Walter Grassie-mail: w.grassi@ing.unipi.it

Daniele Testie-mail: d.testi@ing.unipi.it

LOTHAR (LOw gravity and THermal AdvancedResearch Laboratory),

Department of Energetics,“L. Poggi,” University of Pisa,

via Diotisalvi 2, 56126 Pisa, Italy

Heat Transfer Augmentation byIon Injection in an Annular DuctThe thermofluid-dynamic effects of ion injection from sharp metallic points added per-pendicularly to the inner wire of a short horizontal annulus were experimentally inves-tigated. A dielectric liquid (FC-72 by 3M) was weakly forced to flow in the duct, whichwas uniformly heated on the outer wall. A dc voltage as high as 22 kV was applied to theinner electrode, while the heated wall was grounded. Both the laminar and the turbulentmixed-convection regimes were obtained, varying the imposed flow rate. Once an electricfield is applied, the flow is dramatically modified by the jets of charged particles, whichtransfer their momentum to the neutral adjacent ones. Different injection strengths wereobtained on the emitters, because the shape of the point tips was not controlled at themicroscale. Nusselt number distributions were obtained azimuthally and longitudinally,monitoring the wall temperatures. In all cases, heat transfer turned out greatly enhancedin the proximity of the emitters, without a significant increase in pressure drop throughthe test section and with a negligible Joule heating, making this technique very attractivefor application in compact heat exchangers. �DOI: 10.1115/1.2150838�

Keywords: EHD, heat transfer enhancement, ion injection, mixed convection

BackgroundThe work presented herein is part of a broader research aimed

at the design of an electrohydrodynamic �EHD� high efficiencycold plate for space applications. Cold plates are contact-typecompact heat exchangers, developed for electronic equipment andpayload thermal control, which remove the generated heat by afluid cooling loop. Typically, a weakly forced flow �Re�103� isneeded to provide the necessary heat sink performance �1�. EHDenhanced heat transfer rates would result in a diminished pumpingpower or in a more compact structure for the cold plate.

Theoretical Aspects. For the sake of clarity, we briefly sum-marize the main theoretical aspects related to the matter dealt within the following. In the presence of an electric field, a body forcearises in a dielectric medium, given by:

fE = �EE −1

2E2 � � +

1

2� ��E2� ��

���

T �1�

We generally refer to the first RHS term as the electrophoreticcomponent, which is the Coulomb force exerted by an electricfield upon the free charge present in the medium, with density �E.The second term represents the dielectrophoretic body force,which is related to a local change of fluid permittivity in an elec-tric field. The third term, called electrostriction, acts on the dielec-tric in a nonuniform electric field. The last two forces act onpolarization charges. In order to solve an EHD problem, oneneeds to insert fE in the momentum equation among the externalforces. Besides, the complete set of Maxwell equations, togetherwith free charge continuity, is required in addition to the mass,momentum, and energy balances.

Taking the curl of the momentum equation, one obtains thevorticity ��=�Ùu� balance. This relation is particularly useful,since it clearly shows which conditions lead to a rotational, con-vective motion. In the Boussinesq approximation �2�, we havecalculated �3�:

Contributed by the Heat Transfer Division of ASME for publication in the JOUR-

NAL OF HEAT TRANSFER. Manuscript received January 21, 2005; final manuscript re-ceived August 23, 2005. Review conducted by John H. Lienhard V. Paper presentedat the 2004 ASME Heat Transfer/Fluids Engineering Summer Conference �HT-

FED2004�, July 11, 2004–July 15, 2004, Charlotte, North Carolina, USA.

Journal of Heat Transfer Copyright © 20

d�

dt= � · �u + ��2� + ��g +

��� � E2

2�� ٠�T +

��E ٠E

�

�2�

where

d

dt=

�

�t+ u · �

is the Lagrange derivative, calculated on the path of the fluidparticle, and

�� = −1

�� ��

�T�

p

is the temperature coefficient of electrical permittivity.The electrostrictive component in Eq. �1� cannot produce a vor-

tical flow, while dielectrophoresis needs both a temperature gra-dient and a non-uniform electric field. A free charge gradient in-stead is necessary for the Coulomb force to generate vorticity.Although free charge can be created in the fluid by temperaturegradients, which induce electrical conductivity gradients, in thepresence of a sharp electrode and at dc electric fields as high as107 V/m, the main mechanism responsible for free charge build-ing up in a liquid is ion injection at the electrode/liquid interfacevia electrochemical reactions �4,5�.

