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ELECTROSTATIC AND HYDRODYNAMIC EFFECTS IN THE ELECTRICALBREAKDOWN OF LIQUID DIELECTRICS

P. K. Watson

Xerox CorporationRochester, New York

ABSTRACT

Optical studies of pre-breakdown events in insulating liquidsenable one to distinguish three stages in the breakdown of negativepoint-plane gaps: (1) The creation of a rapidly expanding vaporcavity adjacent to the point electrode. (2) The instability of thesurface of the cavity, characterized by the growth of wave-likedisturbances, and (3) The runaway growth of the instability leadingto vapor streamers that bridge the gap and cause the actual break-down. This paper contains an analysis of the electrostatic andhydrodynamic forces acting on the cavity and on the resultingstreamers. It is assumed that the vapor in the cavity is ionizeddue to the high Field, so that the potential at the bubble surfaceis close to that of the point electrode. This assunption enablesone to calculate the electrostatic force expanding the cavity. Theelectrostatic force on the cavity wall also leads to an instabilityof the interface, and the runaway growth of the instability leads tothe generation and propagation of streamers; by combining simpleelectrostatic and hydrodynamic concepts, we derive an equation forstreamer velocity that is in approximate agreement with measured v?lues.

INTRODUCTION Based on the photographic evidence, we may distinguishthree stages in the breakdown of negative point-plane

The idea that the electrical breakdown of liquid di- gaps: (1) The generation of a low density region, whichelectrics involves the generation and growth of a vapor we interpret as a vapor cavity, near the point cathode;cavity on a microsecond time scale was suggested many this cavity expands rapidly during the first few micro-years ago [1], but a detailed analysis of the process seconds with an initial velocity of about 5X103 cm/s,has been lacking. Our understanding of breakdown in and slows down as it expands. (2) The expansion isliquids has been advanced considerably over the past accompanied by the appearance of a wave-like instability20 years by the use of pulsed schlieren and shadowgraph on the surface of the cavity, and initially the insta-techniques to photograph the prebreakdown events. Ex- bility grows as A(t)=A0exp(nt), where n defines theperimental work of this type began with the work of growth rate of the disturbance. (3) The runaway growthHakim and Higham [2] and of Farazmand [3], who were the of the instability leads to streamers that fan out fromfirst to publish schlieren photographs that optically the point electrode and propagate across the gap; theresolved the pre-breakdown events in liquids; these subsequent breakdown of the gap then occurs throughphotographs show that a low density region forms in the the low density vapor in the streamer.vicinity of the point electrode a few microseconds be-fore breakdown. Optical studies of this type were con- A system of this complexity cannot be analyzed with-tinued by Chadband and Wright [4], who also used a out makeing some simplifying assumptions. Fortunately,streak camera to measure the velocity of the pre-break- there are a few well-founded assumptions that have adown events. More recently Forster [5], Chadband and major simplifying effect on the calculations. InCalderwood [6], Devins et al. [7], Hebner and Kelley effect, we are able to show that the electrostatic[8], Nelson and McGrath [9], and others have extended field provides the major driving force in the variousthese experimental techniques, and a considerable stages of the breakdown process and that hydrodynamicamount of photographic information of this type is now forces provide the rate-limiting drag.available. The majority of these workers have concen-trated on the positive point-plane system, but thenegative point has been studied in a few cases . A THE VAPOR CAVITYquite different approach was taken by Nelson and Hashad[10] who introduced vapor bubbles into a parallel plate In negative point-plane gaps, a vapor cavity appearsgap and studied the growth of the bubbles experimentally a few microseconds after voltage is applied, and oneand theoretically, under step voltage conditions . may suppose that the cavity is formed as a result of

