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EHD behavior of nitrogen bubbles in DC electric fields
Experimental Thermal and Fluid Science 32 (2007) 174–181 www.elsevier.com/locate/etfs
EHD behavior of nitrogen bubbles in DC electric fields F. Chen, Y. Peng, Y.Z. Song *, M. Chen Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China Received 7 September 2006; received in revised form 25 February 2007; accepted 7 March 2007
Abstract The deformation of nitrogen bubbles injected into transformer oil with various DC electric fields was studied experimentally and theoretically. The bubble deformation was visualized by a high speed camera. The major axis of the bubble was elongated along the direction parallel to the electric field, with the elongation increasing as the electric field strength was raised. The electrical Weber number (We) was used to correlate the electric field strength and the dielectric permittivity of the working fluid to the bubble relative aspect ratio (ARe/ AR0). The experimental results show that the relative aspect ratio increases with increasing We. The electric stresses were calculated on an actual bubble shape including the electrostriction stresses to analyze the bubble elongation. 2007 Elsevier Inc. All rights reserved. Keywords: Electrohydrodynamic (EHD); Bubble deformation; Electric field; Electric stresses
1. Introduction Electrohydrodynamic (EHD) enhancement is an active heat transfer enhancement method, which has attracted considerable interest since Chubb [1] claimed that the water evaporation rate was increased threefold by an electric field in 1916. Extensive experiments have been conducted to study the heat transfer enhancement and the behavior of a single bubble attached to a wall in DC and AC electric fields. The preliminary experimental results include the increases of the bubble departure aspect ratio, the increases of the bubble departure frequency and the increases of the maximum heat flux, the decreases of the bubble departure volume and the elimination of boiling hysteresis by electric fields [2–11]. In these studies, the electric field effects on the bubbles were of particular concern because the bubble behavior changes in the presence of an electric field were believed to be one of the main reasons for the enhancement of the pool boiling heat transfer.
*
Corresponding author. Tel.: +86 10 62772938; fax: +86 10 62795832. E-mail address: [email protected] (Y.Z. Song).
0894-1777/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2007.03.006
In the early years, Melcher and Taylor [12] calculated the electric stresses on a spherical drop or vapor void suspended in a liquid assuming incompressibility. They presented a discrimination factor to distinguish between oblate and prolate shapes of drops or vapor voids with an electric field, which indicated that the electric field led to the bubble deformation. Later, Cheng and Chaddock [13] analyzed the bubble deformation and stability by considering the free energy in a uniform electric field and assumed that the bubble was spheroidal, an approximation to the actual bubble shape. Cho et al. [14,15] numerically and experimentally investigated the effects of a uniform electric field on a bubble attached to the wall without making the spheroidal approximation. The electrostriction force was neglected in their calculation. The results showed that the bubble was elongated parallel to the direction of the applied electric field. The elongation increased as the electric field strength increased. Recently, experiments with a single bubble in a uniform electric field by Dong et al. [16] showed interesting bubble detachment and break-up phenomena. However, the basic mechanisms for the EHD effect are still not completely understood. The present paper describes the behavior of a single nitrogen bubble injected
F. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 174–181
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Nomenclature AR E E L P Q T T U We d r rd w z
bubble aspect ratio, AR = d/w electric field strength (kV/mm) electric field strength vector electrode distance (mm) pressure (Pa) flow rate (mm3/s) Maxwell stress (N/m2) temperature (K) voltage (kV) electrical Weber number major axis of the bubble (mm) radial direction bubble equivalent radius at departure (mm) minor axis of the bubble (mm) symmetry axis
Greek Symbols c surface tension (N/m) e permittivity of the dielectric fluid (F/m) h angle around the bubble (0) into 25# transformer oil in a uniform electric field. The experimental conditions could be well controlled by employing bubble injection rather than boiling. Bubble formation in a DC electric field was visualized by a high speed camera. In addition, the electric field distribution and electric stresses acting on the actual bubble surface were analyzed. 2. Experimental apparatus and procedure The experimental apparatus consisted of a test cell, a high voltage power supply, a nitrogen bubble injection sys-
l qe qm r u
viscosity (kg/(m s)) electric charge density (C/m3) fluid mass density (kg/m3) electrical conductivity (S/m) orifice diameter (mm)
Subscripts DEP dielectrophoretic EHD electrohydrodynamic STR electrostriction e electric field g gas l liquid n normal direction r relative 0 absolute or initial s bubble surface t tangential direction
tem and a recording system as shown in Fig. 1. The experiments were conducted at 281 K and 1.013 · 105 Pa. The working fluid was 25# transformer oil, selected for its high resistivity, non-toxicity and non-flammability at ambient temperature and pressure. The injected gas was 99.999% Table 1 Transformer oil and nitrogen properties at P = 1.013 · 105 Pa and T = 293 K
Nitrogen Transformer oil
qm (kg/m3)
l (kg/(m s))
er
c (N/m)
re (S/m)
1.165 849
– 0.0202
1.0 2.1
– 0.044
1.0 · 1016 3.3 · 1012
Fig. 1. Schematic of the experimental setup.