The injected charges are homocharges �6�, i.e., of the samepolarity as the injecting electrode, to which we generally refer asthe emitter. This phenomenon most often occurs at the sharperelectrode and is mainly controlled by the high field electrochem-istry of the interface, which critically depends on the chemicalnature of the dielectric fluid and on the shape �radius of curva-ture�, composition, and polarity of the electrode.

The injected space charge is driven by the Coulomb force to-wards the facing electrode, while dragging the fluid neutral mol-ecules and giving rise to a jetlike motion �7�. When this jet im-pinges on a heat transfer surface, heat transfer coefficients arehugely enhanced �8�. It has been shown that the thermofluid-dynamic characteristics of the EHD-induced liquid flow are analo-gous to those of submerged impinging jets, with the injectingelectrode considered as the nozzle �9�.

An order of magnitude of the velocities induced in the bulk of

MARCH 2006, Vol. 128 / 28306 by ASME

the fluid can be obtained assuming that the electrostatic energypresent in the proximity of the sharp electrode is converted intokinetic energy of the accelerated particles:

1

2�u2 �

1

2�E2 Þ

u

E��

��3�

Higher heat transfer enhancements by ion injection can be ob-tained with fluids of low viscosity and low electrical conductivity�10�. In fact, an increased viscosity of the fluid delays the onset ofthe electrohydrodynamic motion and the concurrent strong chargeinjection. Besides, with a high electrical conductivity, the chargesin the liquid cross the space between the electrodes before thor-oughly transferring their momentum to the fluid. This lack of mo-mentum transfer decreases the induced velocity and thus the heattransfer augmentation.

Brief Review of Past Results. Notable results from past studieson single-phase forced convection heat transfer enhancement bythe ion-drag mechanism will be shortly listed hereafter, in chro-nological order. Most of them are more deeply reviewed in �11�.

In 1970, Porter and Poulter �12� studied EHD laminar convec-tion of transformer oil in an annulus, enhancing heat transfer of1.9 times by the ion-injection process. A corresponding pressuredrop increase was also reported.

In 1976, Mizushina et al. �13� obtained an increase of both heattransfer coefficients �up to 2.5 times� and friction factors, with airflowing in an annulus in the laminar regime.

In 1979, Velkoff and Godfrey �14� analyzed a low-velocity air-flow along a heated flat plate, obtaining a heat transfer enhance-ment of 3.5 times, together with a friction factor augmentation.The injecting electrodes were an array of fine wires, positioned inparallel with the main flow.

In 1987, Fernández and Poulter �15� tested a high-Prandtl num-ber dielectric oil �Shell Diala “B”� flowing in an annular duct inthe laminar regime. The result was a heat transfer enhancement upto 2000%, with a threefold pressure-drop augmentation and aJoule dissipation lower than 1 W.

In 1989, Fujino et al. �16� studied the effect of the electric fieldin a laminar flow of R113 within two parallel plates, reporting asevenfold heat transfer enhancement as the maximum valueachieved, when a trace amount of an electrolyte solution wasadded to the working fluid. The sum of pumping power and elec-trical power consumed in the test section was less than 1% of theimposed heat flow.

In 1991, Ohadi et al. �17� examined the turbulent flow of air ina circular duct, with two high-voltage thin wire electrodes placedlongitudinally in the proximity of the pipe axis, obtaining a heattransfer augmentation of 2.6.

In 1997, Hasegawa et al. �18� performed a numerical analysisof EHD-enhanced forced convection in a rectangular tube with alarge aspect ratio, at Re=1470 and with “Freonsolve” �in weight,96% of R113 and 4% of ethyl alcohol� as the test fluid. Thenumerical results showed that the heat transfer enhancement, pro-duced by four longitudinal high-voltage wire electrodes placedwithin the channel, was four times larger than the pressure dropincrease.

