ABSTRACT 
 
 
 
 
 
 
 
Title of Document: LIQUID-LIQUID DISPERSION IN BATCH 
AND IN-LINE ROTOR STATOR MIXERS 
  
 Paul Edwin Rueger, PhD, 2013 
  
Directed By: Professor Richard V. Calabrese, Dept. of 
Chemical & Biomolecular Engineering 
 
 
This two-part dissertation investigates the behavior of batch and in-line rotor-stator 
mixers separately. In the first study, water was dispersed into viscous oil using a 
batch Silverson L4R rotor-stator mixer. The flow regime was determined by reference 
to published Power number data and by qualitative differences in drop size data. Drop 
breakup in laminar flow was analyzed by comparison to published single drop 
breakup experiments in idealized flow fields. The breakup mechanism in laminar 
flow was similar to that for simple shear flow and equal to about twice the nominal 
shear rate in the rotor-stator gap. Drop breakup in turbulent flow followed a 
mechanistic correlation for mean drop size for drops less than the Kolmogorov 
microscale, but still large enough that both inertial and viscous effects were manifest. 
  
Surfactants decreased drop size with Marangoni effects observed near the CMC for 
laminar, but not for turbulent flow. Below phase fractions of ? = 0.05, d32 increased in 
a log-linear fashion with phase fraction for all conditions tested including: laminar 
and turbulent flow, presence of surfactant, and hydrophobically treated high-shear 
surfaces. The significant effect of phase fraction was caused by the flow structure 
being locally laminar near the drops, and was permitted by sufficiently low fluid 
viscosities which promoted film drainage. Above phase fractions of ? = 0.1, drop 
sizes plateaued. This was attributed to decreasing coalescence rate and efficiency, 
along with increasing breakup. In the second study, the power consumption of an 
IKA 2000/4 in-line pilot scale rotor-stator mixer was measured with a purpose-built 
torque meter. The power spent by the mixer on pumping was insignificant compared 
to viscous dissipation. A constant power number was obtained for turbulent flow 
using constant power per stage with an empirically determined effective diameter for 
each generator type. For conditions where mean drop size was close to equilibrium, 
as determined by flowrate independence, previously reported mean drop size data 
were calculated using the well-known inertial subrange scaling law along with the 
power draw measurements of the present study. The maximum local energy 
dissipation rate was found to be nine times the average energy dissipation rate. 
  
 
 
 
 
 
 
 
 
 
 
LIQUID-LIQUID DISPERSION IN BATCH AND IN-LINE ROTOR STATOR 
MIXERS 
 
 
 
By 
 
 
Paul Edwin Rueger 
 
 
 
 
 
Dissertation submitted to the Faculty of the Graduate School of the  
University of Maryland, College Park, in partial fulfillment 
of the requirements for the degree of 
Doctor of Philosophy 
2013 
 
 
 
 
 
 
 
 
 
 
Advisory Committee: 
Prof. Richard V. Calabrese, Chair 
Prof. Panagiotis Dimitrakopolous 
Prof. James Duncan 
Prof. Sheryl Ehrman 
Prof. Srinivasa Raghavan 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
? Copyright by 
Paul Edwin Rueger 
2013 
 
 
 
 
 
 ii 
 
Acknowledgements 
The author would like to thank Professor Richard V. Calabrese for his continual 
counsel, direction, and clear insight throughout the duration of this project, and would 
also like to note that he is an excellent advisor because he always makes sure that 
everything is in order and that his students have everything that they need to succeed 
in graduate school as well as in other spheres of life. The author would also like to 
thank Professor Panagiotis Dimitrakopoulos for hiring him as an undergraduate 
researcher and for the encouragement to pursue graduate education. Thanks go to 
Professor Srinivasa Raghavan for the use of his fabulous viscometer and to Mr. Scott 
Anderson of IKA? for doing a significant amount of mechanical work to install the 
torque sensor in-line with the IKA Labor Pilot. The author is grateful for financial 
support from Chevron ETC and the University of Maryland High Shear Mixing 
Research Program. Thanks go to Dr. Lee D. Rhyne of Chevron for many useful 
discussions and his open-minded approach to the project. The author would like to 
extend his thanks to all of his colleagues primarily for their comradeship, but also for 
their advice: Meng Yang, B.N. Murthy, Justin Walker, Derrick Ko, Kanan 
Ghaderzadeh, Emily Chimiak, Jung Kim, and Seong Kim. Finally, the author would 
like to thank his family and friends for their support and encouragement. 
 
