Understanding of electrohydrodynamic conduction pumping phenomenon

Transcription

1 PHYSICS OF FLUIDS VOLUME 6, NUMBER 7 JULY 2004 Understing of electrohydrodynamic conduction pumping phenomenon Yinshan Feng Jamal Seyed-Yagoobi Heat Transfer Enhancement Two-Phase Flow Laboratory, Mechanical, Materials, Aerospace Engineering Department, Illinois Institute of Technology, Chicago, Illinois 6066 Received 24 November 2003; accepted 9 March 2004; published online June 2004 Electrohydrodynamic EHD phenomena can be applied to enhance control mass heat transfer in both terrestrial microgravity environments. The emerging EHD conduction pumping technique shows its potential as an active control method of the flow distribution. The EHD conduction pumping is associated with the heterocharge layers of finite thickness in the vicinity of the electrodes, which are based on the process of dissociation of the neutral electrolytic species recombination of the generated ions. This paper theoretically experimentally studies the EHD conduction phenomenon in a dielectric liquid. The analytical solutions provide the non-dimensional distributions of electric field charge density in the vicinity of the electrodes. The characteristic heterocharge layer thickness is also theoretically predicted. Measured pressure heads current levels are compared with the theoretical results. The EHD conduction pump presented here is capable of electrically driving controlling the dielectric liquid flow motion in a single-phase loop American Institute of Physics. DOI: 0.063/ I. INTRODUCTION As an active control method, electrohydrodynamic EHD technique has presented its unique advantages, such as the simple design, no mechanical parts, low acoustic noise, lightweight, the rapid control of performance by varying the applied electric field, low power consumption. It can be applied to enhance mass heat transfer in both terrestrial microgravity environments. As an emerging EHD technique, the EHD pumping based on the conduction mechanism shows its potential as an active control method for purposes of heat mass transport in single-phase liquid two-phase flows, as well as flow control distribution such as application in parallel evaporators. The EHD pumping phenomena involve the interaction of electric fields flow fields in a dielectric fluid medium. This interaction between electric fields flow fields induces the flow motion by an electric body force, which can be expressed as follows: f e e E 2 E2 2 E2. T The first term represents the Coulomb force, which is the force acting on the free charges in an electric field. The second third terms represent the polarization force acting on polarized charges. The third term, the electrostriction term, is relevant only for compressible fluids. Thus, EHD pumps require either free space charges or a gradient in permittivity within an incompressible liquid. In an isothermal single phase liquid, the gradient in permittivity,, vanishes, resulting in the Coulomb force as the only force for generating a net EHD motion. There are three kinds of EHD pumping mechanisms utilizing the Coulomb force: conduction pumping, induction pumping, ion-drag pumping. The electro-osmotic pumping is not taken into account here due to its specific applications in mini- or micro-scales its dependence on the liquid solid interfacial property. The induction pumping ion-drag pumping have been well studied. The induction pumping is not suitable for pumping of an isothermal liquid, which is almost the case for the flow prior to the evaporators. Ion-drag pumping is not appropriate for any applications since over time it results in the degradation of the working fluid electric properties. The ion-drag pump is also potentially hazardous to operate due to the corona discharge associated with it. Unlike the induction ion-drag pumping, the conduction pumping is associated with the heterocharge layers of finite thickness in the vicinity of the electrodes which are based on the process of dissociation of the neutral electrolytic species recombination of the generated ions. 2 The conduction term here represents a mechanism for electric current flow in which charged carriers are produced not by injection from electrodes, but by dissociation of molecules within the fluid. The heterocharge layer closely depends on the applied electric field. Its thickness can be up to several millimeters, much larger than the double layer thickness in the electro-osmotic flow, at an intense electric field. The EHD conduction pumping can be applied to the isothermal working fluid will not degrade the electric properties of the working fluid since the imposed electric field will be below the intensity level necessary for the ion-drag pumping. The EHD conduction pumping phenomena have been studied only very recently by a few researchers. 2 4 The related research so far has numerically experimentally verified the pressure generation with the EHD conduction phenomenon. Based on the promising initial work, this paper further develops a non-dimensional theoretical model, presents experimental results, connects the theoretical model of the dissociation-recombination process with the EHD con /2004/6(7)/2432/0/$ American Institute of Physics

