the impact of external electrostatic fields on gas–liquid bubbling dynamics

Chemical Engineering Science 59 (2004) 247–258www.elsevier.com/locate/ces

The impact of external electrostatic #elds on gas–liquidbubbling dynamics

Sachin U. Sarnobata, Sandeep Rajputa, Duane D. Brunsa ;∗, David W. DePaolib,C. Stuart Dawc, Ke Nguyend

aDepartment of Chemical Engineering, University of Tennessee, 436 Dougherty Engr Bldg, Knoxville, TN 37996, USAbSeparations and Materials Research Group, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6181, USA

cEngineering Technology Division, Oak Ridge National Laboratory, P.O. Box 2009, Oak Ridge, TN 37831-8088, USAdDepartment of Mechanical Engineering, University of Tennessee, Knoxville, TN 37996, USA

Received 2 June 2003; received in revised form 20 August 2003; accepted 16 September 2003

Abstract

The e9ect of an applied electric potential on the dynamics of gas bubble formation from a single nozzle in glycerol was studiedexperimentally. Dry nitrogen was bubbled into glycerol through a nozzle having an electri#ed tip while pressure measurements weremade upstream of the nozzle. As the applied electric potential was increased from zero, bubble size reduced, bubble shape became morespherical, and bubbling frequency increased. At constant gas ;ow, bubble-formation exhibited a classic period-doubling route to chaos withincreasing potential. We de#ned an electric Bond number assuming that both the liquid and gas phases are conducting. This is in contrastto previous studies where one phase was considered a perfect conductor and the other one a perfect nonconductor or insulator. Althoughelectric potential and gas ;ow appear to have similar e9ects on the period-doubling bifurcation process for this system, the relative impactof electrostatic forces, as measured in terms of electric Bond number for conducting liquid and gas phases, is smaller. However, therelative impact of electrostatic forces for the case of insulating liquid and conducting gas phases is comparable to ;ow forces. Furtherdata collection is required for di9erent nozzle geometries and liquid column heights in order to verify the relative impacts of electrostaticand ;ow forces, and would allow us to ascertain if electrostatic potential is a feasible manipulated variable for controlling this system.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Multiphase ;ow; Nonlinear dynamics; Bubble formation; Separations; Electrostatic spraying

1. Introduction

Although gas–liquid bubbling can seem to be a simplephenomenon, it actually involves complex dynamical inter-actions. Such bubbling is a major component of a wide rangeof chemical and environmental processes. Typically, one ex-pects that the most eAcient gas–liquid processes have smallbubbles, since greater interfacial area increases inter-phaseheat and mass transfer. One might also expect that bub-bles are of uniform size which will simplify control andpredictability. However, in reality, gas–liquid bubbles gen-erally have broad size distributions and complex propertyvariations over time (e.g., Kikuchi et al., 1997; FDemat et al.,1998; Luewisutthichat et al., 1997).

∗ Corresponding author. Tel.: +1-865974-5317; fax: +1-865974-7706.E-mail address: [email protected] (D.D. Bruns).

Bubble formation has been the subject of numer-ous studies (e.g., Tsuge, 1986; Deshpande et al., 1991;Longuet-Higgins et al., 1991; DrahoHs et al., 1992; Terasakaand Tsuge, 1993). One key #nding from those studies is thatbubble-to-bubble interactions create instabilities leading tobifurcations and chaos. At low gas ;ow rates, bubbling isregular and periodic, but it becomes increasingly irregularwith increasing ;ow. In their classic paper, Davidson andSchJuler (1960) were among the #rst to use high-speed pho-tography to study the interaction and coalescence betweenleading and trailing bubbles. More recently Leighton et al.(1991) illustrated the complicated hydrodynamic phenom-ena present in bubbling through high-speed imaging andacoustic signatures. Other investigators identi#ed di9erentregimes of bubbling, de#ned by dimensionless groups andcharacterized by di9erent amounts of interactions betweenforming bubbles (Miyahara et al., 1984; Tsuge, 1986).Predictive models have been developed based on a simple

