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Direct contact, liquid-liquid heat transfer to a vapourizing, immiscible drop. Adams, Arthur Edward Steele 1971

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DIRECT CONTACT, LIQUID-LIQUID HEAT TRANSFER TO A VAPOURIZING, IMMISCIBLE DROP by ARTHUR EDWARD STEELE ADAMS B.A.Sc. , University of British Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER. OF*APPLIED SCIENCE i in the Department of Chemical Engineering We accept this thesis as conforming to the required standard The University of British Columbia July, 1971 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f £/-i£MfCrtt*. £\i6,£ The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date S/sPr ^3 t / 9 7/ i i ABSTRACT This thesis presents a study of some of the factors affecting direct contact, liquid-liquid heat transfer from a continuous phase of 0.0%, 56.02%, 73.07%, and 77.06% glycerine-water solutions to a dis-persed phase, which is vapourizing, of isopentane or cyclopentane. An average heat transfer coefficient based on the i n i t i a l area, the total evaporation time, the total heat transferred, and the .temperature driving force at the end of evaporation was calculated. This c o e f f i c i -ent was correlated to the parameters of the systems by the dimensionless groups of continuous phase Prandtl number, dispersed phase Prandtl number, and a viscosity ratio. The results are compared to the works of Klipstein, Sideman and Prakash. A comparison made between the photographic and dilatometric method of volume measurement showed the dilatometric method to be the best for this type of work. i i i TABLE OF CONTENTS LITERATURE .. . . 1 INTRODUCTION 1 Definitions of Drop, Bubble, and Vapour Bubble . . . . 3 Definitions of Evaporation Ratio and Percent Evaporation 4 GENERAL HYDRODYNAMICS OF DROPS AND BUBBLES 4 Drag Coefficients 4 Internal Circulation 6 DROPS . 7 Introduction 7 Transfer to Solid Bodies .. 8 Transfer to Drops 9 Effect of Internal Circulation on Transfer Processes 9 Effect of Drop Distortion on Transfer Processes 10 Effect of Drop Oscillation on Transfer Processes .. .. 11 BUBBLES . . . . . . 12 Bubble Shape 13 Bubble Measurement 13 SUMMARY .. .. .. 15 MAJOR WORKERS IN THE FIELD 17 INTRODUCTION 17 Page DATA COLLECTION 17 MAIN VARIABLES,STUDIED'AND SYSTEMS USED .. 19 SIDE EFFECTS ELIMINATED 19 Wall Effects 19 Mass Transfer . 19 Stationary Continuous Phase - 20 Surfactants 20 Heat Transfer During Droplet Formation 20 Nucleation 21 INSTANTANEOUS HEAT TRANSFER COEFFICIENT .. 22 Introduction . • ..... 22 Temperature Driving Force 22 The Instantaneous Rate of Heat Transfer 23 Area .. 24 Correlation of the Instantaneous Heat Transfer Coefficient 29 AVERAGE HEAT TRANSFER COEFFICIENT .. • 30 Total Evaporation Time 31 Start of Evaporation 31 End of Evaporation 32 Temperature Driving Force .. 33 Area 33 Values of the Average Heat Transfer Coefficient . . . . 34 INITIAL DECISIONS 36 RECOMMENDATIONS OF OTHER WORKERS 36 V Page INITIAL MODIFICATIONS OF EQUIPMENT 36 SYSTEMS CHOSEN 37 FACTORS NOT STUDIED EXPLICITY 38 Drop Diameter .. . 39 Temperature Driving Force 39 Column Size 39 Surfactants 39 Mass Transfer , . . 39 Effect of Surface and Interfacial Tension 40 EQUIPMENT, EXPERIMENTAL PROCEDURE AND DATA COLLECTION .. 41 EQUIPMENT 41 Column ...... 41 Photography 47 Injection System .. 48 Dilatometer 49 • Zeroing the Dilatometer 51 Calibration of the Dilatometer 51 Set-Up for Determining Bubble Height .. .. .. .. 52 EXPERIMENTAL PROCEDURE 53 Preparation for a Set of Runs 53 Execution of a Run .. 54 Termination of a Set of Runs '. 55 DATA COLLECTED . . 56 PRELIMINARY PROCESSING OF DATA 57 RATIONALE FOR BATS 58 v i Page Photographic Data 60 Dilatometer Data 61 INTERPRETATION OF RESULTS 63 INTRODUCTION 63 VISUAL OBSERVATION OF DROP EVAPORATION 64 RANGE OF VARIABLES .. . . 65 THE DILATOMETER CURVE 67 Advantages of the Dilatometric Method 67 Qualitative Discussion of the Dilatometer Curve . . . . 68 Explanation of the Volume Curve 68 General Equation 68 Equations Applicable to Specific Regions of Evaporation 73 DISCUSSION OF METHODS OF PROCESSING VOLUME-TIME DATA .. 77 DISCUSSION OF THE INSTANTANEOUS HEAT TRANSFER COEFFICIENT AS A CORRELATION PARAMETER 78 DISCUSSION OF THE CURVE FIT FOR THE DILATOMETER CURVE .. 81 AVERAGE HEAT TRANSFER COEFFICIENT 83 Correlation Parameters of the System 96 Correlation of the Results of A l l Workers 117 Summary of Correlation Results 117 EVALUATION OF INITIAL DROP AREA . . . . 122 Comparison of Methods Used to Measure the Mass of the Dispersed Phase .. 123 Causes of Differences .. 127 Conclusion of the Source of Differences 136 Calculation of I n i t i a l Area 136 v i i Page CONCLUSIONS .. 1 4 0 RECOMMENDATIONS 1 4 1 NOMENCLATURE 142 LITERATURE CITATIONS . 148 APPENDIX I Equipment Specifications .. 150 (a) Column Details. .. I 5 0 (b) Exposure Meter 150 (c) Constant Temperature Baths 151 (d) Potentiometer 151 (e) Strip Chart Recorder . . 152 (f) Projector 152 (g) Camera 153 APPENDIX II Dispersed Phase Properties 154 APPENDIX III Continuous Phase Properties 159 (a) Concentration of the Continuous Phase 159 (b) Density .. 160 (c) Viscosity .. •. 161 (d) Specific Heat 165 (e) Thermal Conductivity .. 169 (f) Vapour Pressure-Temperature Relationship 171 V ' i i i Page APPENDIX IV Computer Program .. .. 174 Calculation of the Vapour Bubble Temperature 174 Curve Fitting of the Dilatometer Volume with Time .. .. 175 Description of Input Cards for the Computer Program .. 173 Symbol Listing for Computer Program .. .. 280 Computer Program 187 APPENDIX V Computer Output and Sample Calculation .. .. 204 Computer Output .. .. 204 Sample Calculation - Run 1308 213 APPENDIX VI Calculation of I n i t i a l Area 219 APPENDIX VII Calculation of Average Heat Transfer Coefficient 227 APPENDIX VIII Correlation Equations 240 (a) Trip Definitions 240 (b) Summary of Correlation Equations 246 .(c) Description of " t " Test .. .. 246 (d) Results of TRIP Correlation .. .. 249 ix LIST OF TABLES No. Page I Modes of Behaviour for Moving Drops and Bubbles - Klipstein (1) .. 26 II Ranges of Behavior Indexes Corresponding to Observed Fluid Dynamic Modes of Vapourizing Drops .. .. . . 27 III Equations for the Average Rate of Heat Transfer Presented by Various Workers 35 IV Average Heat Transfer Coefficients by Various Workers 35 V Range of Variables 66 VI Coefficients of Equations used for Curve F i t of Dilatometer Volume 91 VII Average Heat Transfer Coefficients . . , 94 VIII Parameter Values for Calculating the Correlation Equations 95 IX Correlation Equations for Cyclopentane and Isopentane 100 X Correlation Equations using the Parameters Pr c, Pr d, and y c / ( ( uc + y d)) 1 0 6 XI Summary of Correlation Equations for the Average Heat Transfer Coefficient 1° 7 XII Calculated Values of the Average Heat Transfer Coefficient from the Correlation Equations .. .. 108 XIII Correlation Equations Based on the Average Heat Transfer Coefficients of A l l Workers 118 XIV "Calculated Values of the Average Heat Transfer Coefficients from the Correlation Equations Based on the Results of A l l Workers 119 XV Masses of Dispersed Phase Calculated by the Photographic and Dilatometric Methods 124 XVI St a t i s t i c a l Results of the Comparison of the Dispersed Phase Masses Calculated from the Photographic and Dilatometric Methods 128 X No. Page XVII Calculated I n i t i a l Diameters for Use in the Calculation of the Average Heat Transfer Coefficient 139 APPENDIX II I- AII Properties of the Dispersed Phase Liquid 155 II- AII Density of the Dispersed Phase Liquid 157 APPENDIX III I- AIII Determination of Percentage from Density and Viscosity Measurements 160 II- AIII Variation of Density with Temperature 162 III- AIII Variation of Viscosity with Temperature 164 IV- AIII Variation of Specific Heat with Glycerine Concentration 165 V- AIII Variation of the Specific Heat of Glycerine with Temperature 167 VI- AIII Vapour Pressure - Temperature Relationship at Percent Concentrations of 56.02%, 73.07%, and 77.06% 172 APPENDIX VI I- AVI Calculation of I n i t i a l Area for Set 13 220 II- AVI Calculation of I n i t i a l Area for Set 14 221 III- AVI Calculation of I n i t i a l Area for Set 16 222 IV- AVI Calculation of I n i t i a l Area for Set 17 223 V- AVI Calculation of I n i t i a l Area for Set 18 224 VI- AVI Calculation of I n i t i a l Area for Set 19 225 VII- AVI Calculation of I n i t i a l Area for Set 20 226 x i No. Page APPENDIX VII I- AIII Average Heat Transfer Coefficient for Set 13 . . 228 II- AVIII Average Heat Transfer Coefficient for Set 14 .. 229 III- AVII Average Heat Transfer Coefficient for Set 16 .. 230 IV- AVII Average Heat Transfer Coefficient for Set 17 .. 231 V- AVII Average Heat Transfer Coefficient for Set 18 .. 232 VI- AVIII Average Heat Transfer Coefficient for Set 19 .. 233 VII- AVII Average Heat Transfer Coefficient for Set 20 .. 234 VIII- AVIII Average Heat Transfer Coefficient for Isopentane and D i s t i l l e d Water - Prakash .. 235 IX- AVII Average Heat Transfer Coefficient for Furan and Di s t i l l e d Water - Prakash .. .. 236 X- AVI Average Heat Transfer Coefficient for Cyclopentane and D i s t i l l e d Water - Prakash .. 237 XI- AVIII Average Heat Transfer Coefficient for Pentane and D i s t i l l e d Water - Sideman .. .. 238 APPENDIX VIII I- AIII Summary of Physical Constants Used for Correlation 241 II- AVIII Summary of Correlation Equations 242 x i i LIST OF FIGURES Figure Page 1 Area Development versus Fraction Vapourized for Various Drop Sizes as Calculated from Experimental Data . . 28 2 Front View of the Equipment Set-up . . ..... .. 43 3 Side View of the Equipment Set-up .. 44 4 Details of the Top Portion of the Column . . . . 45 5 Details of Brush for Event Marker 46 6 Set-up for Controlling the Dilatometer 50 7 Reproduction of Experimental Dilatometer Curve for Run 1308 59 8 Idealized Example of Dilatometer Curve 69 9 Paths for Heat Transfer to an Evaporating Drop .. 70 10 Comparison of Curve F i t to Experimental Data -Run 1308 84 11 Comparison of Curve F i t to Experimental Data -Run 1402 85 12 Comparison of Curve F i t to Experimental Data -Run 1604 86 13 Comparison of Curve F i t to Experimental Data -Run 1708 .. 87 14 Comparison of Curve F i t to Experimental Data -Run 1803 88 15 Comparison of Curve F i t to Experimental Data -Run 1906 .. • 89 16 Comparison of Curve Fit to Experimental Data -Run 2011 .. 90 17 Variation of U with Continuous Phase Prandtl Number 97 18 Curve F i t of U with Equation (50) .. .. .. .. 98 19 Curve F i t of U with Equation (51) 101 x i i i Figure Page 20 Curve F i t of U with Equation (52) 102 21 Curve F i t of U with Equation (53) 103 22 Curve F i t of U with Equation (54) 104 23 Comparison between Experimental and Correlation Value of U Calculated from Equation (50) 109 24 Comparison between Experimental and Correlation Value of U Calculated from Equation (51) .. .. 110 25 Comparison between Experimental and Correlation Value of U Calculated from Equation (55) I l l 26 Comparison between Experimental and Correlation Value of U Calculated from Equation (52) .. . . . . 112 27 Comparison between Experimental and Correlation Value of U Calculated from Equation (53) 113 28 Comparison between Experimental and Correlation Value of U Calculated from Equation (54) 114 29 Comparison between Experimental and Correlation Value of U Calculated from Equation (56) 115 30 Comparison between Experimental and Correlation Value of U Calculated from Equation (57) 116 31 Comparison between Experimental and Correlation Value of U Calculated from Equation (58) 120 32 Comparison between Experimental and Correlation Value of U Calculated from Equation (59) 121 33 Comparison of the Masses Calculated by the Photo-graphic and Dilatometric Methods for the System of Isopentane and D i s t i l l e d Water . . .. 130 34 Comparison of the Masses Calculated by the Photo-graphic and Dilatometric Methods for the System of Isopentane and 76% Glycerine Solution 131 35 Percentage Difference versus Volume of the Air Bubble for Isopentane in D i s t i l l e d Water 133 36 Percentage Difference versus Volume of the Air Bubble for Isopentane in 76% Glycerine Solution 134 xiv Figure Page 37 Comparison of the Injected and Calculated Mass of the Dispersed Phase for Isopentane i n D i s t i l l e d Water • • . . 135 38 Shape of I n i t i a l Drop of Dispersed Phase 138 39 Effect of Temperature and Percentage Concentration on Dispersed Phase Density • • • . • 158 40 Effect of Temperature and Percentage Concentration on Continuous Phase Density • • • • . 163 41 Effect of Temperature and Percentage Concentration on Continuous Phase Viscosity 166 42 Effect of Temperature and Percentage Concentration on Continuous Phase Specific Heat • •• •• 168 43 Effect of Temperature and Percentage Concentration on Continuous Phase Thermal Conductivity . . . . .. 170 44 Flow Chart for T r i a l and Error Solution of Vapour Phase Temperature 176 45 Output Results from Computer Program for Run 1308 • • 211 ACKNOWLEDGEMENTS I wish to thank Dr. K.L. Pinder, under whose direction this work was conducted, for his continued support and guidance through-out this investigation. I also would like to express my thanks to the faculty and staff of the Chemical Engineering Department of the University of British Columbia. Particularly, I wish to thank Dr. N. Epstein for his ideas on several aspects of the work. I would also like to thank my fellow graduate students for their sympathetic ears and useful suggestions. The co-oper?ation of Mr. R. Muelchen and his staff for their assistance in construction of the apparatus is appreciated. I am indebted to the National Research Council of Canada and the Chemical Engineering Department of the University of British Columbia for financial support. 1 LITERATURE INTRODUCTION Some of the advantages of transferring heat from one immis-cible liquid using another immiscible liquid are: 1. The absence of solid surfaces on which scale deposits can form. These deposits reduce the rate of heat transfer; thus decreasing the efficiency of the process and necessitating frequent cleaning of equipment and otherwise increasing maintenance costs. 2. Since the choice of the material of construction i s not restricted by the material's a b i l i t y to conduct heat, suitable materials may be chosen to reduce corrosion i f one or both of the liquids are corrosive. If in addition the liquid used for transferring heat i s evaporating, further advantages are: 1. The large capacity for the absorption of heat of the' vapourizing liquid obtained by u t i l i z i n g the latent heat of vapouri-zation, enables a much lower throughput of the heat transfer liquid to be used compared to that needed i f only the sensible heat content of the liquid were used for heat transfer. 2. The improved temperature driving force. This i s the result of the evaporating liquid's temperature remaining relatively constant. The potential advantages of this method have inspired several workers to engage in research to understand the basic mechanisms 2 underlying this type of heat transfer and to determine experimental values which may be of use in design. The introductory work in this f i e l d was done by D.H. Klipstein (1) who completed a D.Sc. thesis at M.I.T. under the supervision of Professor G i l l i l a n d . His thesis "HEAT TRANSFER TO A VAPOURIZING IMMISCIBLE DROP" laid the foundations for ensuing workers by determining the important variables and defining the special problems encountered in this subject. Dr. Sideman and coworkers, working at the Israel Institute of Technology in Haifa, produced a series of papers connected with this f i e l d . The work of particular interest in this series was done by Y. Taitel (2) for his M.A.Sc. thesis. Working simultaneously, but completing his study later than T a i t e l , C.B. Prakash (3) finished a Ph.D. thesis at the University of British Columbia under the super-vision of Dr. K.L. Pinder. Dr. Sideman (4) has written a very complete and comprehensive literature review on the subject of direct contact heat transfer between two immiscible liquids and topics related to i t . In deference to his work the literature discussed i n this thesis w i l l emphasize new works and those which are necessary for the development of, and the comparison to, this work. The general method used by a l l three workers to study the transfer of heat between two immiscible liquids by evaporating one of them, was to inject a drop of a f a i r l y v olatile liquid into a column of a more dense, immiscible liquid whose temperature was above the boiling 3 point of the dispersed phase l i q u i d . Heat absorbed from the con-tinuous phase vapourized the dispersed phase l i q u i d , thus causing a vapour bubble to form and grow i n the top of the l i q u i d drop. The vapour bubble continued to grow, as the drop moved up the column, be-coming l a r g e r and l a r g e r , with the dispersed phase l i q u i d forming a puddle i n the bottom of the vapour bubble. At the point where a l l the dispersed phase l i q u i d had evaporated (the end of evaporation) the vapour bubble ceased to grow s i g n i f i c a n t l y i n s i z e and assumed a c h a r a c t e r i s t i c form and v e l o c i t y depending on the parameters of the system. A study of the l i t e r a t u r e shows that the terms: drop, bubble, and vapour bubble are, f o r t h i s f i e l d , to some extent interchangable and a discussion c l a r i f y i n g these terms as used i n t h i s work i s d e s i r -able. D e f i n i t i o n s of Drop, Bubble, and Vapour Bubble The terms drop and bubble w i l l have t h e i r usual meanings with the extension that t h e i r d e f i n i t i o n i s based on t h e i r p h y s i c a l appearance and hydrodynamic behavior rather than the amount of each phase present on a mass basi s . For example, i f the dispersed phase consists of a l i q u i d drop containing a bubble of vapour i t w i l l be r e -f e r r e d to as a drop; whereas a bubble containing a puddle of l i q u i d w i l l be r e f e r r e d to as a bubble. The term vapour bubble w i l l be r e -served f o r a bubble containing only vapour or more s p e c i f i c a l l y i n t h i s work the vapour portion of the dispersed phase bubble. 4 Definitions of Evaporation Ratio and Percent Evaporation Another term which i s common to this type of work i s the variable which defines the amount of evaporation which has taken place up to a certain point. In this work this w i l l be called the evaporation ratio and is defined by evaporation ratio = mass of dispersed phase vapour total mass of the dispersed phase This term i s equivalent to Klipstein's fraction vaporized (f ) , Sideman's evaporation ratio (5), and Prakash's evaporation ratio (m). ? The percent evaporation i s the evaporation ratio on a percentage basis. GENERAL HYDRODYNAMICS OF DROPS AND BUBBLES  Drag Coefficients In general the motion of a drop or bubble is determined by a force balance, based on a free body rising vertically i n a stagnant continuous phase, of the form(5) ma = bouyant force - drag force - mg 2 Vdu y ( , CD P c U _ d 2 p d do = Vg(pd - pc} r n 2 (1) (2) is a dimensionless quantity referred to as the drag coefficient and defined as follows: = drag force ( p c u 2) (Frontal Area) (3) Licht and Narasimhamurty (6) consider their drag coeffi-cient to be a function of the following variables: C D = G ( q , g , d , p d > p c , p d , y c ) (4) and consider the dimensionless groups and We = f 1 (Re, »\d) (5) V c Cn I-1 J -f = f 2 ( e i 5 R e , f- ) (6) CD • c and e, - gd2 (pd - p o 4 a (7) Although in the above cases the acceleration term was assumed to be zero, there is no reason to make the same assumption for the present situation. The most pertinent work which considers the effect of acceleration i s by Hughes and G i l l i l a n d (5). They relate their drag coefficient to the variables of the systems by a series of dimensionles groups one of which contains a term for the acceleration. Internal Circulation Both drops and bubbles may have internal circulation pre-sent. Internal circulation i s defined (7) as a "Hadamard like move-ment of liquid (vapour) within the drop (bubble) in a manner that i s almost laminar". It is caused by the shear stress applied to an interface as the drop (bubble) moves through the continuous phase. A force balance on any point of the f l u i d interface (8) gives ^ du. dti, 3 a c d . . 9 l = y c dy~ " P d dy~ ( 8 ) where x is parallel to and y is perpendicular to the drop interface. This equation may be used to qualitatively explain the effect of some quantities such as viscosity ratio, c r i t i c a l Reynolds number, inter-f a c i a l tension etc. which affect the internal circulation. Surface active agents are thought to retard or stop internal circulation by producing a concentration gradient on the surface under the influence of the drag force exerted by the continuous phase. This concentration gradient opposes the shear stress on the surface causing the interface to become immobile. The physical indicat ions of the presence of internal circulation are a decreased drag coefficient and a higher terminal velocity compared to those for an equivalent solid body. By considering an equation similar to the above equation (8) Hughes and G i l l i l a n d (5) conclude that: 1. Bubbles of gas in liquids are nearly always circulating. 2. Liquid drops in gases are rarely circulating. 3. Liquid drops in liquids are usually in the transition region. Therefore i t would seem that in the present case the vapour in the vapour bubble would be circulating; and, except for the cases of small drops or the presence of surface active agents, the liquid portions of the dispersed phase would also be circulating. DROPS Introduction Calculations made by Klipstein (1) showed that even at seventy-seven percent evaporation the rate of heat transfer through the liquid portion of the dispersed phase was close to ten thousand times more than that through the vapour portion. This ratio increases as the percent--evaporation decreases u n t i l the point, in the early stages of evaporation, where the dispersed phase takes the form of a drop and a l l of the heat transfer i s through the liquid which completely surrounds the vapour bubble. Thus, the study of the literature con-cerning the mechanism of liquid-liquid heat transfer i s quite important. Most of the literature i s concerned with heat transfer to drops, spherical or otherwise, i n which the liquid completely covers the interface. Therefore the results reported in the literature are only valid in this work.for the early stages of evaporation, since for higher percent evaporations the liquid portion of the dispersed phase i s assumed not to cover the entire surface of the bubble. Thus, the changes 8 in transfer rates at localized points on a drop as reported in the literature may not have the same meaning for this work - especially considering the hydrodynamic changes which the vapour bubble portion of the dispersed phase would impart to the continuous phase flowing around i t . Transfer to Solid Bodies Eor small drops or those in which surface active agents occur, internal circulation i s not considered to be present. Thus, the drops in these cases may be considered as having a static surface; thus making applicable some of the sections in the literature concerning trans-fer processes to solid surfaces. Some of the basic or more recent works are: Garner ( 9 ) , who was studying the dissolution of benzoic acid spheres found that the minimum rate of dissolution occurs at the separ-ation point and that about twice as much is dissolved from the front portion of the sphere as from the rear. Hughmark (10) has correlated the exponents on the Reynolds and Prandtl numbers, used in the usual correlation of the Nusselt number, for a wide range of both groups. Lochiel and Calderbank (11) present theoretical equations for mass transfer from any solid or rapidly circulating axisymmetric body of revolution for high Peclet and Schmidt numbers. 9 Transfer to Drops Effect of Internal Circulation on Transfer Processes Internal circulation may occur i f the drops are f a i r l y large and no surface active agents are present. The chief effect of internal circulation on transfer processes i s to increase their rates by mixing the contents of the drop and by causing a thinning of the outside boundary layer (12). In many cases, especially those at low Reynolds number, the drops may be approximated mathematically as spheres and many of the theoretical works are based on this approximation. Johns and Beckmann (13) discuss the mass transfer inside a drop at low Reynolds number using the Hadamard stream function. mass transfer. This i s based on a boundary layer approach at high Reynolds numbers. Their results show satisfactory agreement with experimental results.. Their work accounts for single spherical bubbles as well as for drops. Their correlation i s : Cheh and Tobias (14) present a theoretical correlation for (9) and in their paper they show a plot of (10) against Reynolds number for the cases of drops where p.d = pc and bubbles where u « u d c p. << p d c They also show that their solution reduces to the potential solution as the Reynolds number approaches i n f i n i t y . The Handlos and Baron model (15) has been f a i r l y well established and several workers have used i t as a starting point for more advanced models of transfer mechanisms. Patel and Wellock (16) extended the Handlos and Baron model by expressing the f i r s t eigen value as a function of the continuous phase mass transfer coefficient which was calculated from the Higbie penetration model. To study the internal circulation patterns, Head and Heliums (17) studied, experimentally, the temperature profiles in large drops at low Reynolds numbers. An upper limit for mass and heat transfer from non-oscillating droplets was calculated by Winnikow (18) based on the assumptions that the phase resistances are equal in magnitude and that complete internal circulation occurs. A purely qualitative discussion of mass transfer and wake phenomena, based on their photographic study, has been given by Magarvey and MacLatchy (19) whose studies cover a range of Reynolds numbers from 0 to 2500. Effect of Drop Distortion on Transfer Processes Distortion of liquid drops depends on a force balance be-tween the surface forces, which act to cause a spherical shape, and the hydrodynamic forces. At low Reynolds numbers the surface forces pre-dominate and the drops form the approximately spherical shape discussed previously. However, at higher Reynolds numbers i n e r t i a l forces tend to distort the drops into an oblate ellipsoidal shape. In highly viscous liquids the drops w i l l change from a spherical, to an ovate spherical, to a symmetrical oblate ellipsoidal, to a non-symmetrical ellipsoidal and f i n a l l y to an inverted mushroom shape. In non-Newtonian fluids similar shapes occur except for a tai l i n g filament (20) A theoretical equation for the distortion of the drops is given by Taylor and Acrivos (21). This equation is only valid for Reynolds numbers less than twenty, however they also give empirical correlations of Weber number, Eotvos number and viscosity ratios. The restriction on internal circulation given in the work of Taylor and Acrivos (21) was removed by the work of Pan and Acrivos (22). Effect of Drop Oscillation on Transfer Processes Oscillation between the oblate and prolate forms, which occurs mainly with large drops and low interfacial tension, causes an increase in the rate of mass (23) and heat transfer (24). Although i t is doubtful that the drops in the present work approximate these conditions, i t has been noted that the liquid in the bottom of the vapour bubble has a tendency to slosh around and oscillate from one form to another. Although at this time there i s no apparent connection between the two types of oscillation i t i s possible that a similar en-hancement of the transfer rate may occur in both situations. There are several types of oscillations which can occur with liquid drops;KIntner (20) describes several of these! An oblate ellipsoidal drop usually produces an axially symmetric type*, of oscillation which may be started by vortex shedding into the wake. In some drops random wobbling is encountered and the motion of these drops tends to follow an approximately helical path. Waves have been observed to move over the interface and internal circulation i s damped out. Another form of oscillation which has been observed are surface indentations. Of special interest i s the similarity between the above descriptions and those of Klipstein presented in Table I later in this work, especially his modes two through five where the characteristics expected would be those of a bubble. Combining the work of Schroeder and Kintner(24) on the oscillation of drops with the concepts of interfacial stretch and internal mixing in drops Rose and Kintner(7) have developed a mass transfer model for single drops oscillating from a spherical to an oblate ellipsoidal shape. They note that internal circulation stream-lines are broken up and a type of internal mixing is achieved in the drop. By redefining the Rose and Kintner analysis (7) for the limiting case of a l l resistance being in the droplet, Angelo et al.(25) developed a method for predicting low rates of mass or heat transfer through "a i stretching or shrinking interface over a f i n i t e time interval. BUBBLES As mentioned previously calculations have shown that only a small amount of heat is transferred through the liquid-vapour interf ~- .13 of the bubble. Therefore, since this work is not chiefly concerned with the vapour bubble after the end of evaporation, the study of the literature concerning transfer processes to bubbles is not very re-levant to this work except for those sections concerning the bubble shape and the methods of measuring the bubble volumes. Bubble Shape Bubble shape is important for two reasons: One>the bubble shape along with the Reynolds number and the interfacial characteristics of the bubble determine the motion of the bubble through the liquid. Two, the bubble shape is needed in the calculation of the bubble area either from the volume or the dimensions of the bubble. Although the shape of the bubble is a function of the f l u i d dynamic and interfacial forces; the surface tension concept has not been useful for predicting these interfacial forces. Experimentally, the shapes usually obtained in Newtonian liquids are the sphere, the ellipsoid and the spherical cap (26). Apart from noting the above shapes there appears to be a lack of work relating detailed bubble shapes to the parameters of the systems. Most workers have had to be content with relating the above shapes to the index of equivalent diameter calculated from the bubble volume. For example, Calderbank and Lochiel (27) assumed e l l i p -soidal shape for bubbles 0.2 to 1.8 cm. in diameter and a spherical cap shape for bubbles greater than 1.8 cm. in diameter. Bubble Measurement Calderbank and Lochiel (27) in their work on determining mass transfer coefficients for carbon dioxide bubbles i n water, measured the change in volume of their bubbles by recording the pressure changes of a closed column i n which they were rising. They obtained the cross sectional area of the bubble from a vertical shadow photograph, thus removing the problem of estimating the cross sectional area of the bubble. However, since they did not take a photograph showing the ve r t i c a l dimension of the bubble the height dimension of the bubble was not known accurately. Thus, as mentioned before, they s t i l l had to assume a shape for the bubble before being able to calculate the surface area. The above work was done with an air space above the liquid. This caused d i f f i c u l t i e s in the calculations and there was some chance of the measurements being inaccurate due to the presence of a time lag caused by the air space. In a later study, Calderbank et al, (28), this d i f f i c u l t y was removed and a "constant volume" method was used. In this method the liquid completely f i l l s the column and the tendency for the bubble to expand, due to mass transfer to i t , increases the pressure in the column. Since the column Is sealed the bubble re-mains at a constant volume. A study to compare the experimental techniques of photo-graphy and the pressure change method was made by Garbarini and Tien (29). Although they made the comparison between the photographic and pressure methods under identical operating conditions they did not use the two methods simultaneously, thus i t is possible that they were not measuring the same thing. This may account for the large amount of scatter in their work which tends to make the comparison less valid than i t might be. In their conclusions they state: "The results of the present study seem to indicate that the inherent error involved in the determination of instantaneous mass transfer rates is substantial regardless which particular method i s used for i t s determination. Although many reasons can be advanced for this d i f f i c u l t y , the single most important factor probably i s the failure to estimate the surface area accurately. Any improvement in this re-gard can be made only i f more detailed information on the bubble be-havior such as deformation, oscillation, deviation from r e c t i l i n e motion, etc., becomes available. This seems to underscore the importance of more fundamental work on bubble mechanisms". This statement, especially the parts about estimation of surface area and more work on bubble mechanics should be kept in mind as their im-portance is even greater in this study. SUMMARY The previous sections covered briefly some of the relevant literature available on the topics of drops and bubbles and the trans-fer processes to them. However, even i f the individual processes occurring simultaneously in an evaporating drop are clearly understood the applicability of these processes to the present problem is hampered for the following reasons: 1. The presence of liquid in the bottom of the bubble affects both the velocity of the bubble and i t s shape. In addition the bubble is constantly changing in size due to the evaporation of the dispersed phase liquid. Therefore i t is doubtful that a valid com-parison can be made with the drag coefficients of ordinary bubbles rising in liquids at their terminal velocity. 