THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING ON THE BEHAVIOR OF A PULSED EXTRACTION COLUMN Stanley Co Jones A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Chemical and Metallurgical Engineering 1962 December, 1962 IP-597

ACKNOWLEDGEMENTS I gratefully acknowledge the time and assistance given me by the many persons and organizations who have contributed to this research. Thanks are due particularly to: Professor M. R. Tek, chairman of the doctoral committee, for his personal interest, guidance, and efficient handling of administrative matters. Professor R. H. Kadlec, committee member, for his assistance and constructive criticism in several phases of the work. Professors L. Oo Case, G. B. Williams, and J. L. York, committee members, for their assistance and helpful suggestions, and to Dr. K. H. Coats, who served as a committee member while on the staff of the Department of Chemical and Metallurgical Engineering. Messrs. H. B. Kristinsson, M. CO Miller, R. L. Nielsen, D. A. Saville, and J. R. Street for their interest and constructive criticism throughout the research. My wife, Barbara, and Mr. Russel Okamoto for their generous assistance in the preparation of the manuscript. The shop and secretarial personnel of the Chemical and Metallurgical Engineering Department, who were most cooperative and helpful at all times, and to the Computing Center at the University of Michigan for their donation of computing time and services. Professor D. L. Katz, for his personal interest and assistance. The American Gas Association for financial aid in the form of fellowships. Messrs. R. E. Carroll and D, L. Danford and the personnel of the College of Engineering for their complete cooperation and efficient production of the final form of this dissertation. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS................................................... ii LIST OF TABLES........................................ o.... Vi LIST OF PFIGURES e.............................................. vi LIST OF APPENDICES........... 0...................................... 0 0 ix NOMENCLATURE....................................................... ABSTRACT e o oSt ed.............oO oo......... o.................. XV INTRODUCTION.................................................... 1 SURVEY OF PREVIOUS WORKee....................................... 4 EXPERIMENTAL SCOPE AD EQUIPMENT................................. 10 Variables Studied,.................................... 2410 Equipmentd....Dk oo.m Proedre.......................... o 12 DROP SIZE DISTRIBUTIONSrem................................ 22 Experimental Proceduresn..o..o.........oo................... 22 Plate eteae Pr eament. eers.......................... 22 Photographic Calibrations..........o................O,.... 23 Drop Photographs o...,.........o..,..........o......o..... 24 Film and Dark Room Procedure..o......OOO.OOo o.o. ooioOo. 25 Dro o H up Size Measurements oe............ o..o...ooo.... o 25 Mathematical Treatment.o.oo......o....o. o........o.o.......Oo 26 Computation of Mean Diameters...o...o.......oOO.............oOOO 27 Log-Normal Distribution.O...o..........o............oooo o 28 Results and Discussion......o.o..........o........e.......o....O o 30 HOLDUP AND FLOODING..........o.......o..o..................o o....... 43 Estimation of Holdup from Manometric Measurements ooooo...........oooo 43 Flooding Due to Insufficient Pulsation....................o... 48 Mean Orifice Velocity......................................0000 51 Slip Velocity, Holdup, and Flooding Due to Excessive Pulsation 55 Summary and Conclusions...... o.....oo......................o 64 LONGITUDINAL MIXING............o.................................. 66 Experimental Methods.......o................................ 66 Mathematical Treatment,...oo......O......OO...........O O.....o 74 Results of Diffusivity Experiments....o,...................... 77 Summary and Conclusions...... o.. o....o........o............ 87 iii

TABLE OF CONTENTS (CONT'D) Page STEADY STATE MASS TRANSFER...........,........................ 88 Experimental Methods......................................e 88 Column Startup and Operation........................... 88 Calibration of Pulse Amplitude....................oo...... 93 Calibration of Refractometer............................. 95 Mathematical Representation of the Mass Transfer....o....,,. 99 Results and Discussion of Steady State Mass Transfer Experiments.................................................. 104 Summary and Conclusions.................................... 127 UNSTEADY STATE MASS TRANSFER...........................,.,,,. 129 The Approach to Equilibrium............................. 129 Transient Rates in a Stirred Pot...................o....... 130 Experimental..................................... o.,000 130 Mathematical Treatment of Data....................,... 132 Results......o........................ o............ 138 Summary and Conclusions................ o................. 139 AREAS FOR FUTURE RESEARCH,.o.... o........o...... e....,..,,,0,.0,0 14v BIBLIOGRAPHY.......o............................................ 143 APPENDICES....,.o.........,...........,.... 147 iv

LIST OF TABLES Table Page I Typical Results of Mean Drop Diameter Calculations.... 31 II Results from Drop Size Measurements.................. 33 III Experimental and Calculated Values for Runs at Incipient Flooding Due to Insufficient Pulsation........ 49 IV Comparison of Calculated and Experimental Holdup at Near-Flooding Conditions......................... 57 V Experimental Conditions and Results from Axial Diffusion Experiments................................. 83 VI Experimental Data and Calculated Results for Mass Transfer Experiments.............................. 105 VII Data and Calculated Results for Transient Mass Transfer Experiments.............................. 135 v

LIST OF FIGURES Figure Page 1 Column and Sampling Apparatus........................ 13 2 Pulser............................................... 15 3 Switching Arrangement on Pulser...................... 16 4 Equipment for Mass Transfer Experiments.............. 18 5 General View of the Equipment........................ 20 6 Prints of Typical Drop Photographs................... 29 7 Typical Drop Size Distributions at High Pulse Velocities...........O............................. 35 8 Typical Drop Size Distributions at Low Pulse Velocities....................................36 9 Comparison of Distributions at High and Low Frequencies with Constant Pulse Amplitude............... 37 10 Variation of Sauter Mean Diameter with Experimental Parameters.................................... 39 11 Correlation of Sauter Mean Diameter................. 40 12 Definition of Lengths Used for Manometric Estimation of Holdup...........o o...o............. 44 13 Comparison of Actual- and Manometrically Estimated Holdup............................... 46 14 Comparison of Calculated and Experimental Continuous Phase Superficial Velocities at Incipient Flooding Due to Insufficient Pulsation............... 50 15 Slip Velocity-Holdup Relationship for Countercurrent Flow.................................... 59 16 Variation of Holdup with Dispersed Phase Flow Rate.....o........................................ 61 17 Variation of Holdup with Continuous Phase Flow Rate,.~o...................................... 62 18 Correlation of Slip Velocity......................... 63 vi

LIST OF FIGURES (CONTPD) Figure Page 19 Dye Tracer Apparatus................................ 67 20 Calibration Curve for Dye Tracer Experiments........ 69 21 Beer's Law Plot for Dye Concentration Calibration.... 71 22 Recorder Traces for Dye Tracer Run 61............... 73 23 Input Concentration Profile for Dye Tracer Run 61 Showing Scale Readings....................... 78 24 Output Concentration Profile for Dye Tracer Run 61... 79 25 Output Concentration Profile for Dye Tracer Run 50... 81 26 Effect of Experimental Variables on Effective Diffusivity for Longitudinal Mixing.................. 84 27 Effect of Experimental Variables on Effective Diffusivity for Longitudinal Mixing.................. 85 28 Simplified Correlation of Diffusivity Coefficients... 86 29 Determination of the Mass Transfer Rate at the Principal Interface...............................o 91 30 Refractometer Calibration.......................... 97 31 Comparison of Calculated and Experimental Concentration Profiles.................................. 108 32 Comparison of Calculated and Experimental Concentration Profiles.................................o 109 33 Comparison of Calculated and Experimental Concentration Profiles.................................o 110 34 Comparison of Calculated and Experimental Concentration Profiles................................. 111 35 Comparison of Calculated and Experimental Concentration Profiles.................................. 112 36 Comparison of Calculated and Experimental Concentration Profiles............................... O 113 vii

LIST OF FIGURES (CONT'D) Figure Page 37 Comparison of Calculated and Experimental Concentration Profiles.....o.......o..o................ 114 38 Comparison of Experimental Concentrations with Profiles Calculated from First and Second Order Equations...o......o................................ 115 39 Actual and Effective Holdup Volumes.................. 118 40 Correlation of Specific Interfacial Area............ 120 41 Mass Transfer Coefficient, k2, as a Function of Sauter Mean Diameter.........o..................... 121 42 Correlation of Mass Transfer Coefficient, k2...o.... 124 43 Correlation of Mass Transfer Coefficient, k3......... 125 44 Correlation of Volumetric Mass Transfer Coefficient, kla....................................... 126 45 Illustration of Slow Dissolution Rate of MIBK in Nearly Saturated Aqueous Solutions................... 131 46 Rate Function for Transient Mass Transfer Experiment I.......o...................... o. o....o....o. 136 47 Rate Function for Transient Mass Transfer Experiment IIoo. o... o. 0 o. 0 0.. o 0 o.. 0 00 o o. 0. 0 0 0 o a 137 viii

LIST OF APPENDICES Appendix Page A Derivation of Generalized Mean Diameter from Log-Normal Distribution...,.,.,... o........... 148 B Derivation of Relationship for Flooding Due to Insufficient Pulsation o...... O,................... 151 C Derivation of Expression for Mean Orifice Velocity.............................. 155 D Solution of Diffusion Equation with Generalized Input O o o o oo.................o............. 159 E Tables of Dimensionless Concentration as Functions of Dimensionless Length, Modified Peclet Number, and Mass Transfer Number......o........, 167 F Derivation and Solution of Equation for Transient Mass Transfer in a Stirred, Batch System....... 171 ix

NOMENCLATURE A Cross sectional area of column, cm2 Al Constant defined by Equation (1), cm A1,2 Constants a Pulse amplitude, cm ~a Specific interfacial area, cm2/cm3 B Constant defined by Equation (90) C Continuous phase flow rate, cc/sec Cl2 Constants do Drop diameter at initial time, cm D Dispersed phase flow rate, cc/sec D Drop diameter, cm or mm De Effective diffusivity for axial mixing, cm2/sec Dmn Weighted mean, drop diameter, cm D; Geometric mean drop diameter, cm erf Error function exp(o) Base of natural logarithms, e, raised to the 0 power f Pulse frequency, sec1 or min-1 f(x) Density function, Equation (6) F(t) Function of time f(s) Laplace transform of F(t) g Gravitational acceleration, cm/sec2 x

g(x) Function of x h Perforation diameter, cm hi Length defined by Figure 12, cm Ah Length defined by Figure 12, cm j Variable defined by Equation (10) kl Over-all mass transfer coefficient for piston flow, cm/sec k2 High rate mass transfer coefficient, cm/sec k3 Low rate mass transfer coefficient, cm/sec kc Mass Transfer coefficient, cm/sec K Absorption coefficient in Beer's law K Weight of ketone, g L Length of contacting section of column, cm M Moment, see Equation (A-6) in Number of drops N Mass transfer number, see Equation (68) P Ratio of flow rates, see Equation (32) Pe Modified Peclet number based on column length Q Downflow volume, cc Qcdown Continuous phase effluent volume Qup Volume of fluid pushed through perforations during upstroke, cc QD Effective holdup volume, cc R Variable defined by Equations (28) and (29) RoIo Refractive Index s Parameter in Laplace transform s Standard deviation, Equations (9) and (A-5) xi

S.R. Scale reading t Variable defined by Equation (A-9) t Time, sec At Elapsed period of time during upstroke up T Temperature, ~C u Actual mean linear velocity, cm/sec us Slip velocity, cm/sec v Volumetric displacement of pulser, cc v Superficial fluid velocity, cm/sec vo Mean orifice velocity, cm/sec Vcol Volume of contacting section of column, cc Vr Continuous phase recycle rate, cc/sec w Weight of dye solution injected W Weight of water, g WD Weight of dispersed phase held up, g x Concentration, weight fraction or weight per cent x Natural log of drop diameter, Equation (7) x Mean, defined by Equation (8) y Vertical length, cm y* Equilibrium concentration of MIBK in ketone phase, weight fraction Y Weight of ketone dissolved into aqueous phase, g z Dimensionless length = y/L xii

Greek Symbols a Constant defined by Equation (89) a Variable defined by Equations (19) and (75) Pi ~ Variable defined by Equations (B-8) and (76) 7 Variable defined by Equation (C-2) 6 Variable defined by Equations (93) and (C-7) A Denotes a difference e ~ Holdup, volume fraction of dispersed phase 9 Integral rate function, see Equations (88) and (93) 3 Dimensionless time ~ Sin-1 (p), or Sin1 (5) X Wavelength, millimicrons it 3.14159 p Density, g/cm3 ~0 Cos-1 (6) 0 ~ Fraction free are of perforations in plates with respect to column cross sectional area. Subscripts c Critical, or transition values c Continuous phase D Dispersed phase D Dimensionless f Feed i Inlet j Index m Index xiii

n Index o Outlet o Initial condition 0,1,2 Order of moments, see Equations (50) and (51) 10 Linear mean, Equation (5) 32 Sauter mean, Equation (3) Superscripts e( First derivative f ~Second derivative Transformed variable Denotes dimensional moment xiv

ABSTRACT The purpose of this research was to characterize mass transfer and fluid flow in a pulse column by mathematical models and experimental measurements. The binary system: methylisobutyl ketone/water was employed. Mass transfer from the dispersed, water-saturated ketone phase to the continuous aqueous phase took place in a one inch diameter column containing 15 perforated plates on a two inch spacing. Variables were pulse frequency (12.5 to 200 RPM) and amplitude (0.65, 1.86, and 4.47 cm), perforation diameter (1/16 and 1/8 inch), and flow rates of both phases. Plate free area was held constant at 32.8% Sauter mean drop diameters were computed from high speed photographic measurements. They varied from 0.7 to 2.6 mm, and were correlated by the function vh/', where h is the perforation diameter and vo is the "mean orifice velocity". The log-normal distribution adequately described drop size distributions at high orifice velocities, but not at the lower velocities where bimodal tendencies were observed. Measured dispersed-phase holdup was related to flow rates by the slip velocity concept. Holdup at incipient flooding was shown to be independent of fluid properties and column geometry. A simple technique for estimating holdup by manometric measurements is presented. Effective diffusivity coefficients for axial mixing were determined using a dye tracer technique. A mathematical treatment was developed which permits analysis using a tracer input of arbitrary shape. Two factors were observed to contribute to the effective diffusivity: xv

eddy diffusion was predominant at high pulse rates whereas Taylor mixing prevailed at low pulse velocities. Most of the effective diffusivities were in the range: 0.5 to 3 cm2/sec. Measured concentration profiles for steady-state mass transfer were analyzed by both a "piston flow" equation and equations which allow for back-mixing. Mass transfer rates were adequately described using two rate coefficients, one for dilute concentrations and another, with a much lower value, for concentrations near the equilibrium value. Transient mass transfer measurements substantiated the change in the rate coefficient as equilibrium was approached. Specific interfacial area, a, was evaluated from the Sauter mean diameter and holdup. The mass transfer coefficient k was obtained from the volumetric coefficient, kT, and T. The value of k was found to decrease sharply with decreasing drop diameter. It ranged from.002 cm/sec for the smallest drops to.018 for larger sizes. A fiftyfold variation in a resulted in only a five-fold variation in the volumetric coefficient, ka. However, kt varied approximately as holdup raised to the 0.b power when drop size was held constant. Results were again correlated by the function fh/vo. xvi

INTRODUCTION Liquid-liquid extraction has been utilized for many years for separation processes. The search for improved extraction efficiencies has led to the development of a class of liquid-liquid contactors in which mechanical agitation is provided to increase the interfacial area between the two liquid phases. One of these contactors is the pulse column which was introduced by Van Dijck(60) in 1955. He recommended that perforated (sieve) plates be reciprocated vertically to produce the pulse, but noted that the same effect could be achieved by mechanical pulsation of the fluid. The latter mode of operation is most generally used today due to its design simplicity. The highly successful application of pulsed columns to the atomic energy industry(5) in 1948 for heavy metals recovery and purification has stimulated widespread research in evaluation of column performance for design and scale-up purposes. Many of the investigations have been devoted to relating flooding characteristics and extraction performance to such operating variables as continuous and dispersed phase flow rates; pulse amplitude, frequency, and wave form; the number, spacing, and material of construction of the perforated plates; plate perforation diameter and per cent free area; column height and diameter; and to fluid properties such as viscosities, densities, etc. The complicated nature of the dispersion of the fluids and the large number of variables involved has almost always led to empirical correlations. -1 -

-2 -The effects of the different variables reported by various investigators are not all in agreement. In attempts to resolve the disagreement, factorial experiments in which several variables are considered have been performed. These experiments have all shed light on the state of the art, but leave much room for further work. Nearly all observers agree, however, that high mass transfer rates are associated with large interfacial areas which are attained by producing small drops and high holdups, It is further agreed that research should be directed toward studying the effect of operating variables and fluid properties on drop size and holdup. Due, in part, to technical difficulties, little work has been done in the measurement of drop sizes. This dissertation was undertaken to investigate mass transfer rates in a pulsed, liquid-liquid extraction column, utilizing a semimicroscopic approach; i.e,, through direct measurement of drop size, dispersed phase holdup, and some other factors influencing mass transfer. In this manner the mass transfer problem can be divided into parts, each of which is more amenable to analysis than the problem as a whole. The parts listed below will all be treated in some detail in this dissertationo 1. Mass transfer rates can be characterized by a "volumetric coefficient" which incorporates a specific interfacial area term. It has long been felt that, when operating conditions are varied, most of the variation of this coefficient is due to changes in the specific area available for mass transfer. The evaluation of the coefficient

-3 -when stripped of its area term is one of the main objectives of this research, and gives rise to the other problems. 2. The interfacial area term depends upon the mean diameter of the dispersed phase droplets. Drop size distributions were measured for various modes of column operation. 3. Specific area for a given mean drop diameter is directly proportional to the volume fraction of the column occupied by dispersed phase droplets (holdup). Column flooding characteristics and operational stability are also associated with holdup. These problems will be discussed. 4. Longitudinal mixing or "backmixing" has been shown(49) to have a detrimental effect on the extraction efficiency of a pulse column. A technique for determining an effective diffusivity for backmixing will be presented along with experimental measurements for its evaluation. 5. Finally in an attempt to demonstrate the validity of the model used for the rate expression in the mass transfer equation, results~from independent experiments involving transient mass transfer rates in a stirred tank will be presentedo

SURVEY OF PREVIOUS WORK Considerable effort has been devoted to the development and understanding of the pulsed, sieve plate extraction column since its introduction by Van Dijck(60) in 1935. As normally utilized today a pulse column consists of a vertical, cylindrical column provided with enlarged calming sections at either end. Two liquid phases flow countercurrently through the column. The heavy, generally aqueous, phase is introduced near the top and flows downward contacting the light (organic) phase as the latter ascends the column. Superimposed upon the countercurrent flow is a nearly sinusoidal translation of both phases, produced by a reciprocating piston or bellows which is in fluid communication with the flowing liquids. The line to the pulser is connected near the bottom of the column. Either fluid phase may be dispersed. The dispersion is accomplished through a shearing action by the perforated plates as the liquid is forced through them by the pulser. Sege and Woodfield(42) observed five distinct types of behavior in pulse columns as a function of pulsing conditions and throughput rate: 1) Flooding due to insufficient pulsation. With sieve plates there is little countercurrent flow without the pumping action of the pulser. 2) Mixer-settler operation, characterized by very stable operation, coase drops, nearly complete coalescence and -4 -

-5 -separation of the phases between pulse cycles, and rather inefficient operation. 3) Emulsion-type operation at the higher pulse energy inputs, characterized by small drop sizes and high extraction efficiencies. 4) Unstable operation near the upper flooding point. 5) Complete flooding due to excessive pulsation. Various investigators have correlated column extraction performance as a function of the product of pulse frequency and amplitude.(2,6,10o18228245345148249) It should be pointed out that the frequency x amplitude product is a measure of the mean velocity of the dispersed phase through the perforated plates. However, it will be shown later that this "pulse velocity" is not a complete measure of the mean orifice velocity. Belaga and Bigelow(2) found that mass transfer rates improved with increasing pulse velocity, but passed through a maximum. Further increase in pulse amplitude or frequency had a detrimental effect on mass transfer rates. Griffith, Jasny and Tupper(l8) observed the same phenomenon whereas Cohen and Beyer(lO) found that increasing the pulse velocity beyond the maximum efficiency point had no effect. By contrast, Burkhart and Fahien,(3) who studied pulsed column performance in the low-energy, mixer-settler region, observed a decrease in extraction efficiency with increasing pulse velocity. This was attributed to a high degree of backmixing. Chantry, Von Berg and Wiegandt() utilized the product of amplitude and the square of frequency as a correlating parameter for

-6 -mass transfer rates, whereas Sege and Woodfield(43) suggested the use of pulse velocity raised to some power. Sanvordenker(41) observed no maximum efficienty as pulsation was increased, but rather continuous improvement until flooding occurred. Sanvordenker and others(4,16,51) observed that extraction efficiency and holdup were extremely sensitive to plate wetting characteristics. Sobotic(51) studied plate wetting by using both stainless steel plates (water wet) and polyethylene plates (wet by the organic phase). He found that extraction efficiencies were improved up to 50% by using plastic plates when the transfer was from the organic to the aqueous phase. However, the efficiency was not affected by plate wetting characteristics for mass transfer in the opposite direction. Sanvordenker(41) showed that the pretreatment given to plates (i.e., their chemical cleanliness) is a most important factor which controls transfer rates. Edwards and Beyer(l6) studied holdup and flooding characteristics and presented a theoretical expression for flooding due to insufficient pulsation. Thornton(57) and co-workers(29,30) investigated flooding due to excessive pulsation. Thornton presented generalized correlations for flooding and mass transfer. Pratt(36) pointed out that holdup at incipient flooding is independent of fluid and column properties, and is a function only of the ratio of dispersed to continuous phase superficial flow velocities. Smoot and associates(50) correlated flooding and mass transfer data of four investigators. The average deviation is claimed to be + 20% for the flooding correlation.

-7 -Weaver et al. 61) related holdup to flow rates in a spray tower using the slip velocity concept. Slip velocity is the mean velocity of the dispersed droplets with respect to the continuous phase. These authors utilized the Zenz(66) correlation to predict slip velocities. Longitudinal mixing or back-mixing has been known to have a detrimental effect on mass transfer rates through reduction of the driving force. These effects have been discussed by Newman(34) and Miyauchi.(33) Many workers have investigated eddy diffusion in extraction columns.1,4,l12,15,26,3147,49,52,63) It should be pointed out that longitudinal mixing is caused not only by eddy diffusion, but also by non-uniform velocity and subsequent radial mixing, which is sometimes called Taylor diffusion after G. I. Taylorls analysis of axial mixing in pipes o5354) Mar and Babb determined effective eddy diffusion coefficients for a pulse column using steady state and delta injection techniqueso They performed a factorial design and obtained a generalized correlation for eddy diffusivities. It may be noted that the steady state technique which was utilized for most of the experiments of the above authors 'does not incorporate Taylor diffusion contributions to axial mixing. These will be shown in this thesis to be important at low pulse rates. Smoot and Babb(49) who employed the correlation in a study to determine the effect of back-mixing on mass transfer in pulse columns, reported that the diffusivities appeared to be 20 to 30% too low

-8 -Smoot and Babb(49) demonstrated the importance of including the effects of back-mixing in the calculation of mass transfer coefficients. Under their most severe pulsing conditions the actual coefficient is reported to be three times that calculated assuming no backmixing. The ratios are more typically 1 to 2 throughout most of the pulsing spectrum, however. Reports of direct measurement of drop sizes in pulse columns are meager. Graham and Burkhart(17) made photographic measurements of drop sizes for a column operating in the low pulse energy range (mixersettler operation). They found that bubble size increases with increasing pulse amplitude and/or frequency and perforation diameter under conditions of incipient flooding due to insufficient pulsation. At higher pulse rates they observed decreasing drop sizes with increasing frequencies and amplitudes, with the exception of a few irregularities at low flow rates. Again, drop diameters increased with an increase in perforation diameter. The authors did not report the effect of drop size on mass transfer rates. Ruby and Elgin(40) determined interfacial area in a spray column. They showed that for a given drop diameter, mass transfer coefficients were directly proportional to holdup (hence interfacial area). An extensive review of mass transfer studies involving single drops has been made by Johns and Beckmann.(23) Johnson and Hamielec(24) found, that for equal interfacial areas, higher mass transfer rates were associated with larger drop diameters and low rates with small drops -- for binary systems. This phenomenon was ascribed to greater internal circulation in larger drops. Grober,(19) Kronig and Brink,(27) Handlos

-9 -and Baron,(20) and others have formulated models describing internal circulation in a moving drop. A review of the literature reveals that a considerable number of investigations were undertaken to relate the over-all performance of pulse columns to flow rates.,pulsing conditions, column geometry and fluid properties. The column was considered to be a black box for the majority of the studies, and certain discrepancies have resulted. Quantitative measurements of the effect of the experimental variables on drop dispersion are sparse. This dissertation was undertaken in order to gain a better understanding of pulse column behavior through a study of the dispersion of drops, and to relate mass transfer rates to the degree of dispersion.