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2003.09.001

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248 S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247–258

description of the interaction between a primary bubble andsubsequent bubbles at higher gas ;ow rates (Deshpandeet al., 1991; Ruzicka et al., 1997; Ruzicka, 2000).With the availability of better experimental measurements

and improved nonlinear dynamics analysis tools, additionalprogress has been made in understanding the nature of bub-bling instabilities. Deterministic chaos in bubbling was #rstreported by Tritton and Egdell (1993), who studied air in-jected from a single submerged ori#ce in water–glycerolmixtures. In these studies, Tritton and Egdell reported aperiod-doubling bifurcation with increasing air ;ow. Mittoniet al. (1995) subsequently reported deterministic chaos ina similar system under a range of conditions by varyingchamber volume, injection nozzle diameter, liquid viscos-ity and gas ;ow rate. Nguyen et al. (1996) further identi-#ed the spatio-temporal aspect of chaotic bubbling; notingthat bubble-to-bubble interactions propagate long distances.This research clearly observed period-8 behavior. A simplemodel was also used that gave a good representation of theattractor behavior. Such spatial interactions reduce the di-rectness of the analogy between bubbling and the classicaldripping faucet.In other studies, the e9ects of external perturbations, such

as ;ow and acoustic pulsations, have been examined (e.g.,Fawkner et al., 1990; Cheng, 1996). In some cases suchperturbations have led to control schemes. Most notably,Tufaile and Sartorelli (2000, 2001) reported the capabilityto transform a chaotic bubbling state to a periodic state bythe application of a synchronized sound wave.Zaky and Nossier (1977) #rst reported the e9ect of an

electric #eld on bubbling, noting a decrease in bubble sizeand an increase in pressure with increasing voltage for bub-bling of air into transformer oil and n-heptane through anelectri#ed needle. Further studies by Ogata et al. (1979,1985), Sato et al. (1979) and Sato (1980) showed that bythe application of a few kilovolts, bubble size can be re-duced from a few mm to less than 100 �m in many liquids,including nonpolar ;uids like cyclohexane and polar com-pounds such as ethanol and distilled water. Sato et al. (1993)reported similar results for liquid–liquid systems in whichthe time scale of electrical charge relaxation (i.e., permit-tivity/conductivity) of the injected ;uid is greater than thatof the continuous ;uid. This type of dispersion has beentermed inverse electrostatic spraying (Tsouris et al., 1998)to di9erentiate it from the well-studied normal electrostaticspraying (Grace and Marijnissen, 1994). Several practicalapplications have been suggested for this type of spray-ing, including generating #ne bubbles for ;ow tracers (Satoet al., 1979), enhancing gas–liquid reactions (Tsouris et al.,1995), and producing uniform microcapsules (Sato et al.,1996).Two main contributing phenomena have been identi#ed

for bubble formation in the presence of electric #elds, elec-tric stress and electrohydrodynamic ;ow. Electric stress actsdirectly at the gas–liquid interface of growing bubbles andis directed inward (Tsouris et al., 1994). This force is man-

ifested by an increase in nozzle pressure with an increase inapplied voltage. Above a critical voltage (that depends onnozzle geometry and ;uid properties), electrohydrodynamic;ows are induced in the bulk ;uid (Sato et al., 1979, 1993,1997). These ;ows have a toroidal shape with a high mag-nitude near the injection nozzle and move outward from thepoints of highest #eld gradient. Under conditions of electro-hydrodynamic ;ow, a signi#cant decrease in nozzle pressureis exhibited with increasing voltage (Tsouris et al., 1998).The dynamics of electri#ed bubbling are complicated bythe interactions of these mechanisms. Sato et al. (1979) de-scribed three regimes of bubbling: periodic bubbling, dis-persed bubble production, and a high-voltage region charac-terized by sparking and larger bubble production. Similarly,Shin et al. (1997) outlined three bubbling modes—dripping,an erratic mixed mode, and a spraying mode.To date, no detailed study of the dynamics of electri#ed

