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Electrohydrodynamic Induction Pumping of Liquid Film in Vertical Annular Configuration Salem Ahmed Aldini Jamal Seyed-Yagoobi Department of Mechanical Engineering Mechanical, Material, and Aerospace Engineering Texas A&M University Department College Station, Texas 77843-3123, USA Illinois Institute of Technology salem@tamu.edu Chicago, Illinois 60616, USA yagoobi@iit.edu Abstract— Electrohydrodynamic induction pumping of two- Pr Prandtl number= c p m k . phase medium is attractive for terrestrial and outer space R Radius, m. applications since it is non-mechanical, lightweight, and involves no moving part. In addition to pure pumping purposes, EHD r Radial space coordinate, m. induction pumps are also used for the enhancement of heat Re Real component of a complex number. transfer, as an increase in mass transport often translates to an S Slip coefficient = [o (-Kw)/]. augmentation of the heat transfer. Applications include two-phase T Temperature, oC. heat exchangers (evaporators and condensers), heat pipes and u Velocity in r direction, m/s. capillary pumped loops. A theoretical model for the EHD w Velocity in z direction, m/s. induction pumping of an annular liquid/vapor medium where the charges are induced at the two-phase interface as well as within z Space coordinate in streamwise direction, m. the bulk of the liquid phase in vertical configuration is presented. Liquid film thickness, m. The dimensionless numerical results are obtained and the flow Electric permittivity, F/m. physics are discussed in conjunction with the effect of the Electric permittivity of vacuum= 8.854, F/m. controlling parameters. The controlling parameters include: Wavelength, m. liquid film thickness, voltage, wavelength, frequency, external pressure, and gravity. The dimensionless numerical results Dynamic viscosity, kg/ms. obtained provide an essential tool for designing and optimizing e Electric potential at the electrode, V. ˆ two-phase liquid /vapor EHD induction pumps. r Density, kg/m3. Circumferential angle. Keywords-Electrohydrodynamics; interface; pumping; two- phase flow; Electric conductivity, S/m. Shear stress, N/m2. NOMNUCLUTURE Angular frequency = 2 f , Hz. Cp Specific heat at constant pressure, J/kg.K. Absolute magnitude of a complex number. C Electric potential function, V. Subscripts and Superscripts r E Electric field strength vector, V/m. b0 Bulk value at pump entrance. f Frequency, Hz. e Electric. g Gravitational constant= 9.81, m/s2. i Space coordinate in tensor notation. G Gravitational force density= r g , N/m3. j Space coordinate in tensor notation. I0 Modified Bessel function of the first kind order zero. l Liquid. I1 Modified Bessel function of the first kind order one. r In r direction. j Imaginary number = 1 . zr In z direction acting on a plane with normal r. k Thermal conductivity, W/m.K. T Total. K Wave number = 2 , 1/m. z In z direction. L Pump length, m. v Vapor. m* Dimensionless mass flux. ^ Peak value. - Time-averaged value. M e r b0 e0jˆ e 2m . * 2 2 b0 ' Conjugate complex. m 0 e0 r b0 s b0 d2 . b * Dimensionless value. P Pressure, Pa. I. INTRODUCTION liquid film thicknesses are carried out to study the effect of the Electrohydrodynamic EHD induction pumping utilizes the controlling parameters: vapor height, liquid film thickness, charges induced in the liquid film due to the existence of an applied voltage, wavelength, and frequency. The results of this electric conductivity gradient in the bulk of the film and/or at study are pertinent to the pumping or flow management during the liquid/vapor interface. This gradient exists due to the internal condensation or evaporation processes in the presence temperature within the liquid film and/or discontinuity of and absence of gravity. This study helps in explaining the electric conductivity at the liquid/vapor interface. Upon the physics of EHD induction pumping and serves as a necessary application of an electric field in the form of a traveling wave, tool for optimum design of two-phase EHD pump. the induced charges will get attracted or repelled leading to fluid motion. EHD induction pump is non-mechanical, and II. THEORETICAL MODEL lightweight. It involves no moving parts, produces no vibration, The model