAP SIZE DIST. (SUMO
ESTABLISH CUMULATIVE PERCENTS UP TO
A GIVEN PARTICLE SIZE (PRCUC(D)
PRINT WUNCOR, DMDLD, RDMDLD,
CDMDLD AND DIAM IN TABLE
END LOOP OVER
PARTICLE SIZE BANDS
•112 I
Figure 21. Flow chart for subroutine ADJUST (Sheet 11 of 12).
116
-------
NRUN = NRUN + 1
CALL PRTSUM: PRINT OUT
SUMMARY TABLE
YES
c
END SUBROUTINE
Figure 21. Flow chart for subroutine ADJUST (Sheet 12 of 12).
117
-------
DXS(I) - Number of particles per unit volume of gas for a
given particle size band which are removed from the
gas stream under unadjusted, ideal conditions (#/m3).
XMV(I) - Unadjusted, ideal migration velocity for a given par-
ticle size band (cm/sec).
PCNT(I) - Fraction by mass of a given particle size band in the
inlet particle size distribution.
CCF(I) - Cunningham correction factor for a given particle
size band.
LSECT(J) - Number of increments to be taken in a given electrical
section.
LINGS(J) - Incremental length size to be taken in a given elec-
trical section (ft).
PS(J) - Gas pressure in a given electrical section (atm).
VG - Gas volume flow rate (m3/sec).
ATOTAL - Total collection plate area (m2).
DD - Mass density of the particles (kg/m3).
ETAO - Estimated or design efficiency (%).
DL - Inlet mass loading (kg/m3).
PL - Total electrical length of the precipitator (m).
RHO - Resistivity of collected particulate layer (ohm-m).
NS - Number of size bands in inlet particle size histogram.
ZMMDI - Fitted mass median diameter of the inlet particle size
distribution (m).
SIGMI - Fitted geometric standard deviation of the inlet par-
ticle size distribution.
NONID - Total number of sets of nonideal conditions of gas
velocity nonuniformity and gas sneakage and/or par-
ticle reentrainment without rapping to be considered.
NRAPD - Total number of rapping puff size distributions to
be considered.
TDK - Temperature of the gas stream (°K).
118
-------
NUMSEC - Number of electrical sections in the direction of
gas flow.
NEFF - Indicator which can have the values of 1 and 2. If
NEFF = 1, then the total mass reentrained at the
outlet due to rapping is determined from the mass •
collected in the last field under adjusted no-rap
conditions. If NEFF = 2, then the total mass re-
entrained at the outlet due to rapping is determined
from the mass which would be collected in the last
field under unadjusted, ideal conditions.
NTEMP - Indicator which can have the values of 1 and 2. If
NTEMP = 1, then the mass reentrained due to rapping
is calculated based on equation (24) for a cold-side
precipitator. If NTEMP = 2, then the mass reentrained
due to rapping is calculated based on equation (25).
for a hot-side precipitator.
GFIT - Log-normal goodness of fit parameter for the fitted
inlet particle size distribution.
VOS(J) - Applied voltage in a given electrical section (V).
TCS(J) - Total current in a given electrical section (A).
ENDPT(K) - Endpoints of the particle size band intervals in the
inlet particle size histogram (ym).
NENDPT - Number of endpoints in the inlet particle size dis-
tribution histogram.
ARD50(L) - Specified mass median diameter used to describe a
log-normal particle size distribution for the rapping
puff (ym).
ARSIGM(L) - Specified geometric standard deviation used to de-
scribe a log-normal particle size distribution for
the rapping puff.
ASNUCK(M) - Specified fraction of gas flow which bypasses the
electrified region in each baffled stage of the pre-
cipitator and/or fraction of the mass collected in
each stage of the precipitator which is reentrained
due to factors other than rapping.
AZNUMS(M) - Specified number of baffled stages in the precipitator,
AZIGGY(M) - Specified normalized standard deviation of the inlet
gas velocity distribution.
NPRNT - Indicator whose value must be that which designates
the print unit for a given machine.
119
-------
SCOREF - Overall mass collection efficiency under no-rap +
rap conditions (%).
CZMDL - Fitted mass median diameter of the outlet particle
size distribution under no-rap + rap conditions (ym) .
CSIGMO - Fitted geometric standard deviation of the outlet
particle size distribution under no-rap + rap con-
ditions .
NRUN - Indicator that specifies which set of nonideal con-
ditions is under consideration.
SNUCK - Particular value of ASNUCK(M).
ZIGGY - Particular value of AZIGGY(M).
RMMD - Particular value of ARD50(L) [ym] .
RSIGMA - Particular value of ARSIGM(L).
D50 - Same as ZMMDI (ym).
SIGMAP - Same as SIGMI.
Of the above variables, the values of the following must be
provided by the main program: DIAM, ONO, DXS, XMV, PCNT, CCF,
LSECT, LINGS, PS, VG, ATOTAL, DD, ETAO, DL, PL, RHO, NS, ZMMDI,
SIGMI, NONID, NRAPD, TDK, NUMSEC, NEFF, NTEMP, GFIT, VOS, TCS,
ENDPT, NENDPT, ARD50, ARSIGM, ASNUCK, AZNUMS, AZIGGY, NPRNT, D50,
and SIGMAP. The values of the following variables are determined
in the subroutine: SCOREF, CZMDL, CSIGMO, NRUN, SNUCK, ZIGGY,
RMMD, and RSIGMA. The values of these variables must be supplied
to subroutine PRTSUM. In the above arrays, I, J, K, L, and M
can not exceed values of 20, 10, 21, 10, and 15, respectively.
The restrictions on I, J, K, L, and M limit the number of par-
ticle size bands, the number of electrical sections, the number
of particle diameters in the inlet particle size distribution,
the number of rapping puff particle size distributions, and the
number of sets of nonideal conditions of a and S, respectively.
Subroutine WADJST
This subroutine adjusts the no-rap migration velocities by
applying the empirical correction factors given in Figure 7.
These correction factors and their corresponding particle sizes
are tabulated for 24 particle diameters between 0.2 ym and 4.5 um
in data statements. Table 1 shows the particle sizes and cor-
rection factors which are tabulated. Correction factors for
no-rap migration velocities for particle diameters in the range
0.2 ym - 4.5 ym are found by interpolating the table. No-rap
120
-------
TABLE 1. PARTICLE SIZES AND CORRECTION FACTORS FOR
NO-RAP MIGRATION VELOCITIES TABULATED
IN SUBROUTINE WADJST
Particle Correction Particle Correction
Diameter (ym) Factor Diameter (ym) Factor
0.20 2.430 0.80 1.790
0.25 2.325 0.85 1.760
0.30 2.240 0.90 1.740
0.35 2.170 0.95 1.710
0.40 2.110 1.00 1.685
0.45 2.050 1.50 1.500
0.50 2.000 2.00 1.370
0.55 1.965 2.50 1.270
0.60 1.920 3.00 1.180
0.65 1.885 3.50 1.115
0.70 1.850 4.00 1.050
. 0.75 1.820 4.50 1.000
121
-------
migration velocities outside this range are left unchanged. Based
on the adjusted no-rap migration velocities, the subroutine cal-
culates for each particle size an adjusted no-rap collection
fraction and the number of particles removed.
Figure 22 shows a detailed flow chart for this subroutine.
This subroutine is called by subroutine ADJUST and all information
which is transmitted between these subprograms is transferred
through calling arguments. The following is a sequential list
of the calling arguments and their descriptions.
DIAM(I) - Midpoint of a given particle size band (m).
I - Index which specifies the different particle diameters.
WY - Enters the subroutine as a no-rap migration velocity
and returns as an adjusted no-rap migration velocity
(cm/sec).
ONO(I) - Number of particles per unit volume of gas for a given
particle size band entering the precipitator (#/m5) .
PXS(I) - Number of particles per unit volume of gas for a given
particle size band which are removed from the gas
stream under adjusted no-rap conditions (#/m3).
ATOTAL - Total collection plate area (m2).
VG - Gas volume flow rate (m3/sec).
EFESR - Enters the subroutine as a no-rap collection fraction
and returns as an adjusted no-rap collection fraction.
All of the above variables must be supplied by subroutine ADJUST.
The values of WY and EFESR are replaced by new values for particle
sizes in the range 0.2 - 4.5 ym. In all of the uses above, I can
not exceed a value of 20. The restriction on the value of I
limits the number of particle size bands.
Subroutine LNDIST
This subroutine constructs a particle size distribution
histogram for a specified log-normal distribution. For specified
particle diameters, the fraction in each particle si'ze band and
cumulative fraction less than each particle size are determined.
In order to use this subroutine, subroutine QTFE must be supplied.
The log-normal distribution function [f (z)] is given by
the expression
fL-VZ> = a"
/2TT
EXP
(z-z)
"
(58)
122
-------
START LOOP OVER
PARTICLE SIZE
START SUBROUTINE
DOUBLE PRECISION: EFESR
DIAM(I) > DCHECK(L)
AND
IAM(I) < DCHECKIL + 1
CALC. CORRECTION
FACTOR (WFACT)
DIMENSION: DIAM, ONO, PXS,
CFACT, DCHECK
DATA: CFACT, DCHECK
(ESTABLISH TABLE OF CORRECTION
FACTORS FOR THE NO-RAP MIGRATION
VELOCITIES)
CALC. CORRECTED
MIGRATION VELOCITY (WY)
END LOOP OVER
PARTICLE SIZE
CALC. CORRECTED
EFFICIENCY (EFESR)
YES/"DIAM(I)<2
DIAM(I)
CALC.CORRECTED NUMBER OF
PARTICLES REMOVED (PXS)
CALC. CORRECTED
EFFICIENCY (EFESR)
CALC. CORRECTED NUMBER OF
PARTICLES REMOVED (PXS)
END SUBROUTINE
Figure 22. Flow chart for subroutine WADJST.
123
-------
where
a = In a , (59)
z p
z = In d , (60)
z" = In d50 f
and
d = particle diameter (jam) ,
d50 = mass median diameter for the distribution (ym) ,
a = geometric standard deviation for the distribution,
z = independent variable for the log-normal distribution,.
~z = mean value of z, and
a = standard deviation of z.
z
fT ,.,(z) dz represents the amount of mass (or other variable if
J_i"~N
desired) in the range between z and z + dz. The distribution is
completely described by specifying the values of dso and a .
The subroutine constructs the log-normal distribution histo
gram by (1) determining the total mass contained between z\ =
In 0.01 and z = In 1000.0, (2) calculating the mass contained
in each size band specified by the user, and (3) calculating the
ratios of the mass contained in each size band to the total mass.
The total mass (M) contained in the distribution is obtained in
cumulative steps in the form
M =
Z 2
/ fL-N(z)dZ H
Z 1
Z3
h/ fL-N(z)dZ H
Z2
Zi,
h / fL-N(2)dZ
Z3
z z
n-1 n
fL-N(2)dZ + J fL-N(z)dz • (62)
Zn-2 Zn-l
The integrals in equation (62) are evaluated numerically by
calling subroutine QTFE which utilizes the Trapezoidal Rule. Each
integration is performed by dividing the size band into 99 inter-
vals and evaluating the integrand at 100 points. The value of
124
-------
each integral is stored as well as the cumulative sum. The user
specifies the particle diameters in jam which correspond to the
values of z from z2 to z where z2 > -4.605 and z , < 6.908
n-i n-1
(d2 > 0.01 and dn_1 < 1000.0).
The mass fractions (F^ for the size bands in equation (62)
are obtained from the expressions
Fi =
F, = / fT_kT(z)dz /M
f
•/
Fn-l = fL-N(z)dZ/M
where there are n-1 size bands. Since the size bands specified
by the user are contained in the range from z2 to z .. , the
excess mass fractions in the size bands zi to z2 and z _, to
z are added to the mass fraction in the size band z „ to z ,
n n-2 n-1
for the histogram which is returned from the subroutine. This
is done to ensure that the size distribution used in the model
accounts for 100% of the mass.
The cumulative mass fractions (S.) for the specified diameters
and size bands are obtained from
C? — TJ1
S2 - r i
S3 =
125
-------
3
^
S4 =
n-1 / n-1
S^ = > F, + [ 1 - > , F. | , (64)
where the cumulative mass fraction less than -the largest specified
diameter is constrained to be a value of 1 by adjusting the mass
fraction in the size band from zn_2 to zn_i'
Figure 23 shows a detailed flowchart for this subroutine.
This subroutine is called by the main program and subroutine ADJUST.
Information which is transmitted to and from this subroutine is
transferred through calling arguments and block common statements.
The following is a sequential list of the calling arguments and
their descriptions.
D50 - Specified mass median diameter for a log-normal dis-
tribution (ym) ,
SIGMAP - Specified geometric standard deviation for a log-normal
distribution,
PRCU(I) - Cumulative mass fractions for the log-normal distri-
bution, and
PCNT(J) - Mass fraction in a given size band of the log-normal
distribution.
The following is a list of the necessary variables which are
in common with the main program and subroutine ADJUST.
NS - Number of specified particle size bands in the histogram
for the log-normal distribution,
ENDPT(I) - Specified endpoints of the particle size band intervals
in the histogram for the log-normal distribution (ym) ,
and
NENDPT - Specified number of endpoints in the histogram for the
log-normal distribution.
Of the above variables, the values of the following must be
provided by the calling program or subprogram: D50, SIGMAP, NS,
ENDPT, and NENDPT. PRCU and PCNT are determined in the subroutine.
I and J can not exceed values of 21 and 20, respectively. The
126
-------
c
START SUBROUTINE
DIMENSION: Y, Z, AREA,
PRCU, PCNT
BLOCK COMMON: NS
BLOCK COMMON: ENDPT,
NENDPT
SPECIFY VALUE OF 7T
DEFINE CT,
DEFINE VALUE OF z NEXT
TO THE LARGEST AND THE
LARGEST VALUE (X1, X2)
f
I
YES
ESTABLISH NO. OF PARTICLE
SIZES USED TO CONSTRUCT
HISTOGRAM FROM (N)
ESTABLISH NO. OF POINTS AT WHICH
TO EVALUATE LOG-NORMAL DISTRIBUTION
FUNCTION FOR INTEGRATION OVER THE
DIFFERENT SIZE BANDS (NINO
INITIALIZE CUMULATIVE MASS FRACTION
SUMMATION EQUAL TO 0 (ASUM)
SET INDEX OVER SPECIFIED SIZE
BANDS EQUAL TO 0 (K)
START LOOP OVER PARTICLE
SIZE BANDS IN HISTOGRAM
YES
DEFINE SMALLEST VALUE OF Z
AND NEXT VALUE UPWARD (X1, X2)
DEFINE UPPER AND LOWER VALUE OF
z FOR A GIVEN SIZE BAND (X2, X1)
ESTABLISH STEPSIZE FOR INTEGRATION
OVER A GIVEN SIZE BAND (DX)
-------
ESTABLISH LOWER LIMIT
OF INTEGRATION (D)
EVALUATE THE QUANTITY
[SGT1]
EVALUATE THE QUANTITY
2az2 (SGT2)
START LOOP OVER
INTEGRATION POINTS
"\
J
EVALUATE LOG-NORMAL DISTRIBUTION
FUNCTION AT INTEGRATION POINTS (Y)
ESTABLISH NEXT POINT AT WHICH
TO EVALUATE THE FUNCTION (D)
END OF LOOP OVER
INTEGRATION POINTS
CALL QTFE: INTEGRATE TO FIND MASS
IN A GIVEN SIZE BAND
SUM MASS CONTAINED IN
SUCCESSIVE SIZE BANDS (ASUM)
CALC. CUMULATIVE FRACTION
LESS THAN A GIVEN PARTICLE
SIZE (PRCU)
[
K
= K + 1
ESTABLISH MASS IN A
GIVEN SIZE BAND (AREA)
END OF LOOP OVER PARTICLE
SIZE BANDS IN HISTOGRAM
START LOOP OVER SPECIFIED
SIZE BANDS
CALC. FRACTION IN EACH
SPECIFIED SIZE BAND
END LOOP OVER SPECIFIED
SIZE BANDS
^
J
INITIALIZE CUMULATIVE SUM OVER
SPECIFIED SIZE BANDS TO 0 (SUM)
START LOOP OVER SPECIFIED
SIZE BANDS
CALC. CUMULATIVE SUM OVER
SPECIFIED SIZE BANDS
CALC. DIFFERENCE BETWEEN 1
AND THE CUMULATIVE SUM
OVER THE SPECIFIED SIZE
BANDS (CHECK1)
•®
Figure 23. Flow chart for subroutine LNDIST (Sheet 2 of 3).
128
-------
ADJUST FRACTION IN LAST SPECIFIED
SIZE BAND TO ENSURE CUMULATIVE
SUM OF 1 (PCNT(NS))
CALC. DIFFERENCE BETWEEN 1
AND THE CUMULATIVE SUM
OVER ALL SIZE BANDS USED
IN THE PROCEDURE (CHECK2)
ADJUST LAST SPECIFIED CUMULATIVE
FRACTION TO 1 (PRCU(NENDPT))
END OF SUBROUTINE
Figure 23. Flow chart for subroutine LNDIST (Sheet 3 of 3).
129
-------
restrictions on I and J limit the number of particle diameters in
the inlet particle size distribution and the number of particle
size bands, respectively.
Subroutine QTFE
This subroutine performs the integration of an equidistantly
tabulated function by the trapezoidal rule.37 Cumulative integral
values (Z.) are determined by
x.
/-1
ZL = Zi(x) = / y(x)dx , (65)
where x. = a + (i-l)h and i=l/2,••••,n. The function values y.
are tabulated at the equidistant points x., where h is the incre-
ment size for the integration. Starting with the integral value
Zi = 0, successive integral values Z. (i=2,3,••••,n) are computed
by using the trapezoidal rule in the form
Z. = Z.^ + | (y, + y.^) . (66)
In applying the trapezoidal rule, it is assumed that the function
to be integrated is continuous and can be differentiated at least
twice.
This subroutine is called by subroutine LNDIST. Figure 24
shows a detailed flow chart for this subroutine. All information
which is transmitted between subroutine LNDIST and this subroutine
is transferred through calling arguments. The following is a
sequential list of the calling arguments and their descriptions.
DX - Increment size for the integration.
Y(I) - Table of function values at the equidistant points
used in the integration procedure.
Z(I) - Cumulative integral values.
NINC - Number of points at which the function to be integrated
is evaluated.
Of the above variables, the values of DX, Y, and NINC must be
provided by the calling program (subroutine LNDIST). The values
of Z are returned from subroutine QTFE.
130
-------
c
START SUBROUTINE
DIMENSION:
Y.-2
INITIALIZE CUMULATIVE
INTEGRAL TO 0 (SUM2)
<0
>0
CALC. HALF-INCREMENT
SIZE (DDX)
START INTEGRATION
LOOP
CALC. (i-D-TH CUMULATIVE
INTEGRAL (SUM1)
CALC. i-th CUMULATIVE
INTEGRAL (SUM2)
STORE (i-1)-TH CUMULATIVE
INTEGRAL (2(1-1))
END OF INTEGRATION
LOOP
CALC. FINAL CUMULATIVE
INTEGRAL te(NINC))
END OF SUBROUTINE
Figure 24. Flow chart for subroutine QTFE.
131
-------
Subroutine LNFIT
This subroutine fits a measured or calculated particle size
distribution to a log-normal distribution. A mass median diameter,
geometric standard deviation, and fit parameter are determined in
order to describe the fitted log-normal distribution. In order
to use this subroutine, subroutine CFIT must be supplied.
Using equation (58), we can write the cumulative fraction
(S(X)) up to a given particle size for a log-normal distribution
in the form
S(X) = / fL-N dZ
— 00
a V^TT -°°
z V
where the symbols are as previously defined. By making a change
in variable of the form
t = ^ (68)
z
and
dz = az dt , (69)
we can write equation (67) in the form
S(0=-Lr- ft EXP T-|i] dt . (70)
The cumulative fraction (Q(O) greater than a given particle
size can be expressed in the form
Q(t') = -A- f EXP [-|11 dt . (71)
t-(Q) l 2 J
S(f) and Q(f') are called inverse Gaussian (Normal) integrals.
The variable f(Q) can be approximated by the expression38
132
-------
(72)
where the error in t"(Q) is equal to or less than 0.00045 and
ao = 2.515517, ai = 0.802853, a2 = 0.010328, bi - 1.432788, b2 =
0.189269, and b3 = 0.001308. The approximate expression for t'(Q)
is valid for 0 is defined as
* =
(73)
Since S(f*) + Q(f) = 1, equation (72) can be used for t'(S)
where 0.5 < S(f*)
-------
t' = A + Bz" (77)
by calling subroutine CFIT which uses a linear least squares fit
procedure.
Since t" = 0 at the point where 50% of the distribution has
been accumulated, the fitted, actual d5o can be obtained from
0 = A + Bz' = A + B In d50 . (78)
Thus,
d50 = EXP (- A/B) . (79)
In order to obtain the geometric standard deviation (a ) of
the fitted log-normal distribution, it is recognized from equations
(76) , (77) , and (79) that
— In d5 o A
*--§--- -3— - FT^IT <801
Z Z P
or
a = EXP (1/B) . (81)
Figure 25 shows a detailed flow chart for this subroutine.
This subroutine is called by the main program and subroutine ADJUST.
Information which is transmitted to and from this subroutine is
transferred through calling arguments and block common statements.
The following is a sequential list of the calling arguments and
their descriptions.
PRCU(I) - Measured or known cumulative mass fractions.
D50 - Mass median diameter obtained from the fit of the
actual distribution to a log-normal distribution (ym) .
SIGMAP - Geometric standard deviation obtained from the fit of
the actual distribution to a log-normal distribution.
GFIT - Goodness of fit parameter for the log-normal fit.
The following is a list of the necessary variables which are
in common with the main program and subroutine ADJUST.
ENDPT(I) - Particle diameters corresponding to the measured or
known cumulative mass fractions (ym).
NENDPT - Number of points in the measured or known distribution.
134
-------
START SUBROUTINE ]
DIMENSION: Z, Y, PRCU
BLOCK COMMON: ENDPT, NENDPT
SET INDICATOR WHICH DETERMINES
THE NO. OF POINTS IN THE CURVE
FIT EQUAL TO 0 (NSTAG)
SET INDICATOR WHICH SEQUENCES
THE POINTS IN THE CURVE FIT
EQUAL TO O (J)
START LOOP OVER POSSIBLE
POINTS IN THE CURVE FIT
CONVERT ACTUAL CUMULATIVE %
TO A FRACTION (XY)
EMPLOY SYMMETRY OF
GAUSSIAN FUNCTION AND
CONVERT ACTUAL CUMULATIVE
% TO A FRACTION (XY)
A
A
DETERMINE THE VALUE OF
0 IN EQ. (74) (XYY)
CALC. THE VALUE OF t'(Q)
FROM EQ. (72) (Y(J))
YES
Y(J)
= -Y(J)
NSTAG
NSTAG i- 1
END LOOP OVER POSSIBLE
POINTS IN THE CURVE FIT
CALL CFIT: FIT THE POINTS
(Y, Z) TO A STRAIGHT LINE
CALC. MASS MEDIAN DIAMETER OF
FITTED CUMULATIVE % CURVE (D50)
CALC. GEOMETRIC STANDARD DEVIATION OF
FITTED CUMULATIVE % CURVE (SIGMAP)
END OF SUBROUTINE
Figure 25. Flow chart for subroutine LNFIT.
135
-------
Of the above variables, the values of PRCU, ENDPT, and NENDPT
must be supplied by the calling program. D50, SIGMAP, and GFIT
are determined in the subroutine. I can not exceed a value of 21.
The restriction on I limits the number of particle diameters in
the particle size distribution.
In the calculations, all points which have an actual cumu-
lative fraction of zero are ignored. Since measured cumulative
particle size distributions may tend to flatten out for the larger
particle sizes, the calculation is cut off at the point where 99%
of the distribution is accumulated. This is done in order to
keep the curve fit from being prejudiced towards the flat portion
of the curve even though the majority of the distribution is log-
normal. The cumulative fractions and corresponding particle dia-
meters should be stored in arrays PRCU and ENDPT, respectively,
in order from smallest to largest values. If the goodness of
fit parameter GFIT, which is determined in subroutine CFIT, is
nearly 1, the actual distribution is very close to a log-normal
distribution and the fitted dso and a are meaningful quantities.
If GFIT is much less than 1, the actual distribution should be
examined in order to determine if the fitted d5o and a are
meaningful quantities. "
Subroutine CFIT
This subroutine fits a set of data points to a straight line
by using a least squares fit procedure.3 If the data points
(x,y) are to be fitted to a linear relationship of the form
y = a + bx , (82)
the problem is to find the undetermined coefficients a and b such
that the line is a good fit to the data. In this case, appli-
cation of the principle of least squares results in two normal
equations of the form
ma + > X.b = > y. (83)
and
-jY- , (84)
where (x_.,y..) are the data points and m is the number of data
points. Equations (83) and (84) form a system of two simultaneous
equations in two unknowns. The solutions of this system of equa-
tions are
136
-------
and
where
a =
(85)
m
;>x .y.
L*, 3^3
(86)
A = m
(87)
With the above determination of a and b, the least squares fit to
the data is obtained.
A linear-correlation coefficient r can be constructed in order
to measure the degree of linear correlation or the probability
that a linear relationship exists between the two observed vari-
ables x and y. Since we are interested in the interrelationship
between the variables x and y, we can equally well consider x as
a function of y and ask if the data correspond to a straight line
of the form
x
+
(88)
The values of the coefficients a" and b' will be different from
those of a and b in equation (82) , but they are related if the
variables x and y are correlated.
The inverse slope
is determined in the same manner as b
b' =
where
1_
A'
m
z
.y
r
(89)
A" = m
(90)
If there is no correlation between the variables x and y, the
least-squares fit must yield a horizontal straight line and b =
b' = 0.
If there is complete correlation between x and y, there
exists a relationship between the coefficients a and b of equation
(82) and the coefficients a" and b' of equation (88). From
equations (82) and (88),
137
-------
(91)
jj j->
Equating coefficients gives
a = - | (92)
and
b = £r - (93)
If there is complete correlation, we see that bb" = 1. If
there is no correlation, both b and b' are 0. Thus, the quantity
r = Vbb' (94)
is defined as the linear-correlation coefficient and is used as a
measure of the degree of linear correlation. The value of r
ranges from 0, when there is no correlation, to 1, when there is
complete correlation.
In the context of the model of electrostatic precipitation,
r is called the goodness of fit parameter. If r is much less
than 1, the coefficients a and b, which are also used in sub-
routine LNFIT, may not lead to meaningful information for the
user.
This subroutine is called by subroutine LNFIT. Figure 26
shows a detailed flow chart for this subroutine. All information
which is transmitted between subroutine LNFIT and this subroutine
is transferred through calling arguments. The following is a
sequential list of the calling arguments and their descriptions.
A - Constant term in the fitted linear relationship.
B - Coefficient of the independent variable in the fitted
linear relationship.
R - Linear correlation coefficient (goodness of fit para-
meter) .
NSTAG - Number of data points.
Z(I) - Measured or known values of the independent variable.
Y(I) - Measured or known values of the dependent variable.
Of the above variables, the values of NSTAG, Z, and Y must
be provided by the calling program (subroutine LNFIT). The values
138
-------
START SUBROUTINE J
I
| DIMENSION: Z. Y }
I
INITIALIZE VALUE OF DATA
POINT COUNTER TO O (XN)
INITIALIZE SUM OVER VALUES OF
THE INDEPENDENT VARIABLE
TO O (SUMX)
INITIALIZE SUM OVER VALUES
OF THE DEPENDENT VARIABLE
TO O (SUMY)
INITIALIZE SUM OVER THE VALUES OF
THE PRODUCT OF THE INDEPENDENT
AND DEPENDENT VARIABLES TO
O (SUMXY)
INITIALIZE SUM OVER THE VALUES OF
THE SQUARE OF THE INDEPENDENT
VARIABLE TO 0 (SUMXX)
INITIALIZE SUM OVER THE VALUES OF
THE SQUARE OF THE DEPENDENT
VARIABLE TO O (SUMYY)
/" START LOOP OVER "N
I DATA POINTS J
CALC. VALUES OF SUMX, SUMY,
SUMXY, SUMXX, SUMYY, AND XN
(END LOOP OVEFiN
DATA POINTS ]
CALC. COEFFICIENT OF THE
INDEPENDENT VARIABLE FOR
THE FITTED LINEAR
RELATIONSHIP (B)
CALC. CONSTANT TERM FOR THE
FITTED LINEAR RELATIONSHIP (A)
CALC. LINEAR-CORRELATION COEFFICIENT
(GOODNESS OF FIT PARAMETER) FOR THE
LINEAR FIT OF THE DATA (R)
END SUBROUTINE
J
Figure 26. Flow chart for subroutine CFIT.
139
-------
of A, B, and R are calculated in the subroutine. The index I can
not exceed a value of 21. The restriction on I limits the number
of particle diameters in the particle size distribution.
Subroutines PRTINP, PRTINC, PRTCHG, and PRTSUM
Subroutines PRTINP, PRTINC, PRTCHG, and PRTSUM perform the
function of printing out information of importance to the user.
Since these subroutines do not involve operations based on physi-
cal principles or numerical techniques, they will not be discussed
in detail. However, the output from these subroutines is discussed
in detail in Volume 2. Briefly, PRTINP prints out all the input
data to the program, PRTINC prints out the results of calculations
which are a function of incremental length through the precipitator,
PRTCHG prints out information concerning the charge on each par-
ticle size in each incremental length through the precipitator,
and PRTSUM prints out a summary table of precipitator performance
and operating parameters.
140
-------
SECTION 8
DESCRIPTION OF INPUT DATA
GENERAL DESCRIPTION
The format of the original computer program which performs
the calculations in the model for electrostatic precipitation
has been re-structured to make the inputting of data less cumber-
some. The number of cards which is necessary to input data has
been reduced significantly by allowing different operating
conditions to be analyzed from one basic set of input data. Due
to the fact that several options are available in using the model,
the number of cards and type of information in the input data may
vary from one set of data to the next. Thus, it is necessary for
the user to familiarize himself with the logic associated with the
input data in order to ensure that the desired operations will be
performed.
Some of the input variables are read into the program in
British units whereas others are in MKS units. All input data
which are in British units are converted to MKS units prior to
performing the calculations. The input variables and format
specifications are discussed in detail in the following subsection.
CONSTRUCTION OF THE BASIC DATA SET
The following is a sequential listing of the variables in
the first group of data which is read in, along with the de-
scriptions of the variables and the format specifications.
(1) NENDPT is the number of discrete points on a cumulative per-
cent versus particle diameter curve. NENDPT is
specified by the user and must have a value of at
least 1 but not greater than 21. If NENDPT has a
value of 99, the program terminates. If 21
-------
1 - A complete data set must be inputted. NDATA must
have this value on the first data set.
2 - Only cards 1 and 2 and data concerning size distri-
bution information must be inputted. All other data
remain as defined in the previous data set. NDATA
can have this value only after a basic data set has
been run. This value of NDATA allows one to examine
the effects of particle size distribution on precipi-
tator performance with all other variables held fixed.
3 - Only cards 1 and 2 and information concerning the gas
volume flow and gas velocity must be inputted. All
other data remain as defined in the previous data set.
NDATA can have this value only after a basic data set
has been run. This value of NDATA allows one to
examine the effects of specific collection area (SCA)
on precipitator performance with all other variables
held fixed.
4 - Only cards 1 and 2 and information concerning the
applied voltage and current must be inputted. All
other data remain as defined in the previous data
set. NDATA can have this value only after a basic
data set has been run. This value of NDATA allows
one to examine the effects of the electrical conditions
on precipitator performance with all other variables
held fixed.
NDATA is read in with an 12 format and must be right
justified in columns 3-4. If NDATA ^ 1,2,3, or 4, an
error message will be given by the computer at the point
in the program where NDATA is used in a "computed go
to statement" (line 64). Depending on the particular
computer, the program may or may not terminate at this
point. If the program continues to execute, it may
terminate abnormally at a later point in the program
due to incorrect usage of the input data. If the pro-
gram terminates normally, the calculations may or may not
be correct, depending on the input data and the action
taken by the computer.
The overall format for this group is (212). The data con-
tained in this group is on the first card and this card must
be the first card in each new data set.
Data group 2 is for specifying information which will
identify the data set which is under consideration. All or part
of columns 1-80 on data card 2 can be used in identifying the
data set. The overall format for this card is (40A2). This data
group must be the second card in each new data set.
142
-------
At this point, the third and successive data groups depend
on the choice of the value of NDATA. The basic data set must be
read into the program before shortened data sets can be used.
For NDATA=1, the program reads in the data groups in the basic
data set in the sequence discussed below.
The following is a sequential listing of the variables in
data group 3, along with the descriptions of the variables and
the format specifications.
(1) NEST is an indicator which can have the values of 1 and 2.
If NEST = 1, the program will perform extensive, detailed
calculations in order to determine precipitator per-
formance. If NEST = 2, estimation procedures are used
to determine precipitator performance. Both of these
options have been discussed in detail in Volume 1. Use
of the estimation procedure will result in considerable
savings in computer time and can be used to establish
trends or to establish ranges over which to apply the
more rigorous calculations. NEST is read in with an
12 format and must be right justified in columns 1-2.
(2) NDIST is an indicator which can have the values of 1 and 2.
If NDIST = 1, the user must supply the inlet particle
size distribution. If NDIST = 2, the program will
construct a log-normal particle size distribution
based on parameters provided by the user. The technique
used to construct the log-normal size distribution is
described in Volume 1. NDIST is read in with an 12
format and must be right justified in columns 3-4.
(3) NVI is an indicator which can have the values of 1 and 2.
If NVI = 1, the user must supply known or measured
values of the operating applied voltage and current.
If NVI = 2, the program will construct a voltage-
current curve (or curves) for a specified wire-plate
geometry up to a voltage which is specified by the user.
Both of the techniques for determining the electrical
conditions are discussed in Volume 1. NVI is read in
with an 12 format and must be right justified in columns
5-6.
(4) NX is the number of grid points in the x-direction
(perpendicular to the gas flow) which is used in the
numerical techniques that determine the electrical
conditions. NX can not exceed a value of 15. If
NVI = 1, sufficient accuracy can normally be obtained
with NX >_ 11. If NVI = 2, NX should be set equal to
15. NX Ts read in with an 12 format and must be right
justified in columns 7-8.
143
-------
(5) NY is the number of grid points in the y-direction
(parallel to the gas flow) which is used in the
numerical techniques that determine the electrical
conditions. If NVI = 1, sufficient accuracy can
normally be obtained with NY >_ 9 - If NVI = 2, NY
should be set equal to 15. NY is read in with an 12
format and must be right justified in columns 9-10.