EXPERIMENTAL SCOPE AND EQUIPMENT The experimental objective of this research, the determination of mass transfer rate coefficients for a range of operating variables in a pulsed column, involves several somewhat unrelated measurements. These are measurements of solute concentrations, drop size distributions, dispersed phase holdup, and axial diffusion coefficients. The experimental techniques and mathematical treatment employed for each of these topics will be discussed in succeeding sections. Variables Studied The number and ranges of variables within the operable limits of a pulse column are large. In order to determine column performance in a reasonable length of time, only those variables having the most significant effect, as found from previous studies, were investigated. For the present work, the following ranges of variables were found to be convenient: Continuous phase flow rate: 1,5, and 4 g/sec. Dispersed phase flow rate: 1/4 and 1/2 g/sec. Amplitude:.65, 1.86, and 4.47 cm. Frequency: 12.5, 25, 50, 75, 100, 150, and 200 cycles/min. Perforation diameter:.159 and 0.317 cm (1/16 and 1/8 inch). Most of the mass transfer experiments fell in the "emulsion" range of operation. Many of the possible combinations of the above variables were not used because they either caused flooding or were outside the range of interest. The flow rates listed above are nominal and only approximate the actual rates used. The ranges of flow rates -10 -

-11 - and frequencies utilized for some of the holdup and flooding investigations exceeded those listed above: Continuous phase flow rate: 0 to 5 g/sec. Dispersed phase flow rate: 0 to 4 g/sec. Frequency: 12.5 to 300 cycles/min. Most commercial applications of liquid-liquid extraction involve two relatively immiscible solvents and a distributed solute. These ternary systems undergo simultaneous three-way mass transfer. The transfer of one solvent into the other and vice versa is generally neglected. Hennico and Vermeulen(21) have recently pointed out that serious errors may be introduced by neglecting the solvent-solvent transfer due to changes in their mutual solubility with changes in solute concentration. Colburn and Welsh(ll) suggested that in fundamental investigations the solute be eliminated and transfer limited to two relatively immiscible solvents into one another. In accordance with the suggestion of Colburn and Welsh, the binary system: methyl isobutyl ketone (MIBK) - water was chosen for this investigation. The ketone phase was dispersed and transfer was from the ketone to the aqueous phase. This system afforded certain advantages. It has a high interfacial tension (10.1 dynes/cm at 20~C) which tends to prohibit the formation of stable emulsions. The density of the MIBK is only 80% that of water, permitting high throughput. The concentration of the ketone in the water phase could be measured accurately with a- precision refractometer. The water concentration in the ketone phase had no observable effect on the rate of ketone transfer into the aqueous phase; consequently MIBK saturated with water was used

-12 -throughout. Since the composition of the saturated ketone never changed, it could be reused repeatedly. Finally, little or no data pertaining to mass transfer rates have been reported for the MIBK-water system; thus the opportunity for a contribution to the fund of existing data is available. Equipment The column was constructed from standard "Pyrex" fittings. It consisted of a 34 inch precision bore, 1.000 +.002 inch I.D. tube provided with flanges at either end. Three inch I.D. calming sections were connected to both ends of the one inch tube. A stainless steel flange served as the bottom of the column. The flange was drilled and tapped to accomodate fittings for the pulser line, the ketone inlet line, the aqueous effluent line, and a drain-cock. A polyethylene tube extending from the bottom flange to the one inch I.D. section served as the ketone inlet. The top end of the column was open to the air, and the ketone was withdrawn from a one inch side outlet in the upper calming section. Water was introduced into the top end of the column through a polyethylene tube, the end of which was sealed off. Holes were drilled through the walls of this tube: near the sealed end, causing the entering water to be radially distributed. The aqueous phase was withdrawn from the bottom via a hydraulic leg. The level of the ketone-water interface was controlled by raising or lowering the leg. A simplified sketch of the column is presented in Figure 1. Stainless steel plates were used. They were cut from 000793 cm thick sheet and machined so that the clearance from the column wall

-15 -Polyethylene Tubes From Intermediate Sampling Points Water Feed Air — Ketone Interface Thermometer / Copper Center Tube Ketone Effluent Tube For Sompling Outlet ConcentrotioI r Ketone-Water 'Interface Holes ForRadial -- Distribution Of Water 1, 19 *0 @ C Aqueous Phase E Effluent J o, ^ Adjustable o u, a r Hydraulic Leg @ e o - Spacer 0 Hypodermic Needle For Intermediate 0 / Sampling / Flexible Tube / Y Aqueous Phase Effluent Line Ketone Inlet _ - - To Pulser Figure 1. Column and Sampling Apparatus.

-14 -was less than 0.005 inches. They were supported on a 1/4 inch copper tube which served as a center rod, and were kept in position by 2 inch long spacers made from 3/8 inch stainless tubing. The assembly, consisting of the center tube, plates, and spacers constituted a cartridge which could be removed from the column at will for cleaning purposes. Two sets of plates were fabricated. There were 15 plates per set, Each plate in the first set was drilled to provide 72 - 1/16 inch diameter holes on a triangular pitch. The plates of the second set had 18 - 1/8 inch diameter holes arranged in a symmetric hexagonal pattern. Each plate had a total orifice area of 1.425 sq. cm. The cross sectional area of the column open to flow was 4.345 cm2, thus the per cent free area of each plate was 32.8%. The length of the contacting section for mass transfer was 92.1 cm, providing an effective column volume of 400 cc. The upper end of the ketone inlet tube was considered to be the lower limit of the contacting section. No holdup of the ketone drops occurred below this point, and hence no mass transfer. The top end was defined by the principal ketone-water interface (see Figure 1). The column was provided with three intermediate sampling points. These consisted of No. 22 hypodermic needles with their bases removed. Both ends of each needle were ground flat. One end was forced into PE 100 (.034" I.D. x.060" O.D., polyethylene) tubing and the other end was squeezed nearly closed. Holes were drilled through the spacers and center tube at three locations. The polyethelene tubes were threaded through the holes and up through the 1/4 inch copper center tube, coming out at the top of the column. The tips of the hypodermic needles were

15 Figure '4 Pu~serv I Figure 2. PWlser,

-16 -To Flash Unit Micro- Switch forward Or Backward Figure. Switching Arrangement ission Pulser Figure 3. Switching Arrangement on Pulser~

-17 -located 11,0, 48~8, and 79.1 centimeters, respectively, from the bottom of the contacting section. Sampling was accomplished by siphoning the aqueous phase through the three sample lineso Also, continuous phase effluent samples were withdrawn from the leg used to control the principal interface. Temperatures were also measured at this point. The pulser consisted of a one inch Teflon bellows powered by a Grahm motor and speed reducer. The continuously variable transmission provided a frequency range from 0 to 350 rpm. The pulse amplitude was controlled by an eccentric attached to the shaft of the transmission, Stepwise changes in amplitude were achieved through the use of different eccentrics. The pulser is shown pictorially in Figure 2, A sleeve containing several knobs acting as cams was fit on the transmission drive-shaft. These knobs were attached in a spiral pattern at 30~ intervals, and served to actuate two micro-switches. The setup is illustrated in Figure 35 The switch on the right hand side was used for accurate measurement of pulse frequency. When engaged, the micro-switch was actuated once during each revolution of the shaft. The switch was connected in series with a counter which could be turned on and off simultaneously with an electrical timer. The resulting number of counts during a measured time interval permitted calculation of the pulse frequency, The body of the left hand switch (Figure 3) was threaded such that it was moved forward and backward similarly to a micrometer when a knob was turned. In this manner the switch was aligned with one of the several cams. This scheme was utilized to fire the flash unit, which was used for droplet photographs, at known angles of the pulse cycle,

-18 -RINSE I ORGANIC AQUEOUS WATER FEED FEED _, CENTER ROD Heated Rinse ' Column Column Quick C: o losing il1 ~ 70 Hydraulic Leg Closing Valve Aqueous Organic Rota eters Out Out Plexiglas - Jacket 2000 v, Imf. Jacket _ Power Unit._Flash.Unit Camer Adjustable, Ov T Im lOr Figure 4. Equipment for Mass Transfer Experiments. Height Relay Teflon [wrive and Eccentric Figure 4. Equipment for Mass Transfer Experiments.

-19 -Three stainless steel 55 gallon drums served as feed tanks. One was filled with MIBK, saturated with water, and the two remaining drums contained distilled water. Advantage was taken of gravity feed by locating the drums several feet above the top of the column. The water and ketone feed streams were conveyed to the column by 3/8" polyethylene tubes. The flow rates were controlled by needle valves and measured with rotameters. Distilled water from the third tank was used for rinsing the cartridge containing the plates following pretreatment cycles. These will be discussed later. The water and ketone feed lines and the aqueous phase outlet line were provided with number 460 stainless steel Hoke toggle valves. These valves permitted nearly instantaneous cut-off of the three streams for holdup measurements. A camera for photographic drop size measurements was constructed using an Argus, 75 mm, f/3.5 lens and a Graflex film holder and ground glass focusing screen. A magnification of approximately 2.2 was obtained. Illumination was provided by a General Electric FT 220 flashtube powered by the discharge of a 1 microfarad capacitor, initially charged to 2000 voltso The power unit was designed by Professor J. L. York, and its description and circuitry is given in detail in his thesis.(65) The flash duration of the FT 220 flashtube is rated at 33 microseconds for the above discharge conditions. The apparatus employed for drop photographs is shown schematically in Figure 4. The sequence of the flashtube firing is as follows: A micro-switch is actuated at a desired angle of the pulse cycle by a

20i:X~ i:' 5 eea f h qtrn

-21 -cam on the transmission shaft. The micro-switch is connected to a relay, which in turn triggers the power unit which fires the flashtube, exposing film in the open-aperture camera. A time delay unit served as the relay. The time delay of several microseconds caused by this unit was entirely negligible compared to the maximum pulse frequency of 200/minute. The delay unit had a reset feature which eliminated the problem of undesirable mpultiple flashing at high pulse frequencies. Circuitry is presented in detail in the thesis of Short.(46) Two continuous recording photometers were used in the measurement of dye concentrations for longitudinal diffusion experiments. Each consisted of a 100 watt light source with a focusing lens, filter, and slit andanIP39 high vacuum phototube with a patentiometric type circuit. A Sanborn "Twin Viso" dual channel, oscillographic recorder was used to record the output signals from the photometers. The setup is shown schematically in Figure -9,) Details of the photometers are given in Ebach's(14) thesis. Ketone-water concentration analyses were performed with the aid of a Bausch and Lomb Precision Refractometer, Model 33-45-230 It was used in conjunction with a sodium vapor lamp (X 589.3 millimicrons) and a constant temperature bath maintained at 25 + 0.02~Co

DROP SIZE DISTRIBUTIONS The proper mean drop diameter is required for the calculation of the interfacial area available for mass transfer. Furthermore, it will be seen that the entire performance of the pulse column under study is closely dependent upon the drop sizes produced. This section deals with the experimental measurement and mathematical treatment of drop size distributions encountered in the operation of a pulse column. Drop breakup occurs when the dispersed (ketone) phase is forced through the perforations of the plates. The entire column of fluid is displaced upward during the upstroke of the pulser. An upward displacement is also caused by the flow of the ketone into the column. In general, small drops are associated with small perforation diameters and high orifice velocities. Large drops are produced at low pulse velocities (frequency x amplitude) and also when the dispersed phase wets the plates. (17,1) Experimental Procedures Plate Pretreatment In an attempt to maintain the plate wetting characteristics constant, it was decided to give the cartridge containing the plates a standard pretreatment before each run. A 30 minute passivation in a 50 per cent nitric acid solution at 130~F as suggested by Graham and Burkhart,(l7) followed by immersion in a 15 per cent tri-sodium phosphate solution at 180~F for 10 minutes, with an intermediate, 15 minute rinse in distilled water was tried initially. This was followed by a -22 -

-23 - final rinse in distilled water. The plates treated in this manner produced an extremely fine drop dispersion for a short time. Due to the possibility of trace contamination by the pretreatment chemicals and the large time requirement, the above technique was abandoned. Sanvordenker(41) observed that nearly identical results were obtained by either treating the plates with soap and water, or by boiling them in water. A slight modification of the latter method was adopted: The cartridge was removed from the pulse column with the aid of a pulley and nylon cord (see Figures 4 and 5). It was then lowered into a column of flowing distilled water to rinse off any ketone present. The cartridge was subsequently placed in a heated column containing distilled water. The temperature, controlled with a Variac, was. raised to 200-205~F, A large evolution of gas from the plates was observed to occur at a temperature of 190~Fo Heating (at about 200~F) was continued until the evolution ceased - a period of about 20 minutes. The cartridge was then removed from the hot column and immediately cooled in the flowing distilled water column before it could steam dry. The cooled cartridge was quickly replaced into the pulse column. The drop dispersion was not as fine as that produced after the nitric acid-tri-sodium phosphate treatment, but it remained constant over a long period of time, and was reproducible. Photographic Calibrations Drop diameters were to be measured photographically, but severe optical distortion was caused by the curvature of the cylindrical column walls. This was minimized by placement of a jacket of square cross

-24 -section over the cylindrical column and filling the "annular" space with water. The jacket was constructed from 1/8 inch thick Plexiglas. Magnification and distortion were determined by photographing a glass sphere of known dimensions in various locations inside the column. A grid of 0.050 inch spacing in the horizontal direction and 0.010 in the vertical was engraved on a transparent film which was inserted in the column. The grid was photographed for magnification calibration. Measurements on pictures of both the sphere and the grid indicated that a magnification of 2.20 was attained with little distortion in the vertical direction and virtually no radial distortion out to 80 per cent of the column's inside radiuso Radial distortion gradually increased outward from that point, reaching a maximum of about 10 per cent at the column wall. However drop size measurements were taken in the vertical direction, thereby eliminating the radial distortion problem. Drop Photographs All drop photographs were exposed using transmitted light. An f/6o5 aperture opening gave satisfactory results in conjunction with the 2 watt-second flash. The camera was focused such that its focal plane passed through the axial center of the column. The depth of field was approximately 5 millimeters. Assuming angular symmetry, radial variations in drop size distribution, if any existed, were taken into account with this camera placement. During a typical run, the column was allowed to attain steady state operation. The micro-switch was adjusted so that it would be actuated at a 0~ angle of the pulse cycle (i.e., half way up on the up-stroke). Pictures were then taken in four locations along the

-25 -axis of the column: above plates 3, 6, 10, and 1l4 (plates 1 and 15 are the bottom and top plates, respectively), one picture in each location. Preliminary runs showed that no significant changes in results were obtained using replicate photographs. The number of drops represented by the four pictures of a given run averaged somewhat more than 1000. Film and Dark Room Procedure Film used was 4" x 5" Kodak Contrast Process Ortho. This is a fine-grained orthochromatic film giving sharp contrast. The film was developed for five minutes at 68~F in Kodak D-ll developer, rinsed for 30 seconds, immersed in Kodak Acid Fix for 10 minutes, rinsed for 30 seconds, placed in Kodak Hypo Clearing Agent for 2 minutes, rinsed for 10 minutes, and air dried. Intermittant agitation was supplied during immersion in the solutions. A Wratten series 2 red safelight filter was used to control dark room illumination. Drop Size Measurements The negatives were projected onto the ground glass screen of a Jones and Lamson Bench Comparator at a 10X magnification. The total magnification by the camera and the comparator was 22.0. The drop diameters were grouped into 12 size ranges. These were measured using a template which was engraved on a transparent piece of photographic film. Counting the number of drops corresponding to each size range was facilitated by a 12 channel counter which was constructed during this research. It consisted of twelve telephone relay counters each connected in series with a doorbell push button and a rectifier circuit.

-26 -Photographic drop analyses such as the present one are subject to a certain amount of human judgement and error. One source of trouble is the decision whether or not to include drops which are nearly in focus. Comparison of doubtful images to a "standard" series of photographs of drops which were out of focus by varying degrees helped to lend objectivity to the measurements. Another difficulty arises when drops are partially or totally obscured under high holdup conditions. Exclusion of drops from the large countable population in this case did not markedly affect the value of the mean diameter, however. Mathematical Treatment Kessler(25) and others have observed that dispersed drops in liquid-liquid systems often closely follow a log-normal distribution, i.e., the logarithms of the drop diameters are distributed according to the normal (Gaussian) distribution. Accordingly, size classes based on a geometric- rather than an arithmetic or other series were used in the calculation of the drop distributions. A drop whose diameter lies in the range A,/ < D_ l (1 where Al and b are constants and j is an integer, is defined here to lie in the j-th size class. Values of.005682 centimeters and N2 for Al and b, respectively, were convenient for this work. Consequently the diameter, Dj, of a drop of the j-th size class falls in the range.00 82 (2) < D. 00 82 (2)c (2)

-27 -Computation of Mean Diameters A host of mean diameters can be defined, each having a specific application. Of these, only two are of interest here. The most important is the Sauter mean diameter. This is the diameter of a drop which would have the same volume to surface area ratio as the total volume to total surface of a group of distributed drops, or: O (3) where nj is the number of drops counted in the j-th size interval and D. is the average diameter of drops in that interval. Since we are using a geometric series to represent our classes, the geometric mean diameter of each class is to be utilized. It is represented by - I r ma (4) The other mean diameter of interest is the "linear" mean, defined as D (5) d.o The Sauter mean diameter, D32, is used extensively in later sections in the calculation of specific interfacial area and in correlations of the pulse columnYs performanceo

-28 -Log-Normal Distribution As pointed out before, dispersions in liquid-liquid systems often approximate a log-normal distribution. The chief utility of this distribution is that all the mean diameters, i.e., linear, Sauter, volumetric, etc,, can be expressed in terms of only two parameters: mean and variance (or standard deviation). The log-normal density function appears as ^(- AI- eP -2i (6) where?-. — =.,,, DE (7) - = )v As + ( A - 2 ) b (8) 3 — M -2 1 ';() Y.f^ - 2f () I. _ and * E frs =......... (10) It may be seen that the mean, x, and the standard deviation, s, are defined in terms of an experimental distribution. The general mean drop diameter Dmn for a log-normal distribution can be obtained from the ratio of the m-th to the n-th moments of

— 29 -X~ip.::::::::::-i::::: i:::: pp~ SI --- g ri r B 859 3t:::'-i::::::i:::-i::-i:::j:::T;I' i-~~l::_jy:l:::::-:-:: ~::::::. : -:::::::,,-~cv._::::-:::-:: -~-II~: _::: -:::::_::::::_:_::::::.:-'-::-:::-:::_::::::::-:::::::_::::::: -:::i_ _:::-:_.:::::;:::::::: -:-::::::::-:-:I::-:I:::-:-:::-:-::I::::::::i::::::::-::-:-:-:- -:::_:-::::::s~s —;~:i l:::,.-~~~:-ti-~:a:::::::; —~~:::~x: '::O::i;"l:::::' Ba bs" O 6 Q ss, d a so BI u, ed:~ i: k:: ~:: hO O O::::::::::::::::: r:::: -:::::::l'j:-:i':::::::j::::j::i'j::::::~:.:::::::::::::::::::-:::::::::::::::::::::::::;~38686 rl ~ehB cJ ~p-(l O ~rf P:::-.~:~-:i E-l a i eno P% tZ iJ -t d a e% --- ~d bD d s P Bm, tt re cr B I::_ —:::-i- I:::ll-::-l:il: —:sil:::::i::::;: o I:: ~rl CCI -1-::: I: -: 1:::::-:-I-:::.-:: —::l —li-l::l, —'-':: j;- -:::::: j::: I-:::-_:- j:: I:: -::-:::::I-:i::::::-:-l:ljisi::::. ~ I::: I --- ass d ns %f P-8

350 the density function raised to the l/(m-n) power: Upon simplification this reduces to: D - > ecp \ + ( -m2n) 52] (11) Details of the derivation of Equation (11) are given in Appendix A. It is readily seen from Equation (11) that the Sauter mean (D32) diameter will always be greater than the linear (D10) mean diameter unless all the drops are the same size, in which case the two are equalo The resulting dimension from Equation (11) is the same as that used for Al in the calculation of x. It should be pointed out that x is the natural logarithm of the more commonly defined mean of the distributiono Results and Discussion Positive prints of typical drop photographs are shown in Figure 60 The first three pictures show the effect of increasing pulse frequency at a constant amplitude and. the fourth shows drops formed at a low fret quency but higher amplitudeo Three important qualitative observations may be made from these photographs: (1) Smaller drops are obtained at higher frequencies and/or amplitudes, (2) higher frequencies tend to produce more uniform drop sizes, and (3) the product of frequency and amplitude (pulse velocity) appears to be a controlling factor in the determination of drop sizeso In regard to the last statement it can be noted that the pulse velocities corresponding to the second and fourth pictures are 96 and 93 cm/minute, respectively9 and that the drop distributions for the two are quite similar,

-31 -N oQcoO M Ntcv O ~,-4 M C oM - 00 c N ^ CM ^o M-< m n o C o CV - LCIM N 0~ N W Z o 0 oo _M f V Vco o ~No N ' f- r..,O. 0-4 N M X ~4 Z * * * @ O. Xr. 10 00 Z M 0 ON 0 0- 0% li, < = < Z V) CQ -N WU JZ 0 ~ * * * J '00O C a a - 0 d..,, L) a0 O' * aN O0 0 ^ ~ I I::?? LII,.. 1 0 O0 0 9.., < ' M M 0 0 ~ gJ i CO0 W _ J CO*. M C0-. o- c o * in W O. 0000 - N,: X, -~: | 3 w W -J. O o <tJ 2 o'. S4Is 2:uj^ JWaoon o o o o~~Z U S (I i -Cr C xL o W;. it - w"1=r 0 I in i Q1:&{~~ C~ c: ',i LA U. ~ ~ ', 0 U"2 I*- - 0 E- 14 iv -J W,Jc cr l; I. ' Q CQ )- P Ca. 0-;.- ujc~ uj i Ui oLc ^ 0. <^;Z2WLU> ~% i~~~N 'C i j I o o H;;iZ 0^; L; ^i1?! ~ 3 H w o;o -' U pg! <1<;b ~- ff:;^!u in w O uj:; a: a:e!= r = ~ * o I II 9 0 0 Uit~Ta QI IC ~~32"~:~t ~ g i, i i i *I~; cc; 1 I *J r l 0 0 i,. sn m * IC 4i^ U-% In tFI w a '~r`-(Q ~'