0018-9367/OO/0400-03;95*O1 .010 @_ 1985 IEEE

396 IEEE Transactions on Electrical Insulation Vol. EI-20 No.2, April 1985

local joule heating (due to high field conduction in tension. We find that, over the range of Fig. 1, thethe liquid) in the presence of a negative electrostatic kinetic energy is at least a factor of five greaterpressure. This negative pressure is generated in the than PV and about two orders of magnitude greater thanliquid by the electrostatic field acting on space surface tension. Hence, we are able to justify thecharge in the liquid adjacent to the electrode. The simplifying assumption that, for an ambient pressurespace charge may be due in part to the equilibrium of one atmosphere or less, we can ignore PV and sur-charge that must exist at the interface, but it will face tension.also include space charge that is injected from theelectrode due to the applied field. In any case, one An interesting limiting case of cavity expansion ismay set an upper bound to the electrostatic force, of analyzed in an accompanying paper [11] in which wethe order of F=ccoE2/2, which will act on the liquid show that cavity radius increases as t0> for constantif there is free charge ecoE near the interface. Sim- kinetic energy. A cavity that is expanding fasterilarly, an upper bound may be set to the energy that than t0-4 must, therefore, be driven by a force, sinceis delivered to the liquid when this free charge is it is increasing in energy. There are two major com-moved by the field, and an order of magnitude calcula- ponents of force driving the pre-breakdown cavity,tion indicates that in this limiting case the temper- the electrostatic component Pes, and the vapor pres-ature rise may be sufficient to vaporize a small vol- sure Pv. The equation of motion of the cavity isume of liquid near the cathode. then obtained by equating kinetic energy to the work

done by the driving pressure, which isOnce the cavity is formed, it expands rapidly. The

data in Fig. 1 are taken from streak camera photo- W = - Tr ( + R3-Ro)4 r P(R)R3 (2)graphs published by Chadband and Wright [4] and show 3 es (cavity radius vs. time, measured from the first appear-ance of the cavity. From these photographs one finds Equating Eqs. (1) and (2), we find thatthat during the first microsecond the cavity expandsat a velocity of about 5X103 cm/s. Subsequently, the U2 = (2/3p)P(R) (3)expansion slows down and takes the form R tn, wheren *0.6. From this equation, we see that an expansion velocity

of 30 m/s requires a driving pressure of about 10 at-| ' 'l mospheres, and an order of magnitude calculation shows

0.2 _ 50 that a pressure of that size can be generated by theVELCIY40 electrostatic field. We, therefore, make the assump-

E~ .15 , / ] o tion that the electrostatic field is the major termE.15 w ~~~~~~~~~~~inEq. (2).in os 30 T

a ,! - ^ ¢ To analyze the electrostatic pressure, we need to4 0.l know the electrostatic field and the charge distribu-

F /RADIUS - 20 J tion in the vapor cavity. Chadband and Wright [4]RADIUS > suggested that the region might be an ionized plasma,which would make the surface an equipotential withthe cathode. More recently, Kelley and Hebner [12]have used Kerr effect measurements to observe the time

lr l l evolution of internal fields associated with pre-0.5 1 2 3 4 5 breakdown events in point-plane gaps in nitrobenzene;

TIME (/.SEC) they have confirmed that there is a roughly sphericalregion, which is equipotential with the electrode, ex-

Fig. 1: Cavity Radius and WaZl Velocity vs. Time for tending from the point electrode. We note that thesen-Hexane. 1.5 mm, negative point-pZane gap, 33 kV. conditions are quite different from those studied byThe data for radius vs. time are taken from the Nelson and Hashad [101 who were concerned with thestreak camera record of Chadband and Wright [4] dynamics of an unionized vapor cavity, subject to di-Figure 5b. electric forces.

Assuming then that the cavity is an ionized vapor orFromthi reatioshi beweenF ad twe cn eti- plasma, so that its surface is equipotential with the

mate the magnitudes of the various energy terms in the p

cavity expansion. The major components are: the kin- cathode, and that it is roughly spherical in shape,one may calculate the force on the interface and solve

etic energy of the fluid surrounding the cavity, the fo th eutn oinoftecvt al hPV work done against ambient pressure; and the work frtersligmto ftecvt al h

FVdwork doainet surfag staentspresure;andnte.wor field on-axis at the surface of a spherical conductorradius R, center distance a above a ground plane is

To analyze the kinetic energy of the fluid, we write given to a good approximation [13] by the equationU, the wall velocity, and u(r), the velocity of the Vliquid outside the cavity at radius r. Then u(r)r2= E(R) V (4)UR and we can write the kinetic energy of the liquid Rln2(a/R)0^5with density p,