presented considers a two-dimensional and the electric power input is generally negligible. These developing annular flow of a thin liquid film in an pumps have a great potential for more than just pure pumping axisymmetric vertical tube as illustrated in Fig. 1. The liquid applications. Applications include single-phase and two phase film is separated from the vapor by a flat interface. Boundary heat exchangers, heat pipes, or capillary pumped loops. condition of electric traveling wave is imposed in the liquid Enhancing heat transfer in phase change processes by film adjacent the wall of the tube while symmetric boundary controlling the liquid film flow is one of the practical condition for the electric field is imposed at the tube centerline applications. In addition, mechanical pumping of liquid film, in (i.e. Erv 0 ). In addition to the charges, and therefore electric both terrestrial and outer space applications, is usually, not possible. However, EHD induction pumping provides a shear stress, at the liquid/vapor interface the model considers practical technology. charges throughout the bulk of the liquid film. Due to an electric conductivity gradient in the liquid film, charge The first theoretical model for two-phase attraction induction takes place. This gradient may exist as a result of a induction pumping of a poorly conducting fluid was developed temperature gradient, which is a consequence of viscous in 1966 by Melcher [1]. He considered an analytical domain dissipation (negligible), Joule heating, or cooling or heating of constituted by an electrode generating the electric traveling the boundaries. The following assumptions are made with wave placed above the liquid film, and a conducting plate as a regards to the electric field and liquid film flow field: lower boundary. Only charges induced at the liquid/air interface were considered. He later improved his work by 1) the vapor has properties of the vacuum; expanding it not only to cover both attraction and repulsion 2) the electric field is irrotational due to low electric pumping, but also the pumping of a liquid/liquid interface [2]. current; The improved model was later used by Crowley [3] to 3) all fluid properties in the liquid film are temperature incorporate the effect of temperature induced charges in the dependent; bulk of the liquid for single phase pumping process. For the purpose of latter studies, a fluid exhibiting a continuous 4) charge transport at the interface due to ion mobility and temperature gradient was modeled as having two layers at surface conduction, and bulk conduction in z-direction distinct constant temperature separated by a temperature jump. are negligible; and Wawzyniak and Seyed-Yagoobi [4] further developed an 5) the flow is two-dimensional and in vertical analytical model for EHD induction pumping of a stratified configuration; liquid/vapor medium. They assumed charges to be present only 6) the flow field may be modeled as flow between two at the interface and analyzed four different electrode concentric cylinders, the outer one being stationary and geometries. As an extension to their work Wawzyniak and the inner one moving at the constant speed of the Seyed-Yagoobi [5] considered charges not only present at the liquid/vapor interface in the z-direction; liquid/vapor interface but throughout the bulk of the liquid for only one electrode configuration. Brand and Seyed-Yagoobi 7) the flow is steady state, laminar, incompressible , but [6] further extended the work done by [5] to analyze and not fully developed; investigate the effect of four different electrode configurations. 8) the flow is rotationally symmetric, i.e., the variables do The current study is an extension of the theoretical model not depend upon ; presented by [5] and [6] who studied the EHD induction 9) the pressure in r-direction is uniform, the pressure pumping of a stratified liquid/vapor medium in a horizontal gradient in z-direction is constant; channel. This current paper investigates the EHD induction pumping of liquid film in vertical annular configuration in the The model assumes a flat interface and neglects the presence of gravitational force and external load which has not possibility of surface waves due to forces, which may arise been studied previously