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Abstract

We propose a bi-directional electrohydrodynamic pump developed for transporting dielectric liquid, where the electrodes are symmetrically configured but the applied voltage is non-symmetric. The underlying principle for liquid transport comes from the so-called Onsager effect, which states that the ion concentration is increased as the electric field is increased. Multi-physics software is used to perform numerical simulation for the fluid flow, the electric potential, and the transport of ion concentrations for two kinds of electrode patterns. A flow-visualization experiment is also conducted to verify the physical models and numerical methods employed. It is found that significant reduction of the ion recombination constant is required to get matching of the experimental and simulation results. We demonstrate through a parametric study that there is an optimum distance between two large grounded electrodes for producing a maximum pumping velocity at the diameter of two small electrodes fixed at 0.3 mm. The effect of the size of large grounded electrodes on the pumping performance is also studied in terms of streamlines, electric field, and charge distribution. A general account is also given of the basic ideas of electrode arrangement for the enhancement of pumping.

Keywords

Electrohydrodynamic pump bi-directional pump dielectric liquid Onsager effect

Introduction

How to pump fluids in the transport system of mini- and micro-scales has become a very important issue. In a lab-on-chip or micro-total analysis system ( $μ$ TAS), for instance, the liquid sample must be transported from the inlet reservoir to the places where mixing and chemical reaction of samples and detection of the outcome take place.^1–4 In electronic devices, due to the trend of miniaturization, a more efficient and powerful but small-size pump is demanded for the cooling of electronic components^5–7 or power electronics.^8,9 Mechanically actuated pumps are seldom used because of difficulty in fabrication, flow control, and so on. Electrokinetic pumping then emerges as a practical way to achieve fluid transport in view of its simple structure, ease of fabrication, and controllability. Aside from its compatibility, the electrokinetic method is susceptible to electrode degradation and bubble generation through electrolysis when a polar liquid with high permittivity, such as water, is to be pumped.¹⁰ Various pumping methods and the related technologies in small scales have been reviewed in Iverson and Garimella.¹¹

Recently, pumping of dielectric liquid with low permittivity is suggested as a new method in particular for cooling of micro-electronic components.^12–16 It is free of noise, vibration, bubble generation, and electrode degradation. The underlying principle for the pumping of dielectric liquid comes from the electrohydrodynamic (EHD) flow caused by the Onsager effect. When a certain amount of surfactant is added to a dielectric liquid containing unknown impurities, free ions are created in the form of reverse micelles, the concentration of which increases with the local electric field intensity, leading to the creation of charge (the Onsager effect); for more detailed explanation on the fundamentals of EHD flows and related literature, refer to Suh.¹⁷

Initial studies on EHD pumps driven by the Seyed-Yagoobi group (see the review article¹⁸ and references therein) were motivated by the need to enhance heat transfer in pool boiling and heat pipes, where working fluids are characterized by relatively low permittivity. Recently, they demonstrated the feasibility of applying the EHD pump to heat exchangers under a zero- or micro-gravity environment.¹⁹ Their numerical model for the ion transport includes ion dissociation and recombination, but the Onsager effect has not been considered and the fluid flow is assumed to be driven only by the counter-charge in the dissociation layer (DSL). On the contrary, some fundamental studies on the EHD flows in dielectric fluid have neglected the effect of thin DSLs near the electrodes.²⁰

A variety of parameters are involved in the performance of the EHD pump, such as shape, arrangement and size of electrodes, applied electric field, and various fluid properties. Jeong and Seyed-Yagoobi^21,22 employed a needle-type electrode as the anode and a ring-type electrode as the cathode (also used in Pearson and Seyed-Yagoobi¹⁸). Later, a combination of a ring-type and a perforated electrode was used.^23,24 In their studies, however, the contribution of the bulk-driven flow to the EHD pumping was not considered. The combination of saw-tooth and planar electrodes employed in Seo et al.¹² and Kim et al.¹⁵ also carries a similar concept as to the enhancement of pumping performance. Recently, in view of its easy fabrication, EHD pump with planar electrodes has been studied.^25,26 Even a symmetric pattern of the electrode arrangement is shown to produce a unidirectional pumping, which is attributed to the mismatch of mobility for positive and negative ions.²⁶ The needle-type or perforated or planar electrode somehow involves unavoidable geometrical singularity, which is a disadvantage in developing a mathematical model for the ion or charge creation and transport. Circular electrodes are free from such singularity. In Hojjati et al.,²⁷ a series of symmetrically arranged circular cylinders of equal size were used to pump dielectric liquid in an open circulating channel. It was demonstrated that mismatch of ion mobility between cations and anions can yield a unidirectional flow. Very recently, three identical circular electrodes were used in a straight channel to demonstrate both experimentally and numerically that unidirectional pumping can be achieved if the electrodes are asymmetrically arranged.²⁸ However, the effects of various parameters, not least the geometrical factor, are yet to be explored.

Effect of the external field intensity on the EHD pumping velocity (or pumping flow rate) reported in the literature is found to have a common pattern regardless of the data source. The pumping velocity remains at a low level for small applied-field intensity. After a turning point, however, it increases in a linear fashion with the field intensity.^{12,15,18,26,27} Shown in Gharraei et al.²⁶ and Hojjati et al.²⁷ is the existence of a second turning point beyond which the flow rate tends to a saturated value. No plausible explanation for such variation exists yet. On the other hand, the asymptotic analysis of Suh et al.²⁹ predicts that the bulk-driven flow velocity and the DSL-driven flow velocity increase with field intensity to the cubic and fourth power, respectively, of the field intensity. Since the DSL-driven flow is counter-active to the bulk-driven flow, it may happen that the former overtakes the latter at high electric field so that the rate of increase in the pumping velocity monotonically decreases with the field intensity.