2. The problems a l l workers have encountered in deter-mining the surface area of a bubble illustrates the d i f f i c u l t i e s in the more complicated situation of trying to determine not only the shape but also the area of the liquid puddle in the bottom of the bubble Without this area term no quantitative comparison, as w i l l be shown later, can be made between the work, on the evaporation of the liquid drop and the work on l i q u i d - r l i q u i d heat transfer. Therefore the works on this topic essentially stand by themselves and each of the workers has had to study the evaporation process as an isolated situation. Since the three main workers in this f i e l d followed similiar approaches to the solution of the problem, they w i l l be discussed as a group. 17 MAJOR WORKERS IN THE FIELD INTRODUCTION The i n i t i a l conception of the present research was based on a study of the works of Klipstein (1), Sideman (2), and Prakash (3); whose works were the major sources of reference material for this thesis. In particular many of the details of the experimental equipment and pro-cedures were obtained from Prakash's work. Before discussing i n detail particulars of the methods, procedures, and results of each of the workers i t i s desirable to have an idea of the attitudes of each of the workers and the situations under which the work was done. Klipstein was the f i r s t worker in the f i e l d and although his practical results were limited, his work did much to s i f t through the many variables present and to define many of the experimental pro-cedures and methods used by subsequent workers. Both Sideman and Prakash greatly improved the experimental procedures and because of the work of Klipstein they could concentrate on the important variables and consider a large number of systems. DATA COLLECTION The main source of data i n this work is the measurement of the volume of the evaporating drop with respect to time. The method used by these workers was to obtain photographs of the evaporating drops as they rose up the column. Klipstein used stop action photography and a Polaroid camera. This enabled him to evaluate his results at once, which, because of the exploratory nature of his work had decided advantages. However he recognized the disadvantages of not having a continuous picture of the bubble's growth and one of his recommendations was that "backlighting and motion picture photography should be used to better define the disposition of the unvapourized liquid". Sideman and Prakash both followed this recommendation and used back lighting and motion picture photography. Unfortunately the pictures were not sufficiently clear to determine accurately the shape of the liquid puddle in the bottom of the bubble. In both of the latter cases the motion picture camera was mounted on a movable frame so that i t could follow the drop up the column. The photographic method relies on the measurement of the drop shape and dimensions from a photograph to calculate the volume. These measurements are extremely doubtful above ten percent evaporation since the bubble becomes extremely distorted and may be in the transition region between an ellipsoidal shape and a spherical cap shape and can be approximated by neither. Prakash was the f i r s t worker in this f i e l d to attempt to measure the volume by another method. However, his dilatometric method was.used to determine the end of evaporation and he never attempted to calibrate i t for use in measuring the change in volume of the vapour bubble during i t s rise and evaporation in the column. MAIN VARIABLES STUDIED .AND SYSTEMS USED Klipstein studied the effect of drop size, temperature driving force, continuous phase Prandtl number, and surfactant con-centration on drops of ethyl chloride evaporating i n continuous phases of glycerine -water and Aerosol 22_water mixtures. Sideman and Taitel (2) used butane and pentane as dis-persed phases evaporating in sea water and d i s t i l l e d water. The main feature of their work, apart from the systems studied, i s the large range of temperature driving forces which they studied. Prakash was mainly concerned with.the effect of the dis-persed phase parameters; he used furan, isopentane, and cyclopentane as dispersed phases evaporating in a continuous phase of d i s t i l l e d water. SIDE EFFECTS ELIMINATED  Wall Effects A l l workers attempted to minimize wall effects by having a f a i r l y large column. However, in some cases where the bubble was nearly f u l l y vapourized wall effects may have been important; even though none of the workers reported unexpected shape changes or abnormal velocity readings which might indicate the presence of wall effects. Mass Transfer This variable can be minimized by saturating both phases with each other before contact. Klipstein was the only worker who re-ported a high degree of solubility of his dispersed phase, ethyl chloride, in his continuous phase, glycerine-water. Other workers re-ported no d i f f i c u l t i e s in this respect. Stationary Continuous Phase A l l workers kept the continuous phase stationary, having decided that a non-stationary continuous phase would add com-plications which were not j u s t i f i e d , especially since the cameras used in their studies were able to photograph the drop as i t moved up the column. Surfactants The addition of surface active agents (surfactants) to the system tends to form an immobile interface around the drop, thus re-ducing the internal circulation. Klipstein was the only worker who attempted to study their effect. Sideman and Prakash both attempted to exclude surfactants from their systems, when they used d i s t i l l e d water as a continuous phase, but they made no special checks to ensure that this had been done. Heat Transfer During Droplet Formation If an attempt were made to form a drop in the heated section of the column the dispersed phase liquid would immediately start to vapourize and i t is unlikely that any reasonable drop could be formed. To overcome this problem one solution has been to cool the dispersed phase liquid so that there i s a sufficient sensible heat requirement of the dispersed phase to ensure that the drop w i l l not begin to vaporize before forming the proper shape. Klipstein and Sideman both attempted to do this by cooling their injection nozzle with circulating water. Thus, the dispersed phase liquid was cold enough so that i t would form a stable drop on the nozzle before evaporation became important. Prakash obtained the same result by forming his drop in a section of his column f i l l e d with cooled continuous phase liquid. The drop rose in this section for a short time' before entering the sec-tion containing the heated continuous phase liquid. Prakash's method had the advantage of allowing the drop to come to a known state in the lower section of the column where i t could be photographed. This was especially important in his work since the size, temperature, and pressure of the air bubble in his drop had to be known accurately so that the partial pressure of the air in the vapour bubble further up the column could be calculated. However, his drop entered the heated section of the column at a temperature below the boiling point of the dispersed phase liquid, therefore the time required to heat the drop up to the boiling point must be accounted for when the time for the total evaporation i s calculated. Since a l l three workers obtained photographs of the drop after i t l e f t the nozzle i t was assumed that this eliminated the effect of the type of nozzle used. Nucleation A drop of dispersed phase liquid, i f sufficiently pure, w i l l not start evaporating u n t i l nucleation, by some means, has taken place. To ensure that the drops nucleated at the start of the run several methods were used by the various workers: Klipstein used a hot wire to begin nucleation in his drops. Sideman relied on dis-solved air in the dispersed phase as well as small air bubbles in the continuous phase. Prakash studied several different methods of nucle-ation and decided the best one was to inj ect a small bubble of air into the drop as i t was being formed on the nozzle. He also studied the effect of the "size of air bubble injected but found that there was no significant correlation of q/A with air bubble size. INSTANTANEOUS HEAT TRANSFER COEFFICIENT  Introduction A l l workers have considered the Fourier equation of heat transfer q = U A AT (11) to be a basic starting point in their study of instantaneous heat trans-fer coefficients. However different approaches have been used by each worker to determine each of the elements of the above equation and these differences show up in the calculation of the instantaneous heat trans-fer coefficient. Temperature Driving Force The temperature driving force is the difference between the temperature of the continuous phase and the dispersed phase. A l l workers attempted to keep the continuous phase temperature constant throughout the evaporation section of their columns. They also assumed the temper-ature of the dispersed phase to be the temperature of the vapour in the vapour bubble. This temperature was calculated by comparing the external pressure exerted on the vapour bubble, by the hydrostatic and atmospheric 23 pressure, to the sum of the partial pressures of the ai r , the dis-persed phase vapour, and the continuous phase vapour calculated from the i n i t i a l air bubble, and the vapour pressure-temperature relationship of the dispersed phases. The internal and external pressures are equal only at a unique temperature which can be determined by a t r i a l and error solution. The Instantaneous Rate of Heat Transfer The instantanous rate of heat transfer i s the rate at which heat i s being transferred from the continuous phase to the dispersed phase at any instant. The total heat content of the vapour bubble at any instant may be calculated from the mass of the dispersed phase vapour and the latent heat of evaporation of the dispersed phase. Prakash used the simplest method to calculate the rate of heat transfer with respect to time. This was basically to divide the difference i n the total heat content at two points by the difference in times at these two points. This quantity was used as the rate of heat transfer at the second time. This method of course magnifies any errors in either the experimental measurement of volume or time and thus ine~ creases the scatter i n the calculated values of the rate of heat trans-fer. Klipstein after calculating his total heat content at various times calculated his instantaneous rate of heat transfer by the means of "standard numerical differentiation methods using second order differences". This method would reduce the effect of errors in experimental measurement and is applicable for computer solution. Sideman was the f i r s t worker to plot his total heat con-tent as a function of time and obtain an equation which was then d i f f e r -entiated analytically with respect to time to obtain the instantaneous rate of heat transfer. He plotted his data on a graph of the log of heat content ys the log of time and fitted the data by a linear relation ship. This resulted in an equation of the form Q = Qf - Q± = KGf (12) which was then differentiated to give the instantaneous rate of heat transfer. This method enabled him to use judgement in positioning the linear f i t and thus reduce the effect of scatter in his experimental dat However, this method does require choosing a specific equation to f i t the data, which then determines the shape of the heat content vs time curve without regard to the physical situation. Area The area term is one of the most complex variables in this work. A l l three workers agree with the assumption that the main heat transfer takes place through the liquid-liquid interface and that the correct area is the area of this interface. Prakash and Sideman both f e l t that this area was too complicated to determine. Prakash used an instantaneous area calculated from the equivalent spherical diameter based on the sum of the volumes of the dispersed phase liquid and vapour Sideman used the instantaneous area as defined above and also used the area of the i n i t i a l drop. Klipstein used a more complex .method for calculating the area. This is discussed below. The f i r s t stage in Klipstein's analysis of the area term for ethyl chloride drops and bubbles in d i s t i l l e d water was to divide his experimental bubble shapes, taken from his photographs, into six modes given in Table I. He then attempted to correlate these modes with the indices of equivalent diameter, Reynolds number, and the Eotvos number. The range of these indices for the various modes is shown in Table II. He f e l t that there was "sufficient consistancy to demonstrate that the index approach has value". The next stage of the solution was to decide what was the form of the liquid-liquid heat transfer area i n the various shapes which the bubble could form. Klipstein considered three basic shapes: the sphere, the ellipsoid, and the spherical cap with a dimpled bottom. , He presents equations for the calculation of the liquid area having assumed the dispersed phase liquid l i e s i n a simple puddle at the bottom of the vapour bubble. Since a drop undergoes several shape changes as i t vapourizes, different equations, depending on the stage of evaporation, are used to describe the drop. Klipstein organized his information by dividing the drops into large, medium, and small sizes and calculating what shape changes they would undergo by means of the behavioral indices. The area i s then calculated from the equation most closely representing the particular shape under consideration. He shows these results in a graph of interfacial area vs fraction vapourized with drop size as a parameter. This graph i s reproduced i n Figure 1. Sideman (30) attempts to evaluate the instantaneous liquid-liquid transfer area by assuming both a spherical and an ellipsoidal shap for the bubble with the dispersed phase liquid forming a symmetrical cup TABLE I - Modes of Behaviour for Moving Drops and Bubbles - Klipstein (1) Mode Description Equivalent Spherical Diameter Reynolds No. Eotvos No. 1 Spherical bubbles traveling in rectilinear paths 2 Oblate spheriod, rectilinear motion 3 Oblate spheriod, helical motion 4 Oblate spheriod but shape becomes increasingly irregular with increasing Reynold;s number 5 Transistion-oblate spheriod to spherical cap. Shape i s very irregular and fluctuates continuously. Motion is almost rectilinear. 6 Spherical caps, rectilinear motion <.04 .04-.077 ,062-.077 ,077-.24 .24-.35 .35-.88 >.88 <70 70-400 400-500 500-1000 1100-1600 1600-5000 >5000 <1 1-13 1-13 1-13 13-40 >40 ON 27 TABLE II - Ranges of Behavior Indexes Corresponding to Observed Fluid Dynamic Modes of Vapourizing Drops Behavior Indexes Eb'tvos No. 1) Bubbles 2) Small drops 3) Medium drops 4) Large drops Et Cl - Water System 1 2 3 <1 <1 1 <4 1-4 <1 4-6.4 4-6 1-3 13 6.4-11 6-28 3-11 13-40 a l l 28-60 11-45 >40 >60 >45 Equivalent Spherical Diameter 1 <.06 .06-.08 .08-.24 .24-.35 .35-.88 >.88 2 <.3 .3-.4 .4-.53 .53-1.+ 3 <.15 .15-;3 .3-.4 .4-.7 .7-1.05 >1.05 4 <.2 .2-.3 .3-.6. .6-1.1 >1.1 Reynolds No. Equ. Sph. Dia. 1 <400 400-500 500-1100 1100-1600 1600-5000' >5000 2 350-1000 1000-1400 1400-3000 3000-5400 >5400 3 <300 300^950 950-1100 1100-3000 3000-7000 >7000 4 340-850 850-1200 1200-3000 3000-6500 >6500 Reynolds No. Pro j . Sph, Dj,a. 1 2 450-1300 3 <250 250-1100 4 350-1100 No Data Available 1300-2000 2000-4300 4300-8000+ 1100-2100 2100-5000 5000-10500 1100-2000 2000-4500 4500-10000 >10800 >10000 w 00 0.1 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9 FRACTION VAPORIZED Figure 1 - Area Development versus Fraction Vapourized for Various Drop Sizes as Calculated from Experimental Data. oo of constant thickness in the bottom of the bubble. Unfortunately this analysis cannot be taken seriously, at least not near the end of evaporation, since the Figure 3 in his paper showing "Effective liquid-liquid heat transfer area of pentane 'drop'"does not approach the value of zero as the evaporation ratio approaches 1 0 0 percent. Correlation of the Instantaneous Heat Transfer Coefficient Since a l l the terms are known the instantaneous heat trans-fer coefficient may be found from: U = 1 AT <13> This i s usually included in a Nusselt number and related to the dimen-sionless groups of the parameters of the system. Klipstein reported his results to be: Nu = 2 . 0 + 0 . 0 9 4 Re 0 , 9 3 Pr 1 / 3 ( 1 4 ) c c c Sideman reported his results as a function of the Peclet number and where 8^ I s t n e opening angle of the vapour phase i n the drop. His theoretical equation i s : Q FI 3 Q _ I _ O 0 . 5 p 0 . 5 Nu = / 3 cos B 2 - cos p + 2 Pe c ( c ( ) n Prakash reported his results for his system as: Pe 0 . 4 1 7 p 1 . 2 5 . Nu, = 0 . 0 5 0 5 ( I ) (-£)  U b J d 1 + ^ d _ p t y c Prakash used only one continuous phase and therefore he was forced to use the properties of the dispersed phase liquid i n his dimensionless groups of Nusselt number and modified Peclet number. AVERAGE HEAT TRANSFER COEFFICIENT By redefining the terms in the equation q = U A A T (11) a different form of the heat transfer coefficient can be arrived at. This form is called the average heat transfer coefficient, U, and is de fined by: U 0 A AT (17) or 6 d AT (18) It derives i t s applicability from the unique situation occurring in this study when the components of the system are fixed. This situation is that there are only two parameters: i n i t i a l dia-meter and continuous phase temperature, which can be controlled in any set of runs having the . same dispersed phase and continuous phase. Furthermore, i f the temperature of the continuous phase is held con-stant any change in the rate of dispersed phase evaporation must be due to a change in the i n i t i a l diameter (mass) of the dispersed phase. Although this definition of the heat transfer coefficient gives less information about the mechanism of heat transfer at any instant than the instantaneous heat transfer coefficient; i t partly overcomes the problem of determining the interfacial area and results in a variable which may be correlated with the parameters of the system more easily. Also i t is much more useful as a design parameter than the instantaneous heat transfer coefficient. As in the case of the instantaneous heat transfer coeffic-ient each worker has defined certain of the parameters in a different way. The differences in the definitions and their effect on the average heat transfer coefficient are discussed below. Total Evaporation Time Start of Evaporation Klipstein appears to have started measuring his evaporation time from the instant the drop was released from the nozzle. As his drop was formed in the heated section of the column the temperature of the drop was probably very close to the boiling point of the dispersed phase, thus i t should have started to vapourize as soon as i t l e f t the nozzle. In addition there is a possibility that some evaporation took place as the drop formed. This would reduce the total evaporation time and would result in a larger average heat transfer coefficient. Prakash started his time measurement from the moment his drop entered the heated portion of the continuous phase from the cooler portion of the column. Thus, the temperature of his drops were below the boiling point of the dispersed phase and some of the time was spent heating the.drop up to this temperature. This would make his times longer than those of Klipstein and Sideman and cause his average heat transfer coefficient to be lower. Sideman chose his evaporation time to start above one per-cent evaporation and thus had no problem with sensible heat or end effects. End of Evaporation Klipstein calculated the end of evaporation times by ob-serving the end of the linear portion of his volume vs time curves. Sideman used his equation Q = Q F _ Q. = K 0 f (12) and for Q_ substituted Q calculated from the i n i t i a l volume of the f max drop and the latent heat of vaporization, thus: V = ^^ir) f ( 1 9 ) Prakash used a.dilatometric method and obtained a curve of volume vs time from which the end point of evaporation could be seen quite clearly. In summary we could expect Klipstein's average evaporation times to be about the same as Sideman's with the possibility of being shorter i f very much evaporation took place on his nozzle. Prakash's times would most certainly be longer than those of either Klipstein's or Sideman's because of the sensible heat transfer to the drop. The total heat transferred, Q.j was calculated between the two times representing the start and end of evaporation. Klipstein calculated his on an average of bubble volumes taken at times greater than his end of evaporation time. Sideman and Prakash calculated their total heat transferred from the mass of the dispersed phase and the latent heat of vapourization. Temperature Driving Force The temperature driving force was calculated as an average of the temperature driving forces at various points i n the column. Both Klipstein and Prakash are vague in their discussion of their calculation of an average temperature driving force for a complete run. Sideman used an average value of the temperature driving forces at the start and end of his runs. In any case the-difference i n the temperature driving forces, for the heights of the columns used in these studies, between the start and end of evaporation are small compared to the temperature driving forces themselves. Thus, any differences resulting from the different methods of calculation used w i l l not greatly affect the value of the average heat transfer coefficient. Area A l l workers have considered the area to be a function of the i n i t i a l diameter of the drop as obtained from their photographs. Before any correlation can be made between the physical characteristics of the liquid systems, the relationship between the rate of heat transfer, the i n i t i a l drop diameter and the temperature driving force must be determined. This was done by Klipstein and Prakash and i s shown in Table III. Although Prakash does not give confidence intervals for his exponents on the i n i t i a l diameter and the temperature driving force, i t does not seem unlikely that with the usual range of error in this type of work the confidence intervals might include two for the exponent on the i n i t i a l diameter and one for the exponent on the temperature driving force as in Klipstein's correlation. On this basis i t seems simpler to use the i n i t i a l area instead of the i n i t i a l diameter and to take the value of II out of the constant. Values of the Average Heat Transfer Coefficient The average heat transfer coefficients are defined as * = O^TAT ( 1 7 ) by the different workers are given i n Table IV. The value for Sideman's work was calculated from the data presented i n his paper.(2). 35 TABLE III - Equations for the Average Rate of Heat Transfer Presented by Various Workers Worker System Equation Klipstein ( 1 ) Ethyl chloride-distilled water q = 2 . 8 4 d 2 AT l Prakash ( 3 ) Furan-distilled water i M A 2 - 2 4 q = 1 . 1 4 d. AT' 9 4 Prakash ( 3 ) Isopentane-distilled water n * A 1 - 9 3 q = 0 . 5 d. l AT'9 3 Prakash ( 3 ) Cyclopentane-distilled water q = 1 . 0 2 d . 2 , 1 9 AT* 9 4 TABLE IV - Average • Heat Transfer Coefficients by Various Workers Worker System Coefficient2 (cal/sec cm °C) Klipstein (1 ) Ethyl chloride-distilled water 0 . 9 0 4 Sideman ( 2 ) Pentane-distilled water 0 . 5 8 5 Prakash ( 3 ) Furan-distilled water 0 . 3 1 6 Prakash ( 3 ) Isopentane-distilled water 0 . 1 4 6 Prakash ( 3 ) Cyclopentane-distilled water 0 . 2 3 9 INITIAL DECISIONS RECOMMENDATIONS OF OTHER WORKERS The study of the literature revealed recommendations of other workers which formed a basis for parts of this work. One of Klipstein's recommendations not followed up by either Sideman or Prakash was that "If possible fu l l y vapourized bubbles should be caught and measured to provide an independent size value". Prakash set forth the following recommendations: "The experimental data were limited to ten percent of the vapourization due to the inability to estimate the correct vapour volume using two dimensional photography. If the amount of vapourization could be known more accurately during the whole process a better correlation could be found. Therefore in any further work a dilatometric or any other suitable method should be tried for estimating vapour volume." also "D i s t i l l e d water was used as the only continuous phase liquid with a l l three dispersed phase liquids in the present study. It would be interesting to see the affect on the present correlation of using different continuous phase liquids. The liquids should be chosen to give a good range of Prandtl number." •INITIAL MODIFICATIONS OF EQUIPMENT It was i n i t i a l l y decided to carry out the tests in an apparatus similar to that used by Prakash with the following modifications Since the inner perspex column used by Prakash had been corroded by the dispersed phase liquids, destroying i t s c l a r i t y , the equipment was redesigned and rebuilt with the inner column made of glass. This eliminated any possibility of corrosion by the continuous phase. 37 A new injection system was designed and built in an attempt to increase the reproducibility of the drop size. Since the photographic method using motion picture photo-graphy and back lighting had been effectively used by Sideman and Prakash i t was chosen as the major source of experimental data. A dilatometer, connected to a strip chart recorder, was set up to measure the total evaporation time and was to be used in conjunction with the photographic method. An i n i t i a l series of runs made with the equipment showed that besides being suitable for measuring the end point of evaporation the dilatometer could be calibrated to measure the volume of the bubble. Indeed, the accuracy of the dilatometer was much greater than the photo-graphic method for a l l except very low percent evaporations. Thus the dilatometer could be used to measure the entire range of evaporation quantitatively, including the f i n a l volume of the bubble. With this in mind a series of modifications were made to improve the performance of the dilatometer and to allow a closer comparison to be made between the time measured by the recorder chart and the camera. SYSTEMS CHOSEN For the continuous phase a series of glycerine and water solutions were chosen for the following reasons: - A large range of Prandtl numbers is easily available due to the large change in viscosity of the solutions with a moderate change in glycerine concentration. - Nearly a l l the physical properties of the solutions are available in the literature, see Appendix III. The dispersed phases chosen were isopentane and cyclo-pentane. Furan was considered as a possibility but was rejected since i t failed to pass the following tests which were used to check for immiscibility of the dispersed phases in the glycerine-water solutions. These tests were: - No visual solubility when the dispersed phase liquid was mixed with solutions of varying concentrations of glycerine and water in the range of temperatures to be used in the study. - No significant change in the surface tension of the dis-persed phase liquids when they were mixed with the different solutions of the continuous phase at room temperature. - No colour in the continuous phase when the dispersed phase containing an o i l soluble dye, was mixed with the continuous phase at a range of temperatures including the boiling point of the dispersed phase. Values and equations for predicting the physical properties of the dispersed phases are given in Appendix II. FACTORS NOT STUDIED EXPLICITY The following factors were not studied as a main part of thi work: 39 Drop Diameter Because of the work of K l i p s t e i n and Prakash i t was f e l t that the e f f e c t of the i n i t i a l drop diameter on the average heat transfer c o e f f i c i e n t had been adequately studied. Therefore the power of two on the i n i t i a l diameter, indicated by t h e i r work, was accepted and only a small range of drop diameters was used. Temperature Driving Force As i n the case of the drop diameter i t was f e l t that the temperature d r i v i n g force had been studied s u f f i c i e n t l y and the power on the temperature d r i v i n g force term was assumed to be one. Column Size It was f e l t that the column s i z e used eliminated w a l l e f f e c t s except f o r the very large bubbles obtained at the end of evaporation. In any case no correction was attempted. Surfactants The e f f e c t of surfactants was not studied and although pre-cautions were taken to avoid contamination no checks were made to ensure that t h i s was the case. Therefore there i s a p o s s i b i l i t y of surfactants being i n the system. However, on the l i g h t of the precautions taken i f there are any surfactants present they would most l i k e l y be present i n any system unless extreme precautions were taken. Mass Transfer Even though the dispersed phase l i q u i d s and the continuous phase solutions are considered to be immiscible in each other and there should be no mass transfer between them, both phases were saturated with each other before starting a set of runs. Effect of Surface and In.terfacial Tension The possibility of the presence of surfactants, the d i f f i -culties in measuring the surface and interfacial tension, especially at the boiling point, coupled with the lack of literature data and valid empirical correlations presented many problems, the solution to which was f e l t to be beyond the scope of this work. Therefore the effects of the surface and interfacial tensions were not studied and the variables were not included in any correlations. 