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                                 Electrohydrodynamic Pressure
                                 of the Point-to-Plane Corona
                                                   a                  a
                                 E. J. Shaughnessy & G. S. Solomon
                                  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:


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Electrohydrodynamic 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

                                                                                 P+              I
                                                                                                 -   '

                                                                                                                               2-D Plate Traverse
                                                                               Oscilloscope              Pressure

                                                                                n                         Sensor         Reference Pressure

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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;
                                                              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

                                                                                                            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
                                                                                                             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 viscosity
Pressure 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)
                                                                                                           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
                                                               1 4
                                                          0    12-
                                                                                                                                         %    0.015-
                                                                                                                                                                     *   m        *   m       -
                                                          e          a.

                                                          x    10-
                                                          N,                                                                                  0.010-                     .'                   -
                                                          f    8-

                                                          2.                                                                                                *::
                                                          %    6-

                                                                                                                                              0.005 -                                         -
                                                                          :                                                      -
                                                               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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                                                          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-
Downloaded by [] at 09:06 28 February 2015

                                                          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.
                                                          REFERENCES                                                    Ph.D. thesis. Ohio State University, Colombus, Ohio.
                                                          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.
                                                                                                                    Received April 20, 1988; accepted November 7, 1988.
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