Effect of other fluid properties, such as ion concentration, ion-recombination constant, and mobility on the EHD pumping, has not been treated yet. The concept proposed in Suh et al.²⁹ and Suh and Baek,³⁰ together with the corresponding formula for the velocity, that the EHD flow field is determined from the competition between the bulk-driven and DSL-driven flow should be useful in exploring the qualitative and quantitative features of EHD pumps in terms of parameter variation.

In this study, we propose a bi-directional EHD pump with a simple arrangement of electrodes of a circular cylinder. The feasibility of dual-direction pumping has been demonstrated in, for example, Lemke et al.³¹ and Chee et al.,¹⁶ but these studies required mechanical actuators and valves, fabrication of which may not be so easy on a small scale. Use of induced-charge electro-osmosis via biased direct current (DC) combined with alternating current (AC) voltage is proposed in Islam and Reyna³² for a bi-directional pump. In that study, however, deionized (DI) water is used as the pumping medium and thus, the mechanism of the pumping is of course different from the present study. We in this study propose a bi-directional EHD pump composed of circular-cylindrical electrodes in a circulating channel of annular shape and explore the effect of key geometrical parameters, that is, electrode arrangement and size, on the pumping velocity. The rest of this article is organized as follows. Section “Geometrical configuration and flow-visualization experiment” describes the geometrical configuration of the electrodes employed in the design of our proposed bi-directional pump and the experimental methods used for flow visualization. Section “Numerical analysis” presents the governing equations for the fluid flow and ion transport and the numerical methods for solving them. In particular, we address how the parameters associated with the ion transport are set for the best matching between the experimental and numerical simulation results. In section “Basic idea on the principle and development of EHD pump,” basic ideas to be applied for developing new EHD pumps or for enhancement of their performance are explained, which are also useful in understanding the numerical results as well as the experimental data presented in section “Numerical and experimental results and discussion” for two kinds of electrode patterns. Finally, we summarize important findings of the study in section “Conclusion.”

Geometrical configuration and flow-visualization experiment

Figure 1 shows the experimental arrangement for visualization of EHD flow of the dielectric liquid dodecane (No. 297879; Sigma-Aldrich) inside an annular channel having several cylindrical electrodes symmetrically arranged so as to make the bi-directional pumping possible. The surfactant Span80 (No. 85548; Sigma-Aldrich) was added to the liquid in 0.5 wt% in order to control the concentrations of the free ions. For flow visualization, we also mixed in the liquid a small amount of nylon fluorescence particles with the average diameter of 7 $μ$ m (Model No. 10456; Kanomax, Inc.). The whole channel including a top plate is made of acrylic, and two small breathing holes are drilled on the top plate for intake of the liquid and air removal. The height, inner diameter, and outer diameter of the channel are 10, 60, and 80 mm, respectively.

Figure 1.

Schematic diagram of the experimental arrangement and two kinds of electrode patterns used for visualization of the EHD flows inside a annular channel: (a) experimental apparatus: two kinds of electrode pattern with (b) four (pattern A) and (c) seven (pattern B) electrodes, where the filled circles indicate the anode; (d) top view of the annular channel used as a bi-directional pump, where the diamond shape described by dash-dot lines around the electrodes indicates the boundary of image frames the camera has taken for the streamline patterns and the line denoted by the coordinate s in the upper side indicates the section at which the velocity data are calculated from the PIV technique.

Tungsten wires are inserted through drilled holes into the top and bottom plates to be used as electrodes. The flow rate and the direction of liquid pumping within the channel mainly depends on the pattern of electrode arrangement as well as the way electrical potentials are applied on multiple electrodes. In this study, we consider two kinds of electrode patterns, A and B, as shown in Figure 1(b) and (c). The pattern A consists of two small electrodes, one on the left-hand side (LHS) as the anode and the other as the cathode, and two large electrodes, both as the cathode. The diameter of the small electrodes (e.g. P_C and G_C for pattern A; see Figure 1(b)) is fixed at $d = 0.3 mm$ . In the experiment, the gap between large electrodes (G_B and G_T) is fixed at $H = 3 mm$ and the gap between small electrodes at $W = 0.6 mm$ in both patterns. In the experiment with pattern A, the diameter of G_B and G_T takes three values, D = 0.3, 1.0, and 2.0 mm. In pattern B, the diameter of the large cylinders is fixed at D = 1 mm. The two small cylinders on the LHS of Figure 1(b) and (c) serve as anodes and all the others as cathode (grounded). It is obvious that simply switching the polarity of the external voltage on the small electrodes for both patterns will make the pumping flow reverse.

