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Electrohydrodynamic Pressure
of the Point-to-Plane Corona
Discharge
a a
E. J. Shaughnessy & G. S. Solomon
a
Department of Mechanical Engineering and Materials
Science , Duke University , Durham, NC, 27706
Published online: 08 Jun 2007.
To cite this article: E. J. Shaughnessy & G. S. Solomon (1991) Electrohydrodynamic
Pressure of the Point-to-Plane Corona Discharge, Aerosol Science and Technology, 14:2,
193-200, DOI: 10.1080/02786829108959482
To link to this article: http://dx.doi.org/10.1080/02786829108959482
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www.tandfonline.com/page/terms-and-conditionsElectrohydrodynamic Pressure of the
Point-to-Plane Corona Discharge
E. J. Shaughnessy and G. S. Solomon*
Department of Mechanical Engineering and Materials Science, Duke University,
Durham, NC 27706
-
INTRODUCTION charges. Most analyses of corona wind in
precipitators (Ramadan and Sou, 1969; Ya-
The electric or corona wind discovered by mamoto, 1979; Yamamoto et al., 1980;
Hauksbee (1719) was regarded as a curiosity
Ushimaru et al., 1982) have assumed that
for many years. In fact, the investigation by
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the discharge on the wire is uniform. Al-
Chattock (1899) of the needle-plate and though this may sometimes occur with posi-
needle-ring electrode geometries was con- tive corona, the current density distribution
cerned primarily with determining the ion on a negative corona wire is highly nonuni-
mobilities in air. His curve I, which gives form and fluctuating. A recent paper by
the pressure distribution on the plate for Yamamoto and Sparks (1986) proposes a
small currents, can be looked upon today as vertical model of the corona wind created by
the first quantitative investigation of corona a single tuft and discusses the effects of this
wind. Chattock's measurements suggest that flow on particle sneakage. To our knowl-
the corona wind is a jet of gas whose diame- edge, however, no one has solved the elec-
ter is comparable to the diameter of the trohydrodynamic equations for a tuftlike dis-
current density distribution on the plate. In charge.
the needle-ring electrode geometry, he esti- In this article we discuss the electrohy-
mated that the wind velocity was on the drodynamics of the point-to-plane discharge,
order of 1 m/s. which is geometrically somewhat simpler
In recent years, investigators have fo- than the wire-plate electrode arrangement.
cused on the effect of corona wind on the Our interest was stimulated by shadowgraph
performance of electrostatic precipitators studies by Solomon (1983) in air which em-
(Robinson, 1976; Shaughnessy et al., 1985; ployed currents and point-to-plane spacings
Yamamoto and Sparks, 1986; Atten et al., much larger than those used by Chattock
1987). Corona wind is significant because (1899). These shadowgraphs show a very
the discharge in a wire-plate precipitator stable laminar corona wind jet whose radius
with negative corona is highly nonuniform. is comparable to the radius of curvature of
The discharge occurs as physically small, the point itself, and thus much narrower
unstable sources of current that create an than the radius of the current density distri-
intense corona wind. The effect of corona bution in the far field. The jet remains lami-
wind on the particle deposition process is of nar and thin all the way to the plane, bends
great interest. when subjected to a weak transverse airflow,
Until recently little was known about the and at large spacings develops the kind of
velocity field induced by these localized dis- sinuosity one observes in a low Reynolds
number laminar jet. The laminar column of
*Current address: Research Triangle Institute, Re- smoke rising from a cigarette is strikingly
search Triangle Park, NC 27706. like the shadowgraphs, though it must be
Aerosol Science and Technology 14:193-200 (1991)
O 1991 Elsevier Science Publishing Co., Inc.E. J. Shaughnessy and G. S. Solomon
High Voltage
1 Needle
Voltmeter
P+ I
- '
Plate
4
2-D Plate Traverse
Oscilloscope Pressure
n Sensor Reference Pressure
II
Micro-
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computer Correlator
FIGURE 1. Experimental arrangement.