41 EQUIPMENT, EXPERIMENTAL PROCEDURE AND DATA COLLECTION EQUIPMENT  Column The experimental equipment was rebuilt along the lines of that used by Prakash. The main modification was the replacement of his perspex inner column by one of glass. The 2,5 inch diameter glass tube was enclosed in a square plexiglass column which was divided into two compartments 43.5 and 5.5 inches long, see Figures 2 and 3. Constant temperature water was circulated in the two sections to control the temperature in the upper and lower portions of the tube. The square plexiglass column also minimized optical distortion which would have been caused by the round glass tube. For details of the column, see Appendix I(a). A plexiglass, doughnut shaped disk with a 1.0 inch dia-meter hole was secured by an "0" ring at the junction of the hot and cold portion of the column. This reduced mixing between the warmer liquid in the upper portion of the glass tube and that in the lower section. The top of the glass tube was secured by an "0" ring and a threaded aluminium top, see Figure 4. A polyethylene plug was i n -serted into this top and sealed by means of another "0" ring. This plug contained the glass dilatometer tube, a brass tube containing six copper-constatan thermocouples and an o i l f i l l e d glass tube containing a paper scale attached to a plexiglass strip. This vertical height scale was graduated to 0.1 inch and was marked with numbers at each inch. Legend for Figures 2, 3, 4 and 5 1 Glass tube 2 Upper perspex square column 3 Lower perspex square column 4 Upper column constant temperature bath 5 Lower column constant temperature bath 6 Perspex doughnut shaped disk 7 "0" ring holding glass column 8 Aluminium screw top 9 Polyethylene plug 10 "0" ring holding polyethylene plug 11 Glass dilatometer tube 12 Brass tube containing six copper-constatan thermocouples 13 O i l f i l l e d glass tube containing scale 14 Movie camera 15 Portable light stand (four 150 watt flood lamps) 16 White diffuser screen 17 Bevel on underside of polyethylene plug 18 Photoelectric strip 19 Fluorescent lamp 20 Hewlett-Packard recorder 21 Biasing device 22 Metal strip 23 Solenoid switch 24 Dispersed phase 25 Ice bath 26 Syringe 27 Check valves 28 Air injection system 29 Calibration burette 30 Main supply tank 31 Tee between dilatometer tubes 32 Slurp tube 33 Mercury Reservoir 34 Portion of metal strip 35 Non-conducting tape 36 Carbon brush 37 Movable camera platform Figure 2 - Front View of the Equipment Set-up . 44 Figure 3 - Side View of the Equipment Set-up. 45 Figure 5 - Details of Brush for Event Marker. 47 Five thermocouples were positioned at heights of 0.7, 6.3, 16.7, 28.6 and 39.5 inches above the doughnut shaped disk; a sixth one was placed in the lower portion of the glass tube. Standard thermo-couple tables and a calibrated thermometer were used to check the thermo-couples for accuracy. A Leeds and Northrup m i l l i v o l t potentiometer, see Appendix 1(d), was used to measure the thermocouples. The temperatures in the two sections of the column were re-gulated by circulating water from the two constant temperature baths. Specifications for the baths are given in Appendix 1(c). Photography A 16 mm H 16 RX Paillard Bolex movie camera, see Appendix 1(g), operated at approximately 32 fps with Kodak Plus - X Reversal Film, type 7276, was used in a l l photographs. The camera was mounted on a movable platform similar to that used by Prakash. Back lighting was provided by four 150 watt flood lamps mounted on a portable stand at the rear of the column. The light shone through a white diffuser screen which was marked by quarter inch wide strips of black masking tape at intervals of one quarter inch. These vertical strips aided in defining the interface between the vapour, the drop liquid and the continuous phase liquid. To determine the exact number of frames per second at which the film was run, a stop watch, making one revolution every six seconds, was photographed for approximately one revolution. The focusing dis-tance of the camera was reduced to about four inches by a five millimeter extension tube placed between the camera turret and the lens. The light intensity was measured by means of a Honeywell Pentax 1°/21° Exposure meter, see Appendix 1(b). Since this type of meter allows a light reading to be made on an exact spot i t proved to be very satisfactory for this type of work. Further details on the photographic equipment may be found in Appendix I. Injection System The injection system is shown i n Figure 2. It consists of a test tube containing the dispersed phase liquid surrounded by a beaker of water and ice, a 1.0 ml gas syringe, two 1/8" check valves, and a rubber bulb connected to a very fine stainless steel tube which ran through the nozzle and extended about 1/32" past the tip of i t . Withdrawing the plunger in the syringe caused liquid to be drawn up through the f i r s t check valve and into the body of the syringe. When the plunger is pushed in the liquid i s forced through the second check valve into the nozzle where i t formed a drop on the end of the nozzle. The rubber bulb was then squeezed u n t i l a small bubble of air was formed inside the drop. Although the injection system was satisfactory i t did not work as well as expected. The main reason for this was a small vapour bubble formed in the syringe (and other parts of the system) due to the reduction i n pressure required to open the f i r s t check valve. The compressibility of this vapour bubble made i t impossible to obtain good control over the size of the drop formed on the tip of the nozzle. Dilatometer The dilatometer consisted of a one centimeter, inside dia-meter, glass tube located in the plug at the top of the column, see Figure 4. A photoelectric strip was placed behind the tube and a six inch fluorescent lamp in front; both were placed as close as possible to the glass of the tube. As the liquid rose in the tube, because of the volume expansion of the bubble, less light reached the photoelectric strip and caused a decrease in the output voltage. The change in the output was recorded on a Hewlett Packard strip chart re-corder, see Appendix 1(e). A biasing device was used to remove the voltage produced by the photoelectric strip when there was no liquid in the dilatometer tube. Vapour bubbles were prevented from collect-ing at the entrance to the dilatometer tube by a bevel on the bottom of the top plug. The setup, which controlled the dilatometer, is shown in Figure 6. Closing valve B and opening valve A allows the pressure exerted by the liquid i n the supply tank to force continuous phase liquid into the calibration burette, which in turn forces mercury into the mercury reservoir. If valve A is closed, valve B may be used to control the amount of liquid forced into the dilatometer by the pressure in the mercury reservoir. Any liquid in the dilatometer tube reach-ing the slurp tube located at the top of the dilatometer w i l l be sucked up by the vacuum produced by the water ejector. 50 SLURP TUBE WATER EJECTOR G B A SUPPLY TANK MERCURY RESERVOIR BURETTE LEVEL I LEVEL 2 1/ J >i 11 m i ? i ) J > t i t Figure 6 - Set-up for Controlling the Dilatometer. Zeroing the Dilatometer •' If the mercury reservoir i s set at level 1 and valves A and B are opened, continuous phase liquid w i l l flow from the supply tank into the dilatometer where i t w i l l be taken off by the slurp tube. Simultaneously, the liquid in the burette w i l l be forced down to a level determined by the height of the mercury reservoir. If valve A is then closed the mercury level w i l l rise slightly in the calibration burette, forcing liquid, which w i l l be taken off by the slurp tube, into the dilatometer tube. If the mercury reservoir i s then lowered to level 2 the liquid in the dilatometer w i l l drop to a level pre-determined by the height of level 2. Since this level can be re-produced, i t was chosen as the zero point for the dilatometer. The biasing device and the zero suppression control, on the recorder, were used to set the recorder pen on the f i r s t division from the edge of the chart paper. The reason for setting the pen in slightly from the edge of the paper was to check for column leakage which would drop the level in the dilatometer tube below the zero point and would thus be detected on the recorder. It was not always necessary to have valve A and B open together, as there was usually enough liquid and pressure in the burette (built up by opening valve A, with valve B closed) to cause the liquid in the dilatometer to rise to the bottom of the slurp tube when valve A was closed and valve B opened. Calibration of the Dilatometer Before starting a set of runs the dilatometer, or rather the output reading on the recorder, was calibrated by injecting measured volumes of the continuous phase into the dilatometer tube. 52 A graph, which related the volume injected to the m i l l i v o l t reading on the recorder, was drawn and used to transform the recorder output of mill i v o l t s versus time to one of volume versus time. The calibration was rechecked at the end of each set of runs. The procedure followed for the calibration was as follows: The dilatometer was zeroed and the mercury reservoir raised from level 2 to level 1. Valve B was closed and valve A was opened forcing the level of the mercury lower in the burette. Valve A was then closed and the level of the mercury i n the burette read and recorded. Valve B was then opened slightly to allow a small volume of liquid, usually about 0.5 ml, into the dilatometer tube and then closed again. The change in the recorder chart reading was then measured and recorded along with the actual change in the volume of the liquid in the burette. This procedure was repeated u n t i l the entire section of the dilato-meter tube that was to be used in the subsequent set of runs, had been calibrated. Set Up for Determining Bubble Height Correlation between the height of the bubble and the change in the dilatometer reading on the recorder was made possible by a device which caused the event marker pen on the recorder to record the time at which the camera lens reached various heights as the camera was made to follow the bubble up the column. Since the camera was moving with the drop, the event marks related the drop height to the dilatometer read-ings of volume and time, the time being found from the recorder chart speed. The device consisted of a metal strip insulated with tape at six inch intervals, a carbon brush attached to the camera platform, and the event marker in the recorder, see Figure 5. The intervals, marked by the tape, corresponded to the markings on the scale inside t. the column. The carbon brush and the metal strip were connected to the event marker of the recorder. As the contact between the brush and the metal strip was broken by the brush passing over a strip of the one inch tape the event marker recorded this on the chart. EXPERIMENTAL PROCEDURE The main objectives of this research were: 1. To use a dilatometric method to produce a record of the volume and height change vs time for a drop evaporating in the column. 2. To obtain photographs of the drop and bubbles moving up the column from which the time, height of drop in the column and the dimensions of the drop could be determined. Preparation for a Set of Runs The following steps were taken in the preparation for a set of runs: 1. The column was drained of d i s t i l l e d water and f i l l e d with the glycerine-water solution to be used in the set of runs. 2. The electrical equipment, except for the photographic lights, was turned on, and the water baths were adjusted to the desired temperatures. The column temperature was steady within half an hour. 3. The thermocouple cold junction bath was f i l l e d with ice. The injection system was attached and primed with dispersed phase liquid and a stream of drops was introduced into the column to ensure a saturated continuous phase and promote mixing of the con-tinuous phase. 4. As the drops, those that did not vapourize, rose in the column their rise was used to check that the column was vert i c a l and that the drops rose in the middle of the column away from the thermocouples, the scale and the sides of the column. 5. The camera was loaded and a sheet, on which was printed the date and the particulars of the system, was photographed. The stop watch used for calibrating the frame speed was photographed and the camera was then mounted on i t s stand. The i r i s settings of the movie camera were made from measurements of the light made by the ex-posure meter. The camera spring was rewound often since Prakash had found that the filming speed depended on the tension of the winding spring. 6. The dilatometer was calibrated as described in the section "Calibration of the Dilatometer". Execution of a Run 1. The recorder was brought to zero as mentioned in the section "Zeroing the Dilatometer". 2. Thermocouple readings were taken and noted down. 3. A small drop of liquid was squeezed out of the nozzle and a small air bubble injected into the drop. If the drop did not release immediately more dispersed phase liquid was injected u n t i l i t did. 4. As the drop began to rise in the column the camera was trained on i t and the filming begun. The chart drive motor in the re corder was switched on automatically by a microswitch as the camera began to move upwards. The bubble was followed un t i l i t reached the top of the heated portion of the column. 5. At the end of the run the recorder chart drive was switched off and the speed, run number, and calibration marked on the chart paper. 6. The next run number was photographed and the frame number was read from the camera and recorded. 7. The camera was rewound and positioned at the bottom of the column in readiness for the next run. Termination of a Set of Runs Basically the same procedure was followed as for the start up, except of course, in the reverse order. To prevent clogging of the injection system i t was removed and cleaned. The screw top of the column was loosened to allow for the thermal contraction of the column after the water baths were shut off. The column was drained, rinsed out, and then f i l l e d with d i s t i l l e d water. 56 DATA COLLECTED The data collected were as follows: 1. One thermocouple reading of the lower portion of the column, and five thermocouple readings of the upper portion of the column taken at heights of 0.7, 6.3, 16.7, 28.6, and 39.5 inches above the doughnut shaped baffle. 2. Barometric pressure and room temperature taken shortly before a set of runs. 3. The height of water in the dilatometer tube when i t is at the zero point. 4. A photographic film containing the date, place, the number of the set of runs, the run number i t s e l f , the photographs of the bubbles rising up the column beside a marked scale, and a series of frames of a running stop watch used for calibrating the filming speed, 5. A recorder strip chart containing the results to be used for: the calibration of the photoelectric strip output against the volume for each set of runs, the photoelectric output of the dilatometer versus time for each run and the marks of the event marker corresponding to the time at which the camera lens passed the heights of 0.0, 6.0, 12.0, 18.0, 24.0, 30.0, 36.0, and 42.0 inches as measured on the scale inside the column. The chart also contained the date, the run number, the voltage setting, the f u l l scale voltage setting and the speed of the recorder. 6. The density and viscosity of the continuous phase, which were determined with a Westphal balance and Cannon-Fenske viscometers respectively. PRELIMINARY PROCESSING OF DATA Before the data were processed by the computer the follow-ing operations were performed on i t : 1. By use of standard tables the thermocouple readings were converted to °C. 2. The photographic film was projected on a screen by a Specto Motion Analysis, single frame, projector, (Appendix 1(f))and the following measurements made: The frame speed was obtained by counting the number of frames needed to photograph approximately one revolution (six seconds) on the stop watch. A magnification factor for the photographs was obtained by measuring, on the photographs, a one inch interval on the scale inside the column. This factor was used to determine the true size of the other dimensions in the photographs, namely: the i n i t i a l diameter of the drop, the dimensions of the air bubble inside the drop, and the horizontal and.vertical dimensions of the vapour bubble. In addition, the height of the bubble as measured from the scale was recorded. 58 The frame number, corresponding to the photograph being measured, was obtained from the frame counter on the projector. 3. Using the experimentally measured values of the density and viscosity of the continuous phase, the composition of the con-tinuous phase was determined by interpolation of literature data of glycerine-water solutions. 4. Figure 7 is a reproduction of the recorder chart output for Run 1308. The calibration curve made at the start of this set of runs was used to convert the m i l l i v o l t readings, at the points marked on the chart, into their equivalent volume in m i l l i l i t e r s . These points represented the entire curve, with the second last point, placed at the start of the linear portion of the curve, corresponding to the end of evaporation. The time at each point was calculated from the recorder chart speed (inches per second) and the distance (inches) measured from the event mark corresponding to the time at which the camera lens passed the zero position of the scale inside the column. The times at which the camera lens passed the six inch intervals, shown by the event marks, were calculated in the same manner. RATIONALE FOR DATA This section gives an insight into the purpose of the data collected and outlines i t s use in further calculations which are performed by the computer. i 1 1 r v*END OF EVAPORATION END OF TIMING RUN 1308 CHART SPEED 2 IN./SEC O POINTS CHOSEN TO REPRESENT CURVE START OF TIMING J l _ I _ I _ i _ i i _ l l i _ 42 36 30 24 18 H 6 0 HEIGHT OF BUBBLE (IN.) Figure 7 - Reproduction of Experimental Dilatometer Curve for Run 1308. 60 Photographic Data 1. The temperature and the pressure measured in the lower portion of the column are used to calculate the i n i t i a l properties of the liquid drop and the air bubble contained in i t . 2. The temperature measured in the upper portion of the column w i l l be curve fit t e d by the computer as a function of height. From the resulting correlation equation the continuous phase temperature at various heights in the column w i l l be calculated and w i l l be used in the calculation of the temperature driving force between the temperature of the continuous phase and the temperature of the vapour inside the drop. 3. The atmospheric pressure, the i n i t i a l height of water in the dilatometer, and the height of the drop are used to calculate the external pressure on the bubble, which w i l l be used, in conjunction with the vapour pressure data of the dispersed phase, the continuous phase, and the partial pressure of the ai r , to determine the temperature of the vapour inside the bubble. This is possible since only at one unique temperature w i l l the partial pressure of the a i r , continuous phase and dispersed phase vapour add up to a pressure equal to the total pressure outside the drop. 4. The i n i t i a l diameter and the volume of the air bubble is used in the calculation of the mass of the dispersed phase liquid. Also the volume of the air is used to calculate the partial pressure of the air used in the calculation mentioned in section 3 above. 61 5. The dimensions of the vapour bubble are used to cal-culate the amount of dispersed phase liquid which has been evaporated between any two observation points, This i s used only for low evapora-tion ratios. 6. The height is used in determining the total external pressure on the drop; and after the time has been calculated from the frame number and the frame speed, i t is used to calculate the velocity of the drop or bubble. Dilatometer Data 1. The event marks, which give the height of the camera in relation to the time determined from the recorder chart, allow a correlation to be made between the time found by the photographic method and that found by the dilatometric method. 2. The point marking the end of evaporation is used to determine the total evaporation time and the f i n a l volume of the vapour bubble, from which may be determined the mass of the dispersed phase vapour and the total amount of heat absorbed by the dispersed phase. 3. The recording of the dilatometer reading is basically one of volume versus time which is proportional to the amount of heat transferred from time zero to any other time. The slope of this curve at any instant i s therefore proportional to the rate of heat transfer at that point. 4. By knowing the total amount of heat absorbed by the dispersed phase, the total evaporation time, the i n i t i a l area of drop, and the temperature driving force the average heat transfer coefficient may be calculated from: U = Q/0 A± A T INTERPRETATION OF RESULTS INTRODUCTION At the start of this work the plan of attack was to pro-ceed as Prakash had done and rely heavily on the photographic method, which had also been used by Sideman, to obtain most of the data. A dilatometer, used concurrently with the camera, would measure the total evaporation time. The preliminary work with the dilatometer suggested that the emphasis should be changed from the photographic method to the dilatometric one, as i t showed that the dilatometer was sufficiently accurate to measure the volume of the bubble for a l l except very low (1 - 3%) percent evaporations. It was therefore possible to measure the f i n a l volume of the vapour bubble - something which had not been done before accurately - and to obtain a continuous re-cording of the growth of the vapour bubble as i t expanded. Naturally, since the photographic method is not valid above, at most, ten percent evaporation, a study of the f u l l range of evaporation, which means the drop must completely evaporate before reaching the top of the column, reduces the time during which the photographic method may be used. Thus, the importance of the photo-graphic method, as a source of data was diminished. By changing the main source of data from the photographic to the dilatometric method a great deal of time and effort was consumed in modifying both the equipment and the computer program. This, coupled with the formulation of a new outlook based on the dilatometer data, which resulted in the rejection of the instantaneous heat trans-fer coefficient for correlating the results to the parameters of the systems, accounted for a large proportion of the time spent on this thesis. VISUAL OBSERVATION OF DROP EVAPORATION 1. Drop Formation and Release Ejection of a small amount of dispersed phase liquid formed a small liquid drop on the end of the nozzle. Injected into this drop was a minute bubble of air. The drop f i n a l l y released as more liquid was injected into i t . 2. Rise of the Drop in the Lower Cooled Portion of the Column Shortly after i t s release from the nozzle the drop seemed to rise steadily in the lower cooled portion of the column. There was no noticeable change in the size of the air bubble in this section of the column. 3. Rise of the Drop in the Lower Portion of the Column Until the Onset of Evaporation As the drop entered the lower part of the heated section of the column i t s velocity immediately increased due to the decreased viscosity of the continuous phase _ caused by the higher temperature. The air (vapour) bubble increased slowly in size. However, since most of the heat absorbed was used to heat up the cool dispersed phase liquid, very l i t t l e evaporation took place. 4. Region of Main Evaporation As soon as the liquid became heated to i t s boiling point evaporation started and the vapour bubble grew quite rapidly. The characteristics of the dispersed phase changed from those of a drop to those of a bubble. In a l l the runs the liquid puddle remained at the bottom of the vapour bubble and did not separate from i t . The bubble continued to rise and expand, forming various shapes depending on the parameters of the system. 5. End of Evaporation Near the top of the column the bubble ceased to increase in size and in most cases the shape and velocity of the bubble be-came constant. The end point of evaporation was very d i f f i c u l t to see visually and there was no possibility of not accepting the point shown by the recorder as the end of evaporation. The bubble then made i t s way out of the column through the dilatometer tube. RANGE OF VARIABLES The ranges of important variables covered in this study are given in Table V. It was necessary to s p l i t them into two sections The f i r s t section contains those which are related to an entire run or are used for the calculation or correlation of the average heat trans-fer coefficient. The second section contains those variables which were determined from measurements made at various times during the evaporation and which are therefore related to the percent evaporation. TABLE V - Range of Variables Section I Average Those variables considered constant for an entire run. Temperature Driving Force Continuous Phase % Composition Temperature Prandtl number Viscosity cp Thermal Conductivity cal/(sec)(cm) (°C) Heat Capacity % Glycerine by weight °C Density Dispersed Phase I n i t i a l Diameter Prandtl number Viscosity cal/(gm)(°C) gm/cm"^  cm cp Thermal Conductivity cal/(sec)(cm) (°C) Heat Capacity cal/(gm)(°C) Density gm/cm^  2.31 - 8.77 0.0 - 77.06 31.5 - 55.2 3.36 - 105.4 0.517- 13.3 0.600807 - 0.00154 0.661 - 1.00 0.9877 - 1.174 0.164 - 0.336 3.33 - 6.10 0.273 - 0.322 0.000255 - 0.000301 0.3113 - 0.568 0.610 - 0.668 Section II Instantaneous Those variables determined at any instant during the evaporation. Diameter cm 0.177 - 1.94 Velocity cm/sec 5.13 - 27.09 Reynolds number - 14.2 - 9059 Peclet number - 939 - 47349 Tempatature Driving Force °C 2.31 - 32.12 THE DILATOMETER CURVE Advantages of the Dilatometric Method Prakash (3) was the f i r s t worker to use a dilatometric method in this type of work. However his dilatometer could not.measure volume expli c i t l y and he used i t only to determine the time at which the evaporation ended. Although this was an improvement over previous work, (1) and (2) ,in which the end of evaporation had been estimated from extrapolation of experimental data or by visual observation, his main work was s t i l l limited to a low range of evaporation (< 10%), due to the d i f f i c u l t y in determining the volumes of the distorted bubbles-from photographs. One of the main contributions of the present work is the extension and development of a dilatometric method that would measure the volume of the bubble over the f u l l range of evaporation. With the calibrated dilatometer used in this study i t is possible to obtain the following: 1. The entire volume vs. time relationship for an evaporating drop. 2. The time at which the evaporation has effectively ceased. 3. The volume of the vapour bubble at the end of the evaporation. 4. An alternate method to measure the mass of the dispersed phase via the vapour volume at the end of evaporation. The above information presents a much more complete picture of the evaporation process than has previously been given. 68 From the "S" shaped curve i t appears that the slope of the curve at any point is a function of the time (percent evaporation); and the change in the slope from point to point is due mainly to a change, resulting from the evaporation of dispersed phase liquid, in the area of the liquid-liquid interface. If the above is true, i t points out the importance of the liquid-liquid area term in determining the instantaneous heat transfer coefficient and the fallacy of attempting to relate this coefficient to the parameters of the system, without considering the amount of evaporation which has taken place or using a limited region of evapora-tion as occurs in a photographic study. Qualitative Discussion of the Dilatometer Curve The shape of the "S" shaped curves representing the volume -time relationship of the evaporating drop, shown in Figure 8, can be explained by the following semi-qualitative discussion of the evaporative process. Explanation of the Volume Curve  General Equation Let us assume a spherical shape of the bubble as shown in Figure 9. There are two pathways for the heat to enter: One is through the vapour portion of the bubble, say at a rate q^. The second is through the liquid at a rate q . Therefore the total heat transferred i s : Figure 8 - Idealized Example of Dilatometer Curve. Figure 9 - Paths for Heat Transfer to an Evaporating Drop. 71 q = + q v (20) Using the usual assumption of the rate of heat transfer being d i r e c t l y proportional to the area of heat transfer and the temperature d r i v i n g force, we have: \ " U* \ AT* ( 2 1 ) and q = U A AT (22) v v v v where U. and U are the o v e r a l l heat transfer c o e f f i c i e n t s f o r the £ v l i q u i d and vapour r e s p e c t i v e l y . Since the heat transferred to either the vapour or l i q u i d i s dependent on the i n t e r n a l and external heat transfer c o e f f i c i e n t s of each, the o v e r a l l heat transfer c o e f f i c i e n t i s r e l a t e d to these by: U A - - i (23) 1 1 Kl AoSL h i £ A i £ U A = (24) v v ( 1 + 1 ) K h A h, A. ' ov ov i v i v and the rate of heat transfer i s AT AT q v (26) ' ^ nTV +H-JA-> I V X V ov ov or AT AT H . v q = + (. 1 + 1 ) ( 1 + — i ) hp A h.„ A./ V A. h A 1 oX oi ±H iv iv ov ov (27) where the subscripts o, i , stand for the outside and inside of the drop or vapour bubble respectively. To simplify the above expression the following assumptions are made: X ) Ao* = A i £ = A £ (28) and A. = A = A (29) iv ov v \*-->J 2) AT^ = AT v = AT (30) The last equation assumes the liquid temperature is the same as the vapour temperature. So the expression become A A q = ( + - ) AT (31) 1 1 1 + 1 h . h.„ h h. osi il ov xv In the case of the gas phase, the controlling resistance would most li k e l y be the internal one and we can assume 3) h. is much less than h xv ov therefore we have: or where and An q = ( ^ ~ + - 2 — ) AT (32) 1 + 1 1 h . h.