Chemical Engineering Science 59 (2004) 247 – 258
www.elsevier.com/locate/ces
The impact of external electrostatic #elds on gas–liquid
bubbling dynamics
Sachin U. Sarnobata , Sandeep Rajputa , Duane D. Brunsa;∗ , David W. DePaolib ,
C. Stuart Dawc , Ke Nguyend
a Department
of Chemical Engineering, University of Tennessee, 436 Dougherty Engr Bldg, Knoxville, TN 37996, USA
and Materials Research Group, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6181, USA
c Engineering Technology Division, Oak Ridge National Laboratory, P.O. Box 2009, Oak Ridge, TN 37831-8088, USA
d Department of Mechanical Engineering, University of Tennessee, Knoxville, TN 37996, USA
b Separations
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 studied
experimentally. Dry nitrogen was bubbled into glycerol through a nozzle having an electri#ed tip while pressure measurements were
made upstream of the nozzle. As the applied electric potential was increased from zero, bubble size reduced, bubble shape became more
spherical, and bubbling frequency increased. At constant gas ;ow, bubble-formation exhibited a classic period-doubling route to chaos with
increasing potential. We de#ned an electric Bond number assuming that both the liquid and gas phases are conducting. This is in contrast
to previous studies where one phase was considered a perfect conductor and the other one a perfect nonconductor or insulator. Although
electric potential and gas ;ow appear to have similar e9ects on the period-doubling bifurcation process for this system, the relative impact
of electrostatic forces, as measured in terms of electric Bond number for conducting liquid and gas phases, is smaller. However, the
relative impact of electrostatic forces for the case of insulating liquid and conducting gas phases is comparable to ;ow forces. Further
data collection is required for di9erent nozzle geometries and liquid column heights in order to verify the relative impacts of electrostatic
and ;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 simple
phenomenon, it actually involves complex dynamical interactions. Such bubbling is a major component of a wide range
of chemical and environmental processes. Typically, one expects that the most eAcient gas–liquid processes have small
bubbles, since greater interfacial area increases inter-phase
heat and mass transfer. One might also expect that bubbles are of uniform size which will simplify control and
predictability. However, in reality, gas–liquid bubbles generally have broad size distributions and complex property
variations 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: dbruns@utk.edu (D.D. Bruns).
0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2003.09.001
Bubble formation has been the subject of numerous studies (e.g., Tsuge, 1986; Deshpande et al., 1991;
Longuet-Higgins et al., 1991; DrahoHs et al., 1992; Terasaka
and Tsuge, 1993). One key #nding from those studies is that
bubble-to-bubble interactions create instabilities leading to
bifurcations and chaos. At low gas ;ow rates, bubbling is
regular and periodic, but it becomes increasingly irregular
with increasing ;ow. In their classic paper, Davidson and
SchJuler (1960) were among the #rst to use high-speed photography to study the interaction and coalescence between
leading and trailing bubbles. More recently Leighton et al.
(1991) illustrated the complicated hydrodynamic phenomena present in bubbling through high-speed imaging and
acoustic signatures. Other investigators identi#ed di9erent
regimes of bubbling, de#ned by dimensionless groups and
characterized by di9erent amounts of interactions between
forming bubbles (Miyahara et al., 1984; Tsuge, 1986).
Predictive models have been developed based on a simple
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S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
description of the interaction between a primary bubble and
subsequent bubbles at higher gas ;ow rates (Deshpande
et al., 1991; Ruzicka et al., 1997; Ruzicka, 2000).
With the availability of better experimental measurements
and improved nonlinear dynamics analysis tools, additional
progress has been made in understanding the nature of bubbling instabilities. Deterministic chaos in bubbling was #rst
reported by Tritton and Egdell (1993), who studied air injected from a single submerged ori#ce in water–glycerol
mixtures. In these studies, Tritton and Egdell reported a
period-doubling bifurcation with increasing air ;ow. Mittoni
et al. (1995) subsequently reported deterministic chaos in
a similar system under a range of conditions by varying
chamber volume, injection nozzle diameter, liquid viscosity and gas ;ow rate. Nguyen et al. (1996) further identi#ed the spatio-temporal aspect of chaotic bubbling; noting
that bubble-to-bubble interactions propagate long distances.