(6) NITER is an indicator which serves two different purposes.
If NVI = 1, the value of NITER determines the maximum
number of iterations the program will make on a loop
which converges on overall mass collection efficiency.
If the overall mass collection efficiency converges
within 0.05% before NITER iterations, the calculation
of collection efficiencies is completed at this point.
NITER serves the purpose of cutting the calculation off
in a reasonable amount of time when convergence requires
more iterations and computer time than is warranted.
For normal inlet mass loadings and particle size dis-
tributions a value of NITER - 2 is sufficient. For
high inlet mass loadings or very fine particle size
distributions a value of NITER = 3 or 4 may be necessary
to provide sufficient accuracy. If NVI = 2, the value
of NITER determines the number of iterations which will
be performed over each incremental length of the pre-
cipitator in order to obtain self-consistent solutions
for the electrical conditions. In its present stage
of development, the calculation procedure yields the
same results for all values of NITER. Thus, in this
case, set NITER = 1. The calculation procedure is
discussed in Appendix A of Volume 1. NITER is read
in with an 12 format and must be right justified in
columns 11-12.
(7) NCALC is an indicator which can have the values of 0 and 1.
If NCALC = 0, particle charge is determined by using
equation (12) in Volume 1. Due to the number of times
particle charge must be calculated and the use of
numerical techniques in order to solve the charging
equation, the particle charging calculations for
NCALC = 0 take a considerable amount of computer time.
If NCALC = 1, particle charge is estimated empirically
by using the sum of the charges predicted from classical
field and thermal charging theories [see equation (15)
in Volume 1]. In this case, particle charge can be
determined very rapidly from analytical expressions.
Thus, in those cases where a significantly shorter run
time is more important than the best accuracy possible,
NCALC should be set equal to 1. If NEST = 2, particle
charge will be performed as if NCALC = 1 regardless of
the value of NCALC. NCALC is read in with an 12 format
and must be right justified in columns 13-14.
144
-------
(8) NRAPD is an indicator which specifies the number of rapping
puff particle size distributions which will be utilized
by the program in predicting the effect of rapping
reentrainment on overall mass collection efficiency-
NRAPD must have a value of at least 1 and can not exceed
a value of 10. If NRAPD = 1, the program will deter-
mine the rapping puff particle size distribution based
on the average of data obtained from several field tests
on full-scale precipitators. These tests yield an
average rapping puff particle size distribution with a
mass median diameter (HMD) of 6.0 ym and a geometric
standard deviation (
-------
(11) NONID is an indicator which specifies the number of com-
binations of normalized gas velocity standard deviation
(ag) and gas bypassage fraction and/or particle re-
entrainment fraction without rapping (S) which are to
be used to simulate the possible nonideal conditions.
The procedures used to account for these nonideal
effects are described in Volume 1. NONID must have a
value of at least 1 and can not exceed a value of 15.
Each set of nonideal conditions is used in conjunction
with the same basic ideal calculation and its effect
is determined with very little expenditure of computer
time. NONID is read in with an 12 format and must be
right justified in columns 21-22.
The overall format for this data group is (1112) and all the
data are contained on the third data card.
The next data group which is read in depends on the values
of NCALC and NVI. If NCALC = 0, the rigorous charging theory is
used. In this case, the following is a sequential listing of
the variables in the next data group which is read in, along
with the descriptions of the variables and the format specifi-
cations .
(1) NN is the number of increments in the Runge-Kutta inte-
gration of equation (12) in Volume 1. If NVI = 1,
a value of NN = 10 normally provides sufficient accuracy
when the precipitator is divided into incremental
lengths of approximately 0.305m or less. If NVI = 2,
a value of NN =5 normally provides sufficient accuracy.
NN is read in with an 12 format and must be right
justified in columns 1-2.
(2) NUMINC is the number of increments in the Simpson's Rule
integration over 8 in equation (12) in Volume 1.
NUMINC must be an even number and a value of NUMINC =
20 normally provides sufficient accuracy. In order to
speed up the calculations, NUMINC can be reduced to a
value as low as 10 without causing too great a change
in the results. The use of values of NUMINC which are
less than 10 is not recommended. NUMINC is read in
with an 12 format and must be right justified in
columns 3-4.
The overall format for this data group is (212) and all the
data are contained on a single card. If NCALC = 1, the above
data group is not read into the program.
If NVI = 2, the model must calculate a voltage-current curve.
In this case, the following is a sequential listing of the variables
in the next data group which is read in, along with the descrip-
tions of the variables and the format specifications.
146
-------
(1) IFINAL is an indicator which causes the calculation of suc-
cessive points on the voltage-current curve to cease
after IFINAL points. This indicator allows the user
to have the V-I calculation terminated at a point
before the specified operating voltage is reached
whenever it is taking an excessive number of points to
reach the specified operating voltage. IFINAL is read
in with an 12 format and must be right justified in
columns 1-2.
(2) JI1 is an indicator which allows the initial increment
size on current density in the calculation of the
voltage-current curve to be changed after JI1-1
points are determined on the curve. Since the voltage-
current calculation finds the applied voltage corre-
sponding to a specified value of current density, this
indicator allows the user to cover a large range of
current densities without using an excessive number
of points. JIl is read in with an 12 format and must
be right justified in columns 3-4.
(3) JI2 is an indicator which allows the second increment size
on current density in the calculation of the voltage-
current curve to be changed after JI2-1 points are
determined on the curve. JI2 serves the same function
as JIl and JI2 must have a value greater than JIl for
proper usage. JI2 is read in with an 12 format and
must be right justified in columns 5-6.
(4) VISKIP is an indicator which may have the values of 0 and 1.
If VISKIP = 0, a voltage-current curve will be calcu-
lated up to a specified operating voltage for each
successive length increment of the precipitator. If
VISKIP = 1, only the operating current density which
corresponds to a specified operating voltage will be
calculated based on an estimation procedure discussed
in Volume 1. In most cases, the user will want to set
VISKIP = 1 since this will result in a prediction of
the operating current density in each increment of
length of the precipitator without using the large
amounts of computer time required by the calculation of
a voltage-current curve. Only extremely detailed
analysis would warrant setting VISKIP = 0. VISKIP is
read in with an 12 format and must be right justified
in columns 7-8.
(5) VISAME is an indicator which may have the values of 1 and 2.
The proper use of VISAME can result in significant
savings in computer time whenever the applied voltage
is the same in each electrical section. If the applied
voltage is the same in each electrical section, set
VISAME = 1 and only one "clean" voltage-current curve
147
-------
will be calculated. If VISAME = 1, as many data sets
as desired can be read into the program and all calcu-
lations will be based on the one "clean" voltage-
current calculation. The use of VISAME = 1 is especially
beneficial in studying hypothetical cases due to the
large savings in computer time. If the applied voltage
differs from one electrical section to the next, the
user must set VISAME = 2. Whenever the operating voltage
and current are unknown and the user must specify the
use of the voltage-current calculations (NVI = 2), the
quickest run time will occur when VISKIP = 1 and
VISAME = 1. The longest run time will occur when
VISKIP = 0 and VISAME = 2. VISAME is read in with an
12 format and must be right justified in columns 9-10.
The overall format for this data group is (512) and all data
are contained on a single card. If NVI = 1, the above data group
is not read into the program.
The following is a sequential listing of the next data group
which is read in, along with the descriptions of the variables and
the format specifications.
(1) PL is the inlet particulate mass loading in units of
grains/ft3 . DL is read in with a F8.0 format and
must be left justified in columns 1-8.
(2) PL is the total electrical length of the precipitator
in units of feet. PL is read in with a F8.0 format
and must be left justified in columns 9-16.
(3) ETAO is the overall mass collection efficiency in units
of percent and it has two different interpretations
depending upon the value of NVI. If NVI = 1, ETAO
represents the measured or estimated overall mass
collection efficiency and is used as a test for con-
vergence in an iteration loop on overall mass collection
efficiency. If NVI = 2, ETAO simply represents the
desired design efficiency and is not used in the
calculations. ETAO is read in with a F8.0 format
and must be left justified in columns 17-24.
(4) DD is the density of the particles in units of kg/m3 .
DD is read in with a F8.0 format and must be left
justified in columns 25-32.
(5) EPS is the dielectric constant of the particles for use in
the particle charging calculations and is dimensionless.
Values of EPS must be equal to or greater than 1. In
most industrial applications, the flue gas is sufficiently
humidified so that the particle surface becomes con-
ductive and a value of EPS = 100 can be used to simulate
148
-------
a conductor. EPS is read in with a F8.0 format
and must be left justified in columns 33-40.
(6) VRATIO is the ratio of the peak voltage to the average voltage
and is dimensionless. In the calculation of particle
charge, it is assumed that the particles will charge
to an extent determined by the peak voltage rather than
the average voltage. For industrial applications,
VRATIO has a value around 1.2. VRATIO is read in with
a F8.0 format and must be left justified in columns
41-48.
(7) US_ is the ionic mobility at standard temperature (273°K)
and standard pressure (1 atm) and is in units of
m2/(V-sec). This mobility is referred to as the
"reduced mobility". Values to use for reduced ionic
mobilities for flue gas compositions are not well-
established at the present time. The reduced ionic
mobility for air is in the range 1.2-2.1 x 10~"m2/(V-sec)
Reduced ionic mobilities for flue gas compositions have
been reported that are considerably larger than those
reported for air. These values cover the range of
2.2-5.4 x 10~4m2/(V-sec). Some reported values of
reduced ionic mobility for various gas compositions
are given in Table 2. Since the ionic mobility has
a strong influence on the electrical conditions through
the current and electric field distributions, this is
an important parameter in determining precipitator
performance. A value of 3.0 x 10~4m /(V-sec) should
provide a representative value to use for flue gases
emanating from coal-fired boiler applications. US is
read in with a F8.0 format and must be left justified
in columns 49-56.
(8) FPATH is a parameter which is used in the field charging
equation and is dimensionless. FPATH represents the
number of ionic mean free paths over which the momentum
of the ions will persist and allow the ions to reach
the surface of the particle even though the saturation
charge has been reached. The effect of this parameter
is to increase the saturation charge. FPATH normally
should have a value in the range 0-2. It is recommended
that FPATH be assigned a value of 1. FPATH is read in
with a F8.0 format and must be left justified in
columns 57-64.
(9) EBP is the electrical breakdown strength of the gas or the
particulate layer in the region near the plate and is
in units of V/m. The value of this parameter is a
strong function of the resistivity of the collected
149
-------
TABLE 2
REDUCED EFFECTIVE NEGATIVE ION MOBILITIES FOR VARIOUS GAS COMPOSITIONS
Reduced Effective
Gas Composition Ion Mobility
(Volume Percent) (cm2/V-sec)
N2 COj 0_2_ SQ2 Ha 0
100.0 0.67 + 0.17a
i_
100.0 2.46 + 0.06
100.0 1.08 + 0.03b
100.0 0.35C
(Laboratory Air) 1.03
(Laboratory Air) 1.26 - 1.96
79.
73.
65.
71.
75.
75.
78.
78.
77.
77.
a.
b.
c.
d.
4
5
9
0
7
1
5
3
9
6
J
E
E
B
14.
13.
12.
11.
11.
11.
10.
19.
10.
10.
. J.
. W.
. W.
.Y.H.
7 4.
6 4.
2 3.
2 3.
6 3.
5 3.
9 3.
8 3.
8 3.
7 3.
6
2
8
7
2
2
6
6
6
7
Lowke and
McDaniel
McDaniel
Liu,
K.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
J.
and
2
2
2
0
0
1
0
1
3
7
A.
H.
and M.
T.
0.
8.
17
14
9.
9.
7.
7.
7.
7.
6
4
.
.
4
9
0
0
0
0
Rees
5.
2.
8 2.
0 2.
3.
2.
3.
2.
2.
2.
, Australian J
39
93
23
35
02
74
36
67
70
43
*
R. Crane, Rev. Sci.
R.
C
Whitby,
. McDowell, Phys
and H.H.S. Yu,
J
f
f
f
f
f
f
f
f
f
f
Phys. 16, 447 (1963).
Instru. 28, 684 (1959).
. Rev. 114, 1028 (1959)
. Appl. Phys. 38,
1592 (1967).
e. J. Bricard, M. Cabane, G. Modelaine, and D. Vigla, Aerosols
and Atmospheric Chemistry. Edited by G. M. Hidy, New York,
New York, 27 (1972).
f. H. W. Spencer, III, "Experimental Determination of the Effective
Ion Mobility of Simulated Flue Gas." In Proceedings of 1975
IEEE-IAS Conference, September 28, 1975, Atlanta, Georgia.
150
-------
particulate layer and the condition of the collection
plates. At present, mathematical techniques which are
based on physical principles do not exist for pre-
dicting the value of EBD under differing conditions.
Thus, experimental data and prior experience must be
used to choose appropriate values of EBD. In practical
applications, EBD fails in the range of 2-15 kV/cm.
A value of 2 kV/cm should provide a conservative
estimate of EBD whereas a value of 15 kV/cm would in
most cases provide the most optimistic value. The
value of EBD is used whenever NVI = 2 and a voltage-
current curve is generated. If the field at the
plate exceeds the value of EBD at any point on the
curve, a message to this effect is printed out with
the V-I calculation terminating at the corresponding
applied voltage and current density. These values of
voltage and current are then used in the projection pf
precipitator performance. EBD is read in with a F8.0
format and must be right justified in columns 65-72.
(10) RHO is the resistivity of the collected particulate layer
and is in units of ohm-cm. The resistivity to be used
must be determined experimentally by either in situ
or laboratory methods. RHO is used in the model only
to estimate the average electric field in the collected
particulate layer. It is not used to determine allow-
able electrical operating conditions. The effect of
RHO on the allowable electrical operating conditions
must be reflected in the input data for the operating
voltages and currents. RHO is read in with a E8.2
format and must be right justified in columns 73-80.
The above data group has an overall format of (9F8.0, E8.2)
and is contained on a single data card. This data set must be
read in with each basic data set, i.e. when NDATA = 1.
The next data group which is read in depends on the value
of NRAPD. If NRAPD is greater than 1, the following is a
sequential listing of the variables in the next data group,
along with the descriptions of the variables and the format
specifications.
(1) ARD50(I) is an array containing the mass median diameters in
ym for log-normal particle size distributions of the
different rapping puff distributions which will be
utilized in the model. The values of this variable
are read in with a F4.0 format and must be left
justified in columns 1-4, 9-12, 17-20, 25-28, 33-36,
41-44, 49-52, 57-60, 65-68, and 73-76.
(2) ARSIGM(I)is an array containing the geometric standard devia-
tions for log-normal particle size distributions of
the different rapping puff distributions which will
151
-------
be utilized in the model. Values of ARD50(I) and
ARSIGM(I) with the same index are used together to
construct a log-normal particle size distribution.
The values of this variable are read in with a F4.0
format and must be left justified in columns 5-8,
13-16, 21-24, 29-32, 37-40, 45-48, 53-56, 61-64,
69-72, and 77-80- ARSIGM(I) can not have a value
less than 1.
The above variables must be read in for 1=2 up to I=NRAPD
where NRAPD can not exceed a value of 10. The overall format for
this data group is (10(2F4.0)) and is contained on a single data
card. If NRAPD=1, this data group is not read in. In this case,
only one rapping puff particle size distribution will be considered
where ARD50(1) = 6.0 in ym and ARSIGM(l) = 2.5. This case is
built into the program and relates to experimental data discussed
in Volume 1.
The following is a sequential listing of the variables in the
next data group which is read in, along with the descriptions of
the variables and the format specifications.
(1) ASNUCK(I) is an array containing different fractions of gas
flow which bypass the electrified region in each
baffled stage of the precipitator and/or different
fractions of the mass collected in each stage of
the precipitator which are reentrained due to
factors other than rapping. The values of this
variable are read in with a F4.0 format and must
be left justified in columns 1-4, 13-16, 25-28,
37-40, 49-52, and 61-64 of the first two data cards
in the group and in columns 1-4, 13-16, and 25-28
of the third data card in the group. ASNUCK(I)
must lie in the range 0.0 to 1.0.
(2) AZIGGY(I) is an array containing different normalized standard
deviations for the inlet velocity distribution of
the gas flow. The values of this variable are read
in with a F4.0 format and must be left justified in
columns 5-8, 17-20, 29-32, 41-44, 53-56. and 65-68
of the first two data cards in the group and in
columns 5-8, 17-20, and 29-32 of the third data card
in the group. AZIGGY(I) must be equal to or greater
than 0.0.
(3) AZNUMS(I) is an array containing the number of baffled stages
in the precipitator. The values of this variable
are read in with a F4.0 format and must be left
justified in columns 9-12, 21-24, 33-36, 45-48,
57-60, and 69-72 of the first two data cards in the
group and in columns 9-12, 21-24, and 33-36 of the
third data card in the group. The values of AZNUMS(I)
must be whole numbers.
152
-------
The values of ASNUCK(I), AZIGGY(I), and AZNUMS(I) with the
same index are used together to simulate one set of nonideal
parameters and to produce one set of no-rap efficiencies. The
values of I are determined by NONID which must have a value of
at least 1 and can not exceed a value of 15. Thus, at least one
set of these parameters must be read in. It is recommended that
the user take the first set of these variables to be ASNUCK(l) =
0.00, AZIGGY(l) = 0.00, and AZNUMS(1) = actual number of stages
so that efficiencies under ideal conditions will be obtained. In
practical situations, a well-operating precipitator will have
values of ASNUCK and AZIGGY of approximately 0.1 and 0.25,
respectively.
The overall format for this data group is (6(3F4.0)) and the
data group is contained on 3 or less cards. For NONID < 6,
6
-------
The next data group which is read in depends on the value
of NDIST. If NDIST = 2, the following is a sequential listing
of the variables in the next data group, along with the descrip-
tions of the variables and the format specifications.
(1) D50 is the mass median diameter of a log-normal inlet
particle size distribution and is in units of ym.
The value of D50 must lie between 0.01 and 1,000 pm.
The value of D50 is read in with a F8.0 format and
must be left justified in columns 1-8.
(2) SIGMAP is the geometric standard deviation of a log-normal
inlet particle size distribution and is dimension-
less. The value of SIGMAP must be equal to or
greater than 1. The value of SIGMAP is read in
with a F8.0 format and must be left justified in
columns 9-16.
The program uses the values of D50 and SIGMAP to construct a
log-normal particle size distribution over the range and size bands
determined by the values of ENDPT(I). Any mass which is not in
the size range determined by ENDPT(I) will be put into the size
band with the largest midpoint. This must be done to ensure that
the sum over all size bands of the percentage of total mass in
each size band will equal 100%.
The above data group has an overall format of (2F8.0) and is
contained on a single data card. This data set is not read in if
NDIST = 1.
If NDIST = 1, the next data group which is read in consists
of a single array- The description of this array and its format
specification are given below.
(1) PRCU(I) is an array containing values of cumulative percents
corresponding to points on a curve of inlet mass
cumulative percent versus particle diameter. The
number of cumulative percents that must be read in
depends on the value of NENDPT which can not exceed
21. The cumulative percents must match the particle
diameters specified in the array ENDPT(I). The
cumulative percents are inputted in units of %.
The first value of PRCU(I) must be 0% and the last
value must be 100%. The program determines the
percentage by mass in each particle size band from
the values contained in ENDPT(I) and PRCU(I). The
user must supply values of PRCU(I) based on measured
or known particle size information for the particular
application under consideration. The values of
PRCU(I) are read in with a F8.0 format and must be
left justified.
154
-------
The overall format for this data group is (10F8.0) and the
data group is contained on 3 or less data cards. For NENDPT <_ 10,
10
-------
(3) TCS(NSECT)
along with the corresponding current will be used
in the calculation of precipitator performance.
The values of this variable are read in with an
Ell.4 format and must be right justified in
columns 12-22.
is the total current in a given electrical section
and is in units of amperes. If NVI = 1, the value
of TCS(NSECT) corresponds to a measured or known
value. If NVI = 2, TCS(NSECT) has no meaning in
terms of input data since it will be calculated in
the program. In this case, the appropriate columns
on the data card can be left blank or any arbitrary
number can be entered. The values of this variable
are read in with an Ell.4 format and must be right
justified in columns 23-33.
(4) WLS(NSECT) is the total effective wire length in a given
electrical section and is in units of feet. The
values of this variable are read in with an
Ell.4 format and must be right justified in
columns 34-44.
(5) ACS(NSECT) is the corona wire radius in a given electrical
section and is in units of inches. The values of
this variable are read in with an Ell.4 format
and must be right justified in columns 45-55.
(6) BS(NSECT)
is the wire-to-plate spacing in a given electrical
section and is in units of inches. The values of •
this variable are read in with an Ell.4 format and
must be right justified in columns 56-66.
(7) NWS(NSECT) is the number of discharge electrodes per given
electrical section per gas passage and is dimen-
sionless. The values of this variable normally
should not exceed 20. If the values do exceed 20,
use 20 in the program. These values are used to
determine the number of terms in a series summation
which determines the electrostatic electric field
distribution and 20 terms are more than sufficient
to reach convergence. The values of this variable
are read in with an Ell.4 format and must be right
justified in columns 67-77.
(8) SYS(NSECT) is one-half of the wire-to-wire spacing in a given
electrical section and is in units of inches. The
values of this variable are read in with an Ell.4
format and must be right justified in columns 1-11.
156
-------
(9) VGS(NSECT) is the total gas volume flow rate in a given
electrical section and is in units of actual
ft3/min. The values of this variable are read
in with an Ell.4 format and must be right justi-
fied in columns 12-22.
(10) VGASS(NSECT) is the gas velocity in a given electrical section
and is in units of ft/sec. The values of this
variable are read in with an Ell.4 format and
must be right justified in columns 23-33.
(11) TEMPS(NSECT) is the gas temperature in a given electrical
section and is in units of °F. The values of
this variable are read in with an Ell.4 format
and must be right justified in columns 34-44.
(12) PS(NSECT) is the gas pressure in a given electrical section
and is in units of atmospheres. The values of
this variable are read in with an Ell.4 format
and must be right justified in columns 45-55.
(13) VISS(NSECT) is the gas viscosity in a given electrical section
and is in units of kg/(m-sec). Table 3 gives
values of viscosity for different temperatures
and water contents for a gas composition whose
components are those of air. This table provides
values of viscosity which cover most cases found
in practice although some extrapolation is
necessary for those cases involving hot precipi-
tators where temperatures are greater than 300°C.
The values of this variable are read in with an
Ell.4 format and must be right justified in
columns 56-66.
(14) LINCS(NSECT) is the incremental length size which will be
taken in a given electrical section and is in
units of feet. If NVI = 1, LINCS(NSECT) should
be given a value of approximately one foot
although larger values can be used with some
loss in accuracy in order to save computer run
time. If NVI = 2, LINCS(NSECT) must be given a
value equal as near as possible to the wire-to-
wire spacing in order for the numerical proce-
dure to be valid. In any case, the product of
LSECT(NSECT) and LINCS(NSECT) must equal the
total length of the given electrical section.
The values of this variable are read in with an
Ell.4 format and must be right justified in
columns 67-77.
The overall format for this data group is (7(E11.4)) and the
data group is contained on two data cards. This data group must
be read in with each basic data set.
157
-------
TABLE 3. VALUES OF VISCOSITY FOR AIR AT VARIOUS TEMPERATURES AND WATER CONTENTS*
Percent H20
CD
I£
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
0
1.767
1.810
1.854
1.900
1.938
1.979
2.020
2.059
2.099
2.137
2.175
2.213
2.250
2.286
2.321
2.356
2.390
2.424
2.457
2.489
2.521
2.552
2.583
2.613
2.642
2.671
2.699
2.727
2.754
2.780
1
1.758
1.801
1.844
1.887
1.929
1.970
2.011
2.050
2.090
2.129
2.167
2.204
2.241
2.277
2.313
2.348
2.382
2.416
2.449
2.482
2.513
2.545
2.575
2.606
2.635
2.664
2.692
2.720
2.747
2.773
2_
1.748
1.792
1.835
1.878
1.920
1.961
2.002
2.042
2.081
2.120
2.158
2.195
2.232
2.269
2.304
2.339
2.374
2.408
2.441
2.474
2.506
2.537
2.568
2.598
2.628
2.657
2.685
2.713
2.740
2.767
3_
1.739
1.783
1.826
1.869
1.911
1.952
1.993
2.033
2.072
2.111
2.149
2.189
2.224
2.260
2.296
2.331
2.366
2.400
2.433
2.466
2.498
2.530
2.560
2.591
2.621
2.650
2.678
2.706
2.734
2.761
4_
1.730
1.774
1.817
1.860
1.902
1.943
1.984
2.024
2.063
2.102
2.140
2.178
2.215
2.252
2.288
2.323
2.358
2.392
2.425
2.458
2.490
2.522
2.553
2.583
2.613
2.643
2.671
2.700
2.727
2.754
X ]
5_
1.721
1.765
1.808
1.850
1.892
1.934
1.975
2.015
2.054
2.093
2.132
2.169
2.207
2.243
2.279
2.315
2.349
2.383
2.417
2.450
2.482
2.514
2.545
2.576
2.606
2.636
2.664
2.693
2.720
2.748
L0~5 kg/
6
1.712
1.755
1.799
1.841
1.883
1.925
1.966
2.006
2.046
2.085
2.123
2.161
2.198
2.235
2.271
2.306
2.341
2.375
2.409
2.442
2.475
2.507
2.538
2.569
2.599
2.628
2.657
2.686
2.714
2.741
(m-sec)
7_
1.702
1.746
1.790
1.832
1.874
1.916
1.957
1.997
2.037
2.076
2.114
2.152
2.189
2.226
2.262
2.298
2.333
2.367
2.401
2.434
2.467
2.499
2.530
2.561
2.592
2.621
2.650
2.679
2.707
2.734
8_
1.693
1.737
1.780
1.823
1.865
1.907
1.948
1.988
2.028
2.067
2.105
2.143
2.181
2.218
2.254
2.289
2.325
2.359
2.393
2.426
2.459
2.491
2.523
2.554
2.584
2.614
2.643
2.672
2.700
2.728
9_
1.684
1.728
1.771
1.814
1.856
1.898
1.939
1.979
2.019
2.058
2.097
2.135
2.172
2.209
2.245
2.281
2.316
2.351
2.385
2.418
2.451
2.483
2.515
2.546
2.577
2.607
2.636
2.665
2.694
2.721
10
1.675
1.719
1.762
1.805
1.847
1.888
1.930
1.970
2.010
2.049
2.088
2.126
2.164
2.201
2.237
2.273
2.308
2.343
2.377
2.410
2.443
2.476
2.507
2.539
2.570
2.600
2.629
2.658
2.687
2.715
*Calculations according to:
C.R. Wilke. A Viscosity Equation for Gas Mixtures.
J. Chem. Phy., J^8_(4) :517-519 (April, 1950)
-------
The next data group which is read in depends on the value of
NVI. If NVI = 2, the following is a sequential listing of the
variables in the next data group which is read in, along with the
descriptions of the variables and the format specifications.
(1) RFS(NSECT) is the roughness factor for the wires in a given
electrical section and is dimensionless. In
precipitation practice, if the wires are scratched
or dirty but not completely coated with air, then
the values of RFS(NSECT) lie in the range 0.5-
1.0. A value of 1.0 corresponds to wires which
are in perfect condition. The effect of decreas-
ing the roughness factor is one of increasing the
current that can be achieved at a given voltage
level. If the wires are completely covered with
dirt, then the effect may be one of increased
wire diameter with a roughness superimposed.
This situation would lead to compensating effects.
The values of this variable are read in with an
Ell.4 format and must be right justified in
columns 1-11.
(2) STARTl(NSECT) is the chosen initial current density at which
the calculation of a voltage-current curve starts
in a given electrical section and is in units of
A/m2. In generating the voltage-current curve,
the current density increments in steps of STARTl
(NSECT) until a change is specified. The values
of this variable are read in with an Ell.4 format
and must be right justified in columns 12-22.
(3) START2(NSECT) is a chosen increment in current density which
is used in place of STARTl(NSECT) when the Jll-th
point on the voltage-current curve is reached and
is in units of A/m . The values of this variable
are read in with an Ell.4 format and must be
right justified in columns 23-33.
(4) START3(NSECT) is a chosen increment in current density which is
used in place of START2(NSECT) when the Jl2-th
point on the voltage-current curve is reached and
is in units of A/m . The values of this variable
are read in with an Ell.4 format and must be
right justified in columns 34-44.
(5) VSTAR (NSECT)
is an estimate of the applied voltage correspond-
ing to the first point on the voltage-current
curve as defined by STARTl(NSECT) and is in units
of volts. If VSTAR(NSECT) is close to the actual
applied voltage, the calculation will be performed
faster. However, whatever the choice of VSTAR
(NSECT), it will not affect the accuracy of the
159
-------
calculation. The values of this variable are
read in with an Ell.4 format and must be right
justified in columns 45-55.
The overall format for this data group is (7(Ell.4)) and
the data group is contained on a single data card. If NVI = 1,
this data group is not read in.
The data input starting with AS(NSECT) above must be repeated
for each electrical section of the precipitator, proceeding from
the inlet to the outlet of the precipitator. Thus, the data
group containing AS(NSECT) and possibly the data group containing
RFS(NSECT) must be read in NUMSEC different times.
At this point, the basic data set has been entered into the
program and precipitator performance will be projected based on
the inputted data. The last card in the data section must have
a 99 in columns 1-2. This causes the program to terminate normally.
CONSTRUCTION OF SHORTENED DATA SETS
Once the basic data set is processed, then all the parameters
which are needed by the program to calculate precipitator perfor-
mance are stored in memory. By using values of NDATA equal to 2,
3, or 4, shortened data sets can be entered after the basic data
set in order to analyze the effects of particle size distribution,
specific collection area, and electrical conditions on precipitator
performance. In the shortened data sets, the values of a small
number of variables which are stored in memory are changed to
new values in order to produce a new set of data.
In each shortened data set, the first two data groups and
data cards which are read in are the same as those discussed for
the basic data set. The value of NDATA on the first data card
determines the variables in memory that will be changed. The
effects of particle size distribution on precipitator performance
can be analyzed by setting NDATA =2. In this case, the third
data group which is read in depends upon the value of NDIST which
is stored in memory. If NDIST = 2, an inlet mass median diameter
(D50) and geometric standard deviation (SIGMAP) must be read in
according to the same specifications discussed for the basic
data set. If NDIST = 1, cumulative percents (PRCU(I)) correspond-
ing to the particle sizes (ENDPT(I)) stored in memory must be read
in according to the same specifications discussed for the basic
data set. After the third data group is read in, the shortened
data set is complete. By repeating this type of shortened data
with different choices of D50 and SIGMAP or PRCU(I), the effects
of particle size distribution can be analyzed with the use of
only a few data cards.
The effects of specific collection area (SCA) on precipitator
performance can be analyzed by setting NDATA =3. In this case,
160
-------
the following is a sequential listing of the variables which
are inputted in the third data group, along with the descriptions
of the variables and the format specifications.
(1) VGS(I) is the total gas volume flow rate in a given
electrical section and is in units of actual
ft3/min. The values of this variable are read
in with an Ell.4 format and must be right justi-
fied in columns 1-11, 23-33, and 45-55.
(2) VGASS(I) is the gas velocity in a given electrical section
and is in units of ft/sec. The values of this
variable are read in with an Ell.4 format and
must be right justified in columns 12-22, 34-44,
and 56-66.
The overall format for this data group is (3(2E11.4)) and
the data group is contained on 4 or less cards depending on the
value of NUMSEC which is stored in memory. For NUMSEC £ 3,
3
-------
terms of input data since it will be calculated
in the program. In this case, the appropriate
columns on the data cards can be left blank or
any arbitrary number can be entered. The values
of this variable are read in with an Ell.4 format
and must be right justified in columns 12-22,
34-44, and 56-66.
The overall format for this data group is (3(2E11.4)) and
the data group is contained on 4 or less cards depending on the
value of NUMSEC which is stored in memory- For NUMSEC <. 3,
3
-------
/READ: NENDPT. NDATA/
READ: NEST, NDIST, NVI, NX
NY, NITER, NCALC, NRAPD,
NEFF, NTEMP, NONID
/READ: NN, NUMINC /
/READ:
' ^
(VOS(I), TCS(I)
NUMSEC)
, / / READ:
/ / i-i.r
(VGS(I), VGASS(I)
NUMSEC)
7
READ: IFINAL, JI1, JI2,
VISKIP, VISAME
Figure 27. Flow chart for the input data logic (Sheet 1 of 2).
163
-------
1
1
/READ: DL, PL, ETAO, DD, EPS, /
VRATIO, US, FPATH, EBD, RHO /
<^NRA
PD>> N°»
YES
/ READ: (ARD500), ARSIGM(I), 7
/ 1 2, NRAPD) /
/READ: (ASNUC
AZNUMS(I), 1 -
/ READ; (ENDPTl
<^NDIE
^
K(l), AZIGGY(I), /
1.NONID) /
U\ 1 wpwnpT /
«fl 1 5
^XMO .»
>T - 2^» g»'
YES
/READ: D50, SIGMAP /
1^ ». / 3
[NO
/READ: NUMSEC, (LSECT(I), /
I = 1, NUMSEC) /
(START DO LOOP OVER THE NUMBER A
OF ELECTRICAL SECTIONS )
/ READ: AS, VOS, TCS, WLS, ACS, BS, NWS, /
/ SYS, VGS, VGASS, TEMPS, PS, VISS, LINCS /
<^ NV
= 1 j]> » [
NO
/READ: RFS, START1, START2, /
/ STARTS, VSTAR /
(END DO LOOP OVER THE NUMBER A
OF ELECTRICAL SECTIONS )
d
^ f ? 1
)
Figure 27. Flow chart for the input data logic (Sheet 2 of 2).
164
-------
SECTION 9
MACHINE-DEPENDENT ASPECTS OF THE COMPUTER PROGRAM
The computer program, presented and discussed in this report,
has been developed on a Digital Equipment Corporation (DEC) POP
15/76 computer. By changing only two statements, the program has
been executed successfully on an IBM 370/158 computer and on a
UNIVAC 1100 computer. By changing the same two statements and
certain output formating, the program has been executed success-
fully on a Control Data Corporation (CDC) 7600 computer. Although
the program should compile successfully with only minor changes on
most computers with a FORTRAN compiler, there are certain
machine-dependent aspects of the program that should be discussed.
These machine-dependent properties can be utilized to make the
usage of the program more general and to extend the application
of the program.
In order to use the program on most computers, the first two
executable statements in the program must be changed. These state-
ments define the input (read) and output (write) unit numbers.