2 Phys. Fluids, Vol. 6, No. 7, July 2004 Understing of electrohydrodynamic conduction 2433 E denotes the applied electric field. When the dielectric liquid in Fig. is static u0 the diffusion of charges is negligible, the current density based on the pure conduction phenomenon in the dielectric liquid can be expressed as jk pk ne. 4 The steady-state behavior of the ions in the dielectric liquid bulk is governed by the following conservation equations: "K Epk D ck R pn, 5 duction pumping phenomenon,, for the first time, experimentally verify the electrically driven flow by the EHD conduction phenomenon. The non-dimensional theoretical model described here has been normalized by the characteristic length of the heterocharge layer, instead of the electrode spacing proposed by Atten Seyed-Yagoobi. 2 Such improvement of the non-dimensional model leads to a generalized analytical solution independent of the fluid properties the applied electric field. The non-linear theoretical model has been analytically solved to obtain the exact normalized profiles of electric field charge density in the vicinity of electrodes further predict the associated pressure head consumed current. Experiments have been conducted to compare with the theoretical predictions. The electrically driven flow generated by the EHD conduction pumping mechanism has been experimentally verified. II. THEORETICAL MODEL OF EHD CONDUCTION PHENOMENON The basic concept of the EHD conduction pumping can be illustrated using two parallel electrodes immersed in a dielectric liquid as shown in Fig.. The electric conduction mechanism in a pure dielectric liquid can be explained with a simple model, which assumes that there is a reversible process of dissociation-recombination between a neutral species denoted AB) its corresponding positive A negative B ions: AB FIG.. Parallel electrodes immersed in dielectric liquid. Dissociation Recombination A B. These generated ions, A B, function as charge carriers between the parallel electrodes. The current density passing through the dielectric liquid is expressed as jk pk nepnud pn, 3 where the three terms on the right-h side represent the contribution of the charge mobility, convection, diffusion, respectively. n p denote the negative positive charge density. K K denote the negative positive charge mobility. D denotes the charge diffusion coefficient 2 "K Enk D ck R pn, "E pn, 7 where k D k R denote the dissociation rate the recombination rate of charges, respectively. c represents the concentration of neutral species in the dielectric fluid represents the electric permittivity of dielectric fluid. At a thermodynamic equilibrium, the ion conservation allows k D ck R p 0 n 0 k R p 0 2 k R n 0 2, where the subscript 0 indicates the equilibrium state. Assuming the positive negative ions have the identical mobility, K K K, the recombination rate constant becomes 5 k R 2K/. 9 To simplify the analysis, the parallel electrodes in Fig. will be treated one-dimensionally. Let x*x/, E* Ed/V, p*p/n 0, n*n/n 0, C n 0 d/v. denotes the characteristic thickness of the heterocharge layer it is assumed that at x characteristic thickness of the heterocharge layer, the non-dimensional positive ion density, p*, is equal to 0.99, which means that the charge concentration almost reaches the equilibrium level at that location. Atten Seyed-Yagoobi 2 proposed the electrode spacing, d, as the characteristic length for the normalization of the variable, x. In this paper, the heterocharge layer thickness is applied as a characteristic length in order to let C be a constant obtain a generalized analytical solution. The corresponding non-dimensional governing equations based on Eqs. 5 7 are dp*e* 2C dx* p*n*, dn*e* 2C dx* p*n*, de* dx* C p*n*. 2 Atten Seyed-Yagoobi 2 obtained approximate solutions of the above governing equations by ignoring the non-linear recombination term, (p*n*), in Eqs. 0. However, such treatment of the recombination term leads to the