„ h. ol i£ i v q = (U. A + U A ) AT (33) X, X, V V U 4 = 1 . (34) o & i £ U = h. (35) v iv Klipstein has shown that q^ is small compared to q^. This w i l l hold except at or near the end of evaporation when A^ becomes very large compared to A£ and the term A^ U becomes significant. Equations Applicable to Specific Regions of Evaporation Consideration of specific areas of the evaporation curve, shown in Figure 8, yield further simplifications. Region 1 - Lower Heated Portion of the Column While the drop is heating up to i t s boiling temperature and before any significant evaporation has taken place A^ may be assumed small and the equation may be written as: q = Ufc A £ AT (36) 74 and since is f a i r l y constant even though the vapour volume has i n -creased slightly A^ may be replaced by the i n i t i a l area q = U £ A. AT (37) The temperature driving force w i l l of course be changing quite rapidly due to the liquid heating up to the boiling temperature. Region 2 - Main Evaporation Region In most of the main evaporation region the surface area of the vapour bubble A is nearly equal to the total surface area of the bubble A^therefore: * " ( U v \ + U* V A T ( 3 3 ) This equation can be l e f t as i t is or U A^ can be assumed to be small and the equation further reduced to: q = A £ AT (36) The term A^ is very d i f f i c u l t to evaluate since i t is a function of the spreading tendencies of the dispersed phase liquid, the amount of dispersed phase liquid l e f t , and the total volume and shape of the bubble. Klipstein has attempted to evaluate this area by assuming various shapes for his bubbles aid relating i t to the percent evaporation. By assuming the shape to be either a sphere, an ellipsoid, or a spherical cap he found that the rate of increase of the liquid-liquid inter-f a c i a l area, decreases as the vapour bubble's growth approaches forty percent evaporation, where i t ends. From there on the area decreases u n t i l i t reaches zero at total evaporation. His work discounts Sideman's (30) suggestion that the liquid-liquid interfacial area i n -creases un t i l the end of evaporation. Klipstein (1) also observed that "the dependence of area on i n i t i a l diameter which was demonstrated on an average basis in Figure 4.8 continues to hold on a differential basis, i.e., instantaneous area values for the three drop sizes are related to each other according to the relation: \ = c d. 2 (38) where e is .a function of fraction vapourizedl' • Region 3 - Upper Evaporation Region Considering the equation: « " ( U v A v + U* V- A T ( 3 3 ) i t can be seen that as becomes small the equation approaches: q = U A AT (39) v v and since A^ = A^ q = U v A^ AT (40) Since the evaporation has effectively ceased, the rate of change of A^ is due to changes in the hydrostatic pressure and the temperature of the vapour bubble. However, this temperature w i l l soon come to equilibrium with the continuous phase and hence the volume-time curve is expected to be a straight line after evaporation has ended.". In summary we have three equations: 1) q = A. AT where since therefore 2) q - \ A £ AT h = C d i (37) (36) (38) c = f (% evaporation) A. = g (Jo evaporation) 3) q - U v A , , AT. (40) These equations should be kept i n mind when the discussion of the r e s u l t s , derived from the curve f i t t i n g process, i s presented. DISCUSSION OF METHODS OF PROCESSING VOLUME-TIME DATA The basis for a l l work in this f i e l d has been directly or i n -directly, a relationship between the volume of the dispersed phase eva-porated and the time taken for the evaporation. This relationship usually consists of sets of discrete points, whether due to the technique of photography or, in the case of this work, choosing discrete points from the dilatometer curve for use in a computer calculation. The f i r s t stage in processing the data is to evaluate the rate of change of volume with respect to time. This may be done in two ways: The f i r s t one, used by Klipstein and Prakash, is to use a type of numerical differentiation of the discrete sets of points of volume and time to obtain the instantaneous rate of heat transfer. The second method, used by Sideman and in this work, is to f i t a curve to the discrete sets of points and then differentiate the re-sultant equation analytically. The second method has a number of advan-tages to recommend i t s use. These are: 1. Less dependence on scatter in the experimental data. 2. The rate of heat transfer may be determined at any point on the curve by substitution of time values into the equation. 3. The equation of the curve is representative of the entire evaporation process and not just a part of i t . Given the constants and this equation one can reproduce the evaporative process, as well as the rate of heat transfer at any point. Therefore the entire evaporation process can be related to the parameters of the system by correlating the constants in the equation to those parameters. DISCUSSION OF THE INSTANTANEOUS HEAT TRANSFER  COEFFICIENT AS A CORRELATION PARAMETER The f i r s t method discussed above that used by Klipstein and Prakash, gives an instantaneous rate of heat transfer at a specific time. The second method can give the instantaneous rate of heat trans-fer at any time, simply by substituting the time into the differentiated equation. This instantaneous rate of heat transfer can be used in the following equation to obtain the instantaneous heat transfer coefficient: U = — 3 — (41) A ± AT or U = '—3— (42) Again the problem of the correct value to be used for the area occurs. Klipstein has been the only worker to assume various bubble shapes and distributions of the liquid in the bubbles and to calculate the area to be used in the above equation. Other workers have resorted to the use of an area term based on the instantaneous area of the bubble or the i n i t i a l area of the drop. The area based on the i n i t i a l area of the drop may be adequate for very low percent evaporation as shown in equation (37), however, above this point i t has no valid basis. The instantaneous area term also takes no account of the change in area of the liquid as the bubble expands and the volume of liquid decreases. Thus, these two concepts, that of using the i n i t i a l area and that of using the instantaneous area of the vapour f a i l for the following reasons: 1. They do not account for the decrease in the rate of heat transfer after a certain point, that is most lik e l y caused by the de-crease in area of the liquid-liquid interface. 2. They do not consider that the rate of heat transfer i s a function of the amount of evaporation. Since the instantaneous heat transfer coefficient is forced to be a function of the percent evaporation, and since i t must vary to take into account the change in the liquid-liquid heat transfer area, calculation of a Nusselt number and i t s relationship to the variables of the system in the usual form of: Nu = G + E Rep Pr S (43) i s misleading. For example, a bubble having the same outside diameter as another-in the same type of systen^ might have the same values of Reynolds and Prandtl numbers but have a different instantaneous heat transfer coefficient because i t has undergon-e a different amount of evaporation. The only way this might be handled is to add another d i -mensionless group which has the percent evaporation as a parameter. Klipstein's approach is the only one which seems reason-able and is the one which might lead to an understanding of the basic 80 processes. To use his method one must use the following steps for each instant: 1. By means of certain indicesj Klipstein suggested the Eotvos number, the Reynolds number and the equivalent spherical dia-meter, the f l u i d mode of the bubble is determined. From the f l u i d mode the shape of the bubble, either spherical, ellipsodal, or spherical cap with a dimpled bottom is determined. 2. From this shape and the percent evaporation the heat transfer area i s obtained based on the assumption that the liquid forms a definite shape in the bottom of the bubble. 3. This heat transfer area is then used in the usual equation (13) to obtain the instantaneous heat transfer coefficient at one parti-cular instant. Although this method is certainly the most reasonable way to proceed; the basic foundation work on the f l u i d dynamic modes and their relationship to certain indices has not yet been done. Also there could be a great deal of discussion about the actual shape of the liquid puddle in the bottom of the vapour bubble. Klipstein had hopes that with backlighting and motion picture photography the liquid area and shape could be determined. However, this has not been the case and the correct value for the area i s s t i l l open to assumptions and speculations. In the i n i t i a l stages of the work i t was planned to use the instantaneous heat transfer coefficient for correlation, however, as the above discussion shows, i t was found to be an unsatisfactory con-cept and a study of other parameters to be used for correlation was undertaken. This led to a closer examination of the relationship be-tween the volume and the time - which in this work is easily obtained usi the dilatometer. DISCUSSION OF THE CURVE FIT FOR THE DILATOMETER CURVE As explained in the "Experimental Procedure" the dilato-meter curve from the recorder chart was converted into a series of dis-crete values of volume and time. The differential of an equation f i t t i n g these points relates the rate of heat transfer to the time. Thus, i f an equation could be found that would satisfactorily f i t these points and i t s constants could be related to the parameters of the system this would provide a means of correlation. In other words, given the equation and the relationships between the constants of the equation and the parameters of the system, i t would be possible to determine the rate of heat transfer at any instant without becoming involved in the con-cept of interfacial area. However, there were restrictions on the type of models which could be chosen for the curve f i t s . These were: 1. Due to the large number of points and the s t a t i s t i c a l nature of the curve f i t t i n g , a computer calculation was the only feasible way of processing the data. 2. The computer programs available, which would s t a t i s t i c a l l y determine the coefficients of the chosen equation, were found to be very unstable when certain models containing exponential functions were used, unless the i n i t i a l c o e f f i c i e n t s were chosen very c a r e f u l l y . With the large number of data points obtained i n t h i s work, c a r e f u l control over the i n i t i a l c o e f f i c i e n t s was not p r a c t i c a l and therefore i t was decided that the equation chosen be one that could be transformed into a l i n e a r equation which could be e a s i l y processed on the computer. The equation selected was a growth rate equation of the form: (44) V - V. x which could be converted into a l i n e a r equation of the form: w = ctz + 3 (45) by the transformation (46) z = lti9 (47) a n (48) B. = In C (49) Since the V\ term was small the equation was written i n the form: V = / (1 + C0 n) (44a) 83 For further discussion of the above equation and i t s use i n the computer program, see Appendix IX. This produced an "S" shaped curve s i m i l a r to the experi-mental ones. However, t h i s type of curve could not f i t the l i n e a r portion of the experimental curve, which occurred a f t e r the end of evaporation; t h i s decreased the p o s s i b i l i t y of r e l a t i n g the c o e f f i c i e n t s to the parameters of the system. The Figures 10 through 16 show, for selected runs, experimental points with the curve f i t equations trans-posed on them. In nearly a l l cases there i s good agreement between the experimental points and the f i t t e d curve. The calculated curve was terminated when the time became greater than t o t a l evaporation time and the experimental points were used to determine the l i n e a r portion occurring a f t e r t h i s point. The c o e f f i c i e n t s used i n each run are given in'<Table VI along with the mean value and standard deviation for each of the sets of runs. Unfortunately there was a good deal of scat t e r i n the c o e f f i c i -ents, e s p e c i a l l y the "C" term and no serious e f f o r t was made to c o r r e l a t e these with the parameters of the system. AVERAGE HEAT TRANSFER COEFFICIENT The average heat transfer c o e f f i c i e n t gives a parameter which i s a function only of the i n i t i a l mass or diameter, the continuous phase temperature, and the ph y s i c a l properties of the system. As mentioned before i n the section "Factors Not Studied" the r e l a t i o n s h i p between the Figure 10 - Comparison of Curve Fit to Experimental Data - Run 1308. oo -IN T 1.8-16-1.4-0.0 1.0 2.0 3.0 4.0 5.0 60 7.0 TIME (SEC) Figure 11 - Comparison of Curve Fit to Experimental Data - Run 1402. oo 1-8 16 1.4 O EXPERIMENTAL CURVE FIT RUN 1604 3.0 4.0 5.0 TIME (SEC) 6.0 70 Figure 12 - Comparison of Curve F i t to Experimental Data - Run 1604. CO Figure 13 - Comparison of Curve Fit to Experimental Data - Run 1708. 4.0 3.0 2.0 1.0 0.' O EXPERIMENTAL — CURVE FIT RUN 1803 0.0 1.0 2.0 3.0 4.0 5.0 6.0 TIME (SEC) 14 T- Comparison of Curve Fit to Experimental Data - Run 1803 1-8 1.6 i12 UJ 1.0 | 0.8 o 0.6 0.4-0.2 0.0 1 O EXPERIMENTAL — CURVE FIT RUN 1906 3.0 4.0 50 TIME (SEC) 60 7.0 Figure 15 - Comparison of Curve Fit to Experimental Data - Run 1906. 0 0 VO 1.8-1.6-14-3 1.2-2 ^ 1.0-LLI §,0-8-O 0.6-0.4-0.2-0.0*-0.0 O EXPERIMENTAL — CURVE FIT RUN 2011 1 i 1 3.0 4.0 5.0 TIME (SEC) 6.0 7-0 Figure 16 - Comparison of Curve Fit to Experimental Data - Run 2011. O TABLE VI - Coefficients of Equations used for Curve F i t of Dilatometer Volume v = — i + ce11 Set No. 13 14 16 17 Run No. V f C n V f C n V f C n V f C n 1 3.23 76.1 -5.42 2.45 54.4 -5110 3.46 346.4 -6.10 2.60 796 -7.79. 2 2.17 30.7 -5.84 2.25 18.8 -4.55 2.50 438.0 -5.59 2.72 573 -7.90 3 4.50 111.1 -5i20 2.18 148,0 -5.84 3.25 63.0 -4.75 - - -4 2.02 53.9 -5.76 2.30 20.6 -5.01 2.00 114.0 -4.69 3.06 336 -7.19 5 2.25 14.2 -4.13 2.70 75.3 -5.85 - - - 2.41 216 -7.92 6 2.72 23.4 -4.21 2.65 416.0 -7.17 3.35 85.5 -5.29 1.60 1669 -7.50 7 2.31 46.9 -5.14 2.67 681.0 -6.74 2.62 167.0 -5.53 3.10 323 -7.13 8 1.81 31.0 -5.22 2.35 30.8 -4.93 2.66 128 -5.40 2.75 180 -7.47 9 2.65 11.9 -4.06 2.68 351.0 -6.99 1.70 346 -5.03 2.51 559 -8.77 10 - - - - - - 3.15 785 -7.00 2.51 353 -7.60 11 2.28 17.5 -4.90 3.67 839 -7.72 2.90 409 -6.01 2.64 117 -7.39 12 3.25 47.0 -4.62 1.85 15.6 -4.61 2.90 468 -5.66 2.78 369 -7.68 mean value 2.65 42.2 -4.96 2.52 241 -5.87 2.77 305 -5.55 2.61 499 -7.67 standard deviation 0.766 30.0 0.630 0.463 293 1.23 0.554 220 0.661 0.398 435 0.449 TABLE VI - Coefficients of Equations used for Curve F i t of Dilatometer Volume V = i + ce n Set No. 18 19 20 Run No. V f C n V f C n V f C n i 1 3.94 43.7 -5.39 4.00 205 -5.89 2.97 1671 -7.84 2 2.94 30.7 -4.99 3.60 212 -6.54 2.63 "286 -7.00 3 4.80 46.1 -4.70 3.45 301 -6.71 3.10 915 -7.37 4 2.99 50.7 -6.04 1.35 279 -5.74 2.90 256 -6.33 5 3.54 58.6 -5.84 3.70 286 -6.61 2.75 224 -6.75 6 3.77 44.2 -5.79 2.98 125 -6.35 2.71 320 -6.51 7 3.14 29.6 -4.56 - - - 3.00 394 -6.95 8 - - - 3.11 147 -6.47 3.20 582 -7.35 9 - - - 3.40 265 -6.12 3.00 1178 -7.85 10 4.20 23.5 -4.18 3.65 332 -6.87 2.95 371 -6.59 11 2.15 35.9 -4.83 3.08 160 -6.18 2.92 855 -7.41 12 - - — 2.46 42.5 -5.25 _ _ _ mean value standard deviation 3.50 40.3 0.789 11.25 -5.15 3.16 214 -6.25 2.92 641 0.648 0.734 88.5 0.478 0.169 465 -7.09 0.519 93 i n i t i a l mass and temperature is assumed to be of the form: 5 = A. 9 AT ( 1 7> i t The total heat transferred, Q, i s calculated from the mass of the dispersed phase, determined from the volume measured by the d i l a -tometer at the end of evaporation, and the latent heat of vaporization of the dispersed phase. Measurement of the total evaporation time, &^,, is started when the drop has definitely begun to evaporate (3-5%) and ends when the evaporation is complete. A discussion of the measurement and calculation of the i n i t i a l area term, A^, is given in the section "Calculation of I n i t i a l Drop Axea". The temperature driving force, AT, i s the temperature difference between the continuous phase and the dispersed phase temperatures at the end of evaporation. Klipstein has presented values directly comparable to the average heat transfer coefficient; however, those presented for Sideman and Prakash had to be calculated from data published in their works. The average heat transfer coefficient was calculated for a l l runs - see Appendix V for a sample calculation and Appendix VII for the calculations. The results are given in Table VII with the mean value and standard deviation for each set of runs. Table VIII shows the values TABLE VII - Average Heat Transfer Coefficients SET NUMBER Run No. 13 14 16 17 18 19 20 1 0.314 0.561 0.265 0.273 0.169 0.374 0.288 2 0.288 0.588 0.231 0.318 0.142 0.501 0.298 3 0.385 0.607 0.263 - 0.150 0.489 0.291 4 0.338 0.645 0.247 0.400 0.166 - 0.281 5 0.373 0.588 - 0.311 0.179 0.402 0.301 6 6.303 0.694 0.253 0.316 0.137 0.422 0.263 7 0.372 0.508 0.213 0.331 0.152 - 0.296 8 0.346 0.546 0.244 0.328 - 0.556 0.305 9 0.375 0.651 0.197 - - 0.340 0.312 10 - 0.501 0.299 0.321 0.137 0.445 0.289 11 0.450 0.647 0.242 0.389 0.117 0.382 0.292 12 0.299 0.717 0.223 0.337 - 0.388 -mean value 0.349 0.604 0.243 0.332 0.150 0.430 0.292 standard deviation 0.0479 0.0726 0.0278 0.0371 0.0192 0.0673 0.0130 TABLE VIII - Parameter Values for Calculating the Correlation Equations Pr c P r d y c V d U Prakash (furan and water) 3.95 4.19 0.599 0.340 0.316 Prakash (isopentane water) 3.95 6.10 0.599 0.273 0.224 Prakash (cyclopentane water) 3.95 3.33 0.599 0.322 0.436 Sideman (pentane and water) 4.16 3.48 0.555 0.203 0.516 Klipstein ,ethyl chloride, glycerine 5.13 3.83 0.759 0.271 0.916 Klipstein ^ethyl chloride, glycerine 6.22 3.83 0.902 0.271 0.859 Klipstein ,ethyl chloride, glycerine 7.45 3.83 1.05 0.271 0.930 Klipstein ,ethyl chloride^ glycerine 63.6 3.83 10.0 0.271 0.382 Klipstein ,ethy.l chloride, glycerine 86.8 3.83 13.8 0.271 0.436 This work (Set 13) 3.37 3.33 0.517 0.322 0.349 This work (Set 14) 5.31 6.10 0.775 0.273 0.604 This work (Set 19) 22.7 3.33 2.93 0.322 0.430 This work (Set 20) 39.4 6.10 5.08 0.273 0.292 This work (Set 17) 56.1 3.33 6.90 0.322 0.332 This work (Set 16) 75.0 3.33 9.14 0.322 0.243 This work (Set 18) 105.3 6.10 13.3 0.273 0.150 of the parameters used for correlating the average heat transfer coefficients. Correlation Parameters of the System The reasons for choosing these parameters w i l l be given in the following discussions: The continuous phase Prandtl number was chosen as the basic parameters for correlation since i t contains the main variables which were considered important in heat transfer: those of viscosity, thermal conductivity, and heat capacity. In Figure 17 the average heat transfer coefficients are plotted against the continuous phase Prandtl number. A linear equation was calculated to correlate the average heat transfer coefficient with the continuous phase Prandtl number. This equation is shown in Figure 18 and and found to be: U = 0.701 p r " ° - 2 4 5 (50) c This equation as well as a l l other correlation equations was obtained by a stepwise multiple regression technique using the UBC TRIP (Triangular Regression Package) program (31) supplied by the Computing Center at the University of British Columbia. Terms used from this program, such as RSQ, are defined in Appendix VII along with the TRIP results and a summary of the correlation equations. Equation (50) does not take into account the differences 1.0, o o C M o o 111 w 0 .1 < u 0 . 0 1 1 I I I ) 1 1 k 1 | 1 1 1 l _ A — O o A ° O — • • A O C Y C L 0 P E N T A N E — - A ISOPENTANE — i i i l l I I i i 1 1 I I I 1.0 1 0 1 0 0 CONTINUOUS PHASE PRANDTL NUMBER 1 0 0 0 Figure 17 - Variation of U with Continuous Phase Prandtl Number VO I.Oi o o C M 2 o o LU _1 < O I T T 1 — T T OCYCLOPENTANE A ISOPENTANE 0.01" 1 1 F*0.70l P re 0 , 2 4 5 I l_L i J I L 1.0 10 100 CONTINUOUS PHASE PRANDTL NUMBER 1000 Figure 18 - Curve Fit of U with Equation (50) VO 00 between the physical properties of the dispersed phases. The para-meter most lik e l y to accomplish this was f e l t to be a dispersed phase Prandtl number. Table IX shows both the main equation as well as the ones applicable to the individual systems. Equation (51) shown in Figure .19 simply changes the intercept of the equation and does not result in a much more satisfactory f i t than Equation (50). In order to allow the dispersed phase Prandtl number to have an affect on the slope of the line i t must be put in the exponent of the continuous phase Prandtl number. This was done in Equation (52) and is shown in Figure 20. This equation is poor since the intercept i s held constant for both dispersed phases. Equation (53) shown in Figure 21 allows both the intercept and the slope to vary and results in a reasonable f i t with the RSQ value equal to 0.761. In Equation (54).- shown in Figure 22, the slope of the line was allowed to be a linear function of the dispersed phase Prandtl number. This resulted in an increase in the RSQ value to 0.864. Since the main difference between the continuous phases was the continuous phase viscosity, the ratios of viscosities, expressed as p,/p and p /(p + p,)., might be important over and above their use on d c c c d the Prandtl numbers. The density ratios of P / p . and (p - p. ) p were c d c d c also considered as important variables. A stepwise regression, including the above variables and with the F ratio set at 0.5, resulted in the significant variables being the continuous phase Prandtl number, the dis-persed phase Prandtl number, and the viscosity ratio of p /(p + P-,). TABLE IX - Correlation Equations for Cyclopentane and Isopentane Cyclopentane Isopentane Equation Pr, = 3.33 Pr„ = 6.10 • • • • d d fj = 0.82 P r c - ° ' 2 4 3 P r / ' 1 1 2 (51) 0.717 P r ^ 0 ' 2 4 3 (51a) 0.670 P r ^ 0 " 2 4 3 (51b) ti - 0.572 P r c - 0 - 0 4 0 1 P r d (52) 0.572 P r J 0 ' 1 3 4 (52a) 0.572 P r ^ 0 ' 2 4 5 (52b) U = 0.241 P r / ' 7 9 5 t r ' 0 - 0 6 0 2 P r d (53) 0.627 P r . " 0 " 2 0 0 (53a) 1.015 P r J 0 " 3 6 7 (53b) U - 0.0439 P r / ' 8 9 1 Pr (0.367-0.134 P r d ) Q p r -0.0792 1 > 3 4 2 p r -0.450 d c c c A 0.670Prc O CYCLOPENTANE I- A ISOPENTANE TT- 0.82 Prl0,243PrL0,112 10 100 CONTINUOUS PHASE PRANDTL NUMBER Figure 19 - Curve Fit of U with Equation (51) 1.0 C M O o O o U J C O < 0.1 o ZD 0.01 i — i r i — i r I— i—r 572 Pr" 0- 1 3 4 0.572Pr" 0* 2 4 5 OCYCL0PENTANE I- AIS0PENTANE J I L U = 0 .572Pr - a 0 4 0 , P r d l J I L I -I I 1.0 10 100 CONTINUOUS PHASE PRANDTL NUMBER Figure 20 - Curve Fit of U with Equation (52). mnoo o r-o l — i r -0.200 1.015 Pre0'3 6 7 OCYCLOPENTANE \- A ISOPENTANE T>0.24l P r J 7 9 ^ " 0 - 0 6 0 2 ^ i J I L J 1—1 10 100 CONTINUOUS PHASE PRANDTL NUMBER 10 Figure 21 - Curve F i t of U with Equation (53). 10 10 100 1000 CONTINUOUS PHASE PRANDTL NUMBER Figure 22 - Curve F i t of U with Equation (54) 105 These groups were used in the three equations shown in Table X. A summary of the correlation equations produced is pre-sented in Table XI. Using the experimental data of other workers, the average heat transfer coefficients were calculated using the correlation equations presented in Table XI. The calculated results and the experimental values are compared in Figures 23 through 30. Prakash's results are low compared with the values which the correlation equations predict. However, in his total evaporation time calculations he measured the starting time from the time his drop entered the warmer portion of the column. This does not take into account the time taken to heat up the drop to i t s boiling point. Based on the present work the average of the time taken to heat up the drop to about three to five percent evaporation was 38% for eyclopentane and 47% for isopentane. If Prakash's results are modified by these amounts his average heat transfer coefficients would give a value of 0.436 for 2 -cyclopentane and 0.224 cal/(sec) (cm.*) (°C) for isopentane. This improves the agreement between his results and the present correlation. Sideman's data needed no adjustments and was quite close to the results predicted from the equations. Klipstein's data seems quite high compared to the results from the correlation equations; however, he did have d i f f i c u l t i e s with the ethyl chloride dissolving in the continuous phase. This would re-duce the amount of dispersed phase evaporated and therefore the total evaporation time would be shortened, resulting in a higher apparent average heat transfer coefficient. Another reason for these higher 106 y TABLE X - Cor r e l a t i o n Equations using the Parameters Pr , Pr,, and (- c Equation RSQ - _ 9 , -0.304 _ -0.921 , M c .5.354 U - 21.67 P r d P r c ( + ) °« ( 5 5 ) 0 > 9 4 6 c d TT - n 9fto 1 A 9 B -0.113 Pr, / _ _ M c ,2.19 U - 0.262 P r d P r c d ( + ) ( 5 6) 0.958 c d B.2.20P r /-™S r c<-°-«S-°-°«^V (57, c d 107 TABLE XI - Summary of Correlation Equations for the Average Heat Transfer Coefficient Equation RSQ U = 0.701 P r c " 0 , 2 4 5 (50) 0.548 U = 0.820 Pr " ° ' 2 4 3 P r , " 0 ' 1 1 2 (51) 0.555 "c . d U = 21.67 P r c ° ' 9 2 1 P r d " C - ^ ( ^  + ^ ) (55) 0.946 0.921 „ -0.304 , y c 5 - 3 5 4 U = 0.572 P r ^ 0 - 0 4 0 1 P r d (52) 0.558 U - 0.241 P r d 0 - 7 9 5 P r c - ° - 0 6 0 2 P r d (53) 0.761 U = 0.0439 P r / ' 8 9 1 (0-367-0.134 Pr d) ( 5 4 ) ^ U = 0.262 P r d 1 , 4 9 P r c " ° ' 1 1 3 P r d ( — — ) (56) 0.958 c d U = 2.20 P r , 0 " 7 0 6 Pr ("0.445-0.0642 Pr d) ( ^ _ ) 3 . 9 , 1 d c y . + y, TABLE XII - Calculated Values of the Average Heat Transfer Coefficient from the Co r r e l a t i o n Equations This Work Prakash Sideman K l i p s t e i n Pr c 3. 37 5.31 22.7 39.4 56.1 75.0 105.3 P r d 3. 33 6.10 3.33 6.10 3.33 3.33 6.10 0. 517 0.775 2.93 5.08 6.90 9.14 13.3 vd 0. 322 0.273 0.322 0.273 0.322 0.322 0.273 c c 0. 616 0.740 0.901 0.949 0.955 0.966 0.980 Equation (50) 0. 521 0.466 0.326 0.285 0.261 0.243 0.224 Equation (51) 0. 533 0.446 0.336 0.274 0.269 0.251 0.216 Equation (55) 0. 368 0.534 0.485 0.321 0.2S9 0.234 0.154 Equation (52) 0. 486 0.380 0.377 0.233 0.334 0.321 0.183 Equation (53) 0. 492 0.550 0.335 0.263 0.280 0.264 0.184 Equation (54) 0. 388 0.632 0.333 0.256 0.310 0.303 0.165 Equation (56) 0. 345 0.633 0.387 0.275 0.313 0.287 0.150 Equation (57) 0. 348 0.600 0.437 0.297 0.303 0.261 0.148 U 0. 349 0.604 0.430 0.292 0.332 0.243 0.150 3.95 3.33 0.599 0.322 0.650 0.501 0.513 0.424 0.476 0.476 0.382 0.366 0.387 0.436 3.95 4.19 0.599 0.340 0.638 0.501 0.500 0.356 0.454 0.532 0.505 0.432 0.391 0.316 3.95 6.10 0.599 0.273 0.687 0.501 0.480 0.473 0.409 0.613 0.722 0.661 0.576 0.224 4.16 3.48 0.555 0.203 0.732 0.494 0.504 0.752 0.469 0.482 0.403 0.485 0.605 0.516 5.13 3.83 0.759 0. 271 0.737 0.470 0.474 0.623 0.445 0.481 0.438 0.489 0.556 0.916 6.22 3.83 0.902 0.271 0.769 0.448 0.452 0.656 0.432 0.460 0.426 0.494 0.575 0.859 7.45 3.83 1.05 0.271 0.795 0.429 0.433 0.663 0.420 0.441 0.414 0.491 0.578 0.930 63.6 3.83 10.0 0.271 0.974 0.253 0.257 0.273 0.302 0.269 0.303 0.303 0.290 0.382 86.8 3.83 13.8 0.271 0.981 0.235 0.238 0.213 0.288 0.250 0.290 0.269 0.241 0.436 U=0.70l P r g 0 , 2 4 5 Figure 23 - Comparison between Experimental and Correlation Value of U calculated from Equation (50). o VO Tj s 0520Prd 0 l l 2 Prc 0 - 2 4 3 Figure 24 - Comparison between Experimental and Correlation Value of U calculated from Equation (51). O CYCLOPENTANE THIS WORK ® ISOPENTANE THIS WORK 6 PRAKASH A SIDEMAN • KLIPSTEIN I i i i 1.2 OS 10 Figure 25 - Comparison between Experimental and Correlation Value of U calculated from Equation (55). 1-4 1 1 1 • 1 1 A — © A / — — • o/e X e° o CYCLOPENTANE THIS WORK ~ © © ISOPENTANE THIS WORK ° e e PRAKASH A SIDEMAN • KLIPSTEIN i i 1 1 I 1 1 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 U=0.572 Pr- 0 0 40Pr d Figure 26 - Comparison Between Experimental and Correlation Value of U calculated from Equation (52). 0.8 z 0.6 Ul £0 .4 0.2 0.0 0.0 1 1 1 i / i 1 I • / — — — • o • y ° e O CYCLOPENTANE THIS WORK ~ °/ 0 ISOPENTANE THIS WORK e 9 PRAKASH A SIDEMAN • KLIPSTEIN 1 1 1 1 1 1 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 U = 0.24IPi£ 0.795p-00602Prd Figure 27 - Comparison between Experimental and Correlation Value of U calculated from Equation (53). O CYCLOPENTANE THIS WORKl (D ISOPENTANE THIS WORK e e PRAKASH A SIDEMAN • KLIPSTEIN L _ 1.0 12 1.4 U=0£439 P r 11-891^0367-0134 Prd) Figure 28 - Comparison between Experimental and Correlation Value of U calculated from Equation (54). O CYCLOPENTANE THIS WORKl <D ISOPENTANE THIS WORK e 9 PRAKASH A SIDEMAN • KLIPSTEIN I I I l I 0.4 0-6 1.4 9 0 8 o , J ° 2 . 1 9 1.2 1.4 Figure 29 - Comparison between Experimental and Correlation Value of U calculated from Equation (56). I—' Cn U=2.20Pr d a 7 0 6 [ - T Z^Za] 3 ' 9 ' P r ^ 0 - 4 5 5 " 0 - 0 6 4 2 ^ Figure 30 - Comparison between Experimental and Correlation Value of U calculated from Equation (57). 117 results might be the extra energy supplied by his hot wire nucleation system. Correlation of the Results of A l l Workers As mentioned previously the correlation equations con-taining Pr c, ^ T£> and y c / ( y c + U^) gave the best f i t s t a t i s t i c a l l y . These equations were based only on £he results of this work. To see how these equations would apply to the results of a l l workers the con-stants appearing in the Equations (58), (59) and (60) were recalculated using the results of a l l the workers. The resulting equations are presented in Table XIII. The values of the heat transfer coefficients calculated from these equations are shown in Table XIV and are plotted against the experimental heat transfer coefficients in Figures 31 and 32. There was found to be very l i t t l e difference between Equation (59) and (60) and therefore only one graph is given for both equations - (Fig-ure 31) . Summary of Correlation Results In summary, the two equations considered to provide the best correlations are: U = 0.241 Pr 0.795 Pr - 0.0602 Pr (53) c and U = 0.262 Pr 1.49 Pr - 0.113 Pr d ( c .2.19 c (56) 118 TABLE XIII - Correlation Equations Based on the Average Heat Transfer Coefficients of A l l Workers Equation RSQ U - 0.972 Pr ° ' 7 5 8 ( - ^ — ) 2 ' 9 2 4 Pr - ° ' 1 1 5 5 P r d (58) 0.510 d u + y c c d U = 52.20 P r / 0 , 7 7 3 (—^ ) 6 ' 1 4 p r ~ 0 ' 9 1 9 (59) 0.686 d y + y , c c d U - 52.33 P r / 0 ' 7 7 4 ( - ^ — ) 6 ' 1 4 Pr ("0.919 + 0.000116 Pr d) d y + y , c TABLE XIV - Calculated Values of the Average Heat Transfer C o e f f i c i e n t s from the Correlation Equations Based on the Results of a l l Workers This Work Prakash Sideman K l i p s t e i n Pr c 3.37 5.31 22.7 39.4 56.1 75.0 105.3 3.95 3.95 3.95 4.16 5.13 6.22 7.45 63.6 86.8 P r d 3.33 6.10 3.33 6.10 3.33 3.33 6.10 3.33 4.19 6.10 3.48 3.83 3.83 3.83 3.83 3.83 0.517 0.775 2.93 5.08 6.90 9.14 13.3 . 0.599 0.599 0.599 0.555 0.759 0.902 1.05 10.0 13.8 vd 0.322 0.273 0.322 0.273 0.322 0.322 0.273 0.322 0.340 0.273 0.203 0.271 0.271 0.271 0.271 0.271 u c / ( y c + 0.616 0.740 0. 