-32 -A fourth observation can be made in reference to the left-most photograph of Figure 6: at low pulse velocities the drops tend to move in clusters. This is a condition approaching 'mixer-settler" type operation. Large drops having comparatively high terminal velocities are formed. They rise rapidly from one plate and collect under the next plate until they are forced through it on the next upstroke of the pulse cycle. If the pulse frequency is very low relative to the bubble rise velocity complete coalescence can occur. Low extraction rates are generally associated with mixer-settler operation.(17,28) It can be seen from Figure 6 that the drop spatial distribution becomes more uniform with increasing pulse velocity. The column operating at high pulse velocities appeared to be filled with a homogeneous mass of drops. On the quantitative side, several mean drop diameters were computed for each of the four locations photographed. Sample results are presented in Table I. A "LOC. " of 3,5 in the table indicates that the picture was taken between the third and fourth plates from the bottom of the cartridge. XBAR and STD.DEV. are x and s and were calculated from Equations (8) and (9), respectively. D1, D2, D31, and D32 are linear, area, volume, and volume/area based mean diameters, and the LN preceding these denotes mean diameters computed from the log-normal distribution using the experimentally obtained values of x and s. Average values for the four locations are given on the bottom line. Table I shows that the diameter of an average drop changed very little from the bottom-most observation point to the top of the column. This was found to be the case for nearly all the runs; i.e., the majority of the drop breakup occurred during passage through the first two or three sieve plates. The above

-33 -\0 88880088888888888888888888880088 0 o 000o'-*c u0-)-00 ( 0 22 0' 0 c ct o H HH H HHH01 0C: ~~o O00 OOOH -UU\ O ~8 8 8 ~~o~~i~~ 0 8 \o \o 8\D. 88.80 \0 O — LrN 0 8 m \O Ca rqO ~lL~ cu 8 e 8 a g 0; 14-% 0 0 0%\ \ 8 0 CU~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~u O(( KO(A%~ A M uisO8 9 0 0 r-IK-%0 0 r- co )-%%'O c? N m CN U H. ~~~ ~- 0h O)- (A O3 id~~~~~~~~ a- O M 88 m\O 0 \O0 o0 0 t -- gN s LC Ch rl CM ~ ~ o\ o a,,,-o m o o,, m22.jmm o-aa m *~~~~~~~~~~~~~~~~~~~0 0\ 0'0\ H0a0\ 0\ u u \ - ` 2 24 O'% t H~~~~~~~~CCL u O.c`~tc a) co CU( - L t-\OLL ~ Lr (AK0%CHU~* NCUO CU Q 0 ( U 0-I OH - g t. t4 a ~ CA co5? (B t Mr —N,, t- L-' " 0w 3;. t -m m:t U\; t 0 4:; CO H\~ 8 ~ -'2'.Urr ~ ~4 O- \O 0\ 88~ IX3cU s O OHH%0 p H%\c\O\ \ 0\ 0 Q O\ K%\ r 4q Ch G \ M O 0\ O\ Ov Ch \c C) H t 0\ \ K\ pc~~ K\ ~ea\ K\ a \0 "_pt ND co q o 0\ 8;9 ` r-l\t- ( uN 'O 8 — t' zt Z.K0\-t r OaOc OH\ ~ \0 IC%0\ If\p -* (A'0- I t %0 P -'. -* (A U IC \ L;,ra o-N4 H-K CUO 'O K C.t\u-'. O rC)( K\ % K\- t 0 H-c u KC K O - H H CU H O t- CU\\ H H L(\H Ct O\ K\ H O\ 0 UN W\ r- I rO -I 0 Is t- to-\9L 8 lij tI Ir C.I rI t C- IL\:CU r lt q \0a C m' K\ K t c O tl -'t c Vsr s ~~~9 Oa ~ \ nio pcVp UN 00 U P-\ CW r k M O rj O\ CU ~ us \b ~~ — - t - C l AOD m OO bk co \ \ 6; CY\ PC~ \ \ 0 V) CM O 4 K r\ Lr\ \D C\ (j o;0 rdr\ u r\ Os 0\ \ 0 ~ o 0) H H ~ H OCCHK0 Ut —U 0 * (AL~~~~~~l P\I- CiC)H KK\ \ UK-K' - %o cu H 0 K9 tw -- ~ O"'.-* U -CU H H 0 H H 1 0OC U -*OHt HO K %O ~t aO P —~ P — 00 CU -t.\CO CU ic LC..- I'O n Pc KI~~~~~~~~~~ O~L\ a\ a\ i* _t co Ow us\ r\ L- CO \0 l n "~t U lrlO 0 '-0 Lf\ Lr\ CU HO OOOOOOOOOOHOOOOOOOOO 00 00 00 Lr r ( N-I r \n C N\3U \0 P N O &\0 \ -- U O\ 0 H 0\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a UN Cu u\ Lr\ \. U PC\ LA -I r - 0 co cu O\O c CM 0 HH n a) L 0\QOHC0 U M GOCU 0 CU (0 (M \0 0 t- UN -- A t- EJ 0 C ID C30 0S t- CID Ci F'1 CU~~u'cm \-o 1%?C 0' \ 0) LL%~ c IC\N '-.0(AL-\0 \ 0\C t-\ r-*H av '0\ Ft-i 00 -d a) * ~ ~ ~ ~ ~ ~ ~ 4-H a) a) 43 CMrg- tIr c \ C\\ C to - C) H) 0C0 co~~~~~~~~~~~~~~~~~~~~~~~c (30 rco U- tCrIUO L- -Hco.-* e - iCU c0 Mt'-0 0 o- I- H'cC rCUIOt-0 u-t L a 0 o c 01 C Lci \'D(A rtI lt-'OO C-OnLr nCr) W L \0~ v t — -— Oi \ \t- -t -.00- OH '.0.)J CO E: a) CUUUCCCCUUCUUUUCCCCUUUCCCCUUUUCUK IO o s s0 la ~ JP \rl\ 0 i \CIF% El000 u V c In 4'a 0)O n np CO UQ ~ ~ P ~V \0 - -- -tcoa M n \ DO aUN0 -tU\ r-\ Cy L'- -zt 0 Cm Lf\ r( d f cm \Cc3 0; \0M0 0\ l 0 " cu\; g 0\ L- m \0 u 00 u t c A n 'I. \ 0 p a H CU K- 0 C O\ CUh UN WH C\ 0 CY\ a CO \ \ U -1CO - - HO \\H O H \CM t 0CI CU " O DCU \0 CU NU 0 " C — \ H K- -0 Hf \%0 rL - t OH\ \0 - CU H \V - H.. a) cINcovM- CU \0oJ E-r\CU U\Oi' CU00)t \\L0\CUH.0\c \ t —0 CU C —\co (\ H H U a) CUGO)t I IHICU OI HCIHOIUI IHIHIHIHI I I I I IO OIIIIU HO OIII I I I II il 4) 43 H a)j '.O-*c v~O vicO U%-* C —'.0* Hl CU Hl Oi\iA. (\~)aK'ic0' 0 (AH 0 CU'. 0);~ a\ M —t cm'CY\H - 00 "CC) W rc% -t -t --- -t pM co rl 0\ \0 n fZ \ Cin (30 I CM Y\ li 0 CMO- S-* 'i r - E -c 0 a\ t: ( LfH (M 0) — Q Pr\ I\ t01) (aoC CM —UN O o) \- Ot-coOCU _ -\ C- -' Ln\I\o. (A t —1 \i a\ H 0L)\O 0 c4 r4.4 c4 r4 r4 c r4 d d r4 4 r r4 0" d d c r4 o O 4 r 4 o o o H co 1~~~~~~~~~~~~~~~~~~~~~~~~o a) I\\ 0 J,46)o r- O' O-' O' O4 4 64 8 O' 6 6 i ' 4 u ' O' O'i O' O 3 HH CUHH H H HHCU H HHCU a) H0 800 H0 0C 0 0 0CU COA 0\ CU *I\. \,-COUN0N 0 0( 0 to OOOOOOOOHHHHH\HH H2CU CUCUI CU CU CUNCU2U CUHHHHHH HH H4 4 HHHHHHH HHHHHH

-34 -statement must be qualified: varying degrees of coalescence can occur, followed by re-formation of drops of nearly the same diameter. This was particularly evident at low pulse velocities. Due to the longituditudinal uniformity of the mean drop diameter only average values were used in subsequent calculations of specific interfacial area. The log-normal distribution proved to be an excellent approximation to the actual drop size distribution for the example shown in Table I, as it was for all the high pulse-velocity runs. However, large deviations appeared at the lower pulse velocities. These will be discussedo Results from the drop size measurements are tabulated in Table II. These include the linear and Sauter mean diameters, the experimentally obtained mean and standard deviation for the distributions as defined by Equations (8) and (9), and the cumulative drop size distributions. Some of these distributions are plotted on log-probability paper in Figures 7, 8, and 9. A plot of drop diameter vs. cumulative per cent of the drops smaller than the given diameter on log-probability paper results in a straight line for a log-normal distribution. Figure 7 shows distributionsof drops produced at high pulse velocities; at a low amplitude but high frequency in one case, and at a high amplitude and lower frequency in the other. Both of these are seen to be fairly well represented by log-normal distributions, with the exception that both show bimodal tendencies indicated by the two slopes for each curve. The excess of small drops is caused, in part, by the inability of these drops to proceed up the column as rapidly as their larger counterpartso At very high aqueous phase velocities some of the

-35 -99.99 - 99.9 99 - -_- - 98 95/ S 50 ---- - ---- _ - E ' _I I I I so 0 50, 306 /I Run 103 Run 121 3 Yll l l | )5 V L Frequency, RPM 200 75 02 c 20 / n Amplitude, cm. 0.65 4.47 CI I Perforation Diameter, In. 1/16 1/16 I0 0 1 | H1120 Flow Rte, cc./sec. 1.02 2.95 5 - - -- MIBK Rate, cc./sec. 0.32 0.32 0.1 - --.01 0.1 1.0 10.0 Orop Diameter, mm. Figure 7. Typical Drop Size Distributions et High Pulse Velocities.

-36 -99.99 99.9 _ 99 —.8 / / c 80 C so 70 60. 50 I -I__ 30 O210 < 2 / l Frequency, RPM 50 12.5 90 ) I Amplitude, cm. 0.65 4.47 a 5 - - - --- - Perforation Diameter, in. 1/16 1/16 2 8 5g l [ Ll H20 Flow Rate.,cc./sec. 3.90 2.86 2 /- - - l | 1 | MIBK Flow Rate, cc./sec. 0.58 0.58. 0.1 1.0 10.0 Drop Diameter, mm. Figure 8. Typical Drop Size Distributions at Low Pulse Velocities.

-57 -99.99 99.9 - - - - 0. 98 o.._ 95 C so C i3 60 / I-~~~n~-, _ 3 0, _Oiomtr 1/8ich.___ o Water Flow Rat = 2. cc./s c. o 3~ --- -- - _ Proai D e /i,20 I / | A. { S t~ Ml MIBK Flow Rate =0.65 cc./sec. 70 -.-. 60I / I- - - - - - 30.01 Drop Diame ter, mm2. Figure 9. Comparison of Distributions at High and Low Frequencies with Constant Pulse Amplitude. a. ^/,, '~y- 0 — with Constant Pulse Amnplitude.

-38 -smaller droplets were actually carried downward and entrained in the aqueous effluent stream. The marked deviations from the log-normal distribution at lovo pulse velocities are illustrated in Figure 8. The same amplitudes as were represented in the previous figure are shown here, but the corresponding frequencies are much lower. Figure 9 shows the effect of increasing frequency at a constant pulse amplitude, and is typical of all the runs. In all cases the distributions deviated considerably from the log-normal at low pulse velocities, but appeared to approach it in the limit as pulse velocities were increased. All the mean diameters used in subsequent calculations were computed from the actual experimental distributions rather than the lognormal. The variation of the Sauter mean diameter with experimental parameters is shown in Figure 10. Pulse frequency and amplitude had the most pronounced effect on the mean diameter, while the flow rate of either phase had little effect. However, the increase of any of these variables tended to diminish the mean drop diameter. As might be expected larger drops were produced by the larger perforation diameter. The experimental variables: perforation diameter, h, pulse frequency, f, pulse amplitude, a, dispersed phase volumetric flow rate, D, continuous phase volumetric flow rate, C, plate fraction free area, <, and column cross sectional area, A, were combined into one dimensional group, Jh/Vo, to correlate the Sauter mean diameter (see Figure 11). The denominator represents the time-average velocity of the dispersed phase through the plate perforations, and will be called "mean orifice velocity."

-39 -3.0 2.8. 2.6 C eT~ tFLOW RATE cc/sec 2.4! --- Water MIBK 0 1.08 0.324 2.21 \ 2- -- -- X 2.98 0.324 t\\ tt! "^^ e 2.98 0.648 2.0 — X A 3.96 0.324 O 3.96 0.648 1.8 1.l x8]' U a0 = Pulse Amplitude, cm. cE |t \\ _' _< \ h= Plate Perforation Diameter, in. E 3.6 1.4' ( 1.864: 6 h = I/8 1.2. 2 T -- --- _ _- - x 1.0.8sC- Pulse Frequency, RPM a = 0.65 Figure 10. Variation of Saut.6er Mean l=O. 447 h =1/16.4.2 0 20 40 60 80 100 120 140 160 180 200 220 240 260 Pulse Frequency, RPM Figure 10. Variation of Sauter Mean Diameter with Experimental Parameters.

-40 -2.8 _______ 2.6 0 2.4 t 2.0 dX 1.8 1.6 -- - -/ ^ / Amplitude Hole Size i_.4 O ___ 0 0.65 cm. 1/16 inch. J IO') I / A X 1.86 cm. 1/16 inch. g1.2 4.4:. 7|. A 1 4.47 cm. 1/16 inch. A^y El Di 0.65 cm. 1/8 inch. 1.0 r-/Lx.6,.4.2 0.02.04.06.08.10.12.14.16.18.20.22.24 F/ Vo, sec/cm1/2 Figure 11. Correlation of Sauter Mean Diameter.

-41 -As used in Figure 11, it is defined by: o -? ia + 22 MA D C] (12) This is a simplification of a more rigorous expression for the mean orifice velocity, but was found to correlate the Sauter mean drop diameter fairly well. The rational for the mean orifice velocity expression will be presented in the next section. The Sauter mean diameter appears to vary as the square root of the perforation diameter and inversely as the mean orifice velocity. This is similar to the functional dependence of the mean drop diameter reported by Putnam, et al. (8) for the solid injection of liquid fuels. Upper and lower bounds for drop diameters seem to exist. Beyond these limits, changes in orifice velocities produce little change in the mean drop diameter, i.eo, the drops have a maximum stable size and also a minimum diameter whose reduction requires the expenditure of large amounts of energyo The most important variables affecting these boundaries are probably interfacial tension and liquid viscositieso Since only one system was investigated here the exact dependence on these fluid properties cannot be ascertained. In summary, the following observations have been made concerning the dispersion of drops in the system studied: 1) The material bf construction and history of a plate can be important in the determination of drop size. (17'41 51) This variable was eliminated through the use of a single material and a standard plate pretreatment.

-42 -2) Small drops are associated with small perforation diameters and high orifice velocities, and vice versa. 5) Drop diameters become more uniform and tend to conform more closely to the log-normal distribution as orifice velocities are increased. Marked deviations from this distribution are noted at low orifice velocities, and there is a tendency toward the formation of bimodal distributions 4) Between upper and lower limits the Sauter mean diameter can be correlated as a linear function of Vh/vo.

HOLDUP AND FLOODING Holdup, the volume fraction of the contacting section of the column occupied by the dispersed phase, is important in two respects. First, the interfacial area available for mass transfer depends on the holdup; and second, any counter-current column is limited by a maximum holdup beyond which unstable operation and flooding result. In particular, pulse columns can flood under two extremes of operation: flooding due to insufficient pulsation and to excessive pulsation. Both cases are treated in this section along with discussion of a simple approximate method for estimating holdup. Estimation of Holdup from Manometric Measurements Droplets rising at a constant velocity in a second liquid exert a "lifting" force on that liquid exactly equal to the gravitational or bouyancy force due to the density difference of the two fluids. Furthermore, the direction of the fluid velocity with respect to the column wall periodically changes sign in a pulse column. Consequently the wall shear is approximately canceled out during a complete pulse cycle (the pulse velocity is generally high compared to the bulk fluid velocity). From these considerations the time-average pressure drop over the length of a column is approximately equal to the effective density (times an appropriate conversion factor) of the two liquid phases in the column, and the holdup can be estimated from a simple manometric measurement. Neglecting inertial forces and wall shear and referring to Figure 12, a pressure balance yields: ( L -a ^ ) C _ ^^ CO CD C(/-~) \C -43 -

-44 -Air-Liquid Interface Continuous Phase Feed Liquid Hih Dispersed Phase " — ";T"^ ^ ^"//W ^ Effluent Ah lih ~~I Liquid Height 0 * Dispersed- Continuous * Phase Interface e Manometer, ~ 0 | l Cross Sectional Area _~ ---~*^,/' ~Of Column Is Measured *o 0 Here 0 Capillary For Damrping Pulsation Continuous Phase Effluent Dispersed Phase Feed Figure 12. Definition of Lengths used for Manometric Estimation of Holdup 0 Capillary For Damping Pulsation _- Continuous Phase Effluent L-Dispersed Phase Feed Figure 12. Definition of Lengths used. for Manometric Estimation of Holdup.

-45 -where g = gravitational acceleration c = holdup PD'PC = dispersed and continuous phase densities, respectively LAhh1 = lengths defined by Figure 12. Solving Equation (13) for holdup, we obtain hC ( -6D/1'C) l 1^(14) L Equation (14) was tested experimentally using the following procedure. The column was provided with a manometer similar to the one shown in Figure 12, except that a needle valve rather than a capillary tube was used to damp pulsation in the manometero The column was allowed to attain steady-state operation, and hi and Ah were observed and recorded. Then the continuous and dispersed phase feed lines and the aqueous effluent line were cut off simultaneously using toggle valves. The dispersed ketone drops rose to the top interface where they coalesced, thereby lowering the position of the interface. Just enough water was admitted to raise the interface back to its original position. The ketone thus displaced was collected and weighed, and the holdup evaluated from 14 (15) where WD = weight of dispersed phase collected, g Vcol = volume of contacting section of column, cc

-46 -1.0.9 0 f=100 RPM.8 Af = 200 RPM.7 - MIBK Flow Rate =2.98 cc./sec. H20 Flow Rate- Variable -UT.6.5 E.3 x W.2.1 0.1.2.3.4.5.6.7.8.9 1.0 Calculated Holdup Figure 13. Comparison of Actual- and Manometrically Estimated Holdup.

-47 -Holdup was also calculated from the manometric measurement using Equation (14)o The results are plotted in Figure 13. The comparison of calculated and experimental holdups lie very nearly on a straight lineo However, it may be noted that the slope is not unity, nor does the line pass through the origin, indicating that some of the forces disregarded in the derivation of Equation (14) are not entirely negligible. On the other hand the worst error incurred in any experimental point is about 25%. If a liquid level controller is used to maintain h1 constant, Equation (14) may be written in the form ~ C / h -C2 (16) where C1 and C2 are empirically determined constants which are approximated by 1 ~ —,1 __f (17) C and C -... (18s) A manometer can be calibrated with a linear scale to read holdup directly with an error of 5 to 10 per cent. It has the advantage of being continuously readable, and is a sensitive indicator of transient column behavior. The holdup indicated is the average value over the entire contacting section, and pointwise deviations from the average are not reflected.

-48 -Flooding Due to Insufficient Pulsation Countercurrent operation cannot be realized in asieve-plate pulse column without the action of the pulser: the light phase collects under a plate and the combination of surface and bouyancy forces constitutes an effective barrier which prevents the heavy phase from passing through the pla-ieso These forces must be overcome by energy supplied by the pulsero At low frequencies the pulser behaves much as a pump~ the light phase is pushed up through the plate perforations on the upstroke and the heavy phase is pulled down on the downstroke. If the pulse rate is decreased sufficiently, or the liquid flow rates increased the "pump" capacity will be exceeded and an accumulation of one or both phases takes place, ultimately resulting in floodingo As used here flooding is defined as "the flow condition when the fluid of one phase entering at one end of the column cannot leave at the opposite end and must exit through the effluent line intended for the second phase. t(50) An expression relating flow rates to pulsing conditions at incipient flooding due to insufficient pulsation can be derived from a volumetric rate balance. The following represents an extension of the work of Edwards and Beyer (16) h 2 ) 2 D \ T where vc = superficial continuous phase velocity, cm/sec vD = superficial dispersed phase velocity, cm/sec f = pulse frequency, seci a = pulse amplitude (in the column), cm

-49 -TABLE III EXPERIMENTAL AND CALCULATED VALUES FOR RUNS AT INCIPIENT FLOODING DUE TO INSUFFICIENT PULSATION Run f, sec-1 VD, cm/sec VC,cm/sec V calc,* 1 o.643.518.163 o194 2.0873.680 o306.274 3 0 928.666 o316 312 4 1,118.837.395.361 5 1o114.836.399.361 6 1,438.979 487 502 7 1o438.979.491.502 8 o,674.505.225.218 9 Oo675.500.224.221 10 o 804 0343 344.364 * Calculated from: VD-Uv c, 1 Ve [f a cosh (D fa ) vD] / (1 - ) where a= 0 a = 0.654 cm

-50 -6..5 ________ 4 0 0 ' 0 C..I Q. 2 0 0.1.2.3.4.5.6 Calculated Vc, cm./sec. Figure 14. Comparison of Calculated and Experimental Continuous Phase Superficial Velocities at Incipient Flooding Due to Insufficient Pulsation.

