THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING SOME PROPERTIES OF SPRAYS FORMED BY THE DISINTEGRATION OF A SUPERHEATED LIQUID JET William Lo Short A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Chemical and Metallurgical Engineering 1962 October, 1962 IP-583

ACKNOWLEDGEMENTS The author is particularly indebted to many people who rendered assistance during the course of the doctoral programo Sincere appreciation and thanks should go to: Professor Jo Louis York, the chairman of the doctoral committee, for his guidance, assistance and suggestions throughout the course of the worko Professors Ao Go Hansen2 Ro Bo Morrison, Mo Ro Tek, and Go Bo work and in the preparing of this dissertationo Mro Gordon Ringrose, who constructed much of the electronic equipment used in the researcho The Allied Chemical Company for their support during the second year of studyo EoIo du Pont de Nemours and Company, who very generously donated the Freons used in this studyo Do Po Kessler who as a CM 690 project, wrote a portion of one of the computer programs used. The Industry Program of the College of Engineering who prepared the final form of this dissertationo ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS..................................... ii LIST OF TABLES. oooo...............................................ooo o o ooo oo. v LIST OF FIGURES o.ooooo.............. o o o............oooooooo..oo o...... vili LIST OF APPENDICESo00.0000000000000...........o........00, 000., x NOMENCLATUREo.ooooooooooooo......... o.o.o.o...o.. xi ABSTRACT.o....oo.o... ooooooooooooo........................... Xiv I INTRODUCTIONo....... 0... 0.0.................0 1 II THEORY AND MECHANISM OF LIQUID JET BREAKUP BY PRESSURE ATOMIZATIONo....... 0000000 000000000000000 4 1o The Breakup Regimes..... o...o,......,ooo.. 4 2o The Breakup Mechanism,,. o.....o.ooo.oooo 6 30 Other Drop Formation Processeso..o... oo..o... 11 301 Secondary Atomization....... o........o o 11 302 Drop Coalescenceo......o o o....o... o. 12 III EXPERIMENTAL APPARATUS AND PROCEDURES....... ooo.... 13 1o The Injection System.......... o...o...... 14 2. Spray Analysis................................ 16 53 Dark Room Procedure........ oo....o.......... 23 4o Drop Analysis and Counting..,o...... ooo.o..oo... 23 5. Experimental Determination of Shatter Temperatures............................... 26 IV DISCUSSION OF RAW DATA............................... 28 1. Range of Variables Studied................. 28 2o Size Distribution Data.................o..... 30 35 Shatter Temperature Datao..........o......... 30 V THE BREAKUP MECHANISMo ooo...o....ooo........oo o...... 33 1o The Breakup Mechanism o.................... oo 33 2o Effect of Fluid Properties on the Breakup Mechanism o o o......O o..... o o o o o o.. 38 35 Effect of Physical Variables on the Breakup Mechanism 0o o.o..o.oooo............ o o o...... 48 4o Shatter Temperature Correlation...... o........ o o 51 5o Summary..oo...o.o.o o...oooo.......o.o.ooooooo 58 iii

TABLE OF CONTENTS (CONT'D) Page VI SPRAY EVAPORATION... o o............................ 59 1. Theory....................................... 59 2. Prediction of Drop Size Using a Measured Initial Size Distribution................... 64 3. Flow Rate Checks Based on Evaporation........ 79 3.1 Evaporation Due to Sensible Heat Loss.... 79 3.2 Convective Evaporation.................. 81 4. Summary e.................................. 84 VII DROP VELOCITY PROFILES WITHIN THE SPRAY.............. 85 1. Equation of Motion of a Particle............ 85 2. Corrections to the Drag Coefficient......... 88 2.1 Corrections for Non-Creeping Flow........ 89 2.2 Corrections for Accelerated Motion...... 89 2.3 Corrections for Evaporation of the Droplet......o.................. o..... 90 2.4 Corrections for Interaction of Particles. 92 5. Induced Air Velocity......................... 93 3o1 Analysis of Induced Air Flow in a Hollow Cone Spray....................... 94 3.2 Solid Cone Sprays....................... 98 4. Discussion of Data............................. 104 5. Summary................................... 107 VIII SPRAY CHARACTERISTICS............................ 108 1. Drop Size Distribution Functions............. 108 1.1 Distribution Function................... 108 1.2 Test of Distribution Functions to Fit Experimental Data.................... 112 1.3 Discussion of the Utility of the Three Functions....................... 115 2. Experimental Mean Drop Diameters............. 116 30 Reproducibility of Experimental Data,....... 120 IX CONCLUSIONS....................................... 122 X RECOMMENDATIONS O.................................... 124 APPENDICES............................................ 126 iv

LIST OF TABLES Table Page I Physical Properties of Water, Freon 11 and Freon 113.o 29 II Experimental Nozzle Diameters o...........,.....o. 29 III Typical Drop Size and Velocity Data; Water System at a Distance of Four Inches for 120 Psig Injection.oo 31 IV Measured Shatter Temperatures o......................... 32 V Temperatures for Use with Equation (5o8).............. 44 VI Values of Dimensionless Constants in Equation (5o8) for Figure 10,ooo..o...........o............. 45 VII Calculated Values of Dimensionless Groups for Breakup Datao oooooooooooooo.....o o..o..o............o. 53 VIII Comparison of Calculated and Experimental Values of Shatter Temperature Groupo...................o. 57 IX Travel Times for Spray Droplets..................... 69 X Summary of New Average Diameters 00...0...00.......0... 70 XI Calculated Per Cent Evaporation of Sprays.o...... 82 XII Per Cent of Total Flow by Location in Spray......... 94 XIII Calculated Induced Air Velocities oo.......,...o 96 XIV Commonly Used Mean Diameters........................ 110 XV Transformations and Integrals for Density Functionso.. 112 XVI Size Distribution for Total Spray —Run 2........o.... 115 XVII Mean Drop Sizeso....o............ o o.... o... oo....... 117 XVIII Comparison of Mean Diameters with Brown's Data...... o 121 XIX List of Input Variables for Computer Program..o.....o 128 XX Computer Program. o........o................. o o o 130 v

LIST OF TABLES (CONTVD) Table Page XXI Drop Size Ranges...o................................. 137 XXII Drop Size Distribution and Velocity Data............. 138 XXIII Minimum Initial Radius for Bubble Growth Under One Atmosphere.................................... 154 XXIV Ratio of Resistance to Gravity Forceso o....o....... 164 XXV Calculated Values of Constants in Distribution Functions........................................ 166 vi

LIST OF FIGURES Figure Page 1 Stages of Jet Disintegration in Relation to the Reynolds Number and the Ohnesorge Number.....oo..... 7 2 Liquid Injection Systemo,,o oooooo........... o ooo. 15 5 Camera Arrangement for High Speed Photographso...o... 19 4 Time Delay Unit o........oo..........o.....oo..... 20 5 Power Supply for Time Delay Unit......o o........... 21 6 Sample Locationso,, o ooo............................ 24 7 Flashing Jet (lOX), 135~F................. ooooo o ooo 35 8 Flashing Jet (lOX), 138~F........................o oo 36 9 Flashing Jet (1OX), 140oFo o, o o................... o o 37 10 Plot of Dimensionless Time versus Dimensionless Radius for Water, Freon 11 and Freon 113......0 0 0 0 46 11 Growth Rate Curves for Water, Freon 11 and Freon 113 47 12 Shatter Temperature CorrelationO o o o o.... 0 o o 56 15 Velocity Profiles for Water System Location 1, Injection Temperature 287~F, 120 Psig...............o 67 14 Velocity Profiles for Water System, Location 1, Injection Temperature 287~F, 120 Psig o...o.......o.. 68 15 Comparison of Measured and Predicted Average Drop Diameters for Water, Location 1..........ooo..oo...... 71 16 Comparison of Measured and Predicted Average Drop Diameters for Water, Location 2...0 0........0 0.... o 72 17 Comparison of Measured and Predicted Average Drop Diameters for Water, Location 3~o........ 72 18 Comparison of Measured and Predicted Average Drop Diameters for Water, Location 4...................... 73 vii

LIST OF FIGURES CONT'D Figure Page 19 Comparison of Measured and Predicted Average Drop Diameters for Water, Total Spray... oo o......o o. 74 20 Comparison of Measured and Predicted Average Drop Diameters for Freon 11, Locations 1 and 2,,,00 0 0 0 76 21 Comparison of Measured and Predicted Average Drop Diameters for Freon 11, Location 3000000000o oo oo o. 77 22 Comparison of Measured and Predicted Average Drop Diameters for Freon 11, Total Spray.,....0000..0.... 78 23 Comparison of Experimental and Measured Flow RateSoo 83 24 Typical Velocity Profiles -- Water at a Distance of Four Inches from the Nozzleoo..oo..oo o,,ooo....o 99 25 Velocity Profiles for Water, Locations 1 and 2.. 000 100 26 Velocity Profiles for Water, Locations 3 and 4o..0.. 101 27 Velocity Profiles for Freon 11, Locations 1 and 20o. 102 28 Velocity Profiles for Freon 11, Location 3,,5.0000 o 103 29 Comparison of Water and Freon 11 Drop Velocitiesoo.. 105 30 Typical Fit of Log Normal Distribution -- Run No 2. 114 31 Comparison of Experimental Data with Brown's Data -- Freon 11,o oooo ooooooo,, ooo.. o o...oooo. oo 118 32 Comparison of Experimental Data with Brownis Data -- Water, oo 0.........oo..o.o.oo...............o 119 33 Distribution Plot for Run 1oooo..,, o.ooooooooooo0000 141 34 Distribution Plot for Run 35o.... oo..,ooo........ o 142 35 Distribution Plot for Run 5oo.o00...o........oooo00 143 36 Distribution Plot for Run 7o oo...0 o..oo000o...... 144 37 Distribution Plot for Run 800....00 000000000000...00 145 viii

LIST OF FIGURES CONT'D Figure Page 38 Distribution Plot for Run 9,000...............000000. 146 39 Distribution Plot for Run 10..................... 147 40 Isoclines for Water................................. 156 41 Per cent of Spray Unevaporated as a Function of Evaporation Index.................................. 161 ix

LIST OF APPENDICES Appendix Page A Computer Program..oo................. o oooooooooooo 127 B Raw Data.O O... O O.. OOO.. o......... Oooooo o.. 136 C Bubble Growth in a Superheated Viscous Liquid....... 148 D Probert's Method of Calculating Spray Evaporation oo 158 E Determination of Magnitude of Gravity Effectso....., 163 F Calculated Values of Constants in Distribution Functions o o e o o........... oo... o.o.o oo o... o o 166 G List of References o,.......,............o o 168 x

NOMENCLATURE A Acceleration modulus c A Area C Growth rate constant [see Equation (5o4)] C Dimensionless variable [see Equation (5o8)] CD Drag coefficient Cf Skin friction coefficient Cp Specific heat D Diameter of jet or drop D Dimensionless variable [see Equation (5~8)] Dm Molecular diffusivity Dt Thermal diffusivity Dmn Mean drop diameter dw/dQ Evaporation rate E External force f(x) Probability distribution function F(x) Probability density function gc Conversion factor hc Convective heat transfer coefficient I Integral [see Equation (C-8)] k Thermal conductivity kg Constant [see Equation (5 5)] kc Correction factor [see Equation (7o16)] kg Mass transfer coefficient K Evaporation constant [see Equation (6o10)] xi

mk k-th moment about origin M Mean molecular weight M~ Mass n Number of droplets N Total diffusion rate P Pressure Pf Vapor pressure Pr Prandtl number r Radial distance [see Equation (7o21)] R Radius R Gas law constant R Rosin-Rammler distribution Re Reynolds number s Radial distance [see Equation (7.21)] Sc Schmidt number t Time T Temperature T Dimensionless variable [see Equation (7o5)] tm Mean temperature difference V Velocity We Weber number X Mean of X Z Ohnesorge number Greek p Density X Wavelength xii

~ Time, spray cone angle Xs Latent heat p Viscosity ~a Surface tension AT Superheat a See Equation (5o5) T Dimensionless time Pi ~ Dimensionless radius r,() S ti -1 e-tt Uniformity parame Uniformity parameter Strain [see Equation (C-3)] Subscripts 1 Liquid 0 Initial value v Vapor x X direction y Y direction xiii

ABSTRACT Classically sprays are formed by pressure atomization, spinning disks and air atomizationo The purpose of this study was to study the properties of sprays formed by the flashing of superheated liquid jets and to investigate the variables which control the breakup of a superheated liquid jeto Sprays were produced by the flashing of superheated water, Freon 11 and Freon 113 using simple orifice type nozzles, having small length to diameter ratiosO The breakup of the superheated jets, as characterized by a shattering temperature, was found to correlate using a simple model relating a shatter temperature group to a function of the Weber number, Ohnesorge number and a vapor to liquid density ratio, The drop size distributions may be satisfactorily represented either by a log normal or a Rosin-Rammler distributiono Mean drop diameters are strongly dependent upon the surface tension of the fluid injected. Spray evaporation rates were calculated using the method of Probert with satisfactory resultso The drop velocities were found to asymptotically approach a limiting value which is equal to the induced air velocity. The breakup mechanism is described in terms of the growth of a vapor bubble in a superheated liquid. The most important variables are surface tension and nozzle diametero It is pointed out that viscosity likely plays an important role but the range over which the viscosity may be varied at the breakup point is necessarily smallo xiv

CHAPTER I INTRODUCTION The purpose of any atomization or spray process is to break up a continuous liquid jet into a discontinuous series of liquid droplets of varying sizeso Spray formation is most commonly carried out by pressure injection, by a swirl chamber nozzle or by air atomization. In the first two of these processes the liquid is broken into droplets by the creation of an unstable jet or an unstable sheet of liquid which must disintegrate under the action of pressure and surface tension forceso The sprays studied in this investigation were created, at least in part, by the flashing or vaporizing of a portion of the liquid jet, after it issued from the nozzleo The purpose of this study is threefold. Firstly, to investigate the effect of fluid properties on the breakup of a superheated liquid jet, and to discover what properties of the injected fluid are most important to the breakup mechanism; secondly, to investigate the drop size distributions which arise from this method of spray creation; and finally to study the effect of distance from the nozzle and position in the spray, upon the average drop diameter; io.e. to determine the effect of evaporation on the drop size distribution and on the average diameter. Thermodynamically, flashing of a liquid occurs when it is at a temperature greater than the equilibrium temperature corresponding to the pressure of the surroundingso For example, in this sense, liquid water at a temperature above 212 degrees Fahrenheit will "flash" when it is exposed to atmospheric pressureo Under truly adiabatic conditions all of -1

-2 the vapor formed will receive its heat by conduction through the liquid, at the expense of the enthalpy of the unvaporized portion of the liquid phase. Equilibrium will be reached when the residual liquid phase has cooled to its saturation temperature (212 degrees Fahrenheit for water at atmospheric pressure)o Flashing, in the sense defined above, could also result from the vaporization of a gas dissolved in the liquid phaseo It has, however, been demonstrated experimentally and with some theoretical justification(5) that this method of spray formation is not as effective as flashing by superheating techniqueso Sprays formed by the "flashing" technique have much different characteristics than sprays formed by pressure injection at a comparable pressure levelo The sprays resulting from flashing contain many more small droplets and have a narrower drop size distributiono For example, at an injection pressure of 120 psig, a typical linear average drop diameter reported in this study might be 50 microns, with 90 per cent of all of the drops included within the range of 20 to 120 microns. As a consequence of the larger number of smaller drops, one would expect that the spray would evaporate more rapidly at a given set of conditions of the surroundings than a spray formed by pressure injection and hence one which contained larger dropso The results reported here consider the breakup of three liquids: water, trichloromonofluoromethane ("Freon 14) and trichlorotrifluoroethane (Freon ll3)o The breakup mechanism is described by the measurement of a "shatter temperatureo" The meaning of this term and a description of its measurement will be given in considerable detailo A measure of the effectiveness of the breakup of a liquid jet is given by the drop size distribution,

-5These distributions were analyzed by a photographic techniqueo The variation of diameter throughout the spray results directly from the study of the drop size distribution throughout the sprayo The flashing process occurs in at least one application since it is likely that the fuel injected into the afterburner of a jet engine is superheated before it leaves the nozzleo An application of a very similar technique is the "aerosol bomb" in which the material to be sprayed is stored in a pressure vessel with a portion of propellanto When the mixture of the material and propellant is injected into the atmosphere, a fine spray results from flashing of the propellant.

CHAPTER II THEORY AND MECHANISM OF LIQUID JET BREAKUP BY PRESSURE ATOMIZATION lo The Breakup Regimes In the past fifty years many excellent papers have been published on the mechanism of spray formation and on the influence which the various fluid properties, such as density or viscosity, have on the breakup of liquid jets. It is not the purpose to present here a complete or even a semi-complete summary of the literature of spray formation. Such literature surveys are available.(9'43) It will be useful to review briefly some of the pertinent references relating to sprays formed by pressure or hydraulic atomizationo Pressure or hydraulic atomization may be defined as a process in which the liquid is forced through an orifice or a nozzle to form an unstable liquid jet or sheet which disintegrates upon leaving the atomizer. Holfelder and Haenlein(20) have observed liquid jet disruption using spark photography, and reported four stages of jet disintegration. (1) At verylow injection velocities the breakup into drops is caused by rotationally symmetric oscillations of the jet surface due to the effect of primary disturbances and surface tension forceso (2) At higher jet velocities the breakup into drops results from oscillations with the additional effect of air friction. (3) At larger velocities the breakup occurs through waviness of the jet assisted by air friction,,4~

-5(4) Finally at still larger velocities there is immediate and complete disruption of the jet, which takes place at or near the orifice. The exact breakup mechanism in this regime is still not known, although one or two theories will be discussed later. Nukiyama and Tanasawa(37) in a similar photographic study of spray formation distinguished these stages of jet disintegration: (1) Dropwise splitting of the jet due to surface tension forces. (2) Twisted, ribbon-like atomization which corresponds to (2) and (3) above. (3) Filmwise atomization corresponding to (4) above. All of these three regions were observed with the jet velocity increasing from an initially low value. Ohnesorge(38) found similar stages of jet disintegration and hypothesized that the different stages occur at different values of the Reynolds number and that these values are determined by a characteristic viscosity number. This number is sometimes called the Ohnesorge number, Z, and is given by Z7 _ A. (2.1) where p is the liquid viscosity p is the liquid density a is the interfacial tension D is the jet diameter.

-6Figure 1 is a plot of his relationship between the Ohnesorge and Reynolds numbers. The regimes in this figure labeled I, II and III correspond approximately to the three regions of breakup described by Nukiyama and Tanasawao Although the boundaries between the regimes have been shown in Figure 1 as having a sharp division, in actual fact there is no sharp or clear cut transition from one breakup mechanism to another. The loci of the transition points could perhaps be more accurately described as a band rather than as a lineo For illustrative purposes, the injection conditions for the nozzles and fluids used in this study are shown in the shaded regions of Figure 1. The information portrayed in this figure is taken from Ohnesorge's paper, and it will be utilized later in the description of the breakup of flashing liquid jets. 2o The Breakup Mechanism The mechanism of the disintegration of a mass of liquid into small drops has been the subject of many theoretical and experimental investigations since the pioneer work of Rayleigh.(46) His efforts were mainly confined to a study of the stability of low speed, non viscous jets; i.e., region (1) described above. Rayleigh concluded that surface tension would cause a small disturbance on the jet surface to propagate and that the wave length of these disturbances having the maximum growth rate would be about 4-1/2 times the jet diameter, and that the drops formed would be slightly less than twice the diameter of the jet itself. His solution neglects any aerodynamic or viscous forces and he therefore has only dealt with the injection of "inviscid liquids" at low velocities.

-7-I ____ 10 5 2,j ] \ | X NOZZLES USING CN \ / II F-II AND F-113 REGN FO REGION FOR 103 2 5 104 2 5 105 2 5 Re= DvP -3.~ ~R. Figure 1. Stages of Jet Disintegration in Relation to the Reynolds Number and the Qhnesorge Number. Reynolds Nunmber and the Ohnesorge Number.

-8Haenlein(20) among others, showed experimentally that for very viscous liquids the wavelength of the disturbance with the maximum growth rate would be much larger than that predicted by Rayleigh (for example, up to 30 or 40 times the jet diameter for the case of caster oil). Weber(59) studied the same problem theoretically and was able to show that with the inclusion of viscous effects the wavelength of the disturbance with the maximum growth rate is: S7D -/ +1) (2.2) where X is the wavelength D is the jet diameter. is the viscosity p is the density a is the interfacial tension. In the limit of zero viscosity this gives excellent agreement with Rayleighs' results. Haenlein's experiments were confirmed theoretically by Weber for the first three stages of jet breakup (two stages according to the description of Nukiyama and Tanasawa), but not for the final stage of random and immediate disruption. Weber, from a study of the aerodynamic forces acting on the jet, demonstrated that the breakup time (length) decreases with increasing jet velocity. He also showed that the critical value of the wave length of the disturbance decreased with a dimensionless

-9constant, which is now called the Weber number, It is //w PI= S D — (253) where PG is the density of the receiving medium V is the jet velocity D is the jet diameter a is the interfacial tension gc is a conversion factor to engineering units (in an FMLT system) The liquid velocity at which the aerodynamic forces become important depends upon the physical properties of the liquid and is given, for instance, by Haenlein(20) as 26 feet per second for water. The third breakup regime, often called wave=like breakup, has been the study of few experimental and/or theoretical investigations. The limited evidence available indicates that it first becomes important at a velocity of about 80 feet per second for water. This critical velocity is dependent upon the physical properties of both the injected and receiving fluidso Miesse(34) in a survey of his own and other data, concluded that the last stage of disintegration was determined chiefly by the Weber number and very slightly by the jet Reynolds number The theoretical studies described above assume some slight imperfection in the liquid surface, at which point the instability propagates itself to eventually cause the jet disintegrationo Some workers have claimed that the important mechanism in atomization is the manner in

-10which these originally small imperfections are formedo Mehlig(33) cited the importance of the radial components of the velocity arising from turbulent flow through the injector. Thiemann(56) held forth that liquid turbulence is a primary factor in jet disruption. In fact, Schweitzer(50) has shown that a jet can disintegrate without air action, if the turbulence level on leaving the orifice is sufficiently higho To date, the only theory suggested for the last, and most important region of jet breakup, is the ligament theory first suggested by Castleman (6) He proposed that disturbances on the liquid surface are acted on by the air stream and that the disturbance is caught up and drawn out as a fine ligament, one end of which remains anchored to the liquid jeto The ligament is then "cut off" by a rapid growth of a dent in its surface and the small detached mass quickly collapses to form a dropo Thus atomization occurs at the gas liquid interface due to the relative velocity between the two phaseso Castleman also proposed that in high velocity gas streams ligaments collapse as rapidly as they are formedo A consequence of this line of argument is that a continuing increase in relative velocity will at some point cause no further decrease in the size of the drop formed. Littaye,(28) while agreeing with the theory of Castleman in most respects, does not report a minimum drop size in an atomization process. Hinze(22) has pointed out that even though turbulence may cause the initial disturbances in the et the these disturbances are most likely amplified by air friction to form ligaments which are torn off to form drops. Thus he, in effect, agrees with the ligament theoryo

-11 3o Other Drop Formation Processes The above discussion has not included two other possible mechanisms of drop formation: secondary atomization and coalescenceo While neither of these two methods of drop formation is too important in spray analyses, they will be discussed briefly. 31l Secondary Atomization Secondary atomization may be defined as the formation of drops from drops which have previously been broken off liquid jets. It is a well known fact that a spherical drop, when subjected to a relative velocity in an air stream can, under certain conditions, be unstable and shatter to form two or more smaller dropso The mechanism of secondary atomization has been studied by Lane,(26) Baron,(3) Balje and Larson,(l) Littaye,(29) Siestrunck(52) Hinze,(21) and Dodd.(10) Lane9 in a photographic study, observed that liquid drops were blown into the shape of a liquid ring with a thin film in the centero This film then expanded to form a hollow bag with a liquid torus rim, and the bag finally shattered to form fine dropletso This was followed by the disintegration of the torus to form larger dropletso Whether or not a drop will be blown into this shape and disintegrated depends upon interfacial tension, drop size, air velocity and the nature of the exposure of the drop to the air (transient or steady flow). The hollow bag shape observed by Lane was predicted theoretically by Baron. Dodd developed a similar theory to predict the distortion of a water drop exposed to a stream of air with a continuously increasing relative velocityo Probably the most important conclusion to