901 0.949 0.955 0.966 0.980 0.650 • 0.638 0.687 0.732 0.737 0.769 0.795 0.974 0.981 Equation (58) 0.368 0.489 0.537 0.297 0.450 0.415 0.136 0.405 0.398 0.485 0.567 0.534 0.556 0.565 0.396 0.353 Equation (59) 0.345 0.436 0.616 0.247 0.385 0.315 0.158 0.415 0.309 0.364 0.792 0.631 0.687 0.713 0.345 0.271 Equation (60) 0.345 0.436 0.616 0.247 0.385 0.315 0.158 0.415 0.309 ' 0.364 0.792 0.631 0.687 0.713 0.346 0.272 U 0.349 0.604 0.430 0.292 0.332 0.243 0.150 0.436 0.316 0.224 0.516 0.916 0.859 0.930 0.382 0.436 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 U-0.972Prd [-TZ^J P'c d Figure 31 - Comparison between Experimental and Correlation Value of U calculated.from Equation (58). < 0 .6 Figure 32 - Comparison between Experimental and Correlation Value of U calculated from Equation (59). The basic difference between the equations is the inclusion of a viscosity ratio in Equation (56). Equation (53) provides the best correlation, with a minimum number of terms, for this work and best shows the effect of the continuous phase on the average heat transfer coeffic-ient. Equation (56) gives a better correlation to the results of other workers and therefore i s the better correlation to use in calculating the average heat transfer coefficient for systems other than the ones used here. EVALUATION OF INITIAL DROP AREA As mentioned before, the dilatometric method can measure the volume of the bubble at the end of evaporation. From this volume the mass of the dispersed phase, present at the end of evaporation, can be calculated. Similarly we can, from the measurements taken from the photo-graphs of the drop in the lower section of the column, determine the mass of the dispersed phase in that section. Since the basic assumption is that there i s no mass transfer between the dispersed phase liquid and the continuous phase liquid, the mass of the dispersed phase calculated from the i n i t i a l drop dimensions should be the same as that calculated from the vapour volume at the end of evaporation. However, a comparison between these methods showed there was a significant difference between them and a series of experiments were conducted to determine the more reliable one. 123 On the basis of these experiments the mass of the dispersed phase calculated from the dilatometer results was f e l t to be more accurate; and this value was chosen to calculate an equivalent spherical diameter from which the i n i t i a l area was calculated. Comparison of Methods Used to Measure the Mass  of the Dispersed Phase In this experiment two different methods were used to measure the mass of the dispersed phase. The f i r s t method is based on the photograph of the drop taken in the lower portion of the column. From measurements taken from this photograph the total volume of the drop and the volume of the air bubble may be calculated. By subtracting the volume of the air bubble from the total volume of the drop the volume of the dispersed phase liquid may be obtained which, when multiplied by the density of the dis-persed phase liquid, gives the mass of the dispersed phase. This method neglects the very small amount of the dispersed phase which has been vapourized into the air bubble in the lower portion of the column. The second method is based on the volume of the bubble as measured by the dilatometer at the end of evaporation. This volume is multiplied by the ratio of the partial pressure of the dispered phase to the total pressure to give the partial volume which, when multiplied by the vapour density, gives the mass of the dispersed phase. The results of these calculations are shown in Table XV where the mass calculated by each method as well as the difference be-tween them is given. If the mass calculated by the dilatometric method TABLE XV - Masses of Dispersed Phase Calculated by the Photographic and Dilatometric Methods SET NUMBER 13 14 16 RUN : NO. Mass* Masst Difference Mass* Masst Difference Mass* Masst Difference 1 0.00879 0.00737 0.00142 0.00570 0.00386 0.00184 0.00624 0.00649 -.00025 2 0.00569 0.00537 0.00032 0.00428 0.00345 0.00083 0.00382 0.00373 0.00009 3 0.00833 , 0.00832 0.00006 0.00470 0.00327 0.00143 0.00634 0.00602 0.00022 4 0.00420 0.00434 -.00014 0.00418 0.00361 0.00057 0.00281 0.00273 0.00008 5 0.00293 0.00350 -.00057 0.00555 0.00470 0.00085 - - -6 0.-00525 0.00538 -.00013' 0.00436 0.00462 -.00034 0.00633 0.00661 «.00028 7 0.00320 0.00425 -.00105 0.00576 0.00502 0.00074 0.00630 0.00557 0.00073 8 0.00413 0.00366 0.00047 0.00398 0.00359 0.00039 0.00633 0.00567 0.00063 9 0.00405 0.00426 0.00021 0.00447 0.00451 -.00004 0.00195 0.00179 0.00016 10 - - - 0.00563 0:00545 0.00018 0.00633 0.00578 0.00055 11 0.00279 0.00354 -.00075 0.00699 0.00739 -.00040 0.00633 0.00532 0.00101 12 0.00502 0.00580 —.00078 0.00333 0.00226 0.00107 0.00637 0.00488 0.00149 * Mass from photographic method t Mass from dilatometric method XV - Masses of Dispersed Phase Calculated by the Photographic and Dilatometric Methods SET NUMBER 17 18 RUN — ' — NO. Mass* Masst Difference Mass* Masst Difference 1 0.00799 0.00630 0.00169 0.01887 0.01054 0.00833 2 0.00638 0.00619 0.00019 0.01508 0.00749 0.00759 3 - - 0.01887 0.01166 0.00721 4 0.00636 0.00621 0.00015 0.01533 0.00781 0.00752 5 0.60636 0.00605 0.00031 0.01533 0.00815 0.00718 6 0.00210 0.00168 0.00042 0.01819 0.01079 0.00740 7- 0.00799 0.00634 0.00165 0.01471 0.00694 0.00777 8 0.00796 0.00649 0.00147 - - - -9 0.00796 0.00647 0.00149 - - -10 0.00799 0.00571 0.00228 0.01826 0.00970 0.00856 11 0.00799 0.00635 0.00164 0.01197 0.00520 0.00677 12 0.00633 0.00620 0.00013 _ — _ * Mass from photographic method t Mass from dilatometric method TABLE XV - Masses of Dispersed Phase Calculated by the Photographic and Dilatometric Methods SET NUMBER RUN NO. 19 20 Mass* Masst Difference Mass* Masst Difference 1 0.01132 0.00783 0.00349 0.00997 0.00603 0.00394 2 0.00951 0.00671 0.00280 0.00783 0.00459 0.00324 3 0.00951 0.00641 0.00310 0.00816 0.00593 0.00222 4 0.00699 0.00301 0.00398 0.00815 0.00508 0.00307 5 0.00943 0.00684 0.00259 0.00802 0.00494 0.00308 6 0.00928 0.00531 0.00397 0.00815 0.00516 0.00299 7 - - - 0.00827 0.00576 0.00251 8 0.00943 0.00524 0.00419 0.00820 0.00621 0.00199 9 0.00943 0.00613 0.00330 0.00823 0.00576 0.00247 10 0.00951 0.00692 0.00259 0.00826 0.00524 0.00302 11 0.00772 0.00523 0.00249 0.00826 0.00521 0.00305 12 0.00928 0.00420 0.00508 - - -* Mass from photographic method t Mass from dilatometric method 127 i s greater than that calculated by the photographic method the value of the difference i s given a minus sign. The results of a s t a t i s t i c a l t test performed on the differences by the program, see Appendix VIIl(d), supplied by the Computing Center (32) is shown in Table XVT. The table shows that in six of the seven cases the mass calculated from the photographic method is significantly greater than that calculated from the dilatometric method. Causes of Differences The f i r s t factors to be considered as sources of the d i f f e r -ences were: 1. The Volume of Air A small error in the measurement of the volume of the air bubble would result in a f a i r l y large error in the mass of the dispersed phase calculated from the photographs. It was f e l t that an error in the dimensions of the air bubble could be caused by diffraction of the light rays passing through the liquid drop which surrounds the air bubble. This might cause the air bubble to appear smaller than i t actually was and would result in an apparent larger mass for the dis-persed phase. 2. The Final Volume of the Bubble Since the volume of the bubble was determined when i t was s t i l l rising in the column there was a chance that an error occurred due to the dynamic response of the dilatometer and recorder. This might make the bubble seem smaller than i t actually was, thereby resulting in a smaller calculated mass. TABLE XVI - Statistical Results of the Comparison of the Dispersed Phase Masses Calculated from the Photographic and Dilatometric Methods Set No. Photographic Dilatometric ' Difference f t T T f , Sig.** Diff. No. of Mean Obs.* SD No. of Obs.* Mean SD Mean SD .05 13 11 0.00495 0.00202 11 0.00507 0.00159 -.00012 0.000676 10 -.58856 2. 23 NO 14 12 0.00491 0.00102 12 0.00431 0.00131 0.00060 0.000672 11 3.09744 2. 20 YES 16 11 0.00537 0.00166 11 0.00496 0.00156 0.00041 0.000548 10 2.48334 2. 23 YES 17 11 0.00686 0.00177 11 0.00582 0.00139 0.00104 0.000798 10 4.32455 2. 23 YES 18 9 0.01629 0.00238 9 0.00870 0.00210 0.00759 0.000564 8 40.37086 2. 31 YES 19 11 0.00922 0.00110 11 0.00580 0.00138 0.00342 0.00820 10 13.82920 2. 23 YES 20 11 0.00832 0.00056 11 0.00545 0.00052 0.00287 0.000540 10 17.61591 2. 23 YES * Number of Valid Comparisons f t Degrees of Freedom ** Significant Difference i—• oo To study these factors a procedure similar to that in the main work was used. A drop of liquid was- injected into the lower portion of the column, photographed, and allowed to rise in the continuous phase. However, instead of a dilatometer setup an inverted graduated cylinder was used to catch the partially evaporated drop. This drop then completely vaporized and when steady state had been achieved i t s volume was measured. Since the temperature and pressure were known, the amount of dispersed phase vapour could be determined from the measured volume by correcting for the partial pressure of the air and the vapour of the con-tinuous phase. When this volume of dispersed phase i s multiplied by the density, the mass of the dispersed phase is obtained. The size of the air bubble was varied from zero volume to nearly the f u l l volume of the liquid drop. The dispersed phase used was isopentane and two continuous phases, d i s t i l l e d water and a 76% glycerine-water solution,were tested. Figure 33 shows the comparison of the mass of the dispersed phase calculated from the photographic data for the continuous phase consisting of d i s t i l l e d water. Figure 34 shows the similar graph for the 76% glycerine-water solution. These graphs show the same character-i s t i c s as those noted in the main work. The masses calculated from the photographic method were higher than those calculated by the dilato-metric method, and the difference seems to be higher with the glycerine solution than with the d i s t i l l e d water. These results seem to remove the measurement of the volume of the bubble by the dilatometer as a source of the difference since the same results were obtained by another method for measuring the f i n a l volume. 0.0 01 0.2 0.3 MASS OF VAPOUR (GM) Figure 33 - Comparison of the Masses Calculated by the Photographic and Dilatometer Methods for the System of Isopentane and D i s t i l l e d Water. Co O ISOPENTANE a 76% GLYCERINE 0.2 0.3 MASS OF VAPOUR (GM) Figure 34 - Comparison of the Masses Calculated by the Photographic and Dilatometric Methods for the System of Isopentane and 76% Glycerine Solution. 132 Figures 35 and 36 show the percentage d i f f e r e n c e based on both the mass of the vapour and the mass of the l i q u i d , against the volume of the a i r bubble for the two continuous phases, These graphs do not seem to show a s i g n i f i c a n t increase i n error with increasing a i r bubble volume and even i n cases where no a i r bubble was used, and the l i q u i d drop was nucleated by contact with the sides of the graduated c y l i n d e r , there was a s i g n i f i c a n t d i f f e r e n c e . This seemed to eliminate an error i n the measurement of the a i r bubble as a source of the d i f f e r -ence . The next experiment was made to check the volume measurement and i t s reduction to the mass of the vapour. This was done by i n -j e c t i n g a large known volume (0.5 to 2 ml.) of dispersed phase l i q u i d (isopentane) into a large graduated c y l i n d e r . This was then placed i n the heated portion of the column. The drop evaporated and formed a large vapour bubble i n the tube. A stopcock i n the lower part of the tube was open so the displaced water i n the tube could escape into the column. The volume of the vapour was measured as w e l l as the con-tinuous phase temperature and the pressure obtained from the height of water and the atmosphic pressure. Using the same method of c a l c u l a t i o n as that used before t h i s volume was reduced to the mass of the dispersed phase. The mass of the l i q u i d i n j e c t e d into the cy l i n d e r was obtained by m u l t i p l y i n g the volume in j e c t e d by the density of the l i q u i d . Figure 37 shows the masses pl o t t e d against each other and within ex-perimental error they seem to agree with each other. It would appear therefore that the major source of error 200 — UJ o UJ 150 cc UJ u. u. 5 100 50 0.0 o o O A A A o Q O A 2 A A 0 % VAPOUR BASIS A % MASS BASIS ISOPENTANE a WATER 0.002 0.004 0-006 VOLUME OF AIR (ML) Figure 35 - Percentage Difference versus Volume of the Air Bubble for Isopentane in Di s t i l l e d Water. UJ o z LU cr LU u_ LL O jo 200 150 100 5qfcr CD 0.0 A * A * O % VAPOUR BASIS A % MASS BASIS — ° ISOPENTANE 8 76% GLYCERINE O -O O O O ° A A A A A o o A A 2 0.002 0.004 0006 VOLUME OF AIR (ML) Figure 36 - Percentage Difference versus Volume of the Air Bubble • for Isopentane in 76% Glycerine Solution. CO ISOPENTANE a WATER .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 IB MASS OF LIQUID INJECTED (GM) Figure 37 - Comparison of the Injected and Calculated Mass of the Dispersed Phase for Isopentane in D i s t i l l e d Water. 136 must be in the measurement of the liquid drop from the photographs. To check the obvious sources of error which might occur in this method, two transparent rods, one of glass and one of perspex which had two lines inscribed on i t , exactly one inch apart, were suspended in the column and photographed. These photographs, along with ones j taken of the scale,were projected using the same projecter as was used in the rest of the work. Measurements were made on an inch graduation on the scale and the distance between the two inscribed lines on the perspex rod. These measurements agreed with each other thus eliminating the possibility of an inaccurate scale marking affecting the calibration of the magnification factor. Measurements of the rod diameters were made on the photo-graphs of the perspex and glass rods and after reduction by the magni-fication factor these diameters were found to be the same as those made on the rods by a micrometer. Conclusion of the Course of Differences The above.results seem to leave the determination of the drop diameter by the photographic method suspect. The cause of this error i s most lik e l y the diffraction of light at the drop surface which shows up on the photographs as a band of light. The exact position of the interface, therefore, i s not definite and leads to an incorrect inter-pretation of the drop diameter. Calculation of I n i t i a l Area Since a l l the tests showed that the mass calculated by the dilatometer method was the correct one, the i n i t i a l volume of the dis-persed phase liquid was determined by dividing the mass by the density of the dispersed phase liquid. Since the i n i t i a l shape of the drop was as shown in Figure 38 the volume of ai r , calculated from measurements of the photo-graphs , was added to the corrected volume of the dispersed phase to give the i n i t i a l volume of the drop. The equivalent diameter, which i s used in the calculation of the i n i t i a l area, was calculated from this volume. These corrected i n i t i a l diameters are shown in Table XVII, the calcula-tions for which are shown-in Appendix VI. 138 Figure 38 - Shape of I n i t i a l Drop of Dispersed Phase. 139 TABLE XVII- Calculated I n i t i a l Diameters for Use in the Calculation of the Average Heat Transfer Coefficient RUN SET NUMBER NO. 13 14 16 17 18 19 20 1 0.267 0.234 0.257 0.253 0.325 0.279 0.265 2 0.248 0.229 0.213 0.251 0.304 0.262 0.248 3 0.284 0.217 0.251 - 0.336 0.258 0.264 4 0.224 0.234 0.193 0.252 0.304 0.246 0.251 5 0.208 0.251 - . 0.250 0.308 0.264 0.251 6 0.246 0.242 0.258 0.164 0.334 0.247 0.252 7 .0.223 , 0.248 0.243 0.254 0.302 - 0.260 8 0.216 0.233 0.245 0.256 - 0.243 0.267 9 0.228 0.241 0.170 0.256 - 0.256 0.260 10 - 0.255 0.247 0.245 0.323 0.264 0.252 11 0.218 0.283 0.240 0.254 0.254 0.242 0.251 12 0.255 0.195 0.232 0.252 - 0.230 CONCLUSIONS The dilatometric method is superior to the photographic method for measuring the rate of change of volume during the complete evaporation of an immiscible drop. The concept of an instantaneous heat transfer coefficient was shown to be unrealistic unless a definite effort is made to determine the liquid-liquid interfacial area between, the dispersed phase liquid and the continuous phase liquid. The concept of an average heat transfer coefficient was found to adequately correlate the rate of heat transfer with the para-meters of the system and allow comparison with the result of other workers. The best equation for correlation to the results of other workers was found to be: — 1 49 -0 113 Pr '"V 2.19 U = 0.262 P r / ' ^ Pr U , - L i j r r d ( f ^ — ) (56) d c u + u, c d A comparison between the photographic technique, used in the lower portion of the column, and the dilatometric technique showed a discrepancy between the two methods. Further work indicated that the difference was due to the photographic method. RECOMMENDATIONS 1. Further work should take two directions: (a) A study should be made to determine the liquid-liquid area; , in a gas-liquid^liquid system. This area should be correlated to the indices suggested by Klipstein or to similar ones. A similar re-commendation was made by Klipstein in which he states"Hydrodynamic behavior of a i r - o i l bubbles should be studied to identify separately, aspects of behavior due to the presence of two phases in the bubbles". (b) Using only the dilatometric method a large range of dis-persed phase and continuous phase liquids should be tested and the re-sults presented in the form of an average heat transfer coefficient. The elimination of the photographic part of the study should greatly re-duce the time needed for the experiment. 2. Further work should be done to attempt to determine a more suitable equation for f i t t i n g the dilatometer curve, the coefficients of which could then be correlated to the parameters of the systems. 3. If any further photographic work is attempted, a study of the method of measurements of the liquid drop from the photographs should be made to determine and eliminate the descrepancy found in this study. 142 NOMENCLATURE a acceleration , 2 cm/sec A area 2 cm \ surface area of bubble 2 cm A . 1 surface area based on i n i t i a l drop diameter 2 cm \ interfacial area of dispersed phase liquid 2 cm A V interfacial area of dispersed phase vapour 2 cm inside interfacial area of dispersed phase liquid 2 cm Ao£ outside interfacial area of dispersed phase liquid 2 cm A . IV inside interfacial area of dispersed phase vapour 2 cm A ov outside interfacial area of dispersed phase vapour 2 cm c constant in equation (38) dim* C constant in equation (44) -I / n 1/sec C constant ±i equation (18) cal/(sec)(cm 2) (°C) CD drag coefficient, defined in equation (3) dim. S V drag coefficient of a r i g i d sphere dim. c p heat capacity cal/(gm)(°C) C P C heat capacity of continuous phase cal/(gm)(°C) c p d heat capacity of dispersed phase cal/(gm ) (°C) d diameter cm d. 1 i n i t i a l diameter of the drop cm E constant in equation (43) dim. * dimensionless 143 f exponent of time, equation (12) dim, 2 g acceleration of gravity cm/sec G constant in equation (43) dim. 2 h individual heat transfer coefficient cal/(sec)(cm )(°C) h±z inside "h" for "A i £" cal/(sec)(cm2)(°C) hQl outside "h" for "A o £" cal/(sec)(cm2)(°C) h. inside "h" for "A. " cal/(sec)(cm2)(°C) iv iv h outside "h" for "A " cal/(sec)(cm2)(°C) ov ov k thermal conductivity cal/(sec)(cm) (°C) k^ thermal conductivity of dispersed phase cal/(sec)(cm)(°C) k^ thermal conductivity of continuous phase cal/(sec)(cm)(°C) K constant in equation (12) cal/sec^ m mass of the drop (mass of the dispersed phase) gm n exponent on time, equation (44) dim. Nu Nusselt number, (hd/k) dim. Nu £ Nusselt number based on continuous phase properties, (h cd/k c) dim. Nu^ Nusselt number based on dispersed phase properties, (h,d/k,) dim. d a p exponent on "Re", equation (43) dim. P pressure mm Hg sum of the partial pressures inside the bubble mm Hg P Q pressure exerted on outside of bubble mm Hg Pe Peclet number, (dCppu/k) dim. Pe^ Peclet number based on continuous phase pro-perties, (dCp cP cu/k c) dim. Pe^ Peclet number based on dispersed phase pro-perties, (dCp^p^u/k^) dim. 144 Pr Prandtl number, (Cpu/k) dim. Pr c Prandtl number based on continuous phase properties, (Cp y /k ) dim. c c c Pr^ Prandtl number based on dispersed phase properties, (Cp^y^/k^) dim. q instantaneous rate of heat transfer cal/sec q^ instantaneous rate of heat transfer through liquid portion of bubble cal/sec q instantaneous rate of heat transfer through vapour portion of bubble cal/sec Q amount of heat transferred during time 0 ; Q=Qf-Qi cal heat content of drop at 6 = 0. cal heat content of drop (bubble) at 0 = 0^  cal Q total heat absorbed by the dispersed phase: max J Q_ = m ^  cal max Re Reynolds number, (dup/y) dim-Re £ Reynolds number based on continuous phase properties, (dup /y ) dim. c c Re^ Reynolds number based on dispersed phase properties, (dup,/y ) dim. a d s exponent on "Pr", equation (43) dim-t temperature °C T temperature °K AT temperature driving force °C or °K AT^ temperature difference between continuous phase and dispersed phase liquid °C or °K 145 AT temperature difference between continuous phase and dispersed phase vapour °C or °K u v e l o c i t y of drop or bubble cm/sec u £ v e l o c i t y of continuous phase, equation (8) cm/sec u^ v e l o c i t y of dispersed phase, equation (8) cm/sec U heat transf e r c o e f f i c i e n t cal/(sec)(cm^)(°C) 2, 2 U\ instantaneous heat transfer c o e f f i c i e n t cal/(sec)(cm )(°C) - 2 U average heat transfer c o e f f i c i e n t cal/(sec)(cm )(°C) o v e r a l l heat transfer c o e f f i c i e n t between the continuous phase and the dispersed phase 2 l i q u i d cal/(sec)(cm )(°C) U o v e r a l l heat transfer c o e f f i c i e n t between the v continuous phase and the dispersed phase 2 vapour cal/(sec)(cm )(°C) 3 V volume of drop or bubble cm 3 V. i n i t i a l volume of bubble cm x 3 f i n a l volume of bubble cm 3 V£ constant i n equation (44a) cm w v a r i a b l e i n equations (45) and (46) dim. 2 We Weber number, (du p/u) dim. x distance, p a r a l l e l to drop i n t e r f a c e , equation (8) cm y distance, perpendicular to drop i n t e r f a c e , equation (8) cm z v a r i a b l e , equations (45) and (47) sec 1 1 146 Greek Letters a constant i n equations (45) and (48) l / s e c 1 1 g constant i n equations (45) and (49) dim. 6^  constant i n equation (6), defined i n equation (7) dim. $2 opening angle of the vapour phase, equation (15) deg G time sec 0^ s t a r t of the evaporation time sec 6^ end of the evaporation time sec X lat e n t heat of vaporization cal/gm u v i s c o s i t y cp u c v i s c o s i t y of the continuous phase cp v i s c o s i t y of the dispersed phase cp p density gm/cm3 3 P c density of continuous phase gm/cm 3 p^ density of dispersed phase gm/cm 3 p o v e r a l l density of drop or bubble gm/cm a surface or i n t e r f a c i a l tension dynes/cm Miscellaneous % percentage, percent evaporation Subscripts c continuous phase; i f used with a dimensionless number itmeans the physical properties are based on the continuous phase 147 d dispersed phase; If used with a dimensionless number i t means the physical properties are based on the dispersed phase I liquid, refers to the dispersed phase liquid v vapour, refers to the dispersed phase vapour LITERATURE CITATIONS 1. K l i p s t e i n , D.H., D.Sci. Thesis, Mass. Int. Technol., Cambridge, Massachusetts, 1963. 2. Sideman, S. and T a i t e l , Y. , Int. J . Heat Mass Transfer, ]_, I 2 7 3 (1964). 3. Prakash, C.B., Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C., 1966. 4 Sideman, S., Advan. Chem. Eng., J5, 207 (1966). 5. Hughes, R.R. and G i l l i l a n d , E.R., Chem. Eng. Progr., 48, 497 (1952) 6. L i c h t , W. and Narasimhamurty, G.S.R., A.I.Ch.E.J., _1, 366 (1955). 7. Rose, P.M. and Kintner,R.C., A.I.Ch.E.J., 12, 530 (1966). 8. Grifflth.R.M., Chem. Eng. S c i . , l l , 198 (1960). 9. Garner, F.H., Chem. Ind. , 141, (1956). 10. Hughmark, G.A., A.I.Ch.E.J., 13, 1219 (1967). 11. L o c h i e l , A.C. and Calderbank, P.H., Chem. Eng. S c i . , 19, 471 (1964) 12. Bowmann, C.W., et aL, Can. J . Chem. Eng., 39, 9 (1961). 13. Johns, L.E. J r . and Beckmann, R.B., A.I.Ch.E.J., 12, 10 (1966). 14. Cheh, H.Y. and Tobias, C.W. , Ind. Eng. Chem. Fundam. , 48 (1968). 15. Handlos, A.E. and Baron, T., A.I.Ch.E.J., 3, 127 (1957). 16. P a t e l , J.M. and Wellock, R.M., A.I.Ch.E.J., 13, 384 (1967). 17. Head, H.N. and Heliums, J.D., A.I.Ch.E.J., 12, 553 (1966). 18. Winnikow, S., Chem.Eng. S c i . , 22, 477 (1967). 19. Magarvey, R.H. and MacLatchy, C.S., A.I.Ch.E.J., 14, 260 (1968). 20. Kintner,R.C. , Advan. Chem. Eng., 4_, 51 (1963). 21. Taylor, T.D. and Acrivos, A., J . F l u i d Mech., 18, 466 (1964). 22. Pan, F.Y. and Acrivos, A., Ind. Eng. Chem. Fundam., 1_, 227 (1968). 23. Garner, F.H., Chem. Ind., 141, (1956). 149 24. SehroedervR.R. and Kintner,R.C., A-I.Ch.E.J.,.11, 5 (1965). 25. Angelo, J.B., Lightfoot, E.N. and Howard, D.W., A.J.Ch.E.J.,.12,. 751 (1966). 26. Astarlta, G. and Apuzzo, G., A.I.Ch.E.J., 11, 815 (1965). 27 Calderbank, P.H. and Lochiel, A.C., Chem. Eng. Sci., 19, 485 (1964). 28. Calderbank, P.H., Johnson, D.S.L. and London, J., Chem. Eng. Sci., 25, 235 (1970). 29. Garbarini, G.R. and Tien, C., Can. J. Chem. Eng., 47, 35 (1969). 30. Sideman, S. and Hirsch, G. , Isr. J. Technol., 2_, 234 (1964). 31. U.B.C. TRIP (Triangular degression /Package), Computing Centre, University of British Columbia, Vancouver, B.C., January (1968). 32. U.B.C. SIMCORT, (Means, Standard Deviations, Simple Correlation Coefficients, and T -Tests), Computing Centre, University of British Columbia, Vancouver, B.C., June (1969). 33. Beall, I.N., Refiner Natural Gasoline Mtr., 14, 437 (1935). 34. Mc Cullough, J.P., et. a l . , J. Amer. Chem. Soc, 81, 5880 (1959). 35. Willingham, C.B. et a l . , J. Research Natl. Bur.Standards, j$5, 219 (1945). 36. Timmermans J., Physico-Chemical Constants of Pure Organic Compounds, Vol.1, Page 38, Elsevier Publishing Co. Inc., Amsterdam, 1950. 37. Timmermans,J., Physico-Chemical Constants of Pure Organic Compounds, Vol. 1, Page 185, Elsevier Publishing Co. Inc., Amsterdam, 1950. 38. Gerlach, G. Th., Chem. Ind. 7_, 277, as cited in Timmermans, J., Vol. IV, Page 256, Interscience Publishers, Inc., New York (1960). 39. Handbook of Chemistry and Physics, 44th Edition, Page 2273, The Chemical Rubber Publishing Co., Cleveland, Ohio (1961). 40. Ernst., R.C., et a l . , J. Phys. Chem., 40, 627 (1936) as cited in Timmermans, J., Vol. IV, Page 266, Interscience Publishers, Inc., New York (1960). 41. Handbook of Chemistry and Physics, 44th Edition, Page 2375,The Chemical Rubber Publishing Co., Cleveland, Ohio (1961). 42. Bates, O.K., Ind. Eng. Chem., 28, 494 (1936). 43. Von Mayer-Bugstrom, Zeit. Deut. Oel und Fettind. , 44, 417 (1924), as cited in Timmermans J. Vol. IV, Page 254, Interscience Publishers, Inc., New York (1960). APPENDIX I Equipment Specifications (a) Column Details Material of Construction Perspex and glass Diameter of Inner Column 2.50 in. Outer Square Column 4.81 in. Length of Upper Portion 43.5 in. Length of Lower Portion 5.5 in. Volume of Circular Column 240.0 cu in. Volume of Annular Space 895.0 cu i n . Diameter of Baffle Hole 1.0 in. (b) Exposure Meter Honeywell Pentax 1°/21° Exposure Meter Chem. Eng. No. Viewing Angle Vertical Horizontal Diagonal Light Sensitive Element Shutter Speed Range Diaphragm Range ASA Film Speed Range 2359 12° 17° 21° Cadmium Sulfide 4 min-1/4000 sec f/l-f/128 6-6400 151 (c) Constant Temperature Baths Ultra-Thermostats Type K Type NB Chem. Eng. No. Thermometer Operating Range Temperature Control Bath Capacity Pump Capacity Head Attainable Heating 2259 No.1143 -30 to +150°C ± 0.02 1.6 l i t e r s 10 liters/min 4 m of water 300 watts 2423 No.1133 -60 to +180°C ± 0.01 14.7 l i t e r s 12 liters/min 4 m of water 500/1000/1500 watts (d) Potentiometer Leeds & Northrup Company Range Limits of Error Reference Junction Com-parison Range Standard Cell Adjustment Range No.8686 M i l l i v o l t Potentiometer -10.1 mv to 100.1 mv ± (0.03% of reading + 3yv) without reference junction compensator ± (0.03% of reading + 6yv) with reference junction comparison 0 to 5 mv adjustable to ± 2yv 1.0170 to 1.0200v (e) Strip Chart Recorder Hewlett Packard Recording Mechanism Response Time Chart Speed Power Requirements Voltage Source Event Marker Input Resistance Dimensions of Chart Model 71018-12 Servo-actmated ink-pen drives, manually operated pen l i f t 0.5 sec. for f u l l scale 1.2 in./hr 0.1, 0.2, 0.5, 1,2 in./min 0.1, 0.2, 0.5, 1,2 in./sec 115 or 230 v 60 c/s 42 volt-amperes 60 ± 2 cycles 105 to 125 v remote contact closure (current self supplied) 10 + 6 ohms @ 0.1% f u l l scale off balance 12 ins. wide with a main 1.0 in. grid and a secondary 0.1 in. grid. (f) Projector Specto Motion Analysis Projector Mk III Ele c t r i c a l Power source, drive and blower motors 110 - 120 a c 50-60 c/s Control apparatus - 12 and 9v d-c supply Mechanical - 2,4,8,16 frames per sec - single shot forward and reverse - remote control box - 800 f t of 16 mm sound or silent stock Lenses 2" f2 lens Bolex-Paillard SA H-16 - 16 mm film - reflex viewing - variable shutter - turret - 3 interchangeable lenses - 50-100 f t spools - 12, 16, 18, 24, 32,64 frames/sec Used with the camera was a 5 mm extension tube 154 APPENDIX II Dispersed Phase Properties The nomenclature used in this section is given below: P d vapour pressure of the dispersed phase mm Hg t temperature °C T temperature (absolute) °K A d latent heat of vapourization cal/gm U d viscosity cp k d thermal conductivity cal/sec cm °C C P d heat capacity cal/°C gm P d l liquid density , 3 gm/cm Pdb liquid density at the boiling point gm/cm3 Pdv vapour density gm/cm M molecular weight gm/gm-mole % boiling point °C P total pressure mm Hg This section is meant to serve as a convenient reference for the properties of the dispersed phases. Most of the information, except for a few cases where new information is presented, was obtained from Prakash's work (3), which should be consulted for details of the equations and the experimental values reported here. The values of the following properties are shown in Table I - A l l . 1. Normal boiling point 2. Density 3. Viscosity TABLE I-All - Properties of the Dispersed Phase Liquid (taken from Prakash (3), Table II) Properties at Normal Boiling Point Isopentane Cyclopentane Normal boiling point °C 27.95 49.6 Density gm/cm3 0.61 0.668 Viscosity c p 0.273 0.322 Thermal conductivity cal/sec cm °C 0.255(10~3) 0.30K10" 3) Specific heat cal/°C gm 0.568 0.3113 Latent heat of vapourization cal/gm 83.3 100.0 Prandtl number - 6.10 3.33 Ul Ln 156 4. Thermal conductivity 5. Specific heat 6. Latent heat of vapourization 7. Prandtl number In addition the following equations, taken from Prakash's work were used in the computer program. Vapour Density Isopentane Beall (33) P d v = -0.000479 + 0.0001233t (1-AII) Cyclopentane McCullough et al (34) p = 70.13 P  { 62300T+ [-192.0-59.59 exp (^) ] P } (2-AII) Vapour Pressure-Temperature Relationship Isopentane Beall (33) log $ = 7.37 - 1351.35/(t + 273) (3-AII) Cyclopentane Willingham (35) log p, = 6.87798 - 1119.208/(t + 230.738) (4-AII) d Lastly, the density of the liquids was needed at the temper-ature in the lower portion of the column. A graph, plotted from data given in Table I I - A l l , of p d £ against t is shown in Figure 39. The values used were: Isopentane Timmermans (36) p d £ = 0.628 gm/cm3 @ 12°C Cyclopentane Timmermans (37) p d x = 0.744 gm/cm3 @ 22°C TABLE II-AII - Density of the Dispersed Phase Liquid Cyclopentane (37) Temperature (°C) Density (gm/cm ) 0 0.765 15 0.750 20 0.745 25 0.740 30 0.735 Isopentane (36) 3 Temperature (°C) Density (gm/cm ) 0 0.639 15 0.625 20 0.620 25 0.615 TEMPERATURE (°C) Figure 39 - Effect of Temperature and Percentage Concentration on Dispersed Phase Density. APPENDIX III Continuous Phase Properties The following symbols apply to this Appendix: vapour pressure of continuous phase mm Hg t temperature °C T temperature (absolute) °K y c viscosity c p k c thermal conductivity cal/sec. c i "C C P C specific heat cal/°C gm P c ' density gm/ cm P total pressure mm Hg % percent concentration (by weight) of glycerine The continuous phase properties which were needed for the computer calculations are: 1. Density 2. Viscosity 3. Heat capacity 4. Thermal conductivity 5. Vapour pressure - temperature relationship (a) Concentration of the Continuous Phase The above properties were needed as a function of temper-ature at the following percent concentrations (weight basis), as dete mined from the viscosity measurements. 160 1. 0.0% 2. 56.02% 3. 73.07% 4. 77.06% There was some discrepancy between the concentrations as determined from the density measurements and those determined from the viscosity measurements. The values for each method are given in Table I-AIII . TABLE I-AIII - Determination of Percentage from Density and Viscosity Measurements Density Temp. (gm/cm3) (°C) 1.1956 20 1.1850 20 1.1399 19.6 % Concentration % 75.82 71.90 55.07 Viscosity Temp, c p (°C) 36.012 25.06 24.393 25.06 7.429 25.06 % Concentration % 77.06 73.07 56.02 Although this discrepancy i s disturbing the differences were not considered large enough to influence the results; especially since the same tables which were used to determine the percent concentration from the experimental density and viscosity data were used to determine the den-sity and viscosity values at various other temperatures. (b) Density The values for the density were taken from Gerlach (38) and, as mentioned above, the value of the percent concentration used was that determined by the density measurement. To obtain the values of density as a function of temperature at the required concentration the table was interpolated at the desired concentration and these values were then used with the temperature to determine the curve fit t e d equation. The values used for the curve f i t are found in Table II-AIII. The effect of temperature and concentration is shown in Figure 40. The curves which were f i t t e d to the data and their sum of squares are: 0.0% P c = 1.001+0.35286(10"4)t-0.63181(10"5)t'2+0.18859(10_7)f3 (1-AIII) sum of squares = 0.15856(10 55.07% P c = 1.1491-0.4l389(10"3)t-0.29206(10~5)t2+0.12107(10"7)t3 (2-AIII) sum of squares = 0.35766(10 ^) 71.90% P c = 1.1937-0.52547(10~3)t-0.11921(10"5)t2+0.27940(10~8)t3 (3-AIII) — 6 sum of squares = 0.69105.