The value of the variable NREAD specifies the input unit number
and the value of NPRNT specifies the outlet unit number. These
two changes should normally be all the modifications which are
necessary to allow successful compilation of the program. However,
in order to execute the program on the CDC 7600 computer, it was
also necessary to change single quotes to double quotes in output
format statements. The approximate times required to compile the
entire program on the DEC POP 15/76, IBM 370/158, UNIVAC 1100, and
CDC 7600 computers were 1575, 51, 95, and 5 seconds, respectively.
Although these times can not be compared directly due to software
differences and the fact that an overlay was necessary on the DEC
PDP 15/76, they do give some indication of the relative compile
times.
Once the program is compiled, it will execute provided that
enough core is available to store the program. The total core
requirements on the DEC PDP 15/76 are 86,334 octal words (36,060
decimal words) for the program plus 10,276 octal words (4,286
decimal words) for system software necessary to implement the
program. Table 4 lists the various segments of the program and
their core requirements.
165
-------
TABLE 4. CORE REQUIREMENTS FOR VARIOUS SEGMENTS OF THE COMPUTER PROGRAM
RESIDENT CODE
ESPM
CMAN
BLK1
BLK2
BLK3
BLK4
BLK5
BLK6
BLK7
BLK8
BLK9
BLK10
BLK11
BLK12
BLK13
BLK14
BLK15
BLK16
BLK17
BLK18
BLK19
BLK20
System Software
Octal
Words
11,113
573
502
62
16
1
15
1,354
3,410
170
74
74
53
202
702
3
71
5
2
17
57
263
7,515
Decimal
Words
4,683
379
322
50
14
1
13
748
1,800
120
60
60
43
130
450
3
57
5
2
15
47
179
3,917
LINK1
SPCHG1
EFLD1
LINK2
SPCHG2
EFLD2
LINK3
ADJUST
WADJST
CFIT
LNFIT
QTFE
LNDIST
PRTSUM
System Software
LINK4
CHARGN
RATE
ARCCOS
ZERO
System Software
LINKS
PRTINC
PRTCHG
PRTINP
CHGSUM
System Software
Octal
Words
407
13,663
732
15,774
7,156
610
467
616
160
1,567
1,540
437
343
1,244
200
130
12
1,744
1,562
5,113
1,115
110
Decimal
Words
263
6,607
474
7,164
3,694
392
311
398
112
887
864
287
228
676
128
88
10
996
882
2,635
621
72
166
-------
Due to the fact that the particular DEC PDF 15/76 which has
been used to develop the program has only approximately 55,714
octal words (23,500 decimal words) of core that can be accessed
at any given time, it was necessary to overlay subroutines in
order to fit the program into core. The main program (ESPM) and
subroutine CMAN were kept in resident core and the overlay was
established by setting up the following five links:
LINKl = SPCHG1, EFLD1
LINK2 = SPCHG2, EFLD2
LINK3 = ADJUST, WADJST, CFIT, LNFIT, QTFE, LNDIST, PRTSUM
LINK4 = CHARGN, RATE, ARCCOS, ZERO
LINKS = PRTINC, PRTCHG, PRTINP, CHGSUM
With the above overlay, the required core is 55,633 octal words
(23,451 decimal words) including system software. The core require-
ments were determined by the core utilized in resident core and
the largest link (LINK2). In this particular overlay, LINK2
had 4,707 octal words (2,503 decimal words) of core which were
not utilized. Also, the link table required an additional 323
octal words (211 decimal words) of core.
In order to get the program to execute on computers with
small storage capacities, an overlay similar to the one discussed
above may be possible. On computers with large memories such
as the IBM 370/158, UNIVAC 1100, or CDC 7600, no such action is
necessary.
Without changing the fundamental operations of the program,
the dimensions of certain arrays can be decreased or increased
if necessary- The dimensions of these arrays may be decreased
in order to fit the program on a small computer or they may be
increased to give greater flexibility on a large computer. In
the version of the program presented in this report, the following
quantities determine array sizes which may be changed:
• number of increments along the length of the precipi-
tator
• number of particle size bands
• number of electrical sections in the direction of gas
flow
• number of grid points used in the calculations of
electrical conditions
• number of rapping puff particle size distributions
167
-------
• number of sets of nonideal conditions of nonuniform gas
velocity distribution and gas sneakage and/or particle
reentrainment without rapping.
The above quantities have maximum values of 45, 20, 10, 225, 10,
and 15, respectively -
The number of increments along the length of the precipitator
that can be utilized can be changed by changing the dimension of
DW and the dimension of the first subscript of XDC. DW appears
in COMMON/BLK6/ and XDC appears in COMMON/BLK7/. COMMON/BLK6./
appears in the main program and subroutines PRTINP, CHGSUM, PRTINC,
PRTCHG, ADJUST, and PRTSUM. COMMON/BLK7/ appears in the main
program and subroutines SPCHG2 and PRTCHG. DW also appears in the
dimension statement in the subroutine SPCHG1. If the storage
capacity of the computer is large enough, the program should be
modified to handle more than 45 increments. Although 120 incre-:
ments should be sufficient to handle most cases, as many as 180
increments may be necessary in certain cases.
The number of particle size bands that can be utilized can be
changed by changing the dimension of CHKSUM, DIAM, ONO, DXS, XMV,
PCNT, RAD, CCF, VOL, XNO, Q, WS, QSAT, OLDQ, OLDXNO, XDC, OLDQF,
OLDQT, SOLDQF, SOLDQT, YY, RPCNT, DMDLD, WUNCOR, RDMDLD, CDMDLD,
PCTOT, CPCTOT, WSL, PXS, EUNCOR, and AREA. In addition, changes
must be made to those variables which depend on the number of
particle diameters in the particle size histogram. These variables
must have a dimension which is a value of 1 greater than those
which depend on the number of size bands. These variables include
PRCU, ENDPT, PRCUNR, RPRCU, PRCUC, Z, and Y. CHKSUM appears in
the dimension statement in the main program. DIAM, ONO, DXS, XMV,
PCNT, RAD, CCF, and PRCU appear in COMMON/BLK1/. VOL, XNO, Q, WS,
QSAT, OLDQ, AND OLDXNO appear in COMMON/BLK6/. XDC appears in
COMMON/BLK7/. ENDPT appears in COMMON/BLK11/. OLDQF, OLDQT, SOLDQF,
and SOLDQT appear in COMMON/BLK20/. COMMON/BLKl/ appears in the
main program and subroutines PRTINP, PRTCHG, and ADJUST. COMMON/
BLK6/ and COMMON/BLK7/ appear in those locations previously
designated. COMMON/BLKll/ appears in the main program and sub-
routines PRTINP, ADJUST, LNFIT, and LNDIST. COMMON/BLK20/ appears
in the main program and subroutine CHGSUM. QSAT and XNO appear in
the dimension statement in subroutine SPCHG1. XNO, RAD, CCF,
OLDQ, and Q appear in the dimension statement in subroutine SPCHG2.
YY appears in the dimension statement in subroutine PRTCHG.
RPCNT, DMDLD, WUNCOR, RDMDLD, CDMDLD, PCTOT, CPCTOT, WSL, PXS,
PRCUNR, RPRCU, PRCUC, and EUNCOR appear in the dimension state-
ment in subroutine ADJUST. DIAM, ONO, and PXS appear in the
dimension statement in subroutine WADJST. Z and Y appear in the
dimension statement in subroutine CFIT. Z,Y, and PRCU appear in
the dimension statement in subroutine LNFIT. AREA, PRCU, and
PCNT appear in the dimension statement in subroutine LNDIST. In
changing XDC, it is the second subscript which accounts for the
maximum number of size bands which can be considered.
168
-------
The number of electrical sections in the direction of gas
flow that can be utilized can be changed by changing the dimension
of LSECT, LINCS, PS, AS, VOS, TCS, WLS, ACS, BS, SYS, VGS, VGASS,
TEMPS, VISS, RFS, START1, START2, STARTS, VSTAR, and NWS.
LSECT, LINCS, and PS appear in COMMON/BLK2/. AS, VOS, TCS, WLS,
ACS, BS, SYS, VGS, VGASS, TEMPS, VISS, RFS, STARTl, START2,
STARTS, and VSTAR appear in COMMON/BLK6/. NWS appears in
COMMON/BLK19/. COMMON/BLK2/ appears in the main program and in
subroutines PRTINP and ADJUST. COMMON/BLX6/ appears in those
locations previously designated. COMMON/BLK19/ appears in the
main program and subroutines PRTINP, PRTCHG, and ADJUST. LSECT
appears in the dimension statement in subroutine SPCHG1.
The number of grid points that can be utilized in the calcu-
lation of electrical conditions can be changed by changing the
dimensions of VCOOP, RHO, EX, OLDRO, OLDV, CDNSTY, V, EY,
EAVGS, CHFIDS, ECOLLS, EAVG, CHFID, and ECOLL. VCOOP appears •
in COMMON/BLK13/. EAVG and CHFID appear in COMMON/BLK8/. ECOLL
appears in COMMON/BLK9/. COMMON/ELKS/appears in the main pro-
gram and subroutines CMAN, EFLDl, and EFLD2. COMMON/BLK8/ appears
in the main program and subroutines SPCHG2, EFLD2, and PRTCHG.
COMMON/BLK9/ appears in the main program and subroutine EFLD2.
RHO, EX, OLDRO, OLDV, CDNSTY, V, and EY appear in the dimension
statement in subroutine EFLDl. RHO, EX, OLDRO, OLDV, CDNSTY, V,
EY, EAVGS, CHFIDS, and ECOLLS appear in the dimension statement
in subroutine EFLD2. VCOOP, RHO, EX, OLDRO, OLDV, CDNSTY, V,
and EY are doubly subscripted variables with the first subscript
referring to the number of grid points in the direction perpen-
dicular to the gas flow and the second subscript referring to
the number of grid points in the direction parallel to the gas
flow. EAVG, CHFID, ECOLL, EAVGS, CHFIDS, and ECOLLS are singly
subscripted variables whose dimension must be a value of two less
than twice the dimension of the second subscript in the variables
VCOOP, RHO, EX, OLDRO, OLDV, CDNSTY, V, and EY.
The number of rapping puff particle size distributions that
can be utilized can be changed by changing the dimension of ARD50
and ARSIGM. ARD50 and ARSIGM appear in COMMON/BLK12/. COMMON/
BLK12/ appears in the main program and in subroutines PRTINP
and ADJUST.
The number of sets of nonideal conditions of nonuniform gas
velocity distribution and gas sneakage and/or particle reentrain-
ment without rapping that can be utilized can be changed by
changing the dimension of ASNUCK, AZIGGY, and AZNUMS. These
variables appear in COMMON/BLK12/. COMMON/BLK12/ appears in
those locations previously designated.
If any changes are made that affect arrays, it should be
pointed out that these changes will also affect the limitations on
the input data discussed in Section 8. The limitations on the in-
put data discussed previously are only applicable to the version
of the program presented in Appendix C of Volume 1. If changes are
made, new limitations on the input data must be established.
169
-------
REFERENCES
1. Gooch, J. P., J. R. McDonald, and S. Oglesby, Jr. A Mathe-
matical Model of Electrostatic Precipitation. EPA-650/2-75-037,
U.S. Environmental Protection Agency, Raleigh Durham, North
Carolina, 1975. pp. 78-79.
2. Gooch, J. P-, and J. R. McDonald. Mathematical Modelling of
Fine Particle Collection by Electrostatic Precipitation. At-
mospheric Emissions and Energy-Source Pollution, AICHE Sym-
posium Series, 73(165):146, 1977.
3. Gooch, J. P., and J. R. McDonald. Mathematical Modelling of
Fine Particle Collection by Electrostatic Precipitation.
Conference on Particulate Collection Problems in Converting
to Low Sulfur Coals, Interagency Energy-Environment Research
and Development Series. EPA-600/7-76-016, U.S. Environmental
Protection Agency, 1976. 68 pp.
4. Gooch, J. P., and G. H. Marchant, Jr. Electrostatic Precipi-
tator Rapping Reentrainment and Computer Model Studies. Final
Draft Report prepared for the Electric Power Research Insti-
tute, 1977.
5. Pauthenier, M., and M. Moreau-Hanot. Charging of Spherical
Particles in an Ionizing Field. J. Phys. Radium, 3 (7):590-
613, 1932.
6. White, H. J. Particle Charging in Electrostatic Precipitation.
Trans. Amer. Inst. Elec. Eng. Part 1, 70:1186-1191, 1951.
7. Murphy, A. T., F. T. Adler, and G. W. Penney. A Theoretical
Analysis of the Effects of an Electric Field on the Charging
of Fine Particles. Trans. Amer. Inst. Elec. Eng., 78:318-
326, 1959.
8. White, H. J. Industrial Electrostatic Precipitation. Addison-
Wesley, Reading, Massachusetts, 1963. p. 157.
9. Fuchs, N. A. The Mechanics of Aerosols. Chapter 2. Macmillan,
New York, 1964.
10. White, H. J. Reference 8, pp. 166-170.
11. White, H. J. Reference 8, pp. 185-190.
170
-------
12. Penney, G. W., and S. Craig. Pulsed Discharges Preceding
Sparkover at Low Voltage Gradients. AIEE Winter General
Meeting, New York, 1961.
13. Pottinger, J. F. The collection of Difficult Materials by
Electrostatic Precipitation. Australian Chem. Process Eng.,
20(2):17-23, 1967.
14. Spencer, H. W. Electrostatic Precipitators: Relationship
Between Resistivity, Particle Size, and Sparkover. EPA-
600/2-76-144, U.S. Environmental Protection Agency, Raleigh
Durham, North Carolina, 1976.
15. Bickelhaupt, R. E. Surface Resistivity and the Chemical
Composition of Fly Ash. APCA Journal, 25 (2) :148-152, 1975.
16. Bickelhaupt, R. E. Volume Resistivity - Fly Ash Composition
Relationship. Environmental Sc. & Tech., 9 (4):336-342, 1975.
17. Selle, S. J. , L. L. Hess, and E. A. Sondreal. Western Fly
Ash Composition as an Indicator of Resistivity and Pilot
ESP Removal Efficiency. Paper 75-02.5 presented at the 68th
Meeting of the Air Pollution Control Association, Boston,
Massachusetts, 1975.
18. Contract No, 68-02-2114, Task IV between E.P.A. and So.R.I.
19. White, H. J. Reference 8, pp. 238-293.
20. Preszler, L., and T. Lajos. Uniformity of the Velocity
Distribution Upon Entry into an Electrostatic Precipitator
of a Flowing Gas. Staub Reinhalt. Luft (in English),
32(11):l-7, 1972.
21. Spencer, H. W. A Study of Rapping Reentrainment in a Nearly
Full Scale Pilot Electrostatic Precipitator. EPA-600/2-76-140,
U.S. Environmental Protection Agency, Raleigh Durham, North
Carolina, 1976.
22. Leutert, G., and B. Bohlen. The Spatial Trend of Electric
Field Strength and Space Charge Density in Plate-Type Electro-
static Precipitators. Staub, 32(7):27, 1972.
23. McDonald, J. R., W. B. Smith, H. W. Spencer, and L. E. Sparks.
A Mathematical Model for Calculating Electrical Conditions
in Wire-Duct Electrostatic Precipitation Devices. J. Appl.
Phys., 48 (6) :2231-2246, 1977.
24. Smith, W. B., and J. R. McDonald. Development of a Theory
for the Charging of Particles by Unipolar Ions. J. Aerosol
Sci., 7:151-166, 1976.
171
-------
25. Hewitt, G. W. The Charging of Small Particles for Electro-
static Precipitation. AIEE Trans., 76:300, 1957.
26. Oglesby, S., and G. B. Nichols. A Manual of Electrostatic
Precipitator Technology: Part I, Fundamentals. NTIS PB
196380, APTD 0610, National Air Pollution Control Adminis-
tration, Cincinnati, Ohio, 1970. pp. 57-66.
27. McDonald, J. R. Mathematical Modelling of Electrical Con-
ditions, Particle Charging, and the Electrostatic Precipita-
tion Process. Ph.D. Dissertation, Physics Dept., Auburn
University, Auburn, Alabama, 1977. pp. 47-54.
28. Cooperman, P. The Dependence of the Electrical Character-
istics of Duct Precipitators on Their Geometry. Unpublished
Report, Research Corp., 1952.
29. Flugge, S. Handbuch der Physik (Handbook of Physics).
Springer-Verlay, Berlin, 16:248 ff., 1958.
30. Burns, K. J., and P. J. Lawrenson. Analysis and Computation
of Electrical and Magnetic Field Problems. Pergamon Press,
Oxford, 1963. p. 251 ff.
31. Ralston, A., and H. S. Wilf. Mathematical Methods for
Digital Computers. Wiley and Sons, Inc., New York, 1960.
p. 144 ff.
32. Young, D. Iterative Methods for Solving Partial Difference
Equations of Elliptic Type. Trans. Amer. Math. Soc.,
76:92-111, 1954.
33. Nielsen, K. L. Methods in Numerical Analysis, 3rd Edition.
The MacMillan Company, New York, 1964. pp. 236-239.
34. Nielsen, K. L. Reference 32, p. 122.
35. CRC Standard Mathematical Tables. Fourteenth Edition, edited
by Samuel M. Selby, The Chemical Rubber Co., Cleveland, Ohio,
1965. p. 409.
36. Murphy, A. T. Charging of Particles by Random Motion of
Ions in an Electric Field. Ph.D. Dissertation, Dept. of
Electrical Engineering, Carnegie Institute of Technology,
Pittsburgh, Pa., 1957. p. 59.
37. Hildebrand, F. B. Introduction to Numerical Analysis.
McGraw-Hill, Inc., New York, 1956. p. 75.
38. Hastings, C., Jr. Approximations for Digital Computers.
Princeton University Press, Princeton, New Jersey, 1955.
p. 192.
172
-------
39. Bevington, P. R. Data Reduction and Error Analysis for
the Physical Sciences. McGraw-Hill, Inc., New York, 1969.
pp. 99-122.
40. Spencer, H. W. Experimental Determination of the Effective
Ion Mobility of Simulated Flue Gas. In: Proceedings of
1975 IEEE-IAS Conference, Atlanta, Georgia, 1975.
41. White, H. J. Reference 8, p. 92.
173
-------
APPENDIX A
DEVELOPMENT OF NEW PROCEDURE FOR
DETERMINING SPACE CHARGE EFFECTS
174
-------
When particles are introduced into a precipitator, the mech-
anisms of particle charging and particle collection come into play.
In order to account for the dynamics of these mechanisms, it is
necessary to determine the ion density and electric field distri-
butions to which the particles will be subjected. These are ob-
tained for the flue gas without particles by calculating a voltage-
current curve using the technique discussed earlier. The reli-
ability of this calculation will depend to a large extent on the
choice of ion "effective mobility", used to represent clean flue
gas, and the condition of the discharge electrodes. Representative
values of ion "effective mobility" should be obtained from in. situ
measurements, or laboratory measurements made on gases of similar
composition in the proper environment.^° The condition of the
discharge wires with regards to roughness is accounted for by a
roughness factor f.41 This factor normally lies in the range of
0.5-1.0 and has a significant effect on the space charge density
near the discharge electrode.
For the desired operating voltage and current which are ob-
tained from the "clean" voltage-current curve, the corresponding
current density and electric field distributions are used to de-
termine average current densities j and average electric fields
E. for n incremental lengths A£ contained in wire-to-wire spacings
A/
centered on the wires. This formulation is depicted in Figure 28.
The incremental lengths A£ are the same size as the grid spacings
in the direction of gas flow used in the calculation of the
electrical conditions. Using symmetry considerations, we can
obtain all the information shown in Figure 28 from calculations
based on the area enclosed in the dashed lines. Although this
formulism does not provide a complete positional description, it
does allow for the effects of nonuniform current density and elec-
tric field on particle charging and.particle collection.
The values of the J{ and E. and the designated particle
charging equation are used to calculate the charge q. on each
i, x,
particle size i at the end of the £-th incremental length. In
the regions midway between wires,_the par_ticle charging rate will
be lowest due to lower values of j£ and E^. As uncollected par-
ticles move toward regions directly between a wire and the plate,
the cha£ging rate will tend to increase due to higher values of
j£ and Er
The average charge density p\ due to the total particulate
loading in the £-th incremental length is given by
X q. , (95)
175
-------
nA£ 2Sy
GAS FLOW
=n-1
ln-1
CORONA WIRES
\
COLLECTION PLATE
n NUMBER OF INCREMENTAL LENGTHS CONTAINED IN ONE
WIRE-TO-WIRE SPACING
Sy ONE-HALF THE WIRE-TO-WIRE SPACING
A£ = INCREMENTAL LENGTH
Eg AVERAGE ELECTRIC FIELD IN 2-TH INCREMENTAL LENGTH
7g AVERAGE CURRENT DENSITY IN £-TH INCREMENTAL LENGTH
Figure 28. Nomenclature used in the procedure which determines
paniculate space charge effects.
176
-------
where p. 0 = average charge density for the i-th particle size
i , x
at the end of the £-th incremental length (coul/m3),
and
X. 0 = number of particles per unit volume of gas of the
1 f A/
i-th particle size entering the £-th incremental
length (m~3) .
A weighted particulate mobility b,, due to all particles in
the £-th incremental length can be defined as
y;
r- i (Xi,£ <*!,*
07)
177
-------
where b" = molecular ion "effective mobility" (m2/V-sec),
p"" = average ionic charge density without mass loading in
A/
the £-th incremental length (coul/m3), and
Ap"0 = average charge density shifted from molecular ions to
A/
particles in the £-th incremental length (coul/m3) .
An effective mobility b^ due to both ions and particles in
the £-th incremental length is found from
b'F + b p
p£ + p£
Since a certain number of particles will be removed from the
gas stream in the £-th incremental length, it is necessary to
calculate collection efficiencies for the different particle sizes.
The collection efficiencies are calculated using equation (5) ,
where the migration velocities are calculated from
Wi,£ "" ^M* ' (99)
and E^ is the average electric field at the collection plate in
the £-th incremental length. Thus, the size distribution entering
the (£+l)-th incremental length is obtained from
- r) ) X . (100)
J. , X, J- , X,
An "effective mobility" b, is calculated for each of n suc-
cessive incremental lengths over a total length equal to the wire-
to-wire spacing. Then, the "average effective mobility" be for
ions and particles over a length equal to the wire-to-wire spacing
is calculated from
n
b? . (101)
£=1
The value of b is used to generate a voltage-current curve
for the particular wire-to-wire length under consideration in order
178
-------
to describe the effect of particles on the electrical conditions.
In this calculation, it is assumed that over several wire-to-wire
spacings symmetry in the electric field and space charge density
distribution is essentially preserved. Also, strictly speaking,
it is not valid to generate the entire voltage-current curve with
Q
a constant value of b when particles are present. This is because
Q
b is a function of applied voltage and current density since it
depends on particle charging and particle collection. However, at
the operating applied voltage, the correct current density will
be predicted and, in fact, the entire voltage-current curve will
be approximately correct since b normally will not vary enough
over practical voltage ranges to produce significant differences.
In any case, the generation of a voltage-current curve can be
viewed as a systematic procedure for searching for the current
density that would exist at the operating applied voltage.
In lieu of using the time-consuming, voltage-current calcu-
lation, it is possible to estimate the operating current density
for a given applied voltage from a simple relationship, provided
certain considerations are made. If particles are introduced into
a precipitator and the applied voltage is held fixed, then the
current density at any location and the effective charge carrier
mobility will be lowered. Since the product of total space charge
density and electric field strength at any location is equal to
the ratio of current density to effective mobility, the product
tends to remain constant. If it is assumed that the limited
regions of ionization near the corona electrodes are unchanged
by the presence of particles, then, even though charge is trans-
ferred to particles in these regions, the space charge density
and electric field near a wire will both remain essentially
constant. Thus,
jw/b' = jw/be , (102)
where
J = average current density near the wire without particles
(A/m2), and
j" = average current density near the wire with particles
(A/m2).
The average current density at the wire is related to the average
current density at the plate by
)T ' U03)
179
-------
where
A' = collection plate area receiving current from a single
wire (m2),
A = surface area of a single wire (m2), and
w
3" = average current density at the collection plate without
particles (A/m2).
Using equations (102) and (103) ,
Tp - (be/b")Tp , (104)
where J"* (A/m2) is the average current density at the collection
plate with particles present. Equation (104) provides a simple
way to estimate the reduction in current density at the plate due
to the presence of particles.
The operating current densities are determined for successive
wire-to-wire spacings throughout the length of the precipitator.
Since the space charge scheme incorporates the dynamics of the
precipitation process, particle collection efficiencies are pre-
dicted as well as operating voltages and current densities. In
this scheme, no estimate of overall mass efficiency is necessary
as is the case in the procedure using equation (16) and this is
advantageous in designing a new precipitator.
Figures 29-35 show some theoretical trends predicted by using
the new space charge scheme. In these figures, the ion "effective
mobility", at standard temperature and pressure, and the roughness
factor were taken to be 2.2 x lO"4 m2/V«sec and 1.0, respectively.
The parameters used in the calculations are typical of full-scale,
cold-side precipitators. The electrode geometry consists of
plate-to-plate and wire-to-wire spacings of 22.86 cm and a wire
radius of 0.138 cm. The inlet particle size distribution (MMD =
25 urn, a = 2.8) is characteristic of fly ash obtained from the
combustion of Eastern coal.
Figures 29-31 show the variation of average current density
along the length of the precipitator for different inlet mass
loadings, specific collection areas, and voltage levels. The
different specific collection areas were obtained by leaving the
plate area fixed and varying the volume flow rate. Near the inlet
of the precipitator, the curves show a minimum in the average
current density at the plate. This behavior might be expected
since the "effective mobility" should initially decrease due to
the charging process and then, as the charging process slows down
180
-------
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Figure 29. Theoretical variation of average current density at the plate with
precipitator length for different specific collection areas and inlet
mass loadings at 33 kV.
181
-------
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Figure 30. Theoretical variation of average current density at the plate with
precipitator length for different specific collection areas and inlet
mass loadings at 35 kV.
182
-------
URRENT DENSITY WITHOUT LOADING
19.7 m2/(m3/sec)
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Figure 31. Theoretical variation of average current density at the plate with
precipitator length for different specific collection areas and inlet
mass loadings at 40 kV.
183
-------
and charged particles are collected, it should reach a minimum
and begin to increase.
Figures 32-35 show results obtained by dividing the pre-
cipitator into four electrical sections and calculating voltage-
current characteristics for each section. The calculations are
for an inlet mass loading of 9.16 x 10~3 kg/m3 and a range of
specific collection areas. The curves indicate the effect of
particulate space charge on the operating voltage-current char-
acteristics in different electrical sections of the precipitator.
In Figure 36, the predictions of the model with the new
space charge scheme are compared with field test data from a
full-scale precipitator with wire-duct geometry and the predic-
tions of the model with the old space charge scheme. The figure
also shows the effect of the roughness factor in determining
current density and collection efficiency. The results indicate
that the condition of the wire plays a very important role in '
the theoretical determination of voltage-current characteristics.
A roughness factor of 0.9 yields an average operating current
density of 25 nA/cm2 at the operating voltage of 33 kV. The
actual average operating current density was 20 nA/cm2. The
roughness factor could be adjusted to yield 20 nA/cm2 but this
refinement would probably not be meaningful due to the uncertainty
in the ion mobility of the flue gas which was taken to be 2.2 x
10-k m2/V«sec at standard temperature and pressure. A roughness
factor of 0.9 would not be unreasonable since the electrodes were
known to be in good condition at the time of the test. If the
wires are specked with dirt or scratched, then it is appropriate
to use a roughness factor. However, if the wires are uniformly
coated with a layer of dirt, then the effect is one of increasing
the radius of the discharge electrode which has a different
effect on the voltage-current characteristics.
184
-------
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SCA = 19.7 m2/(m3/sec)
INLET MASS LOADING = 9.16 x 10'3
33 34 35 36 37
APPLIED VOLTAGE, kV
38
39
40
Figure 32. Theoretical voltage-current curves for a specific collection area of
19.7 m^/(mj/sec).
185
-------
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SCA = 59.1 m2/(m3/sec)
INLET MASS LOADING = 9.16 x 10'3 kg/m3
33 34 35 36 37
APPLIED VOLTAGE, kV
38
39
40
Figure 33. Theoretical voltage-current curves for a specific collection area of
59.1 m2/(m3/sec).
186
-------
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SCA = 98.4 m2/(m3/sec)
INLET MASS LOADING = 9.16
I
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33 34 35 36 37
APPLIED VOLTAGE, kV
38
39
40
Figure 34. Theoretical voltage-current curves for a specific collection area of
98.4 m2/m3/sec).
187
-------
tM
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cc
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INLET MASS LOADING = 9.16 x 10'-3 kg/m
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33 34 35 36 37
APPLIED VOLTAGE, kV
38
39
40
Figure 35. Comparison of theoretical voltage-current curves for different
specific collection areas.
188
-------
99.9
99.8
99.5
99
£ 98
o
95
90
80
70
60
I
n
m
13
MODEL WITH NEW SPACE CHARGE SCHEME
MODEL WITH OLD SPACE CHARGE SCHEME
f = 0.8, j = 40.4 nA/cm2
f = 0.9, j = 25.6 nA/cm2
j = 20 nA/cm2
f = 1.0, j = 13.3 nA/cm2
' I ' I I I I,
0.1
1.0
PARTICLE DIAMETER,
10.0
Figure 36. Comparison of model predictions using the different space charge
schemes with field test data from a full-scale precipitator. Model
predictions are for unadjusted, no-rap efficiencies where og = 0.25
and S = 0.
189
-------
APPENDIX B
DEFINITIONS OF VARIABLES USED IN
THE MAIN PROGRAM AND SUBROUTINES
190
-------
LIST OF VARIABLES, DEFINITIONS, AND UNITS
FOR THE MAIN PROGRAM OF THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
NWIRE - Number of wires per electrical section per gas
passage in a particular electrical section
LTHICK - Thickness of the collected particulate layer in a
particular increment of length (mm/min)
JPART - Current density due to particles in a particular
increment of length (A/m2)
JION - Current density due to ions in a particular incre-
ment of length (A/m2)
LING - Length of the increments taken in a particular
linear electrical section (m)
NWS(I) - Number of wires per electrical section per gas
passage for the different linear electrical sections
LINGS(I) - Lengths of the increments taken in the different
linear electrical sections (ft)
VISKIP - Indicator which determines whether or not a dirty-
gas voltage-current curve is calculated in each
increment of length
VISAME - Indicator which determines whether or not a clean-
gas voltage-current curve is calculated for each
of the electrical sections or just the first elec-
trical section
CHKSUM(K) - Fractional increase in charge from one increment to
the next for the different particle sizes
DIAM(K) - Diameters of the different particle sizes (urn and m)
ONO(K) - Initial number of particles per cubic meter of gas
in each particle size band (#/m3)
DXS(K) - Total number of particles removed per cubic meter of
gas in each particle size band under ideal conditions
and with no empirical corrections (#/m3)
191
-------
XMV(K) - Effective migration velocities for the different
particle sizes under ideal conditions and with no
empirical corrections (m/sec)
PCNT(K) - Percentage or fraction by mass in the inlet particle
size distribution of the different size bands (% and
decimal)
RAD(K) - Radii of the different particle sizes (m)
CCF(K) - Cunningham correction factor for the different
particle sizes
PRCU(L) - Cumulative percent by mass up to each particle size
in the inlet particle size distribution (%)
LSECT(I) - Number of length increments in the different linear
electrical sections
PS (I) - Gas pressure in the different electrical sections
(atm)
VG - Gas volume flow rate in a particular electrical
section (m3/sec)
ATOTAL - Total collection plate area of the precipitator
(m2)
DD - Mass density of the particles (kg/m3)
ETAO - Estimated or design overall mass collection
efficiency (%)
DL - Inlet mass loading (grains/ft3 and kg/m3)
PL - Total electrical length of the precipitator (ft
and m)
RHO - Resistivity of the collected particulate layer
(ohm-cm and ohm-m)
NS - Number of different particle size bands in the inlet
particle size distribution
ZMMDI - Specified or fitted mass median diameter of the
inlet particle size distribution based on a log-
normal distribution (ym)
SIGMI - Specified or fitted geometric standard deviation of
the inlet particle size distribution based on a log-
normal distribution
192
-------
NONID - Number of nonideal conditions of gas velocity non-
uniformity and gas sneakage and/or particle reen-
trainment without rapping to be considered
NRAED - Number of rapping puff particle size distributions
to be considered
TDK - Temperature of the gas in a given electrical sec-
tion (°K)
NUMSEC - Number of linear electrical sections in the precip-
itator
NEFF - Indicator which determines whether the unadjusted,
ideal or adjusted, no-rap efficiency is used to
determine the mass reentrained due to rapping
NTEMP - Indicator which specifies whether the precipitator
is cold or hot side
GFIT - Linear-correlation coefficient obtained in the log-
normal fit of the inlet particle size distribution
VOL(K) - Total volume of particles per cubic meter of gas
in the different size bands (m3/m3(gas))
XNO(K) - Number of particles per cubic meter of gas in each
size band at the start of each increment (#/m3)
Q(K) - Charge on each particle size at the end of a partic-
ular increment (coul)
WS(K) - Total weight of material per cubic meter of gas
removed in each size band in a particular incre-
ment (kg/m3)
ITL(M) - Identifying label for the calculations
DW(J) - Amount of material removed per increment on a total
weight basis (kg)
AS(I) - Collection plate areas for the different linear
electrical sections (m2)
VOS(I) - Applied voltages for the different linear electrical
sections (V)
TCS(I) - Total current for the different linear electrical
sections (A)
WLS(I) - Total wire length for the different linear electri-
cal sections (ft2)
193
-------
ACS(I) - Corona wire radii for the different linear electri-
cal sections (in.)
BS(I) - Wire-to-plate spacing for the different linear
electrical sections (in.)
SYS(I) - One-half the wire-to-wire spacing for the different
linear electrical sections (in.)