This research clearly observed period-8 behavior. A simple
model was also used that gave a good representation of the
attractor behavior. Such spatial interactions reduce the directness of the analogy between bubbling and the classical
dripping 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 such
perturbations have led to control schemes. Most notably,
Tufaile and Sartorelli (2000, 2001) reported the capability
to transform a chaotic bubbling state to a periodic state by
the 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 size
and an increase in pressure with increasing voltage for bubbling of air into transformer oil and n-heptane through an
electri#ed needle. Further studies by Ogata et al. (1979,
1985), Sato et al. (1979) and Sato (1980) showed that by
the application of a few kilovolts, bubble size can be reduced from a few mm to less than 100 m in many liquids,
including nonpolar ;uids like cyclohexane and polar compounds such as ethanol and distilled water. Sato et al. (1993)
reported similar results for liquid–liquid systems in which
the time scale of electrical charge relaxation (i.e., permittivity/conductivity) of the injected ;uid is greater than that
of the continuous ;uid. This type of dispersion has been
termed inverse electrostatic spraying (Tsouris et al., 1998)
to di9erentiate it from the well-studied normal electrostatic
spraying (Grace and Marijnissen, 1994). Several practical
applications have been suggested for this type of spraying, including generating #ne bubbles for ;ow tracers (Sato
et 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, electric stress and electrohydrodynamic ;ow. Electric stress acts
directly at the gas–liquid interface of growing bubbles and
is directed inward (Tsouris et al., 1994). This force is man-
ifested by an increase in nozzle pressure with an increase in
applied voltage. Above a critical voltage (that depends on
nozzle 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 magnitude near the injection nozzle and move outward from the
points of highest #eld gradient. Under conditions of electrohydrodynamic ;ow, a signi#cant decrease in nozzle pressure
is exhibited with increasing voltage (Tsouris et al., 1998).
The dynamics of electri#ed bubbling are complicated by
the interactions of these mechanisms. Sato et al. (1979) described three regimes of bubbling: periodic bubbling, dispersed bubble production, and a high-voltage region characterized 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 been
veri#ed that the regimes characterized as periodic are truly
periodic, nor are there any detailed analyses and/or means
of prediction of the transitions from periodic bubbling. Beyond its intrinsic scienti#c value, such information would
be highly valuable in guiding the development of methods
for controlling bubbles size distribution.
In the present study, the e9ects of an applied electrostatic
potential on bubbling dynamics were determined experimentally. Bubbles were formed in a viscous liquid (glycerol) such that electrohydrodynamic ;ows were negligible
and the main electrostatic mechanism a9ecting bubbling
was the electric stress at the gas–liquid interfaces of the
forming bubbles. To keep the analysis reasonably straightforward, bubble formation from only one submerged nozzle was examined. The dynamics were characterized using
pressure measurements upstream from the injection nozzle.
The combined e9ects of gas ;ow and applied voltage were
evaluated.
In the following section we describe the experimental
setup. Subsequently, we show results from the data analysis
and discuss their implications. Finally we o9er conclusions
and 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 a
glass bubbling column, a gas metering system, a pressure
transducer for monitoring the response of the system, a
high-voltage, direct-current power supply for maintaining
a potential di9erence between the nozzle and a ground
electrode immersed in the liquid, and a data acquisition
system.