3 2434 Phys. Fluids, Vol. 6, No. 7, July 2004 Y. Feng J. Seyed-Yagoobi TABLE I. Coefficients for Eqs Coefficient Value Coefficient Value a a a a a a a a linear profiles of p*e* n*e*. The corresponding charge electric field distributions become non-smooth at the outer edge of the heterocharge layer, which is not true in the real case. To precisely capture the dissociationrecombination behavior of charges, the solution should include the effect of the recombination. Appendix A presents the derivation procedures for the exact solutions with the recombination term considered. As described in Appendix A, the general solutions of Eqs. 0 2 are p*e* C3 d 2C gc 2 C x*, n*e*p*e*c, 3 4 p*e*p*e*c E* 2 gc 2 p*e*p*e*c, 5 where the function, g, is defined as gye y y. If the heterocharge layer thickness is rather small compared with the electrode spacing (d), the boundary conditions require p*0 at x*0, 6 p*n* at x*, 7 E* at x*. 8 The constants can be determined as C 2, C 2 e, C FIG. 2. Non-dimensional profiles around heterocharge layer adjacent to anode. Finally the solutions become 0 p*e* d 2C ge 2.0 x*, n*e*p*e*2.0, E* 2 p*e*p*e*2.0 ge p*e*p*e* Since g is an implicit function, it is not possible to directly calculate the above equations. An alternative way is to obtain an explicit expression of g through curve-fitting further simplify Eqs Following the procedures in Appendix B, the above equations can also be expressed as p*e* a a 2 a a 3 /a 2 n*e*p*e*2.0, a 2 a e 2C a 2 a a 3 x* a 3, E* 2 p*e*p*e*2.0 a 4 a 5 p*e*a 6 p*e* 2 /a 7 p*e*a 8 p*e* 2, 25 where the coefficients, a i, are listed in Table I. The definition of requires that p* x * p*e* x* E* 2 x * Based on Eqs. 23, 25, 26, the non-dimensional parameter, C, can be determined as a constant C.8, 27 which indicates that the non-dimensional parameter, C n 0 d/v, is independent of the properties of dielectric fluid the applied electric field if d. Figure 2 illustrates the profiles of the non-dimensional parameters adjacent to the positively charged electrode. Since C remains a constant, the normalized profiles shown in Fig. 2 are generally valid for the dissociation-recombination conduction phenomenon in any dielectric liquid at any applied electric field if d. However, the theoretical model also indicates

4 Phys. Fluids, Vol. 6, No. 7, July 2004 Understing of electrohydrodynamic conduction 2435 TABLE II. Properties of saturated liquid R-23 at 22 C. Fluid e S/m K m 2 /V s Pa s kg/m 3 R a 2.6E8 a 4.59E8 b 435.7E6 c c a Data from Bryan Ref. 9. b Based on Walden s Rule: K(20 )/m 2 /V s Crowley et al. Ref. 0. c Data from NIST Ref.. FIG. 3. Differences between the local properties their corresponding equilibrium values. that when x, the charge densities electric field reach the equilibrium state very quickly. Figure 3 shows the differences between the local properties their corresponding equilibrium values. At x3, all differences are approximately 0.000%, which indicates that if d/3, the theoretical model will give the almost identical solutions. Equation 27 also denotes that the characteristic thickness of the heterocharge layer is a function of the electric properties of the dielectric liquid the applied electric field intensity.8 V n 0 d, 28 where the equilibrium negative charge density, n 0, can be estimated as n 0 e /2K, assuming that the equilibrium negative positive charge densities are identical the measured electric conductivity of dielectric liquid, e, is purely due to the charge mobility. Table II lists the properties of liquid R-23 at 22 C. Equation 28 indicates that the heterocharge layer thickness is proportional to the applied electric field for a given dielectric fluid. When the electric field increases from 00 V/m to MV/m, the heterocharge layer in liquid R-23 will grow from 0.03 m to 0.3 mm. The electric body force density in the heterocharge layer can be expressed as the product of the net charge density the local electric field, f e EpnE. 29 For a symmetric electrode configuration, such as the parallel electrodes in Fig., no net electric force will be generated since the electric forces in the heterocharge layers, adjacent to the positively negatively charged electrodes, balance each other. However, if the electrode configuration is nonsymmetric, a net electric force may be generated. Figure 4 shows the schematic picture of a non-symmetric electrode design studied in this paper. The non-symmetric electrode pair consists of a ring ground electrode a perforated disc electrode. The applied electric field generates heterocharge layers adjacent to the ring ground electrode the perforated electrode. The two-dimensional electric field distribution between the perforated electrode the ring ground electrode has been numerically calculated. Figure 5 shows the two-dimensional non-uniform electric field, which is mainly in the axial direction, in the vicinity of the perforated electrode with the dissociation-recombination process not taken into account. The illustrated electric field has been normalized as E*Ed/V, where d4.3 mm denotes the distance between the surface of the perforated electrode the center of the ring ground electrode. The electric field decreases significantly around the holes of the perforated FIG. 4. Non-symmetric electrode design, a schematic b picture.