VGS(I) - Gas volume flow rate for the different linear
electrical sections (ft3/min)
VGASS(I) - Gas velocity for the different linear electrical
sections (ft/sec)
TEMPS(I) - Gas temperature for the different linear electrical
sections (°F)
VISS(I) - Gas viscosity for the different linear electrical
sections (kg/m-sec)
QSAT(K) - Saturation charge for the different particle sizes
(coul)
U - Ion mobility adjusted for temperature and pressure
(m2/V-sec)
E - Elementary charge unit (coul)
EPSO - Permittivity of free space (cou!2/nt-m2)
PI - Value of the constant IT
ERAVG - Average electric field used for particle charging
(V/m)
BC - Boltzmann's constant (J/°K)
TEMP - Gas temperature in a particular linear electrical
section (°R)
EPS - Relative dielectric constant of the particles
VAVC - Root mean square velocity of the ions (m/sec)
OLDQ(K) - Charge on the different particle sizes in the incre-
ment prior to the one under consideration (coul)
OLDXNO(K) - Number of particles per cubic meter of gas in each
size band at the start of the increment prior to
the one under consideration (#/m3)
-------
RFS(I) - Roughness factor for the corona wires in the dif-
ferent linear electrical sections
STARTl(I) - Specified initial current density at which the
calculation of a voltage-current curve starts in
a given electrical section and the initial current
density increment size (A/m2)
START2(I) - Specified increment in current density which is
used in place of STARTl(I) when the Jll-th point
on the voltage-current curve is reached (A/m2)
STARTS(I) - Specified increment in current density which is
used in place of START2(I) when the Jl2-th point
on the voltage-current curve is reached (A/m2)
VSTAR(I) - Estimate of the applied voltage corresponding to
the first point on the voltage-current curve as
defined by STARTl(I) (V)
XDC(J,K) - Charge on each particle size at the end of each
increment (coul)
EAVG(N) - Average electric fields for particle charging in
subincremental lengths (V/m)
CHFID(N) - Average free ion densities for particle charging
in subincremental lengths (#/m3)
ECOLL(N) - Average electric fields at the plate in subincre-
mental lengths (V/m)
ECLEAN(N) - Average electric fields at the plate for clean gas
in subincremental lengths (V/m)
ENDPT(L) - Particle diameters in the inlet cumulative percent
by mass distribution (ym and m)
NENDPT - Number of particle diameters in the inlet cumulative
percent by mass distribution
ARDSO(II) - Rapping puff mass median diameters (ym)
ARSIGM(II) - Rapping puff geometric standard deviations
ASNUCK(JJ) - Fractions of gas sneakage and/or particle reentrain-
ment without rapping
AZNUMS(JJ) - Number of stages over which gas sneakage and/or
particle reentrainment without rapping occur
195
-------
AZIGGY(JJ) - Normalized standard deviations of the gas velocity
distribution
VCOOP(KK,LL) - Values at different grid points of the electric
potential in a wire-plate geometry under conditions
of no space charge (V)
TMFP - Ionic mean free path multiplied by a factor (m)
NVI - Indicator which specifies whether to base the elec-
trical calculation on known voltages and currents
or on calculated voltage-current characteristics
NPRINT - Indicator which designates when to print certain
sectionalized data
NSECT - Indicator which keeps track of which electrical
section the calculation is in
SLNGTH - Length of a particular electrical section (m)
A - Collection plate area of a particular linear elec-
trical section (m2)
VO - Applied voltage in a particular linear electrical
section (V)
TC - Total current in a particular linear electrical
section (A)
B - Wire-to-plate spacing in a particular linear elec-
trical section (m)
AC - Corona wire radius in a particular linear electrical
section (m)
WL - Total wire length in a particular linear electrical
section (m)
CL - Total current per length of corona wire in a partic-
ular linear electrical section (A/m)
CD - Average current density at the plate in a particular
linear electrical section (A/m2)
ET - Average electric field in the deposited particulate
layer in a particular linear electrical section (V/m)
SY - One-half the wire-to-wire spacing in a particular
linear electrical section (m)
VGAS - Gas velocity in a particular linear electrical sec-
tion (m/sec)
196
-------
P - Gas pressure in a particular linear electrical
section (atm)
VIS - Gas viscosity in a particular linear electrical
section (kg/m-sec)
W - Total weight of particles per second passing into
a particular linear electrical section (kg/sec)
XPI - Overall mass collection efficiency per increment
based on the estimated or design efficiency (%)
RIOVR - Ratio of the ionic space charge density to the total
space charge density
EPLT - Absolute value of the average electric field at the
plate in a particular length increment (V/m)
AFID - Average reduced free ion density for particle
charging in a particular length increment (#/m3)
XCD - Average current density at the plate in a particular
length increment (nA/cm2)
ZMD - Interpolated mass median diameter of the collected
particulate layer (m)
WT - Total weight of material per cubic meter of gas
removed in all particle size bands in a given length
increment (kg/m )
I - Index which runs over the different incremental
lengths in its major usage
ROVRI - Ratio of the total space charge density to the ionic
space charge density
NCALC - Indicator which determines whether to use equation
(12) for particle charging or the sum of the clas-
sical field and diffusion charges
NI - Number of subincremental lengths into which the
incremental length is divided
VRATIO - Ratio of the peak applied voltage to the average for
use in particle charging
NF - Number of increments taken along the length of the
precipitator
NREAD - Indicator which specifies the unit number of the
input device for reading data into the program
197
-------
NPRNT - Indicator which specifies the unit number of the out-
put device for printing data from the program
SCOREF - Overall mass collection efficiency under no-rap +
rap conditions (%)
CZMDL - Fitted log-normal mass median diameter of the outlet
particle size distribution under no-rap + rap
conditions (ym)
CSIGMO - Fitted log-normal geometric standard deviation of
the outlet particle size distribution under no-rap
+ rap conditions
NRUN - Indicator that specifies which set of nonideal
conditions is under consideration
SNUCK - Particular value of ASNUCK(JJ)
ZIGGY - Particular value of AZIGGY(JJ)
RMMD - Particular value of ARDSO(II)[ym]
RSIGMA - Particular value of ARSIGM(II)
LK - Indicator which determines whether or not the input
data are printed at a certain location in the program
DV - Total volume per cubic meter of gas occupied by
particles [m3(particles)/m3(gas)]
NN - Number of increments in the Runge-Kutta integration
of equation (12)
NUMINC - Number of increments in the Simpson's Rule integra-
tion over 0 in equation (12)
NX - Number of grid points in the x-direction for the
numerical calculations of electrical conditions
NY - Number of grid points in the y-direction for the
numerical calculations of electrical conditions
NDATA - Indicator which determines the type of data set that
is to be read into the program
NEST - Indicator which specifies whether to use extensive
calculations or estimation procedures in determin-
ing precipitator performance
NDIST - Indicator which specifies whether the user is to sup-
ply the inlet particle size distribution or the pro-
gram is to calculate a log-normal distribution
198
-------
NITER - Indicator which determines the maximum number of
iterations over a loop that converges on overall
mass efficiency or the number of iterations that
will be performed over each incremental length of
the precipitator in order to obtain self-consistent
solutions for the electrical conditions
IFINAL - Indicator which causes the calculation of successive
points on the voltage-current curve to cease after
IFINAL points
JIl - Indicator which allows the initial increment size
on current density in the calculation of the voltage-
current curve to be changed after JIl-1 points are
determined on the curve
JI2 - Indicator which allows the second increment size on
current density in the calculation of the voltage-'
current curve to be changed after JI2-1 points are
determined on the curve
US - Ion mobility at standard temperature and pressure
(reduced ion mobility)
FPATH - Factor which scales the ion mean free path
EBD - Electrical breakdown strength of the gas near the
collection electrode or the collected particulate
layer (V/m)
NDSET - Counter which keeps track of the number of the
particular set of nonideal conditions which is under
consideration
D50 - Same as ZMMDI (ym)
SIGMAP - Same as SIGMI
SCHARG - Saturation charge number from the field charging
equation
CHRFID - Average free ion density for particle charging
(#/m3)
TIMEI - Initial value of time for particle charging (sec)
TIMEF - Final value of time for particle charging (sec)
V - Value of the quantity [e2/4Tre 0akT] found in equa-
tion (12)
FACTRE - Value of the quantity [Trva2/2] found in equation
(12) [m3/sec]
199
-------
RSIZE - Radius of a particular particle (m)
CNUMBR - Charge number of a particular particle at time TIMEF
J - Index which runs over different particle size bands
II - Index which runs over subincremental lengths
ITER - Counter which keeps track of the number of itera-
tions which is limited by NITER
OLDQF(K) - Value of field charge on the different particle sizes
at the end of a given increment or subincrement
(coul)
OLDQT(K) - Value of diffusion charge on the different particle
sizes at the end of a given increment or subincre-
ment (coul)
SOLDQF(K) - Value of field charge on the different particle
sizes at the start of an increment which must be
saved for the iteration procedure over subincrements
in a given increment (coul)
SOLDQT(K) - Value of diffusion charge on the different particle
sizes at the start of an increment which must be
saved for the iteration procedure over subincrements
in a given increment (coul)
CMKS - Value of the quantity [4Tre0] found in equation (12)
[cou!2/nt-m2]
KA - Index which runs over the different linear electrical
sections
ZWT - Total weight of material per cubic meter of gas
removed up to a given increment (kg/m3)
RATIO - Value of the quantity [(K-l)/(K+2)] found in the
particle charging equations
G - Value of the quantity [K+2] found in the particle
charging equations
INDEX - Indicator which keeps track of how many increments
the calculation is into a particular linear electri-
cal section
NCOOP - Indicator which allows certain calculations to be
made only at the start of a new linear electrical
section
200
-------
SX - Wire-to-plate spacing in a particular linear electri-
cal section (m)
RF - Roughness factor for the discharge wires in a partic-
ular linear electrical section
START - Particular value of STARTl(I) [A/m2]
DSTART - Particular value of START2(I) [A/m2]
CSTART - Particular value of STARTS(I) [A/m2]
VSTART - Particular value of VSTAR(I) [V]
ZMFP - Ionic mean free path (m)
VAVG - Root mean square velocity of the ions (cm/sec)
VC - Value of the quantity [e2/kT] found in the charging
equations (coul/V)
FACTRC - Value of the quantity [irv/2] found in the charging
equations (m/sec)
COEFFC - Value of the quantity [eirb] found in the charging
equations (coul-m2/V-sec)
TINC - Time interval for the gas to travel one increment
(sec)
DTINC - Time interval for the gas to travel one subincrement
(sec)
L - Index which runs over the different particle size
bands
R - Value of the quantity [eE0/kT(K+2)] found in equa-
tion (12) [m~f]
RR - Value of the quantity [eE0/kT] found in equation
(12) [m-1]
RG - Same as RR
VW - Operating applied voltage corresponding to a spec-
ified current density (V)
UEQ - Effective charge carrier mobility (m2/V-sec)
NEC - Indicator which determines whether or not the average
current density, average electric field, and average
electric field at the plate are to be calculated in
the subincremental lengths
201
-------
AEPLT - Average electric field at the plate in a particular
increment (V/m)
ACDNTY - Average current density at the plate in a particular
increment (A/m2)
NZ - Index which runs over subincremental lengths
CDCLN - Average current density at the plate when the gas
is clean (A/m2)
USUM - Sum of effective charge carrier mobilities over the
subincremental lengths in a particular incremental
length (m2/V-sec)
WSSUM - Total weight of material per cubic meter of gas
removed in a particular size band in a particular
subincrement (kg/m3)
RHOSUM - Sum of the ratio of the ionic space charge to the
total space charge over the subincremental lengths
in a particular incremental length
SW - Cumulative sum of estimated amount of material
removed per second in successive length increments
(kg/sec)
OROVRI - Ratio of total charge density to ionic charge dens-
ity in increment prior to the one the calculation
is in
XS - Computed value of the exponential argument in the
Deutsch equation for the estimated or design overall
mass collection efficiency
ETAPF - Overall mass collection fraction per increment based
on the estimated or design efficiency
FID - Average free ion density (#/m3)
AVGFID - Average reduced free ion density for particle charg-
ing (#/cm3)
PROT - Total charge density due to particles that remain
after passing through a given increment (coul/m3)
SERAVG - Average electric field in a particular increment
(V/m)
XIPC - Initial value of charge number on a given particle
size at the start of a new increment
202
-------
H - Increment size for the Runge-Kutta integration of
equation (12) [sec]
DCONST - Value of the quantity [ (K-l)a3/(K+2)] found in
equation (12) [m3]
CONST - Value of the quantity [2(K-l)a3E0/(K+2)] found in
equation (12) [V-m2]
S - Value of the quantity [3a] found in equation (12)[m]
ECONST - Value of the quantity [3eE0a/kT(K+2)] found in
equation (12)
FCONST - Value of the quantity [ (K-l)eE0a3/kT(K+2)] found in
equation (12) [m2]
COEFF - Value of the quantity [bqs/4e0] found in equation '(12)
[m3/sec]
CTIME - Time at the end of a given increment (sec)
EMV(K) - Unadjusted, ideal migration velocities for the dif-
ferent particle sizes in a given increment (m/sec)
X - Exponent used in the Deutsch equation to determine
the unadjusted, ideal collection fractions for the
different particle sizes in a given increment
EFF - Unadjusted, ideal collection fraction for a given
particle size band in a given increment
DXNO - Number of particles per cubic meter of gas removed
from a given particle size band in a given incre-
ment (#/m3)
DNS 101-7 - Ion density in the absence of particles (#/m3)
DELTNP - Number density of charges transferred from ions to
particles in a given subincremental length (#/m3)
SUMMOB - Weighted summation of particle mobilities (m2/V-sec/
m3)
PNUM - Total number of particles per unit volume of gas
entering a given subincremental length (#/m3)
RHOP - Total average particulate charge density in a given
subincremental length (coul/m3)
TCHRG - Average particle charge density for a given particle
size in a given subincremental length (coul/m3)
203
-------
PMOB - Weighted particulate mobility in a given subincre-
mental length (m2/V-sec)
TDNSP - Total average particulate charge number density in a
given subincremental length (#/m3)
RDNSI - Average reduced ion density in a given subincremental
length (#/m3)
SUMCD - Sum of the average current densities at the plate from
the different increments in a particular linear elec-
trical section (A/m2)
SUMVO - Sum of the applied voltages from the different incre-
ments in a particular linear electrical section (V)
SKIP - Electric field at the plate in the increment prior
to the one the calculation is in (V/m)
SIGMA - Difference between the ratio of the total space
charge density to the ionic space charge density in
the (I+l)-th and I-th increments
VERGE - Initial estimate of the space charge density at the
corona wire to start the calculation of the electric
field at the plate (coul/m3)
CVERGE - Converged value of the space charge density at the
wire in calculating the electric field at the
plate (coul/m3)
ZTM - Cumulative sum of the weight of material per cubic
meter of gas collected up to a given particle size
in a given increment (kg/m3)
CZA - Ratio of the partial sum of the weight of dust re-
moved per cubic meter of gas up the K-th particle
size in a given increment to the total weight of
dust removed per cubic meter of gas in a given
increment
CZB - Ratio of the partial sum of the weight of dust re-
moved per cubic meter of gas up to the (K-l)-th
particle size in a given increment to the total
weight of dust removed per cubic meter of gas in
a given increment
TL1 - Difference between CZA and CZB for use in interpolat-
ing to find the mass median diameter of the collected
dust
204
-------
TL2 - Difference between 0.50 and CZB for use in interpolat-
ing to find the mass median diameter of the collected
dust
KJ - Index which runs simultaneously with the index which
runs over the different particle sizes and keeps
track of the (K-l)-th particle size
ETC - Ideal, unadjusted overall mass collection efficiency
for the entire precipitator (%)
DIFF - Difference between the calculated ideal, unadjusted
overall mass collection efficiency and the estimated
value
205
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LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE PRTINP USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
LK - Indicator which determines whether or not the input
data are printed at a certain location in the program
NPRNT - Indicator which specifies the unit number of the
output device for printing data from the program
NDSET - Counter which keeps track of the number of the
particular set of nonideal conditions which is
under consideration
DL - Inlet mass loading (kg/m3)
DLB - Inlet mass loading (grains/ft3)
PL - Total electrical length of the precipitator (m)
PLB - Total electrical length of the precipitator (ft)
RHO - Resistivity of the collected particulate layer
(ohm-m)
RHOCGS - Resistivity of the collected particulate layer
(ohm-cm)
NCARD - Counter which keeps track of the number of each
successive imput data card
NENDPT - Number of particle diameters in the inlet cumulative
percent by mass distribution
NDATA - Indicator which determines the type of data set that
is to be read into the program
ITL(M) - Identifying label for the calculations
NEST - Indicator which specifies whether to use extensive
calculations or estimation procedures in determin-
ing precipitator performance
NDIST - Indicator which specifies whether the user is to
supply the inlet particle size distribution or the
program is to calculate a log-normal distribution
206
-------
NVI - Indicator which specifies whether to base the elec-
trical calculation on known voltages and currents
or on calculated voltage-current characteristics
NX - Number of grid points in the x-direction for the
numerical calculations of electrical conditions
NY - Number of grid points in the y-direction for the
numerical calculations of electrical conditions
NITER - Indicator which determines the maximum number of
iterations over a loop that converges on overall
mass efficiency or the number of iterations that
will be performed over each incremental length of
the precipitator in order to obtain self-consistent
solutions for the electrical conditions
NCALC - Indicator which determines whether to use equation'
(12) for particle charging or the sum of the clas-
sical field and diffusion charges
NRAPD - Number of rapping puff particle size distributions
to be considered
NEFF - Indicator which determines whether the unadjusted,
ideal or adjusted, no-rap efficiency is used to
determine the mass reentrained due to rapping
NTEMP - Indicator which specifies whether the precipitator
is cold or hot side
NONID - Number of nonideal conditions of gas velocity non-
uniformity and gas sneakage and/or particle reen-
trainment without rapping to be considered
NN - Number of increments in the Runge-Kutta integration
of equation (12)
NUMINC - Number of increments in the Simpson's Rule integra-
tion over 0 in equation (12)
IFINAL - Indicator which causes the calculation of successive
points on the voltage-current curve to cease after
IFINAL points
JI1 - Indicator which allows the initial increment size
on current density in the calculation of the voltage'
current curve to be changed after JI1-1 points are
determined on the curve
207
-------
JI2 - Indicator which allows the second increment size on
current density in the calculation of the voltage-
current curve to be changed after JI2-1 points are
determined on the curve
VISKIP - Indicator which determines whether or not a dirty-
gas voltage-current curve is calculated in each
increment of length
VISAME - Indicator which determines whether or not a clean-
gas voltage-current curve is calculated for each
of the electrical sections or just the first elec-
trical section
ETAO - Estimated or design overall mass collection
efficiency (%)
DD - Mass density of the particles (kg/m3)
EPS - Relative dielectric constant of the particles
VRATIO - Ratio of the peak applied voltage to the average for
use in particle charging
US - Ion mobility at standard temperature and pressure
(reduced ion mobility)
FPATH - Factor which scales the ion mean free path
EBD - Electrical breakdown strength of the gas near the
collection electrode or the collected particulate
layer (V/m)
ARDSO(II) - Rapping puff mass median diameters (ym)
ARSIGM(II) - Rapping puff geometric standard deviations
ASNUCK(JJ) - Fractions of gas sneakage and/or particle reentrain-
ment without rapping
AZIGGY(JJ) - Normalized standard deviations of the gas velocity
distribution
AZNUMS(JJ) - Number of stages over which gas sneakage and/or
particle reentrainment without rapping occur
NDCARD - Indicator which determines how the arrays ENDPT(L)
and PRCU(L) should be printed
ENDPT(L) - Particle diameters in the inlet cumulative percent
by mass distribution (ym and m)
208
-------
D50 - Specified or fitted mass median diameter of the
inlet particle size distribution based on a log-
normal distribution (ym)
SIGMAP - Specified or fitted geometric standard deviation of
the inlet particle size distribution based on a log-
normal distribution
PRCU(L) - Cumulative percent by mass up to each particle size
in the inlet particle size distribution (%)
NUMSEC - Number of linear electrical sections in the precip-
itator
LSECT(I) - Number of length increments in the different linear
electrical sections
AS(I) - Collection plate areas for the different linear
electrical sections (m2)
VOS(I) - Applied voltages for the different linear electrical
sections (V)
TCS(I) - Total current for the different linear electrical
sections (A)
WLS(I) - Total wire length for the different linear electri-
cal sections (ft2)
ACS(I) - Corona wire radii for the different linear electri-
cal sections (in.)
BS(I) - Wire-to-plate spacing for the different linear
electrical sections (in.)
NWS(I) - Number of wires per electrical section per gas
passage for the different linear electrical sections
SYS(I) - One-half the wire-to-wire spacing for the different
linear electrical sections (in.)
VGS(I) - Gas volume flow rate for the different linear
electrical sections (ft3/:min)
VGASS(I) - Gas velocity for the different linear electrical
sections (ft/sec)
TEMPS(I) - Gas temperature for the different linear electrical
sections (°F)
PS(I) - Gas pressure in the different electrical sections
(atm)
209
-------
VISS(I) - Gas viscosity for the different linear electrical
sections (kg/m-sec)
LINCS(I) - Lengths of the increments taken in the different
linear electrical sections (ft)
EPS(I) - Roughness factor for the corona wires in the dif-
ferent linear electrical sections
STARTl(I) - Specified initial current density at which the
calculation of a voltage-current curve starts in
a given electrical section and the initial current
density increment size (A/m2)
START2(I) - Specified increment in current density which is
used in place of STARTl(I) when the Jll-th point
on the voltage-current curve is reached (A/m2)
START3(I) - Specified increment in current density which is
used in place of START2(I) when the Jl2-th point
on the voltage-current curve is reached (A/m2)
VSTAR(I) - Estimate of the applied voltage corresponding to
the first point on the voltage-current curve as
defined by STARTl(I) (V)
210
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LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE SPCHG1 USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
I - Index which runs over the incremental lengths
SW - Cumulative sum of estimated amount of material
removed per second in successive length increments
(kg/sec)
ROVRI - Ratio of the total space charge density to the ionic
space charge density
OROVRI - Ratio of total charge density to ionic charge dens-
ity in increment prior to the one the calculation
is in
ETAO - Estimated overall mass collection efficiency (%)
XS - Computed value of the exponential argument in the
Deutsch equation for the estimated overall mass
collection efficiency
LING - Length of the increments taken in a particular
linear electrical section (m)
PL - Total electrical length of the precipitator (m)
ETAPF - Overall mass collection fraction per increment based
on the estimated efficiency
W - Total weight of particles per second passing into
a particular linear electrical section (kg/sec)
DW(J) - Amount of material removed per increment on a total
weight basis (kg)
CD - Average current density at the plate in a particular
linear electrical section (A/m2)
E - Elementary charge unit (coul)
U - Ion mobility adjusted for temperature and pressure
(m2/V-sec)
211
-------
ERAVG - Average electric field used for particle charging
(V/m)
FID - Average free ion density (#/m3)
SUM - Total particulate charge density in a given incre-
ment based on saturation charges (coul/m3)
NS - Number of different particle size bands in the inlet
particle size distribution
L - Index which runs over the different particle size
bands
QSAT(L) - Saturation charge for the different particle sizes
(coul)
XNO(L) - Number of particles per cubic meter of gas in each
size band at the start of each increment (#/m3)
LSECT(I) - Number of length increments in the different linear
electrical sections
NSECT - Indicator which keeps track of which electrical
section the calculation is in
TC - Total current in a particular linear electrical
section (A)
VG - Gas volume flow rate in a particular electrical
section (m3/sec)
ZC - Ratio of the particulate charge density to the ionic
charge density (ratio of 200 times the particulate
current to the total current)
AFID - Average reduced free ion density for particle
charging in a particular length increment (#/m3)
AVGFID - Average reduced free ion density for particle
charging (#/cm3)
XCD - Average current density at the plate in a particu-
lar length increment (nA/cm2)
UEQ - Effective charge carrier mobility (m2/V-sec)
XPI - Overall mass collection efficiency per increment
based on the estimated efficiency (%)
212
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE SPCHG2 USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
I - Index which runs over the incremental lengths
ETAO - Design overall mass collection efficiency (%)
XS - Computed value of the exponential argument in the
Deutsch equation for the design overall mass col-•
lection efficiency
LINC - Length of the increments taken in a particular
linear electrical section (m)
PL - Total electrical length of the precipitator (m)
ETAPF - Overall mass collection fraction per increment
based on the design efficiency
DELTNP - Number density of charges transferred from ions to
particles in a given subincremental length (#/m3)
SUMMOB - Weighted summation of particle mobilities (m2/V-sec/
m3)
PNUM - Total number of particles per unit volume of gas
entering a given subincremental length (#/m3)
RHOP - Total average particulate charge density in a given
subincremental length (coul/m3)
J - Index which runs over the different particle size
bands
XNO(J) - Number of particles per cubic meter of gas in each
size band at the start of each increment (#/m3)
XDC(I,J) - Charge on each particle size at the end of each
increment (coul)
TCHRG - Average particle charge density for a given part-
icle size in a given subincremental length (coul/m3)
213
-------
CCF(J) - Cunningham correction factor for the different
particle sizes
VIS - Gas viscosity in a particular linear electrical
section (kg/m-sec)
RAD(J) - Radii of the different particle sizes (m)
DIFF - Difference between the charge on a given particle
size in the (I+l)-th and I-th increments (coul)
II - Index which runs over the subincremental lengths
Q (J) - Charge on each particle size at the end of a partic-
ular increment (coul)
OLDQ(J) - Charge on the different particle sizes in the incre-
ment prior to the one under consideration (coul)
PMOB - Weighted particulate mobility in a given subincre-
mental length (m2/V-sec)
TDNSP - Total average particulate charge number density in
a given subincremental length (#/m3)
CHFID(II) - Average free ion densities for particle charging
in subincremental lengths (#/m3)
DNSION - Ion density in the absence of particles (#/m3)
RDNSI - Average reduced ion density in a given subincremental
length (#/m3)
PIR - Ratio of the total charge density which can be
accepted by particles in a given subincrement to
the available free ion density
NPRNT - Indicator which specifies the unit number of the
output device for printing data from the program
AFID - Average reduced free ion density for particle
charging in a particular length increment (#/m3)
AVGFID - Average reduced free ion density for particle charg-
ing (#/cm3)
U - Ion mobility adjusted for temperature and pressure
(m2/V-sec)
E - Elementary charge unit (coul)
UEQ - Effective charge carrier mobility (m2/V-sec)
214
-------
RIOVR - Ratio of the ionic space charge density to the total
space charge density
XPI - Overall mass collection efficiency per increment
based on the design efficiency (%)
215
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE CMAN USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
NX - Number of grid points in the x-direction for the
numerical calculations of electrical conditions
NX1 - Number of grid intervals in the x-direction for the
numerical calculations of electrical conditions
NY - Number of grid points in the y-direction for the
numerical calculations of electrical conditions
NY1 - Number of grid intervals in the y-direction for the
numerical calculations of electrical conditions
SX - Wire-to-plate spacing in a particular linear elec-
trical section (m)
AX - Interval size in the x-direction (m)
SY - One-half the wire-to-wire spacing in a particular
linear electrical section (m)
AY - Interval size in the y-direction (m)
I - Index which runs over grid points in the x-direction
J - Index which runs over grid points in the y-direction
X - Value of x used in equation (26) [m]
Y - Value of y used in equation (26) [m]
VW - Electrical potential at the wire (V)
VCOOP(I,J) - Array containing the values of electric potential
given equation (26) at the different grid points
(V)
NWIRE - Number of wires per electrical section per gas
passage in a particular electrical section
M - Series sum in equation (26) is taken from -M to M
216
-------
NUM - Sum in the numerator of equation (26)
DENOM - Sum in the denominator of equation (26)
PI - Value of the constant TT
El - Arguments for the hyperbolic cosine functions in
the numerator of equation (26)
Fl - Arguments for the cosine functions in the numerator
of equation (26)
Gl - Arguments for the hyperbolic cosine functions in
the denominator of equation (26)
Hi - Arguments for the cosine functions in the denom-
inator of equation (26)
E2 - Hyperbolic cosine functions in the denominator of
equation (26)
F2 - Cosine functions in the denominator of equation (26)
G2 - Hyperbolic cosine functions in the denominator of
equation (26)
H2 - Cosine functions in the denominator of equation (26)
TT - Argument for the logarithmic function in the num-
erator of equation (26)
TB - Argument for the logarithmic function in the denom-
inator of equation (26)
F - Logarithmic function in the numerator of equation
(26)
G - Logarithmic function in the denominator of equation
(26)
217
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE EFLDl USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
VO - Applied voltage (negative value used in calculations)
[V]
PI - Value of the constant IT
EPSO - Permittivity of free space (cou!2/nt-m2)
AC - Radius of the corona wires (m)
RO - Radius of the corona wires (m)
ROC - Radius of the corona wires (cm)
RF - Roughness factor for the corona wires
TDK - Temperature of the gas stream (°K)
P - Pressure in the gas stream (atm)
RELD - Relative air density [6 = (T0/T)(P/P0)1
EORO - Product of the corona starting electric field and
the wire radius
CD - Average current density at the plate (A/m2)
UEQ - Effective charge carrier mobility (m2/V-sec)
VERGE - Initial estimate of the space charge density at the
corona wire to start the calculation of the electric
field at the plate (coul/m3)
QZERO - Space charge density at the corona wire (coul/m3)
I - Index which runs over grid points in the x-direction
NX - Number of grid points in the x-direction for the
numerical calculations of electrical conditions
