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Vibration-induced droplet atomization and dropwise condensation

University of Florida Institutional Repository

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VIBRATION-INDUCED DROPLET ATOMIZATION AND DROPWISE CONDENSATION By ADAM K. ROBINSON A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Adam K. Robinson

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This document is dedicated to my mother, Lorraine L. Parker.

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iv ACKNOWLEDGMENTS I thank my advisor, Dr. Jacob N. Chung, w ithout whom the project would not have been possible. I also am grateful to Dr. James Klausner and Dr. William Lear for their input on my committee. Dr. Wei Shyy and Dr. Mark Sheplak gave valuable input in the early stages of the project. Dr. Louis Cattafesta gave much appreciat ed advice on data acquisition and signal conditioning. Dr. David W. Hahn gave excel lent counsel on mode ling heat conduction. Dr. Bojan Vukasinovic, of the Georgia In stitute of Technology, graciously granted copyright permission along with copies of some figures from his dissertation on Vibration Induced Droplet Atomization. Several of my colleagues have given valuab le input relating to their experience in different areas of this proj ect including Ryan Avenall, Venkat Chandrasekaran, Quentin Gallas, Ryan Holman, Stephen Horowitz, Landon Tully, Chris Velat, Renqiang Xiong, and Kun Yuan. Their advice undoub tedly saved weeks of effort. I am very grateful to Ron Brown for his indispensable he lp and advice in fabrication of the experimental apparatu s. Becky Hoover, Pam Simon, and Teresa Mathia provided crucial help and were al ways pleasant in helping me to navigate administrative requirements. Several professionals outside of the university system ha ve also been remarkably helpful in completing this project. Pam Bi shop, from MTI Instruments, was especially prompt and courteous in helping me to le arn about optical pos ition sensing. Richard

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v Brooks and Jeff Zahnd from American Piezo Cor poration provided crucial insights on the function of their piezoelectric products, ad vice on signal processing, and development samples. Doug Hand from Thales-Optem, Inc. gave excellent advice on optics. John Vickers from Watlow Heating produc ts also gave crucial advice in the early stages of this project. Finally, but most importantly, I thank my Mother for her years of support.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix LIST OF EQUATIONS....................................................................................................xii NOMENCLATURE........................................................................................................xiii ABSTRACT.....................................................................................................................xv i CHAPTER 1 INTRODUCTION........................................................................................................1 2 LITERATURE SURVEY.............................................................................................5 Droplet Ejection............................................................................................................5 Filmwise Condensation (FWC)..................................................................................10 Dropwise Condensation (DWC).................................................................................11 Main Theory........................................................................................................11 Functional Behavior of Rose Theory..................................................................17 Competing Models..............................................................................................27 Droplet Size Distribution.....................................................................................33 Effect of Maximum Droplet Size........................................................................34 General Results on Drop-w ise Condensation (DWC).........................................35 Constriction Resistance.......................................................................................38 Drop-wise Condensation (DWC) Promotion......................................................40 Effects of Non-Condensable Gases (NCG).........................................................44 Heat Transfer with Vibration......................................................................................49 Piezoelectric Droplet Ejection with Heat Transfer.....................................................53 3 THEORETICAL ANALYSIS....................................................................................54 4 APPARATUS.............................................................................................................63 Equipment...................................................................................................................73 Additional Testing......................................................................................................80

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vii 5 PROCEDURE.............................................................................................................84 6 RESULTS AND DISCUSSION.................................................................................88 Initial Testing..............................................................................................................88 Intermediate testing....................................................................................................92 Final testing..............................................................................................................101 Data Summary and Comparison with Theory..........................................................119 Additional Testing....................................................................................................123 7 UNCERTAINTY......................................................................................................131 Main Study................................................................................................................131 Additional Testing....................................................................................................133 8 CONCLUSIONS......................................................................................................135 Main Study................................................................................................................135 Trend Explanation....................................................................................................136 Additional Testing....................................................................................................137 Improvements...........................................................................................................138 Apparatus...........................................................................................................138 Procedure...........................................................................................................140 APPENDIX A MATLAB CODES...................................................................................................142 DWC_HTC.m...........................................................................................................142 Dropwise_Codensation_Model.m............................................................................144 Theoretical_Heat_Rate_Grid.m................................................................................146 DWC_Behavior.m....................................................................................................146 Get_Theoretical_THR_and_r_eff.m.........................................................................149 Heat_Rate_Archive.m..............................................................................................150 PROPS.m..................................................................................................................159 exp_profile.m............................................................................................................162 simpsons_rule.m.......................................................................................................163 B TOTAL HEAT RATE TABLES..............................................................................165 C DETAIL DRAWINGS.............................................................................................170 LIST OF REFERENCES.................................................................................................178 BIOGRAPHICAL SKETCH...........................................................................................186

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viii LIST OF TABLES Table page 6-1 Experimental and theo retical data for THR...........................................................123 6-2 Data summary from Additional Testing.................................................................129 7-1 Uncertainty in measured values.............................................................................132 B-1 Theotretical THR with theoretica l maximum departing droplet size.....................166 B-2 Theotretical THR with 4 mm ma ximum departing droplet size............................166 B-3 Theotretical THR with 2 mm ma ximum departing droplet size............................167 B-4 Theotretical THR with 1 mm ma ximum departing droplet size............................167 B-5 Theotretical THR with 0.5 mm maximum departing droplet size.........................168 B-6 Theotretical THR with 0.25 mm maximum departing droplet size.......................168 B-7 Theotretical THR with 0.1 mm maximum departing droplet size.........................169 B-8 Theotretical THR with 0.01 mm maximum departing droplet size.......................169

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ix LIST OF FIGURES Figure page 2-1 Dimensionless critical accelerat ion vs. dimensionless drop diameter.......................7 2-2 Droplet atomization by frequency modulation...........................................................8 2-3 Heat transfer coefficient vs. saturation temperature.................................................18 2-4 Heat transfer coefficient vs. subcooling...................................................................19 2-5 Heat flux vs. saturation temperature........................................................................20 2-6 Heat flux vs. subcooling...........................................................................................21 2-7 Heat transfer coefficient vs. ma ximum departing droplet radius for Tsat = 380K....22 2-8 Heat transfer coefficient vs. maximum departing droplet radius.............................23 2-9 Heat transfer coefficient vs. ma ximum departing droplet radius for Tsat = 280K....24 2-10 Heat flux vs. maximum depa rting droplet radius for Tsat = 380K............................24 2-11 Heat flux vs. maximum de parting droplet radius at T = 1K..................................25 2-12 Heat flux vs. maximum departing dr oplet radius for extreme parameters...............26 2-13 Smallest thermodynamically viable droplet radius vs. Tsat......................................27 3-1 Assumed radial surface temperature distribution.....................................................56 3-2 Radial HTC and HF distributions.............................................................................57 3-3 Block diagram of iterative loop to determine theoretical THR................................59 3-4 Theoretical THR vs. maximum droplet radius.........................................................62 4-1 Corner view of experimental apparatus...................................................................66 4-2 Sectioned view of experimental apparatus...............................................................67 4-3 Insulation applied to and removed from diaphragm frame......................................68

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x 4-4 Exploded view of experimental apparatus...............................................................69 4-5 Diaphragm clamp, lower face..................................................................................70 4-6 Schematic of equipment setup with labels...............................................................75 4-7 Schematic of vacuum apparatus adjustments for Additional Testing......................81 4-8 Placement of surface thermocouples on the piezoelectric diaphragm.....................82 6-1 Video frames of DWC without (left) and with (right) VIDA..................................89 6-2 Temperature data from 1/26.....................................................................................89 6-3 Total heat rate data from 1/26..................................................................................90 6-4 Temperature data from 2/17.....................................................................................92 6-5 Total heat rate data from 2/17..................................................................................94 6-6 Two frames taken from videos on 2/17....................................................................95 6-7 Temperature data from 2/24.....................................................................................97 6-8 Total heat rate data from 2/24..................................................................................99 6-9 Typical droplet distribution for DW C on the test diaphragm without VIDA........100 6-10 Video frames of t ypical droplet distributi on for DWC with VIDA.......................100 6-11 Temperature data from 3/4.....................................................................................102 6-12 Total heat rate data from 3/4..................................................................................103 6-13 Temperature data from 3/13...................................................................................104 6-14 Total heat rate data from 3/13................................................................................105 6-15 Temperature data from 3/14...................................................................................106 6-16 Total heat rate data from 3/14................................................................................107 6-17 Temperature data from 3/15-16..............................................................................108 6-18 Total heat rate data from 3/15-16...........................................................................109 6-19 Droplet departure and regernat ion in DWC without VIDA, 3/15-16....................110 6-20 Video frames of VIDA onset, front view...............................................................113

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xi 6-21 Calibration curve for peak-to-peak displacement vs. input voltage.......................115 6-22 Peak-to-peak displacement vs. frequency for used driver......................................116 6-23 Video frames of VIDA process, corner view.........................................................118 6-24 Experimental THR vs. theoretical THR.................................................................120 6-25 Total heat rate vs. maximum subcooling...............................................................122 6-26 Temperature data from 5/21...................................................................................124 6-27 Comparison of 5/21 surface temperature data........................................................127 6-28 Comparison of 5/21 VIDA and non-VIDA temperature data for shim surface.....128 C-1 Bottom Cover detail drawing.................................................................................171 C-2 Cooler Barrel detail drawing..................................................................................172 C-3 Diaphragm Clamp detail drawing..........................................................................173 C-4 Diaphragm Frame detail drawing...........................................................................174 C-5 Optical Probe Guide detail drawing.......................................................................175 C-6 Optical Probe Seal detail drawing..........................................................................176 C-7 Top Cover detail drawing.......................................................................................177

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xii LIST OF EQUATIONS Equation page 2-1 Nusselt correlation for FWC on a vertical flat plate..............................................10 2-2 Modified heat of vaporiz ation for Nusselt correlation...........................................10 2-3 Thermal resistance in a droplet due to conduction................................................13 2-4 Thermal resistance in a droplet due to interphase matter transfer resistance........14 2-5 Finite temperature dr op required for equilibrium..................................................14 2-6 Heat flux through a drop........................................................................................14 2-7 Fraction of condenser surface area covered by drops larger than r.......................15 2-8 General expression for fractional area coverage of condenser surface..................15 2-9 Specific expression for fractional area coverage of condenser surface.................15 2-10 Average heat flux over an isothermal surface undergoing DWC (Rose Theory)..16 2-11 Maximum departing droplet radius........................................................................16 2-12 Minimum thermodynamically viable droplet radius..............................................16 2-13 Empirical relation for DWC...................................................................................16 2-14 Interfacial mass transfer resistance........................................................................27 2-15 Theoretical HTC accounting for constriction resistance.......................................28 3-1 Heat conduction equation in cylindrical coordinates.............................................55 3-2 Assumed exponential form of radial temperature distribution..............................56 6-1 Energy equation.....................................................................................................91

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xiii NOMENCLATURE A area B a constant C a constant cP,f mass specific heat of fluid state dr differential radius E energy E time rate of energy change f fraction of condenser surface coverage g acceleration of gravity h convection coefficient hfg heat of vaporization K a constant k thermal conductivity L characteristic length n droplet size dist ribution exponent q” heat flux R gas constant R” thermal resistance r droplet radius ˆ r maximum departing droplet radius

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xiv r smallest thermodynamically viable droplet radius T temperature t temperature in Celsius Greek Letters T difference between saturation and condenser surface temperatures t change in time density specific heat ratio surface tension absolute viscosity kinematic viscosity pi Subscripts avg average CV control volume cond conduction f fluid g gas gen generation in in L length mattertransfer interphase matter transfer out out s surface (condenser)

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xv sat saturation Operators partial differential

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xvi Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science VIBRATION-INDUCED DROPLET ATOMIZATION AND DROPWISE CONDENSATION By Adam K. Robinson December 2005 Chair: Jacob N. Chung Major Department: Mechanic al and Aerospace Engineering This thesis studied the process of drop wise condensation (DWC) of water vapor on a non-isothermal condenser surface. Specifical ly, the object was to quantify the effects of Vibration Induced Droplet Atomization (VIDA) on this pr ocess. The working fluid was water vapor in the saturation temperat ure range from 24 to 48C (75 to 118F), corresponding to 3.0 to 11.2 kPa (0.89 to 3.30 in-Hg). The condenser surface was the clean metal side of a brass shim, of 19 mm (0.75 in) diameter and 0.4 mm (0.016 in) thickness. A piezoelectric ceramic disk was att ached to the other side of this shim. The resonance frequency of this arrangement in a clamped edge boundary condition ranged from 2.1 to 4.0 kHz with the temperature of the disc. The driving voltage signal was a sinusoidal wave form of about 35 Vp-p. Theoretically, any reduction in the maximu m aggregate size reached by the drops of condensate should result in an increase in the heat transfer coefficient, or HTC. If vibration was applied to a sessi le droplet in a certain rang e of frequency and amplitude,

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xvii the droplet surface developed capillary waves. Given enough intensity of vibration, these waves outstretched the viscous forces holding the drop together and the drop disintegrated into hundreds of small drops. These small drops then moved away from the original, static drop surface at r oughly perpendicular trajectories. Direct case-by-case comparison of the eff ects of VIDA on DWC was not possible. The control model assumed an exponential radial temper ature distribution in the diaphragm. Direct surface temperature measurements of the diaphragm during application of vibration indica ted that the assumed profile di d not apply. The area above the piezoelectric patch on the back of the diaphragm was found to have a higher temperature than the saturated vapor. Theref ore the active heat transfer area of the VIDA case was smaller than that of the non-VIDA case. However, the acceptable data points from the former set still showed comparable to improved overall HT rates over the latter. Therefore some advantage was shown, but it could not be quantified. Maximum peak-to-peak displacement of th e diaphragm was as high as 40 m for a new driver, while for an old driver may have been about 10 m. Edge temperature readings for the shim ranged from 6 to 19C ( 43 to 66F) with coolant inlet temperatures near 4C (40F). The bulk of the original da ta showed much lower than predicted overall heat transfer rates, or THR. This was due to the extreme sensitivity of low pressure DWC to the presence of non-condensable gases, or NCG, namely air. A final round of testing was able to clear up this problem enough to clea rly examine VIDA effects on DWC.

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1 CHAPTER 1 INTRODUCTION Heat dissipation has been a major barrier to increases in processor speed and decreases in length scale of electronic devices The Semiconductor Industry Association report of 2001 [1] projected cooling loads that would be necessary in the near future for new generations of electronic devices. At the time of their study, 40 W/cm2 was given as a typical required heat flux (HF) for the high performance electronics category, which included for example military avionics. This cooling requirement, projected to the year 2016, more than doubled to 93 W/cm2. The difficult aspect of dissipating this level of heat flux for cooling electronic devices was that it had to be accomplished ove r a relatively small temperature difference, since the maximum temperature of the electron ic devices must stay safely below 85C, the melting point of the silicon substrate, wh ich leaves at most only about a 40 to 50C window between the temperature of the h eat source to the ambient atmosphere. Mudawar, in 2001 [2], examined strategies for dissipating high heat fluxes. He compared the reported heat transfer coe fficients (HTC) for the three most common coolant fluids from literature, air, water, and fluorochemical liquids. The three heat transfer (HT) regimes examined were natural convection, single-phase forced convection, and pool boiling. The former two methods are currently the mo st common and easily implemented methods of electronics cooli ng. However, only the latter method was capable of reaching the needed 100 W/cm2 over a temperature diffe rence on the order of

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2 10C. However, boiling presents a problem in steady cooling due to temperature shocks that accompany the formation and departure of vapor bubbles. Significantly higher HTCs were reporte d for more involved cooling scenarios such as forced convection bo iling, spray, and jet cooling, however these methods also added significant complexity, weight, and re liability issues to the overall design of a given device. Still, the main conclusion of the paper was that a two phase cooling solution was the most likely choice for higher le vels of heat dissipation, but that this would likely add significant co mplexity to the devices. Thus a problem has emerged of how to integrate a two-phase, high heat flux cooling scheme into electronic devices of ever decreasing length scale, while not greatly increasing the complexity. Francois, Shyy, and Chung, in 2000 [3], offe red a potential solution for such a two phase cooling device. They proposed a theo retical model of a ME MS (micro-electromechanical-systems) device, completing a vapor compression cooling cycle with a fluorinert chemical as the working fluid. The proposed system was to be driven by a piezoelectric diaphragm that would also act as the condensing surface. This would allow for surface tension control of the drops of conde nsate. This step was taken so that the device would not be gravity dependent. Th is analysis presumed dropwise condensation (DWC) on the diaphragm surface, which has been shown to offer HTC an order of magnitude higher than filmwise condensa tion (FWC). The relationship between maximum droplet size and HTC for DWC w ill be explicitly examined after the introduction of the theory. Since the length scale of this device was very small, the droplets involved were also very small and therefore very high HF was anticipated.

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3 In Francois, 2002 [4], a numerical simulation looked in detail at the droplet impact on a heated wall and HT aspects of the ev aporative side of the proposed device. However, analysis of the conde nsation side of the cell was left as an assumption that DWC would offer a certain orde r of magnitude of HF. Deta iled consideration of this process was left for further studies. This model assumed bulk ejection of the condensate droplet from the membrane, a significant obstacle considering the increas ing dominance of surface tension over body force with a decrease in length scale, a lthough an energy balance model was offered for the energy required for ejection. Friedman and Chung, in 2001 [5], made an initial investigation into the droplet ejection process. The device used was in the meso-length scale since the piezoelectric dr iver had about a inch diameter. The prefix meso refers to a characteristic length scale in the range of 1 mm to 1 m. Primary droplet volumes investigated ranged from 1 to 5 L. Even at this relatively large scale of 30 to 150 times the volume of the largest drop considered in the current study, th ey still encountered significant difficulty achieving droplet detach ment from the diaphragm surface. When detachment was achieved, most often the pr imary droplet divided into several smaller drops. The difficulty in drop detachment was due to the high surface tension forces with respect to the momentum forces of the fl uid mass. It was because of these surface tension related problems that good repeatability was not attain ed. Thus, further work was needed to determine an appropriate method of droplet ejection from a piezoelectric membrane surface. This project was undertaken to study the process of condensation and subsequent droplet ejection of coolant va por on a vibrating piezoelectri c disk. This effort was

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4 needed to clear up the cooling side of the de vice proposed in [3]. The following criteria were set as goals for the development of the investigation: Meso length scale for ease of study. Dropwise Condensation mode fo r its expected high HF. Water as the working fluid for its high heat of vaporization. However, this would necessitate sub-atmospheric working pressures in order to lower the saturation temperature below 85C. Piezoelectric disc for both the droplet ejection mechanism and condensing surface. Some form of droplet ejection woul d be used to remove condensate

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5 CHAPTER 2 LITERATURE SURVEY Droplet Ejection A literature search for droplet ejection methods involving piezoelectric devices yielded several potential methods Percin et al., in 1997 [6], tested a piezoelectric droplet ejector. The piezoelectric diaphragm had a small hole at the center, 50 to 100 m in diameter. It supported a rese rvoir of liquid. The hole wa s small enough that the surface tension of the liquid across the hole kept it from leaking out. Dr oplet ejection occurred when the oscillation frequency and amplit ude caused the momentum of the fluid to overcome the surface tension. Flow rates of up to 100 mL/s and ejection frequencies of up to 32 kHz were cited. The device was r oughly 8 mm in diameter 8 mm in length, and operated on a 32 Vp-p sinusoidal voltage. Th e working fluid was a special, low-viscosity ink. Higher viscosity fluids were tested, requiring voltages up to 200 Vp-p. This design was promising for the purposes of this study, except that the orie ntation of the liquid surface had to be away from the direction of ejection, not leav ing much surface area available for removing the heat that would induce condensation on the liquid surface. Zhu and Kim, in 1998 [7], reported a ME MS device which achieved significant fluid mixing using loosely focused acoustic wa ves from a piezoelectric diaphragm. The diaphragm bore ceramic patches arranged specifically to accomplish the focusing by constructive interference. The focal length wa s designed to be just above the liquid/vapor interface. The device was small enough, 1.5 x 1.5 x 0.5 mm, that the top could be open and still be insensitive to gr avity effects because of the su rface tension. The authors

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6 reported negligible heating of the fluid from driver operation. This device showed much promise for fluid motivation on a micro-scale bu t lacked the droplet ejection required for the present device. In 2001, Huang and Kim [8] used a similar arrangement to [7] to achieve droplet ejection. Moving the surface of the fluid a little higher, to coincide with the focal point of the acoustic waves, was the main differe nce between the two devices. These waves were again created by a specially patterned piezoelectric ceramic. The ejected droplet velocity, caused by constructive interference, was found to be between 2 and 3 m/s, with operating frequencies up to 10 kHz. Dropl et diameter was re portedly around a few microns with uniform size and direction. Th e device was less than 2 mm in diameter and 0.5 mm in length. The ejection characteristic s and the length scale of this device appear well suited for the needs of the proposed HT cell. The limitation here was the conduction resistance of the fluid bulk. At this point, it was assumed that DWC would provide more intense cooling than could be achieved by direct contact condens ation, due to the relatively slow conduction of h eat through the liquid phase. Szita et al., in 2001 [9], reported the de velopment of a micropipettor, using the capacitance of one piezoelectric disc for feedback informati on to very precisely control the dispensed volume and another disc for pum ping force. Volume control was in the sub-L range. This design however, lacked the sufficient velocity of ejected droplets desired for the current applica tion. This velocity was re quired to break through a vapor layer on the hot surface of the proposed HT cell. The most promising method of droplet eject ion that came up in the literature search was Vibration Induced Droplet Atomization (VIDA) [10]. A horizontal piezoelectric disc

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7 with diameter on the order of 1 inch was used as the supporting base for a single drop of water placed at its center. As the perpendi cular vibration of the diaphragm approached resonance, standing waves developed on the su rface of the sessile droplet. A critical acceleration and frequency existed for a given drop of fluid, above which the momentum of the fluid in these standing waves overcame the surface tension. This caused the droplet to effectively burst. The order of the critical acceleration was given as roughly 20 times standard-earth gravity. The length of time of the bursting event ranged from Figure 2-1. Dimensionless critical acceleration vs. di mensionless drop diameter. Acceleration was determined by 3 / 1 3 / 4ˆ af a and diameter by 3 / 1 3 / 2ˆ df d Used with permission from [10]. essentially instantaneous for a la rge step input change in eith er amplitude or frequency to up to a minute for small changes in these para meters. Excellent correlation was found for the dimensionless critical acceleration versus dimensionless drop diameter involving the fluid properties and the frequenc y of oscillati on (Figure 2-1).

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8 Note the inverse exponential manner in which the dimensionless critical acceleration required for atomization increases as the droplet diameter decreases. This implies an effective lower size limit for ach ievable atomization events. Primary droplet diameters used in this study ranged from 10 to 150 L while the resonant frequencies of the drivers used were 1, 3, 5, 7, 9, and 15 kH z. Generally the larger the diameter or the thickness of the piezoelectric disc driver the lower the resonant frequency. When the amplitude of the input signal to the drive was fixed and the frequency incrementally increased from 0 towards res onance, the atomization process had roughly three steps (Figure 2-2). When vibrations commenced at low frequency, standing lateral Figure 2-2. Droplet atomization by frequency modulati on, 0.1 mL drop. Used with permission from [10]. a) No applied acceleration. b) Lateral standing waved develop on drop surface. c) Lateral st anding waves grow in amplitude. d) Axial waves become visible over lateral wa ves, creating a cros s-hatch pattern. Wave momentum begins to overcome surface tension, as drops begin to issue from the outer wave peaks. e) Amplit ude of oscillations grows, more drops issue from primary drop. f) Drop ejecti ons increase in number and intensity. g) Primary drop no longer visible due to number of secondary drop ejections. h) Ejections of secondary drops continue. i) Primar y drop mostly dissipated and secondary droplet ejection complete.

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9 waves appeared around the circumference of the sessile droplet. As the frequency approached the resonant frequency of the dr iver-drop system, axial waves also appeared, making a cross-hatched pattern with the lateral waves. As input frequency came closer still to resonance, the peak of the wave os cillations began to overcome the surface tension forces keeping the primary drop intact, causi ng secondary drops to pinch-off from the wave peaks. Near the resonant frequency and still with fixed am plitude of the input signal to the driver, the pinching-off of sec ondary droplets became much more intense. As this process progressed, the primary drop became depleted until there was only a small fraction of it left. The diameter of the ejected droplets was reported to be relatively independent of both the time during the atomization event a nd the volume of the primary droplet. The ejected velocity was said to range from 0.2 to 2.0 m/s. The droplet velocity was measured by a particle tracking velocimetry algorithm, developed by the investigator, and digital video images. A peak in ejection velocity was found at the onset of atomization. This quantity then decreased to be roughly equi valent to the velocity of the resonating diaphragm, just below the point of issuance. The trajectory of the ejected droplets wa s roughly perpendicular to the surface of the undisturbed primary drop at the given location, and, on average, was axisymmetric. Viscosity and surface tension were also sligh tly varied to probe for an effect on the atomization process, though no majo r quantitative change was found. On the operation of the piezoelectric dia phragms, Vukasinovic [10] noted that the resonance frequency was extremely sensitive to temperature as well as mass loading of

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10 the diaphragm, both of which would no doubt be important considerations in any application of VIDA to DWC. Filmwise Condensation (FWC) Filmwise condensation is the default mode of condensation and has therefore been taken as the design assumption in industrial de sign. Filmwise condensation will indicate the failure of the DWC mode, due to several possible factors includ ing contamination of the vapor, contamination of the condenser su rface, too large of a contact angle for the working fluid with the condenser surface, or too high HF leading to flooding of the condenser surface. Nusselt provided the most successful correla tions for FWC on a vertical flat plate in a pure vapor (Eq. 2-1) [11]. This correla tion used a modified heat of vaporization, which accounted for subcooling of the c ondenser surface (Eq. 2-2) [11]. s sat f f fg g f f f L LT T k L h g k L h Nuavg 3' 943 0, 2-1 s sat f P fg fgT T c h h ,68 0 ', 2-2 In this Nusselt correlation, h is the HTC, L is the length of the plate, g is the acceleration of gravity, is density, is the dynamic viscosity, k is the thermal conductivity, T is temperature, and cP is the constant pressure mass specific heat of the condensate. The subscr ipt ƒ denotes fluid, sat is for saturated, s is for surface, and g is for gas. One of the seminal investigator s of DWC, Dr. John Rose, commented on the remarkable accuracy of the Nusselt theory for FWC in an address on developments in condensation theory, in 1988 [12] He said that further tw eaking of this Nusselt relation

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11 for FWC was essentially “an academic exercise,” since this theory was correct to within a few percent for most cases. Dropwise Condensation (DWC) For decades now, researchers have sought to master the process of DWC. The high HF values attainable with this mode of c ondensation offered attract ive potential increases in efficiency for industrial HT processes. However, this phenomenon has proven quite resistant to efforts at precise characterizati on. The main performance difference in DWC and FWC is generally accepted as a potential order of magnitude increase in HTC. This holds possibly dramatic cost sa vings in many applications in cluding power plants that run steam turbines as well as seawater desalina tion plants. These cost savings would exist specifically in terms of lower capital and maintenance costs for a given output capacity. Main Theory Early investigations, however, showed widely ranging trends of relationship of heat flux (HF) to degree of subcooling of the condenser surface, Tsat – Ts = T. This variation in reported results wa s so significant that, the first authors to offer a viable theory on DWC HT, said they were, “of very little use even as a check,” [13]. These investigators asserted that the massive varian ce in the reported trends in DWC HT as due to the presence of non-condensable gases (NCG) in the test vapor. In most cases, this non-condensable species was air. To prove this theory, these authors chose to study DWC of steam near 1 atm. They used an isothermal copper block as a condensing surface and a heat flux meter. Some of th e main efforts undertaken to control the influence of NCG on the experime ntal results were as follows: Careful attention was paid to the vacuum integrity of the apparatus.

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12 Long blow-off periods were employed prio r to taking data, usually 3-4 hours, to allow sufficient degassing of the supply wate r and of the walls of the stainless steel apparatus. For these periods the boilers ran at full power and the vessel was vented to atmosphere in a controlled manner, to prevent backflow. A small vent tube was placed near the conde nser surface. Differ ent rates of steam venting were tested for effect on temp erature changes of the condensing surface and also for visual disturbance of the conde nsate. A venting rate was then chosen based on a balance between these competing considerations. The authors also took extreme care to minimize uncertainty in temperature measurement, which they had also noted as a major source of error in previously published studies. Several temperatur e measurements were taken along the centerline of a copper block to determin e the heat flux and then the gradient extrapolated to the condensing surface. A summary of their key findings follows: NCG had the significant effect of depr essing the surface temperature for a given HF and saturation temperature, in other words, increasing T. Significant difference was noted in effec tiveness of the four promoters used, though the underlying surface of polished copper and the cleaning regimen remained constant. No dependence of HF on plate height was found. This quantity was measured 1 inch and 4 inch from the top of the test surface with no discernable difference in the data. Sweeping frequency and maximum drop si ze did not affect HTC in the range studied. The first finding helped cement the noti on that NCG had an important effect on condensation HT. The second became important later in the debate over the effect of surface thermal conductivity on overall HT in DW C, which will be discussed at length in the Constriction Resistance section of the literature surv ey. The third finding of no height dependence has also been contentious, as several other rese archers have noted a height dependence, for example, Koch, Kraft, and Leipertz, in 1998 [14] and in the much more extreme cases studied by Yamali and Merte, 1999 and 2000 [15,16]. There were several contemporary studies of Le Fevre a nd Rose that found similar magnitude HF, and these will be discussed further in the Genera l Trends section below. For now, consider

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13 the theory offered by the same pair of resear chers the following year, Le Fevre and Rose, 1966 [17]. The theory has been republished numerous times, though in its most compact form in Rose, 1988 [12]. Some corrections a nd coefficient values were suggested in a 2002 presentation [18]. The Rose theory, as it will henceforth be referred to, was based on the assumption that all the HT in DWC eff ectively occurred on the surface of the liquid droplets rather than the bare condenser surface between drops. The theory assumed only three significant thermal resistances, with each repr esented by a temperature difference. The first was conduction through the liquid of the sessile droplet. The second was interfacial matter transfer resistance. The third assu med-significant thermal resistance in DWC was the finite decrease in saturation temperature required for equilibrium of a saturated vapor with the curved surface of a liquid drop. From these resistances, an expression was derived for the HF through a given drop of ra dius r. This expression for HF, based on fluid-vapor properties and drop radius, was then weighted by an assumed distribution of drop sizes. The weighted HF was then inte grated over a range of drop sizes from the minimum thermodynamically viable drop radi us to the maximum. The maximum drop radius was determined from dimensional analysis and experiment. The thermal resistance due to heat co nduction through the drop was calculated by neglecting internal circulati on or convection (Eq. 2-3). K1 is a shape factor for the ratio of surface areas between th e base and the top of a hemispherical drop. r is a given drop radius. kƒ is the thermal conductiv ity of the condensate. f condk r K R1" 2-3

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14 The expression for interfacial matter tran sfer resistance was calculated as a thermodynamic function (Eq. 2-4). K2 is a constant, experimentally determined and relatively constant for different fluids. g is the kinematic viscosity. T is the saturation temperature of the vapor. R is the gas constant of the vapor. is the specific heat ratio of the vapor. hƒ g is the heat of vaporizat ion of the working fluid. 1 1 2 "2 1 2 2 RT h T v K Rfg g sfer mattertran, 2-4 Finite temperature drop associate with a saturated liquid and va por in equilibrium at the curved surface of a drop was also a th ermodynamically derived quantity (Eq. 2-5). is the surface tension of the fluid. fg f sat curveh r T T 2 2-5 The thermal resistances in (Eq. 2-3,4,5) were then combined with the temperature difference between the condenser surface and th e saturated vapor to form an expression for the HF through a single drop of radius r (Eq. 2-6). The correction factor in the interphase matter transfer resistance te rm was offered as a correction for the polyatomicity of water. 2 / 1 2 2 12 1 1 664 0 627 0 2 RT h T K kT r K h r T T r qfg g sat fg f sat, 2-6 Now a statistical distribution for drop si zes, and limits on those sizes had to be chosen in order to integrate over all drop sizes to obtain the total HF for the surface. Le Fevre and Rose chose the distri bution in (Eq. 2-7), based on an expression for the fraction

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15 of the condenser surface that was covered by drop s larger than r. This fraction went to 1 as the given radius went to zero and went to 0 as r we nt to the maximum radius, rmax. In other words, all of the surface was covered by drops larger than zero, and none of the surface was covered by drops larger than rmax. When differentiated with respect to r, (Eq. 2-7) gives the area covered by a differ ential drop radius, dr (Eq. 2-8). nr r f ˆ 1, 2-7 r dr r r n dr r Anˆ ˆ1 2-8 For the original theory, Le Fevere and Ro se assumed from observations that n was roughly equal to 1/3, an assumption that was later vitiated by the results of Rose and Glicksman, in 1973 [19]. There has been much discussion of this topi c in the literature which will be addressed later in this paper. With n = 1/3, (Eq. 2-8) gave (Eq. 2-9). r dr r r dr r A ˆ ˆ 3 13 / 2 2-9 It should be noted that this equation gave an exponent of -2/3 as opposed to the -1/3 given in [12]. This was decided a typo in th e original, since the derivative of (Eq. 2-7) with respect to r, yields the (Eq. 2-9) as given and the expression given for the q”avg in the same reference follows from the value of the exponent used in (Eq. 2-9), which was arrived at when equation (Eq. 2-6) was mu ltiplied by the differential area (Eq. 2-9) and integrated from the minimum to the maximum dr op radius. The result was the HF for the entire condensing surface, averaged over all drop radi i from the minimum to the maximum (Eq. 2-10). The maximum droplet radius was determined by dimensional analysis in (Eq. 2-11) [12]. Here K3 is an empirical constant. The minimum

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16 thermodynamically viable drople t radius was given, finally in a dimensionally correct form, in (Eq. 2-12) [20]. Note: some of the notation above and throughout this paper has been changed to be homogenous with that of this study, but the meaning has been retained. In the same reference, [20], Ro se offered an empirical relation for DWC of steam “near and below atmospheric pressure” (Eq. 2-13). r r sat fg f sat fg f sat avgdr r RT h T K k r K h r T T r qˆ 3 / 2 2 / 1 2 2 1 3 / 22 1 1 2 ˆ 3 1 2-10 2 / 1 3 maxˆ g K r r 2-11 T h T r rfg f 2min 2-12 2 8 03 0 5 T T t q 2-13 In (Eq. 2-13) q” is in kW/m2 and t and T in C. No uncertainty was explicitly stated. However, the presented data were claimed to be ‘quite well repr esented’ by the given relation. A graphical estimation of the quality of this representation gave an uncertainty of 10%. Likewise, no explicit statement of unc ertainty for (Eq. 2-10) was given. This was most likely due to the myriad of factors th at could potentially ye t be proven to be of import to the phenomenon of DWC. A brief lis ting of these factors follows here, each of which was furthered explored in this literature survey: General comments on DWC trends in T, Tsat, and rmax Drop size and Nucleation site density Effect of maximum dropl et size on overall HF Constriction resistance debate Promotion methods

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17 Condensation coefficient, as it affects the interfacial mass transf er resistance term Effects of NCG on both DWC and FWC Functional Behavior of Rose Theory One main advantage of the Rose theory wa s that it presented a closed-form relation for HF and surface temperature in a compact form. Most of the competing theories, which will be discussed in the next section, are presented only as numerical simulations, which can be enlightening but also daunti ng to implement. The following figures examine the behavior of the mathematical func tion of the Rose theory, specifically in the ranges of parameters relevant to this study, sligh tly extended to cover most applications of DWC of wate r vapor. Mathematical function was emphasized in this section, and therefore only small effort was made to confirm the achievability of the extreme values presented. That is, no consideration for transition to FWC or the existence of corresponding experimental values, as these concepts were addressed in the General Results section. In fact, only the more moderate ranges of saturation temperat ure and maximum droplet size will be used in this study, for comparison to experimental data. In the many journal presentations of his theory, Rose had yet to offer a quantif iable maximum error for his theory, merely saying often that good or general agreement exis ted with available experimental data that he considered reliable, as in Styl ianou and Rose, 1980 & 1983, and Rose, 1981 [21,22,23], or simply presenting a figure of the theory graphed over the experimental results of several investiga tions and offering no explicit co mment on uncertainty, as in Rose, 1998 & 2002 [24,18]. The farthest Rose ha s thus far ventured in explicit comments on the accuracy of his theory wa s in [12], where he noted excellent agreement of data and theory for 380 data points from 5 different investigations of DWC. These 5 different

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18 studies used 8 different promot ers on copper and steam near 1 atm. In that figure, the apparent maximum deviation was ro ughly +1K (30%), in a graph of T vs. HF. The variation appeared well centered about the theory line. This T vs. HF relation has been his preferred method for examination of DWC performance. This choice was due to the large uncertainty in measuring the condensi ng surface temperature and the consequently much larger uncertainty in HTC since it i nvolves the reciprocal of this quantity. Additionally, the choice of HF as the independent and the surface subcooling as the dependent variable more accurately represents the experimental circumstances. This topic will be addressed in more detail in the General Trends section. Of particular importance for the following figures was that the theory concerns only an isothermal surface while the current study involves only a condenser surface with a variant temperature distribution over its surface. 250 300 350 400 450 500 10-2 10-1 100 101 Saturation Temperature, [K]HTC, [MW/m2K] Increasing T Figure 2-3. Heat transfer coeffici ent vs. saturation temperature, Tsat, for subcooling, T = 0.1, 1.0, 10, 50.

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19 The HTC determined from the Rose theo ry was somewhat flat relative to increasing saturation temperature and for small surface subcooling (Figure 2-3). However, as surface subcooling increased the HTC predicted from 300 to 500C saturation temperature experien ced almost a two order-of-magn itude increase. Note that the effects of subcooling are increasingly mitigated for lower vapor pressures/ temperatures. Also, as subcooling increas es for a given saturation temperature, successive increases in HTC diminish in magnitude. 10-1 100 101 102 10-2 10-1 100 101 Tsub, [K]HTC, [MW/m2K] Increasing Tsat Figure 2-4. Heat transfer coefficient vs. subcooling, T, for saturation temperatures of 280, 320, 380, 430, 500K. All analyses involving the theory in this study were conducted in MATLAB, and their corresponding code files can be found in appendix A. Specifically, th e calculations of (Eq. 2-10,11,12) were accomplished by the subroutine DWC_HTC.m, which can be

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20 found on page A-1 of the appendix. Th e following figures were generated by DWC_behavior.m, page A-9. For a given subcooling, si gnificant vapor pressure de pendence was noted in the subatmospheric saturation pressure range (Fig ure 2-4). However significant increases in HTC were absent above the 1 atm level. Th is was the precipitous drop predicted in the original work of HTC with vapor pressure below 1 atm [17]. The subcooling range was of more importance in this study than the satura tion temperature. Here the saturation temperature ranged only from about 24 to 49C but was constant for each data point. The surface subcooling, due to the temperature di stribution on the diaphragm, ranged from 0 to as high as 30K across the radius of the diaphragm for each test. 250 300 350 400 450 500 10-3 10-2 10-1 100 101 102 103 Saturation Temperature, [K]HF, [MW/m2] Increasing T Figure 2-5. Heat flux vs. saturation temperature, Tsat, for subcooling, T = 0.1, 1.0, 10, 50K.

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21 The predicted HF is relatively flat versus saturation temperature for a given surface subcooling (Figure 2-5). Howe ver, the relative differences in predicted HF for rising subcooling are roughly order-of-magnitude for order-of-magnitude, at a given saturation temperature (Figure 2-6). These results em phasize the dependence of HF on subcooling, clearly indicating a potential 4 orders-of-magnitude increase from near zero subcool to 50K, the highest considered, with 100 MW/m2 peak HF predicted for a Tsat of 500K and T of 50K. 10-1 100 101 102 10-3 10-2 10-1 100 101 102 103 Tsub, [K]HF, [MW/m2] Increasing Tsat Figure 2-6. Heat flux vs. subcooling, T, for saturation temperatures of 280, 320, 380, 430, 500K. Heat transfer coefficient s howed a tight distribution for subcooling from 0.1 to 50K (Figure 2-7). If it were possible to direct ly control the maximum droplet size, then further increases in the alrea dy high HTC of DWC could be theo retically attained. A few

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22 10-5 10-4 10-3 10-2 10-1 100 101 10-2 10-1 100 101 102 Max Droplet Radius, [mm]HTC, [MW/m2K] Increasing T Figure 2-7. Heat transfer coefficient vs. maximum depa rting droplet radius for Tsat = 380K and T = 0.1, 0.5, 1.0, 5.0, 10, 20, 50K. investigations have used different methods to attempt this, as did the present study. Most of the investigations by Rose have focused on a vertical plate in essentially quiescent vapor. Therefore these studies did not distin ctly examine the effect of maximum droplet size. The following studies howev er have all examined maximu m droplet radius effect in one way or another: Citakoglu and Rose, 1969 [25] – varied surface inclination Tanasawa et al., 1978 [26] – periodic wiper applied to the test surface Tanasawa and Utaka, 1983 [27] – shear flow of steam Tanaka and Tsuruta, 1984 [ 28] – shear flow of steam Yamail and Merte, 1999 [15] – centripetal acceleration, i.e. body force with high subcooling range The results of these studies will be examin ed in relation to the Rose theory in the Departing Droplet Size secti on of the literature survey.

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23 10-5 10-4 10-3 10-2 10-1 100 101 10-2 10-1 100 101 102 Max Droplet Radius, [mm]HTC, [MW/m2K] Increasing Tsat Figure 2-8. Heat transfer cofficient vs. maximum departing droplet radius. Surface subcooling was constant at T = 1K. Saturation temper atures used were 280, 320, 380, 430, 500K. Examining the effects of maximum droplet radius (reached by growing drops just prior to departure) for a constant T of 1K again exhibits th e dip in HTC and therefore HF for sub-atmospheric pressures (Figure 28). Comparing HTC fo r a given departing droplet radius, over a range of subcooli ng showed significant dependence on this quantity, as well as on saturati on temperature (Figure 2-9). This significant decrease of HTC with pressure was noted by Rose to be primarily due to the increase in the interphase matter transfer resistance. This quantity was the second term in the denominator of the expression for the HF through a single drop (Eq. 2-6). The Rose theory predicts linear HF, in l og-log coordinates, fo r any practical range of maximum departing droplet radii (greater than 10-3 mm) (Figure 2-10). Of course, smaller maximum departing droplet radius yiel ds larger overall HF. Also, many smaller

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24 10-5 10-4 10-3 10-2 10-1 100 101 10-2 10-1 100 101 102 Max Droplet Radius, [mm]HTC, [MW/m2K] Tsat = 280K Tsat = 380KT = 50K Tsat = 500K To = 50K Figure 2-9. Heat transfer coefficient vs. maximum depa rting droplet radius for Tsat = 280K, subcooling is 0.1 and 5.0K. For Tsat of 380 & 500K, subcooling is 0.1 and 50K. 10-5 10-4 10-3 10-2 10-1 100 101 10-3 10-2 10-1 100 101 102 103 Max Droplet Radius, [mm]HF, [MW/m2] Increasing T Figure 2-10. Heat flux vs. maximu m departing droplet radius for Tsat = 380K and for T = 0.1, 0.5, 1.0, 5.0, 10, 20, 50K.

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25 drops provide less conduction resistance and la rger total surface area for direct contact condensation than fewer large drops. When theoretical HF is compared to maxi mum departing droplet radius for several saturation temperatures, successive increases in Tsat have less effect while the linear (in log-log coordinates) trend with droplet size remains for each case (Figure 2-11). 10-5 10-4 10-3 10-2 10-1 100 101 10-2 10-1 100 101 102 Max Droplet Radius, [mm]HF, [MW/m2] Increasing Tsat Figure 2-11. Heat flux vs. maximu m departing droplet radius at T = 1K and for saturation temperatures of 280, 320, 380, 440, and 500K. Comparing overall HF with maximum depa rting droplet size (Figure 2-12) gives special emphasis to the dominating effect of surface subcooling. In fact, the overall HF function appears more sensitive to subcooling than either of the other two control factors, maximum departing droplet radius and saturatio n temperature. To illustrate this, a steady 4 order of magnitude incr ease over the range of T considered can be observed here for a given maximum droplet radi us and for the vapor temperatures considered.

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26 10-5 10-4 10-3 10-2 10-1 100 101 10-3 10-2 10-1 100 101 102 103 104 Max Droplet Radius, [mm]HF, [MW/m2] Tsat = 280K, T = 0.1K Tsat = 380 & 500K Tsat = 280K, T = 5K Tsat = 500K, T = 50K Tsat = 380K, T = 50K Figure 2-12. Heat flux vs. maximum departi ng droplet radius for extreme parameters. For Tsat of 280K, subcooling is 0.1 and 5.0K. For Tsat of 380 and 500K, subcooling is 0.1 and 50K. The smallest thermodynamically viable drop let (Eq. 2-12) was dependent on fluid properties, saturation temperat ure, and the condenser surface temperature (Figure 2-13). Note that the scale was in nanometers, an indication of the 106 to 109 ratio of droplet radii active in the phenomenon of DW C. The range of the maximu m departing droplet radius (Eq. 2-11) only changed a fraction from 1. 1 mm at 280K to 0.77 mm at 500K. It is indeed the high number of these small dr ops combined with their small conductive resistance to HT that affects the relati vely high HT efficiency of DWC over FWC.