 iii 
 
Table of Contents 
 
 
Acknowledgements ....................................................................................................... ii 
Table of Contents ......................................................................................................... iii 
List of Tables ............................................................................................................... vi 
List of Figures ............................................................................................................. vii 
Nomenclature ............................................................................................................. xvi 
Chapter 1: Introduction ................................................................................................. 1 
1.1 ? Motivation & Purpose ...................................................................................... 1 
1.2 ? Approach .......................................................................................................... 3 
1.3 ? Organization of the Dissertation ...................................................................... 6 
Chapter 2: Relevant Physical Properties and their Measurement ............................... 10 
2.1 ? Index of Refraction ........................................................................................ 10 
2.1.1 ? Definition of Index of Refraction ........................................................... 10 
2.1.2 ? Relevance of Index of Refraction: Determination of Oil for Use in 
Dispersion Experiments ...................................................................................... 11 
2.2 ? Viscosity ........................................................................................................ 13 
2.2.1 ? Definition of Viscosity ............................................................................ 13 
2.2.2 ? Relevance of Viscosity ........................................................................... 14 
2.2.3 ? Experimental Measurement of Oil Viscosity .......................................... 15 
2.2.4 ? Verification that Crystal Oil is Newtonian ............................................. 15 
2.2.5 ? Viscosity as a Function of Temperature and Viscosity Grade ................ 16 
2.3 ? Interfacial Tension ......................................................................................... 19 
2.3.1 ? Definition of Interfacial Tension ............................................................ 19 
2.3.2 ? Relevance of Interfacial Tension ............................................................ 20 
2.3.3 ? Scatter in the Published Values of Interfacial Tension ........................... 21 
2.3.4 ? Alternate Methods to Measure Interfacial Tension ................................ 22 
2.3.5 ? Description of the Pendant Drop Method ............................................... 25 
2.3.6 ? Pendant Drop Method as Used in this Study .......................................... 29 
2.3.7 ? Calibration of the Pendant Drop Method ................................................ 40 
2.3.8 ? Clean Interfacial Tension Results ........................................................... 41 
2.3.9 ? Dynamic Interfacial Tension Results ...................................................... 42 
2.4 ? Physical Property Ranges .............................................................................. 46 
Chapter 3: Batch Dispersion Theory........................................................................... 48 
3.1 ?Dilute Theory .................................................................................................. 48 
3.1.1 ? Dilute, Turbulent Theory ........................................................................ 49 
3.1.2 ? Dilute, Laminar Theory .......................................................................... 54 
3.2 ? Surfactant-Laden Systems ............................................................................. 57 
3.3 ? Hydrophobic Treatment Theory .................................................................... 58 
3.3.1 ? Surface Free Energy ................................................................................ 58 
3.3.2 ? Surface Modification .............................................................................. 60 
3.4 ? Non-dilute Theory .......................................................................................... 64 
Chapter 4: Experimental Methods for Batch Dispersion Experiments ....................... 70 
 
 iv 
 
4.1 ? Batch Dispersion Process ............................................................................... 70 
4.1.1 ? Experimental Setup ................................................................................. 70 
4.1.2 ? Experimental Procedure .......................................................................... 75 
4.1.3 ? Microscope Technique ............................................................................ 78 
4.1.4 ? Cleaning Procedure ................................................................................. 82 
4.2 ? Recycling of Oil ............................................................................................. 84 
4.3 ? Hydrophobic Treatment of Surfaces .............................................................. 85 
4.3.1 ? Experimental Method.............................................................................. 85 
4.3.2 ? Results & Treatment Selection ............................................................... 87 
Chapter 5:  Flow Regime Determination for Batch Experiments ............................... 95 
5.1 ? Power Draw vs. Reynolds Number Relationship .......................................... 95 
5.2 ? Padron?s (2001) Power Draw Work .............................................................. 97 
5.2.1 ? Brief Summary of Methodology ............................................................. 97 
5.2.2 ? Results & Application to the Present Study ............................................ 98 
5.3 ? Qualitative Differences in Drop Size Behavior ........................................... 100 
5.3.1 ? Data Range of Physical Properties ........................................................ 100 
5.3.2 ? Evidence from Outlier Data .................................................................. 100 
5.3.3 ? Capillary Number Behavior .................................................................. 102 
Chapter 6:  Dilute, Batch Experimental Results ....................................................... 106 
6.1 ? Drop Size Characterization .......................................................................... 106 
6.1.1 ? Use of d32 to Characterize DSDs .......................................................... 106 
6.1.2 ? Relationship of d32 and dmax .................................................................. 108 
6.2 ? Reproducibility of Experiments ................................................................... 111 
6.3 ? Laminar Flow Results & Analysis ............................................................... 112 
6.4 ? Turbulent Flow Results & Analysis ............................................................. 117 
Chapter 7:  Non-dilute, Batch Experimental Results ................................................ 121 
7.1 ? Clean Systems .............................................................................................. 121 
7.2 ? Clean Systems with Treated Mill Head ....................................................... 126 
7.3 ? Surfactant-Laden Systems ........................................................................... 129 
7.3.1 ? Behavior of Surfactant-Laden Systems ................................................ 129 
7.3.2 ?Surfactant-Laden Systems? Complete Data ........................................... 133 
7.4 ? High Concentration Systems........................................................................ 140 
7.5 ? Tests for Drop Adherence on Low-Shear Surfaces ..................................... 141 
Chapter 8: In-line Mixer Power Draw Experiments ................................................. 145 
8.1 ? In-line Theory .............................................................................................. 145 
8.1.1 ? Power Draw .......................................................................................... 145 
8.1.2 ? Single Pass Drop Size ........................................................................... 149 
8.2 ? In-line Experimental Procedure ................................................................... 152 
8.2.1 ? Mixer Details ........................................................................................ 152 
8.2.2 ? Power Draw .......................................................................................... 157 
8.3 ? Power Draw Results & Analysis.................................................................. 162 
8.3.1 ? Mixer Pumping ..................................................................................... 162 
8.3.2 ?Dissipated Power ................................................................................... 169 
Chapter 9: In-line Mixer Power Draw Comparison with Drop Size Data ................ 184 
9.1 ? Single Pass Drop Size Measurement ........................................................... 184 
9.2  ? Drop Size Results & Correlation with Power Draw ................................... 186 
 