with charges existing not only at the from the radial electric field, the gravity, or the surface tension. liquid/vapor interface but also throughout the bulk of the fluid. It was reported by Benjamin [7] that films falling vertically In addition, three different temperature profiles are specified at solely due to gravity are unstable, even at small Reynolds the entrance of the pipe to investigate their effect on the electric numbers. However, for the case presented in the presence of conductivity gradient, thus the electric shear stress. A gravitational force, one can assume that the flat interface theoretical model in a dimensionless cylindrical coordinates is corresponds to long wave length associated with liquid film presented. A numerical parametric study for two different The time-averaged electric shear stress and the Joule heating RT expressed in terms of the electric potential are [8] Rv dC ' * * Re jK *C * * * (5) e ,rz dr liquid vapor 1 2 2 2 dC * E * K * C * * *2 (6) 2 dr * traveling wave electrodes symmetry The non-dimensional slip coefficient is g * K * w* S* (7) * z The corresponding dimensionless boundary conditions for the flow field are r u * 0 @ r* RT * (8) Figure 1. Schematic of the vertical analytical domain w* 0 @ r* RT * (9) instability corresponding to relatively flat interface. It is also u * 0 @ r* Rv* (10) important to mention that the film thickness is assumed to be constant in this study. In the absence of any phase change, the C * I1 K * Rv *C 2 2 M e * K * 2 * *R only source of potential heat is the Joule heating. However, the C mass evaporation rate generated due to the Joule heating under * w* I 0 K * Rv * (11) the conditions considered here is three orders of magnitude less r * 1 * *2 S* S r* Rv * than the mass flow rate generated due to the electric field. This suggests that it is acceptable to assume no phase change, thus constant film thickness. z * 0, z L * * u* 0 @ (12) The flow is governed by conservation equations of mass, momentum, and energy. The momentum and energy w* 0 @ z * 0, z L * * (13) conservation equations contain terms based on the electric field z * solution. The governing equations can be expressed in a dimensionless form as (see [8] for more details) Equation (11) states the balance of viscous and electric shear stresses at the interface. * . * v * 0 (1) For the energy equation, the dimensionless boundary conditions are *v * . v * p* . * *G* T * Prbo " Me 1 * (2) k @ r* RT * * q (14) * * * r* rz r * bo e ˆ 2 wall N e r r T * Prbo " k @ r* Rv * 1 * * * * q (15) c u .T * * p * * k T * E *2 (3) r* boeˆ 2 int erface Prbo To solve the above equations, the electric shear stress in (2) The imposed dimensionless temperature profile at the channel and the Joule heating in (3) must be expressed in terms of the entrance becomes dielectric potential. The basic electrostatic relations for the bo c pbo T * r * T r @ z 0 * complex amplitude of the electric potential can be reduced in a (16) dimensionless form to [8] bo e2 ˆ The dimensionless boundary conditions for the electric * j* * w* K * 1 d * r * dC* d * dC* * * * potential equation are r dr * dr dr dr (4) C * 1 @ r* RT * (17) j w K K C 0 * * * * * *2 * * where C is the complex amplitude of the electric potential [8] N e K *C * * * I1 K * Rv * j * governing equations iteratively, using the penalty method to * ensure the satisfaction of the continuity equation. Each C * I 0 K * Rv S (18) iteration starts by updating all the fluid properties based on the r * 1 2 temperature of the last step (see [8] for further details). r* Rv * *2 * S Computations were carried out on a domain with dimensionless liquid height and axial length of one and The dimensionless fluid properties are given below, twenty, respectively, on a 20x100 grid. The parameters are varied over a range corresponding to stable operation of EHD c k * , * , c* p , k * , * , induction pumping. Instability of EHD induction pumps can bo bo bo p c pbo kbo manifest itself in a sudden drop/jump in pump output. The (19) instability can also result in alternating/bi-directional flow. 2 , * where c bo bo 0 The stability criteria defined in [9] shows that the erratic c bo Ne behavior of the unstable pump can be eliminated by a proper and the dimensionless length scales are as