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The equilibrium shapes of liquid menisci emitting liquid and charges in steady cone-jet mode
~ Pergamon
J AtrOJoi Sd.• Vol. 27. 5uppl. I. pp. 5187-5ISS. 1996 Copyrishl iCl 1996 Ebevier Science Ltd
PU: SOO21-8502(96)OO16tHl
Printed in G....l Britain. All riSblJ reserved 0021-8502196$15.00 + 0.00
THE EQUILIBRIUM SHAPES OF LIQUID MENISCI EMITTING LIQUID AND CHARGES IN STEADY CONE-JET MODE ALFONSO M. GANAN-CALVO, CARLOS PANTANO, and ANTONIO BARRERO Dept. de Ingenieria Energetica y Mecanica de Fluidos, E.S.I., Universidad de Sevilla, Avda. Reina Mercedes sin, 41012 Sevilla, Spain. KEYWORDS Taylor Cone; Electrohydrostatics; Electrospray; Electrohydrodynamics; Electrostatic Atomization One of the main problems associated to the electrohydrodynamic atomization of liquids in steady cone-jet (SCJ) mode is the determination of the equilibrium shapes of the cone-like charged menisci (Taylor's cones) that give rise to the steady, thin ligaments whose breakup yields the charged spray. The analysis of the parametric ranges of existence and stability of these menisci is essential for the analysis of the SCJ mode. The first cone-like equilibrium solution was given by Taylor; some other authors refined or raised some controversy about that first solution since it was not capable to deal with the usual boundary conditions (presence of a feeding needle, fiat electrodes, etcetera). The effect of the space charge in the form of a conical spray emerging from the cone's tip on Taylor's solution was analyzed by F. de la Mora (1994). Other authors (e.g. Joffre and Cloupeau 1985) have analyzed the shape of rounded-tip ended, captive menisci under electric fields. The equilibrium shapes of sharp tip-ended, axisymmetric capillary menisci placed at the tip of a feeding needle, under intense electric fields and in the absence of space charge effects owing to liquid and charge emissions where firstly obtained by Pantano et al. (1994) using a Boundary Elements Method for the electric field and a Galerkin scheme for the non-linear capillary equation at the free surface. In that work, the solutions are assumed to be the first order approach to the SCJ mode, necessary for the study of the electrohynamics of the emitted microjet under the influence of the electric field provoked by the cone. In the present work, those solutions obtained by Pantano et al. are shown to be the asymptotic limits of the actual shapes of tip-ended menisci emitting charges when the liquid flow rate is slowly decreased, until a minimum flow rate is achieved. In our experiments, a flow rate Q( m 3 Is) of octanol dopped with a small amount of HCI is injected through a stainless steel, capillary tube with inner and outer radii R e = 0.45mm and R; = O.8mm, respectively. The tube is placed perpendicularly to a horizontal flat metallic grounded electrode (see Fig. 1) at a distance H = 10mm. The measured liquid density, electrical conductivity and permittivity, surface tension, and viscosity at the working temperature (20°C) are P = 824kglm3 , K = 6.5 X 1O-4Sl m, e, = 9.98co , 'Y = O.0264Nlm, and p == O.Olkg/m.s, respecti~ely. The mouth of the feeding metallic needle, a~ which the electrified meniscus If formed, IS carefully shaped to form a sharp plane perpendicular to the tube's axis. A microscope with a video camera is horizontally placed with its optical axis in the tube's mouth plane and focused on the tube's axis (Fig. 2). Horizontal and vertical scales are verified to be identical in a monitor's screen. The feeding metallic tube is connected to a voltage 4>0 supplied by a BERTAN 250B·10R High Voltage Power Supply, and the liquid is feeded through a teflon line of inner diameter O.33mm from a presurized srynge. This line is calibrated so that the flow rate is known for a given srynge pressure. The experiment is carried out at standard atmospheric pressure. On the other hand, the theoretical tip-ended meniscus profile in the absence of space charge effects are calculated for those experimental conditions, using Pantano's et al. code. An easy to measure and sensitive parameter for an experimental comparison is the resulting elongation e = hiRe of the profile, where h is the distance from the meniscus tip to the plane fanned by the feeding tube's mouth. 5187