 v 
 
9.2.1 ? Time-Averaged Drop Size Data............................................................ 187 
9.2.2 ? Transient Drop Size Data ...................................................................... 195 
9.3  ? Drop Size Correlation with Power Draw .................................................... 199 
Chapter 10:  Summary & Conclusions ..................................................................... 212 
10.1 ? Interfacial Tension ..................................................................................... 212 
10.2 ? Flow Regime Determination ? Silverson L4R Batch Rotor-Stator Mixer 212 
10.3 ? Dilute Systems ? Silverson L4R Batch Rotor-Stator Mixer ...................... 213 
10.4 ? Non-Dilute Systems ? Silverson L4R Batch Rotor-Stator Mixer .............. 214 
10.5 ? Non-Equilibrium Systems ? IKA Labor Pilot 2000/4 In-line Rotor-Stator 
Mixer ..................................................................................................................... 217 
Appendix A: Catalog of Literature Data for Interfacial and Surface Tension for some 
Common Fluids ......................................................................................................... 219 
Appendix B: Computer Programs Used in Measuring Interfacial and Surface Tension
 ................................................................................................................................... 224 
Appendix C: ImageJ Macro Used to Measure Drop Sizes from Drop Images ......... 235 
Bibliography ............................................................................................................. 239 
 
 
 vi 
 
List of Tables 
Table 2.4-1: Range of physical properties which are used in the dispersion 
experiments. Yellow = symbol, green = adjustable parameters, and grey = units. 
 
Table 3.1.1-1: Derivation of Turbulent Scaling Laws. * Shinnar?s sub-Kolmogorov viscous 
scaling law was derived using an alternative method in which the characteristic shear rate, ?
  
  
was defined to be the ratio of the Kolmogorov velocity to the Kolmogorov length scale. The 
stress then follows as ?    ?   This approach does not yield a value of E(k). 
Table 7.1-1: Table of approximate values of collision efficiency. The table shows 
that the viscosities play a controlling role in determining the probability that a 
collision event will result in coalescence. Bolded columns indicate the results that 
correspond to Figures (7.1-1) & (7.1-2). 
Table 7.3.1-1: The effect of surfactant concentration on the value of a in laminar flow 
experiments. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
Table 8.2.1-1: Mixer geometries? physical dimensions.  
Table A-1: Literature values of surface and interfacial tension for some relatively 
common substances. This table illustrates the significant amount of scatter present in 
the reporting of these values. 
 
 
 vii 
 
List of Figures 
Figure 1.2-1: Batch Silverson L4R mixer. The top image is the entire setup with 
attached mixing vessel. The bottom two images depict the high-shear mill head which 
is composed of a rotor surrounded by a stator with very small clearance. 
Figure 2.1.1-1: Illustration of Snell's Law. The angle of incidence, a, is shallower 
than the angle of refraction because the index of refraction in material a is greater 
than that of material b. 
Figure 2.1.2-1: Refractive index of various oil's that were considered for use in the 
dispersion experiments. The difference between the oil's index of refraction and 
water's determines the phase contrast for imaging purposes. 
Figure 2.2.4-1: Viscosity vs. shear rate for Crystal Oil 500 FG at 20.0? C. This 
shows, as expected, that Crystal Oil is a Newtonian fluid - that shear stress is linearly 
related to shear rate. 
Figure 2.2.5-1: Viscosity as a function of temperature for the four grades of Crystal 
Oil as well as water.  
Figure 2.2.5-2: Viscosity of all the grades, batches, and blends of Crystal Oil 
including all the temperatures at which they were used in the dispersion experiments 
as well as that of water. 
Figure 2.3.1-1: Spreading of a soap film in air. As the red wire is pulled to the right 
by an external force the free energy of the system rises as a result of the greater 
interfacial area. 
Figure 2.3.5-1: The two configurations of the pendant drop method. The yellow fluid 
is less dense than the blue fluid. The measured value of the surface tension is the 
same regardless of which configuration is used. 
Figure 2.3.5-2: The shape factors used to calculate interfacial tension via the pendant 
drop method. De is the equatorial diameter and Ds is the diameter of the drop at a 
height (measured from the bottom of the drop) equal to the equatorial diameter. 
Figure 2.3.6-1: Glass cuvette constructed out of 2? x 3? microscope slides. Used to 
contain the surrounding phase in liquid-liquid pendant drop experiments to measure 
interfacial tension. 
Figure 2.3.6-2: Acceptable pendant drop image. The criteria that are met are: sharp 
contrast between phases, needle in the image, drop not touching image edges, and 
lack of (non-drop) dark spots in the image. 
Figure 2.3.6-3: Thresholded pendant drop image using straightforward thresholding. 
Figure 2.3.6-4: Pendant drop image after the ?Enhance Contrast? and ?Find Edges? 
plug-ins have been executed. 
Figure 2.3.6-5: Pendant drop image after the ?Enhance Contrast? and ?Find Edges? 
plug-ins have been executed. 
 