-12arise from all of these investigations was the result that a drop will be unstable if its Weber number exceeds a certain limiting valueo The numerical values reported for this critical Weber number vary somewhat from investigator in investigator. Richardson(48) reports a value of about 20 and Hinze(22) a value of about 22 for the critical value for watero Two difficulties arise when attempting to apply the critical Weber number concept to the stability of drops normally encountered in sprayso Firstly, all of the above studies were carried out with drop sizes in excess of one thousand microns, and secondly all of the critical Weber numbers cited refer to non viscous liquidso Certainly an increase in viscosity should make a drop more stable, but no quantitative studies have been made on this pointo 3o2 Drop Coalescence It is possible that at some point fairly far removed from the spray nozzle that the droplets may be moving slowly enough that any "large" drop formed by this mechanism would prove stable. Collision of drops can occur as a result of eddy diffusivity or through varying axial velocitieso Very little experimental or theoretical work has been done on the problem of coalescenceo The major difficulty lies in the fact that even if a good manner of describing the collision frequency is found, that one still must find some additional information which will determine whether or not drops which do collide in actual fact coalesceo Gorbatschew(l8) has shown that the coalescence of colliding liquid drops is dependent upon the liquid properties and the drop size and also upon the angle and velocity of impacto

CHAPTER III EXPERIMENTAL APPARATUS AND PROCEDURES In the preceding chapter an outline of the present state of knowledge of spray formation by pressure atomization was giveno It is clearly evident that the theories and mechanisms discussed can not apply to sprays formed by a flashing liquid jet, at least not without major modifications It is then necessary to devise experimental methods which can be used to investigate this type of spray formationo Three distinct areas of study immediately suggest themselves: (1) The mechanism of disintegration of a superheated liquid jet. Brown(5) has suggested a theory for this breakup, but because he studied only one system (water), his theory is not of general useo (2) The heat source for evaporation of sprays formed by this method will be different from those generally reported in the literature and for this reason the means of handling this problem analytically should be investigatedo (3) The drop size distributions and drop velocities should be analyzed and the results compared to any existing datao With these points in mind the experimental equipment described below was developedo Also various analytical techniques were considered and it was found that photographic analysis would be most suitable for the purposes of this studyo The reasons for this choice will be brought out in the succeeding pageso -13

lo The Injection System A diagram of the liquid injection system is shown in Figure 2. The system is designed so that "hot" liquids (in this case superheated water) and "cold liquids" (Freon 11 and Freon 113) may all be injected. In both cases the liquid to be injected is fed into the ten gallon tank, and injected as follows: (a) Superheated Water The tank is filled to about the mid point with water and steam passed into the tank until the pressure is raised to the pressure of the steam line (about 130 psig)o The superheated water is then passed through the double-pipe heat exchanger and out of the nozzleo The spray temperature is controlled by means of the heat exchanger, using cooling watero The nozzle is connected to the piping by means of a heavily insulated flexible coupling. (b) Freon 11 and Freon 113 These two liquids are injected in essentially the same manner as water, except that the driving pressure is maintained by a nitrogen supply, as indicated in Figure 20 The Freon is superheated as it passes through the heat exchanger, using 130 psig steam as the heating mediumo The flow rate of the injected fluid may be metered using a Fischer-Porter variable area flow meter having a range of 0.026 gallons per minute to 0.211 gallons per minute, or the flow rate may be calculated from the well known orifice equation and from a knowledge of the

STEAM INLET (125 PSIG) VENT AIR INLET (100 PSIG) 7i | |~10 | ~STEAM INLET IO GALLON TANK ROTAMETER WATER FLEXIBLE INLET TUBING FREON SUPPLY UTANPK PRESSURE j | —-| ~~~ TAN ~K IGAUGE NITROGEN CYLINDER 7MIXERTHERMOWELL ~r I / /T NOZZLE _HEAT THERMOCOUPLE DRAIN' EXCHANGER DRAIN Figure 2. Liquid Injection System.

-16discharge coefficiento This latter method was nearly always employed. The value of the discharge coefficient was determined experimentally to be about 0.8. The injection pressure is measured with a Jo Po Marsh Mastergauge Type 103, which was calibrated by Brown(5) using a dead weight testero By means of this gauge it was possible to determine the injection pressure to the nearest one half pound. Injection temperatures are measured by a copper-constantan thermocouple, which makes contact with the pipe about one half inch above the nozzleo Both the pipe and as much of the nozzle as possible are heavily insulated to reduce the experimental error to a minimum, In addition, a thermowell is located just downstream from the pressure gauge which can be used as a check on the accuracy and reliability of the thermocoupleo 2. Spray Analysis There have been many attempts made to develop a completely satisfactory method of spray analysis. The large concentration of effort in this area is due, in part, to the very great and numerous experimental difficulties involvedo The methods which are normally used can be separated into the following groupso (a) Slide and cell collection (b) Size discriminating collectors (c) Photographic methods (d) Other optical methods The most commonly used method for collecting and analyzing drops is coated slideso Such slides are coated with a soft material such as magnesium oxide and are then exposed to the spray for a short

-17period of time, When the drops strike the coating they leave an impression whose diameter is related to the drop diameter. Analysis may then be carried out by use of a microscopeo The primary disadvantage of this method of drop size analysis is that if the spray is carried by a gas which is moving relative to the slide, the slide discriminates against collecting small drops. Hence, the measured average drop diameter will always be too largeo A second and obvious difficulty arises from the fact that the impressions left on the slide are not the true drop diameters, and the relationship between the impression diameter and drop diameter must be determined by experimento A third disadvantage is that some of the large drops may shatter upon contact with the slide. Two other important drawbacks are the problems of coalescence on the slide and evaporation of drops from the slide before the actual counting takes place, if some material other than magnesium oxide is used to coat the slideo A similar technique is the use of a collection cello It simply consists of a receptacle filled with a liquid into which the spray droplets fallo This method of analysis has most of the drawbacks associated with a coated slide, but their magnitude is usually decreased. As pointed out above, the major disadvantage of slides and cells is their tendency to discriminate against collecting small droplets. This phenomenon, known as collection efficiency, is made use of in a sampling device called a jet impactoro This method is not too widely used and it will not be described hereo Aside from photography there are several other optical methods available for the sampling of sprayso One of the very well known of

-18these techniques is the photometer method of Sautero(49) By means of this method the volume to surface mean diameter is determined by measuring the decrease in intensity of a light beam passing through the spray. This method gives no information concerning the drop size distribution. In addition to the method of Sauter, there have been attempts to analyze spray droplet sizes by light scattering techniqueso These have not proved to be effective for other than a few specialized cases, The method used in this investigation is photographic analysis. This technique was first successfully used by York(60) in 1949, and has been further developed and utilized by York and Stubbs(61) and Brown,(5) among many otherso This method of analysis was chosen for the following reasons: (a) It is possible to obtain drop velocity measurements which is essential if one is to carry out a study of the spray evaporation, In addition, the drop size distribution should be velocity weighted so that it will be representative of the temporal rather than the spatial distribution. (b) The analytical technique of counting drops on a photographic negative does not discriminate against small drop sizes (except perhaps for drops below a diameter of ten microns) and the sprays investigated contain a very large number of drops below fifty microns in diametero The camera arrangement for the high speed photographs is shown in Figure 3. Light illumination is supplied by two General Electric catalog number 9364688G photolightso These give an extremely intense

TIME DELAY PHOTOLIGHT CIRCUIT 50mm PHOTO- 3LIGHT JMR V DEPTH oA OF FIELD HALF- SILVERED MIRROR Figure 3. Camera Arrangement for High Speed Photographs.

0 +255 v. 270K 82K SPARK SPARK COIL LEADS MAY 1^rK BE REVERSED iLMFD SCREW DRVER A BY DP DT SWITCH ADJUST 100K POTS POS PULSE TRIGGER 3 E LMFD (* -.2050 _______ _ _ - _NEG PULSE %~ / —- - 1 MEG - |00' 00.1 1 o 30 60 150, 750 1500 rB VEG MEG RESET G RANGE iI -.51K v2.2K 2250 0lo 0.03 EXTC.00 0 i0 2, 2K OFF V T OLTS 10 J70 T 0^ * 270K 290K -150 v. W -150 v. 5K 5K I2j POTENTIOMETER STEPPED VERNIER Figure 4. Time Delay Unit.

5Y3GT 350-D-350 l g 4 2 _4^ o400-2 10K,1 W 100o I I [ \ I + —-- — 2 55O^ —T^ 1T^>+255v I 1-C I IVQ'='* -- __;120/450 20/450 25 020/4 50 10K,1W 100Q A/4 - 20/450 0/450 6.3 AC Figure 5. Power Supply for Time Delay Unit.

-22flash which has a duration of one or two microseconds. The lights are fired by means of the two time delay units shown in Figures 4 and 5. The firing source is a number 2050 thyraton and the time delay may be varied from about 5 to 1500 microseconds. The two delay units are wired in series with the first unit firing the first light and simultaneously supplying an impulse to the second unit which in turn fired the second light. The desired time interval between the two light flashes was set on the second delay unit. The time delay units were calibrated by photographing the moving teeth on a band saw, Since the linear velocity of the saw blade was known it was a very simple calculation to measure the actual time delay. In order to ensure that the delay circuits remained accurate at all times the time delay was measured for each double flash picture using a Hewlett Packard Model 524B counter (with a model 52CB time interval unit). Also checks were made using photocells to ensure that the response times of the lights were small compared to the time interval between receipt of the high voltage impulse and the flash. The two lights are placed at right angles to each other, with a half-silvered mirror positioned to provide silhouette illumination from each light. As explained above, if a double flash picture is required, they are discharged with a controlled and measured time interval between flashes. On the other hand only one light is triggered if a single flash picture is required. The latter are used to measure the drop size distributions while the double-flash pictures are used to determine the drop velocitieso Double-image negatives were never used to measure drop size distribution.

-23The camera used was fitted with a 50-mm, f 3.5 lens which had a magnification of 10 X. Each photograph provides an image of the spray in a finite volume, which is about 0.4 X 0o5 X 0.06 inches. The film used in all cases was Kodak Contrast Process Ortho. In the spray analyses, the nozzle is placed on a movable stand so that samples may be photographed at various spray locations. These locations are shown in Figure 6. 3. Dark Room Procedure In order to ensure results that are as reproducible as possible, the developing technique was very carefully controlled. The developing tanks were immersed in a bath of running water maintained at 680F. The films were developed for five minutes in Kodak D-ll developer with fairly continuous agitation, particularly during the first minute. After a 30 second rinse in water they were immersed in Kodak Acid Fix for ten minutes. Following another 30 second water rinse they were placed in a tank of Kodak Hypo Clearing agent for two minutes and then into a water bath for five minutes. All of the solutions were renewed in accordance with the manufacturers' recommendations. A Wratten series 2 red safelight filter was used to control illumination in the dark room. 4. Drop Analysis and Counting An analysis of a spray location consists of four single-flash and four double-exposure photographso Only the single-exposure photographs are employed to provide drop size distributions since the double exposure reduces the resolution of the smaller drops. The negatives are projected at an additional magnification of 10 X onto the ground glass

I INOZZLE 0.5 in. 0.5 ho^~ in.PHOTO- LIGHT *C^S^ 0.4 in. i NUMBERS REFER TO. SAMPLE LOCATIONS Figure 6. Sample Locations.

-25screen of an optical comparator, making a total image magnification of 100 Xo Since each photograph contains drops which are in sharp focus and drops which are blurred because of displacement just outside the sample volume, a judgment must be made of which drop images are to be considered as part of the sampleo To help overcome this difficulty a series of standard drop images has been developed to determine whether individual drop images on the negatives are to be countedo These standard drop images were prepared by Brown(5) and are fully discussed by himo A word should be said here about the determination of the drop velocitieso When measuring the velocity of a particular drop only the linear distance was determined without regard to its actual direction; ioeo, no correction was made if the drop was not moving vertically downwards. The drop count by sizes provides a spatial distribution of the spray, which can be multiplied by the velocities in each size range to obtain a temporal distributiono From this information the surface area flow. volume flow (which may be checked against measured flow rates) and various mean diameters were calculated. A computer program for use with the IBM 709 at the University Computing Center was used for the numerical calculations mentioned aboveo This program is described in Appendix Ao The question naturally arises, whether the average diameters are dependent upon the number of drops counted; ioeo how many drops must be counted in order to have a statistically meaningful sample? This problem was investigated by Brown(5) and he reached the conclusion that

-26spray analyses data of this type would be meaningful, in a statistical sense, if a minimum of 200 to 300 drops were countedo Nearly all of the data taken here reports drop counts well in excess of the minimum, The exceptions are for the outer periphery of the spray where the drop number density is too low to fulfill the above requirement with a reasonable number of photographso 5o Experimental Determination of Shatter Temperatures The term "shatter temperature" as used here may be defined in the following manner. When the superheated jet leaves the nozzle, its temperature will decrease because of vaporization of part of the liquid and because of convective heat transfer. After a vapor bubble is first formed it will continue to grow as long as the liquid temperature is above the equilibrium temperatureo There are then two possible cases: (1) The vapor bubble will not grow large enough to disrupt the jet and will undergo the so-called growth-collapse phenomenon described by Bankoff and Mikesello(2) (2) The vapor bubble will grow large enough to completely disrupt the jet. The lowest temperature at which this occurs is defined to be the shatter temperature of the jeto In order to measure experimentally the shatter temperature the following procedure was adoptedo The fluid to be used was loaded into the receiving tank and it was then ejected through the orifice at the desired pressure but initially at a temperature below the shatter

-27temperature. The temperature of the fluid was then slowly raised. The occurrence of the shatter temperature was marked by an abrupt change from a continuous liquid jet to a discontinuous liquid jet, i.eo, a spray. This change occurred over a range of about five degrees and for this reason the location of a shatter temperature is somewhat arbitrary. The shatter temperatures were all measured using the thermocouple located at the nozzleo In no case was it possible to observe the growthcollapse phenomenon, even though the temperature may be below the shatter temperatureo The entire procedure was repeated several times in order to get the best possible estimate of the true shatter temperatureo Great care was taken to raise the fluid temperature as slowly as possible to ensure that the thermocouple reading was as accurate as could be obtained.

CHAPTER IV DISCUSSION OF RAW DATA In this study three fluids, water, Freon 11 (trichloromonofluoromethane, CC13F) and Freon 113 (trichlorotrifluoroethane, CC12F-CC1F2) were used. The first two of these were used in the work on velocity profiles, drop size distributions and evaporation effects. All three were used in the investigation of the shatter temperature (this term is defined in a preceding section). The choice of fluids was, in the main, governed by the following considerations: (1) The fluids must not be flammable or present an explosion hazard. (2) The fluids must not be toxic. (3) They should exhibit as wide a range as possible of the significant fluid properties (surface tension, viscosity, density, etc.). The three fluids used are consistent with these requirements. The selection of the important variables a priori, is a very difficult problem and is best done by good hindsight. The relative roles of the variables will be brought forth in succeeding chapterso Let it suffice to say at this point that the fluids do have a large range of all of the important variables except for viscosity. The reason for this last statement is fully developed in Chapter Vo 1. Range of Variables Studied Table I summarizes the values of the physical properties of water, Freon 11 and Freon 115o It should be noted that in some cases the -28

-29TABLE I PHYSICAL PROPERTIES OF WATER, FREON-11, FREON-113 Property Water Freon-l1 Freon-113 Liquid Density (lb/ft3) 62 4 91o38 96.96 Viscosity ( 1)(200C) 1.00 - Viscosity (B. Pto) 0.21 0o405 0o619 Surface Tension (Dynes/cm) 7 7F) 19 (77F) 19 (77OF) Specific Heat (liquid)(Btu/lbtF) loOO 0.209 0o218 Thermal Conductivity (Btu ft/hr ft2 F) 0,394 0.0609 0.0521 Heat of Vaporization (Btu/lb) 970.5 78.o31 63 12 Vapor Density at Bo Pto (1b/ft3) 0o0373 05365 o.461 TABLE II EXPERIMENTAL NOZZLE DIAMETERS Nozzle Number Diameter, Inches 1 0.0124 2 o00577 3 0o0425 4 0,0247 5 oo0166 6 0,oi85 7 0 0322 8 0 0216 9 0 0168 10 o00662 11 0 o016 12 00510

-30values of the properties are given at the normal boiling point as well as at room temperature. Table II depicts the range of nozzle diameters used in the study of the shatter temperature. Only one nozzle diameter was used in the evaporation studies (number 12, 0.031 inches diameter). The nozzle diameters were determined using a Bausch and Lomb microscope equipped with a fylar eyepiece. 2o Size Distribution Data Table III shows a typical set of raw data which was used to calculate the size distributions and the various mean diameters. The data shown are for the water system injected at a temperature of 287~F and a pressure drop across the nozzle of 120 psi, The data were taken at a distance of four inches from the nozzle. The number of drops and their average velocities are given for the drops which lie within any given size range. The size ranges referred to are those given in Table XXI of Appendix B. The sample locations are those shown in Figure 6, All of the additional data of this type is included in Appendix B. 35 Shatter Temperature Data Table IV depicts all of the measurements of the shatter temperatures for the three fluids under various flow conditions and using the various nozzleso A part of the data is taken from Brown(5) and is repeated here in order to include it in the correlative and interpretative work, which is discussed in the next chapter. As can be seen from Table IV, the shatter temperatures were measured for as wide a variation of flow conditions as was possible using the existing experimental equipment.

TABLE III TYPICAL DROP SIZE AND VELOCITY DATA WATER SYSTEM AT A DISTANCE OF 4 INCHES FOR 120 PSIG INJECTION Location Photos Number of Drops in Each Size Range 1 2 5 4 5 6 7 8 9 10 11 12 1 2 5 52 125 195 92 28 22 14 2 2 2 12 78 251 256 100 46 24 20 11 2 5 2 44 258 504 182 155 55 22 7 1 71~~~~~~~~~~~~~~~~~~~~~ 4 4 5 54 71 70 96 105 22 6 Average Velocities in Each Size Range (Ft/Sec) 1 4 321.6 1.8 59.9 41.6 97.5 117.5 127 155 2 4 15.9 18.3 33.4 8,.4 48.3 85.2 lo6 110 125 125 5 4 7.44 13.4 18.3 22.0 36.6 55.8 85.7 95.9 4 4 4.35 7.17 11.5 16.9 27.5 47.5 64.8

-32TABLE IV MEASURED SHATTER TEMPERATURES Run Fluid Measured Nozzle Pressure Dif- Shatter TemDiameter (inches) ference (Psi) perature ~F 1 F-ll.161 60 158 2 F-ll.0161 90 153 3 F-ll.0425 78 108 4 F-ll.0425 45 110 5 F-ll.0247 60 152 6 F-ll.0247 89 147 7 F-ll.0247 40 155 8 F-ll.0662 48 85 9 F-ll.0577 45 93 10 F-ll.0310 80 141 11 F-ll.0510 120 1354 12 F-ll.0250 94 152 13 F-ll.0577 78 90 14 F-ll.0124 74 164 15 F-ll.0577 62 92 16 F-ll.0577 35 105 17 F-113.0124 55 150 18 F-113.0124 75 145 19 F-113.0247 79 138 20 F-113.0247 46 144 21 F-113.0322 58 133 22 F-113.0322 38 138 25 F-113.022 80 1355 24 F-113.0425 80 125 25 F-113.0425 60 132 26 F-113.0577 55 123 27 F-113.0577 80 120 28 F-113.0577 70 122 29 F-113.0667 58 120 30 F-113.0667 80 119 31 W.0247 120 280 32 W.0247 131 280 33 W.0322 100 272 34 W.0322 120 268 35 W.0322 130266 36 W.0662 80 237 37 W.0662 120 215 38 W.0310 120 273 39 W.0310 93 272 40 W.0310 134 270 41 w.0550 84 268 42 w.0350 128 237 43 W.0557 60 270 44 W.0557 80 258 45 W.0557 96 235 46 w.0557 120 223 Runs 31 to 46 are from R. Browns data and were experimentally verified in this study.

CHAPTER V THE BREAKUP MECHANISM In Chapter II a brief review was given of the theory of the disintegration of liquid jets and it was pointed out that Ohnesorge has presented a convenient graphical method of summarizing the location of the various breakup regimes. Figure 1 depicts this relationship expressed as the Ohnesorge number as a function of the Reynolds number. The approximate positions of the injection conditions for the nozzles used in this study are shown on this figureo Most of the locations corresponding to the injection conditions are seen to be either in region II or on the"boundary" between regions II and III. In other words, the jets will not disintegrate at or very near to the orifice. If disintegration does occur at all, it will result from wavelike breakup at some fairly large distance away from the nozzle. This was borne out experimentally in all cases as none of the jets did disintegrate without the addition of some superheat, but the amount of heat required is strongly dependent upon the nozzle diameter, One must therefore search for a different mechanism of disintegrationo 1o The Breakup Mechanism For the range of nozzle diameters and flow rates studied the controlling mechanism of breakup is bubble growth resulting from the superheat of the liquid, or in other words, "flashing" of the superheated liquid jeto Thermodynamically, flashing occurs when the liquid is at a temperature above the saturation temperature corresponding to the -~55'

pressure of the surroundings. Under truly adiabatic conditions all of the vapor formed will receive its heat by conduction through the liquid, at the expense of the enthalpy of the unvaporized phase. Equilibrium will be reached when the residual liquid phase has cooled to its saturation temperature. Bubble growth can also result from vaporization of a dissolved gas, but it has been shown that this means of breakup is not as effective as superheat.(5) When the superheated liquid jet leaves the nozzle, its temperature will decrease because of convective heat transfer and vaporization of part of the liquid. The heat loss by convection is small relative to that due to vaporizationo The vaporization process generates two types of bubbles: surface bubbles which are easily observed and bubbles which grow in the interior of the liquid jet and which are not observable. Figures 7, 8 and 9 are photographs of a Freon 11 jet taken just below its shatter temperature. The jet diameter is.031 inches and the injection pressure is 90 psigo The temperatures are 135, 138 and 140~F, respectively. It is easy to see that there are a large number of bubbles present on the jet surface and that the number density of these bubbles increases as the temperature increases. These bubbles do not play a major role in the breakup of the liquid jet. Despite their high frequency of occurrence, their only role seems to be to tear off small ligaments of liquid from its surface, by means of small explosions. These ligaments then form a fine spray or mist which surrounds the remaining, intact portion of the liquid jet. This mist may also be seen on the three photographs. The vapor bubbles appear on the surface (at

Figure 7. Flashing Jet (10X), 135~F.

-56Figure 8.Flashing Jet (lox), 158~F.