follows selection of geometric and liquid film parameters as well as the traveling electric wave frequency. The convergence criteria of z r Rv RT L the numerical results presented with respect to dimensionless z* , r* , Rv * , RT * , L* (20) velocity, potential, and temperature is 10 -5. It is important to mention that all the numerical results presented are The dimensionless electric parameters are given below corresponding to laminar flow with Reynolds number less than 2500. E ˆ E* , where Ec e , Ec IV. NUMERICAL RESULTS C C ' The factors controlling the performance of a two-phase C* , C '* , induction pump can be summarized in the following three ˆe e ˆ categories: K 1 K* , where K c , 1. Thermal Kc (21) a) Entrance temperature profile * , where c bo 2 , c bo b) Heat flux at the wall of the pipe and interface; '' '' S qwall , qint erface S* , where N e bo 2 0 , * Ne* bo bo c) Transport properties of the working fluid; e ,rz bo bo ˆ 2 , , k , c p *,rz , where e ,rzc e e ,rzc 2 bo e 2. Electrical and finally, the dimensionless flow parameters are defined as a) Applied voltage; ˆ follows b) Frequency; f u u* , where uc bo , Wavelength, uc bo c) w d) Electrical properties of the working fluid, , w* , where wc bo , wc bo 3. Physical p 2 p * , where pc bo 2 , a) External load or pressure drop pc bo (22) b) Liquid film thickness, vapor radius, the length and g 2 the radius of the pumping section. G * , where g c 2 bo 3 , gc bo The effect of the above parameters on the performance of a T 2 ˆ 2 M * vertical induction pump is discussed in this section. Since the T * , where, Tc bo e bo bo e transport and the electrical properties of any working fluid to Tc bo c pbo o c pbobo be considered in this study are function of temperature and require temperature dependent relations, a dimensionless parametric study is conducted using refrigerant R-123 as III. NUMERICAL METHODS reference working fluid. The transport and the electrical A finite element code was developed using the finite properties of the working fluid (R-123) are function of element software Fastflo Version 3.0 (developed at CSIRO temperature and are given in [10]. Three different temperature Mathematical and Information Sciences) to solve the profiles at the entrance corresponding to a uniform profile of 20 o C (Profile No. 1), and a two linear temperature profiles film thickness of 33 % of the total radius of the pipe responded producing an average entrance temperature of 20 oC (Profile more positively to the effect of temperature gradient. I t is also No. 2) and 22.0 oC (Profile No. 3), respectively, are considered. important to not that for Case 1 the dimensionless radius of 9 On the other hand the heat fluxes at the wall of the tube and at and 10 correspond to the liquid/vapor interface and the pipe the interface are taken as zero because no phase change is wall, respectively. On the other hand, dimensionless radius of 2 taking place as it was justified in Section II. and 3 correspond to the liquid/vapor interface and pipe wall, respectively, for Case 2. The effect of the parameters such as the applied voltage , ˆ electric wave angular velocity wave number K, the vapor Fig. 2 displays the dimensionless mass flux as a function of radius Rv, and gravity g are investigated through the following the dimensionless electric wave number, which is inversely dimensionless parameters Me*, *, K*, RV*, and G*, proportional to the wave length, for Case 1. The dimensionless respectively. Each of these parameters is studied separately mass flux is calculated at the pipe exit by integrating *w* over while the others are maintained constant. Since the working the dimensionless liquid film thickness. The behavior of all the fluid is fixed, the electric permittivity through * and the four curves presented in Fig. 2 is similar. The mass flux rises to its peak by increasing the electric wave number and then starts electric conductivity * through Ne* will not be varied to to decrease. The reason of such behavior is that at large wave accommodate for different working fluids, however, they are number (i.e., short wave length), the electric force is limited to function of temperature. To account for the effect of the liquid a smaller space and the electric field is confined to the vicinity film