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Table 2 Variation of the flow rates with increasing electric field strength (L = 10 mm, u = 1.5 mm) E0 (kV/mm) Q (mm3/s)
0.0 21.86 ± 0.13
0.5 21.52 ± 0.13
1.0 20.80 ± 0.09
1.5 20.92 ± 0.11
volume percent nitrogen, selected for its chemical stability. The physical properties of the transformer oil and the nitrogen gas are listed in Table 1. The nitrogen gas flowed through a pressure regulator with an exit pressure of 0.21 MPa into a cylindrical vessel 45 cm long and 2.5 cm in diameter to stabilize the gas flow before entering the orifice. The nitrogen volume flow rate ranged from 20.06 ± 0.09 to 21.86 ± 0.13 mm3/s as given in Table 2. The last entry in Table 2 gives the flow rate after the high voltage power supply was shut off to show that the flow rate was quite stable. A schematic of the test cell was shown in Fig. 2. The test cell volume was 324.8 cm3 (11.2 cm · 14.5 cm · 2 cm). The front and back walls of the test cell were made of glass to facilitate the visualization. The sidewalls were made of epoxy resin. A brass plate served as the grounded electrode (the negative electrode). A 1.5 mm diameter orifice was drilled at the center of this plate (u = 1.5 mm). The brass plate was embedded into an epoxy resin base. The positive electrode was made of brass mesh to allow the bubbles to pass through and connected to a 50 kV high voltage power supply (Glassman ER50R06-DM22). The electrode system was shown in Fig. 3, where L was the distance between the positive and negative electrodes. The oil depth was kept at 20 mm. The parallel positive and negative electrodes generated a quasi-uniform electric field. A high speed camera (AOS VITcam high speed digital camera) was placed in front of the test cell. The frame rate was 1000 fps and the resolution was 640 · 512 pixels. Two fluorescent lamps were used to illuminate the nitrogen bubbles from the back of the test cell.
2.0 20.58 ± 0.09
2.5 20.06 ± 0.09
3.0 20.27 ± 0.09
0.0 21.22 ± 0.13
Fig. 2. Schematic of the test cell.
3. Results and discussion 3.1. Experimental observations and results The bubble growth process is shown in Fig. 4 as a function of time for electric filed strengths of 0.0 kV/mm, 1.0 kV/mm, 2.0 kV/mm and 3.0 kV/mm. Fig. 4a shows the growth of a bubble attached to the surface without an electric field. The bubble is spherical in the beginning; the lower half of the bubble gradually changes into a cone while the upside remains spherical as the bubble grows.
Fig. 3. Schematic of electrode system.
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Fig. 4. Bubble growth for various electric field strengths (L = 10 mm, u = 1.5 mm).
The bubble neck forms as departure occurs at 0.046 ± 0.001 s. Fig. 4b–d show that the bubble growth times increase as the bubbles elongate along the electric field direction with increasing electric field strength. Fig. 5 shows the bubble shapes at departure for various DC electric fields. As the electric field strength increases from 0 to 3.0 kV/mm the aspect ratio increases by 36.9%. The bubble injection was uniform in the experiments. At 0.0 kV/mm, the bubble departed at about 0.046 s with the next bubble beginning to form at 0.9 s, long after the previous bubble departed. At 3.0 kV/mm, the bubble departed at 0.306 s with the next bubble forming at 0.59 s. The major axis length, d, the minor axis length, w and the aspect ratio, d/w of the bubble at departure are illustrated in Fig. 6.
The bubble aspect ratio (AR), electrical Weber number (We) and the bubble equivalent radius at departure (rd) are defined as [13] AR ¼ d=w; We ¼
ð1Þ
e0 erl E20 rd =c; 2 1=3
rd ¼ ðdw Þ
=2;
ð2Þ ð3Þ
where e0 is the vacuum permittivity (e0 = 8.854 · 1012 F/m), erl is the relative permittivity of the transformer oil, and c is the surface tension of the transformer oil. E0, the initial electric field strength, is E0 ¼ U =d;
ð4Þ
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Fig. 5. Bubble detachment shapes for various electric field strengths (L = 10 mm, u = 1.5 mm).
Fig. 6. Schematic showing the bubble dimensions.