-37Figure Flashing Jet (10X), 140N OF.

least only visually) over a very small temperature range. This temperature range usually extends from the shatter temperature downwards for five or so degreeso From observation of the surface bubbles it appears most likely that their growth is initiated by some micro disturbance on the jet surface, perhaps due either to roughness or vibration in the nozzle orifice. This supposition is supported in part by the observations of Brown(5) who reported that their occurrence was much less frequent for a sharp edged orifice than for any other type, all other conditions being equal. These bubbles were observed for all three fluids, water, Freon 11 and Freon 113. The bubbles which are nucleated within the liquid jet and which grow to the required size are those which cause the disintegration of the liquid jet. Considering only these interior bubbles, after one is first formed it will continue to grow as long as the liquid temperature is above the equilibrium temperatureo As previously discussed there are two possible cases, and the minimum temperature necessary to shatter the jet is the variable used to define the disintegration point. When the fluid is sprayed into the atmosphere with an initial temperature which is above the shatter temperature, vapor bubbles which are nucleated will grow to a sufficient size to "explode" the liquid jet and form a very fine cloud of liquid droplets. 2. Effect of Fluid Properties on the Breakup Mechanism If one were to write down, a priori, the various fluid properties which affect this breakup mechanism, the following would certainly

-539be included —interfacial tension, liquid density, liquid thermal conductivity, vapor density, latent heat of vaporization, degree of superheat (this can be measured in many ways), liquid viscosity and specific heat of the liquid. The most useful manner in which to illustrate the relative role of each of these fluid properties would be to formulate and solve the appropriate differential equation which describes the growth of a vapor bubble in a superheated cylindrical jet. To do this at the present state of knowledge of this problem is exceedingly difficulto Much useful information may be gained from a study of an associated problem —the growth of a vapor bubble in a semi-infinite liquid which has a constant temperature. It is obvious that this immediately rids us of two difficulties: (1) the non-uniformity of the temperature field (2) the necessity of having a knowledge of the bubble spacing in the liquid jeto In recent years a great deal of work has been published in the literature on the growth of vapor bubbles in a superheated liquido Theoretical studies have been carried out by Plesset and Zwick,(40) Forster and Zuber,(l5) Griffith,(l9) Poritsky(41) and Shu (51) Plesset and Zwick and Forster and Zuber have attacked the problem in essentially the same manner, although the mathematical details are differento They both began with the Rayleigh equation for the motion of a bubble in a non viscous, incompressible liquidS d2R 3 I

-4owhere R is the bubble radius t is the time ~AP is the pressure difference between the bubble cavity and the surroundings at an "infinite" distance p is the density of the liquid and extended it to include the effects of surface tension R z ( P-2 B ) (5.2) where a is the surface tension. In order to solve Equation (5.2) the pressure difference must be related to the temperature difference. This may be done by assuming that the temperature at the expanding vapor surface is the same as the temperature within the bubble cavity. Both of the authors conclude that there are two distinct regions of bubble growth. In the first of these the bubble radius is of the same order of magnitude as the initial radius and the growth is quite rapid. In the second stage (asymptotic stage) the surface tension and dynamic effects become less important and the growth rate is controlled by the rate of heat conduction from the bulk of the liquid to the bubble wall. They concluded that this growth rate is given by R = Ro + Ct (

-41where Ro is the bubble radius at the beginning of the second stage C is a proportionality constant, The proportionality constant C in Equation (5~3) is known as the growth rate constant and contains most of the important fluid properties c F where C (54) where AT is the superheat Hfg is the latent heat of vaporization at the saturation temperature k is the thermal conductivity Cp is the specific heat and the subscripts refer to liquid or vaporo It will be noted that the first term in brackets in the above equation is equal to the weight percent flashingo This solution assumes a perfect fluid and negligible compressibility effects, as is reflected by the terms which appear in the bubble growth rate constant. Griffith formulated a mathematical model for the growth of vapor bubbles on a heated surface, assuming hemispherical bubbles. His approach was to investigate the problem of conductive heat transfer from the liquid to the growing bubbleo Assuming a laminar flow field and

-42constant fluid properties surrounding the bubble, the equation for the heat transfer process is V?-__ f c, C /T, r- (5r ) VT ( + v ) where V is the velocity vector, Griffith solved the problem numerically with the result that R='t" where k' is a constant a is a number between 1/3 and 1/4. He also concluded that the average growth rate of a bubble decreased with increasing maximum size and decreased with increasing pressure. At high pressures the maximum size of the bubble was found to be independent of pressure and primarily a function of the thickness of the superheated layer near the surface. This latter is in agreement with Forster and Zuber who state that the thickness of the thermal boundary layer is a very important considerationo Poritsky has approached the problem in a much different manner, and for demonstrating the role of the fluid properties, a more useful oneo Beginning with the Navier-Stokes equations he derived the equation below for the growth of a vapor bubble in a superheated viscous liquid. n dt6 * 2tdt/' PR ( R II

-43where. is the liquid viscosity. It is of interest to note in passing, that with a suitable choice of the pressure difference term in Equation (5.6), that it may be also used to describe the collapse of a vapor bubble (i.e., cavitation). Equation (5.6) may be expressed in the equivalent form (Po -fRo 3 R3) - (R - R14) ~XR ) + ~ R/ R St - (5.7) where Po - P = P and the subscript "o" refers to the initial conditions. By defining the dimensionless variables e(- F. (PPR) C = 4P__O R,. (P,.o - P, )

Equation (5.7) becomes i2 -Dt @2-1) d = (5.8) The "C'" in Equation (5.8) is not to-be confused with the C previously defined to be the bubble growth rate constant. Equation (5.8) Was solved by the method of isoclines for the three fluids, water, Freon 11 and Freon 113. In order to do so a value of (Po - Pa) must be assigned. This was done by using the arithmetic average between the shatter temperature and the saturation temperature for each of the three fluids. For illustrative purposes a nozzle diameter of 0.031 inches and an injection pressure of 120 psig was choseno This corresponds to the temperatures given in Table V. The mathematical details leading up to Equation (5.8), and for the solution by the method of isoclines is given in Appendix C. TABLE V TEMPERATU ESFOR USE WITH EQUATION (5.8) Fluid Shatter Temp. Saturation Temp. Assumed* Temp. Water 270 210 240 Freon 11 134 72 103 Freon 113 134 116 125 * These temperatures were used in Equation (5.8) to arrive at the curves in Figure 10.

-45Figure 10 shows the results of the calculations. This is a plot of dimensionless time (T) versus dimensionless radius (P) for the assumed conditions in Table V. Figure 11 shows the conversion of this plot to a more understandable one of bubble radius versus time. From this it will be observed that the vapor bubble grows most rapidly in the Freon 113 and least rapidly in the Freon 11 fluids at the temperature which is assumed to describe its growth throughout its lifetime. The curves on Figure 10 correspond to the values of C and D given in Table VI below. TABLE VI VALUES OF DIMENSIONLESS CONSTANTS IN EQUATION (5.8) FOR FIGURE 10 Fluid C D Water.0445.201 Freon 11.140.1345 Freon 113.0661.0728 Changes in the constants C and D reflect a change of viscosity and surface tension as well as other fluid properties. A value of C = 0 represents an inviscid fluid and a value of D = 0 a fluid with negligible surface tension. From a study of the properties of the solutions to Equation (5.8) Poritsky has shown that an increase in D merely decreases the initial growth rate but after a short period of time the bubble growth rate becomes equal to that of a fluid having the same value of C, but a lower value of D. An increase in C causes an abrupt decrease in the rate of growth of the vapor bubble. Poritsky has presented

-4617 14 1 /3 12.D 12 L R U, F /10 0 8 z w 7 o6 4 A I 2.3 4 5 6 7 8 9 10 II 12 13 14 15 16 DIMENSIONLESS RADIUS,j Figure 10. Plot of Dimensionless Time versus Dimensionless Radius for Water, Freon 11 and EFreon 113.

.04 PREDICTED BY EQUATION 5.3.03.02 C/).01 --------.01 - 0 20 40 60 80 100 120 140 160 180 200 220 240 TIME (MICROSECONDS) Figure 11. Growth Rate Curves for Water, Freon 11 and Freon 115.

_48curves which show this effecto The effect of the fluid properties may be summarized as follows: (1) An increase in viscosity will cause a marked decrease in the bubble growth rateo (2) An increase in fluid density will cause a slight increase in the bubble growth rateo (3) An increase in surface tension will cause a slight decrease in the growth rate but at large bubble diameters (say.05 or more inches), this effect will not be very noticeable, (4) An increase in the vapor pressure difference term will result in an increase in the growth rateo This effect will be partially offset by a decrease in the minimum bubble radius. It is interesting to determine the agreement between Equations (5.8) and (5o3) and between Equation (5o8) and the experimental data of Brown(5) for the growth of vapor bubbles on the surface of a superheated liquid jet. This comparison is shown in Figure 11 and it will be seen that the deviation is not too large considering the assumptions involved in determining the initial radius. 35 Effect of Physical Variables on the Breakup Mechanism The term physical variable is used here to represent any variable which may be changed by experiment, e.og, pressure drop, diameter of the nozzle etc., as opposed to the other variables such as surface tension which were called fluid properties, These latter, of course, may not be altered at will.

-49As would be expected from the introductory material of Chapter II,, the physical variables must also play a role in the shatter of the jet, albeit a sometimes passive roleo This is clearly demonstrated from the data of Table IVo Consider for example runs number 5, 6 and 7o These three runs are all at the same nozzle diameter and using Freon 11l The only variable present is the pressure drop across the orificeo A decrease in the pressure in all cases causes an increase in the shatter temperatureo The magnitude of the change of temperature is dependent upon the fluid and upon the jet diameter, but in general a lowering of the pressure by 60 psi results in an increase in shatter temperature of from five to ten degreeso To phrase it in a different manner, at the same bubble growth rate one liquid jet will be disintegrated while the same jet, but at a lower pressure, will not be brokeno From the data of the same table it will also be seen that at any given pressure level the shatter temperature decreases rapidly with increasing nozzle diametero Both of these effects can be predicted in a semiquantitative way, using jet stability theoryo Consider for example Figure lo At one given diameter as the pressure drop decreases the Reynolds number also decreases, tending to move the location of a point horizontally to the lefto Also as diameter increases for any given value of the pressure levelJ, the Reynolds number increases directly, tending to move a point on Figure 1 downward and to the righto We may think of any action which causes a point to move towards the boundary between regions II and III as reducing the jet stability and bringing it nearer to immediate disintegrat ion. Such a movement should require less energy to be imparted to the jet by vapor

-50evolution in order to disintegrate ito Since this latter energy may be characterized by the shatter temperature, the effects described above are precisely what would be expected. Attention should also be drawn to runs 3 and 24, of Table IVo These two runs are for the same nozzle diameter and at essentially the same injection pressure, but the two fluids are differento The shatter temperature for Freon 11 is 108~F and for Freon 113 is 125~Fo These two temperatures represent superheats of 36 and 8~F, respectively, and a weight per cent flashing of 14o5 and 3075 per cent. Also the temperature required to shatter the smallest jet represents much less superheat for Freon 113 than it does for Freon 11, This occurs in spite of the fact that the important fluid properties of surface tension, liquid and vapor density, latent heat of vaporization, thermal conductivity and viscosity are not too much different from each other, This large difference in the weight per cent flashing is attributable to the fact that the vapor bubbles grow more rapidly in Freon 1153 than in Freon 11 at the shattering conditions, as is illustrated by Figure 11, and to the effect of temperature on the liquid viscosityo This further serves to point out the interplay between all of the fluid properties in affecting the disintegration of the jet by flashing. One might well ask at this point what effect if any does the level of turbulence have upon the shatter temperature (or upon the bubble growth rate since the two are interrelated)? It will be recalled that the solution to the bubble growth rate problem assumes a laminar flow field immediately surrounding the expanding bubble wallo If this

-51assumption is not valid then turbulence should be accounted for in some manner; ioeo, the exclusive consideration of the heat conduction problem is invalido It can be shown that the characteristic diffusion length is about 5 to 10 per cent of the film thickness for heat transfer and thus the thickness of the layer surrounding the bubble in which the major temperature drop occurs is much less than the thickness of the laminar boundary layero As a consequence the effect of turbulence upon the bubble growth rate should be negligible for the cases studiedo 40 Shatter Temperature Correlation A correlation of the shatter temperatures was developed, using the approach of dimensional analysis Assuming that the shatter termperature? designated by T, is a definitive characteristic of the jet breakup, a relationship may be sought of the form:?(TDV P, P,, P., ) HFI,-) = 0 (5.9) where Hfg is the latent heat, and, f(TkDjVHAP~p9,jo>kHjg) represents an unknown function to be determined Examination of Bernoulli's equation for flow through a nozzle shows that V and AP are not independent and only one of them need be considered. Eliminating the variable V and applying the standard method of dimensional analysis and assuming a power relationship yields the equation -. - a-(N we) z) (510o)

-52 - The constants a,bc were determined by regression for the three fluids individually and for the three fluids combined. For the individual cases Equation (5.10) becomes, for water, T*A / o-560 0o.I -3.5 I = 0.705 (Nwe) (Z) (5011) for Freon 11, -0 21 -_0,608 T = 26 3( e) (Z) (512) and for Freon 115 - o -.875' o o 8399 TA 0 914 Nwe) Z) (5.13) The over-all correlation coefficients for the three equations are 089, 0o84 and 0o98, respectively. The values of the dimensionless groups are given in Table VIIo The exponent on the Weber number in the above equations is negative for all three fluids which is to be expected from the remarks previously made about the stability theory of liquid jets. The exponent for the Ohnesorge number is also negative, except for Freon 113 and for this latter case the value is very near to zero. As a part of the computer program used to determine the best equations (in a statistical sense), the correlation coefficients between the Weber and Ohnesorge numbers were calculated. These are 021, 073

-53TABLE VII CALCULATED VALUES OF DIMENSIONLESS GROUPS FOR BREAKUP DATA Run Fluid Tk NWe Bubble Growth Tk Calculated Eq. V MLHfg (x 10-3) Rate Constant p5 Hf PL (Ft/Hrl2) (5.1l, 12 or 13) x o10-5 (Ft/Hrl/), 1 F-ll.648 5.49 2.96 6.35.674 15.8 2 F-ll.637 8.16 2.99 5.95.659 14.75 3 F-ll.483 17.95 2.18 2.75.514 7.23 4 F-ll.503 10.42 2.15 2.90.530 7.47 5 F-ll.633 8.43 2.42 5.90.578 14.62 6 F-ll.621 12.3 2.45 5.55.567 13.5 7 F-ll.643 5.54 2.40 6.10.593 15.2 8 F-ll.429 16.8 1.89 1.10.473 4.93 9 F-ll.451 13.85 1.95 1.70.489 5.53 10 F-ll.601 13.81 2.23 5.10.531 12.37 11 F-ll.577 20.6 2.27 4.6o.521 11.05 12 F-ll.637 13.3 2.41 5.90.558 14.6 13 F-ll.443 27.6 1.971 1.50.468 5.25 14 F-ll.656 5.23 3.48 7.15.746 17.38 15 F-ll.449 19.0 1.985 1.60.484 5.44 16 F-ll.487 10.85 1.855 2.50.483 6.86 17 F-113.496 4.07 4.95 2.50.501 8.60 18 F-113.480 551 5.15 2.17.490 7.65 19 F-113.456 11.55 3.78 1.70.447 7.02 20 F-113.477 6.77 3.65 2.10.466 7.77 21 F-113.442 11.02 3.39 1.38.444 6.46 22 F-113.456 7.25 3.32 1.70.460 7.02 23 F-113.448 13.91 3.35 1.50.435 6.68 24 F-113.415 19.91 3.08 1.30.418 5.60 25 F-113.434 14.95 2.96 1.60.427 6.32 26 F-113.408 18.50 2.65 o.65.415 5.40 27 F-113.398 26.7 2.74 0.42.403 5.11 28 F-113.411 23.4 2.68 0.58.407 5.31 29 F-113.398 22.5 2.53 0.42.406 5.11 30 F-113.396 30.9 2.54 0.30.395 5.03 31 W.736 5.63 0.903 16.3.638 2.06 32 W.729 6.14 0.915 15.9.649 1.995 33 W.665 4.78 0.931 14.0.746 1.760 34 w.641 7.17 0.899 13.1.596 1.655 35 W.627 7.75 0.915 12.6.569 1.60 36 W.483 9.71 0.793 5.8.511 0.983 37 W.404 14.4 0.925 0.7.401 o.684 38 W.673 6.94 0.877 14.3.609 1.792 39 W.667 5.38 0.883 14.0..703 1.76 40 W.654 7.75 0.899 13.6.571 1.705 41 W.642 5.47 0.860 13.1.698 1.655 42 W.483 8.25 1.08 5.8.537 0.940 43 W.654 5.92 0.687 13.6.688 1.705 44 W.579 7.85 0.760 10.8.579 1.402 45 W.473 9.34 0.899 5.4.514 0.949 46 W.427 11.6 0.973 2.5.449 0.767

and 0.91 for water, Freon 11 and Freon 1139 respectively. These values indicate that almost as good a correlation could be obtained for Freon 113 using either group alone, but for water and Freon 11 no single group correlation will exist. For the particular case of water this has been demonstrated by Brown(5) who has shown that no linear correlation exists using the Weber number alone, When Equation (5o10) was tested using all of the experimental data, the over-all correlation coefficient fell to Oo65, indicating that the relationship does not adequately describe all of the data together. This implies that a variable necessary to interrelate the fluids was not present in Equation (5o10)o An interesting fact which arose while trying to correlate all of the data using the two-group model was that the equation would not accurately describe any two of the data sets taken together except for the Freon pairo Inclusion of a vapor density ratio in Equation (5o10) results in OL Nwe (5.14) The constants a,b,c and d were determined by regression on the IBM 709 computer. Equation (5.14), then becomes A o l cn-o4 te.31 -orh ao.. A ahe ov r-a 0344Ne)i c Z )n f)o139 (5.15) The over-all correlation coefficient for the above model was Oo86o At

-55" a 95 per cent confidence level the exponents are given by: b = -0o231 + 0.03 c = -0,272 + 0.054 d = 0.139 + 0.02 The correlation coefficients between the groups were 0,070, 0.26 for 0.079, substantiating the above remarks that a correlation for all of the data of the form of Equation (5.10) does not exist. The results indicated by Equation (5.15) are shown graphically in Figure 12 and in a tabular form in Table VIIIo The signs of the exponents of Equation (5.15) are worthy of some comment. The negative sign on the exponent of the Weber number is entirely consistent with the stability theory of liquid jets since at increasing values of the Weber number, the jet tends to break up under the action of inertial and surface tension forces alone, without the addition of superheat. The negative sign on the exponent of the Ohnesorge number is very interesting, since either its value or magnitude would be very difficult to predict. The Ohnesorge number contains four very important variables, viscosity, diameter and surface tension and density. At a constant level of Weber number and density ratio The negative sign then indicates the very strong importance of the diameter as a variable controlling jet breakup, since based on surface tension and density alone the sign would have been positive.

0.86 _ _ _ __ _ _ _ _ _ _ KEY 0 WATER A F-II 0.76: 0.5 0 0O c 0.3 I ULU 2 0.2 I L) 0.3 3: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.344 (Nwe)-0'23' (Z)-O'172 (Pv/PL) 0.39 Figure 12. Shatter Temperature Correlation.

-57TABLE VIII COMPARISON OF CALCULATED AND EXPERIMENTAL VALUES OF SHATTER TEMPERATURE GROUP Run Tk Tk RT. Exp. Tk Calc. (Eq. 5.15) Pjrfg ^PHfg 1 0.648 0.613 2 0.637 0.551 3 0.483 0.456 4 0.503 0.522 5 0.633 0.589 6 0.621 0.530 7 0.643 0.655 8 0.429 o.616 9 0.451 0.458 10 o.601 0.528 11 0.577 0.470 12 0.637 0.532 13 o.443 o.405 14 0.656 0.594 15 0.449 0.443 16 0.487 0.537 17 o.496 0.541 18 0.480 0.489 19 o0.456 0.454 20 0,477 0.528 21 0.442 0.470 22 o.456 0.530 23 0.448 o.450 24 0.415 0.414 25 0.434 0.458 26 o.4o8 0.442 27 0.398 0.397 28 0.411 o.415 29 0.398 0.424 30 0.396 0.394 31 0.736 0,660 32 0.729 0.640 33 0.665 o0.66 34 0.641 0.602 35 0.627 0.584 36 0.483 0.537 37 0.404 o.436 38 0.673 0.620 39 0.667 0.654 40 o.654 0.594 41 0.642 0.651 42 0.483 0.495 43 0.654 0.696 44 0.579 0.608 45 0.473 0.515 46 0.427 0.459

-585o Summary From the discussion of the previous sections of this chapter we may conclude that the shatter temperature is definitive of the breakup of a superheated liquid jet and that this shatter temperature is unique for any given set of physical and fluid variables. The relationship expressed by Equation (5.15)9 which relates the shatter temperature to the variables of the system can be used to calculate the breakup temperature of other superheated liquid jets. This relationship would perhaps not be completely accurate at very high liquid viscosities or in cases where the receiving pressure is very far removed from atmospheric. High liquid viscosities could occur if liquids containing impurities (slurries) or artificial thickeners were injectedo High viscosities can not result from the use of other liquids and hence Equation (5015) may be considered to apply to all pure liquids.

