Electrical Charge Transport and Energy Conversion with Fluid Flow During Electrohydrodynamic Conduction Pumping

Transcription

1 Worcester Polytechnic Institute From the SelectedWorks of Jamal S Yagoobi May, 2007 Electrical Charge Transport Energy Conversion with Fluid Flow During Electrohydrodynamic Conduction Pumping Jamal S Yagoobi, Worcester Polytechnic Institute Yinshan Feng Available at:

2 Electrical charge transport energy conversion with fluid flow during electrohydrodynamic conduction pumping Yinshan Feng Jamal Seyed-Yagoobi Citation: Phys. Fluids 19, (2007); doi: / View online: View Table of Contents: Published by the AIP Publishing LLC. Additional information on Phys. Fluids Journal Homepage: Journal Information: Top downloads: Information for Authors:

3 PHYSICS OF FLUIDS 19, Electrical charge transport energy conversion with fluid flow during electrohydrodynamic conduction pumping Yinshan Feng Component Group, Thermal Fluid Sciences, United Technologies Research Center, 411 Silver Lane, MS , East Hartford, Connecticut Jamal Seyed-Yagoobi Heat Transfer Enhancement Two-Phase Flow Laboratory, Department of Mechanical, Materials Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois Received 27 July 2006; accepted 1 March 2007; published online 7 May 2007 The electrohydrodynamic EHD conduction pumping takes advantage of the electrical Coulomb force exerted on dielectric liquid by externally applied electric fields. The conduction term here represents a mechanism for electric current flow in which charged carriers are produced not by injection from electrodes or induction from electric fields, but by dissociation of neutral electrolytic species within the dielectric liquid. The EHD conduction pumping can be applied to drive both isothermal liquid two-phase fluids without the degradation of the working fluid electric properties. Such nonmechanical low-power-consumption pumping mechanism can be utilized for active flow generation/control under both terrestrial microgravity conditions. So far, the majority of conducted studies has been focused mainly on the experimental realization of the EHD conduction pumping phenomenon the computational fluid dynamics simulation verification. More fundamental studies, such as theoretical analysis with convection terms included, generalized nondimensional modeling, pumping efficiency prediction, are required for a complete understing of this new EHD pumping phenomenon. An asymptotic nondimensional theoretical model for the EHD conduction pumping has been presented in this paper, with the fluid convection taken into account. The theoretical analysis provided here reveals the effects of flow convection on the EHD conduction pumping the associated energy transport/conversion during the pumping process. Based on the asymptotic model, the pumping efficiency of the EHD conduction pumping is analytically derived compared with the experimental data. Such results help clarify the capabilities limitations corresponding to the nature of the EHD conduction pumping American Institute of Physics. DOI: / I. INTRODUCTION When a dielectric liquid is exposed to electric fields, an electric force will be exerted on the liquid bulk. Such an electric force can be expressed as a function of dielectric liquid properties, charge density, electric fields: 1 f e = e E 1 2 E E 2 / T. The associated interaction between dielectric liquid electric field can cause various electrohydrodynamic EHD phenomena, such as ion-drag motion, EHD induction, EHD extraction, dielectrophoresis, EHD instability, so on. A properly imposed electric field can even pump the dielectric liquid without the assistance of any moving parts. Ion-drag pumping EHD induction pumping are two well-studied EHD pumping mechanisms. Both of them take advantage of the Coulomb force between charges electric field, which is represented by the first term of Eq. 1. Ion-drag induction indicate the generation modes of electrical charges. The ion-drag pumping requires charge injection from sharp edges of the electrode, as shown in Fig. 1a. The associated pumping direction is always from the sharp electrode to the smooth electrode. Bryan Seyed-Yagoobi 2 1 conducted experimental theoretical studies on the integration of ion-drag pumping with capillary pumped loop to actively control improve the thermal management performance of spacecraft systems. However, unexpected pumping direction was observed when an electrode design as shown in Fig. 1b was utilized. Such an electrode design in Fig. 1b is very similar to that in Fig. 1a except for the round needle tip. The observed pumping phenomenon can be explained by neither the ion-drag pumping mechanism nor the induction pumping mechanism. Further studies on the above-mentioned extraordinary pumping phenomenon have been carried out by Atten Seyed-Yagoobi, 3 Seyed-Yagoobi et al., 4 Jeong Seyed-Yagoobi, 5 Jeong, 6 Feng Seyed-Yagoobi, 7,8 Yazdani. 