bubbling has been conducted. For example, it has not beenveri#ed that the regimes characterized as periodic are trulyperiodic, nor are there any detailed analyses and/or meansof prediction of the transitions from periodic bubbling. Be-yond its intrinsic scienti#c value, such information wouldbe highly valuable in guiding the development of methodsfor controlling bubbles size distribution.In the present study, the e9ects of an applied electrostatic

potential on bubbling dynamics were determined experi-mentally. Bubbles were formed in a viscous liquid (glyc-erol) such that electrohydrodynamic ;ows were negligibleand the main electrostatic mechanism a9ecting bubblingwas the electric stress at the gas–liquid interfaces of theforming bubbles. To keep the analysis reasonably straight-forward, bubble formation from only one submerged noz-zle was examined. The dynamics were characterized usingpressure measurements upstream from the injection nozzle.The combined e9ects of gas ;ow and applied voltage wereevaluated.In the following section we describe the experimental

setup. Subsequently, we show results from the data analysisand discuss their implications. Finally we o9er conclusionsand propose future directions for research.

2. Experimental setup

A schematic of the experimental apparatus (Sarnobat,2000) is shown in Fig. 1. The apparatus consisted of aglass bubbling column, a gas metering system, a pressuretransducer for monitoring the response of the system, ahigh-voltage, direct-current power supply for maintaininga potential di9erence between the nozzle and a groundelectrode immersed in the liquid, and a data acquisitionsystem.The experiments were conducted in a square glass col-

umn (4 cm × 4 cm in cross-section, 27 cm in height) intowhich gas was injected through a central vertical nozzlehaving an electri#ed metal tip. The column was #lled with

S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247–258 249

8.2 8.25 8.3 8.35 8.4 8.45 8.5 8.55x104

-1

-0.5

0

0.5

1

1.5

Time series for pressure

Pre

ssur

e

(1)

(2)

(3)

(4)(7)

(6)

(10) (11)

(12)

(13)

(14)

(5) (8) (9)

time

Fig. 1. Schematic of experimental bubble system: (1) High-voltage power supply; (2) ground electrode; (3) electri#ed nozzle; (4) drain valve; (5) meteringvalve; (6) pressure transducer; (7) rotameter; (8) pressure reducer; (9) pressure regulator; (10) nitrogen gas cylinder; (11) data acquisition system.

99.98% pure glycerol to a level 21:6 cm above the tip ofthe nozzle which protruded 3:5 cm from the column base.Dry nitrogen from a compressed gas cylinder formed thebubbles. The column was operated at atmospheric condi-tions, and the gauge pressure measured in the tubing im-mediately upstream of the nozzle served to characterize thebubbling.The gas train used in the experiments was similar to those

used in previous studies (Nguyen et al., 1996). It consistedof two pressure regulators, a rotameter with a stainless-steel;oat (Shor-Rate II, tube number R-2-15-D, Brooks In-struments), piezo-electric valve (MaxTek MV-112), ;owsensor (Cole-Parmer 8168), high-speed pressure transducer(Setra Systems 228), a Nupro double cross metering valve(with a maximum Cv of 0.004 for ;ow control), and aball valve connected in series. This system supplied mea-sured gas ;ow rates covering a range of 10–500 cc=min.The tubing volume from the piezo-electric valve to thenozzle was minimized to reduce the dynamics of gascompression.Details of the nozzle design are shown in Fig. 2. The

nozzle was constructed from 6.35-cm-o.d. Lucite tubing. A6.35-cm-o.d., 1.0-mm thick brass ori#ce plate was securedat the end of the nozzle. This plate could be electri#ed bya 22-gauge, uninsulated copper wire passing through thenozzle to a connector on the gas inlet line.An ori#ce diameter of 0:75 mm was used to avoid indis-

tinct return maps that have been obtained with larger ori-#ce diameters (Mittoni et al., 1995). Preliminary experi-ments with applied electrostatic potential conducted in thisstudy using a 1.0-mm ori#ce obtained results with irregularspikes and peaks in the pressure time series. Three designfeatures allowed the system to be operated such that liq-

uid that may have weeped through the ori#ce during setupcould not adversely a9ect the results by causing intermittentbubbling. These were the relatively large inside diameterof the nozzle tube, a tee-bend in the gas inlet, and a drainvalve.A high-voltage, direct current power supply (model

