United States
Environmental Protection
Agency
Industrial Environmental
Research Laboratory
Research Triangle Park NC 27711
Research and Development
EPA-600/S7-84-069 Aug. 1984
Project Summary
A Mathematical Model of
Electrostatic Precipitation
(Revision 3)
M.G. Faulkner and J.L DuBard
The objectives of this research pro-
gram were: to upgrade the model of
electrostatic precipitation developed
under the sponsorship of the Environ-
mental Protection Agency (EPA), to
make the computer program which
performs the calculations required by
the model more user oriented, and to
fully document computer program
subroutines that perform fundamental
calculations or utilize numerical tech-
niques.
Model improvements include: a new
method of calculating solutions to the
electric field equations, a dynamic
method of describing the effects of
rapping reentrainment, a procedure for
calculating effluent opacity, and a
routine that checks the input data.
Revision 3 of the model calculates
efficiency in about 10% of the time
required by Revision 1. The option to
use input data expressed entirely in the
metric system is provided, as are
options to terminate the calculation
after the calculation of V-l curves and to
use an internal data set to estimate pre-
cipitator efficiency based on the resis-
tivity of the dust particles.
This Project Summary was developed
by EPA's Industrial Environmental
Research Laboratory. Research Triangle
Park, NC. to announce key findings of
the research project that is fully docu-
mented in a separate report of the same
title (see Project Report ordering infor-
mation at back).
Introduction
Revision 3 of the electrostatic precipi-
tator (ESP) performance model developed
by Southern Research Institute (SRI) for
the EPA has been completed. Since the
model was released in 1975, it has been
widely used to study and troubleshoot
existing ESPs and to validate proposed
new ESPs. The two revisions to the
original model increased its usefulness
and convenience of operation. Revision 3
further increases the model's utility by
offering a significant reduction in required
computation time plus several new
features.
Collection Processes Used in
the Model
The model predicts ESP performance
by first performing the collection efficien-
cy calculation under ideal conditions and
then correcting the results for non-ideal
conditions and unmodeled effects. For
the calculation, the ESP is divided into
sections (permitting different electrical
conditions in each section) and then
further divided into a maximum of 99
incremental lengths on the order of the
ESP's wire-to-wire spacing. Efficiency is
calculated sequentially on each incre-
mental length to determine total collec-
tion efficiency. To compensate for the
different charging and collection rates of
large and small particles, the input dust
load is divided into a maximum of 20 size
bands, each handled separately in the
efficiency calculation.
The ESP model uses the Deutsch
equation to calculate the collection
efficiency for each particle size band in
each incremental length.
/? = 1 - exp(-Aw/Q) (1)
where
/7=the fractional collection efficiency,
A=the collection plate area,
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W =the migration velocity of the charged
particles, and
Q=the gas volume flow rate.
The migration velocity is calculated from:
w= QEPC (2)
where
q=the charge on the particle,
EP=the electric field at the plate,
C=the Cunningham slip correction
factor,
a=the radius of the particles, and
u=the viscosity of the gas.
Thus the calculation of collection efficien-
cy requires calculating the charge on
each size particle in each incremental
length and of the electric field at the plate.
This can be done simply by estimation
formulas or more rigorously by using a
more complete charging theory and a
detailed analysis of the field at all points
in the ESP. The more rigorous process
yields better accuracy over a wide range
of applications and is used morefrequent-
ly. However, it involves many lengthy
calculations and requires much more
computer time than the estimation
process. The changes resulting in reduced
computer time for Revision 3 affect the
electric field and particle charge calcula-
tions of the rigorous calculation.
Features of the ESP Model
Table 1 summarizes ESP model features
including the changes due to the various
revisions. These features are discussed
below in the order in which they are
presented in Table 1.
1 . The estimation procedure for calcu-
lating particle charges and the electric
field at the plate remains unchanged.
2. The rigorous calculation of the
electric field at the plate has been
changed in Revision 3. Although the
general procedure used to solve the field
equations remains the same, the conver-
gence of the iterative technique used to
achieve a solution is accelerated using
the dominant eigenvalue method. This
method was previously applied to various
chemical processes and to the electric
field calculation.