The relative aspect ratio (ARe/AR0) increases with increasing electric field strength for various electrode distances as shown in Fig. 7. The subscripts e and 0 denote the departure aspect ratio in the presence of electric fields and without electric fields respectively. The elongation is closely associated with the initial electric field strength. We is used to correlate the changes of ARe/AR0, as shown in Fig. 8. The small amount of scatter of the data around the correlation indicates that the Weber number has a significant effect on the aspect ratio. Fig. 8 shows that ARe/AR0 is linearly related to We. In some cases, instabilities in the air–oil interface caused by
Fig. 8. Variation of the relative aspect ratio at departure (u = 1.5 mm).
the electric field affected the bubble shapes. The interface instabilities intensified with increasing electric field strength. At L = 18 mm, the upper electrode was only 2 mm below the air–oil interface. The instabilities were so strong that the air–oil interface was below the upper electrode at times, which led to a marked difference between the experimental data and the curve fit in Fig. 8 for the strongest electric field strength. A linear least square fit of the experimental data gave: ARe =AR0 ¼ 0:05We þ 0:99:
ð5Þ
Note that the coefficients in Eq. (5) will depend on the system geometry. Regardless, We is a crucial parameter in the industrial application when the electric field is imposed. 3.2. Theoretical analysis of the electric field and electric stresses The effect of the electric field on the bubble growth process is further analyzed by calculating the electric field distribution around the bubble and the electric stresses on the actual bubble surface at departure.
Fig. 7. The aspect ratio changes with the initial electric field strength E0.
3.2.1. Electric field distribution around a bubble The electric field distribution around the bubble is calculated by solving the Laplace equation for the voltage field:
F. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 174–181
r ðerU Þ ¼ 0;
ð6Þ
where e is the permittivity, U is the applied voltage. Half of the axisymmetric bubble is analyzed in a domain with dimensions of 10 mm · 10 mm. The top and bottom boundary conditions are U ¼ 0 kV at z ¼ 0 mm; U ¼ 30 kV at z ¼ 10 mm:
ð7Þ ð8Þ
The side boundary conditions are n rU ¼ 0 at r ¼ 0; 10 mm:
ð9Þ
In Eqs. (7)–(9), r and z are the radial and symmetry axis coordinate of the bubble, respectively. n is the outer normal vector of the bubble surface.
179
The experimental photograph in Fig. 5 for an electric field strength of 3.0 kV/mm was first processed by Adobe Photoshop as shown in Fig. 9. The bubble interface line was then extracted and the electric field was calculated for this interface shape with MATLAB. All the calculations are mesh-independent. Fig. 10 shows the voltage distribution around the bubble. The equipotential lines bend near the bubble surface, which indicates that some induced charges occurs on the bubble surface; however, the equipotential lines are parallel to each other in most of the domain. Fig. 11 shows the electric field distribution, where the electric field is defined as E ¼ rU :
ð10Þ
The arrows in Fig. 11 indicate the direction from the stronger electric field to the weaker electric field. The direction of the electric field gradient is the same with the outer normal of the bubble surface on the bubble poles. However, the directions of the electric field gradient and the outer normal of the bubble surface are opposite on the bubble equator. 3.2.2. Electric stresses on the bubble surface The distribution of the electric field strength, E, around the bubble and the permittivity of the dielectric fluid, e, are then used to calculate the electric body force, fe, [17]. 1 2 1 2 de fe ¼ qe E E re þ r E q ; ð11Þ 2 2 dqm m Fig. 9. Bubble shape for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
where qe is the volume charge density, qm is the fluid mass density. The first term on the right side of Eq. (11) represents the Coulomb stress, which is the stress exerted on the free charges. The second term stands for the dielectrophoretic
Fig. 10. Voltage distribution around a bubble at departure for a 3.0 kV/ mm electric field strength (L = 10 mm, u = 1.5 mm).
Fig. 11. The electric field distribution around a bubble at departure for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
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stress, which is generated by space changes of dielectric permittivity of the fluids. The third term is the electrostriction stress, which primarily occurs when the dielectric permittivity varies with the mass density of the fluid. Eq. (11) can be rewritten by using the Maxwell stress tensor [18] ð12Þ
The total electric stresses on the interface of the nitrogen bubble are then [19] el q del Ts ¼ el E2ln nl þ el Eln Elt tl ðE2ln þ E2lt Þnl þ ml E2l nl 2 2 dqml þ
eg þ eg Egn Egt tg Þ ðE2gn þ E2gt Þng þ 2
! # qmg 2 deg ng ; E 2 g dqmg
ð13Þ
where the subscripts g and l represent the parameters in the gas and liquid respectively. The direction vectors ng, nl, tg and tl are described in Fig. 12. The dielectrophoretic stresses, TDEP, in Eq. (11) can be simplified to 1 TDEP ¼ e0 ðE2g eg E2l el Þnl : 2
ð14Þ
For an incompressible fluid, the electrostriction stresses were frequently omitted [14,15]. However, the electrostriction stresses were induced at the bubble surface, since e varied significantly between the inside and outside of the bubble in this analysis. For non-polar fluids, the electrostriction stresses, TSTR, can be simplified by applying the Clausius–Mossotti law [18] 1 ðel 1Þðel þ 2Þ nl ; TSTR ¼ e0 E2l 2 3
ð16Þ
Therefore, the electrohydrodynamic stresses, TEHD, on the bubble surface are [19] 2
TEHD ¼ e0
e q de 2 T ¼ eðn EÞ E E2 n þ m E n: 2 2 dqm
ðeg E2gn ng
el Eln ¼ eg Egn ; Elt ¼ Egt :
ðel 1Þ ð2E2ln E2lt Þnl : 6
ð17Þ
The distribution of the electrohydrodynamic stresses, TEHD along the bubble surface, at 0.305 s are shown in Fig. 13 for a 3.0 kV/mm electric field strength. In Fig. 13, the positive direction of the electric stress is from the liquid to the gas along the normal line, h is the angle between the normal and the electric field directions on the bubble surface. Tse and Tsp are the maximum electrohydrodynamic stress on the bubble equator and the minimum electrohydrodynamic stress on the bubble poles respectively. The electrohydrodynamic stresses compress the bubble around the equator, but elongate the bubble on the poles. The compressive stresses below the equator of the bubble pinch the neck as the bubble grows. The compressive stresses reach the maximum of 25.03 N/m2 at h = 138.3. The expanding stress then amounts to a maximum of 11.2 N/m2 at h = 180.0 (the opposite direction). The maximum expanding stress is smaller than the maximum compressive stress. Moreover, the regions of the expanding stresses exerted on the bubble surface are also much smaller than those of the compressive stresses. Consequently, the variation of electrohydrodynamic stresses leads to the bubble elongation.