-51 -and where a is the fraction of the over-all average continuous phase flow rate produced during the downward, movement of fluid in the column. It is defined precisely by Equation (B-i) in Appendix B. Details of the derivation of Equation (19) are also given in Appendix Bo Equation (19) was tested experimentally using the following procedure- Dispersed and continuous phase flow rates were set and then measured by weighing samples collected over known intervals of time. Frequency was slowly diminished until the column started to flood. The frequency was then slightly increased until stable operation was restored. This frequency was measured using the clock-counter system described in a previous section. No continuous phase production occurred during the downstrokes, hence a f 0. Ten different runs were performed. The measured and calculated results are presented in Table III. For purposes of comparisons experimental superficial aqueous velocities are plotted in Figure 14 against values calculated from a rearrangement of Equation (19). The agreement is favorable. The apparently random scatter is probably due to the difficulty in measuring the inherently unstable mode of operation, and not to the equation. Within close limits Equation (19) specifies either maximum flow rates or minimum pulsing conditions when one or the other is seto The value of a can vary from 0 to 1, but it is usually near 0 at low continuous phase flow rates and pulse frequencies, and near 1 at high values of these parameters. Mean Orifice Velocity The "mean orifice velocity" was the basis of the Sauter mean drop diameter correlation presented in the previous section. However it

-52 -was stated that its defining Equation (12) only approximated the "true" mean orifice velocity. The latter is readily obtained from an analysis similar to the flooding derivation. We are interested in the average velocity during the upstroke portion of the cycle only because this is the time when the majority of the dispersed phase passes through the plates. Three factors contribute to this velocity. The upward sinusoidal displacement by the pulser and the continuous injection of ketone into the bottom of the column are additive contributions whereas the removal of the water phase from the bottom of the column diminishes the upward velocity. The mean orifice velocity during the upstroke is defined as the volume of fluid pushed up per unit cross sectional area divided by the elapsed time, or v~~b~~~~~z~ L Q-~f~~~~~ - -^ _iw_.L:.(20) where. = fraction free area of the plates 0 = sin-1 6 6 = (Vc-vD)/lxaf 7 = the average relative rate of production of the continuous phase during the upstroke with respect to the average rate of continuous phase production for the entire pulse cycle. The derivation of Equation (20) is presented in Appendix C. As 6 approaches zero, Equation (20) reduces to /21f~ = (2 i /)/<5 (21)

-55 -At high frequencies and in the absence of check valves y ->1, therefore (21) becomes X~ = (2 at+ 2- )/ (22) The product of frequency times amplitude is generally large compared to the superficial velocity of either phase in emulsion type operation. Therefore the group (fa/0) is of primary importance in determining pulse column performance. Equation (20) is a rigorous expression for the mean orifice velocity during upward movement of fluid with respect to the column walls, but various degrees of simplification such as that represented by Equation (22) can be utilized with fair accuracy under certain conditions. Note that in the above equations the dispersed phase was taken to be the light phase. Modifications must be applied if the heavy phase is dispersed. Slip Velocity, Holdup, and Flooding Due to Excessive Pulsation The concept of slip velocity is certainly not new(37'40) but to the author's knowledge it has rarely been exploited to its full utility. Slip velocity is defined as the average velocity of the population of droplets rising or falling in a column with respect to the mean velocity of the continuous phase flowing in the column. It can be considered to be somewhat analogous to the terminal velocity of a single drop, and is a useful parameter for describing holdup. Consider now a simple column operating under steady-state counter-current flow. Drops of ketone enter the bottom of the column at a constant rate, D cc/sec, and rise at a uniform velocity. Water enters

-54 -+he top of the column at a constant rate, C cc/sec, and flows down. Assume for the moment that the drops are uniformly distributed throughout the entire column. By virtue of the ketone injection and water effluent at the bottom, an element of fluid in the column will undergo a -teady vertical translation at the rate t) -C/5 (25) (2.5) where A = cross sectional area of column. Tn addition to this translation the ketone drops will move upward with respect to the fluid element at a rate determined by the balance of bouyancys drag, and inertial forces. Assume the absence of mass transfer. Since steady-state operation prevails, the rate of ketone production must equal the injection rate, and this production rate must equal the volume of ketone passing an arbitrary plane lying normal to the bulk flow. The relationship expressed in symbols is D = A C _ tc) (24h wxhe re - the volume fraction of fluid space occupied by ketone drops vD - superficial ketone velocity - D/A, cm/sec vc - superficial water velocity = C/A, cm/sec vt - velocity of drop with respect to the element of fluid But the term, vt, is loosely defined, because an upward movement of the drop must cause an equal downward volumetric displacement of water. The relative motion of the two phases gives rise to slip velocity, Us: si/ =it t (,_) ~'(2$ t

-55 -The first term on the right-hand side of (25) represents the velocity of a drop with respect to the water if the water were stationary and the second term is the mean downward water velocity caused by the rise of the drop. Eliminating vt between Equations (24) and (25) U, = -^ u (26) In terms of the definition of slip velocity, Equation (26) is a rigorous expression for any differential volume element, Aody, in the column. If the parameters involved are fairly constant throughout the column, Equation (26) may be used to represent the entire column, and even if rather marked inhomogenities exist it is still a convenient correlating parameter. Equation (26) can be de-dimensionalized by dividing both sides by Us. Its rearrangement for an equation explicit in holdup appears as (l -(.C e lax +/i? + )2 4 )- (27) where ~= l' AUC ((28) (~..~ - A UL and D? = A- D (29) ^o - U, A Us For physical reality the expression under the radical in Equation (27) cannot be less than zero. Partial differentiation of Equation (27) with respect to either RD or Rc results in expressions involving the square

-56 -root term as a denominator. Hence as the square root term approaches zero the rate of change in holdup with respect to RD or Rc becomes infinite. Physically, this corresponds to the flooding point. Thus at incipient flooding conditions -/ C- -:,) =4 PoD( l/ a (30) Also s t/,,d 2(-^Ro) 2 i D(31) If we define P ~' l0 (32) and combine Equations (30), (51), and (52), we find that 2 J P= 1 2 This equation implies the rather remarkable conclusion that the holdup at incipient flooding in countercurrent flow is a function only of the ratio of the volumetric flow rates and is independent of fluid or column properties. The preceding statement strictly applies to local holdup and local incipient flooding if the holdup is non-uniform throughout the column. Pratt(36) also concluded that holdup at incipient flooding is independent of fluid and column properties, but he presented a

-57 -different relationship. In terms of the present nomenclature Pratt's equation is C1 (n ) (L4) Equations (33) and (34) are not equivalent. This is a disconcerting situation; therefore the equations were subjected to an experimental test: the measurement of holdup at flooding or near flooding conditions. Measurement of the exact point of incipient flooding is difficult due to the transient nature of flooding. Visual observations of the onset of flooding were aided by manometric measurements as described above. The approach to flooding is accompanied by a sharp increase in holdup. This was indicated by the manometer. At the onset of the rise, the operation was "frozen" using the toggle valves, and holdup was measured gravimetrically as described before. Results are listed in Table IV. TABLE IV COMPARISON OF CALCULATED AND EXPERIMENTAL HOLDUP AT NEAR-FLOODING CONDITIONS Amplitude = 0.654 cm for all runs f, C, D,, Eflood, Eflood, Run RPM cc/sec cc/sec Experimental Eq. (33) (Pratt) 92 200 3548 2.24.452.446.302 36 100 3.96 2o24.413.429.291 55 150 4.49 2.24.302 o415.280 44 200 2.52 2.24.408.484.324

-58 -Runs 92 and 36 listed in Table IV were at incipient flooding, and Rans 55 and 44 were conducted at about the limits of stable operation, but somewhat below their respective flooding points. It can be seen that the experimental values lie above Prattgs limiting holdups in every case, even for stable operation, whereas they are approximately equal to the values predicted by Equation (33) for the incipient flooding runs, and properly below the limiting values in the non-flooding runs. It would appear, then, that the data for this system support Equation (33) rather than Equation (34), The slip velocity relationship, Equation (27), is of utility because the variation of holdup with the flow rates of both phases can be expressed in terms of only one parameter, US. Equation (27) is plotted in parametric form in Figure 15. If the slip velocity were held at constant levels the parameters represented by the solid lines in Figure 15 can be interpreted to be the variation in holdup with continuous phase superficial velocity at constant values of dispersed phase superficial velocityo It is seen that as the continuous phase velocity is increased a point is reached where the holdup rises sharplyo Further increase ultimately results in flooding. The flooding line is a plot of Equation (31). A cross-plot of Figure 15 would reveal that for constant slip velocity conditions, holdup is nearly proportional to the dispersed-phase superficial velocity when the continuous-phase rate is held constant. An approximate "limit of stable operation" line has been included in Figure 15. The line intersects the ordinate at holdup somewhat less than 0.5. This is based on the observation that a high degree of

-59 -1.0 =.G6 J6/-f Xr Lines Of Constant VD/VC ___ _.6 VD.,C 5. a.4 7 -0. 0.1.2.3.4.5.6.7.8.9 1.0 Vc/us Figure 15. Slip Velocity-Holdup Relationship for Countercurrent Flow.

-60 -coalescence and phase inversion occurs at higher holdups. The intersection with the abscissa at vc/Us =.85 is based on the limitation imposed by droplet entrainment. Since a distribution of drop sizes are present during operation of a pulse column, some of the smaller drops may be swept out the continuous phase effluent line when their terminal velocities are exceeded. When the slip velocity is known or can be estimated for a given pulsing condition, throughput for a predetermined ratio of dispersed to continuous-phase flow rates can be determined from Figure 15. Since mass transfer considerations generally dictate high holdup operation, a point on one of the broken lines, corresponding to a given flow ratio, near the limit of stable operation will define a given vc/Us value. Knowledge of the slip velocity then permits evaluation of the continuous phase superficial velocity. The fixed ratio of superficial velocities in turn sets the dispersed-phase rate. Experimental measurements of holdup are compared to the prediction of Equation (27) in Figures 16 and 17. Figure 16 shows the variation of holdup with ketone flow rate at a constant water rate and constant pulsing conditions. One would expect a fairly constant slip velocity under these conditions, and indeed the line (Equation 27) representing a constant slip velocity shows good agreement with the points. As mentioned before, holdup is nearly proportional to dispersed phase flow rate until flooding is approached and the curve rises sharply. Figure 17, showing the variation of holdup with continuous phase flow rate at a constant ketone rate, indicates that slip velocity does not remain constant as the water flow rate is increased. The broken lines represent

-61-.7.6 __ Points Are Experimental Line Is Calculated From Eq. 27 With Us= 3.25 cm/sec..5 Frequency = 200 RPM___ Amplitude =0.65 cm. Continuous Phase Rate = 1.08 cc./sec. Perforation Diameter= 1/16 inch..4 I 4o I ly.3.2 0 0 I 2 3 4 5 Dispersed Phase Flow Rate, cc./sec. Figure 16. Variation of Holdup with Dispersed Phase Flow Rate.

-62-.6 Frequency, RPM 0 100 A 150 X 200.5 MIBK Rate =2.24 cc./sec. Amplitude =0.65 Perforation Diameterz 1/16".4 /' /.3 Lines Of Constant__ /Fw__ Slip Velocity of li o,Z 0 2 3 4 5 Continuous Phase Flow Rate, cc./sec. Figure 17. Variation of Holdup with Continuous Phase Flow Rate.

-63 -V + 0 o X ~ A0 0 0.65 0.32 2.93 1/16 0/ 3 *_____ i^ — KEY: 1.86 0.64 2.93 1/16 ^I 0 c 1 4.47 0.32 2.93 1/16 * 4.47 0.64 2.93 1/16 V 0.65 0.32 2.93 1/8 / 0.65 0.64 2.93 1/8 A 0.65 0.60 2.93 A/8 0.04.08.12.16.20.24 1/2 Ti/Vo., sec/cm. Figure 18. Correlation of Slip Velocity.

-64 -constant slip velocities. It is seen, for example, that the slip velocity at the 200 RPM frequency decreases from 3.2 to 2.6 cm/sec as the water rate is increased, showing that slip velocity is a function of holdup as well as pulse velocity. Holdup was measured for each of the mass transfer experiments (to be described later). Slip velocities were calculated as a by-product using Equation (26). The results were approximately correlated by the same function utilized to correlate Sauter mean diameters, fh/vo, and are shown in Figure 18. It should be pointed out that this correlation may be a gross over-simplification, and is included only to demonstrate the trend in slip velocity as operating conditions are varied. It does not, for example, take into account the effect of holdup on Us. Nevertheless the trend is clear: slip velocity varies approximately in the same manner as terminal velocity. This would be expected from the terminal velocity analogy for single drops. Summary and Conclusions Holdup affects pulse column performance in two ways: through its determination of interfacial area, and its effect on flooding characteristics. A simple manometric technique for estimating holdup is presented, yielding values to within about 25% of the actual holdup by direct application of a pressure balance. Better results can be obtained from a calibration. An equation relating flow rates to pulsing rates at incipient flooding due to in iien ain insufficient pulsation s discussed. Its generality has been extended beyond the original derivation. (16) The equation appears to be well supported by experimental data.

-65 - A rigorous expression is developed for the mean orifice velocity. The group fa/0 is shown to be the largest contributing factor in most instances, but not a complete measure of the average local velocity at the sieve plates. Holdup at the upper flooding limit was shown to be independent of fluid and column properties. Discrepancy was noted between an equation appearing in the literature and the one developed here, but the present data support the latter equation. Holdup is conveniently related to flow rates by a parameter called slip velocity. Solution of the slip velocity equation is presented graphically and approximate limits of stable operation are shown. A slip velocity correlation is also given.

LONGITUDINAL MIXING The high rate of mass transfer in pulse columns can be attributed in part to the high degree of turbulence and mixing producedo Unfortunate] _, however, this same mixing in the axial direction has a detrimental effect on mass transfer rates through reduction of the concentration gradient. Backmixing has been blamed( to have such a profound effect on extraction efficiency that it may mask improvment due to the greater interfacial area associated with the smaller drops produced at high pulse rateso Effective axial diffusion coefficients to be incorporated in the mass transfer equations (next section) were determined using a dye tracer techniqueo The experimental methods, mathematical treatment and resulting coefficients will be presented in this sectiono Experimental Methods The experimental setup for axial mixing experiments is shown schematically in Figure 19o The technique consists essentially of injecting a pulse of dye solution into the top of the column and observing its concentration from light transmittance near the top of the column and again near the bottom, after it has diffused and mixed while being carried down the column by the continuous phaseo For the majority of the experiments only the continuous (aqueous) phase was employed, but the effect of the dispersed phase on axial mixing was tested by several experimentso Approximately 0O2 cc o66 -

-67 -Center Rod Dye Injection Point Feed Water Inlet\ Photocell Input Photometer Lamp -- Lens Filter Slit __ f_// Sanborn Dual Channel Oscillograph Perforated Plates _ _ OpOutput Photometer Figure 19. Dye Tracer Apparatus.

-68 -of concentrated dye solution was rapidly injected at a point below the ketone-water interface for each run after the column had attained steady-state operation. The injection was accomplished through the use of a length of number 100 PE polyethylene tubing connected to a standard hypodermic needle and syringe. A 1/8 inch polyethylene tube served as a guide for the small tube to insure reproducibility of the injection point. The concentrated dye solution was made from Fast Green (Allied Chemical, Cat. No. 537) biological stain powder added to distilled water at an approximate concentration of 3 g/l. In all subsequent statements this will be considered to be the unit concentration of the dye solution. An in situ calibration of the dye concentration, photocells, and Sanborn dual channel oscillographic recorder was accomplished by a method of successive dilutions. Water was added to 8 cc of concentrated dye solution to make one liter of dilute solution. This solution was poured into the empty column and the output signals from both photometers were recorded by the Sanborn recorder. The column was then drained, and the drained dye solution was diluted by a factor of two. The 4 cc/1 solution was poured into the column and the process repeated through several successive dilutions. The zero adjust and the sensitivity of the recorder were always set to produce a full scale reading when the column was filled with clear water and a zero reading when an opaque card was placed in front of the photometer, blocking out all light to it. In this manner the readings were always normalized to cancel any variation in the light-source intensity or the photometer sensitivity. Results of the "system" calibration are shown in Figure 20.

-69 -a 0 1 2 3 4 5 6 7 8 9:r2 0 0 2 3 4 6 7 8 9

-70 - The dye calibration data were checked for conformance to Beer's law: I/ I = & (35) where I = intensity of the incident light I = intensity of the transmitted light c = concentration of absorbing solution K = absorption coefficient. For convenience Io was taken as the intensity of the transmitted light from a clear solution containing no dye and the output signal was assumed to be proportional to the transmitted light intensity. The scale was reversed for all readings such that zero dye concentration (maximum signal) was read as 0 scale reading, and zero light transmittance (no signal) was read as 5. Hence, in terms of scale readings f 5 R(36) 0 and Beer's law may be written as /z ( i S.. - - C(37) The dye calibration data are plotted in accordance with Equation (37) in Figure 21. The system obeys Beer's law in the dilute range. Most of the dye concentrations measured in the axial diffusivity experiments fell in the range: 0 - 2 cc/1. The Sanborn recorder was allowed to warm up for at least four hours before any run in order to attain relatively drift-free operation. The instrument was also re-normalized before each run if necessary.

-71 -1.0 0.9 --- —, 0.8 0.7 0.6 0.5 = 0.4 0.3 0.2 0. I_ 0 I 2 3 4 5 6 Dye Concentration cc. of dye/s Figure 21. Beer's Law Plot for Dye Concentration Calibration.

-72 -Typical traces for a run are presented in Figure 22. The upper trace corresponds to the top photometer and the lower trace to the bottom. The passage of the concentrated dye zone past the photocells is clearly shown by the dip in each curve, and the time displacement between the two traces provides a measure of the flow velocity since the photocells were separated by a known vertical distance. The fine oscillations in the trace are from the sinusoidal pulse motion superimposed upon the steady translational flow. The dye slug moves down the column rapidly on the downstroke of the pulser attenuating the light transmission. But it is pushed back up the column during the upstroke, thus producing the sawtooth trace. This illustrates the strictly unsteady-state behavior of the pulse column. The operation is periodic and "'steady-state" applies only when one refers to the same phase angle of each period. A continuous recording technique such as the one employed is a requisite for accurate determinations in a periodic system since it permits comparison of reading at the same phase angle. The bottom of the stroke was chosen for the present work, i.e., all readings were taken from an imaginary smooth curve drawn from peak to peak of the saw-tooth traceo This corresponds to the bottom-most points of the trace on the left hand side of the major dip and to the uppermost points on the right hand side, since in one case the dye pulse is coming toward the photocell and in the other it is going away. The heavy line at zero scale reading in Figure 22 was caused by 60 cycle flicker of the light sources.

t_ o al"Hilmi M 311!!13, ' -....... X rfii 11 mIas l S4Tlt-t-i ---- --- ttliSf iii 3 f ~ 113~ i L -- - 'ic f~i ~ 'lf -~t f ' 3 ' ' '.1 " ' " t| —3 ---1 --- — I fil --- - --— il-il --- I — i ~~titit i_,t._...._....... _ _ _ _ a f~tt --- i g g ifE IO"' -; --- -- - - -I-OL W:1 41:j;:1 161,i__ 1 OD j indu I* - @DJj IndinO iUOIDJIUUOo *LU BUISDSJOUI-_ 4 4i t toit ---- -- z:t~~~~~~~;-,;t:t~~~~~~4 0~~~~~~Uftf'!',~itt~ iit~it- ~ -_t~i FH~~ ir t~t O n In O cu n a~~ I.1-4 tALeoil~n~n T Jlu~u 3sQ Ib!oru( ---+

-74 -Mathematical Treatment If it can be assumed that Fick's law adequately describes longitudinal mixing in the present system, then a material balance around a differential section of the column, assuming no radial or angular variation, yields the well known diffusion equation D -& __ - _ (38) e cd 2~ dd t where D = effective diffusivity for axial mixing, cm /cm ue = continuous phase velocity, cm/sec x = dye concentration, weight fraction y = vertical length, cm t = time, seco It should be pointed out here that we are interested only in longitudinal mixing of the continuous phase since the dispersed phase is always at a uniform concentration (saturated) throughout the length of the column, hence its axial mixing is of no consequence. Boundary conditions for Equation (38) have been discussed at length in the literature(l912926) and the ones generally used are not the boundary conditions to be employed here. However all concentration sampling is performed in the column, inside the entrance and exit boundaries, and the boundary conditions presented below are entirely proper: 1) x(y,0) = 0 (39) 2) x(0,t) = F(t) (40) 3) lim x = 0 (41) y - 00

-75 -For convenience the physical system is defined by the following: 1) y = 0 at the top photocell slit. 2) y = L at the bottom photocell. 3) t commences from 0 at the instant the first dye particle reaches the top photocell. Thus Equation (39) states that the test section contains no dye initially; (4o) denotes that the dye concentration at the top photocell after zero time is an arbitrary, but measurable function of time; and Equation (41) states that the dye concentration must approach zero at an -infinite distance down-stream, implying that the dye injection must be of finite duration and not a step function, for example. All of these conditions are met experimentally. It is convenient to cast Equations (38) through (41) into dimensionless form by the following substitution of variables 7 ~a ^//' 4(42) (; = "t/ (45) '0 e 6.-."./ {z.,// (44) Equation (38) through (41) now become,: ) xz 0)x _ -9 (45) 1) x(z,O) = O (46) 2) x(o,@) = F( ) (47) 3) lim x = 0 (48) Z -> oo

-76 -Application of two theorems relating to Laplace transforms permits calculation of the effective axial diffusion coefficient from the equation: De4 L. [ — (M )-M (0)) ('~()] (49) 2 LF 2 where O o - All: — t X ^(ot0)t (50) and / (GL) C t' (L,t) cIt (51) 0 The moments Mn are obtained from numerical integration of the input and output concentration profiles. The solution to Equation (45) appears as o 2 Pe ((2 aC/ -r cJ'7 (52) or in terms of the dimensional variables, as (I' t) 2e^r^ t3/ 2 1/ ^'e2) 2)j (55) 2l it ' I /7- where T is a variable of integration, and x(0,t-T) is recognized to be the concentration profile at the top photocell, or F(t-T) corresponding to Equation (40). All numerical integrations were performed using Simpson's rule. Nearly all computation throughout this entire thesis was carried out on either an IBM 709 or IBM 7090 digital computer1

-77 -Details of the solutions to the diffusion equation are outlined in Appendix D. Equation (49) represents a simple, but powerful tool for obtaining longitudinal diffusion coefficients from experimental concentration profiles. All data points are employed in the evaluation of De and the experimentally difficult production of a tracer input of some exact shape such as a sine-wave or square-wave is not required. Instead, a pulse of an arbitrary shape, such as can be produced by the manual discharge of a hypodermic syringe, is employed. Results of Diffusivity Experiments Scale readings and the corresponding concentration profile for the input trace given before are presented in Figure 23. The output concentration profile for the same run is shown in Figure 24. The points are concentrations calculated from the recorder trace, and the curve was calculated using Equation (53)* The effective diffusivity was obtained from Equation (49). Three independent checks on the velocity used in Equation (49) were available. The first was from a rotameter reading; the second was calculated from the time displacement of the concentration peaks on the traces, and the third from U = ( M~ ) L (54) Equation (54) is a by-product of the mathematical analysis presented in Appendix D. The three velocities were 0.686, 0.689, and 0.723 cm/sec, respectively. The average of the last two values, 0.706 cm/sec., was utilized in Equation (49). All three values are seen to lie within about 2 1/2% of this average.

-78 -O0 * 0, c i 1"1 I). ii cI o 4 H I II, c -rl.o 4 - - I I / / a/ 0 O 4~ C ~4 o 4' Co C) 6uipo9^ o eeoS jlpjoelj u1oquoS JO T/'0 * uoiiojtuou *AQ o H. a., ~, ~/' * o ~

-79 -m 0 0o.. o ~0 0 3| E e / o " C).t 0 C 0. 11 /" -- CoU~~~ ~~ Io 4- o- e 0,,- o.o 0._ ____ O_ 0 0 ~ ^ 0.0f 0u,~~~~~~~~~ o ^ —~o o C D IE: *~- ~i-: 4' ''3 0 LL 7: d * Oi o0 W O 0 -t/oo 'uo!fDl.ueouo o *AO

-80 -The close agreement between the calculated curve and experimental points, which was typical for all high pulse velocity runs shows good evidence for the validity of the mathematical treatment and assumptions involved in the description of longitudinal mixing as a diffusional process. Another phenomenon was observed to occur at the lower pulse velocities. Instead of the axial mixing decreasing at lower and lower turbulence levels as one might expect, it increased sharply. With no pulsing and completely laminar (single phase) flow the axial mixing was greater than for the most severe pulsing conditions measured. This observation is quite in accordance with a well known analysis.(5 ) Axial mixing can be caused by non-uniform velocity and subsequent radial mixing, which is sometimes called Taylor diffusion after G. I. Taylor's analysis of axial mixing in pipes. Tichacek and associates(58) analyzed axial mixing in straight pipes by a modification of Taylor's work. Their analysis indicates that axial mixing increases rapidly as the flow approaches the laminar region. Since the mechanism here is a sort of "variable radial retention" rather than eddy diffusion, the diffusion model cannot be expected to apply. If one attempted to interpret the results from a diffusion model it would appear that little forward diffusion takes place, but that it is rapid in the backward direction with respect to the bulk flow. This is illustrated in Figure 25, an illustration of a low pulse velocity run. The points and broken line are experimental and the solid line is calculated from the diffusion model. It is

-81.O -C~~~~~~~~~~~~~~C O 0 cr 0 I O) LlS II C~O o * 4 — 4- 6 ' CO uo 0. 0D 0D " (" S h. C' -,-U 0 0-I *- H e o —: d.(~'^^~ ~~~~~~~~~~ ~ 4c e0 0 -. w E* "ACM 0 C CM* CM ~~~~- 0 ijy/o *9O R ~''-44D -,-I ~t 0 0) NU 0) 0 N/~~~ Nuo~lor~ue~uo3 - - 7/00 6 UO!IDJIUS0UO~~~..a

-82 - evident that there is less "diffusion" on the leading edge of the dye pulse (at small time values) than is predicted by the model. The method of determining a mixing coefficient from the moments of the distributions (Equation 49) probably yields the best equivalent diffusion coefficient to represent combinations of eddy and Taylor diffusion since all data points are utilized in the calculation. The use of the second moments in the computation places a heavy weighting on the tail, or backmixing portion of the profile since the square of time appears in the second moments. A mixing coefficient computed from a single concentration corresponding to the peak of the output profile gives a measure of the mixing due to pure eddy diffusion. Such coefficients were calculated for all runs by trial and error insertion of De into Equation (53), until the calculated and experimental peak concentrations agreed to within one per cent. Experimental conditions and effective diffusivity coefficients for axial mixing, computed using both the peak concentrations and moments (Equation 49) are summarized in Table V. The "peak" diffusivities can be interpreted as the contribution from eddy diffusion, and the "moment" values as the sum of both eddy and Taylor diffusion. It can be seen from Table V that the entire axial mixing at zero pulse frequency is due to Taylor diffusion, whereas it is accomplished completely through eddy diffusion at very high pulse velocities. The entire spectrum of dual contributions appears in the intermediate pulse-velocity range.

