Department of Mechanical Engineering HEAT POST

HEAT TRANSFER TO IMPACTING DROPS AND POST CRITICAL HEAT FLUX DISPERSED FLOW Gail E. Kendall Warren M. Rohsenow Report No. 85694-100 Contract No. NSF Grant ENG 76-82564 Heat Transfer Laboratory Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts March 1978 ENGINEERING .NGINEERING IGINEERING INEERING NEERING 'EERING ERING RING PROJECTS PROJECTS PROJECTS PROJECTS PROJECTS PROJECTS PROJECTS PROJECTS LABORATORY LABORATOR LABORATO' LABORAT' LABORA LABOR LABO' LAB' INGPROJECTSLA4G PROJECTS L PROJECTS PROJECTF ROJEC)JEr 02139 I -EN111101", - 1 - TECHNICAL REPORT NO. 85694-100 HEAT TRANSFER TO IMPACTING DROPS AND POST CRITICAL HEAT FLUX DISPERSED FLOW by Gail E. Kendall Warren M. Rohsenow Sponsored by National Science Foundation Contract No. NSF Grant ENG 76-82564 D.S.R. Project No. 85694 March 1978 Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 - 2 - ABSTRACT Heat transfer to drops impacting on a hot surface is examined in context of dispersions of flowing, boiling fluids. The liquid contribution to heat transfer from a hot tube to a two-phase dispersion is formulated in terms of heat transfer contributions due to surface impacts of individual drops. High heat transfer rates are associated with liquid wetting of the surface at surface temperatures between saturation and the minimum stable film boiling (transition) temperature. Low heat transfer rates are associated with non-wetting, or dry, impacts at surface temperatures above the transition temperature. In the wetting region, experimental measurements of heat transfer rates to sparse streams of drops impacting on a hot surface showed complete evaporation of the drops. In the non-wetting region, an analysis of impact dynamics and heat transfer for deformable drops was performed using simple, idealized shapes to model the deformation. Lagrangian methods were used to derive equations of motion and deformation for impacting drops. Resulsts compare well with available information on drop dynamics and heat transfer. The analysis of heat transfer to impacting drops was formulated for incorporation into dispersed flow analysis, given the statistical distribution of drops in the dispersion. Applications include the prediction of local heat transfer and vapor generation rates. The liquid contribution to dispersed flow heat transfer must be included for the prediction of rewet in nonuniformly heated tubes. - 3 ACKNOWLEDGEMENTS The authors wish to express their than ks to Professors Peter Griffith, Bora Mikid, and Hank Paynter , and Mr. Grady Yoder, for their careful discussions, sugges tions, and review of this work. Technical assistance was prov ided by Mr. Fred Johnson and Mr. Dave Hart; typing by Ms. Gis ela Ri nner, Ms. Leslie Regan, and Ms. Joan Gillis; drafting by Mr. Arthur Giordani; and filming with the help of Mr. Ben Hal prin, Dr. Harold Edgerton, and Mr. Charlie Miller. is sincerely appreciated. Their as sistance - 4 - This research was supported by the National Science Foundation. - 5 CONTENTS Abstract Acknowledgements Lists of Figures Lists of Tables Nomencl ature 1 INTRODUCTION Heat Transfer in Two Phase Flow Systems: Flow Regimes Review of Related Work Scope: Liquid Contribution to Dispersed Flow Heat Transfer 2 PROBLEM FORMULATION: TRANSFER Flow Boiling Heat DISPERSED FLOW HEAT Transfer Dispersed Flow Structure Total Heat Transfer: A Sum of Contributions Knowns and Unknowns 3 EXPERIMENTAL DESIGN: IMPACTING DROPS HEAT TRANSFER Dispersed Flow Conditions and System Design Liquid Supply System Drop Generation and Charging System Page Heat Transfer Target System 65 Temperature and Heat Transfer Interpretation System 67 Independently Selected Parameters and System Checks Measurement Technique Data Processing 4 5 HEAT TRANSFER DATA AND INTERPRETATION Drop Heat Transfer Measurements: Some Observations 79 Drop Heat Transfer Effectiveness Data 84 The Effect of Entrained Air at H gh Superheat 91 ANALYSIS OF DYNAMICS AND HEAT TRANSFER IMPACTING DROPS Background and Previous Efforts Drop Dynamics Constraints and Equations of Motion Boundary Conditions External Force in Drop Motion Near a Plane Surface Evaporation and Heat Transfer Effectiv eness: Dry Impact Nondimensional Equations Dynamics of Dry Impacts: Model and Dat a 126 - 1 Page A Simplified Model: Transfer 6 7 Dry Collision Heat 170 Heat Transfer with Surface Wetting 175 Surface Wetting Tra nsition Temperature 183 LIQUID DROP CONTRIBUTION TO DISPERSED FLOW HEAT TRANSFER 186 Distribution Functions for the Dispersed Phase 187 Incorporation of Drop Heat Transfer Effectiveness 191 The Role of Drop Heat Transfer in Dispersed Flow Analysis 193 CONCLUDING REMARKS 195 Summary 195 Conclusions 197 Recommendations 199 REFERENCES 200 APPENDICES A3 CIRCUITRY 206 A4 DATA SUMMARY 211 - - . ! %4 4 pmq-"- o-- - 8 Page A5 DETAILS OF DROP DYNAMICS ANALYSIS Equations of Motion for Drop Models 232 232 Dynamic Scaling Functions for Two Drop Shapes - 237 Integration of Velocity Profile Function 244 Nonlinear Temperature Profile 246 Temperature Correction for Vapor Density 251 Property Data 252 Maximum Pressure Rise and Reynolds Number in the External Flow 254 Maximum Extension Radius, Cylindrical Model 256 IMMI., - 9 - LIST OF FIGURES Figure 1-1 Steady, Heated Two Phase Flow 2-1 Boiling Curve 2-2 Flow Structure 2-3 Drop Heat Transfer Effectiveness and Target Superheat 3-1 Drop Heat Transfer Experiment Schematic 3-2 Liquid Supply System 3-3 Target Assembly 4-1 Drop Heat Transfer Effectivness Data Summary 4-2 Drop Heat Transfer Effectiveness for Two Target Materials 4-3 Drop Heat Transfer Effectiveness wi th Liquid Subcooling 4-4 Drop Heat Transfer Effectiveness: Transition 4-5 Target Heat Transfer and Weber Number 5-1 Cylindrical Model of Drop Position - Equilibrium Cylindrical Model of Drop Position - Deformed. 5-2 62 107 108 5-3 Truncated Sphere Model of Drop 112 5-4 Steady Axisymmetr ic Laminar Flow between Parallel Disks 127 - 10 - Fi qure Page 5-5 Free Drop Oscillation Period 146 5-6 Drop Impact Period 147 5-7 Maximum Drop Extensio n 148 5-8 Minimum Drop - Wall Separation vs. 151 Weber Number 5-9 Minimum Drop - Wall Separation vs. Wall Superheat 5-10 Maximum External Forc e 5-11 Maximum Vapor - Flow Reynolds Number 5-1 Drop Thickness Duri ng Impact 5-1 Drop Bottom Radius During Impact 5-1 Drop - Wall Separat ion During Impact 5-1 External Force Duri ng Impact 5-1 Heat Transfer Rate During Impact 5-1 Drop Bottom Radius Profiles for Three Shapes 162 Drop Bottom Radius Profiles for Two Weber Numbers 163 Radial Dimension During Impact: Model and Data 164 Dry Impact Heat Transfer Effectiveness vs. Weber Number 167 Dry Impact Heat Transfer Effectiveness vs. Wall Superheat 168 5-18 5-19 5-20 5-21 ~Il' - 11 - Fi gure 5-22 Dry Impact Heat Transfer Effectiveness: Model and Data 169 Simplified Drop Heat Transfer Effectiveness vs Weber Number 173 Simplified Drop Heat Transfer Effectiveness vs Wall Superheat 174 5-25 Nucleation and Evaporation Times 178 5-26 Initial Nucleation Buble Size 179 5-27 Bubble Growth in Nonuniform Temperature Field 180 5-28 Instability in Phase Transition 184 A3-1 Pulse Selection Circuit Schematic 207 A3-2 Pulse Selection Circuit Signals 208 A3-3 Drop Charge Vol tage Switching Circuit Schematic 209 Drop Charge Vol tage Switching Circuit Signals 210 Effect of Superheat on Temperature Profile Coefficients 250 5-23 5-24 A3-4 A5-1 - 12 - LIST OF TABLES Table Page 3-1 Data Interpretation: Sample Calculation 5-1 Dynamic Scaling Functiona for Two Shapes 115 5-2 Dimensionless Dynamic Groups 138 Variables and Characteristic Dimensionless Groups and Parameter Ratios 141 A4-1 Data Summary 212 A5-1 - Property Units Conversion Factors 252 A5-2 Property Data 253 5-3 - 13 - NOMENCLATURE Symbol s drop diameter area (m) (m ) bubble radius (m) tube diameter (m) specific heat (J/kg-C) constant differential drop spacing (m) electric potential mechanical energy frequency force (V) (J) (1/s) (kg-m/s2 ) mass flux (kg/s-m ) specific enthalphy (J/kg) integral constant thermal conductivity (J/s-m-C) kinetic energy scaling function kinetic energy (J) potential energy scaling function drop charging selection - 14 - M center of mass position scaling function n number density N drop bottom radius scaling function p distribution function P pressure PE potential energy q heat per unit area Q heat r radial coordinate R radius S slip ratio t time T temperature U velocity v specific volume V volume W mass flow rate x dummy variable X thermodynamic quality y drop-wall z axial coordinate z momentum flow (1/m3 ) (kg/m-s 2 ) (J) (J/m2) (J) (m) (m) (s) (K or C) (m/s) (m3/ kg) (m3 ) (kg/s) separation (m) (m) (kg-m /s2) hi 15 - - Greek and Other Symbols a. void fraction coefficient Y boundary layer thickness 6 virtual change E effectiveness roughness (m) e angle X geometric length viscosity (m) (m) (kg/s-m) iT 2 [arcsine (1)] p mass density a surface tension (kg/rm 3 (kg/s 2) sum -r impact period (s) electrical resistance (2) friction factor partial differential --- ! 11-- - -- 0"Q"" ronI P- - Subscripts a diameter c characteristic cm center of mass cr critical e equilibrium f liquid g vapor i interface L laminar max maximum min minimum 0 initial p pressure r radial s saturated u velocity v volume w wall z axial frictional perpendicular 16 - M - 17 - Superscripts (above a variable) indicates the time derivative the variable: g (above a variable) variable: = it indicates the time average of the f g dt 0 T (above and to the right of a function) indicates the derivative of the function with respect to its argument: g (x) -dx (above and to the right of a function) indicates weighted average of the function with respect to distribution (p) of one of its arguments: f x m g(x) p(x) / x m p(x) - 18 - Groups AR - a diameter ratio T. c Tr s ) -T 1 1/2 b hf P (T - T hfg subcooling group cpg (T - T ) hf g. superheat group yi 9 = = RE P 9 C k. ' Prandtl number k 9 B U 9 P9 RHO bubble growth group ) . s- PR P f Cpf kf t Reynolds number density ratio (vapor-liquid) Pf Pq RR = density ratio (vapor-saturated vapor) 9 Pg ,s kfPfcp.f 1/2 thermal k wp~c k (T T TR UR - = - T s) sT) hfg property ratio (liauid-solid) 1/2 p aa g conductivity group - T temperature ratio WT s U velocity ra t io III - 2 vi p aa -g9 p U a WE =-- 19 - 1/2 viscosity group Weber number - 20 - Units (metric) C degre es Celsius (temperature) f farad (electrical capacitance) J joule (energy) K degre es Kelvin (temperature) kg kilog ram (mass) m meter s secon d (time) V volt 2 ohm ( electrical resistance) (length) (electrical potential) (British) BTU British Thermal Unit (energy) F degrees Fahrenheit (temperature) ft foot Clength) hr hour (time) in inch (length) lbf pound (force) lbm pound (mass) Unit Prefixes c centi (10-2) m mili p micro (10-6) (10-) III - 21 - Unit Identities j = kg-m2 /s f = s/2 K = C + 273 = BTU 778 ft-lbf hr 3600 s ft = 12 in = 2 = 2.51x10 0.3048 m 4 f -t2-bm/s2 - 1055 J - 22 - Abbreviations AECL Atomic Energy of Canada Limi ted AEE (United Kingdom) Atomic Energy Establishment AERE (United Kingdom) Atomic Energy Research Establishment ANL Argonne National Laborat ory ASME American Society of Mech anical Engineers BNL Brookhaven National Labo ratory CNEN Comitato Nazionale Energ ia Nucleare CSChE Canadian Society for Chem ical Engineering CSME Canadian Society for Mech anical Engineering JSME Japan Society of Mechani cal Engineers MIT Massachusetts Institute of Technology NRC (United States) Nuclear Regulatory Commission UC University of California mllilillmllil mill HillIla III I] - 1 23 - INTRODUCTION Heat Transfer in Two-Phase Flow Systems: Flow Regimes Heat transfer in sys tems experiencing liquid-to-vapor phase change is a governi ng factor in the design and use of many types of equipment, especially those associated with power generation and cool ing. Convective boiling, in which heat is transferred from hot flow channel walls to a flowing two-phase fluid is an effective method for continuous heat removal. T he quanti ty of heat exchange in convective boiling is heavi ly depend ent on the geometric distribution of the gas and 1iquid pha ses in the flow. The concept of flow regime is used to describe the different gas-li quid configurations in two-phase flows. As heat is conti nuously added to a fl ui d undergoing phase change, the flui d will experience a transition from mostly liquid to mostly vapor phase. In this transition,the type and sequence of flow regimes depe nds on the flow and boundary conditions. Som e typica 1 flo w regimes are depicted in Figure 1-1. Of parti cul ar in terest is the location at which the liquid is no 1on ge r in direct contact with the tube. This is c alled th e dry out point. on a heated wall In general, liquid is asso ciate d wi th effecti ve heat transfer at low temperatu re di fference s, whil e vapor on a heated wall is associated with less effective heat transfer, requiring large temperature differences for heat removal. - 24 - (MOSTLY VAPOR) B INEFFECTIVE HEAT TRANSFER REQUIRES HIGH WALL TEMPERATURE POST CRITICAL HEAT FLUX DISPERSED FLOW HEAT ADDITION q(z) (J/s-m2) DRYOUT EFFECTIVE HEAT TRANSFER AT LOWI WALL TEMPERATURE CONSTANT MASS FLOW RATE W (kg/s) (MOSTLY IQUID) Z FIGURE 1-1 STEADY, HEATED TWO-PHASE FLOW - 25 - Thus hi gh tube temperatures are often found beyond the dryo ut poi nt. Post dryout temperatures in heated tubes can be so high that the tube material may be in danger of mel t ing. Therefore heat transfer beyond the dryout point is a subje ct of great interest, particularly in the field of nucl ear reactor safety. The dispersed flow regime consists of a continuous vapor phase which carries with i of liquid drops. a fine disp ersion or mist This flow regime virtually always exists downstream of the dryout point and may persis t for appreciable lengths. Because of therma 1 nonequili brium, dispersed flow can exist even beyond the equilib rium quality region: liquid drops can still be present whe n the bulk vapor is superheated above the sat uration (bo iling) temperature. The vapor volume fraction, or void fr action, in dispersed flow is usually high: ty pically gre ater than .95 but often closer to 1.0. Because of the large difference between the liquid and vapor densi ties, an appreci able amount of liquid may be present in the flow ( for example, half the mass) even though the voi d fraction is close to one. It is important that both the liquid and the vapor phases be accounted for in heat transfer and flow development calculations for dispersed fl ows. 00 WIMP WN - 26 - Review of Related Work Heat transfer in forced convection boiling is the subject of many investi gations. Typically, a single study focuses on a particular flow regime, that is, a particular geometric distribution of liquid and gas phases in the flow channel. In experimental studies the flow regime is often inferred from the heat transfer data: lower wal 1 temperatures in dicati ng wet wall and higher indicating dry, and the calc ulated e qui 1ibrium vapo r mass fraction or quality (determi ned from the average enthalpy of the flow) indicating the relati ve propo rtions of liquid and vapo r present in the flow. Di ffi cul ties in i nterpretation of such data arise for seve ral reas ons. The calculated vapo r quality is frequently differen t from th e can appear wel 1 before the bulk and liquid can exist complete vapor ization. well flow quality : bubbles liquid reaches saturation beyo nd the calculate d point of The rel ative velocity o f the phases (a phenomenon called s lip) cann ot be determined from heat transfer data without making as sumptions about the flow quality and either the relati forces between the phases or the depende nce of the heat ransfer on phase velocity. The effects of axi al conductii are often unacc ounted for in transient-type (nonsteady state) experiments. Thus heat flux and wall temperature measurements are often associated - 27 - with an approximation to the true flow conditions. The difference between the actual flow conditions and the nferred is not, ondi ti ons depends on flow history in a way that in general, accounted for. Analytical work in forced con vection boil ing must deal with all the problems of data i nterpretati on and provide explanations for the observed sequence of flow regimes and the associated heat transfer as well. The standard approach is the specification of a heat transfe r prediction scheme for a given flow regime base d on those parameters which describe that flow regime (ph ase ve locity, temperature, etc.) and a best guess as to when a particular flow regime is encountered (based primar ily on experience under matching conditions). A number of predi c tion schemes introduce a slip ratio, and a vapor mass fraction X differ- The dependence of the ent from the equilibrium quality. nonequilibrium quality and slip rat io on flow parameters is sometimes inferred from data, an d such correlations may not be valid outside the data base. One of the earlier approaches to di spersed flow heat transfer analysis was presented by Dougal et al. [1]I (MIT) as a limiting case: the heat transf er was assumed equal to single phase vapor heat transfer at the same vapor velocity. INumbers in brackets indicate numbered references listed separately under REFERENCES. - 28 In conjun ction with this, it was usually assumed that there was no sl ip between the phases, and that thermodynamic equilibrium existed. Laverty et al. [2] reported their observation of the transition from annular to dispersed flow, and formulated the two step heat transfer process: step one, from wall to vapor, step two, from vapor to drops. This model allowed for both slip and thermal nonequilibrium. Forslund et al, [3] extended this analysis to include droplet breakup and a wall-to-liquid heat transfer term. Hynek et al. [4] further extended this work, includ- ing additional effects such as twisted tapes in the tube. Plummer et al. [5] introduced a simplified approach to thermal non-equilibrium and a wall-to-liquid heat transfer term. Illoeje et al. [6] developed a detailed model for heat transfer from the wall to liquid drops touching the wall or in the boundary layer, attempting to predict the minimum point on the boiling curve. Ganic et al. [7] calculated trajectories of liquid drops near the tube wall in order to determine the rate of liquid deposition. This series of investigations at MIT was supported by concurrent experimental investigations of heat transfer to upflowing nitrogen in a uniformly heated tube. The research discussed in this report is a continuation of the study of post critical heat flux heat transfer at MIT. Other research in dispersed flow heat transfer - 29 - provides important information in the forms of both meas urements and modeling of the heat transfer process. detail ed photographic study of drop si z e and velocity distributions in post dryout dispers ed flows of Freon was performed by Cumo et 1. [8, 9] of CNEN, Rome. A two-step heat transfer model fir post-dryout, di s persed flow, similar to that deve loped by Laverty et al. was developed independent ly by Bennet et al. [10] of AERE, Harwell, Recently, new models have been presente d by Jones et al. [11], and Saha et al. [12], of BNL New York. Vario us aspects of dispe rsed flow heat transfer have been specifi cally addressed by a number of investig ators. - The therma nonequilibirum and phase velocity slip phenomena are fundamental to a 11 of the two step heat transfer models. A review of rese arch addressed to the q uestion of thermal nonequilibrium i n heated two-phase flow is given by Jones et al.LiJ1 . The question or phase velocity slip and phase volume fraction is discussed by Butterworth [14] (AERE, Harwell). Groeneveld[15] (AECL, Chalk River) gives an extensive summary of data and heat transfer predictions for post critical heat flux dispersed flow. Most heat transfer analyses and correlations neglect the liquid contribution to heat transfer at the tube wall in post dryout dispersed flows. however, Several investigations, have been addressed specifically to this point. - 30 - Brevi et al. [16] and Cumo et al, [17] (CNEN, Rome) attempted to infer the liquid conbtri bution to heat transfer from their data. Harwel al [18] (AERE, Keeys bserved rewet following dry out in t ubes with cosine heat flux distribution, indic ating th e importance of liquid heat transfer. Bailey [ 19] (AEE , Winfrith) addressed some comments to the cou nter-eff ects of underestimating total heat transfer by ne gl ecting tube-to-liquid heat transfer, and overestimating total heat transfer by assuming thermal equilibrium resulti ng in an increased vapor velocity and tube-to-vapor heat transfer ; effects may be important. Forslund, his data suggested both Illoeje, and Ganic (MIT) each presen ted models or correlations for the liquid contibution to heat transfe r in post dryout dispersed flow. (Lehigh University, Chen et al. [20] NRC) Pennsylvania, for presented a liquid dro p heat transfer model based on Illoeje's work. These mode ls are unsuccessful in predicting what is known about hea t transfer from hot surfaces to impacting drops in disperse d flow conditions. Some useful informati on regarding the nature of the solid-liquid heat transfer process is obtained from research on topic s related to but not primarily concerned with post dryout dispersed flow. cooling and surfa ce rewetti ng. made to measure and Two such areas are spray Numerous attempts have been model the surface heat transfer to - - impacting drops. 