kept in mind that shadowgraphs show den- EQUIPMENT AND METHODS
sity variations rather than the velocity field
itself. Adachi et al. (unpublished observa- The point-to-plane electrode arrangement
tions) have obtained laser-Doppler velocime- shown in Figure 1 employs a 90-cm-long,
ter measurements at a 4-cm point-to-plane 0.5-cm diameter stainless steel needle sus-
spacing which support a much more diffuse pended up to 30 cm above a 1 m2 aluminum
model of the corona wind jet than has been plate. The plate is used with its mill finish
described here. Their electrode apparatus except for an occasional cleaning to remove
was inside a relatively small chamber which any oily film on the surface. Two smaller
by confining the jet may have caused signif- plates were employed in preliminary experi-
icant recirculation and interference. The fact ments to establish the plate size required to
that Solomon's studies were conducted in an avoid edge effects at the largest point-to-
open system free of recirculation may ex- plane spacing. The needle is tapered over
plain these differing observations. Finally, the lowest 5 cm to a point radius of curva-
we note that in the shadowgraph studies the ture of 0.57 mm. The position of the point
corona wind jet remained laminar and slen- relative to the plate is adjustable along three
der even for large point-to-plane spacings. axes, and well-rounded aluminum corona
In the experiments reported here the hy- suppressors are used as required to prevent
drodynamic variable of interest is the static discharge from any location but the needle
pressure on the plane at the point of intersec- tip. The power supply is a type 3PN171
tion of the axis of a sharp needle. This rated at 60 kV and 7 mA (Hipotronics, Inc.,
pressure is much easier to measure than the Brewster, N.Y.) with manually reversible
velocity field, yet provides a means of esti- polarity. The output voltage and current are
mating the maximum gas speed which might measured using multimeters (model 179A,
occur in the jet. The experimental results Keithley Instruments, Inc., Cleveland,
also include current-voltage (I-V) data and Ohio). The output voltage is taken across a
thus provide a very simple test case for built-in metering circuit; the current is mea-
future numerical and analytical models of sured at ground potential.
this electrohydrodynamics problem. Pressures are measured with a type 523-Pressure of Point-to-Plane Corona Discharge 195
15 electronic barocell. (Datametrics, Inc., scribed by the reduced Maxwell equations
Wilmington, Mass.) with a 1-rnrn Hg range, and the Navier-Stokes equation:
connected differentially to the atmosphere
and to a 0.5-mm diameter static pressure tap Ji=Pl pClEi; (1)
in the ground plate. The barocell provides a
proportional DC output between 0 and 1 V
for each of seven ranges. On the lowest
range 1 V corresponds to lop3 rnm Hg. A
correlator and signal averager (model SAI-
48, Honeywell, Inc., Pleasantville, N.Y .)
connected to a Radio Shack TRS-80 model
111 computer is used to obtain mean values
of fluctuating pressure signals. The correla-
tor is operated in the probability density
mode which allows a selection of the resolu-
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tion, sampling rate, and total number of and
samples. The completed probability density
function is transmitted to the computer for aui ap a2ui
p u . = - -+ p - +~cEi, (6)
analysis of the mean and mean square val- Ja~j axi ax;
ues.
A great deal of effort was expended in where Ji is the current density, 6 is the
investigating the drift and scatter in the cur- mobility, p, the space charge density, Ei the
rent, voltage, and pressure measurements. electric field, 4 the potential, E the permit-
For example, in obtaining I-V curves it was tivity, ui the gas velocity, p the gas density,
noted that there was some overshoot of the p the absolute viscosity, and p the pressure.
eventual steady-state current. This led to the The needle has a point radius of curvature
selection of a 200-s settling time after each R , and the point to plane spacing is L.
current change in either an I-V or pressure The electric equations are nondimension-
measurement. Once the steady-state current alized using scales J,, E,, A+, and R , for
was attained, the barocell output exhibited current, field, potential, and the spatial coor-
large-amplitude, low-frequency fluctuations dinates. The scale for space charge is taken
which required an excessively long averag- as J, / P E,, we take E, L = A 4 and define
ing time to obtain a stable mean. A compro- a total current I = J, R ~ The . nondimen-
mise was therefore reached in selecting the sionalized equation for the potential 4 takes
sample increment and total number of sam- the form:
ples in the probability density function. The
procedure used was to take 8192 samples 10
ms apart. Without going into great detail, it
can be stated that the procedure becomes where
less adequate as the current and pressure
increase.