Before doing the main experiment, we washed the channel and electrodes by immersing them three times in water mixed with a detergent and then in pure water, after which they were dried. Electrodes were inserted into pre-drilled holes and the electric wires were connected. The whole apparatus was then put on a larger container and the liquid was poured. The DC voltage was applied using a high-voltage power supply (Model 610E; TREK Inc.). As soon as a DC voltage difference $V_{0}$ is applied to the electrodes, EHD flow is created around the electrodes causing circulating flow through the channel. After about 20 s, when the flow is assumed to have arrived at a fully steady state, we took videos of the particle motions representing the EHD flow, shown on a thin laser-light sheet, using a high-speed charge coupled device (CCD) camera (Model GX1050; Allied Vision Technologies, Inc.) for more than 15 s at the rate of 100 frames per second (see Figure 1(a)). The streamline patterns were obtained by superimposing consecutive images taken for the frame indicated by dash-dot lines in a diamond shape in Figure 1(d). The stream-wise component of the fluid velocity u was measured by applying a standard particle image velocimetry (PIV) technique to the images taken for the region around the line indicated by the coordinate s in Figure 1(d). The mean velocity across the channel section, U, was then obtained from integration of the velocity distribution $u (s)$ measured along the s-axis.

Numerical analysis

Governing equations

Equations governing the steady flow in the annular channel read

\nabla \cdot u = 0

(1)

ρ (u \cdot \nabla) u = - \nabla p + η \nabla^{2} u + q E

(2)

where $u$ is the velocity, p is the pressure, $ρ$ is the density, $η$ is the viscosity, $q = e (z_{1} c_{1} + z_{2} c_{2})$ is the space charge density, $E = - \nabla V$ is the electric field, and V is the electric potential. Furthermore, $c_{1}$ and $c_{2}$ are number concentrations of cations and anions, respectively; $z_{i}$ is the corresponding valence; and e is the proton charge. In this numerical analysis, both the cation and anion are assumed to have a unit valence, $z_{1} = - z_{2} = 1$ .

Electric potential V is determined by the Poisson equation

ε_{0} ε_{r} \nabla^{2} V = - q

(3)

where $ε_{0}$ and $ε_{r}$ are the vacuum permittivity and the relative permittivity of the liquid, respectively.

The ion concentrations are determined by the Nernst–Planck equations

\nabla \cdot j_{i} = R_{i}

(4)

where $j_{i}$ is the ionic flux defined as

j_{i} \equiv u c_{i} + z_{i} μ_{i} E c_{i} - D_{i} \nabla c_{i}

(5)

In addition, $D_{i} = k_{B} T / (6 π η a_{i})$ is the diffusivity, $μ_{i} = e / (6 π η a_{i})$ is the mobility, $a_{i}$ is the ionic radius, $k_{B}$ is the Boltzmann constant, and T is the temperature. The source term on the right-hand side (RHS) of equation (4) reflects the ion dissociation and recombination

R_{i} = α [c_{0}^{2} F_{O} (b) - c_{1} c_{2}]

(6)

where $α$ is the recombination constant and $c_{0}$ is the zero-field concentration. The Onsager function is defined as $F_{O} (b) = I_{1} (2 b) / b$ . Here, $I_{1}$ is the modified Bessel function of the first kind and order one and $b^{2} = 4 γ E$ . Furthermore, $γ = e^{3} / (16 π ε_{0} ε_{r} k_{B}^{2} T^{2})$ is a constant with the unit m/V and $E = | E |$ is the field intensity. The Onsager function can be written in an ascending series for small b as follows

F_{O} (b) = \sum_{k = 0}^{\infty} \frac{b^{2 k}}{k! (k + 1)!} = 1 + 2 γ E + O (γ^{2} E^{2})

(7)

In this numerical simulation, we use for simplicity the two-term expansion, that is, RHS of equation (7), in evaluation of the Onsager function. As usual, the recombination constant $α$ may have to be reduced from the Langevin value

α_{L} = \frac{e (μ_{1} + μ_{2})}{ε_{0} ε_{r}}

(8)

with the reduction factor $β_{L}$ ; $α = β_{L} α_{L}$ . It was previously shown by comparing the experimental and numerical results that $β_{L}$ must be taken as significantly smaller than 1.^29,30 For more detailed explanations on the formulation for the ion dissociation and recombination, refer to Suh.¹⁷ In section “Parameter setting” under this section, we also discuss the way in which $β_{L}$ is determined in this study in relation to the convergence problem of numerical simulation.

Numerical methods

We used the multi-physics code COMSOL to numerically solve the system of equations (1) through (4). Solving the given multi-physics problem using other kinds of program, including the in-house code, may not be practically relevant, because so many variables involved in the formulation would definitely tend to cause the numerical instability, which is not thought to be easily resolved. Even with COMSOL, we underwent significant trial-and-errors to overcome such numerical instability, part of which has been explained in our previous studies.^29,30

When the original form of the ion transport equations, that is, equation (4), has been used in the numerical simulation, it is found to cause a numerical instability.³⁰ Thus, following our previous study,³⁰ in order to overcome such instability problem, we set the concentrations $c_{i}$ as the sum of the zero-field concentration $c_{0}$ and the perturbed one $Δ c_{i}$ , say $c_{i} = c_{0} + Δ c_{i}$ , and consider $Δ c_{i}$ as the main variable instead of $c_{i}$ . Such a successful treatment implies that the code instability must be due to the explicit treatment of the source terms $R_{i}$ in equation (4). Aside from such fundamental instability, the code also fails to give converged solutions when $β_{L}$ is taken much smaller than 1 for the better agreement between the numerical and experimental results, the reason for which is yet unknown.

The triangular and tetra meshes are used for the two-dimensional (2D) and three-dimensional (3D) calculations, respectively. Very fine grids are built near the electrodes in such way the thin DSLs can be well captured. No-slip and impermeable conditions are applied as the velocity boundary conditions on the channel walls and electrode surfaces. The anodes are applied with a constant DC voltage $V_{0}$ while the cathodes are grounded. Regarding the boundary conditions for the concentrations, we specify insulation condition (zero ionic flux) on the channel walls. On the surface of electrodes, the co-ion concentration and the normal gradient of the counter-ion concentrations are set at zero.