-85 -TABIE V EXPERIMENTAL CONDITIONS AND RESULTS FROM AXIAL DIFFUSION EXPERIMENTS Run f,RPM a* Vc, vD, ht De, cm2/sec De, cm2/sec cm/sec cm/sec calculated calculated from peaks from moments 43 0 -.700 0 1 0 5.87 44 50 1.732 o 1 0.60 1.35 45 100 1.694 0 1 0.70 1.66 46 150 1.704 0 1 0.54 1.27 47 200 1.696 0 1 0.43 0.80 48 0 -.950 0 1 0 5.04 49 50 1.9500 1i 0.66 3.50 50 100 1.950 0 1 1.01 1.69 51 150 1.951 0 1 0.69 1.54 52 200 1.945 0 1 0.59 1.21 53 200 1.257 Q 1 0.32 0.41 53a 50 1.257 0 1 0.21 0.42 54 50 3.653 0 1 1.38 1.50 55 100 3.573 0 1 2.72 2.26 56 150 3.675 0 1 4.19 3.70 57 100 3.814 0 1 2.79 2.61 58 50 3.245 0 1 1.63 1.29 59 150 3.231 0 1 5.7 77.77 60 50 2.674 0 1 0.46 o.64 61 100 2.706 0 1 0.79 0.79 62 150 2.658 0 1 0.97 0.86 ~6I3 50 2.958 0 1 0.50 o.84 64 100 2.887 0 1 0.72 1.02 65 150 2.897 0 1 1.22 1.44 66 50 2.238 0 1 0.28 0.46 67 150 2.232 0 1 1.03 0.86 73 50 2.648 0 2 0.50 9.55 74 100 2.611 0 2 0.65 0.69 75 150 2.559 0 2 1.14 1.42 76 50 2.850 o 2 0.55 0.79 77 100 2.857 0 2 0.52 0.89 78 25 3.847 0 2 o.91 1.17 79 50 5.814 o 2 1.27 1.78 80 100 3.585 0 2 3.04 2.69 81 50 3.607 0 2 1.22 1.41 81a 25 3.632 0 2 0.72 0.93 82 50 1.661 0 2 0.57 0.86 85 100 1.658 0 2 0.58 0.85 84 200 1.651 0 2 0.37 0.73 85 300 1.617 o 2 0.49 0.75 86 200 1.840 0 2 0.50 L.26 87 100 1.876 0 2 0.48 1.04 88 100 1.855.147 2 1.28 1.60 89 200 1.897.147 2 1.51 2.82 90 200 1.847.074 2 1.81 2.46 91 100 1.852.074 2 o.84 1.51 92 25 3.849.074 2 0.73 1.14 95 25 3.868.147 2 0.86 1.38 94 25 3.886.216 2 0.92 1.72 95 10 3.859.147 2 1.17 1.58 96 10 3.891 0 2 0.73 1.56 * Amplitude values: 1 = 0.65 cm tPerforation Diameters: 1 = 1/16 inch 2 = 1.86 cm 2 = 1/8 inch 3 = 4.47 cm Plate spacing = 2 inches, and per cent free Area = 52.8%6 for all runs.

-84 -Amplitude =0.65 cm. Perforation Diameter = 1/8" H20 Rates, cc./sec. 3 X 2.98 i' A $.6,~.,," ou 0 3.96 0 50 100 150 200 250 300 Frequency, RPM ~ | \ c~~MIBK Flow Rate =0 c0 E 4 0-1.08 —_____0 108 o \.. x 298 O J 3.96 0 50 100 150 200 250 300 Frequency, RPM Figure 26. Effect of Experimental VAriables on Effective DiffPusivity for Longitudinal Mixing. E04 0 1.08 S.0 50 100 150 200 250 300 Frequency, RPM Diffusivity for Longitudinal Mi~xing.

-85 -8 7 Both 1/16" And 1/8" Perf. Diameters Are Represented Flow Rates, cc./sec. 6 Symbol C D -- 0 1.08 0 m X 2.98 0 E 5 A 3.96 0 0! 3.96 0.64._ 4l * 3.96 0.94 4') 3~ L^ ]I,~ z~ Amplitude = 4.47 cm. 0 Amplitude =1.86 cm. 00 5 100 150 200 250 Frequency, RPM Figure 27. Effect of Experimental Variables on Effective Diffusivity for Longitudinal Mixing.

-86 -6 5 54)::~ MIBk Flow =O Amplitude, cm. 4 | 0 00.65,1 1 8 6.86 = iiii _....-.-. —.-.-.............-'..-...'. I I................................,..0.:.......::: ' ' ' ' /........:'."'.'"'" '''".'X' 77 0 2 3 4 5 6 7 8 9 10 11 12 fxa, cm./sec. Figure 28. Simplified Correlation of Diffusivity Coefficients.

-87 -The over-all coefficients are plotted in Figures 26 and 27. The effect of dispersed phase flow is shown in the upper portion of Figure 26 and in Figure 27. Increasing flow rates of either phase tend to increase the effective diffusivities. A simplified correlation which entirely neglects the effect of flow rates is presented in Figure 28 for illustrative purposes. The general effect of pulse velocity is shown. Fortuitously the pulse velocities most commonly utilized for convenient column operation occur at the minimum of the band! For design purposes, a value for De of 2-3 cm'/sec. in this range should yield conservative results. Since the goal of the present research was not to exhaustively study axial mixing, but rather to measure specific values for use in the mass transfer equation a more complete correlation was not attempted. Summary and Conclusions A simple but powerful technique was developed for the measurement of axial diffusion coefficients. The method assumes no prescribed shape of the tracer input pulse. Significant contributions to axial mixing by Taylor diffusion were observed to occur at low pulse velocities. These gradually diminish with increasing pulse velocity, and at very high pulse rates all axial mixing is by eddy diffusion. Molecular diffusion appeared to be negligible. Increasing flow rates of either phase increased the diffusion coefficient. For the major portions of available operating ranges, De varied from about 0.5 to 3 cm2/sec.

STEADY STATE MASS TRANSFER While the diameter of a column is often governed by throughput and flooding considerations, the height required to perform a given job 3 a function of the rate of mass transfer in the column. The high rates attainable in a pulse column and the correspondly shorter lengths required were major factors for their extensive adaptation by the atomic energy industry. Pulse columns provided sufficient height-reduction in a uranium extraction process to permit installation of the process in an existing building at Hanfordo (9) Experimental details and mathematical treatment of mass transfer data taken during the present research will be presented in this section along with a discussion of the resultso Experimental Methods Column Startup and Operation Following the pretreatment procedure (described before) the cartridge was lowered into the column which was full of aqueous phase up to the principle (top) interface, and organic phase thereabove. The pulser was started and the frequency adjusted to the desired value, the amplitude having been preset by insertion of the proper eccentrico The line connecting the pulser to the column was provided with a valve at its highest point which was opened in order to purge any air trapped in the lineo The presence of air or other -88 -

-89 -non-condensable vapors in the pulser line would cause changes in the pulse characteristics. Needle-valve settings were approximately adjusted before a run was started. The water flow was started first by simultaneously opening toggle valves on the feed and effluent lines. The watersaturated MIBK feed was started by opening another toggle valve. Final, careful adjustments were made to the needle valves controlling both feed rates. The rates were set to within about 1 to 2% of desired nominal values indicated by the rotameters. The ketone-water interface was always maintained at the same level for all runs. This level was indicated by a mark located near the top of the one inch diameter section of the column. The interface was controlled by vertical adjustment of the hydraulic leg with a pulley and wire-cable arrangement such that the pulsating interface dropped to the mark at the bottom of each stroke. Preliminary runs were carried out with the interface well up in the enlarged calming section as recommended by a previous investigator(41), but it was lowered for two reasons. First, the concentration of MIBK in aqueous-phase samples taken at a point just below the calming section was nearly identical to the effluent concentration, showing that almost all the mass transfer had taken place in the calming section and very little in the column-proper. This is not entirely unreasonable to expect since the volume of this portion of the calming section was greater than the remaining volume of the column. Thus the residence time was greater in the enlarged portion

-90 -than in the remainder of the column, and the mass transfer rate was higher due to the larger driving force and also due to a peculiarity of the system. The second reason for lowering the interface was to increase the accuracy of holdup measurements. A nine to one vertical-height-tovoluJei- ratio advantage was achieved at the smaller diameter. An experiment to determine the rate of mass transfer at the ketone-water interface in its new position was performed. The column was pulsed at 200 RPM. The aqueous flow was stopped and a high ketone rate was employed until all the aqueous phase contained in the column was nearly saturated. The ketone flow was then discontinued and all the droplets were allowed to rise to the top interface and coalesce leaving the column filled only with the nearly saturated watero The interface was adjusted to its normal level and a constant water flow rate was started at time zero. The aqueous effluent was then periodically sampled. The results are plotted in Figure 29, where %o represents the initial concentration of ketone in the aqueous phase, Figure 29 shows that, under the experimental conditions employed9 mass transfer from the upper interface was negligibleo It also illustrates the length of time required for the outlet concentration to reach steady state after the fluid flow has attained a steady condition. The slow decay of the effluent concentration as illustrated by Figure 29 was caused by the large volume and long residence time in the bottom calming sectiono As a result of the slow approach to

1.0.8 I. l W Frequency =200RPM * | Amplitude =0.65 cm. x% I~ ~MIBK Flow Rate =0 x W\ Water Flow Rate = 1.08 cc/sec..4 ------.2 0 - 0 0 20 40 60 80 100 Time, Minutes Figure 29. Determination of the Mass Transfer Rate at the Principal Interface.

-92 -steady state, the volume of the bottom section was reduced by about 2/3 for experimental convenience. The approach to steady state operation was accelerated correspondingly. However, during mass transfer runs at least one hour of steady operation elapsed before any concentration samples were taken. The effluent flow rates of both phases were determined by weighing samples collected over measured intervals of time. Both weights and times were measured to four significant figures for all runs. The gravity-feed system delivered extremely constant flow rates Material balances were checked by comparing the ketone feed rate, calculated from IDc - Do ~t~I DcO/' (55) where D. = ketone (saturated with H20) feed rate, g/sec. D = ketone outlet rate, g/sec. x, = weight fraction of MIBK in the outlet aqueous phase. C0 = outlet aqueous rate, g/sec y* = equilibrium weight fraction of MIBK in watersaturated ketone phase = 0o9817 at 25 C. to the ketone feed rate estimated from the rotameter reading. In nearly all cases the flow rates checked to within one or two per cento Closer checks were beyond the accuracy to be expected from the rotameter readings. Photographs were taken at four locations along the length of the column. Aqueous effluent concentration samples were collected at intervals during operation of the column, well after steady-state

-93 -operation had been attained. The temperature of this outlet stream was observed and recorded. Attempts were made to sample the three intermediate locations via the hypodermic needles and siphon tubes while the column was in operation, but droplets of ketone were siphoned over, along with the desired continuous phase. These droplets transferred mass into the aqueous phase causing high readings and considerable scatter in the concentration profiles. The droplets were entrained even after nearly closingoff the ends of the sample needles and collecting the samples at rates of less than one cc/minute. Much better results were obtained by stopping both feed rates instantaneously, allowing the drops to rise to the principal interface, and immediately withdrawing samples. No drops were entrained in this manner, and the results proved to be much more satisfactory. The initial portion of the samples delivered by the tiny polyethylene tubes were discarded, then 3-5 cc samples were collected and analyzed on a refractometer. The samples collected in this manner had concentrations which agreed well with a few samples collected during steady state operation in which the droplet entrainment was avoided. Holdup measurements were taken by volumetric displacement and subsequent weighing of the coalesced keytone drops. Calibration of Pulse Amplitude Measurements of the amplitude of the fluid pulsation were first attempted utilizing a method suggested by Sanvordenker(41) and Thornton.(56) A pointed stainless steel probe was silver-soldered onto a micormeter screw, which was connected in series with a NE-51

-94 -neon glow lamp and suitable resistance. The tip of the probe was moved vertically in the region of the principal ketone-water interface by turning the micrometer screw. The aqueous phase was conductive and caused the lamp to glow when in contact with the probe, but the ketone phase was not sufficiently conductive to activate the lamp. When a measurement was made, the probe was lowered until the tip just contacted the interface on the upstroke of the pulse, indicated by a flash of the lamp on the peak of the stroke. The probe was then lowered until the lamp was extinguished only instantaneously at the bottom of each stroke. The difference in the two micrometer readings provided a measure of the pulse amplitude. This method is subject to error, however, because the interface acts as flexible membrane and was observed to be convex upward at the peak of the upstroke and concave upward at the bottom of the downstroke, thus increasing the apparent magnitude of the amplitude. The apparent increase was a function of the frequency. At the resonant frequencies of the "membrane" its oscillation was strikingly evident. Manipulation of the micrometer was also tedious, especially for high amplitudes. The method was discarded for a simpler, more accurate measurement. The method finally adopted consisted of filling the column and pulser with water, and checking to insure the absence of any air in the system. A manometer tube of accurately known volume per unit length was connected to the bottom of the column. The top of the column was securely plugged with a rubber stopper and all lines to or from the column except the manometer line were closed off by their

-95 -respective valves. The pulser was then started at a slow rate and its volumetric displacement was determined by observing the height of fluid in the manometer tube at the top and bottom of the pulse cycle, Fourplace readings were obtained by estimating the fluid levels to the nearest tenth of a millimetero Pulse amplitudes were obtained by dividing the volumetric displacement by the cross. sectional area of the column. The three eccentrics utilized in the present research produced amplitudes of 0.65, 1.86, and 4.47 centimeters in the contacting section of the column. Calibration of Refractometer Utmost care and precision were demanded in the analyses of sample compositions, since the concentration of methyl-isobutyl ketone in a saturated water solution is only about 1.7 weight per cent. The refractive indices vary from 15533651 for pure water at 250C to about 1533845 for the saturated solution. The original gaskets for the sample bottles used were removed and replaced with Teflon gaskets. The bottles were always baked in a drying oven between useso Standard samples were prepared gravimetrically for the calibration of the refractometer. It was observed, however, that the ketone contained in samples which should have been nearly saturated (had the ketone all dissolved) refused to go into solution, even after 48 hours of continuous agitation. In prepared samples containing enough ketone for 75% of saturation and greater, the refractometer readings asymptotically leeleved off indindiag an apparent pseudoequilibrium valueo These samples contained undissolved ketone drops.

-96 -On the other hand, when a large excess of ketone was present the sample reached the true equilibrium concentration rapidly. Consequently calibration of nearly saturated solutions was accomplished by diluting a saturated solution by known amounts so that some of the resulting concentrations fell on the established portion of the calibration curve. The concentration of the parent solution was then computed from these "known" concentrations, estimated from the curveo The remaining concentrations lying beyond the established curve were calculated from the now-known value for the parent solution and the measured dilutionso Results of the refractometer calibration are presented in Figure 30 and may be represented by the linear equation '/X S" - O 2 (56) 0,/171/ where Y = weight per cent MIBK in H20 SoRo = refractometer scale reading. Refractive indices relative to air are related to the scale readings by RoIo = 1o33376 + o000647. (SoR.) (57) The ketone was found to have an inverse temperature-solubility relationship, ioeo solubilities decreased with increasing temperatures in the aqueous phase. The opposite was true for the ketone phase. The solubility of MIBK in water is satisfactorily represented by )X = 1.920 - Oo0080 T (58)

-97 -0.80.............. Saturated at 25~ C 0.70 - _ 5 — All Reading Taken At 25 C - 0.2.4.6.8 1.0 1.2 1.4 1.6 1.8 Weight Percent MIBK In H20 CFiure 0 Refractometer Cal n. 0.60.... 0.2.4.6.8 LO 1.2 1.4 1.6 1.8 Figure 50. Refractometer Calibration.

-98 -where T = temperature, ~C for a small range of temperatures near 250C. The relationship represented by Equation (58) was found by direct measurement of samples in equilibrium at various temperatures. The prisms of the refractometer were always maintained at 25 C, hence cold, saturated solutions precipitated droplets of ketone when they contacted the warm surfaces, but the kinetics were such that a few seconds were available for a reading. The precipitation was indicated by the loss of the sharp optical "interface" on the instrument. Prism temperatures were maintained at 25.00 + 0.020C. Variations in temperature beyond these limits caused detectable changes in readings. Samples were delivered to the prisms by means of a short section of 4 mm OoDo glass tubing. In a typical analysis the outside of this tube was wiped off and it was then inserted into a sample bottle. A finger placed over the protruding end held the fluid in the tube. This liquid was then discarded, followed by three other discards The fifth sample was injected into the instrument and a reading taken. The procedure was repeated, and if any disagreement appeared between the two readings, a third was taken. All readings were estimated to the nearest quarter of a vernier unit, and the prisms were carefully dried between all readings. Analyses should have been accurate to within + 0.02 weight % units.

-99 -Mathematical Representation of the Mass Transfer Based on the assumption that the rate of mass transfer is proportional to a driving force, here concentration difference from saturation, a material balance around a differential section of a column yields various equations depending upon the system and assumptions invoked. Only binary, counter-current flow systems are considered here. If longitudinal mixing is negligible the rate equation based on the above considerations is A-% ~: ' = ~' (T~ — C) ~ (59) where y = vertical distance from the bottom end of the contacting section. The solution of Equation (59) with the boundary condition -{(L} = 7-X, (60) is 4D 1 a 3C _= TIC ( %X) e c(61) Since the specific interfacial area for spherical drops is E ~ -..D 5 (62) the mass transfer coefficient can be evaluated from concentration measurements and: I2 = Q Dr 3 -- C (ef-^) ( —^ (63)

-100 -Outlet concentrations are generally usedo Since our feed concentration, xf, was zero (i.e. pure water), k1 evaluated at outlet conditions is ~e~4~~~~~~~ X x(64) Whenever axial mixing is considered, the following equation applies +~ D + D;(65) Equation (65) was derived by incorporating Fick's law and the mass transfer rate expression into a differential material balance, It is convenient to introduce new variables and de-dimensionalize Equation (65); these are (66) De - Jo (67) //=- /Z'J^nf (68) D e"- (69) In terms of the new variables, Equation (65) becomes __ D2' PS(: D) (70)

-101 -The general solution to Equation (70) is ')% (z) =: 1. -A, eP( 2 fi t / / e - -'? ei2 ) z 2 1 - 2 2 F,- I (71) The boundary conditions applicable to this problem were first proposed by Danckwerts(l2) on intuitive grounds. Wehner and Wilhelm(62) presented theoretical justification for their use. They are obtained by writing material balance equations at each end of the column, and appear as,, - ( f - - ) at z=1 (72) and -~:~ ^ at z=0 (73) Application of these boundary conditions to Equation (71) results (since xf = O) in 1. _ -= e ((74) l e - Pe 0 where ( 2 12 (75) and Pe + (y2+ FeN (76)

-102 -Values of xD were calculated for wide ranges of the modified Peclet number, Pe, and the mass transfer, N, from Equation (74). They are presented in tabular form in Appendix F. The experimental concentration profiles did not correspond to those predicted by Equation (74) when a single, constant mass transfer coefficient was employed, unless unreasonably high effective diffusivities were used. Independent experiments indicated, however, that mass transfer in the pulse column could be described through the use of two rate constants: a high coefficient, k2, for the top portion of the column, and a low rate coefficient, k3, for the remaining portion where solute concentrations approached the equilibrium value. This description requires the utilization of two sets of equations and boundary conditions coupled at some intermediate length and concentration coordinates, zc and Xc, respectively. These equations are -I -' ~ t + X/ +,- -- f~ - DC&?1) ~x-(77) o _ C C (78) -(0) - o ) The solutions to these sets of equations are straightforward and appear as

-103 -, [(7Q-l) e - (>#-l) e z Cue - (lec-l^eI'l2 x e ma w %~ -1 e (c-i ) r_7 4 c)eo ( )e where -(2C'z.2) -(2 2c,)] -e %? = 6 -^ -f —.L_.R.-..L.- - -. -.-..-...:-e.,..-.-J_.........^.-.k ^,- -. -. - — 7 =e-(f 2__ _2 e-t_(80)) and.- = T -V + N2 (8z) j t r^ 2 = (81) i _ V \2 2eN (82) N\/2 (85) for 1 > z > z. For the low-rate portion of the column (zc > z > 0), the solution is 2^_ = I + (^- e - an(e.d-7) (84) 'x:no* -~~~~~~~x -- ' 3c" ^e^^-^e~ " S -, ~~~

-104 -where a5 and P are defined as in Equations (81) and (82), respectively except they are evaluated using N3 instead of N2, where >?~^:,'~~~ -- f(85) Results and Discussion of Steady State Mass Transfer Experiments Forty-two steady-state mass transfer experiments were conducted in which frequency, amplitude, flow rates, and perforation diameter were varied. The experimental conditions and calculated results for these runs are summarized in Table VIo Mass transfer coefficients were calculated using terminal concentrations in the "piston flow" equation, io.e the first order equation which neglects longitudinal mixing. These coefficients are denoted by kl, and they provide a superficial measure of the' mass transfer rates. The piston flow model failed rather badly, however, in describing the experimental concentration profiles. Accordingly, the model incorporating axial mixing was tried. This model is represented by Equation (74). Experimental profiles were approximated only when axial diffusivity coefficients in the order of 10 to 50 cm'~,/sec were used. On the basis of the dye tracer experiments and other measurements of effective diffusivities in pulse columns(49), these values are an order of magnitude too high. However as a double check more diffusivity measurements -were taken under conditions favorable for extensive axial mixing, viz, high flow rates and