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27 250 300 350 400 450 500 10-1 100 101 102 103 Saturation Temperature, [K][nm] Increasing T Figure 2-13. Smallest thermodynamica lly viable droplet radius vs. Tsat and for T = 0.1, 1.0, 10, 50K. Maximum droplet radius essentially constant between saturation temperatures of 280 and 500K, for which it is 1.1 and 0.77 mm, respectively. Competing Models Rose’s theory was not the only one offe red to model the ph enomenon of DWC. Just a few years after his initial publication, in 1969, Mikic [29] offered a theory of DWC HT that attempted to include the effect of condenser surface thermal conductivity. Mikic looked at the distributio n of drop sizes qualitatively and t ook that in conjunction with a general expression for the interf acial mass transfer resistance, he, (Eq. 2-14) similar to that considered by Rose (Eq. 2-4). r r T R P h g hg sat fg e* 1 2 2 22 / 5 2 / 3 2 / 1 2-14 is the condensation coefficient. This quantity represents th e fraction of vapor molecules striking the vapor/li quid interface which remain in the liquid. This value has

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28 been a topic of considerable uncertainty until the last decade. g is the acceleration of gravity. hfg is the heat of vaporization. Psat is the saturation temperature. R is the gas constant of the vapor. Tg is the vapor temperature. r* is the critical drop radius spoken of above. R is a given droplet radius. The aut hor assumed that this expression for the HTC on the surface of a drop was cons tant over its surface area. He also noted that there was a critical drop radius, r*, above which the thermal resistance of conduction in the drop dominated the HTC. By considering a drop suffi ciently larger than th e critical size to be adiabatic along with the relatively high HF in small drops immediately around it, he sought to estimate the resulting lateral temper ature gradients in the condenser surface, of a given thermal conductivity, and from that estimation, the effect on overall HTC (Eq. 2-15). 1 2 / 3 2 / 1 2 2 17 3 1 1 / 2 1 4 1 1 A c k A r h k r k hi i s j j e f j f 2-15 r is the drop radius. kf is the liquid thermal conductivity. he is the interfacial matter transfer resistance (Eq. 2-14). is the fraction of area cove red by visible – those with radius above critical – drops. ks is the surface thermal conductivity. C is the radius of an inactive drop. A is the total area of the surface. The major assumptions of this theory are as follows: Known droplet size distribution, which at that time was only being qualitatively guessed at HTC independent of subcooling (contrary to Rose Theory) Negligible circulation in the drop Negligible drop in saturation temperatur e due the curved su rface of the drop (contrary to Rose theory)

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29 A pure vapor, i.e. no NCG present No experimental verification of this theory was offered. In fact, pains were made to note that this relation was not intended to be calculable, merely representative. However, based on analysis of published result s, the author did venture to estimate that the constriction resistance in a stainless stee l was about 84% of the total resistance to HT versus 20% for copper. This theory was not specifically tested in any experimental studies reviewed for this paper. However the topic of constricti on resistance continues to be contentious in current discussions of DWC. This topic was addressed in more detail in the Constriction Resistance section of the literature survey. Few alternatives have thus far been offe red to the Rose theory. Subba and Murthy in 1983 [30] offered a model for DWC HT based on the opposite assumption of Rose. They assumed that the drops only served to in sulate the surface and therefore all the HT took place on the bare surface between drops. Their investigation wa s purely analytical. The results were compared to those of Le Fevre and Rose [31], Citakoglu and Rose [32], and one other. The analysis also attempted to predict the effect of NCG on DWC HT. Both of the comparisons were claimed to be good, but again no offer was made of uncertainty. The shape of the curve was ad justed by assuming different functional forms of the condensation coefficient, some depe ndent on HF and some not, which caused the theoretical trace to a lternately bound the data and to cr oss through it, however, which of the three adjustments gave the best fit was not noted. No other instances of this theory were found in the cited sources for this paper. Tsuruta and Tanaka, in 1991 [33], pres ented a numerical simulation somewhat based on the same arguments as [29]. A dditionally, they offered a non-dimensional

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30 model that accounted for temporal varia tion in drop size and departure as well as considering the area underneath large drops as an adiabatic cylinder. Experimental verification of that theory in was offered in [34]. This theory was considerably more complex than the Rose theory, as it attempte d to model the temporal variations in surface temperature as well as the overall HT. It was based on the Biot number of the surface and the non-dimensional drop radius. The experiments used thin film resistive thermometers to directly measure the conde nser surface temperature of carbon steel, stainless steel, and quartz glass. The i nvestigators found decreasing HTC for decreasing surface thermal conductivity. They also offe red a Nusselt number correlation involving only three variables, maximum droplet radi us and surface and liquid thermal conductivity ratio. The reported agreement of the correlation with their data and that of Hatamiya and Tanaka, 1987 [35] was 20%. The condenser surface was pure copper. The experimental portion was conducte d at low vapor pressure to limit the lower drop radii to a size visible with micro-optics. Results showed condenser surface temperature fluctuations of up to 3C which increased in magnitude with HF. The troughs and peaks corresponded to a droplet reaching its maximum radius and then departing as confirmed by filming of the surface during test runs. The coldest surface temperatures occurred under large drops, since the su rface was insulated. Conversely, the warmest temperatures occurred just after a drop de parture or merger exposed bare surface, indicating a spike in HT. The temperature sensors were deliberately made smaller than the expected drop radius, and were clearly visi ble as completely covered by a sessile drop on the condenser surface. This study considered only small surface subcooling, between 1.0 and 3.5C and

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31 assumed no dependence of HTC on subcooli ng. Comparison with data was limited, though agreement of theory and experiment was demonstrated in the examined parameter range. Song et al., 1991 [36], asserted that a f ilm of condensate existed between drops, which would contribute to heat transfer. Spectral reflectance measurements taken before and during condensation were pur ported to prove this existenc e due to their difference. No estimate of the thickness of this layer was offered, but it was estimated to be on the order of a few monolayers, that is a few times the width of a single condensate molecule. Curiously, spectral measurements were presen ted, with wavelength on the abscissa, as evidence of the occurrence of a single tempor al event, pointing out a loosely repeating spectral pattern as evidence of droplet departure. The fact that the coordinate domain of the analysis was in question some what shadows the conclusions. Gavrish et al., 1993 [37], presented a nu merical simulation that modeled the process of DWC considering only the c onduction through the dr op, surface thermal conductivity, and the effects of sweeping of departing droplets. This therefore neglected both the interfacial matter transfer resistance and the small drop in Tsat due to the curved droplet interface considered in the Rose theo ry. The results, while in agreement with their own data for conductivity vs. HTC, were lower than most of the compared studies. The correlation of data with the model wa s labeled only as good with no quantitative measure of agreement offered. Yamali and Merte, in 2002 [16], conducte d an involved numerical simulation of DWC, considering high body force (1 to 100g) as a control of maximu m droplet diameter and high subcoolings, from 10 to 100K. Th is followed their experimental study of

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32 similar conditions in 199 9 [15]. Their model assumed that most of the HT took place at the edge of the droplets, where the temperatur e gradient was largest and that furthermore most of the HT also took place through drops of very small radius due to their shorter conduction path and therefore higher gradie nt in temperature along with their high population numbers. They started with a complex relation for HF through a droplet based on Tsat, T, and contact angle and other elements similar to the Rose theory. They then modeled the temporal growth and de parture process over th e entire condenser surface, accounting for the total volume of c ondensate flowing down the surface. Most previous studies had assumed that departed drops simply disappeared, a measure largely taken because of computational limitations. They also accounted for coalescence of growing drops and looked at the results of the model with and w ithout the effects of sweeping. They then tracked the overall HT by the total volume of condensate at each step. The result of this simulation was to co rroborate several of the trends predicted by Rose, namely that HF quickly reached an asymptotic value as a function of surface length, x, and so was essentially not depende nt on this quantity. Another similar conclusion was that HTC was indeed dependent on HF instead of constant, as assumed by all the competing models mentioned a bove. A unique finding for very large body force (above 50 g), large subcooling (above 70K ), and large x, a peak was found in HF that was analogous to a critical heat flux in pool boiling. However, the parametric boundaries for this phenomenon were so high as to not be of much practical concern. The simulation results compared well with th e earlier experimental results [15], within 15%.

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33 Droplet Size Distribution A governing assumption of the Rose theory was existing knowledge of the population distribution of dropl et radii. Much work ha s been done to estimate a sufficiently accurate model of this characteri stic since then. The numerical simulation results of Glicksman and Hunt, in 1972 [38] and Burnside and Hadi, in 1999 [39], supported a simple negative exponential rela tion for the droplet size distribution, with much higher populations for smaller drop size s. The former offered -1/3 as the exponential value, later selected by Rose for application with his theory. Some recent studies have employed more complex methods to model the droplet size distribution, such as the random fractal method [40] and mathematical morphology [41]. The resulting comparison to experiment was not significantly improved over the assumed simple exponential size dependence. Rose and Glicksman, in 1973 [19], performe d a close visual study of the temporal development of the droplet size distribution and had these main conclusions to offer: There were distinct generati ons of drops based on what areas were swept by falling drops. Drops of any generation were more or less uniform in size and spacing. The density of nucleation sites was independe nt of droplet size and the same for all generations. The first assertion was backed up by the findings of Burnside and Hadi, in 1999 [39]. “Nucleation sites” refe rs to the microscopic surface defects that serve as bases for droplet nucleation. Numerical si mulations have estimated them to be closely packed in either square or triangular arrays at a spacing of roughly one minimum thermodynamically viable droplet diameter with good comparison of results with

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34 experimental data. The resulti ng site density estimates were 109 and 1011 sites/cm2 [38,28]. Wu et al., in 2001 [42], conducted a micr oscopic study, at several magnifications, of low pressure DWC and found that a self-simila r pattern in relative drop sizes existed at all magnifications. That is, that pictures at different magnifi cations looked almost identical but for different exact positions of drops. As far as other parameters that might a ffect the distribution of drop sizes, HT parameters or surface roughness were eliminat ed [19,35], except for the conclusion that an increasing HTC with incr easing subcooling was due to increasing activation of nucleation sites, specifically below 1K subcooling [43]. Rose [23,44] offered some modifications of the bounds of integra tion in his theory for a better fit to experimental data. Specifically, he suggested replacement of the lower bound of the integral in his theory with 10 r instead of 1 r The upper bound adjustment was two part with 1.5 r ˆ replacing 1 r ˆ below 1 atm while substituting 2 r ˆ for pressures near or above atmospheric. However in later issuances of the theory [12,18,20,24,45,46] these suggested adjustments were not repeated. Effect of Maximum Droplet Size Overall HT has been found to depend on the maximum, or departing, droplet size [15,27,37,38,39,42,44]. The only citations not to note this dependence were [25,26], however the range of drop sizes in these stud ies was relatively small (much less than an order of magnitude change). Various met hods of controlling the maximum droplet size have been employed, ranging from numerical simulation [38,39] to a periodic wiper on the condenser surface [26] and condenser surf ace inclination adjustment [25]. However, the clearly dominant method employed was to induce significant va por shear flow on the

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35 condenser surface, such that the drag on th e sessile droplets would limit their maximum size [27,37,47]. This method had the additional benefit of limiting the influence of NCG in the test vapor. Yamali and Merte, in 1999 [15], noted an increase in HTC with body force, but only for very high accelerations (i.e., above 5 g). This dependence was most likely due to a corresponding decrease in maximum droplet size, which was determined by the balance of body force and surface tens ion on the drop. These authors found a 6fold increase in HTC for a 100-fo ld increase in body force. In support of the idea that smaller droplets are more active than larger ones in HT [28], Tanasawa et al., in 1978 [ 26], noted a peak HTC of 1 MW/m2K just after the passing of the wiper, which was 2.5 to 3.5 times the time averaged value. This peak then decreased linearly until the next wiper passi ng. In a numerical simulation [39], a peak HTC of 2 MW/m2K was found. These high peak HTC values indicate the potential benefit of controlling maximum drop size in order to induce more intense HT than standard DWC. General Results on Drop-wise Condensation (DWC) Lorenz and Mikic, in 1970 [48], offere d a numerical simulation of internal thermocapillary circulation in a liquid drop i nvolved in DWC. The results showed that, for the applicable size range, in ternal circulation in the drop lets was negligible. This finding justified one of the main assumptions of the Rose theory. Another main assumption of the Rose theory – that promot er thermal resistance was negligible – was confirmed by several other studies [17,23,49,50]. Two other studies found no dependence of DWC HT on tilt a ngle of the condenser surface for small deflections from vertical [25,31]. Neither of these two st udies or [17,19] found any dependence on surface height for HTC. However, surface height dependence of HTC for DWC was noted [14]

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36 merely by using a rectangular condenser surface for a series of resu lts, then rotating the apparatus 90 about its centra l, horizontal axis and finding different results. It was unclear whether the orientation of the steam flow remained the same relative to the condenser surface or not, which may have accoun ted for the difference in results between the two cases. The condensation coefficient, contained in an empirical constant in the Rose theory, was defined as the fraction of vapor molecules striking the liquid surface which remain in the liquid phase, was the subj ect of much contention in the early decades of DWC research. Prior to 1985, most esti mates of this fraction ranged from 0.45 to 1 [28,51,52], which left a considerable gap in th e understanding of th e process. Roughly after this date, reported resu lts coalesced to unity [20,24,53]. Dr. Rose often argued that distinct differences existed between the DWC performance of various promoter me thods and chemicals [13,21,31], which was corroborated by the work of others [54,55]. This assertion was fundamental to his argument against most of the evidence presented for a dependence of DWC HTC and HF on the condenser surface thermal conductivity, or constriction resistance, which will be discussed below. An observed result of the efforts of Le Fevre and Rose to minimize the presence of NCG in the test apparatus was that HTC incr eased with increasing HF, or, less precisely, with subcooling [31]. This trend has b een reported elsewher e [15,32,45,54]. Necmi and Rose noted [56], however, that for DWC of mercury, HTC was essentially independent of HF, due to the high thermal conductivity of th e condensate. In [31,43] this trend was specifically labeled as an indicator of th e NCG content of the test vapor. If HTC increased with HF, then NCG was termed su fficiently minimized. If, however, the

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37 opposite were found true, then this would i ndicate poor control of NCG in the test apparatus. The latter assertion made sense, since the concentration gradient set up by the presence of NCG in the condensing satura ted vapor, would tend to be exacerbated by greater relative concentration of NCG. This problem will be discussed in more detail in a section dedicated to the eff ects of NCG on condensation. However, some results have asserted HTC to be independent of HF [4,35]. It should be noted that studies making this case often tested only over a s hort range of HF or subcooling. With such small parameter ranges, a linear relation might be easily inferr ed between these primary variables. Those experiments covering large subcooling ranges, cited above, along with other theories [16,17], did show this trend. The high HF and HTC values reported for DWC [31] were repor ted in a range of 0.3 to 1.8 MW/m2 at 1 atm, with a maximum HTC of 0.3 MW/m2K. Tanasawa and Utaka, in 1983 [27], reported HF from 0.5 to 10.0 MW/m2 at subcooling from 0.8 to 50K, near 1 atm. As predicted by the Rose theory, a decrease in HTC with decrease in saturation pressure was reported by [17,18,24,35,55,56] This decrease was noted to be up to 6-fold for DWC of steam at the lowe st saturation pressures, as compared to atmospheric saturation pressure [35,55]. Rose, in 1998 [24], attributed this behavior to an increase in the interphase matter transf er resistance between the decreasingly dense vapor phase and the liquid phase. Signifi cantly lower HTC was also noted at lower pressures [28,47]. For comparison to FWC, the results for DWC HT range from just 2 to 3 times higher [57], to a range of 5 to 10 times [37], to very high values of 20 to 30 times the FWC, as found in experiment and agr eement with Nusselt correlation (Eq. 2-1)

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38 [58,59]. The lowest DWC HF values were attributed by Rose to poor quality DWC promotion. Some unique findings were those of Koch et al., in 1998 [14], that partial condenser surface coverage (of only 20%DWC/80%FWC) produced almost 90% of the full DWC coverage value of HF. Also, Citakoglu and Rose, in 1969 [25], used their surface inclination dependent measurements of DWC HTC to conjecture that the average HTC for a horizontal circular tube would be about 80% of a ve rtical plate under the same conditions. Finally, a point of note in all of the presentations which involved Rose, the HTC was only plotted as a function of HF and not surface subcooling. This distinction was made because of the error inherent in measuring this often small difference between the vapor saturation temperature and the su rface temperature and then relying on its reciprocal for the HTC. For the same r eason, he advocated only plotting subcooling on the abscissa or y co-ordinate and the HF on the ordinate or x axis for graphs comparing the two quantities [12,13,17,18,20–25,31,32,45, 46,52,53,56]. Constriction Resistance There has been considerable contention, in the discussion of DWC, over whether the overall HT was at all dependent on the th ermal conductivity of the condenser surface, that is whether a constriction thermal resistance existed or not. The Rose theory [17,24, 45], employed in the present work, assumed a ny dependence of HF on this property to be negligible, and obtained “excelle nt” agreement with experiment al data for an array of promoters and for a few different surfaces. One contention of Rose, stemming from the observed differences in promoter effectiv eness [31,60], was that surface chemistry was critical to the performance of any DWC scheme and might easily be the contributing factor in those studies which claimed affirm ative evidence of such a dependence [21].

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39 The authors used a thin, on th e order of a few microns, coppe r layer on a stainless steel condenser block and compared the results for DWC with a solid copper block, finding a difference in HTC only on the order of magnitude of the uncertainty in this value. The thin copper layer on the steel block was assu med to offer identical surface chemistry to solid copper block, which was the key to Ro se’s argument that surface chemistry was the actual critical factor and not the substrate thermal conductivity. Over the decades of research on this t opic, various researchers have found their results to indicate a dependence of overall HT in DWC on the condenser surface thermal conductivity. Most did not account for potential differences in surface chemistry [33,34,37,50,54,61]. The latter pair of references offered a Nusselt correlation for DWC dependent only on thermal conductivity of th e substrate and condens ate and on the given non dimensional droplet radius that ag reed within 20% of the accompanying experimental data. The same two studies also noted an increased importance of constriction resistance at s ub-atmospheric pressures. Tw o other studies [14,43] found a significant constriction resistance while controlling for surface chemistry. However, discussion of these findings did not include any mention of the inherent uncertainty in the small temperature difference measurements and the resulting implications when used in the denominator of the HTC. Mikic offered a theory including a constriction resistance [29] and then a study looking fu rther at this theoretical dependence [62]. He then compared the theory to experi ment [61], with visually agr eeable results. However, the discussion of uncertainty, as mentioned above, wa s also omitted from this presentation. In his 1998 address on the topic of DWC, Rose reiterated his opposition to the importance of a potential constriction resist ance, while still acknowledging the existence

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40 of debate on the topic [24]. In 2002, he went fu rther to offer that it may be significant, in an additional T of 1 to 2K, but only for very low conductivity surfaces and low HF, less than about 0.1MW/m2, and small condenser surface lengths, less than about 10 mm [18]. In that talk, and also [21], he dispatched th e findings of most of the studies cited above, which had all found evidence of a constriction resistance. He did this by questioning their methods of surface temperature estima tion in relation to uncertainty and the magnitude of the difference in HT supposedly caused by the constriction resistance. Another issue he cited to question their asse rtions on constriction resistance was surface chemistry differences, which none had addressed. Thus the literature was still quite divided on the existence and importance of constriction resistance. Howeve r, the author of the main theory, the investigator most often invited to speak on the topic of DWC, maintained that the potential importance of constriction resistance need only be considered under certain, limited conditions. Drop-wise Condensation (DWC) Promotion For most applications of DWC, some so rt of promotion method was necessary to effectively reduce the wettability of the conde nser surface, in other words to decrease the contact angle of the vapor-liquid-solid interf ace or contact line. Three main modes of surface preparation were us ed in the literature: 1. Direct application of or ganic or other compounds to the condenser surface 2. Addition of these same promoters to the bulk liquid reservoir supplying the steam to the condenser surface 3. Physical or electro-chemical manipulation of the surface makeup Organic promotion, as the first group will be referred to, yielded the highest HTC values. This promotion mode was durable but only under clean la boratory conditions.

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41 This was the mode most often employed in the study of DWC phenomenon itself. However, this method later fell out of use due to its sensitivity to contamination usually present in industrial conditions. The substanc es employed in organic promotion included montan wax [55], montanic acid [55], dioctadecyl disu lfide [31,55], oleic acid [21,25,32, 63], and stearic acid [64]. Self assembled organic monolayers of 1-octadecanethiol and 16-mercaptohexadecanoic acid were also tested [65], but the lifetime of DWC promotion was only on the order of a few hours. Tanasa wa et al., in 1978 [ 26], employed a silicon resin with some success as a promoter. Some of the same substances were applie d by addition to the bulk liquid reservoir, by simply dropping a few ounces on the liquid surface, as was done with dioctadecyl disulfide [61], 2-ethyl-1-hexanol [66], and with partially-fluorin ated carbon disulfide [37]. The latter case was tested with several condenser surf ace substrate materials. The investigators reported lifetimes of up to 4200 h. Furthermore, with this promoter the DWC promotion regenerated simply by adding more of it to the reservoir. This regeneration however could onl y be induced for a few cycl es before disassembly and cleaning were required. The bulk addition was also coupled with the direct surface application of the organic promoter w ith good results by othe r researchers [55]. According to Dr. Rose, in 2002 [18], dur able promotion of DWC still remains a “key barrier to commercial implementation.” Modification of the physical makeup of the condenser surface has been the focus of most of the work in recen t years pursuing this goal. A thin plating of noble metals ha s shown durable results in a laboratory environment using gold [15,35] and silver [67]. The latter showed a considerable 2400 h lifetime. The effectiveness of these noble metals as promoters has been attributed to their

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42 adsorption of trace organic elements from the air, as their promotion ability was notably absent when air exposure was not allo wed between cleaning and testing. Chromium was studied as an alternative to the noble me tals for a condenser surface [35,63]. The best results came from a pow der formed chromium condenser block, vacuum impregnated with oleic acid. This surface preparation s howed a 3000 h life of DWC, under laboratory conditions. Koch et al., in 1998 [14], used amorphous hydrogenated carbon films, described as a diamond-like coating. The physical toughne ss of these layers was held to offer potential for a durable promotion met hod, though no lifetime tests were conducted. Contact angle was also purportedly controllab le in a small range, through the processing conditions of the coating. What was described as a Langmuir-Blodgett (L-B) film showed much promise for a durable promotion method [59]. Several m onolayers of barium stearate were overlaid on the condenser surface in such a way that the negative pole of the molecules all pointed away from the surface, resulting in a highl y non-wettable surface. DWC heat transfer coefficients were reportedly up to 30 times higher than FWC. The coating was also reported as very physically tough and insoluble in both oil and wa ter. This fact was quite significant since barium stearate was soluble in both when not prepared in this manner. The coating was also reported to be resist ant to frictional wear and evaporation in a vacuum. No lifetime tests were conducted. Dimensional modification of the condenser surface to increase sweeping frequency was employed by cutting verti cal half-cylindrical secti on grooves in the condenser

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43 surface [68]. However the increas e in HT was at most 25% a nd was attributed only to the increase in surface area, despite a noted increase in sweeping frequency. Teflon coating has been studi ed as a DWC promoter [47] with propylene glycol as the working fluid. Bochagov and Dor okhov, 1983 [57], induced DWC of steam on a Teflon coating with less success, likely due to presence of NCG in the vapor. More recent studies have employed techniques involving ion implantation, to reduce the wettability of Teflon [69, 70], and copper [58] The Teflon results were poor due to bad physical bonding of the coating with the substr ate, but the copper re sults showed promise in effectiveness and durability. Here a beam of ions was reflected from a target, made of a given material, onto the condenser surface to be modified, where the reflected ions combined with the metal atoms of the surface to form an ‘amorphous crystalline state’. This amporphous surface had a much lower su rface energy than a pure metal surface and therefore showed much lower we ttability. Ma, et al. [71], cited a study maintaining that minimization of surface energy was a more reliable method of assessing potential promotion effectiveness than contact angl e or wettability. Species tested for implantation were He, Ar, N, and H. The H surface was reportedly the least wettable, with He and Ar next in line and roughly e qual, and with N being the most wettable and therefore least effective DWC promoter. The HT results for the tubes used were on the same order of performance as organic promot ers such as oleic acid, but with much longer lifetimes and better resilience. In 1994, Zhao and Burnside [72] conducte d the only successful test of a DWC promotion scheme under industrial conditions An 800 tube bundle, treated in the manner described above, replaced a 1600 tube bundle of similar tube diameter and length

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44 in a power station in China. Cr and O were the elements chosen for implantation in this test on Cu. Over the 15,600 h length of the test, DWC was never lost and the test was purportedly only curtailed due to the primary re searcher’s graduation. The authors of this study felt secure enough in their results to main tain that they indicat ed the possibility of permanent and reliable DWC promotion. The processing of the surface coating, however, was as yet prohibitively expensive. In a review paper in 1999, Rose noted th ese promising developments [45], but he also noted was the downward trend in HTC with increasing HF. Rose had long held this decrease as an indicator of the presence of NCG in the condensing vapor, its presence thereby calling into question th e accuracy of the reported data. By 2002, he had reverted to his contention that, “...reliable means fo r promoting dropwise condensation has yet to be convincingly demonstrated,” [18]. In 2004 [20] he elaborated that more stable promoters had been developed but the reporte d HTC from these studies were still well short of the order of magnitude increase over FWC expected from monolayer organic promotion. As a side note on DWC of liquid metals, Rose noted that promotion was often not necessary for these species [24] He also noted however th at the typical advantage of DWC over FWC was greatly reduced due to the very high therma l conductivity of the condensate, relative to a typical wo rking fluid such as water. Effects of Non-Condensable Gases (NCG) Condensation is ideally studied with only a pure, saturated vapor supplied to the condenser surface. This state can be quite di fficult to achieve in reality, since removal of non-condensable gases, or NCG, usually air, is effectively an asymptotic process rather than a discrete one. At equilibrium, the condensable vapor would have the NCG

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45 molecules evenly distributed through the volume. However, once the process of condensation is initiated, a net mass flow to ward the condensing surface is induced. The resulting movement of the bulk sweeps the NCG molecules with the vapor flow toward the surface. However, at the surface, the vapor molecules condense, but the NCG molecules cannot. In this manner, a conti nuous process builds up the concentration of NCG at the condensing surface [73]. Since pre ssure is a continuous property, the partial pressures of the two species must add up to th e vessel pressure at a ny point in the vessel volume. Therefore, the increased partial pr essure of the NCG at the condenser surface forces a decrease in the partial pressure of the vapor species at the condenser surface. Since, for a saturated liqui d-vapor state, pressure a nd temperature are dependent quantities, the reduced partial pressure of th e vapor species at the liquid-vapor interface leads to a decrease in the apparent saturation temperature. If the driving force of the NCG buildup at the condenser surface were removed, that is, the cooling of the condenser c eased, the NCG molecules would disperse by diffusion, until they were evenly distributed in the vapor bulk again. However, if the cooling of the condenser proceeds at a steady rate and the geometry is 1-dimensional, then an equilibrium partial pressure, higher than that in the bulk of the vapor, will develop. The resulting concen tration gradient then driv es the diffusion of the NCG molecules against the net mass inflow of vapor. If the geometry of the problem is 2 or 3 dimensional, the NCG build-up and departure causes significant transients in the HT readings. In 1965, Kroger and Rosenhow [74] mode led the 1-dimensional diffusional resistance problem of FWC in the presen ce of a non-condensable species, namely

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46 potassium vapor in the presence of He and Ar, separately. Their simple model showed agreement with experiment within 25% for He Results were less positive for the runs using Ar for the NCG. Steady state was neve r satisfactorily reached, the theorized reason for which was a periodic buildup and departur e of Ar at the condenser surface, which would serve to insulate the condenser surface as they buildup and then allow HT to spike when they depart. Many simulations of FWC have been conduc ted, growing from models similar to the diffusional one offered by Kroger and Rosenhow. These studies showed that geometry was a major factor in the sensitiv ity of a given condensation process to the presence of NCG. For closed channel flow FWC, reductions in HT from the case with no NCG content of 30% to 60% were repor ted for simulations of 1% to 10% NCG concentration, respectively [75]. In a simila r case to these studies, it was reportedly necessary for NCG concentrations to be as low as 0.1% in order for there to be no discernable effect upon the HT [76]. For FWC in a porous medium, a 50% decrease in HT for 5% NCG concentration was reported [ 77]. For closed channel flow geometry, only 3.3% NCG reportedly produced the same re duction [78]. A theoretical investigation of FWC on the inside surface of a rotating dr um in [79] showed a process much more sensitive to NCG relative to those discusse d above. Here a 1% NCG concentration produced an 80% reduction in HT. This in creased sensitivity was explained as the centripetal acceleration due to the rotation of the drum increasing the resistance to diffusion of the NCG away from the liquid-va por interface. The bottom line to all the above findings was that the effect of NC G in the condensing vapor on FWC was to impede the condensation process. Furthermor e, this impediment was exacerbated when

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47 geometry of the situation restricted diffusi on of the NCG molecules. This conclusion makes sense in light of the diffusional problem described above. Parametric effects noted by these studies were as follows: More pronounced effect of NC G at lower pressures [75] More pronounced effect of NCG fo r restrictive geometry [75,77,79] Vapor with NCG has lower saturation temperature [76] Le Fevre and Rose, in their 1965 presenta tion of experimental results on DWC of steam near 1 atm with organic promoters [ 31] and also the ensuing presentation of a theory on DWC [17], noted an extreme varian ce in the reported HT characteristics found for DWC. They attributed this variance to the presence of NCG in the test vapor. While not controlling directly by addition of NC G to the steam flow, the experimental study instituted a “close venting” scheme. This e ffectiveness of this scheme was demonstrated by varying the venting rate, to simultaneously minimize variation in the recorded surface temperature and raise the recorded surface te mperature. These two effects indicated higher HT. In this case the surface temp erature was found by extrapolation of the internal temperature distributi on of the copper condenser block. Also in 1965, Tanner et al. [54] studied DWC near 1 atm with the controlled addition of NCG to the steam flow. These i nvestigators reported a strong dependence on HF of the sensitivity of DWC to the presence of NCG and also showed that DWC was significantly more sensitive to NCG than FWC. For example, for an addition of 90 ppm of NCG to the steam flow, or 0.009%, a 50% reduction in HF was noted for 0.1 MW/m2. However, for the same addition, but lower HF, 0.03 MW/m2, no reduction was noted over the case without added NC G. For the higher concentr ation of NCG, 500 ppm or 0.05%, still very small relative to those c oncentrations generati ng a noticeable reduction

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48 in FWC, a 70% decrease was noted for the higher HF of 0.1 MW/m2 and 20% decrease for 0.03 MW/m2. In 1968, Tanner et al. again studied the e ffect of NCG on DWC [55], but this time for reduced saturation pressures from 0.03 to 0.1 bar or 0.8 to 3.0 in-Hg-abs. This pressure range almost precisely coincided with the pressure range of the present investigation. NCG concentration was verifi ed as less than 1 ppm for the inlet steam without addition, and varied up to 15000ppm, or 1.5%. The effect of this controlled NCG addition on HTC was noted. The results showed that lower saturation pressure made the HTC significantly more sensitive to NCG concentration. In some cases, any concentration above 0.1% NCG yielded more than 50% reduction in HTC, clearly showing the following: DWC is more sensitive to NCG presence than FWC. DWC HTC is more sensitive to NCG for lower saturation pressures. DWC HTC is more sensitive to NCG presence for higher HF. There were several different methods enc ountered in the literature for controlling NCG concentration in the study of DWC. Th e close venting strate gy used in [13,31] was also employed in [21,22,32,25], all of which invol ved Dr. Rose. His experience lead him to note in a 2002 review [18] the ‘extreme se nsitivity’ of DWC to the presence of NCG, even down to a concentration of a few pp m. The most common method was simply using vapor shear from the excess steam flowing past the condenser surface to sweep away any accumulating NCG [27,28 ,34,35,37,43,47,54,55,61,63]. A drawback to using vapor shear to control NCG was that, as dem onstrated by the cooling curves constructed by Tanasawa and Utaka, in 1983 [27], was that steam velocity had a convective enhancing effect on the HT, impeding a pure an alysis of the process of DWC itself.

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49 Some, however, made no effort to contro l NCG [14,59,67], though most in this case did not indicate any appreciation fo r the importance of NCG in this process. Yamali and Merte [15] admittedly made no effort to control NCG, but it was estimated that the extreme body forces used in shear on the condenser surface in their study effectively accomplished what vapor shear would, by sweeping away NCG. Additionally, increased sensit ivity of DWC to the presence of NCG for increasing HF was noted [45,47]. Rose asserted that a downward trend in HTC with increasing HF (or subcooling) indicated a significant presen ce of NCG in the condensing vapor [24,31]. He later employed this assertion as a litmus test for the validity of DWC HT results [45]. This trend was corroborated by the contemporary results of Griffith and Lee [43]. Finnicum and Westwater, in their 1989 st udy of DWC on chromium [63] noted an important point about verifying the effec tiveness of DWC by c onsideration of the increase in HTC over the FWC case. They noted that, if air was allowed to back up into the test cell, by decreasing th e heater input, the appearance of the condensate in the FWC test changed to DWC, however the HTC was even lower than before the allowed lapse. Heat Transfer with Vibration Not much study has been made into the eff ect that vibration in general may have on phase change HT. In fact only one study was found looking specifically at the interaction of vibration with DWC. Some work has been done on vibration and FWC, but mostly in the range of low frequency os cillations, such as would be found in power plant pipes or on an ocean platform. In 1969, Dent offered e xperimental results of FWC of steam near 1 atm on a horizontal tube, subjected to ver tical vibrations [80]. The frequency range was from 20 to 80 Hz with amplitudes up to 0.17 inch. In this case vibration was found to increase HT, but only by a maximum of 15%. That same year, Dent also offered

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50 results for FWC of steam near 1 atm on a verti cal tube, subjected to horizontal vibrations in the frequency range from 22 to 98 Hz, with a resonance at 38 Hz. Amplitudes ranged up to 0.5 inch at resonance, which also show ed the peak HT increase of 55% over the non-vibrational case [81]. In 1977 Brodov et al. [ 82] also looked at FWC of steam near 1 atm on a horizontal tube, subjected to vertical vibrations. The frequency investigated was from 0 to 50 Hz, with amplitudes rangi ng up to 1.5 mm. They found a maximum increase in HT over the non-vibr ational case of 20%, and also noted that no change in HT was observed without visually significant dist urbance of the condensate film. In 1980, Savel’yev et al. [83] looked at the effect of vertical vibrat ions on both a horizontal and a vertical tube undergoing FWC of steam near 1 atm. The intens ity of the oscillations was given as up to 0.12 m/s, where intensity was calculated from 2 f A, f being the frequency and A being the amplitude of oscillation. The results were notable in that certain intensities showed a signifi cant decrease in HT due to th e addition of vibration, up to 40%. This decrease was noted as due to a vibrational anti-nod e causing the condensate layer to thicken significantl y, thereby impeding HT. The maximum increase in HT was noted at 20%, which likewise corresponded to a node in vibrational mode that caused significant thinning of the condensate layer. In 1986, Nishiyama et al. [84] looked at the FWC of R-11 on a vertical tube subjected to oscillations likely to be experienced on an ocean mounted condenser. They looked at 3 modes of oscillation, heave – up and down motion, sway – side to side motion, and roll – pivoting relative to vertical. The frequency range investigated was from 0.1 to 1 Hz. The results showed no significant effect for the first two modes, but up to a 15% increase in HT for the roll case, also described as a pendulum motion of the vertical tube. This significant in crease for the roll case was

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51 notably accompanied by thinning of the condens ate film due to the centripetal force. This corroborated the findings of Brodov et al., 1977 [82], that significant visual distortion of the condensate film was necessa ry for significant change in HT. In 1988, Nishiyama [85] used the same apparatus and experimental parameters, this time adding vertical flutes to the tube surface. The aut hor again reported negligible effects on HT in the heave and sway vibration modes and so me increase in the roll mode, due to the thinning of the condensate film. The effect of vibration on phase change wa s also investigated by Oh et al., in 2002 [86]. This author looked at the influence of ultrasonic vi bration on the me lting process of solid paraffin wax. In a rectangular block, heated from one side and insulated on all others, the melting time was found to be decr eased by up to 2.5 times. This increase in efficiency of the melting process was found not to be due to increased HT from the heater surface, but rather to be the result of ac oustic streaming. Acoustic streaming is a phenomenon in which the compression and expa nsion waves of the vibrations induce the rapid formation and collapse of gas from the liquid, exciting bulk fluid motion which thereby increased mixing of the liquid volume. In 2004, Kim et al. [87] looked at the effect of ultrasonic vibration on pool boiling of FC-72 at 1 atm on a nichrome wire. The results showed an increase of 80% in HT in the natural convection range, 5 to 15% increase in the nucleate boiling range, and 5 to 30% increase in the saturated boiling range. Acoustic streaming was again deemed responsible for the no ted increases in HT efficiency. The relatively more significant e nhancement in the natural convection regime was explained by the fact that, in the nonvibration case, there was no induced mixing from vapor bubbles, while this effect wa s present due to cavitation and acoustic

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52 streaming in the vibrational case. Since the nucleate and the sa turated boiling regimes already had the vapor bubble induced mixi ng present, the enhancement of mixing by introduction of the ultrasonic vibration was less intense. There was one study found, Shiraiwa et al., 2004 [88], which focused on DWC of steam near 1 atm and the effect of perpendicu lar mechanical vibrati on of a vertical plate on this process. Promotion was accomplishe d with oleic acid. The oscillations were driven by an eccentric crank mechanism from 0 to 15 Hz with a fixed amplitude of 5 mm peak-to-peak. The analysis was limited to thr ee coolant flow rates, in a close cluster, and only a small range of steam pressure. Also, no mention was made of the surface temperature of the condenser plate or the ma terial, leaving important ambiguity in the analysis of the quality of the DWC occurring in the test. No specific attempt was made to control NCG in the apparatus, a fact whic h showed in the low HTC attained of 11.7 kW/m2K. To attain a HTC this low, the Rose theory would require a surface subcooling much less than 0.1C, and much lower than any probable uncertainty in the calculated quantity. This discrepancy between the Rose theory and the results of this study imply a significant, depressing influen ce of NCG on HT in these expe rimental results. Despite this discrepancy, the results cl early showed an increase in HTC of up to 40% due to the application of mechanical vibration, while not ing a measurable decrease in the maximum droplet size and accompanying increase in sweeping frequency. An attempt at a numerical model was made, based on severa l simplistic assumptions of proportionality between maximum droplet diameter, growth time, and HTC, which predicted a much smaller increase in HTC than actually found, and relied on the measured HTC as a base. In sum, a positive increase in DWC HTC was shown in this study, but the quality of the

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53 DWC was very low, asserted by comparison he re to the Rose theory, and therefore the magnitude of the effect was questionable. Also of note in this study was a finding that temperatures at 15, 45, and 75 mm from the t op of the vertical condenser plate were essentially equal for all the vibrations, co rroborating a central assumption of the Rose theory, that HTC for DWC on a vertical plate is independent of surface height. Piezoelectric Droplet Ejection with Heat Transfer Some noted attempts have been made to establish a HT cell using a piezoelectric disc as the driving force to create a spray which then cools a heated surface. Heffington et al. [89,90] described a method of droplet ejection using a piezoelectric diaphragm which created a spray of wate r droplets by using a cover pl ate with holes in it which acted like nozzles. By using th is setup for spray creation a nd direct contact condensation on a large pool of liquid, they auth ors achieved a HF greater than 100 W/cm2 with a heater surface temperature of 85C and a si nk at room temperature. The design was highly gravity dependent to collect the liquid re servoir and keep the pr oper liquid level so that the driver cover wa s not flooded with liquid. Xia, 2002 [91] also used a piezoelectric dr iver with a cover as a nozzle array. In this study, the nozzles were etched in the cover material in orde r to create jets of water to cool a target surface. The nozzle array wa s fed by pump and the liquid reservoir was subcooled. The frequency of dr iver operation was chosen such that a new droplet arrived as soon as the previous one had finished spreading on the heated surface. Especially of note in this case was the extrem ely high HF achieved of 1 kW/m2.

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54 CHAPTER 3 THEORETICAL ANALYSIS The theoretical analysis of this study was based on the Rose theory for DWC. The two main arguments of this func tion are saturation temperature, Tsat, and surface subcooling, T. The theory assumes a simplified thermal circuit of the HT through a single drop of radius r, with direct contact condensation occurring only on the curved surface of the drop. Then a statistical dist ribution of the drop population by radius is assumed. This distribution is then used to we ight the expression for thermal resistance of a given drop of radius r. Fi nally, the weighted expression is integrated over the assumed distribution of drop sizes, yiel ding a theoretical prediction fo r the average HF of a surface undergoing DWC of a pure vapor. Manipulation of the upper bound of the integral, maximum departing droplet radius (rmax), was used to estimate an effective maximum droplet radius for the measured experimental conditions. A comparison of this value for rmax was then used to assess the effects of NCG on that particular experimental run and also the effects of the presence or absence of VIDA on HT. This section details th e process used in the theoretical analysis of experimental data. Since the intent of the design was to dire ct all HT absorbed by the heat sink through the brass shim of the piezoelectric diaphragm, the te mperature gradients in this thin piece of metal would likely be significant. In fact, with a nominal 1K subcooling, the Biot number for the driver is on the order of 100 for the radial direction, also implying significant temperature gradients. Since the Rose theory depended on the

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55 surface subcooling, it would then require an estimate of the surface temperature distribution in order to determine the theoretical HF distribution. So initially, a boundary value problem was c onstructed to model the heat conduction in the brass shim. Axisymmetric heat conducti on, cylindrical coordinates, and steady state conditions were assumed. Then the Heat Conduction Equation (Eq. 3-1) was solved using the following boundary conditions for a cylinder of brass with radius b and thickness c. 0 12 2 2 2 r T r T r r T 3-1 1. r T as r 0 is finite 2. b r T = To, the measured edge temperature 3. 00 zz T, insulated condition at r ear surface of driver 4. k q z Tavg c z" assumed average HF at front surface of driver This assumption of average HF over the face of the driver greatly overestimated the actual temperature gradients. The input average HF had to be lowered to around 1W in order to lower the center temperature of the diaphragm to anything in the realm of the study. Even with this significan t overestimate of the temper ature gradient in the shim, the greatest temperature difference in the ver tical direction was less than 1C, an amount considered negligible in the face of a maxi mum difference between the center and edge temperatures of almost 40C. Armed with justification for neglecting the temperature gr adient in the z direction, different functional forms were assumed for the likely radial surface temperature distribution of the shim. An exponential relation was deemed most likely to satisfy the

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56 experimental conditions due to the implied zer o slope at the origin (central axis) and ability to fix the slope at the edge (heat flux) The distribution was then defined with the following form (Eq. 3-2) and boundary conditions. B Cr r Tn 3-2 1. To b r T 2. Tatm r T 0 3. bc k THR k q r Tc r2 0 1 2 3 4 5 6 7 8 9 10 5 10 15 20 25 30 35 40 45 50 radius, [mm]Temperature, [C]Representative Radial Temperature Profiles T(r) = C/rn+B Tatm = 45C, To = 18C, q = 55W Tam = 40C, To = 15C, q = 30W Tam = 40C, To = 15C, q = 30W Tatm = 30C, To = 8C, q = 7W Figure 3-1. Assumed radial surface temperat ure distribution. Distributions based on three sets of experimental parameters re presenting a low, middle, and high HF case. A theoretical HF distribution on the shim surface was estimated for each set of experimental parameters. First a radial -exponential temperature distribution was determined from a MATLAB subroutine, exp_p rofile.m (Appendix A) (Figure 3-1). For low overall HT rates (e.g., 7 W) the radial temperature distribution smooth and gradual.

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57 However, for higher HT rates (e.g., 30 W) the temperature gradient at the edge of the shim was steep, and the majority of the shim surface area was near to the assumed atmospheric temperature, meaning that this portion of the surface would be relatively inactive in the DWC process. Next, the assumed temperature distributi on was fed into a MATLAB function that would determine the corresponding HF dist ribution, DWC_HTC.m (A ppendix A), using the Rose theory (Figure 3-2). For the highe r HF considered (e.g., 55 W), the majority of the diaphragm surface was relatively inactiv e in the HT. The higher the THR was the smaller the predicted active ar ea of HT on the diaphragm surf ace. This was most evident from the HTC distribution in th e top section of the figure, wh ich clearly shows that the 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 HTC, [MW/m2K] 0 1 2 3 4 5 6 7 8 9 10 10-10 10-5 100 105 HF, [MW/m2]radius, [mm] Tatm = 45C, To = 18C, q = 55W Tatm = 30C, To = 8C, q = 7W Tatm = 40C, To = 15C, q = 30W Figure 3-2. Radial HTC and HF distributions to follow the assumed temperature distributions. higher HF-case HTC was more than an order of magnitude less than the peak value for about 95%, 60%, and 40% of the radius for the 7, 30, and 55 W cases, respectively.

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58 The calculated radial HF distribution wa s next numerically integrated over the surface of the diaphragm, using a Simpson’s Rule algorithm, simpsons_rule.m (Appendix A). The resulting value was the theoretical THR based on the inputs. This theoretical THR was then fed back into the temperature profile determination step to create an iterative scheme (Figure 3-3). Convergence of this scheme was quite slow and dependent on the initial guess for the THR input. Therefore a step wa s added after the determination of the theoretical THR which av eraged the current iter ation value with the value from the last iteration. The result of this additional step was that the number of iterations required for convergen ce to the desired error was re duced by a factor of about 20 and the dependence on initial guess was eff ectively eliminated from the scheme. This adjustment to the iteration was evaluate d for accuracy by comparison to an actual converged result. Also a rela xation factor was added, which va ried from 1 to 0. Using a relaxation factor equal to 1 resulted in th e original scheme, without relaxation. While using a relaxation factor of 0 resulted in th e current scheme. Zero was determined the fastest rate of convergence with no impact on the resulting answer. Convergence acceleration aside, the com putation was still time consuming, taking about 5 minutes of processor time per data va lue. Thus, for efficiency in determining uncertainty in THR from uncertainty in the m easured values, a data table was constructed of theoretical THR values for a grid of saturation temperature and edge temperature values. The function was esse ntially linear in subcooling, de spite the dependence of the lower bound of the HF integral on this valu e, and only slightly non-linear in saturation temperature for the ranges considered. Because of this, large steps were possible in the table construction. The i nput values used were Tsat of 24 to 50C, in 2C increments and

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59 Figure 3-3. Block diagram of iterative loop to determine theoretical THR. A theoretical THR was estimated in this manner for each set of experimental conditions.

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60 edge temperature (To) of 4 to 20C in 4 C increments. One table was constructed assuming the theoretical maximum droplet ra dius (Eq. 2-11), which was essentially 1.1 mm for the range of Tsat used. Seven more tables were then constructed, each assuming a different value for rmax, from 0.01 to 4 mm. For each of these tables of theoretical THR values for the given experimental variables and geometry, linea r interpolation was used to determine the THR for the measured parameters of a given data point. This resulted in what was the official, theoretical THR, using the theoretical maximum droplet size from the Rose theory. This value was computed from the first table. Th en the experimental parameters were used with the seven other THR tables to predict wh at would have been th e results if the given maximum drop radius value had been in effect (Appendix B). The seven rmax values were then fit to an exponen tial curve, similar to the one used for the temperature profile, by the least s quares method. The least squares method was not able to be directly solv ed, as would be the case for a polynomial function. Therefore a brute-force method was used, by simply comp uting the sum of the squares of the error for sets of values that were likely to cont ain the appropriate coefficients. The minimum sum of the squared errors was chosen. Th en several steps of range and increment reduction in the coefficient inputs were made to increase the precision of the coefficient estimates and decrease the overall error of the fit. The fitted curve was extended to a factor of 10 smaller than the lowest rmax used and to a factor of 10 larger than the largest rmax used. The experimental THR was then compared to this fitted curve to determine an effective rmax for that data point. The effective values were then compared with the

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61 theoretical rmax to infer whether the results of that test had achieved appropriate HT for DWC (i.e., NCG presence was reduced enough that it did not appreciably affect the HT results). For those data points which had an rmax larger than 10 times the largest table value, 0 was assigned and these points were th en disregarded. The intent was that those runs with VIDA present would sy stematically show effective rmax values smaller than those runs without VIDA present. This w ould result from showing higher THR for the same temperatures. The Data Summary and Comparison with Theory section of Chapter 6 addressed the results of the above theoretical analysis for this study. Uncertainty was determined in these inte rpolations by taking the combination of the saturation temperature and edge temper ature uncertainties th at would produce the largest difference from nominal. These limiting values were then interpolated from the tables. The difference between these extrem e theoretical THR values the nominal one, determined from experimental conditions, was reported as the uncertainty in theoretical THR. For the effective rmax uncertainty, the uncertainty in the experimental THR was used in the same manner discussed above to calculate the high and low values, then the difference reported. For example, the best data point attained from the experimental results was put through this process, and th e results follow. For Tsat of 24.700.17C, To of 13.80.2C, and an experimental THR of 332 W, the theoretical THR returned was 21.70.8 W and the effective rmax was 0.080.04 mm. An exponential curve fit to the range of theoretical THR values, determined from the consider ed range of possible maximum departing droplet radii, shows an excelle nt fit (Figure 3-4). The squa re root of the sum of the squares of the residual error to this fit was 0.24 W, or about 1% of the theoretical THR.

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62 Total estimation errors from using the tables could be expected to be of similar magnitude for all other experimental data point s. This data was discussed further in the Data Summary and Comparison to Theory section of Chapter 6. The MATLAB programs used for this interpolation and curve fitting process were called Get_Total_Heat_Rate_and_r_eff.m and Heat_Rate_Archive.m (Appendix A). 10-3 10-2 10-1 100 101 102 10 20 30 40 50 60 70 [mm][W] Figure 3-4. Theoretical THR vs. maximum droplet radius. THR is in W. Radius is in mm. Here, Tsat is 24.700.17C and To is 13.80.2C. Circles mark the seven theoretical THR values for the given te mperature conditions, determined from the seven different maximum drop sizes considered. The curve is the least squares fit to those seven data points.