Parameter setting

There are several geometric parameters associated with the electrode arrangement. In order to make the bi-directional pumping possible, we must keep the symmetry of the electrode pattern. Thus, we have three parameters to be considered as shown in Figure 1(b) and (c): the diameter of large electrodes, D; the gap between large electrodes, H; and the gap between small electrodes, W. We fix the diameter of the small electrodes at $d = 0.3 mm$ in this study.

There are also several parameters related to the material properties to be specified for the calculation. Following our previous studies,^29,30 we determine the relative permittivity, ionic radius, and zero-field concentration for the dielectric liquid, that is, dodecane mixed with Span80 in the 0.5 wt%, as $ε_{r} = 2$ , $a_{1} = a_{2} = 2.25 nm$ and $c_{0} = 3.28 \times 10^{19} / m^{3}$ (or $5.45 \times 10^{- 5} mol / m^{3}$ ), respectively. Density and viscosity of the liquid are set at $ρ = 750 kg / m^{3}$ and $η = 0.00134 Pa s$ , respectively. Temperature is set at $T = 295 K$ . The zero-field electric conductivity is calculated to be $σ_{0} = 2.96 \times 10^{- 8} S / m$ . The voltage difference applied to the electrodes is 1 kV in all cases, except when the effect of its variation is considered.

The reduction factor for the recombination constant, $β_{L}$ , is to be determined as usual in such way that the numerical results best fit the experimental data. However, as expected, the 3D computation for the whole domain requires very large number of grids and thus huge amount of CPU time. Furthermore, even a slight reduction of $β_{L}$ from 1.0 in 3D calculation is also found to lead to numerical instability, the reason for which is also unknown. Since performing a parametric study is one of the important objects of this work, 3D calculation cannot be employed for the general purpose. Fortunately, however, 3D numerical results reveal that the flow around the electrodes is almost uniform along the axis parallel to the electrode posts, except very near to the top and bottom walls of the channel due to the no-slip conditions. In addition, the fluid-flow resistance exerted by the channel walls turns out to be very small, so that the pumping operation is close to an unloading state. This then implies that 2D calculation should serve as a good approximation to the original 3D problem as far as the correlation between the mean velocity $U_{3 D}$ obtained from 3D calculation and that from 2D calculation $U_{2 D}$ is correctly established. For this, we first performed 3D and 2D calculations with $β_{L} = 1$ to obtain the ratio of the mean velocities, $U_{3 D 1} / U_{2 D 1} = 0.784$ . We assume that even with decreased value of $β_{L}$ , the ratio would not be changed because of the 2D nature of the driving flow near the electrodes. On the other hand, from the experiment with the pattern A under the parameter set-up, D = 2 mm, W = 0.6 mm, and H = 3 mm, we measure the mean velocity, U_ex = 1.76 mm/s, indicating that the target velocity to be obtained from 2D calculation with adjusted $β_{L}$ must be $U_{2 D} = 2.24 mm / s$ . So, 2D calculation is performed at the identical geometrical parameter set but with $β_{L}$ continually reduced from 1. Figure 2 shows the variation of $U_{2 D}$ (in Figure 2, $U_{2 D}$ is simply denoted as U) with $β_{L}$ obtained in this way. $U_{2 D}$ is shown to decrease as $β_{L}$ is decreased in accordance with the prediction of Suh et al.²⁹ through asymptotic analysis; the decrease in $β_{L}$ causes increase in the slip velocity at the edge of DSLs near the electrodes acting against the bulk-driven flow, which results in an overall decrease in flow velocity in the bulk or decreased flow rate. However, the numerical simulation becomes unstable at $β_{L}$ less than 0.025 at which we have U_2D = 2.43 mm/s still larger than the target value $U_{2 D} = 2.24 mm / s$ . Thus, it is concluded that exactly matching the experimental data by the adjustment of $β_{L}$ is not possible. We therefore finally set $β_{L} = 0.04$ which turns out to be the smallest value at which the 2D simulation can give converged solutions over the whole range of parameter values considered in this work. For some cases, however, we use, if possible, decreased values of $β_{L}$ in order to attain a better comparison between the numerical and experimental results.

Figure 2.

Effect of the reduction factor $β_{L}$ on the mean velocity U obtained from 2D simulation with D = 2 mm, W = 0.6 mm, and H = 3 mm for the electrode pattern A.

Basic idea on the principle and development of EHD pump

Before presenting the experimental and numerical results, we address the fundamental mechanism of EHD pumping, which can be utilized in development of new EHD pumps or their enhancement. First, we must be able to estimate the charge distribution in the bulk. Following the asymptotic analysis,²⁹ under the assumption of equal mobility and small values of the dimensionless parameters, $ε^{*} = ε_{0} ε_{r} V_{0} / (e c_{0} d^{2})$ (= 0.037 in this study) and $γ^{*} = γ V_{0} / d$ (= 0.926 in this study), the bulk is almost neutralized and $c_{1} \approx c_{2}$ . However, the Onsager effect that larger local electric field intensity brings about larger ion concentration gives rise to a non-zero concentration gradient in the bulk. This in turn causes a slight difference in the local concentration between the cation and anion, which corresponds to net charge. The space charge density created thereby in the bulk can be formulated as $q = - ε_{0} ε_{r} E \cdot \nabla c / c$ , where $c = (c_{1} + c_{2}) / 2$ is the averaged concentration. Applying an approximate expression for the field-enhanced concentration, $c = c_{0} (1 + 2 γ E)$ , we derive