225-50R, Bertan High Voltage Corp.) was connected to thenozzle tip with positive polarity. A 3.2-mm-diameter stain-less steel rod immersed 2:5 cm into the upper surface of theglycerol served as the counter electrode and was connectedto electrical ground. The electrode was not inserted fartherinto the liquid in order to minimize the electrode surfacearea in contact with the liquid. With this con#guration, apotential of approximately 13 kV could be attained at thenozzle tip before the 0.3-mA current safety limit of thepower supply was reached.A capacitive transducer (model 228, Setra Systems Inc.)

having a range from 0 to 1 psig was used to measure thepressure in the gas line upstream of the nozzle. The out-put from the pressure transducer was a 0–5 V DC ana-log signal, which was fed through a signal conditioningcard signal ampli#er to a data acquisition board (modelsSC-2043-SG and PCI-MIO-16E-50, National Instrument) ina 300-MHz Pentium IITM-based personal computer. NationalInstruments LabviewTM 5.1 was used as the data acquisitionsoftware.For consistent experimental results, a fresh batch of glyc-

erol was used for each run. Each run consisted of data col-lected at a speci#c ;ow rate and changing the applied voltagein increments of 1 kV from 0 V to 10 kV. Time intervalsof 300 s were provided between successive readings. Thepressure data were collected for 50 s at 2000 or 5000 Hz ateach set of experimental conditions.


(9)

(10)

(8)

(1)

(2)

(3)

(4)(5)

(7)

(6)

Fig. 2. Details of bubble nozzle construction: (1) Gas inlet; (2) ball valve; (3) connection for wire to nozzle tip; (4) glass column; (5) nozzle tip;(6) liquid drain; (7) drain for accumulated glycerin; (8) metal cap; (9) 0.75-mm-i.d. ori#ce; (10) copper wire.

3. Data analysis

The following data analysis techniques were used for thecharacterization of nonlinear bubble-formation dynamics.

Power spectra: The classical linear method of Fourieranalysis was used to transform the time-series informationinto the frequency domain, which has been shown to besensitive to changes in periodicity (FDemat et al., 1998). Al-though not de#nitive for nonlinear time series, the powerspectra yield useful information for continuous physical sys-tems as a pointer to the relevant time scales and the choiceof parameters in nonlinear time-series analysis.Delay embedding: We used the delay embedding

(Takens, 1981) of pressure measurements to charac-terize the attractor. Given the set of measurements{xi|i = 1; : : : ; n}, the sequence of vectors formed asxj=[xj; xj+; xj+2; : : : ; xj+(m−1)]T (where j is a time index)can be used to reconstruct the dynamic trajectory of thesystem. Takens (1981) proved that if some m is suAcientlylarge, the embedding vectors preserve the geometrical prop-erties and invariants of the system. Our observations indi-cated that typically an embedding dimension of 5 and delayof 35 are appropriate embedding parameters for resolvingthe injected bubble patterns. We also apply local principalcomponent analysis (PCA) to the embedded data to allowthe resulted to be projected into two or three dimensionsfor easier viewing.Period-doubling bifurcation and route to chaos: Bifurca-

tion plots have been widely used (e.g., Nguyen et al., 1996;

Tufaile et al., 1999; Tufaile and Sartorelli, 2001) to illus-trate the onset of complex dynamics as a result of varia-tion in some process input variable. For the bubbling pro-cess, the changes were quanti#ed in terms of the periodof formation of the bubble. In the context of this experi-ment, a plot of bubbling rate (or period of bubble formation)against the gas ;ow rate or the electrostatic potential is abifurcation diagram. We also generated three-dimensionalbifurcation plots involving both system variables—gas ;owrate and electrostatic potential—to illustrate the simultane-ous e9ect of electrostatic potential and ;ow rate on bubbleformation.Time return maps: Time return maps (Moon, 1992) were

used to condense the information of time series and to helpdetermine the periodicity of the system. Period-of-formationintervals were generated by measuring the peak-to-peak timeintervals of the pressure time-series data. A peak in the pres-sure trace corresponds to the beginning of bubble growth atthe nozzle, and the peak-to-peak time-interval correspondsto the bubble formation time.