ð15Þ
where e0 is the vacuum permittivity. The gas electrostriction stresses are neglected in Eq. (15) because the relative dielectric permittivity of the gas is close to 1.0. Assuming without electrical charges on the bubble surface, the electrical displacement continuity is [17]
Fig. 12. The schematic of the liquid–gas interface.
Fig. 13. Variation of the electrohydrodynamic stresses along the bubble surface at 0.305 s for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
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Transfer and Energy Conversion – the Key Laboratory of Beijing Municipality for their financial support for this project. References
Fig. 14. Electric stresses along a bubble at departure for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
The dielectrophoretic stress, TDEP, the electrostriction stress, TSTR, and the electrohydrodynamic stress, TEHD, are shown in Fig. 14. Fig. 14 shows that the electrostriction stress compresses the bubble while the dielectrophoretic stress expands the bubble all along the bubble surface. The electrohydrodynamic stress, TEHD, makes the bubble more slender in the electric field than without an electric field. 4. Conclusions Bubble deformation in a uniform electric field was studied to analyze the effect of the electric stresses on the bubble shape. The electric stresses around the bubble were calculated to show the deformation mechanism. The results show that: (1) The major axis of the bubble is elongated at departure along the direction parallel to the applied electric field. The bubble elongation increases with increasing electric field strength. (2) The bubble relative aspect ratio, ARe/AR0, can be linearly correlated to We, with ARe/AR0 increasing with We. (3) The electrostriction stress compresses the bubble while the dielectrophoretic stress expands the bubble all along the bubble surface. The electrohydrodynamic stress exerted on the bubble surface leads to the bubble elongation at departure. These experiments were all performed with isothermal conditions. Studies of bubble shapes during nucleate boiling heat transfer in an electric field will be conducted in the future. Acknowledgements The authors are grateful to the National Basic Research Program of China (Grant No. G2000026301) and Heat
[1] L.W. Chubb, Improvements relating to methods and apparatus for heating liquids, UK, Patent No. 100796, 1916. [2] P. Cooper, EHD enhancement of nucleate boiling, ASME Journal of Heat Transfer 112 (1990) 458–464. [3] J. Ogata, A. Yabe, Augmentation of boiling heat transfer by utilizing the EHD effect-EHD behavior of boiling bubbles and heat transfer characteristics, International Journal of Heat and Mass Transfer 36 (3) (1993) 783–791. [4] P.H.G. Allen, T.G. Karayiannis, Electrohydrodynamic enhancement of heat transfer and fluid flow, Heat Recovery Systems & CHP 15 (5) (1995) 389–423. [5] M.C. Zaghdoudi, M. Lallemand, Analysis of the polarity influence on nucleate pool boiling under a DC electric field, ASME Journal of Heat Transfer 121 (1999) 856–864. [6] Y.C. Kweon, M.H. Kim, Experimental study on nucleate boiling enhancement and bubble dynamic behavior in saturated pool boiling using a nonuniform DC electric field, International Journal of Multiphase Flow 26 (2000) 1351–1368. [7] C.C. Pascual, S.M. Jeter, S.I. Abdel-Khalik, A statistical analysis of EHD-enhanced nucleate boiling along a heated wire, International Journal of Heat and Mass Transfer 44 (2001) 1201–1212. [8] C. Herman, E. Lacona, I.B. Foldes, G. Suner, C. Milburn, Experimental visualization of bubble formation from an orifice in microgravity in the presence of electric fields, Experiments in Fluids 32 (2002) 396–412. [9] J. Madadnia, H. Koosha, Electrohydrodynamic effects on characteristic of isolated bubbles in the nucleate pool boiling regime, Experimental Thermal and Fluid Science 27 (2003) 145–150. [10] P. Di Macro, A. Faini, W. Grassi, Electric