q = - 2 ε_{0} ε_{r} γ E \cdot \nabla E / (1 + 2 γ E) = - K E \cdot \nabla E

(9)

where $K = 2 ε_{0} ε_{r} γ > 0$ is a constant and the second equality is valid for small values of $γ E$ . Thus, we can see that positive (negative) charge is created when the field intensity decreases (increases) along the direction of $E$ . Very near to the electrode, $E$ is directed almost normal to the electrode surface (toward the liquid for the anode and toward the electrode for the cathode) and E decreases with the normal distance from the surface. Thus, positive (negative) charge is created near the anode (cathode). Figure 3 illustrates the schematic of charge distribution around a pair of circular electrodes with different sizes, which is taken as a simplest configuration for illustrating the principle of EHD pumping. The dash-dot line in Figure 3 corresponds to the border between the regions of positive and negative charge. We can see that the positive charge around the anode is denser than the negative charge around the cathode, whose size is set to be larger than the anode, because both the field intensity and its gradient are larger near the anode than near the cathode (refer to equation (9)).

Figure 3.

Schematic of electric field lines (thin solid lines with open arrows), net-charge distribution (circles with + and − signs), and forces (filled arrows) acting on the charges around a pair of electrodes of different sizes illustrating the mechanism of EHD pumping from the smaller to larger electrode. Also shown are the local unit vector set $(t, s)$ .

Next, we must be able to estimate the fluid flow and EHD pumping from the charge and electric field distributions. The Coulomb force density, the driving term in the momentum equation (2), $q E = - K (E \cdot \nabla E) E$ , is directed toward the fluid side from the electrode surface regardless of the electrode polarity. If its magnitude was uniform over any concentric circle around the electrode, we cannot expect fluid flow but only positive radial pressure gradient required for the force balance. Since the field intensity is usually higher in the facing side than the back side of the electrode, the net flow around the electrode must be from the back to facing side. This concept applies to both electrodes, so the force from the anode opposes that from the cathode. However, overall, the field intensity (and so the charge density) is higher near the smaller than the larger electrode, so the net flow must be from the anode to the cathode side in the case of Figure 3, which corresponds to the pumping flow. The creation of EHD flow can be also understood in terms of the vorticity source, which is given in Appendix 1.

The dominant parameter affecting the increase in the pumping capacity is of course the electric field. However, under the assumption of fixed level of electric field, the pumping capacity depends on how the electrodes are spatially arranged. When we endeavor to enhance the pumping capacity under a fixed potential $V_{0}$ , we must consider two aspects associated with this. The first one is increasing not only the magnitude of $| \nabla E |$ itself around the driving electrode (the anode in this study) in the direction of the field, so that the co-charge density is increased as seen from equation (9), but also enlarging the space where the co-charge (positive charge in this study) density is dominant. This can be achieved by making the field lines more expanded, as possible, toward the fluid side from the electrode. In the pattern A of this study, for instance, field lines from the single small anode are made expanded by the other three grounded electrodes so that the level as well as the region of the co-charge density is increased as possible. Enlarging the ground electrodes surely helps such field expansion, but there must be a limit, because such enlargement will make the flow passages relatively narrowed. The second aspect contributing to the increased capacity of the EHD pump is making the co-charge more concentrated in the downstream (facing) side of the driving electrode than the one in the upstream (back) side so that a larger fraction of the Coulomb force contributes to the creation of fluid flow than to the build-up of osmotic pressure. For the pattern A, for example, arranging all the ground electrodes on the downstream side follows this concept. Optimum arrangement of electrodes for maximum capacity for a given level of field intensity, however, cannot be determined intuitively but needs numerical or experimental work.

On the other hand, DSLs near electrodes exert adverse influence on the pumping effect because much more concentrated counter-ion concentrations in the thin DSL give slip velocity opposing the bulk-driven flow. The effect of DSL-driven flow is increased as the recombination constant $α$ is decreased, which is controlled by the reduction factor $β_{L}$ . A more systematic treatise on the solutions in DSL can be found in Suh et al.²⁹

Numerical and experimental results and discussion

Parametric study with electrode pattern A

We conducted the grid-dependence test to get a suitable mesh system. For this, we performed 2D simulation with the electrode pattern A at the parameter set, $D = 2 mm$ , $H = 3 mm$ , $W = 0.6 mm$ , and $β_{L} = 1$ with various grid resolutions. We get the mean velocity $U = 4.04 mm / s$ (in the following, the mean velocity obtained with 2D simulation is simply denoted as U) with 31,000 elements, 4.14 mm/s with 71,000 elements, 4.17 mm/s with 89,000 elements, and 4.17 mm/s with 120,000 elements, among which the final mesh system with 120,000 elements was selected as a suitable one for this study. The Reynolds number based on d = 0.3 mm, $η = 0.00134 Pa s$ , $ρ = 750 kg / m^{3}$ , and the typical velocity $U = 4.17 mm / s$ is 0.7 indicating that the flow should be of a low-Reynolds-number character.

