VIBRATION INDUCED DROPLET GENERATION FROM A LIQUID LAYER FOR EVAPORATIVE COOLING IN A HEAT TRANSFER CELL. A Thesis Presented to The Academic Faculty

Transcription

1 VIBRATION INDUCED DROPLET GENERATION FROM A LIQUID LAYER FOR EVAPORATIVE COOLING IN A HEAT TRANSFER CELL A Thesis Pesented to The Academic Faculty By Fank Pytle III. In Patial Fulfillment Of the Requiements fo the Degee Docto of Philosophy in Mechanical Engineeing Geogia Institute of Technology Decembe 2005 Copyight Fank Pytle III i

2 VIBRATION INDUCED DROPLET GENERATION FROM A LIQUID LAYER FOR EVAPORATIVE COOLING IN A HEAT TRANSFER CELL Appoved by: D. William Z. Black Co-adviso School of Mechanical Engineeing Geogia Institute of Technology D. James G. Hatley Co-adviso School of Mechanical Engineeing Geogia Institute of Technology D. Kenneth A. Cunefae School of Mechanical Engineeing Geogia Institute of Technology D. William D. Hunt School of Electical and Compute Engineeing Geogia Institute of Technology D. Lauence J. Jacobs School of Civil and Envionmental Engineeing Geogia Institute of Technology Date Appoved: August ii

3 To my wife Ashanti Thank You. and In dedication to the memoy of my loving mothe Lona J. Pytle ( ). ii

4 ACKNOWLEDGEMENTS Fist I would like to give hono to God fo He is the head of my life. In Him I live I move and I have my being. Without Him none of this would be possible. This dissetation is epesentative of not only had wok but is also the esult of the encouagement and suppot of many people. I would like to thank my co-advisos D. William Z. Black and D. James G. Hatley fo helping me navigate this couse. Thank you D. Black fo all the time and suppot you gave me even afte etiement. You commitment to my academic development duing my gaduate expeience is geatly appeciated. D. Hatley thank you fo shaing you insight and wisdom that led to geat ideas. I also thank D. Hunt fo stepping in and offeing suppot when I was unning out of options. D. Jacobs thank you fo you time and willingness to help when I asked. Finally D. Cunefae I thank you fo you acute attention to the details of this wok. I am also thankful fo my lab mates fellow gaduate students and othes who have helped me immensely: D. Sam Heffington Sang Hun Lee Desmond Stubbs Anthony Dickhebe Pasto James and Wanda Tune membes of the Black Gaduate Student Association Jackie Wothy Cosetta Williams Noma Fank Tudy Allen Glenda Johnson Teddy and Pam Taylo and Paul Smith. To Willie Cayton thank you fo encouaging me to choose to attend college and to boaden my hoizons instead of pusuing my bugeoning caee as a daiyman. I thank each of you fo all you help getting me fom the beginning to the end of this leg of my jouney. I would like to ecognize the following entities fo poviding funding: the National Consotium fo Gaduate Degees fo Minoities in Engineeing and Science iii

5 Inc. the Geogia Institute of Technology Cente fo the Enhancement of Teaching and Leaning and Sigma Xi The Scientific Reseach Society. Of all the peson I am most thankful fo is my wife D. Ashanti J. Pytle. Without he love patience and suppot I cetainly would not have finished. Thank you fo you faithfulness commitment and willingness to postpone you caee while I attended Geogia Tech. You ae cetainly my cown jewel. Finally I thank my natual and spiitual family. God placed you in my life to assist in the fulfillment of His odained pupose and destiny fo me. Thank you all fo being His instuments to help eveal to me my pupose. iv

6 TABLE OF CONTENTS Acknowledgements List of Tables List of Figues Nomenclatue Summay iii vii viii xi xiv Chapte 1 Intoduction Backgound Reseach Objectives 7 Chapte 2 Liteatue Review Spay Cooling Doplet Cooling Vibation Induced Doplet Geneation 16 Chapte 3 Methodology Dive-Doplet Geneation Coelation Dive Chaacteization Appaatus and Pocedues Results Spay Chaacteization Appaatus and Pocedues Results Doplet Geneation Analysis Fist Appoach 69 v

7 3.4.2 Second Appoach 73 Chapte 4 Heat Tansfe Cell Model Fomulation Results 94 Chapte 5 Conclusions 103 Chapte 6 Futue Wok and Recommendations 107 Appendix 109 Refeences 111 Vita 116 vi

8 LIST OF TABLES Table 3.1: Dive test conditions fo measuements made with lase vibomete 39 Table 3.2: Dive voltages (V pp2 ) fo constant V pp1 duing dive chaacteization 41 Table 3.3: Measued aveage doplet diametes 60 Table 4.1: Actual heat tansfe cell dimensions and attibutes 95 vii

9 LIST OF FIGURES Figue 1.1: Mooe's Law pedictions of micopocesso tansisto doubling (Wijetunga 2002) 2 Figue 1.2: Micopocesso powe equiements as a function of yea (Wijetunga 2002) 3 Figue 1.3: Convective and boiling heat tansfe coefficients (Mudawa 2001) 4 Figue 1.4: Simple heat pipe 5 Figue 1.5: Schematic of a doplet cooling heat tansfe cell (Heffington 2000) 7 Figue 3.1: Schematic of doplet geneation test setup 24 Figue 3.2: Dimensionless acceleation as a function of dimensionless angula fequency (Goodidge et al. 1997) 27 Figue 3.3: Schematic of liquid body 30 Figue 3.4: Schematic of piezoelectic diaphagm and components 34 Figue 3.5: Schematic of piezoelectic diaphagm deflection 35 Figue 3.6: Schematic of powe/monitoing cicuit of dive 37 Figue 3.7: Maximum velocity pe volt as a function of fequency 42 Figue 3.8: Maximum displacement pe volt as a function of fequency 43 Figue 3.9: Maximum acceleation pe volt as a function of fequency 43 Figue 3.10: Ovehead and side views of contou plot of dive velocity at 400 Hz and V pp1 = 15 V 45 Figue 3.11: Ovehead and side views of contou plot of dive velocity at 400 Hz and V pp1 = 40 V 45 Figue 3.12: Ovehead and side views of contou plot of dive velocity at 500 Hz and V pp1 = 40 V 45 Figue 3.13: Photogaph of dive secued in enclosue 46 Figue 3.14: Photogaph of dive in enclosue in liquid pool 48 viii

10 Figue 3.15: Schematic of ovehead view of photogaphic setup fo depth of field A- nea distance B-focus distance C-fa distance 50 Figue 3.16: Photogaph of spay geneated at 500 Hz fom 2.4 mm liquid laye 50 Figue 3.17: Photogaph of vetical stage and open-cell sponge 52 Figue 3.18: Doplet geneation fequency ange as a function of liquid laye thickness in an open pool 54 Figue 3.19: Doplet geneation fequency ange as a function of liquid laye thickness in an open pool with indicated maximum geneation fequencies 54 Figue 3.20: Ratio of maximum dive displacement to citical amplitude fo wave fomation as function of fequency in wate 56 Figue 3.21: Doplet geneation fequency ange as a function of liquid laye thickness in a 38.1 mm constained pool 57 Figue 3.22: Aveage doplet diamete (D b ) as a function of fequency in wate fom Table 3.3 and Equation Figue 3.23: Doplet mass flow ate as a function of dive fequency fo constant dive voltage and liquid laye thickness t in an open pool 63 Figue 3.24: Doplet mass flow ate as a function of dive fequency fo constant dive voltage and liquid laye thickness t in a 38.1 mm constained pool 64 Figue 3.25: Doplet mass flow ate as a function of dive fequency fo constant dive voltage and liquid laye thickness t in a 50.8 mm constained pool 65 Figue 3.26: Doplet mass flow ate as a function of dive fequency fo constant inteception distance and liquid laye thickness t in an open pool 66 Figue 3.27: Doplet mass flow ate as a function of dive fequency fo constant inteception distance and liquid laye thickness t in a 38.1 mm constained pool 67 Figue 3.28: Acceleation necessay fo doplet geneation as a function of fequency in an open pool 69 Figue 3.29: Acceleation necessay fo doplet geneation as a function of liquid laye thickness in an open pool 70 Figue 3.30: Acceleation necessay fo doplet geneation as a function of fequency fo Equation Figue 3.31: Schematic of vetically oscillating vibation of liquid laye 75 ix

11 Figue 3.32: Dimensionless acceleation (a*) as a function of fequency in an open pool 77 Figue 3.33: Bounds of dimensionless acceleation (a**) as a function of fequency in an open pool 79 Figue 3.34: Dimensionless acceleation (a**) as a function of fequency at intemediate fequencies in an open pool 79 Figue 3.35: Bounds of dimensionless acceleation (a**) as a function of fequency in a 38.1 mm constained pool 81 Figue 4.1: Schematic of modeled heat tansfe cell 84 Figue 4.2: Heat souce oientation to uppe wall of heat tansfe cell 85 Figue 4.3: Themal esistance diagam fo heat tansfe cell model 93 Figue 4.4: Photogaph of actual heat tansfe cell (Heffington) 95 Figue 4.5: Actual and pedicted (mist cooling) heat tansfe esults as a function of heat souce tempeatue 97 Figue 4.6: Actual and pedicted (ideal evapoation) heat tansfe esults as a function of heat souce tempeatue 97 Figue 4.7: Cell heat tansfe ate as a function of dive fequency fo diffeent heat souce tempeatues and liquid laye thicknesses 99 Figue 4.8: Cell heat tansfe ate as a function of cell inne height 100 Figue 4.9: Cell heat tansfe ate as a function of dive voltage 101 Figue 4.10: Cell heat tansfe ate as a function of heat souce tempeatue fo aluminum and coppe cells 102 x

12 a Acceleation (m/s 2 ) NOMENCLATURE A Dimensionless amplitude a* Dimensionless acceleation [a t/(f d) 2 ] a** Dimensionless acceleation [a t/(f d u w ) 2 ] A H Aea of heat souce (m 2 ) A s Aea of sink (m 2 ) a t Theshold acceleation (m/s 2 ) Bo c p c pl d D b d c f F Bond numbe Specific heat at constant pessue (J/(kg C)) Specific heat at constant pessue of liquid (J/(kg C)) Displacement of dive (m) Aveage doplet diamete (m) Citical wave amplitude (m) Fequency (Hz) Foude numbe g Gavitational acceleation (m/s 2 ) G h Liquid mass flux (kg/(m 2 s)) Heat tansfe coefficient (W/(m 2 C)) h fg Modified latent heat of vapoization (J/kg) h avg h fg h mist Aveage heat tansfe coefficient (W/(m 2 C)) Latent heat of vapoization (J/kg) Heat tansfe coefficient fo doplet mist cooling (W/(m 2 C)) xi

13 I 1 k k l k s L ṁ P RMS q Resisto cuent (A) Wavenumbe (1/m) Themal conductivity of liquid (W/(m C)) Themal conductivity of sink (W/(m C)) Chaacteistic length (m) Mass flow ate (kg/s) Root mean squae powe (W) Heat tansfe ate (W) q Heat flux (W/m 2 ) R R 0 R 1 R1-R11 Re R s t T T1-T6 T s t s T sat U s u w Electical esistance (ohms) Aveage themal esistance ( C/W) Electical esistance of esisto (Ω) Themal esistances in heat tansfe cell model ( C/W) Reynolds numbe Two-dimensional speading esistance ( C/W) Liquid laye thickness (m) Tempeatue ( C) Tempeatues in heat tansfe cell model( C) Tempeatue of suface ( C) Thickness of sink (m) Tempeatue of fluid at satuation ( C) Scaling velocity (m/s) Wave speed (m/s) xii

14 V pp1 V pp2 We Peak-to-peak voltage dop acoss dive and esisto (V) Peak-to-peak voltage dop acoss dive (V) Webe numbe Geek Symbols T θ λ Tempeatue diffeence ( C) Phase angle (degees) Wavelength of suface waves (m) µ Dynamic viscosity (kg/(m s)) µ l Dynamic viscosity of liquid (kg/(m s)) ν π Kinematic viscosity (m 2 /s) Constant ( ) ρ l Density of liquid (kg/m 3 ) ρ v Density of vapo (kg/m 3 ) ρ Density (kg/m 3 ) σ ϕ ω Suface tension (N/m) Constant Angula fequency (ad/s) ω* Dimensionless fequency Ω Dimensionless fequency xiii

15 SUMMARY Cuent pedictions suggest that themal dissipation fo high-pefomance micopocessos will appoach 160 W with egional suface heat fluxes that exceed 100 W/cm 2. One solution fo effective themal management of those micopocessos is the use of two-phase heat tansfe devices. To dissipate the anticipated heat fluxes at pescibed opeating tempeatues high-pefomance themal management devices will be essential fo the continued development of moe poweful computes. Dissipation of high heat fluxes at low tempeatue diffeences is possible using themal management devices that incopoate boiling and condensation. In this investigation vibation induced doplet geneation fom a liquid laye was examined as a means fo achieving high heat flux evapoative cooling. Expeiments wee pefomed in which doplets wee geneated fom a liquid laye using a submeged vibating piezoelectic dive. Duing this investigation the dependence of doplet geneation on conditions elated to paametes descibed duing spay and dive chaacteization was detemined. Those paametes include doplet mass flow ate doplet size dive fequency dive voltage and liquid laye thickness. The esults showed that as the liquid laye thickness was inceased the fequencies at which doplet geneation occued deceased. Doplet mass flow ates wee vaied by adjustment of the liquid laye thickness dive fequency and dive voltage. Afte detemining the dependence of the dive s displacement velocity and acceleation on fequency and voltage the dive s chaacteistics wee elated to the occuence of doplet geneation. As a esult a fequency-dependent dimensionless paamete was developed fo xiv

16 detemination of conditions necessay fo doplet geneation on the suface of the liquid laye. The dimensionless paamete is a combination of the Foude numbe and the dive acceleation. Results show that doplet geneation occus when the calculated dimensionless paamete is between distinct uppe and lowe bounds. An analytical heat tansfe model of a doplet cooling heat tansfe cell was developed to simulate the pefomance of such a cell fo themal management applications. Using doplet flow ates detemined as functions of dive voltage dive fequency liquid laye thickness and inteception distance the heat tansfe ate of a doplet cooling heat tansfe cell was pedicted fo vaied heat souce tempeatues and cell conditions. The heat tansfe model was fomulated in such a way as to accommodate a geat numbe of paamete vaiations that can be used fo the design of a simple heat tansfe cell. The model was used to detemine the effect of doplet cooling on the heat tansfe ate fom a heated suface but it can also be used to detemine the influence of othe opeating conditions that may be of inteest fo themal management applications. xv

17 CHAPTER 1 INTRODUCTION 1.1 Backgound As the semiconducto industy continues to make advances in the design of micochips computation times continue to decease theeby enabling computes to incease computational speeds. Those design advances enable the chips to un at inceasingly faste speeds but with the dawback of inceased heat poduction fom smalle size components. Because of inheent intenal electical esistances in the mateials used to build cicuit pathways heat geneation is unavoidable. As the computing powe of the industy's chips has inceased so have the heat fluxes and chip tempeatues. Accoding to the Intenational Technology Roadmap fo Semiconductos pedicted themal dissipation fo high-pefomance micopocessos will appoach 160 W within the next five yeas (Joshi 2001). The esulting heat fluxes ae anticipated to be well ove 100 W/cm 2 in cetain egions on the chip. To dissipate these heat fluxes while still maintaining tempeatues below pescibed limiting tempeatues moe aggessive themal management solutions will be essential fo continued development of high-powe computes. In 1999 the Semiconducto Industy Association identified themal management as a key challenge duing the next decade fo achieving pojected pefomance goals of the semiconducto industy. Once employed only as a means to 1

