Applied Mechanics and Materials Vols. 325-326 (2013) pp 180-185 Online available since 2013/Jun/13 at www.scientific.net © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.325-326.180
Numerical Simulation of Electrohydrodynamic(EHD) Atomization in The Cone-jet Mode Abdolkarim Najjaran 1+, Reza Ebrahimi 1 ,Morteza Rahmanpoor 1 , Ahmad Najjaran 2 1
Aerospace Engineering Faculty
Khaje Nasiradin Toosi University Of Technology 2
Mechanical Engineering Department
School of Engineering, Shiraz Azad University +
[email protected].
Keywords: Electrohydrodynamic, Atomization, Electrostatic, Simulation.
Abstract. Electrohydrodynamic (EHD) has been applied in many areas, such as EHD atomization, EHD enhanced heat transfer, EHD pump, electrospray nanotechnology, etc. EHD atomization is a promising materials deposition technique as it allows uniform and regular deposition, and offers a range of other advantages, such as low cost compared with other current techniques, easy set-up, high deposition rate, and ambient temperature. Simulation is carried out using ANSYS FLUENT system. The approach in this work was to simultaneously solve the coupled (EHD) and electrostatic equations. The fields of velocities and pressure, as well as electric characteristics of EHD flows, are calculated. The model does not include a droplet break-up model. Introduction The flow of a liquid jet from an electrically-charged orifice to a grounded surface can result in a unique phenomenon. Given the proper combination of electrical, rheological and geometric properties, a conical meniscus will form at the exit of the charged orifice and a very thin jet (thin relative to the diameter of the orifice) will emanate from the tip of the cone. Inherent instabilities in the jet ultimately lead to its breakup into droplets, in which the droplet size tends to be on the order of the jet diameter rather than the capillary diameter. This phenomenon was first reported by Zeleny (1914, 1917) with subsequent analysis by Taylor (1964, 1969) and is often referred to as a “Taylor cone” or “cone-jet” [1]. Most previous modelling attempts have been based on an assumed initial shape of the liquid cone. In many cases, a certain charge distributaries has been assumed. In this model, no assumptions have been made for the shape and charge distribution, i.e., the initial shape is a flat liquid and with no charges present. The shape and charge distribution are been developed as an effect of the applied electric field . the objective was to investigate the parameters Leading to conejet formation and assess the predict ability of the voltage operating window of an EHD atomization. The model will not describe the spray behaviour after the forming of a stable cone-jet mode. Numerical Approach A model of the EHD atomization based on CFD provides numerous advantages over the approaches discussed above. If a free charge moves at a far slower speed than light, the EHD equations can be summarized as follows according to the theories of electrodynamics and Navier-Stokes equations for on incompressible, viscous dielectric liquid: Q ∇E = (1) ε
E = −∇φ
(2)
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∂Q + ∇. j = 0 ∂t ∇.v = 0 dv ρ = −∇P + µ∇2v + f e dt
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(3) (4) (5)
where E is the electric field , Q the free charge density, ε permittivity, φ the electric potential, j The current density, t time, ν the media velocity, ρ the mass density, p the pressure, µ the viscosity and f e the electric force . Within the dielectric liquid, the current density can be written as: (6) j = QkE − DQ ∇Q + Qv where k and DQ are ion mobility and molecular diffusion Coefficients respectively. If the charge transport due to an external field is faster than that due to its thermal motion, i.e. the applied voltage is higher than the thermal voltage, the charge diffusion can be ignored. According to the theory of Laudau[3], an electric force can be expressed as: f e = QE − 1 E 2∇ε + 1 ∇ ρ d ε E 2 2 2 dp
(7)
where the first term on the right hand side represents Coulomb force. The second and third represent dielectric and electrostrictive force respectively. Because the permittivity of most materials can be considered as constant and the liquid is incompressible, the second and third terms of eq(7) can be ignored. This indicates that the Coulomb force is the dominant force, thus Eq (7) can be reduced to f e = QE (8) The electric body force, QE, is a function of the electric field. In a conducting liquid, the charges and electric body forces are concentrated to the surface boundary. In a CFD model, the liquid surface boundary is not sharp but is defined as a volume density. This means that the surface charge density will be spread over the interface region, giving rise to columbic forces deforming the liquid. Various kinds of forces acting on the Taylor cone are indicated in figure 1. In this approach the hydrodynamics of the liquid are included. in this electrostatic solution, no current is present and current is not included in the governing equations. To model the complete EHD atomization process together with current conservation, all the Maxwell’s equation must be solved for the two-phase system of liquid and air.