field effects on bubbles of nitrogen in FC-72 originating from a flat plate, in: 3rd International Symposium on Two-Phase Flow Modeling and Experimentation Pisa 22–24 September 2004. [11] C. Herman, E. Iacona, Z. Liu, S. Chang, Bubble formation and detachment in variable gravity environment under the influence of electric fields, ASME Journal of Heat Transfer 127 (2005) 804. [12] J.R. Melcher, G.I. Taylor, Electrohydrodynamics: a review of the role of interfacial shear stresses, Annual Review of Fluid Mechanics 1 (1969) 111–146. [13] K.J. Cheng, J.B. Chaddock, Deformation and stability of drops and bubbles in an electric field, Physics Letters 106A (12) (1984) 51–53. [14] H.J. Cho, I.S. Kang, Y.C. Kweon, M.H. Kim, Study of the behavior of a bubble attached to a wall in a uniform electric field, International Journal of Multiphase Flow 22 (5) (1996) 909–922. [15] Y.C. Kweon, M.H. Kim, H.J. Cho, I.S. Kang, Study on the deformation and departure of a bubble attached to a wall in D.C/ A.C electric fields, International Journal of Multiphase Flow 24 (1) (1998) 145–162. [16] W. Dong, R.Y. Li, H.L. Yu, Y.Y. Yan, An investigation of behaviours of a single bubble in a uniform electric field, Experimental Thermal and Fluid Science 30 (6) (2006) 579–586. [17] J.R. Melcher, Continuum Electromechanics, MIT Press, Cambridge, 1981. [18] X.C. Huang, Electromagnetism Energy and Electromagnetism Force, People’s Education Press, Beijing, 1981 (in Chinese). [19] M.C. Zaghdoudi, M. Lallemand, Study of the behaviour of a bubble in an electric field: steady shape and local fluid motion, International Journal of Thermal Science 39 (2000) 39–52.
EHD behavior of nitrogen bubbles in DC electric fields F. Chen, Y. Peng, Y.Z. Song *, M. Chen Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China Received 7 September 2006; received in revised form 25 February 2007; accepted 7 March 2007
Abstract The deformation of nitrogen bubbles injected into transformer oil with various DC electric fields was studied experimentally and theoretically. The bubble deformation was visualized by a high speed camera. The major axis of the bubble was elongated along the direction parallel to the electric field, with the elongation increasing as the electric field strength was raised. The electrical Weber number (We) was used to correlate the electric field strength and the dielectric permittivity of the working fluid to the bubble relative aspect ratio (ARe/ AR0). The experimental results show that the relative aspect ratio increases with increasing We. The electric stresses were calculated on an actual bubble shape including the electrostriction stresses to analyze the bubble elongation. 2007 Elsevier Inc. All rights reserved. Keywords: Electrohydrodynamic (EHD); Bubble deformation; Electric field; Electric stresses
1. Introduction Electrohydrodynamic (EHD) enhancement is an active heat transfer enhancement method, which has attracted considerable interest since Chubb [1] claimed that the water evaporation rate was increased threefold by an electric field in 1916. Extensive experiments have been conducted to study the heat transfer enhancement and the behavior of a single bubble attached to a wall in DC and AC electric fields. The preliminary experimental results include the increases of the bubble departure aspect ratio, the increases of the bubble departure frequency and the increases of the maximum heat flux, the decreases of the bubble departure volume and the elimination of boiling hysteresis by electric fields [2–11]. In these studies, the electric field effects on the bubbles were of particular concern because the bubble behavior changes in the presence of an electric field were believed to be one of the main reasons for the enhancement of the pool boiling heat transfer.
*
Corresponding author. Tel.: +86 10 62772938; fax: +86 10 62795832. E-mail address: [email protected] (Y.Z. Song).