In 2000, Molki et al. �19� documented a numerical and experi-mental investigation of airflow in a square duct, with the high-voltage electrode situated at the centerline along the channel axis.The heat transfer rate at most increased by a factor 2 in the entry-region and by a factor 3.4 for the fully-developed flow. Owing tothe EHD-induced motion, turbulence was taken into account evenat Re=500.

In 2005, Grassi et al. �20� examined the flow of a fluorocarbon�FC-72� in a vertical annulus, under the regime of turbulent aidedmixed convection. Heat transfer was augmented up to five times,with a negligible power input. Ions were injected by point elec-

trodes fixed perpendicularly to the inner wire, along its length.

284 / Vol. 128, MARCH 2006

Experimental ApparatusIn this work we examine the effect of a dc electric field on

developing mixed convection in a heated horizontal annulus withsharp points added to the inner electrode. In the absence of theelectric field, according to the map proposed by Metais and Eckert�21�, the transition from the laminar to the turbulent regime occursin the Reynolds number range 700–1000. It is reasonable to ex-pect the transition to be anticipated even more in the EHD case,having added a cause of instability �22�.

The Working Fluid. The dielectric fluid chosen for the experi-mental campaign is the Fluorinert™ Electronic Liquid FC-72�manufactured by 3 M, St. Paul, MN�. FC-72 is thermally andchemically stable, compatible with sensitive materials, nonflam-mable, nontoxic, colorless, and has no ozone depletion potential.This valuable combination of properties makes FC-72 suitableeven for the strict safety specifications needed in space applica-tions. The physical properties of the fluid at 25°C are reported inTable 1.

Polar molecules either have a high intrinsic conductivity due toself-dissociation or are very sensitive to impurities, being goodsolvents. In fact, for a small increase of relative electrical permit-tivity around the value of 2, the resistivity can drop of manyorders of magnitude �23� and Joule heating losses in the dielectriccould become relevant at high applied voltages. For this reasonwe chose a nonpolar fluid as FC-72, with extremely low electricalconductivity.

Furthermore, the low viscosity of FC-72 is very attractive foran EHD heat exchanger, not only because of the already men-tioned high efficiency of the ion-injection process, but also from apurely hydraulic point of view. In fact, in a pipe of length L andcross section area A, the pumping power needed to compensatefor pressure drop can be expressed as:

�P = V�p =1

2�Au3f

L

Dh�4�

At a given Reynolds number, Eq. �4� can be replaced by:

�P =1

2�A

Re3 �3

Dh3 f

L

Dh= ���3 �5�

being � dependent only on geometrical parameters and on theReynolds number. Thus, it must be taken into account that a re-duction in viscosity can greatly diminish the pumping powerneeded for the fluid loop of the heat exchanger.

Description of the Apparatus. A test loop was built as shownin the schematic of Fig. 1. The fluid, moved by a gear pump,circulated through the test specimen at volumetric flow rates in therange 0.10–2.25 l /min. A flowmeter measured the flow rate withan accuracy of ±0.0086 l /min. The fluid, uniformly heated in itsflow along the test section, was cooled to the desired inlet tem-perature by means of a shell-tube heat exchanger, with counter-current chilled water flowing in the shell. The annular test speci-

Table 1 FC-72 physical properties at 25°C „by courtesy of3 M…

chemical formula C6F14electrical resistivity 1013 � mdielectric strength 1.5107 V m−1

relative permittivity 1.75boiling point �at 1 bar� 56°C

density 1.68103 kg m−3

kinematic viscosity 3.810−7 m2 s−1

specific heat 1.1103 J kg−1 K−1

thermal conductivity 5.710−2 W m−1 K−1

coefficient of expansion 1.5610−3 K−1

men was connected with a 0–30 kV dc reversible polarity high

Transactions of the ASME

voltage power supply. An expansion vessel was used to compen-sate the volume increase in the loop due to temperature variationsand to set the pressure of the position it was mounted on. Anabsolute pressure transducer was placed right upstream of the gearpump, in order to check that the lowest pressure of the loop wasabove the room pressure, thus avoiding air from being sucked inby the pump. All the tests were performed at an absolute pressureof about 2 bars.