Figure1. Forces acting in the liquid cone[8]. Geometry and Boundary Conditions An axi-symmetric geometry was generated based on the setup described above. The capillary geometry was modeled as infinitely thin cylindrical walls. This gives infinitely high potential gradients in the computation. The effect of this is that as the cone and jet is formed. The computational domain can be seen in figure 2.
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Figure 2. Computational domain. The boundary conditions are divided into electrostatic and hydrodynamic conditions. On the symmetry boundaries, Neuman conditions were used. Dirichlet conditions were used on the remaining boundaries. The initial liquid geometry is a flat surface at the rim of the capillary. During the iterations of the computational cycle, the shape of the liquid transforms into a conical shape. In this state, the shape is similar to the steady-state atomization in the cone-jet mode as shown in the literature [2]. Modes of Spraying The jet can disintegrate into droplets, while issuing from a capillary maintained at high potential, in many different ways. Several attempts, based on different criteria, have been undertaken to classify the modes of EHD spraying. The spraying modes can be divided into two groups. Into the first group are included the modes in which only fragments of liquid are ejected from the capillary directly. This group comprises: the dripping mode, micro dripping mode, spindle mode, multi-spindle mode, and ramified-meniscus mode. Into the second group are included the modes in which the liquid issues a capillary in the form of a long continuous jet which disintegrates into droplets only in some distance, usually a few mm, from the outlet of the capillary. The jet can be stable or move in a certain manner. This group includes cone-jet mode, precession mode, oscillating-jet mode, multi-jet mode and ramified-jet mode. The following considerations classify the modes of spraying and characterize most of them in terms of the jet and drop formation. Cone-jet mode In the cone-jet mode the liquid forms a regular, axisymmetric cone with a thin jet at its apex. According to Cloupeau and Prunet-Foch [7] the cone can assume three different forms: with linear sides, convex or concave. The jet flows along the capillary axis or deflects from it only on small angle usually smaller than 10 . The jet at its end undergoes instabilities. Two types of instabilities are known: varicose [7] and kink [7]. In the case of varicose instabilities, the waves are generated on the surface of the jet, but the jet does not change its linear position. In the nodes of the wave the liquid contracts and the jet disintegrates into equal droplets, which further flow close to the capillary axis. In the case of kink instabilities the whole jet moves irregularly off the axis of the capillary, with high amplitude and breaks up into series of fine droplets due to electrical and inertial forces. The aerosol is spread out off the axis, but nearly uniformly in the spray cone of an apex angle of 50 − 60 . CFD Model In [2] Lastow and Balachandran presented a novel CFD simulation method of the EHD atomization process. The model can be used to predict the shape, stability regions and droplet sizes. The approach is similar to the Lastow and Balachandran. The heat conduction equation, which is solved by the CFD solver, can be used to solve the electric field equation. Where as in this approach, the liquid conductivity and electric field equation are included (with ANSYS FLUENT 12.1).