(10 u) 75.82% p = 1.2065-0.51498(10~3)t-0.17881(10"5)t2+0.74506(10"8)t3 (4-AIII) c sum of squares = 0.29176(10~5) (c) Viscosity The concentrations used in the interpolation of the data, found in the Handbook of Chemistry and Physics (39), were those deter-mined by the actual viscosity measurements. The values used to curve TABLE II-AIII - Variation of Density with Temperature Temperature (°C) Percent Concentration of Glycerine 0.0% 55.07% 71.90% 75.82% 0 10 20 30 40 50 60 70 80 90 100 0.9999 0.9998 0.9983 0.9957 0.9923 0.9882 0.9834 0.9777 0.9717 0.9652 0.9588 1.1488 1.1447 1.1397 1,1342 1.1284 1.1225 1.1164 1.1100 1.1034 1.0973 1.0907 1.1936 1.1878 1.1828 1.1766 1.1706 1.1651 1.1584 1.1519 1.1452 1.1389 1.1321 1.2067 1.2001 1.1956 1.1894 1.1832 1.1776 1.1715 1.1636 1.1572 1.1515 1.1449 30 4 0 50 60 TEMPERATURE (°C) Figure 40 - Effect of Temperature and Percentage Concentration on Continuous Phase Density. as O J TABLE III-AIII Variation of Viscosity with Temperature Temperature Percent Concentration (°C) 0.0% 56.02% 73.07% 77.06% 0 1.792 23.8 110.4 182.7 10 1.308 14.1 55.0 86.1 20 1.005 8.89 30.5 45.6 30 0.8007 6.00 18.6 26.4 40 0.6560 4.29 12.00 16.6 50 0.5494 3.21 8.23 11.04 60 0.4688 2.46 5.93 7.77 70 0.4061 1.99 4.54 5.81 80 0.3565 1.61 3.45 4.35 90 0.3165 1.33 2.75 3.42 100 0.2838 1.13 2.24 2.74 f i t the data are shown in Table III-AIII. The effect of temperature and concentration on the viscosity is shown graphically in Figure 41. The curve f i t t e d equations with their sum of squares are: 0-0% lnU c = 0.57794-0.32369(10-1)t'+0.21046 (10~ 3)t 2-0. 7085 (10" 6)t, 3 (5-AIII) _3 sum of squares = 0.15934(10 ) 56.02% l n u c = 3.1685-0.56072(10"1)t.+0.38044(10"3)t2-0.1247: (10" 5)t 3 (6-AIII) sum of squares = 0.29396(10~3) 73.07% l n y e = 4.6975-0.73161(10_1)t-l-0.51147(10"3)t2-0.16962(10~5)t'3 (7-AIII) sum of squares = 0.59695(10~3) 77.06% InV c= 5. 2020-079329 (10-1)t+0.55796 (10~ 3)t 2-0.18463(10~5)t 3 (8-AIII) _3 sum of squares = 0.43685(10 ) (d) Specific Heat The following data, taken from Ernst (40), were used in cal-culating the specific heat: TABLE IV-AIII - Variation of Specific Heat with glycerine Concentration % C P C t 0 1.000 25 20 0.930 25 40 0.810 25 60 0.715 25 80 0.610 25 100 0.555 .25 TEMPERATURE (°C) Figure 41 - Effect of Temperature and Percentage Con-tration on Continuous Phase Viscosity. The following data, taken from the Handbook of Chemistry and Physics (41), were also used: TABLE V - AIII - Variation of the Specific Heat of Glycerine with Temperature % cpe t 100 0.540 0 100 0.600 50 100 0.669 100 Although these data are not complete, i f a linear relation-ship i s assumed between the specific heat and the concentration and be-tween the specific heat and the temperature, a graph, Figure 42, may be drawn from which the linear equations may be taken. These equations were found to be: 0.0% Cpc= 1.0+0.0t (9-AIII) 56.02% Cpc= 0.711+0.00914t (10-AIII) 73.07% Cpc= 0.618+0.001157 (11-AIII) 30 40 50 60 TEMPERATURE (°C) 70 80 Figure 42 - E f f e c t of Temperature and Percentage Concentration on Continuous Phase Specific Heat. 77.06% Cp = 0.597+0.00120t c (12-AIII) (e) Thermal Conductivity An excellent, detailed study on the thermal conductivity of glycerol and water solutions has been made by Bates (42). He gives the thermal conductivity as a function of temperature for the following ranges of concentration. Figure 43 shows the effect of temperature and concentration on the thermal conductivity. 0.0% kQ= 0.00134+0.00000367t (13-AIII) 55.0% 0.00093+0.00000116t (14-AIII) 75.0% 0.00080+0.00000030t (15-AIII) 80.0% 0.00077+0.00000030t (16-AIII) These were li n e r a l l y interpolated to give: 0.0% k = 0.00134+0.00000367t (17-AIII) THERMAL CONDUCTIVITY (I03) (CAL/SEC CM °C) I I I Oil 56.02% kc= 0.00092+0.OOOOOlllt (18-AIII) 73.07% kc= 0.00082+0.00000038t (19-AIII) 77.06% k c= 0.00079+0.00000030t (20-AIII) (f) Vapour Pressure-Temperature Relationship The equation for the vapour pressure-temperature relationship for water was that used by Prakash(3) which he gave as: log pc= 9.754 - 2500.0/(t + 273.0) (21-AIII) The values for the other concentrations were linearly ex-trapolated from the data of Mayer - Bergstrom (43), which is shown in Table VI-AIII. These values were fitt e d to a form of Antoine's equation and were found to be: 56.02% In pc= 18.239 - 3959.8/(t + 233.41) -4 sum of squares = 0.30723(10 ) 73.07% In pc= 17.055 - 3331.9/(t + 203.77) -3 sum of squares = 0.10854(10 ) (22-AIII) (23-AIII) TABLE VI-AllI - Vapour Pressure - Temperature Relationship at Percent Concentrations of 56.02%, 73.07%, and 77.06%. Pressure Percent Concentration mmHg 56.02% 73.07% 77.06% Temperature °C 40 38.70 45.57 47.60 100 57.08 63.70 66.29 150 65.94 72.83 75.46 200 72.54 79.66 82.33 250 77.97 85.22 87.97 300 82.47 89.82 92.57 350 86.37 93.85 96.64 400 89.83 97.38 100.21 450 93.05 100.61 103.48 500 95.85 103.64 106.55 550 98.55 106.34 109.25 600 100.95 108.84 111.75 650 103.25 111.24 114.15 700 105.41 113.37 116.32 760 107.81 115.87 118.82 In pc= 17.677 - 3739.0/(t + 219.72) (24-AIII) sum of squares = 0.28852(10~4) APPENDIX IV Computer Program The computer program i s f a i r l y straightforward, except for two areas: One, the calculation of the temperature inside the vapour bubble. Two, the curve f i t of the dilatometer volume - time data by the equation. V. - V V - V. c u 1 These areas are discussed i n the next two sections, which are then followed by the symbol l i s t i n g for the computer program, the description of the input cards, and the l i s t i n g of the computer program. The results, for Run 1308, and a sample calculation of the average heat transfer coefficient i s given in Appendix V. Calculation .of the Vapour Bubble Temperature An estimation method had to be used to calculate the internal temperature of the drop or bubble. The basic assumption i s that the bubble is at equilibrium; i n which case the partial pressures of a i r , continuous phase vapour, and dispersed phase vapour would have to equal the total pressure on the outside of the drop. This outside pressure can be calculated from the atmospheric pressure and the height of water above the bubble. The partial pressure of the a i r is calculated from the i n i t i a l volume of the air bubble in the lower portion of the column by correcting for the increase in temperature and the decrease in pressure. The partial pressure of the continuous phase and the dispersed phase are calculated from equations (Antoine's equations), the constants of which given in Appendices A l l and AIII . The t r i a l and error solution proceeds as follows: At a certain height in the column the outside pressure on the drop is calculated from the height of water and the atmospheric pressure -the height of water takes into account the increase in height of water in the dilatometer tube caused by the volume expansion of the bubble. A temperature i s then assumed and by substitution into the appropriate equations, the partial pressures exerted by the vapour of the continuous and the dispersed phases calculated. Using this temperature and the volume of the vapour bubble at this height the partial pressure of the air i s calculated from the i n i t i a l volume determined in the lower portion of the column. These partial pressures are then summed and compared to the total pressure. If these two pressures are within two mm of Hg of each other then that temperature is used; i f they are not, an ad-justment is made in the temperature and the calculation repeated. Figure (44) shows a flow chart for the calculation. Curve Fitting of the Dilatometer Volume with Time To allow i t s use i n the computer program the dilatometer curve was broken down into a series of points of volume and time. The rate of heat transfer i s related to the derivative of the volume-time relationship (the slope of the volume-time curve) by the following equation: 176 P .CALCULATE T PA P PD PW PI 1 ASSUME / External pressure Temperature of vapour P a r t i a l pressure of a i r P a r t i a l pressure of dispersed phase P a r t i a l pressure of continuous phase Internal pressure 5 CALCULATE PA CALCULATE PD, PW CALCULATE PI=PA+PD+PW T=T+ fr"(0.005T) Figure 44 - Flow Chart f o r T r i a l and Error Solution of of Vapour Phase Temperature. 177 q = Xp dv v ~z— de In order to reduce the effect of experimental scatter, the volume-time curve was fitted by an equation which, when differentiated, would give the rate of heat transfer, at any instant. However, i t should be realized that the type of equations chosen w i l l automatically deter-mine the relationship between the rate of heat transfer and the time, and this must be taken into account when the rate of heat transfer i s correlated to the properties of the system. A large number of equations were considered and after care-f u l consideration the type of curve chosen was one that i s known as a growth curve and i s of the form: V, - V _ i = C 0 n V - V x or transformed into a linear equation of the form: In (V f - V/(V - V ±)) = In C + nln G which may be easily calculated on the computer. Although this curve approaches a constant value as the time goes to i n f i n i t y , i t cannot f i t the experimental data after the end of evaporation where the volume continuously increases in a linear fashion; this linearity i s likely caused by the decrease in pressure and the additional heat transfer to the vapour bubble. 178 Therefore to obtain the best f i t of the portion of the curve before the end of evaporation the value for the f i n a l volume was i n -creased in 0.01 ml increments u n t i l the error of the curve f i t was re-duced to a minimum. The resulting equation was of the form: V = V£/ (1 + C0n) which was differentiated and used to give the rate of heat transfer at any instant. Description of Input Cards for the Computer Program  The cards needed for a set of runs are: First Card This card determines which physical properties to use for the dispersed and continuous phase for each set of runs. The physical pro-perties of the system and their dependence on temperature and pressure, i f significant, have been incorporated into the program. ..Second Card This card gives the computer the number of runs in each set. The cards needed for each run are: An i n i t i a l card giving: - the temperature of the lower portion of the column - the i n i t i a l height of water in the dilatometer - the atmospheric pressure - the frames per minute 179 - the dimensions of the air bubble - the i n i t i a l diameter of the drop Following this card i s a series of cards giving the data from the selected frames of the cine photograph, namely: - the height of the drop or bubble - the dimensions of the vapour bubble - the frame number This group of cards i s terminated by a -1.0 card. The next group of cards gives a series of points of volume 'vers-es time representing the recorder chart of the dilatometer reading. This group is also terminated by a -1.0 card. The second last card for each run gives the eight time readings from the recorder chart corresponding to the height readings in the column of 0, 6, 12, 18, 24, 30, 36, and 42 inches. The last card for the run contains the five continuous phase temperature readings for the top portion of the column. 180 Symbol L i s t i n g f o r Computer Program A Constant i n LQF of HTWC versus TIME A l Constant i n LQF of VEft versus TIME A2 Constant i n LQF of VEL versus TIME A3 Constant i n LQF of VEL versus TIME A4 Constant i n LQF of VEL versus TIME AA Average area between any two points from photographic data AC F i n a l value of volume i n LQF f i t of volume versus time for dilatometer data ACC Acceleration ADIA Dimension of a i r bubble from photographs AHTWC Curve f i t t e d value of HTWC used to f i t TB AMASS Mass of drop or bubble from photographic data AMASSL Mass of l i q u i d i n drop from photographs AT External function for LQF AUB External function f o r LQF AUC External function f o r LQF AUT External function f o r LQF AUW External function f o r LQF AUX External function for LQF AUZ External function f o r LQF B Constant i n LQF f i t of HTWC vs TIME BDIA Dimension of a i r bubble from photographs BHTWC Curve f i t t e d height calculated by su b s t i t u t i n g TIME2 into LQF equation used to f i t HTWC versus TIME obtained from the photographic data BMA.SS Mass of vapour bubble from dilatometer data BPCEVP (100 - PCEVAP) BTIME Curve fit t e d time from LQF of TIME versus HTWC at various heights corresponding to HTWC1 CC Constant in V0L1 versus TIME2 for dilatometer volume CCP Specific Heat of continuous phase CDEN Density of continuous phase CK Thermal conductivity of continuous phase CNU Nusselt ftumber based on continuous phase properties COR TIME1 - BTIME CPE Peclet number based on continuous phase properties CPR Prandtl number based on continuous phase properties CRE Reynolds number based on continuous phase properties CVIS Viscosity of continuous phase CVOL1 LQF f i t of V0L1 versus TIME2 D Constant in LQF f i t of TB versus HTWC3 DCC Constant in LQF f i t of VOL1 versus TIME2 DDEN Density of dispersed phase liquid i n lower section of the column DDENV Density of dispersed phase liquid at normal boiling point DDROP Diameter of liquid drop in the lower section of the column DDROPT Volume of i n i t i a l drop whose diameter = DDROP DELT Temperature driving force from photographic data DEN Overall density of the drop or bubble from photographic data DEN1 = CDEN/DEN DEN2 = (CDEN-DEN)/DEN DTC Thermal conductivity of dispersed phase DTIME Time difference between photographic measurements DVA Dimension of vapour bubble from photographs DVB Dimension of vapour bubble from photographs DVIS Viscosity of dispersed phase E Constant in LQF f i t of TB versus HTW3 EP LQF parameter El LQF parameter E2 LQF parameter F Drag Coefficient related to photographic results FEVAP Percent evaporation at end of evaporation FN Frame number FPM Frames per minute G Constant in LQF f i t of TOTALV versus TIME H Constant in LQF of VAP versus TIME HHG Atmospheric Pressure HT LQF f i t of HTWC versus TIME HTWC . Height of vapour bubble as measured by the scale. Changed to HT and AHTWC after curve f i t and redefined as below HTWC Height of water above vapour bubble corrected for expansion of vapour bubble HTWC1 Height at which TIME1 on the recorder chart was taken HTWC3 Height at which TB was taken HW I n i t i a l height of water in the dilatometer KKK Constant used to determine which properties for the dispersed phase to use 1 = Furan, 2 = Isopentane, 3 = Cyclopentane LAMMDA Latent heat of vapourization of the dispersed phase MMM Constant used to determine which properties to use for the continuous phase 183 MODCPE Modified Peclet Number = CPE(I) / (1.0 + (DVIS(l)/CVIS(I) ))) M Parameter in LQF program N Integer counter NA NA = NX1 - 1 ND Parameter in LQF program NDATA Number of data cards of photographic data Nl Parameter in LQF NNEW Number of data entries in photographic section containing data on the vapour volume NRUN Number of runs in one set NVOL Number of data cards in dilatometer data NX1 Dilatometer data at which TIME 2 i s greater than TIME^ NNEW.) NX2 NVOL - NXl. + 1 NY1 NVOL - 1 NY 2 NX2 - 1 P Parameter in LQF PA Partial pressure of air in vapour bubble PAI Partial pressure of air in bottom section of column PCEVAP Percent evaporation PD Partial pressure of dispersed phase PI External pressure outside bubble PSUM Sum of partial pressures PW Partial pressure of continuous phase Q Rate of heat transfer RATDIA Rate at which the diameter of the bubble,; measured photographically, is changing with respect to time 184 RATEVP Rate of evaporation calculated from photographic data RATVAP Rate of evaporation calculated from dilatometric data RATVOL Rate of change of volume calculated from dilatometric data RUNNO Number of the run T Temperature of vapour inside vapour bubble TA Temperature of continuous phase in lower section of the column TB LQF fitte d value of temperature versus height for the continuous phase in the heated section of the column TCOR Average correction between the time basis of the photographic data and the dilatometric data TIME Time measured from the photographic data TIME1 Time from recorder chart at HTWC1 TIME2., Time from recorder chart readings at which the value for V0L1. was taken TOTALA Area calculated from TOTALD TOTALD Equivalent spherical diameter based on photographic measurements TOTALQ Total heat transferred based on photographic measurements TOTALV Volume calculated from the TOTALD TT Temperature used in subroutine UA Instantaneous rate of heat transfer divided by the temperature driving force UINST Heat transfer coefficient based on instantaneous area from photographic data UTOTAL Heat transfer coefficient based on area of the i n i t i a l drop VACC Acceleration based on TIMEl VAP Volume of dispersed phase vapour 185 VAPDEN Density of vapour VAREA Area calculated from VDIA VBPEVP ( 100 - VPCEVP ) VBUB Volume of dispersed phase and continuous phase vapour VBUBA Volume of air bubble in lower section of column VBUBAC Volume of air bubble in upper column corrected for temper-ature and pressure VCNU Nusselt number based on continuous phase properties and dilatometric data VCPE Peclet number based on continuous phase properties and dilatometric data VCPR Prandtl number based on continuous phase properties and dilatometric data VCRE Reynolds number based on continuous phase properties and dilatometric data VDELT Temperature driving force calculated from dilatometric data VDEN AMASS/CV0L1' VDENl, CDEN/VDEN VDEN2 (CDEN-VDEN)/VDEN VDIA Equivalent spherical diameter based on volume measured by dilatometer VDROP Volume of dispersed phase liquid in lower column VDVA ) ) Pseudo - diameter based on dilatometer volume VDVB ) VEL Velocity based on TIME VF Drag coefficient related to dilatometric data 186 VHTWC Height of water above bubble based on TIME2 VLIQ Volume of dispersed phase l i q u i d l e f t i n the bubble based on photographic measurements VMODPE Modified Peclet .number based on dilatometric data VOL Volume of vapour bubble based on photographic measurements V0L1 Volume of bubble measured by dilatometer VPCEVP Percent evaporation based on dilatometer data VQ Rate of heat transfer based on dilatometer data VT Temperature of vapour based on dilatometric data VTB Temperature of continuous phase based on dilatometer data VUA = VQ/VDELT WEL V e l o c i t y based on dilatometer data WLIQ Volume of l i q u i d i n bubble rel a t e d to dilatometer data W Parameter i n LQF WZ Parameter i n LQF X A r b i t r a r y Variable XI A r b i t r a r y Variable X2 A r b i t r a r y V a riable Y A r b i t r a r y Variable YF Parameter i n LQF Y l Parameter i n LQF Y2 Parameter i n LQF Z Constant for LQF of TB versus HTWC3 ZC Estimate of Error f o r LQF 187 Computer Program X X X X X X X X X X X X X X X X X X X X X X X X X X X > < X X X X X X X X X X X X > X > X X X X X X X X X X X X X X y X X X X X X X X X X X X X X X X XX XX XXX RF S N O , 0 2 3 2 0 0 U N I V E R S I T Y OF B C C O M P U T I N G CL N TRE M T S ( A N 1 2 0 ) P L E A S E R E T U R N TO C H E M I C A L E N G I N E E R I N G * * * * * * * * * * * * * * * * * * * * $ S I G AESA * * L A S T S I G N O N K A S : 1 4 : 0 4 : 0 6 0 6 - 0 2 - 7 1 USER " A E S A " S I G N E D ON AT 1 5 : C 7 : 5 5 ON C 6 - C 4 - 7 1 $ L I S T F H A I N .1 0 1 P E N S I O N H TWC( 5 0 ) , CVA( 5 0 ) , C V 6 ( 50 ) , F N ( 5 0 ) , P<50 ) . YE < 50 ), E !. ( 5 0 ) , 2 1E2 ( 5 0 ) , VEL ( 50 ) , ACC( 5 0 ) , T( 50 ) , TT( 50 ) ,TI 'ME 1 50) . VHUP.AC ( 5 0 ) , 3 2 V D U P ( 5 0 ) , V A P ( 5 0 ) , V A P D E N ( 5 0 ) , V C R O P ( 5 0 ) , V L I Q ( 5 0 ) ,TOTAL V ( 5 C ) , 4 3 T 0 T A L A ( 5 0 ) , A I NI ( 5 0 ) .DTI ME ( 50 ) ,AA( 50 ) , RATE VP ( 50 ) , 0 ( 50 ) ,OELT ( 5 0 ) , 5 4 U A ( 50 ) t <\!" A S S L ( 50 )» 3 C E V A P ( 5 0 ) , UI D A ( 50) , U I N S T ( 50 ) , 6 5 C R E ( 5 0) , F ( 5 0 ) ,TOT ALO I 5 0 ) ,DE LV A P ( 5 0 ) , A T I M E ( 5 0 1 . H T ( 5 0 ) 7 6 . H T W C K 8) , iH TWC3I 5) , VOL 1 ( 50) , TI MH2 ( 5 0 ) , T I M E l ( ",o ) , 8 7T £ ( 5 0 ) , B T I M E ( 5 0 ) ,COR( 5 0 ) , C D E N ( 5 0 ) , C V I S ( 50 ) 9 D I M E N S I O N AHTWC ( 5 0 ) T RAT VOL ( 5 0 ) ,P ATC I A 1.5 0 ) , BHTUC (50 ) ,UTOT AL ( 50 ) 10 D I MENS ION D E N ( 5 0 ) , C M U ( 5 0 ) , C P R ( 5 0 ) ,OEN 1 ( 5 0 ) , D E N ? ( 5 0 ) i C P E ( 5 0 ) 1 1 . D I M E N S I O N MUDCPE 150 ) 12 D I M E N S I O N VCT A( 5 0 ) , V D E N ( 5 0 ) , V C N U ( 5 0 ) , VCPR ( 50 ) , V O E \ ' l < 50 ) 13 D I M E N S I O N V DEN 2 ( 50) , V C P E l 50 1 , V.MQDPF ( 5 0 ) , V P C E V P ( 5 0 ) 14 D I M E N S I O N AT (50 ) ,01 ( 5 0 ) , C 2 ( 5 0 ) 115 D I M E N S I O N C V O L 1 ( 5 0 ) , R A T V A P ( 5 0 ) , V ( . ) ( 5 0 ) , V U A ( 5 0 ) , V A R E A ( 5 0 ) 1.6 D I f ENS 10M V V - E H 5 0 ) , V A C C ( 5 0 ) 1 7 D I M E N S I O N C K I 5 0 ) ,CC.P( 50 ) 1 8 D IMENS ION A H T ( 5 C ) , V T ( 5 0 ) » V T R ( 5 0 ) , V D E L T < 5 0 ) 1 9 C I M E N S I G N V C R E I 5 0 ) 2 0 D IMENS ICN V H T w C ( 5 0 ) , V C V A ( 5 0 ) , V C V E ( 50 ) 21 D I M E N S I O N X ( 5 0 ) , Y ( 5 0 ) 2 2 01 M E N S I C N V OL (50 ) 2 3 D I M E N S I O N V F ! 5 0 ) 24 D IMENS I ON X l ( 5 0 ) , X 2 ( 5 0 ) , Y 1 < 5 0 ) . Y 2 ( 5 C ) 2 5 C IMENS ICN H ( 5 0 ) , G ( 5 0 ) 2 6 D I N E N S I O N F E V ^ P I 50) t B P C E V P ( 5 0 ) , V B P E V P ( 5 0 ) T W L I Q ( 5 0 ) 2 7 INTEGER RUNNO 28 PE A L LAM MCA 2 9 R E A L MGOCPE 3 0 1 3 0 C O N T I N U E 31 32 R E A D ( 5 , 1) K K K , M M M 3 3 R E A C ( 5 , 5 0 ) NrtUN . 34 3 5 C A L L I L O F ( P ,WZ,E P ,NI ) 3 6 C A L L D P R O P ( K K K , DOEN,DV I S , D T C , D D E N V) 3 7 C A L L HE AO( K K K , M M M ) 38 3 9 CO 9 KK=1,.MRUN 40 4 1 C DATA I N P U T 4 2 4 3 REAO( 5 ,2 )TA , H W , H H G , F P M , A D I A , 6 0 1 A , R U N N U , DOROP 4 4 4 5 1 = 1 4 6 4 7 15 P . F A l ) ( 5 , 7 2 ) HTl-;f. ( i ) ,OVA ( I ) ,OVB I i ) , F N ( I ) 4 8 I F (FiTWCn ) . £ 0 . - 1 . 0 ) GO TO 34 4 S 1=1+1 50 GOTO 15 51 34 t ! C A T A = I - l 52 188 5 3 0 0 £C7 6 K = 1 i N D A T A 5 4 1 P ( C V A I K ) . D O . . 0 0 0 ) GO T O 8 0 7 7 5 5 V O L ( K ) - O V A I K ) * D V A ( K ) * C V P ( K> * 0 . 5 2 3 6 5 6 8 0 7 6 C G N T I N U t 5 7 5 0 8 C 7 7 N = K - 1 5 9 C 6 0 C C U R V E F I T T I N G O F P H O T O G R A P H I C D A T A 6 1 C 6 2 D O 8 0 7 8 K= 1 , N 6 3 V ( K ) = A L Q G ( V O L (K ) ) 6 4 X ( K ) = H T W C < K > 6 5 8 G 7 8 CCN n \ u e 6 6 6 7 6 8 I F I M . C T . N / 2 1 M = 3 -6 9 E X T E R N A L AUW 7 0 C A L L L Q F ( X , Y , Y F , W , E 1 , E 2 , P , W Z , N , N , N I , N 0 , E P , A U W ) 7 1 7 2 DO ' 3 0 7 9 K = 1 , N 7 3 V O L ( K ) = E X P ( Y F ( K ) > 7 4 D V A ( K ) = ( 1 . 9 0 9 9 * V C L ( K ) ) * * . 3 3 3 3 7 5 D V B ( K ) = D V A ( K ) 7 6 8 C 7 9 C O N T I M E 7 7 7 3 C A L L W A T H T ( H T W C 1 t H T W C 3 ) 7 9 8 0 1 = 1 8 1 8 2 3 0 0 0 R E A D ( 5 , 7 5 ) V O L 1 ( I ) , T I N'E2 ( I ) S 3 I F ( V O L 1 ( .1 1 . EO 1 . 0 ) G O TO 4 7 8 4 1 = 1 + 1 6 5 G O TO 3 0 0 0 8 6 4 7 N V 0 L = I - 1 • 8 7 8 8 R E A 0 I 5 , 7 2 ( ( T I M E 1 ( 1 1 , 1 = 1 . 8 ) 8 9 R E A D ! 5 , 2 0 0 3 ) I Tii I I 1 , I = 1 , 5 ) 9 0 9 1 9 2 0 0 7 7 K = 1 , N O A TA 9 3 T IT-' E (K ) = ( F N ( K ) / F P M ) * 6 0 . 0 9 4 7 7 C C N T I . M E 9 5 9 6 9 7 9 8 C T I M E C O R R E C T I C N B E T W E E N F I L M A N D C I L A T G M E T E R 9 9 1 0 0 h = N O A T A 1 0 1 f' = 4 ' 1 0 2 F XT E R N A L AUW 1 0 3 C A L L L C F ( H T W C . T I M E , Y F , W , E 1 , E 2 i P « W Z , N , H , N l , N 0 , E P , A U W ) 1 0 4 1 0 5 C O 7M K = l , 3 1 0 6 T.TT ME ( K ) = P ( 1 ) + F 12 ) * h T W C I [K ) +P ( 3 ) * H T W G 1< K ) * * 2 * P ( 4 ) * H TWC 1( K ) ' * * 3 1 0 7 C O R ( K ) = ( f I Mr I ( K 1 T I ME (KI ) 1 0 8 fcRITE ( 1 , 1 4 ) H T W C 1 ( K ) , T I M t 1 ( K ) , B T l M E ( K ) , C O R ( K ) 1 C 9 7 8 C O N T I f,'UE 1 1 0 l i t TCOR-=< C O R ( I )< -COR ( 2 1 + C O R ( 3 ) + C O R ( 4 ) +COR ( 5 ) + C O R ( 6 ) +C C P ( 7 1 + C O R ( 8 ) 1 / 8 189 1 1 3 0 0 79 K = 1 , N C A T A 1 1 4 T I M E ( K ) = T I M E ( K ) + T C 0 R 1 1 5 I F ( T I M E ( K ) . L E . 0 . 0 1 ) T I M E ( K ) = 0 . 0 1 1 1 6 WR ITE ( 1 , 7 5 ) T I M E ( K ) .TCCIR 1 1 7 7 9 CCN T INUE 1 1 8 1 19 1 2 0 C ... . 12 1 C CURVE F I T T I N G OF H E I G H T . . . . . . . 1 2 2 C 1 2 3 132 N = N 0 AT A 1 2 4 M = 3 1 2 5 EXTERNAL. AU X 1 2 6 C A L L L Q F ( T I M E , H T W C , Y F , W , E 1 , E 2 , P , W Z . , N , M , N I , N O , E P , A U X ) 1 27 A = P ( 1 ) 1 2 8 D = P ( 2 ) 129 C = P ( 3 ) 1 30 131 DO 99 K = 1,N DA TA • 1 32 AHTV lC tK )=YF ( K ) _ . - 1 3 3 H T ( K ) = Y F ( K ) 13 * . HTWCIK ) = H W - Y F ( K ) + D V A ( K ) * D V A ( K ) * D V B ( K ) # 2 . 0 / ( 3 . 0 * 2 . 5 4 ) 1 3 5 99 C C N T I N U E 1 3 6 C 1 3 7 c C A L C U L A T I O N OF V E L O C I T Y AND A C C E L E R A T I O N 1 3 8 c . _ 1 3 9 DO 17 K=1 ,N 1 4 0 VEL (K. )= (P ( 1 ) + 2 . 0 * P ( 2 ) *T IME( K ) + 3 . 0 * P ( 3) * (T IME( K ) * * 2 ) ) * 2 . 54 141 1 7 C C N I I N C E 1 4 2 c 1 4 3 c. CURVE F I T T I N G OF V E L O C I T Y 1 4 4 c l ' « 5 M=4 1 4 6 E X T E R N A L AUZ 1 4 7 C A L L L C F (T I M E , V E L , Y F , W , E l , E 2 , P , W Z , N , M , N I , N D , E P , A U Z ) 1 4 8 A l =P ( 1) 1 49 A 2 = P ( 2 ) 1 5 0 A3 = P ( 3 I 15 1 A4=P ( 4 ) ... •_. .. .. 1 5 2 1 5 3 DO 135 K = 1 ,NDAT A 1 5 4 A C C ( K ) = ( P( 2 ) + 2 . 0 * P ( 3 ) * T I M E ( K ) + 3 . 0 * P ( 4 ) * ( T I M E ( K ) * * 2 ) ) 1 5 5 135 C C N T I N U E 156 1 5 7 1 5 8 1 5 9 C C U R V E F I T T I N G OF BULK T E M P E R A T U R E 1 6 0 161 N = 5 1 62 M=3 1 6 3 E X T E R N A L AUT 1 6 4 C ALL L O F ( H T W C 3 » T B , Y F , W, E 1 , E 2 , P , W Z , N , M , N I , N O , E P , A U T ) 1 6 5 0 = P ( 1 ) 1 6 6 E = P < 2 ! 1 6 7 Z = P ( 3 ) 1 6 6 169 0 0 13 1 1=1 , N D A T A 1 7 0 TR ( I ) = P ( 1 ) + P ( 2 ) *A HT WC( I ) + P ( 3 ) * AH T WC ( I 1 * * 2 171 131 C C N T I N U E 1 7 2 C 190 173 C CALCULATION O F TEMPERATURE FOR' PHOTOGRAPHIC DATA 174 c 1 7 5 CONTINUE 176 1 7 7 00 6 0 7 1 1 =1 , NDATA 178 CALL CPROPIN'.MM, I , CDEN , C VI S i CC P f C K, TP > 179 6 C 7 1 CCNTINUE 180 1 8 1 V EUR A= (Af.IA*ADIA*BO IA *0.5 2 3 6 ) 182 183 CALL TEMP (K,NDATA,KKK,T,PI,HHG,CDEN,HTwC,HW, 1 8 4 IPO,PW,PSUM,TT,VBUbA,PA,OVA,DVB,TA) 1 6 5 C 1 8 6 C CURVE FITTING OF TEMPERATURE FOR DILATCfETER DATA 187 C 188 r = 4 189 N N E W = K - 1 1 9 0 1 9 1 IF!M.GT.NNEW/2) f=NNEW/2 192 DO 9 0 8 0 K = l , N N E W 193 X !K) = ALCG(TIME IK) ) 1 9 4 V(K)=A LOG(T (K ) ) 1<35 WRITE!2f75) TIME(K) ,T(K) 1 9 6 9 0 3 0 CCNTINUE 1 9 7 198 EXTERNAL A U 3 1 9 9 CALL I L(vF ( P , W Z , EP» N I ) 2 0 0 CALL LCF<X,Y ,YF ,W»E1,E2 , P ,WZ,NNEW .M .N I ,ND ,EP ,AUB ) 201 2 0 2 CC .3 8 K=1,NNEW 203 Tl K) =E X P ( YF (K ) ) 2 0 4 A T(K) = T(K1 + 2 7 3 . 1 6 205 3 8 CCNTINUE 2 0 6 207 DO 1 0 0 1 1=1,NDATA 208 I F (OVA! t ) .EQ . 0 . 0 ) C O TO 5 1.2 8 209 PA I= riHC +C DE N( 1)*HW*l.Sh7 210 P1= FHG + CDEN( I )*HTWC( I ) * 1.867 2 1 1 CALL PPRESS (PD ,PW , I , T , KKK , -MMM) 2 1 2 V B U B A C ( I ) = V 3 U 3 A * ( ( T ( 1 1 + 2 7 3 . 1 6 ) / ( T A + 2 7 3 . 16 1 )*(PA I/PI ) 213 VBUR! 1 > = DVA( I )*DVA( I)*DVB( I ) * 0 . 5236-VBUBAC( I) 2 1 4 V A P < I ) = VRUP. U )*PCY ( PD+PW ) 2 1 5 1 0 0 1 CONTINUE 216 2 1 7 N=I 218 GO TO 5 1 2 9 219 5 1 2 8 N=I-1 2 2 0 5 1 2 9 CGNTI NLlE 2 2 1 C.= 5 2 2 2 223 DG 1 0 0 5 K = l ,N 2 2 4 VAP ( K ) =A8 S ( VA P! K ) ) -2 2 5 VAP (K ) = Al O G (V iP ( K ) ) 2 2 6 100 5 CONTINUE 2 2 7 2 2 8 C CURVE FITTING C F V A P O U K VOLUME 2 2 9 230 I F ( " . C I . N / 2 ) M=N/2 231 EXTERNAL A U H 2 3 2 C A L L L OF ( TI '•IE ,VAP ,YF , W , E 1 , E 2 , P ,WZ , N, M, NI , ND, FP, AU B ) 191 233 M = M 234 DO 1006 3 K=1,M 235 H(K) = P(K) 236 10063 CCNTINUE 237 238 DO 1002 K=1,.M 239 VAP <KI=EXP(YF(K) ) 240 1002 CONTINUE 241 _ . . .. ..... . _ .. _ ' 242 DO 40 1 = 1 ,N 243 CALL PPRESS(PD,FW,I ,T,KKK,MMM) 244 CALL DEN V AP ( KKK , I , VAPDEN, P! , T ) 245 VOPOPI I )=CDROP**3*0 .5236-V3UBA 246 VLI Oil > =VDROP( I l-VAPI I ) *VAPDF..N( I ) /DDENV 247 TOTALVII ) = VBUR( I l+VLIUI I)+VaUBAC( I) . . . _._ 248 TOT ALC(I ) = (1 .9099*TOTAL V( I) )**. 33 33 249 TOTALAI I ) = 3.Ill59* (TOTA ID I I )) **2 250 251 IF{1-1)45,45,46 252 45 DT IME( I )= 0.0 253 A A ( I )=TOT AL A ( I ) _ 254 RAT EVP. 1 )=0.0 255 GO TO 10 1 256 4 6 CT T ME( I) = TI ME(I)-TIMg( I- I ) 257 AA(I)=(TOTALAII)+TOTALA(1-1))/2.0 258 RAT EVP(I ) = H(2) 2 59 CO 10030 K = 2,Ml 260 RATEVPI I ) =R A TEVP( I)+ K *H (K+l)*TIME(I)**(K-l) 261 100 30 CONTINUE 262 P. A T E V P ( I )=RAThVP( I ) *VAPOEM I )*VAP( I ) 263 101 CONTINUE 264 265 CALL LAM(KKK,I ,LAM MCA,T) 266 A MA SS=VDROP ( I )*D0E.N 267 01 I ) = R ATEVP ( I ) *L AM.MDA 268 DELT(I ) = TB( I)-T( I ) 269 UA( I) =<j< I )/'JcLT( I ) 270 AMASSL( I )=VL I0( I )*DDEN 271 PCEVAPII )=( \MAS S-AMAS S L(I))* 100.0/AMASS . . 272 B PC E V P ( I ) = 1.0 0. 0- P C E V A P ( I ) 273 TCT ALQ = V A P ( I)*V AP DEN( I )*L A'iMnA 274 275 C DIMENSIONLESS GROUPS FOR CORRELATION OF PHOTO OATA 276 2 77 UINST(I )= (U A (I)* 10.0)/TCTAL A( I ) 273 DEN( I )= A MAS S / TO T A L V ( I ) 279 CNU I I > =UINST( ()*TOTAl0 < I )/(CK( I )*360C0.0) 2R0 C PR(I)= C V1S( I )* CC P( 1 ) /CK( I ) 28 1 DEN 1 ( I >=COEN( I)/UEM( I ) 2 82 CfcN2(I ) = (COCN( I )-OEN( I ) l/CDEN ( I ) 283 CPE( I ) = TO TA LO( I )* VE L( I )*CDE NI I )*CC P 11 )/CK (I ) 234 MODC.PE ( I)=CP E(I )/( 1 .C + < DVI S/C VI SI I ) ) ) 285 CKE ( I ) = TO TALI) I I )*V'EL I I ) *CDEN( I ) / CV IS I 1 ) 2 86 F ( I ) = ( TO UL V( I ) *r.r)F M I ) *9B0. 9-AMASS* ( ACT. I I ) +98 0 .9 ) ) 2 87 1/ { < 3.1 416*1 f>T 4LD( I ) ** 2* CDEN( I ) *VEL( I ) **2)/8.0) 283 40 CONTINUE 289 C 290 C CURVE FITTING OF TOTAL VCLUME FROM PHOTO CATA 291 C 2 92 DO R081 K=1,N 293 V(K)=ALOG(TOTAL V(K) ) 294 X(K) = TIME(K) 295 8C81 CONTINUE 296 297 M=4 298 I F I'M. GT .N/2 1 M=N/2 '  299 • EXTERNAL AUW 300 CALL LQF(X,Y,YF,W,E1,E2,P,WZ.N,M.NI,ND»EPtAUW) 30 I DO 10031 K=l ,M . 302 G(K)=P(K) 303 10031 CONTINUE 304  30 5 DO 8C82 K=1,N 306 TOTALV(K)=EXP(YF(K) > 307 8C82 CONTINUE 3C8 309 N1=N + 1 310  311 DO H78 I = I'M 1 , NDATA 312 TOT AL C( 11 = 0.0 313 ' T 0 T A L A ( I 1=0.0 314 PCEVAPf I 1=0.0 315 CELT( I 1=0.0 316 TOT A L V ( I 1=0.0  317 378 CONTINUE 318 319 879 CCN TINUE 320 321 322 IF( V0L1 ( U . EQ.-l 1 GO TO 2002  32 3 324 C CALCULATIONS USING 0 IL ATC-I ET ER DATA 325 326 C CURVE FITTING OF DILATOMETER VOLUME 327 328 DO 9085 K = l ,NVCL  329. IF<<TIMfc2(K>.GE.TIME!NNEW)I 33 0 1.AND.(V0L1(K).NE-0.0011 GO TO 90 8 7 331 9C85 CONTINUE 332 333 9087 NX1=K 334 NX2=NVCL-NX1+1  335 NY2=NX2-l 336 L=l 337 AC = VOL 1 (NVOLI+0.01 3 38 10001 CONTIKUE " 33 9 340  34 1 00 10C02 K= 1 ,\Y2 342 Y(K 1 = VOL I(N X 1+K-11 343 X (K ) = TIfcE2 < NX 1 + K-l )' 344 DDR0PT=DDPOP-*3*0.5236 345 IFl {Y(K l-QDROPT ) . LE.0.0) DD RO PT=Y(K1 — 0.Cl 346 Yl ( K) = ( AC-Y ( K) ) / ( Y ( X) -DDROPT 1 347 Y2( K )= ALliGI Y 1 IK 1 1 34 8 X2(K)=ALGG(X(K1) 349 1C002 CCNTINUE 350 351 EXTERNAL AUC 352 CALL OPI.CF ( X2 ,Y2, YF . A ,£1 , F2 ,? ,*Z , NY 2 ,2, 10 , ND , .01, AUG) 193 3 53 354 IF (L.EQ.ll GO TO 10003 355 IF ( E2(2).GT.ZC) GO TO 10004 356 IF(L.EG.lOO) GO TO 10004 35 7 10003 CC=EXP(P(1)) 358 DCC=P<2> 359 AC=AC+0.01 360 L = L H 361 ZC=E2(2> . _ . _ .. .' 362 GO TO 10001 363 100C4 CONTINUE 364 WRITEI9, 10051 ) RUNNO.L 365 10061 FORMAT!14,15) 366 NA=NX1-1 . . 367 36b 00 8026 K=l,NA 369 CVOL H K)=G! 1) 3 7 0 CO 10033 L=2,M 37 1 CVfSLl (K) =CVGL1 (K)+G(L)*T I ME2 ( K ) ** (L-1 ) 372 100 33 CONTINUE 373 CVOL 1 (K ) =EXP ( CVOL 1! K ) ) 3 74 RAT VOL!K)=H(2) 375 00 IOC 34 L=2,Ml . 