The experiments were conducted in a square glass column (4 cm × 4 cm in cross-section, 27 cm in height) into
which gas was injected through a central vertical nozzle
having an electri#ed metal tip. The column was #lled with
S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
249
Time series for pressure
1.5
(14)
1
Pressure
(2)
0.5
0
-0.5
-1
8.2 8.25 8.3 8.35 8.4 8.45 8.5 8.55
time
x104
(12)
(6)
(1)
(3)
(5)
(4)
(7) (8) (9)
(10)
(11)
(13)
Fig. 1. Schematic of experimental bubble system: (1) High-voltage power supply; (2) ground electrode; (3) electri#ed nozzle; (4) drain valve; (5) metering
valve; (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 of
the nozzle which protruded 3:5 cm from the column base.
Dry nitrogen from a compressed gas cylinder formed the
bubbles. The column was operated at atmospheric conditions, and the gauge pressure measured in the tubing immediately upstream of the nozzle served to characterize the
bubbling.
The gas train used in the experiments was similar to those
used in previous studies (Nguyen et al., 1996). It consisted
of two pressure regulators, a rotameter with a stainless-steel
;oat (Shor-Rate II, tube number R-2-15-D, Brooks Instruments), piezo-electric valve (MaxTek MV-112), ;ow
sensor (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 a
ball valve connected in series. This system supplied measured gas ;ow rates covering a range of 10 –500 cc=min.
The tubing volume from the piezo-electric valve to the
nozzle was minimized to reduce the dynamics of gas
compression.
Details of the nozzle design are shown in Fig. 2. The
nozzle was constructed from 6.35-cm-o.d. Lucite tubing. A
6.35-cm-o.d., 1.0-mm thick brass ori#ce plate was secured
at the end of the nozzle. This plate could be electri#ed by
a 22-gauge, uninsulated copper wire passing through the
nozzle to a connector on the gas inlet line.
An ori#ce diameter of 0:75 mm was used to avoid indistinct return maps that have been obtained with larger ori#ce diameters (Mittoni et al., 1995). Preliminary experiments with applied electrostatic potential conducted in this
study using a 1.0-mm ori#ce obtained results with irregular
spikes and peaks in the pressure time series. Three design
features allowed the system to be operated such that liq-
uid that may have weeped through the ori#ce during setup
could not adversely a9ect the results by causing intermittent
bubbling. These were the relatively large inside diameter
of the nozzle tube, a tee-bend in the gas inlet, and a drain
valve.
A high-voltage, direct current power supply (model
225-50R, Bertan High Voltage Corp.) was connected to the
nozzle tip with positive polarity. A 3.2-mm-diameter stainless steel rod immersed 2:5 cm into the upper surface of the
glycerol served as the counter electrode and was connected
to electrical ground. The electrode was not inserted farther
into the liquid in order to minimize the electrode surface
area in contact with the liquid. With this con#guration, a
potential of approximately 13 kV could be attained at the
nozzle tip before the 0.3-mA current safety limit of the
power supply was reached.
A capacitive transducer (model 228, Setra Systems Inc.)
having a range from 0 to 1 psig was used to measure the
pressure in the gas line upstream of the nozzle. The output from the pressure transducer was a 0 –5 V DC analog signal, which was fed through a signal conditioning
card signal ampli#er to a data acquisition board (models
SC-2043-SG and PCI-MIO-16E-50, National Instrument) in
a 300-MHz Pentium IITM -based personal computer. National
Instruments LabviewTM 5.1 was used as the data acquisition
software.
For consistent experimental results, a fresh batch of glycerol was used for each run. Each run consisted of data collected at a speci#c ;ow rate and changing the applied voltage
in increments of 1 kV from 0 V to 10 kV. Time intervals
of 300 s were provided between successive readings. The
pressure data were collected for 50 s at 2000 or 5000 Hz at
each set of experimental conditions.
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S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
(9)
(8)
(4)
(5)
(10)
(2)
(6)
(1)
(3)
(7)
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 the
characterization of nonlinear bubble-formation dynamics.
Power spectra: The classical linear method of Fourier
analysis was used to transform the time-series information
into the frequency domain, which has been shown to be
sensitive to changes in periodicity (FDemat et al., 1998). Although not de#nitive for nonlinear time series, the power
spectra yield useful information for continuous physical systems as a pointer to the relevant time scales and the choice
of parameters in nonlinear time-series analysis.