31 011111111,111i - One of the recent works by Hall [21] (Berkeley Nuclear Lab) concludes that the major cooling effect in certain types of sprays is due to air entrainment by the drops, which far exceeds the actual surface-to-drop heat transfer. A review of the rewet literature is contained in a recent report by Elias et al. [22] (UC, Berkeley). The bulk of the analyses of the rewetting phenomenon treat the surface temperature at the rewet front as a constant determined by surface properties (both thermal and surface finish) for the fluid in question. The conclusions of these research efforts, in addition to those of dispersed flow boiling research, are reflected in this study. - 32 Scope: Liquid Contribution to Dispersed Flow Heat Transfer The objective of research in post critical heat flux heat transfer is the prediction of steady heat flux and flow conditions beyond the dryou t point, based on an understanding of the importa nt physic al mechanisms controlling heat transfer an d flow devel opme nt. This study focuses on a particular pos t dryout flow regime, dispersed flow, with the goal of inve stigating the liquid contribution to the total heat trans fer. It is impo rtant to account properly for the liquid contribution to heat transfer (frequently omitted in post dryout heat tran sfer calculations) not only for the predicti on of the total heat transfer to the flow, but also for the prediction of the vapor generation rate which determines the vapor cooli ng rate further downstream. The scope of this work is outlined as follows: 1) The fo rmulation of the dispersed flow heat transfer problem, 2) meas based on local con ditions; The design and cons truction of an apparatus to re heat transfer to impa cting drops under conditions si mu ati ng po st dryout disper sed flow; 3) The measurement of heat transfer to impacting drops in the wetting region (where data are unavailable in the literature) for the conditions of interest; 4) The modeling of heat transfer to impacting drops in the non-wetting region for the conditions of interest; 1, - 5) 33 - The incorporation of information on drop heat transfer into the analysis of post dryout dispersed flow heat transfer. This outline is the framework for the presentation of the results of this study. - 2 34 - PROBLEM FORMULATION: DISPERSED FLOW HEAT TRANSFER Post dryout dispersed flow heat transfer analysis is discussed in the framework of the complete flow boiling system. given. Details of the structure of dispersed flow are The flow structure suggests the separation of the liquid and vapor contributions to the total heat transfer from a hot tube to the dispersed flow. The flow structure also suggests the further breakdown of the liquid heat transfer contribution to a sum of contributions due to individual drop - surface interactions. The heat transfer associated with a single drop - surface interaction represents a major unknown. - 35 - Flow Boiling Heat Transfer Heat transf er to di spersed flow beyond dryout is usually encountered in the context of a complete flow boiling system, in which several regimes are present. The dispersed flow heat tra nsfer analysis is formulated in terms of the local conditi ons (such as liquid and vapor temperatures, velocities, and v olume fractions, and tube temperature). But the local con ditions depend on a number of overall flow parameters includ ing the mass flux and flow quality (vapor flow mass fractio n), as well as the flow history. A boiling c urve is the standard means of representing heat transfer in two-phase systems in both flow and pool boiling. A boili ng curve is a plot of heat flux (J/s-m 2 ) vs. wall superhea t (wall temperature in excess of the saturation temperatur e, C), constructed for a fixed set of conditions, parti cularly for a fixed mass flux and equilibrium quality shown in Figure 2-1. Xe . G (kg/s-m ) A typical boiling curve is Some important features of the curve are: th e critical heat flux (CHF) or local maximum in the curve; the trans ition region, or region of negative slope; and the minimum heat flux or local minimum. Not all boiling curves exhibit t hese phenomena; in particular, some high qual ity (mostly vapor) boiling curves show no transition CRITICAL HEAT FLUX DRYOUT LLJ G (kg/s-m 2 ) Xe WALL SUPERHEAT, Tw-Ts (C) FIGURE 2-1 BOILING CURVE FIXED Nil'J" - 37 . - region, though there may be an inflection point. In general, however, the negative slope in a heat flux vs. wall temperature plot is associ ated with two-phase heat transfer. Since boiling curves are constructed for fixed quality, a single boil ing curve applies to only one location in a heated tube which corresponds to the local quality. As the quali ty changes (increases) with length, other boiling curves are needed. The local heat flux can be deter mined from the local wall temperature, or vice versa, by locating the appropriate point on each boiling Thus a set of boiling curves is needed to describe curve. the heat transfer in a finite length channel. The local equili brium quality is determined from the kno wn inlet condition and the heat addition along the leng th of the t ube. /s-m2) If the heat addition per unit area is q (z) (a function of the axial coordinate a tube of diameter (z) (J/kg) B (m), z the local specific the tube) ow enthalpy is : h (z) = Lq 7 Bdz W (2-1 ) - where W (kg/s) 38 - is the total mass flow rate. bri urm quality is defined as : Xe h (z) Xe The equili- (z) h - (2-2) , - = hfg where and is the specific enthalphy of saturated liquid hf h i s the latent heat of vaporization (J/kg). Note Xe may be less than zero (for subcooled liquid) or greate r than unity (for super- that the equilibrium quality heat ed vapor). The flow quali ty (the ratio of vapor X flow to the total flow) is alway s betwe en zero and one. The local equilibrium qual ity for s teady heating of a constant flow. qual i ty and other related local , is easily evaluat ed However, the flow condi ti ons which depend on the flow history are not easily determi ned. Thermal nonequi 1ibrium (superheated vapor in the presence of 1iquid) and phase velocity slip (liquid velocity different from vapor velocity) are usually presen t. In post dryout dispe rsed flow, the liquid is usually at the satur ation tempe rature. In this case the vapor superheat quality : (T - Ts) flow quality X and the are related to the equilibrium - 39 Xe - X X Note that The slip X < Xe r a tio S c -T (T f p g ) 5 (2-3) hfg when the vapor is superheated. is defined as : U (2-4) Uf where U and Uf are the mean axial vapor and liquid velocities, respectively (m/s). The local vapor volume (void) fraction the slip ratio and the (1 - flow (1 a.) pg and pf Uf < Ug X) p U -- Qf Uf (2-5) are the vapor and liquid densi ties (kg/m If quality : X a where - is related a 3 ), respectively. (as in upflow), and X < Xe , then a It is useful to express the phase velocities in terms of the mass flux fraction a U G (kg/s-m2 ),quality > a, e and Uf X, and void : GX (2-6) U 9 r P. M IMI "Noloploll p aL M-4 - Rm.-Q.--.'---"PF R" - -- --- -- --- - G(1 U = where the mass flux flow W (kg/s) 40 - - X) G (kg/s-m 2 ) to the flow area G (2-7) P f (1 is the ratio of the mass A (m2) = (2-8) If the thermal nonequi librium and phase vel oci ty slip are known, the other local parameter descri bed in this paragraph can be evaluated. (For a discussion of these relations see e.g. [23]). . .... I........... , a"111M i A - ' -milli, - 41 'k,INI I,- ,,IWA IuilmiliNilllmilolii I" - Dispersed Flow Structure Dispersed flow heat transfer is formulated in terms of local flow con ditions characterized by local flow parameters which reflect the equilibrium. deviation from thermal and velocity Details concerning the actual structure of the flow regim e are essential to the understanding of the heat transfer process. The structure of the nearly continuous vapor phase is considered very similar to single phase vapor flow. This assumption is supported by the observati ons of Cumo et al. [ 8] in post dryout, turbu lent dispersed flows of Freon, in which the drop velocity profile match ed the single phase velocity profile and the s lip ratio was estimated to be one. Soo et al.[24] showed mean gas velocity profiles in turbulent flow were- unchanged with the additi on of solid particles (I0 to 250 ytm diameter). Hutchinson et al. [25] used single phase gas boundary layer thickness for turbulent flow to predict deposition of drop s over a wide range of conditions. Ganic et al. [7],,in the analysis of dry wall dispersed flow, used the boundary layer thickness used by Kirillov et al. [26] for annular mist flow with a turbul ent vapor core(liquid film on wall). This'boundary 1ayer, which Ganic assumed to be laminar, is significantly larger than the si ngle phase laminar subla yer thickness. [27] and Namie et al. Both Gill et al. [28], based their own observations - 42 - the marked difference in velocity profiles for dry wall and annular mist flow, remarked that the increased boundary layer thickness may be due to the liquid-gas intera cti on Therefore it is considered in- at the film interface. appropriate for use in analysis of dry-wall dispers ed flow. The laminar sublayer thickness1 TL ( in turbulent single phase flow in a tube is estimated as a function of tube Reynolds number ~- 25 ) (RE B RE : 8 75 (2-9) g and is plotted in Figure 2-2. The laminar sublayer thickness in turbulent flow is generally given by: P9 U YL ~ ~ 5 with - U (2) 2 (friction velocity, m/s) .079 and (REg) 0. 25 (friction factor for smooth tubes) See, for example [29]. Turbulent flow in tubes is general ly found for Reynolds numbers greater than about 104. witiliAlwilildli, - 43 BV 10~ B IIA hhI.,ILi I1 - H2 0, frcr = 0 .31 (1.25 cm ID) 102 (1.0 cm ID) L B 10~ 4 -,SMOOTH 104 TUBE 1 cm I.D. 105 106 REYNOLDS NUMBER, RE FIGURE 2-2 FLOW STRUCTURE 107 , w ili 1,dim - - 44 The Reynolds number is : P U B (2-10) where is the vapor viscosity (kg/s-m). A typical - ratio is also shown in Figur e 2-2 for B smooth finish 1 cm ID tube. a The tube is considered smooth for all Reynolds numbers where yL Average drop diameters in dispersed flow may vary over a wide range of sizes, from drops nea rly the s ize of the tube in low quality dryout, to vanishi ngly smal 1 drops as the flow completely vaporizes. Tatters on et al a relation for drop sizes in annular mist v - 1.6 x 10-2/ flow : 1/2 2a Pg p g2U92 $PB B with the recommended friction factor [30] give (2-11) 'p: .046 (RE )0.2 (2-12) ,I - 45 - so that where a .106 B (RE ) 9 9g ay B p /2 (2-13) q2 9 is the volume average diameter (m) (the diameter of a drop such that half the liquid in the flow is contrained in drops with diameter smaller than and a is the surface tension (kg/s 2 a ) ). This compar es reasonably well with drop sizes measured by Cumo et al. [8] in dry wall dispersed flow. This func ti on is al so plo tted in Figure 2-2 as a function of Reynold numbe r. s The conclusion is that typical drop sizes are often much larger than the sublayer thickness. Ganic et al. performed point mass trajectory cal cul ations for small drops rotating in and traversing a laminar sublayer and evaporating unevenly due to a temperature gradient. Some trajectories showed small drops rejected from the boundary layer under the action of a force due to drop rotation and a force due to evaporation. The cut-off size for rejected drops was extermely sensitive to the assumed value of slip ratio, and drop rejection could only: be*predidted for drop velocities slower than the gas velocity (that is, for upflow). The cut-off deposition diameter was assumed to - be temperature dependent. 46 - In seven of ten cases considered no temperature dependence was observed; in the other three, the possibilit y of temperature dependence was not investigated because the computer program could not handle some trajectories. The program's inability to handle some trajectories was interpreted as indicative of the temperature dependence of the trajectories; the reported temperature dependence was this range of unresolved trajectories. In all cases, no drop above about 10 yim was rejected from the boundary layer For drop s 1arger than the boundary layer thickness, trajectory cal C ulati Povarov et al. [ 31] does show that aerodynamic ef fects may ons need not be performed. However, be important very close to t he wall as a drop approaches a surfac e at a high relat ive speed and shallow rel ative angle. Under certain con dit ions a drop may ski d off an unheated surface without wet ti ng. Drops in dispersed fl ow have a radial com ponent of velocity . Some measurem en ts [32] suggest a dependence of radial v elocity, U (m/s) on frictional veloci ty U9 (m/s) in the range of conditions considered : Ur .L U. 1 .15 (2-14) WId , - or 1 47 U1 .030 U9 (RE )0.125 9 This relation is (2.15) also plotted in Figure 2-2. Due to the lack of conclusive da ta, the radial velocity may just as easily be taken as a constant percentage (e.g. 2% ) of the throughput gas velocity. The details of flow struct ure allow reasonable estimates for the drop size and velocity as well as the vapor velocity profile. The frictional velocity U is given by; 1/2 U = U and the friction factor (2' $ is given in this case by: 0.079 RE9 0.2 5 - 48 Total Heat Transfer : A Sum of Contributions This analysis of heat transfer from a hot tube to a dispersed flow begins with the assumption that the total local heat flux Q A (J/s-m2 ) can be constructed as a sum o ne due to the interaction of the vapor of two components with the hot wall, an d the other due to impingement of liquid drops : +- A wall to flow A wall to vapor A wall to drops (2-16) is further assumed that the heat transfer from the tube the vapor can be described by the same relations used des cribe heat transfer to single phase gas flow: Q = A where m kg C RE n PR Bg all to Ivapor (Tw - C is a constant (e.g., C = .023 ka is the thermal conductivity of the vapor PR is the vapor Prandtl number, m and n (2-17) T ), for tu rbul ent flow) are exponents (e.g., m = 0.8 (J/s-m-C), n = 0.4 for turbulent flow )' and Tw and T g w are the temperature of the tube wall and bulk vapor, respec tivel y -~-~ ~- (C) . - The vapor Prandtl number is defined as : R PR where 49 = , i24-. (2-18) cp19 is the specific heat of the vapor at constant pressure (J/kg-C). Vapor properties are usually evaluated at a temperature halfway between the wall and the bulk vapor temperatures. Since the heat transfer to the vapor is assumed known, the real focus of the study is the liquid contribution to the total heat transfer. The liquid heat transfer can be represented as a product of a mass flux of liquid drops to the wall and a heat transfer per unit mass : _ wall A drops where (1 - ) Pf U h to mass flux heat transfer per unit mass is the vapor volume fraction, C (1- c) (2-19) S so that is the liquid volume fraction * is the component of a characteristic drop U velocity normal to the wall (m/s) and * E: is a characteristic fraction of the drop which is evaporated on impact with the hot wall. * The qua ntity E ness. is a characteristic heat transfer effective- If no heat is transferred to a drop, E = 0 ; if - 50 - enough heat is transferred to the drop to completely vaporize it, 1 . E = If there is li quid subc ooling prior to impact and/or vapor superheati ng upon i mpact , it is possible to have E > 1 However, the heat asso ciated with typical amounts of subcooling and su perh eati ng is generally small compared to the latent heat, so that the heat transfer effectiveness is generall y not much more than unity. liquid contr ibution to total broken into a sum of heats hea t transfer can be further Q. ( J) transferred to all drops intera cting with the hot wall all drops transfer A wall to to a drop, i i drops heat -z where n (Q.) U like i to the tube wa flux of drops (2- 20) ( (n. U i ) (2- 21) i is the (spatially of drops like and The i uniform) number density (1/im 3) is the component of drop vel ocity of drop i perpendicular to the tube wa 1 (m/s) . The drop heat transfer effectiveness for drop i is E. defined as the ratio of the heat transfer red to the drop Q. , to the latent heat of the drop : Q. Tra i3 6 Pf h fg (2-22) - where a. 51 - is the diameter of a (spherical ) drop such that the volume V V. i (m) of the drop is : (2-23) Tr a 6 The heat transfer from the wall to the drops can be rewritten in terms of the heat transfer effectiveness and the number ensity size and velocity n. of the drops with a given (a, U ) : 7Ta. Q A wall f h g9 to E, 3 U 6' n (ag , U )' E i drops (2-24) The information necessary to evaluate the heat transfer to the drops is the density of drops of given sizes and velocities, and the heat transfer effectiveness for those drops, given the condi ons of impact. Some additional relationships among the defined variables are useful. The liquid volume fraction can be determined from the drop volume and drop number density (1 -Ca) = ZTra. 6 3 n (2-25) i This relation is used to specify the characteristic velocity and heat transfer effectiveness : - 52 - 3 ra. U U . Z . n. (a., U .) 3 Tra 1 n (2-26) The breakdown of dispersed flow heat transfer into its vapor and liquid contributions, and the further breakdown of the liquid heat transfer into contributions of individual drops, is in keeping with the philosophy behind past research in dispersed flow. The result of a set of related concurrent events is taken to be the same as the net result of a set of similar, but independent, events. This is appropriate in dispersed flow because the void fraction is nearly one, so that the vapor phase may be treated as a continuum and first order drop-to-drop interactions may be neglected. - 53 - Knowns and Unknowns Details of the flow structure and assumptions about the separabil ity of the liquid and vapor contributions to heat transfer allow the formulation of the dispersed flow heat transfer in terms of local conditions and one major unknown, the drop heat transfer effectivenes e . The loc al conditions at dryout may be estimated for known conditi ons of flow and heating. The change in flow conditions with axial position beyond dryout may be estimated on the basis of a heat transfer scheme [eg. 10, 12] which can be adapted to include wall-to-drop heat transfer and the assoc iated vapor generation. Informa tion on drop heat transfer effectiveness is neede d to com plete the analysis. Some information is avail able in the literature for heat transfer to impacting drops under various conditions. study of heat Wachters' et al. [33, 34] transfer to impacting drops at high wall super heats is the closest to dispersed flow conditions; no data for heat transfer with wetting is available under these conditi ons. Pederson [35] presented both wetting and non-wetting data for impacting conditions. drops in spray cooling The drop velocity normal to the impact surface in that experiment was much higher than usually found in dispersed flow, and the drops were subcooled, as in sprays, as opposed to saturated in dispersed flow. Both Pederson's - 54 - and Wachter's data for drop heat transfer effectiveness are shown in Figure 2-3. The large discrepancy in data from these two sources has been attributed to the additional (and dominant) effect of air entrainment [21] in Pederson's experiment. Measurements performed in this study support this conclusion. The measurement of drop heat transfer effectiveness in the wetting temperature zone, the modeling of the heat transfer process to predict heat transfer effectiveness for non-wetting impacts, and the distinction between the roles of drop heat transfer in spray cooling vs. dispersed flow are presented in this study. 1,0 SPRAY COOLING DATA (INAIR) (SEE PEDERSON 351) + o A1 0,8 (INSTEAM) (SEE WACHTERS [34]) 0,6 -o LI) u) u- O.4 Cu) 0.2 + ++ +0 0 100 200 300 TARGET SUPERHEAT, Tw-Ts (C) 400 FIGURE 2-3 DROP HEAT TRANSFER EFFECTIVENESS AND TARGET SUPERHEAT - 56 - 3 EXPERIMENTAL DESIGN: HEAT TRANSFER TO IMPACTING DROPS Simulation of dispersed flow conditions and identification of the conditions of individual drop - surface impacts were governing criteria for the design of the heat transfer experiment. The experiment focused on drop - surface heat transfer in the wetting region, where data are not available in the literature for dispersed flow conditions. Measurements were made of heat transfer rates to sparse streams of drops impacting on a heated target surface. Measurements were made in a quasi-steady state, that is, with constant target heating rates at constant target temperatures. The system was designed for independent control of most parameters, checks and calibrations. with several sel f-cons istency Data processing invol ved very few steps; sample calculations are given. - 57 - Dispersed Flow Conditions and System Design The heat transfer experiment was designed to simulate post dryout dispersed flow conditions for drops impacting on a hot surface. The conditions considered most important in determining the heat transfer to impacting drops in post dryout dispersed flow are : the wall and drop temperatures, the drop size and velocity, the angle of impact, and the separation between the drops. In dispersed flow, the drops are typically at saturation temperature; drop diameters are typically a few hundred microns (though much larger and smaller sizes are frequently encountered); and drop velocities in the direction perpendicular to the mean flow are a few percent of the throughput velocity, or on the order of 1 m/s. Dispersed flows are characterized by large vapor volume (void) fractions, so that average drop spacing is many drop diameters and drop - wall impacts may be considered independent. The flow channel wall may be at any temperature above saturation, and the angle of impact is usually shallow. These conditions provided the design basis. Actual dispersed flow conditions would have made identification of a quantity of heat transfer associated with a single drop - surface impact extremely difficult. Therefore dispersed flow conditions were simulated. In order to apply information gained from measurements of heat - 58 - transfer to impacting drops under simulated dispersed flow conditions to real dispersed flows, the details which characterize the impact had to be known. A quantity of heat transferred to an impacting drop was to be associated with a drop of given size and velocity, approaching a surface of a given temperature through a known angle of approach. Since the quantity of heat transferred in a single impact is small, however, a heat transfer measurement for multiple impacts was performed. The multiple impacts occurred under identical conditions. The experiment was designed to measure the rate of heat transfer to a sparse stream of identical drops impacting on a target surface. Drop sizes, velocities, and temperatures were consistent with dispersed flow conditions beyond the dryout point. Pederson's [35] experiments, designed to measure heat transfer to impacting drops in spray cooling, had many of the features desired for dispersed flow drop heat transfer measurements. Some basic elements of Pederson's experiments were reproduced for the dispersed flow drop heat transfer experiment. As in Pederson's experiments, a sparse stream of drops was generated from the vibration of a jet flowing from a thin tube, with selective charging and deflection of most of the drops to render the stream sparse. This stream of drops impacted on a hot surface, which was independently heated and - 59 - instrumented with a thermocouple for temperature sensing. The heat transfer measurements were made in a quasi-steady state. A constant target heating rate was set and measured at a constant target temperature (unlike Pederson's quench experiments). An overall schematic of the apparatus as built and operated is given in Figure 3-1. The experimental system was divided into subsystems which performed specific functions. subsystems There were four major the liquid supply system, the drop generation and charging system, the heat transfer target system, and the temperature and heat transfer interpretation system. Their functions are described briefly in the following sections. - 1,,___ 4RI4 1 t- - FIGURE 3-1 DROP HEAT TRANSFER EXPERIMENT SCHEMATIC - 61 - Liquid Supply System The liquid sypply system supplied distilled water und-er pressure (typically 25 psig) to a stainless steel, 30 gage hypodermic needle. Steady liqui d flow was achieved through system pressurization. Pressure was supplied from a high pressure nitrogen source through a pressure regulator to a liquid supply tank. When the liqui d supply valve was opened, liquid was forced out of the tan k and through a flow meter, through a heated section, through a filter holder, and finally through the hypodermic needl e. Figure 3-2 is a schematic diagram of the liquid supply system. In the heated section, the liquid temperature was raised almost to saturation temperature. The liquid temperature was monitored with a thermocouple at the entrance to the filter holder. The filter holder contained a 5 micron teflon filter to prevent small particles in the flow from entering and clogging the needle. The needle itself was specially made with the tip cut square (rather than at a sharp angle as with most hypodermics), and then chemically etched so that the tube exit would be as smooth as possible. The needle inside diameter was checked by viewing through a microscope; the manufacturer specified diameter of 0.004 in was verified. The needle was soldered to the exit of the filter holder to prevent leakage. HEATING TAPE FLOW METER N2 30 GAGE STAINLESS STEEL NEEDLE VARIAC, 0 - 110 (V) FIGURE 3-2 LIQUID SUPPLY SYSTEM U1111110111IM1111101 - 63 .11, - Drop Generation and Charging System The sybsys tem which gene rated a sparse stream of drops of uniform size and veloc ity used the same technique as that used by Ped erson and deve loped by Schneider et al. (see also [37]; [38]). [36] L iquid was forced through a small tube and exited in a lami nar j et. The tube was vibrated with a piezoelectric bimorph t ransd ucer at a characteristic frequency which caused the jet to break up into uniform drops. One drop is prod uced per vibration period T = 1/f, and f T (s), where is the v ibrat ion frequency (1/s). Axial vibration of the tube is recommended; however, for these a transverse vibra tion was sufficient. condition s, were then s electively charged : 1 out of every was left uncharged, where m m The drops drops was any integer between 1 and 99 The drops were charged by pla cing a ring around the jet at the point 0 f break-up and app 1ying a voltage to the ring, while the 1iquid stream itsel f was grounded (by grounding the metal tube through which it passed). m One out of every drops was left uncharged by pulsing the charging ring voltage t o zero once in every m cycles for the duration of one cycle The electronic ci r cuits which performed these functions are shown schematic a lly in Appendix A3, together with samp le signal diagrams. The charged drops were then deflected by a potential fiel d between a pair of deflection plates. The uncharged drops did not deflect, a nd instead . - formed a sparse stream. 