18 pevent themal failue themal management technology is now a necessay enable fo high-pefomance high-powe and high-tempeatue electonics systems. The need fo moe innovative themal management solutions is diven by the polifeation of devices that incopoate the use of micopocessos in thei designs. The gowth of mobile computing electonic hand-held devices telecommunications systems and automotive electonics systems ae some of the applications that ae diving the effot towad the miniatuization and design optimization of micopocessos. In 1965 Godon Mooe obseved an exponential gowth in the numbe of tansistos pe integated cicuit and pedicted that this tend would continue. As depicted in Figue 1.1 Mooe s Law the doubling of tansistos placed in micopocessos evey couple of yeas has been maintained and still holds tue today. Figue 1.1: Mooe's Law pedictions of micopocesso tansisto doubling (Wijetunga 2002) As the density of tansistos continues to incease powe dissipation becomes a citical facto in the eliability of the device. Figue 1.2 indicates that micopocesso powe equiements will appoach 1000 W by 2010 if cuent tends continue. 2

19 Miniatuization of micochip dies and inceasing tansisto densities poduce highe heat fluxes necessitating moe sophisticated and highe pefomance themal management solutions. Figue 1.2: Micopocesso powe equiements as a function of yea (Wijetunga 2002) One solution fo effective themal management of high heat flux micopocessos is the use of two-phase heat tansfe. Though boiling and condensation two-phase heat tansfe enables the dissipation of high heat fluxes at elatively low tempeatue diffeences. As shown in Figue 1.3 the heat tansfe coefficients possible with cooling schemes that involve phase change ae much geate than taditional foced convection heat tansfe which has been widely used fo low-heat flux applications. 3

20 Figue 1.3: Convective and boiling heat tansfe coefficients (Mudawa 2001) Evapoating liquid in a hot section and condensing vapo in a cold section of a heat tansfe device which is placed in contact with a micopocesso has been poven to be an effective themal management solution. These types of heat tansfe devices in the fom of heat pipes have been successfully used to povide high heat fluxes in a numbe of applications including the cooling of micopocesso packages. Heat pipes ae themal devices that opeate with the use of two-phase heat tansfe and ae often used to dissipate heat in vaious applications including micoelectonics. As shown in Figue 1.4 a simply configued heat pipe is a patially evacuated vessel with its intenal walls lined with a capillay stuctue that is satuated with a woking fluid. Since ai in the heat pipe is evacuated and the heat pipe is chaged with woking fluid befoe it sealed the intenal pessue is detemined by the vapo pessue of the woking fluid. 4

21 Adiabatic section Condense Evapoato Figue 1.4: Simple heat pipe When heat is tansfeed into the evapoato section the fluid is vapoized ceating a pessue gadient in the pipe. This pessue gadient foces the vapo to flow along the pipe to the condense section whee it changes phase giving up its latent heat of vapoization. The woking fluid is then pumped to the evapoato by the capillay foces developed in the capillay stuctue (Gane 1996). Heat pipes can be designed to opeate in a boad ange of tempeatues but have limitations that estict thei effectiveness fo heat tansfe. These limitations include viscous sonic capillay pumping entainment/flooding and boiling limits which ae all dependent on the heat pipe opeating tempeatue. Most limiting in the tempeatue ange of electonics cooling applications is the capillay pumping limit (Faghi 1995). This limitation is eached when nucleate boiling occus in the capillay stuctue ceating vapo bubbles that block the flow of liquid fom the condense to the evapoato. Nucleate boiling initiation in the capillay stuctue is the esult of locally highe heat fluxes in a paticula aea of the heat pipe. 5

22 The poposed eseach will study an altenative heat tansfe device that elies upon the highly efficient phase change pocess but does not suffe fom the issues that limit the opeation of heat pipes. The altenate cooling scheme that will be examined in detail will be based on doplet geneation fom a liquid laye. An altenative mechanism fo deliveing fluid to the evapoato section of a heat tansfe device is liquid laye doplet geneation which avoids the limitations associated with capillay pumping. Vibation of a dive beneath a laye of liquid can cause doplet geneation at the suface of the liquid laye. The doplets can be diected towad the heated suface of a micopocesso chip whee they impinge on that suface. Heat is dissipated as the doplets contact the suface and change phase fom liquid to vapo cooling the suface. Since the phase-change heat tansfe coefficient is much highe than that which occus duing single-phase foced convection a moe effective method of heat tansfe fo the same tempeatue diffeence esults. One design of a heat tansfe cell based on doplet cooling shown in Figue 1.5 has poven to be a viable solution fo cooling sufaces that equie high-heat flux dissipation ates. As shown by Heffington in 2000 heat tansfe cells that use doplet cooling have been constucted and ae able to achieve heat fluxes of geate than 100 W/cm 2 while the heated suface is maintained at tempeatues less than 100 C. Theefoe peliminay eseach into the capability of simple heat tansfe cells based on doplet geneation technology has shown that they can meet the cooling equiements of futue micopocesso designs. 6

23 Figue 1.5: Schematic of a doplet cooling heat tansfe cell (Heffington 2000) The pupose of this wok was to detemine the basic mechanisms that lead to doplet geneation fom a liquid laye and to identify the measuable physical paametes that ae significant contibutos to the phenomenon of doplet geneation. The esults wee incopoated into a heat tansfe cell model to pedict heat tansfe pefomance when a doplet geneation-based cell is attached to a high heat flux souce. Pedictions of heat tansfe pefomance wee compaed to the esults of expeimental wok pefomed by Heffington. Desciption of the heat tansfe cell and opeating conditions ae pesented in Chapte Reseach Objectives The hypothesis of this dissetation is: The measued doplet geneation fequency ange and dive displacement can be used to develop a method fo detemination of conditions necessay fo doplet geneation. The motivation fo this wok comes fom the need fo a modeling tool that can be used to design heat tansfe cells that ae based on doplet cooling technology. The objectives of this eseach ae to: 7

24 Detemine the paametes that influence the geneation of doplets fom the suface of a liquid laye. Develop a method to detemine the conditions necessay fo doplet geneation. Develop an analytical model that can be used to pedict the local and global heat tansfe ates in a doplet cooling heat tansfe cell. To detemine the anges at which doplet geneation occus expeiments wee pefomed to measue the piezoelectic dive s vibation fequency ange and voltage needed to geneate doplets fo seveal liquid laye thicknesses. The velocity at the cente of the piezoelectic dive was also measued to detemine the displacement and acceleation as a function of fequency. Coelations wee developed to detemine the conditions necessay fo the geneation of doplets fom the liquid laye using the dive and spay chaacteization esults. This infomation was used as input to an analytical heat tansfe model to pedict the pefomance of a doplet cooling heat tansfe cell. 8

25 CHAPTER 2 LITERATURE REVIEW 2.1 Spay Cooling Cooling by means of small doplet spays is of inteest because the cooling capacity of liquid spays often exceeds that fo pool boiling o fo liquid cooling by advection. Spay cooling effectiveness is highe because a thin laye of liquid can be maintained so that phase change occus and the themal esistance of the liquid laye is minimized. When the liquid laye emains thin and the suface tempeatue is below the Leidenfost tempeatue a vapo blanket which would insulate the suface is not allowed to fom. Instead as vapo is fomed in nucleation sites o othe impefections on the heated suface it is moe easily expelled fom the suface without eve foming an insulating vapo. Those potential bubbles if allowed to coalesce beneath the liquid laye esult in an insulating vapo blanket inhibit the heat tansfe ate fom the heated suface. In 1969 Kopchikov et al. wee able to dissipate 500 W/cm 2 fom a suface with only 20ºC of excess tempeatue using wate spays with doplets measuing 0.02 to 0.2 mm in diamete. Excess tempeatue is the diffeence between the suface tempeatue and satuation tempeatue of the liquid used to cool the suface. This achievement is significant because 500 W/cm 2 is moe than fou times geate than the heat flux possible at the same excess tempeatue fo cooling by pool boiling of wate at atmospheic pessue. 9

26 In an effot to detemine the dependence of the citical heat flux on spay chaacteistics Toda used spay nozzles with diametes anging fom 0.34 to 0.61 mm. His esults published in 1972 showed that the heat flux fo a given doplet size tempeatue and velocity inceased as the suface excess tempeatue inceased. He also showed that the excess tempeatue at which citical heat flux occued inceased as the doplet size and velocity wee inceased. A spay-cooling cuve simila to the satuated pool-boiling cuve was geneated fo wall-liquid tempeatue diffeences in excess of 200ºC. The cooling cuve could be subdivided into thee distinct egions: a low tempeatue egion a tansitional tempeatue egion and a high tempeatue egion. In the low tempeatue egion heat tansfe was a esult of evapoation of the liquid film at the liquid-vapo inteface. In the high tempeatue egion heat tansfe occued though a film-boiling-like state in which a vapo laye was fomed between the liquid film and the heated suface and evapoation was induced by convective heat tansfe though the vapo laye in addition to adiation fom the heated suface. Toda s futhe mist cooling eseach published two yeas late esulted in the identification of spay doplet paametes that govened fomation of a liquid film and fomulation of equations quantifying heat tansfe ates in the thee egions of the spaycooling cuve. In the low tempeatue egion heat flux is geatly affected by the tempeatue mass flow ate and velocity of the subcooled doplets. In the tansitional egion heat flux inceases with mass flow ate and velocity but is not appeciably affected by the subcooled doplet tempeatue. Results of a heat tansfe study that used mist cooling at low suface supeheat wee published by Bonacina et al. in Doplets measuing 90 µm in diamete with 10

27 velocities up to 2 m/s impinged on a heated suface yielding a maximum heat flux of 215 W/cm 2. Assuming dopwise evapoation a mathematical model was used to coelate the expeimental esults. An investigation into nucleate boiling on a heated suface sustained by wate doplets impinging on a heated suface was conducted by Monde. Heat tansfe was measued as a function of the volumetic flow ate and mean velocity of the doplets. Boiling cuves wee geneated and maximum heat fluxes of appoximately 1000 W/cm 2 wee attained at 40ºC of suface supeheat. Shoji et al. investigated mechanisms of cooling by subcooled doplets in the non-wetting egime fo a wide ange of doplet conditions. In this egime the suface tempeatue is high enough that the doplets evapoate befoe actually contacting the suface. Wate and ethanol wee the liquids used in the expeiments. Duing this expeimental study doplet diametes anged fom 0.88 to 2.54 mm doplet velocities extended between 0.89 and 1.80 m/s and the wall supeheat tempeatues wee measued between 250 and 800 C. Fo the investigated conditions using wate the non-wetting egime began at a suface tempeatue of about 300 C. Due to the natue of the nonwetting conditions that existed duing the expeiments the maximum heat tansfe ates wee seveely esticted and wee less than 10 W. At Canegie-Mellon Univesity Yao and Choi investigated the effects of liquid mass flux doplet size and doplet velocity on heat tansfe ates. The mono-size spay was ceated using a piezoelectic tansduce to contol the Rayleigh jets which wee used to ceate the doplets. An ai supply steam was used to dispese the doplets ove the heated suface because the ejected doplets tended to tavel in staight paths and 11

28 impacted only on fixed aeas of the heate. The doplet diametes wee held constant at mm and velocities anged fom 4.02 to 5.84 m/s fo suface tempeatues of 100 to moe than 400 C. The investigatos found that when the liquid flux on the suface was kept low the film boiling heat tansfe ate was affected by doplet velocity and size. On the othe hand when the liquid flux was high the doplet velocity and size had little affect on heat tansfe ate. Liquid mass flux was detemined to be the dominant paamete affecting the impacting spay heat tansfe ate. The film boiling heat tansfe of impacting spays had a powe-law dependency on the liquid mass flux with the powe vaying fom unity to a faction as the liquid mass flux inceased. Because the attainable heat flux is not govened solely by chaacteistics of the spay it is also impotant to conside the condition of the heated suface. A study conducted at Pudue Univesity in 1995 by Benadin et al. found that suface oughness had a damatic affect on the citical heat flux. In this study wate doplets with Webe numbes of 20 and 60 wee spayed onto thee coppe disks with diffeent suface pepaations. The citical heat fluxes obtained in ode fom highest to lowest wee achieved fo the paticle-blasted polished and ough-sanded sufaces. The esults indicated that the maximum citical heat flux occued on the modeately ough paticleblasted suface. In a study by Pais et al. the effect of suface oughness on heat tansfe was studied. With ai-atomized nozzles using wate a thin liquid film was poduced on the polished heated suface. With suface oughnesses of less than 1 µm heat fluxes in the ange of 1200 W/cm 2 wee achieved. Based on esults of the study a new method fo detemining and designating the suface textue was poposed. 12

29 2.2 Doplet Cooling While spay cooling and doplet cooling have many simila chaacteistics they diffe in one main attibute. Spay cooling is used to descibe a heat tansfe method in which the liquid doplets ae popelled towad the heated suface by a caie fluid often pessuized ai. Doplet cooling on the othe hand is the tem used to descibe the heat tansfe that esults when the liquid is deliveed to the heated suface by its own momentum. The vibation induced doplet atomization pocess studied in this thesis is classified as a doplet cooling technique. Wachtes et al. studied the heat tansfe fom a suface by the impingement of wate doplets measuing 60 micons in diamete taveling at a velocity of 5 m/s. Expeiments wee pefomed with suface tempeatues up to 400 C. Results showed that the themal popeties and suface pepaation of the heated suface as well as the Webe numbe of the impacting doplets affected the tempeatue of the heated suface. In 1970 Pedeson investigated doplet cooling using doplet diametes fom 200 to 400 µm and doplet velocities fom 2.44 to m/s. He concluded that doplet velocity was the dominant paamete affecting the heat tansfe ate. Pedeson found that the suface tempeatue had little effect on the heat tansfe ate in the non-wetting egime the same finding late made by Shoji while conducting spay cooling expeiments. The evapoation of lage wate doplets (2.5 to 4.5 mm) on smooth sufaces of coppe bass cabon steel and stainless steel at tempeatue anging fom 80 to 450 C was investigated by Makino and Michiyoshi. High-speed photogaphy was used to study the tansient behavio of a wate doplet on a heated suface initially at tempeatues above the satuation tempeatue and finally at tempeatues geate than the Leidenfost 13

30 tempeatue. The measued data wee used to calculate the time-aveaged heat flux and the suface tempeatue fo the heated suface placed in the path of the liquid doplets. Xiong and Yuen investigated cooling of a hot stainless steel plate by small liquid doplets of wate and hydocabon fuels. The study used doplet diametes that anged fom 0.07 to 1.8 mm and heate suface tempeatues that anged fom 63 to 605 C. The citical heat flux fo wate was found to be 500W/cm 2 at a suface tempeatue of 155 C. Results showed that the maximum heat tansfe ate is independent of doplet sizes fo all tested hydocabon fuels and that the maximum heat tansfe ate deceased slightly with inceasing doplet sizes fo wate. Kuokawa and Toda pefomed expeiments using high-speed photogaphy to study the impact of single doplets on glass. Using wate ethyl alcohol and mecuy coelations wee poposed fo the maximum film diamete as functions of Webe numbe Reynolds numbe and doplet diamete. Doplet defomation and the subsequent fomation of liquid film fo doplet diametes between 1.3 and 2.3 mm and velocities between 2.1 and 2.3 m/s wee analyzed. Duing impact the heat tansfe between the liquid film and the hot wall was dominant fo the duation of evapoation of the doplet. The maximum citical heat flux fo a steam of monodispesed wate doplets impacting a heated suface was studied by Halvoson et al. In this study doplets having diametes between 2.3 and 3.8 mm a velocity of 1.3 m/s and fequencies of 2 and 15 dops pe second impacted a heate suface at a tempeatue of 124 C. The maximum citical heat flux was 325 W/cm 2. The measuements showed that the citical heat flux inceased with deceasing doplet diamete and inceasing fequency fo a given mass 14