0894-1777/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2007.03.006
In the early years, Melcher and Taylor [12] calculated the electric stresses on a spherical drop or vapor void suspended in a liquid assuming incompressibility. They presented a discrimination factor to distinguish between oblate and prolate shapes of drops or vapor voids with an electric field, which indicated that the electric field led to the bubble deformation. Later, Cheng and Chaddock [13] analyzed the bubble deformation and stability by considering the free energy in a uniform electric field and assumed that the bubble was spheroidal, an approximation to the actual bubble shape. Cho et al. [14,15] numerically and experimentally investigated the effects of a uniform electric field on a bubble attached to the wall without making the spheroidal approximation. The electrostriction force was neglected in their calculation. The results showed that the bubble was elongated parallel to the direction of the applied electric field. The elongation increased as the electric field strength increased. Recently, experiments with a single bubble in a uniform electric field by Dong et al. [16] showed interesting bubble detachment and break-up phenomena. However, the basic mechanisms for the EHD effect are still not completely understood. The present paper describes the behavior of a single nitrogen bubble injected
F. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 174–181
175
Nomenclature AR E E L P Q T T U We d r rd w z
bubble aspect ratio, AR = d/w electric field strength (kV/mm) electric field strength vector electrode distance (mm) pressure (Pa) flow rate (mm3/s) Maxwell stress (N/m2) temperature (K) voltage (kV) electrical Weber number major axis of the bubble (mm) radial direction bubble equivalent radius at departure (mm) minor axis of the bubble (mm) symmetry axis
Greek Symbols c surface tension (N/m) e permittivity of the dielectric fluid (F/m) h angle around the bubble (0) into 25# transformer oil in a uniform electric field. The experimental conditions could be well controlled by employing bubble injection rather than boiling. Bubble formation in a DC electric field was visualized by a high speed camera. In addition, the electric field distribution and electric stresses acting on the actual bubble surface were analyzed. 2. Experimental apparatus and procedure The experimental apparatus consisted of a test cell, a high voltage power supply, a nitrogen bubble injection sys-
l qe qm r u
viscosity (kg/(m s)) electric charge density (C/m3) fluid mass density (kg/m3) electrical conductivity (S/m) orifice diameter (mm)
Subscripts DEP dielectrophoretic EHD electrohydrodynamic STR electrostriction e electric field g gas l liquid n normal direction r relative 0 absolute or initial s bubble surface t tangential direction
tem and a recording system as shown in Fig. 1. The experiments were conducted at 281 K and 1.013 · 105 Pa. The working fluid was 25# transformer oil, selected for its high resistivity, non-toxicity and non-flammability at ambient temperature and pressure. The injected gas was 99.999% Table 1 Transformer oil and nitrogen properties at P = 1.013 · 105 Pa and T = 293 K
Nitrogen Transformer oil
qm (kg/m3)
l (kg/(m s))
er
c (N/m)
re (S/m)
1.165 849
– 0.0202
1.0 2.1
– 0.044
1.0 · 1016 3.3 · 1012
Fig. 1. Schematic of the experimental setup.
176
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Table 2 Variation of the flow rates with increasing electric field strength (L = 10 mm, u = 1.5 mm) E0 (kV/mm) Q (mm3/s)
0.0 21.86 ± 0.13
0.5 21.52 ± 0.13
1.0 20.80 ± 0.09
1.5 20.92 ± 0.11
volume percent nitrogen, selected for its chemical stability. The physical properties of the transformer oil and the nitrogen gas are listed in Table 1. The nitrogen gas flowed through a pressure regulator with an exit pressure of 0.21 MPa into a cylindrical vessel 45 cm long and 2.5 cm in diameter to stabilize the gas flow before entering the orifice. The nitrogen volume flow rate ranged from 20.06 ± 0.09 to 21.86 ± 0.13 mm3/s as given in Table 2. The last entry in Table 2 gives the flow rate after the high voltage power supply was shut off to show that the flow rate was quite stable. A schematic of the test cell was shown in Fig. 2. The test cell volume was 324.8 cm3 (11.2 cm · 14.5 cm · 2 cm). The front and back walls of the test cell were made of glass to facilitate the visualization. The sidewalls were made of epoxy resin. A brass plate served as the grounded electrode (the negative electrode). A 1.5 mm diameter orifice was drilled at the center of this plate (u = 1.5 mm). The brass plate was embedded into an epoxy resin base. The positive electrode was made of brass mesh to allow the bubbles to pass through and connected to a 50 kV high voltage power supply (Glassman ER50R06-DM22). The electrode system was shown in Fig. 3, where L was the distance between the positive and negative electrodes. The oil depth was kept at 20 mm. The parallel positive and negative electrodes generated a quasi-uniform electric field. A high speed camera (AOS VITcam high speed digital camera) was placed in front of the test cell. The frame rate was 1000 fps and the resolution was 640 · 512 pixels. Two fluorescent lamps were used to illuminate the nitrogen bubbles from the back of the test cell.
2.0 20.58 ± 0.09
2.5 20.06 ± 0.09
3.0 20.27 ± 0.09
0.0 21.22 ± 0.13
Fig. 2. Schematic of the test cell.
3. Results and discussion 3.1. Experimental observations and results The bubble growth process is shown in Fig. 4 as a function of time for electric filed strengths of 0.0 kV/mm, 1.0 kV/mm, 2.0 kV/mm and 3.0 kV/mm. Fig. 4a shows the growth of a bubble attached to the surface without an electric field. The bubble is spherical in the beginning; the lower half of the bubble gradually changes into a cone while the upside remains spherical as the bubble grows.
Fig. 3. Schematic of electrode system.
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Fig. 4. Bubble growth for various electric field strengths (L = 10 mm, u = 1.5 mm).