The group IR / E P A+' will be called the
DIMENSIONAL ANALYSIS
nondimensional current. The effect of geom-
The presentation of the results reflects a etry is described by the ratio L / R .
dimensional analysis of the governing equa- The Navier-Stokes equation is nondimen-
tions and experimental data in the following sionalized using scales J,, L , and PE, for
manner. In steady state the problem is de- current, length, and velocity, respectively.196 E. J. Shaughnessy and G . S. Solomon
The selection of L and PE, as the length so the hydrodynamic characteristics are ex-
and velocity scales, respectively, for the flow pected to depend on the electrical parame-
problem is not unique, other possibilities ters via the groups E /pP2, IR / E PA + ~and ,
include R and (E/p)'12fiE,. AS usual the p / 3 E o L / ~ .Typical values for these nondi-
pressure is nondimensionalized by density mensional groups are l o p 4 , lo-', and lo5,
times velocity scale squared, i.e., by respectively. For values of L / R of - 50,
p ( p ~ , ) ~ There
. are two dimensionless the electric body force is balanced by iner-
groups in the resulting nondimensionalized tial forces and viscous forces are negligible.
Navier-Stokes equations. The first is an elec- This can be seen by inserting these estimates
tric Reynolds number Re = pPE, L /p, into the nondimensional Navier-Stokes equa-
which appears in front of the viscous term as tions.
'.
(Re) - The second group, which appears A different nondimensionalization of the
in front of the electric body force term, is fluid dynamics problem results from using
the velocity scale suggested by Atten et al.
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(1987). They sug est that fluid velocity
k
scales with ( ~ / p ) l2 ~ using
0 the argument
Using Eq. (8) and the definition of v, the that electrostatic energy 112E E ~ is con-
verted to kinetic energy 1/2pU2. This
electrohydrodynamic coupling constant
produces an electric Reynolds number
(Flippen, 1982), we may write this as
( p c)'12~,L / p and a second dimensionless
group in front of the electric body force
term which is simply times the w
given in Eq. (9). This appears to be an
where incorrect scaling for corona wind effects in
gases not only because of a low conversion
efficiency (Robinson, 1961) but because it
leaves the electric body force unbalanced in
Flippen (1982) has shown that, when v, the scaled Navier-Stokes equations. In liq-
6 1, the Maxwell equations are uncoupled uids the situation is quite different since v,
from the Navier-Stokes equations. This is one or two orders of magnitude larger.
means that the electrical problem may be A classic nondimensional analysis of the
solved independently of the fluid dynamics current-voltage characteristic of the point-
problem with the presence of the gas being to-plane discharge may be performed by
felt only through a constant mobility 0. A assuming that the current I depends on R ,
different interpretation from Robinson (1961) E, P, L, A@, p, and p. The Buckingham a
is also instructive, as he points out that the theorem then asserts that the problem is
efficiency of conversion of electrical energy described by four independent dimensionless
to fluid motion is proportional to v,. groups. These are L / R , v,, A, and a new
In these experiments in air the values of group which can be written as X'I2v;~e.
v, are -0.013 and 0.019 for positive and Thus the nondimensional current X can be
negative polarity, respectively, Conse- written as
quently, we expect to observe that the cur-
rent-voltage characteristics depend on ge-
ometry through the group L / R but not on
the groups ~/p/3' and ppE,L/p, which where f , is a function of the three argu-
contain the density and viscosity of the gas. ments to be determined by experiment. The
Conversely, the Navier-Stokes equations are presence of V, and Re in Eq. (12) reflects
strongly coupled to the Maxwell equations the inclusion of gas density p and viscosityPressure of Point-to-Plane Corona Discharge 197
p in the analysis. Our hypothesis that the
electrical problem is uncoupled implies that
the experimental data will depend only on
L / R . Thus, if the data collapse satisfacto-
rily to the form
the hypothesis is confirmed.