The logarithmic term in Eq. (4) makes the integrationrO ~~~~~~~~~ofthe equation of motion very difficult and, withoutK.E. = 1/2 PI 4rr r2u2(rdr = 2TUpU2R3 (1) introducing too large an error, the logarithm can be

treated as a constant. A preferrable way to treatF this term is to note that, over the limited range of

the data in Fig. 1, the tern lm2(a/R) °3 can be approxi-We insert typical values for U and R from Fig . 1, mated to better than 5% accuracy by (a/R)05. The

in Eq. (1) and compare this with the FV work (assuming electrostatic component of pressure is then given byF is one atmosphere) and the work done against surface

Watson: Electrostatic and hydrodynamic effects in liquid dielectrics 397

500 cm-'. The field at the surface of the cavity re-CCOV2 quired to drive this EHD instability at a growth rate

P = 1/2 secE2(R) = 2 5 (5) of 5x1O5 s-1 is, therefore, about 8x105 V/cm. FromEq. (4), we find that a field of this magnitude inFarazmand's experiment corresponds to a cavity radius

The work done by the electrostatic force in expanding of 0.2 mm. This is consistent with the observationthe cavity is then given by that the instability is well developed when the cavity

has expanded to 0.4 mm radius.

R PEV0V2 42dR= 4 V2 R15 (6) Kath and Hoburg's equation is based on a two-dimen-dPesd J 4rRdR -(

sional geometry in which the applied field is constant,es 2a05R15 3 a°05 whereas in the point-plane gap, as the cavity expands,

field decreases and so, therefore, must the growth

Equating work done, from Eq. (6), to kinetic energy, rate n. Now, the growth rate varies linearly with thefrom Eq. (1), and integrating, we obtain the required unperturbed field at the interface, and the field isequation for cavity radius vs. time given by combining Eqs. (4) and (7). Hence, the time-

4h/ dependent growth rate can be written

[7 [2ssol ~~0.5R(t) = Vt - ] (7) [03 5] F p 02 8

4

L3a0.5p n(t)-

k [-v [ j(8)Comparing this equation with the results shown in Fig.1, we note that the time dependence predicted by theelectrostatic force model is in close agreement with The amplitude of the instability is given bythe experimental data (the line through the data hasthe theoretical slope of 4/7). Inserting appropriate Atvalues for voltage, density, and gap length, Eq. (7) A(t) = A0 J exp(n(t).t)dt. (9)predicts a cavity radius of 0.15 mm at 5 us: this is 0close to the average experimental value of 0.18 mm.In this particular case, therefore, the electrostatic 05 16 025\field appears to be the major force expanding the A(t) = 2Ao ep) (10)cavity. In other cases, however, we find that the e ()=2oepk3 LPJ /radius predicted by Eq. (7) is too low, and one con-cludes that a complete explanation of cavity expansion From Eq. (10) we note that the log of the amplitudewill have to take vapor pressure into account, in of the instability scales with k/t and in Fig. 2 weaddition to electrostatic pressure. have plotted the data from Farazmand [3Land Chadband

& Wright [4] in the form logA(t) v's. k/Vt. Inevitablythe scatter is very large, but it is encouraging to

INSTABILITY OF THE VAPOR-LIQUID INTERFACE note that the theoretical line, based on Eq. 10,passes close to the data.

We now examine the role of the electrostatic forcein the growth of the wave-like instabilities on thesurface of the vapor cavity. These instabilities canbe seen increasing in amplitude with time in many A FARAZMANDphotographs of negative discharges [3,4]. One would Aexpect the instability to grow initially, as ACt) = AAoexp(nit), where n is the growth rate. From an analy- os5 JCABNsis of Farazmand's data [3], we find that this growth X/rate is of order 5±2xlO5s- for the experimental con-ditions, gap length 1.5 mm, and applied voltage 45 kV.