thickness, two base cases corresponding to two different liquid film thicknesses of 10% and 33 % of the total radius of the pipe are considered. However, for Case 2 only the results TABLE I. BASE CASES IN DIMENSIONAL FORM FOR R-123 that show different behavior than Case 1 are presented. Rv f e ˆ The dimensional and the dimensionless parameters of the (mm) (mm) (mm) (Hz) (V) two base cases for Profiles No. 1 and No. 2 are given in Tables Case 1 12.6 1.4 4.4 19 501 I and II. The effect of the external pressure gain or load on the Case 2 9.34 4.66 9.8 19 501 pumping section of EHD system or loop is also studied. This pressure gain or load, which plays a critical role in operating the EHD pump, is present in the liquid film only, for example, TABLE II. BASE CASES IN NON- DIMENSIONAL FORM FOR R-123 AT 20 O due to the existence of other EHD (or mechanical) pumping C sections or upstream and downstream connecting lines. In general a negative pressure gradient is a favorable load that R*v M*e N*e assists the pumping, whereas the positive gradient is unfavorable and is expected to work against the pumping Case 1 9.0 1.0 2.0 720 7000 6.0e-5 direction. However, in the results presented here the opposite is Case 2 2.0 1.0 3.0 8000 7000 5.4e-6 true since the pump is operating in repulsion mode (i.e. negative velocity, corresponding to negative z in Fig. 1). of the electrodes, in effect limiting its penetration depth into the In each of the following figures four cases are compared: 1) liquid film, resulting in a lower mass flux. In addition, slip numerical solutions with a uniform temperature profile at the coefficient (7), which is the measure of the lag between the entrance and the Joule heating is set to zero; 2) numerical traveling electric wave and the charges induced in the liquid solutions with a uniform temperature profile at the entrance film at the interface, is small. This also means that large wave (Profile No. 1) and the Joule heating present; 3) numerical numbers correspond to small wave speeds. On the other hand, solutions with a linear temperature profile at the entrance at small wave numbers (i.e., long wave length) the distances (Profile No. 2) with 21.0 oC at the pipe wall and 19.0 oC at the between the electrodes are long; hence the electric field is not liquid/vapor interface, and producing an average entrance weak resulting in small electric shear stress, therefore, small temperature of 20.0 oC; 4) numerical solutions with a linear mass flow rate. Furthermore, at a given frequency and very temperature profile at the entrance (Profile No. 3) with 25.0 oC small wave numbers the slip coefficient grows too large again at the pipe wall and 19.0 oC at the liquid/vapor interface, and producing small electric shear stress (7). producing an average entrance temperature of 22.0 oC. While the first two cases are presented to study the effect of the Joule Fig. 2 also reveals that the inclusion of the Joule heating is heating on the pump performance, the last two are presented to insignificant regardless of the value of the wave number. This illustrate the effect of the temperature gradient coupled with is due to the proportionality of the Joule heating to the electric Joule heating on the pump performance. conductivity which, under the operating conditions considered, is on the order of 10-8 (S/m). However, the presence of the When the film thickness is larger, the electric shear stresses temperature gradient (Profiles No. 2 and No. 3) becomes are smaller because the dominant induced charges at the significant only when the wave number is greater than unity. At liquid/vapor interface are far away from the electrodes. This is larger wave numbers, the bulk electric shear stress within the why the results for Case 1 show higher mass fluxes than those liquid film is responsible for the increase in the mass flow rate. of Case 2. It is noteworthy to mention that the effect of the This is due to the fact that the electric shear stress is temperature gradient is more noticeable when the generated proportional to the square of the wave number. In addition, the velocity falls in the lower range of the Reynolds number electric shear stress is a function of the electric conductivity (~<600). For this reason the numerical results produced for