I. J. Trans. Phenomena, Vol. 5, pp. 247-258 Reprints available directly from the publisher Photocopying permitted by license only © 2003 Old City Publishing, Inc. Published by license under the OCP Science imprint, a member of the Old City Publishing Group. 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@example.com 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.. 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). 254 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  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 Improved VOF Method and Its Applications to Phase Change Problems, Proceedings of the 5th ASME/JSME Thermal Engineering Joint Conference (CD-ROM), AJTE99-6424 Yabe, A. (1991), Active Heat Transfer Enhancement by Applying Electric Fields, Proceedings of the ASME/JSME Thermal Engineering Joint Conference 1991, Vol.3, pp.xv-xxiii.