The procedure used in the rigorous
calculation of the field at the plate
consists of simultaneously solving two
field equations relating the electric
potential and the space charge density at
every point inside the ESP:
(3)
(4)
where
V=the potential,
x=the coordinate from wire to plate,
y=the coordinate along the gas flow,
p=the space charge density, and
£o=the permittivity of free space.
In the model, the calculation is performed
at every point on the grid shown in Figure
1. The initial estimate of the potential at
each grid point, V,,, is made using Cooper-
man's solution to the field equations. Fol-
lowing this, the space charge density at
each grid point, p,,, is calculated from V,,.
V,, is then recalculated from p,,. The pro-
cess of alternately calculating p,, and V,,
continues until the change in V,, between
iterations is less than a preset value at
every grid point. At this point, the calcu-
lated current density at the plate is com-
pared to the measured current density. If
the values differ by more than 1%, the
space charge at the corona is adjusted
and the calculation of />,, and V,, begins
again. If these values are within 1%, the
potential solution is considered to have
converged and the value of the electric
field at the plate is obtained.
In Revision 3, the convergence rate has
been accelerated by the dominant eigen-
value method. Using the new procedure,
two criteria are applied to the calculated
potential after each iteration: (1) at least
five iterations must have taken place
since the last acceleration step, and (2)
the change in the potential between the
last two iterations must be less than a
present value at every grid point. These
criteria ensure that each acceleration
step has sufficient time to produce a sta-
ble solution before a new acceleration
step is applied. If both criteria are met, the
potential is adjusted at every point by a
Table 1. ESP Model Revision Summary
Process
factor derived from the potential changes
due to the last two iterations. The poten-
tial adjustment has the form:
V,,ln+11 = V,,'"-1' + a(V,,lnl -
V,,'"'1')/!! - ^) (5)
where
V],n+1'=the adjusted potential for use in
the next iteration,
Vlr1'=the potential calculated in the
previous iteration,
a=a damping factor, 0
-------
rigorous field calculation is now as fast at
the approximation. Consequently, the
approximation procedure was not included
in Revision 3.
3. The speed of the particle charging
routine is also reduced. In order to
calculate the charge on each size particle
in each incremental length using the
more exact ionic charging theory, many
integrations are required. The change to
the Gaussian integration scheme in
Revision 2 greatly reduced the integra-
tion time.
The net effect of the changes to the
electric field calculation and the particle
charging routine is a reduction in
computer time of about 90% with no loss
in accuracy.
4. As previously mentioned, after the
ideal collection efficiency is calculated,
the model corrects the efficiency for
various non-ideal effects. Corrections for
sneakage, non-rapping reentrainment,
and non-uniform gas velocity are applied
by adjusting the migration velocity for each
size particle by a factor based on
theoretical and experimental studies of a
pilot scale precipitator. This method of
correction has not been changed in
Revision 3.
Also unchanged is the empirical
calculation of losses due to rapping
reentrainment. This efficiency correction
is based on the amount of mass collected
in the last section of the ESP. A fraction of
the collected mass is fit to a log-normal
size distribution and added by size bands
to the ESP penetration dust predicted by
the ideal efficiency calculation. The
collection efficiency is then recomputed
based on the combined effluent. The log-
normal size distribution used to describe
the reentrained particles has default
parameters of a 6 //m mass median
diameter with a standard deviation of
2.5. Other sized distribution data may be
used in addition to these values if
desired. However, the total mass of the
reentrained particles is fixed by a
relationship based on data derived from
a study of rapping reentrainment in six
power plants. The subroutine which
performs these operations requires a
negligible amount of computer time
relative to the ideal calculation and
occurs automatically unless the new
dynamic rapping routine is selected.
The new dynamic rapping routine uti-
lizes a different process to calculate the
effects of rapping reentrainment. This
routine keeps track of dust layer growth at
every point in the ESP as a function of
time. At a user specified time of rap, the
collected dust is removed from the rapped
/
2,
o
4 i
I1
X
D
27 22
3/32
42
. »
--If
23
33
43
-^
4 - - - X ^X'S
t y
.. . 4 . -
i
— 4- —
i i
^
.
i
\
\
\
C. m.
'
S
/
S
c
Area of
Integration
S, = one half wire-to-wire spacing
Sx = wire-to-plate spacing
axay = increment sizes for integration
V0 = applied voltage
Ex = component of electric field perpendicular to plate
£y = longitudinal component of electric field
j - average current density
Figure 1. Nomenclature used in the numerical analysis of the electrical conditions in wire-
plate precipitators.