A detailed schematic of the test section is shown in Fig. 2. Theinner wire, of diameter d=1.6 mm, was made of stainless steel.The outer electrode, also made of stainless steel, had an innerdiameter D=17 mm and a wall thickness s=2 mm. The fluid waswarmed up by an electrical resistance heater applied to the outerwall for a length Lh=50 cm. 20 thermocouples were placed on theouter wall, under the heater, at five cross sections, 12.5 cm apartalong the heated length. Each section had four thermocouples, oneevery 90 deg. To minimize heat losses, the entire specimen wascovered with a 34 mm thick thermal insulator �=0.038 W m−1 K−1�. Two more thermocouples were placed at thefluid inlet and outlet, to evaluate the heat flow supplied to the testsection in steady state through the energy balance:

�h = mcp�Tout − Tin� �6�

having assumed that the electrically generated heat within thefluid is negligible, since the currents measured in the experimentalcampaign were in the order of 10−7 A.

All the thermocouples were type-T and used a zero-point refer-ence cell, whose temperature was controlled by a reference resis-tance thermometer. The overall accuracy of each thermocoupleafter calibration in the 15–60°C range was ±0.2 K.

A differential pressure transducer measured the pressure dropthrough the test section. At the application of the electric field, wecould not appreciate order of magnitude variations in the pressuredrop, with measured values below 10 Pa.

Metallic points were welded perpendicularly to the inner elec-trode by means of a silver alloy. The points were simply tin-coated copper wires, of diameter 0.6 mm and length 4 mm, cut atthe tip. The shapes and the radii of curvature of the point tips werenot controlled at the microscale and may have differed from eachother significantly.

In a first series of tests, to which we shall refer as configuration1, two opposite points were fixed in the horizontal direction andmoved from 11 cm upstream to 6 cm downstream of the midwaysection, which we will call Sec. C. Further tests �configuration 2�

Fig. 1 Schematic of the test loop

were conducted with two opposite points positioned in the vertical

Journal of Heat Transfer

direction at one quarter of the heated length �Sec. B� and fourpoints, one every 90 deg, at three quarters of Lh �Sec. D�, asillustrated in Fig. 3.

Test Procedure and Data Analysis. In a preliminary operation,we tested the dielectric strength of the medium. Slowly increasingthe applied voltage, we observed that electrical breakdown oc-curred at 24 kV. Therefore, we decided to safely operate at HV=22 kV and chose to use the positive polarity, which gave stableionic currents.

Every test was run under constant volumetric flow rate and heatflow and a steady state condition in the presence of the electricfield was awaited. The thermocouples’ signals were acquired ev-ery 6 s by a scanner mounted on a digital multimeter �Model2000, manufactured by Keithley, Cleveland, OH�; then the datawere sent to a PC in order to be recorded and processed by thesoftware Labview™ �a trademark of National Instruments, Austin,TX�, using Matlab® �a registered product of The MathWorks,

Fig. 2 Detail of the EHD test section

Fig. 3 Position of the points in configuration 2

MARCH 2006, Vol. 128 / 285

Natick, MA� as the interface for mathematical calculations.The bulk temperature of the fluid at a given axial position was

calculated considering a temperature of adiabatic mixing for theexamined cross section, assuming a linear trend, from the inlet tothe outlet temperature, on the heated length. The temperatures ofthe inner side of the wall were obtained from the ones measuredon the outer side, properly accounting for one-dimensional heatconduction within the wall thickness:

Tw�in = Tw�out −�h

2��Lhln�D + 2s

D� �7�

The dimensionless numbers describing mixed convection inuniform heat flux conditions �i.e., Nu, Re, and Grh� were calcu-lated as:

�Nu =

Dh

f=

qDh

f�Tw�in − Tb�=

�h

f�Tw�in − Tb��Lh

Re =u�inDh

�in=

4V

��D + d��in

Grh = Gr Nu =� fgqDh

4

f� f2 =

� fg�hDh3

f� f2�Lh

� �8�

The relative errors on the derived parameters, erF, were evalu-ated by the propagation law:

erF = i� xi

F�xi�·�F�xi�

�xi· erxi

2

�9�

where F is the parameter depending on the variables xi, which aremeasured with a known relative error, erxi

, calculated by means ofthe above-mentioned accuracies.