64 - Drop charging voltage was about 300 V; drop deflection plate voltage was about 3,000 V. supplies provided these voltages. Two power A signal generator, with a signal of no more than a few volts, provided the signal for both transducer vibration and for the drop charging sel ecti on It circuit. was found that the jet would break up into uniform drops at several different frequencies below the characteristic frequency recommended by Schneider and associates. These frequencies were distributed in a range up to a factor of nearly 10 below the suggested vibration frequency. It was easy to see when drops (on the average of 250 ym in diameter) were formed by illuminating the stream with a strobe set at the vibration frequency (or some integer division of it). Selective drop charging simplified this visualization by dividing the drop stream into a major and a sparse stream. This process is illustrated in the overall schematic of the experiment, Figure 3-1. mm Iin~, - 65 - Heat Transfer Target System Heat transfer measurements were made in a quasi-steady state, that is, at constant target tempe rature. This eliminated several problems associated with interpre tation of transients and comparisons of transients observed under different conditions. The heat transfer target was a sma 11 metal (steel or copper) cylinder, 6.4 mm in diameter and 3.2 mm in length. The flat side on which drops impacted wa s finished smooth. A small ther moco uple hole was drilled from the cen ter of the opposite sid e through almost the entire length placing The cyl inder the thermoco upl e close to the impact surface. was wrapped with a short (10 cm) length of 30 gage fiberglass insula ted nickel-chromium thermocouple wire which served at a res istance heater. Voltage was suppli ed to the heater termi nal s at a constant level from a supply. 0 to 9 V power The hea ter resistance was about 2 Q. The target - heater - the rmoc ouple assembly was cemented into a cyl indri cal ceramic bead wit h the target impa'ct surface facing out and flush wi th t he bead. in Figure 3-3. This assembly is shown schematica 1ly The ceramic bead fit into a Lexan hol der which was set into the test section at various angles with respect to the drop stream. - 66-- CERAMIC BEAD SHRINK FIT TUBING TARGET HEATER FIBERGLASS INSULATION MOCOUPLE TARGET FIGURE 3-3 TARGET ASSEMBLY IW' - 67 - Temperature and Heat Transfer Interpretation System The liquid temperature and target surface temperature were monitored via nickel-chromium - nickel-alumel thermo- couples (30 gage fiberglass insulated). Thermocouple wires were extended from the test section to a thermocouple jack panel (< .5 m distance). The jack panel was wired with a heavier gage thermocouple wire to a thermocouple switch; only one input to the switch could be monitored at one time. The switch (0 - 26 mV Liquid output was connected to a chart recorder in five overlapping 6 mV ranges). and target temperatures were maintained constant by adjusting heat supplied until a steady chart reTemperatureswere interpreted corder reading was observed. from chart recorder readings from published tables of therm 0couple votages vs temperature. Power was supplied to the target heater (- 22) from a 0 - 9 V variable power supply. The voltage supplied to the heater was monitored with a digital multimeter. A switch disconnected the voltage supply so the heater resistance at elevated temperatures could be measured by the multimeter. $ at voltage The power supplied to the heater of resistance E is simply , Q (J/s): .E2 Q .10".14.100, N I M RW (3-1 ) - - 68 Independently Selected Parameters and System Checks The experiment was designed to allow independent control of several parameters the liquid flow rate Wf(kg/s), the liquid temperature Tf(C), the jet vibration frequency f(1/s), the drop charging selection m ture Tw (C), and the impact angle $ . , the target tempera- For a given needle size, certain discrete frequencies f broke up a jet of flow rate Wfinto drops of uniform size. The drop separation D (m) in the sparse stream depended on the flow rate Wf , the vibration frequency f , and the drop charging selection m. The drop spacing was used to calculate the drop velocity in the sparse stream. If fewer drops were left in the sparse stream, they tended to slow down more (due to air drag). Various targets and heaters were used, so that for identical impact conditions (drop size, velocity, temperature, and angle of approach, and target temperature) the heater voltage E(V) and resistance Independent checks meters. $ (Q) were different. were made on most system para- The manufacturer's flow meter calibration was checked by measuring the actual flow rate (i.e. by catching the flow in a graduated cylinder for a fixed period of time). The frequency f of the signal generatorwas verified by displaying the signal on an oscilloscope. The drop charging selection was, evident from the strobe image of the streams, - 69 - with 1 out of every m drops in the spar se stream. This also veri fi ed that one drop wa-s produced per oscillation. The two thermo couples (detecting liquid and target temperatures) werechecked for prope r output at 100 C. The energy input to the target was determine d by measurements of elect ric potential and resi stance. However, when complete evaporation of the drops took pla ce, an effectiveness of 1 sho uld be calculated from the heat input to the target. The data points were checked by rerunnin g the same conditions with dif ferent heaters, since the drop heat transfer effectiveness should not depend on the targe t heater. Variations of the number of drops impacti ng on the target per unit time (by varying m) also provi ded a means of checking the data since the heat transfer to a single drop should not depend on the number of drops striking the surface, impacts were independent. if the - 70 - Measurement Technique The decision to make steady state measurements simplified not only the interpretation of data, but also the measurement procedure. Steady conditions were established in the various subsystems in about the same sequence as they have been described. The liquid supply system was first activated to provide a steady flow. The pressure drop through the filter and needle depended on the water temperature, so the water was first heated (in a large heated section, 'u.5 m long, , 1 cm internal diameter) and then allowed to flow under constant pressure. The desired flow ratewas obtained by adjusting the back pressure on the liquid storage tank. The liquid flow rate W and temperature Tf (thermocouple output), were recorded. Next, the jetwas vibrated by the piezoelectric bimorph transducer . A frequencywas selected to produce drops of a given size; this frequencywas recorded. the frequency was not entirely abitrary, range of possible frequencies. drops was established, The choice of but there was a When a steady stream of the charging and deflecting electronics were activated from the warm-up state. The charging ring and deflection plates (held in a Lexan frame) were positioned so that the ring surrounded the jet break-up I - 71 - It was easy to see when the ring was properly posi- point. tioned because the uncharged drops de fi ne d a very clean path which was the same as the jet path pri or to charging. The charging pulse.width was adjust ed t o exactly one cycle width by (simultaneously) displa yi ng the signal on Another method was to obs erve the sparse an oscilloscope. stream and adjust the pulse width so that a time was uncharged, uncharged jet. If only one drop at and following the pat h of the original the pulse width was too long, more than one drop out of every m would be in the sparse stream. If the pulse width was too narrow, the spar se stream would be slightly charged, and slightly deflected. were used and found satisfactory. Both methods The drop spacing D was measured (with a transparent ruler) for the drop charging selection m, and both D and m were then recorded. The target was introduced into the test section at an angle 6 to the steady sparse stream of drops. The voltage to the target heater was adjusted until the desired temperature was maintained steadily with drops impacting on the front surface. The angle heater resistance 8 , the heater voltage E , the $ , and the target temperature (thermocouple output giving target temperature Tw) were recorded. Next, the entire stream was charged so that all drops deflected away from the target, were and the heater voltage Eo , - 72 - required to maintain the target at the same target temperature T , wa-s recorded. The heat transfer to the sparse stream was taken as the difference between the heat supplied at E and Eo . With a steady drop stream established, additional measurements were easily taken by varying the number of drops in the sparse stream (1/m) and recoding new values of heater voltage E , (E0 heater voltage with no drops on the target, , is the same) and drop spacing: maintained the same. 0 , with all other conditions Similarly, additional measurements were quickly made at different temperatures by adjusting the heater voltage so that the desired temperature was steadily held. Or the impact angle was changed by rotating the target assembly holder, and new values of heater voltage E and Eo_ (with and without drops) and heater resistance $ (which-varies with temperature)were recorded at each target temperature. Some time and care weretaken in estab- lishing a steady flow of drops, with the proper selective charging pulse width. a temperature Tf pulse width T = Once the liquid flow rate Wf at the vibration frequency f , and the , /f , were properly established, the other parameters, (drop charging selection m , target temperature Tw , and impact angle e) were easily and quickly varied. In addition, target-heater assemblies could be changed in a few minutes. - 73 - Data Processing The heat transfer experiment was designed to require relatively few steps between actual measurement and interpreted data. listed i n Tabl e 3-1 Measured quantities a together with sample values. Three ties (evaluate d at saturation condi nown flui d prop erons) were needed to interpret the data; they are listed in Table 3-1 as Known Quantiti es. For each measure ment , four quantities were calculated : drop diameter a (m) ; drop impact velocity (normal to the target) U (m/s) ; impact Weber number WE and drop heate r transfer effectiven es tions were used to compute each one. 6 . Simple equa- Th e calculated parameters, the equations used for comp utat ion, and sample values are listed in Table 3-1 unde r Calculation. 'A drop heat transfer effectiveness of 1 at a l ow wall superheat provided an added check on the sys tem : it was expected that complete vaporization of the drops would occur under these conditions. - TABLE 3-1 74 - Data Interpretation : Sample Calculation Measurement units) Flow Meter (Mass Flow Rate) Wf kg/s) 4.0 x 10-5 Frequency f 1/s) 4600 Drop Selection m Target Thermocouple (Temperature) Tw (C) 147 Heater Vol tage (wi th drops) E (V) 2.89 Heater Vol tage (no drops) E (V) 1.92 40 Heater Resistance 1 .-93 Liquid Thermocouple (Temperature) Tf (C) Drop Separation D (m)m Liquid Density Pf (kg/m Latent Heat of Vaporization hfg (/kg) 86 2.6 x 10-2 Impact Angle Known Quantities Surface Tension 3) 958 2 26 x 10 6 (kg/in2- s) 0584 Calculation 6 W Drop Diameter Drop Impact Velocity , 1/3 -a D f sin ($) a/i WE Pf U, Weber Number 2.6 x 1 (m) = U (m/s) 2.11 - 20 Heat Transfer Effectiveness E2 - E02 = Wfhfg t (T , Tf, a, U.) 1.02 - 75 - The data reducing equations are easily explained. First the drop diameter is calculated. each vibration produces one drop. It is known that The mass of the drop times the production frequency f must equal the mass flow rate Wf, or : Tf a 3 f = 6 (3-2) W 1/3 a = 6Wf Next the drop volo city is calculated. (3-3) Suppose a stream of drops with spacing D is traveling at a velocity U . An ob- server at a fixed location would co unt a rate of U/D drops In the s parse stream, the number passing by per uni t time. of drops generated per unit time is fixed and equal to f/m so that U (3-4) D or U = f/m (3-5) The component of this vel ocity normal to the impact target is MM". P.Mro - - 76 - Df UL = U sin $ = (3-6) sin $ - m The Weber number, based on the normal component of impact velocity, is: 2 (3-7) a Pf WE The heat removed by the sparse stream of drops is the difference between the target heats required to maintain constant target temperature with and without impacting drops: E2 -E 02 . Qdrops -E(3-8) The latent heat of the sparse stream is the product of the Im Wf liquid mass flow rate, of vaporization, (kg/s) and the latent heat , hfg Qlatent - m . f1h (3-9) heat The drop heat transfer effectiveness e is the ratio of the drop stream heat transfer to the drop stream latent heat: 77 - - latent heat drops E2 (3-10) E0 2\ (3-11) W hfg - 4 78 - HEAT TRANSFER DATA AND INTERPRETATION The drop heat transfer data clearly show the distinction between wetting and non-wetting temperature zones. Complete evaporation of drops was observed in the low superheat region, while very little heat transfer took place in the high superheat region. Calculations of heat transfer to air entrained by the drop stream show that this effect can easily account for the entire heat transfer rate measured at high superheat. prevents While this the determination of drop heat transfer from measurements in the non-wetting region, it offers an explanation for the large discrepancy (factors of 10 and more) in the data reported in the literature. - 79 - Drop Heat Transfer Measurements : Some Observations A summary of drop heat transfer data for all conditions is given in Figure 4-1. The major factor influencing drop heat transfer effectiveness is the wall superheat. At low superheats, the heat transfer effectiveness is about 1 , indicating total evaporation of the drops. At high superheats the effectiveness is very low, indicating very little evaporation. This gentral observation is consistent with what is known about boiling heat transfer, and with what was observed in the process of making these measurements. In the 1ow superheat region, the drops seemed almost to disappear into the targe t. In the high superheat region, rebo unded very regularly off the target. the drops Some additional ob servations are useful in interpreting finer details. At low superheats and high impact rates (achieved with by i ncreasing the number of drops in the sparse stream), stea mwas obser ved from the target (visib le steam means wet stea m due eith er to incomplete evaporati on or condensation in t he cooler air) and sometimes liquid build-up on the surf ace (obvio us evidence of incomplete evaporation). Hence the tendency of the drop heat transfer effectiveness to drop off at very low superheats was expected. The temperature region between high and low effective- 1,4 0 * S 0 0. * 0 1.2 S S S 1.0 * 0 * * 1* 30 0 S I * 0e 5 S S S 0.8 LU Lu 0.6 WElTING REGION NON-ETTING REGION 0.4 0.2 . . 0 0 00 50 200 100 150 TARGET SUPERHEAT, Tw-Ts (C) 250 FIGURE 4-1 DROP HEAT TRANSFER EFFECTIVENESS DATA SUMMARY 0 300 - 81 - ness also manifest.ed itself in three ways. Fir st, it was not possible to take steady heat transfer data in this region, because slight perturbations resulted in quick t emperature 1 second response time) in either direction excursions (,,, (towards quench or high superheat). This excur si on in either direction indicated a steady heating rat e som ewhat below the total evaporation rate (hence the ten dency to quench when perturbed), but above the low effec ti ven ess rebounding drop rate (hence the tendency to qui ckly superheat when perturbed). Second, the visual obser vati o n revealed a strong image of the impacting stream, but a faint, slightly irregular image of a rebounding stream (unl ike image of the regul ar reboundi ng stream at high the stron superheat ). Third, a definite hum was heard from the target impact ar a; the pitch of the hum increased when the number of drops n the sparse streams was increased. The three phenomena suggest that what was observed as a quasi-steady response t an average target surface temperature w.as really a highly insteady respo nse as far as each drop was concerned. Some drop might have completely evaporated, might ha rebounded with very little while others evaporation. One f nal visual ob servation was useful not only in interpret ng data from this experiment, some light in the but also in casting discrepancy between Pederson's [35] (high - 82 - ness) and Wachters' [33,34] (low effectiveness) data at high target superheat. established With a steady sparse stream of drops a small smo king torch (cotton swab with a drop of titanium tetrachl orid e) was placed in the test section. The observe d pattern of smoke motion revealed a large jet (many drop diameters) of entrained air around both the sparse and main streams. It will be shown that this air entrainment is sufficien t to account for the entire heat transfer rate measured at high target superheat. While the exact condi tions of Pede rson's experiments are not known (drop spaci ng, drop trav el length, and drop momentum loss), it is likel y from his ra nge of conditions that significant air entrain ment occured, and that his measurements also included signif icant hea t transfer to entrained air. Hall [32] came to the same conclusion in his experimenral work, and based his anal ysis of water spray cooling on air entrainment, negl ecti ng direct drop - surface heat transfer as a secondary effect. Wachters' two experiments were both different from the techn ique used in this and Pederson's experiments, two ways. ge tting aro und the air entrainment problem in In the first [33] a "large" drop (2.3mm) at low impact velocity was cons idered, so the air entrained by the single large drop was le ss, and the latent heat of the drop (which scales the heat transfer) was greater. In the second [34] th e drops (60 im) were actually carried by dry steam, MMINW - 83 IwilwfiM ulh - and measurements with and without drops were all performed in the presence of flowing steam. dispersed flow conditions.) (This is very close to Both of Wachters' measurements showed drop heat transfer effectiveness at high target superheats much less that that experiment. measured by Pederson or in this For these reasons it is concluded that the cooling of the target due to the observed air entrainment prevents a realistic interpretation for heat transfer to impacting drops at higher superheat in this experiment. from the measurements - 84 - Drop Heat Transfer Effectiveness Data The heat transfer data were shown to be repro- ducible by repeated steady measurements under identical conditions. Several interesting aspects of the heat transfer measurements are illustrated in Figures 4-2 through 4-5. In Figure 4-2, drop heat transfer effectiveness data for two targets are shown for nearly identical conditions. The data match quite well in the high effectiveness and low effectiveness regions. However, the transition temperature region is lower for the clean copper surface than for the oxidized steel surface. Moreaux et al. [39] also observed an early transition to rewet in the quench of a metal sample w.hen an oxide layer separated the metal surface from the liquid; that is, they observed transition to a higher heat transfer mode at a higher metal temperature with the oxide on the metal. The effect of liquid subcooling is shown in Figure 4-3. There is some scatter in the data (probably due to error in determining the liquid flow rate from the flow meter reading) but it does not seem directly associated with the subcooling. An estimate of the scatter in the data may be obtained directly from the subcooling group ... APOW" CS : M I y Ingligiiii,I1"I 1.4 '.o0x COPPER TARGET o STEEL TARGET 1.2 ~ x/X w 1.0 -x (OXIDIZED) a =250 pm WE= 14 0.8 = 86 C ~.T LuJ H uc 0.6 C-) c Lu x 0.2 x 0 0 50 100 150 200 250 TARGET SUPERHEAT Tw-Ts (C) FIGURE 4-2 DROP HEAT TRANSFER EFFECTIVENESS FOR TWO TARGETS 300 1,4 1.2 w c/) LUj 0 0 0 Tf = 86 C o NO STEAM A SOME STEAM IN STEAM * Tf = 23 C A 110 a = 260 pm WE = 13 STEEL TARGET 0.8 1- 0.6 LLi UlULLUJ 0.4 0.2 0. .50 100 150 200 TARGET SUPERHEAT, Tw-Ts (C) 250 FIGURE 4-3 DROP HEAT TRANSFER EFFECTIVENESS WITH LIQUID SUBCOOLING 300 Mwill. IW1111WISM - 87 Ccpsf(Ts hfg - - Tf) The maximum value of the subcooling rates is CS = 0.14 at Tf = 23 C; at Tf = 86 C, CS = .03 . The amount of heat assiciated with the subcooling under normal experimen(Tf = 86 C) tal conditions latent heat. is small with respect to the Therefore subcooling significant parameter. is not considered a No noticeable difference in transition temperature was observed with the increased subcooling. There is no systematic change in drop heat transfer effectiveness with Weber number in the low superheat region. The data show an effectiveness of about 1 for the range of Weber numbers considered : 3 < WE < 40 . (The most significant change in Weber numbers was achieved by rotating the target, thus varying the impact angle also reported an effectiveness of of 480 . e.) Pederson .9 for a Weber number This is an important conclusion of these experiments. The data in Figures 4-4 and 4-5 are for the dry impact region. In Figure 4-4, the difference in transition to the low effectiveness region for copper and oxidized steel is still evident. On the whole higher heat transfer is also associated with a higher drop impact Weber number, as is 0,2 COPPER: x WE = 13 STEEL (OXIDIZED): w C/) C/) * WE = 13 o WE = 3 LLu 0.1 LJ a = 250 pm Tf = 86 C e I I 100 200 TARGET SUPERHEAT, Tw-Ts (C) FIGURE 4-4 DROP HEAT TRANSFER EFFECTIVENESS: TRANSITION 300 ,12 .10 c./) C.) LUJ LUJ A a = 240 pm A .08 v a = 260 pm )( a = 290 + a 350 pm = .06 pm + L)J Tf = Tw-Ts .02 86 C = 195 C COPPER TASGET 0 30 20 WEBER NUMBER, WE 40 FIGURE 4-5 TARGET HEAT TRANSFER AMD WEBER NUMBER 90 evident in both Figure 4-4 and Figure 4-5 (except for one a = 290 