CHAPTER VI SPRAY EVAPORATION lo Theory The problem of calculating or predicting the evaporation rate of liquid droplets contained in sprays has received fairly wide attention in recent yearso To gain an appreciation of the factors involved in the problem, let us first consider the simplest possible case —a spherical drop of a pure liquid of diameter D and temperature T in still airo For this case the total diffusion is given by N = 2r D7 t 61- (6o1) R T where N is the total diffusion rate D is the drop diameter Dt is the diffusivity T is the drop temperature R is the gas constant AP is the vapor pressure at T minus the pressure of the diffusing liquid at an infinite distanceo Froessling(l6) has studied this problem theoretically and has obtained some interesting and useful results, particularly in the limiting case of zero Reynolds number; ioeo no relative velocityo This case is of practical use in representing a finely dispersed spray system. =59

-60He was able to predict the relationship: he \ -It I r\/lD P~ (6.2) hcD D g 2 0 (602)?Dt9 where h is the convection heat transfer coefficient c kf is the thermal conductivity of the gas film surrounding the evaporating drop kg is the mass transfer coefficient M is the mean molecular weight of the gas vapor mixture in the boundary layer surrounding the drop Pf is the average value of the vapor pressure of the non diffusing gaseous component surrounding the drop p is the density of the liquid For the more general case of forced convection, many investigators have validated the semi-empirical expressions: D 20 + 0.60 (NPr) (iNiR) (6o3) ^$. N ~qF~~~~~~~~~~~/ * - 2,0 + O.6O(N )J (NSCe) (6N) Dt - (6 4) where Npr is the Prandtl number NSc is the Schmidt number NRe is the Reynolds number

-61An expression equivalent to (6.1) is d~ -- 2 D D p _ p \) (6.5) in which p/Pf has been substituted for RT in (6,1). In the use of the above equation Ranz and Marshall(45) have shown experimentally that the drop temperature is essentially that of the temperature of the surface of the drop and that this surface temperature is the wet bulb temperature for the humidity conditions involved. They also developed and discussed the significance of Equations (6,3) and (6.4), They argued as follows: In still air the drop evaporates uniformly from all portions of its surface. When the drop is in a moving gas stream, this symmetry must be destroyed except at very low relative air velocities. The equations account for this loss of symmetry, resulting from the changed pattern of air flow around the drop, by modifying the coefficients for evaporation in still air. Although the same theoretical considerations apply to clouds of drops as to a single drop, the problem is complicated by additional factors. One of these is that the drops may be dispersed in a turbulent gas stream. Liu(27) and Soo(54) from a study of this aspect of the problem concluded. that the eddy diffusivity of the particles and of the gas are almost equal for small particles and at low intensities of turbulence. This means that small drops, in a turbulent gas stream, should evaporate at a rate corresponding to the rate which occurs at zero relative velocity. Kessler(23) studied a similar problem and showed experimentally that for alcohol drops in the 14 to 30 micron size range that the

-62evaporation rate was that predicted for stagnant conditions. Mirsky(35) in a recent study arrived at essentially-the same conclusion —viz. that small drops evaporate as though stagnant conditions prevail. A factor which must be considered in this study is that nearly all of the data were taken in a region in which the great majority of the liquid drops were decelerating. A few studies have been carried out in which the effects of acceleration or deceleration on the rate of evaporation were studied. Manning and Gauvin(31) in a series of interesting experiments demonstrated that the correlations of Ranz and Marshall accurately describe their measured heat transfer (or mass transfer) coefficients, in the zone of deceleration. The drop sizes, velocities and deceleration rates correspond roughly to those encountered in this investigation. Crowe(7) in his recent PhoDo thesis was able to show on a theoretical basis that deceleration will affect the evaporation rates only at magnitudes of deceleration much above those encountered here, On the basis of this theoretical and experimental evidence Equations (6.3) or (6.4) may be assumed to give a reasonable description of the evaporative process of the spray dropletso Rewriting Equation (605) in a different form leads to: dw _ -c A(At) (6.6) de hs where A is the area for heat transfer At = ta - ts where ta and ts are the air and surface temperatures respectively Xs is the latent heat of vaporization corresponding to ts G is the time

-63If the two relations for he at either zero or finite relative velocity are substituted into (606), there results~ dw 2, A (At) (6~7) de - cD As for zero relative velocity, and deW A(__tt ( 2 o + 0.60(Ne ) (6p8) d e --?s E) for finite relative velocityo Equation (6o7) may be integrated directly to give -8A, t,, ( - 8) (6 9) where DO is the initial drop diameter at some arbitrary time zero D is the drop diameter at any other time 9 Rearranging Equation (609) D - (6.10) where e s For the case of finite relative velocity, substitution and subsequent integration yields Do - _ At s f ( d0 1 4e -t *r Oi3(N, e)2 (iNpf<'3 (6.11)

-64Integration of Equation (6o11) can be carried out by a stepwise process having a knowledge of a relationship between the drop diameter and drop velocity. However it has been demonstrated(32) that for drops below 100 microns in diameter the effect of air velocity on the rate of evaporation is negligible and the use of Equation (6,10) is justifiedo For drops larger than this size, Equation (6o11) should be utilizedo For the particular case under investigation all of the drops of 100 microns and above are traveling at velocities of the order of 50 to 100 feet per second. As a result they require a time of about 10-3 seconds to traverse the distance from four to seven inches away from the nozzle. Consequently, their contribution to the rate of evaporation is small enough so that Equation (6o10) can be used for all drop sizes, without introducing a serious error, 2, Prediction of Drop Size Using a Measured Initial Size Distribution The experimental data afford an excellent opportunity to study evaporation rates of sprays formed by the flashing process. The evaporation of the spray is reflected in the variation of the average drop diameter as a function of location in the spray and distance from the nozzleo This variation is tabulated in Table XVII, Chapter VIII and is shown graphically in Figures 15 through 22~ It will be observed that there is no simple relationship between average diameter and distance from the nozzleo In estimating the evaporation times and/or rates, it is necessary to be able to specify or estimate the droplet temperature in order to evaluate the term Atmo One manner in which to do this is to measure

the humidity and temperature of the air in the spray and from this information determine the wet bulb temperatureo This requires fairly elaborate experimental equipment which was not availableo An estimated value of Atm of 20OF was used in these calculations for the evaporation of the water sprayso This value was arrived at from two considerations (1) An ordinary mercury thermometer was placed in the spray and the temperature difference between it and the air temperature was very near 20'Fo This is admittedly not an extremely accurate determination, but it will give a very good estimateo (2) The air temperature is of the order of 80~F and the droplet temperature cannot be below about 400F3 as an extreme; this gives the maximum attainable Atm as about 40 Fo Fortunately, it was discovered from the calculations that the choice of either 20 or 40~F for Ltm made very little difference to the final answero Having estimated Atm as 20OF then eK _ 8 ^f Z-tM, AS becomes equal to 1000 microns2/sec. for water. The calculations were then carried out in the manner described below. To illustrate these calculationsi location 1 (center line) has been chosen as a specific example. The drop size distribution at a distance of four inches from the nozzle was assumed to be correct and using this as a starting point,

-66the drop number distributions at distances of five, six and seven inches were calculated. Time zero was assigned to the distance of four inches and. the timesfor the drops to reach a distance of five, six and seven inches were calculated using the known velocity data. Figures 13 and 14 depict the drop velocity profiles for this locationo The size ranges referred to are those tabulated in Table XXI of Appendix B, Table IX summarizes the results of these calculations, both for this particular example and for all of the other spray locations. It now is a fairly easy matter to utilize Equation (6o10) to determine the new mean diameters at distances of five, six and seven inches0 In carrying out the calculation it was assumed that the average diameter of any size range was representative of all the drops in the size intervalo This, or some similar assumption, is necessary, because of the method used in taking the raw datao When the drops were counted no indication was made as to where they fit into the size interval; ioe,, whether they are near to one extreme or the other or near the average diametero On this basis the data given in Table X was calculated. Examination of Table X illustrates vividly the well known fact that smaller diameter drops evaporate much more rapidly than larger diameter dropso Using the information in Table X and the known number distribution at a distance of four inches., the drop size distributions were then calculated at the three other distances from. the nozzleO From this the new average drop diameters were calculatedo Figure 15 shows a comparison between these "calculated " average diameters and the experimentally observed valueso Figures 16, 17 and 18 show similar comparisons for the other three spray locations

-67SIZE RANGE 6 45 DA 48.2 MICRONS 40 35 30 25 SIZE RANGE 5 35 DA= 34.1 MICRONS 30 ~ ~ ~~~~~~~~~~ C) 25 I 20 —---— 1 u,.J 30 > N DAV= 24. I1 MICRONS 25 20 15 -- 10 0 SIZE RANGE 3 25 DAV 17.05 MICRONS 20 A SIZE RANGE 2 15 DAV= 12.05 MICRONS 10 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 13. Velocity Profiles for Water System Location 1, Injection Temperature 287~F, 120 Psig.

-68115 11___I__________D0 -1SIZE RANGE 10 DAV =193 MICRONS 105 I00 95 90 105 ^~100I^~~~~ \SIZE RANGE 9 i00 DA= 136.5 MICRONS 95 70 90 - 695 ------ o, U.. 85 ——.... —: ___I - 90 w > I SIZE RANGE 8 85 DS =96.5 MICRONS 80 75 70 65 55 4.0 5.0perature 6.7~F, 120 7.0g. SIZE RANGE 7 50 DAv =68.2 MICRONS 45 ---- 40 35 30 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 14. Velocity Profiles for Water System, Location 1, Injection Temperature 287~F, 120 Psig.

-69TABLE IX TRAVEL TIMES FOR SPRAY DROPLETS (Times Seconds x 10"3) Run Loc, Size Range 2 3 4 5 6 7 8 9 10 2 1 00000 0o000 0o000 0000 0o000 0o000 0 o000 0o000 0o000 5 1 12.20 8.450 3 990 3 550 2.800 2.210 1o190 1 o020 0.920 4 1 2.73 17~07 8.52 7,74 5.95 4.67 2.54 2.12 1o90 5 1 52.2 41.26 1354 12.6 9.40 8.29 4.02 33.1 2.94 2 2 000 0o00 00o0 0.o00 0.00 0.00 0,00 0,00 0,00 3 2 - 10o2 618 3064 2,91 2,30 4 2 - 27o6 l38 9.25 6,42 5o00 5 2 48.7 221 14o7 10,7 8o00 2 3 0,00 OoOO OoOO OoOO OoOO OoOO 3 3 17o01 7.28 4,67 4.29 4 3 42,6 17.47 10o8 9.53 5 3 - 81o8 29,47 17.7 o5o6 2 4 0,00 0OO Oo00.OoOO OoOO 0.00 3 4 - 11o54 9.70 9.17 7o41 4,67 4 4 23o.1 204 20o4 16,7 1o2. 5 4 28,8 31 6 3350 27.3 19o6 7 1 0.00 Oo00 0,00 0,00 0.00 0,00 8 1 6.40 4.14 1.96 1lo 51.1.00 1o48 9 1. 2 16 1. o73 lo05 1o63 10 1 - - 2,18 o81 1o08 1.74 7 2 0,00 0.00 0.00 0.00 0o00 0o00 8 2 - 4,00 2.31 1o87 1o12 0.944 9 2 - - 2o56 2o02 122 lo04 10 2 - - 2.74 2.20 1o30 1oll 7 3 0.00 oO 00 0,00 oOO 0o00 0o00 8 3 - 4.57 2.73 2o12 1.54 lol. 9 3 -.5533 237 1.70 1o28 10 3 - - 4ol.6 2.60 1,81 1.49 Note~ Times are not given past the point of evaporation for the drop,

TABLE X SUMMARY OF NEW AVERAGE DIAMETERS (Diameters in Microns) Size Initial WATER Range Diameter Location 1 2 5 4 Distance From Nozzle, Inches 5 6 7 5 6 7 5 6 5 2 12,05 5 17.05 16.7 16. 158 6.7 16,2 156 6 15.7 14,4 l6.o 11.1 4 24.1 24.0 253.9 23,8 25.9 28 25.6 23,9 25,8 25.5 25.5 22.7 21,5 5 4.1 4.0 3355.9.9 4.o0 3355.9 3355,8 54.1 54.0 3355.9 3355.8 55.5 55.0 6 48.2 48.1 48.0 48.o 48.2 48.2 48.1 48.2 48.1 48.0 48.1 47.8 47.6 7 68,2 68.2 68.2 68.2 68.2 68.2 68.1 68.2 68,2 68,2 68,2 68.1 67.9 8 96.5 96.5 96.5 96.5 96.5 96.5 96,5 96.5 96,5 96.5 96.5 96.5 96,4 FREON-11 5 17.05 12.1 - - 11.6 - - 11.0 7.40 4 24,1 22,5 20.4 18.1 22,2 19.7 16,.7 21.9 18,4 15.1 5 54.1 3355.2 52.2 51.2 55.1 51.8 50.4 52.9 51.4 29.7 6 48,2 47.7 47.2 46,7 47.7 47.2 46.6 47.5 46.7 46.0 7 68.2 68.0 67.7 67.5 68.0 67.6 67.5 67.8 67.5 67.1

-71100 () Z -- PREDICTED FROM 0 g 90 - 4 INCH DATA C) o 80 i-. - 7C / 70 o / "'-PREDICTED o 60 w / I I I^~~~~~ \^~~I. DATA > 50 cr - 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 15. Comparison of Measured and Predicted Average Drop Diameters for Water, Location 1.

-720) 0 8 0 --- PREDICTED FROM 0 70 - l l l 4 INCH DATA | PREDICTED l C3 > 4 Z 3 -J 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 16. Comparison of Measured and Predicted Average Drop Diameters for Water, Location 2. U) z 0 0 ____________ ______________________________ C) C 70__ -— PREDICTED FROM 70 0__ I \4 INCH DATA 0 0- 50 0 CD 440 w 3 z 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 17. Comparison of Measured and Predicted Average Drop Diameters for Water, Location 5.

-73 105,_cn z 0 r ---- PREDICTED FROM 0 100 o2) /IOC 4 INCH DATA 0 D- 90 / \X DATA 0 80, o / w- DATA 0 70 \ > 60 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 18. Comparison of Measured and Predicted Average Drop Diameters for Water, Location 4.

-74Cn z 0 0 o O 4 I NCH DATA.I 0 - Q: w 40 Uli w DISTANCE FROM NOZZLED, INCHES z Figure 19. Comparison of Measured and Predicted Average Drop Diameters for Water, Total Spray.

-75and Figure 19 depicts the comparison for the linear average diameter for the total sprayo Similar calculations were carried out for the Freon 11 spray using a value of Atm of 80~Fo This value was determined in a manner similar to the Atm for waterO The value of the evaporation coefficient K for Freon 11 is 40600 microns2/sec. This is much larger than the value of the coefficient for water, in part because of the larger Atm, but primarily because the latent heat of vaporization is an order of magnitude less (78.5 Btu/lb as compared to 970.3 Btu/lb). The results of the calculations are shown in Figures 20 through 22. Attention should be directed in particular to Figure 17 which shows the worst agreement between experiment and calculations. This afforded an excellent chance to investigate the effect of a variation in Atm and hence in the value of the evaporation coefficient K. (Remember that only a small range of Atm is permissible, because of the physical limitations imposed by the system.) Reference to Figure 17 demonstrates that changing the value of Atm from 20~F to 40~F has a relatively small effect on the calculated drop distribution, In fact any error in the choice of Atm is far outweighed by any error in the experimental data which is used as a starting point in the calculations. This observation is further borne out by the data given in Figure 21, where once again a change in the value of Atm does not significantly alter the predicted drop distributionso

-7640 LOCATION 2 z 30 0- 20 20 ac -— PREDICTED FROM n.F-~~~~~~ 104 INCH DATA w 10 2 4.0 5.0 6.0 7.0 -- ~DISTANCE FROM NOZZLE, INCHES a. 0 50................... W 0S~~~ r ~~LOCATION I 4 -- E —- PREDICTED FROM 40 4 INCH DATA W; DATA 30__ - 20 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 20. Comparison of Measured and Predicted Average Drop Diameters for Freon 11, Locations 1 and 2.

-7750. LOCATION 3, AT =60 40 30 6 7 0 rIO v PREDICTED FROM o.. 4 INCH DATA 10 L 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES I — 0 0. 0 5crQs~Do D sfrLOCATION 3, AT =40 s 4 3 J 4.0 5.0 6.0 7.0 )PREDICTED FROM' 4 INCH DATA DISTANCE FROM NOZZLE, INCHES Figure 21. Comparison of Measured and Predicted Average Drop Diameters for Freon 11, Location 3.

-78C) z50 o FREON II TOTAL SPRAY,, AP = 120 40 —---—. w 30 Q PREDICTED FPREDICTED FROM 0 I I 1 ~~~~4" DATA 20 0 0 QM LJ C3! 0 z 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 22. Comparison of Measured and Predicted Average Drop Diameters for Freon 11, Total Spray.

-793. Flow Rate Checks Based Upon Evaporation One manner in which the over-all accuracy of the spray analyses may be measured is by comparing the computed flow rate of the spray from the drop size analyses and their velocities, and the flow rate of the injected liquid jet. This procedure immediately poses a problem —how much of any given spray will evaporate between the nozzle and the point at which the experimental data were taken? Evaporation, in this case. will result from two things. Firstly, evaporation due to loss of sensible heat in the liquid drops which must cool from their boiling point down to the wet bulb temperature and.secondly the type of evaporation discussed in the previous section, 3.1 Evaporation Due to Sensible Heat Loss When the droplets are formed at the nozzle their temperature will be very close to the saturation temperature. There is considerable evidence in the literature to indicate that once dynamic equilibrium conditions have been attained, evaporating spray drops remain at the wet bulb temperature to the drying air. (2) Near the nozzle an unsteady state period can be expected to occur before the drops reach a constant temperature. The time required to reach this steady state condition is very strongly dependent upon the drop sizes present at atomizationo Coarser droplets will require a longer distance than finer droplets. From theoretical calculations El Wakil, Uyehara and Myers(l1) estimated that a 50 micron drop of octane issuing from a pressure nozzle with an initial velocity of 100 ft/sec. would require 0.23 seconds to reach a steady temperature.

-80 Lyons(3) in 1951 supplied some very useful experimental information. Using a copper constantan-thermocouple probe located 1/8 inch away from the nozzle, he reported the following observations: (1) With a feed water temperature of 81~F the wet bulb temperature of the air (61~F) was reached in 1/8 inch. (2) With increasing feed water temperature and fine atomization the wet bulb temperature was again reached if the feed water temperature did not exceed 102~Fo The spray temperature at a distance of 1/8 inch was 2~F higher than the wet bulb for a feed water of 114OF and 12~F higher for a feed water of 164:F, An estimate of the time required for a droplet to reach a steady temperature, for the conditions of interest here, can be estimated by solution of an equation developed by Froesling: d_ d = ^V, ^ D ( 1 -- 0 - a 7 p6 (6.12) where Dm is the molecular diffusivity of water or Freon through air D is the drop diamneter Py is the vapor pressure of the vaporizing liquid at the drop surface temperature. Equation (6012) may be integrated to give an estimate of the time required for this processO It can be shown that Equation (6.12) for the case of water becomes drm = O5S Dm RvD dcl t - O D-F-D(6013) dt -.

-81The results of integrating this equation predicted that the drops considered would lose their sensible heat and reach the wet bulb temperature in a distance of about one incho This prediction was verified qualitatively by means of a thermocouple placed in the spray at this pointo It was found that the spray temperature at this distance did approximate the wet bulb temperatureo In a similar manner it was estimated that Freon 11 drops would reach a steady temperature in about the same distanceo On the basis of this evidence it was deemed safe to assume that by the time the droplets were four inches away from the nozzle they were at their wet bulb temperatureo 352 Convective Evaporation There are three techniques reported in the literature for the estimation of the evaporation rate of sprayso These were developed by Sjenitzer(53) Fledderman and Hanson(l4) and Proberto(42) Marshall in his recent symposium recommends the use of Proberts method, which will be used hereo The details of his development of the analytical technique are given in Appendix Co In it there are assumptions made which are worthy of discussion at this pointo (1) Probert found it necessary to assume zero relative velocity between the droplet and the air stream. The justification for this same assumption for the particular case under study has already been discussedo (2) Probert further assumed a constant value of the evaporation coefficient Ko Recalling that the evaporation coefficient K is, K = _ this assumption is tank P As tamount to assuming a constant value of Atmo

-82 (3) He assumed that the spray drop size distribution could be represented by the Rosin-Rammler equation. As is shown in Chapter VIII, this equation does fit the data as well as the log normal distribution, so any per cent error resulting from its use will be negligible. Using the numerical integration chart given by Probert (shown in Figure 41) and the values of the constants in the Rosin-Rammler equation found in Chapter VIII9 estimates of the evaporation losses from the spray were then prepared. Adding together this loss with the loss due to evaporation from the sensible heat, the flow rates obtained from the drop count were compared with the measured flow rateso The total evaporation is given in Table XI below and the results of the flow rate comparison are shown graphically in Figure 235 The per cent error in most cases is of the same order of magnitude as those reported by Browno (5) The largest deviation occurred at a distance of 7 inches from the nozzle for Freon 11o Examination of the data revealed that this was caused by two or three large drops present in the sample which were not present at the other three distanceso The occurrence of these drops alters the calculated flow rate by a considerable margin0 TABLE XI CALCULATED PER CENT EVAPORATION OF SPRAYS Distance from Nozzle Per Cent Evaporated (Inches) Water Freon 11 4o0 16 70 5o0 17 72 6~0 18 80 7O0 19 85

-83 - 1.0 FREON II a: 0.75 I(+35%/ w 0.5 o 0.25 (-50%) (-40%) DATA (-30%) 0 1. -.. — 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES 2.5 WATER (+36.8/%) 2.0 LOFLOWE E (+7.35%) FLOW RATE BY CALCULATION 735) 1.5 w n.a~ct44r.:~ 0(-37.8 %) 1.0 0 -i (-62.5%) / 0.5 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 23. Comparison of Experimental and Measured Flow Rates.

4o Summary The data presented in this chapter support the conclusion that the evaporation of this type of spray may be calculated using the drop size distribution and the standard rate equations for heat and mass transfero However, the sensible heat content of the droplets at their injection must be accounted foro Another important conclusion is that evaporation must be taken into account when attempting to check a measured flow rate (i.e., using the drop count and velocities taken from the photographic negatives) with that given by the orifice equationo This must be done not only because of the high injection temperatures but also because of the large number of small drops present and the small average drop diameterso

CHAPTER VII DROP VELOCITY PROFILES WITHIN THE SPRAY As explained in a preceding chapter, drop velocity measurements were taken at each of the spray locations and for both water and Freon 11, This data is tabulated in Appendix Bo The drop velocities were measured at distances from the nozzle of four, five, six and seven inches from the nozzle for an injection pressure of 120 psigo Velocities were also determined for a distance of four inches at an injection pressure of 90 psigo The drop velocity at any point within the spray would be expected to be a function of the initial velocity (injection pressure), the velocity of the gas stream through which it is traveling (induced air velocity) and of the so-called drag coefficient applicable to the drop under question. Before discussing the data, a brief review of the ballistics of droplets will be given,,1 Equation of Motion of a Particle We can write down the equation of motion for a droplet using Newton's second law and a defining equation for the drag forces acting on a particle. The hydrodynamic drag acting on a body moving in an infinite fluid field may be written as D = C, (_ ) A (7.1) where CD is a dimensionless drag coefficient p is the density of the fluid -85

-86V is the velocity of the particle A is the projected area of the particle on a plane oriented at right angles to the direction of motiono With this definition for the drag forces, the equation of motion is - - cD()(X ) A + EX ) iM F =~a\t-d A ~+ E (7-2a) where M~ is the mass Vx, Vy are the components of the stream velocity in x and y directions, respectively Ex and Ey are the components of any external forces acting on the droplets, It is clearly apparent that Ey is zero and it is demonstrated in Appendix E that gravity forces acting on the droplets may be neglected relative to the drag forceso Thus Ex may be set equal to zero. Let us assume radial symmetry within the spray, or in other words, we shall consider that the spray has ceased to expand in radius and has taken on a cylindrical rather than a conical shape. This situation occurs about four inches from the nozzle for the fluids usedo Equation (7~2) then becomes, at low Reynolds numbers i)odt2 3 4T V )- (7 \5)

-87where use has been made of the fact that CD -24 for Re less Re than 20o. In Equation (7o3) the Reynolds number is defined using the relative velocity between the particle and the gas stream, For the sake of convenience, let us correct for any deviations from the relation in (7~3) by writing Mo dA Tr(6- 1NR - ) dxt (7 4) The correction factor k will not be the same for all drop diameters, but over a reasonably small range of Reynolds numbers could be considered to be constant. Defining the dimensionless parameters c _ X TV- V t T Equation (704) becomes d + d (5) d d which has as its solution E(tr) (o0) + r + ()e'o - +o~ez e - (6)

-88Equation (7.6) is of the form -('r = A + Be + z (7~7) where the terms A and B have an obvious meaning, and E'(O) is the value of d/ dT T= 0 Differentiation of (707) then yields an expression for velocity as a function of time: d = - Be (708) Equation (7.8) predicts that velocity decreases exponentially with time and hence will,decrease with distance in a manner approximating an exponential decay. 20 Corrections to the Drag Coefficient In the step between Equation (703) and (704) a correction term was applied to the so-called Stokes law drag coefficient. The question naturally arises as to what considerations are involved in this correction. Corrections to the Stokes law relation should be made if any of the following situations arise: (1) If the fluid is not infinite in extent (2) If the motion is not dominated entirely by viscous forces; io.e, if vortices and/or separation occur behind the sphere (3) If the fluid is not continuous —for the case of very small spheres the fluid must be regarded as a discontinuous molecular field

-89(4) If the sphere undergoes accelerated or decelerated motion in the fluid (5) If the sphere is evaporating to a significant extento Clearly (1) and (3) above do not apply to spray systems such as this. The remaining corrections will be briefly discussed. 201 Corrections for Non-Creeping Flow At values of the Reynolds number above 2o0, inertial forces can no longer be neglected in determining the drag coefficient. This correction is nearly always made by finding a function which will fit the experimentally determined datao A typical function is that of Langmuir 0,6 3 +.4 B 1.38 24 C~ - (I + 0+197N63 + XI6X Re (7.9) This equation describes the experimental data up to a Reynolds number of 2000 within an error of about seven per cento 202 Corrections for Accelerated Motion The drag coefficient is a function of the Reynolds number alone only when the motion is steadyo There is very little experimental data available in the literature concerned with the acceleration effects on the drag relationso Most investigations which have been carried out point toesi the oncusion that the drag coefficient increases for both low and high Reynolds numbers whenever the density of the body approaches the density of the fluid0 Basset(4) and Pearcy and Hill(39) have both

-90concluded from their studies that the effect of acceleration is small for small spheres and increases with increasing fluid density. Crowe(7) has conducted a very interesting theoretical analysis of the effect of a constant linear acceleration on the drag coefficient. He was able to show that C X -_ 7 Ac (7.10) where _ dv Ac - V2 dt By use of Equation (7.10) it is readily shown that acceleration will have no effect on the drag coefficients in sprays since the value of the acceleration modulus) AC) is of the order of 10-4 2.3 Corrections for Evaporation of the Droplet The most useful work carried out on the effect of evaporation on the drag coefficient was published by Crowe(7) He began with the momentum equation describing the system and included in it the term for mass flux at the surface of the dropo The resulting integro-differential equation was solved by a perturbation technique to yield the relation: Cwh Sc23il- cs23 (Ce -) (711) where Cfo = skin friction coefficient for the case of no evaporation Cf = skin friction coefficient for the case of evaporation Sc = Schmidt number

,91Cs concentration of the evaporating component at the surface Ce concentration of the evaporating component in the free stream AC Cs - Ce Crowe was further able to demonstrate that the effect of evaporation in the wake region is negligibly small compared to the effect predicted by Equation (7o 11)o Using Equation (711) as a basis an estimate was made of the probable effect which evaporation will have for the specific cases of water and Freon 11 sprays. Assuming that the mass fraction in the free stream is near zero, the mass fraction at the surface may be estimated from Cs P _I _P - lB/If, / _ (7012) where MA9 MB are the molecular weights of air and the evaporating component respectively Pv is the partial pressure of the evaporating vapor A semi quantitative measurement of the drop temperatures indicated that they were about 60~F for water and about -20OF for Freon 11o At these conditions Pv/P is small relative to one and. Equation (7o12) becomes CS- U p P i' M (7.13)

-92Using data available for water and Freon 11(1l'l2) the values of Cs were calculated for both systemso Substitution of these values into Equation (7.11) subject to the assumptions given above yields: cp < K 4+ 0.1+ (7,14) CF for Freon 11, and R~ < I + 0.005 (7015) CF for water. Equations (7o14) and (7o15) state that the drag coefficient for water should be decreased by less than 0,5 per cent and for Freon 11 by less than 12 per cento 2.4 Corrections for Interaction of Particles The previous sections of this chapter have been concerned with the motion of individual droplets. If at some point in the spray the concentration of drops becomes large, they will exert a mutual influence upon the flow patterns around each other, The effect of concentration in streamline flow may be allowed for by an equation of the form D = 3rtc VD-k (7.16) where kc is a factor dependent upon particle concentration. Two correlations of kc have been publishedo Steinour(55) found in a study of hindered settling that =, J 2 - 181-e (7.17)

-93and Birgers(8) in a theoretical study predicted that for relatively dilute "solutions -ac - I +- 6.875( -<e) (717a) In the above two expressions e is the fraction of fluid phase presento An estimation of ev for the sprays studied and use of Equation (7o17) predicted values of kc of about lo01 to lOOl In other words this correction, based upon Equation (7ol7) or (7.17a) may be considered negligible. 35 Induced Air Velocity In Equations (7.2) and (7o3), which describe the motion of a droplet, one of the important parameters is the velocity of the gas stream. For a spray this velocity results from the phenomenon known as induced air flowo When the spray leaves the nozzle the leading droplets impart their energy to the surrounding air causing its forward motion and thereby reducing the air resistance for the following dropletso Ranz and Binark(44) have conducted an experimental and theoretical investigation of induced air flows in hollow cone and solid cone sprays. The sprays formed by flashing jets are neither truly hollow cone or solid cone. This can be seen by reference to Table XII which summarizes the per cent of total flow by location ini the spray (the locations referred to are those described in Chapter III).