31 flux. The citical heat flux also inceased with inceasing mass flux egadless of the doplet size o velocity. Heat tansfe fom 2.58 mm diamete monodispesed wate doplets impinging on a heated suface was investigated by Sawye et al. (1993). The velocity ange of the doplets was 2.64 to 4.55 m/s and a unifom mass flow ate of g/s was maintained. The citical heat flux deceased significantly as doplet velocity inceased. Sawye et al. (1997) continued investigating doplet cooling using doplet diametes between 1.5 and 2.7 mm and velocities between 2.4 and 4.6 m/s with fequencies between 12 and 42 doplets pe second. Maximum citical heat fluxes nea 500 W/cm 2 wee eached with suface supeheats of 20 C when the heat flux was based on the initial wetted aea. Using monodispesed doplets of diametes between 52 and 287 µm and independent contol of mass flow ate doplet diamete velocity and fequency Sheffield (1994) investigated heat fluxes fom a heated suface. The maximum heat flux of 430 W/cm 2 esulted when the doplet diamete was 260 µm. Fo flow ates less than 0.05 g/s the heat flux inceased with deceasing doplet diametes and inceasing velocities. Fo flow ates above 0.1 g/s the heat flux inceased with inceasing doplet diametes and deceasing velocities. Continuing the doplet cooling investigation initiated by Sheffield Denney (1996) used a dop-on-demand geneato capable of low doplet geneation fequencies. Low citical heat fluxes wee obtained due to the fomation of a liquid pool on the heate suface which pevented the doplets fom impacting the suface. Citical heat fluxes of up to 270 W/cm 2 wee achieved. 15

32 Using the same device used by Sheffield Selles (2000) added an electostatic deflecto to contol doplet spacing and velocity as they impacted a heated suface. Heat fluxes as high at 297 W/cm 2 at a suface tempeatue of 126 C wee obtained using doplet diametes between 97 and 392 µm that wee poduced at fequencies of 2.5 to 38 khz. Results indicate a slight dependence of heat tansfe on doplet size and velocity while the doplet spacing was found to influence the heat flux by as much as 25 pecent. Heffington (2000) investigated the use of a vibating piezoelectic diaphagm as a doplet geneato within a heat tansfe cell. Duing the development of this device measuements showed that many paametes affect the heat tansfe ates when a vibating diaphagm is used to geneate small liquid doplets. The anges of fequencies of vibation that would ceate doplets wee dependent on diaphagm attibutes and liquid loading. Heat fluxes as high as 120 W/cm 2 wee obtained at a heate suface tempeatue of appoximately 95 C when the pessue of the heat tansfe cell was educed to 23.1 kpa. Results of the investigation showed that vibation-induced doplet atomization heat tansfe cells povide an altenative means fo cooling micoelectonics. The pupose of Heffinton s wok was not to chaacteize doplet geneation but to demonstate the effectiveness of doplet geneation fo two-phase cooling. It is the pupose of the pesent investigation to detemine the mechanisms that lead to doplet geneation and to identify physical conditions necessay fo the geneation of doplets. 2.3 Vibation Induced Doplet Geneation To capitalize on the advantages of doplet cooling small doplets of liquid must be geneated in a epoducible and eliable fashion. The dops must then be popelled 16

33 towad the heated suface whee they can emove heat fom the suface as they evapoate. One way to geneate doplets is to eject them fom waves on a liquid laye by vibation of the laye at discete fequencies. At cetain fequencies the waves on the liquid suface incease in amplitude and doplets ae ejected fom the wave cests. Faaday (1831) pefomed qualitative expeiments that showed that suface wave esponse was subhamonic to the focing fequency. He discoveed that the vibation peiod in the waves that fomed on a thin liquid laye on a vibating suface was twice the peiod of the diving vibation. The esults of his wok have been confimed in subsequent Faaday wave investigations and expeiments by othe eseaches. Late Rayleigh (1896) showed that the suface tension of a liquid could be detemined fom the capillay wavelength obseved on the suface of the vibating liquid. He developed an equation elating the capillay wavelength vibation fequency and suface tension of the liquid. His wok was late used by Lang (1962) to pedict the mean diametes of doplets geneated fom the cests of capillay waves. In 1927 Wood and Loomis descibed the atomization of a liquid by ultasonic vibation. Using a quatz piezoelectic oscillato atomization at the suface of an oil bath was geneated at a diving fequency of 300 khz. Late wok by Sollne (1936) in which the physical natue of the atomization of a liquid was investigated esulted in the hypothesis that cavitation was the mechanism fo atomization by ultasonic vibation. Doplet geneation by cavitation was not the method by which doplets wee poduced in the pesent investigation. In 1957 Sookin published eseach wok in which he found that a patten of standing waves was geneated on the suface of the liquid when the focing vibation 17

34 amplitude exceeded a cetain theshold value. When the amplitude of vibation was futhe inceased doplets fomed at the wave cests and wee ejected. In Sookin s study a vessel of wate was vetically vibated between 10 and 30 Hz to ceate the standing waves and doplets. Soon afte in 1959 Eisenmenge also investigated the mechanism of the paametic excitation of capillay waves and the conditions fo the geneation of doplets. His study was pefomed in the fequency ange of 10 to 150 khz. The esults of Sookin s wok and Eisenmenge s wok identified the ejection of doplets fom capillay waves fomed on the liquid suface. The fequency ange used in the pesent eseach investigation falls between the anges studied by these two eseaches. The esults of thei wok indicated that a minimum focing vibation amplitude would be necessay fo the onset of doplet geneation in the pesent investigation. Fo fequencies of 10 Hz to 30 Hz doplet geneation began when the focing amplitude was 7 to 8 times the amplitude needed fo wave geneation. Fo fequencies of 10 khz to 1.5 MHz doplet geneation began when the focing amplitude was 4 times the amplitude needed fo the geneation of waves. Building on the wok of Rayleigh (1896) Lang (1962) ascetained that thee was a elationship between the aveage diamete of the ejected doplets (D b ) and the capillay wavelength (λ). This elationship D b = 0.34 λ was obseved fo woking fluids including wate oil and molten waxes at foced vibation fequencies of 10 khz to 800 khz. Lang postulated that atomization involved the uptue of capillay waves and subsequent ejection of suface paticles fom the wave peaks. In the pesent study doplets wee also geneated by the detachment of liquid fom the cests of waves thus a elationship of doplet diamete to wavelength was expected to exist. The applicability of 18

35 Lang s elationship to the pesent investigation is discussed in the esults section of Chapte 3. Though thei theoetical investigation Peskin and Raco (1963) povided insight into the natue of atomization of liquid by means of vibation. They studied the mechanism of finite amplitude excitation of capillay waves on the suface of a liquid laye and detemined the influence of the bottom suface of the liquid laye on the doplet chaacteistics. A dependence of the aveage doplet diamete on the amplitude of foced vibation and liquid laye thickness was deived Db π d 2σ = ρω d 2tanh 2 3 π d Db t d 1 3 In the pesent investigation the fequency ange in which doplet geneation occued was dependent on the liquid laye thickness and foced vibation amplitude but the dependence of doplet diamete on liquid laye thickness was not obseved. Because the atio of the liquid laye thickness to the dive displacement (t/d) was in the ange of 50 to 150 in this investigation accoding to Peskin and Raco s equation the liquid laye thickness would not measuably affect the aveage doplet diamete. Accoding to thei equation the effect of the liquid laye thickness on doplet diamete is significant fo atios less than unity. As Lang had aleady shown Pohlmann and Stamm postulated in 1965 a elationship between doplet diamete and capillay wavelength. Motion pictues wee used to obseve the poduction of individual doplets duing the atomization of the liquid laye. These motion pictues veified the poposed physical beaking away of doplets 19

36 fom the cests of waves on the liquid suface. In the pesent investigation the same method fo doplet geneation was obseved. Vinig et al. (1988) investigated wave motion in a ectangula containe that was patially filled with wate and vetically oscillated at fequencies less than 5 Hz. Thei investigation did not include the geneation of doplets but only the geneation of waves. Fo seveal wate depths (4.27 to cm) the dependence of wave amplitude on both excitation fequency and excitation amplitude was found to be in good ageement with peviously studied nonlinea theoy pedictions. Results of the expeiments indicated the existence of a citical wate depth below which the fequency dependence of wave amplitude is qualitatively diffeent fom the fequency dependence fo depths geate than this citical value. When the wate level in the containe was above this citical depth the wave amplitude deceased with inceasing fequency. When the wate level in the containe was below this citical depth the wave amplitude inceased with inceasing fequency. The citical depth was pedicted by analysis of thee-dimensional nonlinea waves unde vetical excitations to be 7.70 cm. The esults of thei expeiments indicated that the citical depth was between 6.60 and 7.47 cm. The waves poduced in Vinig et al. s investigation wee gavity waves which wee pedominantly estoed by the foce of gavity. The waves poduced in the pesent investigation wee capillay waves which wee pedominantly estoed by suface tension foce. Although the liquid laye thicknesses of the Vinig et al. investigation (4.27 to cm) and of the pesent investigation (1.3 to 6.1 mm) wee damatically diffeent the indication of an obseved dependence of wave amplitude on liquid depth diving fequency and diving displacement was impotant. Because doplets detach 20

37 fom waves on the liquid suface conditions that affect wave fomation also indiectly have an effect on doplet geneation. In the pesent investigation as the liquid laye thickness deceased the fequency ange in which doplet geneation occued inceased. In 1994 Kuma and Tuckeman investigated the linea stability of Faaday waves. They studied the stability boundaies of the inteface between two fluids that wee vibated in a vetically oiented vessel at fequencies that lead to the geneation of standing waves. They found that fo a given diving fequency the instability occus only fo cetain combinations of wavelength and diving amplitude. While Kuma and Tuckeman s investigation was analytical Bechhoeffe et al. (1995) expeimentally detemined the linea stability of Faaday waves. Theshold acceleations and wavelengths necessay to excite suface waves in a vetically vibated fluid containe wee measued and found to be in ageement with theoetical pedictions. Based on thei esults the investigato s expectation fo the pesent investigation was that the dive acceleation would be an impotant facto fo doplet geneation. As Bechhoeffe et al. had also done Goodidge et al. (1996) pefomed expeiments in which a containe patially filled with liquid was vetically oscillated to geneate doplets fom the liquid suface and the citical amplitudes fo doplet ejection wee measued. Fo low-fequency focing (<60 Hz) gavity waves dominated doplet fomation and fo high-fequency focing capillay waves dominated. The citical acceleation to poduce doplets was found to be dependent on the focing fequency and liquid suface tension. Using the same appaatus Goodidge et al. (1997) investigated the effects of viscosity on the ejection of doplets. The scaling law used to descibe the citical acceleation needed to poduce doplets was extended to include viscous effects. 21

38 In the pesent eseach doplets wee poduced by the shedding of liquid fom the cests of capillay suface waves. Goodidge et al. s detemination of a citical dimensionless acceleation fo the poduction of doplets povided insight and a possible diection fo detemination of a dimensionless acceleation fo the doplet geneation configuation used in the pesent investigation. Ceda and Tiapegui (1998) analytically investigated the effect of viscosity on the stability of a liquid laye. They confimed that the esponse was subhamonic fo lowviscosity fluids which is a conclusion that is in ageement with the esults of Faaday s expeiment. Fo high viscosity fluids the esponse could be hamonic esulting in diffeent mechanisms fo poducing instabilities. In 2000 James investigated dop atomization that is a esult of vibation of a single wate dop placed on a piezoelectically actuated flexible diaphagm. This expeimental investigation was pefomed fo fequencies between 600 to 1200 Hz. An analytical model of the expeiments was fomulated to descibe doplet ejection. Additionally an analytical investigation of vetical oscillation of a liquid laye was pefomed. Results indicated that theshold values of a dimensionless fequency and dimensionless amplitude ae necessay fo doplet ejection. Dimensionless paametes wee developed that descibed dop dynamics duing atomization. Despite the diffeences between atomization of a single dop and doplet geneation fom a liquid laye James investigation was anothe that identified the existence of a theshold displacement fo doplet geneation. In 2002 Vukasinovic investigated the dynamics of the vibation induced doplet atomization pocess. Using wate he studied the membane-dop inteaction instability 22

39 modes and suface dynamics of the dop. In 2004 he also chaacteized the doplet spay that esulted fom atomization of the oiginal lage liquid dop. Atomization of the dop occued as a esult of amplification of capillay waves that fomed on the fee suface of the dop. Investigation of the fee suface dynamics of the dop also evealed thee pimay tansitions befoe atomization. Atomization begins with axisymmetic standing waves that tansition into azimuthal waves which evolve at a sub hamonic of the diving fequency. The azimuthal waves then tansition into a lattice mode just pio to a disodeed pe-ejection state. In this eseach the obsevation of a lattice aangement of suface waves was also obseved on the suface of the liquid laye. The lattice aangement was visible at fequencies just below the lowest doplet geneation fequency and just above the highest doplet geneation fequency. 23

40 CHAPTER 3 METHODOLOGY 3.1 Dive-Doplet Geneation Coelation In this eseach small doplets of wate ae geneated by a vibating dive that is submeged in a liquid pool. By applying appopiate voltages and fequencies of a sinusoidal input to the dive waves ae geneated on the liquid suface. Small doplets ae ejected fom the wave cests and they tavel upwad away fom the liquid suface. Figue 3.1 shows a schematic of the configuation used to geneate the doplet spays. Doplets ejected fom suface Suface waves Wavelength Figue 3.1: Schematic of doplet geneation test setup Doplet geneation fom the suface of the liquid laye is caused by the vibation of the dive which is secued at the bottom of the liquid pool. Chaacteistics of dive vibation influence the anges at which doplets will be poduced. By using the dive s displacement and acceleation as functions of fequency and voltage elationships can be identified between the obseved doplet geneation and dive chaacteistics. 24

41 Insight into the vibation chaacteistics of the dive necessay to geneate doplets can be deived fom the wok of seveal eseaches. Notable wok in the aea of geneation of wate and ethanol doplets was pefomed by Goodidge et al. (1996). Thei investigation focused on the ceation and chaacteization of doplet geneation fom capillay waves on the fee suface of a liquid that wee caused by vetical oscillation of a patially filled liquid containe. Infomation about Goodidge et al. s wok is povided as backgound knowledge fo analysis methods that wee used in this eseach. To chaacteize the tansition to doplet-ejecting waves the theshold diving acceleation was measued as a function of focing fequency. In thei investigation an electodynamic shake was used to povide excitation of the liquid and containe. A theshold condition was detemined by supplying sufficient acceleation to poduce doplets and then the acceleation was loweed until doplet geneation ceased. Using an acceleomete which was attached to the amatue of the shake the acceleation of the liquid and containe was measued. Based on esults of his expeiments it was detemined that the theshold acceleation fo the geneation of doplets was dependent on the fequency of the vetical excitation and the suface tension of the fluid. Equation 3.1 is the obseved dependence of theshold acceleation a t as a function of angula vibation fequency ω accoding to esults published by Goodidge et al. 4 / 3 a t ~ ω 3.1 whee ω = 2 π f

42 A dimensionally consistent fom which combined the dependent paametes of fequency and suface tension was detemined by Goodidge et al. and is shown in Equation 3.3. The constant was empiically detemined fom the expeimental data. 4 / 3 1/ 3 a t = 0.239ω ( σ / ρ) 3.3 Goodidge et al. used Equation 3.3 to pedict the theshold acceleation necessay fo doplet geneation fom a vetically oscillating patially filled liquid containe. Goodidge et al. s late investigation (1997) into the effects of viscosity on doplet geneation yielded two dimensionless quantities that they used to descibe the theshold acceleation necessay fo doplet geneation. These quantities of dimensionless acceleation a* and dimensionless angula fequency ω* ae expessed in Equations 3.4 and aν a * = ( σ / ρ) 3 ων ω * = ( σ / ρ) The esulting dimensionless acceleation gaphed as a function of dimensionless angula fequency in Figue 3.2 epesents the theshold values fo which doplet geneation will occu fom vetical oscillation of a patially filled liquid containe. The cicula makes in the figue epesent the dimensionless acceleation and dimensionless angula fequency detemined fo the expeimental data. In the lowe-viscosity egion (ω* < 10-6 ) and the highe-viscosity egion (ω* > 10-3 ) the theoetical theshold values ae below the expeimentally detemined theshold values. Conditions fo which the calculated dimensionless acceleation and dimensionless angula fequency appea above the line indicate the geneation of doplets. Conditions fo which the calculated dimensionless 26