pim). contradiction for Also larger heat transfer seems to be associated with smaller drops (again except for the contradiction in Figure 4-5 at WE = 10 for a = 290 ypm Inaccuracies are expected in the calculation of heat transfer effectiveness in this region because the volt ages required to maintain steady temperatures with and without impacting drops werevery close in value, and the ca1cul ation procedure involves taking the difference in the squares of these voltages. The average effectiveness value seems to be between and values of .02 .04 numbers (20 to to .1 which is consistant wit h Perdeson's .25 700). for a much higher range of Weber - 91 - Superheat Wall The Effect of Entrained Air at Hih The entrai nment of air by the drop stream was clearly shown test section. by a visual demonst rati on wi th smoke in the The effect of air entrainment on target heat transfer is of interest. A sim p1e observati on shows that cooling by air jet can easily account for the entire increase in heat transfer from the targe t when the sparse stre am is directed at the target at high superheats. Let Q9 9 the heat required to maintain the target- temperature a static environment; let Q9 be in Tw be the heat required to mai ntai n this temperature when a jet is in the test section but all drops are deflected below the target; let Q be the heat required to maintain this tempe rature when a sparse stream of drops is allowed to impinge on the target. T measurements show, for a target at Q9 - Q 0 = A typical set of = 367 C, .52 (J/s) (4-2) and for 1 out of 10 drops in the sparse stream, Q - = - (J/s) (4-3) the sparse stream, or for I out of 5 drops Q .36 Q9o = .58 (J/s) (4-4) - 92 - The conclusion of this example is that the heat removal due to the sparse stream of drops (.36 to 58 J/s) is of the same order of magnitude as the heat removal due to an air jet induced by the main stream which passes near the target (.52 J/s). The entrainment of air by the sparse stream of drops is fully expected because, as the drops slow down due to air drag, momentum is transfered to the air. The rate of momentum loss of the stream can be calculated because both the drop exit velocity from the needle and the'drop approach velocity at the target surface are known. The exit velocity is just the liquid volumetric Wf flow rate -T divided by the needle cross-sectional pf area A0 U0 = (4-5) p A The needle internal diameter is 100 pm manufacturer and checked in a microscope ). velocity tum flow U Z (specified by the The approach is determined by equation (3- 5 ). The momen- at the needle exit (in the sparse stream) is: z0 The momemtum flow U Z (4-6) of the sparse stream at the target 16 - 93 - approach ( Wf m - (4-7) The momentum lost by the drop stream is gained by the air. Consider a simple model of an air jet with average velocity U = Cl U (where C1 < 1) and average area A9 The momentum flow of this jet (4-8) (A9 U9 ) P9 and is equal to the momentum flow loss of the drop stream. (with Z. - Z = Wf (U 0 U) - Z = The mass flow of air (Uo0 pg C1 2 A U2 U0) Wf = PfA pfm A0o0 U (4-9) - U) W, (4-10) (4-11) = p 2 C2 A U 9g1 9 for the simply modeled (4-12) - air jet is 94 - : W = p = pf U A 1 or, heat T = Ag C 1U (4-13) A U (4-14) U - U 0 mnC The p = 0 U capacity Q 0 of the induced air stream at 25 C , wi th respect to the target temperature T = 367 C is W c 1 (Tw - T 0) 4-15) U - U A0 U0 cp)g (T fmC1 U T ) (4-16) For the same conditions discussed above, and for a sparse stream of U0 of 1 out of every 5.3 m/s drops with an exit velocity 10 , and an approach velocity and with an assumed value of jet mass flow is : = and the heat capacity is C1 U 3.7 m/s 1 , the estimated air 1.78 x 10-6 (kg/s) (c of = 1000 J/kg- C) 10g (4-17) I - 95 .61 Actually Cl is less than - (4-18) J/s 1 , so the calculated heat capacity of the air jet would be greater than .61 J/s . The important point i s that heat capacity the jet with respect to the target rger than the umed heat removal rate due to the jet A more detailed analy s is of heat tran sfer to an obstruction is not pe rfo rmed because vi su om a jet observations with smoke show that the natural convection pl ume around the target ass embl with the air jet, mak ing provides considerable interferance low analysis quite difficult. How- ever, the ove rall calculati ons, all based on actual measurements of heat and momentum exchange, show that sufficient momentum is transferred to the air by the drops to induce a flow which could account for the observed heat removal. Measurements of additional heat removal when the target is placed near a similar air jet support this conclusion. These argumen ts, supportet by the visual observation of the entrained air jet, sug gest that the heat removal due to entrained air prevents further interpretation of the heat transfer measurements of high superheat with respect to drop heat transfer alon e. At low superheat the amount of heat due to air jet cooling (measured at high superheats) - - - 96 - amounts to only a few percent of the heat removal due to complete vaporization of the drops. This effect should be proportional to the temperature difference between the wall and the jet, and therefore the heat transfer to the jet of entrained air is reduced at low superheats. - 97 - 5 ANALYSIS OF DYNAMICS AND HEAT TRANSFER TO IMPACTING DROPS An analysis of drops impacting on a hot surface is presented for the non-wetting region. Simple modeling of the drop shape during impact leads to the derivation of equations of drop motion and drop deformation, using Lagrangian methods. The predictions of drop impact dynamics and heat transfer compare favorably with available information on this process. The results are relatively insensitive to the assumed shapes for drop deformation. The analysis lends itself to application in more complex systems. A few brief claculations of surface - liquid heat transfer in the wetting region suggest that a combination of conduction and nucleation may be the mechanism responsible for the total evaporation of drops. Some comments are addressed to the topic of the wetting - non-wetting transition temperature. - 98 - Background and Previous Efforts Numerous research efforts have focused on the evaporation of small drops suspended by vapor over hot surfaces, a pheno menon associated with the observations of Both Wachters et al. [40] and Gottfried et al. Leidenfrost. [41, 42] model ed the vapor cushion between the drop and the surface to estimate evaporation times for drops set on hot surfaces. Baumeister et al. [43] larger, deformable drops, did similar work for as did Sho essow et al. [44] for drops on rotating plates. Heat transfer to non-wetting impact ing dr ops was measured and analyzed by Wachters et al. [33, Reasonably successful pre dictions of heat 34]. trans fer during impacts were obtained usi ng the same vapo r cush ion model as for the stationary drop, and supplying an accel eration force (instead of gravita tional force for the s tationary drop) based on high speed photos of the impacti ng drop. The radius of the drop as a function of time, the photos, t aken from was also requ ired for the ana lysis. Since this study, several other heat transfer models have been presented by Cumo et al. [ 45], McGinnis et al et al, [47], Holman et al.[48], and Hall .[4 [21]. 6], Sarma Cumo, in particular, observed an impact time that was constant for a MAIININh - 99 - range of temperatures above the transition temperature, which increased tremendously in the wetting zone. but McGinnis and later Holman presented a correlation for total heat transfer on impact. Sarma did an overall heat transfer calculation based on the max imum drop defor mation, a dropwall separation given by the force on the drop, and a constant impact period.Inste ad of the force necessary to rebound the drop during the impact period, the liquid stagnation pressure force was used, resulting in significantly higher heat transfer rates. rson' s [35] spra due to effects The results compared well with cool ing data, which are thought to be f entr ained air in the experiment. 's model allowed for no deformation of drops. A straight momentum cal cul atio n was performed based on an external force due to a vapor cushion under the rigid drop. Of the heat transfer models, that presented by Wachters is the most appealing, since it accounts for the significant physical mechanisms, it is not restricted to any data base (though it was derived for a certain range of dynamic impacts and is not valid, for example, for drops which shatter on impact), and is reasonably justified by two independent groups of measurements. The disadvantage of Wachters' technique is the need for information on drop dynamics which must be supplied from photographs. Relatively little work has been done on heat transfer - 100 to drops which impact and we t. - A heat transfer model for wetting-impacting drops was developed by Illoeje [6] and has been extended by Chen et al. [20]. The model assumes that a fraction of the drop is rejected from the wall after a brief nucleation period; thi s is inconsistent with measurements of this study showing complete vaporization and prono unced nucleation o f imp acting drops which wet the surface. Illoeje's model al so does not distinguish between wetti ng and non-wetting surf ace temperatures, it and in fact is most often used in the non-wetting region. Virtually all of the experimental studies with impac ting or stationary drop s identify the transition tempe rature between wet ting and non-wetting. Some investi- gatio ns report surface prope rty and surface finish effects on tr ansition temperatu re. Wachters' study suggests that the non-wetting regime is no t fully developed for a significant temperature range above the initial transit ion temperature. This analysis of heat transfer to impacting drops is in two parts; the non-wetting and wetting impacts are considered separately. For non-wetting impacts, the approach of Wachters is combined with a dynamic model of a deforming drop, thus eliminating the need for deta ils taken from photographs. In the wetting region, calculat ions for conduction, nucleation, and bubble growth are cons istent - 101 - with the observed complete evaporation of the drop. Some comments about the basic approach to the heat transfer analysis are appropriate. The analysis can be described as a coupling of quasi-steady heat transfer and dynamics. The total heat transfer to a dynamic system over a time period T(s) is the time integral of the product of 2 the instantaneous heat flux q (J/s-m2) and the instantaneous heat transfer area A(m ). Q q A dt(5-1) 0 The instantaneous heat flux is assumed for a quasi-steady geometry because the thermal response is much quicker than the dynamic response. For the dry impact, the heat is transferred through a vapor cushion of thickness y(m) so that the instantaneous heat flux is approximately: k (_T q g T ) (5-2 wy where k9 is the vapor thermal conductivity (J/s-m C) and Tw and Ts are the surface and saturation temperatures (C), respectively For a drop wetting the surface, the average heat flux by conduction in a time t (s) is approximately: q (k fPf c )12 (Tw-Ts) (t )1/2 (5-3) - where k P and c 102 - is the liquid thermal conductivity (J/s-m2-C) is the liquid density (kg/m3 ) is the liquid specific heat at constant pressure (J/kg-C). The impact dynamics (and/or bubble growth dynamics) must supply the time period T (or t ), and both the heat transfer area A and the drop-wall separation y as functions of time during the impact. Thus the combination of dynamics with quasi-steady heat transfer analysis allows the computation of the total heat transfer. Mi - 103 - Drop Dynamics Two important types of energy storage for a liquid drop are kinetic energy (due to either center of mass or internal motion) and surface energy (due to surface Certain types of drop-surface impacts are tension). governed by a transfer of energy between these energy storage modes. The kinetic energy of the drop (due to the center of mass motion toward the surface) prior to impact can be stored as kinetic energy of internal motion and increased surface energy as the drop deforms against the surface during impact. This type of impact can occur when the initial kinetic and surface energies of the drop are The kinetic energy KE (J) of a spherical comparable. drop moving with uniform velocity is equal to the product of its mass and velocity squared: 1 ra 3 KE pf - 2 U ; (5-4) and the surface energy PE (J) is equal to the product of the surface tension and the surface area: PE where a is = a r a , the drop diameter (m), Pf is the liquid density' (kg/m3 U is the drop velocity (m/sec) and a is the surface tension (kg/sec 2 (5-5) - A Weber numberS 104 - which is proportional to the ratio of the initial kinetic and surface energies in the undeformed drop, is useful in characterizing the nature of the impact: WE pf U2 a a = (5-6) For low Weber numbers, the kinetic energy is small compared to the surface energy, and a small deformation will accomodate the energy storage mode transfer during impacts. For moderate Weber numbers, a significant deformation* is required. occur. For large Weber numbers, drop break-up will Pederson [35] observed drop break-up on impact for Weber numbers over about 75, in agreement with Wachters et al. [33]. The energy conservation principle prescribes the exchange of energy in the storage modes and the work done by the system against an external force F (kg-m/s2) through a displacement y (see e.g. [49]): t2 [6(KE-PE) + F6y] dt = 0 (5-7) tI This Weber number is different from another Weber number WE' p (Uf - U ) a commonly used to estimate drop break-up (at WE' ~ 7) due to drag forces in gas-liquid dispersions in which the two phases travel at different velocities. Notice the use of the gas density rather than the liquid density. - 105 - The interchange of mean motion kinetic energy, kinetic energy of internal motion, and surface potential energy for a deforming drop can be a complicated process dependent on internal and external flow patterns, internal and external pressure gradients, variations in surface curvature, interficial phenona, and other effects. However, for this application, a reasonable description of the drop configuration as a function of time during impact can be obtained by simple modeling of the important phenomena. - 106 - Constraints and Equations of Motion Suppose the drop deformation can be approximately described by a family of shapes. A corresponding internal flow field can be specified for the drop to describe a progression through the selected famil y of shapes. The kinetic and potential energies of this syst em c an be computed, because the drop shape and internal flow fiel d have been specified. The energy conservation principle can be used to determine -1 the rate of exchange of energy betw een the storage modes for the system model under the geometri c constraint. Consider a simple example of* this techniqu . dro p geome try throughout impac t be node led as a lar cyl inder. ight circu- The cylinder axis and dr op center of mass vel ocity a re perpendicular to the F Let the i npac t surface and a force acts on the bottom of the drop as shown in Fig ure 5-1. As the drop deforms, the height A change while the drop volume and the radiu s emains constant V 3 V = Tra 6 (sphere) 1 (5-8) 2 (equivalent cylinder) The cyl indrical drop is shown in a deformed configuration in Figure 5-2. The radius R of the disk which forms the bottom of the drop (facing the wall) is equal to 1/2 R = A 2 -4- 1M 0 INGWRIMAR - X2 )(5-9) (i~l iil41 !N44i o"4 - 107 - f z cm FIGURE 5-1 CYLINDRICAL MODEL OF DROP EQUILIBRIUM POSITION - 108 - i T IF' Zi m / I r so ,/7// //// FIGURE 5-2 CYLINDRICAL MODEL OF DROP DEFORMED POSITION IN, - 109 - The position of the center of mass (measured perpend icul ar The location of the bottom to the impact surface) is zcm of the cylinder y can be determined from the cente r of mass coordinate and the cyl inder height: y= (5-10) 2 cm The internal flow fie which correspo nds to the assumed geometry of deformati is given by: axial velocity Uz ~ dzcd dtcm 1 + - - dX 1 dt d (5-11) radial velocity Ur The velocity d zcm dt r 2 dX 1 dt (2) (5-12) is the center of mass veloci ty (normal to the impact surface) and the velocity dt is the rate of change of the cylinder height. The kinetic and surface potential energies of the cylindrical system can be computed : These and other computations are given in detail in Appendix 5. - KE - (U + U ) dV p = 110 1 2~ Of [/d zcm-2 j dt (5-13) dA+ d t) 2 I2 + 12 + 8 2 (5-14) RE = a A (5-15) dA surface surface = a [2 A A2 + 2Tr A22] (5-16) Equa tions of motion for the deforming cylinder are obtained by using these relations with the energy conservation principle, Equation (5-7). The resulting equations of motion are: F pf V = F =- dt + (( 1 2 pfV[ A2 3 + 8 + a( Trr2 - I- I 4**M"".."pWO..W.".."**"O- -- - (5-17) d zm NPIN M11110 oil IMIN 0 IN 11-1---- 2 d2 d 1 dA dX1 ] 2 13 2 ) (1 ) (5-18) ( IIWIIIIWI , - 111 - Two coordinates are necessary to specify the cylindrical system configuration. or One shape coordinate, either A1 A2 , determines the shape (since the second can be calculated from the known volume); one position coordinate, either zcm or y, determines the location (since the other can be calculated from the known shape coordinate). A and zcm are used. The two equations of motion can be integrated to determine the cylinder height, center of mass position For this example, zcm Al and cylinder as functions of time during the impact. Other families of shapes may also be described by different sets of two coordinates. example of another A sphere, shown in two-coordin ate system is the truncate Figure 5-3. two-coordin ate system \ length, and tion can be derived for a general Equations , zcm), where A 1 is a geometric is t he cen ter of mass position. Note that zcm the equatio ns of motio n are derived on the basis of expressions for kinetic and po tential ene rgy. between the mass cente r on which th e force that 6y F zcm and the 1ocation of the plane acts (the drop bottom) is needed so can be deter mined in terms of Later, the expression for th e force depend on the drop bot tom radi us separation y, R for a drop with a fl fore, in addition A relationship 6 1 and 6z cm' will be shown to F nd the drop-wall disk bottom. to the expression for y, There- a general ex- - z~ 112 - cm z -cm F y y FIGURE 5-3 TRUNCATED SPHERE MODEL OF DROP EQUILIBRIUM AND DRFOR'ED' POSITIONS - - in terms of R pression for 113 X1is also needed. These relations can be represented in general form for a drop of constant volume = KE 1i -p-2 V: d m V [ L + + ( K( 2 d t dt dz 2] ) (5-19) ,J (5-20) a PE = aa Zcm = y + a M( R = a N( al (5-21) a (5-22) ) where the functions K(x), L(x), M(x), and N(x) are determined (The dummy variable x for a selected geometric constraint. represents the ratio of the geometric length X. ,to the diameter a 6 of a sphere of the same volume V, 1\/3 a = the The general equations of motion for the two parameter system with fixed volume are: d2 z dt 2cm P'f FK( + V [ aa L'( (5-23) X a X d2 X a )V M'( dt dX 2a a ) K a dt 2 J (5-24) - 114 - These can be rewritten in form for integration: d z dt d2 cm 2 d A1 F pVy . = - (5-25) F M' dt 2 1 + oaL a + p _ 22fa K KA' ( A1 a a d ' it ( a 3 [f V K( Aal ] (5-26) The details of the derivations f or the expressions for the kinetic energy, potential energy , center of mass posit ion (with respect to drop bottom), and drop bottom radius for both the cylinder and the sphere appear in Appendix 5, together with the derivation of th e equations of motion for the general two-coordinate syste m. These results are sum- marized in Table 5-1. An additional geometric constraint of some interest can be introduced: that the drop bottom maintain contact with the impact surface. The constra int for maintaining contact is that the drop-wall separation be and remain zero: y = 0 (5-27) This implies a relationship between the center of mass posi- DYNAMIC SCALING FUNCTIONS FOR TWO SHAPES TABLE 5-1 SCALING (CONDITIONS) TRUNCATED SPHERE CYLINDER FUNCTION K C C1 + C 2 1+ C2224x 3 31 C + C2 + 2)2 )( ( l2 c> 3(-- C -4 8x C2 1) - x K' -2c 3x 3 x<x x 13+ 2)( C2 4 ( 1 x (3 1/2 L 2nC 3 [-I- + (-) L' -1 2,fC3[-X- + 1 ( 1 1/2 ] (x + M'1 0 N ( 6x x .87 1 (1 1 + 2x + 2 C5 xC4 x4 ) 2x2 1/2 x 3 1.0 1/2 3 1 1 no contact C5 ~ 1 4 contact C3(-C4 7 M m + C5 2 7 T xc x 2 C3 (C4 )6 13 - 4) _ 12 4 2 ] = 0.63 * X X 2 3x3 )] = 5, Cs = 2 - tion 116 - and the geometric length zcm Equation (5-21) A . y = 0, With becomes: z (5-28) =a M( a and the velocity and acceleration relations are: d z cm d and d2 z dt2cm = = A a a Mi( 1 a dA ( dt 1 dt 2 ) (5-29) A + M'() d2A1 (5-30) dt2 Since the addi tional constraint of drop-wall contact introduces a known relationshir between center of mass position and geometric length, one equation of motion can be eliminated. F The force can be eliminated by subtracting Equation (5-24) from Equation (5-23), and the center of mass acceleration can be replaced by the right hand side of Equation (5-30) to yiel d the following equation of motion for drop- surface contac d2 dt2 {M A - A1 a )Mi + a a Pf V 2~- a ja ~ lM'( ) J al a d a dt 2 + K( 1 (5-31) 117 - - Equation (5-31) can be integrated, and the center of mass velocity can be calculated from Equation (5-29) and the surface force F required to support drop-surface contact can be calculated from either Equation (5-23) or Equation (5-24). The integration of the equations of motion is the similar both with and without drop-surface contact. If no contact occurs, two equations of motion must be integrated to determine the drop position and shape as a function of time, given the external force F. For the case of drop-wall contact, the additional geometric constraint reduces the twocoordinate system to a one-coordinate system, such that only one equation of motion need be integrated. The equations of motion, derived from the energy conservation principle, provide that the total energy of the drop EE (J): (5-32) + PE EE = KE is conserved, except when reduced by the amount of work done against an external force (the drop does work against through a displacement -6y F F (EE - EE) when y - decreases) f F 6Y = 0. (5-33) The two-coordinate system with constant volume is the simplest model of the impacting drop which allows for both center of mass motion and deformation motion. ____ - -ft-. ! More sophis- - 118 - ticated models might include additional position coordinates (for motion in directions besides that perpendicular to the impact surface), additional geometric parameters (for more complicated shapes and variable volume), and/or additional forces (in other directions at other locations). The derivation of the equations of motion for the more elaborate system would proceed in the same way: 1) the derivation of energy expressions in terms of the position and geometric coordinates (Xi , zi) 2) the derivation of the relation between the force 'displacement coordinates (yk) ordinates (X., z ); and 3) and the system co- the determination of some geometric parameters (Rm) which affect the magnitude of the external forces (Fn): VFK~j ~ KE a. ... L) KE X. + K z. , ( dz.E dt dA.d dt ., dt (5-34) PE = aa 2 L( (5-35) a ) x. S Rm aMk( aN ( z. , X. 1 ) (5-36) (5-37) - 119 - No more than three orthogona be chosen. center of mass coordi nates can The equations of motion can be derived from energy conservation principle. of motion for this system. There wil 1 be (i + j) equations - 120 - Boundary Conditions The equations of motion for the two-coordi nate system model of the deforming drop can be integrated to determine t he boundary (or, the impact dynamics if in this case, initial) conditions are sp ecif ied. At the beginning of an impact, the drop is ini ti ally undeformed (spherical). For the drop models, the undeformed or equilibrium position can be determined by minimiz ing the potential (surface tensi on) energy function: d PE d X1 = (5-38) 0 Xl1e The equilibrium geometric lengths Xle are shown in Table 5- 1 for the cylinder and truncated sphere. These equilibrium configuratons represent the drop shape with maximum symmetry and minimum surface area, given a fixed volume V and the geometric constraint on drop shape. Three cases of drop deformation are discussed: 1) drop oscillation in an open environment, 2) drop impact close to a surface, and 3) surface contacting drop. For an oscillating drop, there is no center of mass motion, and the drop is far from the wall. The drop is initially in its equilibrium position, but has an initial deformation velocity. The freely oscillating drop experiences no external force. 