5 2436 Phys. Fluids, Vol. 6, No. 7, July 2004 Y. Feng J. Seyed-Yagoobi FIG. 5. Electric field in the vicinity of the perforated electrode. electrode. The associated heterocharge layer is mainly formed around the unpunctured area of the perforated electrode as shown in Fig. 4a. The associated electric force exerted on the heterocharge layer adjacent to the ring electrode is mainly in the radial direction, while the electric force on the heterocharge layer of the perforated electrode is in the axial direction. Consequently, the dissociation-recombination conduction phenomenon in the non-symmetric electrode configuration, such as the electrode design in Fig. 4, results in a net axial electric force. This net axial force can be utilized to electrically drive the dielectric liquid to effectively control the flow distribution among pipe lines. Figure 5 also shows that in the vicinity of the unpunctured area, the magnitude of the applied electric field, ranging from 76% to 50% of V/d, has the same order of the nominal electric field, V/d. Since the heterocharge layer thickness is expected to be up to mm at the maximum applied voltage of 7 kv, the electric field distributions at various distances up to.25 mm from the perforated electrode are shown in Fig. 6. Within.0 mm from the perforated electrode, the magnitudes of the applied electric fields near the unpunctured area have the similar order 60% 50% of the nominal electric field. To simplify the theoretical analysis for the ring-perforated electrode configuration, the electric field in the vicinity of the perforated electrode is onedimensionally treated as the electric field between parallel electrodes the equivalent electrode distance is set as d 4.3 mm. Based on the above assumptions, the associated heterocharge layer thickness,, in the vicinity of the perforated electrode is determined with Eq. 28 using the nominal electric field, EV/d. As the radial electric force exerted on the heterocharge layer around the ring ground electrode balances by itself, the generated pressure by the axial electric force can be estimated as Pm 0 pnedx.8m V2 n*e*p*e*dx*0.85m V2 d 2 0 d, 2 30 FIG. 6. Electric fields within.25 mm from the perforated electrode. where m denotes the number of electrode pairs 0 (n*e*p*e*)dx*0.474, which can be numerically obtained from the profiles in Fig. 2. The perforated disc electrode studied here has an openness of 4%. The heterocharge layer may not cover the punched area. If the electrode openness is taken in account, Eq. 30 becomes P0.59m pnedx0.5m V2 0 d. 3 2 In the following text, the experimental data in the absence of a net flow will be compared with the theoretical predictions of Eqs III. EXPERIMENTAL SETUP The schematic of the EHD conduction pump is shown in Fig. 7. The EHD conduction pump consists of three pairs of stainless steel electrodes with gold coating. The electrode edges have been well smoothed to prevent charge injection. Each pair of electrodes has a perforated disc electrode with.59-mm-diam holes a ring ground electrode. The open area of the perforated disc electrode is 4% of the total crosssectional area. Spark plugs are used to connect a high voltage power supply Model EW, Glassman with the perforated disc electrodes. The main body of the EHD conduction pump is made of brass is grounded together with the ring electrodes. Teflon is utilized as the insulation material due to its electric properties its chemical compatibility with the refrigerants. A single-phase liquid only loop as shown in Fig. 8 is built to investigate the electrically driven net flow generated by an EHD conduction pump. HCFC refrigerant R-23 is used as the working fluid. The single-phase loop consists of an EHD conduction pump, a turbine flow meter Model FT4-8, EG&G Flow Technology, a regulating valve, a pressure transducer model DP5, Validyne, an Omega T-type thermocouple. The regulating valve is adjusted to control the flow rate in the loop. The flow meter, the pressure transducer, the thermocouple are used to measure the flow rate, the generated pressure head, the operating temperature, respectively. The pressure head without the net flow can be obtained simply by closing the regulating valve.