4. Results and discussion

Previous experimental work (e.g., Mittoni et al., 1995;Tritton and Egdell, 1993; Nguyen et al., 1996; Tufaile andSartorelli, 2000) has shown that as gas ;ow is increased,bubble formation undergoes a bifurcation process that isreadily detected by observing variations in bubble size andspeed. At low ;ow, it is observed that a regular train of iden-tical bubbles is produced. In addition to being of equal size,


(a)

(b)

(c)

Fig. 3. Bubble interaction with increase in ;ow rate: (a) No interaction;(b) slight interaction; (c) signi#cant interaction in period-2 bubbling.

the bubbles are formed and released in equal time intervals.In our apparatus, the sequence of identical bubbles producesa train of identical pressure peaks in the nozzle (Fig. 3a).Increasing gas ;ow increases the frequency of bubbling andleads to interaction between bubbles (Fig. 3b). Speci#cally,trailing bubbles are accelerated by the wakes of precedingbubbles. At the nozzle, this gives rise to the formation oftwo alternating size bubbles (period-2), and the pressurepulses contain two distinct alternating peaks (Fig. 3c).With still greater gas ;ow, the system enters a state inwhich four distinct bubble types are formed (period-4).This period-doubling process continues progressivelywith increasing ;ow rate, #nally leading to deterministicchaos.Fig. 4 shows examples of pressure time-series data for

four di9erent ;ow rates that resulted in periods 1, 2, 4 andchaos. Subplots 4 (a)–(d) contain 2-s pressure traces. Asthe ;ow rate is increased, the bubbling frequency increases,and the number of distinct peaks appearing in the pressuretrace increase. In 4(d), the number of distinct peaks has be-come e9ectively in#nite due to the onset of deterministic

chaos. Under these conditions, there are never two bubblesproduced with exactly the same characteristics. Figs. 4(e)–(h) show the three-dimensional delay embedding formedwith the time-series segments shown in Figs. 4(a)–(d), withembedding delay of 35. The single band in Fig. 4(e), givesway to two distinct bands in 4(f)—leading from period-1 toperiod-2 behavior. Fig. 4(g) has four distinct bands, indi-cating period-4, and Fig. 4(h) has many bands with fractalstructure—a sign of chaotic dynamics.When an electrostatic potential is applied to the injection

nozzle, electric stresses cause a “pinching” action on thebubbles during formation (Tsouris et al., 1994). The visuale9ects of applied voltage in our experiments are illustratedin Fig. 5, which compares the images of bubbles formed ata #xed ;ow rate under applied potentials of 0 and 12:5 kV.The images shown are at a point in time 4 ms before thebubbles detached from the nozzle. It is seen that the bubbleformed with an applied voltage is smaller and more spheri-cal. Increasing electrostatic potential hastens the detachmentof bubbles, which leads to greater and more signi#cant in-teractions between bubbles. These results demonstrated howincreasing gas ;ow a9ects the bubbling dynamics. Now weexamine the impact of increasing the electrostatic potentialat a given ;ow rate.Fig. 6 shows pressure traces collected at 2 kHz for

bubbling at constant gas ;ow of 335 cc=min and multi-ple applied potentials of 1–9 kV (increments of 1 kV).A subtle variation can be observed in the time series interms of an increase in the number of peaks observed, adecrease in the peak heights, and increasing irregularity inthe peak heights with increasing potential. On average, thebubbling frequency also increases with increasing poten-tial. The system is chaotic in subplots (h) and (i). Fig. 7shows the local principal component scores for the timeseries in Fig. 6. Note that with increasing electrostatic po-tential, the banding in the plots becomes more complex andthe system is chaotic for potentials of 8 and 9 kV. Fig. 8shows the spectral densities for the time series shown inFig. 6. The distribution becomes broader with increasingpotential.Fig. 9 illustrates an alternative approach for observing the