43 acceleation and dimensionless angula fequency ae below the solid line shown in Figue 3.2 indicate conditions at which doplet geneation does not occu. Figue 3.2: Dimensionless acceleation as a function of dimensionless angula fequency (Goodidge et al. 1997) In thei investigation Goodidge et al. detemined the theshold acceleations fo lowviscosity liquids whee suface tension effects dominate to be expessed as 4 / 3 1/ 3 a t = 0.261ω ( σ / ρ) 3.6 and theshold acceleations fo high-viscosity liquids to be expessed as 1/ 2 3 / 2 a t = 1.306ν ω 3.7 whee the constants wee empiically detemined. As in Goodidge et al. s wok the pesent investigation epoted in this dissetation involves the geneation of doplets fom the suface of a vibating liquid laye. Although thee ae similaities between the mechanisms fo doplet geneation in the wok of Goodidge et al. s and in the pesent study distinctly diffeent methods fo excitation of the liquid wee implemented. Due to the diffeent methods of excitation used to geneate doplets Equations 3.3 to 3.7 wee not used fo analysis of this 27

44 investigation s expeimental esults. Key diffeences between the expeimental investigations ae as follows. Goodidge et al.: The entie body of liquid and containe wee vetically oscillated The liquid depth was held constant fo all expeiments Multiple liquids wee used so that the effects of suface tension and viscosity could be detemined Fequency ange 20 to 80 Hz Liquid laye thickness 10.0 cm Cuent study: The liquid laye above the dive was vibated in an open pool and in a pool with constained boundaies The liquid depth was vaied to detemine its effect on doplet geneation Distilled wate was the sole liquid used Fequency ange 360 to 575 Hz Liquid laye thickness 1.28 to 6.1 mm The diffeences itemized above have implications fo the expected esults and the expeimental methods used to geneate doplets. The diffeences would affect the doplet diametes equied foced displacements and suface wave amplitudes. In elationship to doplet geneation the longe wavelengths of gavity waves poduce lage doplets than those poduced fom shote wavelength capillay waves. In the pesent investigation a piezoelectic diaphagm clamped aound its peimete was used to vibate the liquid laye. Theefoe the acceleation and 28

45 displacement of the lowe suface of the liquid laye wee not unifom. The distibution of acceleation and displacement ove the suface of the vibating dive was nealy paaboloid in shape with the maximums occuing at the cente of the dive. Goodidge et al. used an electodynamic shake to oscillate a solid piston-like dive which ceated unifom displacement and acceleation of the liquid laye. The nealy paaboloid shape of the velocity distibution on the suface of the dive is shown in Figues 3.10 to The thickness of the liquid laye was vaied in the pesent investigation which led to vaiations in the induced capillay wave speeds that ae geneated on the fee suface of the liquid. This wave behavio is discussed late in Section In Goodidge et al. s expeiments whee the liquid laye was of a constant thickness the calculated paametes used to define a theshold fo doplet geneation wee not functions of the thickness of the liquid laye thickness. Doplet geneation fom a liquid laye thickness simila to that used in Goodidge et al. s investigation (10.0 cm) could not be epoduced in the pesent investigation because the necessay displacement could not be poduced by the piezoelectic dive. Diffeences between the configuations used in the pesent investigation and the configuation used by Goodidge et al. ae seen in the bounday conditions applied to the momentum equations fo a Newtonian fluid with constant density and constant viscosity. These equations could be solved to detemine the velocity distibutions in the liquid body above the dive. The following momentum equations wee simplified assuming the pessue field thoughout the fluid was unifom and no body foces in the and θ diections esulting in the equations below. 29

46 + + = θ v z v θ v v µ z v v v θ v v v v t v ρ θ z θ θ ) ( = θ v z v θ v v µ z v v v v θ v v v v t v ρ θ θ θ θ z θ θ θ θ θ ) ( 1 z z z z z z z θ z z ρg z p z v θ v v µ z v v θ v v v v t v ρ = z θ R H z θ z θ R H Figue 3.3: Schematic of liquid body Figue 3.3 is the epesentative liquid body to which the following bounday conditions ae applicable and fom which doplets would be geneated at its liquid-gas inteface (z = H) in the pesent investigation and in Goodidge et al. s investigations. Fo Goodidge et al. s investigation in which a containe of liquid was vetically oscillated the following initial and bounday conditions would be applicable. 0 0) ( 0) ( 0) ( = = = z v z v z v z θ θ θ θ Initial conditions t H ai ai t H z v z v t v 0 ) 0 ( θ θ µ µ θ = = Radial bounday conditions 0 ) ( 0 ) 0 ( = = t z R v t v θ θ θ θ Pola bounday conditions 30

47 0 ) sin( ) ( 0 max = = t z z z θ v t V t z R v θ ω θ Axial bounday conditions The adial velocity of the liquid v is zeo at the bottom of the containe due to the no-slip condition. At the liquid-gas inteface on the suface of the liquid body the shea stess in the liquid equals the shea stess in the ai. At the bottom and at the sidewalls of the containe the pola velocity of the liquid v θ is zeo due to the no-slip condition. At the sidewalls of the containe the axial velocity of the liquid v z equals the velocity of the containe due to the no-slip condition. V max is the maximum velocity of the containe duing oscillation. Along the axial centeline of the liquid body the axial velocity is independent of the pola angle. In the pesent investigation fo expeiments pefomed in an open pool the initial and bounday conditions fo the momentum equations follow. 0 0) ( 0) ( 0) ( = = = z v z v z v z θ θ θ θ Initial conditions t H ai ai t H t z z v z v θ v 0 0 θ θ θ µ µ = = Radial bounday conditions t H ai ai t H z v z v t v 0 ) 0 ( θ θ θ θ θ µ µ θ = = Pola bounday conditions 0 ) sin( 1 ) 0 ( 0 2 max = = t z z z θ v t R V t v θ ω θ Axial bounday conditions 31

48 Along the axial centeline of the liquid body the adial and the axial velocities ae independent of the pola angle. At the liquid-gas inteface on the suface of the liquid body the adial and pola shea stesses in the liquid equal the shea stesses in the ai. Due to the no-slip condition the pola velocity of the liquid in contact with the dive suface is zeo and the axial velocity of the liquid in contact with the dive equals the velocity of the vibating dive. As the dive vibates its displacement pofile is nealy paaboloid in shape. Fo the pesent investigation expeiments wee also pefomed in a pool constained by a solid cicula bounday. Fo a configuation in which the inne diamete solid bounday is equal to the diamete of the dive the following initial and the bounday conditions fo the momentum equations apply. 0 0) ( 0) ( 0) ( = = = z v z v z v z θ θ θ θ Initial conditions t H ai ai t H t z z v z v θ v 0 0 θ θ θ µ µ = = Radial bounday conditions 0 ) 0 ( 0 ) ( = = t v t z R v θ θ θ θ Pola bounday conditions ) sin( 1 ) 0 ( 0 ) ( 2 max t R V t v t z R v z z ω θ θ = = Axial bounday conditions The adial velocity at the cente of the liquid body is independent of the pola angle. At the liquid-gas inteface on the suface of the liquid body the adial shea stess in the liquid equal the shea stess in the ai. The pola velocities of the liquid in contact with the constaining bounday and in contact with the dive ae zeo due to the no-slip 32

49 condition. The axial velocity of the liquid in contact with the constaining bounday is also zeo due to the no-slip bounday condition. The axial velocity of the liquid in contact with the dive equals the velocity of the vibation dive. The cicula ings used in this investigation wee not the same diamete as the dive. The velocity field esults fo this constained pool and fo the open pool would epesent the two extemes between which would be the expected velocity field esults fo the constained pool configuations used in this investigation. Fo all thee configuations the initial conditions and liquid-gas inteface bounday conditions ae the same. Howeve the bounday conditions at the solid-liquid intefaces ae diffeent which means the solutions fo the motion of the liquid laye diffe. The velocity of the fluid in all diections is initially zeo and the shea stess at the liquid suface equals the shea stess in the ai. Solution of the momentum equations is not made in this dissetation but diffeences in the bounday conditions indicate that a diffeent solution exists fo the time-dependent velocity fields within the liquid body. Diffeences in the velocity fields at the liquid suface would indicate vaiations in suface wave motion and thus diffeent doplet geneation chaacteistics. 3.2 Dive Chaacteization The dive used in the pesent investigation is a piezoelectic diaphagm which consists of two pats; a piezoelectic element and a thin metal plate onto which the piezoelectic element is bonded as shown in Figue 3.4. The piezoelectic element is a sinteed body of poly-cystals that distots when a voltage is applied to it. The specific piezoelectic element used thoughout this investigation was made of PZT (Lead Ziconate Titanate ) and the metal plate was made of an ion nickel alloy. The metal 33

50 plate was mm in diamete and 0.10 mm thick. The diamete and thickness of the piezoelectic element used in the expeiments epoted in this dissetation measued 19.7 mm and 0.22 mm espectively. Figue 3.4: Schematic of piezoelectic diaphagm and components Deflection of the piezoelectic diaphagm is the esult of shea stesses exeted by the piezoelectic element on the metal plate when a voltage is applied. When DC voltage is applied the piezoelectic element distots by adial expansion causing the metal plate to bend as shown in Figue 3.5(a). By evesing the polaity of the DC voltage the piezoelectic element contacts causing the metal plate to bend in the opposite diection as shown in Figue 3.5(b). When an AC voltage is applied acoss the electodes the diaphagm altenates bending in both diections at the fequency of the applied voltage and the epeated bending of the diaphagm motion poduces vibation. 34

51 Figue 3.5: Schematic of piezoelectic diaphagm deflection Knowledge of the displacement and acceleation of the dive is essential fo the analysis of doplet geneation fom a liquid laye. As evidenced by esults of pevious eseach expeiments that involve doplet geneation fom Faaday to Goodidge the focing amplitudes acceleations and fequencies ae cucial fo doplet geneation. Unlike pevious eseach investigations into doplet geneation fom a liquid laye by vetical oscillation of a igid plate the doplet geneation in this eseach was caused by vibation of a flexible membane. To detemine the dive displacement and acceleation a lase vibomete was used to measue the velocity at the cente of the dy dive fo the ange of fequencies and dive voltages used duing the expeiments. Using the velocity measuements displacement and acceleation at the cente of the dive wee calculated. Velocity measuements at diffeent azimuthal locations on the suface of the dive wee used to veify the axisymmety of the dive vibation. 35

52 3.2.1 Appaatus and Pocedues The lase vibomete used to measue the velocity of the dive was a Polytec PSV-200 scanning vibomete. Lase Dopple vibomety (LDV) is a non-contact vibation measuement technique using the Dopple Effect. Lase vibometes can be used to make measuements of many of types of sufaces without influencing them by mass loading. The PSV-200 uses a low-powe Helium-Neon lase and is a two-beam intefeometic device that detects the phase diffeence between an intenal efeence and a measuement beam. The measuement beam is focused on the taget and scatteed back to the intefeomete. Using the heteodyne pinciple detection of the vibation diection is possible by demodulation of the fequency-modulated caie signal. The Polytec PSV-200 consisted of a scanning head vibomete contolle emote focus contol pesonal compute system video contol box pan/tilt head and tipod. Positioning and focusing of the lase by the scanning head is contolled using the popietay softwae installed on the pesonal compute system. This system was also used to contol the signal geneato that poweed the dive. Data acquisition was also pefomed using the popietay softwae povided on the compute system. The dive mounted in its enclosue was secued to a tipod and positioned appoximately 1.5 metes fom the scanning head of the vibomete. The lase light spot fom the scanning head was positioned on the suface of the dive. The exact position of the lase was then adjusted to the cente of the dive face using the compute softwae. Afte positioning the lase spot at the cente of the dive the focal point of the lase light beam was adjusted until maximum signal stength was indicated in the softwae contol pogam. The signal stength is a measuement of the amount of lase light eflected 36

53 fom the dive face back to the eceive which is housed within the scanning head. Because the dive face is a highly eflective nickel-alloy the signal stength was indicated as maximum duing all expeiments Electical Powe Measuement The voltage acoss the dive was measued and the powe calculated fo each expeiment. The dive was poweed with an AC sine wavefom using a signal geneato. A step-up tansfome was used to incease the voltage fom the signal geneato to meet the desied powe equiements of the dive. To calculate the cuent though the dive the voltage was measued acoss a esisto that was placed in seies with the dive as shown in Figue 3.6. Using the oscilloscope the voltage dops acoss the dive (V pp2 ) and the dive-esisto combination (V pp1 ) and the phase between the voltages signals wee measued. A schematic of the electical cicuit used to powe and monito the dive is pesented in Figue 3.6. Signal Geneato I1 Resisto R1 Dive Tansfome Vpp1 Vpp2 Oscilloscope Figue 3.6: Schematic of powe/monitoing cicuit of dive 37

54 The fist step in calculating the powe equied to opeate the dive was to monito the voltage dops acoss the dive-esisto combination (V pp1 ) and the dive (V pp2 ). The diffeence between the two voltage dops yielded the voltage dop acoss the esisto (R 1 ). The phase angle (θ) between V pp1 and V pp2 was ecoded and the esisto powe (P RMS1 ) was calculated. P RMS1 ( V = pp1 V R 1 pp2 ) 2 π cos( θ ) The cuent though the esisto (I 1 ) which is the same as though the dive was then calculated. I 1 = P ( V V 2 ) pp1 RMS 1 pp 3.9 Finally the powe dissipated by the dive (P RMSdive )was calculated. P = ( I V ) 2 RMS dive 1 pp The voltages V pp1 and V pp2 wee ecoded fo calculation of the powe equied to opeate the dive and fo use in elating doplet geneation conditions to the displacement and acceleation of the dive. The displacement and acceleation is dependent on the fequency and voltage applied to the dive (V pp2 ). The fequencies and voltages wee tabulated and efeenced to detemine dive velocities displacements and acceleations to elate dive and doplet geneation chaacteistics Dive Velocity Measuement To measue the velocity at cente the dive the fequency and voltage supplied by the signal geneato to the dive-esisto combination (V pp1 ) wee adjusted to the 38

55 desied values. The time-dependent velocity at the specified voltage and fequency wee ecoded along with dive voltage (V pp2 ) to a data file fo each test condition. The sampling ate and test duations wee set and held constant fo all tests at 256 khz and 8 milliseconds. To ceate the contou plots used to veify the axisymmety of dive vibation the scanning featues of the lase vibomete system wee used. A gid was geneated that coveed the suface of the dive using the video camea that was built into the lase vibomete. The fequency and voltage of the signal geneato wee set and the scanning head of the lase vibomete was moved though the points of the pescibed gid and the maximum velocity was ecoded at each point. This pocedue was used to ceate the contou plots pesented in Figues 3.10 to 3.12 of Section The maximum velocity at the cente of the dive was detemined fom the timedependent velocity measuements fo the ange of test conditions indicated in Table 3.1. Using the velocity measuements the maximum displacements and maximum acceleations wee detemined. Table 3.1: Dive test conditions fo measuements made with lase vibomete Wavefom Fequency (Hz) V pp1 (V) Sine The dive velocity v vaies with time accoding to Equation v( t) = Vmax sin(2πft) 3.11 The displacement of the dive d is 39