9 Both experimental theoretical results confirm that such a liquid pumping phenomenon relies directly on the electrical conduction nature of dielectric liquid its associated free charge generation, which is completely different from the existing ion-drag induction pumping mechanisms. Consequently, this EHD pumping phenomenon is referred as the EHD conduction pumping, to be distinguished from the EHD ion-drag/induction pumping. The conduction term here represents a mechanism for electric current flow in which charged carriers are produced not by injection from /2007/195/057102/11/$ , American Institute of Physics

4 Y. Feng J. Seyed-Yagoobi Phys. Fluids 19, FIG. 1. Reversed pumping directions with different needle shapes: a sharp needle b round needle. electrodes or induction from electric fields, but by dissociation of neutral electrolytic species within the dielectric liquid. The EHD conduction pumping can be applied to drive both isothermal liquid two-phase fluids without the degradation of the working fluid electric properties. At the current stage, experimental studies have shown that the EHD conduction pumping successfully generated pressure up to several kpa, drove dielectric liquid, controlled two-phase refrigerant flow distribution. 5 9 The associated electrical current was on the order of microamperes. Such a nonmechanical low-power-consumption pumping mechanism can be utilized for flow generation active control in either ground or microgravity environment. Nevertheless, most studies conducted have focused on the realization of the pumping phenomenon. More fundamental/ analytical studies, such as unified nondimensional modeling, theoretical analysis with convection effects considered, pumping efficiency prediction, are required for a thorough understing of this new EHD pumping phenomenon. In the following text, an asymptotic nondimensional EHD conduction pumping model with the convection term incorporated will be developed discussed in details. The theoretical analysis will reveal the effects of flow convection on the EHD conduction pumping the energy conversion during the pumping process. Based on the nondimensional model, the pumping efficiency of the EHD conduction pumping will also be analytically derived compared with the experimental data. II. MODELING OF EHD CONDUCTION PUMPING WITH FLUID MOTION The EHD conduction pumping phenomenon stems from the dynamic dissociation/recombination process of ions in a dielectric liquid. At a thermodynamic equilibrium state, a neutral electrolyte, denoted as AB, dissociates into counterions, denoted as A + B, while these ions can recombine back into AB. The corresponding dynamic process can be expressed in the following form: AB Dissociation Recombination A + + B. 2 With an external electric field applied to a dielectric liquid, existing ions due to the dissociation of neutral species experience the electrical Coulomb force, which concentrates the positive negative ions around counter-polarized electrodes. The ion density distributions in the dielectric liquid bulk can be determined on the basis of the conservation law of neutral species ions. Under a steady-state condition, the governing equations of ion distribution have the following expressions: K + E p + up D p = k D c k R pn, K E n + un D n = k D c k R pn, E = p n/, where c, n, p, D denote the concentration of neutral species, the charge density of negative ions, the charge density of positive ions, the charge diffusion coefficient, respectively. K, k R, k D denote the ion mobility, the recombination rate, the dissociation rate, respectively. At a thermodynamic equilibrium, the ion conservation allows k D c = k R p 0 n 0 = k R p 2 0 = k R n 2 0, 6 where p 0 n 0, respectively, represent the positive negative ion densities at an equilibrium state n 0 = p 0. Assuming the positive negative ions have identical mobilities, i.e., K + =K =K, the recombination rate constant becomes 10 k R =2K/. The electrical current density in the dielectric liquid has the following expression of j = K+ p + K ne + p nu D p n, where the three terms on the right side are named as the mobility term, the convection term, the diffusion term, respectively. Since the electrical conductivity of a static dielectric liquid can be expressed as e =2n 0 K =2p 0 K, Eq. 8 becomes 1 j = 2 ep/p 0 + n/n 0 E + e u D e, 10 where e = p n denotes the local net charge density. The dot product of the electrical field intensity the electrical current density represents the localized electrical power consumption