6 Phys. Fluids, Vol. 6, No. 7, July 2004 Understing of electrohydrodynamic conduction 2437 FIG. 7. Schematic of EHD conduction pump. A 586 PC Strawberry Tree software hardware are used as the data acquisition system to obtain all measured data. A total uncertainty analysis is performed for all the measured data quantities based on methods described by Kline McClintock. 6 The maximum total uncertainties are shown in Table III. IV. COMPARISON OF EXPERIMENTAL DATA AND THEORETICAL PREDICTION IN THE ABSENCE OF A NET FLOW The experiments of the single-phase loop, with the regulating valve closed, were first conducted to investigate the EHD conduction pump capacity in the absence of a net flow. Various dc voltages, ranging from 0 to 7 kv, were applied to the EHD conduction pump. The regulating valve was kept closed to provide the static i.e., no net flow operating condition. Figure 9 shows the generated pressure heads in the absence of a net flow the corresponding theoretical predictions of Eqs at an operating temperature of 22 C. The electrode spacing of 4.3 mm, which denotes the distance between the surface of the perforated electrode the center of the ring ground electrode, is used for the theoretical prediction. Since three pairs of electrodes were used, the coefficient m in Eqs is set as three. Equation 30 treats the perforated disc electrode as a solid disc electrode, while Eq. 3 takes the effects of the perforation into account. Both equations are only valid for the EHD conduction phenomenon in the absence of a net flow. The comparison of the experimental results the theoretical predictions shows that most data points locate between the predictions of Eqs When the applied voltages are below 7 kv, both equations over-predict the generated pressure. It may be due to the neglect of the diffusion term in the theoretical model. As the applied voltage exceeds 9 kv, the measured pressure heads are always higher than the predictions of Eq. 3. It can be caused by the extension of the heterocharge layer to the punched area of the perforated disc electrode at an intense electric field during the experiments. Based on the theoretical model, the consumed current level can also be easily determined. Due to the continuity of the charge motion, the consumed current is equal to the total current at the positively charged or ground electrode. With the positively charged electrode monitored, the total current is Imr 2 nek pek x0 mr 2 n 0 V d n*e*k p*e*k x *0. 32 TABLE III. Experimental uncertainties. Total uncertainty Measurements Average Maximum FIG. 8. Schematic of single-phase loop. Pressure head 8.2 Pa 20.5 Pa Flow velocity 0.06 cm/s 0.2 cm/s Temperature 0. C 0. C Applied voltage 0.5 V.2 V Consumed current 0.2 A 0.6 A

7 2438 Phys. Fluids, Vol. 6, No. 7, July 2004 Y. Feng J. Seyed-Yagoobi FIG. 9. Comparison of experimental pressure data theoretical predictions in the absence of a net flow at 22 C. FIG. 0. Consumed current vs applied voltage for EHD conduction pump. Figure 2 shows that n*e*2.0 p*e*0 atx*0. If the mobility of the positive negative charges is identical n 0 e /2K, the consumed current can be estimated as V Imr 2 e d. 33 Equation 33 denotes the consumed current due to the dissociation-recombination process. Gallagher 7 mentioned that the general shape of a current-field characteristic for a dielectric liquid can be roughly divided into three regions: low field region 0. MV/m, intermediate region 2 MV/m, high field region up to 00 MV/m. In the low field region, the current rapidly rises linearly with field. At the intermediate region, the current increases slowly with field tends toward saturation. At the high field region, the current rises up rapidly again breakdown may take place in the region of 00 MV/m. With the electrode spacing of 4.3 mm the applied voltage ranging from 0 to 7 kv, the electric field imposed to the EHD conduction pump covered all three regions the corresponding consumed current data in Fig. 0 show the similar trend described by Gallagher. The dissociation-recombination model is not suitable for the description of the initial rapid rise of current at the low field region. In the intermediate high field regions, the dissociation-recombination process becomes dominant if no charge injection especially at the high field. To reasonably compare the experimental data the theoretical prediction of current level, the initial rise of current at the low field region should be incorporated into prediction based on the dissociation-recombination model in the following expression: I Total I Initial mr 2 e V d if V 0. MV/m. d 34 Similarly, if the punched area of the perforated disc electrode is taken into account, the total consumed current approximately becomes I Total I Initial 0.59mr 2 e V d if V 0. MV/m. d 35 Based on the experimental data shown in Fig. 0, the initial rise of current at the low field region is around 8 A. Figure shows the comparison of the experimental data the predictions of the consumed current. The experimental data follow the trend predicted by Eq. 34. When the applied voltage exceeded 2 kv, the experimental data become higher than the theoretical prediction non-linear to the applied voltage. The charge injection in high field region the internal circulating flow motion, within the pumping section, driven by the electric force may cause the discrepancy. The effects of the flow motion charge diffusion on the current level will be discussed in Sec V. V. ELECTRICALLY DRIVEN FLOW UTILIZING EHD CONDUCTION PUMPING MECHANISM High dc voltages at 0 5 kv were applied to the EHD conduction pump to circulate the working fluid along the loop. At a given applied voltage, the various electrically driven flow rates were achieved by adjusting the regulating valve in the loop. Figures 2 3 show the trend of the flow velocity versus the generated pressure head at 0 5 kv, respectively, for the R-23 single-phase liquid loop. Due to the internal friction of the EHD conduction pump the FIG.. Comparison of experimental current data theoretical predictions in the absence of a net flow at 22 C.