the gradient. A greater temperature gradient gives a greater Figure 3. Dimensionless electric shear stress distribution at the entrance, for conductivity gradient. Therefore, as can be seen from Fig. 3, Case 1 the electric shear stress is the largest with Profile No. 3, which in turns gives the highest mass flux. Note that the electric increase in the frequency. It is noteworthy to mention that this shear stress is the largest close to the electrodes where the critical value of the angular velocity is lower with higher electric field is the strongest. Fig. 2 also shows that as the wave wavelength (i.e. small wave number). The trend shown in Fig. number increases, the inclusion of the temperature Profiles No. 4 depicts the role of the electric shear stress within the liquid 2 and 3 yields higher mass flow rate compared to the case of film. Figure 4 indicates an increase in the mass flux compared Profile No. 1. For example, at wave number of 2.0, the mass to the case of Profile No. 1 of about 4.7% and 15% for Profiles flow flux with temperature Profiles No. 2 and No. 3 increased No. 2 and No. 3, respectively. These values are approximately by 4% and 14.5%, respectively. With the wave number of 4.0, constant for the range of frequency considered here. the mass flux increased by 19% and 88% with Profiles No. 2 The influence of Me*, which is proportional to the square of and No. 3, respectively. the applied voltage, is shown in Fig. 5 for the Case 2. Similar The effect of the electric wave angular velocity on the mass results can be obtained for Case 1.The mass flux increases with flux is presented in Fig. 4. In light of the slip coefficient, it is increasing Me* as expected. For example, at Me* of 7000 obvious that the mass flux has analogous dependency on the Profiles No. 2 and No. 3 result in an increase in the mass flux electric wave angular velocity as it has on the wave number. As of about 9% and 32%, respectively. In addition, Fig. 5 reveals the frequency increases, the mass flow rate increases until it an important observation that for high values of Me* (grater reaches a critical value, and then it falls gradually with further than 1000), inclusion of Joule heating alone can enhance the -1500 0 Profile No. 1, w/o Joule heating Profile No. 1, w/ Joule heating -500 -1750 Profile No. 2, w/ Joule heating -1000 Profile No. 3, w/ Joule heating -2000 -1500 Profile No. 1 w/o m* -2000 m* -2250 Joule heating -2500 Profile No. 1, w/ Joule heating -2500 -3000 Profile No. 2, w/ Joule heating -2750 -3500 Profile No. 3, w/ Joule heating -4000 -3000 0 1 2 3 4 5 0 500 1000 1500 2000 K* * Figure 2. Dimensionless mass flux as a function of dimensionless electric Figure 4. Dimensionless mass flux as a function of dimensionless electric wave number for Case 1. wave angular velocity for Case 1. 10 the mass flux. 9.9 Fig. 6 presents the mass flux as a function of the dimensionless vapor radius for Case 1. The liquid film 9.8 thickness is kept constant. Varying the vapor radius only and 9.7 keeping the liquid film thickness fixed is identical to varying the pipe diameter. For example, for Case 1 with R v* of 9.0, the 9.6 dimensionless liquid film thickness is 1.0 resulting in the tube * r 9.5 dimensionless radius of 10.0 (implying that the film thickness 9.4 is 10% of the total radius). However, if Rv* is 1.0 and the dimensionless liquid film thickness is 1.0 the tube 9.3 Profile No. 1 dimensionless radius becomes 2.0 implying that the film 9.2 Profile No. 2 thickness is 50% of the total radius. Figure 6 states that the Profile No. 3 inclusion of the Joule heating only results in negligible effect 9.1 on the mass flux under the operating conditions considered. 9 According to Fig. 6, the mass flux initially increases with Rv* -1000 -950 -900 -850 -800 -750 -700 and then it reaches an asymptotic value. e* 1). 