5 Electrical charge transport energy conversion Phys. Fluids 19, E j = 1 2 ep/p 0 + n/n 0 E 2 + e u E DE e, 11 where on the right-h side of the above equation, the first term denotes the electrical Joule heating the second term denotes the power to induce dielectric fluid motion. The third term is related to the electrical energy change for local net charges caused by the ion diffusion. The external electrical power input finally transforms into the pumping work output the heat generation due to the viscous dissipation generally negligible the Joule heating in the liquid dielectric. The third term on the right side of Eq. 11 vanishes when the charge diffusion coefficient D approaches zero or the charge distribution becomes homogeneous. As discussed by Feng Seyed-Yagoobi, 7 the electric current density due to the charge diffusion becomes negligible, which is less than 0.1% of the total electric current density according to the Nernst-Einstein formula, 11 when the applied voltage exceeds 12.5 V at the operating temperature around 295 K. Since the voltages of interest in this paper always exceed 1000 V, Eqs. 3 4 are further simplified by neglecting the diffusion term as FIG. 2. Schematic of perforated disk-ring electrode pair described in Feng Seyed-Yagoobi Ref. 7. KE p + up = k D c k R pn 12 KE n un = k D c k R pn. 13 Equations 5, 12, 13 constitute the theoretical model for the EHD conduction pumping process in the absence of charge diffusion. Along with proper boundary conditions given velocity profile, these governing equations are sufficient to determine the charge electric field distributions. With the solved charge electrical field distributions, the total consumed electric current associated with the EHD conduction pumping phenomenon can be obtained by integrating Eq. 10 along the cathode or anode boundary 14 where the ion diffusion term is neglected. The EHD conduction pumping pressure generation can also be expressed as 15 where A denotes the cross-sectional area of liquid flow path E denotes the electric field perpendicular to the crosssectional surface. Hydrodynamic pumping efficiency is a key parameter to evaluate the performance of an EHD conduction pump. Generally, the pumping efficiency is defined as the ratio of work output to power input. For an EHD conduction pump, its work output is the product of pressure generation volumetric flow rate, while the corresponding electric power consumption represents its power input. With the flow treated as homogeneous, the EHD conduction pumping efficiency can be simplified as 16 where U is the average flow velocity. The above equation implies that at a given external voltage potential, the resultant dielectric flow velocity affects the EHD conduction pumping efficiency in both direct indirect manners, as the flow convection transports ions in the dielectric liquid influences the local net charge distribution. Solving Eqs. 5, 12, 13 with convection terms becomes the key step to get the distribution profiles of ion electrical field for quantitative analyses of the EHD conduction pump performance. The theoretical analyses in this paper are presented in a nondimensional form to unify the EHD conduction pumping phenomenon under various conditions. Due to the complexity of theoretical analysis, the theoretical analyses are carried out mainly within a one-dimensional spatial domain. Similar to the theoretical model in the absence of fluid flow previously applied by Feng Seyed-Yagoobi, 7 the perforated disk-ring electrodes as shown in Fig. 2 are treated as the left half of a parallel permeable or perforated plate electrode pair, in order to one-dimensionally formulate the electric field the flow field. Initially, the electrode gap distance is utilized to normalize the one-dimensional theoretical model with convection term included. Later, the heterocharge layer thickness, instead of the electrode gap, will be utilized as the length scale to obtain the asymptotic solutions. Such asymptotic theoretical model will also be extended to twodimensional flow field with the electric field treated onedimensionally. MATLAB Version 7.0 has been utilized to numerically calculate the ion electrical field distribution profiles, if the analytical solutions are unavailable.