56 Vmax d( t) = v( t) dt = cos(2πft) πf The dive acceleation a is dv a( t) = = 2πfVmax cos(2πft) 3.13 dt Thus the maximum displacement and maximum acceleation at the cente of the dive ae scala multiples of the measued maximum velocity at the cente of the dive as shown in Equations 3.14 and V d = max max πf amax = 2πfV max 3.15 The maximum velocity pe volt is gaphed as a function of fequency fo the dive in Figue 3.7. The maximum displacement and maximum acceleation pe volt ae gaphed as functions of fequency in Figues 3.8 and 3.9. The shapes of the gaphs in Figues 3.7 to 3.9 ae simila because the maximum displacement and maximum acceleation ae scala multiples of the maximum velocity. The data epoted in Table 3.2 ae the dive voltages (V pp2 ) applied duing measuement of the velocity at the cente of the dive. The dive-esisto voltage (V pp1 ) was held constant at each fequency and the dive voltage was ecoding duing velocity measuement Results Gaphs of the maximum velocity displacement and acceleation pe volt as functions of fequency ae pesented in Figues Each gaph epesents the elationship of the independent vaiable (velocity acceleation o displacement) pe unit input voltage in the fequency domain. The ange of fequencies in which doplets wee 40

57 geneated in this investigation was 360 to 575 Hz. Repoted in Table 3.2 ae the dive voltages (V pp2 ) that wee ecoded at each fequency fo constant dive-esisto combination voltages (V pp1 ). Table 3.2: Dive voltages (V pp2 ) fo constant V pp1 duing dive chaacteization V pp2 (V) Fequency (Hz) V pp1 (V)

58 A gaph of the maximum velocity pe volt measued at the cente of the dive as a function of fequency is shown in Figue 3.7. At a specific fequency the velocity is detemined by multiplying the velocity pe volt by the measued dive voltage. The ange of fequencies in which doplets wee geneated in this investigation was 360 to 575 Hz Velocity pe volt ((m/s)/v) Fequency (Hz) Figue 3.7: Maximum velocity pe volt as a function of fequency A gaph of the maximum displacement pe volt as a function of fequency is shown in Figue 3.8. The displacement at a specific fequency is detemined by multiplying the displacement pe volt by the measued dive voltage. 42

59 6 5 Displacement pe volt ( µm/v) Fequency (Hz) Figue 3.8: Maximum displacement pe volt as a function of fequency A gaph of the maximum acceleation pe volt as a function of fequency is shown in Figue 3.9. The acceleation at a specific fequency is detemined by multiplying the acceleation pe volt by the measued dive voltage Acceleation pe volt ((m/s 2 )/V) Fequency (Hz) Figue 3.9: Maximum acceleation pe volt as a function of fequency 43

60 The displacements and acceleations at the cente of the dive wee used fo evaluation of the conditions necessay fo doplet geneation. The ange of fequencies in which doplet geneation was poduced in this investigation was 360 to 575 Hz. Fom the gaphs shown in Figues 3.7 to 3.9 the highest velocity pe volt displacement pe volt and acceleation pe volt fo the conditions tested in this investigation wee (m/s)/v 1.18 µm/v and (m/s 2 )/V. Figues 3.10 to 3.12 ae contou plots of the velocities at selected azimuthal points on the suface of the dive fo diffeent fequencies. These plots show the axisymmety of the velocity distibution ove the suface of the dive duing vibation. These contou plots wee made using the scanning capability of the lase vibomete. A ectangula gid consisting of 542 points coveing the suface of the dive was pogammed to poduce contou plots of the local vetical velocity of the dive. Using the lase vibomete the peak velocity at each point was ecoded and the contou plots wee geneated. The contou plots confimed that the maximum velocity of the dive was at its cente. The velocity scales used in each of the contou plots wee diffeent so the heights indicated in the side views cannot be compaed. These plots wee made at fequencies that fall within the ange of fequencies in which doplet geneation occued duing this investigation. 44

61 Figue 3.10: Ovehead and side views of contou plot of dive velocity at 400 Hz and V pp1 = 15 V Figue 3.11: Ovehead and side views of contou plot of dive velocity at 400 Hz and V pp1 = 40 V Figue 3.12: Ovehead and side views of contou plot of dive velocity at 500 Hz and V pp1 = 40 V 3.3 Spay Chaacteization Chaacteization of the spays ceated by doplet geneation on the suface of a liquid laye is essential fo the calculation of pefomance of a heat tansfe cell that incopoates doplet cooling. Because doplet cooling heat tansfe ates ae dependent upon popeties of the liquid doplets it is impotant that doplet sizes and flow ates ae 45

62 known fo each doplet geneation condition. The pocedues used to collect spaychaacteizing data ae descibed in the next section Appaatus and Pocedues An enclosue to secue the dive shown in Figue 3.13 was fabicated to maintain the depth of the dive elative to the suface of the pool to keep the electical teminals on the backside of the dive fom coming in contact with wate and to clamp the edges of the dive in a epoducible fashion. The enclosue was fabicated fom stainless steel to pevent oxidation when it was submeged in wate. Dive White elbow Oute shell Enclosue bottom Figue 3.13: Photogaph of dive secued in enclosue The dive is secued in the enclosue by sliding it into the oute shell of the enclosue. A secuing colla is then slid into the oute shell sandwiching the dive between the colla and a lip in the oute shell. The lip and font edge of the secuing colla secuely clamp the oute 0.79 mm of the peimete of the dive. The electical leads connected on the back of the dive ae then fed though aligned holes in the secuing colla and the oute shell of the enclosue. The bottom of the enclosue is then 46

63 scewed into place and toqued to 0.56 N-m. This toque adequately secues the dive and povides a watetight seal without defoming the metal plate of the dive. The two white elbows shown in Figue 3.13 on the peimete of the oute shell of the enclosue ae passageways that allow passage of the electical leads though the enclosue. They potude above the suface level of the liquid pool in which the enclosue is placed duing expeiments. The passageways also povide fo atmospheic pessue equalization on the font and back sufaces of the dive. Duing Heffington s (2000) development of heat tansfe cells it was found that atmospheic pessue equalization was needed in ode to educe the influence of the closed volume that existed behind the vibating dive. Inside a closed heat tansfe cell the pessue on the liquid side of the cell inceased as doplets evapoated at the heated suface. Without equalization of the inceased pessue that was caused by the evapoation of wate doplets the pessue diffeential on the font and back of the dive loaded the dive enough to significantly decease its displacement. Afte connecting the electical leads to the signal geneato the dive and its enclosue wee placed in a pool which is filled with wate to the pescibed level fo the expeiment. The liquid pool enclosue is squae was made of 6.4 mm thick Plexiglas sheets and measues 620 x 620 x 145 mm. Placement of the dive and enclosue in the liquid pool is shown in Figue To investigate the influence of constained pool boundaies on doplet geneation two cicula ings with inside diametes of 38.1 mm and 50.8 mm wee centeed ove the dive to constain the liquid pool. 47

64 Liquid pool Dive and enclosue Plexiglas enclosue Figue 3.14: Photogaph of dive in enclosue in liquid pool Fequency Range Measuement To detemine the fequencies at which doplet geneation fom the liquid laye occu the height of the pool was vaied theeby changing the thickness of the liquid laye on the dive. Based on findings by Vinig et al. (1988) it was expected that the fequencies at which doplets would be poduced would vay with the liquid laye thickness and dive displacement in the pesent investigation. As shown in Section the dive displacement is dependent on the fequency and applied voltage. Once the dive and enclosue wee placed in the pool the depth of the pool was measued and the liquid laye thickness was detemined. The signal geneato was then adjusted until the desied voltage was measued on the oscilloscope. The fequency of the electical signal fom the signal geneato to the dive was then inceased fom a minimum value of 300 Hz until waves could be obseved on the suface of the liquid laye. The fequency was then futhe slowly inceased until doplets began to be ejected fom the suface. The 48

65 fequency was slowly inceased until doplet geneation fom the suface ceased. The fequency ange of doplet geneation was then defined by the fequencies at which doplet geneation began and ended fo that paticula liquid laye thickness and dive voltage. The pocedue above was followed fo an open pool and fo pools with constained boundaies. Constained pools wee poduced by enclosing the liquid suface above the dive using PVC pipe sections. Pipe sections with inne diametes of 38.1 mm and 50.8 mm wee used. The ings wee centeed above the dive and elevated above 88the pool bottom allowing the wate level inside the ing to emain the same as the wate level in the est of the pool Doplet Diamete Measuement The diametes of the doplets ejected fom the suface of the liquid laye wee measued using high-speed photogaphy. Accoding to the esults of Lang s wok (1962) it was expected that the aveage doplet diamete would decease as the dive fequency inceased. Afte setting the pescibed conditions still images of the spays wee ecoded using a high-speed digital camea. The camea was positioned and its apetue was adjusted so that the depth of field included only the space bounded by the nea and fa edges of the dive. In Figue 3.15 is a schematic (not to scale) of the position of the camea elative to the dive fo optimum depth of field adjustment. The photogaphic setup was adjusted so that the depth of field included only the space above the dive so that the doplets that wee in focus in the still images wee the only ones included in the size measuements. Establishment of the camea depth of field was impotant fo this pocedue because all of the geneated doplets did not tavel pependicula to the pool 49

66 suface. Figue 3.16 is an example of a digital still image that was used fo doplet measuement. If the depth of field had encompassed a lage aea doplets moving towad o away fom the camea would appea lage o smalle than they actually appea in the digital image. Dive Camea A B C Figue 3.15: Schematic of ovehead view of photogaphic setup fo depth of field A-nea distance B-focus distance C-fa distance Piezoelectic dive lead Ejected doplets Liquid laye Suface waves Figue 3.16: Photogaph of spay geneated at 500 Hz fom 2.4 mm liquid laye To calculate the diametes of the doplets fom the still images aluminum cylindes wee included in the field of view of the dive. The diametes of those cylindes wee measued and thei dimensions wee ecoded. Once the still images wee 50

67 obtained imaging softwae was used to detemine the diamete of the cylindes in pixels. The cylinde diamete measued in pixels and in millimetes was used to calculate a atio of pixels pe millimete. The esulting atio was used to convet the diametes of the in-focus doplets fom pixels to millimetes. In-focus doplets wee detemined by visual inspection of the doplets in the digital image. All in-focus doplets wee measued to detemine the aveage doplet diamete. The numbe of doplets in each digital image anged fom 50 to 200 depending on the test conditions. The aluminum cylindes wee ecoded in each still image so that convesion atios wee calculated fo each image. Because the position of the camea was the same fo each ecoded expeiment the atio calculated fo each digital image and used to detemine the sizes of individual doplets was 17 pixels pe millimete Doplet Flow Rate Measuement Measuements of doplet mass flow ates wee detemined as a function of height above the liquid laye using a vetical stage onto which an open-cell sponge was attached. The sponge was cicula in coss-section with a 69.2 mm diamete esulting in an inteception aea of cm 2. The inteception aea of the sponge was lage and absobent enough so that all doplets ejected to the height of the sponge wee absobed. The stage shown in Figue 3.17 was adjusted to pescibed heights above the dive so that it intecepted doplets as they wee ejected fom the liquid suface. The esolution fo adjustment of the vetical position of the sponge was 0.79 mm pe evolution of the cank handle. 51

68 Cank handle Guide ods Theaded od Open cell sponge Figue 3.17: Photogaph of vetical stage and open-cell sponge The fist step in detemining the flow ate of doplets was to measue the initial mass of the sponge. The vetical stage and sponge wee then placed ove the dive and the signal geneato was activated fo a pescibed duation of 5 to 15 seconds depending on the doplet geneation stength. Shote duations wee used fo high atomization ates and longe duations fo lowe atomization ates. The sponge was then emoved fom the pool and its mass was measued. The mass flow ate was calculated by dividing the change in mass of the sponge by the duation of the test. Finally the aveage doplet diamete measued in the digital image fo the same doplet geneation condition was used to calculate the doplet flow ate Results Fo the open and constained pools as the liquid laye thickness was inceased the fequencies at which doplet geneation occued deceased. In addition the ange of fequencies at which doplet geneation occued deceased as the liquid laye thickness 52

69 inceased. These esults indicated that not only was thee a dependence of doplet geneation on the fequency of dive vibation but also a dependence on the thickness of the liquid laye. Additionally in this investigation the geneation of doplets was dependent on the dive voltage due to the dependence of the dive displacement and acceleation on voltage. A dependence of wave amplitude on liquid laye thickness was indicated in the esults of Vinig et al. s wok in Fo the fequency ange that Vinig et al. investigated (2.5 to 5 Hz) they obseved that fo constant liquid laye thickness and excitation amplitude that the wave amplitude deceased as the dive fequency inceased. The wave amplitude was not measued in the pesent investigation howeve because doplets ae ejected fom the wave cests Vinig et al. s esults indicated that doplet geneation would be indiectly affected. Fo the fequencies used in the pesent investigation (360 to 575 Hz) the fequency ange in which doplet geneation occued deceased as liquid laye thickness inceased as shown in Figues 3.18 and Open Pool Fequency Ranges A gaph of the maximum and minimum fequencies at which doplet geneation occued as a function of liquid laye thickness in an open pool fo a constant dive voltage of 50 V is shown in Figue Figue 3.19 includes points that indicate the conditions that existed fo maximum doplet geneation that lie between the uppe and lowe bounds of the fequency ange fo doplet geneation. Detemination of maximum doplet geneation was made by obsevation of the doplet ejection density as doplet geneation vaied with dive fequency. 53

70 lowe bound uppe bound Dive voltage - 50 V Fequency (Hz) Liquid laye thickness (mm) 7 Figue 3.18: Doplet geneation fequency ange as a function of liquid laye thickness in an open pool lowe bound uppe bound max. geneation Dive voltage - 50 V Fequency (Hz) Liquid laye thickness (mm) 7 Figue 3.19: Doplet geneation fequency ange as a function of liquid laye thickness in an open pool with indicated maximum geneation fequencies The lowe bound data points indicate the lowest fequencies at which doplet geneation occued. The uppe bound data points indicate the highest fequencies at which doplet geneation occued. Detemination of the bounding fequencies was made 54

71 based on the obsevation of constant doplet geneation fom the standing waves on the liquid suface. The uppe and lowe doplet geneation fequency bounds ae explained in consideation of findings made by Lieke et al. (1967) and Rozenbeg (1973). Befoe doplets ae geneated capillay waves must be pesent on the liquid suface. As shown by Lieke et al. thee is a minimum vibation amplitude that must be pesent fo the initiation of waves on a liquid suface. The elationship between this citical amplitude fo wave fomation and the vibation fequency is given in Equation / 3 µ ρ d c = ρ fπσ In the same study Lieke et al. also detemined that the doplet geneation inception amplitude was 3 to 6 times the citical amplitude fo wave fomation. In Figue 3.20 is a gaph of the atio of maximum dive displacement to citical amplitude fo wave fomation as a function of diving fequency in wate. The dive displacement was the maximum dive displacement fo a dive voltage of 50 V fo the ange of fequencies in which doplet geneation was poduced in this study. The citical amplitude fo wave fomation was calculated using Equation 3.16 fo wate at ambient pessue and tempeatue. As is shown the citical amplitude fo wave fomation is invesely popotional to the cube oot of fequency of vibation. Within the fequency ange fo which doplets wee geneated in this investigation the maximum displacement of the dive anged fom appoximately 5 to 10 times the citical amplitude fo wave fomation on the liquid laye. The esults wee in ageement with the findings of Sookin (1957) and Eisenmenge (1959) who found that the amplitude equied fo doplet ejection was 4 to 8 times the amplitude equied fo the geneation of suface waves. 55