 viii 
 
Figure 2.3.6-6: Standard deviation between identical interfacial tensions as a function 
of adjustment factor. The common minimum verifies the validity of this approach. 
Figure 2.3.7-1: Calibration curve for pendant drop method as developed in this study. 
Adjusted, uncalibrated interfacial tension vs. literature interfacial tension. 
Figure 2.3.9-1: Dynamic interfacial tension of water in Crystal Oil 500FG at a 
Tergitol NP-4 concentration of 1x10-4 M. Approach to equilibrium is illustrated for a 
single drop. 
Figure 2.3.9-2: Equilibrium interfacial tension vs. Tergitol NP-4 concentration for 
water and Crystal Oil 500FG. 
Figure 3.1.2-1: From Marks (1998). Critical Capillary number for the breakup of an 
individual drop in SSF as a function of the ratio of the drop viscosity to the matrix 
viscosity. 
Figure 3.1.2-2: From Grace (1982). Critical Capillary number for the breakup of an 
individual drop in SSF (upper curve) and SEF (lower curve ? flat line) as a function 
of the ratio of the drop viscosity to the matrix viscosity. 
Figure 3.3.2-1: Qualitative difference between an untreated solid surface and a 
treated solid surface. 
Figure 3.3.2-2: Contact angle: the angle an adhering drop makes with its solid 
surface in the presence of the particular surrounding fluid. 
Figure 3.3.2-3: Illustration of work of adhesion: work of adhesion is the difference in 
surface free energy after specified areas are created relative to the initial state of the 
system. 
Figure 4.1.1-1: Silverson L4R Batch rotor-stator mixer used in this study. a) mixer 
and stand, b) sideview of the mixing head, and c) bottom view of the mixing head. 
Figure 4.1.1-2: Experimental setup. Copper coil is used to transfer heat between the 
water bath and the refrigeration unit?s working fluid. Small unit on the right is a 
stirrer to keep the temperature in the water bath uniform. 
Figure 4.1.1-3: Experimental schematic. Yellow represents the emulsion, and blue 
represents the temperature bath water. 
Figure 4.1.3-1: Schematic illustration of an emulsion sample set-up to be imaged by 
an optical microscope. The drops rest on the microscope slide at the bottom of the 
sample because they are more dense than the surrounding oil. 
Figure 4.1.3-2: Prepared 1? x 3? microscope slide. Each slide was rendered 
hydrophobic, cleaned, taped carefully, and then the sample was deposited with a 
wooden dowel rod and the cover slip was placed atop it. 
Figure 4.1.3-3: Typical microscope image used to measure drop size. Crystal Oil 
500FG surrounded by water under a 60x microscope objective with a ?Watec 
America Corp. LCL-902K.? 
 
 ix 
 
Figure 4.3.2-1: Work of adhesion for various solid substrate treatment methods for 
water in oil on glass. The arrow and the underline indicate which method was 
eventually selected. 
Figure 4.3.2-2: Work of adhesion for various solid substrate treatment methods for 
water in oil on steel. The arrow and the underline indicate which method was 
eventually selected. 
Figure 4.3.2-3: Contact angle for various solid substrate treatment methods for water 
in oil on glass. The arrow and the underline indicate which method was eventually 
selected. 
Figure 4.3.2-4: Contact angle for various solid substrate treatment methods for water 
in oil on steel. The arrow and the underline indicate which method was eventually 
selected. 
Figure 4.3.2-5: Drops sticking to the underside of the mixer lid. This figure shows 
that the hydrophobic treatment alone was insufficient to solve the drop sticking 
problem. 
Figure 4.3.2-6: Effect of Polishing the mixer lid. 2 experiments were performed 
without polishing and then 2 experiments with polishing. There appears to be a slight 
difference caused by polishing. 
Figure 5.1-1: Rushton (1950) plot of Power number vs. Reynolds number. This type 
of plot, specific to each mixer, can be used to coarsely identify the transition between 
flow regimes. 
Figure 5.2.2-1: Padron?s (2001) plot of Power number vs. Reynolds number for the 
batch Silverson L4R mixer. The range of Reynolds numbers for the dispersion 
experiments is bounded by the vertical lines. 
Figure 5.3.2-1: Drop size vs. rotor speed. The two notable outliers have been pointed 
out with arrows. This plot provides evidence that those points are in a transitional 
flow regime. 
Figure 5.3.3-1: Drop Capillary number as a function of rotor speed as a function of 
viscosity ratio (?) and Reynolds number (Re) for clean (no surfactant) systems. 
Dotted = laminar, dashed = transitional/unknown, solid = turbulent. 
Figure 5.3.3-2: Drop Capillary number based on Sauter mean diameter as a function 
of oil viscosity ( c) and Reynolds number (Re) for clean (no surfactant) systems. 
Figure 6.1.1-1: Sample DSD. This plot illustrates the usage of Fibonacci sequences 
to define discrete size categories for use in the definition of d32. 
Figure 6.1.2-1: Relationship between d32 and dmax for laminar flow data. 
Figure 6.1.2-2: Relationship between d32 and dmax for turbulent flow data. 
Figure 6.3-1: Critical Capillary vs. viscosity ratio for this study compared to Grace's 
(1982) curve for simple shear flow with ?  <  ?0.05. 
 
 x 
 
Figure 6.3-2: Empirical correlation for drop size scaling in rotor-stator mixers. This 
plot is comparable to equation (6.3-4) for clean and low surfactant concentration 
systems in laminar flow. 
Figure 6.4-1: Empirical correlation for drop size scaling in rotor-stator mixers. This 
plot is comparable to equation (15) for clean and low surfactant concentration 
systems in laminar flow. 
Figure 7.1-1: Equilibrium Sauter mean diameter vs. water volume fraction for clean 
systems in laminar flow. All experiments use Crystal Oil 500FG. T = 41?,  c = 93.2 
cP, ? = 0.0069, and Re: 470-1100. 
Figure 7.1-2: Equilibrium Sauter mean diameter vs. water volume fraction for clean 
systems in turbulent flow. All experiments use Crystal Oil 70FG. T = 60?,  c = 18.1 
cP, ? = 0.026, and Re: 2400-5600. 
Figure 7.1-3: Dynamic Sauter mean diameter for a clean (surfactant free) system in 
laminar flow. This experiment uses Crystal Oil 500FG. ? = 0.01, T = 41?,  c = 104.6 
cP, ? = 0.0062, and Re: 420 and 980. 
Figure 7.2-1: Equilibrium Sauter mean diameter vs. water volume fraction for clean 
systems in laminar flow with treated rotor-stator mill head. All experiments use 
Crystal Oil 500FG. T = 41?,  c = 93.2 cP, ? = 0.0069, and Re: 470-1100. 
Figure 7.2-2: Equilibrium Sauter mean diameter as a function of phase fraction for 
clean systems in laminar flow with an untreated rotor-stator mill head. All 
experiments use Crystal Oil 500FG. T = 41?,  c = 93.2 cP, ? = 0.0069, and Re: 470-
 1100. 
Figure 7.2-3: Equilibrium Sauter mean diameter as a function of phase fraction for 
clean systems in laminar flow with a treated rotor-stator mill head. All experiments 
use Crystal Oil 500FG. T = 41?,  c = 93.2 cP, ? = 0.0069, and Re: 470-1100. 
Figure 7.3.1-1: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 0.1xCMC (equilibrium interfacial tension = 23.8 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
Figure 7.3.1-2: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 7300 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 890. 
Figure 7.3.2-1: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 3900 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 470. 
Figure 7.3.2-2: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 5600 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 680. 
 