011101 - Oscillating drop: zcm = - initial conditions large far from wall) (5-39) equilibrium position) (5-40) ale d z cm dt dX I dt 121 U1 10 = U 2 ,o = 0 (no center of mass motion) (5-41) (initial internal motion)(5-42) For the impacting drop, the center of mass velocity is perpendicular to and toward a wall. The drop is initially undeformed and has no deformation velocity. The initial separation should not be chosen so large that substantial integration time is spent prior to the beginning of deformati on. Impacting drop: zcm = d z cm dt dAI dt zcmo 1 cm A1 - initial conditions 0 (5-43) equil ibrium position) (5-44) (initial center of mass velocity) (5-45) U 1o 2,o ini tial separation) = 0 (no initial deformation velocity) (5-46) - 122 - An important distinction between these two cases is the development of the deformation velocity. For the oscillating drop, the deformation velocity is at a maximum when the drop is in its equilibrium position. For the impacting drop, however, the deformation velocity increases from zero (when the drop is in equilibrium position), to a maximum, and returns to zero at the point of maximum extension. The contacting drop can only be approached with a pre determined relationship between the center of mass and deformation velocities, and of course the integration must be performed for y = 0, i.e..at the surface. surface contacting drop: initial contitions cm = a M( ae le dz cm dt dX dt = M'( U2 ,o l,e)U a 2,o (drop -surface contact) (5-47) (equi librium position) (5-48) (cont act maintained) (5-49) (initial deformation velocity) (5-50) One way to combine the impacting drop with the contact position is to proceed with the integration of the equations - 123 - of motion for the impacting drop until the contact velocity condition is reached: d z cm A d = (5-51 M( a dt dt and then set y = 0 and proceed with the drop contact calcu 1ation. The deformation dynamics for oscillating, impacting, and surface contacting drops are not easily compared except at the point of maximum deformation. In all three cases the deformation velocity is zero at this point; for the freely oscillating and surface-contacting drops, the center of mass velocity is exactly zero at the maximum extension, and for the impacting drop it is approximately zero. Thus for all three cases, all or virtually of the drop's mechanical energy is stored as surface potential energy at the point of maximum extension. Therefore a comparison of drop dynamics for a given size and shape is best done by selecting cases in which the maximum potential energy of the drop (i.e. the total mechanical energy) is the same. For all cases considered, the drop is initia lly undeformed. For a given size and shape of drop, the initial potential energy is the .same for all cases. Thus fixing the total mechanical energy is equivalent to fixin g the initial kinetic energy. - -_-1 - EE = 124 + KE 0 fixed fixed RE0 (5-52) constant The initial kinetic energy of the drop in all cases can be calculated: KE0 =- pf V (U, For fixed values ,0 2 + K U2,o 2). (5-53) let the initial velocities be K evaluated: oscillating drop U1 = 0 (5-54) KE U2 ,o 1 = (5-55) V K p impacting drop KE U1 ,0 U2 ,o 0 = (5-56) 21E = (5-57) 0 surface contacting drop U = (5-58) M' U2,o KE U2 ,o (5-59) f V[(M')2 + K] - 125 - Each of these conditions satisfies Equation (5-53) for any specified value of KE0 , so that the dynamics of the three cases can be compared for fixed energy values of a drop of given size and shape. - 126 - External Force in Drop Motion Near a Plane Surface As a drop approaches a wall through a gaseous medium there is an increase in pressure between the drop and the wall, because gas must be forced aside in a small space. This effect is more pronounced if vapor is generated on the wall side of the drop. This increased pressure causes the drop to begin its deformation before strik ing th e wall. The derivation of the expr ession of the fo rce due to the pressure rise between the drop and the wall there are two major conbribution s to the f orce: sho ws that one due t 0 increased pressure required to accelerate the vapor in a radial direction out from betwee n the drop and t he wall, and one due to increased pressur e required to overc ome the viscous drag associated with thi s radial f low. An estimate of the magnitu de of the pressu 'e rise is obtained by modeling the vapor flow field betwee and the wall. The simplest appr opriate model is flow between two disks of radius tance y, as shown in Figure 5-4. R, the drop axisymmetric separat ed by a di s- One disk repres en ts the flattened bottom surface of the drop, and the other projection of the drop-bottom onto the wall. radi us R and the separation distance of time during the impact. y is the The botto will be functions However, a qua si-steady flow will be used to est imate the pressure rise between the drop and the wall. The quasi-steady flow will later be as - Ur (R) WALL R FIGURE 5-4 STEADY AXISYMMETRIC LAMINAR FLOW BETWEEN PARALLEL -DISKS - 128 - sociated with a quasi-steady heat transfer, both dete rmi ned for an instantaneous configuration of the drop during the course of its impact. The mass continuity equation for steady uniform density flow between two parallel disks is: aUr + r ar Ur r r + aU z 3z =0 wh ere Ur is the radial gas velocity (m/s) and Uz is the axial gas velocity (m/s). (5-60) The appro pri ate boundary conditions are Ur = 0 at z = 0 and z = y (5-61) (no slip at the impact surface or drop bott om) U0 at z = y U= (5-62) (uniform axial velocity at the drop bottom) and = 0 at z = 0 U (5-63) (zero axial velocity at the impact surface) A solution (whi satisfies the continuity equation) of the following form assumed: Ur _ 12y r U z = -U 0 p' Uo p''( yz ) ( z-), y (5-64) (5-65) where the function P( z ) must meet the boundary conditions: y INI" - 129 p''(O) = p''(l) - (5-66) 0 p ' (1) = 1 (5-67) p' (0) (5-68) = 0 Order of magnitude calcul ations show that the drop does not begin to deform until it approaches very close to the wall, satisfying the condition R >> y. The momentum conservation for this condition is: Ur P Ig g( r + + Ur 3r 3r P+ 3 2 Ur 3r ;z y9 z 2 (5-69) The pressure gradient in the axial direction is neglected, and the radial momentum equation is integrated with respect to the axial coordinate, after substitution of the velocity relations in Equations (5-64) and (5-65): PgU pUo 2 1 4 r p 22 I U.. @P+ 0 2 p 1 2yr p pd r y3 piU z y z y (5-70) - 130 -- Letting 1 1 I2 2 2 - d (5-71) 0 ''' d(- and - -) (5-72) the integration of Equati on (5-70) yields the integralaveraged radial pressure distribution1: P(R) - P(r) With P(R) [pU 0 (411 ) =- + p U 0 R2_ (4I21 R2y 2 2 (5-73) P . the net force due to the increased pressure between the disks can be calculated: F F = f 2Trr[P(r) 2 4 p U0 R - P (5-74) ]dr r y U R4 (5-75) y The first term is the radial acceleration term, and the second is the viscosity term. The integrals the assumed form of the function p. and I2 depend on One possible form is IThe radial pressure distribution is nonuniform, suggesting that the bottom surface of the drop is not really flat. This complexity was omitted from this analysis. Note also that Pmax = 2F/7R2. - p(x) = x3 for which I 131 3- 1 2 - x4 = .225 and 12 = 1.5. The integration is given in Appendix (5-7,6) - 132 - Evaporation and Heat Transfer Effectiveness: Dry Impact The heat transfer to impacting drops prior to wetting or in non-wetting impacts can be simply estimated when the dynamics of the impact and the boundary conditions are known. Prior-to integration of the equations of motion, an important quantity must be specified in order to evaluate the external force: drop bottom. the velocity of the gas normal to the If no evaporation occurs, this would be taken as the velocity of the drop bottom itself. However, with evaporation, the gas velocity at the drop bottom is related to the rate of vaporization, and hence to the heat transfer to the drop. A simple estimate of the rate of heat transfer Q (J/s) TrR 2 , between two disks of area at temperatures Tw and where k - T w y T k 9 r R2 g y ( (5-77) is the thermal conductivity (J/s-m-C) of the medium between the two disks. In the case considered, the medium separating the two disks is a flowing gas, and the temperature distribution between the two surfaces is nonlinear. y, Ts(C) respectively, is: 2 O separated by a distance The factors sl and 62 are introduced to account for the nonlinearlity in the temperature distribution, so that the heat removed from the wall is: - M - 133 - Tw -Ts S$ 1 kg (5-78) 7TR2 and the heat transferred to the drop bottom is: - Q2 2T - T sT. wr k TR2 2 (5-79) Because the flowing fluid carries heat with it as it f 1ows out from between the drop and the wall, the heat trans ferred to the drop is less than the heat removed from-th e wal l; that is, 2 < The factors 1, 3 or 2 < 6 . and 62 depend on the flow field between the drop and the wall, and on the superheat group: c CT = (T - T ) h'g w s hfg (5-80) They are evaluated in Appendix 5 for the same flow field used in the derivation of the extern al force due to pressure rise between the drop and the wall. As for the pressure calculation, the heat transfer calcula tion is performed for a quas i- static system with the flow assumed steady. For the axisymmetric flow considered, the heat tra nsfer from the wall is always greater than the linear appr oximation, and the heat transfer to the drop is always less (for a fixed temperature difference Tw - Ts); The factors 8I and s2 , that is l < 1 and have been calculated 2 ' - 134 - and curve fit as functions of the superheat group CT. For the range: 0 < CT < 1 the relations (5-81) (1 + 2 CT) and 1 + .8CT (5-82) (11 +2 + T CT) provided a curve fit about 3% or better. superheat group less than 1. CT of the If all used for evaporation Typically, t transferred to the drop bottom is he vapor gene ration rate W (kg/s) is: h2 hfg where is the latent heat of vaporization hf the (5-83) (J/kg) The va por velocity associated with this vapor generati on rate is, wi th Equation (5-79) : U v = W 9 p TrR2 ) 32 k (T - TsT) pg hfg y (5-84) - 135 The resultant gas velocity at the drop bottom, the sum of the vaporization velocity Uv drop bottom) and the drop bottom velocity U = + U U0 ,0 is (relative to the U : (5-85) U The pressure force expression, Equation (5-75), can now be evaluated for the flat bottom evaporating drop model. The total heat Q (J) transferred from the wall during an impact period T is the time integral of the instantaneous heat transfer rate: Q Q (5-86) dt The drop heat transfer effectiveness (from the wal 1 side) is the ratio of Q1 6 l QI pf V h to th e latent heat of the drop: (with Eq (5-78)) k (T - T s) Pf V hfg fT 2 y dt (5-87) The heat transfer effectiveness is calculated from the time integral of an instantaneous quasi-static heat transfer rate which is re-evaluated at small time intervals during the dynamic mpact. role in he calculation, as is evidenced by the appearance of The dynamics of the impact play an important the drop bottom radius R and the drop-wall separation Equation (5-87). y in - 136 - Nondimensional Equations The use of the dimensionless ness c heat transfer effective- in the data reduction scheme greatly simplified the interpretation of the heat transfer measurements. Approp- riate introduction of scaling parameters for the purpose of rendering all dynamic variables dimensionless can also simplify the interpretation of the drop dynamics model. The latent heat of vapori zation of a drop, which effectively scales the drop heat transfer, is much larger than the mechanical energy whi ch the drop has in the form of either kinetic or surface potential energy. Therefore, the latent heat is not useful in scaling the dynamic variables. are, however combined to There three other simple parameters which can be ondimens ionalize all the variables used in the dynamic drop model. They are: the diameter of a spherical drop of volume V (m) the liquid density (kg/m3) and a ch aracteris tic vibration period (s). For a drop of volume a and c V, = ( 6V T/3 ) 3 1/2 f a (5-88) (5-89) Ilk - 137 - a and pf is equivalent to the use (The use of Tc with a with a and Pf, since only a is needed in addition to and pf to determine Tc. However, T was chosen as a scaling parameter because the impact time in particular of interest). The nondimensionalizing of dynamic variables is outlined in Table 5-2. The equations of motion, the boundary conditions, t ogether with force e xpr ession, and heat transfer effectiv eness are now rewrit ten with new, al variables . The dimensionless variables have the same symbol as th eir dimensional coun ter -part; is replacing n on-dimension- ,(both e.g. dim ensional). a and X X(dimensionless) When the equations of motion are rewritte n i n terms of dimensionless variables, s everal dimensionless groups appear as scaling factors for various terms in the equations. are named and defined in Table 5-2. These groups The dimensionless equations of motion, force, and heat transfer effectiveness are: d2 Z2 m F F . (5-90) V dt 2 dX 2L [F M' d2A + + dt 26 V K' VK (-dt 2 ] (5-91) dt 2 F = 16 TH2 I (1 + U U .(1 +~Y-) U Uv + S2 TH VI 4 R4 4Y y4 12 (5-92) - TABLE 5-2 138 - DIMENSIONLESS DYNAMIC VARIABLES AND GROUPS dimensionless variable replacing ratios of dimensional variables length a t time T F force pa a) 'r2 KE (or PE) KE5energy Pf a 2 dimensionless groups replacing combinations of dimensional constants 0 RHO WE 9 density ratio Pf pf U2a 2 Weber number 1/2 VI ga viscosity group k9 (T -T TH \ h 1 1/2 pg aa conductivity group CT c p (Tw -Ts) hfg superheat group INWI - 1/26 = 139 THJ (RHO) 2r (5-93) y 0 where 2 6 = (1 + - 2 (5-94) T CT 1 + .8 CT (1 + 2 CT) 1 (5-95) and the initial conditions are: oscillating drop (5-96) large = zcm 1 (5-97) e dz cm (5-98) 0 - dt dA I 7T T dt (WE 1/2 (5-99) \K impacting drop zcm xI (5-100) = Xicm,o e (5-101) x~ = dz T= dt 1/2 (WE ) (5-102) dAX - dt --- pj"i-pp (5-103) 0 -- N-f *-qi- B -Iljllqqq - 140 - surface contacting drop zcm = M(X1 (5-104) )e (5-105) X1 e 1dX dz cm dtA dX 1 dt (5-106) dt ,1/2 ( _ 4 Note the use of the fixed Weber WE [ (M') 2 + K] ) (5-107) number in eva luating all velocities, such that the ratio of init ial kinetic energy to initial (equilibrium) potenti al ener gy is constant for all sets of initial conditions. In order to simplify the computat ions of drop deformation dynamics for the various condi tions and fluids of interest, some characteristic dime ns ionI ess groups are defined. The useful dimensionless group is calculated from the characteristic value by mult iplyi ng by appropriate ratios This procedure is of actual to characteristic para meters. outlined in Table 5-3. In this way, for example, a calculation is performed for several drop di ameters (all else constant) by simply introducing sev eral di ameter ratios. drop diameter appears in three d imens io nless groups. temperature variation of gas den sity p g The The is also accounted for in an approximate way by introducing a temperature ra tio,. TABLE 5-3 CHARACTERISTIC DIMENSIONLESS GROUPS AND PARAMETER RATIOS CHARACTERISTIC GROUP AND PARAMETER RATIOS OBTAINED FROM PARTICULAR GROUP S9 9, s Pf *9 Pf pg,s - RHO RHO RR 2 2 a WE k (T 9 v 2 1/2 pPfcaa) p g' aac VI V - TS) S h fg 1 (1 p gcaa 1/2 k Ts (Tw -T s) hfg CT ac - a 1/2 pa Pga9 1 1 Ac 1/2 RR 1 _ P ) - T T 1/2 aa T . TR Ts Tw TS hf Ts CTc TR 1/2 1/2 p 1 THc c AR 1/2 p 1/2 hfg TH cP c 2 UR - WEc 2 2 U PfUcac PfU a 1/2 -a~I 1/2 1 1/2 - 142 as described in Appendix 5. - Thus the characteristic groups, which depend on fluid properties and some arbitrarily selected parameters, need not be recalculated for each set of new parameters, but only multiplied by the appropriate parameter ratios. "limp0 go "NOWNINION __ ____ -- I - --- - 143 - Dynamics of Dry Impacts: Model and Data The equations of motion for the two-coordinate model of the deforming drop have been integrated for a variety of conditions. Property data are given in Appendix 5. A range of drop sizes and surface temperatures were considered. Two different shape constraints were used: the truncated sphere and the right circular cylinder. sphere, oscillating, impacting, For the truncated and surface contacting cases were examined; for the cylinder, oscillating and impacting cases werB considered. Most cases were calculated for water at atmospheric pressure; one set of calculations was performed for nitrogen over a range of temperatures, using the cyl indrical is well model. The dynamic behavior of all systems behaved and results compare favorably with details taken from Wachters ' [33] photographs of impacting drops. The heat transfer c alculation yields the correct order of magnitude for drop heat transfer effecti veness according to data from two significantly different ments also by Wacht ers [33, 34], able data, no final types of measure- but due to limited avail- conclusion may be drawn. Several of the interesting results of the dynamic model calculations are presented in Figu res 5-5 to 5-22. One test of the sui tability of the model ing is the predicted value of the period of free oscillation. These models are appropriate only for the compression hal f of the oscilla- - 144 - tion period, so a half-oscillation (equilibrium position to maximum deformation back to equilibr ium position) was calculated. period T(s) The ratio of th e c alculated vibration hal f t o the characteristic period of oscillation -c is plotted in Figure 5-5. For both the cylinder1 and the sphere, above a Weber number of about 5, the ratio is about 0.5 as i t sho ul d be. The impact periodwas also calculated for the cylin der and the sphere,2 and is shown in Figure 5-6 together with rough data tak:en from Wachter's photographs showing remar kably good agreement. It was found that the Weber number is the single most important model parameter for determini ng the period T, and results did not change noticeably with large changes in surface temperature, drop size,or veloc ity at fixed Weber numbers. calculated for the deforming drop model function of Weber number. The impact period is not a strong The calculated impact period 3 -Eis approximately 60% of the characteristic vibration period Tc for Weber numbers between 10 and 100. 1 Most of the cylinder calculations were done with the surface area reduced by a factor of .87 so that the potential energy of the drop in its equilibrium position was the same as for a spherical drop of the same volume. 2 For the impacting sphere, a minimum drop bottom radius of R =.laL was used as an initial drop bottom area to scale the external force F. The choice of a minimum area was necesary for the calculation, but the selected value of the minimum area did not affect the calcualted results. HH, - 145 - Drop size at maximum extension is plotted as a function of Web er number for oscillating drops and the surface contacting drop in Figure 5-7. The maximum drop extension 2Rma not vary with independent changes ratio, 2Ra x, ~- does in drop siz e velocity, temperature, fluid, or even deformation condit ions, (impacting or oscillating), at fixed values of t he Weber number. photos Again data from Wachters' sho ws that the model is quite effective in predicting the maximum extension. The data point taken from Wachter's photo's at WE = 41 is for an impact angle of 300 (from norma 1). The velocity used to evaluate the Weber number is the normal component. It is evident from Figure 5-7 that had the total velocity (giving WE = 56), or any other veloc ity been used, the data point would have appeared to be out of place. component 0f It is important that the velcoity normal to the impact surface be used to evalua te the Weber number for this type of dynamic modeling. Figure 5-7 shows the expected increase in maximum extension with Weber number. Larger deformation is required for drops with greater kinet ic energy because at 3 The impact period can be calculated for these models only in the compression part of the deformation: that is, only for the flattening against the wall, but not for extension away from the wall. The impact period was determined from the difference in time between an arbitrarily chosen 1% initial deformation (Xl~ .99X ) prior to maximum deformation, and the return to that hint after maximum deformation. 0,7 0,6' H 0.5 - == $~~~~~~~~ L 0.4. - Fl- 0.3 (- CYLINDER: S0.2 2 2 xr A bIA A A = 0,87XfA 01- 0 0 .1 SPHERE 110 WEBER NUMBER, WE FIGURE 5-5 FREE DROP OSCILLATION PERIOD 100 .. l. - 147 - 1,0 e0 0,8 F U H 1-' "* 6---A-0 -.... 