SI88
Abstracts of the 1996 European Aerosol Conference
Both experimental and theoretical elongations for three different voltages 4>0 = 3400V, 3291V, and 3034V are compared in Fig. 3. One may observe in Fig. 4 that the cone's angle increases (and the elongation decreases) as the flow rate, and consequently, the electrostatic induction of the microjet on the cone decrease. The minimum flow rate is reached for Q ~ IO-12 m3 Is. One may observe a slow convergence towards the theoretical asymptota in a log -log plot as the flow rate is decreased, until the practical minimum flow rate for which the SCJ mode is stable, is reached. The lines are ideally extended below that physical minimum in the plot only to provide an idea of the convergence pattern. One may also observe the existence of a maximum flow rate for each given voltage (at the standard air pressure at which the experiment is performed) that seems to be related to the meniscus stability since it develops, when the elongation reaches a certain value, by a progresively increasing of a small jittering appearing at the cone's tip area while the jet seems to be perfectly steady. This behavior, opposite to the catastrophic disappearance of the jet together with a strong meniscus vibration when the minimum flow rate is reached, suggests that the minimum flow rate is not related to the meniscus stability (the non-emitting meniscus is theoretically stable), but to the microjet eleetrohydrodynamic stability. Compr....d Air I
1
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Fig. 4
REFERENCES
[1] Fernandez de la Mora, J. 1992 The effect of charge emission from electrified liquid cones. J. Fluid Mech. 243,561. [2) Joffre, G., Cloupeau, M. 1986 Characteristic forms of electrified menisci emitting charges. J. Electrostatics is, pp. 147-161. [3) Pantano, C., Gaiian-Calvo, A. M., and Barrera, A. 1994 Zeroth-order, electrohydrostatic solution for electrospraying in cone-jet mode. J. Aerosol Sci. 25, 1065.
J AtrOJoi Sd.• Vol. 27. 5uppl. I. pp. 5187-5ISS. 1996 Copyrishl iCl 1996 Ebevier Science Ltd
PU: SOO21-8502(96)OO16tHl
Printed in G....l Britain. All riSblJ reserved 0021-8502196$15.00 + 0.00
THE EQUILIBRIUM SHAPES OF LIQUID MENISCI EMITTING LIQUID AND CHARGES IN STEADY CONE-JET MODE ALFONSO M. GANAN-CALVO, CARLOS PANTANO, and ANTONIO BARRERO Dept. de Ingenieria Energetica y Mecanica de Fluidos, E.S.I., Universidad de Sevilla, Avda. Reina Mercedes sin, 41012 Sevilla, Spain. KEYWORDS Taylor Cone; Electrohydrostatics; Electrospray; Electrohydrodynamics; Electrostatic Atomization One of the main problems associated to the electrohydrodynamic atomization of liquids in steady cone-jet (SCJ) mode is the determination of the equilibrium shapes of the cone-like charged menisci (Taylor's cones) that give rise to the steady, thin ligaments whose breakup yields the charged spray. The analysis of the parametric ranges of existence and stability of these menisci is essential for the analysis of the SCJ mode. The first cone-like equilibrium solution was given by Taylor; some other authors refined or raised some controversy about that first solution since it was not capable to deal with the usual boundary conditions (presence of a feeding needle, fiat electrodes, etcetera). The effect of the space charge in the form of a conical spray emerging from the cone's tip on Taylor's solution was analyzed by F. de la Mora (1994). Other authors (e.g. Joffre and Cloupeau 1985) have analyzed the shape of rounded-tip ended, captive menisci under electric fields. The equilibrium shapes of sharp tip-ended, axisymmetric capillary menisci placed at the tip of a feeding needle, under intense electric fields and in the absence of space charge effects owing to liquid and charge emissions where firstly obtained by Pantano et al. (1994) using a Boundary Elements Method for the electric field and a Galerkin scheme for the non-linear capillary equation at the free surface. In that work, the solutions are assumed to be the first order approach to the SCJ mode, necessary for the study of the electrohynamics of the emitted microjet under the influence of the