The maximum relative error on the Nusselt number, corre-sponding to the highest heat transfer rate and the lowest heat flow,was 10.5%, while the maximum error on the Reynolds number,obtained at the lowest flow rate, was 8.6%. As for Grh, the highesterror was 16.8%, obtained at the lowest temperature increasethrough the test specimen, which means at the highest flow rateand the lowest heat flow.

In any case, the fluid temperatures in correspondence of theinner side of the heated wall were fairly below the boiling point,never exceeding 50°C.

Discussion of Heat Transfer ResultsThe Nusselt number values reported hereafter represent the av-

erage of 20 measurements, recorded consecutively once a steadystate condition at HV=22 kV was reached.

In configuration 1, the Nusselt number was monitored by thethermocouples positioned in Sec. C and the two opposite, horizon-tal points were moved from 11 cm upstream to 6 cm downstream.The same results could reasonably be obtained by keeping thepoints fixed in Sec. C and moving the thermocouples along thewall in the axial direction. Thus, a surface distribution of theNusselt number was obtained, interpreting the coordinate � as thethermocouple’s position on the heated wall with respect to thepoint, made dimensionless by means of the hydraulic diameter.

From the distribution of Fig. 4, obtained at Re=2220 and Grh=1.40108, we deduce that the point directed towards the leftside of the pipe works much better than the one on the right-handside. This is not surprising, because the strength of the ion injec-tion depends critically on the curvature of the point tip, which wasnot controlled at the microscale. A schematic drawing of the as-perities generated by the cut at the point tip, as observed under themicroscope, is shown in Fig. 5.

As a confirmation that the higher Nusselt numbers on the leftside are due to a sharper point, we rotated the inner electrode of180 deg and we observed symmetrically high Nu on the opposite

side.

286 / Vol. 128, MARCH 2006

Figure 4 also shows that heat transfer on the bottom of theheated surface is higher than on the top, which is coherent withthe fact that the bottom is helped by buoyancy forces in removinghot fluid from its surface. The heat transfer enhancement effectfades while going away from Sec. C and almost vanishes at about5 hydraulic diameters downstream and 2 Dh upstream of theemitters.

In Fig. 4, as well as in the following figures reporting the ex-perimental results as a function of the longitudinal coordinate, thevalues obtained in the absence of the electric field can be read at�=7. In fact, at this downstream dimensionless distance from thepoint, ion injection is no more effective in the examined Reynoldsnumber range.

The maximum values of the Nusselt number move downstream,from �=0 on the left-hand side to ��2 on the right-hand side. Apossible fluid-dynamical explanation could be the following �seeFig. 6�: the strong jet directed on the left side turns around thecircular wall while being moved downstream by the main forcedflow, reaching the farther sides with weakened intensity �the dy-namics of a jet under the effect of a cross-flow is analyzed, forinstance, in �24��.

Coherently with the above interpretation, as Fig. 7 shows, at alower Reynolds number �Re=740�, the maxima lie closer to Sec.C, from �=0 on the left-hand side to less than �=1 on the oppositeone.

Fig. 4 Nu distribution at Re=2220 and Grh=1.40Ã108

Fig. 5 Drawing of the microasperities at the point tip

Transactions of the ASME

Again, heat transfer is higher on the bottom than on the top.The enhancement effect, with a weaker main flow, almost van-ishes more upstream ���−3� and less downstream ���4� thanbefore.

The Nusselt number distribution appears slightly influenced bya change in the wall heat flux. Figure 8 shows negligible Nuvariations passing from Grh=1.40108 to Grh=2.47108.

The Nusselt number was azimuthally averaged and its distribu-tion was plotted in Fig. 9 at three different Re. In this range,higher Reynolds numbers entail higher heat transfer rates.

Fig. 6 Presumed dynamics of the jets from the points in con-figuration 1

Fig. 7 Nu distribution at Re=740 and Grh=1.40Ã108

Fig. 8 Nu vs � at Re=2220 and Grh=1.40Ã108 and Grh=2.478

Ã10 „dotted curves…

Journal of Heat Transfer

This trend seems in contradiction with the Nusselt number dis-tribution illustrated in Fig. 10, obtained in a previous experimentalcampaign with a heated vertical annulus �3�. In fact, Fig. 10 showsthat the area under the Nusselt curve has almost the same valueindependently of Re. In other words, the longitudinal average ofthe Nusselt number does not vary much with the main flow ve-locity. This latter parameter only affects the wall uniformity of�Nu�, which is larger at the highest flow rates.