0,6 H x LU 0~~ x H0~~ 0.4 a = 2300 pm 0,2 0 a = 220 im A CYLINDER A CYLINDER X FROM WACHTERS' 0 SPHERE * SPHERE PHOTOS [33] (SURFACE CONTACT) 10 WEBER NUMBER, WE FIGURE 5-6 DROP IMPACT PERIOD 100 - b A 148 - CYLINDER o SPHERE *SPHERE (SURFACE CONTACT) XFROM WACHTERS' PHOTOS [33] 5 - a = 2300 pm Tw-Ts =400 C 4- eA CD) Lu0 3e x S2* 2x 0 1 10 WEBER NUMBER, WE 100 FIGURE 5-7 MAXIMUM DROP EXTENSION -- -o" i 11,11 1 1111111IN I OR-I,--------- -- - ---- --- - - ----- - 149 - maximum deformation, most or all kinetic energy has been converted to surface potential energy and is stored in the extended surface. The predicted maximum drop extension radius is consistently slightly larger than the observed extension. In Wachters-' photographs, the drops at maximum extension do look very much like a flattened cylinder, except that there is an indentation at the center of the cylinder. This additional area in the indentation is not accounted for in the cylindrical model. The cylindrical model must show a larger extension to have the same surface area as the deformed drop with the indentation. Calculations performed for the surface contacting drop show both a larger impact period, in Figure 5-6, and a larger maximum extension radius, in Figure 5-7. This is because the surface tension was evaluated so as to exclude the surface contact area between the drop and the wall. Thus the restoring force of surface tension is reduced, giving a longer impact period, and a greater deformation is required to store all the initial kinetic energy as potential energy at the maximum extension. The calculations for the surface contacting drop were performed as an exercise in dynamics. If a small drop, with a Weber number in the range considered, does wet a surface, it does not bounce off. The model does not account for forces 150 - - associated with liquid-sol wetting. The minimum separati distance between the drop and the wall during i pact is plo tted as a function of Weber number in Fi gure 5-8 and wall superh eat in Figure 5-9. The difference in the cal cul ated minimum separation value for the two shapes co nsidered is greater than the variation with Weber number or wall sup erheat. The significance of this obser vati on is discu ssed after considering a few more details of the dr 0 p dynamics. The,.d rop experiences a maximum force at the minimum sepa rati on distance. The maximum force is plotted as a fun ctio n of Weber number in Figure 5-10, and does no t change wi t h wall temperature. The maximum force 1 is not sig nifi cantly different for the sphere and cyl ind e r model s. T between maximum vapor Reynold s number, RE, for the flow e drop and the wall, is plotted as a function of Weber number in Figure 5-11. The Reynolds number increases with increased Weber number and wi increased temperature; the maximum Reynolds number is abo the same for both the 1 The maximum force can be used to calculate the maximum pressure which the drop experiences P = 2F /7rR . For the spherical model with water a 1 atf11pherema9t a Weber number of 100, P ~ 130 psia, and at WE = 10 P ~ 10 psia. This presT9e rise is large enough to affM9 the saturation properties of the fluid. However the duration of the maximum force is extremely brief, and the average force is typically a factor of 100 or more lower than the maximum, so the model is left with constant saturation properties. The calculation of the maximum pressure is outlined in Appendix 5. - 151 - A CYLINDER 0 SPHERE 10-2 E 0 C) 10220 ium TwTS = 264 C a = H20, ATMOSPHERIC 1 PRESSURE 10 WEBER NUMBER, WE 100 FIGURE 5-8 MINIMUM DROP - WALL SEPARATION VS WEBER NUMBER - 152 - H2 0, ATMOSPHERIC PRESSURE: A CYLINDER o SPHERE N2, 10-2 ATMOSPHERIC PRESSURE: A CYLINDER A0000e A E HLLJ C/O 00 0 10~ 3 220 uim a = WE O,1 WALL SUPERHEAT RATIO) FIGURE 5-9 = 25 1 (TWTs)/Ts MINIMUM DROP - WALL SEPARATION VS WALL SUPERHEAT - 153 - H2 0, ATMOSPHERIC PRESSURE a = 220 ulm 10 3 x CL x A* 102 C- aLL. LJ -i 101 CYLINDER: -- A/ + Tw-Ts= 132 C A Tw-Ts = 264 C x Tw-Ts = 660 C SPHERE: o T -TS = 264 C 10 10 WEBER NUMBER, WE FIGURE 5-10 MAXIMUM EXTERNAL FORCE 100 - 154 - 10- 8- a CYLINDER a = 220 pm H20- _TMOSPHERIC PRESSURE 0 110 100 WNEBER NUMBER., WE FIGURE 5-11 MAXIMIM VAPOR - FLOW REYNOLDS NUMBER - 155 - cylindrical and spherical drop models. The Reynolds number is given by: RE = (5-108) TH (The derivation is given in Appendix 5). The maximum Reynolds number is at the minimum drop-wall separation. The calculated Reynolds number does not exceed about 10, which justifies the assumption of laminar flow between the drop and the wall. Details of a typical impact for the cylindrical drop (water at 1 atmostphere, a = 220 pm, T, -Ts = 264K, 5-16. WE = 25) The cylinder height is given as a function of time in Figure 5-12. 5-13. are given in Figures 5-12 through The cylinder diameter is given in Figure The drop-wall separation is given in Figure 5-14. The external force on the drop bottom is given in Figure 5-15. The time dependence of the heat transfer rate is given in Figure 5-16. In Figures.5-12 and 5-13 it is evident that the deformation begins somewhat slowly. How- ever, in Figures 5-14 and 5-15 it is shown that the minimum separation and maximum force occur at the beginning of the deformation process.' It is this initial large force that starts the deformation process. There are three minima in the drop-wall separation, and three corresponding peaks in the force during the impact. The first and most extreme initiates the deformation at the beginning of - impact. 156 - The second reduces the drop momentum in the first half of the impact. The third increases the drop momentum in the opposite direction as the drop rebounds away from the wall during the second half of the impact period. The heat transfer rate, in Figure 5-16, also shows three peaks at the same locations. However, the heat transfer rate is much more uniform than either the force or the drop-wall separation. All of the dynamic calculations exhibited similar behavior. In Figure 5-17, drop bottom radius profiles are shown for three models. Note that the cylinder begins the impact with a finite area botto m, whereas the sphere begins actually begi ns with zer o bottom area, though a small area is arbitraril y assigned so that the external force can be calculated. It is this difference in initial drop bot tom area that res ults in sig nificantly different drop-wall Drop separation distances for the cylinder and the sphere. bottom radius profiles f or the spherical model are sho wn for two Weber numbers in Figure 5-18. The model sugge st s the larger Weber number has a shorter impact period. Data fo r drop exte nsion as a functio from Wachters '-photograp hs, are pl otted wi of time, taken the calculated cylinder radi us for the same condi tions in 'igure 5-19. An arrow marks the end of the compression half of the impact, and the beginning of the drop extension away from - 157 - 1,0 H20, ATMOSPHERIC PRESSURE a = 220 pm WE 25 = T-TS= 264 C 0,8 (CYLINDER) 0,6 LtJ "Z: U.J CD) 0,4 0,2 0 1 0 0,2 0,4 TIME, 0.6 0,8 t/Tc FIGURE 5-12 DROP THICKNESS DURING IMPACT - 158 - 2,0 1,6 v) 1.2 0.8 0,4 0,2 0,4 TIME, 0,6 0,8 t/Tc FIGURE 5-13 DROP BOTTOM RADIUS DURING IMPACT - 159 - 0,10 0.08 2 0,06 - 0,04 CL- 220 WE = 25 a = 0.02 um Tw-Ts = 264 C (CYLINDER) H 0, ATMOSPnERIC PRESSURE 0 I 0 t1 0,2 I u I 0,6 0,4 TIME, I 0,8 t/tc FIGURE 5-14 DROP - WALL SEPARATION DURING IMPACT - 160 4100 (CYLINDER) H2 0, ~.?u - ATMOSPHERIC PRESSURE a = 220 pm WE = 25 T -TS = 264 C 300 - LL3 200- LLJ > 100 0 0 0,2 FIGURE 5-15 0,4 TIME, t/Tc 0,6 0,8 EXTERNAL FORCE DURING IMPACT - 161 - 200 (CYLINDER) H20, ATMOSPHERIC PRESSURE a = 220 um WE = 25 150 T -TS 264 C = 0D 100 LJ L50- 50 Li t 0 0 0,2 A-I 0,4 TIME, 0,6 0.8 t/Tc FIGURE 5-16 HEAT TRANSFER RATE DURING IMPACT - 162 - xI 0.6/ _/ -0,4- -0,4 -0,2 0 0.2 0,4 TIME, (t-tC)/ [ FIGURE 5-17 DROP BOTTOM RADIUS PROFILES FOR THREE SHAPES , Ww*"m - 163 - x 0,6- S0,4- 0,2- -0,4 0 -0,2 TIME, 0,2 0,4 (t-t c)/c FIGURE 5-18 DROP BOTTOM RADIUS PROFILES FOR TWO WEBER NUMBERS - 164 - 3,5 WACHTERS' PHOTOS [33] FROM WE = WE WE 75 3,0 F = = 75 15 a = 2300 4m Tc = 0.011 S TW-TS= 300 C ARROW MARKS END OF COMPRESSION 2,5 F- 2.0 H / 1.5 F- / 1,0 I I I I I -0,2 TIME, I I 0,2 0.4 I I 0,6 t/te FIGURE 5-19 RADIAL DIMENSION DURING IMPACT: MODEL AND DATA .. .......... maMIN1m1 - the wall. 165 I - The simple cylindrical model does a remarkably good job in estimating drop size during the compression phase of the impact. Drop heat transfer effectiveness for water drops at I atmosphere, diameter a = 220 yrm, wall superheat Tw - Ts 264K, is plotted as a function of Weber number in Figure 5-20. In spite of significant differences in the dynamics of the two shapes considered (cylinder and truncated sphere), the values for heat transfer effectiveness are reasonably close. The heat transfer effectiveness is an integral result of a calculation in which a given value of impulse ( fF.dt) is required to cause the drop to rebound. The shapes considered predict about the same total time required, and though the detailed time history of the forceseparation, and shape are different, the integral results (such as heat transfer and impulse) are about the same. It is expected, then, that almost any reasonable shape would give satisfactory results. Drop heat transfer effectiveness is shown as a function of wall superheat ratio for both water and nitrogen (1 atmosphere, drop diameter a = 220 pm, and Weber number WE = 25) in Figure 5-21. The effectiveness increases with both temperature and -Weber number as expected. The effectiveness for nitrogen is significantly larger than for water primarily because the density ratio RHO = g/Qf is larger for nitrogen and the square root of RHO appears - 166 - appears directly in the expression used to evaluate the effectiveness (Equation (5-93)). In Figure 5-22, heat transfer effectiveness (cylindrical model only) is shown as a function of temperature for th ree drop sizes at constant Weber number. ments [33, Data points from Wachters' 34] are also shown. measure- The analysis predicts the correct order of magnitude for heat transfer effectiveness. More data are needed for a rigorous comparison. - H2 0 167 - ATMOSPHERIC PRESSURE a = 220pm/ 0,2 - (ciJ c:, ...-- -. -AT LiU 01 / -- ULU. LlU T-Ts= 264 C A CYL INDER 0 SPH ERE Tw- Ts = 660 A CYLINDER . SPHERE 1 10 WEBER NUMBER., WE C 100 FIGURE 5-20 DRY IMPACT HEAT TRANSFER EFFECTIVENESS VS WEBER NUMBER 0 SPHERE cD w 0 ~ 0,4 A N2, ATMOSPHERIC PRESSURE CYLINDER A A LUL A0 WE 0 1--0 0 .25 .50 .75 1.00 1.25 = 25 1.50 1.75 WALL SUPERHEAT RATIO, (Tw-Ts)/Ts FIGURE 5-21 DRY IMPACT HEAT TRANSFER EFFECTIVENESS VS WALL SUPERHEAT + a = 60 4m 0.8 a = 220 y m x a = 2300 pm WACHTERS [34] @WACHTERS [331 0.6 ~'x C WE ~ 25 H2 0, uj 0. ATMOSPHERIC PRESSURE -J "0.2 - x xx 0' 100 200 300 400 WALL SUPERHEAT 500 Tw~TS 600 (C) FIGURE 5-22 DRY IMPACT HEAT TRANSFER EFFECTIVENESS: MODEL AND DATA 700 - A Simplified Model: 170 - Dry Collision Heat Transfer The remarkable regularity of the modeled dynamics of deforming drops in a dry collision, coupled with the observation that the heat transfer effectiveness is an integral quantity showing little sensitivity to arbitrarily chosen details such as deformation shape, suggests the derivation of a simplified model for dry collision heat transfer. The average force on the drop during the collision time T drop. must be sufficient to reverse the momentum of the If the impact time T is some fraction C2 of Cc' then in dimensionless variables, the average force is: (5-109) 2 or in terms of the Weber number F F= T22 12 C E) 1/ 2 WE2 The drop-wall separation y associated with this force for a drop bottom radius R (5-92), -4 y with - F, is determined from Equation U << U : Uv- = 0 si nce 2 TH 2 1~ + 62 TH VI 12 ]i'R = (5-110) (5-111) The maximum drop bottom radius of the deforming drop can be calculated for any shape by assuming all initial kinetic and potential energy is stored as surface potential energy at maximum deformation: MIM - 171 PE max = KE For the cylindrical model, - + PEO (5-112) . wi th the Weber number greater than 1.741 (again in dimensionless variables), the maximum extension radius is: (1 + R2 acos( 12)cos2 2 (5-113) 12 This value of R can be used to evaluate '. The drop heat transfer effectiveness is given by Equation (5-93); substitution of the characteristic values of period T = C2 rc, R and y for a time the simplified version of the effectiveness is: 3Tr E1 ~~ g RHO a TH R2 C2 (5-114) The dynamic calculations suggest a value of .5 to .6 for C . 2 Calculated values for drop heat transfer are shown as functions of Weber number and wa 11 superheat in Figures 5-23 1 If the initial shape is assumed spherical, a certain minimum kinetic energy is required to account for the difference in surface energy in the undeformed spherical and cylindrical drop. For a drop of the same volume, the cylinder shape has more surface area than the spherical shape. For the derivation of the expression for R, see Appendix 5. - and 5-24, respectively. 172 - The simplified model gives a higher val ue for effectiveness, particularly at high Weber numbers. However, the simplicity of the calculation recom- mends this method for a quick estimate of the heat transfer. All of the heat transfer analysis is based on heat removal from the surface area directly beneath the drop. The sizeable vapor flow rate out from the space between the drop and the wall is expected to provide added heat removal from the surface area near, but not directly shadowed by the drop. Wachters [33] included a factor between 1 and 2 to account for this effect. - 0,5 173 - DYNAMIC MODEL: a CYLINDER o SPHERE + SIMPLIFIED MODEL 0,4 a = 220 ,1m T-Ts= 264 C H2 0, x ATMOSPHERIC PRESSURE - 0,3 + 0,2 0 U- 0.1 ++ 0 1 10 WEBER NUMBER. WE 100 FIGURE 5-23 SIMPLIFIED DROP HEAT TRANSFER EFFECTIVENESS VS WEBER NUMBER 0.4 I CYLINDER SPHERE SIMPLIFIED MODEL x 0.3 SIMPLIFIED- 0.2 15 -SPHERE ....-- 0.12 220 pm a = WE 000, 0 0 FIGURE 5-214 .25 = 25 H20, ATMOSPHERIC PRESSURE .50 .75 1.00 1.25 WALL SUPERHEAT RATIO, (Tw-TS)/TS 1.50 1.75 SIMPLIFIED DROP HEAT TRANSFER EFFECTIVENESS VS WALL SUPERHEAT Nib, - 175 - Heat Transfer With Surface Wetting When two large bodies, each at a uniform temperature Tw and Tf respectively, are brought into contact, the interface temperature T.(C) is (Tw - Tf) T. where - (5-115) RT + 1 Tf is a thermal property ratio: RT 1/2 RT = Pf kf kw (5-116) Pw Cp w The interface temperature is assumed instantly and remains constant as heat flows from the hotter body to the cooler body. a time The heat transferred per unit contact area, q, after t1 is q = 1/2 2 (T - Tf) (kfpfP (5-117) f t0 /7T) (See e.g. [501.) Though the liquid drop is by no means an infinite body, and the target has some nonuniform temperature distribution, this approach~is used in estimating heat transfer to a liquid drop which wets a surface (e.g. Illoeje [6]). is often not as inappropriate as it may seem, This because bubble size associated with initial nucleation at moderate superheats - 176 - (> 10C for water at 1 atmosphere) is usually much small er than the drop size, and the times as soc iated with superheating prior to nucleation at mod era te superheats are short compared with the time required to propagate a temperature change through one drop diameter, so that the drop is effectively a "large" body. liquid temperature Tf In post dryout dispersed flow the is the sat uration temperature Ts(K). The conduction time prior to initial nucleati'on was calculated by Illoeje [6] based on the time dependent temperature profile for two semi-i nfinite bodies and the saturation temperature at the bubb le pressure. The pressure of the vapor in a bubble is larger than that in the liquid because of the added force of surface tension. nucleation, the liquid and vapor are For a higher temperature than the saturation temperature corresponding to the liquid pres sure. Smal ler bubbles requi time before i-ni tial nucleation 1arger (sec) superhe ats The is given by Illoeje [61: / 0 and the bubble 2 .24 pg hfg/ size b0 = 1.06 Pfc f kf T Ts T.i-T )2 , (5-118) ( prior to growth) is: kft0 ( f c ,f ) 1 (5-119) oil - 177 - These functions are shown, for water at atmospheric pressure, in Figures 5-25 and 5-26, respectively. Bubble growth in a nonuniform temperature field, corresponding to the profile in the semi-infinite body has been analyzed by Mikid [51]. The heat transferred to the liquid by conduction from the hot surface is then transferred (by conduction) to the liquid-vapor interface of the bubble nucleus oration takes place. where evap- The bubble radius b(m) has been calculated as a function of time. The introduction of a dimensionless bubble growth group BB simplifies the interpretation of the bubble growth. BB p h b = s (5-120) 1/2 ff The bubble growth as a function of time is then given by: BB BBt= ( 1/2 1t - 1 t0- - 1/2 t t( This function is plotted in Figure 5-27. t equal to the conduction time bubble radius is .58 to )] 1/2- ( 12 (5-121) Note that at a time prior to nucleation, the of its maximum. This shows that the bubble growth time is comparable to the conduction time prior to nucleation. bmax max The maximum bubble radius bmax is: T. - T h Ts T Pg hfgff //2 Pfc k to) . (5-122) - 178 - 10~ 10-2 10~ 4 10~5 10- 6 10 1 100 INTERFACE SUPERHEATJ T -T3 (C) FIGURE 5-25 NUCLEATION AND EVAPORATION TIMES - 179 - H2 0, ATMOSPHERIC o PRESSURE 05 - Lu 10-6- 10 INTERFACE SUPERHEAT, Ti-TS (C) FIGURE 5-26 INITIAL NUCLEATION BUBBLE SIZE 100 1.00 0.75 CD 0.58 0.50 CD Lu LJ 0.25 01 10-2 10~1 101 100 TIME, 102 103 t/t FIGURE 5-27 BUBBLE GROWTH IN NONUNIFORM TEMPERATURE FIELD W I - 181 - This calculated value of maximum bubble radius, with substitution of the waiting time to, does not depend on wall superheat. An estimate of the total time of evaporation is made under the assumption that repeated nucleation and bubble growth is responsible for the evaporation. flow to the drop through a contact area time t TR2 The average heat over a conduction is: Q (5-123) tR = 1/2 Q k 2(T -T = fpc )rR2 (5-124) The product of the average heat flux and the total evaporation time must equal the latent heat of the drop; the total evaporation time t t(s) is: Pf V hfg = , (5-125) Q or 1/2 t 12(.24)C where C3 (-) . pf P aa T 2 kTs (T -T )2 (5-126) This estimate is plotted in Figure 5-25 for a drop diameter of 220 im and a Weber number of 25 (C3 - 2). If the evaporation time is longer than the time - 182 - between impacts there will be liquid buildup. this should increase the value of R Initially giving a larger wette d area for heat transfer, while t he heat transfer effectivene ss remains about the same. As mor e liquid is added, however, the effectiven ess is expected to drop and liquid to buildup continually on the surface which is quenched to saturati on temperature. This phenomenon is known as rewet. This bri ef discussion is supported qualitatively by Cumo [45] observed a sharp in- the observatio ns of others. creasing trend in liquid drop "stay time" with decreasing wa 11 superheat in the wetting region (in contrast to a cons t ant impact p iod ments with heat transfer igher wall superheat) . In experi- to a n impinging liqu id jet, Ruch .1[ 52] observed con stant heat flux at constant target tempe rat ure, independent of j et diameter and velocity, and et al the s i ze of the liquid-s urfac e contact area. Mesler et al. [53] in a recent paper, point s out that nucle ate boiling in thin fi lms differs from pool boiling in that lower superheats are required in thin fi lms than in pool boiling for the same heat tra nsfer rate. Thi s also reflects on heat transfer by nuclea tion of smal drops, suggesting shorter total evapo ration t imes than mi ht be' predicted in pool boiling. The general conclusion is that the nucleati the heat transfer mechani sm which accounts process may be r the complete vaporization of impacting drops which wet a hot surface. NN - 183 - Surface Wetting Transition Temperature The tempe rature associated with transition from wetting to non-wetting is noted in expe riments with flow boiling, pool boiling, rewetting, spray cooling, and drop impact. Recently a study of the so call ed minimum film boiling temperature was reported by Yao Illinois). et al. [54] (ANL, In this work, the authors suggest that the breakdown of the surface wetting con dition may be due to one of two instabilities: thermodynami c instability or hydrodynamic instability. The obser ved minimum film boiling temperature will be the minimum of the two predicted by thermodynamic and hydrodynamic instabilities. The thermodynamic tempera ture limit, called the homogeneous nucleation temperature, represents the maximum temperature to which a liquid can be superheated. For a first order phase transition in a fluid described by Van der Waals' equation of state, the pressure-volume relation at constant temperature resembles the plot in Figure 5-28 (See e.g. [55]). The temperature volume relation at constant pressure can be inferred, and is also shown in Figure 5-28. The homogeneous nucleation temperature is loosely interpreted as the local maximum in the temperaturevolume curve. This description reflects the nature of the analysis of the thermodynamic instability, whereas the actual approach is more complex. There is also a heterogeneous -- ---- -F "' -v "I ____ _ .- - 184 - SATURATION Lu T~ - T -~ SPCIICVOUM0 F T5 ~ 3 I N P SATURATION L-LT2 SPECIFIC VOLUME,. v FIGURE 5-28 INSTABILITY IN PHASE TRANSITION - 185 - nucleation limit, mean ing that nucleation occurs at the liquid-solid interface rather than in the liquid. The hydrodynamic instability limit, based on a vapor removal rate from the sur face, in an earlier paper. is described by Henry [5 6] Som e partial liquid-surface conta ct was observed beyond the h ydrodynamic limit, and it was suggested that this parti al contact accounted for the bul k of the heat transferre d i n the process. The temperature 1imits for wetting are discussed in terms of the liquid. Whe n the liquid and the solid are brought together an inter mediate temperature is present the interface. a For short times, this is the same as the interface temperature for two semi-infinite bodies suddently 1 brought in contact. This interface temperature is given by Equation (5-115). Wetting may be expected if the interface temperature T. is less than the minimum film boiling temperature. For water at 1 atmosphere, the hydrodynamic instability limit predicts a temperature of about 200C, which is lower than the homogeneous nucleation temperature of about 300C. The suggested limiting interface superheat, T -Ts (C), of about 100C is reasonably close to the transition temperature observed in this study, as well as in many other investigations 1 For a diitscus-si-on of the surface - drop contact temperature with surface oxide see Seki et al.[57]. - 6 186 - LIQUID DROP CONTRIBUTION TO DISPERSED FLOW HEAT TRANSFER The details of the statistical distribution of drops of various sizes and velocities can be represented by size and velocity distribution functions. The liquid contribution to total heat transfer in dispersed flow is calculated by integrating the single drop heat transfer effectiveness over the range of the drop distribution functions. The evaluation of the liquid heat transfer is used for predicting local heat transfer and vapor generation rates. 