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Consequently, a very fine mesh together with short time step must be used to obtain a satisfactory solution in terms of convergence and accuracy. The liquid properties used in the CFD calculations are summarized in Table 1. Table1. Liquid properties of heptanes Liquid ρ (kg/m3) k (S/m) γ (N/m) µ (Pa s) properties Heptane 0.021 0.39E-3 684 0.77E-6 The solver used was SIMPLE which is an implicit solver [4]. The momentum equation was solved using a upwind Scheme. This scheme is basically a first-order accuracy approximation. The reason to use a low-order scheme was that no convergence could be reached whit the less stable higher order scheme [4, 5]. Since high accuracy is paramount in this calculation, a very fine mesh, (Dx ~ µm ), and short time steps (Dt~10-5s) were used. The numerical model is axi-symmetric, which means that the mesh density increases closer to the symmetry axis. The liquid flow and the free surface were described in terms of volume of fraction (VOF). This means that for each cell, a ratio of liquid to gas volume was determined. The calculation can have any shape of the liquid as initial condition, but in all the result reported from this work except Lastow and Balachandran, a flat cylindrical shape was used. This approach was chosen to avoid any influence of the conditions on the shape of the liquid. CFD simulations, in general, require information about the precise geometry of the capillary and the contact angle between the capillary and liquid. This information is not normally available in the literature and must be based on assumptions. In this work, only a simplified axi-symmetric model was used. CFD simulation Figure 3 illustrated the geometry that modeled in our simulations. A constant electric potential difference v0 is maintained between the conducting cylindrical needle and the metal plate, which is separated by a distance L. The radius of the needle orifice is R0. A semiinsulating liquid of density ρ , viscosity µ , electric conductivity k and permittivity ε flows through the needle with a constant volumetric flow rate Q. the surface tensions coefficient between the liquid and air is γ . The permittivity of vacuum is ε0 formation of the Taylor cone can be quantified by the jet radius R, and the cone shape.
Figure3. Mechanism and geometry that modeled in simulations. The detailed shape of the nozzle determines the surrounding electric field and therefore affects the formation on the Taylor cone significantly, as experienced in simulations and also reported by experiments [6]. Figure 4 shows a successful generation of a Taylor cone.
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Figure 4. Fully developed Taylor cone of heptanes liquid. Results and Discussion The velocity field vector inside the liquid for the case shown in figure 5 as can be seen in figure 6, a back flow near the apex of the liquid cone forms. The liquid moving down along the main flow direction is divided by a toroid-shaped vortex. In that region, the remaining liquid is drawn into the jet. These CFD results are consistent with the vortex dynamo described by Shtern and Barrero [8], who studied the flow phenomena inside a liquid cone by using lycopodium particles and photographing their dynamics. The researchers observed axisymmetric circulating patterns inside the cone. This backflow was caused by tangential electric stress which is characteristic of the conejet mode. If the strength of the backflow becomes weaker, less tangential electric stress would be necessary to maintain the backflow near the apex of the liquid cone. Therefore, reducing the backflow near the apex of the liquid cone would be helpful for stable jet formation by decreasing the onset voltage (figure 7). Figure 8 shows the effect of the flow rate on the potential needed to realize a stable cone-jet mode. When the flow rate was increased, the necessary potential increased.
Figure 5. Velocity vector of heptane.
Figure 6. Velocity field distribution.
Figure 7. Potential field distribution.
Figure 8. Potential for stable cone-jet mode versus flow rate[8].
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References [1] J.Zeleny. On the conditions of instability of electrified drops, with applications to the electrical discharge from liquid Ioins, Proc. Cambridge Philos. Soc. 18 (1915) 71-83. [2] J O. Lastow, W.Balachandran, Numerical simulation of electrohydrodynamic (EHD) atomization, J. Electrostatic 64 (2006) 850-859. [3] L.D.Laudau, Electrohydrodynamic of continuous media, New York:Oxford, 1960. [4] Manual.ANSYS FLUENT 12.1. [5] H.K.Versteeg, W.Malalasekera, An Introduction to Computational Fluid Dynamics, Longman Group Ltd, Harlow, 1995. [6] J.Zeng, D.Sobek, T.korsmeyer, Electrohydrodynamic modeling of electrospray ionization: CAD for a microfluidic device-mass spectrometer interface. [7] A.Jaworek, A.Krupa, Main modes of electrohydrodynamic Conference, 1998.
spraying of liquids, ICMF
[8] S.Kim, Y.Kim, J.Park, J.Hwang, Design and evaluation of single nozzle with a non-conductive tip for reducing applied voltage and pattern width in electrohydrodynamic jet printing. J. Micromech. Microeng. 20 (2010) 055009.
Manufacturing Engineering and Process II 10.4028/www.scientific.net/AMM.325-326
Numerical Simulation of Electrohydrodynamic (EHD) Atomization in the Cone-Jet Mode 10.4028/www.scientific.net/AMM.325-326.180 DOI References [2] J O. Lastow, W. Balachandran, Numerical simulation of electrohydrodynamic (EHD) atomization, J. Electrostatic 64 (2006) 850-859. http://dx.doi.org/10.1016/j.elstat.2006.02.006