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63 CHAPTER 4 APPARATUS The main aluminum cylinder that defines th e pressure vessel for this apparatus was used from a previous study on a different topic. Theref ore many design choices were made simply to accommodate existing features of this pressure vessel. For example seal grooves already existed on the e nds of the cylinder and so only a smooth surface had to be made on the covers to work with them. The original heater built for this expe riment was taken from a study for spray cooling [89] and thus was intended to evaporate droplets impacted on its surface. The heat transfer was to be measured by the temper ature gradient in the neck by three pairs of thermocouples. This temperature gradient co uld then be extrapolated to determine an average surface temperature for the heater. This choice of heater was made in a misguided attempt to reach toward an overall HT cell on the lines of the cited study above, in the hopes of moving cl oser to the ultimate goal of a two-phase HT cell that would be gravity insensitive. Well into the first round of experiments, it was realized this strategy would not work. The design was inadequate because of the requirement of a saturated vapor environment. Therefore the apparatus had to be redesigned to focus on the condensation phenomenon only. Also, it was realized that a benchmark w ould be needed for condensation without vibration, and the only relations that might apply to this appa ratus were for an isothermal vertical plate. Thus the experimental pre ssure vessel (Figure 4-1) was set on a stand, on

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64 its side, so that the condenser surface was close to vertical. The orientation of the outside of the vessel was checked with a common carpe nter’s level. This method was deemed appropriate based on the experi mental results of [25], whic h showed the DWC process to be relatively insensitive to changes in surf ace orientation. In th at study, a 10% decrease in HTC over the vertical case required a 40 forward pitch or a 10 rearward pitch. Therefore if this condenser su rface was within 3 or even 5 of vertical, then any effect would likely be slight. Ice was added to the coolant reservoir to lower the inlet temperature and provide more driving force for the HT process on th e diaphragm. The coolant used was tap water. A simple model was constructed for desired reservoir temper ature and consequent required mass of ice, and the amount found to be reasonable for acquisition and transport. Fifty pounds of ice would be required for an in itial reservoir temperature just above the freezing point of water. Later on, when it was determined that the experiments needed to be run for long time periods, the procedure for ice handling was refined, as described in the Chapter 5. However, the heater problem still needed a quick fix. By experiment, it was determined that a 50 W halogen lamp would provide both extra light for video records and sufficient heat to drive the experiments. However, this would leave little control or measurement of the heat input to the appara tus. Various placement s of the light were tried, and below the bottom window of the vessel, facing up was selected as the optimal. This would allow the placement of the fluid reservoir in the bottom window and get the bulk fluid as close as possible to the heat source. Initially, a crude reservoir for liquid was fashioned from a piece of aluminum and placed in the bottom window.

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65 Experimental runs with this configurati on, however, rarely show ed any condensation on the windows of the vessel. Si nce the vessel was basically co ld to the touch and the Tatm measured inside the vessel was above ambien t, this implied that the vapor atmosphere inside the vessel was not saturated. In order to channel more of the heat energy from the lamp through the fluid reservoir, further tests used th e window well itself as the rese rvoir. After this, when the test atmosphere temperature rose above the lab ambient temperature, condensation almost immediately appeared on the viewing wi ndows, indicating the vapor was indeed saturated. Additionally, a thermocouple was placed in the liquid reservoir and this reading tracked very closely with the record ed vapor temperature for those runs with good NCG control. In fact, the gap between th ese two values would serve as an excellent indicator of NCG presence and also vapor saturation condition. The liquid reservoir temperature would rise above the measured vapor temperature when significant NCG was present, as part of the depression of apparent saturation temperature accompanying this presence. Further, if the vapor temp erature recorded was higher than the liquid reservoir temperature, then the vapor was superheated. This assumption was backed up by a lack of condensation on the viewing windows despite higher than ambient vapor temperature for these cases. Also, modifi cations were made to increase its liquid capacity to about 40mL. This capacity proved to be suffici ent enough that replenishing the water supply was only necessa ry in one experiment of a ll those performed. The only removal of vapor from the system was through the vacuum pump. An oblique view of the pressure vessel showed the driver surface through the side window of the vessel (Figure 4-1). Since th e original heater was in the place of the

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66 window frame on the top cover, all videos in the first and second stage of experiments were taken from this corner view. Frames taken from those videos can be seen in Chapter 6, along with a full explanation of the three phases of the experimenting. Figure 4-1. Corner view of experimental apparatus. After the discovery of a significant air l eak around the edge of the old, disused heater, it was removed and replaced with a window. The new front window provided a much improved view of the condensation pro cess for videos in the third and final phase of experiments and a better seal. The atmosphere of the pressure vessel was effectively divided into two parts, although they were not sealed separately (Figure 4-2). No method of sealing was available that would be effective and also provide a good thermal connection between the

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67 driver edge and the cooler. Also, sufficient gas flow was n eeded between the two sides to allow for an effective evacuation process. Two small notches were made in the upper edge of the diaphragm frame (the divider betw een the two sections of the pressure vessel) for this purpose. As higher heat rates we re encountered in experiments, condensate droplets on the face of the diaphragm frame were noted. This indicated that significant HT was occurring across the Diaphragm Fram e, separating the two sections of the atmosphere. Insulation was added (about 1 in. thick) to the cold side of the frame and also around the exposed brass surface of the Cooler to impede this process. The insulation used was Permatex brand silicone RTV sealant. Figure 4-2. Sectioned view of experimental apparatus. Further testing showed that additiona l insulation was necessary around the diaphragm clamp, where the copper cooler was the closest to hot side atmosphere. The caps of the machine screws that held down th e clamp and also the supporting posts of the diaphragm frame needed to be covered w ith insulation as well, to reduce obvious

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68 extraneous condensation (Figur e 4-3). The fully insulated diaphragm frame was covered in the orange sealant, but showed little visible condensation duri ng the test runs. Figure 4-3. Insulation applie d to and removed from diaphragm frame. The silicone sealant was applied as in sulation over the metal screw head, the metal support posts and the central area of the Dia phragm Frame face. Visible wires are thermocouples. Left picture is a fresh insulation applica tion, with polished driver surface. Right pi cture is a tarnished driver surface after removal of insulation. Given a few days, the mirror-like finish of the polished brass dr iver surface could become quite tarnished. In this state, th e quality of DWC produced was not good. The normal cleaning regimen was described in Ch apter 5. The polishing was done to aid in the promotion of the DWC mode. However, full FWC was never actually achieved in this apparatus. It was eventually noticed th at globules of oil were nearly always present on the liquid surface of the rese rvoir the day after an experiment. The most likely source for this oily residue, which also covered ev ery other surface inside the vessel, was the silicon sealant and presumably, this contam inant was a dropwise promoter. There was also a silicone based Dow-Corning vacuum grease used on the vacuum seals which may have contributed to the oily residue. It has been proven in several studies that promotion can be effectively achieved by addition of an organic promoter to the fluid reservoir [37,43], as discussed in the Promotion Methods section.

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69 In light of the relatively low saturation temperatures that were used in the experiment, as many components as possible were made of polymers rather than metal in the hopes of decreasing extrane ous heat transfer and machine shop time. The Top and Bottom Covers and the Diaphragm Frame were all made of Delrin (Figure 4-4). The Insulation was made of Teflon. The optic al probe Guide and Seal along with the Diaphragm Clamp were of poly carbonate. The choice of di fferent materials for these different components was based solely on availa bility of stock in appropriate geometry Figure 4-4. Exploded view of experimental apparatus. and a maximum service temperature of at l east 200F. Some items were omitted from the exploded view for clarity, such as the vinyl coolant hoses a nd the Y-Or-Lok bulkhead fittings for them, whose locations can be noted in the cross-sectioned (Figure 4-2). All connecting hardware, pressure fittings, and soft seals were also omitted. The multi-pin connector/vacuum feedthrough was left out as well. It fit into the th ird hole in the bottom cover, was held down by 6 screws, and had a soft seal. All soft seals were manually cut from Buna-N sheet stock. The components that were press-fit, namely the Cooler into

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70 the Diaphragm Frame and the Optical Probe Seal into the Bottom Cover, sustained liberal application of the silicone sealant for adhesive and sealing properties. Figure 4-5. Diaphragm clamp, lower face show ing grooves for placing thermocouples to measure edge temperature, To. There were 8 thermocouples placed internal to the pressure vessel. Four of them were placed evenly around the circumfere nce of the Diaphragm Clamp, in specially machined grooves (Figure 4-5). The gr ooves clamped the thermocouple beads to diaphragm edge and also clamped the edge to the Cooler. This desi gn attempted to place the thermocouple junctions as close as possi ble to the edge of the exposed diaphragm surface without piercing through the wall. In practice, however, this very thin barrier between thermocouple and liquid eventually br oke through in three of the four places on the clamp, allowing visible drainage of condensate through the lower holes. Quantitative estimate was not made of how much this phenomenon might have affected the HT measurements. Qualitatively, it was expected to increase overall HT by decreasing the amount of condensate around th e edge of the diaphragm, which would otherwise serve to insulate th e most active portion of the condenser surface. Three of the

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71 remaining thermocouples were placed in a bundle, with the tips sp read, about inch above the condenser surface and just over the edge of the clamp. These three thermocouple readings were averaged and this value taken as the atmospheric temperature, usually taken as the saturation temperature, as described in detail in the Results and Discussion section. The one re maining thermocouple was placed in the liquid reservoir, with the tip slightly curved up, to hopefully measure the liquid temperature rather than the glass surface temp erature. Note, however, that this liquid temperature measurement was only made for the last five days of testing. Two more thermocouples were employed on the outside of the apparatus, one each just outside the bulkhead connectors for the coolant lines. A hole was punched with a hot nail through the vinyl coolant line. Then the thermocouples were stripped of the clear, plastic outer casing, but not the Teflon insulation. This measure was undertaken because cooling water would be pushed by the flow pressu re back though this casing and into the DAQ board. Next the stripped, but insulated, thermo couples were inserted in the holes in the coolant lines and secured with a liberal a pplication of Copper-Bond brand epoxy. After allowing 20 to 30 minutes for this to cure, the arrangement was leak tested with the flow control valve on the flow me ter both fully open and fully closed. If no leakage was apparent after ten minutes with the valve cl osed, then the seal was deemed sufficient. The cooler was constructed of copper st ock. The central cylinder was made to accommodate the Optical Probe Guide with a pr ess fit and to support the Diaphragm with at least a 1mm contact around the circumferen ce. Two ‘washers’ were then made along with a tube section, all of whic h were soldered together to make the flow chamber of the cooler. Two holes were then drilled to acco mmodate the inlet and outlet tubes, also of

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72 vinyl. Like the coolant thermocouples, these were secured in place by liberal applications of Copper-Bond epoxy. Enough lengt h was left on the tubes to facilitate easy attachment of the bulkhead fittings. The Diaphragm Frame was supported by three lengths of 3/8 coarse all-thread, of galvanized steel. Certainly st ainless would have been better and better yet would have been something less thermally conductive, possi bly another polymer. Eventually, these posts had to be covered in silicone insula tion as well, since signi ficant condensation was observed on their surface in th e test chamber. Before th e application of silicone, however, the height of the Diaphragm Frame had to be adjusted for displacement sensor calibration and then set for th e testing. Two nuts were plac ed on the each post, below the frame to support it and lock each others position, then one on top. Once set, further adjustments were not made until the final calibration and displacement measurements of the used driver, after the conclusion of testing. The pressure tap for the apparatus was even tually located toward the top of the top cover, relative to the side or ientation pictured in the corner view (Figure 4-1). Special attention had to be paid to th e pressure tap and fittings orie ntation since, after repeated evacuation procedures of a hot test envir onment, enough condensate would collect in the vacuum hose to clog the flow and prevent e ffective function of the vacuum pump. The vertical orientation of the fittings allowed th is collected condensate to be emptied back into the test vessel by simply lifting the tube above the fitting inlet. As the importance of NCG effects on DW C became apparent during the course of the investigation, the pr essure integrity of the apparatus became a big issue. An earlier section on NCG effects in this paper looks qualitatively at the dynamics of the problem

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73 and at the reported effects in the literature. Upon the initial assemb ly of the apparatus, prior to any experiments, a bubble test was app lied to test for leaks. Bottled nitrogen was hooked up to apparatus and the internal pressure was raised to about 50 psi. The vessel was then submerged. Several leaks were located and fixed in this manner, however, due to the sensitivity of DWC to the presence of even small amounts of NCG, this method turned out to not be a sufficien t leak control. After data tr ends indicated a significant air leak during the intermediate phase of testing, v acuum grease was applied to all soft seals. Initially this made the leaks worse but then better after the excess grease was wiped away and the apparatus reassembled. However, this still did not fully address the problem. The likely effects of NCG resultant to poor s ealing were covered in more detail in the Results and Discussion section. Detail drawings for those part s listed in the above figure that were manufactured in-hous e can be found in Appendix C. Equipment Several other pieces of equipment, in addition to the main apparatus described above were needed for the conduction of th is investigation. A schematic layout was presented, followed by a description, by lett er, of each component (Figure 4-6). Element A was a Mastercool brand vac uum pump, 2 Stage, 1.5 CFM capacity, and oil lubricated. This pump was intended for air conditioning equipment service, and thus was not intended for much contact with wate r vapor. Significant amounts of water vapor put through the pump would condense in the oil, degrade the performance, and cause it to over heat. For this reason, th e evacuation operations described in the results section were limited to 5 to 10 minutes and the pump oil ha d to be replaced each day. Capacity was a few ounces of standard vacuum pump oil.

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74 Element B was an array of brass ball valves and fittings, setup to allow independent venting of the test vessel and isolation of the pressure gauge from the vessel and also from the pump line. This separation of the gauge was necessary due to a small, constant leak in the gauge diaphragm. Because they were thermally exposed to the environment inside the test vessel, albeit minimally, th e valve configuration was wrapped several times in reflective foam insulating tape to mi nimize heat loss. This potential loss would not have affected the accuracy of the meas ured THR, but rather the insulation was a cautionary measure to prevent any reduction in heat supply capacity to the vapor. Element C was a vacuum gauge, ACSI 1200-VACU, with a 0-30 in-Hg range and an uncertainty of 1% full-scale. This large uncertainty existed despite the two decimal place digital readout. This large uncertainty rendered it essentially useless for any information related to the saturation pressure of the water vapor. It was useful, however, to monitor the relative effec tiveness of the vacuum pum p, i.e. the condition of the lubrication oil, since the vacuum level would rise by up to 1.5 in-Hg when a change of oil was needed. Element D was a 250 W, 120 V – AC, Ushi o Halogen Projector Lamp. A copper tubing coil was fashioned to ac t as a spring to both support the lamp, which got very hot and to press it as close to th e reservoir window as possible. The lamp itself, however, was held in an aluminum frame. The bolts of this frame kept about inch spacing from the window surface. This lack of thermal c ontact was instituted avoid excessive heat transfer to the vessel. In practice the vessel sometimes beca me hot to the touch. This issue is addressed in the Uncertain ty and the Improvements sections.

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75 Figure 4-6. Schematic of e quipment setup with labels. Element E was the Water Reservoir, which was simply the window well on the bottom side of the four-windowed aluminum cy linder. Strips of aluminum were lined around the window well and the corresponding hol e in the insulation sleeve using the silicone insulation as adhesive in order to increase the capac ity of the rese rvoir to about 40 mL.

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76 Element F was the Controlled Atmosphere of the vessel. Idea lly this would be composed of pure water vapor, however, as the test results show, NCG was also a significant presence at times. Element G was the piezoelectric Diaphragm. The supply of circular piezoelectric resonators, APC# FT-20T-3.6A1, was gr aciously supplied by American Piezo Corporation at no charge. The diameter of the brass shim to which the piezoelectric ceramic patch was attached was 0.004 inch (0.1 mm) thick and 0.75 inch (19 mm) in diameter. The positive and negative pow er lines both came through the multi-pin connector. The negative lead was hooked to es sentially a washer on one of the bolts that held down the Diaphragm Clamp. The solder ing of the positive lead onto silvered surface of the piezoelectric ceramic patch on the back of the dia phragm was a delicate operation. If the ceramic was heated by the soldering iron, or a ny source, above its critical temperature, it would lose its piezo electric characteristics. Alternatively, the silver, electrically conductiv e backing might easily come off if over-heated. Only a minimal amount of solder was used to avoid extra mass loading of the vibrations. The soldering point was placed as far to the edge as possible fo r the same reason. Also, the soldered joint was covered with a thin late r of JB Weld epoxy, to physically secure the connection against vibration. It was necessary to be certain that the epoxy extended well onto the Teflon insulation of the power supply wire, or the very fine, 36 gauge wire used would often break at this exposed point. Also, for displacement measurements, a patch of metallic, reflective tape was placed on the back of the diaphragm so that it had sufficient reflectivity. About 1 mm of shoulder surface was required around the circumference to properly secure the diaphragm.

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77 Element H was the Cooler Assembly, desc ribed in detail in the above section, fabricated in-house. Element I was the analog, float-type Flow Meter, a King acrylic flow meter, 0.8 to 7 0.1 gph. It was panel mounted on a piece of sheet metal on the table top, and vertical checked by a common level. Element J was the Coolant Reservoir. This Rubbermaid Agri cultural tank, of 60 gallon capacity was placed underneath the table supporting the expe rimental apparatus and equipment. Ice was placed here to both cool and, as it melted, supply feedwater to the coolant pump. Details about ice preparation we re listed in Chapter 5. Element K was the Coolant Pump, a Little Giant Pond Pump, 4300 gph, actually far oversized for this application. Element L was the Multi-pin Connector from Ceram-tec Products, a 19 pin, #16060-13-A, air-side plug/#16133-02-W single-ende d receptacle. A flange held in place with epoxy and six stainless steel pan No. 8 pan head screws were used for attachment to the Bottom Cover. Element M was the Terminal Block, from Measurement Computing, with 16 positive and negative posts for thermocouple attachment. Cold junction compensation done on-board with all terminals mounted on a single iso-thermal block. The block was designed to mate with Measurement Computing DAQ card. Element N was the Data Acquisition Car d, from Measurement Computing, part # PCI-DAS-TC, installed in a PCI slot of an IBM personal desktop. The card would support Labview, MSExcel or MATLAB contro l, but Excel was chosen for convenience.

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78 Element O was the desktop PC – IBM, Windows XP, in which the DAQ board was installed. Data was captured in Excel using DAS Wizard, supplied by Measurement Computing. All data was taken as a temperat ure reading, in F, converted from voltage by the software. Each measurement represen ted 16 averaged values from in between sampling cycles. Sampling ranged from one per second to one per hour in increasing increments. Channels of the DAQ could be sampled individually or as a continuous set between two chosen limit terminals. Element P was an Amplifier, made by Krohn Hite, model DCA-10, a wide band DC to 10 MHz amplifier with 1 and 10-times amplification, with c ontinuously variable gain. This was the model recommended by the engineering staff of the driver manufacturer, APC. Element Q was a Function Generator by B+K Precision, model 4017A, to 10 MHz with a sweep function. Only sinus oidal wave function was used. Element R was an Oscilloscope by Te ktronix, model 2247A, 100 MHz, with many functions, the most important of which were auto setup, to automati cally find appropriate display settings for an applied input si gnal and a peak-to-peak voltage readout. Element S was a Precision Rheostat from STACO, model # 3PN221B, with output range 0 – 132 V and up to 2.5 A, fused. This element was used to va ry the input of the halogen lamp, especially necessary since, on the highest setting, smoke began to issue from the lamp area. Thus, the power was never set above 70%. Elements 1 – 4 were the thermocouples us ed to measure the edge temperature of the diaphragm, whose placement was described in detail in the apparatus section above. Elements 5 – 7 were the thermocouples us ed to measure atmospheric temperature.

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79 Element 8 was the thermocouple used to measure liquid reservoir temperature. Elements 9 & 10 were the thermocouples used for coolant outlet and inlet temperatures. The following elements were not pictured for simplicity: Element T was an Optical Displacement Sensor & Probe, from MTI Instruments, part number MTI-2000 Fotonic Sensor with MTI-2062H probe. This instrument used a broad spectrum halogen source lamp with prec ision input control along with a fiber optic cable to transmit the light to the sensor head. Half the fibers supp lied the source light and half transmitted the light reflected from th e source. There were two working ranges for each probe head design, on either side of an op tical peak in captured, reflected light. The probe selected, in Range 1 used, had a stando ff distance from the target of 0.76 mm with a range of 0.38 mm, with sensi tivity of 0.12 m/mV. Signal to Noise Ratio at resonance ranged in this study from a low of 13 for an old, used driver, to as high as 61 for a new driver. The resulting measured vibration amplitude, peak-to-peak measured had a minimum of about 7.5 m, a maximum of about 38 m and a noise equivalent to 0.6 m. The main drawback to this instrument was th at it required that refl ective tape be placed on the back of the driver, for sufficient return ed light. The problem then was that tiny air bubbles trapped under the tape would bubble up under the tape when the vessel was evacuated, thereby ruining the signal qualit y. No measurements of displacement were therefore possible during expe rimental runs. This data had to be inferred from a calibration curve constructed under atmospheric pr essure. It could be expected that the vibration amplitude might slightly increase wi th less damping effect from air around it.

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80 However, it was likely that the mass loadi ng from the condensate on the driver surface would have a much more significant de pressing effect on oscillation amplitude. Element U was Camera 1, from JAI, m odel CV-730 color 770 x 500 CCD with a dedicated PC and controller card from Flashbus. This was the camera used for the initial and intermediate phases of experimentation. Element V was Camera 2, the Motionscope from Redlake Imaging, with matching controller card and dedicate d IBM – PC. This camera wa s capable of capturing up to 8000 frames per second. However, this fram e rate required a massive light input to return a usable image and not enough light was available. Therefore a much slower frame rate was used. Some videos were taken at 50, 500, and 1000 frames per second, with good results. The frames from the final phase of testing were taken from videos captured with this camera. Element W was the Heater Cartridge and Controller, from Watlow. A Firerod cartridge, 400 W and a Series 96 controller were taken from [89]. This heater was used in the initial stage of testing but was deemed ineffective and later removed. Additional Testing It was determined that the results of the data analysis warranted further clarification. This clarifi cation would require slight modi fications to the experimental apparatus (Figure 4-7). In this new setup, th e fluid reservoir was separated from the test vessel, so as to isolate extraneous ther mal energy conduction from the halogen lamp heating element to walls of the aluminum te st vessel. Additionally, a phase separator, placed in an ice bath was added to the v acuum tubing arrangement between the vessel and the vacuum pump. This chilled separa tor would serve to minimize the amount of water vapor condensing in the oil of th e vacuum pump by reducing the saturation

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81 Figure 4-7. Schematic of vacuum appara tus adjustments for Additional Testing. temperature/pressure of the vapor bei ng pulled through the tubing. This change improved the vacuum pump performance as we ll as the length of time that it could be operated without overheating, leading to bett er reduction of NCG in the test vessel environment. This arrangement was mana ged by inclusion of a series of manually operated ball valves, denoted in the figure by a standard valve symbol of a circle and an inscribed ‘x’. As an additional attempt to confirm assump tions of the theoretical analysis, three thermocouples were placed along a radial line of the brass shim (Figure 4-8). The same RTV silicone used elsewhere in the cell was used as an adhesive, with the thermocouple beads held in contact with th e brass surface during applicati on and curing. The positions

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82 of the bead centers and corre sponding uncertainties were then estimated from geometric analysis of digital photographs. The first th ermocouple was centered within 4% of the radius of the center (9%) while the sec ond bead was centered at 3313% from the center, and the third at 669% from the center. For the additional testing, all thermocouples were replaced with type T, for its smaller absolute uncertainty. All these ne w thermocouples were Omega brand, made of 36 gauge wire. The part number of these thermocouples was 5TC-TT-36-36. They were all calibrated simultaneously to two laborator y-grade precision mercury thermometers. Figure 4-8. Placement of surface thermocouples on the piezoelectric diaphragm. Single thermocouple in foreground measured atmospheric temperature. This step of direct surface temperature measurement was originally avoided due to concerns that the presence of the prob es would interfere with the condensation

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83 phenomenon taking place on the surface. However, the results of the Additional Testing proved to equal the best results of the previous testing in overall HT rate, indicating that any interference was negligible. The atmospheric temperature estimate wa s now taken from a single thermocouple measurement instead of several, since the en tire batch had been replaced with type T thermocouples. This new batch of thermocoupl es was calibrated toge ther at a few dozen points across the full range of applicable te mperatures. The calibration comparisons were two laboratory grade mercury thermometers with smallest divisions on their scales of 0.2 and 0.1C. The calibration and expected ma ximum uncertainty in the readings were discussed in further detail in the A dditional Testing sect ion of Chapter 7.

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84 CHAPTER 5 PROCEDURE 1. Prepare Coolant Reservoir. a. Drain coolant reservoir level to just above the inlet of the pump, or to a depth of about 2 inch. b. Add 100 to 150 lbs of ice was added. i. If the pump was running, i.e. e xperiments being conducted, this amount of ice would last almost 6 hours. If the experiment was for some reason postponed, a usable port ion of ice would still be in place some 12 hours later. 2. Clean diaphragm. a. Clean diaphragm surface with a cotton swab lightly soaked in Brasso abrasive cleaner. b. Wipe away residual material with two successive, dry cotton swabs. i. Light pressure may be applied to scrub away any oxidation present on the diaphragm. ii. Take care not to pull the diaphragm edges out of the clamp. c. Clean diaphragm surface with a cotton swab lightly soaked in rubbing alcohol. d. Wipe away any residual alcohol or debris remaining with two successive, dry cotton swabs. 3. Prepare working fluid reservoir. a. Remove water in working fluid re servoir inside pressure vessel. b. Clean reservoir with alcohol and a clean piece of cloth. c. Refill reservoir with distilled water. 4. Insert Diaphragm, Cooler, Frame, and Bottom cover assembly into the insulation cylinder.

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85 5. Test thermocouple and driver function, sin ce connections were easily interruptible. 6. Secure bolts attaching Bottom Plate to the Aluminum Cylinder. 7. Re-test thermocouple and driver function. 8. General pressure gauge function: a. Turn on, Allow reading to stabilize, ~ 1-2 min b. Hit zero key, allow reading to stabilize, ~1-2 min i. Gauge will stay on for 3 to 5 min. About 10-15 seconds prior to shut-off, the reading will spike a few percent. The shut-off is automatic. ii. If further readings are desired after automatic shut-off, turn on again. Allow same stabilization pe riod. Zeroing is not necessary again. 9. Evolution of evacuation process duri ng experimental method development: a. Cold evacuation, wait for equilibrium. b. Cold evacuation, heat on. c. Cold evacuation, heat on, evacuate again. d. Hot evacuation, repeat if necessary. e. Heat with open valve to near 170 or 180F, then evacuate with heat on – generally acceptable results were obtained with this method. 10. Evacuate the pressure vessel. 11. Preheat vessel with open valve. 12. Turn on pressure gauge. 13. Adjust valves to desired arrangement. 14. Switch on vacuum pump for roughly five minutes. 15. Close off vessel before shutting off pump. a. Take care not to leave vessel open to pressure gauge section, as the gauge itself leaks.

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86 16. Vent vacuum line immediately after turning off pump or pump o il will be sucked into line. 17. Begin taking temperature data anytime be fore or during the evacuation process. 18. Close the coolant flow control valve. 19. Turn on power to coolant pump. 20. Slowly open the coolant flow cont rol valve until it is fully open. a. Allow any air trapped in the coolant line to escape thr ough the flow-meter. b. This process may be assisted my ma nipulating the coolant lines to allow the air pockets to rise in the fluid. 21. Adjust the flow rate down to the desired level. a. Watch the coolant line connections for leaks. 22. Check flow rate every few minutes and adjust as necessary to ma intain desired flow rate. 23. Continually check ice bath during experiment for: a. Sufficient ice around pump inle t, stir reservoir if not. b. Sufficient ice remains for continued testing, add as needed. c. Inlet temperature was generally related to both of the above. 24. For test sections with vibrations: a. Unhook driver cable from amplifier. b. Turn on amplifier. c. Switch selector to 10 x output. d. Allow amplifier to stabilize for ~ 5 min. e. Hook up driver cable to amplifier output. f. Turn on function generator. g. Select sinusoidal wave form. h. Select 1-10 kHz frequency range.

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87 i. Turn the output knob completely counter-clockwise. j. Turn on the Oscilloscope – set voltmeter function as peak-to-peak. k. Adjust the amplitude on the function generator until the oscilloscope reads near 36.00.4 Vp-p. l. Tune in to resonant frequency of dr iver aurally. It will be readily apparent, even from inside the vessel i. If you cannot easily hear the reso nating driver, then there is a problem. ii. Allow to run for a few minutes, to see if the problem corrects itself with a change in driver temperature. iii. If not, then abort test and di sassemble to determine trouble. iv. Possible causes of weak or non-existent driver function: 1. Loose connection at the multi-pin connector, vacuum side 2. Soldering on positive power lead has come off the back of the driver. 3. Thin wire portion of the driver supply line has broken m. If you can easily pick out a range of resonant frequencies, continue. n. Visually tune the frequency, to that range which minimizes the diameter of the residual drop or drops at the cen ter of the vibrating diaphragm. i. This range could be as much as 30 Hz. ii. Closely monitor the resonance intensity of the diaphragm until edge temperature reading stabilizes. 1. This precaution is necessary due to the temperature and also mass loading sensitivity of the resonant frequency of the piezoelectric diaphragm. 25. Several times per test session, when not at a change point in the data stream, interrupt data acquisition and save file, then re-commence DAQ operation. 26. When shutting down, no certain order is necessary, except to quit the DAQ program first.

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88 CHAPTER 6 RESULTS AND DISCUSSION Initial Testing The experimental portion of this investiga tion developed in three stages. The first stage was the exploratory test ing. In this phase, the coolant reservoir was at roomtemperature, and the original heater (taken from a spray c ooling study) [89] was the only heat source to the vessel. Evacuation was imme diately followed by testing, or perhaps an hour delay was allowed for more evaporation of the fluid. Only about 5 mL of liquid was placed in the pressure vessel at this point. The orientation of the driver was horizontal. This orientation was only po ssible at steady state with VIDA, since something was required to clear away the condensate. A comparison of video frames before and af ter the application of VIDA to a driver surface full of condensate demons trated the ability of this process to effectively clear condensate (Figure 6-1). Cl early, the dropwise mode of condensation was underway on the diaphragm in the frame to the left, howev er, this amount of condensate had taken near ten minutes to build up. A few frames late r, after the quasi-ste ady arrangement of droplets has set in for the VIDA section, it wa s apparent that the major drop sizes were much smaller. Unfortunately, with the cool ant at room temperature, ~ 24C (75F) and the vapor temperature near 35C (90F) and the presence of NCG, which would not be identified until much later, there wa s not much HT going on. In fact, the T across the cooler was drowned in the rather large uncertainty in this small heat rate, since heat rate was determined by the difference of two ther mocouples at the coolant inlet and outlet.

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89 Figure 6-1. Video frames of DW C without (left) and with (r ight) VIDA. The fact that the coolant temp for the in itial testing was about r oom temperature along with simple, cold evacuation being the only mode of removing NCG employed, the HT at this stage was negligible. Diaphragm orientation was horizontal. Results of this first phase of testing or iginally focused on small, roughly ten minute windows, chosen for their low levels of oscilla tion in temperature reading, in pursuit of steady state readings. These data trends were in fact very flat, though this turned out to be due to a very low overall HT rate (Figure 62). At this point, th e variation range in 40 50 60 70 80 90 100 0:00:000:00:500:01:400:02:300:03:20 Figure 6-2. Temperature data from 1/26, in F. Atmospheric temperature, Tatm, average edge temperature of the diaphragm, To, and coolant inlet temperature, Tin, respectively the highest, middle, and lowest readings. Readings were taken in F over a 10 minute period at 5 second intervals.

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90 0 5 10 15 20 0:00:000:00:500:01:400:02:300:03:20 Figure 6-3. Total heat rate data from 1/26. Flow rate was 7.0 gph. No vibrations were applied to the diaphragm. THR was in W. The time interval between measurements was 5 seconds. THR readings was almost 100% of the average reading (Figure 6-3). This situation arose from two things. First, the coolant flow ra te was the highest available, 7.0 gph. Second, the THR was directly proportional to the sm all temperature difference between coolant inlet and outlet temperatures. The higher the coolant flow rate was, the higher the discretization error was relative to the overa ll reading because the measured temperature difference was smaller. It was hoped that this high flow rate, would help to get as close as possible to the ideal condition of a cons tant edge temperature for the diaphragm. As yet, no difference between the vibrat ion case and the non-vi bration case was apparent. Steady state for HT transfer is the ideal condition, since this implies no temporal variation in total energy of the syst em. Thus the energy balance (Eq. 6-1) of the apparatus has one of the two unknowns redu ced to zero. Then, since there was no significant source of energy generation with in the control volum e, the second unknown could be neglected. This left the heat input equal to the heat ou tput. Therefore, the

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91 measured heat output, THR, may be assumed to be made up exclusively of the energy input from the heater. t E E E ECV gen out in 6-1 Here ‘E’ stands for energy. The dot above repr esents time rate of change. ‘t’ refers to time. The subscript ‘in’ refers to inflow to the control volume, ‘out’ to outflow, and ‘gen’ to generation within the control volume – zero in this case. The ‘CV’ subscript referred to the control volume of the pre ssure vessel. Though this relation was not explicitly used in this investigation, it served to clarify that unde r steady state conditions, the only non-zero terms are Ein and Eout, the latter being measured and the former ignored. The important as pect here is that the ECV term does not contribute to the measured Eout and thus, it is assumed that all of th e measured HT is due exclusively to the condensation occurring on the dr iver surface. The accuracy of this assumption will be addressed in the discussion of the results. The temperature signal appeared relatively flat for the time period of display, which would be desirable as an indicator of stead state conditions, however, the THR was quite low, and on the order of the uncertainty in its reading. So it was determined that the likely limiting factor was the heater. An alternative arrangement was sought. The design for the original heater, discu ssed in the apparatu s section, had been taken from a study of spray cooling, and was thus ill-suited for this application. An ideal heater for a condensation study would suppl y vapor in a saturated state with no superheating, so that all HT would be invol ved in the heat of vaporization, although, for small superheats, the effect would be neglig ible. This would be best accomplished by submerging a heater in the liqui d reservoir, as has been the case in every experimental

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92 DWC HT study cited in the lite rature survey. However, this would have required a completely new heater design, further pre ssure vessel modifications, and new vacuum components, which would have been quite tim e consuming. So a method of heating was sought that would direct most or all of its thermal ener gy directly through the liquid reservoir while not requiring significant m odification of the existing apparatus. A halogen lamp was placed below the window of th e liquid reservoir. This would led into the second generation of the testing. Intermediate Testing All tests in this and the remaining phases were conducted with a vertical orientation for the condenser surface. After the heater was discovered as a major limiting factor in 35 45 55 65 75 85 95 105 0:000:591:592:593:594:595:596:598:009:0010:0011:0012:0013:00 Figure 6-4. Temperature data from 2/17 in F The time scale is in hours:minutes. The data traces, from highe st to lowest are Twall – placed between the insulation and the vessel wall, at the top window well, Tatm – the atmospheric temperature, To – the average edge temperature of the diaphragm, and Tin – the inlet temperature of the coolant flow. Discontinuities in the Tatm measurements were artificially pl aced markers between input changes.

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93 the HT process, it was replaced with an external, 50 W halogen lamp. At this point, there was no control on the lamp’s input power. The early results during the intermediate phase of testing were characterized by wild fluctuations in both th e atmospheric and the diaphragm edge temperatures (Figure 6-4). This variation, it was later discovered from the literature, is t ypical of condensation measuremen ts in the presence of NCG [27,31,74]. A possible explanation then for these os cillations was a pocket of NCG collecting at the condenser surface, building up, and peri odically departing. Condensation induces a net mass flow towards the condenser surface. When there are NCG molecules present in the vapor, they are swept toward the condenser surface with this flow. However, since the NCG molecules do not condense, they re main at the condenser surface and impede inflow of new molecules of the condensable vapor. Eventually, the concentration of NCG molecules becomes very high at the condenser surface, relative to th at in the rest of the test vapor. This has the effect of in sulating the condenser surface, allowing it to get colder than it otherwise woul d if fresh condensable vapor were supplied in excess. For a vertical condenser surface, this insulating effect eventually i nduces the mass of NCG collected at the condenser surface to be pulle d away by buoyant forces. This leads to a sudden burst of warm, saturated vapor to the artificially chilled condenser surface, shown by an upward spike in temperature. This is exactly the behavior shown in the temperature charts (Figure 6-4). Correspondi ng spikes in THR were also shown by the data (Figure 6-5). A detailed account of th e justification for this conclusion follows: Periodic temperature and THR spikes corre sponded with each other and therefore were not due to instrument error. THR was a differential measurement and the other temperatures were individual measur ements. Therefore any group upswing in

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94 reading would not have regi stered on the THR, as it would have equally affected both thermocouples. Spikes did not correspond to changes in any of the control parameters of the experiment, those being heat input, coolant flow-rate, or presence of vibration. Over the 14 hours of this test, 7 different flow rates were sequentially tested for half of that flow-rate period with the vibrating driver a nd half without. The lengths of successive flow-rate sections were more or less equal, at about one half hour each. Changes in flow rate roughly corres ponded to the corners in the coolant inlet temperature graph. No effect of vibration was apparent in the DWC HT from these results. 0 5 10 15 20 25 30 35 40 45 0:000:591:592:593:594:595:596:598:009:0010:0011:0012:0013:00 Figure 6-5. Total heat rate data from 2/17 in W. Time scale is in hours:minutes. During this run, 7 different flow rates were tested, beginning at 1.0 gph and increasing in 6 steps to 7.0 gph. At each level of flow rate, conditions were monitored with and without the driver vibrating. Each flow rate stage was held for about 2 hours and was evenly divided between vi bration conditions. The spikes in THR did not correspond to changes in th e experimental conditions and were thus attributed to the effects of NCG. The downward trend in To and Tin was due to the flow rate changes (Figure 6-4). For the lower flow rates, the coolant move d through the feed tubes more slowly and therefore had more time to absorb heat from its environment prior to arriving at the test

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95 vessel, despite attempts at insulation. Th is external heating led to lower inlet temperatures for higher flow rates, movi ng from about 38 to 40F for 7.0 gph to near 50F for 1.0 gph. Since this was the heat sink temp erature, the rest of the temperatures of cell were correspondingl y shifted up for a given increase in Tin. Another temporal trend was apparent in the wall temperature reading. The first 4 hours showed some thermal capacitance as the apparatus heated up. As far as the energy balance and THR readings go, this could be ignored, since the en ergy balance was not being monitored and the addition of thermal energy here was positive. If the addition were negative – that is the apparatus were experiencing a net cooli ng – then this would imply that the heat being absorbed by the coolant included thermal energy from the vessel itself, instead of only the condensa tion phenomenon. However, as long as the apparatus temperature was incr easing or steady, the trend co uld be safely ignored. The visual record of the 2/17 test, in th e intermediate phase of testing showed partial DWC (Figure 6-6). This test occurred before the cleaning regimen of Chapter 5 Figure 6-6. Two frames taken from vide os on 2/17, showing condensation without vibration (left) and with vibration (right). The cleaning regimen had been delayed for a few days to try to induce FWC, which only resulted under conditions of vibration. Though not visible in the ri ght hand side picture, droplets were continuously be ing ejected from the film. Down was to the left.

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96 was established. This partial DWC was the re sult of the driver surface being allowed to tarnish over several days. The frame on th e left shows condensation progressing on a quiescent diaphragm. The frame on the right sh ows the result of the same parameter set except for the application of VIDA. Curi ously, the quiescent mode of condensation remained mostly dropwise although the drop ed ges were more irregular. However, under imposed vibrations at the resonance frequenc y of the diaphragm, a film formed on the center of the driver. Unfortunately, the 30 fp s frame rate of the camera used here was insufficient to capture the continuous ejection of droplets from this liquid film. A wave pattern is slightly noticeabl e on the lower portion of the ri ght picture, as evidence of existent capillary waves. These waves were constantly ejecting droplets, as per the description of Vukasinovi c [10] of VIDA. The data from this set were determined not worthy of comparison as filmwise mode to other data, since the NCG presence was still far from under control. At this early portion of the intermediate phase of testing, the data was still only being examined in small – 10 minute – chunks, so the overall variation (Figure 6-4) was not yet apparent. The portions of data without spikes were simply selected for analysis. THR values were still considered low at this point, and though this was later proved true, it was not yet based on a theoretical model. The theoretical model was quite unfortunately constructed afte r the conclusion of experimental work, since a method of modeling had not yet to been found. In the latter portion of the intermediate te sting phase, a concerted effort at reaching steady state conditions was fi nally undertaken. For the re mainder of the tests, the thermocouple between the wall and the insulatio n was moved to the liquid reservoir, in

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97 order to make comparisons between this temp erature and the atmospheric temperature. The relation between these two temperatures was then used to infer the saturation state of the vapor. The remainder of the tests were conducted for durations of 10 to 18 hours with only a few experimental conditions examine d, in order to more accurately pursue and evaluate the steady state conditi on of the variables. Also for this and further tests, the diaphragm was cleaned with the regimen de scribed in the PROCEDURE section just prior to commencing each day’s experiment. The temperature data from 2/24 (Figure 6-7) showed a gradual increase in the liquid temperature (Tliq) and Tatm. These two values also maintained a near constant difference between them. This apparent increas e in the thermal energy of the hot side of the vessel will be ignored in relation to steady state of the apparatus for reasons discussed above. However the concept of warm vessel wall has potential for parasi tic heat transfer 35 45 55 65 75 85 95 105 115 0:591:592:593:594:595:596:598:009:0010:0011:0012:00 Figure 6-7. Temperature data from 2/24, in F Time scale is in hours:minutes. Values are, in decreasing order, Tliq, reservoir temperature, Tatm, atmospheric temperature, To, average edge temperature, and Tin, coolant inlet temperature.

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98 to the cooler, through the insulation sleeve. This potential error wa s addressed later in this chapter. After an initial test period at 1.0 gph, c oolant flowrate was increased to 7.0 gph and held there for the remainder of the test. A constant inlet condition was difficult to maintain. This fact was firstly due to the nonuniform melting of the ice in the reservoir. The pump was the primary heat load in the reservoir and so melte d the ice immediately around it much faster than the rest. Variations in pump performance with supply voltage were also responsible for inlet temperature fl uctuations. Also, small air bubbles or pieces of debris in the coolant flow at times becam e lodged in the flowmeter valve or otherwise restricted flow, thereby causing the equilibrium inlet temperature to change. These are the sources for the corners apparent in the Tin data later than 3 hour s into the test. Possible direct evidence of vibration impr oving HT was found at 7:15 (Figure 6-7). A slight drop in To was noted here, which corresponded directly to the vibration being turned off and a slight drop in measured TH R. While other noticeable drops in To are present in the data record, they each correspond ed to an adjustment in the coolant flowrate and therefore a drop in si nk temperature. This pairing was readily apparent in the temperature graph. The net effect, however, was slight, and will be discussed further in the next section. Another trend of note was apparent in the gradually decreasing To relative to Tin and the corresponding gradual drop in measured THR. This would later be explained by an air leak introducing more NCG to apparatus with time. As compared to the 2/17 test, the NCG concentration is assumed less in this test, due to the lack of chaotic temperature of THR swings (Figure 6-8). The remaining va riation was due primarily to uncertainty in

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99 the temperature measurements and the data ac quisition system, since this value was based on the differential reading between two thermo couples. This uncertainty was higher for a higher flow-rate, this data being taken at the maximum 7.0 gph, since the temperature difference encountered by a flui d flowing faster through the cooler was smaller than for slower transit. The tradeoff of lower uncerta inty, however, was not deemed justifiable in light of the incurred higher sink temperature inherent in using a lower flow rate. 0 5 10 15 20 25 30 35 40 0:001:002:002:593:595:005:596:598:009:0010:0011:0112:00 Figure 6-8. Total heat rate data from 2/24, in W. Time scale in hours:minutes. Note the gradual decrease in THR for increasing time and the disjoint at 7:15, deemed possible evidence of vibration increasing HT. Later tests in the intermediate phase also saw the additional insulation discussed in the apparatus section, vi sible as the orange colored area in the foreground (Figure 6-9). These photos were taken from the same obli que angle as the first schematic of the apparatus (Figure 4-1). Focus and field of view were difficult to work with at this angle and for this geometry, so the picture quality was not optimal, but the main patterns of DWC are nonetheless apparent.

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100 Figure 6-9. Typical droplet distribution for DWC on the test diaphragm without VIDA, prior to departure (left) a nd after departure of a large drop (right). Down was to the left. Graphical estimations showed a maximu m droplet radius for this photo of 1.050.18 mm. This result was well in line w ith the Rose estimation of maximum drop size (Eq. 2-11) for this temperature range. The final section in this chapter compared the measured THR to that predicted by the Rose th eory for this diaphragm. The theoretical analysis included an effectiv e maximum droplet radius. The maximum drop size under conditions of vibration (Figure 6-10) was much smaller than without applied vibr ation (Figure 6-9). It was exp ected that this lowering of Figure 6-10. Video frames of typical dr oplet distribution for DWC with VIDA. A droplet during the process of bursting (left) and after (right). Down is to the left.

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101 the maximum drop size would increase the overall heat transfer, though only slight evidence of this was noted in th e experimental run of 2/24. Two factors in this arrangement likely se rved to mitigate the potential HT gains from the apparent decrease in maximum drop size. First, maximum droplet size around the edges of the diaphragm was still on the same order as the non-vibration case. And since the edge was where most of the HT t ook place (Figure 3-2), th e same level of HT was expected. Second, NCG presence in the test vessel had yet to be sufficiently controlled. The decline of THR with time indicated a leak in the pressure vessel, since this behavior stopped when the seals were better controlled. Al so, THR was relatively low compared to theoretical estimates, presented later in this chapter. Final Testing Efforts to purge NCG from the vessel becam e more effective in the third and final stage of experiments. Multi-stage evacuations and heating times were instituted, eventually leading to the process describe d in PROCEDURE for the final two days of testing, which yielded the highest THR values of the entire study. The resulting data now warranted a close look at th e relative behavior of Tliq and Tatm. The test data from 3/4 clearly demonstrat ed the effect of NCG on DWC in the HT cell (Figure 6-11). From time 0 to about 50 minutes, the vessel was heating up, after evacuation, with no coolant flow. Following an evacuation ending just after the 1 hour mark, where all the temperatures run together another heating period ensues. At about 2:40, the coolant flow was started and Tin drops almost immediately to relatively steady value. From the gradually increasing difference between Tatm and Tliq, with the former dropping and the latter rising, an ai r leak in the vessel is appare nt. The effect of air here to raise the was liquid temp erature while depressing Tatm, To, and consequently THR.