14, - 187 - Distribution Functions for the Dispersed Phase In the formulation of the dispersed flow heat transfer problem, the total heat transfer is broken down into liquid and vapor contributions. is The liquid contribution further broken down into a sum of contributions due to impacts of drops of various sizes and velocities. density n. A number is used to represent the number of discrete drops 'Of a given drop size and velocity. In order to simplify the use of the heat transfer effectiveness in estimating the liquid contribution to dispersed flow heat transfer, the number density is replaced by a continuous drop size and velocity distribution function. The distribution is assumed spatially uniform, and the drop size and velocity distribution are considered independent (that is drop velocity does not depend on drop size) This assumption is consistent with Cumo's [ 8] detailed photographic study of post dryout dispersed flows of Freon with drop sizes on the order of 50 ypm diameter. The number of drops per unit volume of diameter in a about a mean value range da range dUL about a mean value (1 - a) dn(a , U) with a velocity in a a Ui is a U d =) v a ac Uc -- d U- ac U 6 (6-1) - where 188 - the number of drops per unit vol ume the vapor volume (void) fraction the volume mean drop diameter , some characteristic in drop size some characteristic in drop velo city perpendicular to the tub e wall) , the size distribution function , and the velocity distributi on functi on . The drop size and vel oci ty distribution functions are defi ned such that their integrals over the entire range are unity: Ja amax Umax a 0o ( ) d() Uc = ) d pu . (6-2) 0 The total number of drops per unit volume is the integral of the distribution d2n (a , Us) over the entire range of diameters and vel ocities Umax Uc n = j 0 Ic amax a d (6-3) 0 Umax Uc a amax c Tr a 6 Pad( ) u d J0 (6-4) ff", - = (1 189 - c) - (6-5) 3 ra ( 6 = a3 6 The volume of a drop of diameter a is V so that the volume mean drop diameter is defined as: Tr max a 3 a 6v _0c a 3 The liquid volume fraction, Umax (1 - a) T ac (6-6) ) is given by: a3 (6-7) d2n 0 o0 amax ac (1 a - a), ama x a Uc = ( a S6 a)a - 3 ra Umax_ U Tr a 66v pa -pu Pu d( U0 (1 - a ). ,a av (1 - a) (1 - a) Tr ( f f 0 0 6 d(aa a 6v ) (1) (6-8) (6-9) (6-10) - 190 - This simply illustrates the consistency of the function definitions. The formula for drop number density venient because is con- the density is scaled by the average number of drops per unit volume n, that is by the ratio of the liquid volume fraction (1 - a) (liquid volume per unit volume) to the average volume of a drop (volume per drop). The size and velocity dependence is scaled by distribution functions pa and pu, of which the integrals over the entire ranges are unity. - 191 - Incorporation of Drop Heat Transfer Effectiveness The use of information on drop heat transfer effectiveness presented in this study is accompanied by some comments on the applicability of measurements and modeling of a phenomenon under one set of conditions to another set of conditions. Two important differences in conditions are discussed. The drop heat transfer measurements and modeling are performed for normal impact on a hot surface. The photographs of Wachters [33] for impact at an angle of from normal support the assumption that the normal component of velocity characterizes the impact. In dispersed flow the impact angles are very shallow as drops move almost parallel to the surface. The dynamic effect of a significant velocity component parallel to the impact surface has not been investigated. The vapor velocity field field in dispersed flow is also significantly different from the modeled stagnant field. Povarov et al. [31] show the possibility of significant aerodynamic effects on drop motion for drops approaching a surface with a high relative velocity. Schoessow et al. [44]'measured reduced evaporation times in the non-wetting region for drops placed on a moving surface, These effects are not included in the model. With these reservations, the drop heat transfer effectiveness model for dry impacts is recommended for use - 192 - in dispersed flow heat transfer calculations at wall temperatures above the minimum film boiling temp erature, and an effectiveness of 1 is recommended below this temperature. The contribution of heat transfer due to impacting drops to the total heat transfer in dispersed flow is given by an integral over the range of drop diameters and velocities, which replaces the sum in the original formulati on: a.3 Q Pf hf A wall to drops g . 6 a U max max ac fc0 = Pf hfg U U i Tra U n. £. 1 (1 - at) Pf hfg Uc fc 0 Uu (6-12) o max amax = d2 6b- J~ 0 (6-11) 1 '3'a f c 0 d(y) d(U c (6-13) In the simpl ified drop heat transfer model, the expression for dro p heat transfer effectiveness in the nonwetting region is given explicitely in terms of the dimensionless grou ps which contain drop size and velocity, in addition to flu id properties and wall superheat. - 193 - The Role of Drop Heat Transfer in Dispersed Flow Analysis The analy 'sis of heat transfer to impacting drops i useful in the analysis of post dryout dispersed flows in Firs t, two ways. the liqui d contribu tion is included in the prediction of the total heat tran sfer; second, amount of vapor generated at the wall is included in the prediction of the total vap or generat ion rate. cases the drop contribution to total negligible. However, it is still the vapor gener ation rate. the In many heat transfer may b imp ortant in predictin The heat removal from a hot tube by the vap or in a disp ersed flow is almost directly proportional to the vapor velocity. For a constant mass flow rate, the vapor velocity depends directly on the amount of vapor present, that is, on the length integral of the vapor generation rate. There are two contributi ons to the vapor generation rate vapor generated at the wall, and vapor generated in the core : dW -dz dW - -- I dz wall dW + (6-14) - dz core A number of investigations [e.g. 2, 10, 12] have analyzed the vapor generation rate in the core. rate at the wall is : The vapor generation - dW 194 Q a TrB -- 1- dz - 2 wall hfg SS (6-15) A wall to drop There are a few cases in which the actual contribution of heat transfer to the liquid drops is a significant part of the total heat transfer. One example is slow flow, in which the heat transfer from a hot tube wall to the slowmoving vapor is relatively low. Another example is the possibility of rewet after burnout in tubes with nonuniform heat flux. Keeys et al. [8] observed steady burnout and rewet (and sometimes a second burnout) in a length of tube with a cosine heat flux distribution (simulating reactor heat generation distributions). In this case, the heat removal by the impacting liquid drops is fundamental to predicting tube wall temperatures and the amount of liquid build-up on the tube wall. Any analysis which does not include heat transfer from the wall to the drops cannot predict the rewet phenomenon. M P1101WWW 1111w, - 7 195 - CONCLUDING REMARKS Summary The post dryout dispersed flow heat transfer problem is considered in the context of flow boiling. The heat transfer is analyzed in terms of local conditions and known details of the flow structure. The total heat transfer is constructed as a sum of vapor and liquid contributions. The liquid contribution is further broken down into contributions due to individual liquid drops in the dispersion. The study focuses on the measurement and characterization of heat transfer to impacting drops. Experimental measurements under steady conditions show total evaporation of impacting drops in the wetting region under dispersed flow conditions. The observed transition from wetting to non-wetting is co isistent with transition temperatures observed in other studies. Mea surements of heat transfer to an air jet ent rained by a stream of drops suggest that heat transfer in spray cooling experiments is due almost entirely to cooli ng by entraine d air. Analysis of heat transfer to impacting drops includes modeling of drop deformation on impact. A simple dynamic model gives good estimates for drop shape as a function of time durin phenomenon impact (as compared with photographs of this The drop dynamics are relatively insensitive - 196 - to the assumed family of drop shapes during deformation. Estimation of the heat transfer in non-wetting impacts is of the same order of magnitude as published measurements. A simple model of the overall process allows a good estimate of the heat transfer effectiveness without integrating the equations of motion for the deforming drop to solve for the details of the dynamics. In the wetting region, conduction, nucleation and bubble growth calculations suggest the complete evaporation of the impacting drops, as was observed experimentally. The drop heat transfer effectiveness is easily incorporated into dispersed flow heat transfer analysis. The inclusion of heat transfer from the wall to the liquid more accurately represents the local heat flux and vapor generation rate, and allows for the possibility- of predicting rewet in tubes with nonuniform heat flux distributions. . IN OMNI I'mq NPINNOW - 197 - Conclusions 1. Steady state measurements of heat transfer to impacting drops eliminates uncertainties associated with interpretation of data from quench tests. 2. The wall-to-drop heat transfer effectiveness in the wetting region is approximately one for the conditions and range of Weber numbers under consideration. 3. A drop stream in stagnant air entrains an air jet. The cooling of a hot surface by the entrained air jet may be much larger than the cooling due to the impacting of the drops which entrained the air. This can easily account for the difference in heat transfer effectiveness data reported in the literature. 4. The surface temperature and surface properties control the transition between wetting and non-wetting impacts of liquid drops. 5. A model for the deforming drop gives a good estimate of time dependent drop dynamics during impact. The model has two free parameters: a geometry parameter which determines the drop shape, and a position parameter which determines the drop location. Different families of assumed drop shape during deformation give similar results. 6. A heat transfer model combined with the drop dynamics predicts heat transfer effectiveness of the same - 198 - order of magnitude as measurements in the non-wetting region reported in the literature for two drop sizes. 7. An estimate of heat transfer for drops which wet the surface suggests complete evaporation of the drop. 8. Tube-to-drop heat transfer can be estimated for dispersed flow on the basis of the results of this study. It is useful in predicting local heat transfer and local vapor generation. - 199 - Recommendations At the conclusion of this study, several areas w'hich will benefit from further study are identified briefly The detailed nature of the liquid-surface intera ction on wetting is not well understood. microstructure is unknown. The role of surfac e The added complexities of heat conduction and appearance of the vapor phase are furth er unknowns. The general effects of surface properties are observed in wetting transition, but explanations are incomplete. Further experimental work can provide more drop heat transfer data in the non-wetting region against which the heat transfer analysis may be checked. Evaporation times for impacting-wetting drops can be measured. Liquid spreading on the wetted surface may be important. More information is needed to determine the loca l conditions from the flow history in two phase flow bo il1i ng. k1li. - 200 - REFERENCES DOUGALL, R.S., ROHSENOW, W.M., "Film Boiling on the Inside of Vertical Tubes with Upward Flow of the Fluid at Low Qualities", MIT Heat Transfer Laboratory Report No. 9079-26 (1963) .M., "Film Boiling of in a Vertical Tube", ASME 2 LAVERTY, W.F., ROHSENOW, Saturated Nitrogen Flowi Paper 65-WA/HT-26 (1965) 3 FORSLUND Boi 1ing" (1968) 4 HYNEK, S.J., ROHSENOW, W.M., BERGLES,. A.E., "Forced Convection, Dispersed-Flow Boiling", MIT Hea t Transfer Laboratory Report No. 70586-63 (1969) 5 PLUMMER, D.N., GRIFFITH, P., ROHSENOW, W.M. "Post Critical Heat Transfer to Flowing Liqu id in a Vertical Tube", Society Paper 76-CSME/CSChE-13 (1976 ) 6 ILLOEJE, 0.C., ROHSENOW, W.M., GRIFFITH, P. "ThreeStep Model of Dispersed Flow Heat Transfer (Post CHF Vertical Flow)", ASME Paper 75-WA/HT-1 (1975) 7 GANIC, E.N ., ROHSENOW, W.M., "Dispe rsed Flow Heat Transfer", Internationa 1 Journal of Heat and Mass Transfer, 20, p855-866 (1977) 8 CUMO, M., FERRARI, Study of Two-Phase Report RT/ING(71)8 9 CUMO, M., FARELLO, G.E. FERRARI, G., PALAZZI, "On Two-Phase Highly Di persed Flows", Journal Transfer, 96, p496-503 1974) R.P., ROHSENOW, W.M., "Dispersed Flow Film Journal of Heat transfer, 90, p399-407 FARELLO, G.E ., "A Photographic ly Dispersed Flows", CNEN 1) of Heat 10 BENNET, A.W., HEWITT, G.F., KEARSEY, H.A., KEEYS, R.K.F. "Heat Transfer to Steam-Water Mixtures Flowing in Uniformly Heated Tubes in Which the Critical Heat Flux Has Been Exceeded", United Kingdom Atomic Energy Authority Report AERE-R 5373 (1967) 11 JONES, O.C., ZUBER, N., "Post CHF Heat Transfer: A ASME Paper 77-HT-79 Non-Equil ibri um Relaxation Model" (1977) - 201 - 12 SAHA, P., SHIRALKAR, B,S. DIX, G.E., " A Post-Dryout Heat Transfer Model Based on Actural Vapor Generation Rate in Dispersed Droplet Regime", ASME Paper 77-HT-80 (1977) 13 JONES, 0.C., SAHA, P Water Reactor Safety H y li4 d rau c Aspects of "Non-Equilibrium Aspects of Symposium on the Thermal and uclear Reactor Safet Light Water Reactors, p2 4 9-288, ASME (1977 Volume 1: BUTTERWORTH, D., "A Comparison of Some Void-Fraction Relationships for CoCurrent Gas-Liquid Flow", International Journal of Muliphase Flow, 1, p845-850 (1975) GROENEVELD, D.C., GARDINER, S.R.M., "Post-CHF Heat Transfer Under Forced Convective Conditions", Symposium on the Thermal and Hydraulic Aspects of Nuclear Reactor Safety, Volume 1: Light Water Reactors, p43-73, ASME (1977) 16 BREVI, R., CUMO, M., "Quality Influence in Post-Burnout Heat Transfer", International Journal of Heat and Mass Transfer, 14, p483-489 (1971) 17 CUMO, M., URBANI, G., "Anomalies in Post-Dry Out Heat Tran sfer wit h Steam Water Mixture s", Comitato Nazional e Energia Nucleare Report TR/I NG(72)19 (1972) 18 KEEYS, R.F. K., RALPH, J.C., ROBERTS, D.N,, "PostBurnout Hea t Transfer in High Pressu re Steam-Water Mixtures in a Tube wi th Cosine Heat Flux Distribution", Progress in Heat and Mass Transfer, 6, p99-118 (1971) 19 BAILEY, N.A., "The Inter action of Droplet Deposition and Forced Convection in Post-Dryout Heat Transfer at High Subcritical Pres surel", Atomic Energy Establishment, Winfrith, Report AEEW - R 807 (1973) 20 CHEN, J.C. , SUNDARAM, R.K. , OZKAYNAK, F.T,, "A Phenomenol ogical Correlati on for Post-CHF Heat Transfer" Technical Report for the U .S. Nuclear Regulatory Commission Contract #AT(49 -24)-0180, Department of Mechanical Engineering and Mechanics, Lehigh University (1977) Nli - 202 - 21 HALL, P.C., "The Cooling of Hot Surfaces by Water Sprays" , Report RD/B/N3361, Central Electricity Generat ing Board,Berkeley Nuclear Laboratories (1975) 22 ELIAS, E., YADIGAROGLU, G., "Rewetting and Liquid Entrainment During Reflooding - State of the Art 'A Topical Report EPRI NP-435 Research Project 248-1 University of California, Berkeley, Department of Nuclear Engineering (1977) 23 COLLIER, J.G., Convective Boiling and Condensation Chapter 1, McGraw Hill, (1972) 24 SOO, S.L., TREZEK, G,J,, DIMIC K, R. C., HOHNSTREITER. G.F., "Con centration and Mass Flow Distributions in a Gas-Soli d Suspension", Indus trial and Engineering Chemistry Fundamentals, 3, p98 -106 (1964) 25 HUTCHINSON, P., HEWITT, G.F DUKLER, A.E., "Deposition of Liquid or Solid Dispersi s from Turbulent Gas Streams: a Stochastic Model Chemical Engineering Science, 26, p419-439 (1971 26 KIRILLOW, P.L., SMOGALEV, I.P., " Analysis of HeatTransfer Crisis in Terms of a Drop Depositio'n Model" translated from Teplofizika Vysokikh Temperatur, 11 , p794-804 (1973) 27 GILL, L.E., HEWITT, GF., LACEY, P.M.C,, "Sampling Probe Stu dies of the Gas-Core in Annular Two-Phase Flow - II Studies of the Effect of Phase Flow Rate s on Phase and Velocity Distribution", :Chemical Engine ering Science, 19, p665-682 (1964) 28 NAMIE, S., UEDA, T "Droplet Transfer in Two- Phase Annular Mist Flow" , Bulletin of the JSME, 15, p15681580 (1972) 29 SCHLICHTING, H., Boundary Layer Theory, Sixth Edition, p568, McGraw-Hill (1968) 30 TATTERSON, D.F,, DALLMAN, J.C., HENRATTY, T.J., "Drop Sizes in Annular Gas-Liqui d FlowsAIChE Journal, 23, p68( 1977) 31 POVAROV, 0. et al., "Interaction of Drops with a Boundary Layer on a Rotating Surfac e", Indzenerno Fisicheskie Zhurnal,31, pl0 6 8-10 7 3 (1976) in Russian - 203 32 LIU, B.Y.H., ILORI T,A., "Aerosol Deposition in Turbulent Pipe Flow", Environmental Science and Technology 8, p351-356 (1974) 33 WACHTERS, L.H.J., WESTERLING, NA.J,, "The Heat Transfer from a Hot Wall to Implinging Water Drops in the Spheroidal State", Chemical Eng ineeri ng Science, 21, p1047-1056 (1966) 34 WACHTERS L.H .J., SMULDERS, L, , VERMEULEN, JR,, KLEIWEG, H.C. ,"The Heat Trans fer from a Hot Wall t Impingin g Mis t Droplets in the Spheroidal Staten Chemical Engi neering Science, 21, p1231-1238 (1966) 35 PEDERSEN, C.O., " An Experimental Study of the Dynamic Behavior and Heat Transfer Characteristics of Water Droplets Impinging upon a Heated Surface" International Journal of Heat and Mass Transfer, 13, p369-381 (1970) 36 SCHNEIDER, JM., HENDRICKS, C.D., "Source of Uniform Liquid Droplets", The Review of Scientific Instruments, 35, p1349-1350 (1964) 37 SCHNEIDER, J.M., LINDBLAD, N.R., HENDRICKS, C.D. "An Apparatus to Study the Collision and Coalescence of iquid Aerosols", Journal of Coll oid Science, 20, p610-616 (1965) 38 LINDBLAD, N.R., SCHNEIDER, J.M., "Production of Uniform Sized Liquid Droplets", Journal of Scientific Instruments, 42, p635-638 (1965) 39 MOREAUX, F., CHEVRIER, J .C., BECK, G., "Destabili zati on of Film Boil ing by Means of a Thermal Resistance" Internati ona 1 Journal of Multiphase Flow, 2, p183 -190 (1975) 40 WACHTERS, L.H.J,, BONNE, H., VAN NOUHUY S, H.J., "The Heat Transfer from a Hot Horizontal Pla te to Sessile Water Drops in the Spheroidal State", Chemical Engineering Science, 21, p924-936 (1966 ) 41 GOTTFRIED, BS,, BELL, K.J., "'Film Boiling of Shperoidal Droplets", Industrial and Engineering Chemistry Fundamentals, 5, p561-568 (1966) mill - 204 - LEE, C.J . BELL, K.J., "The Leidenfrost Phenomenon: Film Boiling of Liquid Droplets on a Flat Plate" , International Journal of Heat and Mass Transfer 9, p1167-1187 (1966) 42 GOTTFRIED, B.S,, 43 BAUMEISTER, K.J., HAMILL, T.D., SCHWARTZ, F.L., SCHOESSOW, G.J., "Film Boi ling Heat Transfer to Water Drops on a Flat Plat e1, Chemical Engineering Progress Symposiu m Series No. 64, 62, p52-61 (1966) 44 SCHOESSOW, G.J., BAUMEISTER, K.J., "Vel ocity Effects on Leidenfrost Bo iling of Various Liqui ds", Heat Transfer 1970 V, B3.11 (1970) (Paper presented at the Fourth Intern ational Heat Transfer Conference, Paris-Versailles) 45 CUMO , M., FARELLO, G.E., "Heated Wall-Droplet Inte raction for Two -Phase Flow Heat Transfer in Liquid Def icient Region", European Atomic Energy Community Eura tom Eindhoven, Symposium on Two Phase Flow Dynamics II, pl 325-1357 (196 7) 46 MCGINNIS, F.K. III, HOLMAN, J.P. "Individual Droplet Heat-Tran sfer Rates for Splatteri ng on Hot Surfaces", Internati onal Journal of Heat and Mass Transfer, 12, p95-108 ( 1969) 47 SARMA, P.K., MOHAN, N.V.R.S. RAO, K.V.A., "An Analysis of Heat Transfer to Impinging Water Droplets Upon a Heated Surface Under Film Boiling Conditions", Proceedings of the First Nat ional Heat and Mass Transfer Conference, Madras, Paper HMT-37-71, pX-19 X-25 (1971) 48 HOLMAN, J.P., JENKINS , P,E., SULLIVAN, F.G., "Experiments on Indiv idual Droplet Heat Transfer Rates", International Journal of Heat and Mass Transfer, 15, p1489-1 495 (1972) 49 GOLDSTEIN, H., Classical Mechanics, p38 Addison-Wesley (1965) 50 CARSLAW, H., JAEGER, J., Conduction of Heat in Solids, Second Edition, p87, Oxford Clarendon Press (1959) - 205 - 51 MIKIC, B.B., ROHSENOW, W.M.,"Bubble Growth Rates in Nonuniform Temperat ure Field', Progress in 1eat and Mass Transfer, 2, p2 83 -2 9 3, Pergamon Press (1969) 52 RUCH, M.A. , HOLMAN, J.P. , "Boi li ng Heat Transfer to a Freon-ll 3 Jet Impi ngi ng Upwa rd 0nto a Flat, Heated Surface", Internatio nal Journa 1 of Heat and Mass Transfer, 18, p51-60 (1975) 53 MESLER, R., MAILEN, G., "Nucleate Boiling in Thin Liquid Films", AIChE Journal, 23, p954-957 (1977) 54 YAO, S., HENRY, R.E., "An Investigation of the Minimum Film Boil ing Temperature on Horizontal Surfaces" (accepted for publication in the Journal of Heat Transfer - 1978) 55 CALLEN, H.B., Thermodynamics, p148, John Wiley and Sons (1960) 56 YAO, S.C., HENRY, RE., "Hydrodynamic Instability Indu ced Liquid-Solid Contacts in Film Boi 1ing" ASME Paper 76-WA/HT - 25 (1976) 57 SEKI, M., KAWAMURA, H., SANOLAWA, K., "Transient Temperature Profile of a Hot Wall Due to an Impinging Liquid Droplet", Journal of Heat Transfer, 100, p167-169 (1978) W1 11411IN - 206 - A3 CIRCUITRY Selective Drop Charging Electronics Two circuits were used to prov ide a pulsed drop charging voltage. The selection circuit counted the incoming pulses (one pulse per drop) and outp ut one pulse for every m pulses counted, where m was any integer from 1 to 99. The duration of the output pulse was variable, and was adjusted to the time of one cycle. The switching circuit turned the charging voltage off for the duration of the input pulse. The circuit and signal diagrams are shown in Figures A3-1 through A3-4. WITHINPUT 5 (V) BAT if) ,OUTPUT INPUT SIGNAL SIGNAL COMMON GROUND OUTPUT PULSE WIDTH, = (++2.4x103 )-(O.O1x10 1.66x10~5 s < OR WITH T = T < 7.10z10 6 f)-ln(2) 4 1/f 60,100 1/s > f > 1,410 1/s FIGURE A3-1 PULSE SELECTION CIRCUIT SCHEMATIC f = 4, 000 (1/s) LU LU LU CD CD TIME, t (ms) FIRST INTERNAL SIGNAL TIME, t (ms) INPUT SIGNAL C\j LUJ LUJ LU c-i CD C- 0 L 2 TIME, t (ms) SECOND INTERNAL SIGNAL 6 0 2 4 TIME, t (ms) OUTPUT SIGNAL FIGURE A3-2 PULSE SELECTION CIRCUIT SIGNALS 6 INPUT: LINE VOLTAGE 30 (KQ) WITHINPUT 15 (V) BATTERY OUTPUT: PULSED LINE VOLTAGE INPUT: PULSE SIGNAL COMMON GROUND FIGURE A3-3 DROP CHARGE VOLTAGE SWITCHING CIRCUIT SCHEMATIC - 210 LINE VOLTAGE 300 LtJ Lu PULSE Q- ] 2 4 6 8 TIME, t (ms) INPUT SIGNALS: LINE VOLTAGE AND PULSE 300 LtJ LLJ C.D. u 6 0 1i 2 - I. 4 6 8 . TIME, t (ms) OUTPUT SIGNAL: DROP CHARGING VOLTAGE FIGURE A3-4 DROP CHARGE VOLTAGE SWITCHING CIRCIUT SIGNALS - A4 211 - DATA SUMMARY A summary of the drop heat transfer data is given in Table A4-1. The comments can be interpreted as follows: bubbling hissing liquid build-up rebounding drops - 212 TABLE A4-1 Target DATA SUMMARY s s s s s s s s 45 45 45 45 45 45 45 45 86 86 86 86 86 86 86 86 3.35 ::3.35 3.35 3.35 3.35 3.35 3.35 3.35 3900 3900 3900 3900 3900 3900 3900 3900 20 30 20 30 20 30 258 258 258 258 258 258 258 258 Drop Impact Velocity U (m/s) 1.31 1.31 1.31 1.31 1.31 1.31 1.31 1.31 Weber Number WE 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 Target Temperature 110 110 110 147 147 147 209 209 5.35 4.04 1.36 5.92 4.27 1.74 3.49 3.3E 1.06 1.06 1.1C 1.0C - .13 .15 b,Z b,; h h r r Material s or c Impact Angle 6 Liquid Temperature Tf (C) Liquid Flow Rate Wf(kg/s)x 105 Vibration Frequency (1/s) f Drop Selection - - m Drop Diameter a (Nm) T s_(C) Target Heat E2 / (J/s) Effectiveness E Comments - - 213 - DATA SUMMARY Target s s s s s s s Impact Angle 45 45 45 45 45 45 45 Liquid Temperature 86 86 86 86 86 86 86 3.35 3.35 3.35 3.35 3.35 3.35 3.35 3900 3900 3900 3900 3900 3900 3900 Material s or c Tf (C) Liquid Flow Rate Wf(kg/s)x 105 Vibration Frequency (1/s) f Drop Selection - - 20 20 20 - - m Drop Diameter a 258 258 258 258 258 258 258 1.31 1.31 1.31 1.31 1.31 1.31 1.31 7.3 7.3 7.3 7.3 7.3 7.3 7.3 209 172 172 259 259 295 295 3.01 6.52 2.14 4.56 4.29 5.27 4.891 (um) Drop Impact Velocity (m/s) U Weber Number WE Temerature Ts Target Heat E 2 /9 (J/s) Effectiveness Comments 6 h - .07 r - .10 r - - 214 - DATA SUMMARY Target s s s s s s s 45 45 45 45 45 45 45 86 86 86 86 86 86 86 3.6 F4ow Rate Wf(kg/s)x 105 3.6, 3.6 3.6 3.6 3.6 3.6- 4000 4000 Material s or c Impact Angle 0 Liquid Temperature Tf (C) Liquid Vibration Frequency f 4250 4250 4250 4250 4250 (1/s) Drop 20 20 - 20 -20 256 256 256 256 256 262 262 1.95 1.95 1.95 1.95 1.95 1.81 1.81 16 16 16 16 16 14 14 127 127 127 374 374 110 110 Heat E2 /<p (J/s) 6.24 6.24 1.45 7.99 7.61 4.67 4.67 Effectiveness 1.18 1.18 .84 .84 Selection m Drop Diameter a (vim) Drop Impact Velocity U (m/s) Weber Number WE Target Temperature Ts (C Target Comments - .09 r - - 215 DATA SUMMARY .q Target Material s or c s Impact Angle 45 6 i s s i - 1 i i Liquid Temperature - I- .I 4 I 86 (C) T Liquid Flow Rate Wf(kg/s)x 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 4000 4000 4000 4000 4000 4000 4000 4000 20 30 20 30 262 262 262 262 262 1.81 1.81 1.81 1.81 1.81 14 14 14 14 14 110 110 110 110 4.67 3.70 1.27 .84 .90 105 Vibration Frequency f 45 45 45 45 45 45 (1/s) Drop Selection - 30 - 262 262 1.81 1.81 14 14 14 110 110 147 147 5.33 3.80 1.53 4.96 2.29 .93 .84 - m Drop Diameter a (jjm) Drop Impact Velocity U 262 1.81 (m/s) Weber Number WE Target Temperature Ts (C) Target Heat E 2 /p (J/s) Effectiveness Comments - some steam - .98 _ - - 216 - DATA SUMMARY Target S s s S s s s Impact Angle 45 45 45 45 45 45 45 Liquid Temperature 86 86 86 86 86 86 86 4.0 4.0 4.0 4.0 4.0 4.0 4.0 