0 0 Profile No. 1, w/o Joule heating -1000 -500 Profile No. 1, w/ Joule heating Profile No. 2, w/ Joule heating -2000 -1000 Profile No. 3, w/ Joule heating -1500 m* -3000 m* -2000 -4000 -2500 Profile No. 1, w/o Joule heating Profile No. 1, w/ Joule heating -5000 -3000 Profile No. 2, w/ Joule heating Profile No. 3, w/ Joule heating -3500 -6000 0 10000 20000 30000 40000 50000 60000 70000 -4000 Me* -600 -400 -200 0 200 400 600 (dP/dz)* Figure 5. Dimensionless mass flux as a function of Me* for Case 2. Figure 7. Dimensionless mass flux as a function of dimensionless pressure gradient for Case 1. -500 Profile No. 1, w/o Joule heating Profile No. 1, w/ Joule heating 0 Profile No. 1, w/o Joule heating -1000 Profile No. 2, w/ Joule heating Profile No. 1, w/ Joule heating Profile No. 3, w/ Joule heating -200 Profile No. 2, w/ Joule heating -1500 Profile No. 3, w/ Joule heating -400 m* -2000 m* -600 -800 -2500 -1000 -3000 0 5 10 15 20 25 -1200 Rv* -200 -100 0 100 200 300 400 500 600 (dP/dz)* Figure 6. Dimensionless mass flux as a function of dimensionless vapor redius for Case 1. Figure 8. Dimensionless mass flux as a function of dimensionless pressure gradient for Case 2. As Rv* increases the symmetry boundary condition imposed at According to Fig. 7 the curves are almost linear (properties are pipe centerline is moved further away from the liquid/vapor. temperature dependent) for all three profiles for the two cases. Therefore, it is influence on the electric field distribution in the As an example, according to Fig. 7, Profiles No. 2 and No. 3 liquid film and at the liquid/vapor interface becomes less enhance the mass flux for dimensionless pressure gradient of - significant until it losses any influence. 500 by about 7% and 25%, respectively, compared to Profile The dependency of the mass flux on the external pressure No. 1. Figure 8 presents the results for the negative pressure gradient is shown in Figs. 7 and 8 for the two cases considered, gradient up to -150 for Profile No. 1 and -50 for Profiles No. 2 respectively. The external load plays a crucial role in operating and No. 3. The reason is that decreasing the external negative the EHD pump. As stated previously, generally a negative pressure gradient further results in a bi-directional flow or a pressure is a favorable load which speeds the pumping, flow in the opposite direction (in positive z-direction). This whereas the positive pressure gradient is unfavorable and tends behavior will be explained further in light of the velocity to lower the velocity. However, in the results presented here the distribution. Nonetheless, Profiles No. 2 and No. 3 result in an opposite is true since the pump is operating in repulsion mode increase in the mass flux of about 23% and 62%, respectively, resulting in the fluid flow in the negative z-direction (see Fig. at pressure gradient of -50 compared to Profile No. 1. This enhancement ratio decreases as the pressure gradient increases. To further explain the effect of the external pressure Figure 10. Dimensionless velocity profile at the pipe entrance for gradient on the pump performance, the actual velocity profiles (dP/dz)*=-250 and -400, for Case 2. at the entrance of the pipe for the three temperature profiles and selected pressure gradients are presented in Figs. 9 and 10 for Bi-directional liquid film flow, in general, is not desirable and Case 2. Similar results can be obtained for Case 1. In these in designing an EHD pump, the load should not exceed its limit figures, the velocity profiles are subjected to the effects of the in order to prevent this occurrence. electric shear stress as well as the external pressure gradient. Finally, Fig. 11 presents the effect of the gravitational force The effect of the electric shear stress alone (i.e. zero pressure on the mass flux for Case 1. Note that a dimensionless wave gradient, Fig. 9) on the velocity profile results in a negative number of 0.5 was chosen instead of 2.0 to allow for wider velocity distribution, as expected for EHD pump operating in variation of the gravitational force before the results reach repulsion mode. The positive external pressure gradient is values corresponding to the region of transition-turbulent. expected to assist the electric shear stress resulting in a higher Here, the wave number of 0.5 corresponds to a wave length of negative velocity (Fig. 9). However, when the external pressure 17.6 mm. Figure 11 shows that the gravitational force gradient is negative, the overall effect depends on the dominant accelerates the flow, thus enhances the mass flow rate. Another force. If the electric shear stress and the negative pressure observation is that the gravitational force, as it grows larger, gradient