-105 -00 i 0r 1 u 0 10% 1 -* 0 r it,, 0 ~ 4% ) 8 - - ** -* o u ~ m -** 0 ~E As HOOOH IOOOHIOOOHOOOOO OOOH^OOO.-Id0000H000000 - *eKM S88S O0%-* "- 0%01010-010%-*-*O HO 00101HK%01K~a~ff^.^.%a.8..KN 00101 '0 0 010HH01001',0 'I HHHHH0H... 4'.- H 4pUR (-0 p C' C ' L * -0%D-4' 0 0 0 0 M '0' 0.-I 100 0%01'0-*'0 H KNM K.r.'0 0LM M o4 4 S. 35 I 0% 1t'-'%0 0 0\ 0 - r- o 1 -I c " - o CO uH % '0 - 0 - * o'0 H 0 c - o o m o 0 O 0l l.d......... -~...- ~..........-:'L W% % 0 ~ - U Ua o0 0 oQ.. - - ~- % 0 1 (~1 o, - <i GI o\,-I rl GI ~l r-I O ~1G~-I r-I O l GI uO O C rl.1 O 4 O 1 O % 0 f o C.p ~u 0 0 0 ~~~~~0~~~~~0~~~~~~00~0~88~~ 8 o o o 800.0 40 08 00 8o ~~~~004'~ ~ '. I I i i I I I0 K U^-t! aO U^ K^ l\ r K^ CU ~ CU. I II 40 10 r1 H C 01 C H H C 4 C 0 C J0C ICH i! IC! CH o. 5 U 0000 000000 00000000000 g I) f ~~~~~~~Y~~~~~~................... I I.. o o o... -*O (-0%01-* 0%0%OJ 01.4 0%30 0D1- cHO 0%0 U% cu%1H0 H Ca0is 0%%H W% ch 0* 0 1-% 000000000 000000 0 00 00 000 00 g.... oo. 85333333 o ooo.. oo..So o O.... 5455...o...oo 0 00000. 0 O<OOGHOOOOO OOH O1%H'0H5 pO OOX- XXOOH OOOOXOOOOHOO O0OH01O 4' 54' II 5 -5 "%o- o,u%0 - 0 0 H- U0 0-*0 0 -t H -*" -O CD- 1........... ~a....r, -i o KoJ -l o-.........o..o..... o.,,-......... ~........................... oo o l 020 *Q0 CI00 130( -0. 00...... %0 %-g0 H 0 0 0,o 0 0,0 0 O- 0 U U CZ- MN o U.(3 A %; ~ 4 4 ra....O..U. _ I S 4 000000 -400 000000HHH0t -0 N 001 01 0000cm'00 '00100 + < 0<E 0 i O(-r(i —o4'r\Oi'O0u(0-r5(UT%8 0% C —* I I 000 0 5% UN Cu\M Hu\oo ~*\ T|CD-D P3, Cii,, H,H OO.-IO ---IOOHOHHHHHHHHHHHHHHHHHHHHOHIH 0 — sI H H H 01 5% 01 H H'0'0 gl 1- 01 015c 0' % o.0 %-, -- 01' ( —UN 0 0o o 0% 01 Os\~I I i i I I I I i i i 000000.............0000000000 %000.. I _f -1 -1 ~~~14 0 0 0.... 0 0 H H 0 0% - 5- 5% 5 0. - t. — iu u- 0 11 ' -- Noo t- - aO (- -M -t.-.- - o 0 Uo% - C- 5. o so -*-*-*5% 0 HN 01H0 1H 0% _j _t ~4 cu PC% tI _ —'1'O fO:) '0 5% - t H U%0 %D;)O -I0D00001HH'( — A c..... ~..................... 0 U rUA Ao o 0 o ~ o E-1 o o 1 o CO\ CM (g \~ o o aa (\ t idM (O aQ o N 0: UN * aCU \0o o\ ~o ci) t-CC a-IV aC$ CID ~O " (~t UN 0 O (N H 0%- a '%'0 (-0 0% 0 H 01C a 0 0 -0 \ 0 0f\ HN a 01 5 U 0 '0\ (- UN 0 H 01 Ck C3%0 (- 0 0% 0 H 01 __ 5 I ri-I OX. f\ 0 0 UN Er\ 0 0 Er \ UN 0U\ 0 tr % r\ 0 Lf\ 2 c UN t- UN CM _ cm UN If\ IC) 0 U\ a m9o -1~~~~~~~~~~~~~~oo " ~.......r. i ----.fl

-106 -high pulse velocities. These measurements were qualitative in nature and consisted of measuring the retention times of dye injected into the top of the column. Even near the flooding point these observations confirmed the earlier, more quantitative measurements, and diffusivities in the order of 10 to 50 cm2/sec are unreasonably high for the system used. Several approaches were attempted in order to match the experimental concentration profiles, including a number of different boundary conditions for Equation (70). Some of the changes improved the agreement, but they could not be justified on any other basis and were abandoned. A number of independent observations pointed out that mass transfer in the system investigated cannot be adequately described by a single constant coefficient. These observations and the anomalous behavior of the system will be discussed in the next section. Since a single constant was inadequate, the next simplest approach was tried, and with fair success. This was the use of two rate coefficients, k2 and k3, in conjunction with the systems of Equations (77) and (78). Inspection of Table VI reveals that k3 is only 1/6 to 1/2 the value of k2, and that it is only slightly smaller than the superficial constant for piston flow, klo The k2 and k3 values were found by trial and error solution of Equations (79) through (85), using experimentally determined values of vc, De,, L, XD, a and zo Values of xc and zc were somewhat arbitrarily selected. They are the coordinates where the concentration profile, as computed from

-107 -Equation (74), started to deviate widely from the experimental profile. Equivalently, this is the point at which the k values undergo a marked change. Experimental concentration profiles for the mass transfer runs are compared to profiles calculated from Equations (80) and (84) in Figures 31 through 38. A mass transfer coefficient was computed for each concentration sample. The coefficient, k2, used in Equation (80) for the computation of the profile for z > zc, was the arithmetic average of all values in the range z > zc for a given run. Similarly 1k was evaluated as the arithmetic average of values for z less than z,. It should be pointed out that, in general, the first derivative, dx/dz, is discontinuous at the point z = zc even though the concentrations are continuous. These discontinuities in the slopes of the curves were slightly smoothed in the figures for clarity of presentation. The breaks in the curves are still fairly obvious, however, and they denote the locations of xc and zc. Certain features of the mass transfer equations are evident from the curves presented in Figures 31 through 38. First, note that a discontinuity in the concentration exists at the top of the column (z = 1); i.e., even though the continuous-phase feed was pure water the ketone concentration in the aqueous-phase is well above zero just inside the column. This jump becomes more pronounced as the mixing is increased. The extreme example is a perfectly mixed tank. The feed may have zero concentration, but the concentration at any point within the tank is equal to that of the outlet stream.

-108 -1.6 1.5,,. 1.4 - 1.3X- 1,I c -- - - - 1.2 -— " -- --- 1.1 S_________ ^X__X____ 1.0 I9.E.9 _, __ s_ o.7 I..-_ a. Amplitude =0.65 cm..5x ~ Water Flow Rate =2.9 cc./sec. MIBK Flow Rate =0.32 cc./sec. _.4 Perforation Diameter =1/16 inch. Points Are Experimental Data Lines Are Calculated From Eqs. 80,84..2 0.1.2.3.4.5.6.7.8.9 1.0 Figure 31. Comparison of Calculated and Experimental Concentration Profiles

-109 -1.8 - 1.7 1.6 1.5 ri ^^ -- 1.4 '9 L_ 1. -.2 o ii.7.j A_ mAmplitude =0.65 cm. Water Flow Rate = 3.8 cc./sec..5 __ MIBK Flow Rate =0.65 cc./sec. Perforation Diameter = 1/16 inch..4 _ Points Are Experimental Data Lines Are Calculated From Eqs. 80,84..3.2 0.1.2.3.4.5.6.7.8.9 i.0 Z, Dimensionless Distance From Bottom Of Column Figure 32. Comparison of Calculated andExperimental Concentration Profiles.

-110 -1.6....._-0.5c. 1.5 1.4 1.3 C 1..7.5 --— Water Flow Rate = 2.9 cc./sec. MIBK Flow Rate =0.32 cc./sec..4 Perforation Diameter =1/8 inch. Points Are Experimental Data.3 _ Lines Are Calculated From Eqs. 80,84..2.1 I 1_ 0.1.2.3.4.5.6.7.8.9 1.0 Z, Dimensionless Height Figure 33. Comparison of Calculated and Experimental Concentration Profiles.

17 X137X<0 1.1 0 1.3 1.2 - 1.0 o.9.. I I —7I --- — 1-1 '.7 Amplitude = 1.86 cm..6 Water Flow Rate = 2.8 cc./sec MIBK Flow Rote = 0.65 cc./sec..5 Perforation Diameter = 1/16 inch Points Are Experimental Data.4 Lines Are Calculated From Eqs. 80,84..3 -\.2 0.1.2.3.4.5.6.7.8.9 1.0 Z, Dimensionless Height Figure 34. Comparison of Calculated and Experimental Concentration Profiles.

i. 8 -1127-. 1.5 |.4,\ 1.1 1.2 1 1.0 3.9 I __ I I I __~ C -Y 1.0 U I a.8.7 x Amplitude =1.86 cm..6 Water Flow Rate =2.8 cc./sec. MIBK Flow Rate =0.32 cc./sec..5 Perforation Diameter = 1/16 inch. _ Points Are Experimental Data \.4 _ Lines Are Calculated From Eqs. 80,84..3.2.1 0.1.2.3.4.5.6.7.8.9 1.0 Z, Dimensionless Height Figure 35. Comparison of Calculated and Experimental Concentration Profiles.

-1153 -1.8 ' 1.7..... 1.6 __ X 1.5 1.4 _/ 1.3 1.2 0 C 1.0-.9 *.8.7 j _ Amplitude = 4.47 cm. x Water Flow Rate = 2.9 cc./sec..6 — MIBK Flow Rate =0.32 cc./sec. Perforation Diameter =1/16 inch..5 Points Are Experimental Data Lines Are Calculated From Eqs. 80,84..4.3 -.2 0.1.2.3.4.5.6.7.8.9 1.0 Z, Dimensionless Height Figure 36. Comparison of Calculated and Experimental Concentration Profiles.

1.8 1.6.a1.7 - 1.2. _~i i.I a.8 Q Amplitude = 4.47 cm. a. Water Flow Rate =2.9 cc./sec. 7 _ MIBK Flow Rate =0.65 cc./sec. x Perforation Diameter = 1/16 inch..6 Points Are Experimental Data Lines Are Calculated From Eqs. 80,84..5 Water Flow Rate 3.8 cc./sec. For Run 122.4.2 0 0.1.2.3.4.5.6 7.8.9 1.0 Z, Dimensionless Height Figure 57. Comparison of Calculated and Experimental Concentration Profiles.

-115 -1.5 - --- 1.4 1.3 1.0 6 Amplitude =0.65 cm..5 Water Flow Rate =2.9 cc./sec. 0~0~.6 4Perforation Diameter 1/8 inch.iht Points Are Experimental Data 3 --- Solid Lines Are Calcula ted From Eqs. 80,84\ Broken Lines Are Calculated From Eq. 61. \ 0.1.2.3.4.5.6.7.8.9 1.0 Z, Dimensionless Height Figure 38. Comparison of Experimental Concentrationswith Profiles Calculated from First and Second Order Equations.

-116 -Secondly, the concentrations for this system increase much more rapidly, as the solvent proceeds down the column, than is predicted by the equation for no axial mixing. The profile predicted by this equation is plotted in Figure 38 as a broken line. The piston flow equation is the type commonly employed for calculation of HTU's, and is seen to be a very poor description for this sytem. Only volumetric rate coefficients, ka, can be obtained from concentration measurements alone. One of the major goals of this research was to separate out the changes in the rate constant due to variation of the specific interfacial area, a, and find whether or not k itself was truly a constant. The determination of the Sauter mean drop diameter along with a holdup measurement, discussed in previous sections, permits evaluation of the interfacial area term. A slight complication is introduced at this point. As pulse velocities are reduced, a point is reached where the drops rise sufficiently fast in the column that they are detained under the plates before being pushed through on the next upstrokeo Further reduction in pulse frequency allows more drops to collect under each plate, leaving the space between plates devoid of drops for a large portion of each pulse cycle. This nonhomogeneous spatial distribution of drops is characteristic of mixer-settler operation. The accumulated drops under each plate quickly coalesce and thus contribute little interfacial area for mass transfer. The coalesced mass is included in the holdup measurement, however, and represents an ineffective fraction of the holdup volumeo Since only dispersed drops were counted for the calculation of the mean diameter, only

-117 -the effective holdup volume must be employed for the computation of interfacial area The method used for obtaining the effective volume is illustrated in Figure 39~ The points represent experimentally measured holdup volumeso It can be seen that these volumes increase sharply as the pulse velocity (f x a) decreaseso The rise in holdup volume is most evident for the higher dispersed phase flow rate illustrated in the upper portion of Figure 39~ If the upward progress of the ketone drops were not inhibited by the plates, then the holdup in the column would be a function only of the slip velocity of the drops and the flow rates of both phaseso At the limiting condition of zero pulse velocity the drop size is independent of pulse frequency or amplitude (they are both zero), and the slip velocity is that of a distribution of large drops whose diameters are governed chiefly by dropsize stability considerationso Since all the mass transfer runs are well away from the upper flooding limit, holdup (especially for the high slip velocities associated with large drops) is nearly independent of the continuous phase velocity. Thus at the limit of zero pulse velocity the effective holdup would be a function of the dispersed phase flow rate only. The above argument was the basis for selection of effective holdup volumes. The upper section of Figure 39 corresponds to runs having the same ketone flow rate Simla1-rly the runs re)presented by the lower portion all had the same ketone rate; about half that of the others, All the curves for a given MIBK flow rate vwere extrapolated. to the same point cn.the crdiinate. and. the ef.ec ie. hold.up volumes for given pulse velocities were determined from these extrapolated

-118 -60. 50 -- 2 \ ~ " 0 1 2 3 4 5 6 f a, cm./sec....~ Points Are Experimental Solid Lines Correspond To ) 0 Qb cOEffective Values 0X~ ~See Fig. 40 For Key 2 ) 0 0 2 3 4 5 6 fa, cm./sec. Figure 39~ Actual and Effective Holdu p Volumes a)/ Figure 59. Actual and. Effective Holdup Volumes.

-119 -curves. It may be seen that only the lower pulse-velocity holdups are affected. Specific interfacial areas were then computed from a = (86) D32Vol where i) = effective holdup volume, cc V = volume of contacting section of the column, cc col D32 = Sauter mean drop diameter, cm. The values are listed in Table VI. Slip velocities presented in Table VI and correlated in Figure 18 were calculated on the basis of the effective -- rather than the actual holdups. An approximate correlation for the specific interfacial area is presented in Figure 40. The specific area is approximately proportional to the dispersed phase superficial velocity as is the holdup (see Figure 16). The correlating function is again the square root of the perforation diameter divided by the mean orifice velocity. For practical purposes, better values for a may be obtained by using the separate correlations for D^2 (Figure 11) and for slip velocity (Figure 18)o Holdup may then be calculated using Equation (27), and finally a from Equation (62)o Figure 40 shows that, for a given dispersed phase flow rate, nearly a fifty-fold variation in interfacial area was obtained through variation of the pulsing conditionso The variation of k2 with the Sauter mean drop diameter is shown in Figure 41. Rather than being a constant, the mass transfer coefficient is a strong function of the drop size, and decreases with

-120 -400 -- 300 KEY: 200 F a, cm. D, cc./sec C,cc./sec. h, in. + 0.65 0.32 1.08 1/16 150 _ _ _0 0.65 0.32 2.93 1/16 X 0.65 0.32 3.86 1/16 * 0.65 0.64 3.86 1/16 ___ A 1.86 0.32 2.93 1/16 ~100 A 1.86 0.64 2.93 1/16 80 a - C 4.47 0.32 2.93 1/16 -1 * 4.47 0.64 2.93 1/16 60 e____ V 0.65 0.32 2.93 1/8 NE 60I V 0.65 0.64 2.93 1/8 o 40 - 0... 30 X 20 Is 4 + 0.,04..1.___ 16.2_. 3 V / VO, sec./cm1/2 Figure 40. Correlation of Specific Interfacial Area.

-121-.035.030 0.025./ co /Illl *0 | ^/0 0.65 0.3 2.93 1/16 010 ----- — f --- —--- * 0.65 0.64 3.86 1/16 l /a^A 1.86 0.32 2.93 1/16 A 1.86 0.64 2.93 1/16 0 4.47 0.32 2.93 1/16.005.005 ------------ *4.47 0.64 2.93 1/16.015 V 0.65 0.32 2.93 1/8 v 0.65 0.64 2.93 1/8 0 0.5 1.0 1.5 2.0 2.5 3.0 Sauter Mean D Dp Diameter, mm. Figure 41. Mass Transfer Coefficient, k2, as a Function of Sauter Mean Diameter.

-122 -decreasing diameter. This behavior is in accordance with the observations of Johnson and Hamielec(2 ) who studied mass transfer into single drops in binary systems. It agrees also with the models for mass transfer into moving drops formulated by Grober( 9), Kronig and Brink(27), and Handlos and Baron(2), in that these models predict higher mass transfer coefficients for the larger drops. However, a very important difference exists between this system, the models, and the observations of Johnson and Hamielec. The models describe the effect of internal mixing on concentration gradients within a drop. Large drops, according to the models, permit more mixing and thus enhance mass transfer rates. The experimental observations were also attributed to internal mixing. The drops in this system on the other hand, are presumably saturated at all times. This would preclude the existence of internal concentration gradients and the beneficial effects of mixing. This being the case, we are led to two possible alternatives to explain the attenuation of k with diminishing drop sizes. One possibility is that the continuous phase is adversely affected by the same conditions which are favorable for the formation of small drops. However, small drops are produced under high pulse velocities which promote turbulence in the continuous phase: a favorable condition for mass transfer. The only obvious alternative conclusion is that the chief resistance to mass transfer is truly in the interfacial film. This resistance may be an interfacial tension phenomenon which, in an isothermal, constant composition drop, would be expected to vary inversely as the radius of curvature of the drop.

-123 -The dependence of the mass transfer coefficient on the Sauter mean drop diameter, shown in Figure 41, suggests the form of the correlations presented in Figures 42 and 43. The notable features shown by these figures are: 1) k2 values are considerably higher than k3 values at the same orifice velocities; 2) both coefficients tend to approach zero at high velocities. The figures clearly illustrate the detrimental effect of small drop sizes on mass transfer rates. The large increase in specific interfacial area is counterbalanced by the attenuation of the rate coefficient. The net result Is that the product of k and a is considerably more constant than either of these variables taken separately. This relative constancy is illustrated by Figure 44. Here the volumetric coefficient, kla, corresponding to the piston flow model is presented. The main point of interest shown by Figure 44 is that the volumetric coefficient varies only over a five-fold range whereas the interfacial area varies over a fifty-fold range (see Figure 40). However the implication is made, by Figures 40 and 44, that both interfacial area and the mass transfer coefficient, kla, are directly proportional to the superficial velocity of the dispersed phase at a given pulsing condition and perforation diameter. In other words, since holdup is approximately proportional to the dispersed phase flow rate, other things being equal, the volumetric mass transfer coefficient can be doubled by doubling the flow rates. Ruby and Elgin(40) observed the same direct proportionality between ka and holdup when drop sizes were held constant in a spray

-124-.035.030 ---- 0 V.025 0N C/ KEY:.015 10, 1 a, cm. Dcc./sec. C,cc./sec. h,in 0 0.65 0.32 2.93 1/16 * 0.65 0.64 3.86 1/16 ZA 1.86 0.32 2.93 1/16 A 1.86 0.64 2.93 1/16 010 4.47 0.32 2.93 1/16 E* 4.47 0.64 2.93 1/16 I 4.47 0.64 2.93 1/16 V 0.65 0.32 2.93 1/8 V 0.65 0.64 2.93 1/8.005 0 0.05.10.15.20.25 f-7/Vo, sec./cm.1/2 Figure 42. Correlation of Mass Transfer Coefficient, k2.

-125-.0175;.0150.0125.0100 0 70.0075 V.0050 KEY a, cm. D, cc./sec. C, ccJsec. h,in / V0 0.65 0.32 2.93 1/16 * 0.65 0.64 3.86 1/16.0025 A 1.86 0.32 2.93 1/16 &A 1.86 0.64 2.93 1/16 0 4.47 0.32 2.93 1/16 * 4.47 0.64 2.93 1/16 V 0.65 0.32 2.93 1/8 V 0.65 0.64 2.93 1/8 0.05.10.15.20.25 /h/~,O, sec./cm./2 Figure 435. Correlation of Mass Transfer Coefficient, k3.

-126 -0.6 KEY: a, cm. D, cc./sec. C,cc./sec. h, in. 0.5 a.5 ----- + 0.65 0.32 1.08 1/16 0 0.65 0.32 2.93 1/16 ^ X 0.65 0.32 3.86 1/16 0.65 0.64 3.86 1/16 0.4 -- - - A 1.86 0.32 2.93 1/16 4 1' 1.86 0.64 2.93 1/16 E | x a 4.47 0.32 2.93 1/16 * 4.47 0.64 2.93 1/16 -a 0.3 --- 0 l — -- V 0.65 0.32 2.93 1/8 i i \ I [ "v 0.65 0.64 2.93 1/8 lo_ v a x -~ I 0 0.5..10.15.20.25 00.~~~~~~~ ~1/2 'A x 0 ~'"~ ---- —. — 0 0.5.10 /VO sec./cm.. Figure 44. Correlation of Volumetric Mss Transfer Coefficient, k Figure 44. Correlation of Volumetric Mass Transfer Coefficient, kla.

-127 -column. Smoot and Babb(50) presented a generalized correlation for pulse column performance. Their correlation indicates that ka varies as the superficial dispersed-phase velocity raised to the 0.64 power. In another correlation(49) these authors utilized a 0.56 power 0.6 dependency. For the present data the use of kla/vD for the ordinate of a plot similar to Figure 44 gave a slight improvement in the correlation. Summary and Conclusions A number of steady-state mass transfer experiments were performed in which pulse frequency and amplitude, perforation diameter, and flow rates of both phases of the water-MIBK system were varied. The direction of transfer was from the saturated ketone phase to the aqueous phase. Mass transfer rates were analyzed using several different equations. One, the first order equation for "piston flow," gave a poor match to experimental concentration profiles. The equation which allows for longitudinal mixing yielded somewhat better results, but adequate descriptions of the concentration profiles were obtained only when unreasonably high effective diffusivity coefficients for axial mixing were employed. Strange behavior of the dissolution rate of MIBK into water was observed. The ketone droplets apparently refused to dissolve in nearly saturated aqueous-phase solutions. This observation along with others to be presented in the next section was the basis for the mass transfer equations finally employed. Two rate constants were used in the characterization of the profiles, necessitating the utilization of

-128 -two equations and their associated boundary conditions. These equations applied in conjunction with the measured effective diffusivity coefficients proved to be adequate. Measurements of mean drop diameter and dispersed-phase holdup permitted direct computation of specific interfacial area, and in turn, evaluation of the mass transfer coefficients, kl, k2, and k3, from the volumetric coefficients, kla, k2a, and k3a, respectively. The k values were found not to be constant, but rather they decreased sharply as mean drop diameters decreased. This was the case for all three mass transfer coefficients evaluated: k1 for piston flow; k2 for the high rate portion and k3 for the low rate portion of the mass transfer described by the equations which allow for back-mixing. The net results were that while a, the specific interfacial area, varied over a fifty-fold range, the over-all volumetric coefficient, kla, exhibited only a five-fold variation. Hence the volumetric coefficient (or Height of a Theoretical Unit) is not proportional to specific interfacial area. Experimental results were approximately correlated by the function, ~h/vo, where h is the perforation diameter and vO represents the mean velocity of drops as they pass through the perforations of the plates.