72 Ratio 6 4 Dive voltage - 50 V Fequency (Hz) Figue 3.20: Ratio of maximum dive displacement to citical amplitude fo wave fomation as function of fequency in wate Duing doplet geneation as the dive displacement inceases the amplitude of the excited suface capillay waves eventually eaches a maximum limiting value at which the capillay wave motion becomes stable and peiodic (Rozenbeg 1973). When the wave motion stabilizes doplet geneation ceases which is due to the suface tension of the liquid. In an open pool fo each liquid laye thickness the uppe bound fequency fo doplet geneation coesponds to a condition at which the liquid suface waves become stable and peiodic due to the suface tension foces acting on them. As the dive fequency is inceased the suface wave wavelength and aveage doplet diamete poduced at that fequency decease. When poducing doplets with smalle diametes moe foce is equied to ovecome the inceasing liquid suface tension that acts to keep the doplet pat of the liquid laye. Doplet geneation ceases when the volume of liquid at the cest of the wave lacks the equied enegy to ovecome the suface tension exeted on it by the est of the liquid. 56

73 Constained Pool Fequency Ranges The effect of constaining the liquid laye suface within the 38.1 mm ing was to decease the highest fequencies at which doplet geneation occued also deceasing the fequency ange in which doplet geneation occued. Thee was no significant diffeence between the fequency anges measued in the open pool and those measued in the pool constained by the 50.8 mm ing. A gaph of the maximum and minimum fequencies at which doplet geneation occued as a function of liquid laye thickness using the 38.1 mm ing fo a constant dive voltage of 50 V is shown in Figue lowe bound uppe bound Dive voltage - 50 V Fequency (Hz) Liquid laye thickness (mm) Figue 3.21: Doplet geneation fequency ange as a function of liquid laye thickness in a 38.1 mm constained pool As suface waves wee poduced by the vibating dive they taveled away fom the dive. In an open pool the waves dispese adially and ae attenuated by the dissipative pocess of viscous shea. The pool boundaies ae fa enough fom the dive that the wave enegy eflected back towad the dive by the pool boundaies is 57

74 negligible. Thus thee is no influence of the pool boundaies on doplet geneation in an open pool. When using the 38.1 mm ing the pool bounday was spaced 3.97 mm fom the edge of the dive and suface waves wee eflected back towad the dive. When the dispesing suface waves and the eflected suface waves meet and ae of simila magnitude they inteact to fom standing waves (Hughes et al. 1999). Combined with Rozenbeg s assetion that capillay wave motion stabilizes at some maximum limiting amplitude the standing waves poduced by the inteaction of dispesing and eflecting waves assist in stabilization of suface waves causing doplet geneation to cease. In the 38.1 mm constained pool the uppe bound fequency is due to wave stabilization caused by suface tension and eflection of suface waves fom the solid bounday back towad the dive. The effect of the 38.1 mm constained bounday was to decease the uppe bound fequency fo doplet geneation. When using the 50.8 mm ing to constain the pool the pool bounday was spaced mm fom the edge of the dive. Simila to the case of using the 38.1 mm ing when using the 50.8 mm ing the suface waves wee eflected fom the ing back towad the dive. Due to the dissipation of wave enegy by viscous shea the eflected suface waves do not inteact with the dispesing waves fom the dive to poduce standing waves. Thus the effect of the eflected waves on wave motion and doplet geneation is negligible and stabilization of the waves is pimaily due to the suface tension foce acting on them. Consequently the fequency ange in which doplet geneation occued when using the 50.8 mm ing was the same as fo the open pool. 58

75 Aveage Doplet Diametes Aveage doplet diametes measued fom the photogaphic images ae epoted in Table 3.3 and displayed in Figue 3.22 with aveage doplet diametes calculated using Equation The esulting aveage doplet diametes wee appoximately the same fo the open and constained pools (38.1 mm and 50.8 mm). Because the doplets ae ejected fom the cests of suface waves which ae dependent on dive fequency the esults wee expected. At highe fequencies the aveage diametes of geneated doplets follow a deceasing tend with inceasing fequency of vibation as developed by Lang (1962) which elates the aveage diamete of doplets geneated fom capillay waves to liquid popeties and diving fequency. Lang developed this equation following wok done by Rayleigh (1896) in which he detemined a elationship between capillay wavelength and focing fequency Aveage doplet diamete (micons) Lang Open pool 38.1 mm pool 50.8 mm pool Fequency (Hz) Figue 3.22: Aveage doplet diamete (D b ) as a function of fequency in wate fom Table 3.3 and Equation

76 1/ 3 0 8πσ λ = f ρ D b = 3.17 Accoding to Equation 3.17 developed by Lang the aveage doplet diamete should steadily decease with inceasing dive fequency. Accodingly fo the ange of fequencies epoted in Table 3.3 the aveage doplet diamete pedicted using Equation 3.17 should have anged fom 674 to 533 µm. The aveage doplet diamete measued in this study anged fom 587 to 532 µm. Doplets geneated at fequencies below 460 Hz did not follow the tend as shown in Figue Results show that doplets geneated at fequencies above 460 Hz follow the tend pedicted using Equation Depatue of the aveage doplet diametes fom the pedicted tend may be due to limits on the applicability the equation at such low fequencies. Lang et al. developed Equation 3.17 fom expeiments conducted at fequencies of 10 khz to 800 khz. Its applicability to fequencies as low as those used in this eseach has not been peviously poven. Dive fequency (Hz) Table 3.3: Measued aveage doplet diametes Aveage doplet diamete (µm) Smallest doplet diamete (µm) Lagest doplet diamete (µm) Pool bounday constaint Open Open Open Open Open Open Open Open Open Open mm mm mm mm 60

77 Table 3.3 continued mm mm mm mm Doplet Mass Flow Rates Doplet mass flow ates wee dependent on the dive fequency dive voltage liquid laye thickness and inteception distance above the liquid laye suface. In Figue 3.23 is a gaph of the measued doplet mass flow ate as a function of dive fequency in an open pool. Duing the expeiments the dive voltage was held constant at 50 V which dictated the dive displacement and acceleation and the mass flow ates wee measued at thee inteception distances fo five liquid laye thicknesses. The inteception distance above the liquid laye suface was vaied by changing the position of the sponge elative to the liquid laye suface. As the liquid laye thickness deceases the dive fequency must be inceased to sustain doplet geneation and doplet mass flow ate inceases. The dive displacement is not constant because it is dependent on the dive voltage and fequency. The fequency dependence of the dive displacement is shown in Figue 3.8. The doplet mass flow ates measued fo five liquid laye thicknesses t ae shown in Figue It is shown that the flow ate inceases as the inteception distance is deceased. Fo each liquid laye thickness the doplet mass flow ate was measued at thee inteception distances and at multiple fequencies. The diffeence between the doplet mass flow ates at the inteception distances of 42 mm and 22 mm is geate than the diffeence between the doplet mass flow ates at 61

78 the inteception distances of 62 mm to 42 mm. In Figue 3.23 ae the doplet mass flow ates measued fo five liquid laye thicknesses. As shown in Figue 3.18 doplet geneation occus within an indicated fequency ange fo a given liquid laye thickness. Fo each liquid laye thickness shown ( and 4.51 mm) doplet mass flow ates wee measued at distances of 62 mm 42 mm and 22 mm fom the liquid suface. Thus in Figue 3.23 the symbols beneath each designated liquid laye thickness epesent the doplet mass flow ates measued at each inteception distance. Gaps exist between the sets of mass flow ate data fo each liquid laye thickness because the flow ates wee measued only fo potions of the full fequency anges in which doplet geneation occued. Fo example the fequency ange in which doplet geneation occued fo the liquid laye thickness of 3.61 mm was 390 to 460 Hz as shown in Figue The doplet mass flow ates fo the same liquid laye thickness wee only measued in the ange of 400 to 425 Hz. Measuement of doplet mass flow ates ove the full fequency ange fo each liquid laye thickness was not made because nea the boundaies of the fequency ange in which doplet geneation occued the numbe of and ejection heights of geneated doplets wee too low to be measued using the appaatus depicted in Figue

79 Inteception distance - 62 mm Inteception distance - 42 mm Mass flow ate (g/s) Inteception distance - 22 mm Dive voltage - 50 V t mm t mm t mm t mm t mm Dive fequency (Hz) Figue 3.23: Doplet mass flow ate as a function of dive fequency fo constant dive voltage and liquid laye thickness t in an open pool Because not all of the geneated doplets tavel the same distance fom the liquid laye suface the measued mass flow ates decease as the inteception distance inceases. The mass flow ate vaiation fo liquid laye thicknesses of 4.51 and 3.61 mm is due to the vaiation in the intensity of doplet geneation with fequency fo each of the liquid laye thickness. A decease in mass flow ate is not ecoded fo the thinne liquid laye thicknesses ( and 2.03 mm) because the dop in doplet geneation was so geat that the mass flow ate was not measuable even at the shotest inteception distance of 22 mm. Thoughout the doplet geneation fequency ange a numbe of combinations of liquid laye thickness inteception distance and dive fequency can be used to poduce a specified doplet mass flow ate. Fo instance a doplet mass flow ate of appoximately 0.10 g/s can be poduced by combinations of liquid laye thickness inteception distance and dive fequency of (3.61 mm 22 mm 410 Hz) (2.87 mm 42 mm 450 Hz) (2.48 mm 42 mm 485 Hz) and (2.03 mm 62 mm 550 Hz). 63

80 The maximum doplet mass flow ates poduced when using the cicula ings to enclose the pool wee not significantly diffeent fom those poduced in the open pool. In Figues 3.24 and 3.25 ae the doplet mass flow ates measued fo fou liquid laye thicknesses t in 38.1 mm and 50.8 mm constained pool. Fo each liquid laye thickness the doplet mass flow ate was measued at thee inteception distances fo multiple fequencies. The 38.1 mm pool bounday had the effect of educing the uppe bound of the fequency ange fo doplet geneation. The aveage doplet diametes and mass flow ates wee not significantly affected as shown in Figues 3.22 and The constained boundaies eflect waves back towad the dive assisting in stabilizing the wave motion which affects the fequency anges fo doplet geneation. The aveage doplet diametes and mass flow ates ae simila to those poduced in an open pool because the pool boundaies do not affect suface wavelengths caused by the dive vibation no the mechanisms of ejection of doplets fom the wave cests. Mass flow ate (g/s) Inteception distance - 62 mm Inteception distance - 42 mm Inteception distance - 22 mm Dive voltage - 50 V t mm t mm t mm t mm Dive fequency (Hz) Figue 3.24: Doplet mass flow ate as a function of dive fequency fo constant dive voltage and liquid laye thickness t in a 38.1 mm constained pool 64

81 Inteception distance - 62 mm Inteception distance - 42 mm 0.40 Inteception distance - 22 mm Mass flow ate (g/s) Dive voltage - 50 V t mm t mm t mm t mm Dive fequency (Hz) Figue 3.25: Doplet mass flow ate as a function of dive fequency fo constant dive voltage and liquid laye thickness t in a 50.8 mm constained pool By application of appopiate dive fequency dive voltage liquid laye thickness and inteception distance a wide ange of doplet mass flow ates ae possible. Fo a fixed dive voltage the maximum possible doplet mass flow ate inceases as the liquid laye thickness deceases. To continue geneating doplets as the liquid laye thickness is deceased the dive fequency (displacement and acceleation) must be inceased. Fo a constant liquid laye thickness and dive fequency the dive voltage is inceased o deceased to incease o decease doplet geneation and thus the doplet mass flow ate. Doplet mass flow ate as a function of dive fequency fo multiple dive voltages in an open and a 38.1 mm constained pool ae pesented in Figues 3.26 and Results show that fo a constant dive fequency and liquid laye thickness doplet mass flow ate inceases with dive voltage. This is expected since the dive displacement and acceleation incease with voltage. At the highest fequencies fo which doplet geneation occus highe dive voltages than when at the lowest 65

82 fequencies ae necessay to maintain doplet geneation. At fequencies nea the lowe bound doplet geneation can continue at lowe dive voltages. These esults indicate that the fequency ange fo which doplet geneation occus deceases as dive voltage (displacement and acceleation) deceases. Thus the doplet geneation fequency ange fo a constant liquid laye thickness becomes naowe as the dive displacement is deceased. Mass flow ate (g/s) Dive voltage - 50 V Dive voltage - 45 V Dive voltage - 40 V Dive voltage - 35 V Inteception distance - 42 mm t mm t mm t mm Dive fequency (Hz) Figue 3.26: Doplet mass flow ate as a function of dive fequency fo constant inteception distance and liquid laye thickness t in an open pool 66

83 Dive voltage - 50 V Dive voltage - 45 V Dive voltage - 40 V Dive voltage - 35 V Mass flow ate (g/s) Inteception distance - 42 mm t mm t mm t mm Dive fequency (Hz) Figue 3.27: Doplet mass flow ate as a function of dive fequency fo constant inteception distance and liquid laye thickness t in a 38.1 mm constained pool The measued doplet mass flow ate is dependent on the liquid laye thickness dive voltage dive fequency and inteception distance. Fo constant dive voltage and fequency the doplet mass flow ate inceases as the liquid laye thickness deceases and inceases as the inteception distance deceases. At a constant dive fequency doplet geneation can be vaied by adjustment of the dive voltage that changes the dive s displacement. Deviation fom the dive fequency at which maximum geneation is poduced esults in a decease in the doplet mass flow ate. In Figues 3.23 though 3.25 the doplet mass flow ate ends abuptly because when the fequency was inceased doplet poduction deceased to a level that it could not be measued at the pescibed inteception distance. Doplets continued to be poduced within the fequency anges pesented in Figues 3.18 and 3.21 but did not tavel fa enough fom the liquid laye suface to impinge on the sponge. 67

84 3.4 Doplet Geneation Analysis Using the dive and spay chaacteization esults fom Sections 3.2 and 3.3 the following pocedue was used to analyze the data. Fo each expeimental condition at which doplet geneation occus the following infomation is known fom the esults of the spay and dive chaacteizations. Dive fequency Dive voltage Liquid laye thickness Dive maximum amplitude Dive maximum acceleation In addition to the paametes identified by Goodidge et al. liquid laye thickness is a paamete that was consideed to influence the conditions necessay fo doplet geneation. Theefoe in this investigation the paametes consideed to affect doplet geneation wee dive fequency dive acceleation dive displacement and liquid laye thickness. Two appoaches wee taken in an effot to detemine the functional dependencies among the acceleation necessay to geneate doplets the liquid laye thickness and the dive fequency. Due to poo functional dependences the esults of the fist appoach wee unsatisfactoy and could not be used to detemine the ange of conditions that led to doplet geneation. Results of the second appoach wee successful and yielded a dimensionless paamete that can be used to detemine the ange of conditions that lead to doplet geneation. Results using the second appoach ae pesented fo doplet geneation in the open and 38.1 mm constained pool. Because the 50.8 mm constained 68

85 pool did not affect the doplet geneation fequency ange the esults fo this configuation ae the same as fo the open pool Fist Appoach Detemination of the dependence of the acceleation necessay to poduce doplets on fequency was detemined by gaphing the acceleation as a function fequency shown in Figue 3.28 fo the uppe and lowe bounds of the fequency ange in which doplets wee geneated (Figue 3.18). Figue 3.29 is a gaph of the acceleation as a function of liquid laye thickness fo the same conditions. Both gaphs ae of data collected fo doplet geneation in an open pool fo a dive voltage of 50 V lowe bound uppe bound Acceleation (m/s 2 ) a = f a = f Fequency (Hz) Figue 3.28: Acceleation necessay fo doplet geneation as a function of fequency in an open pool 69