6 Y. Feng J. Seyed-Yagoobi Phys. Fluids 19, FIG. 4. Profiles of p * E * at various C o u * =0. FIG. 3. Permeable electrodes immersed in dielectric liquid. III. NONDIMENSIONAL ASYMPTOTIC ANALYSES OF EHD CONDUCTION PUMPING To obtain a fundamental understing of EHD conduction pumping phenomenon, a one-dimensional permeable electrode pair as illustrated in Fig. 3 allowing for uniform flow motion will be first studied. Usually, the governing equations are normalized with x * =x/d, E * =Ed/V, * =/V, p * = p/n 0, n * =n/n 0, u * =u x d/vk=ud/vk, C o =n 0 d 2 /V. It is noteworthy that C o is based on the electrode gap d. The normalization of governing Eqs. 5, 12, 13 leads to dp * E * + p * u * /dx * =2C o 1 p * n *, 17 dn * E * n * u * /dx * = 2C o 1 p * n *, de * /dx * = C o p * n *, E * = d * /dx *, where the boundary conditions require p * =0 at x * =0, 21 to 12.0 when u * =0. Meanwhile, these curves also indicate that when C o 1.0, the centers of all curves reach 1.0 the middle parts of the curves become flat. The effects of convection u * 0 on ion electric field profiles are further investigated at C o =8.0, which is much greater than 1.0. Figures 5 7 show the curves at various velocities. In the presence of flow motion, all profiles are significantly affected by the convective ion transport. The variations in the profiles depend on both the direction the magnitude of flow velocity. It is also observed that the flow motion is inclined to suppress the forming of a heterocharge layer on the downstream electrode enhance the layer formation on the upstream electrode. Nevertheless, the profiles at the center region remain unaffected with the normalized values staying around 1.0 at C o =8.0. With R-123 as the working fluid in the study of Feng Seyed-Yagoobi, 7 C o = 8.0 corresponds to an applied voltage of 15 kv at d =4.3 mm. Such observations are also valid at even greater C o values suggest the existence of unified asymptotic solutions which are independent of the applied electrical field at C o 1.0, if the governing equations are properly normalized. Therefore, in the following section, the heterocharge layer thickness 0 instead of the electrode gap d, is introduced as the length scale to conduct the asymptotic nondimensional analysis. When the length scale of 0 is utilized n * =0 at x * =1, 22 * =1 at x * =0, 23 * =0 at x * =1. 24 While x * =x/d is applied, the resultant parameter C o =n 0 d 2 /V is a function of n 0,, V/d, d. Thus, C o corresponds inversely to the intensity of applied electric field. When the nominal applied electric field V/d becomes stronger, the value of C o decreases. The variation of C o along with the applied electric field makes it difficult to unify the nondimensional solutions of the EHD conduction pumping model. Figure 4 shows the rather different nondimensional ion electric field profiles at various C o ranging from 1.0 FIG. 5. Profiles of n * E * at various u * C o =8.0.

7 Electrical charge transport energy conversion Phys. Fluids 19, FIG. 6. Profiles of p * E * at various u * C o =8.0. FIG. 8. Schematic of a heterocharge layer formed on a permeable anode in a uniform external electrical field. at C o 1.0, the nondimensional governing equations its solutions become independent of the nominal applied electric field V/d. A. Asymptotic theoretical model using heterocharge layer thickness as length scale As presented above, at C o 1.0, the normalized curves such as p * E *, n * E *, E * between two parallel electrodes become flat reach 1.0 at the middle. For simplicity, the following analyses are focused on the left half of the parallel electrode pair with the constant boundary conditions set at the middle, as shown in Fig. 8. The governing equations will be normalized as those in the above section, except that x * =x/d will be replaced by x * =x/ 0 C =n 0 d 0 /V will be used instead of C o. 0 denotes the characteristic thickness of the heterocharge layer in a static dielectric liquid u * =0. The corresponding nondimensional governing equations become dp * E * + p * u * /dx * =2C 1 p * n *, 25 de * /dx * = C p * n *. 27 If C o 1.0, the applied electric field generates a rather thin heterocharge layer compared to the electrode gap distance, 0 d. Consequently, the boundary conditions can be further simplified as p * =0 at x * =0, 28 p * = n * =1 at x *, 29 E * =1 at x *. 