UNSTEADY STATE MASS TRANSFER The question immediately arises: are the unusually high concentrations attained in the top end of the column during the initial contacting period real or are they a result of error introduced by the sampling technique? This question was a matter of considerable concern during the researcho Independent observations all indicated that the unusual mass transfer rate behavior is indeed a characteristic of the chemical system employed, and not gross experimental error. These observations will be discussed in the present section, and an attempt will be made to justify the use of two rate constants for each experimental conditiono The Approach to Equilibrium The strange behavior of the system was first suspected during the preparation of standard samples for the calibration of the refractometer. It was observed that when methyl-isobutyl ketone and water were mixed together equilibrium was attained in a matter of minutes when a large excess of ketone was presento However in the presence of just enough MIBK to make a saturated aqueous phase solution only about 94% of the equilibrium concentration was attained in the aqueous phaseo The remaining ketone was observable in the form of small drops floating on the surface of the solution. Neither violent agitation nor heating nor cooling the mixture appeared to affect the solution concentration after it had been allowed to settle out and again reach room temperature. At first the existence of a pseudo-equilibrium was assumed. However a series -129 -

-150 -of careful measurements indicated an extremely slow dissolution rate; one which depends on the amount of ketone in the mixture. The results of these measurements are presented in Figure 45, The abscissa represents the fraction of ketone required to make a saturated solution which is present in the mixture (solution plus undissolved ketone drops), and the ordinate indicates the fraction of equilibrium attained by the solution. The parameters show the approch to equilibrium after various intervals of time. The ketone in mixtures which contained enough MIBK for 70% (or less) of saturation dissolved fairly rapidly. Figure 45 shows, on the other hand, that dissolution was incomplete even after 400 hours for a mixture containing 100% of the ketone required to make a saturated solution. Equilibrium was attained after 400 hours for a 140% mixture, however. This study indicates, then, that the rate of approach to equilibrium in nearly saturated aqueous-phase solutions is a function of the amount of ketone present. It also indicates the possibility of two distinct rate constants; a high constant for "dilute" solutions and a low rate coefficient for more nearly saturated solutions. Transient Rates in a Stirred Pot A desire for quantitative information regarding the rate of dissolution of ketone into water led to the formulation of another experiment. Experimental One thousand grams of distilled water were poured into a beaker which was thermostated at 250C, Continuous agitation was supplied by a motor-driven impeller. At time, to, a weighed amount of pure MIBK1 was

-131 ----- --- ------ - _ _. _Line Corresponding To Saturated Solution At 25 1.0 Line Corresponding To.Complete Dissolution / - - -0.9 I I - - I I- /II --- ---- 0/ -x I f _ 0.9 8 - - -.. -c 0.8 Aqueous Solutions. 0.7 0.8 0.9 1.0 I.I 1.2 1.3 1.4 (Wt.% MIBK In Both Phases)/X* Figure 45. Illustration of Slow Dissolution Rate of MIBK in Nearly Saturated Aqueous Solutions.

-132 -poured into the beaker. In order to minimize ketone evaporation the beaker was covered and the annular space between the beaker and its thermostating jacket was filled with MIBKo The "annular" ketone evaporated and maintained a high partial ketone pressure in the atmosphere immediately surrounding the beakero Samples were withdrawn at measured time intervals by means of a very fine hypodermic needle (Noo27) and syringe. For each sample the syringe was flushed out four times. The fifth withdrawal was then immediately injected into the refractometer and analyzedo About 0ol to 0.2 cc samples were taken. Droplet entrainment in the syringe was never a problem, The few times a drop was picked up during a sample withdrawal it rose to the top of the syringe, and this portion of the sample was not injected into the refractometero Solution "ageing" due to the presence of the droplet in the syringe was negligible because the period between sample withdrawal and analysis never exceeded 5 seconds. Mathematical Treatment of Data A mathematical model which relates concentrations to a rate constant for an unsteady-state, batch system must obviously take into account the decrease in interfacial area with timeo In order to keep the analysis simple certain assumptions are involved. We shall assume that the mass of ketone added to the water is broken into a number of drops during the first few seconds of agitation, Furthermore assume that this number does not change for the entire course of an experiment, i.eo, no coalescence nor further breakup occurs. If the diameter of the drops

-13355 -at time t - 0 is do, then the interfacial area at any time t can be expressed in terms of do and the ketone concentration in solution, assuming spherical drops. The equation describing the rate of dissolution based on the resistance-driving force concept, for the present system, is G^it/}/) S q ( - r 87) wh ere W = grams of water in system K = grams of ketone added to system Y = grams of ketone dissolved at time t kc - rate constant, cm/sec Pc - continuous phase density, g/cc oD = density of ketone, g/cc do = diameter of drops at t = 0 x_ = weight fraction of ketone in a saturated solut.ion vation of Equation (87) and details of its solution are given in Appendix G. The solution to Equation (87) is O =_ - t =B 01 A7 (o+1(t( 1a+) + i J c+Y2 -)X -AZf ) 8 a 2 (L (< le2(022a 62) g r- (o3 J 9)

-154 -where a' -- (-t/-, ) - ),89) S _- -(W 3 2 ^-C ('/ —)(.")>'90) (_- '4'- ( ^ (91) WJhenever a - 0, then the solution is,c= e t = (l 62 93) Since Y, the weight of ketone in solution is Y-v ( f Di,) W94 where x concentrat- er of ketone in solution, wt. fraction, all the terms on t he right Land side of Equation (88) or Equation 935) are directly measurable. A plot of points calculated from concentration measurements, using the ' -h.~. members of either of these equations, vs time should result in a 'traight line with slope kc/do provided kc and

-1)5 -%D c) os o\ - 0\ c\0 I f \ CU 0 I f 01cu t — CU D - - rl L H 01 r-i O\0O0 O \-t r- o 0t OC — O r- r-l r-l r-,- i a 2d cu cu a; l r ^r ^ C a; i c4 c4 cUa cu ai a a;; r^ 0 60 ~ 0 -- \0 CD c - 0 ovo\ DoO\ (o - -\ o) t r \ -, ~ r - - t ~0r-I I - r- CD L010\ CM Ch W C 10t- 01 MN t-0 H,-I 0 n O O X|?4 0X-lr-i,, o ~, co nO uo -o aD 0s0 -0000\ 3o H U ^ c II PR / -- 0 CM (M \0 r I -* o O O\Of -t C00I\ O\ r —CO 3D U rfM 3 XI^ vA"* r EI R IC O O\ 0\o O Cr O\ UNt CDO QO ro 3X # * * * - * rn CM U O\ r-II r IC\ UNf\ \0\ \ - tIH a P; a 0 E-1 l 0' O-1 C ) H 1 H - O - 00 0\ r- O0 C) Ui COCO O\ 0) 4 Pd Vo0 0c O 3D - H0 C o aC O\D rH -I r-I r aP; PI e -0- -I- -- Li i u UN \0 \0 \0 \ \v) V) -- ' —f t — -- x R ( m....0.... a..... m -I H H C f. O- )C —,-I O! \ 000000f " t 0 000 r-i ~-i ^Cc Orcm F-0 0_ 0 PH0 O ) O C- K 00) C \ r- O O ON O \ C\ O (D rl(CO C - 1 C r - 'd r: I rN - U \-O - r- CO O O O O r - OO O O\ r-l 4I M \U * D.4- 01 0 0I C) CM 0 01 \0 r CM 0 Lt —4 4 At-f O\ C * \ 001 0 "f '\t-CD 0rCHH0 \ [-4 —CuDCO'C4 P., to 4,..H 0H H 4 00) 1 * x OM -4, -zuif'ifO0\' 0 ' c.0c" c.0a) a) aI ^ 8 |r4^^r4^r4*-o|rrg~ ^ o pH oi-i rI cU r vo oai r- ifC Lv- ui Co oo ar C>O ao 01 0 000000000ir'Qo000 FC H0'. O 0 C- D 0 LOfC C D\O t D- U -O z OO < -- I aCO 001 HHHHul0 K0 \0C —CD0 I rl a) I rI rl rl rI NU nU ra Pf V) V 0 0 a u Es ax I

-136 -o 0 0 (0 o o0 0 0tr~t 0 0 --, 0 0-. 0 4 CC <o ^ 0 (I):4- <D I= o h 00- P, 0 - 0 o uo o m C 0 a E 0.. 0 a 'r ^ -- V --- —- 8 0h -Wa_ 0 * 0 \ IOI _.'X~~~~~~ 0C0 \' a ^r 1 0 0M-) 0 q-uo!~~unj s~o~i oaib l -- e ' U~llOn..I elD~ I UJoe4,

-137 -o o 0 0 r\ |S |IDo F R@? lo M tq ' 4-4 A\E " I | pi l~ O O r U) ~a) e 5n DSe o| \ C 0 ~,ab -,~~a \ CDS 0 Cl o b.C 0 W o4 o 3 I- *. 0 F — oo o o 0 C0 M 00 uoji'oun0l e0, 0o0j6eUl 01 01~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l~ * 'U3$j)fl~*ID& IDISIU

-138 do remain constant throughout the course of the dissolution of the ketone in water. A deviation from a straight line would reflect a change in either kc or do o Results Three transient mass transfer experiments were performed. The first was discarded due to poor experimental techniqueo In the first of the two latter runs, just enough MIBK was added to the water to make a saturated solution - if it had completely dissolved, In the other experiment, twice this amount of ketone was added to the water. Results are summarized in Table VII. A total of about 3 cc was withdrawn for concentration samples for each runo This produced an error of less than 0,3% since more than 1000 g of material were initially present. The experimental results are plotted in Figures 46 and 47. Nearly all the points in Figure 46 lie on a straight line up to a point, indicating a constant rate coefficient for this portion of the dissolution. However at about 87% (see Table VII) of the saturation concentration the rate drops off sharply as indicated by the abrupt change in the slope of the lineo The change in rate is also shown for Experiment II in which an excess of ketone was present (Figure 47). The slight upward curvature before the break in the curve may be due to drop breakup during the initial stages of the experiment. It may be seen from Table VII and Figure 47 that the reduction in the rate coefficient occurs at a concentration corresponding to about 94% of the equilibirum value. The higher value

-139 -(94% compared to 87% for the previous run) is consistent with the "approach to equilibirum" observations in that the approach to equilibrium is closer when an excess of ketone is in the mixture. There is a possibility that the break in the curves presented in Figure 46 and 47 is caused by the improper choice of the equilibrium concentration. This term, x*, is used in the evaluation of the driving force for mass transfer. This possibility was checked out by using several values of x in Equation (88), including the pseudo-equilibrium concentration. In all cases, use of any value for x* other than the "true" equilibirum concentration resulted in upward-curving lines in plots similar to Figure 46. The upward curvature was followed by a sharp downward trend at higher time values. The wild behavior of the curves was considered to be an indication that the experimentally determined x is valid and any deviation therefrom will not yield the correct driving force. The "true" x* as determined in the present study is in exact agreement with the solubility data of Doshi(44) at 25~C. Summary and Conclusions Observations of the dissolution of MIBK into water revealed that the rate is rapid until a certain maximum concentration is attained in the aqueous phase. Beyond this point the rate of dissolution is extremely slow, and weeks or months may be required before saturation is reached. Furthermore the transition point from the high to low rate appears to depend on the mass of ketone per unit weight of water. The transition occurs at higher concentrations when greater amounts of ketone are present, Saturation is attained rapidly with a large excess of ketone o Similar results were observed in this laboratory for the isobutanol-water system.

-140 - Rather than attempting to present some explanation for the unusual behavior of the system it will be considered to be an anomaly, worthy of further research. It may exist in several binary systems and perhaps even in ternary systems. The possibility of this phenomenon should certainly be recognized in mass transfer studies. A survey of the literature revealed no mention of the behavior observed here. An experimental procedure and corresponding mathematical treatment were developed to follow the rate of mass transfer in a binary, batch system. The results revealed a marked decrease of the rate coefficient at a certain critical concentration which depends on the mass of ketone available. All observations and auxiliary experiments indicate that the use of two rate constants for description of mass transfer in the pulse column under study is well justified.

AREAS FOR FUTURE RESEARCH Several phases of the present research have presented many unanswered problems. Perhaps some of the more interesting of these are: 1. What is the mechanism that causes attenuation of the mass transfer coefficient as drop sizes are diminished? Here, the drops have uniform concentration throughout, so the explanation must be other than an internal mixing effect. A mass transfer study in which surface conditions are controlled and varied for different known drop sizes may shed light on the subject. 2. What is the reason for the sudden change in the rate coefficient as saturation is approached? Is this phenomenon unique to a few binary systems or does it prevail for both binaryand ternary extraction systems? An exploitation and extension of the method for studying transient mass transfer, outlined in the previous section, should yield considerable information on many systems. 3. Quantitatively, how does the plate wetting condition influence drop size distributions? This problem was avoided in the present research through the use of a standard plate-pretreatment. 4o How do fluid properties affect drop breakup in a pulse column? An investigation of wide ranges of fluid properties, especially interfacial tension, continuous phase viscosity, and density difference would lead to a generalized correlation of specific interfacial area with pulsing conditions and fluid properties. -141 -

-142 -Finally, the method outlined in an earlier section for analysis of axial diffusion should be applicable to other systems (gaseous diffusion in consolidated media, for example). It provides a simple and accurate means for the study of axial mixing.

BIBLIOGRAPHY 1. A.ris, Ro, and Amundson, N.Ro., A.I.ChEo Journal, 3, (1957), 280~ 2o Belaga, MoWo., and Bigelow, J Eo, USAEC, Declassified Document KT-133, (1952 ). 3. Burkhart, LE,, and Fahien, R.,W, USAEC, Declassified Document, ISC860 1.956) 4. Burkhart, LE,, and Fahien, R.W., USAEC Repto ISC-1095, (1958). 5. Burns. WoA., Groot, C, and Slansky, C.Mo, "The Design and Operation of the Pulse Column," HW-14728, (1949). 60 Callihan CD,, Ph D. Thesis, Michigan State University, Dissertation Abs,, 19, (1959), 1678. 7T Chantry, WA,, Von Berg, R.,L, and Wiegandt, H.Fo, Ind. Eng. Chem., 47, (1955), 11535 8. Chilton, TH., and Colburn, A.Po,!Ind. Eng. Chemo, 27, (1935), 255. 9. Churchill, R.W., Operational Mathematics, 2nd Ed, McGraw-Hill Book Company, Inc,, New York, 1958. 10. Cohen, R.,M,, and Beyer, GHo Chem Eng. Prog., 49, (1953), 279. 11. Colburn, A.P,, and Welsh, D oGo, Trans. Am. Inst. Chem. Engrs., 38, (1942), 1435 12. Danckwerts, P,W,, Chem_. Eng. Sci, 2, (1953), 1, 135 Dwight, HBo, Tables of Integrals and Other Mathematical Data, The Macmillan Company, New 'York.7 1934. 14. Ebach, E.,A,, Ph,D, Thesis, University of Michigan, 1957. 15o Ebach, EA., and White, R.Ro, AIoCh.E Journal, 4, (1958), 161o 16, Edwards, RoBo, and Beyer, G.H,, AoLCh.E Journal, -2, (1956), 148. 17. Graham, HoLo,, and Burkhart, LoEo, "Pulse Column Design for MixerSettler Operation," preprint, Iowa State University, 1961. 18, Griffith, WLo, Jasny, GoR,, and Tupper, H.T., USAEC Report AECD-3440, (1952). 19, Grober, HoZ,, Vero deut, Ing,, 69, (1925), 705. -1435 -

-144 -20. Handlos, A.E,, and Baron, T. A, I.ChE. Journal, 3, (1957), 127. 21, Hennico, A., and Vermeulen A To, A.IoCh.Eo Journal, 8, (1962), 394. 22. Hoel, PoGo, Introduction to Mathematical Statistics, 2nd Edo, Wiley and Sons, New York, 195~8 23, Johns, LE., and Beckmann, R.oB., "Fundamentals of Mass Transfer in Liquid-Liquid Extraction, " preprint, Carnegie Institute of Technology, 1960. 24. Johnson, A T., and Hamielec, A.E., A T.Ch.E, Journal, 6, (1960), 145. 25. Kessler, Do.P, Ph.D. Thesis, University of Michigan, 1962. 26. Kramers, H., and Go Alberda, Chemo Engo Sci., 2, (1953), 1735 27. Kronig, R., and Brink, JoC, J. Appl, Sci, Res., A-2, (1950), 142. 28, Li, WH.,, and Newton, WM., _AoI.oh oE. Journal, q, (1957), 56. 29. Logsdail, DoH,, Thornton, J.D., and Pratt, HR.Lo, Trans. Insto Chemo Engrs, (London), 35, (1957), 301. 30. Logsdail, DOHo, and Thornton, JDo. Trans Inst. Chem Engrso (London), 35, (1957), 33551 31. Mar, BWo, and Babb, A.L., Ind. Eng. Chem, 5, (1959), 1011. 32, Markas, SoE,, and Beckmann, R.B., A.I.ChoE Journal, 3, (1957), 2235 335 Miyauchi, T., USAEC Rept. UCRL 5911, (1957) 34. Newman, M.L,, nd, Eng C.hem-, 44, (1]952), 2457. 35. Pratt, H.R.Co, Ind. Chemist, 31. (1955) 505. 36, Pratt, H.R.C, A, TC.h Journal, 1-957), 144 -37. Pratt, HoRoC., Lewis, JB,, and Jones, K., USAEC Report AWRE CE/R 904, (Harwell, Berks, Engl.and), ' 1951.) 38, Putnam, A.Ao, et al., Injection and Combustion of Liquid Fuelds, Battelle Memorial Insitut, 4 WAD Technical Report 56-44 (1957). 39. Richardson, G.L., "The Design and Operation of Purex Process Pulse Columns," HW-SA-2083, `1961)o 40. Ruby, C.L,, and Elgin, JC., Chem. Eng. Prog, Symposium Series, 51, (1955), 17.

-145 -41. Sanvordenker, K.S., Ph.D. Thesis, University of Michigan, 1960. 42. Sege, G., and Woodfield, F.W., Chem. Eng. Prog., 50, (1954), 396. 43. Sege, G., and Woodfield, F.W., Chem. Eng. Prog., 50,(1954), 179. 44. Seltzer, S., Ph.D. Thesis, University of Michigan, 1951. 45. Shirotsuka, T., and Oya, H., Chem. Eng. (Japan), 22, (1958), 687. 46. Short, W.L., Ph.D. Thesis, University of Michigan, 1962. 47. Sleicher, C.A., Jr., A.I.Ch.E. Journal, 5, (1959), 145. 48. Smoot, L.D., Ph.D. Thesis, University of Washington, 1960. 49. Smoot, L.D., and Babb, A.L., Ind. Eng. Chem. Fundamentals, 1, (1962), 93. 50. Smoot, L.D., Mar, B.W., and Babb, A.L., Ind. Eng. Chem., 50, (1958), 1005. 51. Sobotic, R.H., and Himmelblau, D.M., A.I.Ch.E. Journal, 6, (1960), 619. 52. Swift, W.Ho, and Burger, L.L., USAEC Rept. HW 29010, (1953). 53. Taylor, G.I., Proc. Roy. Soc. (London), A219, (1953), 186. 54. Taylor, G.I., Proc. Roy. Soc. (London), A223, (1954), 446. 55. Taylor, G.I., Proc. Roy. Soc. (London), A225, (1955), 473. 56. Thornton, J.D., Chem. Eng. Prog. Symposium Series, 50, No.13, (1954). 57. Thornton, J.D., Trans. Inst. Chem. Engr. (London), 35, (1957), 316. 58. Tichacek, L.J., Barkelew, C.l., and Baron, T., A.I.Ch.E. Journal, 3 (1957), 439. 59. Treybal, R.E., Ind. Eng. Chem., 54, (1962), 55. 60. Van Dijck, W.J., U.S. Patent 2,011,186, (1935). 61. Weaver, R.E.C., Lapidus, L., and Elgin J.C., A.I.Ch.E. Journal, 5, (1959), 533. 62. Wehner, J.F., and Wilhelm, R.H., Chem. Eng. Sci., 5, (1954), 55. 63. Wehner, J.F., and Wilhelm, R.H., Chem. Eng. Sci., 6, (1956), 89.

-146 -64. Wylie, C.R., Jrc, Advanced Engineering Mathematics, 2nd Edo, McGraw-Hill Book Company, Inc, New York, 1960. 65. York, J L., Ph.D. Thesis, University of Michigan, 1949. 66. Zenz, F.A., Petrol Refiner, 36, (1957), 147.

APPENDICES -147 -

APPENDIX A DER.IVATION OF GENERALIZED MEAN DIAMETER FROM LOG-NORMAL DISTRIBUTION The geometric mean diameter of the j-th size -interval is defined by (A-i) and - a A + (a - ) (A-2) The mean is (see Hoel 22), pI 53) AX, - ^AS +( a -t)2 ) (A-3) where d c ba4.s and the standard deviation can be written as 'L,2 (A-5) = / S(x,-^ _ X)48 A6., -2 (A5 -148 -

-149 -The n-th moment of a function is defined by (A-6) ~-00 In the case of the log-normal distribution, this is h i 5e"2 ts D(x-;l (A-7) - 0; but D,() = e (A-8) With the substitution g = (w; - ) ) /~~ S ~(A-9) Equation (A-7) becomes -2 2 dt - _ ~^ _ ^27r% eg (A-10) =.e. e t (A-l) h. + S12 - "~~~~~~e "~ ~(A-12) Since D W = ( / /, see Reference 25, (A-13)

-150 -w Dhi is npe(ali-zX + d(i - s ] (A-14) ht-.' i ~ 2 = exp (isC t +e) 2 ) (A-15) which is the generalized mean diameter.