Figure 4 shows the numerical results of U as a function of the gap between the large electrodes, H, for various diameters of large electrodes, D = 0.3, 0.7, 1.5, 2.0, 2.5, and 3.0 mm. We find that at a fixed value of D, the mean velocity initially increases with H showing a maximum value at a critical value of H, ranging from 2.5 to 3 mm, and then decreases with H after that. Obviously, when H is smaller, the flow passage between the large and small ground electrodes should be more narrowed leading to increase in the viscous friction and decrease in the flow rate. On the contrary, when the gap is too large, the effect of large electrodes on the electric field will become weaker and the electric field tends to be symmetric yielding decreased net flow along the channel. So, we can expect the existence of a critical value of H at which the mean velocity becomes maximized.

Figure 4.

Variation in the mean velocity with H obtained from 2D numerical simulation for the pattern A at W = 0.6 mm with six values of the large electrode diameter, D = 0.3, 0.7, 1.5, 2.0, 2.5, and 3.0 mm.

Figure 4 also reveals that the maximum mean velocity increases with D. This can be understood with the aid of the discussion given in section “Basic idea on the principle and development of EHD pump.”Figure 5 shows comparison of two streamline patterns obtained at D = 0.3 mm and D = 3 mm. We can see that for the case with D = 0.3 mm, many streamlines recirculate around G_B and G_T, and those streamlines contributing to the pumping are less compared with D = 3 mm. A more fundamental reason for this can be given with the aid of the contours of ion-concentration difference, $Δ c \equiv c_{1} - c_{2}$ , and the field vector plot as shown in Figure 6. We can see that increase in D (Figure 6(b)) results in the increase in the region of positive charge near the anode and simultaneous decrease in the region of negative charge near the cathodes compared with the set at D = 0.3 mm (Figure 6(a)). On the other hand, the downstream-wise component of the electric field on the RHS of the anode is higher than the upstream-wise component on the LHS due to the ground electrode G_C, which is thought to be the fundamental forcing responsible for driving the fluid to the right-hand (downstream) side. However, we cannot see any significant difference in the distribution of $Δ c$ and $E$ between the two D values, except for a slightly larger region of positive $Δ c$ near the anode for D = 3 mm, which is not enough to explain the reason for the higher mean velocity at D = 3 mm. Figure 6, however, shows a significant difference in the distributions of $Δ c$ and $E$ between the two results near the electrodes G_B and G_T. For the case with D = 0.3 mm, the Coulomb force density is radially outward around G_T because $Δ c$ is negative there. More importantly, the LHS of G_T shows overall stronger Coulomb force (directing left-downward) than that on the RHS, which should act to squeeze (or block) the main stream around the anode like Figure 5(a). On the other hand, for the case with D = 3 mm, Figure 6(b) shows insignificant levels of $Δ c$ around G_T, whose impact against the main stream should be much less than with D = 3 mm, as seen from Figure 5(b). The squeezed main stream of Figure 5(a) for D = 0.3 mm turned out to give the mean velocity almost half that for D = 3 mm.

Figure 5.

Streamlines around electrodes obtained numerically at $H = 3 mm$ and $W = 0.6 mm$ for (a) $D = 0.3 mm$ and (b) $D = 3 mm$ .

Figure 6.

Contour lines of the concentration difference (solid lines), $Δ c \equiv c_{1} - c_{2}$ , and the electric field vectors (arrows) obtained numerically at $H = 3 mm$ and $W = 0.6 mm$ for (a) $D = 0.3 mm$ and (b) $D = 3 mm$ . The contour lines with zero level are indicated in each frame. The contour levels are, in mole/m³, $4 \times 10^{- 10}$ , $1 \times 10^{- 9}$ , $2 \times 10^{- 9}$ , $5 \times 10^{- 9}$ , $1.5 \times 10^{- 8}$ , and $5 \times 10^{- 8}$ for the lines around the anode and $- 4 \times 10^{- 10}$ , $- 1 \times 10^{- 9}$ , $- 2 \times 10^{- 9}$ , $- 5 \times 10^{- 9}$ , and $- 1.5 \times 10^{- 8}$ for the lines around the cathode. Magnitude of the electric field vectors is drawn in logarithmic scale, that is, in $\log (E)$ .

We next investigate the effect of W on the mean velocity. The numerical results obtained at two values of D are presented in Figure 7 as a function of H. The plots also reveal that the mean velocity is maximized at a critical value of H, ranging from 2.5 to 4.0 mm. The mean velocity is shown to monotonically increase as W is decreased for both diameters, which is simply attributed to the higher field at closer distance between P_C and G_C leading to larger charge density, larger Coulomb force, and stronger EHD flow.

Figure 7.

Variation in the mean velocity with H obtained from 2D numerical simulation for the pattern A at (a) $D = 1.5 mm$ and (b) $D = 3 mm$ with four values of distance between the small electrodes P_C and G_C, W = 0.4, 0.6, 0.8, and 1.0 mm.