 xi 
 
Figure 7.3.2-3: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 9000 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 1100. 
Figure 7.3.2-4: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 0xCMC (equilibrium interfacial tension = 54.8 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
Figure 7.3.2-5: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 0.01xCMC (equilibrium interfacial tension = 41.4 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
Figure 7.3.2-6: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 1xCMC (equilibrium interfacial tension = 41.4 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
Figure 7.3.2-7: Average slope in Equation (7.1-1) vs. equilibrium interfacial tension 
for surfactant systems in laminar flow. All experiments use Crystal Oil 500FG. T = 
43?,  c = 93.3 cP, ? = 0.0066, and Re: 470-1100. 
Figure 7.4-1: Equilibrium Sauter mean diameter vs. water volume fraction at high 
water phase concentrations for surfactant systems with 1xCMC (equilibrium 
interfacial tension = 6.16 mN/m) in laminar flow. All experiments use Crystal Oil 
500FG. T = 43?,  c = 93.3 cP, ? = 0.0066, and Re: 470-1100. 
Figure 7.5-1: ?Radiator? apparatus. The large amount of hydrophilic surface area 
provided by gluing microscope slides together in such a fashion should serve to 
maximize any possible drop adherence to low shear surfaces. 
Figure 7.5-2: Effect of modifying interior, low shear surfaces. Very little effect is 
observed ? this invalidates the idea that possible drop sticking on the walls plays a 
significant role in the determining the equilibrium drop size. All experiments use 
clean Crystal Oil 500FG. T = 36?,  c = 132.7 cP, ? = 0.0053, and Re: 330-770. 
Figure 8.1.1-1: Maximum stable drop size vs. maximum local energy dissipation rate 
for a variety of types of mixers. From Leng and Calabrese (2004).  
Figure 8.1.2-1: Illustration of the kinetic vs. equilibrium region for DSDs in a 
conventional stirred tank. Source: Chang (1990).  
Figure 8.2.1-1: IKA Labor Pilot 2000/4 in-line mixer. Motor on the left, mixer 
volume on the right. 
Figure 8.2.1-2: IKA Labor Pilot 2000/4 in-line mixer. Motor on the left, mixer 
volume on the right. 
Figure 8.2.1-3: Medium generator. Two rows of teeth on both the rotor and stator. 
Figure 8.2.1-4: Fine generator. Three rows of teeth on both the rotor and stator. 
Figure 8.2.1-5: Ultrafine generator. Four rows of teeth on both the rotor and stator. 
 
 xii 
 
Figure 8.2.1-6: Blank stage as compared to an ultrafine stage. 
Figure 8.2.2-1: Principle of the operation of the torque meter. The torque meter was 
installed in-line on the rotor shaft. 
Figure 8.2.2-2: IKA Labor Pilot 2000/4 in-line mixer with Futek TRS600 torque 
meter installed in-line with the rotor shaft. 
Figure 8.3.1-1: Viscous power dissipation when the mixer is off (the rotor-stator 
geometry acts as a static mixer in this case). 
Figure 8.3.1-2: Contribution to flow work by the mixer for 1 medium stage as a 
function of flowrate and rotor speed. 
Figure 8.3.1-3: Contribution to flow work by the mixer for 1 fine stage as a function 
of flowrate and rotor speed. 
Figure 8.3.1-4: Contribution to flow work by the mixer for 1 ultrafine stage as a 
function of flowrate and rotor speed. 
Figure 8.3.1-5: Contribution to flow work by the mixer for 2 medium stages as a 
function of flowrate and rotor speed. 
Figure 8.3.1-6: Contribution to flow work by the mixer for 2 fine stages as a function 
of flowrate and rotor speed. 
Figure 8.3.1-7: Contribution to flow work by the mixer for 2 ultrafine stages as a 
function of flowrate and rotor speed. 
Figure 8.3.1-8: Contribution to flow work by the mixer for 3 medium stages as a 
function of flowrate and rotor speed. 
Figure 8.3.1-9: Contribution to flow work by the mixer for 3 fine stages as a function 
of flowrate and rotor speed. 
Figure 8.3.1-10: Contribution to flow work by the mixer for 3 ultrafine stages as a 
function of flowrate and rotor speed. 
Figure 8.3.2-1: Power measured by the torque sensor with 1 blank stage. This power 
does not contribute to the rate of viscous energy dissipation and must be subtracted 
from the measured power in future experiments. 
Figure 8.3.2-2: Power dissipated by the fluid as calculated by the measured power, 
the blank stage power, the flow work, and the blank flow work. 1 medium stage at a 
flowrate of 1lpm. 
Figure 8.3.2-3: Power dissipated by the fluid as calculated by the measured power, 
the blank stage power, the flow work, and the blank flow work. 2 fine stages at a 
flowrate of 3 lpm. 
Figure 8.3.2-4: Power dissipated by the fluid as calculated by the measured power, 
the blank stage power, the flow work, and the blank flow work. 3 ultrafine stages at a 
flowrate of 5 lpm. 
 