It has been suggested previously [14] that this in- Ew

stability at the vapor-liquid interface may be cthought of as analogous to the gravity-driven Rayleigh-Taylor instability,but driven by the electrostaticforce acting on the surface of the cavity. One cannot /push this analogy too far, however, as the Rayleigh-Taylor instability is due to a bulk force that is in- oi -

dependent of the position and time, whereas, in the /electrostatic case, charges are free to move in accord-ance with field and will tend to concentrate in thehigh field region - i.e., at the tips of the insta-bilities. 0.05 kfrV

The situation with mobile charges corresponds to the 0 10 20 30 40 50 60 70electrohydrodynamic (EHD) instability that has beenconsidered by several workers, including Zahn andMelcher [15], and by Kath and Hoburg [16], who showthat for superimposed fluids of very different con- Fig. 2: Anr2Ztude of EHD InstailZity vs. kVJ6ductivities, stressed by a field normal to the inter- The data are taken from the photographs offace, the growth rate is given by n={k2es0E2/2p}0'5 Farazmand [3] and of Chiadand and Wrght [4].where k is the wavenumber, 27r/X . From Farzmand's pre-breakdown photographs, the wavenumber k is of order

-9R IEFE Transactions on Electrical insulation Vol. E T-20 No.2. pril 1985

STREAMER PROPAGATION vapor cavity adjacent to the point; the EDH instabilityof the cavity surface; and the growth of vapor stream-

Streamer propagation is the most striking of pre- ers that cross the gaps and cause the actual breakdown.breakdown phenomena in liquids. A characteristic ofstreamers is that, for a given voltage, their velocity Although a system of this complexity would be ex-of propagation is relatively constant as the streamer tremely difficult to analyze in a rigorous sense, we

crosses the gap, and for a range of applied voltage, are able to simplify the analysis by selecting onlythe velocity of propagaiton is proportional to voltage. the leading terms in the various equations. AssumingMoreover, streamers from a negative point are invari- that the vapor cavity is ionized, so that its surfaceably of larger cross section and propagate more potential is at the potential of the cathode, thenslowly than the positive streamers. the electrostatic force can easily be calculated, and

this seems to provide the main driving force in theNow, based on the above discussion of EHD instabil- breakdown process. Other terms, such as vapor pres-

ities, it seems reasonable to treat these streamers as sure and surface tension undoubtedly play a role, butan extension of the vapor cavity, so that the interior their effect is less important.of a streamer is an ionized gas or plasma and its sur-face is approximately equipotential with the point In the initial phase of the cavity expansion, weelectrode. Thus one is able to estimate the field at show that work done by the driving force goes mainlythe tip of the streamer, using Eq. 4 above. Negative to increasing the kinetic energy of the expanding cav-streamers are between SO and 100 um in diameter, so ity, and this leads to an equation for cavity radiusthat for a streamer near the middle of a 1.5 mm gap vs time which is quite good agreement with the ratherthe stress at the tip, ES, is approximately V/2RS. limited experimental evidence.Thus the electrostatic force on the hemispherical tipof the streamer is given approximately by The EHD instability of the cavity surface gives rise

to the most characteristic feature of the breakdownF 2rR 21/2cco[V/2R]2R 4C £0v2.(11) process,viz. the generation of streamers which fan outs s 4 from the point and cross the gap with a steady velocity.

The growth rate of the instability has been calculated,To calculate the hydrodynamic drag on the streamer using a two-dimensional EHD model and the magnitude of

we treat the channel as an equivalent flow problem the growth rate appears to be in reasonable agreement[17]. With this approach, one may show that a hypo- with the experimental evidence -- though again, therethetical point source emitting a volume of fluid Q is a dearth of experimental data.per unit time and located in a stream of velocity Usgenerates a surface of separation similar to a stream- The streamer velocity has also been analyzed, assum-er and that this region experiences a force ing that the streamer channels are ionized and that theF = (2/3)pQUs,due to the stream velocity, where electrostatic pull on the end of the channel counter-Q X 7rR52Us. acts the hydrodynamic resistance to the channel's for-

ward motion. This model indicates that channel veloc-

By superposition, we may treat the stream velocity ity should scale directly with driving voltage andas zero and the streamer tip velocity as Us; hence, inversely with channel radius, and those features arethe retarding force on a streamer moving with velocity in general agreement with the experimental results.Us in a stationary fluid is given by F = (2p/3JRs2u.2.Equating this force to the electrostatic force pullingon the end of the streamer gives the required equation REFERENCESof motion,

[1] A. H. Sharbaugh and P. K. Watson, Progress in

U [3P co/8p]05 . (12) Dielectrics, Haywood London 1962, pp. 242-245.