However, the results obtained at a certain Reynolds number in avertical configuration with aiding flow are not comparable withthe horizontal ones at the same Re, as buoyancy forces play dif-ferent roles. Even the transition from laminar to turbulent flow isanticipated in the vertical case: in the absence of the electric field,the flow is already turbulent at Re=400 �21�.

Besides, in aiding flow, gravity helps taking hot fluid awayfrom the heated surface. Instead, in a horizontal arrangement, aminimum main flow velocity is needed to move downstream thehot fluid, especially on the top, and reach high Nusselt numbers.

The large �Nu� increase in Fig. 9 from Re=295 to Re=740could also be due to the transition from laminar to turbulent EHDflow.

Over a certain Re, the �Nu� in correspondence of the crosssection of the emitters is supposed to decrease, as observed in the

8

Fig. 10 ŠNu‹ vs � in vertical aiding flow at Grh=3.89Ã108†3‡

Fig. 9 ŠNu‹ vs � at Grh=1.40Ã10

MARCH 2006, Vol. 128 / 287

vertical upward flow. In fact, the effect of the induced jets shouldspread longitudinally, being carried downstream by the mainforced flow.

Figure 11 actually shows this trend in configuration 2 at Secs. Band D �the points are mounted as in Fig. 3�, in the Reynoldsnumber range 295–5170. The points of configuration 2 were moreaccurately cut at their tips, at a repeatable angle of 45 deg. In thisway, the Nusselt numbers turned out more uniformly distributedalong the azimuthal coordinate.

In the absence of the electric field, �Nu� increases with Re. Asfor the dependence on the longitudinal coordinate, two opposingeffects have to be taken into account: the steeper wall temperaturegradient in the entry-zone, typical of pure forced convection, andthe development of the buoyancy-driven secondary flow. Theoverall result is higher heat transfer coefficients in Sec. D for thelower Reynolds numbers and, inversely, higher �Nu� in Sec. B forthe higher Re, owing to the diminished weight of the buoyantforces with respect to the inertial ones. Similar trends for mixedconvection in a horizontal duct are documented in �25�.

At the application of the electric field, the higher Nusselt num-bers were obtained in Sec. D, due to the presence of four injectingpoints. In particular, heat transfer was enhanced by three times,with �Nu�=59 at Re=295. In order to obtain this value in pureforced convection, without EHD effects, one needs a turbulentflow at Reynolds numbers around 5000, according to the Dittus-Bölter correlation �26�.

ConclusionsElectrohydrodynamic heat transfer augmentation by ion injec-

tion may yield large reductions in pumping power of compact heatexchangers, showing remarkable heat transfer rates even at verylow Reynolds numbers, without significant penalties associatedwith the pressure drop �in any case, less than 10 Pa� and with anegligible Joule heating. Then, the additional power input intro-duced by the technique can be disregarded, being in the order of10−3 W. The phenomenon is localized in the proximity of theemitters, decaying more rapidly upstream than downstream of thepoints. Different injection strengths were observed from pointsapparently of the same dimensions. A more accurate control of theshape of the point tip is therefore required, in order to guaranteestrong injection from each emitter.

As buoyancy forces play a significant role in differentiating theresults of the horizontal configuration from the ones obtained in

Fig. 11 ŠNu‹ vs Re „Secs. B and D… at Grh=1.40Ã108

upward aided flow �3�, experimentation in microgravity condi-

288 / Vol. 128, MARCH 2006

tions will be helpful to assess the heat transfer behavior in thespace environment. This is needed for designing a compact coldplate for space applications with the optimum longitudinal andazimuthal position of the emitters. As far as the presented resultsare concerned, to place the points at intervals of about one half ofthe hydraulic diameter in the axial direction, with four points percross section, seems a good compromise �see Figs. 9 and 11�.