 xiii 
 
Figure 8.3.2-5: Power dissipated as a function of rotor speed for various flowrates 
using 1 medium stage. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
Figure 8.3.2-6: Power dissipated as a function of rotor speed for various flowrates 
using 1 fine stage. Filled data points indicate the torque measurements and hollow 
data points indicate calorimetric measurements. 
Figure 8.3.2-7: Power dissipated as a function of rotor speed for various flowrates 
using 1 ultrafine stage. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
Figure 8.3.2-8: Power dissipated as a function of rotor speed for various flowrates 
using 2 medium stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
Figure 8.3.2-9: Power dissipated as a function of rotor speed for various flowrates 
using 2 fine stages. Filled data points indicate the torque measurements and hollow 
data points indicate calorimetric measurements. 
Figure 8.3.2-10: Power dissipated as a function of rotor speed for various flowrates 
using 2 ultrafine stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
Figure 8.3.2-11: Power dissipated as a function of rotor speed for various flowrates 
using 3 medium stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
Figure 8.3.2-12: Power dissipated as a function of rotor speed for various flowrates 
using 3 fine stages. Filled data points indicate the torque measurements and hollow 
data points indicate calorimetric measurements. 
Figure 8.3.2-13: Power dissipated as a function of rotor speed for various flowrates 
using 3 ultrafine stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
Figure 8.3.2-14: Power dissipated per stage averaged over all flowrates. For each 
generator type, the power dissipated per number of stages is independent of the 
number of stages. 
Figure 8.3.2-15: Power number vs. Reynolds number for the IKA 2000/4 in-line 
mixer with water as the fluid using outer rotor diameter for D. 
Figure 8.3.2-16: Power number vs. Reynolds number normalized by number of 
stages and using a modified ?equivalent? diameter. This yields a constant Power 
number in turbulent flow. 
Figure 9.1-1: IKA Labor Pilot 2000/4 in-line mixer with Dantec ?FiberPDA? PDA 
system used to measure drop sizes after they exit the mixer. 
Figure 9.2.1-1: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 1 medium stage. 
 
 xiv 
 
Figure 9.2.1-2: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 1 fine stage. 
Figure 9.2.1-3: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 1 ultrafine stage. 
Figure 9.2.1-4: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 2 medium stages. 
Figure 9.2.1-5: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 2 fine stages. 
Figure 9.2.1-6: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 2 ultrafine stages. 
Figure 9.2.1-7: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 3 medium stages. 
Figure 9.2.1-8: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 3 fine stages. 
Figure 9.2.1-9: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 3 ultrafine stages. 
Figure 9.2.1-10: Time-averaged number distribution of Immersion oil drops in water 
with 2 fine stages at a flowrate of 1 lpm. 
Figure 9.2.1-11: Time-averaged volume distribution of Immersion oil drops in water 
with 2 fine stages at a flowrate of 1 lpm. 
Figure 9.2.1-12: Time-averaged number distribution of Immersion oil drops in water 
at a rotor speed of 7000 rpm with 1 and 2 ultrafine generators at a flowrate of 1 lpm. 
Figure 9.2.1-13: Time-averaged number distribution of Immersion oil drops in water 
at a rotor speed of 7000 rpm with 1 and 2 ultrafine generators at a flowrate of 1 lpm. 
Figure 9.2.2-1: Residence time distribution of Immersion oil drops in water with 1 
ultrafine generator. 
Figure 9.2.2-2: Residence time distribution of Immersion oil drops in water with 2 
fine generators. 
Figure 9.2.2-3: Cumulative Sauter mean diameter of Immersion oil drops in water 
with 1 fine generator. 
Figure 9.2.2-4: Cumulative Sauter mean diameter of Immersion oil drops in water 
with 2 ultrafine generators. 
Figure 9.3-1: PN vs. Re showing the region in which the drop size experiments were 
performed. 
Figure 9.3-2: Illustration of the high-shear region used to calculate the energy 
dissipation rate. Red indicates the region of high shear. 
Figure 9.3-3: Kolmogorov microscale as a function of rotor speed for the generator 
configurations for a sample flowrate of 3 lpm. 
 
 xv 
 
Figure 9.3-4: Test of the inertial, inviscid correlation?s capability of fitting the drop 
size data. A straight line with a positive slope represents good correlation. 
Figure 9.3-5: Test of the inertial, viscous correlation?s capability of fitting the drop 
size data. A straight line with a positive slope represents good correlation. 
Figure 9.3-6: Test of the inertial, inviscid correlation?s capability of fitting the drop 
size data. A flat line represents good correlation. 
Figure 9.3-7: Test of the inertial, viscous correlation?s capability of fitting the drop 
size data. A flat line represents good correlation. 
Figure 9.3-8: Replotted version of Figure (8.4.2-7). Data with an effect of flowrate 
for 3 lpm vs. 5 lpm has been removed. A flat line represents good correlation. 
Figure 9.3-9: Relationship of d32 with dmax for all drop size data. 
Figure 9.3-10: Comparison of drop size data with Davies (1987) plot. 
Figure 9.3-11: Comparison of drop size data with Davies (1987) plot including only 
data which survived the purge. 
Figure 9.3-12: Calculation of the maximum local energy dissipation rate in terms of 
the average by means of comparison with the Davies (1987) plot. 
Figure C-1: Calibration standard. Spacing between the small increments is 10  m 
and between the large increments it is 50  m. This image yields a calibration factor of 
6.73 pixels/ m. 
Figure C-2 (Figure 4.1.3-3): Typical microscope image used to measure drop size. 
Crystal Oil 500FG surrounded by water under a 60x microscope objective with a 
?Watec America Corp. LCL-902K.? 
 
