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I. J. Trans. Phenomena, Vol. 5, pp. 247-258
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Numerical Study on Motion of a Single Bubble
Exerted by Non-uniform Electric Field
Y. TAKATA1*, H. SHIRAKAWA2, K. TANAKA1, T. ITO1
2
1
Department of Mechanical Engineering Science, Kyushu University, Japan
Department of Mechanical Engineering, Toyama National College of Technology, Japan
(Received March 15, 2002; In final form July 2, 2002)
Numerical analysis has been performed on a bubble growth and deformation in an electric
field in order to disclose the mechanisms of boiling heat transfer enhancement by EHD
(Electro-Hydrodynamic) effects. Transient Navier-Stokes and Maxwell equations were
solved simultaneously for both liquid and vapor phases in a two-dimensional cylindrical coordinate systems making use of VOF (Volume of Fluid) method. The bubble motion in liquid R113 under atmospheric pressure has been simulated. First, an elongation of a single
bubble in a uniform electric field is simulated and the oblateness of the bubble is in good
agreement with Garton’s analytical and experimental results. Second, the bubble deformation process in a non-uniform electric field was simulated. A bubble, initially attached to the
lower electrode, starts to deform and finally takes off from the lower electrode. The shape of
the bubble depends on the intensity of the electric field. The behavior of bubbles, the velocity vectors, and the contour of the electric field are shown, and an experiment has been performed to verify the results of the numerical simulation.
Keywords: Volume of fluid, EHD, Bubble, Electric field, Boiling heat transfer, Surface tension
INTRODUCTION
and heat transfer surface, the mechanism of enhancement in boiling from a thin wire is essentially different from that in boiling from plane surface.
Therefore, the geometrical configuration of electrodes
and heat transfer surface should be taken into consideration when we discuss the effect of EHD on heat
transfer enhancement. In addition, it should be noted
that the heat transfer is not always enhanced by EHD
(Ito et al., 1994).
We must know the inherent nature of electric force
acting on a bubble to disclose the mechanisms of heat
transfer enhancement. Some numerical analyses
(Ogata and Yabe, 1991; Ito et al., 1991) have been
Heat transfer enhancement by EHD effect has
been of interest in recent years because of the advantages that (1) the consumption of electricity is very
low for its enhancement performance, and (2) heat
transfer rate can be controlled by changing intensity
of electric field. It was reported that a very large
enhancement in heat transfer can be anticipated when
it is applied to boiling heat transfer in dielectric liquids (Yabe, 1991; Ogata and Yabe, 1991; Ito et al.,
1993). Since the effect of EHD on the heat transfer
depends strongly on the configuration of electrodes
* e-mail: [email protected]
247
248
TAKATA, et al.
(a)
(b)
FIGURE 1
Analytical model
(a) Uniform electric field; (b) Non-uniform electric field
conducted from this point of view. Characteristics of
these studies are that (1) simple configuration such as
a sphere or a semi-sphere is treated, (2) it is the solution for the static balance of forces between surface
tension and electric pressure acting on the interface,
(3) flow originated by electric force is not taken into
account, (4) the process of deformation is not clear.
In the present study we performed numerical analysis
of a bubble motion in uniform and non-uniform electric fields using an improved VOF method developed
by Takata et al.[1999]. Mechanism of the origin of
fine boiling bubble is investigated. Experimental
observation of the behavior of air bubble in R113 liquid is conducted and compared with the numerical
result.
is adopted as test fluid because its electric properties
are available. A bubble initially at a sphere shape is
placed in the dielectric liquid between two parallel
disks. At t=0 electric potential is applied to both electrodes, and subsequent behavior of bubble is
observed. The coordinate system is cylindrical, and
axisymmetric condition is set at r=0. As shown in
Figure 1, two types of electrodes are used. Electrode
in Figure 1(a) generates uniform electric field, whereas that in Figure 1(b) generates non-uniform one. We
assumed that electric conductivity of R113 is zero
and that both vapor and liquid are treated as incompressible fluids with constant properties. Temperature
in the whole computation domain is assumed to be
uniform and saturated, and therefore energy conservation equation is not solved.