APPENDIX B DERIVATION OF RELATIONSHIP FOR FLOODING DUE TO INSUFFICIENT PULSATION The following derivation closely follows that of Edwards and Beyer(16), but it has been generalized to allow for the removal of continuous phase from the column during the downstroke of the sinusiodal pulse cycle. Consider the portion of the cycle when the heavy phase is flowing down through the perforated plates. Space for the down-flow is created by the expanding bellows of the pulser, and also by the flow of some of the continuous phase out its effluent line. Some of this volume is used up, however, by the influx of the dispersed (light) phase which flows into the column at a constant rate, D cc/sec. If Qc is the volume of continuous phase which flows out of the column during the downward movement of fluid. and = _QdoW (B-l) C dt Jo.ii where Atdown = period during which fluid is moving downward in column, sec C continuous phase feed rate, cc/sec Qcdw = continuous phase effluent volume during downward movement of fluid in column, cc then the rate of fluid flowing downward in the column past any arbitrary -151 -

-152 -plane is given by J.Q - r 'f cs(2ft) D - C (B-2) ct where v = volumetric displacement of pulser, cc Q = volume of fluid flowing past a plane in the column. Since the liquid is periodically pulsed up and down, the downward rate becomes zero at two times during the pulse cycle. The respective times are found by setting the left-hand side of Equation (B-2) equal to zero: -; L- ^ C C ) (B-3) 1,2 / f The total volume pulled down is given by integration of Equation (B-2) between the limits tI and t2 e Since no dispersed phase flows downward during this period, the volume -Q (negative sign indicates downward direction) must be equal to the continuous phase feed rate plus any continuous phase recycle rate times the length of the pulse period, or -Q= c_^. / (B-4) where V = continuous phase recycle rate, cc/sec. r

-153 - Otherwise flooding would result. Therefore, t C^~ J /r [ Xr>r co (2?r+')+ D - 0(Cl gt (B-5) l v=,i[ J.+ (o- ')X,] (B-6) where;J _ (B-7) and P D -X oC (B-8) Rearranging (B-6): D + (1- C - ++ 6.) (B-9) Expanding the terms in parentheses on the right-hand side of Equation (B-9)f.1 J + _ s 1 ^- 2 / t; o-~ (B-10): ~~~Y- -.f~- 8... '-) = 1_,. (B-, l;* Cosl, 0 J Aw t C i (B-12)

-154 -The approximiation of (B-10) by (B-12) yields a 1.8% error at the worst possible case (p2 = 1). For commonly encountered values of 3, ( << 1), the error is negligible. Therefore Equation (B-9) may be written as 1/- 2 C( I ) = f c4 (B-13) At incipient flooding the recycle rate, Vr, is zero. In terms of superficial velocities Equation (B-13) now becomes 2 vis tD) =i ex2ress) (B-14) which is the desired expression,

APPENDIX C DERIVATION OF EXPRESSION FOR MEAN ORIFICE VELOCITY As defined in the text of the thesis the mean orifice velocity is the average speed of liquid moving through the plate perforations during the part of the pulse cycle when liquid is moving upward. Due to the influx of the dispersed phase into the bottom of the column and the removal of the continuous phase from the bottom, the upward motion of fluid does not necessarily take place only during the upstroke of the pulser, For example, if a very high dispersed phase feed rate with respect to the continuous phase feed rate is employed, the liquid in the column will continue to move upward after the pulser has started its downstrokec It will again move upward before the start of the next upstroke. In order to find the average velocity we must determine the volume of fluid displaced upward during the upward movement and also the period of time involved. If the eccentric and bellows arrangement of the pulser produces a sinusoidal variation in displacement with time, then the volumetric rate of flow past any plane in the column normal to the flow is given by Jt-Q - 7fqc>( (7rft) 4 D-YC,^-l) c/t where y is the relative rate of production of the continuous phase during the upstroke with respect to the average rate of continuous phase -155 -

-156 -production for the entire pulse cycle. It is given in terms of a (see Equation (B-l), Appendix B) by 7 _ 1 (ic t) +c (C-2) JA^ The term, Atu, is the length of time the fluid moves upward in the column each cycle. It is found-from (C-l), as the difference in times when the rate of fluid movement dQ/dt is zero during any given pulse cycle: 9f 0(2rft } + D-7C O (C-3) or I _ _ __7C-D) (c-4) and 2 - t — gr - 2g7- (c-5) where 0= c< 1 (C-6) and.=._?..C...D (c-7) 'ffr'/

-157 -Therefore at _t.t - _ (C-8) qp.wf It follows that t. t (C-9) c/oWh f f since t ~ +. = 1 (C-10) ^P; JO h f The total volume of fluid pushed up on each cycle is t, 9 = ) CE ) (2Tt) -(yC-D)] Jt (C-i) or t =;r ( v - 0-~0) (C-12) ~~~= {tr (~-^ -~0 ) (C-13) which can be shown, as in Appendix B, to be closely approximated by r(j - _. 0 _ $) I rJc Sr - v29r (c-14) ru c^ r 6Q + D_-KC (c-15) 2; 2/ The mean orifice velocity is defined as ~ =r ~. —Q -,-(c-16) A/at " A~~~~dt~~~f

-158 -Thus 0 A= J- [vc^ 6 + o rc)/2j (c17) r pL r1 - ^/i.^J ( C'i f + J?W^)1 (c-18) where 6^ Q -<^ (C-19) In terms of a, the mean orifice velocity is r - - J /S' 2 +f.!~ _ ]r (c-2o) 0~ ^(T -^ )L 2 \ ^ -^ ^.z~~

APPENDIX D SOLUTION OF DIFFUSION EQUATION WITH GENERALIZED INPUT The equation which describes the axial diffusion of a tracer fluid as it flows through the column is given in the text in dimensionless form by Equation (45). It appears as Ig JJ d<.) (D-l) where L /L (D-2) 9S =,ct /L -- (D-3) i - Acd L / (D-4) and where x = concentration of the tracer in the carrier fluid y = distance from inlet sampling station, cm L = distance between inlet and outlet sampling stations in the column, cm uc = actual (or "interstitial") mean continuous-phase velocity in axial direction, cm/sec t = time, sec De = effective diffusivity coefficient for axial mixing, cm2/sec -159 -

-160 -The initial and boundary conditions describing the experimental conditions for the dye tracer experiments are 1) 9 (~, ) = 0 (D-5) 2 oQ — ) 2).(o,.) f') (D-6) 5 UO(D-7) The boundary condition (D-6) implies that the tracer input to the inlet sampling station may have a concentration profile of an arbitrary shape. It must, however, be of finite duration, such as a pulse or a square wave, or an irregularly shaped pulse, but not a step function for example. The Laplace transfrom of Equations (D-l), (D-6), and (D-7) with respect to the independent variable, 9, yields a / (,, 5) _ -(25) -S5'(2,) O (D-8) C (o, s) = f(,s) (D-9) 0CO (D-lo) Equation (D-8) has the general solution i (r, t) = C, exp[(P ( + S ) C2 ep[( 2 -et S )2] (D-ll) Insertion of boundary conditions (D-9) and (D-10) into Equation (D-ll) yields %( 5) = X(s) exp[(e -7(p)2- 5 )z] (D-12) %(~~~~~~~ t2

-161 -The inverse transform of (D-12) is obtained through the use of convolution properties of Laplace transforms and Transform 82 (page 328 of Reference 9). It is Pe )~2 (?, )= - -= - - e (D-13) where T is a dummy variable of integration. In terms of the dimensional variables Equation (D-13) becomes 2 _ R4jF ( J -_X) ehc (D-14) 2 VrI De O In general, the integration indicated by Equation (D-14) or Equation (D-13) must be performed using some numerical technique. Simpson's rule was employed for all numerical integrations in the present work. An IBM 709 digital computer facilitated the computations. The effective diffusivity coefficient, De, can be found from Equation (D-14) by trial and error. For example an assumed value of De can be used for the calculation of tracer concentrations at the output sampling station (y = L) for various times. These calculated values can then be compared to experimental concentrations. The value of De is now changed and the calculations repeated. This procedure is continued until the deviation between experimental and calculated concentrations is minimized. The trial and error procedure can be replaced by a much more direct method which incorporates every experimental point for the evaluation

-162 -of De. This method involves the zeroth, first, and second moments of both the input and output tracer concentration vs. time profiles. The n-th moment is defined by )'[ "x^,e)deI (D-15) for the dimensionless variables, or by 3M (X) i (,) Jt.(D-16) 0o for real time. The n-th moments of the analytical solution are easily obtained through the use of two properties of the Laplace transform. The first of these states that taking the n-th derivative of the Laplace transform of a function with respect to s coresponds to multiplying the function by (-_)n tn. The second (Reference 64, page 306) specifies that the limit as s goes to zero of the Laplace transform of a function is equal to the integral from 0 to oo of the function. Combining these two properties we find that ot =9 3 i )J (1 (D-17) Correspondingly, the zeroth moment is -\)00 (Pe ~ s- o c5s (D-)8)

-lo3 -f~)-/~ ( /' e ' ]=;(o (D-_9) Similarly, the first and second moments are given by, (Z) - (-)[,0) - Z ()] (D-20) and, ( ) - f'o) - 2 fo) + 2 g2 ) 1 (o (D-21) Up to this point the quantities f(0),?'(0), and?"(0) have no particular meaning, and they may be considered to be unknowns. However we can evaluate the zeroth, first, and second moments at both the input and output locations (z equals 0 and 1, respectively) from the experimental tracer-concentration vs time profiles measured at these locations. Solving for the unknown quantities in terms of the dimensionless moments, we have for z = 0 e (o) = M/ (O) (D-22) 2>) (0) (D-24) 2( (D-24)

-164 -and for z- 1: (O) = M (1) (D-25) '(O~) = -A/tt1() + ) (o) (D-26) = -440/I) + AM (1) (D-27) X ) M (1) + 2 /O) +- + ) (o) (D-28) = M 1) -2M ) (D-29)( 2 P i) 0 (D-29) Equating the expressions for f(0), i.e., Equations (D-22) and (D-25) Al (0) = A (1) = (),; fi z (D-30) This result is to be expected since 00 00 A (z) (z 0) 9 - ( OG( t) )Jt (D-31) [o The right-hand side of Equation (D-31) is clearly the total tracer which passes an observation point, y, divided by the volume of the column. Hence Mo represents a material balance, and it must be independent of the location in the column. Eliminating f'(O) between Equations (D-23) and (D-27) we find

-165 -that /2/4 (1 )J - <be (0) = 0 (D-32) but the dimensionless moments are related to the experimental moments by 00.0 (Z) ^( )J - (' t (',t) dt (D-33) h o (l If we represent the experimental moments by Mn(y), then Equation (D-32) can be re-arranged and solved for uc in terms of the Mn values: "-..^ ft-^ ) (D-34) 441,(L,) AM:(0) In other words the mean velocity (interstitial, in the axial direction) of the carrier stream during a given experiment can be obtained directly from moments evaluated from the input and output tracer concentration profiles. Finally, if we eliminate f"(O) between Equations (D-24) and (D-29) and solve for the modified Peclet number, Pe, we obtain i = 1 [11 (I) - () - 2 A4j1) + A] (D-35) Equivalently, if Mo in the numerator is replaced by the expression given by Equation (D-32): 1 = I-1[ M() -M () M- (i) -A1(o)] (D-56) 0~~~~~~~~~~e

-166 -The effective diffusivity for axial mixing may now be determined by replacing the dimensionless terms in Equation (D-56) by their dimensional equivalents: = M()- ()] - [ M(L)+ l )} (D-37) e 2 U d \ L A1 0 The effective diffusivity coefficient calculated from Equation (D-37) may be used in Equation (D-14) for purposes of comparing the "calculated" output profile with experimental data. Such a comparison is shown in Figure 24. The numerical integration of the moments is carried out using Simpson's rule:, (^)ftt~dt - 2j +)(2 +42Xx + (D-38) -w'+'2 2herea '2 j 3 where xj = tracer concentration at time jAt, and at either y = 0, or y = L, whichever moment is desired. At = length of time interval.

APPENDIX E TABLES OF DIMENSIONLESS CONCENTRATION AS FUNCTIONS OF DIMENSIONLESS LENGTH, MODIFIED PECLET NUMBER, AND MASS TRANSFER NUMBER Dimensionless concentrations, x/x*, for mass transfer in a binary system where either one phase is saturated or the resistance to mass transfer is controlling in one of the two phases are presented in the following tables. They are listed for various values of a modified Peclet number, vcL/De, dimensionless length, y/L, and the dimensionless mass transfer number, Lkca/vc, and represent the solution of Equation (68) with boundary conditions (72) and (73)o The tables were evaluated using Equation (74)~ -167 -

-168 -DIMENSIONLESS MASS TRANSFER NUMBER, (L.KCA/V).010 Z PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.010.010.0 010.0 010.010.01 0.010 010.010 010.010..1.010.010.010 010.009.009.009.009.009.009.009 009.1.2.010.010.009.09.009.008.008.008.008.008.008.008.2.3.010.009.009.008.008.007.007..007.00.007 007.007.3.4.009.009.008.007.007.006.006.006.006.006.006 006.4 5.009.008.007.006.006.005.005.005.05.005.005.05.5.6.008.007.006.005.005.004.004.004.004.004.004.004.6.7.008.007.005.004.004.003.003.003.003.003.003.003.1.8.007.006.004.003.003.002.002.002.002.002.002.0O2.8.9.007.005.003.002.002.001.001.001.001.001.001.C1.9 1.0.OC6.0C4.002.001.001.000.0 000.000.000.000.O000 1.0 DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V =.020 Z PE- 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.020.020.020.020.020.020.020.020.020.020.02o.o20.0.1.020.020.019.019.019.018.018.01O.018.018.018.018.1.2.019.019.019.018.017.016.016.016.016.016.016.016.2.3.019.018.017.016.015.015.014.014.014.014.014.014.3 -4.018.017.016.014.013.013.012.012.012.012.012.012.4.5.018.016.014.012.01O.011.010.010.010.010.010. 10.5 6.017.015.012.010.009.009.008.008.008.008.008.008.6.7.016.013.011.008.007.007.006.006.006.006.006.006.7.8.015.012.009.006.005.005.4 00 4.00,4..004.004.004-.8.9.014.010.007.004.003.003.002.002.002.002.002.002.9 1.0.012.009.005.002.001.001.000. 000.00 0.000.CGO 1.0 DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V) =.050 Z PE= I 2 4 8 16 32 64 128 256 512 1000 ZCCO Z.0.048.048.048.049.049.0499.04 9.049.049.049.049.049 O..1.048.048.047.047.046.045.045.044.044 04.0 44 44.C44.1.2.047.046.045.044.042.041.040.040.039.039.039.039.2 =3.046.045 4062.040.037.036.035.035.035.034.034.034.3.4.045.042.039.035.032.031.030.030.030.030.030.030.4.5.043.039.035.030.028.026.025.025.025.025.025.025.5.6.041.036.031.026.023.021.021.020.020.020.020.020.6.7.039.033.026.021.018.016.016.015.015.015.015 015. 7.8.036.029.021 -.016.013.011.011 01.010.lO.010.Ci.010.8.9.033.025.017.011.008.007.006.005.005.005.005.005.9 1.0.030.021.012.006.003.002.001 000. 000.000. 000 CO 1.0 DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V) =.100 Z PE= 1 2. 4 8 6 32 64 128 256 512 1000 2000 Z.0.092.093.094.094.095.095.095.095.095.095.095.C95.0.1.092.092.092.091.090.089.087.087.086.086.086.086.1.2.090.090.088.085.082.080.078.0?78.37.077.077.071.2.3.088.086.082.077.073.070.069.068.068.068.068.068.3.4.086.081.075.069.064.061.060.059.59 9.058.058.058.4.5.082.076.068.060.054.052.050.049.049.049.049.049.5.6.078.070.059.050.045.042.041.040.040.039.039.039.6.7.074.063.051.041.035.032. 03 0.030. 030. 0 30.C30.7.8.069.056.042.032.026.023.021.021.020.020.020.020.8.9.064.048.033.022.016.013.011.011.010.010.010.OIv.9 1.0.058.040.023.012.006.003.002.001.000.0 0 000 000.000 l.0

-169 -DIMENSIONLESS MASS TRANSFER NUMBERK, (LKCA/V) =.200 Z PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.170.173.175.178.179.180.181.181..181.181.181.181.0.1.170.171.173.173.171.169.167.166 '165.165.165.165.1.2.167.167.165.161.156.152.150.149 -.148.148.148.148.2.3.164.160.155.147.140.135.133.132.131.131.131.131.3 4.159.152.142.131.123.118.116.114.114.113.113.113.4.5.153.142.i28.114.105.100.098.096.096.095.095.095.5.6.146.131.113.097.087.082.080.078.078.077.077.G77.6.7.137.118.097.079.069.064.061.060.059.059.058.058.7.8.129.105.080.061.050.045.042.041.040.040.039.039.8.9.119.091.063.043.032.026.023.021.021.020.020.OZO.9 1.0.108.076.045.024.012.006.003.002.001.000.000.000 1o. DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V) =.500 PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.348.357.369.378.385.389.391.392.393.393.393.393.0.1.346.354.363.369.370.367.365.364.363.363.363.362.1:2.342.346.349.347.341.336.333.331.330.330 3333.33C.2 =3.335.333.328.319.309.302.299.297.296.296.296.295.3 4.325.317.304.288.275.267.263.261.260.260.259.259.4.5.313.297.276.254.239.230.226.223.222.222.221.221.5.6.299.275.245.218.201.191.186.184.183.182.182.181.6.7.283.250.212.180.161.150.145.142.141.140.14C.139.7.8.265.222.177.141.119.108.101.098.097 -.096.096.095.8.9.245.193.139.099.075.062.056.052.051.050.049.049.9 1.0.223.162.100.056.029.015.008.004.002.001.000.000 1.0 DIMENSIONLESS MASS TRANSFER NUMBER, (LKCA/V) = 1.000 Z PE= 1 2 4 8 16 32 64 128 256 512 1OOG 2000 Z.0.532.553.576.597.612.621.627.629.631.631.632.632.0.1.530.548.569.585.593.594.594.594.594.594.593.593.1.2.524.537.549.556.556.553.552..55.551.551.551.551.2.3.513.519.521.518.512.508.506.505.504.504.504.503.3.4.499.495.487.475.464.458.455.453.452.452.451.451.4.5.482.467.447.426.411.403.398.396.395.394.394.394.5.6.461.434.402.372.353.342.336.333.331.330.330.330.6.7.437.397.352.313.289.275.267.263.261.260.260.259.7.8.410.356.297.249.218.201.191.186.184.183.182.182.8.9.380.310.237.178.141.119.108.101.098.097 096.096.9 1.0.347.260.171.101.056.029.015.008.004.002.001.000 1.0 OIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V)= 2.000 Z PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.721.751.785.815.836.849.857.861.863.864.864.864.0.1.718.747.778.803.819.827.831.833.834.834.834.835.1.2.710.734.758.776.786.792.795.796.797.798.798.798.2.3.697.713.728.739.745.748.751.752.753.753.753.753.3.4.680.686.691.693.694.696.697.698.698.699.699.699.4.5.659.653.645.638.634.633.632.632.632.632.632.632.5.6.633.613.591.573.562.556.554.552.551.551.551.551.6.7.602.566.527.496.476.464.458.455.453.452.452.451.7.8.567.511.453.405.373.353.342.336.333.331.330.330.8.9.526.449.367.298.249.218.201.191.186.184.183.182.9 1.0.481.378.268.172.101.056.029.015.008.004.002.001 1.0

-170 -DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V) = 5.000 Z PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.902.930.955.972.982.988.991.992.993.0993.993.993.0.1 899.926.951.968.978.983.986.987.988.989.989.989.1.2.892.917.940.956.967.974.978.980.981.981.981.982.2.3.881.902.922.939.951.959.964.967.968.969.969.970.3.4.865.880.897.914.927.937.943.947.948.949.950.950.4.5.844.852.864.878.892.902.909.913.916.917.917.918.5.6.817.815.818.927.838.848.856.860.862.863.864.864.6.7.784.767.757.755.759.765.770.773.775.776.776.777.7.8.744.708.675.653.641.635.633.632.632.o32.632 632 o8.9.696.632.566.508 464.434 415.405.399.396.395.394.9 1.0.638.536.420.303.200.121.068.036.019.010.005.002 1.0 DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V) = 10.000 PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.969.984.993.998.999 1.000 1.000 1.000 1.000 1.OCO 1.000 1.000.0.1.967.982.992.997.999.999 1.000 1.000 1.000 1.000 1.000 1.000.1.2.963.978.988.994.997.999.999.999 1.000 1.000 1.000 1.000.2.3.955.970.982.990.995.997.998.999.999.999.999.999.3.4.944.958.971.982.989.993.996.997.997.997.997.997.4.5.928.940.954.968.979.985.989.991.992.993.993.993.5.6.907.914.927.943.957.967.974.978.980.981.981.981.6.7.879.878.88.5.898.914.927.937.943.947.948.949.950.7.8.842.825.817.818.827.838.848.856.860.862.863.864.8.9.793.750.708.675.653.641.635.633.632.632.632.632.9 1.0.729.642.537.420 303 200.121.068.036.019.010.005 1.0 DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V) = 20.000 Z PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0.993.998 1.000 1.000 1.000 1.000 1.000 1 0 1.000 1.000 1000100 1.000.0.1.993.997.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000.1.2.991.996.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000.2.3.987.994.998.999 1.000 1.000 1.000 1000 1.00 1.000 1.000 1.000.3.4.981.989.995.998.999 1.0001.0 1000 1.000 1.000 1.000 1.000.4.5 973.982 990.995.998.999 1.000 1.000 1.000 1.000 1.000 1.000.5.6.959.969.980.989.994.997.999.999.999 1.000 1.000 1.000.6.7.940.947.958.971.982.989.993.996.997.997.997.997..8.910.908.915.927.943.957.967.974.978.980.981.981.8.9.866.843.825.817.818.827.838.848.856.860.862.863.9 1.0.800.730.642.537.420.303.200.121.068.036.019.010 1.0 DIMENSIONLESS MASS TRANSFER NUMBER, (L*KCA/V) = 50.000 Z PE= 1 2 4 8 16 32 64 128 256 512 1000 2000 Z.0 1. 000 1.000 1.000 1.000 1.000 1.000 1. 000 1.000 10 1.000 1.000 1.000.0.1 1.000 1.000 1,000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000.1.2.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.00G 1.000 1.000.2.3.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.00100 0 1. 1.O0U.3.4.997.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000.4.5.995.998.999 1.000 1.000 100 1.000 1. 000 1.000 1.000 1.000 1.000.5.6.991.995.998 1.000 1.000 1.000 1.000 1.000 1. 000 1.000 1 0 1.000.6.7.982.988.994.998.999 1.000 1.000.1000 1.300 1.000 1.001 1.000 o 7.8.965.970.979.988.994.998.999 1.000 1.000 1.000 1.000 1.000.8.9.932.927.928.936.950.964.976.984.988.991.992.993.9 1.0.868.819,754.672.572,458.340.231.143,082.046,024 1.0

APPENDIX F DERIVATION AND SOLUTION OF EQUATION FOR TRANSIENT MASS TRANSFER IN A STIRRED, BATCH SYSTEM Suppose that K grams of pure MIBK are added to W grams of pure water in a beaker. The mixture is well stirred. Suppose further that n drops of ketone of uniform initial diameter, do, are formed. Since water is nearly insoluble in MIBK, neglect the transfer of water into the droplets. Let Y equal the weight of ketone that has dissolved into the aqueous phase at any time t. Assume that the number of drops in the container does not change throughout the course of the experiment. The number, assuming spherical drops, is given by -' ( (F-l) where pD is the density of the ketone phase, g/cc. As dissolution proceeds the diameter of the drops will decrease as will the specific interfacial area. The latter is given by at= - Ti — - = J2 (F-2) V/ ^ dV< where a = specific interfacial area, cm2/cc d = diameter of drops at time t, cm VT = total volume of ketone plus water, cc At time t, the total volume of ketone drops in the mixture is K-_'._ n(/.rd3 -.1- 7 1-^ -.(F-3) -171 -

-172 -thus 1 i c G(- " /-A ), - ~) 3c ] 7 7[ j K ] o" 0 (F-4) Combining Equations (F-2) and (F-4) 3 Vd (F-5) Assume that the drop - continuous phase interfaces are at the equilibrium concentration, x*. The bulk concentration of MIBK in the aqueous phase at any time, t, is x (weight fraction ketone). Now if the widely used resistance - driving force concept for mass transfer is valid, then At~ f/Y. VV - ) (4 Cc (F-6) Since -X = Y (F-7) and Jc /^ J KY (F-8) then, by inserting Equations (F-7), (F-8), and (F-5) into Equation (F-6) we obtain ~c/ t (wY r) K,& D (XI (F-9) I " - ___ o

-173 -which can be rearranged to C2 / K__ _ _ _ _ _ _ _ _ _ _ t ( F - 1 0 ) f^D- ) - -__/m a =..c X (- ) d- ~t ( / ) #Qf-%: t) Letting (F-11) (F-12) and. K- (F-13) and since as = -d (F-14) if we make these substitutions in Equation (F-10), it becomes ____ - c 1 t (F-15) Equation (F-5) can be integrated directly (Reference 1, p. 2). It Equation (F-15) can be integrated directly (Reference 13, p. 32). It

-174 -becomes, for a L O: _- _ Ct 1 I Jo 2 (O,, )_ $ do o (^_o" 2 2 a"or t t*2 [ LLW t.((Y ')6(qn # c ) + __ -_ { A d17) Whenever ca = 0, (a will equal zero if just enough ketone is added to the water to make up a saturated solution), Equation (F-15) becomes ih t = ' 5 B(-,) (F-18) sThe constants, a and B, can be determined from experimental conditions and properties of the two liquids used, and values of 6 can be calculated from concentration measurements taken at various times during the mass transfer process. A plot of the right-hand member of either Equation (F-17) or (F-18), whichever is applicable, versus time will yield a straight line with slope kc/do if the latter group is constant. Such a plot is shown in Figure 46.