86 lowe bound uppe bound Acceleation (m/s 2 ) a = t a = t Liquid laye thickness (mm) 6 Figue 3.29: Acceleation necessay fo doplet geneation as a function of liquid laye thickness in an open pool The dependence of acceleation on fequency and liquid laye thickness wee detemined by fitting egession lines though the uppe bound and the lowe bound data. As peviously stated the esults of this appoach wee unsatisfactoy due to poo functional dependences. As shown in Figue 3.29 the lines connecting the uppe bound data points and lowe bound data points intesect. The equations descibing the egession lines ae included in the figues. The values of the exponents on the dependent vaiables fequency and liquid laye thickness in the egession equations wee used to detemine the elationship in Equation The symbol a is the maximum dive acceleation f is the dive fequency and t is the liquid laye thickness a ~ f t 3.18 A dimensionally consistent fom shown in Equation 3.19 was developed that descibes the acceleation necessay fo doplet geneation as functions of fequency liquid laye thickness and liquid popeties. 70

87 16 / 9 2 / 3 1/ 9 a ~ f t ( σ / ρ) 3.19 The values of the exponents on the dependent vaiables in Equation 3.18 wee conveted to factions to fomulate Equations 3.19 and 3.20 which ae dimensionally consistent. = ( t σ ρ 16 / 9 2 / 3 1/ 9 a 318.5E 6) f ( / ) 3.20 Equation 3.20 was developed by fitting the fom to the expeimental uppe and lowe bound data. This equation is simila to the one developed by Goodidge et al. fo low viscosity fluids a = 0.239ω 4/3 (σ/ρ) 1/3 (Equation 3.3). The diffeence between Equations 3.3 and 3.20 is the inclusion of the liquid laye thickness in Equation Liquid laye thickness was not included in Goodidge et al. s acceleation equation because it was constant fo thei expeiments. Because both acceleation equations must be dimensionally consistent the exponents on the fequency and fluid popeties ae diffeent in each equation. The calculated constant of Equation 3.20 is dimensionless. Assuming wate as the woking fluid the acceleation equied fo doplet geneation can be calculated using Equations 3.3 and Equation 3.3 (Goodidge et al.) was developed based on esults of expeiments in which the liquid laye thickness was constant at 10.0 cm. Calculation of the acceleation using Equation 3.20 (Pytle) fo the common liquid laye thickness of 10.0 cm yields consideably lowe values than values calculated using Equation 3.3. At a fequency of 100 Hz Equations 3.3 and 3.20 yield values of 53.7 m/s 2 and 1.8 m/s 2 espectively. The diffeences between the calculated acceleations is geat because calculation of acceleation using Equation 3.20 fo a liquid laye thickness of 10 cm is well above the maximum liquid laye thickness (5.24 mm) fo which data was collected to develop Equation At a fequency of 100 Hz and a liquid laye thickness of 4 mm Equation 3.20 yields an acceleation of

88 m/s 2. Because Equation 3.3 was developed fo the esults of expeiments in which the liquid laye thickness was held constant the liquid laye thickness is not included in the equation. Due to diffeences in the expeimental conditions of Goodidge et al. and the pesent investigation the calculated doplet geneation acceleation diffeed as indicated by the vaiables in Equations 3.3 and Figue 3.30 is a gaph of acceleation as a function of fequency fo the uppe and lowe bounds of acceleation fo doplet geneation in an open pool fo each liquid laye thickness. The solid makes ae the same acceleation data as shown in Figue 3.28 while the open makes epesent acceleation data calculated using Equation The lines in the gaph epesent egession lines fit though the calculated acceleation data at the uppe and lowe bounds of the fequency anges in which doplet geneation occued. The desied esult was that the lines epesent uppe and lowe bounds of acceleation necessay fo doplet geneation Acceleation (m/s 2 ) lowe bound uppe bound calculated lowe bound calculated uppe bound Fequency (Hz) Figue 3.30: Acceleation necessay fo doplet geneation as a function of fequency fo Equation

89 The esults in Figue 3.30 indicate that the fist appoach taken to detemine the bounds fo the acceleation necessay fo doplet geneation as a function of fequency is not adequate. Due to poo functional dependence the solid points that epesent actual acceleation values in the gaph do not fall on o between the lines poduced using Equation Thus Equation 3.20 does not povide an adequate means fo detemination of conditions necessay fo doplet geneation within the obseved fequency bounds. This appoach esulted in an acceleation equation that was dependent on fequency and liquid laye thickness but did not indicate boundaies fo a ange of conditions that lead to doplet geneation. Results of the study done by Goodidge et al. wee not consistent with the esults of the pesent study. In thei study Goodidge et al. identified a fequency dependent minimum acceleation necessay fo doplet geneation. In the pesent study doplet geneation occued fo a ange of combinations of acceleation displacement and liquid laye thickness conditions. Thus the appoach Goodidge et al. used to identify conditions necessay fo doplet geneation could not be used to identify doplet geneation conditions fo the expeiments pefomed in the pesent investigation Second Appoach Anothe appoach taken to detemine the ange of conditions necessay fo doplet geneation was based on the detemination of a non-dimensionalized acceleation. Nondimensionalization of the acceleation with consideation of the wave behavio on the liquid fee suface poved to be the appoach needed to detemine conditions fo doplet geneation. 73

90 As peviously stated the pesence of suface waves whethe they ae gavity o capillay waves is necessay fo doplet geneation. The speed of the wave which is the speed of popagation of a distubance on its suface depends on its wave numbe. The wave speed is elated to the wave numbe as u w ω = = k g k σk + tanh ρ ( kt) σk ω = gk + tanh( kt) 3.22 ρ Whee u w is the wave speed ω is the angula fequency of the wave k is the wave numbe g is gavitation acceleation σ is the liquid suface tension ρ is the liquid density and t is the liquid laye thickness. Equation 3.22 which is the dispesion equation fo suface waves was developed fom potential flow theoy by solving Laplace s equations with appopiate bounday conditions (Dean and Dalymple 1991). The equation is applicable to conditions fo which the wave amplitude is small elative to the wavelength and liquid depth. It descibes waves that occu on the suface not accounting fo the geneation of doplets. Fom the cests of these suface waves doplets ae geneated as pat of the wave cest detaches and tavels away fom the liquid suface. The wave numbe is elated to the wavelength shown in Figue 3.31 by Equation π k = 3.23 λ 74

91 λ Liquid fee suface Liquid laye h 0t g + f(t) Figue 3.31: Schematic of vetically oscillating vibation of liquid laye The wave speed was detemined to be an impotant paamete fo nondimensionalization of the acceleation that is necessay fo doplet geneation. The wave speed is useful because as shown in Equation 3.21 it embodies effects of local acceleation changes in liquid laye depth and wavelength; all of which vay at diffeent doplet geneation conditions. In this study because the maximum acceleation at the cente of the dive anges fom seven to fifty times the acceleation of gavity duing doplet geneation the gavity tem in Equation 3.22 is eplaced by the dive acceleation fo calculation of the wave speed. The wavelength of the geneated capillay waves is calculated using the elationship developed by Rayleigh (1896) shown in Equation πσ λ = 2 f ρ 1/ Using a linea stability analysis James (2000) non-dimensionalized the foms of the continuity and Navie-Stokes equations that goven the flow of a vetically oscillating liquid laye shown schematically in Figue The following paametes esulted fom the non-dimensionalization she pefomed which wee scaled on the liquid laye thickness and a scaling velocity. 75

92 ρu s t Re = µ ρu s t We = σ 3.26 gt Bo = U s 2 t aρ A = 3.28 σ tω Ω = 3.29 U s These dimensionless paametes ae Reynolds numbe (Re) Webe numbe (We) Bond numbe (Bo) dimensionless amplitude (A) and dimensionless fequency (Ω). The symbol t epesents the liquid thickness depicted in Figue 3.31 and U s is a scaling velocity. The Reynolds numbe is the atio of the inetial to viscous foces of the liquid. The Webe numbe is a atio of the inetial and suface tension foces of the liquid. The Bond numbe is a atio of gavitational and inetial foces of the liquid which is the ecipocal of the Foude numbe. The Foude numbe is an impotant dimensionless paamete fo use in this study because it is the atio of the inetial foce on an element of liquid to the weight of that element. The acceleation necessay fo doplet geneation is non-dimensionalized as shown in Equation 3.30 whee a* is the dimensionless acceleation a is the dive acceleation t is the liquid laye thickness f is the dive fequency and d is the dive displacement. at a * = 3.30 ( fd ) 2 76

93 The esulting gaph of dimensionless acceleation as a function of fequency is shown in Figue The gaphed expeimental data points now in dimensionless fom povide maginally bette claity of the uppe and lowe bounds of the fequency ange necessay fo doplet geneation compaed to when the acceleation was plotted in dimensional fom in Figue The bounds coespond to the conditions pesent at the uppe and lowe bounday fequencies at which doplet geneation occued fo the liquid laye thicknesses in an open pool. Howeve thee is still consideable scatte in the bounding data points which should epesent distinct doplet geneation boundaies. The egession lines that epesent the uppe and lowe bounds fo the conditions necessay fo doplet geneation pooly fit the data points lowe bound uppe bound a* Fequency (Hz) Figue 3.32: Dimensionless acceleation (a*) as a function of fequency in an open pool In its pesent fom the dimensionless acceleation does not embody the effects of changes in the Foude numbe defined as the atio of acceleation and inetial foces. In this investigation F is defined using the calculated wave speed the acceleation of the dive and the liquid laye thickness. The acceleation and inetial foces ae caused by 77

94 the local acceleation povided by the dive and the geneation of waves on the liquid suface. In Equation 3.31 the dimensionless acceleation is edefined so that it includes the atio of those foces though the Foude numbe. at ** = 3.31 a 2 ( fd ) F whee 2 uw F = 3.32 at Thus the dimensionless acceleation becomes 2 at a ** = 3.33 fdu w a** is a new dimensionless paamete developed in this investigation. It is a atio of the vibating dive s foce to the inetial foce of the suface capillay waves. Figue 3.33 is a gaph of the edefined dimensionless acceleation a** as a function of fequency fo the same expeimental data gaphed in Figues 3.30 and These esults moe clealy define the bounds of dimensionless acceleation as a function of fequency that ae necessay fo doplet geneation fom the liquid suface. The powe egession lines that ae dawn though the dimensionless acceleation points epesent the conditions at the uppe and lowe boundaies of the fequency anges fo which doplet geneation occus. Figue 3.34 includes dimensionless acceleation values calculated fo data collected at intemediate fequencies between the uppe and lowe bounds in which doplet geneation occued. 78

95 lowe bound uppe bound a** Fequency (Hz) Figue 3.33: Bounds of dimensionless acceleation (a**) as a function of fequency in an open pool a** a** = (9.286E+13)f a** = (2.198E+20)f Fequency (Hz) Figue 3.34: Dimensionless acceleation (a**) as a function of fequency at intemediate fequencies in an open pool The dimensionless acceleation data gaphed in Figue 3.34 epesent the ange of conditions fo which doplet geneation occus in an open pool using the dive descibed and chaacteized in Section 3.2. The data points ae dimensionless acceleation values calculated fo doplet geneation that occued within the fequency ange defined by the 79

96 uppe and lowe fequency bounds. Using the powe egession cuves that wee dawn though the data points that epesent the uppe and lowe bounds fo doplet geneation at a dive voltage of 50 V Equations 3.34 and 3.35 can be used to detemine conditions necessay fo doplet geneation. At lowe diving voltages the displacement and acceleation of the dive ae smalle deceasing the magnitude of a**. Fo doplet geneation at a constant dive fequency and liquid laye thickness the dive voltage can be vaied as long as a** is within the bounds indicated in Figue Calculated dimensionless acceleation values that lie between the uppe and lowe bounds shown in Figue 3.33 indicate conditions at which doplet geneation occus in an open pool. Equations 3.34 and 3.35 espectively epesent the lowe and uppe bounds detemined fo wate in an open pool and fo a 50.8 mm constained pool. The liquid laye thicknesses fo which the equations wee developed wee fom 1.88 to 5.24 mm. The piezoelectic dive was mm in diamete and 0.10 mm thick a ** = f a ** = f 3.35 The dimensionless acceleation a** is defined in Equation 3.33 and the fequency of the dive is measued in units of Hz. The dimensionless acceleation a** esults fo the 38.1 mm constained pool ae pesented in Figue The uppe and lowe bounds that ae defined by the dimensionless acceleation values ae diffeent fom the esults fo the open pool. The diffeence is due to the diffeence in the doplet geneation fequency anges shown in Figues 3.18 and

97 lowe bound uppe bound a** a** = (3.165E+20)f a** = (2.451E+30)f Fequency (Hz) Figue 3.35: Bounds of dimensionless acceleation (a**) as a function of fequency in a 38.1 mm constained pool Calculated dimensionless acceleation values that lie between the uppe and lowe bounds shown in Figue 3.35 epesent conditions at which doplet geneation occued in a 38.1 mm constained pool. Equations 3.36 and 3.37 espectively epesent the lowe and uppe bounds detemined fo a 38.1 mm constained pool of wate fo liquid laye thicknesses between 1.28 and 6.1 mm. The same piezoelectic dive as was used in the open pool was used fo the constained pool a ** = f a ** = f 3.37 Equation 3.33 was used to calculate the dimensionless acceleation values fo all data points pesented in Figues 3.34 and Powe egessions wee then calculated fo the uppe bounday data points and the lowe bounday data points. Those egessions wee defined by Equations 3.34 and 3.35 and Equations 3.36 and Equations 3.34 and 3.35 define the lowe and uppe bounds of doplet geneation 81

98 conditions in an open pool. Equations 3.36 and 3.37 define the lowe and uppe bounds of doplet geneation conditions in a 38.1 mm constained pool. The diffeences between the pais of equations exist due to the diffeences in the fequency anges in which doplet geneation occued shown in Figues 3.18 and The diffeences in the slopes of the uppe and lowe bounds indicated diffeent fequency-liquid laye thickness elationships. Consequently those diffeences esulted in vaiation of the fequency dependent dimensionless acceleation equations that define the doplet geneation boundaies. Results show that the dimensionless acceleation can be calculated to define the uppe and lowe boundaies fo the ange of conditions in which doplet geneation occus in an open and in a constained pool. Combinations of liquid laye thickness and dive fequency displacement and acceleation can be used to calculate dimensionless acceleation values. Those values that lie between the lowe and uppe bounds indicated in Equations 3.34 and 3.35 fo the open pool and in Equations 3.36 and 3.37 fo the 38.1 mm constained pool epesent conditions fo which doplet geneation occus. Using the dive chaacteization the appopiate dive fequency and voltage can be selected to poduce the desied displacement and acceleation needed to geneate doplets fom a specific liquid laye thickness. 82

99 CHAPTER 4 HEAT TRANSFER CELL MODEL 4.1 Fomulation The pupose fo fomulating the heat tansfe model was to pedict the pefomance of a heat tansfe cell that used doplet cooling as a means fo heat dissipation. Using the esults pesented in the pevious chapte a steady state model was used to estimate heat tansfe pefomance assuming that doplet geneation could be achieved inside a closed heat tansfe cell. The expeimental data that wee descibed fo the 38.1 mm constained pool wee used in the heat tansfe model. These data include the doplet mass flow ate and doplet geneation as functions of dive voltage and fequency. The esults fom the 38.1 mm constained pool wee chosen instead of the esults fom the open pool because the constained pool bette epesents the physical configuation of the heat tansfe cell. The model is an analytical one that is fomulated to simulate heat tansfe in a heat tansfe cell in which doplets ae geneated fom a liquid laye by a dive mounted in the bottom of the cell. A schematic of the modeled heat tansfe cell is shown in Figue 4.1. The cell is a closed system so that no mass leaves the cell and only heat passes though its boundaies. As doplets ae geneated fom the liquid laye in the bottom of the cell they tavel upwad towad the top of the cell whee the heat souce is attached. The doplets impinge on the inne suface of the top of the cell. As the doplets evapoate on the hot suface heat is emoved fom the heat souce. The esulting 83