BASIC EQUATIONS AND NUMERICAL
METHOD
Basic equations and boundary conditions
Physical model and assumptions
Physical model is shown in Figure 1. R113 liquid
Basic equations in cylindrical coordinate system
are written as follows:
NUMERICAL STUDY ON MOTION OF A SINGLE BUBBLE
249
Equation of continuity:
(8)
(1)
Since the electric conductivity is assumed to be zero,
Coulomb force term does not appear in equations (7)
and (8). The Clausius-Mossoti’s relation (Panofsky
and Phillips, 1962)
Navier-Stokes equation:
(9)
(2)
is used in the second term of equations (7) and (8). In
equation (9) c is a constant. Since the characteristic
time of the EHD phenomena considered here is much
smaller than the relaxation time of electric charges
for R113 (Yabe, 1991), the distribution of electric
field is given by solving
(3)
where τrr, τzz and τrz are stress tensors and expressed
as follows.
(10)
where
(4)
(11)
(5)
(12)
(13)
(6)
In equations (2) and (3), fpsr and fpsz are r- and z-components of surface tension term, respectively, in the
form of body force that appears only at the liquidvapor interface. fer and fez are r- and z-components of
electric force, respectively, and given by
Equations (10) to (13) are fundamental equations to
obtain the electric field.
The shape of interface is traced by solving the following transport equation for F that is defined as the
volume fraction of liquid in the control volume.
(14)
(7)
F =1 for liquid phase, F=0 for vapor, and 0<F<1 for
interface cells. Properties at the interface are averaged with F as follows:
250
TAKATA, et al.
FIGURE 3
Ratio of the major and minor semi-axis
(18)
FIGURE 2
Elongation of bubbles in uniform electric field (no gravity)
(19)
(15)
(16)
(20)
(17)
Boundary conditions are given by the following
equations:
(21)
NUMERICAL STUDY ON MOTION OF A SINGLE BUBBLE
251
FIGURE 4
Bubble motion in non-uniform electric field. (Voltage between electrodes: 5kV, no gravity)
(22)
Since there is no true electric charge in the liquid and
polarity makes no difference, the upper pole is set to
positive for convenience. Electric potential φ is
selected as an unknown quantity and the intensity of
electric field E is calculated using φ by equations (11)
and (12).
252
TAKATA, et al.
FIGURE 5
Pressure distribution on lower plate corresponding to Figure 4(c)
FIGURE 6
Velocity vectors corresponding to Figure 4(c)
TABLE 1
Calculation conditions
Distance between electrodes, l
Radius of upper electrode, ro
Radius of lower electrode
(a) uniform electric field, ro
(b) non-uniform electric field, ri
Initial bubble radius:
Specific dielectric constant of vapor, εv*
Specific dielectric constant of vapor, εl*
Surface tension, σ
Contact angle
Electric potential
1 mm
0.5 mm
0.5 mm
0.2 mm
0.1 mm
1.01
2.41
14.6 mN/m
15°
5, 10, 20 kV
Numerical methods
Fundamental equations are transformed into finite
difference equations and solved numerically by the
SOLA algorithm (Nichols et al., 1980). We used the
Gaussian elimination method to solve simultaneous
equations. It is very important to ensure the sufficient
accuracy in calculation of interface curvature because
the surface tension is dominant for small bubble. The
curvature must be calculated from F. Since the accu-
FIGURE 7
Electric forces fe corresponding to Figure 4(c)
racy of the original calculation procedure (Nichols et
al., 1980) is insufficient for the present problem, we
improved the calculation procedure of interface curvature (Takata et al., 1999). We also improved the
accuracy of the donor-acceptor method (Takata et al.,
1999) in the calculation of equation (14). The compu-
NUMERICAL STUDY ON MOTION OF A SINGLE BUBBLE
253
FIGURE 8
Bubble motion in non-uniform electric field. (Voltage between electrodes: 10kV, no gravity)
tation procedure is as follows:
1) Calculate F at the next time step by equation (14)
2) Calculate properties by equations (15), (16) and
(17).
3) Calculate forces by surface tension, fpsr and fpsz,
for all interface cells.
4) Calculate electric forces, fer and fez, using E at the
previous time step.
5) Solve equations of continuity and Navier-Stokes
by SOLA algorithm to obtain u, v and p.
6) Solve equation (10) to determine the distribution
of electric field.
7) Return to the step 1).
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TAKATA, et al.
FIGURE 9
Bubble motion in non-uniform electric field. (Voltage between electrodes: 20kV, no gravity)
These procedures are repeated until preset time.
NUMERICAL RESULTS AND DISCUSSIONS
R113 is selected as the test fluid and the calculation conditions are summarized in Table 1. Departure
radius of bubble calculated by Fritz’s experimental
correlation is 0.157mm for a contact angle of 15°.