Figure 8 shows the monotonic increase in the mean velocity with decrease in W for a fixed set of D and H. Figure 8 also presents the pump efficiency defined as

η_{e} = \frac{Q Δ p}{V_{0} I}

(10)

where Q is the flow rate per unit depth of the channel and $Δ p$ is the pressure difference across the actuator where electrodes are patterned. We calculate $Δ p$ from $Δ p = p_{d} - p_{s}$ , where $p_{d}$ and $p_{s}$ correspond to the delivery and suction pressure. When the pressure is measured along the centerline between the outer and inner circular walls of the channel, the maximum pressure occurs at the point just downstream of the actuator, which corresponds to $p_{d}$ , and similarly, the minimum pressure occurs at the point just upstream of it, which corresponds to $p_{s}$ . Furthermore, I is the current per unit channel depth given from the integration of the current density over a closed path around the anode

I = e \int (j_{1 n} - j_{2 n}) ds

(11)

Here, $j_{1 n}$ and $j_{2 n}$ are the normal component of the cation and anion flux, respectively, and s denotes the coordinate along the path on which the integration is performed. We take a square path around the anode large enough to fully surround the DSL so that the integration is almost invariant of the square size; we confirmed the invariance of the results for two squares of different size, that is, 0.5 and 0.7 mm. We can see from the result (Figure 8) that the pump efficiency $η_{e}$ also increases monotonically with decrease in W, being contrary to the usual engineering concept that a higher performance (higher mean velocity in this case) of any engineering system is commonly associated with a lower efficiency. The level of the efficiency also turns out to be very small, at most 0.03%. But this is not far from the values reported in the literature, for example, 0.1% in Nourdanesh and Esmaeilzadeh³³ and 0.001% in Pearson and Seyed-Yagoobi.³⁴ Furthermore, the pressure difference $Δ p$ in this study remains very small, typically 0.089 Pa at W = 0.4 mm, and we did not attempt to investigate the influence of $Δ p$ on Q,^{12,15,23,25,34} which requires further simulation, for instance, with a longer channel or with an artificial block on the fluid stream inside the channel.

Figure 8.

Mean velocity U and pumping efficiency $η_{e}$ versus W obtained numerically at D = 2 mm and H = 3 mm.

Figure 9 depicts the dependence of the mean velocity on the applied voltage obtained numerically at D = 3 mm, H = 3 mm, and W = 0.6 mm. At low voltage, the mean velocity is shown to increase with the field intensity like $V ~ E^{n}$ , where the exponent n is larger than 1. At $V_{0} > 700 V$ , however, n tends to 1. Dependence of the pumping velocity on the applied voltage reported in the literature is also shown to be qualitatively similar to Figure 9.^12,15,19,34 According to the asymptotic analysis,²⁹ the bulk-driven flow velocity increases like $V ~ E^{3}$ , whereas the DSL-driven flow velocity behaves like $V ~ E^{4}$ . So, at low voltage, the effect of DSL can be negligible yielding $n \approx 3$ . As the voltage is increased, however, the DSL-driven flow becomes more important. Since the DSL-driven flow opposes the bulk-driven flow, n must be correspondingly decreased. The turning point from the bulk-driven to the DSL-driven flow is dependent on the ion concentration and the factor $β_{L}$ . As shown in Suh et al.,²⁹ lower concentration and/or smaller $β_{L}$ yields higher slip velocity at the edge of the DSL, implying that the turning point of applied voltage should be lower at lower concentrations and smaller $β_{L}$ . On the other hand, in Kim et al.,²⁸ the exponent was close to 3 but the applied power was AC at 110 Hz.

Figure 9.

Mean velocity U versus applied voltage $V_{0}$ obtained numerically at D = 3 mm, H = 3 mm, and W = 0.6 mm.

Comparison between the experimental and numerical solutions with electrode pattern A

A visualization experiment was conducted for the pattern A at H = 3 mm and W = 0.6 mm with D = 0.3, 1, and 2 mm. Figure 10 shows the distribution of the stream-wise velocity component obtained along the channel section denoted by the coordinate s exhibited in Figure 1(d). The experimental data are obtained from the conventional PIV technique applied to the trace of particles of the average diameter 7 $μ$ m submerged in the liquid. We see the increase in the mean velocity upon the increase in D, which is consistent with the results presented in section “Parametric study with electrode pattern A.” We can also see that both the experimental and numerical data are close to the parabolic profile indicating that the velocity is almost fully developed at the selected channel section. The numerical results, however, overpredict the experimental ones, in particular in every case at smaller value of D. As explained in section “Parameter setting,” the bulk-driven EHD flow is hampered by the increase in the slip velocity at the edge of the DSL near the electrodes (more effectively for smaller electrodes, as shown in Suh et al.²⁹ and Suh and Baek³⁰), which can be achieved by decreasing the Langevin’s recombination constant $α$ or the reduction factor $β_{L}$ . Each value of $β_{L}$ presented in Figure 10 corresponds to the lowest limit beyond which the numerical simulation becomes unstable. Obviously, the experimental data would be better matched at lower values of $β_{L}$ if a more stable numerical scheme was available.

Figure 10.

Numerical and experimental results of the stream-wise velocity distribution along the coordinate s (indicated in Figure 1(d)) obtained at H = 3 mm and W = 0.6 mm for the electrode pattern A with D = 0.3, 1, and 2 mm. (a) D = 0.3 mm, β_L = 0.026; (b) D = 1 mm, β_L = 0.024; and (c) D = 2 mm, β_L = 0.024.

Comparison of the fully developed velocity profiles between the experimental and numerical results at the channel section far away from the electrodes is not enough for complete verification of the numerical simulation because even erroneous physical models applied to the EHD pumping can yield a matched fully developed velocity profile, which is parabolic regardless of the physics occurring in the region of the electrodes. Thus, we visualized the streamline patterns around the electrodes as shown in Figure 11 to verify the numerical results. Overall, the numerical streamlines well follow the experimental results. The converging and diverging patterns of the streamlines in the upstream and downstream, respectively, of the electrodes P_C and G_C and the vortical flows around the electrodes G_T and G_B are well captured by the numerical simulations, implying that the physical models for ion dissociation and recombination and the simulation methods are qualitatively correct.

Figure 11.