dynamic changes induced by the electric potential in termsof the characteristic intervals between bubbles. In this #gure,successive time interval values between bubbles are plottedas two-dimensional ‘return maps’ for the same conditions inFig. 6. Initially (at low potential) we observe four distinctpoints corresponding to the four di9erent bubble periods—thus the system is exhibiting period-4 behavior. As the po-tential increases, we observe the four points merge into acurve, representing the chaotic state. It should be noted thatthis curve is actually a trace of the dynamic ‘map’ of thissystem. In fact, a function #t through this map can providean approximate model of the bubbling dynamics. The bubbleformation interval continues to wander along this line dueto the continuous state of instability driving the determin-istic chaos. Although not obvious, the average inter-bubble


Fig. 4. Typical pressure traces for di9erent levels of bubbling instability. Period-1 (a, e), Period-2 (b, f), Period-4 (c, g) and Chaos (d, h).

Fig. 5. E9ect of electrostatic potential in bubble formation dynamicsand bubble shape. Images of bubbles formed at a #xed ;ow rate underconditions of no voltage (left) and with an applied potential of 12:5 kV(right).

interval decreases from 0.11 to 0:08 s as the voltage tran-sition is made, indicating that the bubbles produced in thechaotic state are smaller.Fig. 10 is a bifurcation plot in which bubble formation

intervals are plotted as a function of applied potential. Inthese experiments, gas ;ow was held constant, and voltagewas gradually increased in steps of 1 kV. At low values of

the applied potential, the system exhibits period-4 behavioras indicated by the four distinct bands of observed intervals.With increasing voltage, the period of formation of bub-bles decreases and bubble size distribution becomes broader.At applied voltages between 7 and 8 kV, the bubbling be-comes chaotic and there is almost a continuous range ofvalues along a wide band. Fig. 11 shows a similar bifur-cation diagram obtained at a lower gas ;ow. Note that al-though the same general trends are apparent at the two ;ows,the details of the bifurcations with applied potential aredi9erent.To illustrate the combined e9ects of electrostatic potential

and gas ;ow, data collected from experimental runs at dif-ferent ;ow rates and di9erent applied potentials were plottedon the same graph to generate a co-dimension bifurcationplot. Fig. 12 illustrates such a plot showing bubble intervalas a function of the applied voltage and ;ow rate.Perhaps a more useful way of mapping the two-

dimensional bifurcations is to use the appropriate dimen-sionless ;ow and potentials. Tsuge’s ;ow number representsa ratio of the combined mechanical disruptive forces—inertia and buoyancy—to surface tension, and is de#ned as(Tsuge, 1986)

Nw = BoFr0:5; (1)

where

Bo=d2i �g�

and Fr =u2

dig(2)


Fig. 6. Pressure traces for various potentials at a #xed gas ;ow rate. Subplots (a)–(i) refer to potentials of 1–9 kV in increments of 1 kV. The gas ;owrate was 335 cc=min.

Fig. 7. Principal component scores for various potentials at a #xed gas ;ow rates. Subplots (a)–(i) refer to potentials of 1–9 kV in increments of 1 kV.The gas ;ow rate was 335 cc=min.


Fig. 8. Power spectral density for various potentials at a #xed gas ;ow rate. Subplots (a) through (i) refer to potentials of 1–9 kV in increments of1 kV. The gas ;ow rate was 335 cc=min.

Fig. 9. Bubbling interval return maps for various potentials at a #xed gas;ow rate. Subplots (a)–(i) refer to potentials of 1–9 kV in incrementsof 1 kV. The gas ;ow rate was 335 cc=min.

and di is the inside diameter of the nozzle ori#ce, � is theliquid density, g is the gravitational acceleration constant, �is the liquid surface tension, and u is the gas velocity throughthe ori#ce. Bo and Fr refer to Bond and Froude numbers,

Fig. 10. Bifurcation of bubbling intervals with increasing applied potentialat a constant gas ;ow rate of 335 cc=min.

respectively. However, electric forces are also acting on thebubbles, and should be taken into account.Now we consider the electric stress tensors, assuming the

;uids are linearly polarizable (Landau and Lifshitz, 1960).If Ea is the electric #eld though the ambient liquid and Eg

is the electric #eld across the gas phase, we have

Ea�a = Eg�g; (3)


Fig. 11. Bifurcation of bubbling intervals with increasing applied potentialat a constant gas ;ow rate of 290 cc=min.