 
 xvi 
 
Nomenclature 
a , slope of the log-linear phase fraction function of equation (7.1-1) 
A , Hamaker constant 
A , interfacial area per volume 
b , intercept of the log-linear phase fraction function of equation (7.1-1) 
Bo = ?? g De
 2 / ?, Bond number for the pendant drop method 
? , torque 
c , speed of light in a vacuum 
C , particle collision rate (assuming equally sized particles) 
C1 , constant in equation (3.4-5) 
C2 , constant of order unity in equation (3.4-6) 
Ca =  c ?  d/ 2 ?, Capillary number 
Cac =  c ?  d32 / ?, Capillary number based on continuous phase viscosity and d32 
Cac? =  c ?  dmax / ?, Capillary number based on continuous phase viscosity and dmax 
Cad =  d N D / ?, Capillary number based on dispersed phase viscosity and rotor 
diameter 
CV , heat capacity 
d, drop diameter 
d32, Sauter mean diameter 
di , characteristic diameter of the ith size bin 
dmax , maximum stable drop diameter 
D , impeller or rotor diameter 
De , equatorial pendant drop diameter 
Deq , effective diameter 
Douter , outer rotor diameter 
Ds , upper pendant drop diameter 
E(k) , turbulent energy spectral density function 
g , gravitational constant 
k , wavenumber in the energy spectral density function 
k , constant in equation (3.4-1) 
K , constant of proportionality between the characteristic shear rate and the nominal 
shear rate 
K , constant of proportionality between the effective and outer diameters in the IKA 
mixer 
L , macro length scale of turbulence 
N , rotor rotation rate 
 
 xvii 
 
ni , number of drops in the ith bin 
NP = P / ?c N
 3 D5, Power number 
       , average turbulent Power number 
P , collision efficiency 
P , power 
Pbear , power lost to friction in bearings, etc. 
Pblank , power measured with a blank stage 
Pblank flow work , flow work power with a blank stage 
Pdiss , power disspated into the fluid 
Pflow work , power spent on flow work 
Pmeas , measured power 
Q , flowrate 
(Q?P)0 , power dissipation when mixer is off 
r , lever arm 
R1 and R2, principle radii of curvature 
R2 , coefficient of variance 
Rcap , capillary radius 
Re = ?cND
 2/ c, Reynolds number 
t , time 
tc , time required to drain a film of fluid between two colliding drops 
ti , interaction time for a drop collision event 
T , temperature 
       , turbulent mean-square velocity difference across drop surface 
vi , component of the velocity vector 
Vi = (?c / ?d)
 1/2  d N D / ?, Viscosity group 
w , weight 
We = ?c N
 2 D3 / ?, Weber number 
xj , component of the displacement vector 
   , characteristic shear rate 
    , characteristic shear rate (in laminar flow experiments) 
? , clearance between rotor blade and inner stator wall (shear gap) 
?P , pressure difference across the mixer 
?T , temperature difference across the mixer 
?? , density difference between two immiscible liquids 
? , energy dissipation rate 
? , Kolmogorov micro length scale of turbulence 
?d , index of refraction 
 
 xviii 
 
? =  d /  c, viscosity ratio 
  , viscosity 
 d , continuous phase viscosity 
 d , dispersed phase viscosity 
?c , continuous phase kinematic viscosity 
? , phase fraction 
?c , continuous phase density 
? , equilibrium interfacial tension 
?(t) , dynamic interfacial tension 
?0 , initial dynamic interfacial tension 
?? , long time (equilibrium) interfacial tension in the presence of surfactants 
?uncal , uncalibrated interfacial tension 
? , dynamic interfacial tension time constant 
? , residence time 
? , torque 
?c , disruptive stress acting on drop surface 
?ij , component of the stress tensor 
?s , cohesive stress due to interfacial tension resisting drop deformation 
 
 
 1 
 
Chapter 1: Introduction 
1.1 ? Motivation & Purpose 
Mixing is industrially important in a variety of contexts where emulsions are 
produced including polymer processing, paints, cosmetics, food, and many other 
applications (Paul et al. 2004). Whether the purpose of a mixing process is the 
creation of a stable emulsion or the temporary formation of a large interfacial area per 
volume for chemical reaction or other purposes, it is desirable to obtain scaling laws 
so that multiphase processes' can be scaled up using the results of laboratory-scale 
experiments.  
Early mixing studies tended to use simple impellers such as Rushton (1950) turbines 
in a baffled tank with a low viscosity continuous phase in turbulent flow. However, 
this is not the situation in most industrial processes; often, the equipment is more 
complex, the continuous phase has a high viscosity, and flows are not restricted to the 
turbulent regime. There is a wide variety of emulsification apparatus which provide a 
range of shear rates and have a variety of throughput capacities such as stirred tanks, 
static mixers, valve homogenizers, and rotor-stator mixers. Rotor-stator mixers 
produce more intense shear fields than conventional mixers because the rotor rotates 
at much higher speeds, and more importantly because of the very close clearance 
between rotor and stator. 
For all mixing processes, there is generally a tradeoff between the required power 
input and the resultant drop size distribution (DSD). For turbulent fluids, this 
relationship between power input and DSD has been found to be applicable across all 
 