AcknowledgmentThis work was performed within the ESA Microgravity Appli-

cation Program, Project No. AO-099-045. Funding by the ItalianSpace Agency, ASI �Grant No. I/R/061/01�, is gratefully acknowl-edged. The authors would also like to thank Dr. Mario Saputellifor his valuable contribution to the experimental part of this workand Mr. Roberto Manetti for the essential collaboration in settingup the electronic devices.

NomenclatureA � pipe cross-section area �m2�cp � specific heat �J kg−1 K−1�D � inner diameter of the heated wall �m�

Dh � =D−d, hydraulic diameter �m�d � diameter of the inner wire �m�E � electric field �V m−1�

erF � relative error on the derived parametererx � relative error on the measured parameter

f � Darcy-Weisbach friction factorfE � electric body force �N m−3�Gr � =� fg�Tw�in−Tb�Dh

3 /� f2, Grashof number

Grh � Grashof number based on the wall heat fluxHV � applied high voltage �V�

L � pipe length �m�Lh � heated length �m�m � =�V, mass flow rate �kg s−1�

Nu � local Nusselt number�Nu� � azimuthal average of the Nusselt number

p � pressure �Pa�q � heat flux �W m−2�

Re � Reynolds numbers � heated wall thickness �m�T � temperature �K�t � time �s�

u � fluid velocity �m s−1�V � volumetric flow rate �m3 s−1�

Greek � convective heat transfer coefficient �W m−2 K−1�� � coefficient of cubic expansion �K−1�

�� � temperature coefficient of permittivity �K−1�� � =A Re3 Lf /2Dh

4, parameter defined in Eq. �11� �m−1�� � electrical permittivity �F m−1�� � dynamic viscosity �Pa s�� � heated wall thermal conductivity �W m−1 K−1� � fluid thermal conductivity �W m−1 K−1�� � =� /�, kinematic viscosity �m2 s−1�

�h � heat flow �W��P � pumping power �W�

� � mass density �kg m−3��E � free charge density �C m−3�� � vorticity �s−1�� � dimensionless longitudinal coordinate

Subscriptsb � bulkf � calculated at film temperature

in � inlet

Transactions of the ASME

out � outletw � in � inner side of the heated wallw �out� outer side of the heated wall

References�1� Miedza, B., and Behrens, B., 1997, “Cold Plate Development for Space Station

Application,” ESA SP-400, 6th European Symposium on Space EnvironmentalControl Systems, Noordwijk, The Netherlands 1, pp. 299–304.

�2� Tritton, D. J., 1988, Physical Fluid Dynamics, Oxford University Press, Ox-ford, UK.

�3� Grassi, W., Testi, D., and Saputelli, M., 2005, “EHD Enhanced Heat Transferin a Vertical Annulus,” Int. Commun. Heat Mass Transfer, 32�6�, pp. 748–757.

�4� Felici, N., 1971, “dc Conduction in Liquid Dielectrics: A Survey of RecentProgress �Part I�,” Direct Current, 2�3�, pp. 90–99.

�5� Richter, A., Plettner, A., Hotmann, K. A., and Sandmaier, H., 1991, “A Mi-cromachined Electrohydrodynamic �EHD� Pump,” Sens. Actuators, A, 29�2�,pp. 159–168.

�6� Atten, P., 1996, “Electrohydrodynamic Instability and Motion Induced by In-jected Space Charge in Insulating Liquids,” IEEE Trans. Dielectr. Electr. In-sul., 3�1�, pp. 1–17.

�7� Atten, P., Malraison, B., and Zahn, M., 1997, “Electrohydrodynamic Plumes inPoint-Plane Geometry,” IEEE Trans. Dielectr. Electr. Insul., 4�6�, pp. 710–718.

�8� Grassi, W., Testi, D., Della Vista, D., and Torelli, G., 2005, “Augmentation ofHeat Transfer on the Downward Surface of a Heated Plate by Ion Injection,”ECI-ITP-05-03, in Proceedings of the Interdisciplinary Transport Phenomenain Microgravity and Space Sciences IV, Tomar, Portugal.

�9� Grassi, W., Testi, D., and Della Vista, D., 2005, “Heat Transfer Enhancementon the Upper Surface of a Horizontal Heated Plate in a Pool by Ion Injectionfrom a Metallic Point,” Journal of Electrostatics �in press�.