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102 30 50 70 90 110 130 0:001:002:003:004:005:006:00 Figure 6-11. Temperature data from 3/4, in F. Time in hours:mi nutes. Values from high to low are Tliq – liquid reservoir temperature, Tatm – atmostpheric temperature in vessel, To – averaged diaphragm edge temperature, and Tin, coolant inlet temperature. This effect is unquestionably demonstrat ed when, at about 5:40, the vessel is accidently vented to 1 atm. With no change to the control parameters, coolant flow-rate, Tin, or the lamp power, Tliq rose 25F in about 15 min while the boiling that had been fairly vigorous prior to venting immediately ceased. Simultaneously Tatm and To drop a few degrees and THR drops about 30% (Figur e 6-12). Additional support for this NCG effect conclusion was the fact that during an evacuation process, even though the high temperatures dropped significantly, the THR al ways peaked significantly at the same time. This precipitous drop in HT clearly indicat ed that the suppressive effect of NCG on DWC in this HT cell greatly overshadowed the anticipated im provements of VIDA. This accident clearly demonstrated the importa nce of the NCG effect, which was then researched in earnest. New effort was applied to making good seals at every possible

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103 leak point in the vessel. Dow Corning high vacuum grease was liberally applied to all soft and hard seals. This turned out to be even worse, but after the excess grease was removed, the result was much better than the original dry arrangement. Data from the next experiment, 3/7, showed a much closer and temporally stable relationship between Tatm and Tliq. 0 5 10 15 20 25 0:001:002:003:004:005:006:00 Figure 6-12. Total heat rate data from 3/4, in W. Note the drop at 5:40, which was due to accidental venting of the vessel to 1 atm. The low readings before 3:00 were fluctuations associated with establ ishing steady state after turning on the coolant flow. The seal situation on 3/7 was not pe rfect. Fortunately, a problem with thermocouple deterioration requi red re-wiring of the multi-pi n connector and replacement of the soft seal. This activity apparently fi xed the leak, to the ex tent possible with the current apparatus. All subsequent experime nts, from 3/8 through 3/15-16 showed a very close relation between Tliq and Tatm, indi cating a very low presence of NCG. Next the 50 W lamp was replaced with a 250 W lamp. A precision rheostat was added to control the input voltage to the lam p. Superheating of the vapor was noted after

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104 this addition. This new phenomenon was a ttributed to the significant heating of the vessel walls from the new, more powerful lamp This also raised the possibility of parasitic HT from the walls of the vessel directly to the cooler through the insulation sleeve, see Chapter 7. The last three days of experiments in th e Final Testing phase were the best with respect to elimination of NCG. However some small effects of NCG were still noted. No dramatic improvement in HT due to the ap plication of vibration was readily apparent from the graphic displays of this data. Th e major events marking the data of 3/13 were the initial heating period, followed by coolant flow commencement at 1:20, and evacuations at 2:00, 6:30, and 9:00 (Figure 613). The power supply to the lamp was 40 50 60 70 80 90 100 110 120 0:000:591:593:004:004:596:007:008:009:0010:00 Figure 6-13. Temperature data from 3/13, in F. Time in hours:min. The two highest readings are Tatm and Tliq, which tracked quite closely once the NCG presence in the apparatus had been better controlled. Between 6:30 and 9:00, Tatm is actually higher than Tliq, then it drops below it afte r the heat input was reduced 10%. Tliq, the liquid reservoir temperature, Tatm, the atmospheric temperature in the vessel, To, the averaged edge temp erature, lying in at the middle value, and Tin, the coolant inlet temperature.

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105 kept at 70% until 9:00, when it was switched to 60%, in an effort to reduce the superheat seen between 6:30 and 9:00. The desired eff ect was achieved. The presence of superheat in the second stage but not the first may be e xplained as an effect of vibration, since the driver was turned off at 6:30. That is, ev aporation of spray from the VIDA most likely counteracted that tendency to of the vapor to develop superheat. Curiously, the THR was actually higher in the region with superheat (i.e., without vibration) (Figure 6-14), eventhough the Tatm was higher in the cas e with vibration. 0 10 20 30 40 50 60 0:000:591:593:004:004:596:007:008:009:0010:00 Figure 6-14. Total heat rate data from 3/13, in W. Time in hours:mintues. Low initial readings due to due start-up transients and fluctuations in coolant flow rate. This discrepancy of higher HT with lower Tatm and no vibration was not immediately explained. However, one possi bility was that evacuation temporarily improved HT, an effect that was seen more dramatically under the effects of NCG. It should also be noted that this increase in HT with a decrease in pressure would stand to go against the Rose theory’s predictions of decreasing HTC with pressure. A possible reason for the increase in HT with evacuati on may be the inducement of mixing in the

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106 vessel atmosphere assisting the diffusive pr ocess of vapor moving towards the condenser surface as well as removing a portion of the NCG present. The experiments of the next day (3 /14) showed excellent steady state characteristics (Figure 6-15). This was probabl y due to the lower power level used, 60%. There was still a general heating trend of the atmosphere, but the increase was small compared to others. The only effect of VI DA visible in these results was to slightly increase the difference between the average Tliq and Tatm. The difference however was barely noticeable from 1:00 to 4:00, 7:00 to 8:00, and 10:15 to the close of the experiment. 35 45 55 65 75 85 95 105 115 1250 : 0 0 0:59 1:5 9 3 : 0 0 4 : 0 0 5:00 6:0 0 7 : 0 0 8 : 0 1 9 :01 1 0:01 11 : 0 1 12 : 01 13:01 Figure 6-15. Temperature data from 3/ 14, in F. Time in hours:minutes. Tliq was the highest and most variant value, due to the boiling process. Tatm, the atmospheric temperature, was just beneat h it and at times indistinguishable. To was the averaged edge temperature, which hovered around 56-57. Tin was the coolant inlet temperature, the lowest value shown. The THR values (Figure 6-16) showed only a marginal change due to the application of VIDA. The times of these ch anges were noted in the above paragraph.

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107 The evacuation process, in fact, shows a much more significant and consistent effect on THR, as can be seen as an increase near 8:00. 0 10 20 30 400 : 0 0 0:59 1 : 5 9 3 : 0 0 4:00 5 : 0 0 6:00 7 : 0 0 8:01 9:01 10:01 1 1 :01 1 2:0 1 13:01 Figure 6-16. Total heat rate data from 3/14. Data were in W. Time was in hours:minutes. Only marginal differences appeared for the application of VIDA to DWC. The largest jump show n is due to evacuating the vessel at 8:00. The third and final day of tes ting (3/15-16) also failed to show a dramatic impact of VIDA on HT (Figure 6-17). The first secti on of the test, from 2:00 to 12:00, was run with 50% power setting to the light. It s hould be noted that no attempt was made to quantify the heat input from the light and the percentage input values should merely be taken as relative. The VIDA was switched on a little before 9:00. Consequently, the gap between Tatm and Tliq shrank a little. However, this gap could also be seen to grow with time prior to that period. This observation c oupled with the downward trend in THR with time (Figure 6-18) indicated an air leak in the installation of this day. The effects of this leak, however, were small, since the Tatm remained small (less than a few degrees), especially close to th e evacuations at 1:30 and 12:00. Ac tually, after the readings dip

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108 near 17:00, due to a heat shut down rather than an evacuation as has been seen before, the vapor became superheated by a few F for the remainder of the test. 40 60 80 100 1200: 0 0 1: 0 0 2: 0 0 3:00 4: 0 0 5: 0 0 6:01 7: 0 1 8: 0 1 9 :03 10 : 0 3 11 : 0 4 1 2:04 13 : 04 14 : 04 1 5:04 16 : 04 17 : 04 1 8:05 Figure 6-17. Temperature data from 3/15-16. Data in F. Time in hours:minutes. Tliq – liquid reservoir temperature is the more variant of the two higher temperatures. Tatm – the temperature of the atmosphere on the hot side of the vessel, is the less variant of the two hi gher temperatures. To – the averaged edge temperature of the diaphragm, is the middle temperature value. Tin – coolant inlet temperature, is the lo west reading. The flow-rate was 3.0 gph. In discussing the THR data of 3/15 (Figur e 6-18), it should be noted that the first portion of this test was run with 50% input to the heater/lamp and the second portion, after the discontinuity, at 70%. Fifty percen t input yields about 30 W, 60% input yields about 40 W, and 70% input yields over 50 W, which makes sense when considered in the context of an energy balance. It also would seem to highlight the fact that input power was generally held constant and excess steam supplied to the condenser surface in most experimental studies of DWC. Such an ar rangement would theref ore be preferred for comparability of data in the future.

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109 0 10 20 30 40 50 60 700 : 0 0 1 : 0 0 2:00 3:00 4 : 0 0 5 : 0 0 6:01 7:01 8 : 0 1 9 : 0 3 1 0:0 3 11 : 04 12 : 04 1 3:0 4 1 4:0 4 15 :0 4 16 :0 4 17:04 18:05 Figure 6-18. Total heat rate data from 3/ 15-16. Data was in W. Note the gradual decrease in THR between the first an d second evacuation (1:45 and 12:00), a period of relatively constant input para meters. This indicated a probable air leak in the vessel. The quality of DWC on the diaphrag m surface was good (Figure 6-19). The droplet size distribut ion and appearance was qualitatively typical of those shown in [19, 28,55]. A common description of DWC condensa tion was an array of drops of different sizes that were arranged in a self-similar patte rn at different magnifications. Following a departed drop may provide some insight to the origin and maintenance of this pattern. As a newly departed drop swept down the surface, it absorbed all the smaller drops in its path. Once the falling drop had passed, a clean, bare metal surface was left in its path. Small drops were then seen to be in the ba re path. They quickly began to coalesce as they grew enough to touch each other. The coalescence was visible as a flashing, when new, bare condenser surface was exposed after a smaller drop was absorbed into a bigger one. Eventually, these drops neared the size of the neighboring drops and the swept path of the fallen drop became indistinguishable from the areas around it.

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110 Figure 6-19. Droplet departure and rege rnation in DWC without VIDA, 3/15-16. Conditions were THR of 27 W, Tatm of 35C and To of 16C. a) Initial state. b) A large drop began to depart. c) Departed drop quickly swept across condenser surface. d) Swept path left behind departing drop. e) Nucleating drops appear in the swept path of the departing drop. f) New drops continued to grow. g) Swept path began to disappear amid new drop growth. A clear trend in developing droplet diameter was notic eable. Growing droplet diameter increased with radial position away from the origin and up the swept path. These coupled trends resulted from lowe r surface temperatur e and greater age, respectively. This sweeping function would seem essentia l to the phenomenon of DWC, and indeed it has been called such in the lite rature. Rose justified the omission of this

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111 Figure 6-19. Continued. phenomenon from his theory by saying that the net effect of increased condensate volume for a given time on the lower surface, whic h would serve to insulate, was effectively offset by the increased sweeping frequencie s effect of lowering the maximum droplet size, which would serve to increase HF for a given set of conditions, as shown in the functional behavior analysis section. He al so claimed that this net cancellation justified the lack of surface height dependence of his theory [18,24,45]. Indeed, Yamali and Merte, in their 2002 presentation of a theory to go with thei r experimental results of 1999 [15,16] concluded only slight surface height dependen ce of DWC HTC from their

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112 theoretical analysis and numerical simulation. The parameters of that study ranged from typical to extreme values of subcooling and body force (a control on maximum droplet size and therefore sweeping freque ncy). At any rate, the large radial temperature gradient on the diaphragm surface prevented any specula tion on spatial effects in this study. Video frames taken with a high-speed camer a offer a detailed look at the onset of VIDA for a condenser surface undergoing DWC (F igure 6-20). Before the driver was switched on, the droplet size a nd spatial distribution was typical for quiescent DWC. Unfortunately, condensation on the viewing window somewhat often partially obscured the camera view of the condenser surface. A ttempts were made at heating the window to prevent this, but found only moderate success. With VIDA applied, ejected droplets also contributed to obscuring the camera view. Just after the application of vibration to the diaphragm, the outlines of the individual droplets became le ss distinct, due to their indu ced oscillations. Next the oscillating droplets merged and began to m ove toward the center of the diaphragm. Centripetal acceleration from the diaphragm cause droplets above a certain size to be propelled toward the center of motion. This acceleration was more in tense closer to the center of motion, therefor e the stable drop sizes grew sma ller toward the center. This aided the effectiveness of VI DA of clearing the condensate fro m the diaphragm surface. As the merged droplets moved toward the center position of the resonating diaphragm, their surface was constantly oscill ating and ejecting droplets. At the center, this ejection was most intense. The bulk of fluid was quickly dissipated in this manner, leaving a surface with significantly less conde nsate on it. The final frame qualitatively represents

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113 Figure 6-20. Video frames of VIDA onset, front view. C onditions were THR of 27 W, Tatm of 35C, and To of 16C. Video take n at 500 fps. a) Droplet distribution prior to the application of vibration to the diaphragm. b) Just as vibration was applied, the outlines of the drops be come less distinct due to surface oscillations, frame 2 (4 ms). c) The three drops begin to merge, frame 3 (6 ms). d) The three drops continues to merge. The two large drops at right begin to merge, frame 4 (8 ms). e) All large water groupings begin to move toward the center of vibration, whil e undergoing intense surface oscillations, frame 5 (10 ms). f) Migration to cen ter of motion conti nues as oscillation intensity increases, frame 8 (16 ms). g) All large water drops merged at center, frame 12 (24 ms). h) Virtuall y nothing left of the primary drops by frame 60 (120 ms). a steady state droplet distri bution under VIDA conditions. The process of condensate removal by VIDA was quite fast. The frame rate for this video capture was 500 fps.

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114 Figure 6-20. Continued. Frames a through f covered only 12 image captures (24 ms). The entire sequence presented occurred over only 60 frames, or (120 ms). A calibration curve was constructed for a ne w driver during the initial testing phase (Figure 6-21). This curve showed a peak -to-peak displacement of about 37 m at resonance. By this point in the testing, th e driver was quite old. It had been employed for almost half of the testing. A frequenc y response test (Figur e 6-22) conducted after conclusion of experiments indicated a peak-topeak vibration amplitude at resonance of about 13 m. This was only about 1/3 that of the new driver. Input voltage was held

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115 relatively constant at about 35 Vp-p for this test, with up to 4 Vp-p variance at the peak frequency. y = -0.0038x2 + 1.1399x + 2.4693 R2 = 0.9992 0 5 10 15 20 25 30 35 40 0510152025303540 Vp-pmicrons Figure 6-21. Calibration curve for peak-to-p eak displacement vs. input voltage. The relation was essentially linear. Peak displacement for this new driver was almost 4 times that of a used driver tested after running several weeks of experiments. The displacement discu ssed is that at the center of the diaphragm, which would constitute the maximum. Generally, frequency tuning to find the a ppropriate resonance frequency was done by ear. This was easily done, since the sound at resonance was piercing and differences in intensity quite noticeable. When the reso nating driver was enclosed in the vessel, the intensity of sound was greatly reduced, however ear plugs were still required for long exposure. The peak to peak voltage used in this investigation was 35 to 36 Vp-p, with the exception of one day, when testing was done at 42 Vp-p, with no resulting difference in the data. The general uncertainty in this value was 0.2 V. Because of all the variability in the driver characteristics, temperature dependence of resonant frequency, and varying

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116 0 2 4 6 8 10 12 14 16 270028002900300031003200330034003500 frequency, Hzmicrons Figure 6-22. Peak-to-peak displacement vs. frequency for used driver, constructed after the conclusion of testing. Note that the resonant freque ncy of about 3050 Hz (manufacturer reported was 3.6 kHz) was only applicable to room temperature at the time, as this quantity for piezoelectric elements was temperature dependent. The vertical dispersion at resonanc e represents amplitude variation of the input signal of 2 and 4 Vp-p. The displacement discussed is at the center of the diaphragm, the maximum. intensity of peak response, the focus was kept on the presence or absence of vibration and therefore VIDA. In any case, the maximum droplet size was signifi cantly reduced for the presence of VIDA, even for the older driver. Vukasinovic [10] offered an excellent correlation for non-dimensional droplet diameter and critical accele ration, shown in Figure 1, but th is correlation was not applied to this investigation due to time constraint s. Also, a method for application was not immediately clear. The fact that the corre lation referred only to maximum acceleration at the center of the diaphragm was a complicat ing factor. Could this acceleration be projected to the edged of the diaphragm and its correspondingly lower accelerations through a deflection model? This was not a ddressed in the VIDA study. Also, the main

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117 mechanism that would appear to govern a model as such would depend most on the centripetal acceleration experien ced by the drops at the outer ra dii and at what point this would overcome the surface tension holding them in place. The limit on the drop size at the outer radii would be the cr itical concern since that was the active area of HT (Figure 3-2). Due to these complications, an attemp t to predict the limitation on droplet size was not made, and the VIDA was addressed only qualitatively. A more detailed, oblique vi ew of the VIDA process offered a better visual description of how this phenomenon progresse d (Figure 6-23). This series was taken during steady state conditions a nd at a higher frame rate (1000 fps). This video was not from the onset of vibration, but taken during quasi-steady state conditions more representative of test co nditions. The image focus was again a problem due to condensation on the viewing window and insuffic ient lighting. Attempts were made in all the video attempts to heat the window to reduce condens ation. However these efforts were not always effective. The series showed a single, large drop at th e edge of the diaphragm. This drop was propelled toward the center of motion by th e centrifugal accelerati on of the resonating diaphragm. The drop surface was constantly oscillating and ejecting droplets. At the center of motion, it merged w ith a smaller droplet. After only a few frames, the volume of the merged drop was notably reduced by the continuous droplet ejections. These ejected droplets showed up as streaks on the frames. Unfortunately, the frame rate was not fast enough to capture the moving, ejecte d droplets. There was not enough available light to effectively capture images at a higher frame rate. A given increase in frame rate implies a decrease in exposure time for the CCD and therefore a lower quality image.

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118 Figure 6-23. Video frames of VIDA process, corner view. Conditions were THR of 27 W, Tatm of 35C, and To of 16C. Video ta ken at 1000 fps. a) Note large drop in the background, frame 1 (at the edge of the resonating diaphragm). b) Large drop was propelled toward the center of motion, frame 4 (4 ms). Oscillations of drop surface were ejecti ng secondary droplets. c) Large drop merged with smaller drop at center of motion, frame 7 (7 ms). d) Volume of merged drop was significantly reduced, fr ame 10 (10 ms). e) Volume of large drop completely dissipated, frame 27 (27 ms).

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119 The center droplet in the final frame was actually quite large. It was normal for there to be a few stationary drops near the center of motion of th e resonating diaphragm, however they were usually quite small. Fo r a new driver, these st ationary drops were barely visible. The driver employed for thes e videos was the old driver discussed above, that had only 1/3 of the original peak-to-pe ak displacement of a new driver, for the given application of 35 Vp-p. Larger amplitude oscillations corresponded to more intense sound and smaller droplet sizes at the center. No size estimate is available for this center drop, but it was noted when the video was take n to be somewhat larger than usual. Data Summary and Comparison with Theory This section compares the theoretical TH R, computed for the given experimental conditions, with the measured experimental value. The theoretical value for THR relation to maximum departing droplet radius was also used to estimate an effective maximum droplet radius for each set of experimental conditions. See Chapter 3 for further explanation of the calculation methods. Effective maximum droplet radius values were calculated for those experimental runs considered to most likely be free from the effects of NCG. This determination of acceptability of the results for a given experi mental data set was based on the effective maximum droplet radius being less than 2 mm. This upper bound of acceptable reff was chosen as larger than the greatest theoretical maximum droplet radius considered in this study, 1.1 mm at a Tsat of 280K, by a reasonable uncer tainty and also by existing grouping of the data. In fact, only 18 of the 104 data points used from the third phase of testing returned reasonable values for reff, as judged by the aforementioned criteria. The remainder of those 104 data points returned reff values so large as to not merit consideration. The implication of this fact was that the measured THR was much lower

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120 than that predicted by the Rose theory. Assuming the theory to be accurate for the conditions used and also assuming “good” DWC, a qualitative check of DWC appearance commonly used in the literature [27,31,54,55], th e most likely source of this deficiency was a significant presence of NCG. 20 24 28 32 36 40 2024283236 q theory, [W]q experimental, [ W w/ VIDA w/o VIDA Series3 Figure 6-24. Experimental THR vs. theoretical THR. Both axes in W. Diagonal line represented the point where the theore tical value would coincide with the measured value. Those measured THR values which were de termined to be satisfactorily free of NCG were all higher than th e theory predicted (Figure 624). Ideally, the non-VIDA case would lie directly on the yellow line, which ha s a 1:1 slope. Aside from one outlier in each data series, all points were above the di viding line. This implied that the actual maximum drop size in this experiment was sm aller than the theore tical one, proposed by Rose (Eq. 2-11). This scenario was entirel y plausible, due to the inclusion of the empirical parameter K3 in that equation. This parameter was merely chosen by Rose to provide a good fit to experimental data over the full range of satu ration temperature and subcooling data available. Also, the estimat ed error of the Rose theory from the large

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121 comparison of data in Figure 3 from Ro se, 1988 [12], was about 15%. This error estimation easily accounted for the variation. The error was estimat ed from the given figure because an explicit statement of error for the Rose theory was absent from any publication. The error bars in the figure actually represent the standard deviation in the measured value on the independent axis and the effective variati on of the theoretical value from that of the principl e inputs on the dependent axis. Unfortunately, the sample size in compari ng those data points free of significant NCG influence was small. Therefore significa nt conclusions from the data distribution were not warranted. Examining the meas ured THR with respect to the maximum subcooling (difference between the vapor temp erature and the edge temperature of the diaphragm) did not shed any new light on the presence or absence on a VIDA effect on DWC (Figure 6-25). What was of note, however was that 3 of the 6 data points for cases with the presence of VIDA showed more than a 24% increase over the theoretical prediction. The other th ree values were well within uncertainty of the predicted values, though whether this lower performance was due to the VIDA not performing its assumed function of limiting the maximum droplet size or whether it was due to a small presence of NCG was ambiguous. The non-VIDA experimental THR values were all within appropriate uncertainty of the theoretical va lues, though slightly hi gher, again indicating the possibility of a smaller value for maximu m droplet size than that predicted by the theory.

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122 20 24 28 32 36 40 101112131415161718 Subcooling, Delta T, [K]Total Heat Rate, [ W w/o VIDA, exp w/o VIDA, theory w/ VIDA, exp w/ VIDA, theory Figure 6-25. Total heat rate vs. maximum subcooling, Tmax = Tsat – To, in W and K. Experimental values are single points theoretical values are connected by lines. Cases of both the presence and absence of VIDA shown. Effective maximum droplet radii were calc ulated for given experimental conditions using the program Get_Theoretical_THR_ and_r eff.m (Appendix A), as discussed in Chapter 3 (Table 6-1). The average reff determined from these calculations was 24% less for the VIDA cases than for without. If the one high-end outlier was removed from each data set, then the average reff for the VIDA cases became 46% less than the non-VIDA cases. Such a removal was just ified by the fact that the reff for each was almost 70% larger than theoretical rmax. A maximum departing drop size of this magnitude was could not be supported by experimental observations. The results seemed compelling only outsid e the consideration of the very small sample size involved in the calcu lations. Indeed, almost 84% of the experimental results (those not presented here) appear to have been tainted by the effects of NCG. The main reason for this situation wa s poor pressure vessel design.

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123 Table 6-1. Experimental and theoretical data for THR, effective maximum droplet radius, relevant temperature data. VIDA THR, exp. Std. Dev. THR, theory Std. Dev. Effective Max. Droplet Radius Std. Dev. Max. Subcooling, Tatm-To Std. Dev. Tatm Std. Dev. Y/N W W W W mm mm C C C C N 23.9 03 25.9 1.0 1.72 0.13 11.7 0.3 30.7 0.3 N 29 2 25.0 0.5 0.45 0.24 12.2 0.2 25.6 0.11 N 25.6 08 25.5 1.2 1.07 0.22 12.2 0.4 27.4 0.4 N 27.5 1.6 25.0 0.7 0.63 0.27 12.4 0.2 25.2 0.17 N 26.1 1.6 25.0 0.7 0.86 0.39 12.5 0.2 24.7 0.2 N 30 2 26.4 0.7 0.52 0.27 12.9 0.2 26.1 0.2 N 33.3 1.7 28.0 1.9 0.39 0.15 13.5 0.7 27.2 0.7 N 29.1 1.6 28.1 1.2 0.91 0.35 13.8 0.4 26.1 0.4 N 29.7 1.8 28.4 1.2 0.85 0.37 13.9 0.4 26.2 0.4 N 31 3 27.6 0.9 0.56 0.46 14.3 0.3 27.3 0.3 N 36.9 0.9 33.1 1.1 0.58 0.09 14.9 0.3 30.6 0.3 N 37.4 1.1 36.1 1.8 0.89 0.17 16.0 0.5 31.8 0.5 Avg. 0.79 0.26 Y 26 2 20.9 0.6 0.30 0.19 10.7 0.2 24.2 0.2 Y 33 2 21.7 0.6 0.08 0.04 10.9 0.2 24.7 0.2 Y 24.3 1.6 26.2 1.1 1.70 0.78 12.4 0.4 28.3 0.3 Y 30 3 25.4 1.5 0.42 0.36 12.4 0.4 26.2 0.4 Y 36 2 26.8 0.9 0.18 0.08 12.9 0.3 26.8 0.3 Y 36 2 34.8 1.0 0.90 0.36 17.0 0.3 25.3 0.3 Avg. 0.60 0.30 Additional Testing Further testing was undertaken to clarify any effect that VIDA might have on DWC HT. There were three main aims of this ne w round of testing. The first aim was to better control NCG in the test apparatus by special attention to seals a nd valve arrangements. The second aim was to place thermocouples on the diaphragm surface, in order to confirm the assumed radial temperature profil e. The third aim was to better achieve a saturated atmospheric condition in the te st vessel by making the water reservoir physically separate from the pr essure vessel. A halogen lamp was still used for heat input, but a separate vessel wa s constructed to hold the exces s water and connected to the main vessel by vacuum tubing and valves.

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124 Three days of testing were conducted in this new phase of testing. The results from the first day of testing had low overall TH R values, which was a symptom of bad seals allowing continual leakage of NCG into the pr essure vessel. The pot ential leak points were then sealed to the extent possible w ith the current design. The second day of testing, 5/21, showed overall THR values much more inline with the theoretical predictions. Though the temperature data showed slight increasing trends with time, this increase was caused by drift in the coolant inle t temperature, which was the base point for these values (Figure 6-26). 0 5 10 15 20 25 30 0:001:002:003:004:005:006:027:02 Figure 6-26. Temperature data from 5/21, the second day of additiona l testing. Readings were in C. Time axis in hours:min. Th e top four temperature traces show the atmospheric temperature estimate with three thermocouple readings placed on the condenser (diaphragm) surface. The three thermocouples were placed so as to divide the radius of the shim into equal parts along the same line, in order to confirm the assumed temperatur e profile. The middle temperature value was the average edge temperature. The lowest temperature was that of the coolant inlet. Each major dip in temperature coincides with an evacuation operation on the vessel. No VI DA was induced in this test.

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125 The four highest temperatur e traces from that data represented the atmospheric temperature estimate, a single thermocouple locat ed about inch above the test surface, and three thermocouples attached to the surf ace of the brass shim. The adhesive used was the same insulating RTV silicone employed elsewhere in the apparatus. Adhesive was applied and allowed to cure with the thermocouple bead pressed against the brass surface. After allowing 18 to 24 hours for the silic one to cure, digita l pictures of the thermocouple arrangement were taken from di rectly overhead. Geom etric markers were then added to the pictures and bead center positions and uncertainties then estimated by pixel counting. The center thermocouple cove red the center point of the diaphragm and so its reading was assumed to represent the av erage temperature of the center 10% of the radius. The centers of the other two thermocouples were located at 3313% and 669% of the radius from the center, respectively. Visual appearance of condensate distribution on the condenser surface was essent ially the same as without the attached thermocouples. It was attempted to make the thermocouple bead placement collinear. The placement of this line was roughly horizontal during testing, though de liberate variation of this quality showed no effect in the data. The test surface was vertical and all other thermocouples placed in the same arrangeme nt as in the previous tests. See the Additional Testing section of Chapter 7 for a discussion of calibration and uncertainty for this round of testing. In the first hour of testing, superheating was noted in the vapor. This was corrected by rotating the vessel a few degrees to allow for unimpeded entrance of the vapor to the test vessel atmosphere. This step was necessary because the volume of collected

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126 condensate would sometimes be substantial e nough to clog the inlet tube from the vapor supply, located at the center of the lower window hole. The clogging (or reflux prevention) phenomen on necessitated periodic efforts to drain the condensate, since the vapor velocity issuing from the inch ID vacuum tube coming from the Bulk Fluid Reservoir was sufficient to prevent any backflow of condensate. Drainage was accomplished by simp ly turning off the halogen lamp used for a heater for a few minutes and elevating the vessel slightly to facilitate condensate removal. This process was only necessary once every few hours of testing and only lasted about 5 minutes. Thus it was a notable event in th e temperature record, but had few larger effects. However, the initial clear ing of condensate did have the effect here of eliminating the superheat of the vapor for the remainder of the test, during which the maximum variation between any of the su rface thermocouples and the atmospheric reading was less than about 0.5C. Each major collective drop in the temperature readings corresponded to an evacuation operation, similar to earlier tests. More shallo w and short temperature drops were due to condensate draini ng operations. The significance of the 5/21 test data was that it confirmed that the center 2/3 of the radius of the diaphragm was very near the vapor temperature and was thus relatively in active in HT transfer compared to areas closer to the edge, as the assumed temperat ure profile had predicted. Though there was a distinct decrease in each successive temperat ure reading from atmospheric to center to 1/3 to 2/3 thermocouples, the magnitude of the decreases was less than 1C. A comparison was made of the measured su rface temperature data and the assumed temperature profile for the non VIDA case of 5/ 21 (Figure 6-27). A distinct decrease was

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127 noted in the temperature values, 23.35C at the center, 23.19C at the 1/3 position, and 23.06C at the 2/3 position. However, the uncer tainty in the measured temperature was 1.0C, so these values were essentially equi valent and the model was accurate to within experimental uncertainty. Thus with the major assumption of th e temperature profile prediction had been confirmed at a few poi nts along the radius, for the non-VIDA case only. Figure 6-27. Comparison of 5/21 surface temper ature data for experimental run C with the assumed temperature profile. The solid line represents the assumed profile and the circles the three surface temperat ure measurements, averaged over the given time period. The uncertainty in the measurements was 1.0C, so the agreement was good within experimental uncertainty. The next day of testing (5/24) attempte d a more direct comparison of DWC with and without vibration. The nonvibrational data from this da y looked essentially identical to the 5/21 results. This fact along w ith the large number of reasonable rmax values

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128 attained with this day of testing indicated much better control of NCG as compared to previous testing. A very important event wa s noted in the surface temperature readings after the application of VIDA. The two center temperature readings increased suddenly from their otherwise nominal grouping with th e atmospheric reading (Figure 6-28). This jump indicated that the center and the 1/3 position temperatures read higher than the vapor temperature. This meant that no c ondensation at all was taking place on this portion of the diaphragm surface. 20 22 24 26 2:182:483:183:484:18 Figure 6-28. Comparison of 5/21 VIDA a nd non-VIDA temperature data for shim surface and atmospheric. Data is in C Data shows difference in vibration and non-vibration cases. The two hi gher values were the shim surface temperature at the center and at 1/3rd of the radius away from the center. The next highest value was the atmospheric temperature estimate and the lowest temperature was the shim surface temperat ure taken closest to the edge. Note that the center of the diaphragm was at a higher temperature than the vapor under application of vibration to the shim, indicating that condensation was not taking place over this center area at all, thus inva lidating the surface temperature profile assumed in the theore tical model for the vibrational case. The drop in values at 3:40 corresponde d to draining of the condensate from the pressure vessel.

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129 This elevated center temperature explained the observation that a large dry patch appeared in the center portion of the dia phragm upon application of vibration. This phenomenon had been noted before, but it had be en assumed that this region simply had a maximum droplet size that was too small to be visible to the naked eye. The most important implication of this finding was th at the assumed temperature profile did not apply for the cases with vibration. This fi nding thereby invalidated the theoretical model used in the previous analysis for comparison of data points. Presumably, the increased surface temperature of the center portion of the diaphragm was due to resistance heating of the piezoelectric patch. A major implication of this di screpancy was that the active surface area of condensation was very diffe rent between the tw o fundamental cases, which made direct comparison somewhat dubious. Table 6-2. Data summary from additional testing. VIDA THR, exp. Std. Dev. THR, theory Std. Dev. Effective Max Droplet Radius Std. Dev. Max Subcooling Tatm-To Std. Dev. Tatm Std. Dev. W W W W mm mm C C C C N 32 2 24.1 0.9 0.21 0.10 12.4 0.3 23.06 0.13 N 31.0 1.0 24.5 0.5 0.27 0.06 12.5 0.2 22.96 0.10 N 30.3 1.0 24.5 0.6 0.32 0.07 12.5 0.2 23.29 0.11 N 29.7 0.7 25.0 0.9 0.40 0.06 12.7 0.3 23.5 0.2 N 28.8 0.8 25.4 0.8 0.53 0.10 12.8 0.3 24.11 0.17 N 33 2 24.9 0.6 0.20 0.10 12.6 0.2 23.90 0.09 N 28 3 25.2 0.4 0.6 0.6 12.7 0.10 23.70 0.05 N 25.7 1.2 26.8 0.8 1.4 0.4 13.4 0.2 24.40 0.15 Avg. 0.49 0.19 Y 28.7 1.4 25.6 0.8 0.6 0.2 13.1 0.2 23.6 0.2 Y 26.9 1.7 25.6 1.3 0.8 0.4 13.0 0.4 23.7 0.4 Y 38 3 26.3 0.7 0.12 0.08 13.1 0.2 24.7 0.12 Y 28.6 0.8 24.6 1.8 0.46 0.09 12.5 0.6 23.8 0.4 Avg. 0.50 0.19 The theoretical predictions of THR for th e VIDA cases were included with the data summary for the Additional Testing (Table 6-2) This inclusion was made even though a base assumption of the model has been prove n incorrect for the VIDA cases, simply as a

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130 means for comparison with ear lier data. All data points taken from 5/21 and 5/24 appeared to be effectively free of significant influence of NCG in the test vapor. This conclusion was based on the proximity of the non-vibration THR values to the theoretical prediction of THR. Since the theoretical model for comparing the vibrational and non-vibrational cases had to be set aside, raw comparison of the TH R values for similar cases of each had to be examined. No systematic advantage in HT was noted from the presence or absence of VIDA in DWC on a piezoelectric diaphragm from this comparison.

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131 CHAPTER 7 UNCERTAINTY Main Study Uncertainty is always an important factor in experimental studies. The pressure readings of the vacuum gauge, in this study for example, were accompanied by such high uncertainty (1% of full-scale-vacuum) that they were of little use at the low vapor pressures of water used. Indeed, use of these readings to estimate the NCG partial pressure from temperature readings of the atmo sphere in the test vessel often resulted in negative values, of course a physical impossi bility. Thus values from the pressure readings were not referenced in this pape r and were mainly used as an immediate measure of vacuum pump performance. Thermocouple calibration was also a c oncern. Erroneously, the batch was not calibrated together prior to expe riments, merely used straight out of the box. Also, there were several batches of thermocouples use d, of both type J and type K. Whenever applicable, the settings of the DAQ were changed to accommodate the difference in thermocouple type. By the third stage of testing, the importance of these errors was realized and a step implemented to compen sate, to the extent possible. After the conclusion of each day’s experiments, ever ything was turned off, but the vessel not vented to atmosphere. The apparatus was th en allowed to reach equilibrium over night. Thus, the next day, the temperature distributi on in the vessel could be assumed relatively uniform and any systemic differences in th e thermocouple readings noted. Typically 100 data capture cycles (5 seconds per cycle) were taken in the checks. Then all the averages

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132 and standard deviations we re calculated for the resp ective groups. For the two thermocouples external to the pressure vesse l, the average of their difference was used instead of the average of each individual unit. The maximum standard deviation calculated in this manner ove r 9 checks was 0.15F (0.13C) at an average equilibrium temperature of 79F (26.11C). This of course represents only the relative error and not absolute error. In all but a few cases, for those data with reff near or below the theoretical value, the variation in physical value wa s high enough that the experime ntal uncertainty made only a negligible contribution to the standard devi ation of the reading (Table 7-1). In these calculations, it was the standard deviation whic h was used as uncertainty. The majority of the uncertainty for thermocouples (TC), was constituted by the absolute error reported by the manufacturer for the J and K type ther mocouples used, 2.0C. The relative error of these thermocouples was around 0.15C. Flow rate was a very important quantity for these calculations, since the measured heat out put was directly proporti onal to it. If a more precise way to measure the volumetric or mass flow rate were available, it would drastically improve the uncertainty in this quantity. The voltage, frequency, and pressure readings, however did not directly factor into any calculations. Table 7-1. Uncertainty in measured values. Quantity Unc. Units Thermocouple 2 F Flow-rate 0.1 gph Vp-p 0.3 V Frequency 10 Hz Pressure 0.3 in-Hg Also, due to the steep gradient of the temperature function at the edge of the diaphragm, the numerical integration of the th eoretical THR in the iterative process, was

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133 very sensitive to the number of terms that the radial distri bution was divided into. The number used in constructing the tables wa s 2000, chosen based on the highest precision with a reasonable computation time. Since further increases in the number of te rms continued to change the outcome of the calculations, another method had to be a pplied to estimate the bias error of this method of calculation. A geometric series wa s examined, based on the number of radial terms used to iterate the theoretical THR and the magnitude of the change in the result, for otherwise constant input. By projecting the change in calculat ion result to a high number of radial terms, it was determ ined that the iterations, at 2000 terms, systematically underestimated the actual va lue from 2.4% for the lowest THR, to 4.7% for the highest. Intuitively, it made sense th at the error would be higher at higher THR, since these values would have higher gradients in the temperature function at the edge of the diaphragm and therefore be more sensitiv e to step size changes. This condition was exacerbated by the very high HTC for the high subcooling at the edge. Additional Testing In the new round of testing, all thermocouples used were type T. This change was made because T type was also active in the temperature range in question and the absolute uncertainty in temp erature listed by the manufact urer, Omega Engineering, was half that of the type J and type K that were used in previous test s at 1.0C. Calibration was conducted simultaneously with all ther mocouples and two high-precision, 0.1 and 0.05C, mercury thermometers. Data point s in the calibration were taken every few degrees from 0 to 50C. Linear calibration coefficients were then determined for each thermocouple reading. The readings offered by the data acquisition program, issued with the DAQ card, were only available as temperatur e readings not direct voltages, so this

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134 given value was used as a raw reading for calibration and data collection. After the data was collected, the calibration equations were applied to each thermocouple data stream. The maximum root mean squared of the error for each calibrated thermocouple over the calibration data set was less than 0.2C relative to the thermometer readings. The maximum discretization error of the A/D conversion was 0.02C. Therefore, the maximum uncertainty in the measured temp erature values reported in the Additional Testing section, as determined by the square -root-of-the-sum-of-the-squares method was 1.0C. However, steady state differences in the relative surface temperature and vapor temperature readings were noted to be as little as 0.1C. Special care was taken to insulate the ba re wire lead secti ons of the surface thermocouples by covering them with the same silicone insulation used for adhesion. Insulating the wire leads was r ecommended to keep them from acting as fins to cool or heat the thermocouple bead [46]. Using an insulating adhesive for attaching the thermocouple beads to the diaphragm surface was recommended to again insulate the thermocouple bead from all its surroundings except the diaphragm surface [92]. After all, a thermocouple only reads the temperature of the bead, not that of the intended object.

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135 CHAPTER 8 CONCLUSIONS Main Study Non-condensable gases were not sufficiently co ntrolled in this test apparatus. This oversight was due to a lack of understanding of their very significant effects on DWC, at even small relative concentration to vapor through the design, fabrication, early and intermediate experimental phases. Attempts were made in the third and final phase of testing to reduce the NCG presence as much as possible. However, in many cases, the reduction in overall condensation HT due to th e added diffusive resistance of NCG to the net mass inflow to the condenser surface was still significant enough to deny comparison of the experimental and theoretical values. The greatest decrease in HT from theoretical to experimental found in the data used was 94%. This figure emphasizes the absolutely critical importance to DWC of minimizing the presence of NCG. Unfortunately, due to the availability of only a low precision pre ssure instrument, estimations of NCG partial pressure during the testing were impossible. An attempt was made to model the leak rate over several days and then extrapolate to th e time of the experiment, but precision of the results was unsatisfactory. Out of 104 data points considered, only 18 yielded favorable comparisons to theory. Six of these cases were with VIDA pr esent, 12 without. All but one outlier from each category showed better HT than pred icted by the Rose theory. Although the experimental non-vibration values were slight ly higher than the theoretical prediction, they were still within one experimental uncer tainty. Half of the ‘good’ VIDA, or those

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136 runs with vibration present, measurements were an exception to this, showing 24, 34, and 52% more HT than predicted by theory. While these increases show potential, the overall HT rate was relatively small for the total area of the driver (2.9 cm2), yielding a meager 20.5 W/cm2 average HF for the absolute highest TH R measured. In fact, the average HF for those tests yielding reasonable reff values was only 10.6 W/cm2. For all the measured data, THR ranged from 2.9 to 58.3 W for Tsat from 25.3 to 56.5C and To from 24.2 to 46.7C. In light of the results, three experiment al instances likely indicated a significant increase in HT (of a maximum 50%) due to the application of VIDA. The VIDA served to limit the maximum droplet diameter thereby increasing the HT. However, the majority of the data was rendered usele ss by the presence of NCG in th e test vapor. The effect of NCG in the test vapor was to depress the HT. Because of these problems it was impossible to draw any solid conclusions, ex cept to say that any increase in DWC HT due to the application of VIDA would likely be less than an order of magnitude for the overall THR. Trend Explanation The following points offer possible explanati ons of several trends encountered in the experimental data: THR increased significantly with increasing pressure, even despite the presence of NCG in unknown concentration. This follows directly from the prediction of the Rose theory. The application of VIDA had the immediate effect of lowering Tatm for a given set of conditions. This was most likely due to the spray of ejected droplets evaporating wholly or partially in the air, rather than flowing directly into the liquid reservoir. The ejected droplets may also have imp acted the thermocouple beads directly, thereby cooling them.

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137 Once sufficient effort was focused on mini mizing NCG, superheat was noted on the vapor over the liquid reservoir temperature, where most of the heat was coming in. This implied significant heating of the aluminum vessel body. This implication was confirmed by experimental notes that at times the metal vessel was nearly too hot to touch. Possible rami fications of this would be: o Inaccurate saturation pressure estimation from Tatm. In the cases where superheat was noted of magnitude gr eater than the uncertainty in Tatm, the Tliq measurement was substituted for Tatm in theoretical calculations. However, this situation affected only one data point used in reff comparisons. o Potential for significant HT to the cool side of th e vessel through the Teflon insulation cylinder, though th is potential was not quantified. A given increase in NCG concentration in the vessel during DWC would cause an increase in the liquid reservoir temperature (Tliq), a proportionately smaller decrease in atmospheric temperature (Tatm) and edge temperature (To), and also a decrease in the measured tota l heat rate (THR). These trends were totally in line with the literature survey findings under the headi ng Effect of Non-condensable Gases (NCG). Excellent DWC quality (droplet symmetry and size distribution) was noted throughout the testing despite no intentional applicati on of a dropwise promoter, purportedly essential on a clean metal surf ace. An obvious conclusion was that the diaphragm was not sufficiently clea ned, especially since FWC was never satisfactorily induced on the surface of the diaphragm, except under VIDA conditions. In addition to potential su rface contaminants remaining through the driver cleaning process, an oily film wa s noted on the inside of the vessel and on the top of the liquid after every test run was allowed to sit over night. This film may have been from the significant amount of silicone sealant used. It may have been a remnant from the oil used for m achining the largely polymer components of the apparatus. A contribution may have also been made by the Dow Corning High Vacuum Grease used on the seals. Additional Testing Further efforts at testing, which eliminated possible parasitic heat transfer from the pressure vessel wall, directly measured su rface temperature data for the brass shim, and improved control of NCG in the apparatus showed illuminating results. The surface temperature readings showed that the theo retical model, which assumed a substantial portion of the center of the condenser surface was very close to the vapor temperature,

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138 applied well for the non-vibrational case but poorly for the case with vibration. This finding invalidated a potentially useful tool of comparison for the two cases. Also, the elevated center temperatures, relative to the vapor condition, indicated that the surface area of the diaphragm active in condensation wa s significantly smaller in the vibrational case than the non-vibrational case, preventing precise comparison even of the measured THR values. Noting, however, that the active HT ar ea on the diaphragm was smaller with applied VIDA, the fact that those data poi nts judged acceptable showed comparable to slightly better overall THR values with appl ied VIDA says that ap plied vibration does serve to increase the effectiveness of DWC HT for a given area. Unfortunately, due to the unknown active HT area change, it was not possible to offer a quantitative estimation of this increase. Of positive note was that the adjustments to the apparatus when combined with established procedures appeared effective at minimizing NCG in the test atmosphere, thereby increasing confidence in the accuracy of the measured results. Improvements The results of this study were so flawed by the presence of NCG, that, if it were to be undertaken again, the appara tus would likely be completely different. Some primary suggestions for improvement of the apparatus are as follows: Apparatus Design seals with large flanges and gasket s specifically designated for steam use, likely Teflon. Minimize potential places fo r leaks in the design. This study had sought to use a pre-existing pressure ve ssel, and the outcome ultimately suffered because of this. Provide ample windows for both lighting and cameras. Make them electrically heated, to prevent fogging [31]. The wi ndows must also be sufficiently far away

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139 from the driver surface that it does not ge t sprayed with ejected droplets. Six to eight inches should be suffi cient. An alternative w ould be to angle the window surface down and coat it with a substance such as Rain-X to encourage the drops to run off, or provide a wiping system on the interior, though the latter option would likely be prohibitively complex when sealing is considered. Provide for a heater to be directly immersed in the liquid reservoir, so that all heat entering the system may do so throug h the liquid, thereby cutting down on superheating of the vapor. External hea ting may also be used, provided that the reservoir is effectively thermally isolated from the test section. Thermally isolate the liquid reservoir and heating apparatus from the test section of the apparatus. This will also significantly cut down on superheating, which when combined with the heater changes above would make saturation temperature measurement more accurate, since the test area atmosphere should be within a very small margin of the liquid reservoir temper ature. This method was used in nearly every experimental study cited. Make the heater output power constant. Fu rthermore, design the heater capacity for at least 60% excess vapor, based on the ma ximum expected HT rate, to be supplied to the condenser surface so that supply would never be the limiting factor. Make the vapor system closed loop for sub-atmospheric pressure tests. Accomplish this by using a cold trap on th e vapor exit line, often cooled by main water for steam tests at 1 atm, but wo uld need significantly cooler than Tsat temperatures to condense as much vapor as possible. Ice or liquid nitrogen would work sufficiently in a ‘jacket’ around the vapor line. The flow would then enter a phase separator hooked to a vacuum pum p, of continuous duty, to remove any NCG. Then condensate would collect in a holding area and be metered to estimate the venting rate, by time average. Use a close venting scheme for removi ng the excess vapor, where the nozzle is located near the condenser surface. An excellent description is provided in Le Fevre and Rose, 1965 [31], and an in-depth analysis of the method provided for a specific geometry in Citakoglu and Rose, 1968 [32]. Take care from the fabrication and asse mbly through pre-test run cleaning to prevent the addition of unwanted organic flui ds to the chamber such as cutting oil, or vacuum grease. Develop a mechanism for driving a solid block of copper in the frequency and displacement range necessary to produce VIDA so that, 1) the condenser surface may be considered effectively isothermal, 2) the temperature gradient in the block may be directly measured and exptrapol ated to the surface value, and 3) the mechanism does not produce significant heati ng of the condenser block. These precautions will allow for direct comparis on to theory and greatly simplify the analysis.