100 vapoized wate then condenses on the inside peimete suface of the cell. At the coole suface aound the inside of the cell the vapo condenses and the liquid falls into the liquid laye ove the dive. The dive ceates doplets that leave the liquid suface and the cycle continues. Heat souce/pocesso Doplets Liquid laye Fins Piezoelectic dive Figue 4.1: Schematic of modeled heat tansfe cell In the modeled heat tansfe cell key tempeatues ae used to evaluate heat tansfe ates themal esistances fluid popeties and tempeatues of othe cell components. Those key tempeatues ae labeled as T1 though T6 in Figue 4.1. The tempeatue T1 is the heat souce tempeatue T2 is the tempeatue of the inne suface on which doplets impinge and T3 is the satuation tempeatue of the fluid within the cell. The tempeatue at T4 is the tempeatue of the inne suface on which condensation occus T5 is the tempeatue of the inne suface at the bottom of the liquid laye and T6 is the ambient tempeatue. Heat tansfe within the cell walls was modeled assuming 84

101 one-dimensional steady-state conduction. Two-dimensional steady-state conduction was modeled in the top suface of the cell whee the heat souce was attached to the cell and in the fins connected to the peimete of the heat tansfe cell. The simulated heat souce at the top suface of the cell is assumed to be a micopocesso so the heat fluxes and tempeatues selected fo model simulation ae compatible with expected values typically expeienced duing the opeation of moden micoelectonic designs. Oientation of the heat souce to the cell is shown in Figue 4.2. Cell uppe wall Heat souce Doplets Figue 4.2: Heat souce oientation to uppe wall of heat tansfe cell As heat is conducted fom the heat souce though the uppe suface of the cell wall into the heat tansfe cell it speads hoizontally. Heat speading themal esistances can be used wheneve heat is conducted fom one egion to anothe with a diffeent coss sectional aea. Since the model assumes one-dimensional conduction a speading esistance can accommodate two-dimensional effects which is a function of the themal popeties and geomety of the uppe suface of the cell. In the case of heat sink applications the speading esistance is used in the cell to appoximate the effects of two-dimensional conduction. This esults in a highe local tempeatue at the location whee the heat souce is placed. In cases whee the suface aea of the heat souce is appoximately equal to the aea of suface to which it is attached the contibution of the 85

102 speading esistance to the souce tempeatue is not significant. Howeve fo the case of a small micopocesso attached to a much lage doplet cooling heat tansfe cell the themal esistance of the uppe suface of the cell can be modeled with an appopiate speading esistance. The elationship used to descibe the added two-dimensional themal esistance is given by Equation 4.1 (Lee 1998). R S A = ks 1/ 2 S A ( πa A ) S 1/ 2 H 1/ 2 H ϕk S AS R 1 + ϕks A 0 S + tanh R tanh 0 ( ϕt ) S ( ϕt ) S 4.1 whee φ is defined as 3/ 2 π 1/ 2 ϕ = + 1/ 2 AH 4.2 AS In Equations 4.1 and 4.2 R S is the themal esistance A S is the aea of the sink aea A H is the heat souce aea k S is the themal conductivity of the sink R 0 is the aveage themal esistance of the sink and t S is the sink thickness. The coelation assumes that the heat souce is centally mounted on the sink and that the heat sink has a unifom suface tempeatue. The speading model addesses neithe the shape of the heat souce no that of the sink but it was found to yield esults with an accuacy of appoximately 5 pecent ove a wide ange of applications with many combinations of diffeent souce/sink shapes as long as the aspect atio of the shapes involved did not exceed 2.5. Selection of a coelation to simulate the phase change heat tansfe ate fom the uppe suface to impinging doplets poved to be a difficult task. Afte an exhaustive eview of doplet and spay cooling liteatue no expeimental o theoetical wok was found that addessed the physical situation encounteed in the pesent model. In the modeled heat tansfe cell doplets impinge on the bottom of a heated suface that povides no means fo escape of the vapo. Fom a heat tansfe pespective this 86

103 oientation is undesiable fo achieving maximum heat emoval fom the suface. In this oientation as the impinging doplets evapoate the vapo tends to stay nea the heated suface held thee by the vapo s buoyancy. Thus vey little eseach has focused on evaluation of convective heat tansfe ates o coefficients fom inveted heated sufaces when cooled by impinging doplets. Due to the lack of heat tansfe coelations fo this oientation altenate phase change elationships wee used to simulate the mechanism of heat tansfe fom the uppe suface of the cell. Two cooling models wee selected fo use in the heat tansfe model to simulate heat tansfe fom the heated inne suface of the heat tansfe cell. They included an ideal doplet evapoation model in which the heat tansfe coefficient is dependent only on the liquid popeties and flow ate and Ohtake s (2003) mist cooling coelation. Desciptions of and the ationale fo selection of each model follows. The ideal model fo evapoative cooling heat tansfe assumed that the doplets geneated on the liquid laye emoved heat at a ate accoding to Equation 4.3. The magnitude of the heat tansfe coefficient is detemined by the liquid popeties and the mass flow ate of the doplets. This heat flux elationship was selected because it epesents the maximum cooling ate fo a given mass flow ate of doplets that stike the heated suface (Ohtake and Koizumi 2003). q = h ( T ) = mh & fg + mc & p A s ( T T ) s sat 4.3 The heat flux q is a suface-aveage quantity h is the convective heat tansfe coefficient T is the tempeatue diffeence between the heated suface and the impinging doplets ṁ is the liquid mass flow ate of the doplets h fg is the latent heat of vapoization of the liquid c p is the specific heat at constant pessue T s is the suface 87

104 tempeatue T sat is the satuation tempeatue of the liquid and A s is the heate suface aea. Fom mist cooling heat tansfe expeiments that involved ai-assisted wate spays Ohtake and Koizumi detemined the individual contibution of the liquid mass flux of wate to the heat tansfe ate achieved duing mist cooling. In the investigation ai-assisted mist cooling was used to cool an upwad-facing coppe heate suface. Thee coelations of the mist cooling ate fo non-boiling doplet evapoation and liquid film evapoation wee developed fo doplet mass flow ates between 0.08 and 2.22 g/s and ai flow ates between 0.67 and 2.0 l/s. In the study the coppe heate suface aea was cm 2 and its suface tempeatue anged fom 50 to 130 C. The heat tansfe ates anged fom 3 to 600 W/cm 2 fo the non-boiling doplet evapoation and liquid film evapoation heat tansfe modes. Afte evaluating the heat tansfe ates of the aiassisted mist cooling and the contibution due to ai-only cooling the diffeence was attibuted to the phase change of the liquid mass flux. The heat tansfe coefficient fo mist cooling by doplet evapoation was detemined to be h mist = G T W whee h mist is the heat tansfe coefficient in W/(m 2 C) G is the liquid mass flux in l/(m 2 min) and T W is the tempeatue diffeence between the heated suface and liquid in C. This coelation was chosen because the tested ange of liquid mass flow ates heate aea heate tempeatue and heat tansfe ates wee compaable to those simulated in this study s heat tansfe cell model. Afte evapoation of the doplets at the heate suface the vapoized liquid condenses on the cool inside peimete wall of the cell. The latent heat of vapoization is 88

105 eleased into the wall as vapo contacts the wall that is below the satuation tempeatue of the vapo. Heat is tansfeed to the wall and the vapo condenses back to liquid phase. Condensation of the vapo on the peimete wall was modeled using a lamina film condensation coelation fo the heat tansfe coefficient fo a vetical wall. The aveage heat tansfe coefficient fo condensation coelation developed by Sadasivan and Lienhad (1987) is h avg gρl = µ l ( ρl ρv ) kl ( T T ) sat s 3 h L fg 1/ whee L is the chaacteistic length of the condensing suface. The latent heat of vapoization is modified fo inclusion of themal advection effects and takes the following fom. h fg = h fg +.68c p l ( T T ) sat Afte the heat is tansfeed to the oute suface wall of the cell it is conducted along fins that ae attached to the outside of the cell whee the heat is eventually convected to the ambient envionment. The fins ae modeled as ectangula unifom coss-section fins. As peviously stated this heat tansfe cell model assumes one-dimensional steady-state heat tansfe. Use of an equivalent themal esistance cicuit allows calculation of tempeatues heat fluxes o othe paametes of inteest given necessay infomation about the cell and the chaacteistics of the doplet geneation discussed in the pevious chapte. By solving the system of equations that esult fom the simulation of evapoation condensation conduction convection and othe cell conditions in the heat tansfe model the themal pefomance of the cell can be evaluated. s 89

106 In Figue 4.3 is the equivalent themal cicuit that descibes the modeled heat tansfe cell. The nodes between the esistos ae the nodal tempeatues T1 though T6 depicted in Figue 4.1. The themal esistances R1 though R11 ae defined as: t R = k cell cell A top whee R1 is the conductive themal esistance though the cell wall at the top of the cell whee t cell is the thickness of the cell wall k cell is the themal conductivity of the cell wall mateial and A top is the aea at the top of the cell. R 2 = 4.8 R S In Equation 4.8 R2 is the themal esistance caused by two-dimensional speading in the cell wall at the top of the cell whee R S is defined in Equation T 2 T 3 R3 = = 4.9 h A q boil top top In Equation 4.9 R3 is the convective themal esistance fom the suface of the cell wall at the top of the cell to the fluid of the inne volume of the cell h boil is the boiling heat tansfe coefficient and q top is the heat tansfe ate though at the top of the cell. The heat tansfe coefficient h boil is calculated using Equation 4.3 when using the ideal evapoation model o using Equation 4.4 when using Ohtake s mist cooling model. 1 T3 T 4 R4 = = 4.10 h A q condense wall wall The convective condensation themal esistance fom the fluid of the inne volume of the cell to the cicula peimete of the cell is R4 h condense is the condensation heat tansfe coefficient A wall is the aea of the cicula peimete wall and q wall is the heat tansfe ate though cicula peimete wall of the cell. The heat tansfe coefficient h condense is 90

107 calculated using Equation 4.5 whee h condense is equal to h avg. R5 t k A ll = = 4.11 ll bottom T3 T5 q bottom In Equation 4.11 R5 is the conductive themal esistance though the liquid laye at the bottom of the cell t ll is the thickness of the liquid laye k ll is the themal conductivity of the wate A bottom is the aea at the bottom of the cell and q bottom is the heat tansfe ate though the bottom of the cell. Condensation of vapo on the liquid laye suface was assumed not to occu. All condensation was assumed to occu on the cell peimete wall. R = k cell cell t A bottom In Equation 4.12 R6 is the conductive themal esistance though the cell wall at the bottom of the cell. R ( ) ln 2 = 2π k cell H wall In Equation 4.13 R7 is the conductive themal esistance though the cell wall at the cicula peimete of the cell 1 is the inne adius of the cell 2 is the oute adius of the cell and H wall is the height of the cicula peimete wall of the cell. 1 R8 = 4.14 h amb A bottom base In Equation 4.14 R8 is the convective themal esistance fom the un-finned oute suface of the bottom of the cell h amb is the ambient convective heat tansfe coefficient and A bottombase is the un-finned aea outside the bottom of the cell. 1 R9 = 4.15 h amb A side base 91

108 In Equation 4.15 R9 is the convective themal esistance fom the un-finned oute suface of the side of the cell and A sidebase is the un-finned aea outside the side of the cell. θ b bottom 1 R10 = 4.16 q N bottom fin fins bottom In Equation 4.16 R10 is the themal esistance of the fins θ bbottom is the tempeatue diffeence between the tempeatue at the base of the bottom fins and ambient tempeatue q bottomfin is the heat tansfe ate though one fin on the bottom of the cell and N finsbottom is the numbe of fins on the bottom of the cell. θb side 1 R11 = 4.17 q N side fin fins side In Equation 4.17 R11 is the themal esistance of the fins θ bside is the tempeatue diffeence between the tempeatue at the base of the side fins and ambient tempeatue q sidefin is the heat tansfe ate though one fin on the side of the cell and N finsside is the numbe of fins on the side of the cell. Equation 4.18 is used to calculate heat tansfe though individual fins on the side and bottom of the cell. It is the expession fo heat tansfe fom fins of unifom cosssection with a tip condition of convection heat tansfe (Incopea and DeWitt 1990). ( ml) + ( h / mk ) cosh( ml) ( ml) + ( h / mk ) sinh( ml) sinh q = M 4.18 cosh M = θ b hpka c 4.19 m = hp / ka c 4.20 In Equations 4.18 to 4.20 L is the fin length k is the themal conductivity of the fin mateial h is the heat tansfe coefficient at the fin suface P is the fin peimete length A c 92

109 is the coss-sectional aea of the fin and θ b is the diffeence between the fin base tempeatue and ambient tempeatue. T1 R1 R2 T2 qtop R3 R5 T3 R4 T5 T4 qbottom R6 R7 qside R8 R9 R10 R11 qbottomfins T6 qsidefins Figue 4.3: Themal esistance diagam fo heat tansfe cell model Heat tansfe fom the heat souce into the inne volume of the cell is epesented as q top. In the themal esistance diagam it is the heat that flows fom T1 to T3. Heat leaves the cell though the side and bottom of the cell and is epesented as q side and q bottom. The heat q bottom flows fom T3 to T6 via the themal esistances R5 R6 R8 and R10. The heat q side flows fom T3 to T6 via the themal esistances R4 R7 R9 and R11. Each of the esistances ae solved using cell opeating chaacteistics and conditions that ae enteed into the heat tansfe model to find heat tansfe ates and tempeatues. All of the popeties of wate and the cell mateial ae detemined as functions of tempeatue in the model. 93

110 The peceding seies of algebaic equations wee solved using the Windows-based application Engineeing Equation Solve (EES) Academic Vesion D. EES is an equation-solving pogam that is capable of solving lage sets of non-linea algebaic and diffeential equations and has the flexibility to execute customized pogam suboutines. It also contains built-in functions fo themodynamic popeties of many substances including wate steam ai and efigeants. 4.2 Results Veification of the heat tansfe cell model was pefomed by modeling a doplet heat tansfe cell that has peviously been studied by Heffington at the Geogia Institute of Technology in expeimental investigations. This heat tansfe cell was made of aluminum and used wate as the woking fluid. A piezoelectic dive was used to geneate doplets fom a liquid laye. Using the heat tansfe cell to cool an attached heat souce cell heat tansfe ates of 123 to 175 W wee achieved at heat souce tempeatues of 110 to 142ºC. A photogaph of the heat tansfe cell without its cove is shown in Figue 4.4. Using chaacteistics of the actual heat tansfe cell the heat tansfe model was configued to simulate steady state opeation of the actual cell. The heat tansfe ates calculated by the heat tansfe model wee then compaed to the heat tansfe ates measued expeimentally. Dimensions and attibutes of the actual cell that wee simulated in the heat tansfe model ae listed in Table

111 Figue 4.4: Photogaph of actual heat tansfe cell (Heffington) Table 4.1: Actual heat tansfe cell dimensions and attibutes Paamete Values Inside diamete 44.3 mm Inside height 29.7 mm Oveall cell diamete mm Oveall cell height 55 mm Cell wall thickness 1.5 mm Heat souce aea 1 cm 2 Fin length 10.5 mm Fin thickness 1.2 mm Numbe of fins 60 Cell mateial Aluminum Woking fluid Wate The dive voltage dive fequency and liquid laye thickness wee not known fo the povided conditions at which the cell heat tansfe ate was measued as a function of the heate tempeatue. Despite the absence of that infomation the heat tansfe model could be used to simulate opeation of the actual heat tansfe cell because those dive and liquid laye paametes dive the doplet mass flow ate which enables heat tansfe fom the heated suface. Compaable conditions fo doplet geneation wee simulated using the doplet mass flow ate data collected fo the 38.1 mm constained 95