30 Feng Seyed-Yagoobi 7 obtained the following analytical solutions of Eqs in the absence of flow motion u * =0: 0 p * E * dz 2C 1+ge 1 zz 2.0 = x*, n * E * = p * E * + 2.0, dn * E * n * u * /dx * = 2C 1 p * n *, FIG. 7. Profiles of n * E * at various u * C o = E *2 = p * E * p * E * 2.0/ge 1 p * E * p * E * 2.0, 33 where the function g, is defined as gye y =y. Atx * =1.0, Eq. 31 gives that the parameter C has a constant value of 1.8 when u * =0. Since the definition of C excludes the influence of the flow motion, the value of C remains the same even when u * 0. With x * =x/ 0 C =n 0 d 0 /V=1.8 introduced, the asymptotic theoretical model becomes solely dependent on the flow velocity of the dielectric liquid as long as C o 1.0. In the presence of flow motion u * 0, the convection tends to alter the heterocharge layer thickness. Figures 9 10 show the normalized profiles i.e., p * E *, n * E *, E *, around a fully permeable anode at u * = , 0, 0.2, 0.4. For the electrode configuration of Fig. 8, the heterocharge layer becomes thinner around the anode electrode when the dielectric fluid flow is opposite to the x direction vice versa. This is due to the fact that the convective flow in the positive x direction carries the charges away from the anode electrode, resulting in a thicker heterocharge layer. Such a change in the heterocharge layer further affects the pressure

8 Y. Feng J. Seyed-Yagoobi Phys. Fluids 19, FIG. 9. Profiles of n * E * p * E * around a permeable electrode at various u *. FIG. 11. Prediction of EHD conduction pumping pressure generation versus velocity for a permeable electrode. generation of EHD conduction pumping. The influence of convection on pressure generation can be evaluated by a normalized pressure head: P * = P 0 = P n * E * p * E * dx *u* n * E * p * E * dx *u * =0 The predicted curve of normalized velocity u * versus normalized pressure head P * has been provided in Fig. 11 for fully permeable electrodes. The results show that when the pumping velocity reaches its mobility limit i.e., u * = 1.0, the EHD conduction pumping pressure generation decreases to zero as the convective flow blows off the heterocharge layer from the permeable electrode. However, the blow-off phenomenon of the heterocharge layer at high velocities is not true for the perforated electrode design, since the liquid flow stagnates on the solid electrode surface. In the next section, a nonviscous two-dimensional flow assumption will be used to predict the convective flow around the perforated electrode. B. Ideal flow estimation In Sec. III A, the velocity of dielectric fluid between electrodes was assumed to be uniform, since the electrodes were considered as porous permeable. However, this assumption does not properly correspond to the imposed velocity distribution between the perforated disk-ring electrode pair studied here. As illustrated in Fig. 2, the velocity profile varies along x direction consequently affects the ion density electric field distributions accordingly. To predict the velocity variation of dielectric liquid along the x direction, the solid portions of the perforated disk electrode will be treated as flat plates with a length of 2c =0.8 mm. The flow motion in the dielectric liquid is considered as ideal flow with viscosity effects neglected. According to Milne-Thomson, 12 the complex potential flow around a vertical plate with a width of 2c is given as follows: w = Uz 2 + c 2 1/2, 35 where w=+i z=x+iy. The streamline equation has the following form: x/c 2 +1+/Uc 2 /1+xUc/ 2 = y/c The streamlines around the flat plate are depicted in Fig. 12. The local velocity at y=0 can be expressed as u x y=0 = Ux/c 2 + x 2 1/2. 37 FIG. 10. Profiles of E * around a permeable electrode at various u *. For simplicity, the electric field is treated one-dimensionally as E=E x. Within a control volume width=2c illustrated in Fig. 12, the corresponding integral governing equations conservation of charges for the EHD conduction pumping becomes c KEp + u x pdy c d c k D c k R pndy, 38 dx +2u y p = c