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140 Find an instrument to measure displacemen t with sufficient resolution to capture 10 to 40 m amplitude oscillations within a reasonable uncertainty. It may prove quite difficult to find an instrument whic h has both capability and geometric requirements that work well with the pressure vessel. Attention also must be paid that sufficient surface reflectan ce of the back of the driver can be obtained without having to resort to reflective tape, as th is will preclude measurements during low pressure tests, as was the case in this study. Such an instrument will likely be prohibitively expensive as well. Only tw o instruments were found to be acceptable for the present study, one being a Polytech Laser Vibrometer, which had excellent accuracy, range, surface finish, and geomet ric parameters, but cost almost $50,000, the other was less robust, the MTI-2000, used in this study, but still cost $8,000, with possibility of rental as an option. Ho wever, this instrument required reflective tape on the back of the diaphragm wh ich would necessarily ‘bubble’ under evacuation of the cylinder and consequen tly lose signal, so that displacement measurements during experiments were impo ssible. Other means of increasing the reflectivity of the driver backing may be determined which are viable under vacuum conditions. As yet, none have been determined. Provide for different regimes of coolant fl ow rate [31], based on the level of HT predicted and the expected temperature drop across the cooler, which should be minimized to approach a constant temper ature sink. The delta T across the cooler may be monitored for an energy balance, but precision is not required due to the measurement of HF in the condenser block. Procedure At the beginning of every testing sessi on, have a 3 to 4 hour ‘blow-down’ of the apparatus, venting to atmosphere or th rough the vacuum pump with the heater on full power, if it is capable of handling larg e amounts of condensate. The current oil lubricated vacuum pump was not, and thus the oil had to be re placed almost every day. Do a FWC comparison run, requiring a completely clean metal surface, which was not achieved in this study. Extensive cl eaning regimens that may be used or adapted were described in Le Fevre and Rose, 1965 [31], Finnicum and Westwater, 1989 [62], and Vukasinovic, 2002 [10]. Allow ample time for steady state conditions to be reached for measurements. This may be as long as several hours in some cases. There are many potential surface promoti on techniques, several of which are cataloged in the Surface Promotion section of Chapter 2. The best base case, for comparability with the Rose theory woul d be polished copper with an oleic acid coating. Further promoters might then be tested for comparison, such as the L-B film discussed in Zhao et al., 1996 [58]; the diamond like coating (DLC) tested in Koch et al., 1998 [14]; or bulk addition of promoter [37,53,65]. Though proven an

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141 effective promoter, the PTFE coating with ion-implantation of Ma et al., 1999 [68] would likely impede the performance of the driver too much for feasibility.

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142 APPENDIX A MATLAB CODES DWC_HTC.m function [h,heat_flux]=DWC_HTC(Tsat,Ts,r_max) % [h]=DWC_HTC(Tsat,Ts,r_max) % Tsat, [K], a consant value for the far-field vapor % temperature of the steam % Ts, [K], a vector of surface temperature values % r_max, [m], a vector of artificial maximum drop radius, for % theoretical maximum radius, input r_max as zero % h, [W/m2-K], a scalar of the Heat Transfer Coefficient for a % given subcooling and saturated vapor temperature, for dropwise % condensation of pure water vapor on an iso-thermal surface. Theory % offered by Le Fevre & Rose, 1966, and in modified form by Rose, 2002. % % units are m-g-s, except where noted otherwise % This file represents the theoretical model from dropwise condensation % of pure water vapor on an isothermal surface presented by Rose, 2002 % in a review of dropwise condensation. The original theory was % offered by Le Fevre & Rose, 1966, and various adjustments to % experimental constants were gathered in the review address by Rose in % 2002. % Check if Tsat is in K or not if Tsat < 273 Tsat = Tsat + 273; Ts = Ts + 273; end % Constants g = 9.81; % acceleration of gravity, [m/s^2] K1 = 2/3; % shape factor for steam, p.151, (Rose, 1998) K2 = 1/2; % shape factor for steam, p.151, (Rose, 1998) K3 = 0.4; % empirical constant, p.151, (Rose, 1998) R = 461.5; % gas constant for water, RGAS.exe, [J/kg-K] gamma=1.33; % specific heat ratio for steam, [ ] % Properties call PROPS function to interpolate thermophysical % properites from steam tables [rho_f,rho_g,hfg,k,k_g,sigma,Psat,mu_f,mu_g,Cp_f,Cp_g,Beta,alpha_f,… alpha_g,nu_f,nu_g]=PROPS(Tsat);

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143 % Variables deltaT=Tsat-Ts; % vector of surface subcooling values if r_max==0 r_max = K3 (sigma/rho_f/g)^0.5; % Equation (8), maximum drop % radius prior to departure from original nucleation site end % variables which do not involve either r_max, deltaT, or Ts B=( 2*sigma*Tsat )/( rho_f*hfg ); C=K1/k; D=K2*0.627/0.664*Tsat*(gamma+1)/(hfg^2*rho_g*(gamma1))*… (R*Tsat/2/pi)^0.5; % Initialize h h=zeros(length(r_max),length(Ts)); heat_flux=zeros(length(r_max),length(Ts)); % dummy variables to save computations dummy1=2*sigma/rho_f/hfg*Tsat; % Put separate loop out here for r_max variable so it is not recalculated for every Ts value % Define variables which involve r_max for j=1:length(r_max) E(j)=(3*r_max(j)^(1/3))^(-1); end % Loop for Ts values for i=1:length(deltaT) if deltaT(i) > 0 % Define variables which involve deltaT, but not r_max % Equation (7), smallest thermodynamically viable drop radius r_min(i) = dummy1/deltaT(i); % Algebraic Intermediate Variables A=deltaT(i); % Define argument function to work with quadrature (numerical % integration) function, quadl arg= @(r) (A.*r + B)./(C.*r + D)./r.^(5/3); % Loop for r_max values for j=1:length(r_max) % Call function to perform numerical integration of % argument by adaptive quadrature heat_flux(j,i) = E(j)*quadl(arg,r_min(i),r_max(j));

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144 if heat_flux(j,i) < 0 heat_flux(j,i) = 0; elseif Tsat-deltaT(i) < 273 heat_flux(j,i) = 0; end % Convert to Heat Transfer Coefficient, h, [W/m2-K] h(j,i)=heat_flux(j,i)/deltaT(i); end % end of for (j) loop end % end of if statement to make sure end % end of for (i) loop Dropwise_Codensation_Model.m function [q_theoretical,number_of_iterations]=Dropwise_Condensation_Model… (Tsat,To) % The implementation of DWC_HTC has been modified for this program % to be used as a function, and so arguments must be checked with % current iteration of DWC_HTC and this program before implementation alpha=0.0; Tsat=50; % Vapor Saturation Temp, [degC] To=4; % Edge Temp, [degC] c=0.004*25.4/1000; % Shim thickness, [m] b=.75/2*25.4/1000; % Exposed Radius of Shim, [m] len=5000; % Desired number of steps in r-vector factor=50; dr=b/len; err=0.01; % Desired error between steps in iteration q=100; % Total Heat Rate, [W] k=140; % Get Properties based on Tsat [rho_f,rho_g,hfg,k,k_g,sigma,Psat,mu_f,mu_g,Cp_f,Cp_g,Beta,alpha_f,alph a_g,nu_f,nu_g]=PROPS(Tsat); % Set r_max to command for theoretical value use r_max=0; r=0:dr:b; % radius of vector values, [m] h=zeros(length(r)); q_dp_r=zeros(length(r)); Ts=zeros(length(r));

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145 m=length(r)-1; % length of r-vector adjusted for temp profile routine % Initialize dummy variables i=0; number=1; q_theory(1)=q; while i==0 % Execute these commands until the error is below % desired value [Ts]=exp_profile(Tsat,To,b,c,q_theory(number),m); % Obtain radial % temp profile from function % from this point, determine where the subcooling rises above 1.0K, and % only treat the remainder of the radial vector with the smaller dr number=number+1 %Print out the iteration number each time through [h,q_dp_r]=DWC_HTC(Tsat,Ts,r_max); % Normal implemenation % Argument for area integral arg=r.*q_dp_r; % Approximate Integral with Simpson's Rule function [ans,err_order]=simpsons_rule(arg,dr); q_theory(number)=2*pi*(ans); q_theory(number)=(q_theory(number) + q_theory(number-1))/2 + alpha… ( q_theory(number) q_theory(number-1) )/2; %error_order=err_order % Compare differential diff=abs(q_theory(number-1)-q_theory(number)) if err > diff break end end % end of while loop that iterates the theoretical heat flux calculation number_of_iterations=number-1 q_theoretical=q_theory(number) q_measured=q percent_difference=(q_theoretical-q_measured)/q_measured*100 figure(1) subplot(3,1,1) plot(r*1000,Ts),grid ylabel('[\circC]') title('Imposed Ts(r), HTC(r), & q"(r)')

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146 subplot(3,1,2) plot(r*1000,h),grid ylabel('[W/m2-K]') subplot(3,1,3) plot(r*1000,q_dp_r),grid ylabel('[W/m2]') xlabel('r, [m]') % figure(2) % bar([q,q_theory(number),diff]),grid % xlabel('q_measured, q_theory, difference last step') % ylabel('[W]') % title('Total Heat Rate Comparison') Theoretical_Heat_Rate_Grid.m clear all Tsat=22:40; To=10:17; for i=1:length(Tsat) for j=1:length(To) [q_theory(i,j),number(i,j)]=Dropwise_Condensation_Model(Tsat(i)… ,To(j)); end end figure(1) plot(To,q_th),grid xlabel('[\circC]') ylabel('[W]') title('Theoretical Total Heat Rate vs. Edge Temperature and Saturation Temperature') iterations=num DWC_Behavior.m clear all % *********************************************************************** % This program examines the behavior of the Heat Transfer Coefficient for % Dropwise Condensation of pure water vapor as a function of % 1) Saturation Temperature % 2) Subcooling % 3) Maximum Droplet Size % ***********************************************************************

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147 % Saturation Tempuratures to be argued, [K] Tsat=280:10:500; % Subcoolings, [K] deltaT=[0.1:0.1:0.9,1:19,20:10:50]; K3=0.4; % Constant, [-] g=9.81; % Acceleration of gravity, [m/s2] for i=1:length(Tsat) [rho_f,rho_g,hfg,k_f,k_g,sigma,Psat,mu_f,mu_g,Cp_f,Cp_g,Beta,alpha_f,al pha_g,nu_f,nu_g]=PROPS(Tsat(i)); % Theoretical Maximum Drop Radius, Equation (8), (Rose, 2002) r_max_th(i) = K3 (sigma/rho_f/g)^0.5; % Smallest thermodynamically viable drop radius, (Rose, 2004) % only for informational purpose, not fed to any subroutine in this % program r_min(:,i)=2*sigma*Tsat(i)/rho_f/hfg./deltaT; [h(:,i),HF(:,i)]=DWC_HTC(Tsat(i),Tsat(i)-deltaT,r_max_th(i)); end %********************************************************************* % Now look at r-max dependance %********************************************************************* % r_max in [m], Tsat & deltaT in [K] r_max=[1e-8,5e-8,1e-7,5e-7,1e-6,5e-6,1e-5,5e-5,1e-4,5e-4,1e-3,5e-3,1e2]; Tsat_r=[280,320,380,440,500]; deltaTr=[0.1,0.5,1,5,10,20,50]; for i=1:length(Tsat_r) [hr(:,:,i),HFr(:,:,i)]=DWC_HTC(Tsat_r(i),Tsat_r(i)-deltaTr,r_max); end % h vs. Tsat for 5 values of deltaT figure(1) semilogy(Tsat,h(1,:)/10^6,'b',Tsat,h(10,:)/10^6,'r',Tsat,h(19,:)/10^6,' g',Tsat,h(length(deltaT),:)/10^6,'k'),grid title('DWC HTC vs. Tsat for \DeltaT = 0.1, 1.0, 10, 50K') xlabel('Saturation Temperature, [K]') ylabel('HTC, [MW/m^2K]') % h vs. deltaT for 5 values of Tsat figure(2) loglog(deltaT,h(:,1)/10^6,'b',deltaT,h(:,5)/10^6,'r',deltaT,h(:,11)/10^ 6,'g',deltaT,h(:,16)/10^6,'m',deltaT,h(:,length(Tsat))/10^6,'k'),grid title('DWC HTC vs. \DeltaT_s_u_b for Tsat = 280, 320, 380, 430, 500K') xlabel('\DeltaT_s_u_b, [K]')

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148 ylabel('HTC, [MW/m^2K]') % HF vs. Tsat for 5 values of deltaT figure(3) semilogy(Tsat,HF(1,:)/10^6,'b',Tsat,HF(10,:)/10^6,'r',Tsat,HF(19,:)/10^ 6,'c',Tsat,HF(length(deltaT),:)/10^6,'k'),grid title('DWC Heat Flux vs. Tsat for \DeltaT = 0.1, 1.0, 10, 20, 50K') xlabel('Saturation Temperature, [K]') ylabel('HF, [MW/m^2]') % HF vs. deltaT for 5 values of Tsat figure(4) loglog(deltaT,HF(:,1)/10^6,'b',deltaT,HF(:,5)/10^6,'r',deltaT,HF(:,11)/ 10^6,'c',deltaT,HF(:,16)/10^6,'m',deltaT,HF(:,length(Tsat))/10^6,'k'),g rid title('DWC Heat Flux vs. \DeltaT_s_u_b for Tsat = 280, 320, 380, 430, 500K') xlabel('\DeltaT_s_u_b, [K]') ylabel('HF, [MW/m^2]') % h vs. r_max all deltaT, representative Tsat = 380K, Psat ~1atm figure(5) loglog(r_max*1000,hr(:,:,3)/10^6),grid title('DWC HTC vs. r_m_a_x for \DeltaT = 0.1, 0.5, 1.0, 5.0, 10, 20, 50K at Tsat = 380K') xlabel('Max Droplet Radius, [mm]') ylabel('HTC, [MW/m^2K]') n=length(r_max); m=length(deltaTr); o=length(Tsat_r); % Loop to reassign hr variable for easier plotting for s=1:length(Tsat_r) for t=1:length(r_max) hr_DT1(t,s)=hr(t,3,s); HFr_DT1(t,s)=HFr(t,3,s); end end % h vs. r_max all Tsat, representative deltaT=1 figure(6) loglog(r_max*1000,hr_DT1/10^6),grid title('DWC HTC vs. r_m_a_x for Tsat = 280, 320, 380, 440, 500K at \DeltaT = 1.0K') xlabel('Max Droplet Radius, [mm]') ylabel('HTC, [MW/m^2K]') % Compare range of deltaT and Tsat effects with high and low deltaTs for Hi % mid and low Tsat values figure(7)

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149 loglog(r_max*1000,hr(:,1,1)/10^6,'b',r_max*1000,hr(:,4,1)/10^6,'b',r_ma x*1000,hr(:,1,3)/10^6,'r',r_max*1000,hr(:,m,3)/10^6,'r',r_max*1000,hr(: ,1,o)/10^6,'k',r_max*1000,hr(:,m,o)/10^6,'k'),grid title('DWC HTC vs. r_m_a_x for \DeltaT = 0.1 & 5K at Tsat = 280 and \DeltaT = 0.1 & 50K for 380 & 500K') xlabel('Max Droplet Radius, [mm]') ylabel('HTC, [MW/m^2K]') % HF vs. r_max all deltaT, representative Tsat = 380K, Psat ~1atm figure(8) loglog(r_max*1000,HFr(:,:,3)/10^6),grid title('DWC Heat Flux vs. r_m_a_x for \DeltaT = 0.1, 0.5, 1.0, 5.0, 10, 20, 50K at Tsat = 380K') xlabel('Max Droplet Radius, [mm]') ylabel('HF, [MW/m^2]') % HF vs. r_max all Tsat, representative deltaT=1 figure(9) loglog(r_max*1000,HFr_DT1/10^6),grid title('DWC Heat Flux vs. r_m_a_x for Tsat = 280, 320, 380, 440, 500K at \DeltaT = 1.0K') xlabel('Max Droplet Radius, [mm]') ylabel('HF, [MW/m^2]') % Compare range of deltaT and Tsat effects with high and low deltaTs for Hi % mid and low Tsat values figure(10) loglog(r_max*1000,HFr(:,1,1)/10^6,'b',r_max*1000,HFr(:,4,1)/10^6,'b',r_ max*1000,HFr(:,1,3)/10^6,'r',r_max*1000,HFr(:,m,3)/10^6,'r',r_max*1000, HFr(:,1,o)/10^6,'k',r_max*1000,HFr(:,m,o)/10^6,'k'),grid title('DWC Heat Flux vs. r_m_a_x for \DeltaT = 0.1 & 5K at Tsat = 280 and \DeltaT = 0.1 & 50K for 380 & 500K') xlabel('Max Droplet Radius, [mm]') ylabel('HF, [MW/m^2]') figure(11) plot(Tsat,r_max_th*1000),grid title('Maximum theoretical droplet radius vs. Tsat') xlabel('saturation temperature, [K]') ylabel('[mm]') figure(12) semilogy(Tsat,r_min(1,:)*1e9,Tsat,r_min(10,:)*1e9,Tsat,r_min(19,:)*1e9, Tsat,r_min(length(deltaT),:)*1e9),grid title('Smallest viable droplet radius vs. Tsat, \DeltaT = 0.1, 1.0, 10, & 50K ') xlabel('saturation temperature, [K]') ylabel('[nm]') Get_Theoretical_THR_and_r_eff.m clear all % % Feb 21 through Mar 9 data streams out of current range, need to improve

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150 % % range of model before proceeding % Ts = [33.6 27.4 28.3 29.6 29.8 31.2 32.3 30.7 30.2 33.1 35.3 35.7 36.5 37.9 36.9 30.9 31.2 31.1 29.6 29.4 29.7 30.0 30.2 30.3 30.8 30.9 30.7 31.5 29.0 29.3 37.8 36.9 36.2 36.3 35.2 34.8 26.1 25.6 24.9 25.1 26.2 24.2 25.3 24.7 26.8 27.2 26.4 26.1 26.2 24.7 25.2 ]; % % sigma_Ts=[0.22 0.44 0.33 0.33 0.33 0.44 0.78 0.28 0.17 0.22 0.22 0.28 0.28 0.28 0.04 0.07 0.02 0.78 0.11 0.28 0.39 0.39 0.33 0.33 0.22 1.83 0.02 0.02 0.03 0.17 0.39 0.22 0.08 0.28 0.17 0.11 0.22 0.11 0.11 0.17 0.39 0.17 0.28 0.17 0.2 0.67 0.28 0.39 0.39 0.22 0.17 ]; % % Te = [16.1 15.2 15.9 14.7 12.7 11.3 11.1 18.9 12.0 12.3 11.6 11.2 10.3 9.9 14.3 12.8 12.8 8.9 14.5 9.6 8.7 8.3 8.7 8.7 8.7 8.7 13.6 7.8 8.2 7.5 10.8 9.7 8.3 6.1 14.8 14.6 13.2 13.3 11.9 12.2 13.8 13.6 8.3 13.8 13.9 13.8 13.0 12.4 12.4 12.2 12.8 ]; % % sigma_Te=[0.17 0.06 0.11 0.17 0.11 0.17 0.22 0.11 0.07 0.50 0.11 0.11 0.11 0.08 0.09 0.09 0.17 0.09 0.17 0.28 0.28 0.22 0.22 0.17 0.17 0.17 0.03 0.11 0.17 0.02 0.44 0.39 0.11 0.11 0.11 0.17 0.08 0.11 0.07 0.11 0.22 0.11 0.11 0.11 0.17 0.07 0.07 0.09 0.09 0.08 0.11 ]; % % q_exp=[12.2 25.6 24.3 16.6 13.3 12.7 8.7 23.9 22.7 20 19.3 15.9 14.5 13.2 11.9 9.3 9.1 2.9 9.6 9.9 8.2 7 8.5 8.6 8.6 8.7 9.6 6.9 7.3 6.9 13.5 12.8 12.2 8.9 25 26 30 29 20 21.9 30 26 36 33 36 33.3 31 29.1 29.7 26.1 27.5 ]; % % sigma_q=[0.6 0.8 1.6 0.9 0.9 1.1 1.7 0.3 1.9 1.5 1.5 1.8 1.5 1.2 0.5 0.4 0.3 1.6 0.7 1.6 1.4 1.6 1.5 1.5 1.4 1.3 0.3 0.6 0.6 0.6 1.9 1.8 1.5 1.4 2 1.9 2 2 1.8 1.7 3 2 2 2 2 1.7 3 1.6 1.8 1.6 1.6 ]; for i=13:length(Ts) [q_th,r_max,sigma,r_eff,sigma_r]=Heat_Rate_Archive(Ts(i),sigma_Ts(i),Te (i),sigma_Te(i),q_exp(i),sigma_q(i)); q_theory(i)=q_th(1); sigma_q_th(i)=sigma(1); r_effect(i)=r_eff; sigma_r_effect(i)=sigma_r; end q_theory=q_theory.' sigma_q_theory=sigma_q_th.' r_effective=r_effect.' sigma_r_effective=sigma_r_effect.' Heat_Rate_Archive.m function [q_th,r_max,sigma,r_eff,sigma_r]=Heat_Rate_Archive(Ts,sigma_Ts,Te,sigma _Te,q_exp,sigma_q) % [q_th,r_max,sigma,r_eff,sigma_r]=Heat_Rate_Archive(Ts,sigma_Ts,Te,sigma _T

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151 % e,q_exp,sigma_q) % This function uses arrays of theoretical total heat rate values % calculated in another program and archived here, for 5 different values % of maximum drop radius in dropwise condensation mode of pure water % vapor on one face of a vertically oriented, circular brass shim % 0.75 in in diameter and 0.004 in thick, theory used for calculating the % heat flux given by Le Fevre and Rose, 1966 % % given: % % Ts = measured saturation temperature, [C] between 22 and 40 % % sigma_Ts = uncertainty in Ts, [C] % % Te = measured edge temperature, [C] between 10 and 18 % % sigma_Te = uncertainty in Te, [C] % % return: % % q_th [W]= 8 values for the theoretical heat corresponding to the % respective values of r_max, theoretical, 0.01, 0.1, % 0.25, 0.5, 1, 2, & 4 mm % % r_max [mm] = maximum droplet radii corresponding to the values % in q_th % % sigma [W] = uncertainty in theoretical total heat rate due to % argument uncertainty % % r_eff [mm] = estimation of effective maximum droplet diameter % that would yield the experimental total heat rate % under the given conditions % %********************************************************************** % % Test inputs % Ts=33.4; % sigma_Ts=.2; % Te=16.7; % sigma_Te=.1; % q_exp=22.35; % sigma_q=1.5; Tsat=24:2:50; To=4:4:20; A=length(Tsat); B=length(To); % Determine bin number (which arguments it is between) of saturation temperature input dummy=0;

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152 k=0; while dummy==0 k=k+1; if Ts >= Tsat(k) & Ts < Tsat(k+1) Y=k; break end end % Determine bin number of edge temperature input dummy=0; k=0; while dummy==0 k=k+1; if Te >= To(k) & Te < To(k+1) X=k; break end end % 0 = theoretical maximum droplet radius, rest in mm r_max=[0 .01 .1 .25 .5 1 2 4]; C=length(r_max); % Initialize Q array, % (theoretical Heat Rate values specific to this experiment) Q=zeros(A,B,C); % Q for theoretical r_max, which should vastly underestimate HT for this exp. Q(:,:,1)=[ 40.19 31.91 23.67 15.51 7.48 45.86 37.27 28.72 20.23 11.84 51.78 42.89 34.03 25.23 16.51 57.97 48.77 39.61 30.51 21.47 64.71 55.17 45.67 36.22 26.83 71.40 61.57 51.78 42.04 32.35 78.67 68.51 58.40 48.32 38.30 86.16 75.69 65.26 54.86 44.51 93.93 83.15 72.40 61.68 51.02 102.34 91.21 80.10 69.03 58.01 110.63 99.18 87.77 76.40 65.07 119.60 107.81 96.06 84.34 72.66 128.78 116.66 104.57 92.52 80.50 138.23 125.78 113.36 100.98 88.63 ];

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153 % Q for r_max of 0.01mm which should overestimate HT for this exp. Q(:,:,2)=[ 84.58 67.09 49.72 32.52 15.62 96.73 78.55 60.47 42.54 24.84 109.42 90.56 71.80 53.17 34.74 122.69 103.17 83.74 64.43 45.28 137.17 116.90 96.72 76.64 56.71 151.52 130.62 109.80 89.08 68.49 167.14 145.52 123.98 102.53 81.20 183.25 160.94 138.70 116.54 94.51 199.95 176.95 154.02 131.17 108.43 218.05 194.27 170.57 146.95 123.43 235.83 211.39 187.03 162.74 138.55 255.11 229.92 204.81 179.77 154.82 274.83 248.92 223.08 197.32 171.64 295.13 268.50 241.94 215.46 189.07 ]; % Q for r_max of 0.1mm which should overestimate HT for this exp. Q(:,:,3)=[ 59.89 47.54 35.26 23.10 11.14 68.32 55.51 42.77 30.12 17.63 77.12 63.86 50.67 37.56 24.57 86.31 72.61 58.97 45.41 31.95 96.31 82.11 67.97 53.89 39.91 106.23 91.61 77.04 62.53 48.12 117.00 101.90 86.84 71.85 56.94 128.11 112.54 97.02 81.55 66.17 139.61 123.58 107.59 91.66 75.81 152.06 135.50 119.00 102.55 86.17 164.29 147.30 130.35 113.45 96.62 177.54 160.04 142.59 125.18 107.84 191.10 173.11 155.16 137.27 119.44 205.03 186.57 168.14 149.77 131.45 ]; % Q for r_max of 0.25mm which should overestimate HT for this exp. Q(:,:,4)=[ 51.51 40.89 30.34 19.88 9.59 58.75 47.74 36.79 25.91 15.17 66.31 54.91 43.57 32.30 21.14 74.21 62.43 50.70 39.04 27.48 82.79 70.59 58.43 46.33 34.32 91.31 78.74 66.22 53.76 41.37 100.56 87.58 74.65 61.76 48.95 110.10 96.72 83.38 70.10 56.87 119.98 106.20 92.47 78.78 65.16 130.66 116.44 102.26 88.13 74.05 141.17 126.57 112.01 97.49 83.03 152.55 137.52 122.52 107.57 92.67 164.19 148.74 133.32 117.95 102.63 176.16 160.30 144.47 128.68 112.95 ]; % Q for r_max of 0.5mm Q(:,:,5)=[ 45.86 36.41 27.01 17.70 8.54 52.30 42.50 32.75 23.07 13.51 59.03 48.89 38.79 28.76 18.82 66.06 55.58 45.14 34.76 24.46 73.71 62.84 52.02 41.25 30.56 81.29 70.10 58.96 47.86 36.83 89.53 77.97 66.46 54.99 43.58

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154 98.02 86.11 74.24 62.41 50.64 106.82 94.55 82.32 70.14 58.01 116.33 103.67 91.05 78.47 65.94 125.69 112.69 99.73 86.80 73.93 135.83 122.44 109.09 95.78 82.52 146.19 132.44 118.71 105.03 91.39 156.86 142.73 128.64 114.58 100.57 ]; % Q for r_max of 1mm Q(:,:,6)=[ 40.76 32.36 24.01 15.73 7.59 46.50 37.79 29.12 20.51 12.01 52.48 43.46 34.49 25.57 16.73 58.74 49.42 40.14 30.91 21.75 65.54 55.88 46.26 36.68 27.17 72.29 62.34 52.43 42.56 32.75 79.62 69.34 59.10 48.90 38.76 87.18 76.58 66.02 55.50 45.03 95.00 84.10 73.22 62.39 51.60 103.47 92.21 80.99 69.79 58.65 111.81 100.24 88.71 77.22 65.76 120.83 108.92 97.04 85.20 73.41 130.06 117.82 105.61 93.43 81.30 139.55 126.99 114.45 101.94 89.48 ]; % Q for r_max of 2mm Q(:,:,7)=[ 36.20 28.74 21.32 13.97 6.74 41.29 33.56 25.86 18.22 10.67 46.61 38.60 30.63 22.71 14.86 52.17 43.89 35.65 27.46 19.32 58.22 49.64 41.09 32.59 24.14 64.22 55.38 46.58 37.81 29.10 70.74 61.61 52.51 43.45 34.44 77.46 68.05 58.67 49.32 40.02 84.43 74.74 65.07 55.44 45.85 91.97 81.96 71.98 62.03 52.13 99.38 89.10 78.85 68.63 58.45 107.41 96.82 86.27 75.74 65.25 115.62 104.74 93.89 83.06 72.28 124.07 112.90 101.75 90.63 79.55 ]; % Q for r_max of 0.1mm which should overestimate HT for this exp. Q(:,:,8)=[ 32.11 25.49 18.92 12.40 5.98 36.64 29.77 22.94 16.16 9.46 41.36 34.26 27.18 20.15 13.19 46.30 38.96 31.64 24.37 17.15 51.68 44.06 36.47 28.92 21.43 57.01 49.17 41.35 33.57 25.83 62.81 54.70 46.62 38.58 30.58 68.78 60.43 52.09 43.79 35.53 74.98 66.37 57.79 49.23 40.72 81.68 72.79 63.93 55.09 46.30 88.27 79.14 70.04 60.96 51.92 95.41 86.01 76.63 67.28 57.97 102.72 93.05 83.41 73.80 64.21 110.24 100.31 90.41 80.53 70.68 ]; % Interpolation between the indexes determined for Ts and Te

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155 % that is, determine five values of theoretical total heat rate, for % one for each value of r_max for p=1:C q1 = (Ts Tsat(Y)) / (Tsat(Y+1) Tsat(Y)) ( Q(Y+1,X,p) Q(Y,X,p) ) + Q(Y,X,p) ; q2 = (Ts Tsat(Y)) / (Tsat(Y+1) Tsat(Y)) (Q(Y+1,X+1,p) Q(Y,X+1,p)) + Q(Y,X+1,p); q_th(p) = (Te To(X)) / ( To(X+1) To(X) ) ( q2 q1 ) + q1 ; end % Now determine range of q from uncertainty, assumes within same index % range as nominal value, an relatively inconsequential assumption, given % the linearity of the Q array for p=1:C q1 = (Ts + sigma_Ts Tsat(Y)) / (Tsat(Y+1) Tsat(Y)) ( Q(Y+1,X,p) Q(Y,X,p) ) + Q(Y,X,p) ; q2 = (Ts + sigma_Ts Tsat(Y)) / (Tsat(Y+1) Tsat(Y)) (Q(Y+1,X+1,p) Q(Y,X+1,p)) + Q(Y,X+1,p); q_th_hi(p) = (Te sigma_Te To(X)) / ( To(X+1) To(X) ) ( q2 q1 ) + q1 ; end for p=1:C q1 = (Ts sigma_Ts Tsat(Y)) / (Tsat(Y+1) Tsat(Y)) ( Q(Y+1,X,p) Q(Y,X,p) ) + Q(Y,X,p) ; q2 = (Ts sigma_Ts Tsat(Y)) / (Tsat(Y+1) Tsat(Y)) (Q(Y+1,X+1,p) Q(Y,X+1,p)) + Q(Y,X+1,p); q_th_lo(p) = (Te + sigma_Te To(X)) / ( To(X+1) To(X) ) ( q2 q1 ) + q1 ; end unc_lo=q_th-q_th_lo; unc_hi=q_th_hi-q_th; % determine higher uncertainty in q_th and assign to sigma for p=1:C if unc_lo(p) > unc_hi(p) sigma(p)=unc_lo(p); else sigma(p)=unc_hi(p); end

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156 end % now determine the effective maximum droplet radius from exp. q % use a Lagrange Interpoltaion Polynomial, cited at wolfram.com by Eric % Weisstein % exclude theoretical r_max from interpolation formula because of its % varying value with saturation temperature % attempt at least squares fit of q_th(r)=C/r^N+B % Drop theoretical max drop radius values from consideration for finding % r_eff r=r_max(2:length(r_max)); qi=q_th(2:length(q_th)); b=(-100:10); c=(0:100); n=(1:100)*1e-2; dummy=0; counter=1 while counter < 5 Ri=ones(length(b),length(c),length(n)); i=[]; j=[]; k=[]; for i=1:length(b) for j=1:length(c) for k=1:length(n) Ri(i,j,k)=sum( ( c(j)./r.^n(k) + b(i) qi ).^2 ); end end end [JKres,Iint]=min(Ri); [Kres,Jint]=min(JKres); [res,K]=min(Kres) J=Jint(:,:,K); I=Iint(:,J,K); B=b(I); C=c(J); N=n(K);

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157 if Ri(I,J,K) < .001*q_exp dummy=1; break end f=10^counter db=1/f; dc=1/f; dn=1/f; b=(B-20*db):db:(B+20*db); c=(C-20*db):dc:(C+20*db); n=(N-20*db):dn:(N+20*db); counter=counter+1 end % establish data fit to theoretical total heat rate for given input % temperature data x=min(r)/10:min(r)/10:max(r)*10; fit=C./x.^N+B; % figure(1) % semilogx(x,fit,r,qi,'o'),grid % now determine r_effective from q_exp and q_th values % avoid index finding of r_eff if q_exp out of range of fit if q_exp + sigma_q > max(fit) r_eff=0; sigma_r = 0; return elseif q_exp sigma_q < min(fit) r_eff=0; sigma_r = 0; return end % determine index in fit the corresponds to r_eff for i=1:length(x)-1 if q_exp < fit(i) & q_exp >= fit(i+1) r_eff = x(i); break end

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158 end % Now determine uncertainty in r_eff % lower limit for r_eff for i=1:length(x)-1 if q_exp+sigma_q < fit(i) & q_exp+sigma_q >= fit(i+1) r_eff_lo = x(i); break end end % upper limit for r_eff for i=1:length(x)-1 if q_exp sigma_q < fit(i) & q_exp sigma_q >= fit(i+1) r_eff_hi = x(i); break end end % test to see which error is higher and use that as the uncertainty if r_eff_hi r_eff > r_eff_lo r_eff sigma_r = r_eff_hi r_eff; else sigma_r = r_eff_lo r_eff end % % Test outputs % q_theoretiacl=q_th % uncertainty_in_q_theoretical=sigma % r_max_values=r_max % effective_max_drop_radius=r_eff % uncertainty_in_r_eff = sigma_r % % figure(1) % semilogx(r_max,q_th,'o',x,fit),grid % title('Theoretical Total Heat Rate vs. Maximum Drop Radius') % xlabel('[mm]') % ylabel('[W]')

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159 PROPS.m function [rho_f,rho_g,hfg,k_f,k_g,sigma,Psat,mu_f,mu_g,Cp_f,Cp_g,Beta,alpha_f,al pha_g,nu_f,nu_g]=PROPS(Tsat) % [rho_f,rho_g,hfg,k_f,k_g,sigma,Psat,mu_f,mu_g,Cp_f,Cp_g,Beta,alpha_ % f,alpha_g,nu_f,nu_g]=PROPS(Tsat) % % Given Tsat in degrees Kelvin and pure water vapor, return: % % rho_f = density of liquid at Ts, [kg/m3] % rho_g = density of vapor at Tsat, [kg/m3] % hfg = enthalpy of vaporization, [J/kg] % k_f = thermal conductivity, [W/m-K] % k_g = thermal conductivity, [W/m-K] % sigma = surface tension, [kg/s2]=[N/m] % Psat = saturation pressure, [bars] % mu_f = dynamic viscosity, [N*s/m2]=[kg/m-s] % mu_g = dynamic viscosity, [N*s/m2] % Cp_f = Constant Pressure Specific Heat, [J/kg-K] % Cp_g = Constant Pressure Specific Heat, [J/kg-K] % Beta = Coefficient of Thermal Expansion, [1/K] % alpha_f = Thermal Diffusivity, [m2/s] % alpha_g = Thermal Diffusivity, [m2/s] % nu_f = Kinematic Viscosity, [m2/s] % nu_g = Kinematic Viscosity, [m2/s] % % Program valid for Tsat = (275,500] K % % Data taken from Incropera and DeWitt, 1996, p.846 % Tsat, [K] T = [ 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 400 410 420 430 440 450 460 470 480 490 500 ]; % specific volume of saturated liquid, [m3/g] v_f = [ 1.000 1.000 1.000 1.001 1.002 1.003 1.005 1.007 1.009 1.011 1.013 1.016 1.018 1.021 1.024 1.027 1.030 1.034 1.038 1.041 1.045 1.049 1.053 1.058 1.067 1.077 1.088 1.099 1.110 1.123 1.137 1.152 1.167 1.184 1.203 ]*1e-3; % specific volume of saturated vapor, [m3/g] v_g = [ 181.7 130.4 99.4 69.7 51.94 39.13 29.74 22.93 17.82 13.98 11.06 8.82 7.09 5.74 4.683 3.846 3.180 2.645 2.212 1.861 1.574 1.337 1.142 0.980 0.731 0.553 0.425 0.331 0.261 0.208 0.167 0.136 0.111 .0922 .0766 ]; % enthalpy of vaporization, [J/g] hfg_d = [ 2497 2485 2473 2461 2449 2438 2426 2414 2402 2390 2378 2366 2354 2342 2329 2317 2304 2291 2278 2265 2252 2239 2225 2212 2183

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160 2153 2123 2091 2059 2024 1989 1951 1912 1870 1825 ]*1e3; % thermal conductivity of saturated liquid, [W/m-K] k_fd = [ 574 582 590 598 606 613 620 628 634 640 645 650 656 660 668 668 671 674 677 679 681 683 685 686 688 688 688 685 682 678 673 667 660 651 642 ]*1e-3; % thermal conductivity of saturated vapor, [W/m-K] k_gd = [ 18.3 18.6 18.9 19.3 19.5 19.6 20.1 20.4 20.7 21.0 21.3 21.7 22.0 22.3 22.6 23.0 23.3 23.7 24.1 24.5 24.9 25.4 25.8 26.3 27.2 28.2 29.8 30.4 31.7 33.1 34.6 36.3 38.1 40.1 42.3 ]*1e-3; % surface tension of saturated liquid, [kg/s2] = [N/m] sigma_d = [ 75.3 74.8 74.3 73.7 72.7 71.7 70.9 70.0 69.2 68.3 67.5 66.6 65.8 64.9 64.1 63.2 62.3 61.4 60.5 59.5 58.6 57.6 56.6 55.6 53.6 51.5 49.4 47.2 45.1 42.9 40.7 38.5 36.2 33.9 31.6 ]*1e-3; % saturation pressure, [bar] Psat_d = [ .00697, .00990, .01387, .01917, .02617, .03531, .04712, .06221, .08132, .1053 .1351 .1719 .2167 .2713 .3372 .4163 .5100 .6209 .7514 .9040 1.0815, 1.2869, 1.5233, 1.794 2.455 3.302 4.370 5.699 7.333 9.319 11.71 14.55 17.90 21.83 26.40 ]; % dynamic viscosity of saturated liquid, [N*s/m2] mu_fd = [ 1652 1422 1225 1080 959 855 769 695 631 577 528 489 453 420 389 365 343 324 306 289 274 260 248 237 217 200 185 173 162 152 143 136 129 124 118 ]*1e-6; % dynamic viscosity of saturated vapor, [N*s/m2] mu_gd = [ 8.09 8.29 8.49 8.69 8.89 9.09 9.29 9.49 9.69 9.89 10.09 10.29 10.49 10.69 10.89 11.09 11.29 11.49 11.69 11.89 12.09 12.29 12.49 12.69 13.05 13.42 13.79 14.14 14.50 14.85 15.19 15.54 15.88 16.23 16.59 ]*1e-6; % Coefficient of Thermal Expansion, [1/K] beta_f = [-32.74 46.04 114.1 174.0 227.5 276.1 320.6 361.9 400.4 436.7 471.2 504.2 535.5 566.0 595.4 624.2 652.3 697.9 707.1 728.7 761 788 814 841 896 952 1010 0 0 0 0 0 0 0 0 ]*1e-6; % Constant Pressure Specific Heat of Saturated Liquid, [J/kg-K] Cp_fd = [ 4.211 4.198 4.189 4.184 4.181 4.179 4.178 4.178 4.179 4.180 4.182 4.184 4.186 4.188 4.191 4.195 4.199 4.203 4.209 4.214 4.220 4.226 4.232 4.239 4.256

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161 4.278 4.302 4.331 4.36 4.40 4.44 4.48 4.53 4.59 4.66 ]*1e3; % Constant Pressure Specific Heat of Saturated Vapor, [J/kg-K] Cp_gd = [ 1.855 1.858 1.861 1.864 1.868 1.872 1.877 1.882 1.888 1.895 1.903 1.911 1.920 1.930 1.941 1.954 1.968 1.983 1.999 2.017 2.036 2.057 2.080 2.104 2.158 2.221 2.291 2.369 2.46 2.56 2.68 2.79 2.94 3.10 3.27 ]*1e3; % establish length of data record for reference index=length(T); % test Tsat input to see if in celius instead of kelvin if Tsat < 230 Tsat = Tsat + 273.15; % Convert Tsat to Kelvin temp scale elseif Tsat <= 275 print('error = temperature input too low to PROPS subroutine, must be between 275 and 400 K') return elseif Tsat > 500 print('error = temparature input too high to PROPS subroutine, must be between 275 and 400K') return end for i=1:index-1 if Tsat > T(i) & Tsat <= T(i+1) j=i; % sets the j index value to the LHS side of the interval in the data table break end end % use linear interpolation to determine the property value % more accurate method possible, but not warranted given the low accuracy of % the experimental results, unwarranted --what are the error estimates of % each method? i.e. how do you know? I don't know, I'm lazy and can't find an example of log interpolation. rho_f = ( (Tsat-T(j))/(T(j+1)-T(j)) (v_f(j+1)-v_f(j)) + v_f(j) )^(1);

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162 rho_g = ( (Tsat-T(j))/(T(j+1)-T(j)) (v_g(j+1)-v_g(j)) + v_g(j) )^(1); hfg = (Tsat-T(j))/(T(j+1)-T(j)) (hfg_d(j+1)-hfg_d(j)) + hfg_d(j); k_f = (Tsat-T(j))/(T(j+1)-T(j)) (k_fd(j+1)-k_fd(j)) + k_fd(j); k_g = (Tsat-T(j))/(T(j+1)-T(j)) (k_gd(j+1)-k_gd(j)) + k_gd(j); sigma = (Tsat-T(j))/(T(j+1)-T(j)) (sigma_d(j+1)-sigma_d(j)) + sigma_d(j); Psat = (Tsat-T(j))/(T(j+1)-T(j)) (Psat_d(j+1)-Psat_d(j)) + Psat_d(j); mu_f = (Tsat-T(j))/(T(j+1)-T(j)) (mu_fd(j+1)-mu_fd(j)) + mu_fd(j); mu_g = (Tsat-T(j))/(T(j+1)-T(j)) (mu_gd(j+1)-mu_gd(j)) + mu_gd(j); Cp_f = (Tsat-T(j))/(T(j+1)-T(j)) (Cp_fd(j+1)-Cp_fd(j)) + Cp_fd(j); Cp_g = (Tsat-T(j))/(T(j+1)-T(j)) (Cp_gd(j+1)-Cp_gd(j)) + Cp_gd(j); if Tsat <= 420 Beta= (Tsat-T(j))/(T(j+1)-T(j)) (beta_f(j+1)-beta_f(j)) + beta_f(j); else Beta= 0; end % Derived Properties % Thermal Diffusivity, [m2/s] alpha_f = k_f/rho_f/Cp_f; alpha_g = k_g/rho_g/Cp_g; % Kinematic Viscosity, [m2/s] nu_f = mu_f/rho_f; nu_g = mu_g/rho_g; exp_profile.m function [T]=exp_profile(Tmax,To,b,c,q,m)

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163 %********************************************************************** % [T]=exp_profile(Tmax,To,b,c,q,m) % where, % Tmax = max surface temp of diaphragm, [deg C] % Td = edge temp of diaphragm, [deg C] % b = exposed shim radius, [m] % c = shim thickness, [m] % q = total heat rate, [W] % m = length of r-vector in main program, so that T-vector matches % % all inputs are scalars % %********************************************************************** k=140; % Thermal Conductivity of brass shim, [W/m*K] % Tmax=40; % estimated vapor temp % To=15.6; % Measured temperature of shim edge, [degrees C] % b=0.755/2*25.4/1000; % Shim radius, [m] % c=0.004*25.4/1000; % Shim thickness, [m] % q=20; % Total Heat Rate, [W] % m=100; % length of r-vector, [] qdp=q./pi/b/c/2; % Average Heat Flux through shim at edge, [W/m^2] C=To-Tmax; Q=-qdp/k; n=Q*b/C; dr=b/m; r=0:dr:b; T=C.*(r/b).^n+Tmax; % figure(1) % plot(r,T),grid % title('Assumed exponential profile, (Td-Tatm)*(r/b)^n+Tatm') % xlabel('radius, [m]') % ylabel('Temp, [deg C]') simpsons_rule.m function [ans,err_order]=simpsons_rule(integrand,dx) % [ans,err_order]=simpsons_rule(integrand,dx) % % integrand = a vector of argument values for integral to be evaluated % dx = a scalar indicating the step size in the x-vector of f(x) % ans = a scalar indicating the approximate value of the integral % err_order = a scalar indicating the order of magnitude of the error % %*********************************************************************

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164 n=(length(integrand)-1)/2; for i=1:n A(i)=integrand(2*i); end for j=1:(n-1) B(j)=integrand(2*j-1); end ans=dx/3*(integrand(1)+4*sum(A)+2*sum(B)+integrand(2*n+1)); err_order=n*dx^5/90;

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165 APPENDIX B TOTAL HEAT RATE TABLES The theoretical analysis of this investigation consisted of the following: 1. An assumed radial temperature profile for the circular brass shim used as the condensing surface 2. A theoretical estimation of the radial HF distribution, based on the Rose theory (Eq. 2-10) 3. Surface integration of the estimated radial HF distribution The third step gave an estimate of the tota l overall HT rate, or THR, for the given experimental setup and conditi ons. Calculation of this qua ntity was for a given set of experimental conditions was time consuming. However, the function was very close to linear in the two main control parameters edge temperature (To) and saturation temperature (Tsat). This linearity was used to dramatically cut down on processor demands by constructing tables of references values. Linear interpolation between the table values was then used to estimate the THR values from actual experimental condtions (Table B-1 to 8). Chapter 7 discussed the uncertainty added from this estimation process, which was generally the same order of magnitude as that of the measured value. These tables were included in the MATLAB program H eat_Rate_Archive.m (Appendix A).