Fig. 12. Co-dimension bifurcation diagram of bubbling for changes involtage and ;ow rate.

where �g is the permittivity of the gas and where �a is thepermittivity of ambient phase. Since

Ea =V − Vi

daand Eg =

Vi

di(4)

where Vi is the potential at the interface. Eliminating Vi fromEqs. (3) and (4), we obtain

Eg =V

[di + da�g=�a]and

Ea =Vdi

1[�a=�g + da=di]

; (5)

where da is the distance from the submerged electrode tip tothe nozzle and �L is the permittivity of glycerin. Assuming�g ∼= �0 where �0 is the permittivity of air, Eq. (5) reduces to

Eg =Vdi

1[1 + (da=di)1=K]

and

Ea =VdiK

1[1 + (da=di)1=K]

; (6)

where K = �g=�a � �g=�0 is the dielectric constant of theambient phase.If the ambient (liquid) phase is a perfect conductor, i.e.,

K ≈ ∞, and the electric #eld in the liquid will be zeroand the liquid would be isopotential. In that limiting case,Eq. (6) becomes

E′g = lim

K→∞Vdi

1[1 + (da=di)1=K]

=Vdi

and

E′a = lim

K→∞Vdi

1[K + da=di]

= 0: (6a)

The jump in the electric stress tensor at the interface wouldbe

Te =12(�gE2

g − �aE2a) =

V 2

2d2i

(�g − �a=K2)

[1 + (da=di)1=K]2

=V 2�g2d2

i

(1− �g=�a)

[1 + (da=di)1=K]2: (7)

Note that if �a �g and K 1, Eq. (6) reduces to

T ′e = lim

�g��a;K→∞Te =

V 2�g2d2

i

=12�g

(Vdi

)2

≈ 12�gE2

g′ : (7a)

A ratio of this stress to the stress caused by surface tensionforces can be constituted as

Be =2Te�=di

=V 2�g�di

(1− �g=�a)

[1 + (da=di)1=K]2; (8)

which is the electric Bond number for this setup.In the limiting case of the liquid being a perfect conductor,

Eq. (8) reduces to

B′e = lim

�g��a;K→∞Be =

V 2�g�di

=di�gE2

g′

�: (8a)

The di9erence in Eqs. (8) and (8a) (or in Eqs. (6) and(6a)) is a factor for given values of K , di and da. However,the de#nition in Eq. (8) allows us to compare the impactof electric stress as compared to surface tension for all noz-zles. For low to moderately conducting liquids, placing theground electrode closer to the nozzle (decreasing da) wouldallow one to achieve higher electric stresses given the samepotential di9erence.


Fig. 13. Bubbling stability map as correlated with Tsuge ;ow and electricBond numbers. Legend: open circle (period-1), plus sign (period-2), opensquare (period-4) and #lled circles (chaos).

Note that in this derivation, we have assumed that theelectric #eld is normal to the interface, and the tangentialcomponents have been omitted. Regardless of the polar-ity, the electric #eld is normal to the interface and is di-rected inward towards the gas phase. Ideally, we shouldhave conducted the experiment with a very small da so thatthe gas phase would be the dominating resistance. How-ever, that could not be performed as our intention was toreduce the electrode area in contact with the liquid. We alsoignored the aspect ratio, as the #eld is normal to the in-terface, and the shortest path through the gas phase is thenozzle i.d.Fig. 13 presents the bifurcation plot as a stability map

in terms of dimensionless variables; speci#cally, the Tsuge;ow number and the electric Bond number. Since bothdimensionless groups are scaled against surface-tensionforces, they allow comparison of the relative e9ect of forcescaused by gas ;ow and those caused by the applied electric#eld. We note that for chaotic dynamics, the Tsuge numberwas 16.4 and the electric Bond number was greater than0.2. This may indicate that electrostatic forces have greaterimpact on bubbling dynamics than ;ow forces; however,further data obtained at larger electric Bond numbers andsmaller Tsuge ;ow numbers would be necessary to verifythis assertion.It would be desirable to de#ne a system variable incorpo-