J - Index which runs over grid points in the y-direction
218
-------
NY - Number of grid points in the y-direction for the
numerical calculations of electrical conditions
MOBILT(I,J) - Array containing the values of effective charge
carrier mobility at the different grid points (m2/
V-sec)
MAXJ - Upper limit that the calculated average current
density at the plate cannot exceed (A/m2)
MINJ - Lower limit that the calculated average current
density at the plate cannot fall below (A/m2)
NX1 - Number of grid intervals in the x-direction for the
numerical calculations of electrical conditions
NY1 - Number of grid intervals in the y-direction for the
numerical calculations of electrical conditions
SX - Wire-to-plate spacing in a particular linear elec-
trical section (m)
AX - Interval size in the x-direction (m)
SY - One-half the wire-to-wire spacing in a particular
linear electrical section (m)
AY - Interval size in the y-direction (m)
AXS - Value of the quantity [ax2] (m2)
AYS - Value of the quantity [a 2] (m2)
ASP - Value of the quantity [(ax2+a 2)/£0] (m^-nt/coul2)
ASS - Value of the quantity [1/2(ax2+a 2)] (m~2)
Z - Counter which keeps track of the number of times
the calculation iterates due to lack of convergence
in the average current density at the plate
VCOOP(I,J) - Array containing the values of electric potential
given equation (26) at the different grid points
(V)
V(I,J) - Array containing the value of the electric potential
at each point in the grid during an iteration (V)
IZ - Same as Z
NPRNT - Indicator which specifies the unit number of the out-
put device for printing data from the program
219
-------
LL - Counter which keeps track of the number of times
the calculation iterates due to lack of convergence
in the electric potential at each point in the grid
RHO(I,J) - Array containing the value of the space charge
density at each point in the grid during an itera-
tion (coul/m3)
EX(I,J) - Array containing the value of the component of the
electric field intensity perpendicular to the plates
at each point in the grid during an iteration (V/m)
EY(I,J) - Array containing the value of the component of the
electric field intensity parallel to the plates at
each point in the grid during an iteration (V/m)
Ql - Value of the quantity [2b. ] along the line AD
-L i i
where b. .is the effective charge carrier mobility
which is a function of position (m2/V-sec)
Q2 - Value of the quantity [2b. a ] along the line AD
(m3/V-sec) Z/1 X
Q3 - Value of the quantity [2b. a ] along the line AD
(m3/V-sec) lfl Y
Q4 - Value of the quantity [2b. a a ] along the line AD
(mVv-sec) 1/1 x y
Q5 - Value of the quantity [-eoE (2b. a -a b. )]
x i,i y y 1-1/1
along the line AD (cou!2/nt-sec)
Q6 - Value of the quantity [e02E 2(2b. a -a b. )2]
x i/iy y i-iri
along the line AD (coul /nt -sec )
Q7 - Value of the quantity [4b2 a a 2£oE p. ] along
i, i x y x i i, i
the line AD where p. .is the space charge density
11 D
at the different grid points (cou!2-m2/V2-sec2)
Q8 - Value of the quantity [-/Q6+Q7] along the line AD
(coul-m/V-sec)
PI - Value of the quantity [2b. .] along the line AB
(m2/V-sec) lf:i
P2 - Value of the quantity [2b. .a ] along the line AB
(m3/V-sec) 1'D X
220
-------
P3 - Value of the quantity [2b .a ] along the line
AB (m3/V-sec) 1 'D Y
P4 - Value of the quantity [2b .a a ] along the line
AB (mVv-sec) i»D x y
P5 - Value of the quantity [-e0E (2b .a -a b . )]
, Y i ,D x x 1,3-1
along the line AB (coul /nt-sec)
P6 - Value of the quantity [e02E 2(2b .a -a b . )2]
y i ,3 x x i ,-j-i
along the line AB (coul4/nt2-sec2)
P7 - Value of the quantity [4b2 .a 2a e0E p ._ ] along
i j j x y y i / D i
the line AB (cou!2-m2/V2-sec2)
P8 - Value of the quantity [-/P6+P7] along the line AB
(coul-m/V-sec)
Rl - Value of the quantity [2b. ] along the line BC
(m2/V-sec) 1/NY
R2 - Value of the quantity [2b. a ] along the line BC
/3/TT \ l/INxX
(m3/V-sec)
R3 - Value of the quantity [2b. a ] along the line BC
(m3/V-sec) lfNY Y
R4 - Value of the quantity [2b. a a ] along the line
BC (mVv-sec) i,NY x y
R5 - Value of the quantity [-e0Ex(2bi Nya -a bi_i Ny)]
along the line BC (cou!2/nt-sec)
R6 - Value of the quantity [e „ 2Ex 2 (2b± ^Nyay-ayb.._ j ^ Ny) ]
along the line BC (coulVnt2-sec2)
R7 - Value of the quantity [4b2 a a2e0E p. ] along
1 ^ IN i X jf X -L 1 / IN JL
the line BC (coul2-m2/V2-sec2)
R8 - Value of the quantity [-/R6+R7] along the line BC
(coul-m/V-sec)
Dl - Value of the quantity [2b. .] for interior points in
the grid (m2/V-sec) 1' 3
D2 - Value of the quantity [2a b. .] for interior points
in the grid (m3/V-sec)
221
-------
D3 - Value of the quantity [2a b. .] for interior points
in the grid (m3/V-sec) Y 'D
D4 - Value of the quantity [2a a b. .] for interior points
in the grid (mVv-sec)
D5 - Value of the quantity [-e0(E (2a b. .-a b. .) +
x y x / j y i i * j
E (2a b. .-a b. ._ ))] for interior points in the
y xlfD x i, 3 i
grid (cou!2/nt-sec)
D6 - Value of the quantity [D5-D5] (coulVnt2-sec2)
D7 - Value of the quantity [4b. .a a e0(a E p._ .+
i f i x y y x i i / D
a E p. ._ )] for interior points in the grid (coul:
x y i r D i
m2/V2-sec2)
D8 - Value of the quantity [-/D6+D7] for interior points
in the grid (coul-m/V-sec)
OLDV(I,J) - Array containing the value of the electric potential
at each point in the grid during the previous itera-
tion . (V)
OLDRO(I,J) - Array containing the value of the space charge dens-
ity at each point in the grid during the previous
iteration (coul/m3)
CDNSTY(I,J) - Array containing the value of current density at
each point in the grid (A/m2)
ACDNTY - Average current density at the plate (A/m2)
EPLT - Sum of the values of the electric field intensity
at the plate (V/m)
AEPLT - Average electric field at the plate (V/m)
CVERGE - Converged value of the space charge density at the
wire in calculating the electric field at the
plate (coul/m3)
222
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE EFLD2 USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
IVCK - Indicator which terminates the calculation of points
on the voltage-current whenever the specified applied
voltage is reached and interpolated upon
VO - Specified operating applied voltage (V)
VSTART - Particular value of VSTAR(I) [V]
VW - Operating applied voltage corresponding to a spec-
ified current density (V)
AC - Radius of the corona wires (m)
RO - Radius of the corona wires (m)
TDK - Temperature of the gas stream (°K)
P - Pressure of the gas stream (atm)
RELD - Relative air density [6 = (T0/T) (P/P0)]
ROC - Radius of the corona wires (cm)
RF - Roughness factor for the corona wires
EORO - Product of the corona starting electric field and
the wire radius (v)
I - Index which runs over grid points in the x-direction
NX - Number of grid points in the x-direction for the
numerical calculations of electrical conditions
J - Index which runs over grid points in the y-direction
NY - Number of grid points in the y-direction for the
numerical calculations of electrical conditions
UEQ - Effective charge carrier mobility (m2/V-sec)
223
-------
MOBILT(I,J) - Array containing the values of effective charge
carrier mobility at the different grid points (m2/
V-sec)
PI - Value of the constant TT
EPSO - Permittivity of free space (cou!2/nt-m2)
START - Particular value of STARTl(I) [A/m2]
SSTART - Initial value of START which is saved (A/m2)
MINJ - Lower limit that the calculated average current
density at the plate cannot fall below (A/m2)
MAXS - Particular value of current density on the voltage-
current curve (A/m2)
NXl - Number of grid intervals in the x-direction for the
numerical calculations of electrical conditions
NYl - Number of grid intervals in the y-direction for the
numerical calculations of electrical conditions
SX - Wire-to-plate spacing in a particular linear elec-
trical section (m)
AX - Interval size in the x-direction (m)
SY - One-half the wire-to-wire spacing in a particular
linear electrical section
AY - Interval size in the y-direction (m)
AXS - Value of the quantity [ax2] (m2)
AYS - Value of the quantity [a 2] (m2)
ASP - Value of the quantity [(ax2+a 2)/£o] (m"-nt/coul2)
ASS - Value of the quantity [1/2(a 2+a 2)](m~2)
IFINAL - Indicator which causes the calculation of successive
points on the voltage-current curve to cease after
IFINAL points
II - Index which runs over the different current densi-
ties to be used on the voltage-current curve
JI1 - Indicator which allows the initial increment size
on current density in the calculation of the voltage-
current curve to be changed after JI1-1 points are
determined on the curve
224
-------
DSTART - Particular value of START2(I) [A/m2]
JI2 - Indicator which allows the second increment size on
current density in the calculation of the voltage-
current curve to be changed after JI2-1 points are
determined on the curve
CSTART - Particular value of STARTS(I) [A/m2]
MAXJ - Upper limit that the calculated average current
density at the plate cannot exceed (A/m2)
QZERO - Space charge density at the corona wire (coul/m3)
NWIRE - Number of wires per electrical section per gas
passage in a particular electrical section
Z - Counter which keeps track of the number of times '
the calculation iterates due to lack of convergence
in the average current density at the plate
VCOOP(I,J) - Array containing the values of electric potential
given equation (26) at the different grid points
(V)
V(I,J) - Array containing the value of the electric potential
at each point in the grid during an iteration (V)
IZ - Same as Z
NPRNT - Indicator which specifies the unit number of the
output device for printing data from the program
LL - Counter which keeps track of the number of times
the calculation iterates due to lack of convergence
in the electric potential at each point in the grid
RHO(I,J) - Array containing the value of the space charge
density at each point in the grid during an itera-
tion (coul/m3)
EX(I,J) - Array containing the value of the component of the
electric field intensity perpendicular to the plates
at each point in the grid during an iteration (V/m)
EY(I,J) - Array containing the value of the component of the
electric field intensity parallel to the plates at
each point in the grid during an iteration (V/m)
225
-------
Ql - Value of the quantity [2b. ] along the line AD
if i
where b. .is the effective charge carrier mobility
11 D
which is a function of position (m2/V-sec)
Q2 - Value of the quantity [2b. a ] along the line AD
(m2/V-sec) 1,1 x
Q3 - Value of the quantity [2b. a ] along the line AD
(m3/V-sec) '1 y
Q4 - Value of the quantity [2b. a a ] along the line AD
(mVv-sec) 1.1 x y
Q5 - Value of the quantity [-e0E (2b. a -a b. )]
x i, i y y 1-1,i
along the line AD (coul /nt-sec)
Q6 - Value of the quantity [e02Ex2(2bi a -a b_L_ )2]'
along the line AD (coul4/nt2-sec2)
Q7 - Value of the quantity [4b.2 a a 2e0E p. ] along
i» i x y xi 1/1
the line AD where p. .is the space charge density
1 / D
at the different grid points (coul2-m2/V2-sec2)
Q8 - Value of the quantity [-/Q6+Q7] along the line AD
(coul-m/V-sec)
PI - Value of the quantity [2b .] along the line AB
(m2/V-sec) l'3
P2 - Value of the quantity [2b .a ] along the line AB
(m3/V-sec) l'3 x
P3 - Value of the quantity [2b .a ] along the line
AB (m3/V-sec) J ' D Y
P4 - Value of the quantity [2b .a a ] along the line
AB (mv- 1 ' D x Y
P5 - Value of the quantity [-eoE (2b .a -a b . _ )]
Y i / "] x x i / ] i
along the line AB (cou!2/nt-sec)
P6 - Value of the quantity [e02E 2 (2b .a -a b ._ )2]
y i / j x x i , ] i
along the line AB (coul Vnt2-sec2 )
P7 - Value of the quantity [4b2 .a 2a e0E p . ] along
i / D x Y Y i / D- i
the line AB (cou!2-m2/V2-sec2 )
226
-------
P8 - Value of the quantity [-/P6+P7] along the line AB
(coul-m/V-sec )
Rl - Value of the quantity [2b. ] along the line BC
(m2/V-sec) 1'NY
R2 - Value of the quantity [2b. a ] along the line BC
(m3/V-sec) X'NY x
R3 - Value of the quantity [2b. Mva ] along the line BC
(m3/V-sec) 1'NY y
R4 - Value of the quantity [2b. .TVa a ] along the line
BC (mVv-sec) i,NY x y
R5 - Value of the quantity [-e 0Ex (2b± /Nyay-aybi_ ^Ny]
along the line BC (cou!2/nt-sec)
R6 - Value of the quantity [e „ 2E2 (2bi ,Nyay-aybi_ 1/Ny]
along the line BC (coul It/nt2-sec2 )
R7 - Value of the quantity [4b? ^Nyaxaje oExpi_ ^ ^Ny] along
the line BC (coul2-m2/V2-sec2 )
R8 - Value of the quantity [-/R6+R7] along the line BC
(coul-m/V-sec)
Dl - Value of the quantity [2b. .] for interior points in
the grid (m2/V-sec) lf)
D2 - Value of the quantity [2a b. .] for interior points
x i , ]
in the grid (m3/V-sec)
D3 - Value of the quantity [2a b.^ .] for interior points
in the grid (m3/V-sec)
D4 - Value of the quantity [2a a b. .] for interior points
in the grid (mVv-sec) * 'D
D5 - Value of the quantity [-£o(E (2a b. .-a b. .) +
x y j- / j Y-LI/J
E (2a b. .-a b. . ))] for interior points in the
yv x 1,3 x 1,3-1
grid (cou!2/nt-sec)
D6 - Value of the quantity [D5-D5] (coul4/nt2-sec2)
D7 - Value of the quantity [4b? .a a Eo(a E p. . +
i,3xy y x j- i , j
a E p. •_ )] for interior points in the grid (coul2-
x y i, 3 i
m2/V2-sec2)
227
-------
D8 - Value of the quantity [-/D6+D7] for interior points
in the grid (coul-m/V-sec)
OLDV(I,J) - Array containing the value of the electric potential
at each point in the grid during the previous itera-
tion (V)
OLDRO(I,J) - Array containing the value of the space charge den-
sity at each point in the grid during the previous
iteration (coul/m3)
CDNSTY(I,J) - Array containing the value of current density at
each point in the grid (A/m2)
ACDNTY - Average current density at the plate (A/m2)
TEST - Absolute value of the difference between the calcu-
lated average current density at the plate and the
specified value (A/m2)
TEST1 - One percent of the calculated average current den-
sity at the plate (A/m2)
EPLT - Sum of the values of the electric field intensity
at the plate (V/m)
AEPLT - Average electric field at the plate (V/m)
EBD - Electrical breakdown strength of the gas near the
collection electrode or the collected particulate
layer (V/m)
OLDVW - The value of applied voltage at the point prior to
the one under consideration (V)
OLDCD - The value of average current density at the plate
at the point prior to the one under consideration
(A/m2)
NEC - Indicator which determines whether or not the aver-
age current density, average electric field, and
average electric field at the plate are to be calcu-
lated in the subincremental lengths
K - Index which sequences the grid strips in the basic
area for which the calculations are performed
RSUM - Average charge number density in a particular grid
strip (#/m3)
ESUM - Average electric field intensity in a particular
grid strip (V/m)
228
-------
EAVGS(K) - Array containing the average electric field inten-
sities in the different grid strips in the basic area
for which the calculations are performed (V/m)
CHFIDS(K) - Array containing the average charge number densities
in the different grid strips in the basic area for
which the calculations are performed (#/m3)
NYY - Index which renumbers the grid strips so that by
symmetry the area covered by the half-wire spacing
which was not considered in the calculations can be
taken into account
EAVG(L) - Array containing the average electric field inten-
sities in the different grid strips which cover an
area between successive wires (V/m)
CHFID(L) - Array containing the average charge number densities
in the different grid strips which cover an area
between successive wires (#/m3)
L - Index which runs over and numbers the first (NY-1)
grid strips in a given wire-to-wire spacing
KK - Index which runs over the different grid strips in
the basic area for which the calculations are per-
formed
Ml - Number of the first grid strip in the last (NY-1)
grid strips in a given wire-to-wire spacing
M2 - Number of the last grid strip in a given wire-to-
wire spacing
M - Index which runs over and numbers the last (NY-1)
grid strips in a given wire-to-wire spacing
LL - Index which sequences the grid strips in the basic
area for which the calculations are performed
NN - Index which runs over points in the y-direction
ECOLLS(LL) - Array containing the average electric field inten-
sity at the plate in the different grid strips in
the basic area for which the calculations are per-
formed (V/m)
LI - Index which renumbers the grid strips so that by
symmetry the area covered by the half-wire spacing
which was not considered in the calculations can be
taken into account
229
-------
ECOLL(L) - Array containing the average electric field inten-
sity at the plate in the different grid strips which
cover an area between successive wires (V/m)
L2 - Index which runs over the different grid strips in
the basic area for which the calculations are per-
formed
II - Number of the first grid strip in the last (NY-1)
grid strips in a given wire-to-wire spacing
12 - Number of the last grid strip in a given wire-to-
wire spacing
I - Index which runs over and numbers the last (NY-1)
grid strips in a given wire-to-wire spacing
230
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE CHARGN USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
H - Increment size used in the Runge-Kutta scheme (sec)
H2 - One-half the increment size chosen for the Runge-
Kutta scheme (sec)
YI - Time at the start of a given increment or subincre-
ment of the precipitator (sec)
Y - Time at the end of a given increment or subincre-
ment of the precipitator (sec)
XI - Number of charges on a given particle size at the
start of a given increment or subincrement of the
precipitator
X - Number of charges on a given particle size at the
end of a given increment or subincrement of the
precipitator
I - Index which runs over the different points spec-
ified for use in the Runge-Kutta scheme
NN - Number of points specified for use in the Runge-
Kutta scheme
ECHARG - Elementary charge unit (coul)
SCHARG - Saturation charge number from the field charging
equation
NUMINC - Number of increments in the Simpson's Rule integra-
tion over 6 in equation (12)
CONST - Value of the quantity [2(K-l)a3E0/(K+2)] found in
equation (12) [V-m2]
EZERO - Average electric field used for particle charging
(V/m)
231
-------
V - Value of the quantity [e2/47re0akT] found in equa-
tion (12)
RSIZE - Radius of a particular particle (m)
ECONST - Value of the quantity [3eE0a/kT(K+2)] found in
equation (12)
CMKS - Value of the quantity [4TT£o] found in equation (12)
[coul2/nt-m2]
RR - Value of the quantity [eE0/kT] found in equation
(12) [m-1]
FCONST - Value of the quantity [(K-l)eE0a3/kT(K+2)] found
in equation (12) [m2]
FACTOR - Value of the quantity [uva2/2] found in equation
(12) [m3/sec]
COEFF - Value of the quantity [bqs/4eol found in equation
(12) [m3/sec]
AFID - Average reduced free ion density for particle
charging in a particular length increment (#/m3)
Tl - Value of the charge-number rate to the particle
surface at the point (XI,YI) multiplied by the
stepsize H for use in the Runge-Kutta scheme
T2 - Value of the charge-number rate to the particle
surface at the point (XI+H2, YI+T1/2) multiplied
by the stepsize H for use in the Runge-Kutta scheme
T3 - Value of the charge-number rate to the particle
surface at the point (XI+H2, YI+T2/2) multiplied
by the stepsize H for use in the Runge-Kutta scheme
T4 - Value of the charge-number rate to the particle
surface at the point (XI+H, YI+T3) multiplied by
the stepsize H for use in the Runge-Kutta scheme
232
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR STATEMENT FUNCTION RATE USED IN THE
ELECTROSTATIC PRECIPITATOR PERFORMANCE MODEL
ECHARG - Elementary charge unit (coul)
SCHARG - Saturation charge number from the field charging
equation
NUMINC - Number of increments in the Simpson's Rule integra-
tion over 0 in equation (12)
CONST - Value of the quantity [2(K-l)a3E0/(K+2)] found in
equation (12) [V-m2]
EZERO - Average electric field used for particle charging
(V/m)
V - Value of the quantity [e2/4Tre oakT] found in equa-
tion (12)
RSIZE - Radius of a particular particle (m)
ECONST - Value of the quantity [3eE0a/kT(K+2)] found in
equation (12) [V-m2]
CMKS - Value of the quantity [4ireo] found in equation (12)
[coul2/nt-m2]
RR - Value of the quantity [eEo/kT] found in equation
(12) [m-1]
FCONST - Value of the quantity [(K-l)eEOa3/kT(K+2)] found
in equation (12) [m2]
FACTOR - Value of the quantity [Trva2/2] found in equation
(12) [m3/sec]
COEFF - Value of the quantity [bqs/4e0] found in equation
(12) [mVsec]
AFID - Average reduced free ion density for particle
charging in a particular length increment (#/m3)
233
-------
NTIME - Instantaneous charging time (sec)
NUMBER - Instantaneous number of charges on a given particle
size
INTGRL - Value of the integral appearing in equation (12)
NE - Negative of the instantaneous charge on a given
particle size (coul)
THZERO - Maximum angle (60) for field charging in radians
DELTAX - Increment size taken for the integration over the
angle 6 in equation (12)
THETA - Values of the angle 6 taken for the integration
over 0 in equation (12)
SUMODD - Sum of the odd terms contributing to the integral
in the Simpson's Rule integration scheme
J - Index which runs over the different points in the
Simpson's Rule integration
CTHETA - Value of the quantity [cos 8]
TCONST - Value of the quantity [2(K-l)a3E0cos6/(K+2)] (m2-V)
EGOS - Value of the quantity [E0 cos 6] (V/m)
Cl - Value of the quantity [q/4Tre oEocosG] found in
equation (52)
CO - Value of the quantity [(K-l)a3/ (K+2)] found in
equation (52) [m3]
RZERO - Radial distance from the center of a given particle
at which the total radial component of the electric
field is zero (m)
ARG1 - Argument of the exponential function inside the
integral in equation (12)
YVAL - Integrand of the integral in equation (12)
SUMEVN - Sum of the even terms contributing to the integral
in the Simpson's Rule integration
CTZERO - Value of the quantity [cos 60]
ARG2 - Argument of the exponential function inside the
integral in equation (12) for the angle Go
234
-------
ZVAL - Contribution to the integral in equation (12) which
depends on the angle 60
RATE1 - Contribution to the particle charging rate due to
the second term in equation (12) [#/sec]
ARG3 - Argument of the exponential function in the third
term in equation (12)
RATE2 - Contribution to the particle charging rate due to
the third term in equation (12)
RATES - Contribution to the particle charging rate due to
the first term in equation (12) [#/sec]
RATE - Total instantaneous charging rate to the entire
surface of a given particle (#/sec)
235
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE ARCCOS USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
A - Numerator of the ratio A/B whose inverse cosine
is to be determined
B - Denominator of the ratio A/B whose inverse cosine
is to be determined
ACOS - Value of the quantity [cos"1(A/B)] (radians)
RATIO - Value of the ratio A/B
T - Variable used to generate the different numerical
coefficients in the series representation of the
inverse cosine function
SUM - Sum of successive terms in the series representation
of the inverse cosine function
TERM - A particular term in the series representation of
the inverse cosine function
U - Variable used in the generation of the numerical
coefficients in the series representation of the
inverse cosine function
V - Variable used in the generation of the numerical
coefficients in the series representation of the
inverse cosine function
W - Variable used in the generation of the numerical
coefficients in the series representation of the
inverse cosine function
236
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE ZERO USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
Cl - Value of the quantity [q/4Tre 0Eocos6] found in
equation (52)
CO - Value of the quantity [(K-l)a3/(K+2)] found in
equation (52) [m3]
RZERO - Radial distance from the center of a given particle
at which the total radial component of the electric
field is zero (m)
B - Value of the argument of the inverse cosine function
found in equation (55)
C - Value of the inverse cosine function found in equa-
tion (55)
D - Factor multiplying the cosine function found in
equation (12)
I
237
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE CHGSUM USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
NVI - Indicator which specifies whether to base the elec-
trical calculation on known voltages and currents
or on calculated voltage-current characteristics
I - Index which runs over incremental lengths
II - Index which runs over subincremental lengths
OLDQF(J) - Value of field charge on the different particle
sizes at the end of a given increment or subincre-
ment (coul)
OLDQT(J) - Value of diffusion charge on the different particle
sizes at the end of a given increment or subincre-
ment (coul)
ITER - Counter which keeps track of the number of itera-
tions which is limited by NITER
SOLDQF(J) - Value of field charge on the different particle
sizes at the start of an increment which must be
saved for the iteration procedure over subincrements
in a given increment (coul)
SOLDQT(J) - Value of diffusion charge on the different particle
sizes at the start of an increment which must be
saved for the iteration procedure over subincrements
in a given increment (coul)
E - Elementary charge unit (coul)
SCHARG - Saturation charge number from the field charging
equation
SATCHG - Saturation charge for a given particle size from
field charging theory (coul)
CHRFID - Average free ion density for particle charging
(#/m3)
238
-------
U - Ion mobility adjusted for temperature and pressure
(m2/V-sec)
EPSO - Permittivity of free space (cou!2/nt-m2)
TIMEF - Final value of time for particle charging (sec)
TIMEI - Initial value of time for particle charging (sec)
CF1 - Value of the quantity [(N0be/4e„) (tf-t. )] found in
equation (15) 1
CF2 - Value of the quantity [1/d-q^q )] found in
equation (15) S
QF - Charge on a given particle size in a given increment
or subincrement due to field charging (coul)
V - Value of the quantity [e2/4ire oakT] found in equation
(15)
ARC - Value of the quantity [q. e/4ire 0akT] found in
equation (15)
RSIZE - Radius of a particular particle (m)
VAVC - Root mean square velocity of the ions (m/sec)
BC - Boltzmann's constant (J/°K)
TDK - Temperature of the gas in a given electrical sec-
tion (°K)
QT - Charge on a given particle size in a given incre-
ment or subincrement due to diffusion charging
(coul)
CNUMBR - Total charge number on a given particle size in a
given increment or subincrement
239
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE PRTINC USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
NPRINT - Indicator which designates when to print certain
sectionalized data
NSECT - Indicator which keeps track of which electrical
section the calculation is in
NPRNT - Indicator which specifies the unit number of the
output device for printing data from the program
ITL - Identifying label for the calculations
SLNGTH - Length of a particular electrical section (m)
A - Collection plate area of a particular linear elec-
trical section (m2)
VO - Applied voltage in a particular linear electrical
section (V)
TC - Total current in a particular linear electrical
section (A)
B - Wire-to-plate spacing in a particular linear elec-
trical section (m)
AC - Corona wire radius in a particular linear electrical
section (m)
WL - Total wire length in a particular linear electrical
section (m)
CL - Total current per length of corona wire in a partic-
ular linear electrical section (A/m)
CD - Average current density at the plate in a particular
linear electrical section (A/m2)
ET - Average electric field in the deposited particulate
layer in a particular linear electrical section (V/m)
240
-------
SY - One-half the wire-to-wire spacing in a particular
linear electrical section (m)
VG - Gas volume flow rate in a particular electrical
section (m3/sec)
VGAS - Gas velocity in a particular linear electrical sec-
tion (m/sec)
TDK - Temperature of the gas in a given electrical sec-
tion (°K)
P - Gas pressure in a particular linear electrical
section (atm)
VIS - Gas viscosity in a particular linear electrical
section (kg/m-sec)
U - Ion mobility adjusted for temperature and pressure
(m2/V-sec)
VAVC - Root mean square velocity of the ions (m/sec)
TMFP - Ionic mean free path multiplied by a factor (m)
W - Total weight of particles per second passing into
a particular linear electrical section (kg/sec)
LING - Length of the increments taken in a particular
linear electrical section (m)
XPI - Overall mass collection efficiency per increment
based on the estimated or design efficiency (%)
NVI - Indicator which specifies whether to base the elec-
trical calculation on known voltages and currents
or on calculated voltage-current characteristics
RIOVR - Ratio of the ionic space charge density to the total
space charge density
ERAVG - Average electric field used for particle charging
(V/m)
EPLT - Absolute value of the average electric field at
the plate in a particular length increment (V/m)
AFID - Average reduced free ion density for particle
charging in a particular length increment (#/m3)
XCD - Average current density at the plate in a particular
length increment (nA/cm2)
241
-------
ZMD - Interpolated mass median diameter of the collected
particulate layer (m)
WT - Total weight of material per cubic meter of gas
removed in all particle size bands in a given length
increment (kg/m3)
LTHICK - Thickness of the collected particulate layer in a
particular increment of length (mm/min)
JPART - Current density due to particles in a particular
increment of length (A/m2)
JION - Current density due to ions in a particular incre-
ment of length (A/m2)
I - Index which runs over incremental lengths
ROVRI - Ratio of the total space charge density to the ionic
space charge density
242
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE PRTCHG USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
NPRNT - Indicator which specifies the unit number of the
output device for printing data from the program
NCALC - Indicator which determines whether to use equation
(12) for particle charging or the sum of the class-
ical field and diffusion charges
NEST - Indicator which specifies whether to use extensive
calculations or estimation procedures in determin-
ing precipitator performance
JS - Index which is utilized in dividing the output data
for particle charging into sets of eight columns
each with a column for each particle size band
KS - Index which is utilized in dividing the output data
for particle charging into sets of eight columns
each with a column for each particle size band
NS - Number of different particle size bands in the inlet
particle size distribution
DIAM(J) - Diameters of the different particle sizes (ym and m)
J - Index which runs over the different particle size
bands
I - Index which runs over incremental lengths
NF - Number of increments taken along the length of the
precipitator
NVI - Indicator which specifies whether to base the elec-
trical calculation on known voltages and currents
or on calculated voltage-current characteristics
NI - Number of subincremental lengths into which the
incremental length is divided
N - Number of the subincremental strip having the max-
imum values of average electric field and current
density
243
-------
PI - Value of the constant IT
EPSO - Permittivity of free space (cou!2/nt-m2)
RAD(J) - Radii of the different particle sizes (ra)
TMFP - Ionic mean free path multiplied by a factor (m)
EAVG(N) - Average electric fields for particle charging in
subincremental lengths (V/m)
EPS - Relative dielectric constant of the particles
VRATIO - Ratio of the peak applied voltage to the average for
use in particle charging
XDC(I,J) - Charge on each particle size at the end of each
increment (coul)
QSATM - Saturation charge for a given particle size based
on the last electrical section and the subincre-
mental strip containing the largest values of
average electric field and current density (coul)
YY(J) - Array containing the ratio of the charge on a given
particle size to the saturation charge in the last
electrical section for a given increment
QSAT(J) - Saturation charge for a given particle size based
on the last electrical section and the average
electric field for the entire section (coul)
244
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE ADJUST USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
NRUN - Indicator that specifies which set of nonideal
conditions is under consideration
NS - Number of different particle size bands in the inlet
particle size distribution
NS1 - Number of particle size bands plus one
NUMSEC - Number of linear electrical sections in the precip-
itator
NUMS1 - Number of electrical sections less one
TDK - Temperature of the gas stream in the last electrical
section (°K)
PS (NUMSEC) - Pressure of the gas stream in the last electrical
section (atm)
CONVF - Conversion factor which converts kg/ACM to mg/DSCM
NRAPDC - Counter which keeps track of the number of rapping
puff particle size distributions that have been
considered
X - Ideal, unadjusted overall mass collection fraction
(or efficiency) [no units or %]
I - Index which runs over the different particle size
bands
DXS(I) - Total number of particles removed per cubic meter of
gas in each particle size band under ideal conditions
and with no empirical corrections (#/m3)
ONO(I) - Initial number of particles per cubic meter of gas
in each particle size band (#/m3)
EFESR - Ideal, unadjusted mass collection fraction for a
given particle size
245
-------