9 Electrical charge transport energy conversion Phys. Fluids 19, c d c c d c dx KEn + u x ndy dx +2u y n E x dy c = c c = c p ndy k D c k R pndy, with the same boundary conditions indicated by Eqs Assuming n, p, E, u x do not vary along the y direction u x u x y=0, the mass conservation law requires that u y = Uc 3 /c 2 + x 2 3/2, at y =±c, 41 Eqs can be expressed as dkep + FIG. 12. Streamlines of flow around a vertical plate. Ux 1/2p c 2 + x 2 dx + Uc 2 c 2 + x 2 3/2p = k Dc k R pn, 42 FIG. 13. Profiles of n * E * p * E * for ideal flow at various u * c * =0.43. dp *E * + 1/2 u * x * c *2 + x *2 dx * + u * c *2 c *2 + x *2 3/2p* =2C 1 p * n *, 45 dn *E * 1/2 u * x * c *2 + x *2 dx * = 2C 1 p * n *. u * c *2 c *2 + x *2 3/2n* 46 With the effects of nonuniform flow field taken into account the boundary conditions remaining the same, the ion density electric field distributions become different from those discussed in Sec. III A. The profiles with u * = 0.4, 0.2, 0, 0.2, 0.4 for c * =0.43 c * =0.645 are depicted in Figs c * =0.43 c * =0.645 correspond to the applied voltages of kv, respectively, with c =0.4 mm R-123 as the working fluid. Figure 15 shows the pressure generation versus velocity at the applied voltages of 15 kv 10 kv with the fluid motion treated as ideal flow. On the basis of the ideal flow assumption, the d KEn + Ux 1/2n c 2 + x 2 dx + Uc 2 c 2 + x 2 3/2n = k Dc k R pn, 43 de/dx = p n/. 44 Normalization of Eqs with x * =x/ 0, c * =c/ 0, E * =Ed/V, p * = p/n 0, n * =n/n 0, u * =Ud/VK, C =n 0 d 0 /V leads to the following expressions: FIG. 14. Profiles of n * E * p * E * for ideal flow at various u * c * =0.645.

10 Y. Feng J. Seyed-Yagoobi Phys. Fluids 19, FIG. 15. Prediction of EHD conduction pumping pressure generation versus velocity for ideal flow. heterocharge layer stays on the electrode even at the critical fluid velocity of u * = 1.0 more reasonable nonzero pressure generation has been predicted. The ideal flow analysis omits the influence of fluid viscosity on the flow motion. Especially near the perforation edges of electrode, the real velocity profiles are rather different from the ideal flow prediction. Such a discrepancy also affects the downstream fluid motion. Nevertheless, since the predictions of pressure generation electric current consumption are dependent mainly on the solid electrode surface normal to the applied electric field the attached heterocharge layer, the influences of flow in the edge regions are neglected in this paper. IV. COMPARISONS AND DISCUSSIONS As described in the above asymptotic nondimensional analysis, the pressure generation, current-voltage behavior, hydrodynamic pumping efficiency depend closely on the applied voltage, working fluid properties, electrode geometries, the fluid motion. To verify the nondimensional model, comparisons are made between the theoretical predictions the experimental data presented by Feng Seyed-Yagoobi 7 for the perforated disk-ring electrode pair design. The corresponding physical properties of the working fluid are listed in Table I. All experimental results have been normalized for comparison purposes. A. Pressure generation The viscous shear stress occurring during the EHD conduction pumping always causes an internal pressure drop within the pumping section, which lowers the reading of the TABLE I. Properties of saturated liquid R-123 at 295 K. Fluid pf/m e S/m K m 2 /Vs Pa s kg/m 3 R a a b c c a Data from Bryan Ref. 13. b Based on Walden s Rule: K= / m 2 /Vs Crowley et al., Ref. 14. c Data based on NIST Thermodynamic Transport Properties of Refrigerants Refrigerant Mixtures REFPROP FIG. 16. Computational fluid dynamics simulation of frictional pressure drop in EHD conduction pump: a grids of simulation domain b numerically calculated pressure drop. measurable pressure generation. To reasonably compare the theoretical predictions of pressure generation with experimental data, the internal frictional loss of EHD conduction pump needs to be compensated. Figures 16a 16b, respectively, illustrate the meshed flow path containing more than nodes of EHD conduction pump studied by Feng Seyed-Yagoobi 7 the corresponding pressure drop results for liquid laminar flow of R-123 at 295 K simulated by the ANSYS software. The actual pressure generation will be estimated as the sum of the experimental pressure data the calculated frictional pressure drop in the pumping section. Figures 17 18, respectively, show the comparison of the theoretical pressure generation predictions with the raw experimental data the adjusted pressure generation data with internal frictional pressure drop included. The compensation of the internal pressure drop improves the agreement between the experimental data of pressure generation the theoretical predictions. The predicted pressurevelocity trends on the basis of ideal flow match well with the adjusted experimental data. Regardless of the basis of the permeable electrode assumption or the ideal flow assumption, the nondimensional asymptotic theoretical model gives the pressure generation