8 Phys. Fluids, Vol. 6, No. 7, July 2004 Understing of electrohydrodynamic conduction 2439 FIG. 2. Liquid flow velocity vs generated pressure head at 0 kv for R-23 at 22 C. FIG. 4. Current level power consumption vs generated pressure head at 0 kv for R-23 at 22 C. setup of the single-phase loop i.e., the external load on the pump, the maximum flow velocity at the applied voltage of 5 kv was 8.9 cm/s at the generated pressure head of 425 Pa. The temperature of the loop was maintained at around 22 C. The saturated liquid density of R-23 at this temperature was approximately 500 kg/m 3. Therefore, the corresponding maximum mass flux of the electrically driven liquid flow was around G30 kg/m 2 s, which is considerable to control the flow distribution. As indicated by Eq. 3, the presence of a net flow should significantly affect the current level. The EHD conduction pump drives the working fluid provides the pressure head overcoming the frictional force along the loop. It is expected that the generated pressure head should decrease with the increase of the flow velocity due to two causes: the internal pressure loss due to the perforated electrode configuration inside the pump section will increase along with the corresponding flow velocity; 2 the Coulomb force will decrease due to flow effects on the charge distribution, the heterocharge layer thickness, the charge density, which needs to be investigated theoretically later. Figures 4 5 illustrate the corresponding current power consumption curves at 0 5 kv, respectively. In the presence of a net flow, the convection speeds up the motion of charges resulting in an increase in the current level. Figures 2 5 show that at a fixed applied voltage, when the pressure head increases, the current level decreases due to the reduced flow velocity. The current is the lowest in the absence of a net flow. On the basis of the simple electric conduction model presented here, the current density comes from the charge mobility, charge convection, charge diffusion as shown in Eq. 3. The magnitude analysis gives that nn 0, p p 0, EV/d, /d, K K K, n 0 p 0. Consequently, the magnitude of the current density level can be estimated as jn 0 2K V d u D d 2n 0 K V d ud 2KV 2KV D. 36 The Nernst Einstein 8 formula provides that D K kt e, 37 where k denotes the Boltzmann constant ( J/K), e denotes the electron charge (.602 FIG. 3. Liquid flow velocity vs generated pressure head at 5 kv for R-23 at 22 C. FIG. 5. Current level power consumption vs generated pressure head at 5 kv for R-23 at 22 C.

9 2440 Phys. Fluids, Vol. 6, No. 7, July 2004 Y. Feng J. Seyed-Yagoobi 0 9 C), T denotes the temperature of dielectric fluid in degrees Kelvin. The experimental data presented in this paper were measured at a temperature of 295 K. Inserting the Nernst Einstein formula into Eq. 36 leads to j2n 0 K V d ud 2KV 80V, 38 where the three terms in the bracket indicate the contributions of the charge mobility, convection, diffusion, respectively. If the applied voltage, V, exceeds 2.5 V, the contribution of the charge diffusion to the current level falls below 0.%. Thus, with the applied voltage of order of kilovolts, the effect of the charge diffusion on the consumed current is always negligible. The experimental data show that in a single-phase loop, the EHD conduction pumping could generate fluid flow with velocity up to 0 cm/s at an applied voltage of 5 kv. The contribution of convection can be estimated by inserting the velocity the applied voltage into Eq. 38 as j2n 0 K V d 2KV ud, 39 where ud 2KV %. At an applied voltage of 5 kv, the convection in the dielectric fluid can increase the current level by 3%, which qualitatively matches the electric current data in Fig. 5. VI. CONCLUSIONS A non-dimensional theoretical model has been developed to fundamentally underst the dissociationrecombination process of dielectric liquid the associated EHD conduction pumping phenomenon in the absence of a net flow. The analytical solutions indicate that the nondimensional parameter, C n 0 d/v, is a constant if d the