Other properties used in the calculation include those
at the saturation temperature under atmospheric pressure. The radii of lower electrodes differ in uniform
and non-uniform cases.
Most of the calculations have been performed
without gravity to observe pure electric effect. For
confirmation, a bubble attached to the lower surface
NUMERICAL STUDY ON MOTION OF A SINGLE BUBBLE
255
was simulated with gravity and without electric field.
The top of the bubble is lifted up, but it stays on the
surface.
Behavior of bubble in uniform electric field
Figures 2(a), (b) and (c) show the behaviors of
bubble under the uniform electric fields of 5, 10 and
20kV, respectively. The initial bubble is a sphere and
placed at the center of two electrodes. Bubbles in
these figures are in almost steady state. The computation domain is a quarter of these figures and the number of mesh is 40×40. Stripes in the figure indicate
isopotential lines. It is found that the bubble is more
elongated as the intensity of electric field increases.
The relation between the intensity of uniform electric
field, E, and the ratio of the major and minor semiaxis of the elliptic bubble, γ, is shown in Figure 3.
The solid circle ● indicates the result of the present
calculations, and the solid line the theory by Garton
et al.(1964). Both results below 10kV agree with
each other, while in case of 20kV the present result is
larger than that by Garton et al. (1964). This deviation may cause from the fact that they dropped the
second term in righthand-side of equations (7) and
(8) assuming that the bubble does not disturb the
electric field.
Behavior of bubble in non-uniform electric field
Figure 4 shows the behavior of a bubble and the
distribution of electric potential under the voltage
between electrodes of 5kV. The bubble is initially
attached to the lower electrode with contact angle of
15°. Non-uniform mesh is used in the calculation so
as to get a dense number of grids near the lower electrode.
The electric force acts on the bubble and reduces it
to conical shape, and then the bubble finally takes off
from the lower electrode and moves upward with
subsequent deformation. From Figure 4(a)~(i) the
average velocity during 450µs is 0.449m/s, and the
average acceleration calculated from this value is
998m/s2, which is 100 times as much as gravity. The
result from Figure 4 indicates that the bubble takes
FIGURE 10
Experimental apparatus
off and moves upward by the force generated by the
electric field. This implies that in nucleate boiling
bubbles depart before they grow up to usual departure diameter, and in other words bubbles become
finer as compared with normal boiling. Figures 5, 6
and 7 show pressure distribution on the lower plate,
velocity vectors around the bubble and electric force
vectors, fe, respectively, corresponding to Figure 4(c).
Under the non-uniform electric field the electrostriction force increases pressure in the region where the
lines of electric force are intensive. In case of Figure
5 the maximum pressure rise in gauge reaches
260kPa near the edge of lower electrode. It is also
found from Figure 5 that pressure distribution has a
kink at the vapor-liquid interface, which is caused by
surface tension. In Figure 6, large upward velocity
vectors are observed inside the bubble and it is found
that the bubble is going to rise. Figure 7 indicates that
256
FIGURE 11
Bubble behavior in non-uniform electric field
TAKATA, et al.
NUMERICAL STUDY ON MOTION OF A SINGLE BUBBLE
the vectors of electric force, fe, are directed toward
the edge of lower electrode. At a glance it may seem
that the vapor-liquid interface moves to the liquid
phase because the direction of fe is from vapor to liquid. However there is high pressure region near the
edge of lower electrode, then this high pressure blows
the bubble off and consequent flow becomes as
shown in Figure 6.
Figures 8 and 9 show the bubble detachments for
the cases of 10kV and 20kV, respectively. It is understood that the time to detachment becomes short as
the voltage between electrodes increases. Subsequently the bubble is elongated when it reaches at the
center between electrodes. In Figure 9 it is observed
that the bubble splits at 50µs and a small satellite is
created.
Above results explain how fine bubbles appear in
nucleate boiling under the non-uniform electric field.
However it is still unknown how the electric field
works on the heat transfer enhancement and it is
required to solve the energy equation coupled with
the present basic equations.
257
28kV. It is observed that the bubble moves downward
against the gravity changing its shape.
Acrylic resin is used to hold the electrodes and
consequently the lines of electric force can penetrate
into the acrylic resin. On the contrary, the rim of
smaller electrode of numerical model was set to perfect insulator that is impervious to the electric field.