Delay embedding: We used the delay embedding
(Takens, 1981) of pressure measurements to characterize the attractor. Given the set of measurements
{xi |i = 1; : : : ; n}, the sequence of vectors formed as
xj = [xj ; xj+
; xj+2
; : : : ; xj+(m−1)
]T (where j is a time index)
can be used to reconstruct the dynamic trajectory of the
system. Takens (1981) proved that if some m is suAciently
large, the embedding vectors preserve the geometrical properties and invariants of the system. Our observations indicated that typically an embedding dimension of 5 and delay
of 35 are appropriate embedding parameters for resolving
the injected bubble patterns. We also apply local principal
component analysis (PCA) to the embedded data to allow
the resulted to be projected into two or three dimensions
for easier viewing.
Period-doubling bifurcation and route to chaos: Bifurcation plots have been widely used (e.g., Nguyen et al., 1996;
Tufaile et al., 1999; Tufaile and Sartorelli, 2001) to illustrate the onset of complex dynamics as a result of variation in some process input variable. For the bubbling process, the changes were quanti#ed in terms of the period
of formation of the bubble. In the context of this experiment, a plot of bubbling rate (or period of bubble formation)
against the gas ;ow rate or the electrostatic potential is a
bifurcation diagram. We also generated three-dimensional
bifurcation plots involving both system variables—gas ;ow
rate and electrostatic potential—to illustrate the simultaneous e9ect of electrostatic potential and ;ow rate on bubble
formation.
Time return maps: Time return maps (Moon, 1992) were
used to condense the information of time series and to help
determine the periodicity of the system. Period-of-formation
intervals were generated by measuring the peak-to-peak time
intervals of the pressure time-series data. A peak in the pressure trace corresponds to the beginning of bubble growth at
the nozzle, and the peak-to-peak time-interval corresponds
to 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 and
Sartorelli, 2000) has shown that as gas ;ow is increased,
bubble formation undergoes a bifurcation process that is
readily detected by observing variations in bubble size and
speed. At low ;ow, it is observed that a regular train of identical bubbles is produced. In addition to being of equal size,
S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
(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 produces
a train of identical pressure peaks in the nozzle (Fig. 3a).
Increasing gas ;ow increases the frequency of bubbling and
leads to interaction between bubbles (Fig. 3b). Speci#cally,
trailing bubbles are accelerated by the wakes of preceding
bubbles. At the nozzle, this gives rise to the formation of
two alternating size bubbles (period-2), and the pressure
pulses contain two distinct alternating peaks (Fig. 3c).
With still greater gas ;ow, the system enters a state in
which four distinct bubble types are formed (period-4).
This period-doubling process continues progressively
with increasing ;ow rate, #nally leading to deterministic
chaos.
Fig. 4 shows examples of pressure time-series data for
four di9erent ;ow rates that resulted in periods 1, 2, 4 and
chaos. Subplots 4 (a) – (d) contain 2-s pressure traces. As
the ;ow rate is increased, the bubbling frequency increases,
and the number of distinct peaks appearing in the pressure
trace increase. In 4(d), the number of distinct peaks has become e9ectively in#nite due to the onset of deterministic
251
chaos. Under these conditions, there are never two bubbles
produced with exactly the same characteristics. Figs. 4(e)–
(h) show the three-dimensional delay embedding formed
with the time-series segments shown in Figs. 4(a) – (d), with
embedding delay of 35. The single band in Fig. 4(e), gives
way to two distinct bands in 4(f)—leading from period-1 to
period-2 behavior. Fig. 4(g) has four distinct bands, indicating period-4, and Fig. 4(h) has many bands with fractal
structure—a sign of chaotic dynamics.