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166 Table B-1. Theoretical THR with Theore tical Maximum Departing Droplet Size. rmax = To, F Theory (Eq. 2-11) 4 8 12 16 20 24 40.19 31.91 23.67 15.51 7.48 26 45.86 37.27 28.72 20.23 11.84 28 51.78 42.89 34.03 25.23 16.51 30 57.97 48.77 39.61 30.51 21.47 32 64.71 55.17 45.67 36.22 26.83 34 71.40 61.57 51.78 42.04 32.35 36 78.67 68.51 58.40 48.32 38.30 38 86.16 75.69 65.26 54.86 44.51 40 93.93 83.15 72.40 61.68 51.02 42 102.34 91.21 80.10 69.03 58.01 44 110.63 99.18 87.77 76.40 65.07 46 119.60 107.81 96.06 84.34 72.66 48 128.78 116.66 104.57 92.52 80.50 Saturation Temperature, Tsat, F 50 138.23 125.78 113.36 100.98 88.63 Table B-2. Theoretical THR with 4 mm Maximum Departing Droplet Size. rmax = To, F 4 mm 4 8 12 16 20 24 32.11 25.49 18.92 12.40 5.98 26 36.64 29.77 22.94 16.16 9.46 28 41.36 34.26 27.18 20.15 13.19 30 46.30 38.96 31.64 24.37 17.15 32 51.68 44.06 36.47 28.92 21.43 34 57.01 49.17 41.35 33.57 25.83 36 62.81 54.70 46.62 38.58 30.58 38 68.78 60.43 52.09 43.79 35.53 40 74.98 66.37 57.79 49.23 40.72 42 81.68 72.79 63.93 55.09 46.30 44 88.27 79.14 70.04 60.96 51.92 46 95.41 86.01 76.63 67.28 57.97 48 102.72 93.05 83.41 73.80 64.21 Saturation Temperature, Tsat, F 50 110.24 100.31 90.41 80.53 70.68

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167 Table B-3. Theoretical THR with 2 mm Maximum Departing Droplet Size. rmax = To, F 2 mm 4 8 12 16 20 24 36.20 28.74 21.32 13.97 6.74 26 41.29 33.56 25.86 18.22 10.67 28 46.61 38.60 30.63 22.71 14.86 30 52.17 43.89 35.65 27.46 19.32 32 58.22 49.64 41.09 32.59 24.14 34 64.22 55.38 46.58 37.81 29.10 36 70.74 61.61 52.51 43.45 34.44 38 77.46 68.05 58.67 49.32 40.02 40 84.43 74.74 65.07 55.44 45.85 42 91.97 81.96 71.98 62.03 52.13 44 99.38 89.10 78.85 68.63 58.45 46 107.41 96.82 86.27 75.74 65.25 48 115.62 104.74 93.89 83.06 72.28 Saturation Temperature, Tsat, F 50 124.07 112.90 101.75 90.63 79.55 Table B-4. Theoretical THR with 1 mm Maximum Departing Droplet Size. rmax = To, F 1 mm 4 8 12 16 20 24 40.76 32.36 24.01 15.73 7.59 26 46.50 37.79 29.12 20.51 12.01 28 52.48 43.46 34.49 25.57 16.73 30 58.74 49.42 40.14 30.91 21.75 32 65.54 55.88 46.26 36.68 27.17 34 72.29 62.34 52.43 42.56 32.75 36 79.62 69.34 59.10 48.90 38.76 38 87.18 76.58 66.02 55.50 45.03 40 95.00 84.10 73.22 62.39 51.60 42 103.47 92.21 80.99 69.79 58.65 44 111.81 100.24 88.71 77.22 65.76 46 120.83 108.92 97.04 85.20 73.41 48 130.06 117.82 105.61 93.43 81.30 Saturation Temperature, Tsat, F 50 139.55 126.99 114.45 101.94 89.48

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168 Table B-5. Theoretical TH R with 0.5 mm Maximum Departing Droplet Size. rmax = To, F 0.5 mm 4 8 12 16 20 24 45.86 36.41 27.01 17.70 8.54 26 52.30 42.50 32.75 23.07 13.51 28 59.03 48.89 38.79 28.76 18.82 30 66.06 55.58 45.14 34.76 24.46 32 73.71 62.84 52.02 41.25 30.56 34 81.29 70.10 58.96 47.86 36.83 36 89.53 77.97 66.46 54.99 43.58 38 98.02 86.11 74.24 62.41 50.64 40 106.82 94.55 82.32 70.14 58.01 42 116.33 103.67 91.05 78.47 65.94 44 125.69 112.69 99.73 86.80 73.93 46 135.83 122.44 109.09 95.78 82.52 48 146.19 132.44 118.71 105.03 91.39 Saturation Temperature, Tsat, F 50 156.86 142.73 128.64 114.58 100.57 Table B-6. Theoretical TH R with 0.25 mm Maximum Departing Droplet Size. rmax = To, F 0.25 mm 4 8 12 16 20 24 51.51 40.89 30.34 19.88 9.59 26 58.75 47.74 36.79 25.91 15.17 28 66.31 54.91 43.57 32.30 21.14 30 74.21 62.43 50.70 39.04 27.48 32 82.79 70.59 58.43 46.33 34.32 34 91.31 78.74 66.22 53.76 41.37 36 100.56 87.58 74.65 61.76 48.95 38 110.10 96.72 83.38 70.10 56.87 40 119.98 106.20 92.47 78.78 65.16 42 130.66 116.44 102.26 88.13 74.05 44 141.17 126.57 112.01 97.49 83.03 46 152.55 137.52 122.52 107.57 92.67 48 164.19 148.74 133.32 117.95 102.63 Saturation Temperature, Tsat, F 50 176.16 160.30 144.47 128.68 112.95

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169 Table B-7. Theoretical TH R with 0.1 mm Maximum Departing Droplet Size. rmax = To, F 0.1 mm 4 8 12 16 20 24 59.89 47.54 35.26 23.10 11.14 26 68.32 55.51 42.77 30.12 17.63 28 77.12 63.86 50.67 37.56 24.57 30 86.31 72.61 58.97 45.41 31.95 32 96.31 82.11 67.97 53.89 39.91 34 106.23 91.61 77.04 62.53 48.12 36 117.00 101.90 86.84 71.85 56.94 38 128.11 112.54 97.02 81.55 66.17 40 139.61 123.58 107.59 91.66 75.81 42 152.06 135.50 119.00 102.55 86.17 44 164.29 147.30 130.35 113.45 96.62 46 177.54 160.04 142.59 125.18 107.84 48 191.10 173.11 155.16 137.27 119.44 Saturation Temperature, Tsat, F 50 205.03 186.57 168.14 149.77 131.45 Table B-8. Theoretical TH R with 0.01 mm Maximum Departing Droplet Size. rmax = To, F 0.01 mm 4 8 12 16 20 24 84.58 67.09 49.72 32.52 15.62 26 96.73 78.55 60.47 42.54 24.84 28 109.42 90.56 71.80 53.17 34.74 30 122.69 103.17 83.74 64.43 45.28 32 137.17 116.90 96.72 76.64 56.71 34 151.52 130.62 109.80 89.08 68.49 36 167.14 145.52 123.98 102.53 81.20 38 183.25 160.94 138.70 116.54 94.51 40 199.95 176.95 154.02 131.17 108.43 42 218.05 194.27 170.57 146.95 123.43 44 235.83 211.39 187.03 162.74 138.55 46 255.11 229.92 204.81 179.77 154.82 48 274.83 248.92 223.08 197.32 171.64 Saturation Temperature, Tsat, F 50 295.13 268.50 241.94 215.46 189.07

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170 APPENDIX C DETAIL DRAWINGS The main, aluminum pressure vessel cylinder was originally produced for a different study. The rest of the pressure vessel components were manufactured in-house to mate with the available vessel body. Detail drawings for these components follow in this section. The Cooler did not have a detail drawing included in this appendix. This component centered around a hollo w copper cylinder, called the Cooler Barrel. A jacket was then made from two discs of copper and an outer tube. These components were then soldered together. Grooves were machined on the cylinder to ai d the soldering. Two holes of appropriate diameter were drilled in opposite ends of th e outer tube wall to accommodate the coolant tubes. The nylon coolant tubes were then attached to the Cooler with Copper Bond epoxy.

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171 171 Figure C-1. Bottom Cover detail drawing.

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172 Figure C-2. Cooler Ba rrel detail drawing.

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173 Figure C-3. Diaphragm Clamp detail drawing.

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174 Figure C-4. Diaphragm Frame detail drawing.

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175 Figure C-5. Optical Probe Guide detail drawing.

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176 Figure C-6. Optical Prob e Seal detail drawing.

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177 Figure C-7. Top Cover detail drawing.

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178 LIST OF REFERENCES [1] SIA, Semiconductor Industry Associa tion, International technology road map, Assembly and Packaging Section, Technology Requirements (2001) 4–8. [2] I. Mudawar, Assessment of high-heat-f lux thermal management schemes, IEEE Transactions on Components & Pa ckaging Technologies 24 (2001) 122–142. [3] M.M. Franois, W. Shyy, J.N. Chun g, A micro-scale membrane-actuated condenser/evaporator for enhanced ther mal management, ASME Heat Transfer Division, International Mechanical C ongress and Exposition, HDT, Orlando, FL, November (2000) 366-372. [4] M.M. Franois, W. Shyy, Computations of drop dynamics with heat transfer, Doctoral Thesis, University of Florida, Department of Mechanical Engineering, Gainesville, FL (2002). [5] M. Friedman, J.N. Chung, Feasibility study on a piezoelectri c droplet ejector, unpublished paper, University of Florida Department of Mechanical and Aerospace Engineering, Gainesville, FL, November (2001). [6] G. Perin, L. Levin, B.T. Khuri-Yakub, Pi ezoelectrically actuated droplet ejector, Review of Scientific Instruments 68 (1997) 4561–4563. [7] X. Zhu, E.S. Kim, Microfluidic motion ge neration with acoustic waves, Sensors and Actuators A 66 (1998) 355–360. [8] D. Huang, E.S. Kim, Micromachined acoustic-wave liquid ejector, Journal of Microelectromechanical Systems 10 (2001) 442–449. [9] N. Szita, R. Sutter, J. Dual, R.A. Buser, A micropipettor with integrated sensors, Sensors and Actuators A 89 (2001) 112–118. [10] B. Vukasinovic, A. Glezer, Vibration-indu ced droplet atomization, Doctoral Thesis, Georgia Technical University, Department of Mechanical Engineering, Atlanta, GA, March (2002). [11] F.P. Incropera, D.P. DeWitt, Fundamentals of heat and mass transfer, Fourth Edition, John Wiley & Sons, New York (1996) 238-239. [12] J.W. Rose, Some aspects of condensati on heat transfer th eory, International Communications in Heat and Mass Transfer 15 (1988) 449–473.

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186 BIOGRAPHICAL SKETCH Adam Robinson was born in Wichita, KS on January 27, 1979. He received his Bachelors of Science in Mechanical Engineer ing from the University of Florida in May of 2002. He plans to focus his career on re newable energy development in the developed and the developing world.


Permanent Link: http://ufdc.ufl.edu/UFE0010853/00001

Material Information

Title: Vibration-induced droplet atomization and dropwise condensation
Physical Description: xvii, 186 p.
Creator: Robinson, Adam K. ( Dissertant )
Chung, Jacob N. ( Thesis advisor )
Klausner, James ( Reviewer )
Lear, William ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, M.S
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering

Notes

Abstract: This thesis studied the process of dropwise condensation (DWC) of water vapor on a non-isothermal condenser surface. Specifically, the object was to quantify the effects of Vibration Induced Droplet Atomization (VIDA) on this process. The working fluid was water vapor in the saturation temperature range from 24 to 48 degrees C (75 to 118 degrees F), corresponding to 3.0 to 11.2 kPa (0.89 to 3.30 in-Hg). The condenser surface was the clean metal side of a brass shim, of 19 mm (0.75 in) diameter and 0.4 mm (0.016 in) thickness. A piezoelectric ceramic disk was attached to the other side of this shim. The resonance frequency of this arrangement in a clamped edge boundary condition ranged from 2.1 to 4.0 kHz with the temperature of the disc. The driving voltage signal was a sinusoidal wave form of about 35 Vp-p. Theoretically, any reduction in the maximum aggregate size reached by the drops of condensate should result in an increase in the heat transfer coefficient, or HTC. If vibration was applied to a sessile droplet in a certain range of frequency and amplitude, the droplet surface developed capillary waves. Given enough intensity of vibration, these waves outstretched the viscous forces holding the drop together and the drop disintegrated into hundreds of small drops. These small drops then moved away from the original, static drop surface at roughly perpendicular trajectories. Direct case-by-case comparison of the effects of VIDA on DWC was not possible. The control model assumed an exponential radial temperature distribution in the diaphragm. Direct surface temperature measurements of the diaphragm during application of vibration indicated that the assumed profile did not apply. The area above the piezoelectric patch on the back of the diaphragm was found to have a higher temperature than the saturated vapor. Therefore the active heat transfer area of the VIDA case was smaller than that of the non-VIDA case. However, the acceptable data points from the former set still showed comparable to improved overall HT rates over the latter. Therefore some advantage was shown, but it could not be quantified. Maximum peak-to-peak displacement of the diaphragm was as high as 40 microns for a new driver, while for an old driver may have been about 10 microns. Edge temperature readings for the shim ranged from 6 to 19 degrees C (43 to 66 degrees F) with coolant inlet temperatures near 4 degrees C (40 degrees F). The bulk of the original data showed much lower than predicted overall heat transfer rates, or THR. This was due to the extreme sensitivity of low pressure DWC to the presence of non-condensable gases, or NCG, namely air. A final round of testing was able to clear up this problem enough to clearly examine VIDA effects on DWC.
Subject: Atomization, condensation, diaphragm, droplet, dropwise, induced, low, piezoelectric, pressure, vapor, vibration, VIDA, water
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 203 pages.
General Note: Includes vita.
Thesis: Thesis (M.S.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
Original Version: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0010853:00001

Permanent Link: http://ufdc.ufl.edu/UFE0010853/00001

Material Information

Title: Vibration-induced droplet atomization and dropwise condensation
Physical Description: xvii, 186 p.
Creator: Robinson, Adam K. ( Dissertant )
Chung, Jacob N. ( Thesis advisor )
Klausner, James ( Reviewer )
Lear, William ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2005
Copyright Date: 2005

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, M.S
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering

Notes

Abstract: This thesis studied the process of dropwise condensation (DWC) of water vapor on a non-isothermal condenser surface. Specifically, the object was to quantify the effects of Vibration Induced Droplet Atomization (VIDA) on this process. The working fluid was water vapor in the saturation temperature range from 24 to 48 degrees C (75 to 118 degrees F), corresponding to 3.0 to 11.2 kPa (0.89 to 3.30 in-Hg). The condenser surface was the clean metal side of a brass shim, of 19 mm (0.75 in) diameter and 0.4 mm (0.016 in) thickness. A piezoelectric ceramic disk was attached to the other side of this shim. The resonance frequency of this arrangement in a clamped edge boundary condition ranged from 2.1 to 4.0 kHz with the temperature of the disc. The driving voltage signal was a sinusoidal wave form of about 35 Vp-p. Theoretically, any reduction in the maximum aggregate size reached by the drops of condensate should result in an increase in the heat transfer coefficient, or HTC. If vibration was applied to a sessile droplet in a certain range of frequency and amplitude, the droplet surface developed capillary waves. Given enough intensity of vibration, these waves outstretched the viscous forces holding the drop together and the drop disintegrated into hundreds of small drops. These small drops then moved away from the original, static drop surface at roughly perpendicular trajectories. Direct case-by-case comparison of the effects of VIDA on DWC was not possible. The control model assumed an exponential radial temperature distribution in the diaphragm. Direct surface temperature measurements of the diaphragm during application of vibration indicated that the assumed profile did not apply. The area above the piezoelectric patch on the back of the diaphragm was found to have a higher temperature than the saturated vapor. Therefore the active heat transfer area of the VIDA case was smaller than that of the non-VIDA case. However, the acceptable data points from the former set still showed comparable to improved overall HT rates over the latter. Therefore some advantage was shown, but it could not be quantified. Maximum peak-to-peak displacement of the diaphragm was as high as 40 microns for a new driver, while for an old driver may have been about 10 microns. Edge temperature readings for the shim ranged from 6 to 19 degrees C (43 to 66 degrees F) with coolant inlet temperatures near 4 degrees C (40 degrees F). The bulk of the original data showed much lower than predicted overall heat transfer rates, or THR. This was due to the extreme sensitivity of low pressure DWC to the presence of non-condensable gases, or NCG, namely air. A final round of testing was able to clear up this problem enough to clearly examine VIDA effects on DWC.
Subject: Atomization, condensation, diaphragm, droplet, dropwise, induced, low, piezoelectric, pressure, vapor, vibration, VIDA, water
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 203 pages.
General Note: Includes vita.
Thesis: Thesis (M.S.)--University of Florida, 2005.
Bibliography: Includes bibliographical references.
Original Version: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0010853:00001


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VIBRATION-INDUCED DROPLET ATOMIZATION
AND DROPWISE CONDENSATION














By

ADAM K. ROBINSON


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Adam K. Robinson

































This document is dedicated to my mother, Lorraine L. Parker.















ACKNOWLEDGMENTS

I thank my advisor, Dr. Jacob N. Chung, without whom the project would not have

been possible. I also am grateful to Dr. James Klausner and Dr. William Lear for their

input on my committee. Dr. Wei Shyy and Dr. Mark Sheplak gave valuable input in the

early stages of the project.

Dr. Louis Cattafesta gave much appreciated advice on data acquisition and signal

conditioning. Dr. David W. Hahn gave excellent counsel on modeling heat conduction.

Dr. Boj an Vukasinovic, of the Georgia Institute of Technology, graciously granted

copyright permission along with copies of some figures from his dissertation on

Vibration Induced Droplet Atomization.

Several of my colleagues have given valuable input relating to their experience in

different areas of this project including Ryan Avenall, Venkat Chandrasekaran, Quentin

Gallas, Ryan Holman, Stephen Horowitz, Landon Tully, Chris Velat, Renqiang Xiong,

and Kun Yuan. Their advice undoubtedly saved weeks of effort.

I am very grateful to Ron Brown for his indispensable help and advice in

fabrication of the experimental apparatus. Becky Hoover, Pam Simon, and Teresa

Mathia provided crucial help and were always pleasant in helping me to navigate

administrative requirements.

Several professionals outside of the university system have also been remarkably

helpful in completing this project. Pam Bishop, from MTI Instruments, was especially

prompt and courteous in helping me to learn about optical position sensing. Richard









Brooks and Jeff Zahnd from American Piezo Corporation provided crucial insights on the

function of their piezoelectric products, advice on signal processing, and development

samples. Doug Hand from Thales-Optem, Inc. gave excellent advice on optics. John

Vickers from Watlow Heating products also gave crucial advice in the early stages of this

project.

Finally, but most importantly, I thank my Mother for her years of support.














TABLE OF CONTENTS



A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES .......................................... viii

LIST OF FIGURES .................................................. ............................ ix

L IST O F E Q U A T IO N S .................................................. ............................................ xii

N O M E N C L A T U R E .................................................... .............................................. xiii

A B S T R A C T ..................................................................................................................... x v i

CHAPTER

1 IN TR O D U C T IO N ................................................................. .... ..... ...............

2 LITER A TU RE SU R V EY ..................................................................... ...............5...

D ro p let E j ectio n ................................................................................................... 5
Film w ise C ondensation (FW C ) ..................................... ..................... ................ 10
D ropw ise C ondensation (D W C )............................................................ .................. 11
M ain T heory ....................................................................................................... 11
Functional B behavior of R ose Theory ............................................. ................ 17
C om peting M odels .............. ................. ................................................ 27
D roplet Size D distribution ....................................... ...................... ................ 33
Effect of M axim um D roplet Size ................................................... ................ 34
General Results on Drop-wise Condensation (DWC)....................................35
Constriction R resistance ....................... ................ 38
Drop-wise Condensation (DWC) Promotion .................................................40
Effects of Non-Condensable Gases (NCG)....................................................44
H eat Transfer w ith V ibration..................................... .... ................ 49
Piezoelectric Droplet Ejection with Heat Transfer................................................53

3 TH EORETICAL AN A LY SIS ...................................... ...................... ................ 54

4 APPARATUS ................................ ........... ............................. 63

E q u ip m e n t ............................................................................................................... ... 7 3
Additional Testing .......................... ........... ............................. 80









5 P R O C E D U R E ............................................................................................................. 8 4

6 RE SU L TS A N D D ISCU SSION ............................................................ .................. 88

In itia l T e stin g .............................................................................................................. 8 8
Intermediate testing ............. ................. .... ......... ...... ............... 92
F in a l te stin g ............................................................................................................. 1 0 1
Data Summary and Comparison with Theory ..................................... ................ 119
A d d itio n al T e stin g .................................................................................................... 12 3

7 UNCERTAINTY ...... ................................................. 131

M ain Study ................................................................................. ...................... 13 1
Additional Testing ...................... ........... ..................................... 133

8 CONCLUSION S ................... .. ............................... ...... .... ............... 135

M ain Study ................................................................................. ...................... 135
Trend Explanation ............................. .......... ....................... 136
Additional Testing ...................... ........... ..................................... 137
Im p rov em en ts ........................................................................................................... 13 8
A p p aratu s ........................................................................................................... 13 8
P ro c e d u re ........................................................................................................... 14 0

APPENDIX

A M A T L A B C O D E S ................................................. ............................................ 142

D W C H T C .m ......................................................................... .... .... ............ 142
Dropwise_Codensation_Model.m .............................. .........................144
Theoretical H eat Rate Grid.m ..................................... ................. ............... 146
D W C B ehavior.m ............. .................. ... ....................... ....................... 146
Get Theoretical THR and r effm ...... .... ...... ..................... 149
H eat R ate A rchive.m .................... .............................................................. 150
P R O P S .m ..................................................................................................................1 5 9
ex p p ro fi le .m ............................................................................................................1 6 2
sim p so n s ru le .m .......................................................................................................16 3

B TOTAL HEAT RATE TABLES .......................................165

C DETAIL DRAWINGS .........................................................170

LIST OF REFEREN CE S ...........................................................................................178

B IO G R A PH IC AL SK ETCH ...................................................................... ...............186















LIST OF TABLES


Table page

6-1 Experimental and theoretical data for THR. ....................................... 123

6-2 Data summary from Additional Testing.............. ........................ 129

7-1 Uncertainty in m measured values. ...... ........... .......... ...................... 132

B-1 Theotretical THR with theoretical maximum departing droplet size.................. 166

B-2 Theotretical THR with 4 mm maximum departing droplet size. ......................... 166

B-3 Theotretical THR with 2 mm maximum departing droplet size. ....................... 167

B-4 Theotretical THR with 1 mm maximum departing droplet size. ....................... 167

B-5 Theotretical THR with 0.5 mm maximum departing droplet size. .....................168

B-6 Theotretical THR with 0.25 mm maximum departing droplet size. ...................168

B-7 Theotretical THR with 0.1 mm maximum departing droplet size. .................... 169

B-8 Theotretical THR with 0.01 mm maximum departing droplet size. ...................169















LIST OF FIGURES


Figure page

2-1 Dimensionless critical acceleration vs. dimensionless drop diameter ....................7...

2-2 Droplet atomization by frequency modulation...................................... ...............8...

2-3 Heat transfer coefficient vs. saturation temperature........................................... 18

2-4 H eat transfer coefficient vs. subcooling.............................................. ................ 19

2-5 H eat flux vs. saturation tem perature ................................................... ................ 20

2-6 H eat fl ux vs. subcooling .......................................... ......................... ................ 2 1

2-7 Heat transfer coefficient vs. maximum departing droplet radius for Tsat = 380K....22

2-8 Heat transfer coefficient vs. maximum departing droplet radius..........................23

2-9 Heat transfer coefficient vs. maximum departing droplet radius for Tsat = 280K....24

2-10 Heat flux vs. maximum departing droplet radius for Tsat = 380K.........................24

2-11 Heat flux vs. maximum departing droplet radius at AT = 1K...............................25

2-12 Heat flux vs. maximum departing droplet radius for extreme parameters ............26

2-13 Smallest thermodynamically viable droplet radius vs. Tsat .................................27

3-1 Assumed radial surface temperature distribution................................ ................ 56

3-2 Radial H TC and H F distributions................... ................................................. 57

3-3 Block diagram of iterative loop to determine theoretical THR...............................59

3-4 Theoretical THR vs. maximum droplet radius.................................... ................ 62

4-1 Corner view of experimental apparatus. .................................................. 66

4-2 Sectioned view of experim ental apparatus.......................................... ................ 67

4-3 Insulation applied to and removed from diaphragm frame.................................68









4-4 Exploded view of experim ental apparatus .......................................... ................ 69

4-5 D iaphragm clam p, low er face ............................................................. ................ 70

4-6 Schematic of equipment setup with labels ............... .................................... 75

4-7 Schematic of vacuum apparatus adjustments for Additional Testing...................81

4-8 Placement of surface thermocouples on the piezoelectric diaphragm .....................82

6-1 Video frames of DWC without (left) and with (right) VIDA ..................................89

6-2 T em perature data from 1/26 .....................................................................................89

6-3 T otal heat rate data from 1/26 ............................................................. ................ 90

6-4 Tem perature data from 2/17 ....................................... ...................... ................ 92

6-5 T otal heat rate data from 2/17 ............................................................. ................ 94

6-6 Tw o fram es taken from videos on 2/17............................................... ................ 95

6-7 Tem perature data from 2/24 ....................................... ...................... ................ 97

6-8 Total heat rate data from 2/24 ............................................................. ................ 99

6-9 Typical droplet distribution for DWC on the test diaphragm without VIDA........ 100

6-10 Video frames of typical droplet distribution for DWC with VIDA .................... 100

6-11 T em perature data from 3/4 .....................................................................................102

6-12 Total heat rate data from 3/4 .......................................................... 103

6-13 Tem perature data from 3/13 ........................................................ 104

6-14 Total heat rate data from 3/13 ....... .......... ............ ...................... 105

6-15 Tem perature data from 3/14 .................................. ...................... ................ 106

6-16 Total heat rate data from 3/14 ....... ........... ............ ...................... 107

6-17 Temperature data from 3/15-16...... ............ .......... ..................... 108

6-18 Total heat rate data from 3/15-16...... .......... ........ ..................... 109

6-19 Droplet departure and regernation in DWC without VIDA, 3/15-16. ................. 110

6-20 Video frames of VIDA onset, front view. ..... .......... ....................................... 113









6-21 Calibration curve for peak-to-peak displacement vs. input voltage .....................115

6-22 Peak-to-peak displacement vs. frequency for used driver.................................116

6-23 Video frames of VIDA process, corner view ...... .... .................................... 118

6-24 Experimental THR vs. theoretical THR ...... .... ........................................ 120

6-25 Total heat rate vs. maximum subcooling ....... .......... ...................................... 122

6-26 Temperature data from 5/21 ........................................................ 124

6-27 Comparison of 5/21 surface temperature data...... .... .................................. 127

6-28 Comparison of 5/21 VIDA and non-VIDA temperature data for shim surface..... 128

C-i Bottom Cover detail draw ing. ........................................................ 171

C-2 Cooler Barrel detail draw ing. ..... ........... ........... ...................... 172

C-3 Diaphragm Clamp detail drawing. ...... .......... ........ ...................... 173

C-4 Diaphragm Frame detail drawing...... .......... ........ ..................... 174

C-5 Optical Probe Guide detail drawing. ...... ... .......................... 175

C-6 Optical Probe Seal detail drawing ...... ....... ..... ..................... 176

C-7 Top Cover detail draw ing .................................... ....................... ............... 177















LIST OF EQUATIONS


Equation page

2-1 Nusselt correlation for FWC on a vertical flat plate.........................................10

2-2 Modified heat of vaporization for Nusselt correlation.....................................10

2-3 Thermal resistance in a droplet due to conduction ........................... ................ 13

2-4 Thermal resistance in a droplet due to interphase matter transfer resistance ........ 14

2-5 Finite temperature drop required for equilibrium ............................. ............... 14

2-6 H eat flux through a drop .................. ........................................................... 14

2-7 Fraction of condenser surface area covered by drops larger than r ....................15

2-8 General expression for fractional area coverage of condenser surface............... 15

2-9 Specific expression for fractional area coverage of condenser surface .............. 15

2-10 Average heat flux over an isothermal surface undergoing DWC (Rose Theory).. 16

2-11 M axim um departing droplet radius ......................... .................... ..................... 16

2-12 Minimum thermodynamically viable droplet radius.........................................16

2-13 E m pirical relation for D W C .............................................................. ............... 16

2-14 Interfacial m ass transfer resistance ................................................... 27

2-15 Theoretical HTC accounting for constriction resistance .................................28

3-1 Heat conduction equation in cylindrical coordinates........................................55

3-2 Assumed exponential form of radial temperature distribution ..............................56

6-1 Energy equation ................................... ......... .... ............... 91















NOMENCLATURE


A area

B a constant

C a constant

cp,f mass specific heat of fluid state

dr differential radius

E energy

E time rate of energy change

f fraction of condenser surface coverage

g acceleration of gravity

h convection coefficient

hfg heat of vaporization

K a constant

k thermal conductivity

L characteristic length

n droplet size distribution exponent

q heat flux

R gas constant

R" thermal resistance

r droplet radius

r maximum departing droplet radius









smallest thermodynamically viable droplet radius

temperature

temperature in Celsius

Greek Letters

difference between saturation and condenser surface temperatures

change in time

density

specific heat ratio

surface tension

absolute viscosity

kinematic viscosity

pi


Subscripts


avg

CV

cond

f

g

gen

in

L

mattertransfer

out

s


average

control volume

conduction

fluid

gas

generation

in

length

interphase matter transfer

out

surface (condenser)









saturation


Operators

8 partial differential















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

VIBRATION-INDUCED DROPLET ATOMIZATION
AND DROPWISE CONDENSATION

By

Adam K. Robinson

December 2005

Chair: Jacob N. Chung
Major Department: Mechanical and Aerospace Engineering

This thesis studied the process of dropwise condensation (DWC) of water vapor on

a non-isothermal condenser surface. Specifically, the object was to quantify the effects

of Vibration Induced Droplet Atomization (VIDA) on this process. The working fluid

was water vapor in the saturation temperature range from 24 to 480C (75 to 118F),

corresponding to 3.0 to 11.2 kPa (0.89 to 3.30 in-Hg). The condenser surface was the

clean metal side of a brass shim, of 19 mm (0.75 in) diameter and 0.4 mm (0.016 in)

thickness. A piezoelectric ceramic disk was attached to the other side of this shim. The

resonance frequency of this arrangement in a clamped edge boundary condition ranged

from 2.1 to 4.0 kHz with the temperature of the disc. The driving voltage signal was a

sinusoidal wave form of about 35 Vp-p.

Theoretically, any reduction in the maximum aggregate size reached by the drops

of condensate should result in an increase in the heat transfer coefficient, or HTC. If

vibration was applied to a sessile droplet in a certain range of frequency and amplitude,









the droplet surface developed capillary waves. Given enough intensity of vibration, these

waves outstretched the viscous forces holding the drop together and the drop

disintegrated into hundreds of small drops. These small drops then moved away from the

original, static drop surface at roughly perpendicular trajectories.

Direct case-by-case comparison of the effects of VIDA on DWC was not possible.

The control model assumed an exponential radial temperature distribution in the

diaphragm. Direct surface temperature measurements of the diaphragm during

application of vibration indicated that the assumed profile did not apply. The area above

the piezoelectric patch on the back of the diaphragm was found to have a higher

temperature than the saturated vapor. Therefore the active heat transfer area of the VIDA

case was smaller than that of the non-VIDA case. However, the acceptable data points

from the former set still showed comparable to improved overall HT rates over the latter.

Therefore some advantage was shown, but it could not be quantified.

Maximum peak-to-peak displacement of the diaphragm was as high as 40 tm for a

new driver, while for an old driver may have been about 10 am. Edge temperature

readings for the shim ranged from 6 to 19C (43 to 66F) with coolant inlet temperatures

near 40C (400F). The bulk of the original data showed much lower than predicted overall

heat transfer rates, or THR. This was due to the extreme sensitivity of low pressure

DWC to the presence of non-condensable gases, or NCG, namely air. A final round of

testing was able to clear up this problem enough to clearly examine VIDA effects on

DWC.














CHAPTER 1
INTRODUCTION

Heat dissipation has been a major barrier to increases in processor speed and

decreases in length scale of electronic devices. The Semiconductor Industry Association

report of 2001 [1] projected cooling loads that would be necessary in the near future for

new generations of electronic devices. At the time of their study, 40 W/cm2 was given as

a typical required heat flux (HF) for the high performance electronics category, which

included for example military avionics. This cooling requirement, projected to the year

2016, more than doubled to 93 W/cm2.

The difficult aspect of dissipating this level of heat flux for cooling electronic

devices was that it had to be accomplished over a relatively small temperature difference,

since the maximum temperature of the electronic devices must stay safely below 850C,

the melting point of the silicon substrate, which leaves at most only about a 40 to 50C

window between the temperature of the heat source to the ambient atmosphere.

Mudawar, in 2001 [2], examined strategies for dissipating high heat fluxes. He

compared the reported heat transfer coefficients (HTC) for the three most common

coolant fluids from literature, air, water, and fluorochemical liquids. The three heat

transfer (HT) regimes examined were natural convection, single-phase forced convection,

and pool boiling. The former two methods are currently the most common and easily

implemented methods of electronics cooling. However, only the latter method was

capable of reaching the needed 100 W/cm2 over a temperature difference on the order of









100C. However, boiling presents a problem in steady cooling due to temperature shocks

that accompany the formation and departure of vapor bubbles.

Significantly higher HTCs were reported for more involved cooling scenarios

such as forced convection boiling, spray, and jet cooling, however these methods also

added significant complexity, weight, and reliability issues to the overall design of a

given device. Still, the main conclusion of the paper was that a two phase cooling

solution was the most likely choice for higher levels of heat dissipation, but that this

would likely add significant complexity to the devices. Thus a problem has emerged of

how to integrate a two-phase, high heat flux cooling scheme into electronic devices of

ever decreasing length scale, while not greatly increasing the complexity.

Francois, Shyy, and Chung, in 2000 [3], offered a potential solution for such a two

phase cooling device. They proposed a theoretical model of a MEMS (micro-electro-

mechanical-systems) device, completing a vapor compression cooling cycle with a

fluorinert chemical as the working fluid. The proposed system was to be driven by a

piezoelectric diaphragm that would also act as the condensing surface. This would allow

for surface tension control of the drops of condensate. This step was taken so that the

device would not be gravity dependent. This analysis presumed dropwise condensation

(DWC) on the diaphragm surface, which has been shown to offer HTC an order of

magnitude higher than filmwise condensation (FWC). The relationship between

maximum droplet size and HTC for DWC will be explicitly examined after the

introduction of the theory. Since the length scale of this device was very small, the

droplets involved were also very small and therefore very high HF was anticipated.









In Francois, 2002 [4], a numerical simulation looked in detail at the droplet impact

on a heated wall and HT aspects of the evaporative side of the proposed device.

However, analysis of the condensation side of the cell was left as an assumption that

DWC would offer a certain order of magnitude of HF. Detailed consideration of this

process was left for further studies.

This model assumed bulk ejection of the condensate droplet from the membrane, a

significant obstacle considering the increasing dominance of surface tension over body

force with a decrease in length scale, although an energy balance model was offered for

the energy required for ejection. Friedman and Chung, in 2001 [5], made an initial

investigation into the droplet ejection process. The device used was in the meso-length

scale since the piezoelectric driver had about a 3A inch diameter. The prefix meso refers

to a characteristic length scale in the range of 1 mm to 1 m. Primary droplet volumes

investigated ranged from 1 to 5 [tL. Even at this relatively large scale of 30 to 150 times

the volume of the largest drop considered in the current study, they still encountered

significant difficulty achieving droplet detachment from the diaphragm surface. When

detachment was achieved, most often the primary droplet divided into several smaller

drops. The difficulty in drop detachment was due to the high surface tension forces with

respect to the momentum forces of the fluid mass. It was because of these surface

tension related problems that good repeatability was not attained. Thus, further work was

needed to determine an appropriate method of droplet ejection from a piezoelectric

membrane surface.

This project was undertaken to study the process of condensation and subsequent

droplet ejection of coolant vapor on a vibrating piezoelectric disk. This effort was






4


needed to clear up the cooling side of the device proposed in [3]. The following criteria

were set as goals for the development of the investigation:

* Meso length scale for ease of study.

* Dropwise Condensation mode for its expected high HF.

* Water as the working fluid for its high heat of vaporization. However, this would
necessitate sub-atmospheric working pressures in order to lower the saturation
temperature below 850C.

* Piezoelectric disc for both the droplet ejection mechanism and condensing surface.

* Some form of droplet ejection would be used to remove condensate














CHAPTER 2
LITERATURE SURVEY

Droplet Ejection

A literature search for droplet ejection methods involving piezoelectric devices

yielded several potential methods. Percin et al., in 1997 [6], tested a piezoelectric droplet

ejector. The piezoelectric diaphragm had a small hole at the center, 50 to 100 tm in

diameter. It supported a reservoir of liquid. The hole was small enough that the surface

tension of the liquid across the hole kept it from leaking out. Droplet ejection occurred

when the oscillation frequency and amplitude caused the momentum of the fluid to

overcome the surface tension. Flow rates of up to 100 mL/s and ejection frequencies of

up to 32 kHz were cited. The device was roughly 8 mm in diameter, 8 mm in length, and

operated on a 32 Vp-p sinusoidal voltage. The working fluid was a special, low-viscosity

ink. Higher viscosity fluids were tested, requiring voltages up to 200 Vp-p. This design

was promising for the purposes of this study, except that the orientation of the liquid

surface had to be away from the direction of ejection, not leaving much surface area

available for removing the heat that would induce condensation on the liquid surface.

Zhu and Kim, in 1998 [7], reported a MEMS device which achieved significant

fluid mixing using loosely focused acoustic waves from a piezoelectric diaphragm. The

diaphragm bore ceramic patches arranged specifically to accomplish the focusing by

constructive interference. The focal length was designed to be just above the liquid/vapor

interface. The device was small enough, 1.5 x 1.5 x 0.5 mm, that the top could be open

and still be insensitive to gravity effects because of the surface tension. The authors









reported negligible heating of the fluid from driver operation. This device showed much

promise for fluid motivation on a micro-scale but lacked the droplet ejection required for

the present device.

In 2001, Huang and Kim [8] used a similar arrangement to [7] to achieve droplet

ejection. Moving the surface of the fluid a little higher, to coincide with the focal point

of the acoustic waves, was the main difference between the two devices. These waves

were again created by a specially patterned piezoelectric ceramic. The ejected droplet

velocity, caused by constructive interference, was found to be between 2 and 3 m/s, with

operating frequencies up to 10 kHz. Droplet diameter was reportedly around a few

microns with uniform size and direction. The device was less than 2 mm in diameter and

0.5 mm in length. The ejection characteristics and the length scale of this device appear

well suited for the needs of the proposed HT cell. The limitation here was the conduction

resistance of the fluid bulk. At this point, it was assumed that DWC would provide more

intense cooling than could be achieved by direct contact condensation, due to the

relatively slow conduction of heat through the liquid phase.

Szita et al., in 2001 [9], reported the development of a micropipettor, using the

capacitance of one piezoelectric disc for feedback information to very precisely control

the dispensed volume and another disc for pumping force. Volume control was in the

sub-tL range. This design however, lacked the sufficient velocity of ejected droplets

desired for the current application. This velocity was required to break through a vapor

layer on the hot surface of the proposed HT cell.

The most promising method of droplet ejection that came up in the literature search

was Vibration Induced Droplet Atomization (VIDA) [10]. A horizontal piezoelectric disc









with diameter on the order of 1 inch was used as the supporting base for a single drop of

water placed at its center. As the perpendicular vibration of the diaphragm approached

resonance, standing waves developed on the surface of the sessile droplet. A critical

acceleration and frequency existed for a given drop of fluid, above which the momentum

of the fluid in these standing waves overcame the surface tension. This caused the

droplet to effectively burst. The order of the critical acceleration was given as roughly 20

times standard-earth gravity. The length of time of the bursting event ranged from


7-


6-






4-


3 -I I I'
5 10 15 20 25

d


Figure 2-1. Dimensionless critical acceleration vs. dimensionless drop diameter.
Acceleration was determined by a = af4 3 /(o- p)1 and diameter by
d = df2/3/(cr p)1/ Used with permission from [10].

essentially instantaneous for a large step input change in either amplitude or frequency to

up to a minute for small changes in these parameters. Excellent correlation was found for

the dimensionless critical acceleration versus dimensionless drop diameter involving the

fluid properties and the frequency of oscillation (Figure 2-1).









Note the inverse exponential manner in which the dimensionless critical

acceleration required for atomization increases as the droplet diameter decreases. This

implies an effective lower size limit for achievable atomization events. Primary droplet

diameters used in this study ranged from 10 to 150 [tL while the resonant frequencies of

the drivers used were 1, 3, 5, 7, 9, and 15 kHz. Generally the larger the diameter or the

thickness of the piezoelectric disc driver, the lower the resonant frequency.

When the amplitude of the input signal to the drive was fixed and the frequency

incrementally increased from 0 towards resonance, the atomization process had roughly

three steps (Figure 2-2). When vibrations commenced at low frequency, standing lateral


Sb c
















Figure 2-2. Droplet atomization by frequency modulation, 0.1 mL drop. Used with
permission from [10]. a) No applied acceleration. b) Lateral standing waved
develop on drop surface. c) Lateral standing waves grow in amplitude. d)
Axial waves become visible over lateral waves, creating a cross-hatch pattern.
Wave momentum begins to overcome surface tension, as drops begin to issue
from the outer wave peaks. e) Amplitude of oscillations grows, more drops
issue from primary drop. f) Drop ejections increase in number and intensity.
g) Primary drop no longer visible due to number of secondary drop ejections.
h) Ejections of secondary drops continue. i) Primary drop mostly dissipated
and secondary droplet ejection complete.









waves appeared around the circumference of the sessile droplet. As the frequency

approached the resonant frequency of the driver-drop system, axial waves also appeared,

making a cross-hatched pattern with the lateral waves. As input frequency came closer

still to resonance, the peak of the wave oscillations began to overcome the surface tension

forces keeping the primary drop intact, causing secondary drops to pinch-off from the

wave peaks. Near the resonant frequency and still with fixed amplitude of the input

signal to the driver, the pinching-off of secondary droplets became much more intense.

As this process progressed, the primary drop became depleted until there was only a

small fraction of it left.

The diameter of the ejected droplets was reported to be relatively independent of

both the time during the atomization event and the volume of the primary droplet. The

ejected velocity was said to range from 0.2 to 2.0 m/s. The droplet velocity was measured

by a particle tracking velocimetry algorithm, developed by the investigator, and digital

video images. A peak in ejection velocity was found at the onset of atomization. This

quantity then decreased to be roughly equivalent to the velocity of the resonating

diaphragm, just below the point of issuance.

The trajectory of the ejected droplets was roughly perpendicular to the surface of

the undisturbed primary drop at the given location, and, on average, was axisymmetric.

Viscosity and surface tension were also slightly varied to probe for an effect on the

atomization process, though no major quantitative change was found.

On the operation of the piezoelectric diaphragms, Vukasinovic [10] noted that the

resonance frequency was extremely sensitive to temperature as well as mass loading of









the diaphragm, both of which would no doubt be important considerations in any

application of VIDA to DWC.

Filmwise Condensation (FWC)

Filmwise condensation is the default mode of condensation and has therefore been

taken as the design assumption in industrial design. Filmwise condensation will indicate

the failure of the DWC mode, due to several possible factors including contamination of

the vapor, contamination of the condenser surface, too large of a contact angle for the

working fluid with the condenser surface, or too high HF leading to flooding of the

condenser surface.

Nusselt provided the most successful correlations for FWC on a vertical flat plate

in a pure vapor (Eq. 2-1) [11]. This correlation used a modified heat of vaporization,

which accounted for subcooling of the condenser surface (Eq. 2-2) [11].

hL 9 pfg(pf pg )h'fg L3
NuLg= = 0.943 f 2-1
S kf Pf kf (T T)

h'fg = hfg + 0.68cP,f (T,- T), 2-2

In this Nusselt correlation, h is the HTC, L is the length of the plate, g is the

acceleration of gravity, p is density, / is the dynamic viscosity, k is the thermal

conductivity, Tis temperature, and cp is the constant pressure mass specific heat of the

condensate. The subscript f denotes fluid, sat is for saturated, s is for surface, and g is

for gas. One of the seminal investigators of DWC, Dr. John Rose, commented on the

remarkable accuracy of the Nusselt theory for FWC in an address on developments in

condensation theory, in 1988 [12]. He said that further tweaking of this Nusselt relation









for FWC was essentially "an academic exercise," since this theory was correct to within a

few percent for most cases.

Dropwise Condensation (DWC)

For decades now, researchers have sought to master the process of DWC. The high

HF values attainable with this mode of condensation offered attractive potential increases

in efficiency for industrial HT processes. However, this phenomenon has proven quite

resistant to efforts at precise characterization. The main performance difference in DWC

and FWC is generally accepted as a potential order of magnitude increase in HTC. This

holds possibly dramatic cost savings in many applications including power plants that run

steam turbines as well as seawater desalination plants. These cost savings would exist

specifically in terms of lower capital and maintenance costs for a given output capacity.

Main Theory

Early investigations, however, showed widely ranging trends of relationship of heat

flux (HF) to degree of subcooling of the condenser surface, Tsat Ts = AT. This

variation in reported results was so significant that, the first authors to offer a viable

theory on DWC HT, said they were, "of very little use even as a check," [13]. These

investigators asserted that the massive variance in the reported trends in DWC HT as due

to the presence of non-condensable gases (NCG) in the test vapor. In most cases, this

non-condensable species was air. To prove this theory, these authors chose to study

DWC of steam near 1 atm. They used an isothermal copper block as a condensing

surface and a heat flux meter. Some of the main efforts undertaken to control the

influence of NCG on the experimental results were as follows:

* Careful attention was paid to the vacuum integrity of the apparatus.









* Long blow-off periods were employed prior to taking data, usually 3-4 hours, to
allow sufficient degassing of the supply water and of the walls of the stainless steel
apparatus. For these periods the boilers ran at full power and the vessel was vented
to atmosphere in a controlled manner, to prevent backflow.

* A small vent tube was placed near the condenser surface. Different rates of steam
venting were tested for effect on temperature changes of the condensing surface
and also for visual disturbance of the condensate. A venting rate was then chosen
based on a balance between these competing considerations.

* The authors also took extreme care to minimize uncertainty in temperature
measurement, which they had also noted as a major source of error in previously
published studies. Several temperature measurements were taken along the
centerline of a copper block to determine the heat flux and then the gradient
extrapolated to the condensing surface. A summary of their key findings follows:

* NCG had the significant effect of depressing the surface temperature for a given
HF and saturation temperature, in other words, increasing AT.

* Significant difference was noted in effectiveness of the four promoters used,
though the underlying surface of polished copper and the cleaning regimen
remained constant.

* No dependence of HF on plate height was found. This quantity was measured 1
inch and 4 inch from the top of the test surface with no discernable difference in the
data.

* Sweeping frequency and maximum drop size did not affect HTC in the range
studied.

The first finding helped cement the notion that NCG had an important effect on

condensation HT. The second became important later in the debate over the effect of

surface thermal conductivity on overall HT in DWC, which will be discussed at length in

the Constriction Resistance section of the literature survey. The third finding of no

height dependence has also been contentious, as several other researchers have noted a

height dependence, for example, Koch, Kraft, and Leipertz, in 1998 [14] and in the much

more extreme cases studied by Yamali and Merte, 1999 and 2000 [15,16]. There were

several contemporary studies of Le Fevre and Rose that found similar magnitude HF, and

these will be discussed further in the General Trends section below. For now, consider









the theory offered by the same pair of researchers the following year, Le Fevre and Rose,

1966 [17]. The theory has been republished numerous times, though in its most compact

form in Rose, 1988 [12]. Some corrections and coefficient values were suggested in a

2002 presentation [18].

The Rose theory, as it will henceforth be referred to, was based on the assumption

that all the HT in DWC effectively occurred on the surface of the liquid droplets rather

than the bare condenser surface between drops. The theory assumed only three

significant thermal resistances, with each represented by a temperature difference. The

first was conduction through the liquid of the sessile droplet. The second was interfacial

matter transfer resistance. The third assumed-significant thermal resistance in DWC was

the finite decrease in saturation temperature required for equilibrium of a saturated vapor

with the curved surface of a liquid drop. From these resistances, an expression was

derived for the HF through a given drop of radius r. This expression for HF, based on

fluid-vapor properties and drop radius, was then weighted by an assumed distribution of

drop sizes. The weighted HF was then integrated over a range of drop sizes from the

minimum thermodynamically viable drop radius to the maximum. The maximum drop

radius was determined from dimensional analysis and experiment.

The thermal resistance due to heat conduction through the drop was calculated by

neglecting internal circulation or convection (Eq. 2-3). K1 is a shape factor for the ratio

of surface areas between the base and the top of a hemispherical drop. r is a given drop

radius. kf is the thermal conductivity of the condensate.


R"od = Kr 2-3
kf








The expression for interfacial matter transfer resistance was calculated as a

thermodynamic function (Eq. 2-4). K2 is a constant, experimentally determined and

relatively constant for different fluids. Vg is the kinematic viscosity. Tis the saturation

temperature of the vapor. R is the gas constant of the vapor. y is the specific heat ratio of

the vapor. hfg is the heat of vaporization of the working fluid.


?"m K gvT RT)-22^) 2-4
mattertransfer = K2 h 2-4) -

Finite temperature drop associate with a saturated liquid and vapor in equilibrium

at the curved surface of a drop was also a thermodynamically derived quantity (Eq. 2-5).

c is the surface tension of the fluid.

AT, sa 2-5
rp h

The thermal resistances in (Eq. 2-3,4,5) were then combined with the temperature

difference between the condenser surface and the saturated vapor to form an expression

for the HF through a single drop of radius r (Eq. 2-6). The correction factor in the

interphase matter transfer resistance term was offered as a correction for the

polyatomicity of water.