[2] S. S. Hakim and J. B. Higham, "A Phenomenon in

This equation has the general features one finds in n-Hexane Prior to its Electrical Breakdown,"streamer propagation: the velocity scales with voltage; Nature 189, p. 996 (1961).it does not depend strongly on position in the gap,and scales inversely with streamer radius. Moreover, [3] B. Farazmand, "The Study of Electric Breakdownthe absolute magnitude of the negative streamer veloc- of Liquid Dielectrics Using Schlieren Opticality is in order-of-magnitude agreement with experiment. Techniques," Brit. J. Appl. Phys. 12, p. 251Negative streamer velocities are about 100 m/s for an (1961).applied voltage of 33 kV [6]. Substituting theseva-lues in Eq. (12) we calculate a channel radius of [4] W. G. Chadband and G. T. Wright, "A Pre-Break-about 30 vim. The negative streamers shown in [6] down Phenomenon in the Liquid Dielectric Hexane,"appear to have tip radii of this magnitude, but are Brit. J. Appl. Phys. 16, p. 305 (1965).much fatter along their developed length, and it maybe that the propagating tip velocity is limited by a [5] E. 0. Forster, "Research in the Dynamics ofprocess such as the one we describe, but that the chan- Electrical Breakdown in Liquid Dielectrics,," IEEEnel continue to expand as it propagates. Trans. EI-15, p. 182 (1980).

[6] W. G. Chadband and J. H. Calderwood, "The Propa-

CONCLUSIONS gation of Discharges in Dielectric Liquids, " J .Electrostatics 7, p. 75 (1979).

The electrical breakdown of liquids involves thegeneration and propagation of vapor channels through [7] J. C. Devins, S. J. Rzad, and R. J. Schwabe,the liquid, and the development of the vapor channels "tBreakdown and Pre-Breakdown Phenomena in Liquids,"in negative point-plane gaps follow a three-stage pro- J . Appl Phys . 52 (7), p . 4531 (1981) .cess: the generation and expansion of an ionized

Watson: Electrostatic and hvdr-odynamic effect_sin liquzid dielectricS -7

[8] R. E. Hebner and R. F. Kelley, "Observations of [13] R. Coehlo and J. Debeau, "Properties of the Tip-Pre-Breakdown and Breakdown Phenomena in Liquid Plane Configuration," J. Phys. D. Appl. Phys. 4,Hydrocarbons," J. Electrostatics 12 p. 265 (1982). p. 1266 (1971).

[9] P. B. McGrath and J. K. Nelson, "A Divergent [14] P. K. Watson, "Electrohydrodynamic InstabilitiesField Study of Pre-Breakdown Events in n-Hexane," in the Breakdown of Point-Plane Gaps in InsulatingJ. Electrostatics 7, p. 327 (1979). Liquids," 1981 Conference on Electrical Insula-

- tion and Dielectric Phenomena, p. 370.[10] J. K. Nelson and F. M. Hashad, "Cavity Dynamics

in Stressed Dielectric Liquids," J. Electrostat- [15] M. Zahn and J. R. Melcher, "Space Charge Dynamicsics 12, p. 527 (1982). of Liquids," Phys. Fluids 15, p. 1197 (1972).

[11] P. K. Watson, W. G. Chadband and W. Y. Mak, [16] G. S. Kath and J. F. Hoburg, "Interfacial Elec-"Bubble Growth Following a Localized Electrical trohydrodynamic Instability in Normal ElectricDischarge," Proc. 8th Int. Conf. on Conduction Field," Phys. Fluids 20, p. 912 (1977).and Breakdown in Dielectric Liquids, p. 180(1984). [17] B. H. Chirgwin and C. Plumpton, Classical Hydro-

dynamics, Pergamon 1967, p. 95.[12] E. F. Kelley and R. E. Hebner, "The Electric

Field Distribution Associated with PrebreakdownPhenomena in Nitrobenzene," J. Appl. Phys. 52 Thnis panper was presented at the 8th InternatiQnal(1), p. 191 (1981). Conference on Conduction and Breakdown in DieZectric

Liquids, Pavia, Italy, from 24-27 July 1984.

Manuscript was received 10 December 1984.