�10� Paschkewitz, J. S., and Pratt, D. M., 2000, “The Influence of Fluid Propertieson Electrohydrodynamic Heat Transfer Enhancement in Liquids under Viscousand Electrically Dominated Flow Conditions,” Exp. Therm. Fluid Sci., 21, pp.187–197.

�11� Ohadi, M. M., Darabi, J., and Roget, B., 2001, “Electrode Design, Fabrication,and Materials Science for EHD-Enhanced Heat and Mass Transport,” AnnualReview of Heat Transfer, C. L. Tien, ed., Begell House, New York, Vol. 11,Chap. 9, pp. 563–632.

�12� Porter, J. E., and Poulter, R., 1970, “Electrothermal Convection Effects With

Journal of Heat Transfer

Laminar Flow Heat Transfer in an Annulus,” in Proceedings of the 4th Inter-national Heat Transfer Conference, Paris, France, Vol. 2.

�13� Mizushina, T., Ueda, H., Matsumoto, T., and Waga, K., 1976, “Effect of Elec-trically Induced Convection on Heat Transfer of Air Flow in an Annulus,” J.Chem. Eng. Jpn., 9�2�, pp. 97–102.

�14� Velkoff, H. R., and Godfrey, R., 1979, “Low-Velocity Heat Transfer to a FlatPlate in the Presence of a Corona Discharge in Air,” ASME J. Heat Transfer,101, pp. 157–163.

�15� Fernández, J. L., and Poulter, R., 1987, “Radial Mass Flow inElectrohydrodynamically-Enhanced Forced Heat Transfer in Tubes,” Int. J.Heat Mass Transfer, 30�10�, pp. 2125–2136.

�16� Fujino, T., Yokohama, Y., and Mori, H. R., 1989, “Augmentation of LaminarForced-Convective Heat Transfer by the Application of a Transverse ElectricField,” ASME J. Heat Transfer, 111, pp. 345–351.

�17� Ohadi, M. M., Nelson, D. A., and Zia, S., 1991, “Heat Transfer Enhancementof Laminar and Turbulent Pipe Flow via Corona Discharge,” Int. J. Heat MassTransfer, 34�4–5�, pp. 1175–1187.

�18� Hasegawa, M., Yabe, A., and Nariai, H., 1997, “Numerical Analysis ofElectro-Hydrodynamical Enhancement Mechanism of Forced Convection HeatTransfer in Duct Flow,” ASME Proceedings of the 32nd National Heat Trans-fer Conference, Vol. 10, pp. 11–15.

�19� Molki, M., Ohadi, M. M., Baumgarten, B., Hasegawa, M., and Yabe, A., 2000,“Heat Transfer Characteristics of Airflow in a Channel Using Corona Dis-charge,” J. Enhanced Heat Transfer, 7, pp. 411–425.

�20� Grassi, W., Testi, D., and Saputelli, M., 2005, “Heat Transfer Enhancement ina Vertical Annulus by Electrophoretic Forces Acting on a Dielectric Liquid,”Int. J. Therm. Sci., 44�11�, pp. 1072–1077.

�21� Metais, B., and Eckert, E. R. G., 1964, “Forced, Mixed, and Free ConvectionRegimes,” ASME J. Heat Transfer, 86, pp. 295–296.

�22� Metais, B., 1963, “Criteria for Mixed Convection,” Technical Report No. 51,Heat Transfer Laboratory, University of Minnesota, Minneapolis, MN.

�23� Felici, N., Gosse, B., and Gosse, J.-P., 1976, “La Conduction dans les LiquidesDiélectriques—Idées Modernes et Progrès Récents—Aspects Électro-chimiques et Électrohydrodynamiques,” Rev. Gen. Electr., 85�11�, pp. 861–874 �in French�.

�24� Abramovich, G. N., 1963, The Theory of Turbulent Jets, MIT Press, Cam-bridge, MA.

�25� Petukhov, B. S., and Polyakov, A. F., 1988, Heat Transfer in Turbulent MixedConvection, Hemisphere Publishing, New York.

�26� Kakaç, S., Shah, R. K., and Aung, W., 1987, Handbook of Single-Phase Con-vective Heat Transfer, Wiley, New York.

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