ESP increments. A specified fraction of
the removed dust is fit to a log-normal
size distribution and stored for the calcu-
lation of reentrainment. The entire
collection efficiency calculation is then
repeated, except that the reentrained
dust particles are reintroduced into the
gas flow during the calculation. The
mechanism for this is that the number of
particles in each size band per cubic
meter of gas which are to be reentrained
due to the rap of increment i is added to
the number of particles/size band/cubic
meter already present in the gas flow
immediately before the calculations for
increment i+1 begin. Thus the efficiency
in increment i+1 is calculated on an
increased amount of dust due to the
reentrainment from the preceding incre-
ment. Inherent in this mechanism is the
assumption that the reentrained particles
instantaneously acquire the charge
found on particles of the same size at that
point in the gas flow.
The calculation of rapping reentrain-
ment effects using the dynamic rapping
routine is very lengthy because the ideal
collection efficiency must be calculated at
least twice for each rap: the first time, to
find the dust layer thickness at the time of
the rap; and the second, to calculate the
effects of the reentrainment. Depending
on the complexity with which the
reentrained dust is specified, up to seven
additional efficiency calculations may be
required. Therefore it is recommended
that the estimation procedure be used to
reduce computer time. The dynamic
rapping calculation is very flexible in that
the user may vary the times of the raps.
-------
the rapping sequence, the fraction of the
collected dust that gets reentrained, the
size distribution and the density of the
reentrained dust, and the duration of the
rapping puff. This flexibility should be of
use when comparing different rapping
schemes. At present, no experimentally
supported data set, such as that used in
the empirical rapping correction, exists
for the dynamic rapping calculation. A
study, to determine if such a data set can
or should be developed, would be
valuable. Currently, the data for dynamic
rapping must be matched to each
application. Figure 2 shows the results of
using dynamic rapping on power plant
data.
5. A plume opacity calculation has
been added to Revision 3 of the model.
This routine calculates differential and
total extinction efficiencies and the total
opacity of the ESP outlet plume calculated
by the model, using:
Opacity = 1 - exp (-EL) =
1 - exp (-NAQextL), (8)
where
E=the extinction coefficient,
L-the pathlength of the light beam,
N=the particle concentration,
A=the projected area of the particle,
and
Q=the extinction efficiency.
Figure 3 shows the extinction efficiency
as a function of the particle size param-
eter, x = nD/A, for four complex indices of
refraction. D and ^ are the particle diam-
eter and the wavelength of light beam, re-
spectively. The calculation, based on the
Mie theory, is performed at 1 0 wavelengths
weighted according to the photopic re-
sponse of the human eye. EPA-approved
opacity monitors must simulate this color
response, which is maximum at ^ =0.55
yum and has a width at half maximum of
0.1 urn. The required input data are stack
diameter and complex index of refraction.
A single wavelength-dependent index of
refraction or up to 10 values of wave-
length-independent indices of refraction
may be entered. Alternatively, the refrac-
tive index may be omitted, in which case
the calculation is performed using the de-
fault values of 1.5 and 1.5-0.1 i.
6. To facilitate the use of the model in
other countries, the data entry routine
has been modified to receive data in
metric units if desired. However, unless
all-metric data is specified, the data must
be in the mixed metric/English units
used in previous versions of the model.
7. The option of reducing the amount
of printed output data is provided. This
will be of use where only data summaries
are of interest.
w
I
10°
10-
10'
A
Q
V
0 Experimental
Q Dynamic Rapping
A Example 3. No Rap + Rap,
-------
Particle Diameter f/jm) for Wavelength 0.55 ym
0.5 1.0 1.5 2.0 2.5
go
T
T
T
3.0
T
(b) n = 1.5
fa)n=1.33(H2O)
(d) n = 1.5 - 0.1i
(c) n = 1.96-0.66i (carbon)
Scattering Component of (c)
Absorption component of (c)
J I
05 10 15
Particle Size Parameter x = rrD/A
Figure 3. Extinction efficiency as a function of particle size parameter.
20
routine checks each item of input data for
variables which are out of the allowed
range (e.g., having too many size bands)
and for operating conditions which have
allowed but unusually large or small
values, such as found in a laboratory
scale ESP. If anomalies in the data are
discovered, diagnostic messages will be
printed, stating that an error has been
made, or (in the second case) that certain
values are unusual and should be
checked.