AT- 2 -at
q"(r) f2= rpsh/ 2-6
Kr K 0.627 Tat 2 (y -+1 RT/2
kT 0.664 )pghg y-l) 2)

Now a statistical distribution for drop sizes, and limits on those sizes had to be

chosen in order to integrate over all drop sizes to obtain the total HF for the surface. Le

Fevre and Rose chose the distribution in (Eq. 2-7), based on an expression for the fraction









of the condenser surface that was covered by drops larger than r. This fraction went to 1

as the given radius went to zero and went to 0 as r went to the maximum radius, rmax. In

other words, all of the surface was covered by drops larger than zero, and none of the

surface was covered by drops larger than rmax. When differentiated with respect to r, (Eq.

2-7) gives the area covered by a differential drop radius, dr (Eq. 2-8).


f =1- 2-7
r)


A(r)dr = n dr 2-8


For the original theory, Le Fevere and Rose assumed from observations that n was

roughly equal to 1/3, an assumption that was later vitiated by the results of Rose and

Glicksman, in 1973 [19]. There has been much discussion of this topic in the literature

which will be addressed later in this paper. With n = 1/3, (Eq. 2-8) gave (Eq. 2-9).


A (rdr r2/3 dr 2-9
3Ar r

It should be noted that this equation gave an exponent of -2/3 as opposed to the -1/3

given in [12]. This was decided a typo in the original, since the derivative of (Eq. 2-7)

with respect to r, yields the (Eq. 2-9) as given and the expression given for the q"avg in the

same reference follows from the value of the exponent used in (Eq. 2-9), which was

arrived at when equation (Eq. 2-6) was multiplied by the differential area (Eq. 2-9) and

integrated from the minimum to the maximum drop radius. The result was the HF for the

entire condensing surface, averaged over all drop radii from the minimum to the

maximum (Eq. 2-10). The maximum droplet radius was determined by dimensional

analysis in (Eq. 2-11) [12]. Here K3 is an empirical constant. The minimum









thermodynamically viable droplet radius was given, finally in a dimensionally correct

form, in (Eq. 2-12) [20]. Note: some of the notation above and throughout this paper has

been changed to be homogenous with that of this study, but the meaning has been

retained. In the same reference, [20], Rose offered an empirical relation for DWC of

steam "near and below atmospheric pressure" (Eq. 2-13).


AT 2o-at
,q- _3-1 rpfhfg r 2/ 3dr 2-10
3r j K r +K Tsat 2 +1 RTsat1
k Pf hfg 2/ 1) 2z


r =rmax =K3 2-11
P9)


rmin =r 2-- T 2-12
hfgAT'

q"= to[5AT + 0.3AT2 ], 2-13

In (Eq. 2-13) q" is in kW/m2 and t and AT in C. No uncertainty was explicitly stated.

However, the presented data were claimed to be 'quite well represented' by the given

relation. A graphical estimation of the quality of this representation gave an uncertainty

of 10%. Likewise, no explicit statement of uncertainty for (Eq. 2-10) was given. This

was most likely due to the myriad of factors that could potentially yet be proven to be of

import to the phenomenon of DWC. A brief listing of these factors follows here, each of

which was furthered explored in this literature survey:

* General comments on DWC trends in AT, Tsat, and rmax
* Drop size and Nucleation site density
* Effect of maximum droplet size on overall HF
* Constriction resistance debate
* Promotion methods









* Condensation coefficient, as it affects the interfacial mass transfer resistance term
* Effects of NCG on both DWC and FWC

Functional Behavior of Rose Theory

One main advantage of the Rose theory was that it presented a closed-form relation

for HF and surface temperature in a compact form. Most of the competing theories,

which will be discussed in the next section, are presented only as numerical simulations,

which can be enlightening but also daunting to implement. The following figures

examine the behavior of the mathematical function of the Rose theory, specifically in the

ranges of parameters relevant to this study, slightly extended to cover most applications

of DWC of water vapor.

Mathematical function was emphasized in this section, and therefore only small

effort was made to confirm the achievability of the extreme values presented. That is, no

consideration for transition to FWC or the existence of corresponding experimental

values, as these concepts were addressed in the General Results section. In fact, only the

more moderate ranges of saturation temperature and maximum droplet size will be used

in this study, for comparison to experimental data. In the many journal presentations of

his theory, Rose had yet to offer a quantifiable maximum error for his theory, merely

saying often that good or general agreement existed with available experimental data that

he considered reliable, as in Stylianou and Rose, 1980 & 1983, and Rose, 1981

[21,22,23], or simply presenting a figure of the theory graphed over the experimental

results of several investigations and offering no explicit comment on uncertainty, as in

Rose, 1998 & 2002 [24,18]. The farthest Rose has thus far ventured in explicit comments

on the accuracy of his theory was in [12], where he noted excellent agreement of data and

theory for 380 data points from 5 different investigations of DWC. These 5 different










studies used 8 different promoters on copper and steam near 1 atm. In that figure, the

apparent maximum deviation was roughly +1K (30%), in a graph of AT vs. HF. The

variation appeared well centered about the theory line. This AT vs. HF relation has been

his preferred method for examination of DWC performance. This choice was due to the

large uncertainty in measuring the condensing surface temperature and the consequently

much larger uncertainty in HTC since it involves the reciprocal of this quantity.

Additionally, the choice of HF as the independent and the surface subcooling as the

dependent variable more accurately represents the experimental circumstances. This

topic will be addressed in more detail in the General Trends section. Of particular

importance for the following figures was that the theory concerns only an isothermal

surface while the current study involves only a condenser surface with a variant

temperature distribution over its surface.


101





100---------I

E - - - - --- _ T_ _


-







250 300 350 400 450 500
Figure 2-3. Heat transfer coefficient vs. saturation temperature, Tsat, for subcooling, AT
10, 50.
0.1, 10 1 50.




10.2 -----------------------





0.1, 1.0, 10, 50.







19


The HTC determined from the Rose theory was somewhat flat relative to

increasing saturation temperature and for small surface subcooling (Figure 2-3).

However, as surface subcooling increased, the HTC predicted from 300 to 500C

saturation temperature experienced almost a two order-of-magnitude increase. Note that

the effects of subcooling are increasingly mitigated for lower vapor pressures/

temperatures. Also, as subcooling increases for a given saturation temperature,

successive increases in HTC diminish in magnitude.



101

--------- ---- ----- ----- -- --- ------ -----i -- ---




1 0 1 - -
I I I I I II I I I
I-I
L I I I I I I I I -



h ----- ---r--' --- ------------ at
T r[ T L I IIIncreasing Tsat
F T T F I -III ---- I- I I--- I -















All analyses involving the theory in this study were conducted in MATLAB, and
of(Eq. 2-10,11,12) were accomplished by the subroutine DWCHTC.m, which can be
Figur 2-4 Hea trn fe cofIcin I s Iucoig AT Io sauato IemperaturesIof
28 2 ,3 0 4 0 500K. I[I I I II T I F
Il anlye ivlIn Ih thoI in th Itd wer Iodce in IALB Iand I III
Ihi Iorsodn cod Ile Ia be I on in apeni A. Ipcfcly Ih IcalculationsII
ofEq 2-01,2 wer acoplse by Ih surutn [W IT~m whc Ia Ibe I ILI










found on page A-1 of the appendix. The following figures were generated by

DWC_behavior.m, page A-9.

For a given subcooling, significant vapor pressure dependence was noted in the

subatmospheric saturation pressure range (Figure 2-4). However significant increases in

HTC were absent above the 1 atm level. This was the precipitous drop predicted in the

original work of HTC with vapor pressure below 1 atm [17]. The subcooling range was

of more importance in this study than the saturation temperature. Here the saturation

temperature ranged only from about 24 to 490C but was constant for each data point. The

surface subcooling, due to the temperature distribution on the diaphragm, ranged from 0

to as high as 30K across the radius of the diaphragm for each test.



103
- - - - --I- - I - -
----------- -I~~~~-- --------- -- -_ __ __ __ __ __----_ __ __ _

102

101 --------------- ---- ------
L --- --------L-- --I = -
1 0 - - - - -


E
1 100

LLI

101



10-2



10-3
250


350 400
Saturation Temperature, [K]


Figure 2-5. Heat flux vs. saturation temperature, Tsat, for subcooling, AT = 0.1, 1.0, 10,
50K.


Increasing AT










The predicted HF is relatively flat versus saturation temperature for a given surface

subcooling (Figure 2-5). However, the relative differences in predicted HF for rising

subcooling are roughly order-of-magnitude for order-of-magnitude, at a given saturation

temperature (Figure 2-6). These results emphasize the dependence of HF on subcooling,

clearly indicating a potential 4 orders-of-magnitude increase from near zero subcool to

50K, the highest considered, with 100 MW/m2 peak HF predicted for a Tsat of 500K and

AT of 50K.


13
103 --- _- i- -- -i-- --_ --_ ---i -i- -i-_ --_ -_-_ _i _-i-- -i-


10 100 10 102












Heat transfer coefficient showed a tight distribution for subcooling from 0.1 to 50K















(Figure 2-7). If it were possible to directly control the maximum droplet size, then
I I I 1 I I L I I - I I I


^ _ _^ ^_ _^ _ ^ _^ ^ -^ -\~ ^\ ~ - ~ ^ ~ -I --I -- -
E --- I -- P----- F- P --



_ _ _ --I -- I-- I --I -- __--- - -- -- -- I-- m -I --I
0 I I- -I - I_ _I__ I__-I


-2 .. .. .







Figure 2-6. Heat flux vs. subcooling, AT, for saturation temperatures of 280, 320, 380,


Heat transfer coefficient showed a tight distribution for subcooling from 0.1 to 50K


(Figure 2-7). If it were possible to directly control the maximum droplet size, then

further increases in the already high HTC of DWC could be theoretically attained. A few












102




101






c-s
T -I I ti T t t I I I II i I I I III T 1I T I I II I I I- I t I I I III I I I 1 1 1-
I I II IIIIII I I I IIIIII I II IIIIII I II IIIIIII I I IIIIIII I I I III

















10 10 10 10-2 10 100 10
Max Droplet Radius, [mm]
10of the investigations by Rose have focused on a vertI Iplate in essentially quiescent
vapor. Therefore these studies did not distinctly examine the effect of maximum droplet






































* Tanasawa et al., 1978 [26] periodic wiper applied to the test surface

* Tanasawa and Utaka, 1983 [27] shear flow of steam
4 I I 1 t I H l I -I" + H4H1 t +1 4 1











































* Tanaka and Tsuruta, 1984 [28] shear flow of steam
.1 1- 1 1 1 1 HL I I I I HllT ~ I I 4I I.1 .... L I I I




Yamail and Merte, 1999 [15] centripetal acceleration, i.e. body force with high
Departing Droplet Size section of the literature survey.
I :JLI E I 1ll11 L : : 1 111 1-aIDI :

T4 I -- FI 4IIIIII-1 -F F 4-1 -1T -1 I]F I T I I 7 1-
1 I I II H I It n l I I I Ti TI I I T T I I 5 F .
I T tIIlII-It t T rl IrIi t t t T l7 T 7 TI I I II lI T T li IIIIIII I I I IIIII
I I II Hi ll I I I I I I I I I I I -II I III I H II I III H H -I I I I I IIII



10-5 10-4 10-3 10-2 10-1 100 101
Max Droplet Radius, [mm]


Figure 2-7. Heat transfer coefficient vs. maximum departing droplet radius for Tsat =
380K and AT = 0.1, 0.5, 1.0, 5.0, 10, 20, 50K.

investigations have used different methods to attempt this, as did the present study. Most

of the investigations by Rose have focused on a vertical plate in essentially quiescent

vapor. Therefore these studies did not distinctly examine the effect of maximum droplet

size. The following studies however have all examined maximum droplet radius effect in

one way or another:

* Citakoglu and Rose, 1969 [25] varied surface inclination

* Tanasawa et al., 1978 [26] periodic wiper applied to the test surface

* Tanasawa and Utaka, 1983 [27] shear flow of steam

* Tanaka and Tsuruta, 1984 [28] shear flow of steam

* Yamail and Merte, 1999 [15] centripetal acceleration, i.e. body force with high
subcooling range

The results of these studies will be examined in relation to the Rose theory in the

Departing Droplet Size section of the literature survey.












102





10
+ 1- I H + tl +l 4-HI t t t +I41+1- --- 4 + IH I- t l lHHR I- t 141 1-
T --- I ]FI1 T77T rITI F 1- TTIT I TTFITI T I FT IT -T I-I- F17171-








TI- I I I i I increasing Tsat 1_ 1 I-_ I__I r It i r -I rr1_ -
I I I I I 1 1 1 1 1 1 H I I I I I I I1II I I I I I I I I I II I I I I I I I I I I I I I I I I I I







10 10 1 0 i I I I 10 10 0IIII i I I I I i10 I I I i
IL L -] -] 1 1 U l 1 I -L1I LI L L- I J It L- L 1 _1 L
T 2 1- 1 1 n F T -1 -1 T -1 1 ~- TIt ITI F J T FI F T T 1-1 F I- T I 17 171-










Figure 2-8. Heat transfer coefficient 1vs. maximum departing droplet radius. Surface

subcooling was constant t t AT 1K. Saturation temperatures used were 280, 1
1 1 DI -I I I I I H i l l I I I I I I I III I I IT I I I I I I I I II I I









320, 380, 430, 500K.
1 10 -1 1 01 1 Hi ll I 1 0I 1 1 0ill I
















Examining the effects of maximum droplet radius (reached by growing drops just


prior to departure) for a constant AT of 1K again exhibits the dip in HTC and therefore


HF for sub-atmospheric pressures (Figure 2-8). Comparing HTC for a given departing

droplet radius, over a range of subcooling showed significant dependence on this


quantity, as well as on saturation temperature (Figure 2-9). This significant decrease of


HTC with pressure was noted by Rose to be primarily due to the increase in the


interphase matter transfer resistance. This quantity was the second term in the


denominator of the expression for the HF through a single drop (Eq. 2-6).


The Rose theory predicts linear HF, in log-log coordinates, for any practical range


of maximum departing droplet radii (greater than 10-3 mm) (Figure 2-10). Of course,


smaller maximum departing droplet radius yields larger overall HF. Also, many smaller

















Tsat = 500K
To = 50K


101



CM1
E
S100


1I I LI I I J 1 U l11

Tsat = 380K
AT = 50K


Tsat 280K


10-5 10-4 10-3 10-2 10-1
Max Droplet Radius, [mm]


100 101


Figure 2-9. Heat transfer coefficient vs. maximum departing droplet radius for Tsat =
280K, subcooling is 0.1 and 5.0K. For Tsat of 380 & 500K, subcooling is 0.1
and 50K.


CqJ
E
100

LL-
TI


-I---^-Z^-Zrnc^-~ ~ ~ ~ ~ Z IE 33 3 [I I I _^&E DDE^I E 0I
ST T Ti I I T1Ti I I T rI TI I rI r
I I I I I I I I IIH 1 1 1 1 1 1 1 I I I II IIH H I III I I I I I 1I I I I
I- tI I tItIIIIII I I Itil I t lIII t 1 tft t u
I- +I I + H I IIII I I 4 1+ 1 I I I IIIII IIIII
+ + H -- + 1 1 +141i -1 4 + I ++I I H H ~ I 1 141


t I I I l I t I I II I I I I II I I I ti II

I T= 1 1 = E I n cre sin- rA1= TTI- 1 : r r =
+ + -1 1 H 14 -1 1 +14 4 1 4 4 1-+ Hl-^^t+ H H 4 --I-I 1- 14 141!
1_ -l k-l 1 4 14 l l l l 14 l 4 4 1 llll 1 : F I 4 1- -I 14 1 1

== _MI =^ ^ i increasing A l=== `= z


10-3 ..
10-5 10-4 10-3 10-2 10-1
Max Droplet Radius, [mm]


100 101


Figure 2-10. Heat flux vs. maximum departing droplet radius for Tsat = 380K and for AT
= 0.1, 0.5, 1.0, 5.0, 10, 20, 50K.











drops provide less conduction resistance and larger total surface area for direct contact

condensation than fewer large drops.

When theoretical HF is compared to maximum departing droplet radius for several

saturation temperatures, successive increases in Tsat have less effect while the linear (in

log-log coordinates) trend with droplet size remains for each case (Figure 2-11).


102
E I-I aInI-T--rTIrTI FrTITITI ITT ITI] ITTI ]IT I ]I-MM



10
I I I I I I I I I I I I I I I I I II I I I I I I I l l I I I II III I I 1 I I I I I I



1 I 10 -'l I -L I-I+II -1J J- -I_ _- HLLI -l I -dL 1- I H 1
1 1 I I j I I- IIIII I I IIII I I I I IIIII U I t rII I
M I I T4 TIITI T I ITI 4-T 1 1-- -1 1 1-1 T4 I -1-1 1 417-rl
5 I I I H I I I I I I I I I I 2II I I I
I 10 + -1 -1 1 1 H + I +l 11 1 -1 -1 1- I- + + I I+ 1 I I 1 1I H T I-
10 104 10 1 10 I I I I I I 10
Figure 2-11. Heat flux vs. maximum departing droplet radius at AT 1K and for















Comparing overall HF with maximum departing droplet size (Figure 2-12) gives
10function appears more sensitive to subcooling than either of the other two control factors,
maximum departing droplet radius and saturation I t temperature. To illustrate this, a steady

4 ordeT r of magnitude increasing over the range of T Tconsidered can be observed here for a I 1



given10-5 maximum d 0-4 10-roplet3 10-2 10and 10 10considered.
Max Droplet Radius, [mm]


Figure 2-11. Heat flux vs. maximum departing droplet radius at AT = 1K and for
saturation temperatures of 280, 320, 380, 440, and 500K.

Comparing overall HF with maximum departing droplet size (Figure 2-12) gives

special emphasis to the dominating effect of surface subcooling. In fact, the overall HF

function appears more sensitive to subcooling than either of the other two control factors,

maximum departing droplet radius and saturation temperature. To illustrate this, a steady

4 order of magnitude increase over the range of AT considered can be observed here for a

given maximum droplet radius and for the vapor temperatures considered.











104



102


E 10 L L Tsat = 380K, AT = 50K
Tsat = 280K AT1 4 5K -I -
4 -1-1 IH --I-- 4HI4 I r I IT I- 4 4 ITI]] -- IT- Iq I I 1 I14I4-
2 1__I IIIIII I I IIII... I I II I IIIIIII I I I IIIII I I I I1




Tst =- 380 500K I 11
L I LI LII J _LU LJJLII L -IL 11-tI I L LIJI I L L LI_ I-









E 10 I1 Tsat = 280K, AT = 0.1K


10 10-4 10-3 10-2 10 100 10
Max Droplet Radius, [mm]


Figure 2-12. Heat flux vs. maximum departing droplet radius for extreme parameters.
For Tsat of 280K, subcooling is 0.1 and 5.0K. For Tsa of 380 and 500K,

subcooling is 0.1 and 50K.

The smallest thermodynamically viable droplet (Eq. 2-12) was dependent on fluid

properties, saturation temperature, and the condenser surface temperature (Figure 2-13).

Note that the scale was in nanometers, an indication of the 106 to 109 ratio of droplet radii


active in the phenomenon of DWC. The range of the maximum departing droplet radius

(Eq. 2-11) only changed a fraction from 1.1 mm at 280K to 0.77 mm at 500K. It is

indeed the high number of these small drops combined with their small conductive

resistance to HT that affects the relatively high 1T efficiency of DWC over FWC.
resistance to HT that affects the relatively high HT efficiency of DWC over FWC.











103
10 ......... ----- ------- -- ---------- --------- ZZZZ
Increasing AT


102



E 101 4 -




----------250 300 350 400 450 500----------













temperatures of 280 and 500K, for which it is 1.1 and 0.77 mm, respectively.
he- -- -2- I- 2-14
- - - - I - -




10 --------- ---------- ---------- --------- ---------


1-1---------------------------------------------
I - -
0 -- -- -- -- - -




250 300 350 400 450 500












{ ---------- r



















yis the condensation coefficient. This quantity represents the fraction of vapor

molecules striking the vapor/liquid interface which remain in the liquid. This value has









been a topic of considerable uncertainty until the last decade. g is the acceleration of

gravity. hfg is the heat of vaporization. Psat is the saturation temperature. R is the gas

constant of the vapor. Tg is the vapor temperature. r* is the critical drop radius spoken of

above. R is a given droplet radius. The author assumed that this expression for the HTC

on the surface of a drop was constant over its surface area. He also noted that there was a

critical drop radius, r*, above which the thermal resistance of conduction in the drop

dominated the HTC. By considering a drop sufficiently larger than the critical size to be

adiabatic along with the relatively high HF in small drops immediately around it, he

sought to estimate the resulting lateral temperature gradients in the condenser surface, of

a given thermal conductivity, and from that estimation, the effect on overall HTC

(Eq. 2-15).



I-= K 2khJ / JA + -12 -- 2-15
h 4& 2k (2-k) 3.7k A
L hJ + Lr

r is the drop radius. kf is the liquid thermal conductivity, he is the interfacial matter

transfer resistance (Eq. 2-14). B is the fraction of area covered by visible those with

radius above critical drops. ks is the surface thermal conductivity. C is the radius of an

inactive drop. A is the total area of the surface. The major assumptions of this theory are

as follows:

* Known droplet size distribution, which at that time was only being qualitatively
guessed at

* HTC independent of subcooling (contrary to Rose Theory)

* Negligible circulation in the drop

* Negligible drop in saturation temperature due the curved surface of the drop
(contrary to Rose theory)









* A pure vapor, i.e. no NCG present

No experimental verification of this theory was offered. In fact, pains were made

to note that this relation was not intended to be calculable, merely representative.

However, based on analysis of published results, the author did venture to estimate that

the constriction resistance in a stainless steel was about 84% of the total resistance to HT

versus 20% for copper.

This theory was not specifically tested in any experimental studies reviewed for

this paper. However the topic of constriction resistance continues to be contentious in

current discussions of DWC. This topic was addressed in more detail in the Constriction

Resistance section of the literature survey.

Few alternatives have thus far been offered to the Rose theory. Subba and Murthy

in 1983 [30] offered a model for DWC HT based on the opposite assumption of Rose.

They assumed that the drops only served to insulate the surface and therefore all the HT

took place on the bare surface between drops. Their investigation was purely analytical.

The results were compared to those of Le Fevre and Rose [31], Citakoglu and Rose [32],

and one other. The analysis also attempted to predict the effect of NCG on DWC HT.

Both of the comparisons were claimed to be good, but again no offer was made of

uncertainty. The shape of the curve was adjusted by assuming different functional forms

of the condensation coefficient, some dependent on HF and some not, which caused the

theoretical trace to alternately bound the data and to cross through it, however, which of

the three adjustments gave the best fit was not noted. No other instances of this theory

were found in the cited sources for this paper.

Tsuruta and Tanaka, in 1991 [33], presented a numerical simulation somewhat

based on the same arguments as [29]. Additionally, they offered a non-dimensional









model that accounted for temporal variation in drop size and departure as well as

considering the area underneath large drops as an adiabatic cylinder. Experimental

verification of that theory in was offered in [34]. This theory was considerably more

complex than the Rose theory, as it attempted to model the temporal variations in surface

temperature as well as the overall HT. It was based on the Biot number of the surface

and the non-dimensional drop radius. The experiments used thin film resistive

thermometers to directly measure the condenser surface temperature of carbon steel,

stainless steel, and quartz glass. The investigators found decreasing HTC for decreasing

surface thermal conductivity. They also offered a Nusselt number correlation involving

only three variables, maximum droplet radius and surface and liquid thermal conductivity

ratio. The reported agreement of the correlation with their data and that of Hatamiya and

Tanaka, 1987 [35] was 20%. The condenser surface was pure copper. The

experimental portion was conducted at low vapor pressure to limit the lower drop radii to

a size visible with micro-optics. Results showed condenser surface temperature

fluctuations of up to 30C which increased in magnitude with HF. The troughs and

peaks corresponded to a droplet reaching its maximum radius and then departing as

confirmed by filming of the surface during test runs. The coldest surface temperatures

occurred under large drops, since the surface was insulated. Conversely, the warmest

temperatures occurred just after a drop departure or merger exposed bare surface,

indicating a spike in HT.

The temperature sensors were deliberately made smaller than the expected drop

radius, and were clearly visible as completely covered by a sessile drop on the condenser

surface. This study considered only small surface subcooling, between 1.0 and 3.5C and









assumed no dependence of HTC on subcooling. Comparison with data was limited,

though agreement of theory and experiment was demonstrated in the examined parameter

range.

Song et al., 1991 [36], asserted that a film of condensate existed between drops,

which would contribute to heat transfer. Spectral reflectance measurements taken before

and during condensation were purported to prove this existence due to their difference.

No estimate of the thickness of this layer was offered, but it was estimated to be on the

order of a few monolayers, that is a few times the width of a single condensate molecule.

Curiously, spectral measurements were presented, with wavelength on the abscissa, as

evidence of the occurrence of a single temporal event, pointing out a loosely repeating

spectral pattern as evidence of droplet departure. The fact that the coordinate domain of

the analysis was in question somewhat shadows the conclusions.

Gavrish et al., 1993 [37], presented a numerical simulation that modeled the

process of DWC considering only the conduction through the drop, surface thermal

conductivity, and the effects of sweeping of departing droplets. This therefore neglected

both the interfacial matter transfer resistance and the small drop in Tsat due to the curved

droplet interface considered in the Rose theory. The results, while in agreement with

their own data for conductivity vs. HTC, were lower than most of the compared studies.

The correlation of data with the model was labeled only as good with no quantitative

measure of agreement offered.

Yamali and Merte, in 2002 [16], conducted an involved numerical simulation of

DWC, considering high body force (1 to 100g) as a control of maximum droplet diameter

and high subcoolings, from 10 to 100K. This followed their experimental study of









similar conditions in 1999 [15]. Their model assumed that most of the HT took place at

the edge of the droplets, where the temperature gradient was largest and that furthermore

most of the HT also took place through drops of very small radius due to their shorter

conduction path and therefore higher gradient in temperature along with their high

population numbers. They started with a complex relation for HF through a droplet

based on Tsat, AT, and contact angle and other elements similar to the Rose theory. They

then modeled the temporal growth and departure process over the entire condenser

surface, accounting for the total volume of condensate flowing down the surface. Most

previous studies had assumed that departed drops simply disappeared, a measure largely

taken because of computational limitations. They also accounted for coalescence of

growing drops and looked at the results of the model with and without the effects of

sweeping. They then tracked the overall HT by the total volume of condensate at each

step. The result of this simulation was to corroborate several of the trends predicted by

Rose, namely that HF quickly reached an asymptotic value as a function of surface

length, x, and so was essentially not dependent on this quantity. Another similar

conclusion was that HTC was indeed dependent on HF instead of constant, as assumed

by all the competing models mentioned above. A unique finding for very large body

force (above 50 g), large subcooling (above 70K), and large x, a peak was found in HF

that was analogous to a critical heat flux in pool boiling. However, the parametric

boundaries for this phenomenon were so high as to not be of much practical concern.

The simulation results compared well with the earlier experimental results [15], within

+15%.









Droplet Size Distribution

A governing assumption of the Rose theory was existing knowledge of the

population distribution of droplet radii. Much work has been done to estimate a

sufficiently accurate model of this characteristic since then. The numerical simulation

results of Glicksman and Hunt, in 1972 [38], and Burnside and Hadi, in 1999 [39],

supported a simple negative exponential relation for the droplet size distribution, with

much higher populations for smaller drop sizes. The former offered -1/3 as the

exponential value, later selected by Rose for application with his theory. Some recent

studies have employed more complex methods to model the droplet size distribution,

such as the random fractal method [40] and mathematical morphology [41]. The

resulting comparison to experiment was not significantly improved over the assumed

simple exponential size dependence.

Rose and Glicksman, in 1973 [19], performed a close visual study of the temporal

development of the droplet size distribution and had these main conclusions to offer:

* There were distinct generations of drops based on what areas were swept by falling
drops.

* Drops of any generation were more or less uniform in size and spacing.

* The density of nucleation sites was independent of droplet size and the same for all
generations.

The first assertion was backed up by the findings of Burnside and Hadi, in 1999

[39]. "Nucleation sites" refers to the microscopic surface defects that serve as bases for

droplet nucleation. Numerical simulations have estimated them to be closely packed in

either square or triangular arrays at a spacing of roughly one minimum

thermodynamically viable droplet diameter with good comparison of results with









experimental data. The resulting site density estimates were 109 and 1011 sites/cm2

[38,28].

Wu et al., in 2001 [42], conducted a microscopic study, at several magnifications,

of low pressure DWC and found that a self-similar pattern in relative drop sizes existed at

all magnifications. That is, that pictures at different magnifications looked almost

identical but for different exact positions of drops.

As far as other parameters that might affect the distribution of drop sizes, HT

parameters or surface roughness were eliminated [19,35], except for the conclusion that

an increasing HTC with increasing subcooling was due to increasing activation of

nucleation sites, specifically below 1K subcooling [43].

Rose [23,44] offered some modifications of the bounds of integration in his theory

for a better fit to experimental data. Specifically, he suggested replacement of the lower

bound of the integral in his theory with 10 instead of 1 r The upper bound adjustment

was two part with 1.5 i replacing 1 i below 1 atm while substituting 2i for pressures

near or above atmospheric. However in later issuances of the theory [12,18,20,24,45,46]

these suggested adjustments were not repeated.

Effect of Maximum Droplet Size

Overall HT has been found to depend on the maximum, or departing, droplet size

[15,27,37,38,39,42,44]. The only citations not to note this dependence were [25,26],

however the range of drop sizes in these studies was relatively small (much less than an

order of magnitude change). Various methods of controlling the maximum droplet size

have been employed, ranging from numerical simulation [38,39] to a periodic wiper on

the condenser surface [26] and condenser surface inclination adjustment [25]. However,

the clearly dominant method employed was to induce significant vapor shear flow on the









condenser surface, such that the drag on the sessile droplets would limit their maximum

size [27,37,47]. This method had the additional benefit of limiting the influence of NCG

in the test vapor. Yamali and Merte, in 1999 [15], noted an increase in HTC with body

force, but only for very high accelerations (i.e., above 5 g). This dependence was most

likely due to a corresponding decrease in maximum droplet size, which was determined

by the balance of body force and surface tension on the drop. These authors found a 6-

fold increase in HTC for a 100-fold increase in body force.

In support of the idea that smaller droplets are more active than larger ones in HT

[28], Tanasawa et al., in 1978 [26], noted a peak HTC of 1 MW/m2K just after the

passing of the wiper, which was 2.5 to 3.5 times the time averaged value. This peak then

decreased linearly until the next wiper passing. In a numerical simulation [39], a peak

HTC of 2 MW/m2K was found. These high peak HTC values indicate the potential

benefit of controlling maximum drop size in order to induce more intense HT than

standard DWC.

General Results on Drop-wise Condensation (DWC)

Lorenz and Mikic, in 1970 [48], offered a numerical simulation of internal

thermocapillary circulation in a liquid drop involved in DWC. The results showed that,

for the applicable size range, internal circulation in the droplets was negligible. This

finding justified one of the main assumptions of the Rose theory. Another main

assumption of the Rose theory that promoter thermal resistance was negligible was

confirmed by several other studies [17,23,49,50]. Two other studies found no

dependence of DWC HT on tilt angle of the condenser surface for small deflections from

vertical [25,31]. Neither of these two studies or [17,19] found any dependence on surface

height for HTC. However, surface height dependence of HTC for DWC was noted [14]









merely by using a rectangular condenser surface for a series of results, then rotating the

apparatus 900 about its central, horizontal axis and finding different results. It was

unclear whether the orientation of the steam flow remained the same relative to the

condenser surface or not, which may have accounted for the difference in results between

the two cases. The condensation coefficient, contained in an empirical constant in the

Rose theory, was defined as the fraction of vapor molecules striking the liquid surface

which remain in the liquid phase, was the subject of much contention in the early decades

of DWC research. Prior to 1985, most estimates of this fraction ranged from 0.45 to 1

[28,51,52], which left a considerable gap in the understanding of the process. Roughly

after this date, reported results coalesced to unity [20,24,53].

Dr. Rose often argued that distinct differences existed between the DWC

performance of various promoter methods and chemicals [13,21,31], which was

corroborated by the work of others [54,55]. This assertion was fundamental to his

argument against most of the evidence presented for a dependence of DWC HTC and HF

on the condenser surface thermal conductivity, or constriction resistance, which will be

discussed below.

An observed result of the efforts of Le Fevre and Rose to minimize the presence of

NCG in the test apparatus was that HTC increased with increasing HF, or, less precisely,

with subcooling [31]. This trend has been reported elsewhere [15,32,45,54]. Necmi and

Rose noted [56], however, that for DWC of mercury, HTC was essentially independent of

HF, due to the high thermal conductivity of the condensate. In [31,43] this trend was

specifically labeled as an indicator of the NCG content of the test vapor. If HTC

increased with HF, then NCG was termed sufficiently minimized. If, however, the









opposite were found true, then this would indicate poor control of NCG in the test

apparatus. The latter assertion made sense, since the concentration gradient set up by the

presence of NCG in the condensing saturated vapor, would tend to be exacerbated by

greater relative concentration of NCG. This problem will be discussed in more detail in a

section dedicated to the effects of NCG on condensation. However, some results have

asserted HTC to be independent of HF [4,35]. It should be noted that studies making this

case often tested only over a short range of HF or subcooling. With such small parameter

ranges, a linear relation might be easily inferred between these primary variables. Those

experiments covering large subcooling ranges, cited above, along with other theories

[16,17], did show this trend.

The high HF and HTC values reported for DWC [31] were reported in a range of

0.3 to 1.8 MW/m2 at 1 atm, with a maximum HTC of 0.3 MW/m2K. Tanasawa and

Utaka, in 1983 [27], reported HF from 0.5 to 10.0 MW/m2 at subcooling from 0.8 to 50K,

near 1 atm. As predicted by the Rose theory, a decrease in HTC with decrease in

saturation pressure was reported by [17,18,24,35,55,56]. This decrease was noted to be

up to 6-fold for DWC of steam at the lowest saturation pressures, as compared to

atmospheric saturation pressure [35,55]. Rose, in 1998 [24], attributed this behavior to

an increase in the interphase matter transfer resistance between the decreasingly dense

vapor phase and the liquid phase. Significantly lower HTC was also noted at lower

pressures [28,47]. For comparison to FWC, the results for DWC HT range from just 2 to

3 times higher [57], to a range of 5 to 10 times [37], to very high values of 20 to 30 times

the FWC, as found in experiment and agreement with Nusselt correlation (Eq. 2-1)









[58,59]. The lowest DWC HF values were attributed by Rose to poor quality DWC

promotion.

Some unique findings were those of Koch et al., in 1998 [14], that partial condenser

surface coverage (of only 20%DWC/80%FWC) produced almost 90% of the full DWC

coverage value of HF. Also, Citakoglu and Rose, in 1969 [25], used their surface

inclination dependent measurements of DWC HTC to conjecture that the average HTC

for a horizontal circular tube would be about 80% of a vertical plate under the same

conditions. Finally, a point of note in all of the presentations which involved Rose, the

HTC was only plotted as a function of HF and not surface subcooling. This distinction

was made because of the error inherent in measuring this often small difference between

the vapor saturation temperature and the surface temperature and then relying on its

reciprocal for the HTC. For the same reason, he advocated only plotting subcooling on

the abscissa or y co-ordinate and the HF on the ordinate or x axis for graphs comparing

the two quantities [12,13,17,18,20-25,31,32,45, 46,52,53,56].

Constriction Resistance

There has been considerable contention, in the discussion of DWC, over whether

the overall HT was at all dependent on the thermal conductivity of the condenser surface,

that is whether a constriction thermal resistance existed or not. The Rose theory [17,24,

45], employed in the present work, assumed any dependence of HF on this property to be

negligible, and obtained "excellent" agreement with experimental data for an array of

promoters and for a few different surfaces. One contention of Rose, stemming from the

observed differences in promoter effectiveness [31,60], was that surface chemistry was

critical to the performance of any DWC scheme and might easily be the contributing

factor in those studies which claimed affirmative evidence of such a dependence [21].









The authors used a thin, on the order of a few microns, copper layer on a stainless steel

condenser block and compared the results for DWC with a solid copper block, finding a

difference in HTC only on the order of magnitude of the uncertainty in this value. The

thin copper layer on the steel block was assumed to offer identical surface chemistry to

solid copper block, which was the key to Rose's argument that surface chemistry was the

actual critical factor and not the substrate thermal conductivity.

Over the decades of research on this topic, various researchers have found their

results to indicate a dependence of overall HT in DWC on the condenser surface thermal

conductivity. Most did not account for potential differences in surface chemistry

[33,34,37,50,54,61]. The latter pair of references offered a Nusselt correlation for DWC

dependent only on thermal conductivity of the substrate and condensate and on the given

non dimensional droplet radius that agreed within +20% of the accompanying

experimental data. The same two studies also noted an increased importance of

constriction resistance at sub-atmospheric pressures. Two other studies [14,43] found a

significant constriction resistance while controlling for surface chemistry. However,

discussion of these findings did not include any mention of the inherent uncertainty in the

small temperature difference measurements and the resulting implications when used in

the denominator of the HTC. Mikic offered a theory including a constriction resistance

[29] and then a study looking further at this theoretical dependence [62]. He then

compared the theory to experiment [61], with visually agreeable results. However, the

discussion of uncertainty, as mentioned above, was also omitted from this presentation.

In his 1998 address on the topic of DWC, Rose reiterated his opposition to the

importance of a potential constriction resistance, while still acknowledging the existence









of debate on the topic [24]. In 2002, he went further to offer that it may be significant, in

an additional AT of 1 to 2K, but only for very low conductivity surfaces and low HF, less

than about 0.1MW/m2, and small condenser surface lengths, less than about 10 mm [18].

In that talk, and also [21], he dispatched the findings of most of the studies cited above,

which had all found evidence of a constriction resistance. He did this by questioning

their methods of surface temperature estimation in relation to uncertainty and the

magnitude of the difference in HT supposedly caused by the constriction resistance.

Another issue he cited to question their assertions on constriction resistance was surface

chemistry differences, which none had addressed.

Thus the literature was still quite divided on the existence and importance of

constriction resistance. However, the author of the main theory, the investigator most

often invited to speak on the topic of DWC, maintained that the potential importance of

constriction resistance need only be considered under certain, limited conditions.

Drop-wise Condensation (DWC) Promotion

For most applications of DWC, some sort of promotion method was necessary to

effectively reduce the wettability of the condenser surface, in other words to decrease the

contact angle of the vapor-liquid-solid interface or contact line. Three main modes of

surface preparation were used in the literature:

1. Direct application of organic or other compounds to the condenser surface

2. Addition of these same promoters to the bulk liquid reservoir supplying the

steam to the condenser surface

3. Physical or electro-chemical manipulation of the surface makeup

Organic promotion, as the first group will be referred to, yielded the highest HTC

values. This promotion mode was durable but only under clean laboratory conditions.









This was the mode most often employed in the study of DWC phenomenon itself.

However, this method later fell out of use due to its sensitivity to contamination usually

present in industrial conditions. The substances employed in organic promotion included

montan wax [55], montanic acid [55], dioctadecyl disulfide [31,55], oleic acid [21,25,32,

63], and stearic acid [64]. Self assembled organic monolayers of 1-octadecanethiol and

16-mercaptohexadecanoic acid were also tested [65], but the lifetime of DWC promotion

was only on the order of a few hours. Tanasawa et al., in 1978 [26], employed a silicon

resin with some success as a promoter.

Some of the same substances were applied by addition to the bulk liquid reservoir,

by simply dropping a few ounces on the liquid surface, as was done with dioctadecyl

disulfide [61], 2-ethyl-1-hexanol [66], and with partially-fluorinated carbon disulfide

[37]. The latter case was tested with several condenser surface substrate materials. The

investigators reported lifetimes of up to 4200 h. Furthermore, with this promoter the

DWC promotion regenerated simply by adding more of it to the reservoir. This

regeneration however could only be induced for a few cycles before disassembly and

cleaning were required. The bulk addition was also coupled with the direct surface

application of the organic promoter with good results by other researchers [55].

According to Dr. Rose, in 2002 [18], durable promotion of DWC still remains a

"key barrier to commercial implementation." Modification of the physical makeup of the

condenser surface has been the focus of most of the work in recent years pursuing this

goal. A thin plating of noble metals has shown durable results in a laboratory

environment using gold [15,35] and silver [67]. The latter showed a considerable 2400 h

lifetime. The effectiveness of these noble metals as promoters has been attributed to their









adsorption of trace organic elements from the air, as their promotion ability was notably

absent when air exposure was not allowed between cleaning and testing.

Chromium was studied as an alternative to the noble metals for a condenser surface

[35,63]. The best results came from a powder formed chromium condenser block,

vacuum impregnated with oleic acid. This surface preparation showed a 3000 h life of

DWC, under laboratory conditions.

Koch et al., in 1998 [14], used amorphous hydrogenated carbon films, described as

a diamond-like coating. The physical toughness of these layers was held to offer

potential for a durable promotion method, though no lifetime tests were conducted.

Contact angle was also purportedly controllable in a small range, through the processing

conditions of the coating.

What was described as a Langmuir-Blodgett (L-B) film showed much promise for

a durable promotion method [59]. Several monolayers of barium stearate were overlaid

on the condenser surface in such a way that the negative pole of the molecules all pointed

away from the surface, resulting in a highly non-wettable surface. DWC heat transfer

coefficients were reportedly up to 30 times higher than FWC. The coating was also

reported as very physically tough and insoluble in both oil and water. This fact was quite

significant since barium stearate was soluble in both when not prepared in this manner.

The coating was also reported to be resistant to frictional wear and evaporation in a

vacuum. No lifetime tests were conducted.

Dimensional modification of the condenser surface to increase sweeping frequency

was employed by cutting vertical half-cylindrical section grooves in the condenser









surface [68]. However the increase in HT was at most 25% and was attributed only to the

increase in surface area, despite a noted increase in sweeping frequency.

Teflon coating has been studied as a DWC promoter [47] with propylene glycol as

the working fluid. Bochagov and Dorokhov, 1983 [57], induced DWC of steam on a

Teflon coating with less success, likely due to presence of NCG in the vapor. More

recent studies have employed techniques involving ion implantation, to reduce the

wettability of Teflon [69, 70], and copper [58]. The Teflon results were poor due to bad

physical bonding of the coating with the substrate, but the copper results showed promise

in effectiveness and durability. Here a beam of ions was reflected from a target, made of

a given material, onto the condenser surface to be modified, where the reflected ions

combined with the metal atoms of the surface to form an 'amorphous crystalline state'.

This amporphous surface had a much lower surface energy than a pure metal surface and

therefore showed much lower wettability. Ma, et al. [71], cited a study maintaining that

minimization of surface energy was a more reliable method of assessing potential

promotion effectiveness than contact angle or wettability. Species tested for

implantation were He, Ar, N, and H. The H surface was reportedly the least wettable,

with He and Ar next in line and roughly equal, and with N being the most wettable and

therefore least effective DWC promoter. The HT results for the tubes used were on the

same order of performance as organic promoters such as oleic acid, but with much longer

lifetimes and better resilience.

In 1994, Zhao and Burnside [72] conducted the only successful test of a DWC

promotion scheme under industrial conditions. An 800 tube bundle, treated in the

manner described above, replaced a 1600 tube bundle of similar tube diameter and length









in a power station in China. Cr and 0 were the elements chosen for implantation in this

test on Cu. Over the 15,600 h length of the test, DWC was never lost and the test was

purportedly only curtailed due to the primary researcher's graduation. The authors of this

study felt secure enough in their results to maintain that they indicated the possibility of

permanent and reliable DWC promotion. The processing of the surface coating,

however, was as yet prohibitively expensive.

In a review paper in 1999, Rose noted these promising developments [45], but he

also noted was the downward trend in HTC with increasing HF. Rose had long held this

decrease as an indicator of the presence of NCG in the condensing vapor, its presence

thereby calling into question the accuracy of the reported data. By 2002, he had reverted

to his contention that, "...reliable means for promoting dropwise condensation has yet to

be convincingly demonstrated," [18]. In 2004 [20] he elaborated that more stable

promoters had been developed but the reported HTC from these studies were still well

short of the order of magnitude increase over FWC expected from monolayer organic

promotion.

As a side note on DWC of liquid metals, Rose noted that promotion was often not

necessary for these species [24]. He also noted however that the typical advantage of

DWC over FWC was greatly reduced due to the very high thermal conductivity of the

condensate, relative to a typical working fluid such as water.

Effects of Non-Condensable Gases (NCG)

Condensation is ideally studied with only a pure, saturated vapor supplied to the

condenser surface. This state can be quite difficult to achieve in reality, since removal of

non-condensable gases, or NCG, usually air, is effectively an asymptotic process rather

than a discrete one. At equilibrium, the condensable vapor would have the NCG









molecules evenly distributed through the volume. However, once the process of

condensation is initiated, a net mass flow toward the condensing surface is induced. The

resulting movement of the bulk sweeps the NCG molecules with the vapor flow toward

the surface. However, at the surface, the vapor molecules condense, but the NCG

molecules cannot. In this manner, a continuous process builds up the concentration of

NCG at the condensing surface [73]. Since pressure is a continuous property, the partial

pressures of the two species must add up to the vessel pressure at any point in the vessel

volume. Therefore, the increased partial pressure of the NCG at the condenser surface

forces a decrease in the partial pressure of the vapor species at the condenser surface.

Since, for a saturated liquid-vapor state, pressure and temperature are dependent

quantities, the reduced partial pressure of the vapor species at the liquid-vapor interface

leads to a decrease in the apparent saturation temperature.

If the driving force of the NCG buildup at the condenser surface were removed,

that is, the cooling of the condenser ceased, the NCG molecules would disperse by

diffusion, until they were evenly distributed in the vapor bulk again. However, if the

cooling of the condenser proceeds at a steady rate and the geometry is 1-dimensional,

then an equilibrium partial pressure, higher than that in the bulk of the vapor, will

develop. The resulting concentration gradient then drives the diffusion of the NCG

molecules against the net mass inflow of vapor. If the geometry of the problem is 2 or 3

dimensional, the NCG build-up and departure causes significant transients in the HT

readings.

In 1965, Kroger and Rosenhow [74] modeled the 1-dimensional diffusional

resistance problem of FWC in the presence of a non-condensable species, namely









potassium vapor in the presence of He and Ar, separately. Their simple model showed

agreement with experiment within 25% for He. Results were less positive for the runs

using Ar for the NCG. Steady state was never satisfactorily reached, the theorized reason

for which was a periodic buildup and departure of Ar at the condenser surface, which

would serve to insulate the condenser surface as they buildup and then allow HT to spike

when they depart.

Many simulations of FWC have been conducted, growing from models similar to

the diffusional one offered by Kroger and Rosenhow. These studies showed that

geometry was a major factor in the sensitivity of a given condensation process to the

presence of NCG. For closed channel flow, FWC, reductions in HT from the case with

no NCG content of 30% to 60% were reported for simulations of 1% to 10% NCG

concentration, respectively [75]. In a similar case to these studies, it was reportedly

necessary for NCG concentrations to be as low as 0.1% in order for there to be no

discernable effect upon the HT [76]. For FWC in a porous medium, a 50% decrease in

HT for 5% NCG concentration was reported [77]. For closed channel flow geometry,

only 3.3% NCG reportedly produced the same reduction [78]. A theoretical investigation

of FWC on the inside surface of a rotating drum in [79] showed a process much more

sensitive to NCG relative to those discussed above. Here a 1% NCG concentration

produced an 80% reduction in HT. This increased sensitivity was explained as the

centripetal acceleration due to the rotation of the drum increasing the resistance to

diffusion of the NCG away from the liquid-vapor interface. The bottom line to all the

above findings was that the effect of NCG in the condensing vapor on FWC was to

impede the condensation process. Furthermore, this impediment was exacerbated when









geometry of the situation restricted diffusion of the NCG molecules. This conclusion

makes sense in light of the diffusional problem described above. Parametric effects noted

by these studies were as follows:

* More pronounced effect of NCG at lower pressures [75]

* More pronounced effect of NCG for restrictive geometry [75,77,79]

* Vapor with NCG has lower saturation temperature [76]

Le Fevre and Rose, in their 1965 presentation of experimental results on DWC of

steam near 1 atm with organic promoters [31] and also the ensuing presentation of a

theory on DWC [17], noted an extreme variance in the reported HT characteristics found

for DWC. They attributed this variance to the presence of NCG in the test vapor. While

not controlling directly by addition of NCG to the steam flow, the experimental study

instituted a "close venting" scheme. This effectiveness of this scheme was demonstrated

by varying the venting rate, to simultaneously minimize variation in the recorded surface

temperature and raise the recorded surface temperature. These two effects indicated

higher HT. In this case the surface temperature was found by extrapolation of the

internal temperature distribution of the copper condenser block.