Using the same analysis on the centerline
pressure on the plane p yields a functional
Voltage (kV)
relationship
FIGURE 2. Current-voltage characteristic, L =
2.0 cm; (e) negative needle polarity; (W) positive
needle polarity.
where again the actual group in which the
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viscosity occurs is X'12v;~e,which is a
reduced electric Reynolds number. Since X RESULTS
itself is only a function of L / R , v,, and Current-voltage curves were obtained for
X'12v$~eaccording to Eq. (12) we may each polarity for nine point-to-plane spac-
rewrite Eq. (14) as ings over a 2-30-cm range. Figures 2, 3,
and 4 illustrate the curves for point-to-plane
spacings of 2.0, 7.5, and 30 cm, respec-
tively. Equation (13) predicts that the data
Equation 15 shows that the nondimen-
on each I-V curve will collapse to a single
sional plate pressure is a function of v,,
data point on a plot of the nondimensional
L / R , and the reduced electric Reynolds
current IR / A qi2 versus L / R . The best
number. Strictly speaking, to define the
results are obtained by selecting A4 = ( V -
functional relationship between pPR / I and
V,), where V, is the corona starting voltage
L / R , both v, and the reduced electric
for a particular experiment, and V the nee-
Reynolds number must be held fixed while
dle voltage. Other choices for the A4 scale
L / R is varied. Under ambient conditions an are possible; an alternate selection is =
experiment at a given polarity fixes only v ,
i ~ e with the current I.
because ~ ' ' ~ v varies
In our experiments pPR / I is plotted versus
FIGURE 3. Current-voltage characteristic, L =
L / R for each polarity but no attempt is 7.5 cm; (a) negative needle polarity; ( W ) positive
made to hold the value of X'12v;~efixed. needle polarity.
This approach tests the dependence of
nondimensional pressure on the reduced
electric Reynolds number. Using previous
estimates for v i , X and Re, X'12v; ~e is of
the order of 10. If the nondimensional pres-
sure is insensitive to a reduced electric
Reynolds number of this magnitude this
should be evident in the pressure plots. In
that case Eq. (15) reduces to
Voltage (kV)E. J. Shaughnessy and G. S. Solomon
Voltage (kV)
FIGURE 4. Current-voltage characteristic, L =
30 cm; (e) negative needle polarity; (B) positive
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needle polarity.
V( V - V,) . Nondimensionalizing all the I- V
curves in this way yielded the result shown
in Figure 5. Each point in Figure 5 repre- Current (microamps)
sents a complete I-V curve of the type FIGURE 6. Pressure on the plane at the needle
shown in Figures 2-4 for a specific polarity. axis, L = 2.0 cm; positive polarity.
The value in Figure 5 is the numerical aver-
( - vJ2 over all the data
age of IR / ~ / 3V
points in the original I-V curve. The col- the I-V characteristics of the point to plane
discharge may be found in Teal Ferreira et
lapse of data of both polarities to a single
curve confirms the assumption that electric al. (1986).
field is uncoupled from the flow field under Pressure-current curves were obtained for
these conditions. For values of L /R larger each polarity for six point-to-plane spacings
than l3O9 it can be that the data of FIGURE 7. Pressure on the plane at the needle
Figure 5 fit a function f 2 ( L / R ) which is axis, L = 10 cm; positive polarity.
hyperbolic, while at smaller L / R values 0.035
the data fit a function which is parabolic
(Solomon, 1983). An excellent discussion of 0 . 0 3 0 r
FIGURE 5. The influence to point-to-plane spac- 0.025 -
ing on nondimensional current. (e) Negative needle gE
polarity; (B) positive needle polarity. 5 0.020- -
16 I I I I I E
VI
1 4
0 12-
% 0.015-
* m * m -
e a.
t
-
x 10-
N, 0.010- .' -
f 8-
-
2. *::
-E
% 6-
I
-
0.005 - -
4-
: -
:1
2- ' 8 -
' 8 1 I I
'0 IbO 200 3A0 400 ;;5 600 OO
5 10 15 20
LIR Current (microamps)Pressure of Point-to-Plane Corona Discharge
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LIR
FIGURE 8. The influence of point-to-plane spac- FIGURE 9. Dependence of pressure on geometry
ing on nondimensional pressure. (a) Negative nee- and electrohydrodynamic coupling constant.
dle polarity;).( positive needle polarity.