376 RATVOL(K)=RATVOL(K)+L*H(L+1 )*T IME2(K )**(L- 1 ) 377 100 34 CCNTINUE 378 8026 .CUNTINUE .3 79 380 NY1=NVCL-1 381 382 OG 806 3 K=NX1,NY1 383 CVOL I ( K> =i:Oh.OPTt ( ( AC-DORCPT 1/ (1 .0 +CC *T I ME 2 ( K ) **OCC ) ) 38 4 RAT VOL (K )=-( (AC-DOROP T)*CC*OCC*TIME 2t K)**(DCC-l.O) ) . .385 . 1/( < 1.0+CC*T IME2<K)**DCC1**2 ) . . 386 8063 CONTINUE 387 338 CV0L1(NVOL)=VCL1(NVOL) 389 RATVOL! fi VOL 1 =( VCLK NVOD-VOLl (NVOL-1) ) 390 1/(T IME2(NVOLl-TIME2INV0L- 11 1 391 WRITE!1 ,9050) ( T I M E 2 (K) ,V0L1 (K 1 ,CVOL 1!K),K=1,NVCL 1 39 2 9050 FORMAT!3F10.41 3 93 CC 9002 1=1,NVOL 394 WE L(I 1=A1 + A 2 * TI ME 2 (I 1 + A3 * ( T I M E2 ( I 1**21+A4*!TIME2( 11**3) 395 VACCI I 1 = A 2 + 2.0*A3* TIM E2( I 1 + 3,0*A 4*(TI ME 2 ( I 1**2) 396 RAT 0 I A( I) = ( 1 .9J99*CVCL1 (111**.3333 39 7 . 11HTWC! I ) =A* TI ME 21 I 1+6*TI ME2 ( I ) **2+C*T I ME2 ( I )**3 398 VT8!I)=D+F*BHTWC(I>+Z*BHTWC(I)**2 399 CALL CPROP( f-MM , I , CDEN, CV I S, CCP.CK , VTB ) 400 9 00 2 CONTINUE 4Q1 402 DO 900 3 1=1,NVUL 403 VHT'WC ( I ) = HW-BHTWC( I ) + 4. C*C V0L1 (11/(3.1416* 2.54) 4 04 VOkOP ( I ) = r;CK0r'** 3*0 .5236-VHUB A 40 5 IF(CVOL HI ) .LE.VCROf! I) 1 CVOLI ( I 1 =VDROP( I l + .OOOl 406 VOVM 1 1= ( 1 .'l!.^9*( CVOL 1! I l-VOROPl I 1 ) »**. 33 33 4C7 VDVB(I 1 =VCV A( [) 408 9C03 CONTINUE 409 C 4 10 C CURVE FITTING OF TEMP. FROM DILATOMETER DATA 411 C 4 12 " C Al. L T E>-'P ( I , NVOL ,KK K , VT ,P I , HHG ,C()EN , VHT WC ,HW, 194 413 1PD,PW,PSUM,TT.V8UEA,PA,V C V A , V CV 8, T A ) 4 14 415 M = 5 " • 4 16 N=NV0L 417 IF(M.C1.N/2> M = N/2 41 8 00 H030 K=1.NVOL 419 IF< T I ME 2 ( K) . LE. .01 ) T I H E2 (K ) =0 .0 1 420 X(K)=AL0&(TIME2(K)) 421 8030 CCNTINUE _.. 422 423 CALL ILOFIP,WZ,EP ,N I 1 424 EXTERNAL AUB 425 CALL LOF ( X, VT , VT, Vs, FI , E2 , P , W Z , N , M , N I , ND , E P, AU B ) 426 427 DG 9006 I=1,NVCL 428 WRI TE(2,9050) TIME2(I )»VT(I ) 429 CALL PPRESS(PD,PW,I,VT.KKK.MMM) 430 PI=HHG+COEN ( I 1* ( HW-uHTWC (11+4 .0*CVOL 1(1 1/3.14161*1. 86 7 43 1 CALL DENVAP(KKK,I , VAPOEN , PI , VT) 4 32 RATVAP( I l=RATVOLf I 1*PD*VAPDFN( I 1/(PD + PW ) 433 CALL LAM(KK K , I,L AMMC A.VT ) 434 V0( I )=RATVAP( I)*LAM MO A 435 VDELT(I)=VTD(I)-VT(I) 436 IF( I.FO.(NVOL-11 1 BMASS = V0L1( I )*V APCENI 1 )*P0/IPD+PW) 43 7 9006 CONTINUE 438 4 39 A MA S S= B MA SS 440 C DIMENSIONLESS GROUPS FOR COFRELATION 441 442 DO 55 1=1 ,NVGL 44 3 VUA( I )=VQ(I )/VDELT( I I 444 VOIA( I 1 = (1 . 9099 *CVOL 1(1))**.3333 445 V AR E A( I)= 3. 1415 9*(V CI A( I ) 1**2 446 UTOTAL ( I ) = VUA ( I 1 /V.ARE A( I 1 447 VDEN( I) = AMASS/CV0L1( I 1 4 48 VCNUl I ) = UTLiTAL( I ) *Vl?I A( I ) / CK( I ) 449 VCPR( I ) = CVTS( I )*CCP( I ) /CK( I ) 450 CALL PPRESS(PD,PW,I,VT,KKK,NMM) 451 VPCEVP ( I ) =C VCIL1 ( I 1 * VA PDEN (I 1 *PD*100 .0/ ( AMASS * (PD + PW 1 1 452 VBPEVP(I 1= 100 .O-VPCFVP( I 1 453 VVLIQ( I ) = AMASS*VBPEVP(I )/ (100.0*ECENV) 454 VOENK I )=CDEN( I 1 /VDEN ( I ) 455 V0EN2( I )=(COEN( I ) -V DEN( I) )/COEN( I 1 456 VCPt(I)=VCIA(I)*VEL(I)*CDEN(I)*CCP(I)/CK(I) 457 VMOOPEI I )=VCPG ( I )/( 1 . 0+ ( D VI S/C VI S( I 1 1 1 :. 458 VCR c ( I 1=V C I A( 11 *VV EL( I )*C OfcN( I 1 /CV I S( I 1 459 VF( I )=(CVOL 1( I 1*CDtM( I 1*9 30.9-AMASS*(VACGU 1 + 980.91) 460 1 / ( (3. 14 16*VCIA( I ) **2*C0EN( I I*VVEL(I 1**2)/8.0) 46 1 F EV IP ( I 1 = VOL 1 ( I) *VAPUtM( I )*P!l* 100 .0 / ( AM ASS*(P 0+P W 1 ) 462 55 CONTINUE 463 464 C OLTPUT •-- • — - -465 CC 466 2002 CALL EFEAC<KKK.RUNNO) 46 7 468 C ADAMS' RESULTS 409 470 93 WRITE(8,IC0201TA 471 WRI TE( 0, 1CC211HW 4 72 WRITE!8, 100 22 1FHG 195 473 WPI TG(8, 10023)V8UCA 474 WRITE(3, 10024)0OR OP 475 AMASS=VCRGP ( I >*CDEN 476 UP.I TE! 8 , 1 002.5 1 AMASS 477 WRI TE ( H , lC026)iJMASS 478 WRITE( 8 ,10027 ITIME2IN VOL-1) 479 . WRI TE(3 , 100291AC,CC,OCC 480 481 . . WRI TE(8 ,20001 ) . . _ _ 482 98 WRITE(8,124) 433 484 00 51 1 = 1,ME AT A 485 Wkl Tt(3,115)TI ME( I) ,HT< I) ,VEL( I 1 ,ACC(I 1 ,TOT ALD! I ), 486 IT OT AL A 1 I I ,TOTAL V( I ),PCEVAP( I ) ,0E LT(I ) . . ... 487 .5 1 CCNTINUE . . . . 483 439 WRITE! 8, 125-) 490 491 C RESULTS f ROM FILM 492 4 93 DO 52 1 = 1 , N C AT A . .. 494 IF (OVA!I).EQ..001 GO TO 113 495 WRI TE ( 8 t 4 34 ) U A ( I ) ,PCEVAP( I ) ,CRE (I ) , CP P. 1 I) ,CPE( I) 496 1 ,F( I 1,BPCFV P(I 1,VL IQ( I ) 497 WRI Tt! 7,75) F(l > ,CRE(I) 493 52 CONTINUE 499 500 118 CONTINUE 501 5 02 503 504 505 C DILATCMETFR RESULTS 506 507 WRITE(8,20002) 50 8 WRITE!8,124) 509 510 DO 93 1 I=NX 1,NVOL 511 WRITE(8, I 15 )TIME2!I ) ,8HTWC( I),VVEL(I),VACC(I) ,VDI A(I) , VAREA!I ) 512 I.CVOLl! I) ,VPCEVP(I ) ,VEELT (I ) 5 13 931 CONTINUE 514 515 WRI TE(8 ,125 ) 516 517 C RESULTS FROM DILATOMETER 518 5 19 DO 94 1 I-NX 1,NVOL 52 0 WRITE ( 8 ,4 34 )V'J A ( I ), VP C EVP ( I ) , VCR E ( I ) , VC PR ! I ) , VC PE ( I ) , VF ( I ) 521 I,VB PE VP(I) ,VVLIQ 1 1 ) 522 WRITE! 7, 75) VF( I ) , VCRE( I ) 523 941 CCNTINUE . 524 525 9 CONTINUE 526 527 5 0 • FORMAT!13) 528 1 FORMAT! 211) 529 10 FORMAT (6F 10.4, I 3, 14) 530 14 FOP MAT(4F10.5) 531 2 FORMAT 16) 10.4,1 5,F1C.4) 532 72 FORMAT ( BE 10'.'5 ) 196 533 2 00 3 FORMAT!5F10.5) 534 73 FORMATI4F10.5,4F10. 5) 535 75 FORMAT!2F10.5) 536 76 FORMAT!2F10.1) 53 7 20 FORMAT (2OX, F10.4, 10X, F20 .4 ) 538 37 FURMATI20X, 10H T I ME , 1 OX , 10.H TEMP) 5 39 39 FORMAT(2OX,F 10.4, 10X,F10.4 ) 540 3C5 0 FORMAT!' THE LINE REACHED IS 200') 541 115 FORMAT!9(3X,F10.4)) 542 124 FORM AT(I HO,4X,'TI ME(S EC )',3 X, 'HEIGHT!IN) 1 ,2 X, 54 3 1'VEL(CM/SEC)' ,1X,'ACC(CM/S EC2 >',6X, '01 A(CM) ', 544 22X, «APEA( SO CM)', IX,'VOLUME(CU CM ) ' , I X, • P ERC. ENT EVAP', 545 32X, 'DELI (DEG C) •/) 54 6 125 FORMAT( IHO , IX ,'UA(CAL/SEC DFG.C)' ,'PERC ENT EVAP',IX, 54 7 1'REYNOLDS NO', IX , 'PRANDTL NO' , 2X,1PECLET NO',3X,'DRAG CGEFF' 548 2 , 3X , ' (1 00-% ) EVAP', IX , 'I IwU.in VOLUME'/) 549 431 FORMAT!4X, 4 HT I WE, 550 11IX,6HHFIGHT, 551 2 7< , ciHVELOC I TY, 552 32X,12HACCEL ERA TI ON, 553 4 3 X, 8 h 0 I AM b T E R , 554 57X.6HPCEVAP, 555 66 X,4HDE L T /) 556 4 32 FORMAT(7(3X,F10.4) ) 557 433 FOR MAT(3 X.6HUVALUE, 558 17X, 10HNLSSELT NO , 559 23X,1 IHREYNCLDS NC, 560 33X,lOHPRANDTL NO, 561 45X.4HDEN1, 562 511X.4HCEN2 //) 563 4 34 F CR MA T ( 7 ( 3 X , F 10 . 4 ) , 3X , F 10 . 6 ) 564 126 FORMAT(9!3X,F 10.4) ) 565 521 FORMAT (8F10 .4 ) 566 1 8 FORMAT(20X,10H TIME ,10X,20H HEIGHT OF WATER ) 567 100 20 FORMAT ( 'LOWER T EM PER A TORE = • , T 30 ,F 1 0 . 2 »1 X , • DEG C ) 568 1(021 FORMAT l'HEIGHT OF WATER IN C Dl. U MM = ' , T 3C , F 10 . 2, IX, 'IN' ) 569 10022 FOP. MAT I 'ATMOSPHERIC P R E S S UP r. = ' » T 3 C , K 1 0 . 2 , IX, 'MM OF HG' ) 5 70 10023 FORMAT ( 'VOLUME OF AIR B UB BL E = ' , T 30,F 10.8, IX,'CU CM') 571 10024 FCP MAT!' INITIAL DIAMETER CF 0ROP=',T30, F 10 . 5, IX,«CM• ) 5 72 10025 FORMAT( HEIGHT OF DROP=',T30,F10.8,1X,' GM• ) 573 100 26 FORMAT( 'WEIGHT OF VAPOUR BUBBLE^',T30,F 10.8, IX,'GM' ) 5 74 10027 F OR MAT{' TOTAL E VAPOKATICM T I ME = • ,T3C,F 10 . 2 , 1X , ' S EC ' ) 575 10028 FORMAT( ' F INAL PERCENT EVAPORA 110N= ' ,T30 ,F 10.4 ,1X,• l< ) 5 76 100 29 FORMAT ('VOLUME*• , E I 5 . 8 , ' / ( 1 .0 + ' , E 1 5 . 8 , « *T IKE ** < , E I 5. 8 , • ) • ) 57 7 20001 FORMAT! L H 0 , T 3 0 , 'RESULTS FROM PHOTOGRAPHIC DATA'//). 578 20002 F0P.MA1 UH1.T30,'RESULTS FROM DILATOMETER DATA'//) 579 DEBUG UNIT!3).TRACE 580 AT 130 581 TRACE CN 582 END 5 83 SLB ROUT INE OPRDP!KK K, DOE N,D V IS,DTC,DDENV) 5 34 GC TO ( 127, 123, 129),KKK 565 127 CCNTINUE. 5 86 C DATA FOR FljRAN 587 f):1EN=0 .F H3 1 588 DV1 S=.0034 589 f)TC = 0.000 33 2 590 GC TO 130 59 1 12a CONTINUE 592 r. CAT A FOR ISOPENTANE 197 593 DDENV=0.61 594 OVIS=.002 73 595 DTC=.000255 596 t)0EN=.628 597 GO TO 130 598 129 CONTINUE  599 C DATA FOR CYCLOPENTANE 600 0DENV=0.668 6C1 . _ . DVI S = .00322 . .. . „ _ 602 DTC=.000301 603 DDEN=.744 60 4 GO TO 130 . 6C5 130 CONTINUE 606 RETURN 607 . END 608 SUBROUTINE HE AD ( K K K , f-'MM ) 609 GO TO (3,4,5),KKK 61 0 3 WRIT E (8,6)  611 6 FORMAT BOX ,2 IHRESULTS OF FURAN RUNS ///) 6 12 GO TO 100 613 .4 . . KR1TE(8,7) 614 7 F0RMAT(30X,26HRESULTS OF ISOPENTANE PUNS///) 615 GO TO 100 616 5 K * I T F ! 8 , B )  617 8 FORMAT(3CX,28HRESULTS OF CYCLOPENTANE RUNS///) 618 100 CONTINUE 619 GO TO (670,671,672,673,2015,2016,2017),MMM 620 670 WRITE!8.674) 621 674 FORMAT(30X,'PERCENT COMPOSITION IS 0.0'//) 622 CC TQ 6 75  623 671 hRITE(8,676) 624 676 FORMAT(3CX, 'PERCENT COMPOSITION IS 35.5'//) 625 CO TO 675 . ....... 62t> 672 WRITE(8,677) 627 677 FORMAT (3 0 X » 'PERCENT COMPOSITION IS 67.6'//) 62E> GO TO 675 _ _ 629 673 V\R I TE! 8,678 ) 630 678 FORMAT(30X, 'PERCENT COMPOSITION IS 86.0'//) 631 2015 1-.RI TE ( 3 ,201 8) 632 201 8 FORMAT! :SOX, 'PERCENT COMPOSITION IS 56.02?'//) 633 GO TO 675 634 201 6 HP! TE ( 8 , 20 19 )  635 2019 FORMAT!30X, 'PERCENT COMPOSITION IS 73. 072'//) 636 GO TO 675 6 37 2017 hRITF!8.2C2 0) 638 2020 rtlRMAT (30X, 'PERCENT COMPOSITION IS 77. 06?'//) 639 GO TO 675 6 40 675 CONTINUE  64 1 RETURN 64 2 END 643 SUBROUTINE EHEAD(KK K,RUNNO) 644 INTEGER PUNNu 6 45 CO TO III (> i 11 7,11 it), KKK 6 4 6 1 16 WR 1 TF ( 8 , 1 1 9 ) 3 UN NO •  647 1 19 FORMAT ( 1H1, 6h:<UN E- , I 4// ) 648 GO TO 120 649 1 17 . wRITEl 8, 121) RUNNO 650 121 FORMAT(1H1,6HRUN I-.I4//) 65L GO TO 12 0 6 52 1 18 SR. I TP ( 8,122) RUNNO 1 9 8 653 122 FORMAT ( 1 HI ,6HRUN C-,I4//1 654 120 CONTINUE 655 RETURN 656 END 657 SUBROUTINE WATHT ( HTWC1 , HTV.C3 ) 653 DIMENSION H T W C1 ( 5 0) , H T WC 3 ( 5 0 I  659 HTWCl(l)=C.O 660 HTWC1(2) = S.0 661 HTWCl(31=12.0 662 HTWCl(4)=13.0 663 HTWCK 51 = 24.0 66 4 HTWCl (6 1=30 .0  665 HTWCl(7 1 =36 .0 666 HTWC 1(81 = 42.0 667 HTWC3I11=0.7 668 HTWC3(2)=6.3 669 HTWC3I 31=16.7 670 HTWC3 (4 ) = 23 .6  67 1 HTWC3(5 1=39.5 672 RETURN 673 END 674 SUBROUTINE I LQFIP , WI,EP,NI) 675 D IMENSI ON P(10) 676 DO I 1=1,10  677 1 P(I 1=0.0 678 W7.= 0 679 EP=.CP01 630 NI = 10 681 RETURN 6S2 END 68 3 FUNCTION AUX (P,D,TIME,L) 664 DIMENSION P(501,0(50) 68 5 D(l)=TIME 686 DI2)=TIME*TIME 687 D 13 ) = T I M. E * T IM E * T IM E 688 AUX=0.0+P(1 1*Q( 1 )+P(2)»C(2)+P(3)*D<3)  639 RETURN 690 END 691. FUNCTION AUZ (P,D,TIME,L) 692 DIMENSION P(50),0(50) 693 D(l)=1.0 694 D (2 ) =T I ME  69 5 D(3 ) = TIME * TIME 696 C (4 ) = C( 3 ) *T, IME 697 AUZ=P(1 )*D(11fP(2)*0(21+P(3)*D(3)+P(4)*C(4) 69 8 RETURN 699 END 700 FUNCTION AUY (P,D,TIME,L)  701 DIMENSION P(50) ,0(50) 702 C( 1 1 = 1 .0 7C3 AUY=P(1) 704 DO 10 J=2,3 705 D(J )=D(J-1)*TI ME 7C6 10 A UY' ~ A Ll Y + 0 ( J ) " H ( J ) 707 RETURN 7 08 END 709 FUNCTION AUW IP, D, HTWCL 1 710 DIMENSION P(50),D(50) 711 0(11=1.0 712 C (2 1 =HTWC 1 9 9 713 C(3)=HTWC*hTWC 714 0(4 )=HT'..C*HTWC*HTwC 715 AUW=P( 1)*0( 1 )+P<2)*D(2) + P(3)*0(3)+P(4 1*0(4) 716 RETURN 717 END 7 18 FUNCTION AU T ( P , D , H T WC 3 ,1)  719 DIMENSION P(50),D(50) 720 0(1>=1.0 721 0( 2)=HTWC3 .. •„ _ „ . . .'. 722 D!3 )=HTWC3*HTWC3 723 D(4 )=HTWC3*HTHC3*HTWC3 724 AUT = P( 1)*D( 1)+P( 2) *f)( 2) + P(3)*P(3)  72 5 RETURN 7 2 6 E Nil 727 FUNCTION AUR(P,D » TI ME 2» L) 72 8 DIMENSION P(50),C(50) 729 DID =1.0 730 D( 2 ) = T IME 2  731 D(3)=T I,ME2*TIME2 732 0(4)=TI ME 2* T I ME 2 * T IME2 . 7 3 3 AUR=R(1)*D(I)+P(2)*0<2)+P(3)*0(3)+ P(41*0(4) 734 RETURN 7 3 5 END 736 SUB POUT IN F CPROP( MMM. K, CD EN , C VI S . CCP . CK , T ) 73 7 DI MENS I ON CDENI501,CVIS(50),CCP(50),CK(50), T(50) 738 GO TO ( 3 1,82, 83 ,84.«5,86,87) ,MMM .73 9 81. ._ CGNT INUE 74 0 CDENIK)=1.0 001+(0.35286E-04)*T(K)+(-0.6 3181E-05 )*T(K)**2 741 l + ( 0.18859E-C7)*T(K1**3 74 2 CV IS ( K 1 = EXP (0 ,57 794-(0 . 32 369 E-01 )*T( K )  74 3 l + (0 .2 10 46E-0 3I*T(K)*~2 +( -C. 7065GE-06)*T(K )**3 1 744 CVIS (K. ) = CV IS (.<)/ 100 .0 745 CCP ( K) =1.0 . .. . 746 CK(K1=(0 . 134E-02 )+(C.367E-C5)*T(K1 747 GO TO 2000 748 82 CONTINUE  7 49 CDEN(K 1= 1.094 1+(-0. 304 22E-031 *T( K1 7 50 1+(-0.30398E-05)*T(K)**2+(0.76834E-0 81*T(K1**3 751 CVIS(K)=EXP( 1.9217+(-C. 43788E-01>*T(K) 752 1 + (0 .29 691 E-03)*T(K) ** 2+(-0.10 210E-0 5)*T(K 1**31 753 CVIS(K )=CVIS(K)/I 00.0 754 CCP(K) = 0.S27+(0.4S5 7E-03)»T1K)  7 55 CK(K)=(0. 1 0 63E-021+(0.2136E-05)*T(K) 7 56 GO TO 2000 757 8 3 CCNT INU E 758 COE'N (K) =1. 18 41 + 1- 0. 5 064 0E-O3 ) * T ( K ) 7 59 l + (-0.149ClE-C5)*T(K)**2+( 0. 4 423.3E-08)*T(K)**3 760 C.V1S(K) = EXP(4. 03 07 + 1-0. 65976 F-QI)*T(K) 76 1 l + ( 0. 4f:2 53F- 0 3 I *T( K) * *2 + ( -0. 156E;!E-05 1 *T (K 1**3 1 762 CVIS(K)=CVIS(K)/100.0 763 CCP(K) =0. 648+(0. 1C857E-C2 )*T(K 1 764 CK(K1=(0.844C-03)+(0.5 8E-C6)*T(K) 765 GO 10 2000 7 66 8 4 CONTINUE ; 76 7 CO EN (K 1= 1 .2 34 7+(- 0. 5 6 74 4H-0 3 1 * T (.*.) + (- 0. 10729E-05) * T( K 1 **2 768 l+(0.3026KF-08)*T(K)**3 7 69 CVIS(K 1=EXP(6.5329+(-0.955246 — Oil*T(K) 770 1+ O.68239E-03 ) * T I K ) <' * 2 + ( - 0 . 2 2 9 0 3 F. - 0 5 ) * T ( K )**3) 77 1 CVIS(K)= C V I S(K) /lOO. 7 72 CCP ( K 1 = . 56 5 + ( .12 8 57E-02 1 * T ( K. 1  200 773 CMK) = ( . 736E-C3)+(.64E-07 >*T(K) 774 GO TO 2000 775 85 CONTINUE 776 C DE N < K) = 1. . 149 I- ( 0. 4 13 89 E-03 ) *T < K I 777 1+(-0 . 29206E-05)*T(K >**2+( .12107E-07)* T(K)* * 3 778 CVIS(K)= EXP(3.16 35+(-0.5 60 72E-01)*T(K) 779 1 + ( 0 . 3 8 0 4 4 E - C 3 ) * T ( K ) * * 2+ ( — 0. 12 4 7 0 E - 0 5 ) * T ( K ) * * 3 ) 780 CVIS(K)=CVIS(K)/100. 78 1 CCP(K)=.711 + I .914F-03)* T(K) 782 CK IK ) = ( .92E-0 31 +( .1 UE-C5)*T( K> 783 GO TO 2 000 734 86 CCNTI i'J OF 785 CDEN (K ) = 1. . 19 37+ ( 0 .5 254 7E-03 )*T( K ) 786 1+ (0.11921E-05 )*T ( K 1**2+ CO.2 7940E-03 )*T{K)**3 787 CVIS(K)=EXP(4.6975+(-C.73161E-01)*T(K) 788 l + (0 .5114 7E-03 >*T(K)**2+1-0 .16962E-05)*T ( K1**3) 789 CVIS(K )=CVIS<K)/l CO. 790 CCP(K)=.618 + (.11.6E-02)*T(K ) 791 CK(K) = 1 .82E-03) + <.3 8E-06)*T (K ) 792 GO TO 2000 793 87 CONT INUE 794 CDF NIK)=1.2065+1 — C. 51498E-03)*T IK ) 795 I + (- 0 . 1 7 8 8 IC - 0 5) * T I K ) * * 2+ I 0 . 74 506E-08)*T(K)=*3 796 CVIS(K)=EXP(5 .2020 + 1-0.79 329E-01 )~TIK ) 797 1 + t . 557 96E-0 3)* 11K)**2+(-0.184 63 E-05)* T(K)**3) 798 CVISIK)= CV ISIK)/100. 799 CCP(K)=.S9 7+(.120E-C2)*T(K) 800 CK(K)=(.79E-03)+(.3 0E-06)*T(K) 801 GO TO 2C00 802 88 CONTINUE 803 39 CONTINUE 804 90 CONTINUE 805 2000 CONTINUE 806 RETURN 80 7 END 8C8 5. UP. ROUT I ME DENVAP(K KK,1 ,VAPOEN.PI ,T ) 809 DIM ENS I ON VAPOENI 50),T( 50) 810 GO TO (41 ,42.43),KKK 811 4 1 VAPDEMI)=(PI*68.C7)/((62300.0*(T(I)+27 3.077)+ 8 12 l(-2 79.0-22.6*EXP(550.0/(T(I)+273. 0)))*PD) 813 GO TO 44 8 14 42 VAPDEN( I )=-.C004 79+.000 12 3 3*T( I ) 815 GO TO i4 816 43 VAPDEN(I>=(PI*70.131/1(62300.0*(T(I)+273.0))+ 817 1<-192.0-59.59*EXP(8 00.0/(T(I)+273.0)))*PI) 818 44 C C N TINUE 8 19 RETURN 820 END 32 1 SUBkUilT I NE LAMIKKK, I.L AMM DA,T ) 822 DIMENSION T(50) 823 REAL L AMM DA 824 GO TO (110,111.1121,KKK 825 1 10 L AMM DA = ( 1 3 0 . 2 0 5 8 3 - 0 . 0 2 f 6 7 * ( T 1 ! 1+273.0) -0.0002-8* (1(1 1+273 .0 )**2 ) 826 GO T;l 113 8 27 111 L Ar- -1CA = 97 .3-.5*T I I ) 328 GO TO 113 829 L 12 I AMMi)A= I 1 .9364/20 63 .8 >* ( ( T( I ) + 273. 0)**2 ) 830 113 CON T 1 NIJL 831 RE HJ:<r. 832 END 201 833 SUBROUTINE C.PP('KKK,AT,T,CP,OELH,I ) 834 D I MENS ION ATI 50),T( 50) 835 GO TO ( 555 ,556,557 ) ,KKK 8 36 555 IF( I .E0 . 1 ) GO TO 553 83 7 CONTINUE 8 38 ZL=(-1 . 382*2.0-.0 3393*1 AT < I ) + AT < I - 1 ) ) ) /2 .0 839 CP=I35.17*2.0 840 1 + (-0 .I486 ) * (T( I ) +T( I- 1 1 ) 84 1 2+(0.56 95E-03)*(T<I)**2 + T(1-11**2) 842 3 + ( — 0. 5 3 22E- 06 ) * ( T ( I ) * * 3 + T ( I-1)* *3) ) /2 . 0 843 CP=CP+ZL 844 OELH = C P*(T( I )-T <1-1 ) )/6 8.07 845 GO TO 559 846 556 I F ( I . E 0 . i ) GO TO 55 8 847 CONTINUE 840 CP=(EXP(3.2706*2.0 84 9 l+(0.33216E-02)*(T(I)+T(I-1)) 850 2+ (- 0. 1 1.027E-04) *( T(I) **2+ T( I -1 I **2 ) 85 1 3 + ( 0 . 4 7 9 6 3 F - 0 7 ) * ( T ( I 1 * * 3 + T ( I - 1 ) * * 3 ) ) ) / 2 . C 852 0ELH=CP*(T( Il-T( I-I ) J/72. 15 „. 853 GO TO 5 59 854 557 IF( I .E«.l) GO TO 558 655 CONTIMUF 856 CP=(EXP(2.6 79 9*2.0 857 1+(0 .44 3 74E-02)*(T( I )+T ( [-1 ) ) 858 2+(-0. 453C0E-O5)*(T(I)**2 + T(I-1)**2 ) - _ 859 3 + ( - 0.74 506E-C3) *( T( I ) * * 3 + T ( I — 1 ) * * 3 ) ) ) /2 . 0 860 DELH=CP*(T( I )-T ( 1-1 ) )/70. 13 86 1 GO TO 559 862 558 DEIH=0.0 86 3 559 CONTINUE 864 RETURN 865 END 366 Sl.JBFnUTI.Nf TEMP ( I, NCAT A, KKK, T ,P I , HHG, C D E N,H T W C, H W,867 IPO,P W,P SUM,T T,VB UBA,P A,0VA,DVB , T A) 868 0 IMENS I ON T(50),CDENI50),TT(50) ,HTWC(50),DVA(50) ,DVB(50), 869 1 V liU F- ( 5 0 ) 870 DO 21 1=1,NOATA 871 I F(CVA( I ) . EO . .000 ) GO TO 56 872 IF( 1-1 )22 ,22 ,23 873 22 GO TO (11,12,13),KKK 874 1 1 T(1 ) = 32 .0 675 GO TO 24 876 12 T(l)=28.0 877 GO TO 24 • . 878 1 3 T(l)=50.0 8 79 GO TO 24 880 23 T(I ) = T ( 1-1 ) 68 1 WKl TE ( 2 ,902 0) I , T (I) , I I 882 902 0 FORMAT (I 3,F 10.4, I 10) 883 24 CONTINUE 864 P I=T-!HG+COEN( I l*HT,-iC (I ) * 1 . 867 885 PA 1= HHG + CDEN( 1)*HW*1.867 6 66 N2=!000 687 DO 2'5 Ii = l,N2 8 88 689 CALL PA[R(V BUEA, PA I,PI,PW,P D,CVA, CVB,TA,T.KKK,I, VBUB.PA.MMM) 890 89 1 892 CONT I M.J E 202 893 CALL PPRESS(PD,PW,I ,T,KKK,MMN) 894 PSUM=PD+PVi+PA 895 2014 FORMAT (6(IF 10 .4,5X ) ) 896 IF ( ABS( PSUM-PI )-2. 0)29,29,30 897 30 IF(PSUM-PI )3 1,3 1,3 2 898 32 T( I )=T ( I )-(PSUM/P I ) *T ( I 1*0.005 899 GO TO 3 3 9 00 3 1 T ( I ) =T( I)+( P I/PSOM) *T ( I )*0.00 5 901 33 IF(I I-N2)28 ,35 ,36 902 35 TT ( 1 )= T( I ) 903 GO TO 28 904 36 TT(2)=T(I) 905 28 CON TINUE 9 06 1 ( I ) = ITT(l)+TTI2) 1*0.5 907 GO TO 21 908 29 CONTINUE 909 T ( I ) = T ( I > 9 10 21 CONTINUE 911 56 CONTINUE 9 12 RETURN 913 END 9 14 SUB ROUTINE PPRESS(P D,PW, I ,T,KKK,MMM) 915 DI MENSICN T(50) 9 16 Gu TO (25,26,27),KKK 917 25 PD=EXP( 2.303*1 6.575.23-1060. 85 1/1 T( I 1+227.74))) 9 18 GO TO 105 919 26 _ . PO=EXP(2.3O3*(7.3 7-1351.3 5/(T(I)+273.0))) 920 GO TO 105 921 27 PO = EXP(2. 303*(6. 87793-1 119.2C8/(2 30 .738+T( I ) ))) 922 105 GO TO ( 20 1, 20 2, 203, 2C4.2C5, 206,207) ,MMM 92 3 201 PW-EX P(2.30 3*(9.7 54-2 500 .0/(T( I)+2 7 3.0) )) 924 GO TO 20 8 925 202 GO TO 2C6 926 203 GO TO 208 927 204 GO TO 2C8 923 205 PX=EXP(18.2 39-39 59.8/1T(I)+233.41)) 9 29 GO TO 2 08 930 2C6 PW=EXP(17.C55-3331.9/(T(I1+203.77)) 931. GO TO 208 932 2 07 PH=EXP t17.677-3 739.0/ (T (I)+219 .72 ) ) 933 208 CCNTINUE 934 R ET U R N 935 END 9 36 SUBROUTINE PAIR(V3U B A,PAI ,PI ,PW,PD,DVA,EyB,TA,T,KKK,I,V8UB,PA,MM! 937 DIMENSION T (50),CVA(50),DVB(50 ),VBUBI50) 53 8 GO TO (25 ,26 ,27) ,KKK 939 25 PD=EXP(2.30 3*(6.975 23-106C.851/1TA+227.74))) 9<^ 0 GO TO 105 94 1 26 PD=EXP(2.30 3*(7.3 7-1351.35/(TA+273.0))) 942 GO TO 105 943 2 7 PD=EXP(2.30 3*(6.8 7798-1119.20 8/(TA+230.738))) 944 105 GO TO (201,202,203.2C4,2C5,206,2C7),MMM 945 201 PW=EXP( 2.30 3*(9. 7 54-2 500.0/(TA + 27 3.0) ) ) 946 GO TO 2C8 9 47 202 GO TO 2 OF: 948 203 GC TO 208 949 204 GU TO 2C3 950 205 PW=EXP( 18 .2 39-35 59.H/(T A + 2 3 3. 4 1) ) 95 1 GO TO.2 08 952 206 PW=fX»( 17. 0 5 5-33 * t. 9/(TA + 203. 77) ) 203 953 GO TO 208 954 207 PW=EXP<17.6 77-3 739.0/(TA + 219.72)) 955 208 CONTINUE 956 957 VBUB ( I )=DV'A ( I ) *OVA( I ) *OVB ( I ) *0 .5238 958 IF( (VBUB(Il-VBUBA).LT.0.00 ) VBURI I 1 =VBUBA 959 PB=PAI-PC-PW 960 PA = P8*VB1JBA*( Tt I ) +273.16) / (VBUB (I ) * (TA + 273.16 ) ) . . 961 RETURN . „'.._.. _ 96 2 END 963 FUNCTION AUB(PiOiX.L) 964 0 I MENS ION P(5C) ,C(50) 965 c (i) = i . o 966 AUB=P(1) 967 DO 10 J = 2,6 . _ ... 968 D(J (=D(J-1)*X 969 10 AUB=AUB+P(J)*D(J) 970 RETURN 971 END 972 FUNCTION AUC(P.O,X,L) 973 DIMENS ICN P (50),C(50) 974 D1 1) = ) . 0 975 D(2)=X 9 76 AUC=0(1)*P(1)+D(2)*P(2) 977 RETURN 978 END END OF FILE $SIG SHORT 204 APPENDIX V Computer Output and Sample C a l c u l a t i o n As mentioned i n the section "Interpretation of Results -Introduction" the concept of the instantaneous heat transfer c o e f f i c i e n t was not abandoned u n t i l near the end of the work, when i t was apparent that the i n t e r f a c i a l area, between the dispersed phase l i q u i d and the continuous phase l i q u i d , could not be found. However, before t h i s had been determined a computer program had been written to c a l c u l a t e , on an instantaneous b a s i s , some of the important parameters i n the systems. Since only a small proportion of the computer output i s used for the c a l c u l a t i o n of the average heat transfer c o e f f i c i e n t the output for the e n t i r e sets of runs have not been included i n t h i s t h e s i s ; however, i t i s on f i l e with Dr. K.L. Pinder at the Un i v e r s i t y of B r i t i s h Columbia, Vancouver, B r i t i s h Columbia, Canada. A d e s c r i p t i o n of the output from the computer program i s given below along with an example of the output f o r Run 1308. Computer! Output. The following d e s c r i p t i o n gives a l i s t of the output headings with t h e i r meanings and d e r i v a t i o n . LOWER TEMPERATURE This i s the temperature i n the lower portion of the column, based on the reading of the one thermocouple. 205 HEIGHT OF WATER IN.COLUMN This i s the height of water i n the dilatometer tube when the volume reading on the recorder i s zero. ATMOSPHERIC PRESSURE This i s the barometric pressure at the time the run was made. VOLUME OF AIR BUBBLE This i s the actual volume of the a i r bubble i n the lower portion of the column. I t i s calculated from the dimensions of the a i r bubble, taken from the cine photograph, by assuming the a i r bubble i s i n the form of an e l l i p s o i d , INITIAL DIAMETER OF DROP This i s the diameter of the l i q u i d drop measured i n the lower portion of the column. WEIGHT OF DROP This i s the actual weight of the dispersed phase l i q u i d i n i t i a l l y i n the drop i n the lower portion of the column. I t i s the t o t a l mass of the drop neglecting the weight of the a i r and the vapour i n the a i r bubble. I t i s calculated from the i n i t i a l drop diameter and the volume of the a i r bubble taken from cine photographs. WEIGHT OF VAPOUR BUBBLE This i s the weight of dispersed phase calculated from the bubble volume, determined by the dilatometer, at the end of evaporation. TOTAL EVAPORATION TIME This i s the time at which a l l the l i q u i d i n the bubble has evaporated. The f i n a l time i s evaluated from the volume taken from the dilatometer chart at the point where the slope changes. VOLUME This i s the equation of the curve used to f i t the dilatometer volume versus time curve. TIME This i s the time i n seconds of the bubble r i s i n g i n the column. The i n d i v i d u a l times at which the various properties are c a l -culated d i f f e r between the photographs and the dilatometer, however they are both on the same time b a s i s . HEIGHT This i s the curve f i t t e d height as a function of time. By the equation , 2 3 h = t + t + t The same equation but with d i f f e r e n t times substituted i n t o i t i s used i n both the photographic and dilatometric c a l c u l a t i o n s . VEL The v e l o c i t y i s c a l c u l a t e d by d i f f e r e n t i a t i n g the above equation and evaluating i t at various times. The same equation i s used for both the photographic and the di l a t o m e t r i c data. ACC The ac c e l e r a t i o n was calculated by curve f i t t i n g the v e l o c i t y data with time by the equation 3 v = O"Q + t + t + a^ t and then taking the d e r i v a t i v e with respect to time and evaluating i t f o r ei t h e r the photographic or the di l a t o m e t r i c times. DIA The diameter i s calculated from the curve f i t t e d t o t a l volume of the bubble at any in s t a n t . For the photographic portion i t i s based on the photographs.; while f o r the dilatometric region i t i s based on the dilatometer volume taken from the recorder chart. AREA This i s the surface area of the bubble calculated from the above diameter. VOLUME This i s the volume of the drop or bubble a f t e r i t has been curve f i t t e d . In one case i t has been determined by measurements of the photographs and i n the other from the dilatometer r e s u l t s . PERCENT EVAP This i s the amount of evaporation which has taken place up to the present time and calculated by the following formula: 208 „ mass of dispersed phase vapour ., ~_ A = , .-, • .... .r, . . . . jr—, • • • J . — X 100 t o t a l mass of dispersed phase In the photographic case the t o t a l mass i s calculated from the photograph of the l i q u i d drop i n the lower portion of the column. In the dilatometer case i t i s determined from the volume measurement of the bubble at the end of evaporation. DELT This i s the temperature d i f f e r e n c e between the continuous phase temperature and the temperature of the vapour bubble. The temper-ature of the vapour i s calculated by assuming the t o t a l pressure i s equal to the aim of the p a r t i a l pressures of the a i r , continuous phase, and dispersed phase vapour. Both the continuous phase temperature and the vapour temperature were curve f i t t e d as functions of height. UA This i s the heat transfer c o e f f i c i e n t times the instantaneous area calculated from the following formula: q = U A AT or with q calculated from U A - -1 AT PERCENT EVAP This i s simply a repeat of the PERCENT EVAP given above. REYNOLDS NO This i s calculated from the formula: Re -y d - i s the diameter of the bubble v - i s the v e l o c i t y of the bubble p density of the continuous phase y - v i s c o s i t y of the.continuous phase PRANDTL NO This i s calculated from the formula; y - v i s c o s i t y of the continuous phase Cp - heat capacity of the continuous phase k - thermal conductivity of the continuous phas PECLET. NO This i s calculated from the above groups by: Pe = (Pr) (Re) DRAG COEFF This i s calculated from the force balance: 8 C D ma = Vpg - mg ^ j n d pv therefore _ Vpg - m (a + g) ~ 2 2 (n d z P v ) 8 m - mass of the dispersed phase a - acc e l e r a t i o n V - volume of the bubble p - density of the continuous phase g - ac c e l e r a t i o n of g r a v i t y d - diameter of bubble v - v e l o c i t y of bubble Cp - drag c o e f f i c i e n t (100 - %') EVAP This i s simply the r a t i o of the mass of the dispersed phase l i q u i d l e f t i n the drop at any time divided by the t o t a l mass of the drop calculated as mentioned i n the PERCENT EVAP above. LIQUID VOLUME This i s the volume of the dispersed phase l i q u i d l e f t i n the drop at any time. The c a l c u l a t i o n of the average rate of heat transfer makes use of selected values found i n the computer output. The following sample c a l c u l a t i o n i s done for Run 1308. The computer output i s given i n Figure (45) and t h i s should be r e f e r r e d to where appropriate. 211 3 : 3 o o U J o • U a — o 3 0 i/ i o 3 o o o x r\i = 0 o - n o —« •4-• • » -x) •4" • — i o -o —* ^ e g J\ .-o O 3 • -sT •O 3 3 O 2) o 3 3 3 3 II It a . 11 3 3 U J l l _) o II • 3 *3 DJ "3 J _ 3 II 3 ^ L U \S\ S3 . V •A 3 CC 3 •X Xi i l l II 3 —* ±> a: t ~ X O a . •Ll l^i C L < or. < — i . CC .1 L - 1 <; — > L ) a; a •— a 3 . I L CU LL C J L L U i CJ U- 1 c C J C ! > t— L U 1 >— CL L L ' < 1 t»— & X V . y •— X J L _ J U J e> c ¥~ 0 (-.) <l J C — 2. •—• •~ K-r> C a . a U J a * J > J t 7 N O -O M O iJ 1 r\j O i/> OJ ~f CT- ^ • O P- 3 •£> O rO f*\ .p, 1-0 ro O O ~* - • ~* -n o o o 3 O o 3 3 3 O o M * i ' O o -~ O O O O : 3 O 3 O O o O O O O j j O s ? o o 3 3 3 5 M O " 1 •JO O B m fsi ,-sj r \ i 1 M : \ j N ; M - \ i O O 3 o O O* sO "O . M ^ T 3 1> O 1 •O P» C7« O 3 3 3 ) s N <T -C ^ ~o -~o rvj .-%] i> - x C" 0 - f O > 4 -O P - O -O -1 rsi r-o ••o u-s - 0 o o — . 3 ->J T ; > O O 3 —4 .-\J — r", 1 <I N J |fl Ip. o c ^-c c_- o c -n r - o O O O X i <N o o o o Lf\ *1 O O 3 O O 3 - * O 3 3 O —' ~M O O O 3 O 3 o a 3 o 0 o 4 - O 3 O O O O 3 O O O 3 3 "O O C 3 O O O 3 O O 3 N - T J C* 1 C ^ —. -r» ^ cr- r*-N -4 M N ? J N 3 « ^ 3 3 O • -O -O n - J " 1 —1 —* r\) M X X." ^ v.- c:^  C ^ i f i C ^" O ' IT-^* *£• •£> O U"V P^ -„- cu 7« f <.) c c o a — — — o o o o o c O 3 O 3 O O 3 O 3 3 O 3 O o 3 3 3 O 3 O O O O O O O 3 O O 3 3 O O O O O 3 3 O O 3 O O 3 3 3 3 O O 3 O 3 3 O • J - 7* rsj *n O uf\ sT o - r .-o -6 r\j O —* -7 s LT. —1 -NJ - J - r o iT> 7D ^ •> 3 ? - P * S . ' d M r f l C -^ -I 3 - \ X 0 " -J —' o -o "O . 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T t N E ( S f C ) l i t I G t i l t i n ) V C - L I L M / S E C I A C C ( C M / i t C 2 ) U l A t C M ) A K E A t S U C M ) v o L u i i e i c j C M ) t > E K C E W l L V A f D H L T ( D E G C ) / c . S ' . r c 5 . 9 U 9 5 1 9 . 5 3 5 5 6 . 1 8 B 2 0 . 4 4 6 1 0 . 6 3 0 7 0 . 0 4 7 1 - J . 5 u 5 2 4 . 3 9 4 7 < 1 .2500 . 2 1 . 3 0 9 4 5 . 2 5 6 4 0 . 6 9 4 3 1. 5 1 4 6 0 . 1 / 5 2 13. 1 9 / 2 4 . 3 2 6 2 1. ' incn - 9 . / u 6 0 - 2 2 . 0 o 4 0 4 . d 0 5 5 0 . e i 2 1 7 2. 1212 0 . 2 9 0 5 21 .0245 4 . i 5 79 1 . 5 4 0 0 1 1 . 0 0 0 1 2 2 . / 0 / 3 4 . 3 d 4 6 0 . 9 3 7 1 2 . 7 5 a a 0 . 4 3 0 9 3 2 . 3 0 s 3 4 . 4 0 8 6 1 . 6 6 0 0 1 2 . 3 0 5 0 2 3 . 2 1 1 8 4 . 0 2 3 9 1 . 0 2 9 3 3 . 3 2 3 3 0 . 5 7 1 0 4 2 . 7 4 2 3 4 . 4 6 3 2 1 . 7 7 C C 1 . 1 . 0 v : / 5 2 3 . 6 3 u 3 3 . 6 9 3 3 1 . 1 0 6 0 3 . 8 4 2 9 0 . 7 0 3 4 5 2 . 9 4 ^ 3 4 . 5 1 9 / 1 . ' 3 0 0 1 4 , u 0 o 2 2 4 . I d S / 3 . 2 1 2 3 1 . 2 0 1 d 4 . 5 3 / 7 0 . 9 0 3 9 6 7 . 7 4 1 2 4 . 6 0 9 2 2.;??oo 1 / . 5 1 •. 7 2 5 . 0 1 / 2 2 . 3 1 0 6 1 . 3 3 0 7 5 . 5 6 3 4 1 . 2 3 4 0 9 1 . o 2 5 4 4 . 7 9 0 2 ? . 3 ' C O 2 5 . ) 2 9 9 1 . 1 5 9 / 1 . 3 / 4 1 5 . 9 3 1 5 I. 3 5 a 5 1 0 0 . 6 2 u 8 4 . 8 8 3 5 4 . 7 3 0 0 4 t . u ii a 2 1 . 4 0 0 1 - 5 . 2 0 4 2 1 . 4 0 4 3 6 . 1 9 5 0 1 . 4 5 0 0 1 0 2 . 7 0 5 9 6 . 1 5 1 8 U A t C A I . / S F C D E C C 1 P f c l t C E U T E V A P K f c V N J L U S N U 2 K A ; J . 0 T L N O P E C L E T N O U K A C i C O E F F 1 1 0 0 - * ) t V A P U v J U I U V J L U M E 0 . 0 1 * 0 3 . 5 6 D2 1 6 7 0 . u o 5 b 3 . 3 6 5 2 3 9 9 0 . 2 6 9 0 1 . 4 1 3 5 9 6 . * 3 4 U 0 . 0 0 5 2 7 7 C . 0 4 0 7 1 3 . 1 9 / 2 2 J 2 J . 2 4 : 1 7 3 . 3 6 6 4 6 ( i d t . 3(12.3 i . 9 5 7 1 B 6 . 8 0 2 0 O . O C 4 / J 0 0 . 0 5 6 2 2. L . ; i 2 s 5 i 5 a. u •) <., 5 3 - . 3 6 / O « 5 3 5 . 0 5 4 / 2 . 1 / 9 1 7 a . 1 / ->5 0 . 0 0 4 2 la 0 . 0 6 3 3 3 2 . 3 0 3 3 4 0 5 9 . 0 9 / 7 3 . 3 6 7 6 1 0 5 7 / . 1 5 2 3 2 . 3 5 6 3 6 7 . 6 9 1 7 0 . 0 0 3 7 0 4 0 . 0 7 4 6 4 2 . / 4 > ) 4 5 5 6 . DO 72 3 . 3 6 8 2 1 2 6 0 / . 5 1 1 / 2 . 4 8 2 2 5 7 . 2 5 7 / 0 . 0 0 3 i 3 3 0 . 0 7 6 > 5 2 . 9 4 9 J 4'J<1 •>. 3 9 0 6 3 . J ( . » i 1 3 9 / 0 . 3 / 5 0 2 . 5 7 5 6 4 7 . 0 5 0 / U . 0 0 2 5 / 5 0 . 0 7 1 8 o / . f 9 1 2 5 5 4 2 . 9 o 0 5 3 . j 6 9 3 I 6 o 3 9 . 9 l d 0 2 . b / 5 6 3 2 . <:U68 0 . 0 0 1 7 6 3 0 . 0 5 1 7 9 1 . 6 2 1 4 6 3 4 5 . b l 7 2 3 . 3 707 1 9 a 3 B . 3 0 4 7 2 . 7 7 2 7 3 . 3 7 4 6 U . O 0 0 4 5 B 0 . 0 4 0 9 1 0 0 . t>2.la U O 3 3 . 0 2 3 4 3 . 3 / 1 4 2 1«. / 9 . 7 1 0 ' ) 2 . / 9 35 - 0 . 6 2 ( 1 3 - O . u 0 0 0 3 4 0 . 0 0 I 3 1 0 2 . / 0 5 9 5 / 1 J • o O v l u 3 . 3 3 0 6 . 2 2 7 0 5 . 0 0 7 B 4 . 0 0 0 4 - 2 . / 0 ' 5 9 - 0 . 0 0 0 1 4 U Figure 45 ( continued ) - Output Results from Computer Program for Run 1308 Sample Calculation Run 1308 EXTERNAL PRESSURE = Atmospheric Pressure + Hydrostatic Pressure A. Atmospheric Pressure = 767.9 mm Hg B. Hydrostatic Pressure = (Height of continuous phase above bubble)(Density of con-tinuous phase)(Acceleration of gravity) 1. Height of continuous phase above bubble = Height of level in dilatometer tube - Height of bubble a. Height of level in dilatometer tube = height of water in dilatometer tube at zero + increase in height due to change in volume of bubble i . height of water in dilatometer tube at zero =56.0 in. i i . increase in height due to change in volume of bubble = (volume of bubble)/(area of tube) = (1.358 cm3)/(0.786 cm2)(2.54 cm/in.) = 0.68 in. Height of level in dilatometer tube = 56.0 + 0.68 = 56.68 in. 214 b. Height of bubble =19.0 in. Height of continuous phase above bubble = 56.68 - 19.00 = 37.68 in. 2. Density of continuous phase - temperature of the continuous phase = 53.5°C (obtained from Table I-AVIII) - equation for density = 1.001 + 0.35286(10"4)t - 0.63181(10 _ 5)t 2 (1-AIII) + 0.18859(l0" 7)t 3 Density of continuous phase = 0.9877 gm/cm 3. Acceleration of gravity = 980 cm/sec2 Hydrostatic pressure = (37.68 in.)