rating the e9ects of both applied voltage and ;ow rate thatcould be used as a predictor of bubbling regime. In orderto capture the collective e9ect of electrostatic and inertialforces on bubble formation from an electri#ed nozzle, Shinet al. (1997) suggested a modi#ed Weber number that is theratio of the sum of the electrostatic and inertia forces to sur-face tension forces. That modi#ed Weber number (simplythe sum of electrical Bond number and Weber number) is

Fig. 14. Bubbling stability map as correlated with Tsuge ;ow and modi#edWeber numbers. Legend: open circle (period-1), plus sign (period-2),open square (period-4) and #lled circles (chaos).

Fig. 15. Bubbling stability map as correlated with Tsuge ;ow number andelectric Bond number (in limiting case, assuming the liquid phase is aperfect conductor). Legend: open circle (period-1), plus sign (period-2),open square (period-4) and #lled circles (chaos).

de#ned as

WeS =di�aE2 + u2di�a

�; (9)

where �a is the gas density. Fig. 14 replots Fig. 13 withmodi#ed Weber number replacing the electric Bond num-ber as the ordinate. The derivation in Eqs. (7) and (8) is avery important correction in estimating electric stresses atthe interface. If we had assumed the liquid phase to be per-fectly conducting, the electric Bond numbers would havereached up to 20, compared to the maximum value of 0.33 inFig. 13. Fig. 15 replots Fig. 13 but replaces the electric Bond


number de#ned in Eq. (8) by that de#ned in Eq. (8a). Un-der this scaling, it is apparent that electrostatic forces havemuch lower impact on bubbling dynamics than ;ow-relatedforces.

5. Conclusions and recommendations

Our results demonstrate that, at constant ;ow rate, bub-ble formation dynamics from an electri#ed nozzle exhibitsthe classic signs of a period-doubling bifurcation to chaoswith increasing applied potential. With increasing voltage,the bubble frequency increases and the bubbling undergoesperiod-doubling bifurcations to become chaotic, similar tothat for increasing ;ow rate. Similar behavior has been ob-served in liquid–liquid systems under electrostatic spray-ing (Tsouris et al., 1994). Although the basic type of bi-furcation is similar for applied voltage and ;ow, voltagehas a proportionally smaller e9ect than ;ow. Applicationof voltage also reduces bubble size, apparently by promot-ing early bubble release. The smaller bubbles have higherinterfacial surface area for heat and mass transfer, and thismight have bene#cial applications in gas–liquid contactingdevices.Use of applied electrical potential for bubble size control

should be further investigated because it would be possi-ble to manipulate this variable very quickly and thus pro-vide very fast feedback perturbations (e.g., much faster thancan be obtained with gas ;ow perturbations). A possiblelimitation is the relatively large voltage amplitude neededto achieve signi#cant e9ects. This study is the #rst one toconsider the case where both the ambient and gas phaseswere conducting. Other studies have assumed both of theseto be either perfect conductors or perfect nonconductors.An expression for electric Bond number was derived forthe current apparatus, and it was shown that it reducesto the expression for other, limiting cases studied in theliterature.In the present study, the e9ects of electro-hydrodynamic

(EHD) ;ows were assumed to be negligible and were notstudied. Future studies should include EHD ;ows inducedwith di9erent shapes of nozzles, various chamber sizes andchanging the nozzle diameter. The particular liquid usedis another important parameter, and consideration shouldbe given to similar studies in immiscible liquid–liquidsystems.

Acknowledgements

This work was supported by the Division of ChemicalSciences, OAce of Basic Energy Sciences, US Depart-ment of Energy, under contract DE-AC05-00OR22725 withUT-Battelle, LLC. The authors thank Costas Tsouris ofORNL for helpful comments.

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the impact of external electrostatic fields on gas–liquid bubbling dynamics

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