.94TABLE XII PER CENT OF TOTAL FLOW BY LOCATION IN SPRAY (Injection Pressure 120 psig) water Freon 11 Distance, from Nozzle 4 5 6 7 4 5 6 7 Location 1 9.25 2.8 40.0 1036 8o15 7.05 0.6 3.4 2 3504 60.4 40.3 15o35 60.7 59.7 28.1 46.2 3 2508 2359 15.4 4007 31.2 3303 71.3 50o6 4 29.5 12.7 4,07 42,0 301 Analysis of Induced Air Flow in a Hollow Cone Spray An analysis of the induced air flow in a hollow cone spray can be based on a momentum balance between the total drag force operating on all of the droplets passing through an area in a given time interval and the component of air momentum in the direction of drop motion. As an approximation it can be assumed that at any given location on the outer periphery of the spray an average drop diameter and an average drop velocity can be specified together with the appropriate number of drops. This information is readily available from the experimental data. If one assumes that the average values are correct then the drag force on all of the droplets is ( 4 )( 2 ) (7018) where n is the number of droplets pA is the air density

VD is the velocity of the drop of diameter D D is the drop diameter CD is the drag coefficient Ranz has shown that the air enters the spray sheet at right angles to the motion of the dropletso The component of air momentum in the direction of drop motion is %VC ctne (7.19) where ~ is the cone angle Vi is the induced velocity of air entering from outside the sprayo A momentum balance then yields the expression!I'- 4(^ y c" 2 tt(7.20) VD /L V A? CAtn0 Using Equation (7~20) an estimate of Vi can be made and from this the air velocity inside the spray may be calculated, which from continuity considerations must be (Vi/sin 9)o There are some limitations to this model which are enumerated belowe (1) In writing the momentum balance it was assumed that the air entered the spray at right angles to the direction of motion of the dropletso This assumption was partially verified by Ranzgs experiments, but is certainly questionableo

-96(2) The model suggests that once a drop has slowed to its terminal velocity corresponding to the induced air velocity, that it will be blown into the center of the spray. The experimental data do not completely substantiate this conclusion. The drops on the outer periphery of the spray are, in general, larger than those in the center, but small droplets do occur at this location. (3) The model assumes an unidirectional flow for the air in the spray core. Visual observations of the spray showed a tendency for swirling to occur, suggesting the occurrence of air flow in other than a vertically downward direction. Nevertheless, the model is very useful for estimating the induced air velocity. Using Equation (720) the air velocity was calculated at a distance of four and five inches from the nozzle. The velocity at distances of six and seven inches were not calculated because at this point the spray was no longer conical but was cylindrical in shape. The results of these calculations are summarized in Table XIIIo TABLE XIII CALCULATED INDUCED AIR VELOCITIES Distance from Velocity (Eq 7 20) Velocity Nozzle Inches Fluid Ft/Sec (E 72 Ft/Sec (Eqo 7.21) 4 Water 5o4 40 5 Water 2.6 ~40 4 Freon 11 4.9 ~30 5 Freon 11 2o8 ~30 We may summarize this by quoting from Ranz and Binark: "The air velocity outside the spray cone decreases with increasing distance

-97from the orifice and increasing distance from the spray sheeto Velocities inside the spray sheet are higher than the corresponding velocities outside the spray sheet and decrease with increasing distance from the nozzle orificeo In a radial traverse the air velocity shows a maximum on the spray cone axis " Based, on this, inside the spray sheet the following qualitative profile should occur: (1) The air velocity is radially distributed about the center line and the velocity of the small drops should decrease from the center line outwards, since these drops should be carried along with the air streamo (2) On the outer periphery of the spray sheet the drops should behave approximately according to the well known drag relationships if the induced air enters at right angleso The departure from this relationship could perhaps be construed as a measure of whether or not the induced air does actually enter at right angleso (3) The drops in the central core should decelerate more or less uniformly and approach either their "terminal' velocity or the induced air velocity, whichever is largero (The terminal velocity is in a moving air stream and is not to be confused with the more commonly used notion of terminal velocity in a still air stream ) The rate of approach to this terminal velocity will be different for different drop sizeso

-983o2 Solid Cone Sprays Ranz found that for solid cone sprays the air flow was also induced at right angles to the edge of the spray, but on entering the spray turned in the direction of the nozzle axiso He assumed that "average air velocity, drop velocity and liquid flux inside the spray have a normal radial distribution with the same dispersion, and that average drop and air velocities are approximately equal throughout the spray zoneo" Based on this assumption he showed that I + 3a _V _ (7,21) where s is the r distance at which the spray liquid flux equals l/e times the maximum value Do is the orifice diameter. Using Equation (7.21) for the case of r = 0 (center line) and s = 0o8 for water and 04 for Freon 11 air velocities were once again estimatedo They are also given in Table XIIIo It is readily seen that this particular model does not describe the induced air flow in the spray very accurately, since the velocities are too high by an order of magnitudeo

KEY: 140 X CENTER LINE 0 0.4 IN. FROM AXIS * 0.8 IN. FROM AXIS A 1.2 IN. FROM AXIS MC 120'> 100 X I. 0 0 o — o 0. 6 0 40 — /40 DROP DIAMETER, MICRONS Figure 24. Typical Velocity Profiles -- Water at a Distance of Four Inches from the Nozzle.

-100WATER WATER CENTER LINE 0.4 INCHES FROM AXIS (2) N ( ) DATA POINTS NUMBER OF DATA POINTS ( ] SIZE RANGE [ ] SIZE RANGE 110 (3(0 (0)31~ (3)10 (3) 90 \ (3) (91 (12) (6). 80 —------- I8] (4) 80,(3),n________~_______~~__ ~__~~___C>~~ (2) u 70 -- (I) — (4) (6) a. (4) 0: ^^ X(10) 05C lo) F u (15) (3) [7 40-(3) ((6) —-"" (35) (25) )(22) [6] (7~'~" -",^)a' - (l'c ~"J16J' (8) (( 31) [0- - ^ >^ 5^ -- ^: ^ ) C3*) —'" ( 8) ) (44) (30) [ (,)c~ ~..~~[4] >c(2%Yr' —- (41) 2 -l(^ ^1c2^.9) MM _ _ _(5)-( >;(23) ( 4 (28)?((o) o (37. (3)0' —— [3]: (5)x!) 10 (8)7 4.0 5.0 6.0 7.0 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 25. Velocity Profiles for Water, Locations 1 and 2.

-101WATER WATER 0.8 INCHES FROM AXIS 1.2 INCHES FROM AXIS ( ) NUMBER OF DATA POINTS ( ) NUMBER OF DATA POINTS C 3 SIZE RANGE C 3 SIZE RANGE 10 90 -- — L-[9] - (0) 0(7) 80 J;<(9) 70 60 -,_] 5o > (4)(14) 40 C;3 (20)'<. 7...C (22) (43)1 5 -7 —--.. — 6] — oJ (8](8]N(27) 20 161 (24)' c _ [5 >(55) (15)(4 >\ (31) >s,._______(4) ___ (5 7) (31) x (I) (30) (o10)) (30) ^^c10:?r (3I)(;) 1 (5)X ( (4) 57 (3)> 4.0 5.0 6.0 7.0 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE, INCHES Figure 26. Velocity Profiles for Water, Locations 3 and 4.

-10280 - FREON II FREON II CENTERLINE 0.4 INCHES FROM AXIS () NUMBER OF DATA POINTS () NUMBER OF DATA POINTS 70- [ S R E[]] SIZE RANGE SIZE RANGE O(2) 0 60 |o 60T(8e),W, 50 x (I) Iw (18)X D X(I) (6) *N:C w 2 IL 40 - [61 X(8) >0(12) (27)C. (4) 0 30 X(18) [ 1 ___ [5] )(13) (1)X > (7). (5) 3 (23) (17) _ 8) 0 20 45 0 _ 5. 6 7 0r (4)X: (14) X DISAC FrO4M~ ~ X(6) X(4) (7)cFi (3) (V4)c il(4) f o1 aios 1d4 (12) [3J x(l) [2] 4.0 5.0 6.0 7.0 4.0 5.0 6.0 7.0 DISTANCE FROM NOZZLE (INCHES) Figure 27. Velocity Profiles for Freon 11, Locations 1 and 2.

-10370 I "E FREON II 0.8 INCHES FROM AXIS ( ) NUMBER OF DATA POINTS 60 [ 3 SIZE RANGE 60 w0 X L7 " 40 ___ _._-: _(9) LL x (I 0 305. (I) >L IT (27) (12OMO, (7) ~~~> ^^^,^(7) 0 [5] 0: 20 0((227) [4 (5) 101):-.- (6) 4.0 5.0 6.0 70 DISTANCE FROM NOZZLE, INCHES Figure 28. Velocity Profiles for Freon 11, Location 3.

-1044, Discussion of Data The velocity profiles at all injection pressures and at all locations are essentially of the same form. Figure 24 is a typical set of velocity curveso It depicts the drop velocities for water at a distance of four inches from the nozzle and at an injection pressure of 120 psigo The curves are seen to be "S" shaped and the drop velocity for any given size range usually decreases as one moves away from the spray axis. Also at a given position relative to the center line larger drops are traveling at a higher velocity than smaller dropso Similar plots for all the other spray locations and for both fluids exhibited these same general characteristicso A more useful manner of presentation of the velocity data is to construct at each location a plot of drop velocity as a function of distance from the nozzle, with size range as a parametero Figures 25 through 28 are such plots for water and Freon 11o In all cases the drop velocity showed a tendency to decrease with increasing distance from the nozzleo The rate of decrease of the velocity is seen to be largest for larger drops and less for the smaller drops. In fact drops of about 10 to 15 microns do not appear to change velocity at all in traveling from four to seven inches away from the nozzle. This occurs because the drops are traveling at about the same speed as the induced air velocity, as is shown by Table XIII. Figure 29 is a comparison of the drop velocities of water and Freon 11 for two injection pressures of 120 and 90 psig, at a distance of four inches from the nozzle. This figure illustrates the effect of pressure on the drop velocity and the effect of the fluid sprayed on the drop velocity,

140- WATER 120 psig 12 X^ ^ ""-WATER 90 psig lj 1 —------ 50 L + LOWER CORNER EXPANDED 80 F-ll-120 PSi 40 0 LJ.> 60 /-F- II- 120 psig oa F-11-90 psig F-11-90 psig 20 W-90 psig 20- --- 10 O 40___ __________^ 80 10 10 24 20 30 40 50 0 40 80 120 160 200 240 280 320 360 400 DROP DIAMETER, MICRONS Figure 29. Comparison of Water and Freon 11 Drop Velocities.

-106A lowering of the injection velocity is found to decrease the drop velocity as would be expected, since the initial velocity corresponding to the injection pressure is lower (for example in the case of water from 110 ft/sec to 90o5 ft/sec). However, this 25 per cent reduction in initial velocity does not result in a 25 per cent decrease in drop velocity at any location. The reason for this is that a lowering of the injection pressure decreases the cone angle and hence tends to increase the induced air velocity, for the same flow rate. Or in other words the drag coefficient tends to be moved towards a lower value because of the increased gas stream velocity. We may also compare the drop velocities of water and Freon 11 at the same distance from the nozzle and the same injection pressure. The initial velocities are 110 ft/sec for water and 90o5 ft/sec for Freon 11o The cone angles are about 44~ and 22~, respectively, On this basis and considering the flow rates we would expect that for any given location and the same injection pressure that Freon drops will be traveling faster than water drops of the same size. We would further expect this because of the effect of evaporation on the drag coefficiento As was discussed in the previous section the drag coefficient for Freon 11 will be about 10 per cent lower than that for water at the same Reynolds number. Reference to Figure 29 and Figures 25, 26, 27 and 28 shows that this is indeed the case. For smaller droplets this tendency is less pronounced and the data do not support this line of argument too well, At least part of the difficulty in this respect arises from the extreme difficulty of accurately measuring drop velocities for drops of a size

-107below 20 microns, particulary in a very dense spray where it is difficult to separate drop images on a negative. It will also be noted that the velocity profiles plotted as a function of distance from the nozzle do indeed exhibit an exponential type of decay, as was to be expected from Equation (7.8). 5. Summary The preceding information may be summarized as follows: (1) The drag relationships for single droplets may be applied qualitatively to the droplets contained in these sprays after taking into account any correction for the effect of evaporation on the drag coefficient and knowing the pattern of the induced air flow profiles. This same procedure could be applied analytically after an extensive study of these air profiles has been carried out since this is one of the most critical factors which determines the drop velocity profiles. (2) All of the drops tend towards a common terminal velocity, which is the induced air velocity corresponding to the particular location of the drop. Many drops will have evaporated before they reach this terminal velocity. (3) The sprays are more nearly hollow cone than solid cone, although they do not strictly belong to either classification.

CHAPTER VIII SPRAY CHARACTERISTICS 1o Drop Size Distribution Functions The droplet size distribution is certainly one of the most important characteristics of any spray but it is also the spray property most difficult to predict theoretically. The drop size distribution of any spray is usually represented by two parameters —a mean diameter and a parameter which measures the deviation of the drops from this mean 1,1 Distribution Function A probability density function f(x) is said to exist when these criteria are met: if F(x' a 0 (81) oo J>m dx = I (8,2) -0o and if in addition ~P(c X )i =. X(x d* (8.3) then the f(x) is defined as the probability density function. The probability distribution function is defined in terms of the probability density function as X 9F(x- = S 4 dx 0(8.4) -00 -108

-109In Equations (8.1) to (8.4), x is considered to be any random variable. For drop size analyses it is associated with the drop diameter. The methods of expressing distribution functions are usually associated with a number distribution dn/dx or with a volume distribution dv/dx. The term dn/dx may be thought of as representing the number of drops dn in the size interval x to x + dx. Similarly, dv/dx represents the volume of drops in the same size interval. The two distributions are easily related for the case of spherical particles. Statisticians define the k-th moment about the origin of the frequency curve f(x) by >4^ = 5 ^axx (8.5) o. Mugele and Evans(36) developed the expression below which may be used to calculate any mean diameter X" XPx p -%ct Xp% =.xm' (8.6) The most coonly used mean d eters are gven in Table XIV The most commonly used mean diameters are given in Table XIV.

-110TABLE XIV COMMONLY USED MEAN DIAMETERS Name of p q Mean Diameter Diameters Application 1 0 Linear Comparisons, Evaporation 2 0 Surface Surface Area Controlling (Absorption) 3 0 Volume Volume Controlling (Hydrology) 2 1 Surface Diameter Adsorption 3 1 Volume Diameter Evaporation, Molecular Diffusion 3 2 Sauter Mean Efficiency Studies, Mass Transfer 4 3 De Brouckere Combustion Equilibrium There are three commonly used distribution functions in spray analyses — The Rosin-Rammler Equation, the Nukiyama-Tanasawa Equation and the log normal distribution~ The first two of these are completely empirical in nature, while the log normal distribution can be derived theoretically(24) by assuming that the droplet size is the result of a large number of small, independent impulses the effects of which are proportional to the size of the drop. It has been pointed out(4) that all of these distributions are special cases of the equation B=A -- Kxy (807) C(x~ -- ~x X e The distributions are summarized in the following equations: (1) Rosin-Rammler Distribution x - bnx -'e- > (8.8) or as it is often written R = e-bx (8.8a) where R is the volume fraction of drops of size greater than x,

11l(2) Nukiyama-Tanasawa Distribution = ax- Q 2e -bX (8.9) where a, b, n are constants. (3) Log Normal Distribution lo1x- CocJ7s - (xA = 7^- e otcre (8.10) where xg is the geometric mean ag is the geometric standard deviation, Another commonly used form of Equation (8010) is. " ( ~(Sy _2s ) -(8 10a) where y = In x/x, and 6 is a dispersion parameter. The three distributions are integrable as special cases of the incomplete gamma function w4 =,e ~t 5edt (8.11) 0o which is tabulated in most statistics bookso The transformations for the integrations are also readily available and are summarized in Table XVo

112TABLE XV TRANSFORMATIONS AND INTEGRALS FOR DENSITY FUNCTIONS Transformation Distribution Function log normal = 1/2 F(x) = r2/2(1/2) t = x2/2 Rosin-Rammler e = 1 F(x) = rn(l) bxn(1) t = bxn = (1 - e'bxn) Nukiyama-Tanasawa i = 3/n F(x) = r / nb1/ nbxn(3/n) t = bxn 1,2 Test of Distribution Functions to Fit Experimental Data The distribution functions may be tested for their fit of the experimental data in the following ways. The Rosin-Rammler distribution shown is applicable to volumes and R is then associated with the volume fraction, larger than some diameter Xo Taking logarithms twice of Equation (8.8a) gives I9goi (R) -lo)b - nlogx - logloge (8.12) which is a linear relationship between log log R and log Xo The constants b and n may be evaluated either analytically or graphicallyo Table XXV of Appendix F summarizes the values of these two constants for the sprays studied. The numerical calculations were carried out on the IBM 709 computer using standard statistical techniques. With the Nukiyama-Tanasawa distribution we are interested in the number distribution expressed as a per cento Equation (8,9) may

-113be expressed in the form loSF SZ =;< - logc -L bxnloqe (8135) which is a linear relationship in a and b for a predetermined value of n. Nukiyama and Tanasawa have found that values of n between 1/4 and 1/2 give the best fit of drop size data. Three values of the parameter were used —n equal 1/4, 1/3 and 1/2 and in all cases it was found that n equal 1/4 give the best fit, although the difference between any of the three values was small. (Typical correlation coefficients were 0,963, 0.958 and 0.938 for n equal 1/4, 1/3 and 1/2, respectively.) Values of the constants a and b in Equation (8.13) for n equal 1/4 are given in Appendix Fo These were determined using the same type of computer program as was used to find the constants in the Rosin-Rammler equation. The log normal distribution function is most easily tested by plotting on log probability paper the cumulative number per cent versus the drop diameter. (The drop diameter used is the upper limit of the size range.) This plot should yield a straight line if a fit to the log normal distribution is obtained, A typical run is shown in Figure 30 and Table XVIo Similar plots for all of the experimental data may be found in Appendix Bo In testing this distribution the end points should likely be truncated since it is obviously impossible to fit any finite range of drop sizes with an infinite range distribution function. Values of the uniformity parameter [Equation (8.10a)] are readily obtained from the same plots by making use of the relationship 0 3 94 (8.14) _ - 1^,, ( }

-11499.9 99 95 900 N 80-' () w 70 o 60 z/ 5 u, 0. 30 U10 w - 2 0.5 0.2 0. 10 20 50 100 200 500 DROP DIAMETER, MICRONS Figure 30. Typical Fit of Log Normal Distribution -- Run No. 2.