Also in 1965, Tanner et al. [54] studied DWC near 1 atm with the controlled

addition of NCG to the steam flow. These investigators reported a strong dependence on

HF of the sensitivity of DWC to the presence of NCG and also showed that DWC was

significantly more sensitive to NCG than FWC. For example, for an addition of 90 ppm

of NCG to the steam flow, or 0.009%, a 50% reduction in HF was noted for 0.1 MW/m2.

However, for the same addition, but lower HF, 0.03 MW/m2, no reduction was noted

over the case without added NCG. For the higher concentration of NCG, 500 ppm or

0.05%, still very small relative to those concentrations generating a noticeable reduction









in FWC, a 70% decrease was noted for the higher HF of 0.1 MW/m2 and 20% decrease

for 0.03 MW/m2.

In 1968, Tanner et al. again studied the effect of NCG on DWC [55], but this time

for reduced saturation pressures from 0.03 to 0.1 bar or 0.8 to 3.0 in-Hg-abs. This

pressure range almost precisely coincided with the pressure range of the present

investigation. NCG concentration was verified as less than 1 ppm for the inlet steam

without addition, and varied up to 15000ppm, or 1.5%. The effect of this controlled

NCG addition on HTC was noted. The results showed that lower saturation pressure

made the HTC significantly more sensitive to NCG concentration. In some cases, any

concentration above 0.1% NCG yielded more than 50% reduction in HTC, clearly

showing the following:

* DWC is more sensitive to NCG presence than FWC.

* DWC HTC is more sensitive to NCG for lower saturation pressures.

* DWC HTC is more sensitive to NCG presence for higher HF.

There were several different methods encountered in the literature for controlling

NCG concentration in the study of DWC. The close venting strategy used in [13,31] was

also employed in [21,22,32,25], all of which involved Dr. Rose. His experience lead him

to note in a 2002 review [18] the 'extreme sensitivity' of DWC to the presence of NCG,

even down to a concentration of a few ppm. The most common method was simply

using vapor shear from the excess steam flowing past the condenser surface to sweep

away any accumulating NCG [27,28,34,35,37,43,47,54,55,61,63]. A drawback to using

vapor shear to control NCG was that, as demonstrated by the cooling curves constructed

by Tanasawa and Utaka, in 1983 [27], was that steam velocity had a convective

enhancing effect on the HT, impeding a pure analysis of the process of DWC itself.









Some, however, made no effort to control NCG [14,59,67], though most in this case did

not indicate any appreciation for the importance of NCG in this process. Yamali and

Merte [15] admittedly made no effort to control NCG, but it was estimated that the

extreme body forces used in shear on the condenser surface in their study effectively

accomplished what vapor shear would, by sweeping away NCG.

Additionally, increased sensitivity of DWC to the presence of NCG for increasing

HF was noted [45,47]. Rose asserted that a downward trend in HTC with increasing HF

(or subcooling) indicated a significant presence of NCG in the condensing vapor [24,31].

He later employed this assertion as a litmus test for the validity of DWC HT results [45].

This trend was corroborated by the contemporary results of Griffith and Lee [43].

Finnicum and Westwater, in their 1989 study of DWC on chromium [63] noted an

important point about verifying the effectiveness of DWC by consideration of the

increase in HTC over the FWC case. They noted that, if air was allowed to back up into

the test cell, by decreasing the heater input, the appearance of the condensate in the FWC

test changed to DWC, however the HTC was even lower than before the allowed lapse.

Heat Transfer with Vibration

Not much study has been made into the effect that vibration in general may have on

phase change HT. In fact only one study was found looking specifically at the interaction

of vibration with DWC. Some work has been done on vibration and FWC, but mostly in

the range of low frequency oscillations, such as would be found in power plant pipes or

on an ocean platform. In 1969, Dent offered experimental results of FWC of steam near

1 atm on a horizontal tube, subjected to vertical vibrations [80]. The frequency range

was from 20 to 80 Hz with amplitudes up to 0.17 inch. In this case vibration was found

to increase HT, but only by a maximum of 15%. That same year, Dent also offered









results for FWC of steam near 1 atm on a vertical tube, subjected to horizontal vibrations

in the frequency range from 22 to 98 Hz, with a resonance at 38 Hz. Amplitudes ranged

up to 0.5 inch at resonance, which also showed the peak HT increase of 55% over the

non-vibrational case [81]. In 1977 Brodov et al. [82] also looked at FWC of steam near 1

atm on a horizontal tube, subjected to vertical vibrations. The frequency investigated was

from 0 to 50 Hz, with amplitudes ranging up to 1.5 mm. They found a maximum

increase in HT over the non-vibrational case of 20%, and also noted that no change in HT

was observed without visually significant disturbance of the condensate film. In 1980,

Savel'yev et al. [83] looked at the effect of vertical vibrations on both a horizontal and a

vertical tube undergoing FWC of steam near 1 atm. The intensity of the oscillations was

given as up to 0.12 m/s, where intensity was calculated from 27fA, being the frequency

and A being the amplitude of oscillation. The results were notable in that certain

intensities showed a significant decrease in HT due to the addition of vibration, up to

40%. This decrease was noted as due to a vibrational anti-node causing the condensate

layer to thicken significantly, thereby impeding HT. The maximum increase in HT was

noted at 20%, which likewise corresponded to a node in vibrational mode that caused

significant thinning of the condensate layer. In 1986, Nishiyama et al. [84] looked at the

FWC of R-11 on a vertical tube subjected to oscillations likely to be experienced on an

ocean mounted condenser. They looked at 3 modes of oscillation, heave up and down

motion, sway side to side motion, and roll pivoting relative to vertical. The frequency

range investigated was from 0.1 to 1 Hz. The results showed no significant effect for the

first two modes, but up to a 15% increase in HT for the roll case, also described as a

pendulum motion of the vertical tube. This significant increase for the roll case was









notably accompanied by thinning of the condensate film due to the centripetal force.

This corroborated the findings of Brodov et al., 1977 [82], that significant visual

distortion of the condensate film was necessary for significant change in HT. In 1988,

Nishiyama [85] used the same apparatus and experimental parameters, this time adding

vertical flutes to the tube surface. The author again reported negligible effects on HT in

the heave and sway vibration modes and some increase in the roll mode, due to the

thinning of the condensate film.

The effect of vibration on phase change was also investigated by Oh et al., in 2002

[86]. This author looked at the influence of ultrasonic vibration on the melting process of

solid paraffin wax. In a rectangular block, heated from one side and insulated on all

others, the melting time was found to be decreased by up to 2.5 times. This increase in

efficiency of the melting process was found not to be due to increased HT from the heater

surface, but rather to be the result of acoustic streaming. Acoustic streaming is a

phenomenon in which the compression and expansion waves of the vibrations induce the

rapid formation and collapse of gas from the liquid, exciting bulk fluid motion which

thereby increased mixing of the liquid volume. In 2004, Kim et al. [87] looked at the

effect of ultrasonic vibration on pool boiling of FC-72 at 1 atm on a nichrome wire. The

results showed an increase of 80% in HT in the natural convection range, 5 to 15%

increase in the nucleate boiling range, and 5 to 30% increase in the saturated boiling

range. Acoustic streaming was again deemed responsible for the noted increases in HT

efficiency. The relatively more significant enhancement in the natural convection regime

was explained by the fact that, in the non-vibration case, there was no induced mixing

from vapor bubbles, while this effect was present due to cavitation and acoustic









streaming in the vibrational case. Since the nucleate and the saturated boiling regimes

already had the vapor bubble induced mixing present, the enhancement of mixing by

introduction of the ultrasonic vibration was less intense.

There was one study found, Shiraiwa et al., 2004 [88], which focused on DWC of

steam near 1 atm and the effect of perpendicular mechanical vibration of a vertical plate

on this process. Promotion was accomplished with oleic acid. The oscillations were

driven by an eccentric crank mechanism from 0 to 15 Hz with a fixed amplitude of 5 mm

peak-to-peak. The analysis was limited to three coolant flow rates, in a close cluster, and

only a small range of steam pressure. Also, no mention was made of the surface

temperature of the condenser plate or the material, leaving important ambiguity in the

analysis of the quality of the DWC occurring in the test. No specific attempt was made to

control NCG in the apparatus, a fact which showed in the low HTC attained of 11.7

kW/m2K. To attain a HTC this low, the Rose theory would require a surface subcooling

much less than 0. 10C, and much lower than any probable uncertainty in the calculated

quantity. This discrepancy between the Rose theory and the results of this study imply a

significant, depressing influence of NCG on HT in these experimental results. Despite

this discrepancy, the results clearly showed an increase in HTC of up to 40% due to the

application of mechanical vibration, while noting a measurable decrease in the maximum

droplet size and accompanying increase in sweeping frequency. An attempt at a

numerical model was made, based on several simplistic assumptions of proportionality

between maximum droplet diameter, growth time, and HTC, which predicted a much

smaller increase in HTC than actually found, and relied on the measured HTC as a base.

In sum, a positive increase in DWC HTC was shown in this study, but the quality of the









DWC was very low, asserted by comparison here to the Rose theory, and therefore the

magnitude of the effect was questionable. Also of note in this study was a finding that

temperatures at 15, 45, and 75 mm from the top of the vertical condenser plate were

essentially equal for all the vibrations, corroborating a central assumption of the Rose

theory, that HTC for DWC on a vertical plate is independent of surface height.

Piezoelectric Droplet Ejection with Heat Transfer

Some noted attempts have been made to establish a HT cell using a piezoelectric

disc as the driving force to create a spray which then cools a heated surface. Heffington

et al. [89,90] described a method of droplet ejection using a piezoelectric diaphragm

which created a spray of water droplets by using a cover plate with holes in it which

acted like nozzles. By using this setup for spray creation and direct contact condensation

on a large pool of liquid, they authors achieved a HF greater than 100 W/cm2 with a

heater surface temperature of 850C and a sink at room temperature. The design was

highly gravity dependent to collect the liquid reservoir and keep the proper liquid level so

that the driver cover was not flooded with liquid.

Xia, 2002 [91] also used a piezoelectric driver with a cover as a nozzle array. In

this study, the nozzles were etched in the cover material in order to create jets of water to

cool a target surface. The nozzle array was fed by pump and the liquid reservoir was

subcooled. The frequency of driver operation was chosen such that a new droplet arrived

as soon as the previous one had finished spreading on the heated surface. Especially of

note in this case was the extremely high HF achieved of 1 kW/m2.














CHAPTER 3
THEORETICAL ANALYSIS

The theoretical analysis of this study was based on the Rose theory for DWC. The

two main arguments of this function are saturation temperature, Tsat, and surface

subcooling, AT. The theory assumes a simplified thermal circuit of the HT through a

single drop of radius r, with direct contact condensation occurring only on the curved

surface of the drop. Then a statistical distribution of the drop population by radius is

assumed. This distribution is then used to weight the expression for thermal resistance of

a given drop of radius r. Finally, the weighted expression is integrated over the assumed

distribution of drop sizes, yielding a theoretical prediction for the average HF of a surface

undergoing DWC of a pure vapor.

Manipulation of the upper bound of the integral, maximum departing droplet radius

(rmax), was used to estimate an effective maximum droplet radius for the measured

experimental conditions. A comparison of this value for rmax was then used to assess the

effects of NCG on that particular experimental run and also the effects of the presence or

absence of VIDA on HT. This section details the process used in the theoretical analysis

of experimental data.

Since the intent of the design was to direct all HT absorbed by the heat sink

through the brass shim of the piezoelectric diaphragm, the temperature gradients in this

thin piece of metal would likely be significant. In fact, with a nominal 1K subcooling,

the Biot number for the driver is on the order of 100 for the radial direction, also

implying significant temperature gradients. Since the Rose theory depended on the









surface subcooling, it would then require an estimate of the surface temperature

distribution in order to determine the theoretical HF distribution.

So initially, a boundary value problem was constructed to model the heat conduction in

the brass shim. Axisymmetric heat conduction, cylindrical coordinates, and steady state

conditions were assumed. Then the Heat Conduction Equation (Eq. 3-1) was solved

using the following boundary conditions for a cylinder of brass with radius b and

thickness c.

a2T 1 OT 2 T
+-- + = 0, 3-1
Or r Or Or

1. T(r) as r 0 is finite
2. T(r = b) = To, the measured edge temperature
OT
3. -= 0, insulated condition at rear surface of driver


4. T = g assumed average HF at front surface of driver


This assumption of average HF over the face of the driver greatly overestimated the

actual temperature gradients. The input average HF had to be lowered to around 1W in

order to lower the center temperature of the diaphragm to anything in the realm of the

study. Even with this significant overestimate of the temperature gradient in the shim,

the greatest temperature difference in the vertical direction was less than 1C, an amount

considered negligible in the face of a maximum difference between the center and edge

temperatures of almost 400C.

Armed with justification for neglecting the temperature gradient in the z direction,

different functional forms were assumed for the likely radial surface temperature

distribution of the shim. An exponential relation was deemed most likely to satisfy the










experimental conditions due to the implied zero slope at the origin (central axis) and

ability to fix the slope at the edge (heat flux). The distribution was then defined with the

following form (Eq. 3-2) and boundary conditions.

T(r)= Cr" +B, 3-2


1. T(r =b) = To
2. T(r = 0) = Tatm

3. OT q" THR
Or rc k k 2zbc
Representative Radial Temperature Profiles
50
Tatm = 450C, To = 180C, q = 55W
45

40
Tam = 400C, To = 15C, q = 30W
35 ----- ---- -

30 -

25 Tatm = 300C, To = 80C, q = 7W -- -- ------- -
25-
20----- -


15------ -
T(r) = C/rn+B
10 -------- --

5
0 1 2 3 4 5 6 7 8 9 10
radius, [mm]


Figure 3-1. Assumed radial surface temperature distribution. Distributions based on
three sets of experimental parameters representing a low, middle, and high HF
case.

A theoretical HF distribution on the shim surface was estimated for each set of

experimental parameters. First a radial-exponential temperature distribution was

determined from a MATLAB subroutine, exp_profile.m (Appendix A) (Figure 3-1). For

low overall HT rates (e.g., 7 W) the radial temperature distribution smooth and gradual.










However, for higher HT rates (e.g., 30 W) the temperature gradient at the edge of the

shim was steep, and the majority of the shim surface area was near to the assumed

atmospheric temperature, meaning that this portion of the surface would be relatively

inactive in the DWC process.

Next, the assumed temperature distribution was fed into a MATLAB function that

would determine the corresponding HF distribution, DWC_HTC.m (Appendix A), using

the Rose theory (Figure 3-2). For the higher HF considered (e.g., 55 W), the majority of

the diaphragm surface was relatively inactive in the HT. The higher the THR was the

smaller the predicted active area of HT on the diaphragm surface. This was most evident

from the HTC distribution in the top section of the figure, which clearly shows that the


0.2
Tatm= 400C, To = 150C, q =30W
0.15-----


0.05------ --

0
0 1 \ 2 3 4 5 6 7 8 9 10
\ Tatm = 450C, To = 18C, q = 55W

Tatm = 30C, To = 8C, q =7W
E 10 --




1010
0 1 2 3 4 5 6 7 8 9 10
radius, [mm]


Figure 3-2. Radial HTC and HF distributions to follow the assumed temperature
distributions.

higher HF-case HTC was more than an order of magnitude less than the peak value for

about 95%, 60%, and 40% of the radius for the 7, 30, and 55 W cases, respectively.









The calculated radial HF distribution was next numerically integrated over the

surface of the diaphragm, using a Simpson's Rule algorithm, simpsons_rule.m (Appendix

A). The resulting value was the theoretical THR based on the inputs. This theoretical

THR was then fed back into the temperature profile determination step to create an

iterative scheme (Figure 3-3). Convergence of this scheme was quite slow and dependent

on the initial guess for the THR input. Therefore a step was added after the

determination of the theoretical THR which averaged the current iteration value with the

value from the last iteration. The result of this additional step was that the number of

iterations required for convergence to the desired error was reduced by a factor of about

20 and the dependence on initial guess was effectively eliminated from the scheme. This

adjustment to the iteration was evaluated for accuracy by comparison to an actual

converged result. Also a relaxation factor was added, which varied from 1 to 0. Using a

relaxation factor equal to 1 resulted in the original scheme, without relaxation. While

using a relaxation factor of 0 resulted in the current scheme. Zero was determined the

fastest rate of convergence with no impact on the resulting answer.

Convergence acceleration aside, the computation was still time consuming, taking

about 5 minutes of processor time per data value. Thus, for efficiency in determining

uncertainty in THR from uncertainty in the measured values, a data table was constructed

of theoretical THR values for a grid of saturation temperature and edge temperature

values. The function was essentially linear in subcooling, despite the dependence of the

lower bound of the HF integral on this value, and only slightly non-linear in saturation

temperature for the ranges considered. Because of this, large steps were possible in the

table construction. The input values used were Tsat of 24 to 500C, in 20C increments and












Tatm = measured atmospheric temperature
To = measured diaphragm edge temp
q_guess = initial guess at theoretical THR,
Tatm, To, q_guess, usually experimental value
rmax, m r_max = desired maximum droplet diam
zero is used for theoretical value
m = number of divisions in radius vector


Get assumed radial temperature profile
Ts(r) = f(Tsat,Ts,q) exp_profile m






q"(r) = (TsatTsr max) Get theoretical radial HF distribution
S DWC HTC m






Integrate radial HF distribution over the area
q (i ) = integral ( r* q"(r)) of the diaphragm
simpsons_rule m






Accelerate convergence of iteration by averaging
q ( )=(q ()+ q ( 1 ))/2 present and last iterations










| q (i ) q (i 1 ) < desired error? Test iteration step size








Yes

Output q_theory
q ( = theoretical Theoretical THR for this apparatus under the given
C THR Iboundary temperatures




Figure 3-3. Block diagram of iterative loop to determine theoretical THR. A theoretical
THR was estimated in this manner for each set of experimental conditions.









edge temperature (To) of 4 to 200C in 40C increments. One table was constructed

assuming the theoretical maximum droplet radius (Eq. 2-11), which was essentially 1.1

mm for the range of Tsat used. Seven more tables were then constructed, each assuming a

different value for rmax, from 0.01 to 4 mm.

For each of these tables of theoretical THR values for the given experimental

variables and geometry, linear interpolation was used to determine the THR for the

measured parameters of a given data point. This resulted in what was the official,

theoretical THR, using the theoretical maximum droplet size from the Rose theory. This

value was computed from the first table. Then the experimental parameters were used

with the seven other THR tables to predict what would have been the results if the given

maximum drop radius value had been in effect (Appendix B).

The seven rmax values were then fit to an exponential curve, similar to the one used

for the temperature profile, by the least squares method. The least squares method was

not able to be directly solved, as would be the case for a polynomial function. Therefore

a brute-force method was used, by simply computing the sum of the squares of the error

for sets of values that were likely to contain the appropriate coefficients. The minimum

sum of the squared errors was chosen. Then several steps of range and increment

reduction in the coefficient inputs were made to increase the precision of the coefficient

estimates and decrease the overall error of the fit. The fitted curve was extended to a

factor of 10 smaller than the lowest rmax used and to a factor of 10 larger than the largest

rmax used.

The experimental THR was then compared to this fitted curve to determine an

effective rmax for that data point. The effective values were then compared with the









theoretical rmax to infer whether the results of that test had achieved appropriate HT for

DWC (i.e., NCG presence was reduced enough that it did not appreciably affect the HT

results). For those data points which had an rmax larger than 10 times the largest table

value, 0 was assigned and these points were then disregarded. The intent was that those

runs with VIDA present would systematically show effective rmax values smaller than

those runs without VIDA present. This would result from showing higher THR for the

same temperatures. The Data Summary and Comparison with Theory section of Chapter

6 addressed the results of the above theoretical analysis for this study.

Uncertainty was determined in these interpolations by taking the combination of

the saturation temperature and edge temperature uncertainties that would produce the

largest difference from nominal. These limiting values were then interpolated from the

tables. The difference between these extreme theoretical THR values the nominal one,

determined from experimental conditions, was reported as the uncertainty in theoretical

THR. For the effective rmax uncertainty, the uncertainty in the experimental THR was

used in the same manner discussed above to calculate the high and low values, then the

difference reported.

For example, the best data point attained from the experimental results was put

through this process, and the results follow. For Tsat of 24.700.170C, To of 13.80.20C,

and an experimental THR of 332 W, the theoretical THR returned was 21.70.8 W and

the effective rmax was 0.080.04 mm. An exponential curve fit to the range of theoretical

THR values, determined from the considered range of possible maximum departing

droplet radii, shows an excellent fit (Figure 3-4). The square root of the sum of the

squares of the residual error to this fit was 0.24 W, or about 1% of the theoretical THR.







62


Total estimation errors from using the tables could be expected to be of similar

magnitude for all other experimental data points. This data was discussed further in the

Data Summary and Comparison to Theory section of Chapter 6. The MATLAB

programs used for this interpolation and curve fitting process were called

Get_Total_Heat_Rate_andreff.m and Heat_Rate_Archive.m (Appendix A).


70 M l M1


60 I 1 1- -I+ 4 14 H-- I-+ +H I I -- 1- -4 1--H





I L I_ _I L_1 L III II-__ l-_ I -J L J- ll II I __I -_I-
50 -,- -


10 1 0 1
10-3 102 10 100 10 102
[mm]

Figure 3-4. Theoretical THR vs. maximum droplet radius. THR is in W. Radius is in
mm. Here, Tsat is 24.700.170C and To is 13.80.20C. Circles mark the seven
theoretical THR values for the given temperature conditions, determined from
the seven different maximum drop sizes considered. The curve is the least
squares fit to those seven data points.














CHAPTER 4
APPARATUS

The main aluminum cylinder that defines the pressure vessel for this apparatus was

used from a previous study on a different topic. Therefore many design choices were

made simply to accommodate existing features of this pressure vessel. For example seal

grooves already existed on the ends of the cylinder and so only a smooth surface had to

be made on the covers to work with them.

The original heater built for this experiment was taken from a study for spray

cooling [89] and thus was intended to evaporate droplets impacted on its surface. The

heat transfer was to be measured by the temperature gradient in the neck by three pairs of

thermocouples. This temperature gradient could then be extrapolated to determine an

average surface temperature for the heater. This choice of heater was made in a

misguided attempt to reach toward an overall HT cell on the lines of the cited study

above, in the hopes of moving closer to the ultimate goal of a two-phase HT cell that

would be gravity insensitive.

Well into the first round of experiments, it was realized this strategy would not

work. The design was inadequate because of the requirement of a saturated vapor

environment. Therefore the apparatus had to be redesigned to focus on the condensation

phenomenon only.

Also, it was realized that a benchmark would be needed for condensation without

vibration, and the only relations that might apply to this apparatus were for an isothermal

vertical plate. Thus the experimental pressure vessel (Figure 4-1) was set on a stand, on









its side, so that the condenser surface was close to vertical. The orientation of the outside

of the vessel was checked with a common carpenter's level. This method was deemed

appropriate based on the experimental results of [25], which showed the DWC process to

be relatively insensitive to changes in surface orientation. In that study, a 10% decrease

in HTC over the vertical case required a 400 forward pitch or a 100 rearward pitch.

Therefore if this condenser surface was within 3 or even 50 of vertical, then any effect

would likely be slight.

Ice was added to the coolant reservoir to lower the inlet temperature and provide

more driving force for the HT process on the diaphragm. The coolant used was tap

water. A simple model was constructed for desired reservoir temperature and consequent

required mass of ice, and the amount found to be reasonable for acquisition and transport.

Fifty pounds of ice would be required for an initial reservoir temperature just above the

freezing point of water. Later on, when it was determined that the experiments needed to

be run for long time periods, the procedure for ice handling was refined, as described in

the Chapter 5.

However, the heater problem still needed a quick fix. By experiment, it was

determined that a 50 W halogen lamp would provide both extra light for video records

and sufficient heat to drive the experiments. However, this would leave little control or

measurement of the heat input to the apparatus. Various placements of the light were

tried, and below the bottom window of the vessel, facing up was selected as the optimal.

This would allow the placement of the fluid reservoir in the bottom window and get the

bulk fluid as close as possible to the heat source. Initially, a crude reservoir for liquid

was fashioned from a piece of aluminum and placed in the bottom window.









Experimental runs with this configuration, however, rarely showed any condensation on

the windows of the vessel. Since the vessel was basically cold to the touch and the Tatm

measured inside the vessel was above ambient, this implied that the vapor atmosphere

inside the vessel was not saturated.

In order to channel more of the heat energy from the lamp through the fluid

reservoir, further tests used the window well itself as the reservoir. After this, when the

test atmosphere temperature rose above the lab ambient temperature, condensation almost

immediately appeared on the viewing windows, indicating the vapor was indeed

saturated. Additionally, a thermocouple was placed in the liquid reservoir and this

reading tracked very closely with the recorded vapor temperature for those runs with

good NCG control. In fact, the gap between these two values would serve as an excellent

indicator of NCG presence and also vapor saturation condition. The liquid reservoir

temperature would rise above the measured vapor temperature when significant NCG

was present, as part of the depression of apparent saturation temperature accompanying

this presence. Further, if the vapor temperature recorded was higher than the liquid

reservoir temperature, then the vapor was superheated. This assumption was backed up

by a lack of condensation on the viewing windows despite higher than ambient vapor

temperature for these cases. Also, modifications were made to increase its liquid

capacity to about 40mL. This capacity proved to be sufficient enough that replenishing

the water supply was only necessary in one experiment of all those performed. The only

removal of vapor from the system was through the vacuum pump.

An oblique view of the pressure vessel showed the driver surface through the side

window of the vessel (Figure 4-1). Since the original heater was in the place of the









window frame on the top cover, all videos in the first and second stage of experiments

were taken from this corner view. Frames taken from those videos can be seen in

Chapter 6, along with a full explanation of the three phases of the experimenting.


Figure 4-1. Corner view of experimental apparatus.

After the discovery of a significant air leak around the edge of the old, disused

heater, it was removed and replaced with a window. The new front window provided a

much improved view of the condensation process for videos in the third and final phase

of experiments and a better seal.

The atmosphere of the pressure vessel was effectively divided into two parts,

although they were not sealed separately (Figure 4-2). No method of sealing was

available that would be effective and also provide a good thermal connection between the










driver edge and the cooler. Also, sufficient gas flow was needed between the two sides to

allow for an effective evacuation process. Two small notches were made in the upper

edge of the diaphragm frame (the divider between the two sections of the pressure vessel)

for this purpose. As higher heat rates were encountered in experiments, condensate

droplets on the face of the diaphragm frame were noted. This indicated that significant

HT was occurring across the Diaphragm Frame, separating the two sections of the

atmosphere. Insulation was added (about 1 in. thick) to the cold side of the frame and

also around the exposed brass surface of the Cooler to impede this process. The

insulation used was Permatex brand silicone RTV sealant.


PRESSURE TAP
TTIREADS 3/8 FNP'





T I LPROBE


7 -iROUGH










Figure 4-2. Sectioned view of experimental apparatus.

Further testing showed that additional insulation was necessary around the

diaphragm clamp, where the copper cooler was the closest to hot side atmosphere. The

caps of the machine screws that held down the clamp and also the supporting posts of the

diaphragm frame needed to be covered with insulation as well, to reduce obvious









extraneous condensation (Figure 4-3). The fully insulated diaphragm frame was covered

in the orange sealant, but showed little visible condensation during the test runs.














Figure 4-3. Insulation applied to and removed from diaphragm frame. The silicone
sealant was applied as insulation over the metal screw head, the metal support
posts and the central area of the Diaphragm Frame face. Visible wires are
thermocouples. Left picture is a fresh insulation application, with polished
driver surface. Right picture is a tarnished driver surface after removal of
insulation.

Given a few days, the mirror-like finish of the polished brass driver surface could

become quite tarnished. In this state, the quality of DWC produced was not good. The

normal cleaning regimen was described in Chapter 5. The polishing was done to aid in

the promotion of the DWC mode. However, full FWC was never actually achieved in

this apparatus. It was eventually noticed that globules of oil were nearly always present

on the liquid surface of the reservoir the day after an experiment. The most likely source

for this oily residue, which also covered every other surface inside the vessel, was the

silicon sealant and presumably, this contaminant was a dropwise promoter. There was

also a silicone based Dow-Corning vacuum grease used on the vacuum seals which may

have contributed to the oily residue. It has been proven in several studies that promotion

can be effectively achieved by addition of an organic promoter to the fluid reservoir

[37,43], as discussed in the Promotion Methods section.










In light of the relatively low saturation temperatures that were used in the

experiment, as many components as possible were made of polymers rather than metal in

the hopes of decreasing extraneous heat transfer and machine shop time. The Top and

Bottom Covers and the Diaphragm Frame were all made of Delrin (Figure 4-4). The

Insulation was made of Teflon. The optical probe Guide and Seal along with the

Diaphragm Clamp were of polycarbonate. The choice of different materials for these

different components was based solely on availability of stock in appropriate geometry

OPTICAL PROBE SEAL FRAME
OPTICAL PROBE GUIDE

INSULATION COOLER
S-- ---.- ~1TDIAPHRAGM
TOP COVER OPTICAL PROBE


WINDOW \ BOTTOM COVER
GLASS \
DIAPHRAGM STAGE

DIAPHRAGM CLAMP
SW/ HOLES FOR TC

F\

WINDOW FRAME


Figure 4-4. Exploded view of experimental apparatus.

and a maximum service temperature of at least 2000F. Some items were omitted from

the exploded view for clarity, such as the vinyl coolant hoses and the Y-Or-Lok bulkhead

fittings for them, whose locations can be noted in the cross-sectioned (Figure 4-2). All

connecting hardware, pressure fittings, and soft seals were also omitted. The multi-pin

connector/vacuum feedthrough was left out as well. It fit into the third hole in the bottom

cover, was held down by 6 screws, and had a soft seal. All soft seals were manually cut

from Buna-N sheet stock. The components that were press-fit, namely the Cooler into









the Diaphragm Frame and the Optical Probe Seal into the Bottom Cover, sustained liberal

application of the silicone sealant for adhesive and sealing properties.


















Figure 4-5. Diaphragm clamp, lower face showing grooves for placing thermocouples to
measure edge temperature, To.

There were 8 thermocouples placed internal to the pressure vessel. Four of them

were placed evenly around the circumference of the Diaphragm Clamp, in specially

machined grooves (Figure 4-5). The grooves clamped the thermocouple beads to

diaphragm edge and also clamped the edge to the Cooler. This design attempted to place

the thermocouple junctions as close as possible to the edge of the exposed diaphragm

surface without piercing through the wall. In practice, however, this very thin barrier

between thermocouple and liquid eventually broke through in three of the four places on

the clamp, allowing visible drainage of condensate through the lower holes.

Quantitative estimate was not made of how much this phenomenon might have

affected the HT measurements. Qualitatively, it was expected to increase overall HT by

decreasing the amount of condensate around the edge of the diaphragm, which would

otherwise serve to insulate the most active portion of the condenser surface. Three of the









remaining thermocouples were placed in a bundle, with the tips spread, about V2 inch

above the condenser surface and just over the edge of the clamp. These three

thermocouple readings were averaged and this value taken as the atmospheric

temperature, usually taken as the saturation temperature, as described in detail in the

Results and Discussion section. The one remaining thermocouple was placed in the

liquid reservoir, with the tip slightly curved up, to hopefully measure the liquid

temperature rather than the glass surface temperature. Note, however, that this liquid

temperature measurement was only made for the last five days of testing. Two more

thermocouples were employed on the outside of the apparatus, one each just outside the

bulkhead connectors for the coolant lines. A hole was punched with a hot nail through

the vinyl coolant line. Then the thermocouples were stripped of the clear, plastic outer

casing, but not the Teflon insulation. This measure was undertaken because cooling

water would be pushed by the flow pressure back though this casing and into the DAQ

board. Next the stripped, but insulated, thermocouples were inserted in the holes in the

coolant lines and secured with a liberal application of Copper-Bond brand epoxy. After

allowing 20 to 30 minutes for this to cure, the arrangement was leak tested with the flow

control valve on the flow meter both fully open and fully closed. If no leakage was

apparent after ten minutes with the valve closed, then the seal was deemed sufficient.

The cooler was constructed of copper stock. The central cylinder was made to

accommodate the Optical Probe Guide with a press fit and to support the Diaphragm with

at least a 1mm contact around the circumference. Two 'washers' were then made along

with a tube section, all of which were soldered together to make the flow chamber of the

cooler. Two holes were then drilled to accommodate the inlet and outlet tubes, also of









vinyl. Like the coolant thermocouples, these were secured in place by liberal

applications of Copper-Bond epoxy. Enough length was left on the tubes to facilitate

easy attachment of the bulkhead fittings.

The Diaphragm Frame was supported by three lengths of 3/8 coarse all-thread, of

galvanized steel. Certainly stainless would have been better, and better yet would have

been something less thermally conductive, possibly another polymer. Eventually, these

posts had to be covered in silicone insulation as well, since significant condensation was

observed on their surface in the test chamber. Before the application of silicone,

however, the height of the Diaphragm Frame had to be adjusted for displacement sensor

calibration and then set for the testing. Two nuts were placed on the each post, below the

frame to support it and lock each others position, then one on top. Once set, further

adjustments were not made until the final calibration and displacement measurements of

the used driver, after the conclusion of testing.

The pressure tap for the apparatus was eventually located toward the top of the top

cover, relative to the side orientation pictured in the corner view (Figure 4-1). Special

attention had to be paid to the pressure tap and fittings orientation since, after repeated

evacuation procedures of a hot test environment, enough condensate would collect in the

vacuum hose to clog the flow and prevent effective function of the vacuum pump. The

vertical orientation of the fittings allowed this collected condensate to be emptied back

into the test vessel by simply lifting the tube above the fitting inlet.

As the importance of NCG effects on DWC became apparent during the course of

the investigation, the pressure integrity of the apparatus became a big issue. An earlier

section on NCG effects in this paper looks qualitatively at the dynamics of the problem









and at the reported effects in the literature. Upon the initial assembly of the apparatus,

prior to any experiments, a bubble test was applied to test for leaks. Bottled nitrogen was

hooked up to apparatus and the internal pressure was raised to about 50 psi. The vessel

was then submerged. Several leaks were located and fixed in this manner, however, due

to the sensitivity of DWC to the presence of even small amounts of NCG, this method

turned out to not be a sufficient leak control. After data trends indicated a significant air

leak during the intermediate phase of testing, vacuum grease was applied to all soft seals.

Initially this made the leaks worse but then better after the excess grease was wiped away

and the apparatus reassembled. However, this still did not fully address the problem.

The likely effects of NCG resultant to poor sealing were covered in more detail in the

Results and Discussion section. Detail drawings for those parts listed in the above figure

that were manufactured in-house can be found in Appendix C.

Equipment

Several other pieces of equipment, in addition to the main apparatus described

above were needed for the conduction of this investigation. A schematic layout was

presented, followed by a description, by letter, of each component (Figure 4-6).

Element A was a Mastercool brand vacuum pump, 2 Stage, 1.5 CFM capacity, and

oil lubricated. This pump was intended for air conditioning equipment service, and thus

was not intended for much contact with water vapor. Significant amounts of water vapor

put through the pump would condense in the oil, degrade the performance, and cause it to

over heat. For this reason, the evacuation operations described in the results section were

limited to 5 to 10 minutes and the pump oil had to be replaced each day. Capacity was a

few ounces of standard vacuum pump oil.









Element B was an array of brass ball valves and fittings, setup to allow independent

venting of the test vessel and isolation of the pressure gauge from the vessel and also

from the pump line. This separation of the gauge was necessary due to a small, constant

leak in the gauge diaphragm. Because they were thermally exposed to the environment

inside the test vessel, albeit minimally, the valve configuration was wrapped several

times in reflective foam insulating tape to minimize heat loss. This potential loss would

not have affected the accuracy of the measured THR, but rather the insulation was a

cautionary measure to prevent any reduction in heat supply capacity to the vapor.

Element C was a vacuum gauge, ACSI 1200-VACU, with a 0-30 in-Hg range and

an uncertainty of +1% full-scale. This large uncertainty existed despite the two decimal

place digital readout. This large uncertainty rendered it essentially useless for any

information related to the saturation pressure of the water vapor. It was useful, however,

to monitor the relative effectiveness of the vacuum pump, i.e. the condition of the

lubrication oil, since the vacuum level would rise by up to 1.5 in-Hg when a change of oil

was needed.

Element D was a 250 W, 120 V AC, Ushio Halogen Projector Lamp. A copper

tubing coil was fashioned to act as a spring to both support the lamp, which got very hot

and to press it as close to the reservoir window as possible. The lamp itself, however,

was held in an aluminum frame. The bolts of this frame kept about 14 inch spacing from

the window surface. This lack of thermal contact was instituted avoid excessive heat

transfer to the vessel. In practice the vessel sometimes became hot to the touch. This

issue is addressed in the Uncertainty and the Improvements sections.



















-. I,1 ., L .11 1 I 1[










---------- Fi_._I._i -i
I-















I ll 'i l -1 IIri -1I -l I r










I "----i-- -- lit I--
7 117i i i ,.1 -I
rI- 1











Figure 4-6. Schematic of equipment setup with labels.

Element E was the Water Reservoir, which was simply the window well on the


bottom side of the four-windowed aluminum cylinder. Strips of aluminum were lined


around the window well and the corresponding hole in the insulation sleeve using the


silicone insulation as adhesive in order to increase the capacity of the reservoir to about


40 mL.









Element F was the Controlled Atmosphere of the vessel. Ideally this would be

composed of pure water vapor, however, as the test results show, NCG was also a

significant presence at times.

Element G was the piezoelectric Diaphragm. The supply of circular piezoelectric

resonators, APC# FT-20T-3.6A1, was graciously supplied by American Piezo

Corporation at no charge. The diameter of the brass shim to which the piezoelectric

ceramic patch was attached was 0.004 inch (0.1 mm) thick and 0.75 inch (19 mm) in

diameter. The positive and negative power lines both came through the multi-pin

connector. The negative lead was hooked to essentially a washer on one of the bolts that

held down the Diaphragm Clamp. The soldering of the positive lead onto silvered

surface of the piezoelectric ceramic patch on the back of the diaphragm was a delicate

operation. If the ceramic was heated by the soldering iron, or any source, above its

critical temperature, it would lose its piezoelectric characteristics. Alternatively, the

silver, electrically conductive backing might easily come off if over-heated. Only a

minimal amount of solder was used to avoid extra mass loading of the vibrations. The

soldering point was placed as far to the edge as possible for the same reason. Also, the

soldered joint was covered with a thin later of JB Weld epoxy, to physically secure the

connection against vibration. It was necessary to be certain that the epoxy extended well

onto the Teflon insulation of the power supply wire, or the very fine, 36 gauge wire used

would often break at this exposed point. Also, for displacement measurements, a patch

of metallic, reflective tape was placed on the back of the diaphragm so that it had

sufficient reflectivity. About 1 mm of shoulder surface was required around the

circumference to properly secure the diaphragm.









Element H was the Cooler Assembly, described in detail in the above section,

fabricated in-house.

Element I was the analog, float-type Flow Meter, a King acrylic flow meter, 0.8 to

7 + 0.1 gph. It was panel mounted on a piece of sheet metal on the table top, and vertical

checked by a common level.

Element J was the Coolant Reservoir. This Rubbermaid Agricultural tank, of 60

gallon capacity was placed underneath the table supporting the experimental apparatus

and equipment. Ice was placed here to both cool and, as it melted, supply feedwater to

the coolant pump. Details about ice preparation were listed in Chapter 5.

Element K was the Coolant Pump, a Little Giant Pond Pump, 4300 gph, actually far

oversized for this application.

Element L was the Multi-pin Connector, from Ceram-tec Products, a 19 pin,

#16060-13-A, air-side plug/#16133-02-W single-ended receptacle. A flange held in

place with epoxy and six stainless steel pan No. 8 pan head screws were used for

attachment to the Bottom Cover.

Element M was the Terminal Block, from Measurement Computing, with 16

positive and negative posts for thermocouple attachment. Cold junction compensation

done on-board with all terminals mounted on a single iso-thermal block. The block was

designed to mate with Measurement Computing DAQ card.

Element N was the Data Acquisition Card, from Measurement Computing, part #

PCI-DAS-TC, installed in a PCI slot of an IBM personal desktop. The card would

support Labview, MSExcel or MATLAB control, but Excel was chosen for convenience.









Element 0 was the desktop PC IBM, Windows XP, in which the DAQ board was

installed. Data was captured in Excel using DAS Wizard, supplied by Measurement

Computing. All data was taken as a temperature reading, in F, converted from voltage

by the software. Each measurement represented 16 averaged values from in between

sampling cycles. Sampling ranged from one per second to one per hour in increasing

increments. Channels of the DAQ could be sampled individually or as a continuous set

between two chosen limit terminals.

Element P was an Amplifier, made by Krohn Hite, model DCA-10, a wide band

DC to 10 MHz amplifier with 1 and 10-times amplification, with continuously variable

gain. This was the model recommended by the engineering staff of the driver

manufacturer, APC.

Element Q was a Function Generator by B+K Precision, model 4017A, to 10 MHz

with a sweep function. Only sinusoidal wave function was used.

Element R was an Oscilloscope by Tektronix, model 2247A, 100 MHz, with many

functions, the most important of which were auto setup, to automatically find appropriate

display settings for an applied input signal and a peak-to-peak voltage readout.

Element S was a Precision Rheostat from STACO, model # 3PN221B, with output

range 0 132 V and up to 2.5 A, fused. This element was used to vary the input of the

halogen lamp, especially necessary since, on the highest setting, smoke began to issue

from the lamp area. Thus, the power was never set above 70%.

Elements 1 4 were the thermocouples used to measure the edge temperature of

the diaphragm, whose placement was described in detail in the apparatus section above.

Elements 5 7 were the thermocouples used to measure atmospheric temperature.









Element 8 was the thermocouple used to measure liquid reservoir temperature.

Elements 9 & 10 were the thermocouples used for coolant outlet and inlet

temperatures.

The following elements were not pictured for simplicity:

Element T was an Optical Displacement Sensor & Probe, from MTI Instruments,

part number MTI-2000 Fotonic Sensor with MTI-2062H probe. This instrument used a

broad spectrum halogen source lamp with precision input control along with a fiber optic

cable to transmit the light to the sensor head. Half the fibers supplied the source light and

half transmitted the light reflected from the source. There were two working ranges for

each probe head design, on either side of an optical peak in captured, reflected light. The

probe selected, in Range 1 used, had a standoff distance from the target of 0.76 mm with

a range of +0.38 mm, with sensitivity of 0.12 am/mV. Signal to Noise Ratio at resonance

ranged in this study from a low of 13 for an old, used driver, to as high as 61 for a new

driver. The resulting measured vibration amplitude, peak-to-peak measured had a

minimum of about 7.5 am, a maximum of about 38 am and a noise equivalent to 0.6 am.

The main drawback to this instrument was that it required that reflective tape be placed

on the back of the driver, for sufficient returned light. The problem then was that tiny air

bubbles trapped under the tape would bubble up under the tape when the vessel was

evacuated, thereby ruining the signal quality. No measurements of displacement were

therefore possible during experimental runs. This data had to be inferred from a

calibration curve constructed under atmospheric pressure. It could be expected that the

vibration amplitude might slightly increase with less damping effect from air around it.









However, it was likely that the mass loading from the condensate on the driver surface

would have a much more significant depressing effect on oscillation amplitude.

Element U was Camera 1, from JAI, model CV-730 color 770 x 500 CCD with a

dedicated PC and controller card from Flashbus. This was the camera used for the initial

and intermediate phases of experimentation.

Element V was Camera 2, the Motionscope from Redlake Imaging, with matching

controller card and dedicated IBM PC. This camera was capable of capturing up to

8000 frames per second. However, this frame rate required a massive light input to

return a usable image and not enough light was available. Therefore a much slower

frame rate was used. Some videos were taken at 50, 500, and 1000 frames per second,

with good results. The frames from the final phase of testing were taken from videos

captured with this camera.

Element W was the Heater Cartridge and Controller, from Watlow. A Firerod

cartridge, 400 W and a Series 96 controller were taken from [89]. This heater was used

in the initial stage of testing but was deemed ineffective and later removed.

Additional Testing

It was determined that the results of the data analysis warranted further

clarification. This clarification would require slight modifications to the experimental

apparatus (Figure 4-7). In this new setup, the fluid reservoir was separated from the test

vessel, so as to isolate extraneous thermal energy conduction from the halogen lamp

heating element to walls of the aluminum test vessel. Additionally, a phase separator,

placed in an ice bath was added to the vacuum tubing arrangement between the vessel

and the vacuum pump. This chilled separator would serve to minimize the amount of

water vapor condensing in the oil of the vacuum pump by reducing the saturation










To Vacuum
Pump


I: E' [1h


Halogen
Lamp


Figure 4-7. Schematic of vacuum apparatus adjustments for Additional Testing.

temperature/pressure of the vapor being pulled through the tubing. This change

improved the vacuum pump performance as well as the length of time that it could be

operated without overheating, leading to better reduction of NCG in the test vessel

environment. This arrangement was managed by inclusion of a series of manually

operated ball valves, denoted in the figure by a standard valve symbol of a circle and an

inscribed 'x'.

As an additional attempt to confirm assumptions of the theoretical analysis, three

thermocouples were placed along a radial line of the brass shim (Figure 4-8). The same

RTV silicone used elsewhere in the cell was used as an adhesive, with the thermocouple

beads held in contact with the brass surface during application and curing. The positions


Pressure Vessel
with
Test Surface









of the bead centers and corresponding uncertainties were then estimated from geometric

analysis of digital photographs. The first thermocouple was centered within 4% of the

radius of the center (9%) while the second bead was centered at 3313% from the

center, and the third at 669% from the center.

For the additional testing, all thermocouples were replaced with type T, for its

smaller absolute uncertainty. All these new thermocouples were Omega brand, made of

36 gauge wire. The part number of these thermocouples was 5TC-TT-36-36. They were

all calibrated simultaneously to two laboratory-grade precision mercury thermometers.


Figure 4-8. Placement of surface thermocouples on the piezoelectric diaphragm. Single
thermocouple in foreground measured atmospheric temperature.

This step of direct surface temperature measurement was originally avoided due to

concerns that the presence of the probes would interfere with the condensation









phenomenon taking place on the surface. However, the results of the Additional Testing

proved to equal the best results of the previous testing in overall HT rate, indicating that

any interference was negligible.

The atmospheric temperature estimate was now taken from a single thermocouple

measurement instead of several, since the entire batch had been replaced with type T

thermocouples. This new batch of thermocouples was calibrated together at a few dozen

points across the full range of applicable temperatures. The calibration comparisons were

two laboratory grade mercury thermometers with smallest divisions on their scales of 0.2

and 0.1C. The calibration and expected maximum uncertainty in the readings were

discussed in further detail in the Additional Testing section of Chapter 7.