are comparable, then the velocity profile could be bi- dominates the electric shear stress. directional as illustrated in Fig. 10. When the negative pressure is dominant, then the velocity profile is expected to be solely in the positive z-direction (see Fig. 10). -2000 3 -3000 , Profile No. 1 (dP/dz)*=0.0 Profile No. 2, (dP/dz)*=0.0 -4000 Profile No. 3, (dP/dz)*=0.0 2.8 , Profile No. 1 (dP/dz)*=200.0 -5000 Profile No. 2, (dP/dz)*=200.0 Profile No. 3, (dP/dz)*=200.0 m* -6000 2.6 -7000 r* -8000 Profile No. 1, w/ Joule heating 2.4 Profile No. 2, w/ Joule heating -9000 Profile No. 3, w/ Joule heating 2.2 -10000 -5000 -4000 -3000 -2000 -1000 0 G* 2 -250 -200 -150 -100 -50 0 Figure 11. Dimensionless mass flux as a function of dimensionless w* gravitational force for Case 1 (K*=0.5). Figure 9. Dimensionless velocity profile at the pipe entrance for Thus, the presence of temperature profiles within the liquid has (dP/dz)*=0.0 and 200, for Case 2. practically no effect on the resultant liquid film mass flow rate. V. CONCLUSIONS , Profile # 1 (dP/dz)*=- 250.0 , Profile # 1 (dP/dz)*=- 400.0 Profile # 2, (dP/dz)*=- 250.0 Profile # 2, (dP/dz)*=- 400.0 A theoretical model was developed for EHD induction Profile # 3, (dP/dz)*=-250.0 Profile # 3, (dP/dz)*=- 400.0 pumping of liquid film in vertical configuration. The model 3 accounted for the charges not only at the interface but also throughout the liquid film. Three different temperature profiles 2.8 were considered to investigate the effect of the Joule heating and the temperature gradient throughout the bulk of the film on the mass flux for two cases corresponding to two different film 2.6 thicknesses. A parametric study including liquid film thickness, r* voltage, wavelength, frequency, external pressure, and gravity 2.4 was carried out. The numerical results indicated that for the electric conductivity and voltage level of interest, including the 2.2 Joule heating in the analysis proved to be insignificant. In addition, the results showed that higher mass flow rate can be achieved by increasing the voltage, pipe diameter, fluid 2 temperature, and also by increasing the external pressure gain. -40 -20 0 20 40 60 80 100 On the other hand, external pressure load, at certain level and w* beyond, yielded flow reversal or bi-directional flow, which is potentially undesirable. The entrance temperature profile played an important role in the operation of the pump. Finally, the gravitational force could easily dominate the effect of the electric shear stress. REFERENCES [1] Melcher, J. R., “Traveling - Wave Induced Electroconvection,” Physics of Fluids, Vol. 9, No. 8, (1966), pp. 1548-1555. [2] Melcher, J. R., “Continuum Electromechanics,” Cambridge, MA: MIT Press, pp. 5.94-5.54, 1981. [3] Crowley, J. M., “Stability of EHD Induction Pump,” Conf. Rec. IEEE- IAS Annual Meeting, Mexico City, Mexico, (1983), pp. 1149-1153. [4] Wawzyniak, M. and Seyed-Yagoobi, J., “An Analytical Study of Electrohydrodynamics Induction Pumping of a Stratified Liquid/Vapor Medium,” IEEE-IAS Transactions, Ind. Applicat., 35, (1999), pp. 231- 239. [5] Wawzyniak, M. and Seyed-Yagoobi, J., “Electrohydrodynamic Induction Pumping of a Stratified Liquid/Vapor Medium in the Presence of Volumetric and Interface Electric Charges,” IEEE-IAS Transactions, Ind. Applicat., 37, No. 4 (2001), pp. 950-958. [6] Brand, K. and Seyed-Yagoobi, J., “Effect of Electrode Configuration on Electrohydrodynamics Induction Pumping of a Stratified Liquid/Vapor Medium,” IEEE-IAS Transactions, Ind. Applicat., 38, No. 2 (2002), pp. 389-400. [7] B. Benjamin, “Wave Formation in Laminar Flow Down an Inclined Plane,” J. Fluid Mech., Vol. 2, pp. 554-574, 1957. [8] Aldini, S., “Electrohydrodynamic Induction and Conduction Pumping of Dielectric Liquid Flim: Theoretical and Numerical Studies,” Ph.D. dissertation, Dept., Mech. Eng., Texas A&M Univ., College Station, TX, 2005. [9] Aldini, S. and Seyed-Yagoobi, J., “Stability of Electrohydrodynamic Induction Pumping of Liquid Film in Vertical Annular Configuration,” IEEE-IAS Transactions, Ind. Applicat., (in press). [10] J. E. Bryan, “Fundamental Study of Electrohydrodynamic Enhanced Convective and Nucleate Boiling Heat Transfer,” Ph.D. dissertation, Dept., Mech. Eng., Texas A&M Univ., College Station, TX, 1998