11 Electrical charge transport energy conversion Phys. Fluids 19, FIG. 17. Theoretical predictions versus experimental data pressure losses within pumping section are not included. FIG. 19. Comparison of experimental current consumption data with theoretical results for EHD conduction pumping. prediction in a one-dimensional formulation as P C V 2 /d n * p * E * dx *, where the nondimensional integral portion only varies with the flow motion. The model reveals that the pressure generation during EHD conduction pumping depends only on the applied electrical field, the flow velocity, the electrical permittivity of the working fluid. The variation in the electrical conductivity of the working fluid will not affect the pressure generation as long as C o 1.0 the other factors remain unchanged, though higher electrical conductivity may make it difficult to establish intense electric field in the dielectric fluid. B. Consumed electric current For the perforated electrode design in this paper, the consumed electric current associated with the EHD conduction pumping phenomenon can be simplified from Eq. 14 as I = m1 ra 1 2 e p p 0 + n n 0E + e ux * =0 = 1 2 m1 ra V e d n* E * n * u * x * =0, 47 where the ion diffusion term is neglected, p x * =0=0. Normalization of the current level leads to I * = I I u * =0 1 2 m1 ra V e d n* E * n * u * x * =0 = 1 2 m1 ra V e d n* E * n * u * x * =0,u * =0 = n* E * n * u * x * =0 n * E * n * u *. 48 x * =0,u * =0 Figure 19 shows the comparison of theoretical experimental normalized electric current levels. C. EHD conduction pumping efficiency When investigated one-dimensionally, the EHD conduction pumping efficiency can be simplified as EHD pump = AuP VI n pedx Aum1 r0 = Vm1 ra 1 2 V e d n* E * n * u * x * =0 VK d n V 0 d C V n 0 d u* n 0 * p * E * dx * = 1 2 ev V d n* E * n * u * x * =0 = 3.6 e V d K d u * 0 n * p * E * dx * n * E * n * u * x * =0. 49 FIG. 18. Theoretical predictions versus experimental data pressure losses within pumping section are included. Since u * 0 n * p * E * dx * /n * E * n * u * x * =0 is only related to u * in the asymptotic model when C o 1.0, it can be designated by fu *. Equation 49 will be further expressed as EHD pump = 3.6 V K e d d fu*. 50 The above equation indicates that the efficiency of EHD conduction pumping depends on flow motion, working fluid

12 Y. Feng J. Seyed-Yagoobi Phys. Fluids 19, FIG. 20. Theoretical prediction of EHD conduction pumping efficiency at various applied voltages on the basis of a permeable electrode assumption. FIG. 22. Comparison of theoretical predictions experimental data of EHD conduction pumping efficiency at 15 kv. FIG. 21. Comparison of theoretical predictions experimental data of EHD conduction pumping efficiency at 10 kv. properties, applied electric field, the electrode gap. The EHD conduction pumping efficiency can be enhanced by intensifying the applied electric field, selecting proper working fluids with longer relaxation time, decreasing the gap between electrodes, which is favorable in microscale applications. Figure 20 illustrates the influence of applied voltage on the efficiency profile of EHD conduction pump with permeable electrodes at d=4.3 mm for liquid R-123. As the applied voltage increases from 5 to 15 kv, the maximum pumping efficiency goes up from 0.30% to 0.89%. It is also noteworthy that the most efficient EHD conduction pumping always takes place around u * =0.4 on the basis of the permeable electrode assumption. Figures 21 22, respectively, give the comparison of adjusted experimental data with the internal pressure drop included theoretical predictions at applied voltages of 10 kv 15 kv. The predicted efficiency has the same order trend of the experimental results. Both figures show that the experimental efficiency data are lower than the predictions based on the permeable-electrode ideal flow assumptions. Such overprediction partially stems from impurity traces in the working fluid, which increases the electrical conductivity of the working fluid significantly promotes the consumed current level. Feng Seyed-Yagoobi 7 observed similar relatively high electric current levels even for the EHD conduction pumping without net flow. While the pressure generation does not depend on the electrical conductivity, the rise in the electrical conductivity due to the presence of impurity results in higher Joule heating. The