Comparison between the experimental (LHS) and numerical (RHS) streamlines obtained at W = 0.6 mm and H = 3 mm for (a) D = 0.3 mm, (b) 1 mm, and (c) 2 mm (with its magnified view in (d)) around the electrodes of pattern A. The frame taken for showing these plots is indicated by dash-dot lines in Figure 1(d).

We have seen that the rate of ion recombination should be taken usually much smaller than that proposed by Langevin for the enhanced comparison between the experimental and numerical results. The same trend has also been observed in our previous studies on the more fundamental problems.^29,30 As suggested in Suh and Baek,³⁰ however, the reason for the relevance of such reduction of recombination constant should be explored based on statistical and physico-chemical knowledge apart from the scope of mechanical engineering.

Experimental and numerical results with electrode pattern B

The numerical results with the pattern B (Figure 12) also reveal that the mean velocity is maximized at a critical value of H, except for the case with D = 0.3 mm showing a monotonic increase in U with H. Figure 12 also shows increase in U with D, like the case with the pattern A. Most importantly, the magnitude of the mean velocity is more than double that of the pattern A at an equal parameter set; for instance, we get U = 2.01 mm/s with pattern A at D = 0.7 mm and H = 2.5 mm, whereas we get U = 4.91 mm/s with pattern B at the same parameter set, the latter being 2.44 times the former. This suggests that simply building a series of electrode pattern across the channel can enhance the pumping performance.

Figure 12.

Variation in the mean velocity with H obtained from 2D numerical simulation for the pattern B at W = 0.6 mm with three values of the large electrode diameter, D = 0.3, 0.7, and 1 mm.

Experiment for the pattern B was conducted only for the case with D = 1 mm. Figure 13 presents the distribution of stream-wise velocity component along the s-direction depicted in Figure 1(d). The numerical result is very close to the parabolic shape, but the experimental profile is somewhat asymmetric with respect to the centerline of the channel showing a slightly larger magnitude on the outer side (s > 5 mm) than on the inner side (s < 5 mm) of the channel. Maximum deviation of the numerical velocity from the experimental one amounts to 28% near s = 5 mm. Such a discrepancy should be of course decreased at a lower value of $β_{L}$ in so far as the simulation code employed remains stable at such low values of $β_{L}$ .

Figure 13.

Numerical and experimental results of the stream-wise velocity distribution along the coordinate s (indicated in Figure 1(d)) obtained at D = 1 mm, H = 3 mm, and W = 0.6 mm for the electrode pattern B. Here, the numerical results are obtained with $β_{L} = 0.033$ .

Shown in Figure 14 are the experimental and numerical streamline patterns for the electrode pattern B. Here too, the numerical results well predict the experimental ones, such as the diverging and converging structure of the fluid streams and the vortical-flow pattern around the electrodes. In particular, the recirculating flow around the large grounded electrode located at the center is significantly weaker than the ones around the other two near the channel walls, which is mainly responsible for the significant increase in the pumping velocity compared with the pattern A.

Figure 14.

Comparison between the experimental (LHS) and numerical (RHS) streamlines obtained at D = 1 mm, W = 0.6 mm, and H = 3 mm around the electrodes of pattern B. The frame taken for showing this plot is indicated by dash-dot lines in Figure 1(d).

Conclusion

We propose in this study a two-way EHD pump with simple arrangement of circular-cylindrical electrodes. The pumping velocity is obtained with 2D numerical simulations at various sets of geometrical parameters. The numerical results are in good agreement with the experimental ones in terms of the streamline patterns around the electrodes, which turned out to be basically of a 2D character. In the numerical simulations, the recombination constant must be lowered significantly from that of the Langevin’s formula for the qualitative matching between the two, but there is a limit due to the numerical instability. We address the basic concepts to be considered regarding the electrode arrangement so as to achieve higher pumping flow rate. It is shown that under the restriction of symmetrical electrode pattern required for the bi-directional pumping, there exists an optimum distance between the large grounded electrodes that provides a maximum flow rate. Increased size of the large electrodes is shown to enhance the pumping effect because the level as well as the influential range of the Coulomb force created near those electrodes opposing the EHD pumping is decreased with the electrode size. Decreasing the gap between a pair of small electrodes on the other hand results in monotonic increase in the mean velocity simply due to the increase in field intensity leading to the increased Coulomb force. The pumping efficiency remains very low since the annular channel considered in this study requires a very low level of pressure difference across the region of the electrode pattern for the fluid circulation. Numerical and experimental streamlines are also in good agreement with each other for the electrode pattern B. The flow rate with the pattern B turned out to increase more significantly than expected compared with the pattern A, because the recirculating flow around the central large electrode is much weaker than the other two near the channel walls.

Footnotes

Appendix 1 Acknowledgements

The first version of this article has been read by Professor M. Duffy.

Academic Editor: Bo Yu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government Ministry of Knowledge Economy (No. 20134010200550).

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Development of a bi-directional electrohydrodynamic pump: Parametric study with numerical simulation and flow visualization

Abstract

Keywords

Introduction

Geometrical configuration and flow-visualization experiment

Numerical analysis

Governing equations

Numerical methods

Parameter setting

Basic idea on the principle and development of EHD pump

Numerical and experimental results and discussion

Parametric study with electrode pattern A

Comparison between the experimental and numerical solutions with electrode pattern A

Experimental and numerical results with electrode pattern B

Conclusion

Footnotes

Appendix 1

Acknowledgements

Declaration of conflicting interests

Funding

References