-115where D90, D50 are the diameters at cumulative percentages of 90 and 50, respectively. The values of this parameter are given in Table XVII. TABLE XV.! SIZE DISTRIBUTION FOR TOTAL SPRAY RUN 2 Size Per Cent Cumulative Range Per Cent Per Size Range Per Cent 1 0.000 0,00 0.000 2 0.826 0,201 0,826 3 6.88 1.17 7.70 4 18.87 2.30 26.58 5 23.98 2o03 50.56 6 17.79 1.09 68.35 7 14,80 0,627 830116 8 8.83 0.268 91-99 9 5o68 0.121 97.67 10 1.98 0.0301 99.65 11.342 0003 6 100.00 1.3 Discussion of the Utility of the Three Functions Reference to the over-all correlation coefficients given in Appendix F for the Rosin-Rammler and Nukiyama-Tanasawa distribution functions and to the log probability plots shows that all three functions will adequately describe the experimental data. In fact, for an individual location within the spray the Rosin-Rammler distribution is perhaps best, but for purposes of describing the total spray distribution all three appear of equal accuracy. Since the two more complicated distributions do not, in general, give any better fit of the experimental data than the log normal distribution there would appear to

-116be no justification for their use unless for some specific purpose. Also the log normal distribution does have some additional basis in theory(24) while the other two do not, even though all three are special cases of the gamma function, 2. Experimental Mean Drop Diameters The various mean drop diameters were previously defined by Equation (8,6). In order to estimate their value using discrete rather than continuous random variables, the expression Dnn- " _ L __ (8.15) DL: may be used as being approximately equivalent to (8.6). The mean diameters calculated were the linear mean diameter DlO, the surface mean diameter D20, the surface diameter D21 and the Sauter mean diameter D32. The values computed from the data are given in Table XVII. The variation of these mean diameters as a function of distance away from the nozzle is discussed in the chapter on spray evaporation. At any fixed distance away from the nozzle the mean diameter tends to increase with increasing distance from the center line. In most instances for water a minimum appears to exist at a distance of 0,4 inches away from the spray axis. For Freon 11 this does not occur and there is no minimum present in the curve. The presence of the minimum can be explained in part by the breakup mechanism. When the vapor bubbles disintegrate the jet the drops will have two components of velocity, one parallel to and one perpendicular to the spray axis. Under these

-117TABLE XVII MEAN DROP SIZES Run Fluid P T~F Location* Mean Drop Size (Microns) (Psig) 6 D10 D20 D21 D32 1 W 90 287 1 47.3 54.8 63.4 84.0 1 W 90 287 2 46.7 52.2 58.5 73.2 1 W 90 287 3 65.7 69.5 73.4 82.3 1 W 90 287 4 64.1 68.2 72.6 81.8 1 W 90 287 Total 2.09 54.0 59.7 65.9 78.8 2 W 120 287 1 53.8 64.8 78.0 105.0 2 W 120 287 2 58.9 76.3 98.7 144.0 2 W 120 287 3 44.8 53.9 64.9 89.6 2 W 120 287 4 62.6 68.1 74.2 85.4 2 W 120 287 Total 1.56 52.9 65.3 80.6 115.8 3 W 120 287 1 91.1 127.2 177.6 264.1 3 W 120 287 2 57.6 74.5 96.4 146.4 3 W 120 287 3 6o.5 69.2 79.0 93.7 3 W 120 287 4 104.3 108.0 111.9 118.2 3 W 120 287 Total 1.00 63.3 80.6 102.6 159.6 4 W 120 287 1 47.5 55.0 63.7 81.4 4 W 120 287 2 54.8 65.9 79.2 109.4 4 W 120 287 3 66.4 79.6 95.6 131.5 4 W 120 287 4 58.2 68.5 80.6 107.2 4 W 120 287 Total 1.16 59.6 71.0 84.7 115.8 5 W 120 287 1 79.2 120.9 184.7 285.7 5 W 120 287 2 52.0 29.2 67.5 87.9 5 W 120 287 3 52.7 56.7 60.9 69.8 5 W 120 287 4 45.3 48.1 51.2 57.8 5 W 120 287 Total 1.67 52.2 58.8 66.3 97.4 6 F-Il 90 158 1 29.4 31.4 33.6 38.3 6 F-il 90 158 2 30.7 33.1 35.7 40.8 6 F-il 90 158 Total 1.90 29.9 32.1 34.4 39.3 7 F-il 120 158 1 26.9 28.1 29.3 31.9 7 F-il 120 158 2 28.0 29.5 31.0 34.6 7 F-il 120 158 3 28.5 29.8 31.2 33.9 7 F-il 120 158 Total 2.36 28.1 29.5 30.9 34.1 8 F-ll 120 158 1 23.5 24.5 25.6 27.8 8 F-il 120 158 2 26.2 27.9 29.8 34.4 8 F-ll 120 158 3 29.2 30.2 31.3 33.4 8 F-11 120 158 Total 2.43 26.8 28.3 29.8 33.5 9 F-ll 120 158 1 26.2 27.5 28.8 32.3 9 F-ll 120 158 2 30.0 31.6 33.2 6.7 9 F-il 120 158 3 32.7 34.7 36.9 41.4 9 F-ll 120 158 Total 2.03 31.7 33.6 35.6 39.9 10 F-ll 120 158 1 39.5 42.1 44.8 50.5 10 F-ll 120 158 2 45.3 47.6 50.0 54.8 10 F-ll 120 158 3 47.9 50.5 53.2 58.7 10 F-il 120 158 Total 2.10 46.2 48.7 51.3 56.6 Location refers to those given in Appendix B. "Total" means the mean diameter for the entire spray.

-11899.9 PRESENT DATA DATA OF BROWN 95 z 90 z 80 70 o y60 1 /50 (0 "- 40 0 o 30 I20 0 O 2 ----— 20 50 100 I0 20 50 I 00 DROP DIAMETER, MICRONS Figure 31. Comparison of Experimental Data with Brown's Data -— Freon 11.

-11999.9 99 [,s60srr~~~ /^ ~ XPRESENT w- / DATA i.6 -< 95 z 90 0 z 80 H- 70 c) 60 W 50 o 40 30 0 - 20 z 0 10 Ix 10 //^ —DATA OF BROWN 10 20 50 100 200 500 DROP DIAMETER, MICRONS Figure 32. Comparison of Experimental Data with Brown's Data -- Water.

-120conditions the larger drops will, at any given point, have moved furthest away from the spray axis. The drop diameter might be expected to increase at the spray axis because it is at this point that any drops formed from a portion of the jet not shattered by vapor evolution will likely have only one velocity componento If such drops form they will tend to be slightly larger than those formed from a shattered portion of the jeto A second factor to be considered here is that smaller drops evaporate faster than larger drops and any slight minimum existing in the diameter curve will tend to be accentuated as the distance away from the nozzle (and thus evaporation time) increaseso 3. Reproducibility of Experimental Data Whenever experimental techniques involve a human judgement factor the reproducibility of the experimental data is naturally subject to some question. In Chapter VI it was pointed out that the flow rate as calculated from the drop size distributions is a measure of the accuracy of the datao Another means of judging the experimental accuracy is, in this instance, to simply reproduce data taken by Browno Figure 31 is a plot of the log normal distribution function for Freon 11 at an injection pressure of 120 psigo The experimental data of Brown for Freon 11 at 120 psig and the same injection temperature is also showno Similarly, the two sets of data for water at an injection pressure of 120 psig were compared and the results are shown in Figure 32. It will be seen that in both cases the agreement between the two is quite goodo Another valid comparison is between average drop diameters measured by two experiments. The results of this comparison are given

-121in Table XVIIIo Here again the results of the comparison are favorable, particularly for the entire sprayo The largest deviations occur either TABLE XVIII COMPARISON OF MEAN DIAMETERS WITH BROWN'S DATA (Diameters in Microns) Freon 11 Water Location Brown Short Brown Short 1 46.8 26.2 32.2 47.5 2 21.9 30.0 4359 54.8 3 22.4 32o7 60.2 66.4 4 74.9 58.2 Total 37 6 3107 5505 59.6 at the center line of the spray or at the outer periphery of the spray which could be predicted, because these are the most difficult places in the spray to analyze accurately. However, errors in these two locations do not seriously affect a flow rate calculation for the following reasons: (1) The major portions of the drops occur in locations two and threeo (2) Even if a high drop density is found at the spray axis, it represents a very small portion of the total volumetric flow, and an error at this point is not magnified to such an extent, (3) The drop density on the outer edge of the spray is low enough to once again mask any experimental error present when calculating a flow rate. The results of these comparisons was indeed gratifying and would seem to support the reliability of the data,

CHAPTER IX CONCLUSIONS This work has shown that the flashing process is an effective means of producing a fine and uniformly dispersed spray at low values of the Weber number and corresponding low injection pressures. The shatter temperature is a defining characteristic of a flashing spray and it may be represented by the equation -__0 SA-4 \ (90 p (THis 0.34)4(7) ( ) O (9.1) The range of variables represented in the above equation would appear to be wide enough to justify its use with fluids other than water, Freon 11 or Freon 113. The use of Equation (9.1) is not likely justified for flashing sprays containing dissolved or suspended materials. Also it may not prove valid at large values of the liquid viscosity since it is an unfortunate fact that all pure liquids at their normal boiling point have about the same viscosity. For this reason the range of viscosities studied was necessarily small. Equation (9.1) may not prove valid if the receiving pressure is too far removed from atmospheric. The log normal distribution is adequate for the description of drop sizes, particularly if one is interested in the total spray. The other two most commonly used size distributions, the Rosin-Rammler and the Nukiyama-Tanasawa equations, will also give an accurate description of the drop size distributions, but require considerably more expenditure of effort in order to use them, or to evaluate the constants. -122

-123The methods of calculating spray evaporation in the literature(32,42) can be applied to sprays formed by this flashing process with a great deal of success. Care must be taken to insure that the spray droplets have cooled from their saturation temperature to their wet bulb temperature before attempting to use any of the methods. In general, the mean drop diameters increase from the spray axis outwards, at a fixed distance away from the nozzle. No such simple remark may be made about the mean diameter as a function of distance away from the nozzle, at any given location. This functional relationship is strongly dependent upon the drop size distribution and to a lesser extent upon the velocity distribution, both of the air and of the droplets. The drop velocities within the spray follow the expected patterns. The drops tend to decrease in velocity in an exponential manner as they travel away from the orifice with the smaller drops slowing down more rapidly than the larger drops. The drop velocity (for any given size of drop) decreases as the distance away from the spray axis increases. Drops having a diameter of about 20 microns or less approach a common velocity which is the velocity of the induced air flow.

CHAPTER X RECOMMENDATIONS As typically occurs in investigations like this, each problem studied brings forth two or three new and unexpected problems. During the course of the work there were a few questions which arose and seemed worthy of further investigation, at some future dateo 1. In discussing the drop velocity profiles, reference was made to the induced air flowo A more accurate knowledge of this air flow would indeed be very valuable to the interpretation of the velocity profiles. It might prove possible to calculate the induced air velocities from the experimental drop velocities given. Such a calculation would be very involved and require a large number of numerical integrationso Other directions of study in this regard should be to formulate the mathematics necessary to compute the air velocities from the experimental data, and to experimentally measure the air velocities, perhaps in a manner similar to Ranz and Binarko 2. A study of the humidity and temperature profiles within the sprays, particularly Freon 11, would enable one to gain much useful information regarding the evaporation rates taking place. 30 A study of the effects of high viscosity upon the flashing mechanism would be very beneficialo This could be done in at least two ways —either by an artificial thickener -124

-125 such as Methocel or by spraying the liquid into surroundings which are at a pressure below atmospheric. This latter method would effectively change the liquid viscosity by lowering the saturation temperatureo Either of these two studies will require extensive modifications to the experimental system. This particular investigation would also have an additional benefit in testing the effect of the surrounding gas density on the shatter temperature correlation, or in other words to see if the Weber number adequately describes this effecto 4o The most laborious and time-consuming job in a study such as this is the counting of dropso It would prove well worthwhile to try to find some easier and less timeconsuming means of scanning photographic negatives.

APPENDICES -126

APPENDIX A COMPUTER PROGRAM Shown in the following pages (Tables XIX and XX) is the computer program used to analyze the raw data and the input variables associated with ito The program is written in MAD and with the tables its use is fairly obvious. Other computer programs were written to carry out the regression analysis of the shatter temperature data and the least squares analysis of the Rosin-Rammler and Nukiyama-Tanasawa distributions. These programs are not included in this dissertation because they use readily available statistical methods [see for example Volk(58)] and are easily programmedo -127

-128TABLE XIX LIST OF INPUT VARIABLES FOR COMPUTER PROGRAM Height height of sample size (0O5 inches) Width width of sample size (0o4 inches) Depth depth of field Const 1 1,0 Const 2 1o0 Dodia if equal to 0 will not compute any average diameters Docum if equal to 0 will not compute cumulative percentages Dopin if equal to 0 will not compute per cent per size interval Survol if equal to 0 will not compute surface and volume flow rates Dogem if equal to 0 will not use geometric means M number of locations N number of size intervals S highest power on mean diameter (usually 3) VOLV if equal to zero will not compute volume distributions KE if equal to 0 will not compute kinetic energies MOMN if equal to 0 will not compute momentum flux SUPER if equal to 0 will not consider the spray as superheated BAD 0 (controls error check in program) Dens (I) fluid density at i-th location Radius (I) radial vector from nozzle to center of i-th location Angle (I) angle between radius vector and vertical Photos (I) number of photographs at i-th location

-129TABLE XIX (CONT'D) Endpoint (I) endpoint of size range (begins at zero) Cell (I) size interval of each size range NUM (I) numerator in mean diameter calculation DEN (I) denominator in mean diameter calculation eogo, NUM = 1; DEN = 0 is the linear mean diameter SPEED (I)J) velocity of drops in i-th size interval, j-th location NUMBER (IJ) number of drops in i-th size interval, j-th location DM nozzle diameter DELP pressure drop across the nozzle VISC fluid viscosity DG surrounding gas density SIGMA fluid surface tension H21 enthalpy of superheated liquid H2F enthalpy of liquid at saturation point H1F enthalpy of vapor at saturation point RH02 density of liquid RHO1 density of vapor K fluid thermal conductivity SPHT specific heat of liquid

-130TABLE XX COMPUTER PROGRAM DIMENSION SUM(3b), SUMN(30) DIMENSION RADIUS(20), ANGLE(20), PHOTOS(20;, CELL(30), 1 DPK 0 ENDPT (60), TDIAM (1G0), W (30), AREA (20), VOQLLJE_ (20l _ _PK____ 1 G(30), NUM(10), DEN(1), RATIO(30),RAT2(30),DENS(30) 3 DPK DIMENSION SPEED (300,D), NUMBER (300,D), PCT (300,D), 4D —K 0 CUMPCT (30U,D), PCTINT (30OD), DIAM (300,B), F (300,C1), 5 DPK 1 X(300,D), Y(3'u,D1),MASS(3UU,D),MOMl(3UOD),ENERGY(300D) 6 MPK VECTOR VALU ES- B 2t1,VECTOR VALUES Cl = 2,2,0 VECTOR VALUES D = 2,1,0 7 DPK VECTOR VALUES D1=2,2,U 7ADPK START READ FORMAT DCARD1, HEIGTH, WIDTH, DEPTH, CONST1, CONST2, 8 DPK 1 DODIA,DOCUM,DOPIN,SURVOL,DOGEM,M, N S,VOLVKE VECTOR VALUES DCAR.D1 = $5F10.5,10I3*$ READ FORMAT CARD, MOMN, SUPER, BAD VECTOR VALUES CARD =$313*$ INTEGER MN,S,DODIA DOCUM,DOPIN,SURVOL,DOGEM, I,J,R 10ADPK INTEGER NUM, DEN, PHOTOS, VOLV,KE, MOMN, BAD B(1) = M+1 8(2) = M C1(1) = M+1 C1(2) = M D(2) = N 11 DPK D1(2)=N+1 11ADPK READ FORMAT DCARD, DENS(1)*..DENS(M) VECTOR VALUES DCARD=-$(8F10.4)-$ —--- READ FORMAT DCARD2, RADIUS(1)*.*RADIUS(M) 12 DPK ----------— _ —_ —-__-_^^-^-_^^^-^^^_ —^_ V E CTOT-VALUESD C A R D 2- =$( F0-1-*$ 13 D7K —---- READ FORMAT DCARD3, ANGLE(1)..*AN ELE(M) 14 DPK VECTOR VALUES DCARD3 = $(8F10.5)*$ 15 DPK READ FORMAT DCARD4, PHOTOS(1)...PHOTOS(M) 16 DPK ---— _- --— _-_V VALs --— _ —-----— _-___ —-_ —----------------------------- READ FORMAT DCARD5, ENDPT()*...ENDPT(N) 18 DPK' —-- -----------— VE —T V-c —_ —--— T --- -------- ----------- ------- READ FORMAT DCARD6, CELL(1)*.. CELL(N) 20 DPK VECTOR VALUES DCARD6 = $(8F10.5)*$ 21 DPK READ FORMAT DCARD7, NUM(1)**SNUM(S+l) ----------------------- -— _- -- - -- - - — _D K - - VECTOR VALUTS C ARD7 - $1U05*$ 23 DTK READ FORMAT DCARD8, DEN(1).**DEN(S+1) VECTOR VALUES DCARD8 = $105'*$ 25 DpK READ FORMAT MAT1, SPEED(1,1).*.SPEED(MN) 26 DPK VECTOR VALUES MAT1 = $(8F10U5)*$ 27 DPK READ FORMAT MAT2, NUMBER(11)... NUMBER(MN) 28 DPK VECTOR VALUES MAT2 = $(8F10.5)*$ 29 DPK WHENEVER DOGEM E.O, TRANSFER TO RED W(1) = (ENDPT(1)) P.( 5) THROUGH DOG, FOR J=2t1,J.G.N DOG W(J) = (ENDPT(J)*ENDPT(J-1)).P.(.5) TRANSFER TO OUT RED THROUGH.RET4, FOR J=ll9JJG.N W(J)=(ENDPT(J-1)+ENDPT(J))/2. OUT CONTINUE 29FDPK RET4 CONTINUE 29GDPK WHENEVER DOGEME.0O, TRANSFER TO AMEAN1 29HDPK PRINT FORMAT RES23 29IDPK VECTOR VALUES RES23=$38H1THIS COMPUTATION USES GEOMETRIC MEAN 29JDPK OS*$ 29KDPK TRANSFER TO OUT1 29LDPK AMEAN1 CONTINUE 29MDPK PRINT FORMAT RES24 29NDPK

-151TABLE XX (CONT'D) VECTOR VALUES RES24=$39H1THIS COMPUTATION USES ARITHMETIC MEA 290DPK ON S $ 29PDPK ----------------------------------------- 29DPK OUT1 CONTINUE 29QDPK THROUGH RET1, FOR I=1,1,1IGM 30 DPK ANGLE(I)=ANGLE(I)*3,14159/l&1O 30ADPKA WHENEVER ANGLE(I),NE.UQ, TRANSFER TO NOZERO 30ADPK RATIO(I)=1. 30BDPK _ — RAT2_1) —__=-_1. —------ TRANSFER TO ZERO 30CDPK NOZERO RATIO(I)=8.*RADIUS( I )*SIN (ANGLL( I ))/WIDTH 30DDPK RAT2(I) = RADIUS(I)/RADIUS(1) -------------------------------- -------- Y(I,O) =0.0 THROUGH RET1, FOR J=1,1,J.G.N 32 DPK X(IJ)=NUMBER(IJ)*SPEED(I,J)/PHOTOS(I) 33 DPK Y(I,J)=X(I,J)+Y(I,(J-1)) 34 DPK RET1 CONTINUE 35 DPK THROUGH RET2, FOR I=1,1,I.G.M 36 DRK THROUGH RET2, FOR J=1,1,J.GN 37 DPK PCT(I,J)=X( I,J)*100./Y( I N) 38 DPK WHENEVER DOCUM.E.0, TRANSFER TO NOCUM 39 DPK CUMPCT(I,J)=Y(I,J)*100./Y(I#N) 40 DPK NOCUM WHENEVER DOPINE.0O, TRANSFER TO NOPIN 41 DPK PCTINT(I,J)=PCT(I,J)/CELL(J) 42 DPK NOPIN CONTINUE 43 DPK RET2 CONTINUE 44 DPK PRINT FORMAT RES1 45 DPK VECTOR VALUES RES1=$42HOVELOCITY WEIGHTED PERCENTAGES BY LOCA 46 DPK OTION*$ 46ADPK THROUGH RET3, FOR I=1,1,I.G.M 47 DPK PRINT FORMAT RES2,I 48 DPK VECTOR VALUES RES2=$1HOLOCATION I3*$ 49 DPK — ______FR~T- ORMAT RES3 50 DPK VECTOR VALUES RES3=$11H SIZE RANGE,S24,6HNUMBERS510, 51 DPK 0 5HSPEED,S8,7HPERCENT*$ 52 DPK WHENEVER DOCUME.O, TRANSFER TO NOCUM1 53 DPK PRI NT FORMAT RE S4 54 D-K — VECTOR VALUES RES4=$1H+,S73,18HCUMULATIVE PERCENT*$. 55 DPK NOCUM1 WHENEVER DOP-INE*O, TRANSrER TO NOPIN1 56 DPK PRINT FORMAT RES5 57 DPK VECTOR VALUES RES5=$1H+,S93,18HPERCENT/SIZE RANGE*$ 58 DPK NOPIN1 CONTINUE 59 DPK THROUGH RET3, FOR J-=,1fJGeN 60 DPK PRINT FORMAT RES6,ENDPT(J-1),ENDPT(J),NUMBER(IJ),SPEED(IJ), 61 DPK O PCT(I J) 62 DPK VECTOR VALUES RES6=$1H,F10.5,4H TO,F10l5,S5,Fl0o5,S5, 63 DPK O F10.5,S5,,FlO5*$ 64 DPK WHENEVER DOCUMoEO, TRANSFER TO NOCUM2 65 DPK PRINT FORMAT RES7, CUMPCT(IJ) 66 DPK VECTOR VALUES RES7=$1H+,S8 0,F10.5*$ 67 DPK NOCUM2 WHENEVER DOPINNE.O, TRANSFER TO NOPIN2 68 DPK PRINT FORMAT RES8, PCTINT(IJ) 69 DPK VECTOR VALUES RES8=$1H+,S100,F10,5*$ 70 DPK NOPIN2 CONTINUE 71 DPK RET3 CONTINUE 72 DPK PRINT FORMAT RES30 72ADPK PRINT FORMAT RES31 72BDPK VECTOR VALUES RES30-$36H1TOTAL VELOCITY WEIGHTED PERCENTAGES* 72CDPK 0$ 72DDPK

-132ABLE XX (CONT'D) VECTOR VALUES RES31=$11H SIZE RANrE,S23,7HPERCENT*$ 72EDPK __-_WHENEVER DOCUM.E.O, TRAiNSFER_ QQ _ ___________ _________. 72FDPR —___ PRINT FORMAT RES32 72GDPK VECTOR VALUES RES32=$1H+,S46,18HCUMULATIVE PERCENT-$ 72HDPK NOCUM3 WHENEVER DOPIN.E.O, TRANSFER TO NOPIN3 72IDPK PRINT FORMAT RES33 72JDPK VECTOR VALUES RES33=$1H+,Sl1,18HPERCENT/SIZE RANGE*$ 72KDPK NOPIN3 CONTINUE 72LDPK G(O) = 0.0 TOTAL = 0.0_ THROUGH RET11, FOR J=1,1,J.G.N 720DPK G(J) = 0.0 THROUGH RET11, FOR I=1,1,I.G.M 72QDPK X(I,J)=X(I,J)*RATIO(I)*RAT2(I) 72RDPK G(J)=X(I,J)+G(J) 72SDPK TOTAL = TOTAL+X(I,J) 72TDPK RET11 CONTINUE 72UDPK CUMPCT(M,N)=O. 72UDPKA THROUGH RET12, FOR J=l,,J.G.N 72VDPK PCT(MJ)=G(J)O*100./TOTAL 72WDPK CUMPCT(M N)=PCT(MJ)+CUiPCT(M, N) 72XDPK PCTINT(M,J)=PCT(MJ)/CELL(J) 72YDPK PRINT FORMAT RES34, ENDPT(J-l), ENDPTF(J),C iT (,J) 72ZDPK VECTOR VALUES RES34=$1H,F1O.5,4H TO,Flu,55,FFlu.$ - A72ADPK --- ---— WHERTEVER DOCUM.E.O,- TRANSFER TO NOCJM4- A72BDPK PRINT FORMAT RES35, CUMPCT(M,N) A72CDPK VECTOR VALUES RES35=$1H+,S54,,Fl0.5- A72DDPK NOCUM4 WHENEVER DOPIN.E.O, TRANSFER TO tlOPil ____ _ A72EDPK PRINT FORMAT RES36, PCTINT(M,J) A72EDPKA VECTOR VALUES RES36=$1H+,S_/9_,F. __ __________ __ —_ A72FDPK NOPIN4 CONTINUE A72GDPK RET12 CONTINUE A72HDPK WHENEVER VOLV.E.O, TRANSFER T O CAT THROUGH RET20, FOR I=1,1,I. G. ___A72LDPK Y(I,O)=O A72MDPK THROUGH RET20, FOR J=1,1,J.G.N A72NDPK ----------— T —-— J —_X,=U BE( -'; E I, * FJ ) (3 —. _ —---- ) —---— PHTO (I) A720D KY(IJ)=X(IJ)+Y(I,(J-1)) A72PDPK RET20 CONTINUE A72QD-7K THROUGH RET21, FOR I=ll,,I.G.M A72RDPK THROUGH RET21, FOR J=1,1,J.G.N A72SDPK PCT(IJ)=X(I,J)*100./Y(IN) A72TDPK WHENEVER DOCUM.E.U, TRANSFER- TO NOCU,'i,5 A2U K CUMPCT(I,J)=Y(I,J)*1JU./Y( IN) A72VDPK NOCUM5 WHENEVER DOPIN.E.O, TRANSFER TO NOPIN5 A72WDPK PCTINT(I,J)=PCT(I,J)/CELL( )) A72XDPK NOPIN5 CONTINUE A72YDPK RET21 CONTINUE A72ZDPK PRINT FORMAT RES40 B72ADPK VECTOR VALUES RES40=$49HlVOLUME-VELOCITY WEIGHTED PERCENTAGESB72BDPK 0 BY LOCATION*$ B72CDPK THROUGH RET22, FOR I=,11,I.G.M B72DDPK PRINT FORMAT RES41,I B72EDPK VECTOR VALUES RES41=$10HOLOCATION 13*$ B72FOPK PRINT FORMAT RES42 B72GDPK VECTOR VALUES RES42=$11H SIZE RANrES24,6HNUMBERS10,5HSPEEDB72HDPK 0 S8,7HPERCENT*$ B72IDPK WHENEVER DOCUM.E.O0 TRANSFER TO NOCUM6 B72JDPK PRINT FORMAT RES43 B72KDPK