PCNT(I) - Percentage or fraction by mass in the inlet particle
size distribution of the different size bands (% and
decimal)
ARD50(J) - Rapping puff mass median diameters (ym)
ARSIGM(J) - Rapping puff geometric standard deviations
RMMD - Particular value of ARD50(J)[ym]
RSIGMA - Particular value of ARSIGM(J)
RPRCU(I) - Cumulative fraction by mass as a function of particle
size for the rapping puff
RPCNT(I) - Percentages by mass in the different particle size
bands for the rapping puff (%)
NONCK - Counter which keeps track of the number of sets of
nonideal conditions of nonuniform velocity distribu-
tion and gas sneakage and/or particle reentrainment
without rapping that have been considered
ASNUCK(K) - Fractions of gas sneakage and/or particle reentrain-
ment without rapping
SNUCK - Particular value of ASNUCK(K)
AZIGGY(K) - Normalized standard deviations of the gas velocity
distribution
ZIGGY - Particular value of AZIGGY(K)
AZNUMS(K) - Number of stages over which gas sneakage and/or
particle reentrainment without rapping occur
ZNUMS - Number of stages over which gas sneakage and/or
particle reentrainment without rapping occur for
a particular case
NPRNT - Indicator which specifies the unit number of the
output device for printing data from the program
Y - Adjusted overall mass collection fraction (or
efficiency) under no-rap conditions (no units
or %)
XEP - Adjusted mass collection fraction for a given
particle size band under no-rap conditions
246
-------
XMV(I) - Effective migration velocities for the different
particle sizes under ideal conditions and with no
empirical corrections (m/sec)
WY - Adjusted migration velocity for a given particle
size under no-rap conditions (cm/sec)
VG - Gas volume flow rate in a particular electrical
section (m3/sec)
ATOTAL - Total collection plate area of the precipitator
(m2)
Fl - Correction factor for the migration velocity of a
given particle size in order to account for non-
uniform velocity distribution
F2 - Correction factor for the migration velocity of a
given particle size in order to account for gas
sneakage and/or particle reentrainment without
rapping
WYS - Migration velocity of a given particle size cor-
rected only for gas sneakage and/or particle re-
entrainment without rapping (cm/sec)
WYV - Migration velocity of a given particle size cor-
rected only for nonuniform velocity distribution
(cm/sec)
ZNLFF - Combined correction factor for nonuniform velocity
distribution and gas sneakage and/or particle
reentrainment without rapping
WYSV - Migration velocity of a given particle size cor-
rected only for nonuniform gas velocity distribu-
tion and gas sneakage and/or particle reentrainment
without rapping (cm/sec)
WUNCOR(I) - Unadjusted, ideal migration velocities for the
different particle sizes (cm/sec)
EUNCOR(I) - Unadjusted, ideal mass collection efficiencies for
the different particle sizes (%)
DIAM(I) - Diameters of the different particle sizes (urn and m)
PXS(I) - Number of particles per cubic meter of gas for a
given particle size that are removed by the precip-
itator under adjusted, no-rap conditions (#/m3)
247
-------
IDC - Indicator which controls when the summation of
outlet emissions over the different particle size
bands will be performed
SPO - Total outlet emissions under adjusted, no-rap
conditions (#/m3)
SCPO - Total outlet emissions under rap + no-rap condi-
tions (#/m3)
IX - Indicator which determines when the total electri-
cal length up to the last electrical section will
be calculated
SCOREF - Overall mass collection efficiency under no-rap +
rap conditions (%)
XY - Percentage by mass in a given particle size in the
inlet particle size distribution (%)
PENTR - Percentage by mass of a given particle size that
penetrates through the precipitator under adjusted,
no-rap conditions (%)
PCTOT(I) - Percentage by mass in a given particle size band
in the no-rap outlet emissions (%)
CLPTLS - Total electrical length of the precipitator exclud-
ing the last electrical section (m)
IS - Index which runs over the different linear electrical
sections
LSECT(IS) - Number of length increments in the different linear
electrical sections
LINGS(IS) - Lengths of the increments taken in the different
linear electrical sections (ft)
NYX - Index which starts and terminates a loop in which
the mass loss due to rapping and the mass leaving
the precipitator under no-rap conditions are deter-
mined
XEFF - Overall mass collection fraction for either unad-
justed, ideal or adjusted, no-rap conditions
NEFF - Indicator which determines whether the unadjusted,
ideal or adjusted, no-rap efficiency is used to
determine the mass reentrained due to rapping
248
-------
EXPONT - Argument of the exponential function in equation
(2) for either the unadjusted, ideal efficiency
or the adjusted, no-rap efficiency
DL - Inlet mass loading (kg/m3)
PL - Total electrical length of the precipitator (m)
XMELS - Mass entering the last section of the precipitator
from either unadjusted, ideal or adjusted, no-rap
calculations (kg/m3)
XMCLS - Mass collected in the last section of the precip-
itator from either unadjusted, ideal or adjusted,
no-rap calculations (kg/m3 or mg/DSCM)
XMLLS - Mass leaving the last section of the precipitator
from either unadjusted, ideal, or adjusted, no-
rap calculations (kg/m3)
NTEMP - Indicator which specifies whether the precipitator
is cold or hot side
RAPLOS - Mass contained in the outlet emissions due to
rapping (mg/DSCM)
YMELS - Mass entering the last section of the precipitator
from adjusted, no-rap calculations (kg/m3)
YMCLS - Mass collected in the last section of the precip-
itator from adjusted, no-rap calculations (kg/m3 or
mg/DSCM)
YMLLS - Mass leaving the last section of the precipitator
from adjusted, no-rap calculations (kg/m3)
DD - Mass density of the particles (kg/m3)
RNS - Number of particles per cubic meter of gas in a
given size band that are contained in the emissions
due to rapping (#/m3)
EFFWR - Mass collection fraction for a given particle size
containing all corrections and adjustments
CRNP - Number of particles per cubic meter of gas in a
given size band that are collected after rapping
(#/m3)
COREFF - Mass collection efficiency for a given particle
size containing all corrections and adjustments (%)
249
-------
WYP - Migration velocity for a given particle size con-
taining all corrections and adjustments (cm/sec)
CPENTR - Percent penetration of a given particle size con-
taining all corrections and adjustments (%)
CPCTOT(I) - Percentage by mass in a given size band contained
in the no-rap + rap emissions (%)
SL - Number of particles per cubic meter of gas of a
given particle size band exiting the precipitator
under no-rap conditions (#/m3)
RAD(I) - Radii of the different particle sizes (m)
WSL(I) - Weight per cubic meter of gas of particles in a
given size band exiting the precipitator under no-
rap conditions (kg/m3)
ENDPT(I) - Particle diameters in the inlet cumulative percent
by mass distribution (ym and m)
OLD - Value of the quantity [AlogioD] for a given particle
size band in the size distribution histogram
DMDLD(I) - Value of the quantity [AM/Alogi0D] for the different
particle size bands in the outlet emissions under
no-rap conditions (mg/DSCM)
RDMDLD(I) - Value of the quantity [AM/Alogi0D] for the different
particle size bands in the outlet emissions due to
rapping only (mg/DSCM)
CDMDLD(I) - Value of the,quantity [AM/AlogioD] for the different
particle size bands in the outlet emissions under
no-rap + rap conditions (mg/DSCM)
CCF(I) - Cunningham correction factor for the different
particle sizes
ETAO - Estimated or design overall mass collection effic-
iency (%)
ZMMDI - Specified or fitted mass median diameter of the
inlet particle size distribution based on a log-
normal distribution (ym)
SIGMI - Specified or fitted geometric standard deviation of
the inlet particle size distribution based on a log-
normal distribution
NDIST - Indicator which specifies whether the user is to
250
-------
supply the inlet particle size distribution or the
program is to calculate a log-normal distribution
GFIT - Linear-correlation coefficient obtained in the log-
normal fit of the inlet particle size distribution
PRCUNR(I) - Cumulative percentage by mass as a function of
particle size for the outlet emissions under no-
rap conditions (%)
SUMNR - Summation over the different particle size bands of
the percentage by mass contained in each size band
for the outlet emissions under no-rap conditions (%)
ZMDL - Fitted mass median diameter of the outlet no-rap
emissions based on a log-normal distribution (ym)
SIGMO - Fitted geometric standard deviation of the outlet
no-rap emissions based on a log-normal distribution
ZGFIT - Linear-correlation coefficient obtained in the log-
normal fit of the outlet no-rap emissions
COREFW - Precipitation rate parameter under no-rap + rap
conditions (cm/sec)
WZ - Precipitation rate parameter under no-rap condi-
tions (cm/sec)
PRCUC(I) - Cumulative percentage by mass as a function of
particle size for the outlet emissions under no-
rap + rap conditions (%)
SUMC - Summation over the different particle size bands
of the percentage by mass contained in each size
band for the outlet emissions under no-rap + rap
conditions (%)
CZMDL - Fitted mass median diameter of the outlet no-rap
+ rap emissions based on a log-normal distribution
(ym)
CSIGMO - Fitted geometric standard deviation of the outlet
no-rap + rap emissions based on a log-normal distri-
bution
CGFIT - Linear-correlation coefficient obtained in the log-
normal fit of the outlet no-rap + rap emissions
M - Index which runs over the different particle size
bands
251
-------
NONID - Number of nonideal conditions of gas velocity non-
uniformity and gas sneakage and/or particle reen-
trainment without rapping to be considered
NRAPD - Number of rapping puff particle size distributions
to be considered
252
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE WADJST USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
DIAM(I) - Diameters of the different particle sizes (ym and ra)
I - Index which runs over the different particle size
bands
WY - Enters the subroutine as the unadjusted, no-rap
migration velocity for a given particle size and
leaves as the adjusted, no-rap migration velocity
(cm/sec)
ONO(I) - Initial number of particles per cubic meter of gas
in each particle size band (#/m3)
PXS(I) - Number of particles per cubic meter of gas for a
given particle size that are removed by the precip-
itator under adjusted, no-rap conditions (#/m3)
ATOTAL - Total collection plate area of the precipitator (m2)
VG - Gas volume flow rate in a particular electrical
section (m3/sec)
EFESR - Mass collection fraction for a given particle size
under adjusted, no-rap conditions
CFACT(L) - Correction factors for the no-rap migration velo-
cities of the different particle sizes
DCHECK(L) - Particle diameters corresponding to the different
correction factors given by CFACT(L) [m]
L - Index which runs over the different values of
CFACT(L) and DCHECK(L)
WFACT - Interpolated correction factor for the unadjusted,
no-rap migration velocity of a given particle size
253
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE LNDIST USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
D50 - Specified or fitted mass median diameter of the
inlet particle size distribution based on a log-
normal distribution (ym)
SIGMAP - Specified or fitted geometric standard deviation of
the inlet particle size distribution based on a log-
normal distribution
PRCU(I) - Cumulative fractions by mass up to specified particle
sizes
PCNT(J) - Fractions by mass contained in specified particle
size bands
Y(K) - Values of the log-normal distribution function at
different values of the independent variable for use
in integrating the function over the specified size
bands
Z(K) - Cumulative integrals resulting from the integration
of the log-normal distribution function over a spec-
ified particle size band
AREA(J) - Amount of the distribution accumulated in a given
particle size band
NS - Number of particle size bands
ENDPT(I) - Particle diameters specified for use in constructing
the log-normal distribution histogram (ym)
NENDPT - Number of particle diameters specified for use in
constructing the log-normal distribution histogram
PI - Value of the constant TT
SIGMAZ - Value of the quantity [In a ]
N - Total number of particle size bands used in construct-
ing the log normal distribution histogram
254
-------
NINC - Number of points used in the Trapezoidal Rule
integrations over the different particle size bands
ASUM - Value of the integration of the log-normal distri-
bution function over the entire distribution
K - Index which runs over the NS different particle size
bands specified by the user
J - Index which runs over the N different particle size
bands used in the construction of the log-normal
distribution histogram
X2 - Upper limit of integration for a given particle size
band
XI - Lower limit of integration for a given particle size
band
DX - Stepsize taken for the Trapezoidal Rule integration
of the log-normal distribution function over the
different particle size bands
D - Value of the integration variable at different points
in a given particle size band
SGT1 - Value of the quantity [I/a /2¥]
Z
SGT2 - Value of the quantity [2a 2]
£j
I - Index which runs over the different points in a
given particle size band in performing the Trape-
zodial Rule integration of the log-normal distribu-
tion function
SUM - Total fraction by mass contained in the histogram
specified by the user
CHECK1 - Difference between 1 and the calculated total mass
fraction contained in the histogram specified by
the user
CHECK2 - Difference between 1 and the calculated cumulative
fraction by mass up to the largest particle size
specified by the user
255
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE QTFE USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
DX - Stepsize used in the Trapezoidal Rule- integration
scheme
Y(I) - Function values used in the integration scheme
Z(I) - Cumulative integrals resulting from the integration
scheme
NINC - Number of points used in the integration scheme
SUM2 - Cumulative integral up to a given point in the
integration scheme
DDX - One-half of the specified stepsize
I - Index which runs over the different points in the
integration scheme
SUM! - Cumulative integral up to the point prior to the
point under consideration
256
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE LNFIT USED IN THE ELECTRISTATIC
PRECIPITATOR PERFORMANCE MODEL
PRCU(I) - Known or calculated cumulative percentages supplied
by the user (%)
D50 - Fitted mass median diameter based on a log-normal
distribution (urn)
SIGMAP - Fitted geometric standard deviation based on a log-
normal distribution
GFIT - Linear-correlation coefficient obtained in the log-
normal fit
Z(I) - Natural logarithm of the actual particle diameters
corresponding to the known or calculated cumulative
percentages
Y(I) - Calculated natural logarithm of the particle diam-
eters corresponding to the known or calculated
cumulative percentages based on a true log-normal
distribution
ENDPT(I) - Actual particle diameters corresponding to the
known or calculated cumulative percentages (ym)
NENDPT - Number of particle diameters corresponding to the
known or calculated cumulative percentages
NSTAG - Number of points used in the log-normal fit pro-
cedure
I - Index which runs over the different particle diam-
eters corresponding to the known or calculated
cumulative percentages
J - Index which sequences the points which are actually
used in the log-normal fit
XY - Cumulative mass fraction less than a given particle
size
257
-------
XYY - Square root of the natural logarithm of the square
of the reciprocal of XY
A - Y-intercept of the fitted straight line
B - Slope of the fitted straight line
258
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE CFIT USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
A - Y-intercept of the fitted straight line
B - Slope of the fitted straight line
R - Linear-correlation coefficient for the straight
line fit
NSTAG - Number of data points that are fitted to the straight
line
Z(I) - Values of the independent variable
Y(I) - Values of the dependent variable
XN - Running sum over the number of data points
SUMX - Summation over all data points of the values of
the independent variable
SUMY - Summation over all data points of the values of
the dependent variable
SUMXY - Summation over all data points of the values of the
product of the independent and dependent variables
SUMXX - Summation over all data points of the values of the
square of the independent variable
SUMYY - Summation over all data points of the values of the
square of the dependent variables
I - Index which runs over the different data points
259
-------
LIST OF NECESSARY VARIABLES, DEFINITIONS, AND UNITS
FOR SUBROUTINE PRTSUM USED IN THE ELECTROSTATIC
PRECIPITATOR PERFORMANCE MODEL
ATOTAL - Total collection plate area of the precipitator
(m2)
VG - Gas volume flow rate in a particular electrical
section (m3/sec)
SCA - Specific collection area of the precipitator
(mz/m3/sec)
VOSUM - Sum of the applied voltages in the different linear
electrical sections (V)
CDSUM - Sum of the current densities in the different lin-
ear electrical sections (nA/cm2)
NUMSEC - Number of linear electrical sections in the precip-
itator
LSECT(I) - Number of length increments in the different linear
electrical sections
LINGS(I) - Lengths of the increments taken in the different
linear electrical sections (ft)
I - Index which runs over the different linear electri-
cal sections
VOS(I) - Applied voltages for the different linear electrical
sections (V)
TCS(I) - Total current for the different linear electrical
sections (A)
AS (I) - Collection plate areas for the different linear
electrical sections (m2)
AVO - Average applied voltage over the entire precip-
itator (V)
PL - Total electrical length of the precipitator (ft
and m)
260
-------
ACD - Average current density over the entire precipitator
(nA/cm2)
RHO - Resistivity of the collected particulate layer
(ohm-m)
RHOCGS - Resistivity of the collected particulate layer (ohm-
cm)
NPRNT - Indicator which specifies the unit number of the out-
put device for printing data from the program
NRUN - Indicator that specifies which set of nonideal
conditions is under consideration
SCOREF - Overall mass collection efficiency under no-rap +
rap conditions (%)
ZMMDI - Specified or fitted mass median diameter of the
inlet particle size distribution based on a log-
normal distribution (ym)
SIGMI - Specified or fitted geometric standard deviation of
the inlet particle size distribution based on a log-
normal distribution
CZMDL - Fitted log-normal mass median diameter of the out-
let particle size distribution under no-rap + rap
conditions (ym)
CSIGMO - Fitted log-normal geometric standard deviation of
the outlet particle size distribution under no-rap
+ rap conditions
SNUCK - Particular value of ASNUCK(JJ)
ZIGGY - Particular value of AZIGGY(JJ)
RMMD - Particular value of ARD50(II) [ym]
RSIGMA - Particular value of ARSIGM(II)
261
-------
APPENDIX C
COMPLETE LISTING OF
THE COMPUTER PROGRAM
262
-------
*********************************
01 C # *
"2 J; * -E.P.A, ESP MODEL *
03 C *
&a * I.E.R.l.-R.T.P. AND SO. R.I. *
05 C * *
06 C * REVISION I, JAN, I, 1978 *
07 C * *
09 ^ *********************************
09 C
10 HEAL NWIRE,ITHICK, JPART, JION,LTNC,NW$,LINCS
ti INUGF.R VISKIP.VISAME
12 DIMENSION CHKSUMC20)
13 COMMnN/RLKl/01AM(20)iONO(2Q).nyS(20),XMVt20),PCNT(20),RAD(20)f
14 lCCF(2Q),P«rU(21)
15 COMMOfJ/BLK2/LSECT(10), UINC8(in),PS(10)
16
17 COMMON/HLK/I/NS
18
19
20 IVOS(IO), TCS(10),WL8f 10),ACSf J 0) , BS (1 0 ) , SYS f 10) , VGS ( 10) , VGASSf 10)
21 2TF^PS(10),VIS8(lO),08ATf20),U,E,FPSO,PI,ERAVGfBC,TEMP,EPSiVAVC,
22 SOLntjf 20) ,OLDXNOf20)fRFS(10)fSTAPTl(10),START2flO),START7flO),
23
25 CHMMON/RLKB/EAVCfSO) ,CMFIDf 30)
26 COMMOM/RL KQ/ECOLLCIO)
27 CHMMON/BI K 10/ECUFANnn)
30 co^Mnw/BLKlS/vcOOPf 15,15)
32 CHMMON/RLK15/NPRINT, NSECT^SLNGTH, A, vo,Tc,e, AC, WL, CL»CD,ET,
33 IVGAS.P, vTs,w,LiNc,xPi,f»iovRf FPUT,AF:D,XCD,ZMD,
34 2WT,LTH!CK, JPART,
J5
36
37
38
|J9 i JH , JI2,VISKIP,VTSAME,US,FPATH.ERD,NOSET, NW8(10),P50,SIGMAP
, V,FACTRE,HSlZE»CNUM8Rf
MRFADs?
|«5 C
life c CONST AM rs
147 C
WB PI = s.
|19 E s 16
50 .
51 EP80 =
"2 .
153 aooo CONTINUE
'51 C
155 c
'56 NDSFTsO
'57 READfNRF AD, 5) NENDPT,MDATA
'SB 5
'59
'*0 450Q CONTINUE
263
-------
061
062 REAO(NREAD,7) ITU
063 7 FQRMATC40A2)
064 GO T0(9400,9401,9402,9403),NPATA
065 9400 CONTINUE
066 READ(NREAD,4864) NEST,NRIST,NVI,NX,NY,MITER,NCAUC,NRAPO,N|FF,
067 1NTEMP,NONID
068 4864 FQP.MATC11I2)
069 IF(NCALC'.CS,0) READ CNP£AD,5) NN,NUMINC
070 IFfNVI.eO(8) READ(NRIAD,8530) IFINAL,Jit,JI2»VJSKlP,VISAM|
071 8530 PORMATC5I2)
072 REAf)(NREAD,6) DU, PL , ETAO. DD, EPS, VRATIO, US, FPATH, EBD, RHo
073 6 FQRMAT(9F8.0,ES,2)
074 C
075 C CONVERSION
076 C
077 PL = PL * 0.305
078 PL a 01 * 2.29E-03
079 RHO * RHO/100,0
080 DV = DL / DD
081 C
082 iFfNRAPD.GT.l) RF AD (NRE AO, 8331) (ARD50 (I), ARSIGM ( T ), !=i2,NRAPD
0»3 8531 FORMAT(10(2F4.0))
064 RE AD (NPE AD, 8532 5 (ASNUCK (I) , AZICJGY (I) , AZNIJMS f I), I?l ,NONJD)
085 8532 FORMAT(6(3F4,0))
086 READ(NREAD,4) (ENDPT(I),I«1,NENpPT)
087 4 FORMATC10F8.0)
088 DO ^740 1*1,NS
089 DIAMn},((ENDPTfmENDPT(Ht)3/2.)*l,E«06
090 RAD(I)sRIAM(I)/2.
091 «7aO CONTINUE
092 9401 CONTINUE
093 IF(NDIST.EQ,2) READCNREAD,8533) D50,SIGMAP
094 8533 PORMAT<2FB,0)
095 IF(NDIST.EQ,U REAOfNREAD,4) (PRCUtI),1=1,NENQPT)
096 IF(NDI8T.E8,2) GO TO 8521
097 DO 3 IB|,NS
098 PCNTCT)=fPRCUCI+l)"PRCUfI))*ltE*02
099 3 CONTINUE
100 CALL LNFlT(PRCU,D§0,3lGMAP,c;FIT)
101
103 GO TO
104 8521 CONTINUE
105 CALL LNr>ISnD5»0,SISMAP,PRCU,PCNT)
106 ZMMDIsDSQ
107 8IGMI»S!GMAP
108 8522 CONTINUE
109 IF(NDATA.GT,1) GO TO 8534
110 READ(NREAD,770) NUM8EC,(LSFCTtT),Isl,NUMSEC)
111 770 FORMATfI2,lOJ2)
118 DO 1103 NSeCT»l,NUMSEC
113 READ(NR£AU,762) AS(NSECT),VQS(NSECT),TCS(NSECT),
114 1 ACS(NSECT),BS(NSECT),NWS(MSFCT).SVS(NSECT),VG8(NSFCT)
115 ;
lib 762
117 IFfMVI.EQil) RO TO Ha3
118 RE ADfNREAO,762) RFS(NSECT),START 1(NSECT),8TA»T2(NSECT),STARTS
119 i(NSECT),VSTAR(NSECT)
120 11^3 CONTINUE
264
-------
GO TO 8534
CONTINUE
READ(NREAD,9410) (VGS(I).VGASSfH•IB!,NUHSCC)
9410 FORMAT (3C2EU.4))
GO TO flW
9403 CONTIMUF.
REAO(NREAD,94tO) fVOS(I).TCSdt,I«l»NUMSEC)
CONTINUE
LKsO
WRITE(NRRNT,17)
NI a ?*NV » 2
NFBQ
DO 4649 KAsUNUMSCC
NFsMFtl SF-CT(KA)
CONTINUE
C
C
C
DO 1 1 = 1 ,K'S
VOLtn a PCMT(I) * DV
1 CONTINUE
MPRTMTsO
IFfNvi.EQ.J) me = o
305 COMTIMUE
IFfNVI.FQ.t) ITFRsITER+1
nn 9 T = i, MS
n/ c 4./^. * PI * RAnm**3 i
= XMori)
9 CONTTM.IF.
CALL PPTINP
C
C
C***********************************************************************
c
C STA^T INCHEMF.NTAL -ANALYSIS OF PRFCIPIT
C
C***********************************************
C
LKel
PATTP=fE.PS-1 ,
NSECTs}
00 3"0p IslfMF
IFfMV/l.EO,2) ITERsO
D GO TO 761
-LSECT(NSECT
761 CONTINUE
n GO TO 760
760 CONTINUE
^E.l) Gn TO
265
-------
181 A=ASfNSECT)*9.3E.02
182 VQsVOS(NSECT)
183 TCsTCS(NSECT)
184
185
186 SX=BS(NSFCT)*2.54E*02
187 SY»SYSfN8ECT)*2f5aE"02
188 NwjRFsNWS(NSECT)
189 VGsVRSfMSECT)*a,73E"Oa
190 VGAS=VGASS(NSECT)*,305
191 TEMPc:TFHPS(NSFCn+45«>.
192 PsPS(NSECT)
193 VIS=VISS(NSECT)
lOfl Llh'CsLINCS(HSECT)*o,305
195 RFsRFSfNSFCT)
J96 STAPTsSTARTJ (NSECT)
197 PSTARTsSTARTS(NSECT)
198 CSTARTs8TART3fNSFCT)
199 VSTARTsVSTAR(MSECT)
POP SLNRTHsFLHATCLSECTlNSECTl 5*1 INC
?01 BsSX
20? C
2^3 C CALCULATE TON) MEAN FREF PATH
204 C
205
206
207
208
209 VAVQe,snRTf(8(*flt3iaEto7*TnK)/f3. 14*32,))
210 VAVCsVAVR/100.
21? FACTRCs(Pl*VAVC)/2.
213 C
21/4 C COMPUTE. ION MOBILITY CHRRFCTEn FOR TEMPERATURE AND PRESSURE
215 C
216 Us(TDK/273.U)*US*(ltO/P)
217 C
218 COFFFC«PI*U*E
219 TjMCsLTNiC^VGAS
220 IF(NVi.FQBn GO TO U675
221 OTIMC = TINC/FLOAT(NI)
22? «675 CnMTTNUE
223 C
?2« C COMPUTE WEIGHT OF DUST
225 C
226 * s RL * VQ
227 C
228 DO 6^93 J=1»NS
?29 CCFCJ)sl+(ZMFP/RADtJ))*(1.257+.fl*EXP(.l , 1 *RAD C J)
230 h993 CONTINUE
231 IFCNV1.F.Q.2) GO TO a676
23? ERAVGsVtVSX
233 DO 69R9 1=1., MS
235 1 CF.PS-t-2. ) )*f PAD(LW(RAD(L)+TMFPn**3)*VRATIO
236 *>9g9 CONTINUE
237 Rs(E*ERAVG)/fhC*TriK*fEPS + a.n
238 RRsfF*FRAVG)/(BC*TOK)
239 Rt; = R*G
2«0 «676 CONTINUF
266
-------
NCOOPSO
iF(NVI.EQ.a) GO-TO 4677
IF(NEST.EQ,2) GO TO 4676
CALL CMANCVWfNX,NYf SX, SY.PI,AC.NWIRE)
GO TO 4676
4677 CONTINUE
UEQsU
NECsO
IFC(ViSAME.tQ.l).AMD. (NSECT.GT.U) GO TO
TFCfVi8AME,E0.1).AND.fND8ET.OT".l)) GO TO 5564
WF?iTF(NpRNT,7i40) NSECT
7UO FOPMAT(//?3X, "CLEAN GAS VOLTAGE-CURRENT DENSIT Y»F JELO AT THE PLATE
1 RFLATIONSHIP FOR SECTION Mn. ',!?//)
CALL EFLD2CUEQ, AC,VO,SX,SY,MX,NY, AEPLT,TOK,P,RF,START,
lDSTART,CSTART,IFlNAL.V8TARTfVW.ACDNTY,NWTPE,NEC,ERDfJI1fJ!2)
DO 7QP N2 = .tMI
7919
CDCl Ms
CONTINUE
C
f COMPUTE CUPKpNT DENSITY
CD = Tf / A
C
C
C COMPUTE E'LFCTRIC FIELD IN DEPOSIT
f. T = CO * RHO
C
C
C COMPUTF CURRFNT P£R M, OF CORONA
C
CL a TC / WL
C
76/4 CONTINUE
IFfNVI.EQ.n GO TO (1679
ITF-RaTTER-H
USUHsO.
RHOSUMSO.
GO TO Ub&O
afe79 CONTINUE
CALL SPCHG1 (S^,ROVRT,nRnVRT,XS,F.TAPF,DW,nSAT,XNO,W,LSFCT,TC,VG,
1ETAO. FIDf AFlDfAVGFIp, XCn,U.UEO.I,NSECTf LINC,PL»CO,e,ERAVGfNS,XPI)
CHRFlDaAFID
!'0 C
''I C***********************************************************************
!*2 C
"3 C START PARTICLE SIZE LOOP
C
!95 f***********************************************************************
C
!" PROTrO.O
?9R WT = 0.
JPARTsO.
00 c
267
-------
301
302
303
30a
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
32?
323
324
325
326
327
328
329
330
331
332
333
33a
335
336
337
338
339
340
341
34?
345
344
345
346
347
34B
349
350
351
352
353
35U
355
356
357
358
359
360
C
c
COMPUTE CHARGE ON EACH PARTICLE AF«c* UINC
Of TRAVEL
IF(NVI.EO.l) GO TO
II = 0
SERAVG s 0.0
6337 CONTINUE
II a II + 1
CHRFIOsCHFlDCin
ERAVGsEAVGCII)
SERAVG a SERAVG + ERAVQ/NI
DO 9130 L=1,NS
aSATa) = r^.*P!*EPSO*(RAnaHTMFP)**2)*ERAVG*(1.4.2.*C
9130 CONTINUE
WstE*FRAVG)/CBC*TDK*(EPS+2.))
RRsfF*FRAVG)/(BC*TOK)
4681 CONTINUE
DO 2900 J = 1, NS
IF(MVI.EOtl) GO TO 4682
OL HQfJlsOfJ)
IP(II.NE.l) GO TO 426
IF ft. ME. I) GO TO 426
IFCJ.GT.l) GO TO 428
TIMEiso.
XTPCsO.
IF fNVl.En.2) GO TH 4683
TTMFFsTTNC
IFfNCALC.EO.O)
GO TO 4fefl4
4683 CONTINUE
TIMEFsnTIMC
IF(NCALC.EQ,0)
46P4 CONTINUE
GO TO 428
426 CONTINUE
IFfNVT.EQtl) GO TO 4685
IF(J.GT.l) GO TO 429
TIMEIrTIMEF
IF'(ITtR.GT.J)
TIMFIaTIMET-TINC
IF CNCALC.LQ.O) HsDTINC/NN
4?9 CONTTMUE
IFCII.NF.n GO TO 8242
GO TO
CONTINUE
CONTINHF
GO TO 4686
4685 CONTINUE
IFCJ.GT.1) GO TO 4687
TlMEIsFLOATf 1-1 }*TINC
TjMEFsFLOAT CI 5*TINC
IF(NCALC.EQ,0) H=TINC/NN
4686 CONTINUE
CONTINUE
PSI?FsRAD(J)
268
-------
J61 SCHAR(S=QSAT(J)/l.6E-19
J62 DCONST«RATIO*RSIZE**3
363 CONSTs2.*DCONST*ERAVG
36a S=3.+RSTZE
365 VsVC/(R$lZE*CMKS)
367 FCONST»RG*DCON3T
368 FACTRE=FACTRC*RSJZE**2
369 COFFF=(COEFFC*SCHARG)/CMKS
370 IFd.lE.2) GO TO 5850
371 IF(NVI,EQ,1 ) GO TO 5851
372 IF(dl.EG.n.AND.dTER.EQ.n) GO TO 5851
373 GO TO 5852
374 ^851 CONTINUE
375 IEd.EQ.3) GO TO 5853
37fe IFfNSECT.EQ,1) GO TO 5680
377 !FCCTNDEX.Ek.n.AND.(VOS(NSECT).GT.VOSCNSFCT-1 ) )1 GO TO 5850
378 5680 CONTINUE
379 IFfCHKSUM(J),LE,0.005) GO TQ 5R5/4
380 5853 CONTINUE
381 CHKSUM(j) = (XDCf I-1,J)-XDCCI-?,J) )/XDCd-l,>H
383 IF(NSFCT.EQ,!) GO Tn 5681
380 IFfCINDEX.EO.n.ANO. (VOSCNSFCT) .GT.VOS(NSECT-m) GO TO 5850
385 5681 CONTINUE
386 IFCCHKSUMU) ,GT, 0,005) GO To 5850
Jfl7 5854 CONTINUE
388 Q(J)sXPCd-l, J)
389 GO TO "5855
39Q 5850 CONTINUE
39i iff (NCALC.EQ,i),OR.(NEST.EQ.2)) GO TO RISO
392 CALL CHARGN(E,8CHARG,NUMINC,CONST,ERAVG,V,RSIZE,ECONST,CMKS,Ri
393 1FCONST,FACTRE,COEFF,CHRFID,WATE,H,TIMEI,XIPC,NN,CTIMF,CNUMRP)
39« GO TO 8181
395 8180 CONTINUE
396 CALL CHGSUM
397 8181 CONTINUE
399 IF((TIMFI.EQ.O.) .AND,CCNUMRR.GT.SCHARG)) 0(J)sSCHARG*1.6E-19
100 5655 CONTINUE
102 IF(NVI.FQ.i) GO TO ?900
103 C
40a c COMPUTE MIGRATION VELOCITY FOR EACH SIZE RANGE
405 C
406 EMVs(Q(J)*ECOLLCTI))/f6,*PI*RADfJ)*VIS)
407 IFdTER.FO.l) EMVs(0(J)*ECLEANdl))/(6.*PI*»AO(J)*VIS)
40fl EMVsCCE (J )*f-MV
409 XMVfJ)=EMV
110 C
111 C COMPUTE EFFICIENCY FOR EACH SIZE RANGE
112 C
113 Xs(»A*FMV)/(V6*FLQATfLSECTfNSeCT))*FLOATtNI))
115 FFF 3 I. - EXP( X )
'16 C
M7 C COMPUTE NUMBER OF PARTICLES PF.MQVFD IN EACH SIZE RANGE
"18 C
119 IF (ITER.EQ.l ) GO TO 3761
180 TFni.NE.n 60 TO 3763
269
-------
421 XNO(J5sni.DXNO(J)
422 GO TO 3763
423 3761 CONTINUE
42« IFfII.NE.1) GO TO 3763
425 oi.DXNot j)=xNO(j)
426 3763 CONTINUE
427 DXNO«XNO(J)*EFF
42S IFCITER.NF., NITER) en TO 376?
42" DXS(J)*DXS(J)+OXMO
430 wS8UMSDXNO*(1.33333*Pl*RAD(J)**3)*nD
431 t*iS(,n=WS(J
433 C
434 C CALCULATE THE CURRFNT nENSITY AT THE PLATE DUE TO THE PARTICULAR
U 35 JPARTBJPART + (FLOATCUS|:CT(NSECT) ) *VG*DXNQ*Q( J) *Fl_OAT (NI ) I/A
436 3765 CONTINUE
437 XNOf J)sXNO( J3-DXMO
43* C
439 2900 CONTINUE
440 IFfNVI.fQ.l) Gn TO 9131
441 CALL SPCHG2(NS.XNO,VIS,RAD,LTNC,E,U,ERAVG.DNSION,
44? lDELTNp,suMMOB,PNUM,RHOP,TCHRG,PMOB,TDN8P,Rr>NSl,AFlD,UEQf AVGFIDI
443 ?PIOVR,If XS§ETAO,PLfETAPF,CCF,XPI.OLDOf 0,II,NSECT)
444 USUHslJSIJM
446 Sfe76 CONTINUE
4«7 IFfll.LT.NI) GO TO 6337
44S
a50 IF (I.FO.l) GO TO 376
451 IF (RinVR.GT.O.o*)) GO TO 37S
a52 376 CONTINUE
453 IF'UViSKlP.EG.n.OR.tNEST.EQ.Sn GO TO 3187
454 WRITE (HPRNT, 7iai) I
455 7141 FnRMAT(//23X, 'DIRTY GAS VOLTAGE-CURRENT DENSI TV. FIELD AT THE F
456 1 RELATIONSHIP FDR INCREMENT NO. *,I2//)
457 NEC=1
458 STARTsSTARTl (MSF.CT ) * (llEO/ll)
45^ CALL EFLnSfUEO, AC.VO, SX, SV, NX, NY, AePLTfTDK,PfRF,3TART,
460 lOSTART, CSTART, JFJNAt^STARTjVW. ACDNTV, NWTRE,NEC,EBOf JIl, JJ2)
461 GO in 3iBB
46? 31B7 CHNTTNUF
463 ACDWTY*COCLN*tUEn/U)
464 MBfl CONTINUE
465 EPLTs.t.*AEPLT
466 XC&3ACr>NTY*100000t
467 375 CONTINUE
468 IF(ITER.NEtNlTFR) GO TO 1050
469 IF(IMnEX,tQtl) SUMCD=0.
470 IFdNDEX.EO.l) SUMVOsQ.
471 SUMCD=SUMCDtACDNTY
472 SUMVHsSIJMVOoVW
473 IFf IhDEx.EO.LSECTfNSECin TCSCNSFCT )n(SUMCD*A) /FLOAT(L8ECT(N8
474 iFf INDEX. tQ.LSECTf MSEC in VOSCNSECTIBSUMVO/FLOAKLSECTCNSECT)
«75 1050 CONTINUE
476 IFtlTER.LT.NITERj GO TO 76«
477 JPARTsJPART/FLOATfNI)
478 GO TO 46B« \>
47^ 0131 CONTINUE
480 IFfUFQ.LT, 1 ,OE»4)UEQ=1 ,OE»4
270
-------
481 IFnNDFX.EQ.n GO TO 377
i|82 IFfUEQ.NE. J .OE-4) GO TO 9133
483 IFtUEQtEQ.ltOE-«)EPLT«SKIP
484 GO TO 9132
8 9133 CONTINUE
487 STGMAsARS(STGMA)
418 IFfSTGMA.LTf,01) GO TO 9132
469 377 CONTINUE
490 IF(NEST.E0.2) GO TO 8182
W CALL EFLDl(UEQ,CO,AC,\/QfSX,SY,NX.NY,TDK,P,A£PLT.VFRGE,CVERGE)
492 EPlTa.l.*AEPLT
493 GO TO fl!83
494 8162 CONTINUE
495 EPLTeERAVG/1.75
96 R183 CONTINUE
(197
498
499
500
501
50?
503
505
506
507
508
509
510
511
512
51J
511
515
516
517
518
520
521
522
521
524
525
526
527
91
c
c
c
c
c
c
c
c
c
c
c
c
c
Uh
r
SKlPsF.PLT
32 OROVPlsROVRI
DO ?965 .1=1, MS
COMPUTE *!GKATJON VELOCITY FOR EACH SIZE RANRF
EHV=(Q( J)*EPLTj/(6.*PI*RADf J)*VTS)
EMV=CCE(J)*EMV
XMVCJjsEMV
COMPUTE EFFICIENCY FOR FACH SIZE RANGE
Xs(-A*EMV)/CVG*Fl.OAT(LStCT(NSEm ) )
E F E = l . - E X P ( X )
COMPUTE NUMBER OF PARTICLES REMOVED IN EACH SIZE RANGE
Pt'Vkir^Mvx/htfif T^^Ct-C
L^^iNliwwAi'LJ ( *J / " C. • '
DXS(J)rDXSf J)+DXNO
WS(ln=DXNO*( 1 ,33333*Pl*RAr")f,i5**3)*DD
XNO(J)sXNOf J)-DXMU
WTSWT^WSC 5
CALCULATE THE CURRFMT DENSITY AT THE PLATE DUE TO THE PARTICULATE
JPART=JPApTtfELUATCLSECT(NSECTn*VG+DXNO*Q(J))/A
fc5 COHTTMtIF
fl« CONTINUE
ZWTsZWT+WT
8 C CALCULATE THE CURRENT DFMSITY AT THE PLATE DUE TO
W JIONscn-JPART
5JO C
!«1 C
I5J? c CALCULATE THICKNESS OF DUST LAYER IN (MM/MIWJ/INCREMENT
555 c
5J6 c
!J! C CALCULATE MMO AND HEIGHT COLLtCTED FOW EACH INCREMENT
JJB
00
-------
541 ZTM=ZTM+WS(J)
542 CZAaZTM/WT
543 IFCCZA-0.5)2901,2901,2902
544 2901 CONTINUE
545 2902 CZ.Ba(ZTMwWS(J)5/WT
546 TL1=CZA-CZB
547 TL2=0.50-CZB
548 Kjs,l-i
549 IF(KJ)29iO,2910,2911
550 2910 ZMDsOIAMCJ)
551 GO TO 29IP
552 29Jl ZMpsniAM(KJ)i(TL2/TLi)*(D]
553 2912 CONTINUE
554 IF(NVI.fQ,2) ERAVG = SE
555 CALL PRTINC
556 3000 CONTINUE
557 C
558 ETC=(ZWT/DL)*iOO.
559 IFfNVI.F_n.2) GO TO 1620
560 DIFFsFTC-ETAO
561 DIFF=AR$(DTFF)
56? IF(DIFF.O,05)60,300,300
563 300 CONTINUE
564 *IPTTF. (NP«NT,8656) ETAO,ETC ,
565 86^6 FORMATf/' EST, EFFICIENCY r•,F6,2,5*,'UNCORRECTFD COMPUTED EFFK
566 1NCY s',F6,2) ,
567 IFflTER.FU.NITFR) GO TO 60
568 ETAOeETC ,
569 GO TO 305
570 6 0 f. n N T I NI! F
571 GO TO 162t
572 1620 CONTINUE
573 WRITffMpRNT,1622) ETAO,FTC
574 1622 FORHATC/' DESIGN EFFICIENCY s',F6,2t5X,'UNCORRECTFD COMPUTED EFI
575 1IENCY s',F6,2) ' '
576 1621 CONTINUE
577 ATOTALsO.
578 VGsO,
579 DO «98S I=1,NUMSFC
580 ATOTALsATOTAL*ASfI)*(9,3r-02
581 VGsVG+fVGSf n*4.73E-Oa)
582 4985 CONTINUE
583 VGBVG/FLOAT(MUMSEC)
584 CALL PRTCHQ
585 CAiL ADJUST
586 Rn TO aooo
587 Q9Q9 STOP 11111
588 END
PRnr, > aK
272
-------
SUBROUTINE PRTINP
REAL LIMCS,NHS
INTFGER VISKIP,VISAME
DIMENSION IBlNK(ai)
CQMMON/6LK1/DIAM(20),ONO(20),DXS(20),XMV(20) ,PCNT(20),RAD(?0),
lCCF(J?n)fPRCUC21)
CQMMON/BLK2/LSECT(1Q),LINCS(10),PS(10)
COMMON/RL.K WG,ATOTAL,DD,ETAO,DU Plf RHO
COMMON/RLK5/ZMMDT,SIGMI,NONID,NRAPD,TDK,NUMSFC,NFFF,NTFMP,GF.!T
COMMnN/RLK6/VQ.L(20),XNO(20)fG(?0),WS(20),ITUaO),DW(«5),AsnQ)»
IV OS (10) , TCSC1Q),WLS(10), ACS(101,RS(10),8YS(10),Vf;$(10),VGASSUO) »
2TEMPS(tO),VISS(lO),QSAT(20),L'fE,FPSO.PT,ERAVG,BC,TEMP,EPS,VAVC,
30inn(20).OUDXNn(20),RFS(10),STARTl(10),STAHT2(10).START3(10),
aVSTAR(lO)
COMMQN/BLK1 1/ENDPT(21 ) ,NEN|)pT
COMMON /RL.KJ2/ARD50UO) • * »S I GM ( 1 0 ) , AS^UC-K ( 1 5 ) , AZNUMS ( 15) , AZIGGY f 15
COMMON/RLK17/NRFADfNPRNT
CnMMnN/BLKl«»/LK,DV,MM,MllMlMCf NX, NY,NDATAf NF8T,NmsT.N
IJII , JI2,VISKIP,VTSAME,U8,FPATH.EBD,NDSETfNI*S(10),D50,SIGMAP
DATA TRLNK/2t** '/
1F(L K) 111, It 1 , 160
111
FORMAT ( ilOX , '*************************************')
S1 )
, 35X, '*')
S2)
S852 FORMATraOX,'*',9X,'E.P.A. ESP MODEL't 1 OX, '*'!