(0.9877 gm/cm3)(980 cm/sec2)(2.54 cm/in-). *~ A 2 2 = (7.5(10 ) mm Hg cm /dyne)(1 dyne sec /(gm)(cm)) = (37.68)(0.9877)(1.867) = 69.48 mm Hg Therefore the external pressure at the end of evaporation = 767.9 + 69.48 = 837.38 mm Hg Likewise the external pressure in the lower portion of the column = 767.9 + (56)(0.9877)(1.867) = 767.9 + 103.3 = 871.2 mm Hg 215 II. TEMPERATURE OF VAPOUR BUBBLE - assume t = 48.5 °C A. Corrected Volume of Air 3 - volume of air at 871.2 mm Hg and 21.9 °C = 0.000332 cm - corrected volume of air at 834.6 mm Hg and 48.5 °C = (0.000332)(871.2/834.9)(273.16 + 48.5)(273.16 + 21.9)) = 0.000378 cm3 B. Partial Pressures 1. Partial pressure of air = (0.000378)(834.9)/l.359 = 0.232 mm Hg 2. Partial pressure of continuous phase - temperature = 48.5 °C - log p c = 9.754 - 2500/(t + 273.0) (21-AIII) = 95.05 mm Hg 3. Partial pressure of dispersed phase - temperature = 48.5 °C - log p d = 6.87798 - 1119.208/(230.738 + t) (4-AII) = 741.14 mm Hg C. Total Internal Pressure = 0.232 + 95.05 + 741.14 = 836.42 mm Hg D. Difference Between Internal and External Pressure = P - P external internal = 837.38 - 836.42 = 0.96 Since the absolute value of this i s less than 2.0 mm Hg the assumed temperature of 48.5 °C is accepted. 216 III. MASS OF THE DISPERSED PHASE A. Vapour Density of Dispersed Phase - temperature = 48.5 °C and pressure = 836.42 mm Hg - p - ( P ) ( 7 ° - 1 3 ) (2-AII) ° {62300T + [-192.0 - 59.59 exp(800/T)] P} = 0.00304 gm/cm3 B. Mass of Dispersed Phase based on Curve Fitted Volume = (vapour density) (volume) (P<i/ Pi nternal^ = (0.00304)(1.358)(741.14/836.42) = 0.00365 gm C Percent Evaporation Based on Measured Dilatometer Volume = (0.00365)(100)/0.00363 = 100.6% IV. AVERAGE RATE OF HEAT TRANSFER - start: time = 0.94 sec. percent evaporation = 3.57% - fi n i s h : time = 2.38 sec. percent evaporation = 100.63% A. Total Evaporation Time =2.38-0.94 = 1.44 sec B. Total Heat Transferred = (mass evaporated)(latent heat of vaporization) 1. Mass evaporated = (mass of dispersed phase)(difference in percent evapora-tion) = (0.00365)(100.63 - 3.57) = 0.00354 gm 217 2. Latent heat transferred = 100.0 cal/gm Tot a l Heat Transfer = (0.00354)(100.0) = 0.354 c a l Average Rate of Heat Transfer = 0.354/1.44 = 0.246 cal/sec V. INITIAL AREA - mass of the dispersed phase = 0.00365 gm - density of dispersed phase i n lower section of column 3 = 0.744 gm/cm A. Volume of Dispersed Phase Liq u i d = (mass of dispersed phase)/(density) = (0.00365)/(0.744) = 0.00491 cm3 B. T o t a l Volume of Drop = (volume of a i r i n lower section of the column) +.(volume of dispersed phase l i q u i d ) = (0.000332) + 0.00491 = 0.00524 cm3 C. Equivalent Spherical Diameter = (( 6 ) ( v o l u m e ) / n ) 1 / 3 = ((6)(0.00524)/n) 1 / 3 = 0.215 cm 218 I n i t i a l Area 2 = II (diameter) = n (0.215)2 = 0.146 cm2 VI. TEMPERATURE DRIVING FORCE - continuous phase temperature = 53.5 °C - vapour bubble temperature = 48.5 °C Temperature driving force = 53.5 - 48.5 =5.0 5C (the computer program gives 4.88 °C) VII. AVERAGE HEAT TRANSFER COEFFICIENT = (average rate of heat tr a n s f e r ) / ( ( i n i t i a l area)(temperature driving force) = (0.246 cal/sec)/((0.146 cm2)(4.88 °C)) 2 = 0.345 cal/sec cm °C Note: The temperature of the continuous phase i s used i n the computer program as a function of height in the column. For this example an average value of the temperature for the set of runs i s calculated and may be found in Table I-AVIII, Appendix VIII. APPENDIX VI Cal c u l a t i o n of I n i t i a l Ar TABLE I-AVI DISPERSED PHASE = CYCLOPENTANE PERCENT GLYCERINE = 0.0% DENSITY OF DISPERSED PHASE = 3 0.744 gm/cm RUN MASS OF EQUIVALENT VOLUME OF TOTAL EQUIVALENT NUMBER VAPOUR LIQUID VOLUME AIR VOLUME SPHERICAL AREA DIAMETER gm 3 cm 3 cm 3 " cm cm 2 cm 1301 0.007365 0.009899 0.000083 0.009982 0.267 0.224 1302 0.005686 0.007643 0.000344 0.007987 0.248 0.193 1303 0.008327 0.011192 0.000855 0.012047 0.284 0.254 1304 0.004336 0.005828 0.000078 0.005906 0.224 0.158 1305 0.003497 0.004700 0.000441 0.004744 0.208 0.137 1306 0.005376 0.007226 0.000556 0.007782 0.246 0.190 1307 0.004252 0.005715 0.000080 0.005795 0.223 0.156 1308 0.003655 0.004913 0.000332 0.005245 0.216 0.146 1309 0.004259 0.005724 0.000441 0.006166 0.228 0.163 1310 - - - - - -1311 0.003539 0.004757 0.000631 0.005388 0.218 0.149 1312 0.005799 0.007794 0.000855 0.008650 0.255 0.204 N3 O TABLE II-AVI DISPERSED PHASE = ISOPENTANE PERCENT GLYCERINE = 0.0% DENSITY OF DISPERSED PHASE =0.628 gm/cm3 . RUN MASS OF EQUIVALENT VOLUME OF TOTAL EQUIVALENT NUMBER VAPOUR LIQUID VOLUME AIR VOLUME SPHERICAL DIAMETER AREA gm 3 cm 3 cm 3 cm cm 2 cm 1401 0,003865 0.006154 0.000556 0.006710 0.234 0.172 1402 0.003446 0.005487 0.000795 0.006282 0.229 0.165 1403 0.003270 0.005207 0.000119 0.005326 0.217 0.147 1404 0.003608 0.005745 0.000951 0.006696 0.234 0.172 1405 0.004700 0.007484 0.000795 0.008279 0.251 0.198 1406 0.004617 0.007352 0.000113 0.007465 0.242 0.185 1407 0.005019 0.007992 0.000034 0.008026 0.248 0.194 1408 0.003595 0.005725 0.000905 0.006629 0.233 0.171 1409 0.004510 0.007182 0.000113 0.007295 0.241 0.182 1410 0.005450 0.008678 0.000034 0.Q08712 0.255 0.205 1411 0.007391 0.011769 0.000113 0,011882 0.283 0.252 1412 0.002261 0.003600 0.000268 0.003868 0.195 0.119 TABLE III-AVI DISPERSED PHASE = CYCLOPENTANE PERCENT GLYCERTINE = 77.06% DENSITY OF DISPERSED PHASE = 0.744 gm/cirr* RUN MASS OF EQUIVALENT VOLUME OF TOTAL EQUIVALENT NUMBER VAPOUR LIQUID VOLUME AIR VOLUME SPHERICAL DIAMETER AREA gm 3 cm 3 cm 3 cm cm 2 cm 1601 0.006488 0.008720 0.000188 0.008908 0.257 0.208 1602 0.003734 0.005019 0.000071 0.005090 0.213 0.143 1603 0.006016 0.008086 0.000188 0.008274 0.251 0.198 1604 0.002734 0.003675 0.000108 0.003782 0.193 0.117 1605 - - - - - -1606 0.006614 0.008890 0.000071 0.008961 0.258 0.209 1607 0.005574 0.007492 0.000108 0.007503 0.243 0.185 1608 0.005670 0.007621 0.000071 0.007692 0.245 0.188 1609 0.001790 0.002406 0.000188 0.002594 0.170 0.091 1610 0.005783 0.007773 0.000071 0.007844 0.247 0.191 1611 0.005315 0.007144 0.000071 0.007215 0.240 0.181 1612 0.004876 0.006554 0.000013 0.006567 0.232 0.170 N3 TABLE IV-AVI DISPERSED PHASE = CYCLOPENTANE PERCENT GLYCERINE = 73.07% DENSITY OF DISPERSED PHASE = 0.744 gm/cm RUN MASS OF EQUIVALENT VOLUME OF TOTAL EQUIVALENT NUMBER VAPOUR LIQUID VOLUME AIR VOLUME SPHERICAL AREA DIAMETER gm cm cm cm cm cm 1701 0.006300 0.008468 0.000031 0.008499 0.253 0.201 1702 0.006190 0.008320 0.000004 0.008324 0.251 0.199 1703 - - - - - -1704 0.006208 0.008344 0.000031 0.008375 0.252 0.199 1705 0.006046 0.008126 0.000031 0.008157 0.250 0.196 1706 0.001681 0.002259 0.000031 0.002291 0.164 0.084 1707 0.006342 o.008524 0.000031 0.008555 0.254 0.202 1708 0.006491 0.008725 0.000071 0.008796 0.256 0.206 1709 0.006469 0.008696 0.000071 0.008766 0.256 0.206 1710 0.005709 0.007673 0.000031 0.007704 0.245 0.189 1711 0.006347 0.008531 0.000031 0.008562 0.254 0.202 1712 0.006201 0.008335 0.000071 0.008406 0.252 0.200 NS U> TABLE V-AVI DISPERSED PHASE = ISOPENTANE PERCENT GLYCERINE =73.07% DENSITY OF DISPERSED PHASE = 0.628 gm/cm RUN MASS OF EQUIVALENT VOLUME OF TOTAL EQUIVALENT NUMBER VAPOUR LIQUID VOLUME AIR VOLUME SPHERICAL AREA DIAMETER  3 3 3 2 S111 cm cm cm cm cm 1801 0.010545 0.016791 0.001248 0.018040 0.325 0.333 1802 0.007489 0.011925 0.002728 0.014653 0.304 0.290 1803 0.011659 0.018566 0.001249 0.019815 0.336 0.354 1804 0.007809 0.012435 0.002329 0.014765 0.304 0.291 1805 0.008146 0.012972. 0.002329 0.015301 0.308 0.298 1806 0.010790 0.017182 0.002329 0.019511 0.334 0.350 1807 0.006943 0.011055 0.003316 0.014371 0.302 0.286 1808 - - - - - -1809 - - - - - -1810 0.009701 0.015447 0.002222 0.017669 0.323 0.328 1811 0.005204 0.008286 0.003776 0.012060 0.285 0.254 1812 _ _ _ _ N3 TABLE VI-AVI DISPERSED PHASE = CYCLOPENTANE PERCENT GLYCERINE = 56.02% DENSITY OF DISPERSED PHASE = 0.744 gm/cm3 RUN MASS OF EQUIVALENT VOLUME OF TOTAL EQUIVALENT NUMBER VAPOUR LIQUID VOLUME AIR VOLUME SPHERICAL AREA DIAMETER  _ _ 3 ^ gm cm cm cm cm cm 1901 0.007827 0.010520 0.000839 0.011359 0.279 0.244 1902 0.006710 0.009019 0.000392 0.009411 0.262 0.216 1903 0.006414 0.008621 0.000392 0.009013 0.258 0.209 1904 0.003013 0.004050 0.003776 0.007826 0.246 0.191 1905 0.006841 0.009195 0.000493 0.009687 0.264 0.220 1906 0.005313 0.007142 0.000702 0.007844 0.247 0.191 1907 - - - - - -1908 0.005239 0.007042 0.000493 0.007535 0.243 0.186 1909 0.006135 0.008246 0.00Q493 0.008738 0.256 0.205 1910 0.006916 0.009296 0.000392 0.009688 0.264 0.220 1911 0.005228 0.007027 0.000392 0.007419 0.242 0.184 1912 0.004196 0.005640 0.000702 0.006342 0.230 0.166 N3 N3 TABLE VII-AVI DISPERSED PHASE = ISOPENTANE PERCENT GLYCERINE = 56.02% DENSITY OF DISPERSED PHASE = 0.628 gm/ cm3 RUN MASS OF EQUIVALENT VOLUME OF TOTAL EQUIVALENT NUMBER VAPOUR LIQUID VOLUME AIR VOLUME SPHERICAL AREA DIAMETER gm 3 cm 3 cm 3 cm cm 2 cm 2001 0.006029 0.00960 0.000188 0.009789 0.265 0.221 2002 0.004589 0.007309 0.000702 0.008010 0.248 0.194 2003 0.005933 0.009447 0.000188 0.009635 0.264 0.219 2004 0.005080 0.008088 0.000188 0.008276 0.251 0.198 2005 0.004943 0.007872 0.000392 0.008264 0.251 0.198 2006 0.005162 0.008220 0.000188 0.008408 0.252 0.200 2007 0.005757 0.009168 0.000004 0.009172 0.250 0.212 2008 0.006214 0.009896 0.000107 0.010003 0.267 0.225 2009 0.005759 0.009171 0.000071 0.009242 0.260 0.213 2010 0.005240 0.008344 0.000016 0.008360 0.252 0.199 2011 0.005211 0.008298 0.000016 0.008314 0.251 0.198 2012 _ _ _ _ _ _ ho ho APPENDIX VII Ca l c u l a t i o n of Average Heat Transfer C o e f f i c i e n t TABLE I-AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = CYCLOPENTANE PERCENT GLYCERINE =0,0% DENSITY OF DISPERSED PHASE = 0.688 gm/cm3 HEAT OF VAPOURIZATION = 100 cal/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT gm c a l sec cal/sec cm °C cal/sec cm °C 1301 0.0069945 0.69945 1.87 0.374 0.224 5.32 0.314 1302 0.0052464 0.52464 1.83 0.287 0.193 5.15 0.288 1303 0.0074517 0.74517 1.49 0.500 0.254 5.11 0.385 1304 0.0043771 0.43771 1.60 0.274 0.158 5.12 0.338 1305 0.0036260 0.36259 1.39 0.261 0.137 5.11 0.373 1306 0.0050676 0.50676 1.68 0.302 0.190 5.24 0.303 1307 0.0040852 0.40852 1.44 0.284 0.156 4.89 0.372 1308 0.0035481 0.35481 1.44 0.246 0.146 4.88 0.346 1309 0.0039870 0.39815 1.33 0.299 0.163 4.90 0/375 1310 - - - - - - -1311 0.0036310 0.36308 1.14 0.318 0.149 4.75 0.450 1312 0.0056970 0.56968 1.77 0.322 0.204 5.28 0.299 TABLE II-AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = ISOPENTANE PERCENT GLYCERINE =0.0% 3 DENSITY OF DISPERSED PHASE =0.61 gm/cm HEAT OF VAPOURIZATION = 83.3 cal/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT 2 2 gm c a l sec cal/sec cm °C cal/sec cm °C 1401 0.0038520 0.32086 1. 30 0.247 0.172 2.56 0.561 1402 0.0030389 0.25314 1.10 0.230 0.165 2.37 0.588 1403 0.0035669 0.29712 1.29 0.230 0.147 2.58 0.607 1404 0.0030151 0.25116 0.98 0.256 0.172 2.31 0.645 1405 • 0.0047931 0.39927 1.28 0.312 0.198 2.68 0.588 1406 0.0045457 0.37866 1.18 0.321 0.185 2.50 0.694 1407 0.0049225 0.41005 1.47 0.279 0.194 2.83 0.508 1408 0.0031640 0.26357 1.09 0.242 0.171 2.59 0.546 1409 0.0041613 0.34664 1.10 0.315 0.182 2.66 0.651 1410 0.0054950 0.45770 1.46 0.313 0.205 3.05 0.501 1411 0.0074773 0.62386 1.35 0.461 0.252 2.83 0.647 1412 0.0020684 0.17230 0.87 0.198 0.119 2.32 0.717 TABLE III-AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = CYCLOPENTANE DENSITY OF DISPERSED PHASE = 0.688 gm/cm" PERCENT GLYCERINE = 77.06% HEAT OF VAPOURIZATION = 100 cal/gm RUN . MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED" TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT gm c a l sec cal/sec 2 cm °C cal/sec cm^ °C 1601 0.0066079 0.66079 1.74 0.380 0.208 6.89 0.265 1602 0.0038950 0.38955 1.73 0.225 0.143 6.81 0.231 1603 0.0061650 0.61651 1.68 0.367 0.198 7.05 0.263 1604 0.0023850 0.23847 1.20 0.199 0.117 6.88 0.247 1605 - - - - - - -1606 0.0069738 0.69738 1.86 0.375 0.209 7.09 0.253 1607 0.0056208 0.56208 1.99 0.282 0.185 7.16 0.213 1608 0.0054678 0.54678 1.68 0.325 0.188 7.10 0.244 1609 0.0016408 0.16408 1.36 0.121 0.091 6.70 0.197 1610 0.0059155 0.59155 1.51 0.392 0.191 6.86 0.299 1611 0.0053340 0.53340 1.72 0.310 0.181 7.09 0.242 1612 0.0047732 0.47732 1.79 0.267 0.170 7.03 0.223 TABLE IV-AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = CYCLOPENTANE PERCENT GLYCERINE = 73.07% DENSITY OF DISPERSED PHASE = 0.688 .gm/cm3 HEAT OF VAPOTJRIZATION = 100 cal/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT _gm c a l sec cal/sec cm2 °C cal/sec cm2 °C 1701 0.0062730 0.62733 1.69 0.371 0.201 6.77 0.273 1702 0.0062839 0.62839 1.49 0.422 0.199 6.67 0.318. 1703 - - • - - - - -1704 0.0059420 0.59419 1.14 0.521 0.199 6.54 0.400 1705 0.0061220 0.61220 1.50 0.408 0.196 6.70 0.311 1706 0.0015550 0.15553 0.92 0.169 0.084 6.36 0.316 1707 0.0064100 0.64099 1.47 0.436 0.202 6.52 0.331 1708 0.0065124 0.65124 1.44 0.452 0.206 6.69 0.328 1709 1710 0.0056380 0.56385 1.42 0.397 0.189 6.55 0.321 1711 0.0062240 0.62236 1.21 0.514 0.202 6.54 0.389 1712 0.0062710 0.62710 1.42 0.442 0.200 6.56 0.337 TABLE V - AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = . ISOPENTANE PERCENT GLYCERINE = 73.07% 3 DENSITY OF DISPERSED PHASE = 0.610 gm/cm HEAT OF VAPOURIZATION = 83.3 cal/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT 2 2 gm c a l sec cal/sec cm cal/sec cm °C 1801 0.0103600 0.86301 1.98 0.436 0.333 7.76 0.169 1802 0.0073732 0.61419 1.86 0.330 0.290 8.03 Q.142 1803 0.0108310 0.90225 2.1.3 0.424 0.354 7.98 0.150 1804 0.0078630 0.65498 1.65 0.397 0.291 8.23 0.166 1805 0.0084510 0.70398 1.71 0.412 0.298 7.70 0.179 1806 0.0104951 0.87424 2.20 0.397 0.350 8.26 0.137 1807 0.0063039 0.52511 1.48 0.355 0.286 8.16 0.152 1808 1809 1810 0.0096811 0.80644 2.05 0.393 0.328 8.77 0.137 1811 0.0047378 0.39466 1.58 0.250 0.254 8.38 0.117 1812 TABLE VI-AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = CYCLOPENTANE PERCENT GLYCERINE = 56.02% DENSITY OF DISPERSED PHASE = 0.688 gm/cm3 HEAT OF VAPOURIZATION = 100 cal/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT 2 2 gm c a l sec cal/sec cm °C cal/sec cm °C 1901 0.0072007 0.72201 1.59 0.453 0.244 4.96 0.374 1902 0.0062712 0.62712 1.22 0.514 0.216 4.75 0.501 1903 0.0062121 0.62121 1.25 0.497 0.209 4.86 0.489 1904 1905 0.0066283 0.66283 1.60 0.414 0.220 4.68 0.402 1906 0.0054136 0.54136 1.47 0.368 0.191 4.56 0.422 1907 1908 0.0049815 0.49815 1.05 0.474 0.186 4.59 0.556 1909 0.0055310 0.55309 1.63 0.339* 0.205 4.87 0.340 1910 0.0069176 0.69176 1.57 0-.441 0.220 4.50 0.445 1911 0.0047448 0.47448 1.40 0.339 0.184 4.82 0.382 1912 0.0041950 0.41950 1.38 0.304 0.166 4.72 0.388 N3 Co CO TABLE VII-AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = ISOPENTANE PERCENT GLYCERINE = 56.02% DENSITY OF DISPERSED PHASE =0.61 gm/cm3 HEAT OF VAPOURIZATION =83.3 cal/cm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT gm, c a l £££ cal/sec cm2 °C cal/sec r.m2°C 2001 0.0058977 0.49128 1.41 0.348 0.221 5.47 0.288 2002 0.0040297 0.33567 1.12 0.300 0.194 5.19 0.298 2003 0.0058994 0.49142 1.42 0.346 0.219 5.43 0.291 2004 0.0049762 0.41452 1.40 0.296 0.198 5.32 0.281 2005 0.0051357 0.42781 1.32 0.324 0.198 5.44 0.301 2006 0.0054840 0.45682 1.54 0.297 0.200 5.63 0.263 2007 0.0061286 0.51051 1.45 0.352 0.212 5.61 0.296 2008 0.0069097 0.55736 1.46 0.382 0.225 5.57 0.305 2009 0.0059464 0.49534 1.35 0.367 0.213 5.52 0.312 2010 0.0051871 0.43209 1.41 0.306 0.199 5.32 0.289 2011 0.0048986 0.40805 1.28 0.319 0.198 5.51 0.292 2012 - - - - - - -NJ Co -t-TABLE VIII-AVIII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = ISOPENTANE PERCENT GLYCERINE = 0.0% DENSITY OF DISPERSED PHASE =0.61 gm/cm3 HEAT OF VAPOURIZATION [ = 83.3 ca 1/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER • DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT gm c a l sec cal/sec 2 cm °C 2 cal/sec cm °C 4 0.0198 1.64930 3.17 0.520 0.493 9.15 0.115 5 0.0173 1.44109 3.04 0.474 0.450 9.15 0.115 6 0.0226 1.88258 3.31 0.569 0.537 9.15 0.116 8 0.0173 1.44109 2.85 0.506 0.950 9.85 0.114 11 0.0119 0.99127 2.44 0.406 0.351 9.85 0.118 13 0.0212 1.76600 3.10 0.570 0.515 9.85 0.113 14 0.0093 0.77719 2.34 0.332 0.298 9.85 0.113 15 0.0138 1.14954 2.44 0.471 0.351 9.85 0.136 Data obtained from Prakash (3) Table VIII, page 120 TABLE IX-AVII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = FURAN PERCENT GLYCERINE = 0.0% DENSITY OF DISPERSED PHASE = 0.883 gm/cm3 HEAT OF VAPOURIZATION = 98.5 cal/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT gm c a l sec cal/sec 2 cm °C cal/sec cm2 °C 2 0.01602 1.5780 2.74 0.576 0.334 5.3 0.325 6 0.03940 3.8809 4.00 0.970 0.608 5.3 0.301 7 0.03470 3.4179 3.31 1.033' 0.559 5.3 0.349 9 0.02680 2.6398 . 3.06 0.863 0.471 5.3 0.346 10 0.00851 0.8379 2.53 0.331 0.219 5.3 0.285 14 0.00767 0.7552 , 2.63 0.287 0.204 5.1 0.276 15 0.03072 3.0259 3.57 0.848- 0.515 5.1 0.323 16 0.00548 0.5398 2.16 0.250 0.163 5.1 0.301 17 0.02497 2.4595 3.14 0.783. 0.449 5.1 0J342 19 0.01473 1.4509 2.91 0.499 0.316 5.1 0.309 20 0.03281 3.2318 3.59 0.900 0.538 5.1 0.328 Data obtained from Prakash (3), Table VII, page 119 TABLE X-AVI - Average Heat Transfer Coefficient DISPERSED PHASE - CYCLOPENTANE PERCENT GLYCERINE =0.0% DENSITY OF DISPERSED PHASE = 0.668 gm/cm3 HEAT OF VAPOURIZATION = 100.0 cal/gm RUN MASS OF TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER DISPERSED TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER PHASE TIME TRANSFER FORCE COEFFICIENT gm cal sec cal/sec 2 cm °C 2 cal/sec cm °C 1 0.0071 0.71 3.00 0.236 0.234 5.30 0.191 5 0.0217 2.17 4.04 0.537 0.493 5.30 0.206 7 0.0203 2.03 3.85 0.527 0.471 5.30 0.211 10 0.0102 1.02 3.19 0.320 0.298 5.30 0.202 12 0.0131 1.31 2.81 0.466 0.351 6.58 0.202 13 0.0217 2.17 3.29 0.660 0.493 6.58 0.203 16 0.0111 1.11 2.81 0.395 0.315 6.58 0.191 17 0.0247 2.47 3.15 0.784 0.537 6.58 0.222 20 0.0131 1.31 2.91 0.450 0.351 6.58 0.195 Data obtained from Prakash (3), Table IX, page 121 TABLE XI-AVIII - Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = PENTANE PERCENT GLYCERINE =0.0% DENSITY OF DISPERSED PHASE = - HEAT OF VAPOURIZATION = RUN TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER TIME TRANSFER FORCE COEFFICIENT c a l sec cal/sec cm2 °C cal/sec cm2 °C 1 1.270 2.05 0.620 0.408 1.60 0.949 2 1.390 2.11 0.659 0.433 1.60 0.951 3 1.300 2.29 0.568 0.413 1.90 0.723 4 1.234 2.36 0.523 0.400 1.90 0.688 5 1.125 1.55 0.726 0.377 3.70 0.520 6 1.218 1.43 0.852 0.396 3.80 0.566 7 1.290 1.45 0.890 0.412 3.80 0.568 8 1.350 1.17 1.154 0.427 5.50 0.491 9 1.340 1.04 1.288 0.421 5.50 0.556 Continued. Continued. TABLE XI-AVIII -. Average Heat Transfer C o e f f i c i e n t DISPERSED PHASE = PENTANE PERCENT GLYCERINE = 0.0% DENSITY OF DISPERSED PHASE = - HEAT OF VAPORIZATION = RUN TOTAL HEAT TOTAL RATE OF INITIAL TEMPERATURE AVG. HEAT NUMBER TRANSFERRED EVAPORATION HEAT AREA DRIVING TRANSFER TIME TRANSFER FORCE COEFFICIENT c a l sec cal/sec 2 cm °C cal/sec cm2 °C 10 1.300 1.22 1.066 0.413 5.70 0.453 11 1.105 0.935 1.182 0.383 5.70 0.541 12 1.165 0.733 1.589 0.384 8.00 0.517 13 1.220 0.742 1.644 0.415 8.00 0.495 14 0.895 0.745 1.201 0.322 8.00 0.466 15 1.190 0.525 2.267 0.390 12.9 0.451 16 1.010 0.475 2.126 0.349 12.9 0.472 17 0.989 0.339 2.917 0.343 14.5 0.567 18 1.034 0.358 2.888 0.355 14.4 0.565 Data obtained from Sideman (2) Table I, page 1280 240 APPENDIX VIII Co r r e l a t i o n Equations From the ph y s i c a l constants presented i n Table I-AVIII the c o r r e l a t i o n equations were calculated by the TRIP package (31) supplied by the Computing Centre of the Univ e r s i t y of B r i t i s h Columbia. A summary of the c o r r e l a t i o n equations i s given i n Table II-AVIII. The following pages are a condensed version of the TRIP c o r r e l a t i o n sheets produced by the computer program. The terms are e s s e n t i a l l y the same as those used i n the TRIP writeup and are given below with only minor changes: (a) T r i p D e f i n i t i o n s EQUATION This equation i s the equation obtained by reducing the TRANSFORMED EQUATION to i t s o r i g i n a l form. TRANSFORMED EQUATION This equation i s the f i t t e d equation whose c o e f f i c i e n t s were determined by the TRIP c o r r e l a t i o n . TRIP CORRELATION The following are the r e s u l t s of the TRIP c o r r e l a t i o n RSQ - the c o e f f i c i e n t of multiple determination -This i s the proportion of the t o t a l observed variance of Y which i s accounted for by the regression l i n e . Thus: TABLE I-AVIII - Summary of Physical Constants Used for Correlation SET NUMBER 13 16 17 19 14 18 20 DISPERSED PHASE CYCLOPENTANE 1 ISOPENTANE VISCOSITY (y d) cp 0.322 0.322 0.322 0.322 0.273 0.273 0.273 DENSITY (p d) (Cp d) gm/cm^ 0.668 0.668 0.668 0.668 0.610 0.610 0.610 HEAT CAPACITY cal/(gm)(°C) 0.3113 0.3113 0.3113 0.3113 0.568 0.568 0.568 THERMAL CONDUCTIVITY (k d) cal/(sec)(cm)(°C) 0.000301 0.000301 0.000301 0.000301 0.000255 0.000255 0.000255 LATENT HEAT OF VAPOURIZATION (X) cal/gm 100 ' 100. 100 100 83.3 83.3 83.3 PRANDTL NUMBER (Pr d) 3.33 3.33 3.33 3.33 6.102 6.102 6.102 CONTINUOUS PHASE PERCENT GLYCERINE % by weight 0.0 77.06 73.07 56.02 0.0 73.07 56.02 TEMPERATURE (t) °C 53.5 55.2 55.2 53.2 31.5 37.4 34.5 VISCOSITY <vc) cp 0.517 9.14 6.90 2.93 0.775 13.3 5.08 DENSITY ( p c } gm/cm^ 0.9877 1.174 1.162 1.121 0.9964 1.173 1.132 HEAT CAPACITY (Cp ) cal/(gm)(°C) 1.0 0.663 0.682 0.760 . 1.0 0.661 0.743 THERMAL CONDUCTIVITY cal/(sec)(cm)(°C) 0.00154 0.000807 0.000841 0.000979 0.00146 0.00834 0.000959 PRANDTL NUMBER (Pr c) 3.36 75.1 56.0 22.7 5.31 105.4 39.4 VISCOSITY RATIO ( u / u c c + ^ 0.616 0.966 0.955 0.901 0.740 0.980 0.949 AVERAGE HEAT TRANSFER COEFFICIENT (U) cal/(sec) (cm2) (°C) 0.334 0.236 0.325 0.416 0.607 0.153 0.298 TABLE II-AVIII - Summary of Correlation Equation a i U = a Pr, o d ( ^ ) a Pr c ( a 3 + a 4 ao P r d ) a l a 2 a 3 a4 u = 0.701 Pr " ° - 2 4 5 c (50) 0.701 0 0 -i'245 0 u = 0.820 P r " 0 ' 2 4 3 P r / 0 ' 1 1 2 c d (51) 0.820 -.112 0 -.243 0 u 0 1 ^ „ -0.921 „ -0.304 , y c ,5.354 Z1.67 p r Pr, (- ; ) c d y + y c d (55) 21.67 -.304 5. 354 -.921 0 u = 0.572 P r " 0 " 0 4 0 1 P r d c (52) 0.572 0 0 0 -.0401 u = 0.241 P r , 0 ' 7 9 5 P r " 0 ' 0 6 0 2 P r d d c (53) 0.241 0.795 0 0 -.0602 u = 0.0439 P r / ' 8 9 1 Pr (0-367-0.134 Pr,) d c (54) 0.0439 1.891 0 0.367 -.134 u = 0.262 P r / ' 4 9 Pr - ° ' 1 1 3 P r d ( Jf ) 2 ' 1 9 d c y + y, c d (56) 0.262 1.49 2. 19 0 -.113 u = 2.20 P r / ' 7 0 6 ( ) 3 ' 9 1 P r ("0.^-0.0642 d y„ + y, c c d P r d ) (57) 2.20 0.706 3. 91 -.445 -.0642 ,u = 0.972 Pr ° - 7 5 8 ( y ? ) 2 ' 9 2 4 Pr - ° - 1 1 5 5 P r d d \ i + y c c d (58) 0.972 0.758 2. 924 0 -.1155 u = 52.2 P r , " 0 - 7 7 3 ( ) 6 - 1 4 Pr - ° ' 9 1 9 d \ i + y c c d (59) 52/2 -.773 6. 14 -.919 0 u = 52.33 P r / 0 ' 7 7 4 ( ^ ) 6 ' 1 4 Pr ("0.919-0.000116 Pr,) d vy + y c c d (60) 52.23 -.774 6. 14 -.919 -.000116 to 243 n . _ s (y, - y) R 2 = ^ i n - 2 2 (y, - y) i = l where ^ ^ . y\ i s the i predicted value of Y th y. i s the i observed value of Y and y i s the mean of the observed values of Y 2 R , i s always between zero and one; the closer i t i s to one, the better the regression equation f i t s the data. The p o s i t i v e square 2 -root of R i s c a l l e d the sample c o r r e l a t i o n of Y and Y. FPROB - F p r o b a b i l i t y for the e n t i r e equation -This p r o b a b i l i t y i s the p r o b a b i l i t y of obtaining a value of 2 R greater than or equal to the one calculated, given that there i s no assoc i a t i o n between the dependent and independent v a r i a b l e s , that i s , given that the true regression c o e f f i c i e n t 3^,...;'$m are a l l zero. 2 I f t h i s p r o b a b i l i t y i s les s than 0.05, i t i s usually concluded that R i s s i g n i f i c a n t l y d i f f e r e n t from zero. STD ERR Y - standard error of Y (standard error of estimate) -This i s defined by: 244 , . 2 s.e.(y) = / T ^ (y,. - y„.) n - m - i I I x=l n i s the number of the observations, m i s the number of independent v a r i a b l e s , th Y. i s the i observed value of Y, l A and y. i s the i predicted value of Y VARIABLE - v a r i a b l e to which the c o e f f i c i e n t s are rela t e d -COEFFICIENT - c o e f f i c i e n t i n the regression equation -These are the c o e f f i c i e n t s b, i n the regression equation • ^ 0 1 1 m n and are calculated to minimize the sum of squares of the distances of the predicted values of Y from the observed values of Y. STANDARD ERROR - standard error of the regression c o e f f i c i e n t s -I t i s assumed there e x i s t s r e a l numbers B„, 3 n , . . . g , 0 1 m such that the expected value of Y at each of the n points ( x ^ , x^,. . . , x m ^ ) > 1 = l>2,...,n, i s on the hyperplane given by the equation: y > &0 + &± x 1 + , . . . , + 3 m x m Each experiment which produces a value of Y for each of the n fi x e d points provides estimates bQ,b^,...bm of BQ, 3^ ,. m Thus b_, b.. ,.. .b 0 1 m may be considered as variables for which each such experiment provides a value. Being v a r i a b l e s , b_, bn ...b have means (8„, 3, »...3 ) and ° 0 1 m 0 1 m variances. The square roots of these variances are c a l l e d the standard errors of b^, b^,...b m > These standard errors may be used to compute the following 100(l-y)% confidence i n t e r v a l s for each f j \ . b ± = ± t(n - m -1, 1 - h Y) (s.e.(b ±)) where n i s the number of observations, m i s the number of v a r i a b l e s , t (n - m - 1, 1 - y ) i s the 1 - h y percentage point of the t - d i s t r i b u t i o n with n - m - 1 degress of freedom, and s.e. (b.) i s the standard error of b.. 1 x F-RATIO An f - test i s performed to test the s i g n i f i c a n c e of each regression c o e f f i c i e n t b^. The F value i s calculated by the formula: F i = t s . e . V ) ^ FPROB - associated p r o b a b i l i t y for each b^ -This i s the p r o b a b i l i t y of obtaining a value of F^ greater than or equal to the one calculated, given that fi\ = 0. I f t h i s pro-b a b i l i t y i s les s than 0.05 i t i s usually considered that b^ i s s i g n i f i -cantly d i f f e r e n t from zero. (b) Summary of C o r r e l a t i o n Equations The equations given previously may be summarized by the equation: U = a P r ^ l [ — ^ ] % r ( a 3 + a 4 P r d } u o d u + u, c c d which i s given i n Table II-AVIII with the values of the constants f o r the various c o r r e l a t i o n equations. (c) Description of " t " Test The s t a t i s t i c a l s i g n i f i c a n c e of the differences between the masses calculated by the photographic and d i l a t o m e t r i c methods was ob-tained by using the computer program "UBC PAIRC" supplied by the Com-puting Centre at the U n i v e r s i t y of B r i t i s h Columbia. The t i t l e of t h i s program i s "Means, Standard Deviations and T-Test for Paired Comparisons" as the t i t l e i n dicates i t gives the following r e s u l t s : 1. "Means, standard deviation and the number of v a l i d ob-servations of the groups involved i n the paired comparison 2. "The number of v a l i d comparisons, the t-value and t pro-b a b i l i t y of the paired comparison i t s e l f " . The r e s u l t s given under the various headings i n Table XVI were calculated from the following equations: 247 Mean M E x. X = M where M i s the number of v a l i d observations Standard Deviation M M a = / M E x f - ( E x.) i = l 1 1=1 1 M(M-l) For the Differences d. = x . - x 0 . x l i 2x The Degress of Freedom are: M -1 T_ i s calculated from d A M -°d where d = mean of the differences T f ,.05 a , = standard deviation of the differences d i s the value of T for f degrees of freedom and a= 0.05; i f T i s less than T^ ^ there i s not a s i g n i f i c a n t d i f f e r e n c e between the two groups. 248 S i g n i f i c a n t Difference shows whether or not there i s a s i g n i f i c a n t d i f f e r e n c e between the two groups. 249 (d) Results of TRIP Correlation EQUATION (50) U = 0.701 Pr -0.245 TRANSFORMED EQUATION In U=.r-0.3558 - 0.245 l n Pr TRIP CORRELATION RSQ = 0.5478 FPROB = 0.0565 STD ERR Y = 0.3235 VARIABLE CONSTANT l n Pr COEFFICIENT - 0.3558 - 0.2450 STANDARD ERROR 0.3449 0.0996 F-RATIO FPROB 6.0564 0.0565 EQUATION (51) U = 0.820 Pr " ° - 2 4 3 P r / ° ' 1 U c d TRANSFORMED EQUATION In U = - 0.1981 - 0.2432 In Pr - 0.1118 In Pr, c d TRIP CORRELATION RSQ = 0.5545 FPROB = 0.1984 STD ERR Y = 0.3590 VARIABLE CONSTANT In Pr c In Pr, COEFFICIENT - 0.1981 - 0.2432 - 0.1118 STANDARD ERROR 0.7460 0.1107 0.4540 F-RATIO FPROB 4.8262 0.0924 0.0607 0.8014 . 251 EQUATION (55) 0 = 21.67 ; r , - ° - : S I * ( - J ^ ) 5 - 3 5 4 Pr J 0 " 9 2 1 TRANSFORMED EQUATION In U = 3.0761 - 0.3041 In Pr, + 5.3536 In (y / ( y + y j ) d c c d - 0.921 In Pr c TRIP CORRELATION RSQ FROB = 0.9458 = 0.0256 STANDARD ERROR 0.7654 0.1524 0.1875 1.1509 F-RATIO FPROB 35.5333 0.0121 2.6303 0.2033 21.6389 0.0218 STD ERR Y = 0.1446 VARIABLE CONSTANT In Pr c In Pr, COEFFICIENT 3.0761 -0.9290 -0.3041 l n ( y c / ( y c + V d ) ) 5.3536 252 EQUATION (52) U - 0.572 Pr - ° ' 0 4 0 1 P r d c TRANSFORMED EQUATION l n U = -0.5579 - 0.0401 Pr,, In Pr d c TRIP CORRELATION RSQ FPROB STD ERR Y = 0.5582 = 0.0530 = 0.3198 VARIABLE COEFFICIENT STANDARD ERROR F-RATIO FPROB CONSTANT -0.5579 0.2646 Pr, l n Pr -0.0401 0.0160 6.3184 0.0530 d c EQUATION (53) U - 0.241 Pr ° ' 7 9 5 Pr - 0 ' 0 6 0 2 P r d d c TRANSFORMED EQUATION l n U = -1.4243 + 0.7951 l n Pr, - 0.0602 Pr, In Pr d d c TRIP CORRELATION RSQ = 0.7613 FPROB = 0.0583 STD ERR Y = 0.2628 VARIABLE COEFFICIENT CONSTANT -1.4243 In Pr 0.7951 Pr, l n Pr -0.0602 d c STANDARD ERROR 0.5175 0.4309 0.0171 F-RATIO FPROB 3.4042 0.1379 12.4750 0.0252 .254 EQUATION (54) U = 0.0439 P r / ' 8 9 1 Pr ( ° - 3 6 7 " ° - 1 3 4 P rd> d c TRANSFORMED EQUATION l n U = -3.1259 + 1.8909 l n Pr, + 0.3673 l n Pr - 0.1342 P r J l n Pr d c d c TRIP CORRELATION RSQ = 0.8640 FPROB = 0.0842 STD ERR Y = 0.2291 VARIABLE COEFFICIENT .STANDARD ERROR F-RATIO FPROB CONSTANT -3.1259 1.2175 l n Pr 0.3673 0.2441 2.2640 0.2295 c l n Pr, 1.8909 0.8194 5.3247 0.1050 a Pr, l n Pr -0.1342 0.0514 6.8258 0.0808 d c 255 EQUATION (56) - 1 i q U = 0.262 Pr, 0 d .2.19 „ -0-.113'Pr, — ) Pr d y + u c TRANSFORMED EQUATION In U = -1.3399 + 1.4896 In Pr, - 0.1127 Pr, In Pr d d c +2.1880 In y ? ( y c + y,)) TRIP CORRELATION RSQ = 0.9580 FPROB = 0.0189 STD ERR Y = 0.1272 VARIABLE CONSTANT In Pr . P r j In Pr d c COEFFICIENT -1.3399 1.4896 -0.1127 In (y /(y + y,)) 2.1880 c c d STANDARD ERROR 0.2516 0.2790 0.0163 0.5835 F-RATIO". 28.5076 48.1014 14.0623 FPROB 0.0159 0.0091 0.0358 EQUATION (57) U = 2.20 Pr ,°- 7° 6 ( Wc,' ) 3 - 9 0 5 r i (-°.^ 6-0.06-42 Pr.,) TRANSFORMED EQUATION l n U = 0.7899 - 0.4455 l n Pr + 0.7057 l n Pr, -0.0642 Pr, l n Pr c d d c + 3.9050 In (PC/(VC + V d » TRIP CORRELATION RSQ = 0.9879 FPROB = 0.0355 STD ERR Y = 0.0837 VARIABLE COEFFICIENT STANDARD ERROR F-RATIO FPROB CONSTANT 0.7899 0.9730 In Pr c -0.4455 0.2005 4.9344 0.1594 l n Pr, d 0.7057 0.3978 3.1479 0.2197 Pr, In Pr d c -0.0642 0.0243 6.9635 0.1228 l n (v /(v„ + V,)) 3.9050 0.8630 20.4767 0.0536 257 EQUATION (58) U = 0.972 Pr 0.758 , M c N2.924 D -0.1155 Pr ( ; ) Pr d y + y , c TRANSFORMED EQUATION In U = -0.0285 + 0.7580 In Pr, + 2.9244 In (y / ( y + y ,)) d c c d -0.1155 Pr, In Pr d c TRIP CORRELATION RSQ FROB STD ERR Y = 0.5100 = 0.0308 = 0.4034 VARIABLE CONSTANT In Pr, COEFFICIENT -0.0285 0.7580 In (u /(u + u )) 2.9244 c c d Pr, In Pr d c -0.1155 STANDARD ERROR 0.6352 0.6701 1.4430 0.0410 F-RATIO FPROB 1.2796 0.2801 4.1070 0.0632 7.9475 0.0150 258 EQUATION (59) U - 52.20 Pr " ° - 7 7 3 ( - ^ ) 6 - 1 4 Pr - ° ' 9 1 9 d u + y , c TRANSFORMED EQUATION l n U = 3.955 - 0.9188 In Pr -0.7725 l n Pr, c d + 6.1379 l n (y /(y + y ,)) c c d TRIP CORRELATION RSQ = 0.6859 FPROB = 0.0025 STD ERR Y = 0.3230 VARIABLE CONSTANT l n Pr c l n Pr, COEFFICIENT 3.9550 -0.9188 -0.7725 i n (w c/(y c + y d ) ) 6.1379 STANDARD ERROR 1.0514 0.2101 0.3421 1.6519 F-RATIO FPROB 19.1193 5.1001 13.8064 0.0010 0.0416 0.0030 2 5 9 , EQUATION ( 6 0 ) co o Q - 0 . 7 7 4 , M c N 6 . 1 4 _ ( - 0 . 9 1 9 + . 0 0 0 1 1 6 Pr,) U = 5 2 . 3 3 Pr, ( ) Pr d d V + U J C c d TRANSFORMED EQUATION l n U = 3 . 9 5 7 6 - 0 . 9 1 9 4 l n Pr - 0 . 7 7 4 0 l n Pr, + 6 . 1 3 8 7 In (y / ( y + y , ) ) c d c c d + 0 . 0 0 0 1 1 5 9 Pr, l n Pr d c TRIP CORRELATION RSQ = 0 . 6 8 5 9 FPROB = 0 . 0 0 8 4 STD ERR Y = 0 . 3 3 7 3 VARIABLE CONSTANT l n Pr c l n Pr , COEFFICIENT 3 . 9 5 7 6 - 0 . 9 1 9 4 - 0 . 7 7 4 0 In (y / (y + y ,)) 6 . 1 3 8 7 c c d Pr , In Pr d c 0 . 0 0 0 1 1 5 9 STANDARD ERROR F-RATIO FPROB 1 . 6 9 1 5 ' 0 . 3 7 0 4 6 . 1 6 0 7 0 . 0 2 9 3 0 . 8 3 3 7 0 . 8 6 2 0 0 . 3 7 6 2 1 . 7 7 0 1 1 2 . 0 2 7 4 0 . 0 0 5 2 0 . 0 5 7 8 0 . 0 0 0 0 4 4 0 . 9 4 7 0 

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