-153 - TABLE XX (CONTD) VECTOR VALUES RES43=$1H+,S73,18HCUMULATIVE'ERCENT'$ 672LDPK NOCUM6 WHENEVER DOPIN.E. O TRANSFER TO NOPIN6 672 Mi PK PRINT FORMAT RES44 872N PK VECTOR VALUES RES44=$1H+,S93,18HPERCENT/SIZE RANGE-$ B720DPK NOPIN6 CONTINUE 72 PPK THROUGH RET22, FOR J=1,1,J.G.N B72QDPK PRINT FORMAT RES6,ENDPT(J-1 ),ENDPT(J),NUMBER( I,J ),S PEE'D ( I,J ),772RD-PK-. O PCT(IJ) 872SDPK WHENEVER DOCUM.E.O, TRANSFER TO NOCUM7 B72TDPK PRINT FORMAT RES7,CUMPCT(I,J) U72UDPK NOCUM7 WHENEVER DOPIN.E.O, TRANSFER TO NOPIN7 672VDPK PRINT FORMAT RES8, PCTINT(I,J) ---- 2WDPK NOPIN7 CONTINUE 672XLPK RET22 CONTINUE B72YDPK PRINT FORMAT RES45 B72ZLPK VECTOR VALUES RES45=$43H1TOTAL VOLUME-VELOCITY WEIGHTED PEiCE ONTAGES*$ C72BDPK PRINT FORMAT RES31 C72CDPK ---------— CO —M.O, —- T 0RANSr 0 NOCCUM8 C72DDPK PRINT FORMAT RES32 C72EDPK NOCUM8 WHNVR-~DOTNI. —, -TANS-CR T 0 NOP IN8 C72FFDPK PRINT FORMAT RES33 NOPIN8 CONTINUE TOTAL=O. C72IDPK ------------ RR-~RTT — FR'J —I - — JG —--- N C72 JD PK G(J) = 0.0 --------------- - —;- ~-r,- - M - - C- 7 —2 LD7K- --- X(IJ) =X( I,J)*RATIO(I )RAT2( I ) G(J)=X( I, J)+G(J) C72MDPKA TOTAL=TOTAL+X(I,J) C72NODPK RET23 CONTINUE C720DPK CUMPCT(M N) =0. C720L)PKA THROUGH RET24, FOR J=1,1,J.G.N C72PDPK PCT(M J)=G(J)* 100./TOTAL C72QDPK CUMPCT(M N)=PCT(M,J)+CUMPCT(M N) C72RDPK PCTINT(MJ)=PCT(MJ)/CELL(J) C72SDPK PRINT FORMAT RES34, ENDPT(J-1),ENDPT(J),PCT(M,J) C72TDPK WHENEVER DOCUM.E.O, TRANSFER TO NOCUM9 C72UDPK PRINT FORMAT RES35, CUMPCT(MN) C72VDPK NOCUM9 WHENEVER DOPIN.E.O, TRANSFER TO NOPIN9 C72WDPK PRINT FORMAT RES36, PCTINT(MJ) C72XDPK NOPIN9 CONTINUE C72YDPK RET24 CONTINUE C72ZDPK CAT WHENEVER DODIA*.E.O TRANSFER TO NODIA PRINT FORMAT RES9 81 DPK VECTOR VALUES RES9=$33H1DROP DIAMETERS FOR EACH LOCATION*$ 82 DPK THROUGH RET5 FOR R=O,1,R.G.S 89 DPK THROUGH RETS, FOR I=1,1,I.G*M 90 DPK F(RI)=O0 93 DPK THROUGH RET5S FOR J=1,l,J.G.N 94 DPK ____ - - -- - - _______________ —---- ---------------------------- F(RI)=(W(J) P.R)*NUMOER( IJ)*SPEED(IJ)/PHOTOS(I) 95 DPK 0 +F(RI) _ 96 DPK RET5 CONTINUE 97 DPK THROUGH RET6, FOR I=l,1- I*GeM 98 DPK PRINT FORMAT RES14, I 99 DPK VECTOR VALUES RES14=$1lHULOCATION 3*$ 100 DPK PRINT FORMAT RES11.101 PK VECTOR VALUES RES11=$1H,S11,9HNUMERATOR,S9,11HDE!NOMINATOR, 102 DPK 0 S12,8HDIAMETER*$ 103 DPK

-134TABLE XX (CONT'D) PRINT FORMAT RES12 104 DPK VECTOR VALUES RES12=$1H _S1l,2 8HEXPONNT1D5___RKi____EP_ _NI___ 1 P_2_.______. THROUGH RET6, FOR R=U,1,R.G.S 106 DPK DIAM(RI) = (F(NU (R+1),I )/F(DEN(R+1),I )).P.(1. / (NUM(R+1 )-DE 1N!(R+1))) PRINT FORMAT RES13,NUM (R+F1 )DEN(R+1) DIAtMl(RI) ---------— v- VE C'T? —7 - VA —UES —-R- E1 —---- --- 1, — 7I —- -— S —, —-7 —3 —,F —--—.- 10,5 $ 1 -- 1 9 K RET6 CONTINUE 110 DPK PRINT FORMAT RES15 111 DPK VECTOR VALUES RES15=$16H1TOTAL DIAMiETERS*-$ 112 DPK PRINT FORMAT RES16 113 DPK VECTOR VALUES RES16=$1H,S11,9HNUMERATORS9,11HDENOMINATOR, 114 DPK -----— o-^^^^^^- ---------- ------ -------------------- O S15,5HTOTAL-*-$ 115 DPK PRINT FORMAT RES17 116 DPK -------------------- ------------------------- ---- ---------------— v-6-DP VECTOR VALUES RES17=$1H,S12,8HEXPONENT,512,8HEXPONENT, 117 DPK 0 S128HDIAlMETER*$ 118 DPK THROUGH RET7, FOR R=OlR.G.S 119 DPK G(R)=O. 120 DPK THROUGH RET7, FOR I=1,1,I.G.M 122 DPK ---— _ —— _ -------- r G(R )_-=_F(_R I)'RAT IO_(I-'R- AT2(_I )+_ —-- R- -------------------------------- --— R —--- G(R) = F(R,I)*RATIO(I)*RAT2(I) + r-(R) RET7 CONTINUE 125 DPK THROUGH RET8, FOR R=O,1,R.G.S 126 DPK TDIAM(R) = (G(NUM(R+1))/G(DEN(R+l))),P.(10/(NUM(R+l)-DEN(R+i 1))) --- ------ ) —------------------------------------- PRINT FORMAT RES18,NUM(R+1' )DEN(R+l),TI IAM(R) VECTOR VALUES RES18=$1H,S17,I3,S17,I3SSl,3FlC.5*$ 129 DPK ----- RT —---------- -E —------- ------------------- - ---------------------- -D7K ------- RET8 CONTINUE 13 —K NODIA CONTINUE 131 DPK WHENEVER SURVOL.E.LO TRANSFER TO NOSV 132 DPK PRINT FORMAT RES19 133 DPK -------------- ---------- — 7 - R -- -F —E - - ------- OCATION PER UNIT AREA*- 134ADPK.-.-.... —-' —---— R GT-R U-R —--- --------------------------------------------- S-DK —--- PRINT FORMAT RES20,I 136 DPK VECTOR VALUES RES20=$10HUCLOCATION I3'-$ 137 DPK VOLUME(I)=((3.14159)*F(3,I)i*CONST2/(HEIGTH*'WIDTH*DEPTH*6.))_*( ---------------- T T- --------------- ----------------------------------------- ------ AREA(I)=(( 3.14159) *F(2, 2I )*CONST1/(HEIHGTH*WIDTH*DEPTH) )*(541E 1-11) PRINT FORMAT RES21, AREA(I) VOLUME(I) 143DDPK VECTOR VALUES RES21 =$10H SURFACE =E13.5,S108HVOLUME = E13.5 —$ —------- RET10 CONTINUE 143GDPK PRINT FORMAT RES50 143HDPK VECTOR VALUES RES50=$51H1TOTAL SURFACE AND VOLUME RATES (NOT 143IDPK OPER UNIT AREA)*$ 143JDPK THROUGH RET25, FOR I=11,IIG.M AREA(I_) = (AREA(I) -*3.14159*(WIDTHP.2)/4.o*RATIO(I)*RAT2(I))* 11728.0 VOLUME(I) = (VOLUME(I)*-3.14159*(WIDTH.P*2)/4.0*RATIO(I)*RAT2( I — )* 1 728 Ui —---------- PRINT FORMAT RES20, I PRINT FORMAT RES21, AREA(I), VOLUME(I) WHENEVER BAD.NE.O.AND.I.E.1 --------------------------------— T- ------- -- - -------------- -------------- PRINT RESULTS I, RATIO(I), RAT2(I), Fo..F(200), AREA...AREA(2 10), VOLUME...VOLUME(2U) ------------------— EXECUTE ERROR. END OF CONDITIONAL RET25 CONTINUE 143PDPK

-135TABLE XX (CONT'D) ----------------------------- AREA(O) = ( VOLUME(O) = 0,0 AREA(I) = AREA(I) + AREA(I-1) VOLUME(I) = VOLUME(I) + VOLUME(I-1) GREEN CONTINUE PRINT FORMAT GONE, AREA(M), VOLUME(M) VECTOR VALUES GONE =527HL. TOTAL SURFACE, SQ FT/HR = E13,5,S10 ---------------------— Oil_:l52_7^. —_ —_ 1,26HO TOTAL VOLUME, CU FT/HR = E13*5*$ NOSV CONTINUE 144 DPK THROUGH PINK, FOR I=1,1,I.G.M THROUGH PINK, FOR J=1,1,JGN — MASS(IJ) =((((( u(O.u1*W(J))/(12. 0*254),P,3)*3.1415 NUMSER( I1 J))*DENS(I)/6.0)*(SPEED(I J))/PHOTOS(I) ---- WHENEVER KE.E.O, TRANSFER TO NOKE ENERGY(I,J)= (1728.u*MASS(I,J)-*(SPEED( I,J) P*2) )/(64.4*HEIGTH 1*WIDTH*DEPTH) NOKE CONTINUE WHENEVER MOMN.E.O, TRANSFER TO NOMOM'N PINK MOM(I,J) = (MASS(IJ)*SPEE-D(IJ)*t1728.u)/(HLIGTH*'WIDTH*DEPTH) NOMOMN. CONTINUE THROUGH ALPHA, FOR I=1,1,I.G.M PRINT FORMAT ABLEI VECTOR VALUES ABLE=$1OHOLOCATION I3$ PRINT FORMAT BAKER VECTOR VALUES BAKER=$11H SIZE RANrE,S23,8HMOMiENTUiSlOl,4HKIN iETIC ENERGY*$ -~ THROUGH ALPHA, FOR J=1t,1,JGN PRINT FORiMAT CHARLENDPT(J-1),ENDPT(J),DPTJMOM(IJ),ENERGY(IJ) VECTOR VALUES CHARL=$1H,F10.5,4H TO,Fl0.5,S5,E13.5,S5,E13.5 ALPHA CONTINUE __ ___ PRINT FORMAT FRED VECTOR VALUES FRED =$9HOLOCATIONS8,21HKINETIC ENERGY(FT LB), 1S22,19HMOMENTUM(FT LB/SEC)*$ THROUGH BLUE FOR I=,1,__ I G.M SUM(O) = 0.0 SUMN(O) = 00 _ _ _ ________ _____ _____ ___ THROUGH BLUE FOR J=1,1, JG.N ENERGY(I,J) =(ENERGY(IJ)*RATIO(I)*RAT2(I))/24,0 MOM(I,J) =(MOM(IJ)*RATIO(I)*RAT2(I))/24.0 SUM(J) = ENERGY(IJ) + SUM(J-1) SUMN(J) = MOM(IJ) + SUMN(J-1) SUMN(J) = MOM(I,J) + SUMN(J) WHENEVER J.E.N, PRINT FORMAT EASY, I, SUM(J), SUMN(J) VECTOR VALUES EASY =$S5,I3,S8BE13s5,S2UE13.5*$ BLUE CONTINUE WHENEVER SUPERE.O, TRANSFER TO NOSR READ FORMAT DCARD9, DM, DELP, VISC, DGI SIGMA, DENSTY, H21, H 12F READ FORMAT CARD1, H1F, RH02, RH01, K, SPHT VECTOR VALUES DCARD9 =$4F10,5, E10O5, 3F10.5*$ VECTOR VALUES CARD1 =$5F10.5*$ VEL = (64,4*144.0*DELP*DENSTY).P(0,.5) --------------- -- ----- --- -------------------------------- ------ RE = (DM*VEL*DENSTY )/VISC NWE = (DG*VEL*VEL*DM)/(6444*SIGMA) FLOW = (341416*DM*DM/4.0)*VEL*3600 O*^ PDEV = ((FLOW-VOLUME(M))/VOLUME(M)*100O _____ WTPC = (H21-H2F)/(H1F-H2F) GRC = WTPC*(RHO2/RHO1)*(3.1415*K/(RHO2*SPHT)) _________ ___ PRINT FORMAT HE__P ___ VEL, _RE, NW ______________ VECTOR VALUES HELP =$18H VELOCITY(FT/SEC)= E13.4,S8,18H REYN 10LDS NUMBER = E13.4,S8, 15H WEBER NUMBER = E13.4*$ PRINT FORMAT JOHN, FLOW, PDEV, WTPC, GRC VECTOR VALUES JOHN = $7H PDEV = F6.2,SB,18H PERCENT FLASHED,_: 1F6.2,S8,.23H GROWTH RATE CONSTANT = F6.2*$ NOSR CONTINUE TRANSFER TO START END OF PROGRAM 145 DPK

APPENDIX B RAW DATA This appendix contains all of the drop size and velocity distribution data which was taken in the course of the study, The data books and the photographic negatives are located in the Multi-Phase Fluids Laboratory in the Fluids Building at the North Campus of the University of Michigan, Ann Arbor, Michigano -136

-137TABLE XXI DROP SIZE RANGES Size Range Average Diameter Size Range No. (Microns) (Microns) 1 0 - 10.0 5.0 2 loO - 14.1 12.05 3 14.1 - 20.0 17-05 4 20.0 - 28.2 24.1 5 28.2 - 40o0 34.1 6 40.0 - 56.4 48,2 7 56.4 - 80o,0 68.2 8 80.0 - 113 96.5 9 113 - 16o 136.5 10 160 - 226 193 11 226 - 320 273 12 320 - 453 386.5 13 453 - 640 546.5

-1-38TABLE XXII DROP SIZE DISTRIBUTION AND VELOCITY DATA Jet Diameter = 0.031 Inches Run No. 1. Water at 90 psig, 2870F, 4 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 9 10 11 12 1 2 9 67 300 301 172 69 26 10 1 2 3 14 85 239 386 255 93 30 8 3 4 5 25 123 90 21 4 4 4 3 19 31 97 88 14 3 Average Velocities in Each Size Range (Ft/Sec) 1 4 10.0 15.0 20.0 31.4 41.3 55.5 100.2 103.1 111.0 119.5 2 4 10.0 11.6 15.2 23.2 30.4 41.3 66.5 76.1 97.8 3 4 8.3 12.2 18.9 31.3 48.3 73.5 4 4 7.5 10.9 10.3 15.2 26.3 48.5 73.2 Run No. 2. Water at 120 psig, 2870F, 4 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 9 10 11 12 1 2 5 32 132 193 92 28 22 14 2 2 2 12 78 231 256 100 46 24 20 11 2 3 2 44 238 304 182 135 53 22 7 1 4 4 5 34 71 79 96 103 22 6 Average Velocities in Each Size Range (Ft/Sec) 1 4 14.0 16.0 21.6 31.8 39.9 41.6 97.3 105.0 117.5 127.0 133.0 2 4 10.0 13.9 18.3 33.4 38.4 48.3 83.2 106.o 110.0 123.0 125.0 3 4 5.0 7.4 13.4 18.3 22.0 36.6 55.8 83.7 93.9 4 4 3.0 4.4 7.2 11.3 16.9 27.3 47.3 64.8 Run No. 3. Water at 120 psig, 287~F, 5 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 9 10 11 12 1 3 46 171 157 141 126 68 37 15 14 7 9 2 3 43 209 204 147 124 122 41 14 3 3 3 2 4 32 56 92 37 28 23 5 4 2 1 1 Average Velocities in Each Size Range (Ft/Sec) 1 4 7.4 8.7 12.9 17.1 28.4 38.4 85.1 94.0 101 106 110 2 4 6.7 7.9 12.6 13.6 15.6 20.1 37.2 55.1 73.9 86.1 3 4 4.2 4.6 5.8 7.9 11.3 16.1 32.6 41.7 4 4 11.7 25.9 36.7

-139TABLE XXII (CONT'D.) Run No. 4. Water at 120 psig, 2870F, 6 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 9 10 11 12 1 3 12 111 179 187 137 67 23 9 2 3 24 112 138 145 125 84 30 7 6 3 3 16 39 54 90 99 79 39 9 7 4 3 20 75 123 126 114 115 30 12 5 Average Velocities in Each Size Range (Ft/Sec) 1 4 10.9 14.7 18.4 22.5 31.4 44.3 63.9 81.1 2 4 10.0 11.7 14.2 20.4 28.4 38.4 52.3 68.5 84.4 3 4 9.2 10.9 15.2 18.0 23.8 30.9 41.7 54.3 72.7 83.3 4 4 5.9 8.8 10.9 15.8 21.3 27.2 36.0 47.5 65.1 Run No. 5. Water at 120 psig, 2870F, 7 inche's from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 9 10 11 12 1 3 27 72 73 60 44 32 16 6 6 2 5 2 2 4 19 41 101 80 44 15 1 1 3 2 6 18 45 122 207 116 27 2 4 2 1 14 44 54 20 4 Average Velocities in Each Size Range (Ft/Sec) 1 4 10.0 12.0 20.5 24.6 29.2 37.6 58.5 75.1 90.2 99.5 110 2 4 10.4 13.4 15.5 18.4 21.7 27.6 54.3 71.0 81.7 3 4 8.0 8.7 12.1 13.4 16.7 19.2 28.8 45.1 4 4 9.2 10.0 10.1 10.2 10.3 14.2 Run No. 6. Freon 11 at 90 psig, 158~F, 4 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 9 1 4 72 305 536 314 75 14 2 4 15 33 51 31 12 2 Average Velocity in Each Size Range (Ft/Sec) 1 4 7.7 10.0 14.5 22.9 30.0 42.0 2 4 5.0 6.3 10.3 15.9 22.4 28.0 Run No. 7. Freon 11 at 120 psig, 158~F, 4 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 9 1 2 58 374 565 310 35 1 2 2 62 286 631 325 59 6 3 4 27 112 258 162 24 1 Average Velocities in Each Size Range (Ft/Sec) 1 4 9.0 13.9 22.6 32.1 43.4 58.9 2 4 8.5 10.0 17.3 21.7 30.6 42.8 5 4 5.5 12.0 18.5 27.7 39.0 51.0

-140TABLE XXII (CONT'D.) Run No. 8. Freon 11 at 120 psig, 158~F, 5 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 1 1 65 225 263 78 5 2 1 23 149 200 89 9 3 3 2 2 39 111 78 11 Average Velocities in Each Size Range (Ft/Sec) 1 3 8.6 13.0 17.8 25.5 38.5 2 3 8.5 10.3 14.1 20.9 28.7 43.0 3 3 8.0 8.5 11.8 18.1 24.4 38.2 61.7 Run No. 9. Freon 11 at 120 psig, 158~F, 6 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 1 2 8 68 106 44 3 1 2 3 30 232 573 431 89 8 3 3 34 179 450 430 137 26 Average Velocities in Each Size Range (Ft/Sec) 1 3 8.5 9.8 20.5 25.0 41.5 49.0 2 3 9.0 9.8 16.7 21.0 33.0 39.5 3 3 9.0 9.8 13.0 18.0 25.3 30.3 Run No.10. Freon 11 at 120 psig, 158~F, 7 inches from Nozzle Location Photos Number of Drops in Each Size Range 2 3 4 5 6 7 8 1 3 11 133 359 524 316 68 7 2 2 3 36 189 443 418 117 9 3 3 13 133 324 358 120 17 Average Velocities in Each Size Range (Ft/Sec) 1 3 9.0 14.5 21.0 29.2 43.0 63.5 68.0 2 3 6.0 10.0 16.0 22.5 34.0 52.7 57.5 3 1 4.5 7.5 13.0 18.0 27.8 41.7 46.5

99.9 99 95 90 Li 80 z 70 Lu 60 50 40 /) Un 30 Li U) 20 0L 10 0 z ~ Li ir cl 2 0.5 0.2 0.1 0.01 10 20 50 100 200 500 DROP DIAMETER, MICRONS Figure 5533. Distribution Plot for Run 1.

-14299.9........ 99 IJ - 95 z 90 z 80 I 0-_ —--- - 70 cn 60 w 116 -J 50 a. 40 0 o 30 0 20 OI0 10 20 50 100 200 500 DROP DIAMETERibution Plot fICRONS Figure 54. Distribution Plot for Run 3.

-143 - 99.9 99 w W 80 95 70 70 Z 60 cn 50 CI) - 40 a0 30 ii,: 020 0I z 10 C,) 5 a2 0.5 0.2 0.1 10 20 50 100 200 500 DROP DIAMETER, MICRONS Figure 35. Distribution Plot for Run 5.

99.9 99 W 95 N 90 ( /80 I 70 60 W 50.J cn 40 CL 0 I 30 La- 20 0 z (. 5 10 20 50 100 DROP DIAMETER, MICRONS Figure 36. Distribution Plot for Run 7.

-14599.9__ _ _ _ I23 99 z > 90 0 70 cn cn 50 I 70 6 00 -J 0 0 0. / 2 10 20 50 100 DROP DIAMETER,MICRONS

-14699.9 99 95 0:/ W 90 w 0 80: w 70 (D 60 z 50 <I I.- 40 0') U) 40 /). 30 0. 20 0 0 LL 10 -- 0 I — z 5 w 0. 2 0.5 0.2 0.1 0.01 10 20 50 100 DROP DIAMETER, MICRONS Figure 38. Distribution Plot for Run 9.

-14799.9 99 95 Ia 90 _ 80 Z 70 CD 60 z < 50 H- 40 ~) a. 20 0 0 0 \ 5 U 2 0.5 0.2 0.1 aoII -- 10 20 50 100 DROP DIAMETER, MICRONS Figure 39. Distribution Plot for Run 10.