, 'I.E.R.L.-R.T.P. AND SO,R . I , ' ,4X ,
FORMATfaOX,'*',7X, 'REVISION I, JAN, 1, 1 978 ' , 5X , ' * • )
WRTTF(NPRNT,5fi51 )
NDSFTaNDStTtl
L/2.2''E"03
L/0.3n5
,S = 1 00.*RHn
,2000) NDSFT
?000 FORMATt//' PRINTOUT flF INPUT DATA FOR DATA SFT NyMBFR ',I2//)
NCARDsNCARD*!
WRlTFf MPRNT,2001) NCARD
2001 FORHATf//' DATA ON CARn NUMBER f,lJ>//)
WRTTF. (NPRNT, 1000) NFNDPT,NDATA
150 1000 FORMATC NENOPT = ',I2,?X,» NDATA s ',12)
51 NCARnsNCARO+1
5? WRITE(NPRNT,2001) NCARD
WRITE (NPRNT,! 001) TTL
1001 FORHAT(?X,aOA?)
155 GO T0(6000, 6001, 6002, 6002), NDATA
CONTTNUE
NCARDrNCARD+1
WRJTF(NPRNT,2001) NCARD
159 iuPiTF(NpRNT,ioo?) NEST,NDIST,NVI,NX,NY, NITER,NCALC,NRAPO,NEFF,
1NTEMP,NONID
273
-------
061 1002 FORMATf* NEST B '.12,2X,*NpIST s »,I2,2X,*NVJ s »,I2,2X,'NX B •
062 1,I2,2X,'NY = ','12, 2X,'NJTER B »,12,2X,'NCAlC « ',I2,2X,'NRAPD s
063 2I2,2X,'NEFF s *, I2.2X, *NTEHP v *,12,2X,'NONID s ',12)
Q6fl IF(NCALC.NE.O) GO TO 1003
065 NCARDsNCARD+1
066 WRITE(NPRNT,2001) NCAP-0
067 *PIT£(NPRNT,1004) NN,NUMINC
068 1004 FORMATf' NN = ' ,12,2X,'NUMINC s ',12)
06<» 1003 CONTINUE
070 lF(MVJ.En.l> GO TO 1005
071 NCAPOaNCAPOf1
072 WRITEfNPRNT,20013 NCARD
071 WRITE(NPRNT,1006) IFINAL,JII.JT2,VISKIP,VISAME
07CARp
109 101? CONTTMUE «'
110 WRITFCNPRNT, 1015) (IBL WK (I) , I, FNPPT (I) , Is 1 , NENDPT ) I'
111 1015 FriPMATf5flX,Al, 'FNDPTf ',12, ') e *,FS,3,' UM'flX)/)
112 GO TO 1016
113 10i3 CONTINUE
tl« WRlTFfNPRNT,10lS) f I Bl.NK (I ) f I, ENDPT (I) , I a 1 , 1 0 )
115 NCARP=NCARD+I I
116 WRITF.(NPRNT,2001) NCARD
117 WPITFfNPRNT,1015) CIBLNKfI)fT,FNDPT(n,1*11,
118 G H T 0 10 1 6
119 101 a CONTTNUF
101b) (IBLNK(I),1,ENDPT(I),I»1,10)
-------
NCARDsNCARDtl
WRITE(NPRNTf200t) NCARD
WRITE(NPRNT,1015) (I BUNK(I),I,FNDPT(I),I«l1,20)
NCARpsNCARD+i
WRITF(NPRNT,2001) NCARD
WRITE(NPRNT,iOt5) CIBINK(I),I,ENPPTCI),I=21,NENDPT)
1016 CONTINUE
6001 CONTINUE
IFfNDIST.EQ.l) GO TO 1017
NCARDsNCARD+1
WRITE(NPRNT,2001) NCARD
WRITECNPRNT, lOlfl) D50,SIGMAP
1018 FOPMATC 050 = ',F«,4,f 'JM»,2X,»SI
1017 CONTINUE
IFfNDIST.EQ.2) GO TO
NCARHsNCARD+1
WRITE (NPRMT,2001)
GO TO(1020,1021, 1022),
1020 CONTINUE
WRTTEfNPRNT, 10?3) CIHLNKfl),I,PRCU(I),
10?3 FOP-MAT(5(1X, A1.'PRCU(»,I2, ') = ',Ffl.a,' %',1X)/)
GO TU 1021
1031 CONTINUE
WRITF fMPRNT, 1023) f IBL.NK CI) , I, PRCU fl) , I si ,10)
NCARR
,1023) fI8LNK(T),I,PRCU(I),l3l1,NEHDPT'
GO TO 102^
102? CONTINUE
aBLNK(T),I,PPCU(I),T=l,10)
WRITFfNPRNT,2Q01) NCARO
KRTTF(HPRNT,1023) (JBLNK(I),I,PRCU(1),Ic11,20)
NCAKPsNCARD+1
^RITF(NPRNT,?001) NCARO
lMRITF(MPRNTf1023) (IRLNKri),I,PRCU(I),I = 21
102a CONTIMUE
IFfNDATA.GT.t ) GO TO 5000
WRITE(NPRNT,2001) NCARD
62 IF(NUMSEC.GT.S) GO TO 1D26
kJ WRITE frJPRNT, 1025) NUHSEC, f I Bl NK (I ) , I , I SECT f I ) , I = 1,WUMSEC)
W 1025 FORHATf NUMSEC = ' ,IE,2V,5( 1 X,A 1, 'LSFCTf*,12, ' ) = »,I2))
*5 60 TO 1027
^ 10?6 CONTINUE
M WRITFfNPPNT,1025) NUMSEC. (IRLNK(I),I»LSECTfI),1 = 1,5)
WRJTE(MPRNT,8570) f IRLNK(I) , I, L.SECT(I), 1=6, NUMSEC)
8570 FORHAI(/5(1X,A1,'LSFCTf',12,') = ',12))
1027 CONTINUE
DO 102* 1=1,NUMSEC
NCARDsNCARD+1
WRITFCNPWNT, ?001 ) NCARI5
WPITFfMPPNT,1029) I,AS(I),I,VOS(I).I»TCS(I),I,WLSfI)
10?9 FOPMATC ASC',12,') = ',1PF1J.U,' FT**2 ' ,2X, ' VOSf ' ,12, ' ) s
11.a,' V,2X,'TCS(',I2,') - ',1PE11,«»' A',2X,'WLS(',I2,') =
21,«,' FT'/)
WRITEfMPRNT, 1030) I,ACS(I),I,B8(I),I,NWS(I )
ID^n FOPMATf ACS(*,I2,') = ',1'PEll.a,' IN',2X,'BS(',I2,') s »,
275
-------
181 NCARDsNCARD+1
182 HRIT|?(NPRNT, 30015
183 WRITCCNPRNTflbsn I,3YSd), I.VGSd), I, VGASSd)»I,TEMp$f j)
184 1031 FORMAT*' 3YS(',T2,.') s ',IPEU.«,' IN',2X,'VG8f •,12,*) s
185 !<»,» FT**3/MIN'.2X,'VGASS('.!2,') = '»1PEU.4,* FT/SEC '. 2X, 'TE MR
186 2,12,') * MPEilU,' F'/)
187 wRITf(NPRNT,1032) I.PSd), I, VISSdi»1 »LXNCSd)
189 1032 FORMATf P3C.I?,*) = '.1PEH.O,' ATM',2X,'VJSSC',12,'J B «,tp
189 J.a,' KG/M«SEC',2X,'L1NC8(»,I?,') = '.IPEll.a,' FT')
190 IFCNVl'.EG.U GO TO
191 NCAROaNCARD+1
192 WBITE(NPRNT,2001)
193 WPTTC(NPRNT,10?3) I,RF8fI),I.8TART1(I),I,8TART?fIJ
19a 1033 FPRMATf RFS(*,!2,') • ', 1PE1 1 ,«,2X,'START1 ( » , 12, •) * ',1PE11.
195 1 A/M**2',2X,'START2(',I2.'J * '.1PE11.U,' A/M**?»/>
196 WRJTF(MPRNT,I03a) I,START3fI),T,V8TAR(I)
197 103« FORMATC START3 f • , T?, ' 5 = '.iPEll.<»,» A/M**2',2X,'VSTAPf',12,'
19B 1 ', 1PE11.«.' V)
199 1028 CmniNUF
200 GO TO 5000
201 ^002 CONTINUE
202 MCARDeNCAHO-M
203 Wf»ITC(NPRMT,2001) MCARO
20« DO 103S IsljNUMSFC
205 IFfJ.EO.fl) NCARP«NCARD+1
206 IFd'.EO.^) WRITf fNPPNT.2001 5
207 IFCI.CQ.7)
20* IFfI.En.73
209 iFd.EQ.io) NCARD=NCAPD+I
210 IFCI.EQ.105 wPITE(NPPMT,20ni) NCARD
211 IF(MDATA.EOf«) GO TD 1036
212 WRITFfNpRMT,10375 T , VGSd),T,VRASSd)
213 1037 FORMATf VGS(',T2.'3 = ',!PF11.fl,' FT**3/M1M',2X,'VGASS f, 12,'
2H 1 ', IPEll.a,' FT/SEC*/)
215 GO TP 103S
216 }OJ6 CONTINUF
217 to»TTE(NPPNT, 1038). I, V08(I) , T, TrSd )
218 1038 FHRMAT(' VOS(',I2,') * ',1PEH.«,' V',2X, *TC8(', t?,') = ',lPEt
219 l,' A»/5
220 1035 CONTTNllE
221 5000 CONTINUE
222 WRITFfNPRNT,10395
223 1039 FORMATflHn
22^ 160 CONTINUE
225 RETURN
226
276
-------
51 SUBROUTINE SPCHG1 (SW.RDVRI , OROVRI , XS, ETAPF, DW, QSAT, XNQ, W, LSECT,
02 1TC,VG,ETAQ,FID,-AFID, AVGFID,XCO.U,UEQ,ItNSECT,LlNC,PL,CD,E,ERAVG,
OJ 2NS.XPI)
LINC
05 DIMENSION DWC45),QSAT(20),XNOt20),L3ECT(10)
506 IFCI.NF..1) GO TO 1286
j)07 SW = 0.0
08 ROVRI = IO.
OPOVRJB20.0
C
C COMPUTE VAL-UF OF EXPONENT IN DEUTSCH EQUATION FOR THE STATED EFF.
112 C
U3 XSsALOGnOO./nOO.-ETAD))
114 C
15 12B6 CONTJMUE
16 C
H7 C COMPUTF FFFICltNCV PEP LENGTH INCREMENT
C
s l.-tXPf-LINC*XS/PU
i2i c COMPUTE AMOUNT OF MATERIAL RFMOVED PER INCR.OM A TOTAL WFIGHT BASIS
12? C
121 D^fl) s fW - SW) * E
SW = SW + DW(I)
SUMso.n
127 fiO 1300 L=J,NS
128 1300 SUMsSUM4.QSAT(U*XNn(L)
Zcs200.*(DW(I)/W)*(FLnAT(LSECTfNSECT))/TCl*Vfi*SlJM
150 ROVRTsZC + 1.0
131 AFIHcFID/WOVRI
132 AVGFlDaAFir>*l,E-0«>
xcnscn*iooooo,
Ufl C
135 C CO^PUTf EFFECTIVE MOBILITY
36 C
37 UEQslJ/POVHl
138 C
159 XPTsFTAPF*lOO.
HO RETURN
END
277
-------
001 SUBROUTINE SPCHG2 CNS, XNQ, VIS, RAO,LINC , E,U, ERAVG, DNSION,
002 1DELTNP, SI!MM08-,PNUM,RHOP,TCHRG,PM08,TDNSP,RDNSI,AFID,UEQ,AV6FID,
003 2RIOVR. I,XS,ETAO,PL,ETAPF.CCFfXPI.OLDQ,Qf II.NSECT)
ooa PEAL LINC
005 DIMENSION XNO (20 ) , RAD (20 ) » CCF (20) f OUDQ (HO) , Q (20 )
006 cnMMON/BLK7/XDCU5,20)
007 COMMON/BLKB/EAVG(30),CHFID(30)
008 COMMGN/BLMT/NREAOiNPRNT
00? iFfl.NE.l) 60 TO 1286
010 C
Oil C CnMPIJTF VALUE OF EXPONENT IN DEUTSCH EQUATION FOR THE DESIGN FFF.
01? C
013 XSsALOG(100./C10n.»ETAOn
015 !2Hb
016 C
017 C COMPUTF FFFlCItNCY PER LENGTH
018 C
019 FTAPF = l.-EXP(»LINC*X3/PL)
oao c
0?2
023 PNUMsO.
025 nn i J=I,NS
02fc TCHRr,eXNO(,n*XDC(I, J)
027 RHOPaRHOP+TCHRG
03fl 8lJMMOB«SHMMOB+(TCHRG*CCF(J))/(6,*3.1«159*VTS*RAO(J)
029 PWUMaPNUM+XNO( J)
030 DIEFsXDCd. J)
031 IFf (IT.NE.U.OR.d.NE.lJl DlFFaRf J)-OLOQ(J)
032 nELTNP
033 I CONTINUE
035
036
037
038
039 IF(PDNST .GT,0,1 GO TO 10
0«Q PIRsDELTNP/DNSION
oa? WRITECNPRNT, in PIR,I,II
043 11 FOPMATflXt* A FACTOR UF ',Ffi.3,' MORE IONS NEEDED IN INCREMENT
oaa i?,% INTERVAL *,!?,' TO MEET CHARGING RATE')
oas 10 CONTINUE
046 AFIDsRDNSI
047 AVGFTnsAFID*! ,E-»06
049 RIOVRsf AFlD*E)/(AFlD*E-«-RHOP5
050 XPIsFTAPFMOO,
05i RET
052 END
278
-------
OOt SUBROUTINE CMAN (VW, NX,NY, SX , SY>I, AC , N«IRE)
002 C COOPERMAN SERIES DETERMINATION FOR VOLTAGE WIRE TO PLATE
003 C FOR SUBROUTINE EFIFLD
004 REAL NIJM,M,NNIRE
005 COMMON/BI.K13/VCOOPU5, 15)
006 NXlsNX-J
007 MYlsNW-1
008 AXaSX/NXJ
009 AYsSY/NYl
010 DO 40? I=1,NX
Oil (in ajO J=1,NY
012 X=(I-1)*AX
013 Vs(J-J)*A¥
OH IF(X.EO.OtO.AND,Y.EQ.OtO) GO TO
015 GO TO USO
016 ttflO VCOnP(T,J)aVW
017 GO TO iljf)
018 «5n CONTINUE
019 Ms-NWIRE
020 NUMsO.O
021 DfNnMao.n
022 ^90 FtsP!*fY-C2
023 FlaPT*X/f2.*SX)
025 HJ =PI*&C/f ?.*SX)
026 F2a(FXP(En*EXP(-Et))/2.
0?7 F?=roSfF1)
028 G2a(EXP(GU+EXP(.Gl))/2.
029 H2aCOSfHl)
030 TT=(F?»F2)/(E2+F2)
031 TB=(r,2.H2)
032 FsALOGfTT)
033 GsALOGfTR)
0 3 Q N ij H s N U M + F
035 DENOM
036 IFrM.LT.N^JRt) .GO TO «08
037 GO TO ajO
038 /jf>fl MsM+l ,0
039 GO Tf) 090
OflO aiO VCOOPf I, J)sVW*NlJM/DENOM
40? CONTTM.IF
oa3 RETURN
279
-------
001 SUBROUTINE EFLD1 £ UEQ, CD, AC , VQ, SX, SY, NX , NY, TDK .P, AEPLT, VERGE,
002 JCVERGE)
003 C EVALUATION OF FIELDS, SPACE CHARGE DENSITY, POTENTIAL * AND
004 C CURRENT DENSITY FOR A WIRE«"PLATE PRECIPITATOR
005 REAL MAXJ,MINJ,MQBIUT(15,15)
006 DIMENSION RHOnS,15),EXCJ5,l5).OlDROM5,l5),OLDVM5,lS),
007 ICDNSTYMSr 15),VM5, 15),EYM«5,15)
008 CQMMON/BLK13/VCOQP(15,15)
009 COMMON/RLK17/NREAD,NPRNT
010 DATA RHO/22S*0./,V/225*0./,EX/2?t5*0./,EY/22S*Ot/,nLDRO/225*0,/l
Oil lOLDV/2a5*0,/fCDNSTY/225*Ot/,MOBlLT/22b*0./
01? VO=-lt*VO
013 PI = 3.1 fl 16
014 EPSr)B8,8S4E«12
015 HO = AC
016 PQC = 10ft. 0*RO
017 «F = 1.0
oi8 RFLD • c?<53.o/TDK)*(p/i.oi
019 EURO = Rnc*RF*f30,0*RELD + 9. 0*SORT CRELO/ROC) )*1 .nEOJ
020 C
021 c COMPUTE INITIAL ESTIMATE OF SPATF CHARGE DENSITY AT WIRE
02? c
023 VFRGEsr-2,*m*n.ni + 0.<»<>)*SY)/(2.*PI*UF.Q*EORO)
025 QZEROsVERGE
026 On SSO 1=1, NX
027 DO 550 J=1,NY
026 MflRrLTf 1, JJsUEQ
0?9 550 CHNTTNUt
030 MAXJzCD*l,01
03! MINJsCO*0.09
"3? NK1.3NX-1
033 MYlsMY-1
03^ AX=SX/NX1
035 AYsSY/MYl
036 AXS=AX*AX
037 AYSaAY*AY
03B ASP=(AXv«i*AYS)/
03^ ASS=1 ./(?.,*( AXStAYS) )
040 2=0.
041 DO 4615 1=1, NX
04? DO 4M5 J=1,NY
043 4MS V(T,J)8VCnoP(I,Jl
04'4 1 Z = Z+1.
04<5 12 = 7
046 IFf7.Fn.25) wRITECNPRNT,
047 18*,5 FQRflATflX,' CONVERGENCE ON CURRENT DENSITY CAN NOT BE OBTAINED I
048 l?"5 TTFRATIONS')
0^9 IF(Z.FQ.25.) GO TO 700
050 LL=0
051 300 LLsLL+1
052 RHO(1,1 )=QZFRO
053 FX(l,nsO,0
054 EYM , n = o,o
055 DO 201 T=2,NX
056 FYd.nsO.
057 Fyf I, l) = fV(I»l,l>-Vf I.in/AX
058 Ols?.*MOBILT(I,n
059 Q2 = f,ll*AX
060 n 3 s P 1 * A Y
280
-------
061
062 05s.EPSO*EXCI>l)*CQ3«'AY*MORIlTr 1*1,1))
063 06*H5*Q5
064 U7sQl*f3a*EPSO*AY*EX(I,n*RMO(I-l,l)
065 Q8s-SQRT(Q6+Q7)
066 RHO(T, J)
067 201 CONTINUE
068 00 203 J=2,NY
069 EX(1,J)=0.
070 EY(l,sn3(VU,J.n
071 PI*2.*MOBILT(1,J)
072 P2=P1*AX
073 P3sPl*AY
07a P4=P?*AY
07% P5 = -FPSO*EY(i, JJ*(P2"AX*MORH T(Jf J-
076 Pfe=P5*P5
077 P7sPl*Pa*FPSn*AX*EY(J,J)*RHO(l,J»1)
076 P8B.SURT(P«» + P7)
079 RHOn, J) = (P5 + PR) /Pa
OR). 00 ?0? Ia?fNX
OB2 fY(I,NY)sO.
083 EXCI.MY)s(VfI-l,NY)»V(ItNY))/AX
osa
085
086
087
08B
08
-------
121 30« IFCT.EG.l.AND.J.EQ.NY) SO TO 350
122 V( J,J)«A$S*(2j*AY3*V(2»J)+AXS*CV{l,J+mVUiJ«l))+ASP*RHO(i,J))
123 GO TO 301
12« 350 VCi,NY)sASS*(2,*AYS*VC2,NYU?.*AXS*VCliNY-l)+ASP*RHO(l,NYn
125 GO TO 301
126 305 V(I,l)aAS
127 GO TO 301
128 306 V(I,J)
129 10(1,J))
130 301 CONTINUE
131 IFfLL.EO.2000) WRJTE(NPRNTf
132 1«66 FORMATflX,' CONVERGENCE ON POTENTIAL GRID CAN NOT BE OBTAINED
133 1000 ITERATIONS')
134 IF(LL.EQ.2000) GO TO 700
135 DO 320 Ist.NXl
136 DO 320 Jsl.MYl
137 lFfAPSrv(I,J)-OLDV(I,J)).LT.l.) GO TO 320
138 GO TO 300
139 320 CONTINUE
140 nONSTY(NXfJ)*EX(NX,1)*MQ8ILT(NX,1)*RHOfNX, 1)
141 ACDNTYsCDNSTYtNX, I1)
142 950 nn 900 J=2,NY
143 CDNSTY(NX,J)sFX(NX,J)*MOBILT(NX.J)*RMOfNX,J)
l«a ACDNTYzACDNTY+CDMSTY(NX,J)
145 900 CONTINUE
U7 IF(ACnNTY.GT.MAXJ) GO TO 910
1MB IFfAT.nNTY.LT.MlNJ) GO TO 9?0
HP GO TO 980
150 910 GZFROxMINJ/ACDNTY*RZERO
151 Co TO 1
152 9?n G)ZERn
153 GO TO 1
155 DO 1000 J»2,NY
156 EPLTsEPLT+EX(N
157 1000 CONTINUE
158 AEPLTsFPLT/MY
159 7no CONTINUE
160 CVE'PGEs
161 vns-i,*
162 PETUPM
163 PNO
PROG > a K
282
-------
00i SUBROUTINE EPLD2 (UEQ, AC, VO,SX,SY,NX,NY, AEPIT,TDK,P,RF,
002 ISTART.DSTARTfCSTARTiJFlNALiVSTARTfVI^ACDNTYfNwlRE.NECfERD, JI1,JI2)
(OS c EVALUATION OF FIELDS, SPACE CHARGE DENSITY, POTENTIAL , AND
04 C CURRENT DENSITY FQR A WIRE-PLATE PRECIP1TATOR
305 REAL MAXJ,MINJ,MQBILm5,lS),NWIRE(MAXS
00* DIMENSION RHO(15,15),EXn5,l5).QLDRO(15,iS),OLOvei5,l5),
00? ieONSTYn5,lS),VClS,lf),EYn5,l§),EAVG8(30),CHFlDS(S0),ECOLL$<30)
008 CQMMON/BLKS/EAVGOO),CHFID(30)
Df COMMQN/BLK9/ECQLI (30)
0 COMMON/BLK13/VCQQP(15,15)
012 DATA RHOX225*0./iV/a25*0./,EX/225*0,/,EV/225*Oi/,OLDRO/g25*0./,
013 lOLOV/225*0./,CONSTY/2a5*0./,MOBlLT/225*0./
015
016
JIT
018
019 ROC*IQO.*RO
020 EOROaROC*RP*f5fl
021 00 550 lal.NV
022 00 550 Jsi.NY
023 MOBILT(IfJJ"UEO
024 550 CONTINUE
J25 PI»3.iai6
026 EPSO»8.85*»E»12
927 SSTARTsSTART
028 MINJso,
02^ MAXSsO,
030 NXlsNX.i
031
032
OJ3 AysSY/Nyl
U AXSaAX*AX
035 AY8»AY*AY
6 ASPs(AXS*AYS)/EPSO
137 AS8si./(2,*CAX3*AYS3)
J38 DQ 1001 !Isl,IPlNAU
559 IP(II.EO.JM) STARToOSTABT
O iPdl.SE. J!2) STARTsCSTART
141 MAXSeMAXS'fSTART
2 1526 CONTTMUE
143 MAXJsMAXS*1.01
144
146
047 CALL
J48 ZsO
« DO «
550 DO 4
i «6is vd,
2 1 ZsZ +
054 IFCZ.EQ.25) WRITE (NPRNT,
'55 1865 FOP-MATdX,' CONVERGENCE ON CURRENT DENSITY CAN NOT BE OBTAINED IN
'56 125 ITERATIONS')
'57 IFfZ.EQ.25) GO TO 700
058 LL«0
059 300 LL»LL*1
283
-------
061 V(1,1)BVW
062 EXC1,1)«0,0
065 EYC1. 1)80,0
064 DO 201 Is2,NX
065 EYCI,1)»0,
066 Exn,n = (VfI"i,n-V(I,n
067 Qls2t*MOBIUT(I.n
068 Q2SQUAX
069 &3sQUAY
070 Q4=Q2*AY
071 Q5a»EPSO*EX(If 1 ) * (Q3-AY*MnRTl. T ( l-l , I) )
07? Q6=Q5*05
073 Q7 = Ql*Qa*t"PSa*AY*EX(If I J*RHOf 1-1, 1)
075 08 =
076 WHO(T, J )=
077 201 CONTIMUE
078 DO ?03 J = i?,NY
079 FXf1,J)sO,
OBO FYf t, J)s(Vf 1,J-1)-V(1, J) )/AY
081 Pl=
082 P? =
083 P3=
085 PS =
086 PfeEPS*P5
087 P7sPl*pa*tP50*AX*EV(l»J)*RHO(lfJ-l)
088 P7=ABSfP7)
089 P8 = -SQRTf P64-P7)
090 RHO(1, J)=(P5+Pfl)/Pa
091. 203 CONTINUF
092 Hi) ?0? Ts2,NX
093 EY(T,NY}=0,
09« FX(I,NY)=(VfI-1fWY)«VfIfNY))/AX
095 P1=^.*MOBILT(I,NY)
096 R2=P1*AX
097 »3=R1*AY
098 R/J5Rg*AY
099 RS=-EPSO*EX(IfNY)*(R3-AY*MOBTUT(I"l,NY))
100 P6sR5*pc,
101 R7sRl*R«*EPSO*AY*EX(I,NY)*RHO(I-l,NY)
102 R7=ABSfR7)
103 R8S-SORTCR6+R7)
105 202 CQNTTNUE
106 DO 307 1=2, MX
107 DO 307 J=2,NY
108 313 f-Xf T, J) = C-l.)
109 tY(I,J^5f-l.)
110 D1=2.*MOBILT(I,J)
111 D?=D1*AX
112 D3301*AY
113 D«=n2*AY
1H D«.s-FPSO*(EX(I, J)*(n3-AY*MOBILTClM, J))+EY(I, J)*(D2»AX*MQBILT(Ii
115 lim
life Dbsns*n=;
117 07 = Dl*D/i*FPSO*f AV*EX(If Jl*PHO(t-l,J)+AX*EY(I,J)*RHO(I, J»1J)
118 D7
119 nR
120 RHO(I, J)a(D5+D8)/Dfl
284
-------
307 CONTINUE
DO 301 1=1, NX1 '
DO 301 Jel.NV
OLDV(i.J)«Vd,J)
OLUROd. J)sRHOd, J)
IFd.Efl.l,AND,J,EQ.l) GO TO 301
IFd.EQ.l.ANO.J.NE.U GO TO 30«
IFd.NE.l.AND.J.eG.l) GO TO 305
IF(J.EG.NY) GO TO 600
GO TO 306
600 Vd»MY)cASS*(AYS*(Vd«l,NY)+Vdf l»NY))+2.*AXS*Vd,NY.l )+ASP*RHOd,
133 GO Tp 301
134 30+AXS*(VdfJ-l)+Vd.J+l))+ASP*RH
i«2 ion ,jn
143 301 CONTIMHE
TPfLl..F«3.2000) WRITE(NPRNTf 1866)
1*66 FQRMATflX,* CONVERGENCE OM POTENTIAL GRID CAN NOT BE OBTAINED IN 2
|46 1000 ITERATIONS')
147 IFflL.EQ.i?000) Gn TO 700
1»8 DO 3?0 Isl.NXl
00 320 jsl.NY
150 IF(ABS(Vd ,J)-Ol.nv(i, J) ) .LT.l .) GO TO "5?0
151 GO Tn 300
152 320
153
I5S ^50 DO 900 J = 2,NY
156 CDNSTYf NX, J)r£X(NXf J)*MnRILT(MX, J)*RHOfNX, J)
157 '
58 QOO
159
M IFUCD^TY.GT.^AXJ) GO Tn
161 IF f ACDNTY.LT.MINJ) GO TO
6? GO Tn PRO
63 OlO VW = VW^1 ,*V
GO TO 1000
165 920
1000
TFSTlsO.OJ *ACnwTY
69 irf TPST.LT.TFST1) GO TO 9fl0
70 GO TO 1
7) 9flO CONTINUE
FPLTsFXfNX, 1)
173 DO i?QO J = 2,NY
1200 CONTTK'UF
700 CONTIMtlF
WRIT?: (MPRNT,8flB«) VW, ACDNT Y ,
FORMAT(36X, *VW = ', !PFll.a,2X, '*CDNTY = ' , IPE 1 1 , 4f 2X , '
IE 1 !.«//)
285
-------
181 IF(ABS(FX(NX, i)) .IT.EBD) GO TO 1480
182 WRITF. (NPRNT,1U81) VW,ACONTY
183 1481 FORMATf* THE BREAKDOWN FIELD NEAR THE PLATE IS EXCEEDED AT VW a*
184 in. 4. ix, 'AND ACDNTY =',Eii.4)
185 GO TO 1$25
186 148Q CONTINUE
187 IF(IVCK.F.Q.l) GO TO 1525
1BA TFf ABS(VW) .EQ,A8S(VOn GO To 1525
189 TFf ARSfVWJ .6T.ARS(VO)) GO TO 1523
190 Ol_nvw = vw
191 OLDcosAcnwTY
19? GO TO 1524
19? 1523 roNTTNUE
194
195
196 IVCKsl
1.97 GO TO
198 1524 CONTINUE
199 1001 CONTINUE
200 1S2S CONTINUF
201 IF(MFC.NF.O) GO TO JOOO
?02 Kst
203 on ^001 J=i , NY1
204 RSHM=n.
?05 ESUMrn.
206 On -^002 islf^x
207 IFCJ.FQ.1 ) 60 TO 3005
20 8 ESUMsFSUM+f SQPT(EXf Ii JJ**2*FY(Tf J)**2)+80RTfEX(I, J+1 3**2+
20P tFYfI.j+l)**2))/(2.*NX)
210 GO TO 3006
211 3005 CONTINUE
212 ESUMsFSUM + Sf3HT(EX(I, J+1)**2 + EY(T» J + l)**2)/f2.*NX)
213 IFd.EO.NXl FSUMsE8UH.VO/f?,*SX)
214 300b CONTTNUF
215 RSUMBWSUM-fRHUf I, J)+RHO(I, Jf 1 ) 5/(2.*l ,6F-19*NX1
216 3002 CONTINUE
217 tAVGSfk jsESUM
219 K=K+1
220 ?001 CONTIMUF
221 NYY=NY1
222 HO 3003 I =lf NY1
223 FAVr,(L5sFAVGS
224 CHFin(L)=CHFIDS(MYY)
225 NYYarWY-1
226 3003 COMTTNUF
227 KKsl
230 DO 3004 M=M1,M2
231 FAVG(M)sEAVGSfKK)
232 rHFincM)=CHFl[)S(KK)
233 KKsKKtl
234 3004 CONTIHIJE
235 3000 COMTlhUF
236 LL=1
237 DO 3007 NN=1,NY1
240 5007 CONTT^'|JF
286
-------
241 UeNYl
242 HO 3008 L«l»N-Yt
243 eCQUa)
244 Llsti-l
245 3008 CDNTlNLIf
346 L?«l
00 3009 1 = 11,12
250
251 12BL3+1
?5? 3009 CONTINUE
253 V0B-i,*vn
254 STARTsSSTART
255 RETUPN
256
PROG > «K
287
-------
001
002
003
ooa
005
006
007
008
009
010
on
013
013
Olfl
015
nife
017
018
SUBROUTINE CHARGN (ECHARG,SCHARC,NUMINC,CONST,EZERO,V,RSIZE,ECON
*,CMKS,RR,FCONST,FACTOR,CQEFF,AFID,RATE,M,XI,YI,NNfX,Y)
YsYI
XsXI
DO 2 I»i,NN
T1sH*RATE(ECHARC,SCHAR6,NUMINC.CONST,E2FRO,V,RSIZE,ECON8T,CMKS.fi
*FCONST,FACTOR,COFFF,AFID,X,Y)
T^sH*R A TE(ECH A RG.SCHAPG, NIJMJNC. CONST, t ZERO, V,RSTZE,FCDMST,CMKS,R
*FcnNST,FACTOR,COEFF,AFID.X+H2,V+T1/2,)
T3sH*RATE(ECHARG,SCHAf?G,NUHINC.CONST,EZEROf V, R8IZF,ECONST,CMK8,R
*FCONST.FACTOR,COFFF,AFIO.X+H?,V+T2/2,)
T asH*R A TF(F CM ARG.SCH ARC, NUMINC. CONST, EZERO.V.PS I Zf:,FCONST,CMKS,R
*FCONST,FACTORfCOEFFfAFin,X+H,Y+T3)
? CnNTTNUE
Rt'TURN
END
288
-------
(01 FUNCTION RATE (ECNARG,SCHARG,NUMINC,CONST,EZERO,V,RSIZE,ECONST,
,)02 *eMKS,RR,FCONST,FACTOR,COF.FF,AFID,NTIME,NLJMBER)
JOJ REAL INTGRL,NF,NUMBERfNTIME
ME=«NLIMBER*ECHARG
lF(NtlMRER-SCHARG)7005f700fe,700fe
7005 CALL ARCCOSCNUMBER,SCHARG,THZFRO)
|5o7 IF(THZFRO.LE,1.E"05) GO TO 7006
m IFfl.57.THZERO) 7011.70 11,701S
|09 701$ CONTINUE
010 GO TO 7007
|H 7006 THZ£RD=0.