heterocharge layer thickness can be simply expressed as a function of the electric properties of the dielectric liquid, the space between two parallel electrodes, the applied voltage (.8V/n 0 d). It is also theoretically predicted that the generated pressure head is a quadratic function of the applied electric field as P0.85m(V 2 /d 2 ), while the consumed current behaves as a linear function of the applied electric field. To verify the theoretical model, an EHD conduction pump with three pairs of perforated electrodes was designed experimentally investigated. The EHD conduction pump was installed in a single-phase loop with HCFC refrigerant R-23 as the working fluid. To simplify the theoretical analysis for the ring-perforated electrode configuration, the electric field in the vicinity of the perforated electrode is treated one-dimensionally the nominal electric field is used to determine the heterocharge layer thickness. The predicted pressure head generation current level agree with the experimental data in the absence of a net flow. The magnitude analysis of the current level indicates that the charge mobility charge convection can play important roles in the current level while the influence of the charge diffusion is negligible at high voltages. It is also found that the EHD conduction pumping phenomenon has the capacity to successfully drive dielectric liquids, such as refrigerant R-23, for practical applications. ACKNOWLEDGMENT This project was partially financially supported by the American Society of Heating, Refrigerating Air- Conditioning Engineers. APPENDIX A Inserting de* 2 dx* de*2 dp*e* dp*e*, dx* (n*e*)(p*e*)c, Eq. 0 into Eq. 2 leads to de* 2 dp*e* p*e*p*e*c 2p*E*C E* 2. A Rearrange the above equation obtain the following: 2p*E*C p*e*p*e*c E* 2 de*2 dp*e* 0. Timing both sides of Eq. A2 with E* 2 exp p*e*p*e*c 2 E* 2 leads to de* 2 dp*e* E* 2 exp p*e*p*e*c 2p*E*C E* 2 E* 2 p*e*p*e*c de* 2 E* 2 dp*e* 2 de*2 dp*e* E* 2 exp p*e*p*e*c 2 E* 2 A2 0. A3 The above differential equation can be simplified as d E* 2 exp p*e*p*e*c dp*e*0. E* 2 A4 The general solution of the above differential equation is E* 2 exp p*e*p*e*c E* 2 C 2. A5

10 Phys. Fluids, Vol. 6, No. 7, July 2004 Understing of electrohydrodynamic conduction 244 The above equation is an implicit equation of (E* 2 ) (p*e*). It can also be expressed as p*e*p*e*c exp p*e*p*e*c E* 2 E* 2 C 2 p*e*p*e*c. A6 The following explicit expression can be obtained: p*e*p*e*c E* 2 gc 2 p*e*p*e*c, gyey y. A7 Combining Eqs. 0 A7 leads to dp*e* 2C gc 2 p*e*p*e*c dx*. A8 The general solution of Eq. A8 is of the following expression: p*e* C3 APPENDIX B d 2C gc 2 C x*. A9 When 0, ge (2.0) can be curvefitted as ge 2.0a a 2 a 3, B where a , a , a with R Thus, Eq. 20 becomes 0 p*e* d a a 2 a 3 2C x*. B2 Integrating Eq. B2 leads to the expression of (p*e*): p*e* a a a 3 /a 2. B3 a 2 a 2 e 2C a 2 a a 3 x* a a 3 Similarly, ge (p*e*)((p*e*)2.0) can be curvefitted as ge p*e*p*e*2.0 a 4 a 5 p*e*a 6 p*e* 2 /a 7 p*e* a 8 p*e* 2, where a , a , a , a , a J. R. Melcher, Continuum Electromechanics MIT, Cambridge, MA, P. Atten J. Seyed-Yagoobi, Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry, IEEE Trans. Dielectr. Electr. Insul. 0, S. I. Jeong, J. Seyed-Yagoobi, P. Atten, Theoretical/numerical study of electrohydrodynamic pumping through conduction phenomenon, IEEE Trans. Ind. Appl. 39, S. I. Jeong J. Seyed-Yagoobi, Experimental study of electrohydrodynamic pumping through conduction phenomenon, J. Electrost. 56, P. Debye, Reaction rates in ionic solutions, Trans. Electrochem. Soc. 82, S. J. Kline F. A. McClintock, Describing uncertainties in single sample experiments, Mech. Eng. Am. Soc. Mech. Eng. 75, T. J. Gallagher, Simple Dielectric Liquid Mobility, Conduction, Breakdown Clarendon, Oxford, UK, 975, p I. Adamczewski, Ionization, Conductivity Breakdown in Dielectric Liquids Taylor & Francis, London, 969, 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, NIST, NIST Thermodynamic Transport Properties of Refrigerants Refrigerant Mixtures REFPROP NIST, Gaithersburg, MD, 998.