This difference will produce a discrepancy between
the distributions of electric field in experiment and
numerical analysis, and then the quantitative comparison between them is difficult. Referring to Figure 11
the bubble deforms to the lens-like shape immediately after the detachment from upper electrode, and the
tendency of deformation looks like it does in Figure
4. Not shown in Figure 11, the bubble moves upward
by the buoyancy force after 10ms, then deviates from
the electric fields and finally reaches to the liquid surface. This experiment confirms that the bubble under
non-uniform electric field moves to the direction
where the electric field is weaker, and this result
agrees with the numerical analysis.
CONCLUSION
EXPERIMENT
In order to demonstrate the numerical results we
observed the movement of a bubble under non-uniform electric field by the experimental apparatus as
illustrated in Figure 10. Diameters of upper and lower
electrodes are 0.8mm and 4mm, respectively, and the
distance between electrodes is 2mm. The size is
almost double that of the numerical model except for
the lower electrode and the position of two electrodes
is upside down to hold the bubble at the smaller electrode before electric charge because buoyant force
acts on the bubble. Test liquid is R113 and air bubble
is used instead of R113 vapor. The specific dielectric
constant of air is 1.01 and is equal to that of R113
vapor. Temperature of liquid is kept constant at 20°C.
Diameters of air bubble are in the range between
0.4~0.8mm because it is difficult to make fixed size
bubbles. Figure 11 is a series of high-speed motion
pictures of the bubble behavior. The diameter of bubble is 0.54mm and the voltage between electrodes is
Numerical analysis of the behaviors of a bubble
under uniform and non-uniform electric fields has been
conducted. The elongation of bubble under uniform
electric field agrees with the result by Garton and Krasucki [1964] Bubble departure process in a non-uniform field has been simulated and it was found that
under a non-uniform field the size of departing bubble
decreases with the intensity of the electric field. The
present numerical simulation explains the mechanisms
of fine bubble creation. Behaviors of a single air bubble in non-uniform field were photographed by a highspeed video camera and its movement qualitatively
agreed with the numerical results.
NOMENCLATURE
E
fe
intensity of electric field [V/m]
electric force [N/m3]
258
fps
F
g
l
r
ri
ro
p
t
u
v
z
ε
ε0
µ
ρ
τ
φ
TAKATA, et al.
body force by suface tension [N/m3]
Volume fraction of liquid [-]
gravity [m/s2]
distance between electrodes [m]
radial coordinate [m]
radius of lower electrode [m]
radius of upper electrode [m]
pressure [Pa]
time [s]
velocity in r-direction [m/s]
velocity in z-direction [m/s]
vertical coordinate [m]
dielectric constant [C/Vm]
dielectric constant of vacuum [C/Vm]
viscosity [Pa·s]
density [kg/m3]
stress tensor [N/m2]
electric potential [V]
Subscripts
l
Liquid
r
r-component
z
z-component
v
Vapor
+
positive pole
−
negative pole
REFERENCES
Garton, C. G. and Krasucki, Z. (1964), Bubbles in Insulating Liquids: Stability in an Electric Field, Proceedings of the Royal
Society, A280, pp.211-226.
Ito, T., Takata, Y., Tanaka, K. and Irie, Y., (1993), The Effect of
Electric Field on Bubble and Boiling Heat Transfer, 30th
National Heat Transfer Symposium of Japan, pp.829-931 (in
Japanese).
Ito, T., Takata, Y., Tanaka, K., Shirakawa, H. and Katayama, M.
(1994), Heat Transfer Enhancement by Electric Field: Augmentation of Free Convection and Suppression of Nucleate
Boiling by Non-uniform Electric Field, JSME: Thermal Engineering Conference ‘94, pp.119-121 (in Japanese).
Nichols, B. D., Hirt, C. W. and Hotchkiss, R. S., (1980), SOLAVOF: A Solution Algorithm for Transient Fluid Flow with
Multiple Free Boundaries, Los Alamos Scientific Laboratory,
Rep. No. LA-8355
Ogata, J. and Yabe, A. (1991), Augmentation of Nucleate Boiling
Heat Transfer by Applying Electric Fields: EHD Behavior of
Boiling Bubble, Proceedings of the ASME/JSME Themal
Engineering Joint Conference 1991, Vol.3, pp.41-46.
Panofsky, Q. K. H. and Phillips, M. (1962), Classical Electricity
and Magnetism, Addison-Wesley.
Takata, Y., Shirakawa, H., Kuroki, T. and Ito, T. (1999), An
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