3650 3650 3650 3650 3650 3650 3650 20 30 - 20 30 30 280 280 280 280 280 280 280 1.7 1.7 1.7 1.7 1.7 1.7 1.7 13 13 13 13 13 13 13 4.75 3.52 1.23 6.21 4.75 4.75 .78 76 .98 .98 .98 h h h Material s or c e Tf (C) Liquid Flow Rate Wf(kg/s)x 105 Vibration Frequency f (1/s) Drop Sel ection m Drop Diameter a (im) Drop Impact Velocity U (m/s) Nuber WE Target Heat E2 / - 1.79 (J/s) Effectiveness Comments 1 Zj - - 217 DATA SUMMARY Target Material s s s ss 45 45 45 45 45 45 45 86 86 86 86 86 86 4.0 4.0 4.0 4.0 4.0, 3650 4450 3350 4450 4450 - 20 20 30 s or c Impact Angle e _ _ _ Temperature Tf _-I- 86 (C) Liquid Flow Rate ---A -- 4.0 4.0 Wf(kg/s)x 10 5 Vibration Frequency 3650 3650 (1/s) f Drop Selection m Drop Diameter 20 30 280 280 280 261 261 261 261 1.7 1.7' 1.7 1.7 1.7 1.7 1.7 13 13 13 12 12 12 12 134 134 134 134 134 134 J134 5.81 4.31 1.26 7.04 7.04 5.32 1.01 11.01 1.16 1.16 1.17 - a (ym) Drop Impact Velocity U (m/s) Webe r Number WE Target Temperature Ts (C) Target Heat E / 1.79 J/s) Effectiveness Comments withisteam! - - 11 - 218 DATA SUMMARY Target Material s or s s s s s s s c Liquid 86 86 86 86 8686 86 Temperatu Tf (C) Impact Angle 45 45 50 50 50 50 50 Liquid 3.8 3.8 3.8 3.8 3.8 4.0 4.0 Flow Rate 105 Wf(kg/s)x Vibration 4450 4450 4450 4450 4450 445 445 Frequency fDrop (1/s) 30 -20 30 20 15 Selection m Drop 258 258 258 258 258 261 261 Diameter a (lim) _ Drop Impact 1.7 1.7 1.7 1.7 1.7 1.7 1.7 Velocity U (m/s) Weber 12 12 12 12 12 12 12 Number WE Target 122 122 110 110 110 134 134 Temperatu Ts (C) Target 6.67 1.15 6.64 4.77 1.47 7.08 5.0 Heat 21 E /$ (J/s) Effectivene .92 1.16 1.15 -1.26 1.1 I_ 4eam in Comments - 219 - DATA SUMMARY Target Material s S s 50 50 s s or c Impact Angle e Liquid Temperature Tf 86 86 86 86 86 86 86 86 (C) Liquid - 3.8 Flow Rate Wf(kg/s)x 105 Vibration lo Frequency f (1/s) - - 3.8 - - - - 3.8 3.8 -__ 3.8 3.8 3.8 3.8 - - - - 4450 4450 4450 4450 4450 - 20 30 - 20 30 - 30 258 258 258 258 258 258 258 258 1.7 1.7 1.7 1.7 1.7 1.7 1.7 1.7 12 12 12 12 12 12 12 12 122 134 134 134 147 147 147 16 1.68 7.47 5.84 1.93 7.26 5.68 1.96 1.29 1.37 1.23 1.30 h h 4450 _____ 4450 4450 Drop Selection m Drop Diameter a (ym) Drop Impact Velocity U (m/s) Weber Number WE Target Temperature Ts (C)___ Target Heat E2 172 59 (j/s) Effectiveness Comments - - 1.23 h dulin - .1 220 - DATA SUMMARY Target Material s or s s s 50 .50 50 50 50 86 86 86 86 86 s s s s s 50 50 50 86 86 86 c Impact Angle Liquid Temperature Tf (C) Liquid Flow Rate A __ 3.8 3.8 3.8 3.8 4450 4450 4450 4450 3.8 3.8 3.8 4450 4450 4450 14450 30 30 3.8 Wf(kg/s)x 105 Vibration Frequency f (1/s) Drop Selection - 30 30 - - 30 m Drop Diameter a (yrm) 258 258 258 258 258 258 258 258 Drop Impact Velocity 1.7 1.7 1.7 1.7 1.7 1.7 1.7 1.7 12 12 12 12 12 12 12 12 172 197 197 197 247 247 247 1367 2.46 3.40 3.40 2.98 4.32 4.32 4.07 6.52 .15 .15 .09 .09 h,r h,r r r U (m/s) Weber Number WE Target Temperature Ts (C) Target Heat E/ (J/s) Effectiveness Comments- - - - .05 r - 221 DATA SUMMARY Target Material s or c s s s s 50 20 20 20 20 86 86 86 86 86 86 3.8 3.8 3.8 4.1 4.1 4.1 4.1 4450 4450 -- 4450 4450 4450 4450 4450' 20 20 40 s s Impact Angpe 50 50 Liquid Temperature Tf (C) 86 Ae Flow LiquidRate Wf (kg/s)x O ___-- 105 Frequency Vibration f (1/s) - -__ - - L--I Drop Selection - 30 - 258 258 258 264 264 264 264 1.7 1.7 1.7 .91 .91 .91 .91 12 12 12 3.6 3.6 3.6 3.6 367 295 295 122 122 122 122 6.38 5.21 5.04 6.85 6.85 4.32 2.09 1.03 1.03 .96 - m Drop Diameter a (vm) Drop Impact Velocity U (m/s) Weber Number WE Target Temperature Ts (C) Target Heat E2/$ (J/s) Effectiveness Comments 3 - .06 r - - h - 222 - DATA SUMMARY Target Material s s S S s s s s Impact Angle 8 20 20 20 20 20 20 20 20 Liquid Temperature Tf (C) 86 86 86 86 86 86 86 86 Liquid Flow Rate 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 s or c Wf(kg/s)x 105 Vibration Frequency f 4450 4450 4450 4450 4450 4450 4450 4450 (/s) Drop 20 40 - 40 - 40 - 20 264 264 264 264 264 264 264 264 .91 .91 .91 .91 .91 .91 .91 .91 Weber Number WE 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 Target Temperature 147 147 147 172 172 197 197 247 2.09 5.47 2.57 3.30 2.97 3.94 Selection m Drop Diameter a (,pm) Drop Impact Velocity U (m/s) TS (C)___ Target Heat E2 / (Js) Effectiveness Comments ___ 7.72 4.96 1.22 1.24 h - 1.25 h j - .14 r - .03 r - 223 - DATA SUMMARY Target s s s s s s s Angle 20 20 20 20 20 45 45 Liquid Temperature 86 86 86 86 86 26 26 4.1 4.1 4.1 4.1 4.1 3.8 4450 4450 4450 4450 4450 4800 4800 Material s or c Impact e Tf (C) Liquid Flow Rate 3.8 W (kg/s)x 105 Frequency f (1/s) Drop Selection - 10 - 10 - 20 30 m Drop Diameter 264 264 264 264 264 251 251 .91 .91 .91 .91 .91 1.8 1.8 Weber Number WE 3.6 3.6 3.6 3.6 3.6 13 13 Target Temperature 247 295 295 367 367 122 a (Wm) Drop Impact Velocity U (m/s) 122 Ts_(C) Target Heat E 2 / (J/s) Effectiveness Comments 3.81 5.23 4.85 6.65 6.42 - .04 r - .02 r - 5.80 4.23 1.05 1.02 - 224 - DATA SUMMARY Target s s s s s s s s 45 45 45 45 45 45 45 45 26 26 26 26 26 26 26 26 3.8 3.8 3.8 3.8 3.4 3.4 3.4 3.4 4600 4600 4600 4600 4600 4600 4600 4600 40 40 50 - 20 30 40 - Drop Diameter a (pm) 251 251 251 251 245 245 245 245 Drop Impact Velocity 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 13 13 13 13 13 13 13 13 122 122 122 122 147 147 147 147 4.00 3.57 3.24 1.31 5.52 4.21 3.61 1.92 .94 .89 .88 Material s or c Impact Angle Temperature Tf (C) Liquid Flow Rate Wf(kg/s)x 105 Vibration Frequency f (/s) Drop Selection mII U (m/s) Nuber WE Target Temperature T s_(C) Target Heat EE2 / s (J/s)__ Effectiveness Comments __ 1.25 _- 1.05 1.12 - h - - 225 DATA SUMMARY Target Material s s s s s s s s s or c Impact Angle 45 45 45 45 45 45 45 45 26 26 26 26 26 26 26 26 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 Vibration Frequency f (1/s) 4600 4600 4600 4600 4600 4600 4600 4600: Drop Selection 20 40 60 20 40 60 - 2.59 2.59 2.59 2.59 2.59 2.59 2.59 2.59 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 e Liquid Temperature Tf (C) Liquid Flow Rate Wf(kg/s)x 105 - m Drop Diameter a (im) Drop Impact Velocity U (m/s) Weber Number WE I 19 19 I 19 19 19 19 147 172 172 172 172 1.92 6.29 4.54 4.05 12.55 .83 .88 1.00 - Terpertur 147 147 t147 Target Heat 6.55 4.35 3.55 1.02 11.07 1.08 E2 / 19 19 (J/s) Effectiveness Et Comments - 1W. - 226 - DATA SUMMARY Target S S S S c c c Impact Angle 45 45 45 45 45 45 45 Liquid Temperature Tf (C) 26 26 26 26 86 86 86 2.9 2.9 2.9 2.9 4.1 4.1 4.1 3000 3000 3000 3000 5350 5350 5350 10 20 20 - 20 30 50 268 268 268 268 248 248 248 1.6 1.6 1.6 1.6 1.9 1.9 1.9 11 11 11 11 15 15 15 388 388 388 388 110 110 110 6.92 6.77 5.98 4.91 3.71 .95 1.08 1.15 b,t bZ Material s or c e Liquid Flow Rate Wf(kg/s)x 105 Frequency f (/s) Drop Selection m Drop Diameter a (ypm) Drop Impact Velocity U (m/s) Weber Number WE Target Temperature Ts Target Heat E2 / (J/s) Effectiveness Comments _ _ 7.10 7.14 .05 .11 .05 r r r - b,I - 227 DATA SUMMARY Target Material c c 45 45 45 86 86 86 86 4.1 4.1 4.1 4.1 5350 5350 5350 5350 5350 5350 20 30 30 50 248 248 248 248 248 248 248 1.9 1.9 1.9 1.9 1.9 1.9 1.9 15 15 15 15 15 15 110 122 122 122 122 122 c c c c Impact Angle 6 45 45 45 45 Liquid Temperature 86 86 86 4.1 4.1 4.1 s or c Tf (C) Liquid Flow Rate Wf(kg/s)x 105 Vibration - - 5350 Frequency f (1/s) Drop - Selection - - m Drop Diameter a (um) Drop Impact Velocity U (m/s) I Nuber WE Target Temperature Ts__(C)___ Target Heat E 2 /p 1.58 _ __ _ __ 74 7.44 5.26 1.24 1.15 122 1.71 5.69 4.18 1.24 1.25 1.87 (J/s) Effectiveness Comments __ 15 - - - - 228 - DATA SUMMARY Target Material c c c c c c c c Impact Angle 30 30 70 70 70 70 70 70 Liquid Temperature Tf (C) 86 86 86 86 86 86 86 86 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 5350 5350 4250 4250 4250 4250 4250 3100 40 40 s or c e Liquid Flow Rate Wf(kg/s)x 105 Vibration Frequency f (1/s) Drop Selection 40 - 40 - 40 - m Drop Diameter a 244 244 244 244 263. 263 293 1.8 1.8 3.4 3.4 3.4 3.0 3.0 2.8 13 13 46 46 46 39 -39 38 122 122 122 122 122 122 122 122 4.46 1.70 4.46 4.46 1.70 4.37 1.62 4.22 1.25 1..25 (pm) Drop Impact Velocity U 244 (m/s) Weber Number WE Target Temperature Ts (C) Target Heat E2 /4 (J/s) Effectiveness Comments 1.25 - - 1.25 - 1.17 - 229 - DATA SUMMARY Target C c c c c c c c Impact Angle 70 70 70 30 30 30 30 70 Liquid Temperature 86 86 86 86 86 86 86 86 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3100 3100 3100 3100 3100 1900 1900 1900 Material s or c Tf (C) Liquid Flow Rate Wf(kg/s)x 10-5 Vibration7 Frequency f (1/s) Drop Selection - 10 - 10 - 20 - 20 m Drop Diameter a 293 293 293 293 293 345 345 345 2.8 2.8 2.8 1.5 1.5 1.4 1.4 2.7 38 38 38 11 11 11 11 122 295 295 295 295 295 295 295 5.80 6.09 5.44 5.56 5.34 5.87 (urm) Drop Impact Velocity U (m/s) Weber Number WE Temprature _ 41 TsC arget E 2 /4 1.64 6.27 (J/s)- Effectiveness - .05 - .07 - .05 - .06 E Comments r r r - 230 DATA SUMMARY Target c c 30 30 70 86 86 86 86 3.9 3.9 3.9 3.9 5600 5600 5600 C c c c c'C Impact Angle 8 70 70 70 30 30 Liquid Temperature 86 86 86 86 3.9 3.9 3.9 3.9 1900 4250 4250 4250 4250 Material s or c Tf (C) Flow Rate Wf(kg/s)x 105 Vibration Frequency f (1/s) Drop m _ Drop Diameter a _ _ _ _ _ _ 10 - 10 _ 345 263 263 263 263 240 240 240 2.7 2.4 2.4 1.3 1.3 1.3 1.3 2.4 41 25 25 7.3 7.3 6.7 6.7 23 295 295 295 295 295 295 295 295 5.62 6.20 5.62 5.87 5.51 6.09 5.51 6.92 (yrm) Drop Impact Velocity U _ 10 - 10 - Selection (m/s) Nuber WE Target Temperature Ts Target Heat E 2 /9 (J/s) Effectiveness Comments - .07 r - .04 r - .07 r - .11 r - 231 DATA SUMMARY Target Material s or c C Impact Angle 70 Liquid Temperature 86 Tf (C) Liquid Flow Rate Wf(kg/s)x 10 3 Vibration Frequency f (1/s) 5600 Drop Selection ~ m Drop Diameter a Drop Impact Velocity U 241 (lim) 2.4 (m/s) Weber Number WE Target Temperature 23 295 Ts Target Heat E 2 /$ 5.98 (J/s) Effectiveness Comments - - 1,fi, - 232 - DETAILS OF DROP DYNAMIC ANALYSIS APPENDIX A5 Equations of Motion for Drop Models The equations of motion for the drop deformation models are derived for general expressions for the drop kinetic energy, potential energy, center of mass position, and bottom radius. The general expressions for a drop of volume V are: 2 KE = p V 2 Pf L( dtm ) dX1 + K( a t 2 ) (A5-1) potential energy PE = aa2 L( (A5-2) a center of mass position z cm = y + a M( a a (A5-3) ) drop bottom radius R = a N( 1 ) a (A5-4) where a is the diameter of a spherical drop of volume V ( a = V), zcm is the position of the drop center of mass, - ... -m 1 -1 k - P-m - 233 - A1 is a geometric dimension of the drop (depending on the assumed drop shape), and y is the position of the bottom of the drop with respect to the impact surface. The functions K, L, M, and N depend on the assumed shape for the drop. The system has two indepe ndent coordinates, X1 and zcm. Equa- tions of motion are deri ved for these two coordinates using the general functions K, L, and M; the expression for the external force acting on the drop bottom is derived in terms of a general drop bottom radius on the function a N. R, and therefore depends For the sake of brevity, the argument, of the functions K, L M, and N is omitted. The energy conservation principle for this system can be stated as: t2 [6(KE - PE) + F6y]dt = (A5-5) 0. t1 Because of the amount of symbol s involved, the three terms in the energy equation will be considered individually. The integration of the ki netic energy term is detailed as follows. 6(KE) The differential isI: 0 = f V F L( dzdtm c dt ) 2 + K( dA d1 dt 2 12 j (A5-6) For constant density and drop volume, the differential is: Note that d 1 a d 1K a = K - 234 d6z cm dt I PV [2( dz p V dt S(KE) The time integral Pf V (6KE)dt t2 dz dt (A5-7) - Zcm 2( ZCm )zcmdt dt t t2 + 2K dt is obtained by integration by parts: t2 f. dt 1 dt a 6dX ) dcX 2 dX1 +K 1 + 2K dt t2 Jt d2 2K( d 1 2 2 K a )6A tI t2 - 1 t f -a + K' dX dt 1x dt The virtual changes in the coordinates z trary and are customarily set to zero at t the integrated terms are zero. dX1 dt ijdt (A5-8) and X, are arbiand t 2 so that The time integral of the kinetic energy terms is written in terms of the virtual displacements 6zcm and 61 : f2 t (6KE) dt 122{ d2 z cmdt2 V -Pf ft + -K( 2 at ) - K Sz cm dX1 1 dt (A5-9) - 235 The potential energy term is simply: St 2 t2 IL (P E)dt 6(-aa 2L)dt t1 = In the work term, (A5-10) aa f [(-L' )SX ] (A5-ll) FHy, the virtual change 5y must be expressed in terms of the two chosen coordinates, J2 zcm and t [(F) (F6y)dt z cm + (-F M') 5 1 (A5-12) Idt The energy equation for this system is: I PfV ~d 2 z ~ m)(6zcm dt 2c PfV d2 xI K( ..2 ti +.aa[-L']6A1 + [F]Szcm + [-F M']6 dA -K ( dt j dt = 0. (A5-13) The virtual displatements and 6X1 are inde pendent and xI I - 236 integral In order for this arbitrary. efficients of both 6z cm and 6X1 must to vanish, the cozero. The two re- suiting equations of motion are: d2 z m ) + F -pf V( = 0 1 K'(- dt + (A5-14) dA1 dt ) - ca L' FM' = 0, (A5-15) d2z F = pfV( (A5-1 6) dtm) dt2 d2t Pf V K + ( dt 1 K'( ) dX 1 dt aaL M'" (A5-17) - 237 - Dynamic Scaling Functions for Two Drop Shapes The general functions K, L, M, and N are derived for two shapes, the right circular cylinder and the truncated sphere. For a right circular cyl inder of height X1 (m) and radius X2 (m), the volume V(m3 V ) is gi ven by Tr 2 = For a constrant volume V, the radius x2 (A5-18) can be calculated X1 is known: if the height 2 a /6 )1/2 a3 \1 /2 (A5-19) ( The flow field which describes a deforming right ci rcu 1ar cyl inder with center of mass motion in the axial directi on s: _ z z -z dz CM dt z 1 2 1 S 2 X dX 1 dA 1 dt (A5-20) (A5-21 dt The kinetic energy of this flow is KE = -1- (Uz Pf V 2 + Ur 2 )dV (A5-22) - 238 z 1 cm + [dt 2 dz 1 2-Pf cm Z - + A l d1' r I 1 x3 dz dtJ cm 2 2 Tr 2 dz 21T r dr j dt dA 1 +22 2 2xl1 iA2 2 dt 2 ) T A2 A1 4 dAt dt + dz 1 dt 2 2 1 Pf dA; ~2 (z cm + zcm A 1 2 + [ dt dA1 dt (A5-23) The surface tension energy of the cylindrical drop model surface tension and the surface is the product of th area: PE o[27rA = 2 2+ 2 The center of mass cylinder; if X 2 A1 ]1 located at the midpoint is (A5-24) of the y is the location of the bottom of the cylinder, cm = y (A5-25) 1 - 239 - The radius of the bottom of the drop is just the radius of the cylinder: R = X2 . (A5-26) These relationships define the functions K, L, M, and N for A the cylinder. in favor of Equation A5-19 is used to eliminate A1 and the constant volume V(or diameter a): K( a a 1 + -1 l ( +2~7T71 ] (A5-27) 1/2 L( = a 6 M a N( Aa = (i _ + S6 (A5-28) (A5-29) 2- a ) a a 1 1/2 (A5-30) - For 240 - a truncated sphere of height X1 (m) and radius of V(m3 ) is given by: curvature x2(m) , the volume V X1 ( 2 = For a constant volume ) 3 = V, the radius of curvature be calculated if the height X1 S 3 (A5-31) r a3/6 x2 can is known: 1 + 2 (A5-32) 3 1 For the flow field for a deforming truncated sphere i However, the general form of the not easily described. kinetic energy function for the cylinder suggests a similar form for the truncated sphere. The kinetic energy of the cylinder can be written as KE KE = z =1~d V ( c - 2 ) + 2 FdX + C2 dX 2 2 dt d (A5-33) where C = and C2 dX2 1 dX A1 2 and for the cylinder, 2 1 6 a x I 3 1/2 1 2 2 1 (A5-34) - For the s phere, A2 241 - is the radius of curvature which is larger than the maximum radial dimension of the drop when the deformati on proceeds to the extent where A < X2. Let X3 be the maximum radial dimension of the truncated sphere, measured from the symmetry axis. The radial dimension X3 is define d for two regions: = for A 1/2 for and > Al (A5-35) , A2 (A5-36) The kinetic energy for the sphere is assumed to be of the form KE = 12 p V ( 2cm) + CI+ dt C2 dA dX )2] dA dt) 2 (A5-37) The potential energy of the truncated sphere model is: PE = a(41T A2 A1 - Tr xl 2) (A5-38) If the surface area of the drop bottom is omitted (for the surface contacting drop) the potential energy is: PE = a(2 A2Al (A 5- 39 ) (A2-39) - 242 The center of mass of a truncated sphere is found by integration: zcm (z - y)dV J y - (A5-40) The integration is performed with respect to the angle, <e for the range < 0 - 2 (1 The position (z-y) where the angle T, = is expressed in terms of the angle 8: and the differential dV 80is defined: x22Cos 2e z - y = X 2 (cos - (A5-41 ) cose), volume is: 7r(X 2 sine)3d6 = (A5-42) The center of mass location is: J r X2 (sine)3(cos6 - 8 a0 zcm y cose)de (A5-43) iT X1 2 l 2 (A2 4 3 2 ~ Al 1 ~ e 3) (A5 -44) - 243 - The radius of the flat side of the truncated sphere is: 1/2 = R [x1 (2x 2 (A5-45) A 1 )] - The functions K, L, M, and N can now be defined for the truncated sphere, substituting expressions for A2 and A3 in terms of A and the constant diameter, a, of a spherical drop with the same volume, V: 2 ) K( 32 I = Ca + C2 A1 > 2 (A5-46) (2 ) = Cl + C2 K( - 2(-1 1 3 a 3 j ) a 3 2 + 2 2 >X 1 (A5-47) L( Aa ) = L( M( )a 1 ) ( -) [2( ( -) 1 a -) + (- 1 ) 1, no contact (A5-48) A1 ( A1 3a ( + A N( a 2 + 2(a ) ], surface contact(A5-49) a ) ] (A5-50) 3 1/2 (A5-51) - 244 Integration of Velocity Profile Function A velocity profile function which meets the specified boundary condi tion is: =x3 p(x) = y 1 42 (A5-52) The derivatives of p are: p = 3x 2 - 2x 3 (A5-53) p = 6x - 6x 2 (A5-54) p = 6 - 12x (A5-55) - 12 (A5-56) '''= is defined as The integral I =4 [-- (p") - opf tp dx (A5-57) and is evaluated by substitut ion of the derivatives and integration: Il = = 1 (6x 10 = 22 - 1 6x2) 41 2 - 2x3 )(6-12x)ldx (A5-58) (A5-59) 0.225 is defined as The integral I2 - [- f p''']dx (A5-60) - 245 and is evaluated for the selected function I2 20 = = 1.5 ( ) (-12)dx (A5-61 ) (A5-62) IM11", - 246 - Nonlinear Temperature Profile The steady state energy equation for the flow between two parallel disks, of radius tance y, Pg c R and separated by a dis- is: + Uz Urr9r 29 9 kg z 9z( r 9r (r 3r ) + a2 ) (A5-63) Radial variations are neglected for R >> y, and the velocity profile has the assumed form: Uz = Ur = -U 0 p' ( y L) U0 2y (AS-64) (A5-65) p so that the energy equation becomes: -p c U p( y ) 9 p100 d2T 2 k kdz dT dz (A5-66) with the boundary conditions T = Tw at z = 0 (A5-67) at z = y (A5-68) T = Ts One integration of the energy equation yields: dT = C exp U yP k cP'9 p ( (A5-69) and a second integration yields (with the application of the - 247 boundary condition): z/y y p C kg U T = T + y Cf exp p (x) dx (A5-70) Application of the second boundary condition determines the constant of integration T - T U yp k exp w) C= s C ' p(x) dx (A5-71) The gas velocity U0 is given approximately k S2 (T 2 p - T ) hfs (A5-72) so that U0 y pg C k where the superheat group (A5-73) CT Cpg (T, hf The coefficients and is : - T-) 2 are defined as : (A5-74) I - -- - - dT wy dT -dz at z = 0 (A5-75) at z = y (A5-76) -T /T~ ___ - - T T dz 2 248 INII I ) y / y These coefficients are evaluated by substitution of the integ ration constant C into the temperature gradient express ion, so that J A5-7 1exp[S 2 I) CT p(x)]dx 0 and exp( 2 CT) (A5-78) exp[5 2 CT p(x)]dx 0 and S can be 2 evaluated as functions of the superheat group CT by selFor any function p(x), the coefficients a ecting a number to represent the product (s )-(CT) evaluating 2 al and a2 from the relations shown, and finally ca1cul ati ng CT knowing For the function p(x) = 3x2 2* - 2x3 1 and a. are approximately correlated by the relations - (1 + .8CT) (1 + 2 CT) (A5-79) - (1 for 82 + 2 CT) 249 - 2 (A5-80) 0 < CT < 1. These functions, together with the calculated values of and 2 are shown in Figure A5-1. Lu l.2 + 1.o- + x(1 uJ L8 + 0.8 CT) CT) 0.8 0- Liin Li0.20,- 0 1 2 3 SUPERHEAT GROUP CT FIGURE A5-1 EFFECT OF SUPERHEAT ON TEMPERATURE PROFILE COEFFICIENTS - 251 Temperature Correction for Vapor Dens ity be evaluated at a Let the average vapor density temperature = --- T (A5-81 ) (T + T,) The ideal gas law at constant pressure is: P9 =s Pg Is T 2T P g's (T TR If a temperature ratio RR RR P (A5-83) ) - T s w = then the density ratio + T is defined T TR (A5-82) (A5-84) is computed: 9- - g 9s (T T -,,-T w s + 2T s ) (A5-85) or RR = (A5-86) TR 2 + 1) - 252 Property Data Fluid properties used for all calculations are listed in Table A5-2. Metric units are used throughout the analysis. However the property data are also given in British Units, and conversion factors are listed in Table A5-1. TABLE A5-1 CONVERSION FACTORS Property specific. heat c latent heat h thermal conductivity, viscosity density 0 surface tension a British Units Conversion .Factor _ Metric Units (BTU/lbm-F) x (4183) = (J/kg-C) (BTU/lbm) x (2324) = (J/kg) (1.729) = (3/s-m-C) (BTU/hr-ft-F) (1bm/hr-ft) x (4.134x10~ ) = (kg/s-m) (lbm/ft 3) x (16.02) = (kg/m (lbf/ft) x (14.62) = (kg/s A.e, "P.- 2 ) 01940* - - 253 - TABLE A5-2 PROPERTY DATA Fluid: H20 H20 N 0.312 0.0308 0.863 0.616 2 Saturation Point 00458 577 s cr T /Tcr Property (Units) Cpf (J/kg-C) 4180 5020 2050 c (BTU/lbm-F) 1 .00 1 .20 0.491 c (J/kg-C) 1880 5100 1:945 c (BTU/lbm-F) 0.450 1.22 0.250 hfg (J/kg) 2. 25x106 1 .51x10 6 1 .98x105 h fg (BTU/lbm) 970 650 85 kf (J/s-rm-C) 0.678 0.605 0.139 k (BTU/hr-ft-F) 0.392 0.350 0.0805 k (J/s-m-C) 0.0251 0.0622 0.00761 k (BTU/hr-ft-F) 0.0145 0.0360 0.00440 g (kg/s-m) 1 . 30x10-5 1 .98x10- g (lbm/hr-ft) 0.0314 0.0480 0.0131 Pf (kg/rm 3 ) 958 742 801 Pf (lbm/ft 59.8 46.3 50.0 p (kg/m 0.597 35.9 5.25 Pg (ibm/ft 3) 0.0373 2.24 0.328 a (kg/s2 ) .0585 .0181 .00848 a (Ibf/ft) 4. 00 x1 0-3 1 .24x10- 3 3) ) 5 3 5 .42x10-6 5. 0x10 4 III - 254 - Maximum Pressure Rise and Reynol ds Number in the External Flow The maximum pres sure rise in the modeled flow between at r = 0. and separation R two parallel disks of radius occurs is known, the maximum Fmax If the maxi mum force y pressure rise is 2F =-max Tr R Pmax (A5-87) The Reynol ds number which characterizes the flow between the two disks is: = p9. Ur . g_ RE where Ur Or (A5-88) is an average radial velocity. is related to the axial velocity U The radial velocity at the upper disk (by continuity) Urmax =r 2y (A5-89) Uo The maximum radial velocity occurs at r = R. The axial velocity is determined by the evaporation rate: - T ) g hfw y s k (T U o= 2Pg (A5-90) - 255 The Reynolds number is now: (Th REg 2y 2 - T ) (A5-91 ) g hf g or in terms of the conductivity and viscosity groups, TH = k (T g w and - hfg RE = 2 2 VIi 1 s aa 1/2 (A5-92) 1/2 2 p (p 9 T) (A5-93) aa R (A5-94) ;2y The maximum Reynolds number occurs at the minimum separation ymi when the drop bottom radius maximum Rmax' R is less than its - 256 - Maximum Extension Radius, Cylindrical Model The maximum size of a drop during impact can be calculated on the basis of the assumption that all mechanical energy is stored as surfa ce energy at the maximum extension: PE max + KE0 = (A5-95) PE For a spherical drop, the surface energy (in dimensionless variables) is: = PE r 3 (A5-96 ) /16 For the cylindrical drop, the general expression for surface energy is: PE = (27rX2 2 + 2irX2 x ) Tr (A5-97) /16 The constant volume requirement is used to eliminate the cylinder height X1 in favor of the cylinder radius, X2 X (A5-98) 1/(6X 2 ) = The drop bottom radius R is the same as the radius X2 for the cylindrical model: R (A5-99) 2 = The initial value of kinetic energy, KE0 , can be replaced by an equivalent expression involving the impact Weber number: KE0 = (Tr3 WE)/(12-16) (A5-100) With the substitution of these relations into Equation (A5-9 ), the equation for maximum radial extension for the cylindrical drop model is: - 2 (2TrR 16 257 2 + Tr 2R - 3 WE 6_ 16- (A5-101 ) 12 16 or, WE + 2 -24 The solution, fo r WE > 1.74, ) + 1 0 = 6- (A5-102) is (3)1/2 2( 2 R =( (1 + WE + 12 ) acos 1 + WE )3/2 12/ (A5-103)

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