Computer Related Data
Revision 3 requires only 12% more
memory than Revision 1. On the SRI
Digital Equipment Cooperation (DEC)
System 20 computer. Revision 3 uses
206 kilobytes of 7-bit memory. To
facilitate model usage on computers with
less memory, the code is marked so that
some of the less essential functions may
be easily removed. The minimum pro-
gram length, with all of the marked
statements removed, is 135 kilobytes,
which is about 65% of the entire program.
The data format for Revision 3 is largely
the same as for previous versions of the
model. Old data sets may be used without
changes to yield the same results in less
time. Only the additions (e.g., opacity,
dynamic rapping) require additional input
data. Program users may convert Revi-
sions 1 and 2 of the ESP model to the
late$L.y£fsion by making changes in the
routine and seven subroutines
HAN, EFLD1, EFLD2, CHARGN, RATE,
ADJUST, and PRTINP), deleting the data
block and subroutine EFLD3 (Revision 2),
and adding the seven new subroutines.
Conclusions
The EPA/SRI ESP model has proven to
be useful in the design and evaluation of
ESPs. Revision 3 of the model offers
greater utility through greater flexibility
and added features such as opacity
calculation and dynamic rapping. Revi-
sion 3 is also much faster than previous
model versions, which results in signifi-
cantly reduced computer costs that
should encourage wider use of the model.
As with Revision 1, Revision 3 is
complete in two volumes. Volume I
describes the physical processes being
modeled, the algorithms used, and the
necessary input data, and lists the
FORTRAN code. Volume II is a user's
manual that contains complete explana-
tions of the input and output data,
examples of programs which illustrate
the various options available, and a
tutorial of the effects of varying some of
the computer operating parameters, with
supporting examples.
Recommendations
Although the mathematical model of
ESP in this report represents a significant
improvement over the previous version,
additional work would improve the
fundamental basis and user-oriented
aspects of the model. Research of value
would include:
1. Theoretical and experimental studies
of the effects of particles on the electrical
conditions to better describe the effect on
the electric field distribution.
2. Theoretical and experimental studies
of electrical breakdown mechanisms in
the collected paniculate layer to acquire
the capability of theoretical prediction of
when electrical breakdown will ensue for
a given value of dust resistivity.
3. Since the model underpredicts field-
measured collection efficiencies for fine
particles without the use of empirical
correction factors, theoretical and experi-
mental studies could remove the empiri-
cism or explain the discrepancy. These
studies could include a reevaluation of
the theories now used in the model and
an examination of now-neglected effects;
e.g., particle charging near corona wires
and phenomena due to the gas flow field.
There is evidence that free electrons may
play a role in particle charging in negative
coronas for temperatures of 150°C to
350°C. A charging theory that accounts
for this effect would be of value.
4. Theoretical and experimental studies
on the mechanisms involved in separat-
ing the dust layer from the collection
plates during a rap and on the quantity
and nature of the dust reentrained into
the gas flow. This could provide an
analytical method determining the nature
of the reentrained dust used in dynamic
rapping and reduce dependence on
empirically derived data. This change
could increase the usefulness of the
model in optimizing the electrical opera-
ting conditions and the rapping schedule
and intensity of an ESP.
5. Investigations and implementation
of alternative numerical techniques to
make the computer program run signifi-
cantly faster.
-------
M. G. Faulkner and J. L. DuBard are with Southern Research Institute,
Birmingham, AL 35255.
Leslie E. Sparks is the EPA Project Officer (see below).
The complete report consists of two volumes, entitled "A Mathematical Model of
Electrostatic Precipitation (Revision 3):"
"Volume I. Modeling and Programming," (Order No. PB 84-212 679; Cost:
$ 34.00J
"Volume II. User's Manual," (Order No. PB 84-212 687; Cost $28.00)
The above reports will be available only from: (cost subject to change)
National Technical Information Service
5285 Port Royal Road
Springfield, VA22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Industrial Environmental Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
. S. GOVERNMENT PRINTING OFFICE: 1984/759 -102/10636
-------
United States Center for Environmental Research
Environmental Protection Information
Agency Cincinnati OH 45268
Official Business
Penalty for Private Use $300
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