When an electrostatic potential is applied to the injection
nozzle, electric stresses cause a “pinching” action on the
bubbles during formation (Tsouris et al., 1994). The visual
e9ects of applied voltage in our experiments are illustrated
in Fig. 5, which compares the images of bubbles formed at
a #xed ;ow rate under applied potentials of 0 and 12:5 kV.
The images shown are at a point in time 4 ms before the
bubbles detached from the nozzle. It is seen that the bubble
formed with an applied voltage is smaller and more spherical. Increasing electrostatic potential hastens the detachment
of bubbles, which leads to greater and more signi#cant interactions between bubbles. These results demonstrated how
increasing gas ;ow a9ects the bubbling dynamics. Now we
examine the impact of increasing the electrostatic potential
at a given ;ow rate.
Fig. 6 shows pressure traces collected at 2 kHz for
bubbling at constant gas ;ow of 335 cc=min and multiple applied potentials of 1–9 kV (increments of 1 kV).
A subtle variation can be observed in the time series in
terms of an increase in the number of peaks observed, a
decrease in the peak heights, and increasing irregularity in
the peak heights with increasing potential. On average, the
bubbling frequency also increases with increasing potential. The system is chaotic in subplots (h) and (i). Fig. 7
shows the local principal component scores for the time
series in Fig. 6. Note that with increasing electrostatic potential, the banding in the plots becomes more complex and
the system is chaotic for potentials of 8 and 9 kV. Fig. 8
shows the spectral densities for the time series shown in
Fig. 6. The distribution becomes broader with increasing
potential.
Fig. 9 illustrates an alternative approach for observing the
dynamic changes induced by the electric potential in terms
of the characteristic intervals between bubbles. In this #gure,
successive time interval values between bubbles are plotted
as two-dimensional ‘return maps’ for the same conditions in
Fig. 6. Initially (at low potential) we observe four distinct
points corresponding to the four di9erent bubble periods—
thus the system is exhibiting period-4 behavior. As the potential increases, we observe the four points merge into a
curve, representing the chaotic state. It should be noted that
this curve is actually a trace of the dynamic ‘map’ of this
system. In fact, a function #t through this map can provide
an approximate model of the bubbling dynamics. The bubble
formation interval continues to wander along this line due
to the continuous state of instability driving the deterministic chaos. Although not obvious, the average inter-bubble
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S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
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 dynamics
and bubble shape. Images of bubbles formed at a #xed ;ow rate under
conditions 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 transition is made, indicating that the bubbles produced in the
chaotic state are smaller.
Fig. 10 is a bifurcation plot in which bubble formation
intervals are plotted as a function of applied potential. In
these experiments, gas ;ow was held constant, and voltage
was gradually increased in steps of 1 kV. At low values of
the applied potential, the system exhibits period-4 behavior
as indicated by the four distinct bands of observed intervals.
With increasing voltage, the period of formation of bubbles decreases and bubble size distribution becomes broader.
At applied voltages between 7 and 8 kV, the bubbling becomes chaotic and there is almost a continuous range of
values along a wide band. Fig. 11 shows a similar bifurcation diagram obtained at a lower gas ;ow. Note that although the same general trends are apparent at the two ;ows,
the details of the bifurcations with applied potential are
di9erent.
To illustrate the combined e9ects of electrostatic potential
and gas ;ow, data collected from experimental runs at different ;ow rates and di9erent applied potentials were plotted
on the same graph to generate a co-dimension bifurcation
plot. Fig. 12 illustrates such a plot showing bubble interval
as a function of the applied voltage and ;ow rate.
Perhaps a more useful way of mapping the twodimensional bifurcations is to use the appropriate dimensionless ;ow and potentials. Tsuge’s ;ow number represents
a ratio of the combined mechanical disruptive forces—
inertia and buoyancy—to surface tension, and is de#ned as
(Tsuge, 1986)
Nw = Bo Fr 0:5 ;
(1)
where
Bo =
d2i g
and
Fr =
u2
di g
(2)
S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
253
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 ;ow
rate 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.