Taking the values of v, as 0.0134 and
0.0192 for negative and positive polarity,
over a 2-10-cm range, but for a smaller respectively, an attempt was made to corre-
range of current than the I - V curves. As late the data in Figure 8 using a power law
noted earlier, it is difficult to average the of the form
fluctuating wall pressure adequately as the
point-to-plane spacing increases. This is il-
lustrated by Figures 6 and 7, which are for
spacings of 2 and 10 cm, respectively, and Figure 9 shows the result of fitting the
positive polarity. Linear regressions per- data with the correlation
formed on this data (Solomon, 1983) show
that at a given L / R , the pressure is linearly
proportional to the current. It should be
noted that the sparkover currents are roughly where A = - 1.82 x l o p 7 and B = 3.69 x
40 pA for both spacings, hence Figure 7
covers about half the range of possible cur-
rents. Guided by Eq. (16), the pressure-
current data is summarized in Figure 8, DISCUSSION
where each point is the average of all values The point-to-plane corona discharge is a
at a given polarity and value of L / R . The simple electrohydrodynarnic model for the
two distinct sets of data correspond to the localized tuftlike discharges which occur in
two different polarities and consequently to a wire-plate electrostatic precipitator. In both
two different values of v,. As L / R in- systems the discharge causes an intense
creases the nondimensional pressure should corona wind due to the action of the electric
asymptotically approach zero but for the body force on the gas. A straightforward
reasons noted earlier we are unable to obtain dimensional analysis of the Maxwell and
data in this range. Navier-Stokes equations shows that the elec-200 E. J. Shaughnessy and G. S. Solomon
trohydrodynamic problem for the point-to- Flippen, L. D. (1982). Electrohydrodynamics. Ph. D.
plane is characterized by four dimensionless thesis. Duke University, Durham, N.C.
groups including the ratio of gap spacing to Hauksbee, F. (1719). Physico-Mechanical Experiments
on Various Subjects, 2nd edition, London.
point radius, an electric Reynolds number, Ramadan, 0 . E., and Soo, S. L. (1969). Phys. Fluids,
the electrohydrodynarnic coupling constant, 12.
and a dimensionless current. Robinson, M. (1961). Trans. A m . Znst. Electr. Eng.
Measurements taken on a point to plane 80(Part 1): 143- 150.
apparatus for point to plane spacing of 2 -30 Robinson, M. (1976). ERDA Health and Safety Labora-
cm appear to correlate well when plotted in tory. Report HASL-301.
terms of these dimensionless groups for both Shaughnessy, E. J., Davidson, J. H., and Hay, J. C.
(1985). Aerosol Sci. Techno1 4:471-476.
positive and negative polarity. Current-volt-
Solomon, G. (1983). Electrically Driven Flow in a Nee-
age data confirm that the electric field is dle-Plane Electrode Geometry, M.S. thesis. Duke
uncoupled from the hydrodynamic field University, Durham, N.C.
while the pressure data confirms that the Teal Ferreira, G. F., Oliveira, 0 . N. Jr., and Giacom-
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flow field is strongly coupled to the electric metti, J. A. (1986). J. Appl. Phys. 59(9).
field through the coupling constant. Ushimaru, K., Butler, G . W., and Milovsoroff, P. (1982).
Report No. 2 19. Flow Research, Kent, Wash.
Yamamoto, T. (1979). Electrohydrodynamic Secondary
Flow Interaction in an Electrostatic Precipitator.
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Adachi, T., Masuda, S., and Akutsu, K. "Distribution of Yamamoto, T., Nakamura, S., and Velkoff, H. R. (1980).
Negative Ionic Wind Velocity in the Corona Space for In Innovative Numerical Analysis for The Engi-
Needle-To-Plate Electrodes," Unpublished Report. neering Science. University Press of Virginia.
Atten, P., McCluskey, F. M. J., and Lahjomri, A. C. Yamamoto, T., and Sparks, L. E. (1986). IEEE Trans.
(1987). ZEEE Trans. Ind. Appl. IA-23:705-711. Znd. Appl. IA-22:880-885.
Chattock, A. P. (1899). Phil. Mag. S. 5 . Vol. 48, No.
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