theoretical calculation based on the pure properties of working fluids did not take the impurity influence into account. V. CONCLUSIONS This work has provided asymptotic theoretical analyses of the EHD pumping performance in the presence of flow motion. The theoretical study showed that at specific applied electric fields C o 1.0, the nondimensional governing equations boundary conditions can be generalized by applying the heterocharge layer thickness, instead of the electrode gap distance, as the length scale. This nondimensional asymptotic theoretical model depends only on the flow velocity, applied nominal electrical field, the working fluid properties. To simplify the theoretical derivations, the theoretical analyses have been conducted mainly for onedimensional physical domain. The permeable electrode assumption the nonviscous flow around perforated electrode assumption were adopted to estimate the flow field within the pumping section. The comparison between the theoretical predictions the experimental data showed good agreement. The asymptotic theoretical analyses reveal the following: a b c When normalized with the heterocharge layer thickness at u * =0 as the length scale, the asymptotic theoretical model has unified nondimensional solutions, which merely depend on the flow motion. The convection in the fluid significantly affects the charge distribution consequently the EHD conduction pumping pressure head. The pressure generation by EHD conduction pumping depends on the applied nominal electrical field, the flow velocity, the permittivity of working fluid. The electrical conductivity of working fluid does not affect the pressure generation, as long as the required electrical field can be established.

13 Electrical charge transport energy conversion Phys. Fluids 19, d The pumping efficiency can be evaluated with EHD pump =3.6/ e V/dK/dfu *, which indicates that the efficiency of an EHD conduction pump can be improved by applying more intense electric field, selecting working fluids with longer relaxation time, decreasing the gap between electrodes. Finally, it is noted that attention should be paid to the amount of impurity traces in the dielectric liquid. In the asymptotic model, as long as C o 1.0, the presence of extra impurity will not affect the pressure generation, but will increase the electrical current consumption lower the EHD conduction pumping efficiency. However, it has been experimentally observed that the pressure generation electric current both decrease initially with time due to the capture of impurities by the electrodes, where the assumption of C o 1.0 was invalid. 1 J. R. Melcher, Continuum Electromechanics MIT Press, Cambridge, MA, J. E. Bryan J. Seyed-Yagoobi, Heat transport enhancement of monogroove heat pipe with electrohydrodynamic pumping, J. Thermophys. Heat Transfer P. Atten J. Seyed-Yagoobi, Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry, IEEE Trans. Dielectr. Electr. Insul. 10, J. Seyed-Yagoobi, P. Atten, J. E. Bryan, Y. Feng, B. Malraison, Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry experimental study, Proceedings of the 13th International Conference on Dielectric Liquids, Nara, Japan, 1999, pp S. I. Jeong J. Seyed-Yagoobi, Experimental study of electrohydrodynamic pumping through conduction phenomenon, J. Electrost. 56, S. I. Jeong, Theoretical experimental study of electrohydrodynamic pumping through conduction phenomenon, Ph.D. dissertation, Texas A&M University, College Station, TX, Y. Feng J. Seyed-Yagoobi, Understing of electrohydrodynamic conduction pumping phenomenon, Phys. Fluids 16, Y. Feng J. Seyed-Yagoobi, Control of liquid flow distribution utilizing EHD conduction pumping mechanism, IEEE Trans. Ind. Appl. 42, M. Yazdani, Electrically induced dielectric liquid film flow based on electric conduction phenomenon, M.S. thesis, Illinois Institute of Technology, Chicago, IL, P. Debye, Reaction rates in ionic solutions, Trans. Electrochem. Soc. 82, I. Adamczewski, Ionization, Conductivity Breakdown in Dielectric Liquids Taylor & Francis, London, 1969, p L. M. Milne-Thomson, Theoretical Hydrodynamics Dover, New York, 1968, p J. E. Bryan, Fundamental study of electrohydrodynamically enhanced convective nucleate boiling heat transfer, Ph.D. dissertation, Texas A&M University, College Station, TX, J. M. Crowley, G. S. Wright, J. C. Chato, Selecting a working fluid to increase the efficiency flow rate of an EHD pump, IEEE Trans. Ind. Appl. 26,