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S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
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 of
1 kV. The gas ;ow rate was 335 cc=min.
Fig. 10. Bifurcation of bubbling intervals with increasing applied potential
at a constant gas ;ow rate of 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 increments
of 1 kV. The gas ;ow rate was 335 cc=min.
and di is the inside diameter of the nozzle ori#ce, is the
liquid density, g is the gravitational acceleration constant, is the liquid surface tension, and u is the gas velocity through
the ori#ce. Bo and Fr refer to Bond and Froude numbers,
respectively. However, electric forces are also acting on the
bubbles, 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)
S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
255
where da is the distance from the submerged electrode tip to
the nozzle and L is the permittivity of glycerin. Assuming
g ∼
= 0 where 0 is the permittivity of air, Eq. (5) reduces to
Eg =
1
V
di [1 + (da =di )1=K]
Ea =
1
V
;
di K [1 + (da =di )1=K]
and
(6)
where K = g =a g =0 is the dielectric constant of the
ambient phase.
If the ambient (liquid) phase is a perfect conductor, i.e.,
K ≈ ∞, and the electric #eld in the liquid will be zero
and the liquid would be isopotential. In that limiting case,
Eq. (6) becomes
Fig. 11. Bifurcation of bubbling intervals with increasing applied potential
at a constant gas ;ow rate of 290 cc=min.
Eg = lim
V
V
1
=
di [1 + (da =di )1=K] di
Ea = lim
V
1
= 0:
di [K + da =di ]
K→∞
K→∞
be
and
(6a)
The jump in the electric stress tensor at the interface would
(g − a =K 2 )
1
V2
Te = (g Eg2 − a Ea2 ) = 2
2
2di [1 + (da =di )1=K]2
=
V 2 g
(1 − g =a )
:
2d2i [1 + (da =di )1=K]2
(7)
Note that if a g and K 1, Eq. (6) reduces to
Te =
lim
g a ; K→∞
1
= g
2
Fig. 12. Co-dimension bifurcation diagram of bubbling for changes in
voltage and ;ow rate.
where g is the permittivity of the gas and where a is the
permittivity of ambient phase. Since
Ea =
V − Vi
da
and
Eg =
Vi
di
(4)
where Vi is the potential at the interface. Eliminating Vi from
Eqs. (3) and (4), we obtain
Eg =
V
[di + da g =a ]
Ea =
1
V
;
di [a =g + da =di ]
and
(5)
V
di
Te =
2
≈
V 2 g
2d2i
1
g Eg2 :
2
(7a)
A ratio of this stress to the stress caused by surface tension
forces can be constituted as
Be =
V 2 g
(1 − g =a )
2Te
=
;
=di
di [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
Be =
lim
g a ;K→∞
Be =
di g Eg2
V 2 g
=
:
di
(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 impact
of electric stress as compared to surface tension for all nozzles. For low to moderately conducting liquids, placing the
ground electrode closer to the nozzle (decreasing da ) would
allow one to achieve higher electric stresses given the same
potential di9erence.
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S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
Fig. 13. Bubbling stability map as correlated with Tsuge ;ow and electric
Bond numbers. Legend: open circle (period-1), plus sign (period-2), open
square (period-4) and #lled circles (chaos).
Note that in this derivation, we have assumed that the
electric #eld is normal to the interface, and the tangential
components have been omitted. Regardless of the polarity, the electric #eld is normal to the interface and is directed inward towards the gas phase. Ideally, we should
have conducted the experiment with a very small da so that
the gas phase would be the dominating resistance. However, that could not be performed as our intention was to
reduce the electrode area in contact with the liquid. We also
ignored the aspect ratio, as the #eld is normal to the interface, and the shortest path through the gas phase is the
nozzle 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 both
dimensionless groups are scaled against surface-tension
forces, they allow comparison of the relative e9ect of forces
caused by gas ;ow and those caused by the applied electric
#eld. We note that for chaotic dynamics, the Tsuge number
was 16.4 and the electric Bond number was greater than
0.2. This may indicate that electrostatic forces have greater
impact on bubbling dynamics than ;ow forces; however,
further data obtained at larger electric Bond numbers and
smaller Tsuge ;ow numbers would be necessary to verify
this assertion.