The bubble neck forms as departure occurs at 0.046 ± 0.001 s. Fig. 4b–d show that the bubble growth times increase as the bubbles elongate along the electric field direction with increasing electric field strength. Fig. 5 shows the bubble shapes at departure for various DC electric fields. As the electric field strength increases from 0 to 3.0 kV/mm the aspect ratio increases by 36.9%. The bubble injection was uniform in the experiments. At 0.0 kV/mm, the bubble departed at about 0.046 s with the next bubble beginning to form at 0.9 s, long after the previous bubble departed. At 3.0 kV/mm, the bubble departed at 0.306 s with the next bubble forming at 0.59 s. The major axis length, d, the minor axis length, w and the aspect ratio, d/w of the bubble at departure are illustrated in Fig. 6.
The bubble aspect ratio (AR), electrical Weber number (We) and the bubble equivalent radius at departure (rd) are defined as [13] AR ¼ d=w; We ¼
ð1Þ
e0 erl E20 rd =c; 2 1=3
rd ¼ ðdw Þ
=2;
ð2Þ ð3Þ
where e0 is the vacuum permittivity (e0 = 8.854 · 1012 F/m), erl is the relative permittivity of the transformer oil, and c is the surface tension of the transformer oil. E0, the initial electric field strength, is E0 ¼ U =d;
ð4Þ
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Fig. 5. Bubble detachment shapes for various electric field strengths (L = 10 mm, u = 1.5 mm).
Fig. 6. Schematic showing the bubble dimensions.
The relative aspect ratio (ARe/AR0) increases with increasing electric field strength for various electrode distances as shown in Fig. 7. The subscripts e and 0 denote the departure aspect ratio in the presence of electric fields and without electric fields respectively. The elongation is closely associated with the initial electric field strength. We is used to correlate the changes of ARe/AR0, as shown in Fig. 8. The small amount of scatter of the data around the correlation indicates that the Weber number has a significant effect on the aspect ratio. Fig. 8 shows that ARe/AR0 is linearly related to We. In some cases, instabilities in the air–oil interface caused by
Fig. 8. Variation of the relative aspect ratio at departure (u = 1.5 mm).
the electric field affected the bubble shapes. The interface instabilities intensified with increasing electric field strength. At L = 18 mm, the upper electrode was only 2 mm below the air–oil interface. The instabilities were so strong that the air–oil interface was below the upper electrode at times, which led to a marked difference between the experimental data and the curve fit in Fig. 8 for the strongest electric field strength. A linear least square fit of the experimental data gave: ARe =AR0 ¼ 0:05We þ 0:99:
ð5Þ
Note that the coefficients in Eq. (5) will depend on the system geometry. Regardless, We is a crucial parameter in the industrial application when the electric field is imposed. 3.2. Theoretical analysis of the electric field and electric stresses The effect of the electric field on the bubble growth process is further analyzed by calculating the electric field distribution around the bubble and the electric stresses on the actual bubble surface at departure.
Fig. 7. The aspect ratio changes with the initial electric field strength E0.
3.2.1. Electric field distribution around a bubble The electric field distribution around the bubble is calculated by solving the Laplace equation for the voltage field:
F. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 174–181
r ðerU Þ ¼ 0;
ð6Þ
where e is the permittivity, U is the applied voltage. Half of the axisymmetric bubble is analyzed in a domain with dimensions of 10 mm · 10 mm. The top and bottom boundary conditions are U ¼ 0 kV at z ¼ 0 mm; U ¼ 30 kV at z ¼ 10 mm:
ð7Þ ð8Þ
The side boundary conditions are n rU ¼ 0 at r ¼ 0; 10 mm:
ð9Þ
In Eqs. (7)–(9), r and z are the radial and symmetry axis coordinate of the bubble, respectively. n is the outer normal vector of the bubble surface.
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The experimental photograph in Fig. 5 for an electric field strength of 3.0 kV/mm was first processed by Adobe Photoshop as shown in Fig. 9. The bubble interface line was then extracted and the electric field was calculated for this interface shape with MATLAB. All the calculations are mesh-independent. Fig. 10 shows the voltage distribution around the bubble. The equipotential lines bend near the bubble surface, which indicates that some induced charges occurs on the bubble surface; however, the equipotential lines are parallel to each other in most of the domain. Fig. 11 shows the electric field distribution, where the electric field is defined as E ¼ rU :
ð10Þ
The arrows in Fig. 11 indicate the direction from the stronger electric field to the weaker electric field. The direction of the electric field gradient is the same with the outer normal of the bubble surface on the bubble poles. However, the directions of the electric field gradient and the outer normal of the bubble surface are opposite on the bubble equator. 3.2.2. Electric stresses on the bubble surface The distribution of the electric field strength, E, around the bubble and the permittivity of the dielectric fluid, e, are then used to calculate the electric body force, fe, [17]. 1 2 1 2 de fe ¼ qe E E re þ r E q ; ð11Þ 2 2 dqm m Fig. 9. Bubble shape for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
where qe is the volume charge density, qm is the fluid mass density. The first term on the right side of Eq. (11) represents the Coulomb stress, which is the stress exerted on the free charges. The second term stands for the dielectrophoretic
Fig. 10. Voltage distribution around a bubble at departure for a 3.0 kV/ mm electric field strength (L = 10 mm, u = 1.5 mm).