12 7007 DELTAXsn,57-THZFRO)/FLOAT(NUMINC)
13 THF.TAsTHZERn»DFLTAX
\H SlIMOpnsO.
115 HO 7000 J=1,NUMINC»?
116 THETAaTHFTA + DEUTAX*?.
117 CTHETAsCOS(THETA)
TCONSTsCONSTACTHETA
|JJ9 FCn8sEZERO*CTHETA
J22 CALL ZFPOCCS f
123
12 o 7001
US THrTA=THFTAfDELTAX*2.
136 CTHETAsrnStTHETA).
137 TCQNSTsr.OMST*CTHFTA
ECnssEZERO*CTHFTA
IJ9 Cl=-K'F/CCMKS*ECOS)
wo rnsTCnNsT/f2.*Ecns)
Wl CALL ZEPO(C1»CO,RZERO)
APT, is»t
ARSf A«G! ) .GT.30.05 GO in 70?7
GO TO
YVAI.sO.
"8 7028 CONTINUE
IF(.I.EO.WUMINC) GO TO 7Q01
'50 SUMf VN = SUMFVM + YVAL
7001 COMTTNIJF.
IFCTHZFRn.EQ.O.) GH TO 70S1
7050 RZFWOsRSlZF
60 TO 7r»S?
7051 CONTINUE
CT7Ff?OsCOS(THZERO)
'57 TCOM8TsCONST*CTZFPn
ECOS«E7ERO*CTZERn
O = TCOMST/(2,*ECOS)
289
-------
061
062
065
064
065
066
067
068
069
070
071
07?
073
074
075
076
077
078
079
080
081
08?
083
084
085
086
087
088
CALL ZEPO(Cl,Cn,RZERO)
7052 CONTINUE
ARG2»«( NUMBER* V*(RZERO-R3 I ZE)/RZERO+(ECONST»RR*RZe*Q+FCONST/RZER
1*25*CTZERO)
IFC ABSf ARG2).GT,30.0) GO TO 70?9
ZVALsEXP(ARG2)*SlNfTH7ERtn
en TO 7030
7029 ZVAL.SO.
7030 CONTINUE
= INTGRL*FACTOR*AFlf»
GO TO 7012
7011 RATFJzO.
7012 CONTINUE
IF(APSfARG3).GT.30.0) GO TO 7011
RATF2«FACTQR*EXPfAR63)*AFID
GO TO 70J?
7031 RATE?=0.
7032 CONTINUE
7008 RATE3
GO TO 7010
7009 RATE 3=0.
7010 CONTINUE
RETURN
EHP
. -NUMBER XSCH ARC 1 **2*AF!D
290
-------
001 SUBROUTINE ARCCOS (A,B,ACnS)
002 RATlOsA/8
003 T»l.
004 SlIMsO,
005 TERMzRATIQ
006 1 U»2.*T-1.
007 V = 2.*T
008 WB2.*T+1.
009 TERM»TERM/V*U**2/W*RATIO**2
010 SUMaSUH+TERM
Oil TrT + 1.
012 IF(TFRM-S,F«-05)3f 3, 1
01? 3 ACOSBi.5707963.8UM»RATIO
OH RETURN
015 END
291
-------
001 SUBROUTINE ZERO CC1,CO,RZERO)
002 BsSQRTf (?7,*-CO*C(i)/CCl*Ci*Ct))
003 CALL ARCCOSfB,l.,C)
OQ« D»-2.*SQRT(Cl/3.)
OOS
006 PETURN
007 END
292
-------
31 SUBROUTINE CHGSUM
82 PEAL UNC,UTNICK,JPART,JION
03 COMMON/BLK5/ZMMDl,SIGMlfNQNlD,NRAPDfTDK,NUM8EC»NEFF,NTEMP,GFIT
OU COMMON/BUK6/VQL(2Q)»XNO(20),Q(20)» WS(20),ITUUO), nW(«5),AS(10),
05 1 VOSf lfl),TC8CtO)lWL8(lO),AC8(10),BSClO),8V8(10),V6SnO)f VGASS(IO),
06 2TEMPS(10),VIS3(10),QSAT(20),UfE,EPSO,PI,ERAVG,8C,TEMP,EPS,VAVC,
07 SOLDO (20 ), OLDXNO (20), RFS( 10), START iuo), STAR T2( 10), STARTS* jo),
08 QT)/E.
'fcO RFTIJRN
293
-------
001 SUBROUTINE PRTINC
002 REAL LINC,LTHXCK,JPARTiJION
003 COMMON/BLK3/VG,ATOTAL,DD,ETAO,DL«PL»RHQ
004 COMMON/BLK5/ZMMDI,SIGMl,NOMlD,NRAPDlTDKfNUMSEC»NEFFf
005 COMMON/RLK6/VOL(20),XNO(20),G(20),WSC20)tITL(40),nWC«5),ASnO),
006 lVOSn05,TCSnO),WLS( 10),ACS(10)fBS(10), SYSnO),VGSUO).VGASS<10)
007 2TEMPSC10),VI$S(iO),QSAT(2Q),U,F,EPSOfPI,ERAVG,BC,TEMP,EPS,VAVC,
008 30LDQC20),QLDXNOC20),RFSfl01,START 1(10),START2(10),START3UO),
009 4VSTAR(tQ)
OiO COMMOM/RLKltt/TMFP,NVI
Oil eOMHON/RLM5/NPR!NT,NSECT,SLMGTH,A.VO,TC,R,AC,WL,CL,CD,ET,SY,
01? 1VGA3,P,VIS,*,LINC,XPI,RIOVP,FPLT,AFID,*CD,ZMD,
013 2WT,LTHlCKfJPART,JION.I.POVRI
014 rnMMON/Rl.K 1 7/NRFAD.NPRNT
015 IF (NPRTNT.NE.n GO TO 8
-------
61 WRITEfNPRNT,4323) R.IQVR,ERAVG,Ef»LT, AFID,XCD,
62 UPART.J10N.I
63 4325 FORMAT(T2.F6.«.lX,lPEU,3,lX.m.«,lXfEH.
-------
001 SUBROUTINE PRTCHG
002 REAL NWS
003 INTEGER VISKIP,VTSAM£
004 DIMENSION YY(20)
005 COMMON/BLK1/DIAM(20),ONO(20),DXSC20),XMV(20),PCNT(20),RAOC20),
006 1CCE(20),PRCU(2!)
007 COMMON/RLK4/NS
008 COMMriN/BLK6/VOL(20),XNCIC20)fQ(20),WSf20),ITL(4Q),r>Wf45),ASnO)t
009 lVOSnO),TCSnO),WLS(lO),ACSClQ) , RS (1 0) , SYS ( 1 0 ) , VGS (1 0 ) , VGASS ( 10)
010 2TF_MPS(]0),VISS(10) , OS AT f 20 ) , U, E . EPSQ, P I , ERA VG , BC t TEMP , EPS, VA VC,
OH Snunaf 20) .OUDXNOIPO) f RFStlO) ,ST4RT1 f 10) ,START2(lO)fSTART3f10),
012 aVSIAR(lO)
01? CnMMON/BLK7/XDC(fl5f 20)
015 COMMOM/RL.K1 a/TMFP,NVI
016
017
018 CnMMnK,/pLK19/LK,nVf NN,NUMINcf NX,NYf NOATA,NFSTfNDI8Tf MITF",IF1NAL
019 lJIl,JT?.VlSKIPfVTSAMFfUS
020 r
021 C OUTPUT FROM CHARGING ROUTINE
022 C
023 WRITE(NPRNT
024 9992 FORMATflHl)
025 WRTTFfNPRNT
02fe 3S6 FnPMATf/T3, 'CHARGING RATES FOR PARTICLE SIZES FROM SUBROUTINE C
0?7 1GN OR CHGSIJMV)
02B IF(fMCALC.EN.l) ,OP, (NFST.F0.2)! GO TO 1880
0?9 WRTTF. fMPRNT, Ift79)
030 1879 FORMATf /T3, 'SRT THFORY USED FOR PARTICLE CHARGING")
031 Gn Tf) 1881
03? 1880 CfjMTlNUE
033 WRTTF(MPRNT, 1882)
034 1882 FORHATf/TS, 'SUM OF CLASSICAL ETFLD AND DTFFUSTOWAL CHARGES USED
035 i" PARTICLE CHARGING*)
036 iHsi CONTINUE
037 wRiTE(NPRNT,2500)'
038 2500 FOPHAT(//T2, 'INCREMENT NO. *,T20. 'Q/QSATF FOR INDICATED PARTICLE
039 1/F.S')
0«0 JS=1
0^1 KS=8
042 6544 C 0 M T I N 1 1 p
043 IF(KS*NS) 6541,^542,654?
044 ^sa2 CONTINUE
045 KSsNS
046 6541 CONTINIIF
047 ^'RITEfNPRNT,357) (01 AM ( J) , JBJS. KS)
048 357 FORMATf X/T4, 10(Fl 1 .4,?X)//)
049 nn 360 I=lfNF
050 • 00 359 JsJS,«S
051 lF(fJVT.fQ,l ) GO TO 469?
052 N = NI/2
053 OSATMaC«.*PI*tPSn*(RAn(J)+TMFP)**2)*EAV6tN)*(l,+2.*(tePS-l.)/
054 l(EPS-t.?.)5*fRAn(J1/(RAntJ) + TMFPn**
055 YY(.I)eXnC(If J)/QSATM
056 GO TO 1S9
057 ^692 CONTINUE
058 YY(J)sxnC(I,J)/QSAT(J)
059 359 CONTINUE
060 WPITF(MPRMT,3b8) I , ( YY ( J) , JsJS.KS)
296
-------
H 3S8 FORMATfT3»I2»T6,10(F7.4,6Xn
ft 360 CONTINUE
&J IF(KS.EQ.NS) GO TO 6543
(,4 JSaJS+8
b5 KSsKS+8
06 GO TO
&7 65a3 CONTINUE
68 WRITE(NPRNT,
70 «3? FORMAT(/T5, 'CHARGE ACCUMULATFO ON PARTICLE SIZES IN EACH INCREMENT
71 l'//T3. 'INCREMENT', T20,'CHARGF FOR INDICATED PARTICLE SIZES')
12 JSBl
7? KSse
Tfl 6S65 CONTINUE
75 IFfKS.NS)
76 6^67 CONTINUE
77 KSsNR
7P WRITE (NPWKIT, a?S) (DI AM ( j) , JsJS.KS)
80 «25 FnRMATC//TB,10fEll.«,3X)//)
i81 DO U31 IslfNF
(82 WRTTEfNPRNT,«30) I, (XnC(I.J), Jsj3,KS)
I83 430 FnRMAT(T3,I2,Tfe, lOfFll.S, 1^5)
I8A «31 CONTINIJF
185 IF(KS.FO.NS) GO TO
|«7 KSsKS + A
)8B GO TO
)89 656* COW T I Ml IE
IPO RF.TURN
)91 END
297
-------
001 SUPROUTINE ADJUST
002 C
003 C * *
004 c * RAPPING REENTRAINMENT PROCEDURE IN *
005 c * *
006 C * THIS SUBROUTINE WAS DEVELOPED UNDER *
007 C * *
oos c * THE SPONSORSHIP OF E.P.R.I. BY SO.R.I, *
009 C * *
010 c *********************************************
OH DOUBLE PRECISION EFESR,DlOG,EFFWR
012 RFAL LINCS
013 DIMENSION RPCNT(20)fDMOLO(20),WUNCQR(2ft),RnMDLD(20)fCDMPLD(20),
014 iPCTOTf2o),CPCTOT(20),taSL(20),PXS(20),PRCUNR(21)fRPRCIH21),
016 COMMON/BLM/DIAMC20),ONO(20),DXS(20),XMV(20),PCNT(201,RAD(20),
017 1CCF(20),PRCU(2i)
018 COMMON/HI K2/LSECTC10),UNCS(101,PS(10)
019 COMMON/BLK3/VC,.ATOTAL,OD,ETAO,DI, ,Pt,RHO
020 COMMON/BLK4/NS
021 CnMMON/RLK5/ZMMDT,SlGMl,NnNID,NRAPDfTDK,NUMSEC,NEFF,NTEMP,GFIT
022 COMMON/PLK6/VOL(20)lXNOf20J,Q(?0),«S(20),ITL(40),DW(4$),ASnQ),
023 lVOSnoi,TCS(101,WLS(10),ACS(J01,BS(10),SVS(10),Vf;s(10).VGASS(lO)
024 2 TEMP 8 ( 1 0) , VI 88 (10) ,QSAT (20) f U. Ff EPSC1,PI, t'RAVG, BCf TEMP,EPS, VAVCt
025 30LDQ(20).OLDXMO(?0),RFS(10),8TART1(10),START2(10),START3(10),
026 4V$TAP(101
027 COMMON/BLKll/ENpPT(21)«NENOPT
02« COMHON/BLK1?/ARD50(10} ,ARvSlGM(10) , ASNUCK(IS) , AZNUMS ( 1 5) , AZIGGY (1«
029 CnMMON/RLK17/NREAD,NPRNT
030 CPMMON/BLKlfl/SCOPEF,CZMT>l , CSIGMO,NRUN, SNUCK, 2IGGY , RMMO, R9IGMA
031 COMMON/Rl,.Kt9/l,,k,nV,NN,NUMINC, NX,NY,NI3ATA,NEST,NDTST,NITER,IFINAL,
032 JJIl,JT2fVISKIP,VlSAME,U8fFPATHtF.BD,NOSET.NWS(tft)tD50fSTGMAP
033 NfH.IN = 0
034 N S1 a N s + 1
035 NUMSlaNUMSFC"!
036 CONVFs3.67E*-03*(TnK/PS(NUMSFC) 1
037 NRAPDCsO
03fl X=0.0
039 DO 1555 1=1,NS
040 FFFSP=DxS(I)/ONOfT)
041 IF(EFESR.GT.0.999999) EFESRsO,909999
043 1555 CONTINUE
04« 1713 CONTINUF
045 NPAPDCsf-JRAPDC + 1
04ft IF(NRAPI.)C.EU.n GO TO 607P
047 GO TO 6080
oaA 6078 CONTINUE
049 ARD50C1 1=6.0
050 ARSIGMfj )S2,5
051 RMMDsfe.O
053 Gn To fe0
054 6080 CONTINUE
055
056
057 6079 CON I TNI IF
05fl CALL LNrMST(RMMD.RSTGMA,RPRCU,RPCNT)
059 00 7575 Tsl,NS
060 PPCNTf I )sRPCNT (I )*1 ,E-f 0?
298
-------
Ofcl 7575 CONTINUE
062 NONCK = 0
563 1867 CONTINUE
j6« NONCKsNONCK+1
,65 SNUCKsASNl'CKCNONCK)
066 ZIGGYsAZIGGY(NONCK)
,67 ZNLIMScAZNUMSf NONCK)
068 WRITEfNPRNT,l8)
569 18 FORMATUH1,' PARTICLE SIZE RANGE STATISTICS'/)
570 *RITFfNPRNT,1868) NONC*
571 1868 FORMATf/' CORRECTIONS FOR NONIOEALTT IES USING SET NO. ',!?,' OF CO
572 IRRECTJON PARAMETERS'/)
J73 C
574 C PRINT DIAM., PERCENT, AND EFFICIENCY FOR EACH STZE RAWGF
575 C
576 WRITF. fNPRNT, 19)
577 19 FOKMATC4X,'SIZE'.SX,'CCF',?X,'TNLET %',1X,'OUTLET X', IX, 'COR. OUTL
578 JET *', IX, 'MO-RAP EFF ,',1X,'NO-RAP H ' , ?X , 'NO-RAP P',?*,'COR. EFF,',
)79 23X,»CnP. w*,5X,'COR, P')
JBO C
Jfll C
}82 Y = 0.0
563 nn ?Q9n T = I,NS
08U FFeSRsDXS(I)/ONOf I)
085 IF fFFFSR .GT..9Q9999 ) EFESP =
086 XEPsEFF.8RMOO.no
087 IF fXEP ,GE, 99,9999 ) XEP * 9<>
MOBS IF (FFESR.GE,Oi99Q9Q)WY = XMV(I)MOO,
089 IF (FF ESR.l T . 0, 99900 ) WY =(VG/ATO TA L)M^OO.*A LOG (100./( 100..XFP))
I 090 lFf7TGGY-0,0)4704,470 a,4705
1 4704 Fi = l .
|f)92 GO TO 4706
094 M = t. + .766* f.FESK*nGGY**l,7B6+.075S*?IGGY*DL Ofi (1,nO/ft.DO-EFESR))
095 47"6 CONTINUE
596 IFfSNUCK-0.0)4701,4701,<»702
7 4701 F2=l.
8 HO TO 4703
|099 4702 F? = DLOR (1 ,*EFtS&) / (ZNU*S*DLOGf SMUCK+f 1 ,«SNUCK)*(1 . 0-FFFSR ) ** (1 . /
100 JZMIIMS)))
101 a703 CONTTMUE
103 WYVsWY/Fl
1 ZMLFF = Ei*F2
105 WYSVsWY/ZNLFF
|0fe HIIWCORf I )=WY
107 EUNCORf I)=EFFSR*1 00,
108 CALL l^ADJST(DIAM,l,WYSV,nNO,PXS,ATnTAl ,VG,EFESR)
109 TF fFFESR ,GT,.999999 ) FFFSR = ,999999
110 XFP = EFFSR*100,DO
111 IF (XFP ,GE, 99,9999 ) XfP e 99,9999
? IF fEFESR.GE. 0,99999 )WYsyMVf I )MOO,
1*3 IFfFFFSR,LT,0,99999)l«Y=fVR/ATOTAL)MOO.*ALOGf 100,/f 1 00.. XEP))
US Y s Y * EFESR * PCNTfl)
b ?990 CONTINUE
I17 IDC = 0
118 SPnaO.
119 SCPOsO,
120 IX = 0
299
-------
121
122
123
124
125
126
127
128
129
130
151
132
133
134
135
136
137
138
139
\U5
150
151
152
153
15«
155
156
157
158
159
160
161
16?
163
lha
165
166
167
168
169
170
171
17?
173
174
175
I7fe
177
178
179
180
1341 CONTINUE
SCOREF a 0.0
IDCsIDC+1
DO 3540 1=1, NS
IX=IX+1
EFESRsPXSa)/ONO(I)
IF (FFESR .GT,. 999999 ) EFESP = .999999
XEPsFFFSRMQO.no
IF (VEP ,GE, 99.9999 ) XFP = 99.9999
IF f FFE S P.GE, 0,99999) WYSXMV (DM 00,
IF f FFFSP-.LT, 0,99999) WY«(VG/ AT OTAL)*t 00. *ALQG(1 00. /( 1 OO..XEP) )
PFMTPslOO.-XF.P
PCTOT(T)=PENTR*PCMT(I)M.F-02
IFfIX.fiT.1) GO TO 7130
CUPTLScO.
DO 1 ISslfNUMSl
CLPTt SsClPTLS + FLOAT(LSF.CT(JS))*LJNCS (IS 5*0.305
N Y X a 0
c n N T i N 1 1 F.
NYXsNYX+1
IF(NYX.EQ«?) GO TH
EQ.?) XEFFsX
EXPfiNTsALO'5(l,/(1 ,-XEFF
XMr.LSsXMFLS*(l .-EXP(*(EXPONT*Fl fUT (LSETTf MIIMSFC) )*LINCS (NUM8EO*
3053/PD1
XMLLSsXMELS-XMCLS
XMCLS=XMCLS*CONVF
TF (NTFMP.EQ, !) » API OS = 0 . 1 55*XMCLS**0 , 905
GO TO i«32
CONTINUE
FXPn»JTrAl.nG(l./(1 ,
YHCL,SsYMFLS*(l . -FXP f • ( FXPONT* FLOAT ClSEC T f NUMSEC) 1 *L .1 NCS f NUMSEC ) *
305) /PI ) )
YMUSsYMFUS-YMCUS
TH
1U3?
lFfMVV.EO.1)
7130 CONTINUE
R
1 )
^'
EFF^RsfnNnn)*n.-EXPC»tATOTAL*l*(YSV)/(100,*V(5)) ) »PN$ ) /ONO (I >
TPWP s OWOf J)*fl .-FXPr»CATOTAl *WYSV)/(100.*VG) ) )•« RMS
IFfCPHPBLE.O.O) FKFWR = EF ESP
TF fFTFWR.GT, .999999) FFF WW= , 999999
COREFFsEFFWRMOO.DO
IF(CORFFF,6E.99,9999) CORFF F s99 , 9999
lF(EFFUiR.Gt(Of 99999) WYRsWYSV
IF fEFF^P.LT .0,99999) HYPS ( Vc/ATnTAL ) M 00 . *AUOG U On . / ( 1 Oft ,«COREFF
SCORFF s SCOREF ^ COREFF *Pf. WT ( I )
CPFNITRslOO.-COREFF
CPCTrjTfI)aCPF.NTR*PCNT(!)*l.F.O?
TFtinC.NE.l) GO TO
DTf T )
300
-------
1«1 SCPOBSCPn+CPCTOTfl)
182 13«3 CONTINUE
183 SL=d.O-EFE$R)*ONOd)
(84 WSld)«SL* (1.33333*3.14159*RADd)**3)*DQ
185 IFdDC.EO.n GO TO 1344
186 PCTOTd)B(PCTOT(I)/3PO)*lOO.
187 CPCTOTd)s(CPCTOT(I)/SCPO)*lOO.
|88 DUDsALnG10CENDPTd + l) ) "ALOG1 ft (ENDPT (15 )
189 DMOUDd)B(PCTOTd)*YMLLS*CaNVF*l.E»02)/n|.D
190 RnMDLnd)*(RPCNTd)*RAPLOS*l.F-02)/DLD
(91 COMDLDd)=r>MDLDd)+RDMDLDd5
|92 WRITFrNPRNT,2291) DI *M tl) , CCF (15. X Y , PC TOT d ) , CPCTOT d ), XEP, WY,
193 }PENTR,COREFF,WYP
194 ??Gf 1 00 , / (1 00 ,-SCORtF ) 5
230 WZs(Vr,/ATfirAL)*100,*ALOG(l OO./flOO.-Y))
231 WRITF(NpRNT,299fll WZ
232 ?908 FORMATfSX,'PRECIPITATION RATF PARAMETER UNDER NQ-RAP CONDITIONS ='
^33 1,F7.3//1
234 PRCUC(1)=0,
335 suMcsppruccn
236 DO 1751 lsj,NS
237 SUMCaSUMC + CPCTOTd)
238 PRCLICd + 1 )=SUMC
> 1751 CONTINUE
I CAM, LNFIT(PRCUC.CZMDU.CSI6Mn,C6FIT)
301
-------
241 WRITE(NPRNTtil615) ZIG6Y,SNUCK,ZNUMS
?42 4615 FORMAT(SX,'SIGMAGa',2X,F7.3,?X,'WITH',F7,3,' SNEAKAGE
243 1F7. 3. 2X, 'STAGES')
244 WRITFfNPRNT, 7900) NTE.MP
245 7900 FORMAT(5X, 'NTEMP =',I2)
246 WRITE(NPRNT|79G1) RMMD
247 7901 FORMAT(5X,'RMMD =',F6.2)
24R WQITE(NPP.NT,7902) P5IQMA
250 WR1TF(NP«MT,5002) SCOREF
251 5002 FORMAT(«5^, 'COPR. fFF, = SFP.il)
252 HRITE(NPRNT,B352) CZMDL
253 P3S2 FORMATf?^, 'CORRECTED MMp HF EFFLUENT =',JPFU,3)
254 ^RJTF (NPRNT,5800) C8IGMO
255 ^800 FnRMATf5X, 'CORRECTED SIGMAP CF EFFLUENT s»flPE11.3)
25fe WRTTf f ^IPRMT,9^50)
257 WRITF(NPRNT,5003)
258 5003 FORMATC^X, 'CORRECTED PRFC IPIT AT|ON RATE PARAMETER =',F8.?)
259 WRITF(NPRNTl6'5«i5l
260 ^5-S5 FORMATflHl,» UNADJUSTED MIGRATION VELQCITTFS AND FFFTC TE^r IPS, AN
261 I DISCRFTF OUTLET MASS LOADINGS'//)
26? WRITE ;(NPRNTf 19R01
263 1980 FORMATdX, 16HIOEAL UNADJUSTED , ?X , 1 6HinFAI, UN AD JUSTED , 7X , 6HNO-R AP.
264 iOXf 1?HPAPPTMG PUFF,6Xf 15HMO.PAP+RAP PUFF , 5X , 1 ?HR ApPING PUFF,4Xf8H
265 2ARTKLF)
?66 WRITECWPPNT, 19fll)
267 19«1 FUPMATf 5X, 17HMIG. VE'L . CCM/SEC) . «X , 13HEFFICIENCY ( X) . «X, I 7HOM/DLOGO
268 lMG/DSC.M)»2"»17HDM/DLOGt)tMG/DSCM)f?X,17HDM/nLOGO(Mt;/nSCM)f2X, 15HDI
269
270 DU I9fl? M=1. ,NS
271 WPITEfNPRNT,J983)
272
273 1«JR3
274 19fl2
?75 NPIJN s NRUN 4 1
?76 CAl L PPTSUM
?77 IF fNDMCK .UT.NONID) GO TO 1R67
278 IFfNRAPOC.LT.NRAPD) GO TO ?713
?79 RETURN
280 END
302
-------
001 SUBROUTINE QTFE (DX,Y,Z,NINO
002 DIMENSION VCl)fZ(l)
003 8UM2SO.
004 IFCNINC • 1) «,3,1
005 1 DDX».5*DX
006 C
007 C INTEGRATION LOOP
008 002 IS?,NINC
OOP SUM1»SHM2
010 SUM2»5UM2+DDX*(Yfl)tY(!•!))
Oil 2 ZCI-l)asUMl
012 3 Z(NINC)aSUH2
013 4 RETURN!
OH END
305
-------
001 SUBROUTINE UNFIT (PRCU,D50,SIGMAP,GFIT)
002 C THIS SUBROUTINE FITS CUMPFReENT CURVE TO A LOGNQRMAt DISTRIBUTIOI
003 DIMENSION z(2n,Y(2i),PRCu(2n
OOa COMMON/BLKlt/ENDPTf21)»NeNDPT
005 NSTAGsO
006 JsO
007 DO i TSI,NENDPT
OOP IFfPRCU(I).LE.O,fi)eO TO 1
009 J=J+1
010 Z(J)sALOG(ENDPT(in
Oil IFCPRCU(l) .GF-.99.Q) GO TO ' fl
01? IFCPPCl'CD.GT.50.J60 TO 3
OH 2 XYY = SQRTf AL06(1.0/XY**2n
015 Y(J)=XYY-C(2t515Sl7t0.602e53*
017 IFfPRCU(I),GT.50.)GO TO 5
018 Y(J5=-Y(J)
019 GO TO 5
020 3 XYsl.O»(PRCU(I)/100,)
021 GO TO 2
022 5 NSTAGsNSTAG+1
023 1 CONTINUE
Q?.U C CALL CURVE FIT ROUTINE
025 a CALL CFIT(A,R,GFIT,NSTAG.Z,Y)
026 C CALCULATE 050 ANO SIGMAP
027 OSOsFXP(-A/B)
02B SIGMAPsFXPfl.O/H)
029 RETURN
030 HMD
306
-------
001 SUBROUTINE CFIT (A,B,R,NSTAG,Z.Y)
002 C THIS SUBROUTINE FITS A STRAIGHT LINE, YsA+BX, USING LEAST SQUARES
003 DIMENSION Z(21),YC2t)
004 XNaO.O
J05 SUMXaO.Q
006 SUMYBO.O
007 SUMXYeO.O
008 SUMXXsO.O
t) SUMYYaO.O
010 DO 6 Isl,NSTAG
Oil SUMXsSUMX + Zm
Oil 8UMYBSUMY + Y(I)
013
lOlfl
015
016 XMSXN41.0
517 fe CONTTNUe
018 C C41 CULATE A,8
O AB(SUMXX*SUMY-SUMX*SUMXY)/fXN*SUMXX»SUMX**?)
021 R = SQRT(R*( f XN*SllMXY«8UMX*SUMY) / f XN*8UMYY«SUHY**2
022 RETURN
023 PMR
307
-------
001 SUBROUTINE PRTSUM
002 PEAL LINCS
005 CQMMON/BlK2/LSECTn03,LINCSn03,PS(i03
OOa CQMMQN/BLK3/VG.ATOTAL|DD.ETAO,nLfPL»RHO
005 COMMQN/Rl.K5/ZMMDIf9IGMl,NIONID,NRAPD,TDK,NUM$EC»NEFF,NTEMP,GFn
006 COMMQN/RlK6/VOL(203,XNQ(20.3fP(203,WS(203,ITL(^Q)»DW«i5), ASdO),
007 lVOSC103,TCSC10),WLS(lO),ACSM03,BSn03,SYS(103,VGSei03,VGASSUO),
008 2TEMpS(103fVISSao3,QSATC2Q3,U,F,FP$0,PI,ERAVG,BC,TEMP»EPS,VAVC,
OOP 30LPQf20),OLDXNOC203,RFS(lQ),STARTinO),START2n03,$TART3aG3,
010 aVSTAR(lO)
Oil COMMON/HIK17/NRFAD,NPRNT
012 COMMnN/BLKl8/5eOREF,CZMnL,C8IGMQ,NRUN,SNUCK,ZIGGY.RMMO.RSIGMA
013 SCAsATDTAl/VG
OKI VOSLJMrO.
015 CDSUMaO.
oife DO 6571 ISI,NLIMSEC
017 Vn$lJMsvoSm*FLOATaSECTfT33*LINCSm*Ot305 + VnSUM
018 CnSUM=(TCSm/rAS(I3*9.3F.-fl23 3*1 . E + 05*Ft,OAT (LSECT ( 13 1 *LI NCS (I) *
019 10,30S+CDSUM
020 6571 CONTINUE
021 AVOaVOS'JM/PL
022 ACHaCDSHM/PL
023 PHOCGSsRHQ*100.
024
025
026 WRJTF(MPPNTf
027 9520 FORMAT (Q* , *******************************************************
02fi 1*************************************************************')
029 WRITE(NPRHT,JOhOl
030 HPITEfNPRNT,10603
031 1060 FORHAT(9X, •*', ll^X, '*'3
032 WRTTf(UpRNT.95003
033 9500 FOPMAT(9X,'*',39X,'SUMMARY TABIE OF ESP OPERATING*.05*,»*•)
03« WRTTF(NPRNT,95013
035 9501 FOPMATf9X,»*•,«!*,'PARAMETERS AND PERFORMANCE ' fU7X,'*' 3
036 WRJTF(NPRNT,10603
037 WRITF(NPRNT,10603
038 l»RITE (NPRNT, 1Q603
039 WRITF(NPRNT,10603
0«0 ^RITFfNPRNT,95023 NRUM
Oil 9502 FORMATf9X, **',«6X,'DATA SET MIIM8ER ' f I 3 t «9X ,'** )
0«2 WRJTEfKiPRNT,10603
0^3 URITFCNPRNT,10603
Oa« WRTTF(NPRNT,95033 8COPEF,SCA
0«S «?5oi FnRMAT(9X,»*»,12X,'ESP PERFORMANCE I ' ,5X,'EFFICIEMCV » *.F«.«»' «'
O^fe 15^'SCA a %1PF10.3,' M**2/ f M**3/SEC ) ' , 21 X , '* ' 3
047 WBITEfNPRNT, 10603
0«R WRJTFfNPRNT,1060)
049 WRITECNPRNT8950a3 AVO
050 P50^ FOPMATf«»X,«*',j2X,'ELECTRICAL CONDT TIONS I » , 5X , ' A VG . APPLIED VOtTAI
051 IF = *,iPF10,3,' V',«OX,'*'3
052 WRJTF CNPPNJT, 10603
053 WPITFfNPRNT,95053 ACD
05a 9505 FOPMAT(OX,'*»,39X, »AVG. CtJRRFNT DENSITY = *,F7.2»* NA/CM**2", 36X,'
055 1**3
05fc WRITF(NPRNT,J0603
057 URITECNPRNT,95063 RHOCGS
058 P506 F-nRMATf9X, •*',39XI'RF8ISTIVITV B ',1PF10.3,' OHM-CM', a«X ,»** 3
059 WRITE(NPRNT,10603
060 ^PITFfNPRNT, 1060)
308
-------
INLET MMO a »,1PE10.3,
(,3
64
•,68
}69
570
571
j72
173
174
175
]76
l!)77
J78
179
180
181
WRITEfNPRNT,9507) ZMMDI,SIGMI
9507 FQRMAT(9X,'*',-t2Xf "SIZE DISTRIBUTIONS! ' ,5X, *
1' UM',5X, 'INLET SIGMAP a ' , 1PE1 0.3, 33X , '*» )
WRJTE(NPRNT,1060)
WRJTE(NPRNT,9508) CZMDL,CSIQMO
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1 . REPORT NO.
EPA-600/7-78-llla
2.
3. RECIPIENT'S ACCESSION-NO.
Mathematical Model of Electrostatic
Precipitation (Revision 1): Volume I. Modeling and
Programming
5. REPORT DATE
June 1978
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO.
Jack R. McDonald
3540-6
SORI-EAS-78-101
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Southern Research Institute
2000 Ninth Avenue, South
Birmingham, Alabama 35205
10. PROGRAM ELEMENT NO.
1AB012; ROAP 21ADL-027
11. CONTRACT/GRANT NO.
68-02-2114
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Revision; 6/75-2/78
14. SPONSORING AGENCY CODE
EPA/600/13
15.SUPPLEMENTARY NOTES IERL-RTP project officer is Leslie E. Sparks, Mail Drop 61, 919/
541-2925. EPA-650/2-75-037 was the initial report.
16 ABSTRACTThe report briefly describes the fundamental mechanisms and limiting fac-
tors involved in the electrostatic precipitation process. It discusses theories and
procedures used in the computer model to describe the physical mechanisms, and
generally describes the major operations performed in the computer program. It
lists the entire computer program and defines all variables used in the program.
Major improvements to the fundamental basis of the model include: the capability
of generating theoretical voltage-cur rent characteristics for wire-plate geometries,
a new method for describing the effects of rapping reentrainment, and a new proce-
dure for predicting the effects of particles on the electrical conditions. The computer
has been made more user oriented by making the input data less cumbersome, by
making the output data more complete, by making modifications which save computer
time, and by providing for the construction of log-normal particle size distributions.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
Air Pollution
Dust
Electrostatic Precipitation
Mathematical Models
Computer Programming
Air Pollution Control
Stationary Sources
Particulates
13 B
11G
13H
12A
09B
3. DISTRIBUTION STATEMEN1
Unlimited
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES
20. SECURITY CLASS (This page)
Unclassified
330
22. PRICE
EPA Form 2220-1 (9-73)
310
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