It would be desirable to de#ne a system variable incorporating the e9ects of both applied voltage and ;ow rate that
could be used as a predictor of bubbling regime. In order
to capture the collective e9ect of electrostatic and inertial
forces on bubble formation from an electri#ed nozzle, Shin
et al. (1997) suggested a modi#ed Weber number that is the
ratio of the sum of the electrostatic and inertia forces to surface tension forces. That modi#ed Weber number (simply
the sum of electrical Bond number and Weber number) is
Fig. 14. Bubbling stability map as correlated with Tsuge ;ow and modi#ed
Weber 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 and
electric Bond number (in limiting case, assuming the liquid phase is a
perfect conductor). Legend: open circle (period-1), plus sign (period-2),
open square (period-4) and #lled circles (chaos).
de#ned as
d i a E 2 + u 2 d i a
;
(9)
where a is the gas density. Fig. 14 replots Fig. 13 with
modi#ed Weber number replacing the electric Bond number as the ordinate. The derivation in Eqs. (7) and (8) is a
very important correction in estimating electric stresses at
the interface. If we had assumed the liquid phase to be perfectly conducting, the electric Bond numbers would have
reached up to 20, compared to the maximum value of 0.33 in
Fig. 13. Fig. 15 replots Fig. 13 but replaces the electric Bond
WeS =
S.U. Sarnobat et al. / Chemical Engineering Science 59 (2004) 247 – 258
number de#ned in Eq. (8) by that de#ned in Eq. (8a). Under this scaling, it is apparent that electrostatic forces have
much lower impact on bubbling dynamics than ;ow-related
forces.
5. Conclusions and recommendations
Our results demonstrate that, at constant ;ow rate, bubble formation dynamics from an electri#ed nozzle exhibits
the classic signs of a period-doubling bifurcation to chaos
with increasing applied potential. With increasing voltage,
the bubble frequency increases and the bubbling undergoes
period-doubling bifurcations to become chaotic, similar to
that for increasing ;ow rate. Similar behavior has been observed in liquid–liquid systems under electrostatic spraying (Tsouris et al., 1994). Although the basic type of bifurcation is similar for applied voltage and ;ow, voltage
has a proportionally smaller e9ect than ;ow. Application
of voltage also reduces bubble size, apparently by promoting early bubble release. The smaller bubbles have higher
interfacial surface area for heat and mass transfer, and this
might have bene#cial applications in gas–liquid contacting
devices.
Use of applied electrical potential for bubble size control
should be further investigated because it would be possible to manipulate this variable very quickly and thus provide very fast feedback perturbations (e.g., much faster than
can be obtained with gas ;ow perturbations). A possible
limitation is the relatively large voltage amplitude needed
to achieve signi#cant e9ects. This study is the #rst one to
consider the case where both the ambient and gas phases
were conducting. Other studies have assumed both of these
to be either perfect conductors or perfect nonconductors.
An expression for electric Bond number was derived for
the current apparatus, and it was shown that it reduces
to the expression for other, limiting cases studied in the
literature.
In the present study, the e9ects of electro-hydrodynamic
(EHD) ;ows were assumed to be negligible and were not
studied. Future studies should include EHD ;ows induced
with di9erent shapes of nozzles, various chamber sizes and
changing the nozzle diameter. The particular liquid used
is another important parameter, and consideration should
be given to similar studies in immiscible liquid–liquid
systems.
Acknowledgements
This work was supported by the Division of Chemical
Sciences, OAce of Basic Energy Sciences, US Department of Energy, under contract DE-AC05-00OR22725 with
UT-Battelle, LLC. The authors thank Costas Tsouris of
ORNL for helpful comments.
257
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