APPENDIX C BUBBLE GROWTH IN A SUPERHEATED VISCOUS LIQUID The information presented in this appendix is intended to acquaint the reader with the methods used to predict the bubble growth curves shown in Figure 10o The method is due to Poritsky(41) although he did not specifically solve the problem for any of the fluids or conditions given here. The equations of motion are given by: e: g -Vp -+ t (V.V + > 7V (c-l) where v is a velocity vector a is the acceleration vector given by L (-at + v - Wv (C-2) Upon first investigating the problem one is confronted with a paradox: that there is apparently no difference between the equations which govern the growth of a spherical vapor bubble in a perfect (inviscid) liquid and those governing its growth in a viscous liquid~ If the fluid is assumed to be incompressible the second term on the right hand side of Equation (C-l) vanishes. It is also true that the Laplacian of the velocity vector v is zero. This arises from the fact that, because of the spherical symmetry of the problem, a potential function may be defined and hence the Laplacian is zero0 Thus the paradox presents itself, ~l~48

-l49Poritsky resolved this disconcerting dilemma by noting "that while it is true that the effect of viscosity vanishes in the equation of motion, so that the resultant of the viscosity stresses per unit of volume at any point internal to the fluid vanishes, this is not necessarily the case with the stresses themselves." Lamb(25) has shown that at any point the three principal stresses Pi and the three principal strains E. are related by: 1 =-P +2 P. = - — /(c, + C3 + ~5) + 2 i\ (c-5) P2 = -P - |^(e, + Z + 3) + eL~E2 (C-3a) P5 = P - -(f, + 2 + 3) + 3 (C-5b) where the pressure P is the mean of the negative of the three principal stresses P1. P2, P3. Poritsky applies the above three equations to the free spherical surface of the cavity inside which he supposes that a constant pressure PO exists. Note that PO is not the value of P in the above equations, existing at the boundary within the fluid, but is calculated from the negative of the proper principal stress, thus arriving at the relation: P, = p - 2c (C-4) Applying this equation to the problem of determining the rate of growth of the radius R of the bubble one gets: - 3 (d + 4 -dR (C-5) dt2 ^dti PR - \dc t)

-150It should be noted that for the case of an inviscid liquid (l = 0) Equation (C-5) becomes identical to Equation (5.2)o Introducing the dimensionless parameters - P- P V = R Ra Ro/e(P -PO) one obtains the following differential equation: (d2^ )+ ^ 3 ( -~C 0-'-I dP (C-6) d ^- - - -I — If one equates the work per unit solid radian by the pressure PO to the kinetic energy acquired by the fluid, starting from rest, the energy relation given below results: (Po-PoRbo3-R3) + -R ) + 4 fR(dt = O (C-7) 0 Introducing the same dimensionless variables as before leads to the equation: _ 1(do)^ _Cd_= (c-8) _ -_ 0'53l~ 2 d~~~c:

-151If the surface tension term, is included, it is readily shown that equation (C-5) and (C-7) become, respectively: _e_ _P oo _ daR 3 /dR P4 / d. (C-9) dt~- +-~ - - dtl t P Rt dt (Pa o-PR)(R3- R,) _ 2 4) 3 Hd 2 - Zr - t: / 0, (.+4,L R(d^ )t = O (c-lo) Defining a dimensionless variable D Ro(Po - Pa and employing the dimensionless variables C, B and T as before Equation (C-10) becomes _3_ - D(_ -1) - ao - C fdC) (C-ll) The two energy Equations (C-8) and (C-ll) lend themselves easily to numerical integration by the method of isoclines. Denoting the integral in Equations (C-8) and (C-ll) by I and d/dt by a dot, then S e'r

-152d = I and d"- x The following first order equations are derivable from the two integral differential equations: =M A-l (i) ~ g ^^ _"' (d~ -) (C-12) and _ = ( = - Z( l _ R _ (@ \ Idr> c \ (c-13) M -d^ - -s-' ~ > de > The initial condition for the solution to the above equations is, =1 when I = Oo In the solution by means of isoclines, curves of constant slope M = dI/dp are constructed in the (~, I) plane and then the integral curve is drawn corresponding to the initial values l = 1, I = 0o After I has been determined as a function of ~ and the slope M = dI/dp determined graphically, the relationship I versus 3 could be obtained. The dimensionless time T is found from the following relationships: d ( - (l 6 = dz = T (c-14) and r,= _ f d~ (c-_15)

-153Using Equation (C-l5) a curve of dimensionless time versus dimensionless radius was constructed. In order to carry out this solution an estimate of the initial radius RO is required. The surface tension of a liquid exerts a pressure on a spherical bubble in a liquid whose magnitude is given by: P _ 26 (c-16) where P = pressure difference between the inside and outside of the bubble = surface tension of the liquid R = radius of the bubbleo If a bubble is to grow in a superheated liquid, the vapor pressure of the liquid minus the pressure on the liquid must be greater than the pressure given by Equation (C-16). __ P(T) - P R P(To)- P where R = minimum initial radius for bubble growth m P(To) = vapor pressure of a liquid at the arbitrary temperature TO P = pressure on the liquid. Table XXIII gives several values of this minimum initial radius for water, Freon 11 and Freon 1135 There are two assumptions implicit in the calculation of the minimum initial radius in the above

-154manner (1) That the surface tension is applicable down to such small values of the radiuso (2) That the effect of the small radius of curvature of the drop on the vapor pressure is negligible. There is no manner in which the validity of the first assumption can be verified. However, an estimate of the order of magnitude error inherent in the second assumption can be made utilizing thermodynamic principleso Such calculations were carried out and the per cent change in the vapor pressure was found to be negligible, TABLE XXIII MINIMUM INITIAL RADIUS FOR BUBBLE GROWTH UNDER ONE ATMOSPHERE Water R (microns) 5 90 lo71 0o605 0o470 0o378 00300 0.245 T(~F) 220 240 266 275 284 293 302 Freon-ll R (microns) 35435 615 o410 o297 o226 o178 o144 T(~F) 80 100 110 120 130 140 150 Freon-113 Ro (microns) 7o85 220 l.45 o947 o765 0573 o491 T(~F) 120 126 130 136 140 146 150 A detailed summary of the calculations for the growth of a vapor bubble in superheated water is given in the following paragraphso The calculations for the other two fluids are not shown in any detail, but the results are depicted graphically. lo Calculation of Dimensionless Parameters Nozzle diameter = o031 inches, AP = 120 psig Shatter temperature = 268~F

-155Average temperature above saturation 268 + 210 = 259F Liquid viscosity, = 0.25 cp Liquid density p = 58.9 lb/ft3 PO - P = 25o825 - 0o36 = 25.46 lb/in2 Initial radius Ro = 1.71 x 10- (1o71 microns) 4pR \/p(PO - Ij (4)(0.25)(6.72 x 10-4)(2.54)(12) (171 x 10-4 58.9 (25.0)(144)(32.2) = 0o0455 D = RO(PO - poo) Liquid Surface tension = 58.9 dynes/cm. D = (58.9)(6.85 x 10-5)(12)(2.54) (1o71 x 10-4)(25.0)(144) = 0.201 2. Conversion of Dimensionless Time to Microseconds t Po - P g -~ - (t)(12)(2.54) /(25.0)(144) (32o2) 1.71 x 10-4 589 = 7.97 x 106t where t is measured in seconds T = 7.97 t where t is measured in microsecondso

150 - 140 130 120 110 100 90 80 --— __ 70 ( 50 I.____-, 40 30 20 I0 0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 Figure 40. Isoclines for Water.

-15735 Calculation of Isoclines for D = 0.1695, C = 0.0423 O2 = i(-mI) _ T _ D S -) Substitution of the values for C and D yields the equation below, which may be used to calculate the values of the isoclines required: I = 5o90(p5 - 1) - llo9 pM2 Figure 40 shows a graph of the isoclines in the (I, I) plane for values of M of 0, 2, 5, 7 and 10. The corresponding integral curve is also shown. In a very similar manner the corresponding curves were obtained for Freon 11 and Freon 1153 By integration of Equation (C-15) the curves shown in Figure 10 were drawn. It is then a simple conversion to arrive at Figure 11.

APPENDIX D PROBERT'S METHOD OF CALCULATING SPRAY EVAPORATION We assume that the spray has the characteristics such that ~R = e~~ (x)~ (D-l) R = volume or weight fraction of the spray composed of drops greater than x x = size constant n = distribution constant Taking differentials one gets ^ - dx d- n7x l' _(> (D-2) But this represents the small volume fraction of the injected spray in which drops may be taken as of diameter x. The volume of one drop is X'5 6 and therefore the number of drops per unit volume injected of size x is d R - (-n) X e ( dx (D-3) 6 X Under steady state conditions there will be drops of all sizes present at any point. If at any one instant we consider drops of one size, they can be accounted for in two ways —either as drops injected at that moment and of that size or as drops injected earlier of a larger size which have evaporated down to that size. Let us consider at any instant the drops remaining from the spray injected t seconds earliero The -158

-159drops now of size x were then of size Jx2 + Kt, where K is the evaporation coefficient. Therefore the number of drops now of size x per unit volume injected then was n-(-h) I ) i ( -t ) J4 x (D-4) n x But the number of drops of a given initial size remains constant as they evaporate. (This of course assumes no secondary atomization or coalescence and also assumes that the drop has not completely evaporated at the time of later examinationo) Therefore the total number of drops of size x in the steady state is ~t=oo (~ _ri ---- \ ^ A/^n- K-A 2co 6/( n ) + -t dx v dt (D-5) t=o X The total volume of drops of size x is tX c 6 (_ n)_ _ XH'+tt dXvdt (D-6) xT toa voum olqudprset urdx v dti T h t volt e i dt The total volume of liquid present during evaporation is X.C0 0 oo t n-4 J +KYt- \ XrO t'O YX

160o Considering Equation (D-3), after a time interval of t seconds, these drops have decreased in diameter from x to x2 - Kt and the volume of each drop is now. (x2 - Kt)3/ o Thus the volume of drops initially of size x is now 7-4 _(C// X \0 n n e- X (y-KtZ ) dxV (D-8) If the time allowed for evaporation is T, the drops initially of size JKT are just disappearing and only drops initially greater than vKT are now contributing to the remaining volume. Therefore, the total volume remaining after T seconds is X=O n ( \3 -( Ar ) (D-9); -n.x jr - Ht~ e (?) <xv <:- x -t X The volume fraction remaining is vrO Ti itr ortv 2- o x" (X~ t e (D-ll) This integral was evaluated by Probert to give the values shown on

-161100 90 80. 70 La 0 50 EVAPORATION INDEX N8xt) o/D CL Li 40 03 20 I0 O 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 EVAPORATION INDEX (kfLt) 8/D Figure 41. Per Cent of Spray Unevaporated as a Function of Evaporation Index.

-162Figure 41. This figure is reproduced from his article and was the basis of estimating the evaporation of the sprays studied. It is a plot of the unevaporated spray (per cent) versus K- for various x values of n, the dispersion parameter of the spray.

-163APPENDIX E DETERMINATION OF MAGNITUDE OF GRAVITY EFFECTS The purpose of this appendix is to calculate the ratio of the drag forces to the gravity forces which act upon the droplets encountered in the sprays studied in this system. Let us assume that all of the drops are spherical in shape. Then the weight of an individual drop is given by: W = - TT (E-l) The resistance (drag) forces may be calculated by one of the three formulae below,(l7) since for a spherical particle, it is well known that the resistance R is C 8 v DZ (E-2) where PA air density, lb/ft3 CD NRe N < 2 40 CD =0.4+ )2 NRe < o 0 R.e C = O.4 e 4N >500 The above relations may be manipulated to give: R= 3 TCAVD ^ (E-3)

R = f(Co OS AV D tW SLL V ) 2<hi <,500 (E-4) R O.Os5 7tAV%04 NRe> O (E-5) Using Equations (E-3), (E-4) or (E-5), whichever is appropriate, it is a routine calculation to determine the relative value of the ratio of the resistance to the gravity effects. Strictly speaking, Equations (E-3), (E-4) and (E-5) apply only to droplets moving through a quiescent gas stream, and not to a system in which there is a relative air velocity or in which evaporation is significant. The net result of the two effects of evaporation and relative air velocity will likely tend to make the calculated results conservative. Table XXIV below shows the values of R/W, the ratio of the resistance forces to the gravity forces, for a set of typical drop sizes and velocities. A sample calculation is also given. The data in the table are for water droplets, however similar calculations carried out for Freon 11 showed values of R/W of about the same value for each size of drop. TABLE XXIV RATIO OF RESISTANCE TO GRAVITY FORCES Drop Diameter Typical Drop Size Range Microns Velocity (Ft/Sec) NRe R/W 2 12.05 6.0 2.07 79.2 3 17005 7o5 2-76 55.7 4 24.1 11,0 5.73 334 5 34.2 12.0 8.77 189 6 48.2 15.0 15 6 126 7 68 5 25.0 37.0 125 8 96 5 50 o 0 104.1 188 9 136o 5 7500 221 215 10 193 100 447 245

-165For size range 2, taking the density of air as 1.29 x 10-3 grams per cubic centimeter and the viscosity of air as 1.79 x 10-4 poise, then R = i(0.05 x 1.293 x (6.0 x 30.4)2 x (12.05 x 10-4)2 + 5 x 1.79 x 10-4 x 6.0 x 30.4 x 12.05 x 10-4) = 703. x 10-6 dynes = 71.7 x 10-9 gms weight W D3 W= 6 6 tC(12.05 x 104)3(1.0) 6 = 9.05 x 10 10gms weight R/W 71o7 x 109 7902 9.05 x 10-10

-166APPENDIX F TABLE XXV CALCULATED VALUES OF CONSTANTS IN DISTRIBUTION FUNCTIONS 1. Rosin Rammler Equation Values of Constants in Equation (8.8a) Run Location n b Correlation Coefficient 1 1 4.45 4.56x108 0.929 1 2 4.86 1.18x107 0.969 1 3 6.44 1.86x1012 0.975 1 4 5.85 1.37xOll 0.986 1 Total 4.94 3.05xlO9 0.977 2 1 4.40 7.08xlo8 0.961 2 2 3.85 2.01x108 0.959 2 3 3.74 2.61x107 0.966 2 4 5.53 3.58x010 O.985 2 Total 3.67 5.05x107 o.964 3 1 3.38 1.72x108 0.982 3 2 3.32 2.03x107 0.973 3 3 4.82 2.45xl19 O.980 3 4 4.85 2.45xlO9 0.980 3 Total 3.35 250x108 0.970 4 1 4.46 3.12x108 O.965 4 2 3.83 7.27x107 0.974 4 3 3.86 1.87x108 0.983 4 4 4.10 2.31x108 0.977 4 Total 3.81 9.88x107 0.979 5 1 4.46 3.12x108 0.965 5 2 3.83 7.27x107 0.975 5 3 3.87 1.87x108 0.983 5 4 4.11 2.32x108 0.977 5 Total 3.81 9.88x107 0.979 6 1 3.53 1.56x105 0.977 6 2 3.07 3.55x104 0.990 6 Total 3.33 8.14xlO 0.982 7 1 4.15 9.42x105 0.975 7 2 4.11 1.02x106 0.969 7 3 4.07 8.53x105 0.985 7 Total 4.08 8.96x16 0.974 8 1 4.08 3.58xlo5 0.986 8 2 3.49 1.05xl05 0.956 8 3 4.58 7.76x105 0.995 8 Total 3.74 2.46x105 0.965 9 1 3.80 2.23x105 0.971 9 2 3.65 2.50x105 0.925 9 3 3.45 3.50x105 0.980 9 Total 3.75 2.5 x105 0.977 10 1 6.07 2.23x1010 0.971 10 2 7.22 2.65x1012 0.979 10 3 6.87 7.95xllO O 977 10 Total 7.33 4.70x1012 0.977

-167TABLE XXV (CONT'D) 2. Nukiyama Tanasawa Equation Values of Constants in Equation (8.9) Run Location a b Correlation Coefficient (xio-3) 1 1 2.59 5.91 o.963 1 2 3.36 4.86 0.963 1 3 1.44 4.52 0.986 1 4 1.41 0.971 0.971 1 Total 3.58 4.76 0.956 2 1 7.54 5.23 0.956 2 2 15.0 5.63 0.942 2 3 29.7 6.09 0.950 2 4 2.78 4.68 0.959 2 Total 57.5 6.39 0.948 3 1 7.41 5.19 0.792 3 2 40.5 6.26 0.924 3 3 1.18 4.32 0.941 3 4 1.18 4.42 0.941 3 Total 114.0 6.77 0.940 4 1 3.37 4.93 0.951 4 2 4.56 5.05 0.907 4 3 10.2 5.40 0.900 4 4.457 3.33 0.908 4 Total 22.8 5.49 o.888 5 1 0.631 3.98 o.634 5 2 9.88 5.38 0.949 5 3 3.15 4.75 0.958 5 4 1.84 4.68 0.982 5 Total 42.5 6.10 0.935 6 1 1.17 4.52 0.962 6 2.847 4.39 0.937 6 Total 1.015 4.46 0.952 7 1.610 4.16 0.974 7 2 13.07 5.34 0.947 7 3.213 3.64 0.970 7 Total.167 3.48 o.968 8 1 1.51 4.70 0.968 8 2 2.53 4.85 0.971 8 3.209 3.87 o.966 8 Total 3.11 4.97 0.970 9 1 7.08 5.35 0.953 9 2 16.28 5.57 0.974 9 3 2.58 4.81 0.982 9 Total 22.55 5.74 0.970 10 1 2.78 4.87 0.974 10 2 2.32 4.78 0.974 10 3 0.786 4.28 0.971 10 Total 1.07 4.43 0.973

APPENDIX G LIST OF REFERENCES 1. Balje, O.Eo, and Larson, LoV,, "The Mechanism of Jet Disintegration," AAF Air Material Command, Report No. MCREXE-664-531B (1949). 2. Bankoff, SoG,, and Mikesell, R.D,, "Growth of Bubbles in a Liquid of Initially Nonuniform Temperature," Paper No, 58-A-105, Annual Meeting ASME, 1958, 35 Baron, T., Techo Report No, 4, Engo Exp, Station, University of Illinois, (1947)o 4. Basset, ABo, A Treatise on Hydrodynamics, Deighton, Bell and Co., Cambridge, 1888, 5o Brown Ralph, Ph.D. Thesis, University of Michigan, 1960o 6. Castleman, RoA., Journal Res, Nat. Buro Standards, 6, (1931), 369, 7. Crowe, Clayton, PhoDo Thesis, University of Michigan, 1962. 8. Davies, C.N., Symposium on Particle Size Analysis, Insto Chem. Eng. and Soco Chemo Indo, London, Feb. 28, 1947. 9. De Juhasz, KoJo (ed.), Spray Literature Abstracts, ASME, New York (1959), 10, Dodd, KNo, Journal of Fluid Mechanics, Vol. 9, Part 2, (Oct. 1960) 175. 11o E.I. du Pont de Nemours and Co., "Thermodynamic Properties of Freon 11, Trichloromonofluoromethane," 12, E.I. du Pont de Nemours and Co., "Thermodynamic Properties of Freon 113, Trichlorotrifluoroethane," 135 El Wakil, MoM., Uyehara, O.Ao, and Myers, P.S., Nat, Advisory Comm. Aeronaut. Techo Note 3179 (1954)~ 14, Fledderman, R.Go, and Hanson, AoR., University of Michigan Eng. Research Report CM667 (June 1951)o 150 Forster, HoK., and Zuber, N., Jouro Applo Physics, 25, No 4 (1954), 474-478. --- 16, Frossling, No, Gerlands Beitre Geophys., 52, (1938), 170o 17o Giffen, E., and Mursaszew, A,, "The Atomization of Liquid Fuels", John Wiley and Sons Inc., New York, 19535 -168

-16918. Gorbatschew, SoW,, and Nikiforowa, WM., Kollo Zo, 73,(1935), 140 19. Griffith, P., Trans. ASME, 80, (1958), 721. 20. Haenlein, A,, Forsch. Gebiete Ingenieur Forschungsheft, 2, (1931), 139. 21, Hinze, J.0o, AIChE Journal, 1, (1955)o 22. Jimze, J0o,, "On the Mechanism of Disintegration of High Speed Liquid Jets," Sixth IntO Congress Appl. Mechanics, Paris, 19460 23. Kesler, G.H,, ScoDo Thesis, Mass. Inst. Tech., 1952, 24. Kottler, F,, Jo Franklin Inst., 250, (1950), 339, 419. 25. Lamb, Sir Horace, "Hydrodynamics", Dover, 1945, 26. Lane, W.R., Indo Eng. Chem,, 43, (1951), 1312. 27. Liu, Vi-Cheng, Dept. of U,S Air Force, Project 2160, 1955. 28, Littaye, G., Comptes Rendus, 217, No.4, (1943), 99, 340. 29. Littaye, G,, Comptes Rendus, 218, (1944), 440. 30. Lyons, D.B,, Thesis McGill Univ., 1951. 31. Manning, WoPo, and Gauvin, WHo,, AIChE Journal, 6, (1960), 184, 32, Marshall, WRo,, TransASME, 77, (1955), 1377. 33. Mehlig, H., AoT,Z., 37, (1934), 411. 34. Miesse,CCo, Ind. Eng, Chem., 47, (1955), 1690. 35. Mirsky, Wo, Ph.D, Thesis, Univ. of Mich,, 1956. 36. Mugele, RAo,, and Evans, H.D,, Ind. Eng. Chem., 43, (1951), 1317. 37. Nukiyama, So, and Tanasawa, Y., Trans. Soc. Mech. Eng.. (Japan), 4, No. 14, (1938), 86 and No. 15, (1938), 138, 5, No. 18, (1939), 53, No. 22, (1940), 11-7, and No. 23, (1940), II-5. 38. Ohnesorge, Wo, Z. Angew. Math. and Mech., 16, (1936), 355. 39. Pearcey, T,, and Hill, G,W,, Australian Journal of Physics, 9, No. 1, 1956. 40. Plesset, MSo., and Zwick, S.A,, J. Appl. Phys., 25, No.4, 1954.

m170UNI IOF MICHIGAN 170 3 9015 03525 1464 41o Poritsky, Ho, Proc. of First UoS. National Congress of Applied Mechanics, 1951. 42, Probert, RoP,, Philo Mag., 37, (1946), 94, 43. Putnam, AAo,, et al., "Injection and Combustion of Liquid Fuels", WADC Techo Rept, 56-344, March, 1957. 44. Ranz, W.E., and Binark H.o, ASME Paper No. 58-A-284, 1958. 45. Ranz, WoEo, and Marshall, WoRo., Chemo Eng. Progo, 4 8(1952), 141, 1735 46. Rayleigh, Lord, Proc. London Math, Soc., 34, (1892), 1553 47. Rayleigh, Lord, Proc. London Math, Soco, 10, (1978), 4. 48. Richardson, EGo,, Applo Scio Res,, A4, (1954), 374. 49, Sauter, Jo, NACA Techo Memo, No. 390, 1926. 50o Schweitzer, PoHo, PennO State Collo Bullo, No.12, November, 1930, 51, Shu, SoS,, Proc. of First USo National Congress of Applo Mecho, 1951. 52. Siestrunck, R., Comptes Rendus, 215, (1942), 404. 535 Sjenitzer, Fo, Chemo Engo Sci., 1, (1952), 101o 54. Soo, S.L., Chem. Eng. Scio, 5, (1956), 57. 55. Steinour, Ho.I, Indo Engo Chem,, 36, (1944), 618, 840, 901o 56, Thiemann, AoE,, AoToZ.,, (1934), 429, 57. Thiemann, A,E,, AoT.Z,,o 38, (1935), 484. 58. Volk, W., "Applied Statistics for Engineers" McGraw-Hill, 1.958. 59. Weber, C., Zeit, fuer Angew. Matho und Mech,, (1931), 1536 60o York, J.L., Ph.D. Thesis, Univ. of Michigan, 1949, 61, York, J.L., and Stubbs, H.E,, Trans. ASME, 74, (1952), 1157.