Fig. 11. The electric field distribution around a bubble at departure for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
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stress, which is generated by space changes of dielectric permittivity of the fluids. The third term is the electrostriction stress, which primarily occurs when the dielectric permittivity varies with the mass density of the fluid. Eq. (11) can be rewritten by using the Maxwell stress tensor [18] ð12Þ
The total electric stresses on the interface of the nitrogen bubble are then [19] el q del Ts ¼ el E2ln nl þ el Eln Elt tl ðE2ln þ E2lt Þnl þ ml E2l nl 2 2 dqml þ
eg þ eg Egn Egt tg Þ ðE2gn þ E2gt Þng þ 2
! # qmg 2 deg ng ; E 2 g dqmg
ð13Þ
where the subscripts g and l represent the parameters in the gas and liquid respectively. The direction vectors ng, nl, tg and tl are described in Fig. 12. The dielectrophoretic stresses, TDEP, in Eq. (11) can be simplified to 1 TDEP ¼ e0 ðE2g eg E2l el Þnl : 2
ð14Þ
For an incompressible fluid, the electrostriction stresses were frequently omitted [14,15]. However, the electrostriction stresses were induced at the bubble surface, since e varied significantly between the inside and outside of the bubble in this analysis. For non-polar fluids, the electrostriction stresses, TSTR, can be simplified by applying the Clausius–Mossotti law [18] 1 ðel 1Þðel þ 2Þ nl ; TSTR ¼ e0 E2l 2 3
ð16Þ
Therefore, the electrohydrodynamic stresses, TEHD, on the bubble surface are [19] 2
TEHD ¼ e0
e q de 2 T ¼ eðn EÞ E E2 n þ m E n: 2 2 dqm
ðeg E2gn ng
el Eln ¼ eg Egn ; Elt ¼ Egt :
ðel 1Þ ð2E2ln E2lt Þnl : 6
ð17Þ
The distribution of the electrohydrodynamic stresses, TEHD along the bubble surface, at 0.305 s are shown in Fig. 13 for a 3.0 kV/mm electric field strength. In Fig. 13, the positive direction of the electric stress is from the liquid to the gas along the normal line, h is the angle between the normal and the electric field directions on the bubble surface. Tse and Tsp are the maximum electrohydrodynamic stress on the bubble equator and the minimum electrohydrodynamic stress on the bubble poles respectively. The electrohydrodynamic stresses compress the bubble around the equator, but elongate the bubble on the poles. The compressive stresses below the equator of the bubble pinch the neck as the bubble grows. The compressive stresses reach the maximum of 25.03 N/m2 at h = 138.3. The expanding stress then amounts to a maximum of 11.2 N/m2 at h = 180.0 (the opposite direction). The maximum expanding stress is smaller than the maximum compressive stress. Moreover, the regions of the expanding stresses exerted on the bubble surface are also much smaller than those of the compressive stresses. Consequently, the variation of electrohydrodynamic stresses leads to the bubble elongation.
ð15Þ
where e0 is the vacuum permittivity. The gas electrostriction stresses are neglected in Eq. (15) because the relative dielectric permittivity of the gas is close to 1.0. Assuming without electrical charges on the bubble surface, the electrical displacement continuity is [17]
Fig. 12. The schematic of the liquid–gas interface.
Fig. 13. Variation of the electrohydrodynamic stresses along the bubble surface at 0.305 s for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
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Transfer and Energy Conversion – the Key Laboratory of Beijing Municipality for their financial support for this project. References
Fig. 14. Electric stresses along a bubble at departure for a 3.0 kV/mm electric field strength (L = 10 mm, u = 1.5 mm).
The dielectrophoretic stress, TDEP, the electrostriction stress, TSTR, and the electrohydrodynamic stress, TEHD, are shown in Fig. 14. Fig. 14 shows that the electrostriction stress compresses the bubble while the dielectrophoretic stress expands the bubble all along the bubble surface. The electrohydrodynamic stress, TEHD, makes the bubble more slender in the electric field than without an electric field. 4. Conclusions Bubble deformation in a uniform electric field was studied to analyze the effect of the electric stresses on the bubble shape. The electric stresses around the bubble were calculated to show the deformation mechanism. The results show that: (1) The major axis of the bubble is elongated at departure along the direction parallel to the applied electric field. The bubble elongation increases with increasing electric field strength. (2) The bubble relative aspect ratio, ARe/AR0, can be linearly correlated to We, with ARe/AR0 increasing with We. (3) The electrostriction stress compresses the bubble while the dielectrophoretic stress expands the bubble all along the bubble surface. The electrohydrodynamic stress exerted on the bubble surface leads to the bubble elongation at departure. These experiments were all performed with isothermal conditions. Studies of bubble shapes during nucleate boiling heat transfer in an electric field will be conducted in the future. Acknowledgements The authors are grateful to the National Basic Research Program of China (Grant No. G2000026301) and Heat
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