electric field provoked by the cone. In the present work, those solutions obtained by Pantano et al. are shown to be the asymptotic limits of the actual shapes of tip-ended menisci emitting charges when the liquid flow rate is slowly decreased, until a minimum flow rate is achieved. In our experiments, a flow rate Q( m 3 Is) of octanol dopped with a small amount of HCI is injected through a stainless steel, capillary tube with inner and outer radii R e = 0.45mm and R; = O.8mm, respectively. The tube is placed perpendicularly to a horizontal flat metallic grounded electrode (see Fig. 1) at a distance H = 10mm. The measured liquid density, electrical conductivity and permittivity, surface tension, and viscosity at the working temperature (20°C) are P = 824kglm3 , K = 6.5 X 1O-4Sl m, e, = 9.98co , 'Y = O.0264Nlm, and p == O.Olkg/m.s, respecti~ely. The mouth of the feeding metallic needle, a~ which the electrified meniscus If formed, IS carefully shaped to form a sharp plane perpendicular to the tube's axis. A microscope with a video camera is horizontally placed with its optical axis in the tube's mouth plane and focused on the tube's axis (Fig. 2). Horizontal and vertical scales are verified to be identical in a monitor's screen. The feeding metallic tube is connected to a voltage 4>0 supplied by a BERTAN 250B·10R High Voltage Power Supply, and the liquid is feeded through a teflon line of inner diameter O.33mm from a presurized srynge. This line is calibrated so that the flow rate is known for a given srynge pressure. The experiment is carried out at standard atmospheric pressure. On the other hand, the theoretical tip-ended meniscus profile in the absence of space charge effects are calculated for those experimental conditions, using Pantano's et al. code. An easy to measure and sensitive parameter for an experimental comparison is the resulting elongation e = hiRe of the profile, where h is the distance from the meniscus tip to the plane fanned by the feeding tube's mouth. 5187
SI88
Abstracts of the 1996 European Aerosol Conference
Both experimental and theoretical elongations for three different voltages 4>0 = 3400V, 3291V, and 3034V are compared in Fig. 3. One may observe in Fig. 4 that the cone's angle increases (and the elongation decreases) as the flow rate, and consequently, the electrostatic induction of the microjet on the cone decrease. The minimum flow rate is reached for Q ~ IO-12 m3 Is. One may observe a slow convergence towards the theoretical asymptota in a log -log plot as the flow rate is decreased, until the practical minimum flow rate for which the SCJ mode is stable, is reached. The lines are ideally extended below that physical minimum in the plot only to provide an idea of the convergence pattern. One may also observe the existence of a maximum flow rate for each given voltage (at the standard air pressure at which the experiment is performed) that seems to be related to the meniscus stability since it develops, when the elongation reaches a certain value, by a progresively increasing of a small jittering appearing at the cone's tip area while the jet seems to be perfectly steady. This behavior, opposite to the catastrophic disappearance of the jet together with a strong meniscus vibration when the minimum flow rate is reached, suggests that the minimum flow rate is not related to the meniscus stability (the non-emitting meniscus is theoretically stable), but to the microjet eleetrohydrodynamic stability. Compr....d Air I
1
Spr.y
I; !
.l-::',
~;- :::::r-·-----lI,ll
HighVolt.g. Pow.r Supply
F••dlng Lin.
Fig. 1
Fig. 2
2
-e u
~
r:: o
'.;:1 <'I
6
co s:;
o
til I 1==-......- =
Q. 1.1 x IO·nm".
a·
0.9
0.8 IE-013
1V
.-._--._.
- ....-- -.. -.-...-
IE-012
r-'"~--"-
I: '
IE-Oll
-
•-
...
v
IE-olO
Flow rate (ml/s)
Fig. 3
Fig. 4
REFERENCES
[1] Fernandez de la Mora, J. 1992 The effect of charge emission from electrified liquid cones. J. Fluid Mech. 243,561. [2) Joffre, G., Cloupeau, M. 1986 Characteristic forms of electrified menisci emitting charges. J. Electrostatics is, pp. 147-161. [3) Pantano, C., Gaiian-Calvo, A. M., and Barrera, A. 1994 Zeroth-order, electrohydrostatic solution for electrospraying in cone-jet mode. J. Aerosol Sci. 25, 1065.