 2 
 
manner of mixing devices and quite independent of the specifics of the mixer type 
(Davies 1987). For processes which are allowed to proceed to equilibrium, the 
stresses that break up a drop into its smallest size are not due to average energy 
dissipation rate, but to the maximum local energy dissipation rate (Zhou and Kresta 
1998). Therefore what is needed to break up a drop to a specified size is a certain 
intensity of the maximum local shear stress which requires a specified local energy 
dissipation rate. Therein lies the advantage of rotor-stator mixers (pictured in Figure 
1.2-1); most of the energy that is supplied to a rotor-stator mixer is dissipated near the 
mixing head with relatively little energy, except that required for mild recirculation, 
being dissipated far from the mixing head (Yang 2011). This means that more of the 
energy put into a rotor-stator mixer is spent in increasing the maximum local shear 
stress which generally yields smaller drop size distributions. This is why rotor-stator 
mixers are the subject of our interests and are used in this study. However this does 
not mean that the results of this study are necessarily restricted to rotor-stator mixers; 
the arguments that are developed here would apply to any mixing process in which 
drops break up according to the same mechanism(s). The issue of how drops break up 
in rotor-stator mixers is itself a key question in this study. The mechanism(s) of drop 
breakup is/are dependent on the flow regime (laminar, transitional, or turbulent), the 
specifics of the mixing apparatus, and the fluids in question. 
In order to study these issues, a batch and an in-line rotor-stator mixer were 
examined separately. The flow regime for most previous studies has been turbulent 
because of the use of a low viscosity continuous phase. In the second part of the 
study, with an in-line mixer, this was the case. The first part of the study, with a batch 
 
 3 
 
mixer, includes both the laminar and turbulent regimes with a high viscosity 
continuous phase. Switching the phases used in most previous studies, this study used 
an oil continuous phase with water dispersed into it. The increased viscosity of the 
continuous phase allows for a much lower Reynolds number, and hence laminar flow, 
to occur. This also causes the ratio of dispersed to continuous phase viscosity to be 
quite low, which is relevant when comparing the results of the laminar flow 
experiments with controlled laminar drop breakup studies. 
1.2 ? Approach 
For the first part of this study, water was dispersed into Crystal Oil, a food 
grade mineral oil, using a batch Silverson L4R mixer (pictured in Figure 1.2-1) to 
investigate equilibrium drop size distributions as a function of physical properties and 
agitation intensity. The flow regime as a function of Reynolds number and viscosity 
ratio for this mixer was determined both by reference to the work of Padron (2001) 
who performed power draw measurements and by qualitative differences in the 
behavior of the drop size data. Care was taken to ensure that all reported data were 
well within the specified flow regimes. 
 
 4 
 
 
Figure 1.2-1: Batch Silverson L4R mixer. The top image is the entire setup with 
attached mixing vessel. The bottom two images depict the high-shear mill head which 
is composed of a rotor surrounded by a stator with very small clearance. 
The breakage mechanism(s) for this case of very low ratio of dispersed to 
continuous phase viscosity were investigated by performing complete sets of 
experiments for both laminar and turbulent flow using dilute, clean (devoid of 
surfactant) concentrations of water in oil. This is the starting point because not only 
does this allow the flow field to be treated as though it were a single phase, but any 
effect of drop-drop interactions or any coalescence may be treated as negligible. 
After the most straightforward case, complications were added one-at-a-time 
to observe their effect separately. The mixing head was treated with a hydrophobic 
silane to test the effect of the interaction between the drop and the high shear surfaces 
and dilute experiments were performed. Also, in a separate group of experiments, an 
 
 5 
 
oil-soluble surfactant, Tergitol NP-4, was added and dilute experiments were 
performed. 
The next level of complication was that of using a non-dilute volume fraction, 
?, of water. At low volume fractions (but still with ? > 0.001), the effect of this on the 
bulk flow was expected to be minimal, or at least less significant, so experiments 
were first run at low, non-dilute concentrations with clean oil and no treatment on the 
high-shear mixing head. The cases of treated mixing head and added surfactant at 
non-dilute concentrations were performed and analyzed separately. Finally, high 
concentration experiments were performed. 
All of the previously discussed work was done at equilibrium or, in the case of 
dilute systems (since there is no coalescence and only breakage), ultimate times. 
For the second part of this study, Immersion oil, a viscous oil remarkable for 
its high index of refraction, was dispersed into water using a IKA Labor pilot 2000/4 
in-line rotor-stator mixer. Power draw experiments were performed with a torque 
sensor. By constructing a similar plot to Padron?s (2001) plot, it was determined that 
all of the data was within the turbulent regime. 
Non-equilibrium drop size distributions are analyzed and correlated with 
respect to the energy dissipation rate. These data have a high dispersed to continuous 
phase viscosity ratio; Immersion oil is dispersed into water. The experiments were 
performed in an IKA 2000/4 in-line pilot scale rotor-stator mixer. This mixer has 
interchangeable rotors and stators which produce varying levels of shear intensity. 
This study measured the power draw of the mixer as a function of continuous phase 
flowrate, agitation intensity, and continuous phase physical properties with a Futek 
 
 6 
 
TRS600 shear stress sensor that was installed inline with the mixing shaft. The drop 
size distributions were measured separately by B.N. Murthy (2010) who was working 
in the same laboratory concurrently and received as a personal communication. This 
was done through the use of a phase Doppler anemometer (PDA) system. The power 
draw measurements were used to correlate the drop size data. 
In all of the aforementioned cases whether non-dilute or dilute, surfactant-
 laden or clean, hydrophobically treated or untreated mill head, laminar or turbulent, 
equilibrium or non-equilibrium, the goal was always to construct meaningful, 
physically-based correlations for the drop size as a function of physical properties and 
agitation intensity. Specifically, the goal is to define dimensionless groups and 
develop mechanistic correlations that correlate the data, in order to develop general 
rules for scaling up drop size from the laboratory to the industrial scale. 
1.3 ? Organization of the Dissertation 
This dissertation has three main parts. First, the physical properties, and their 
measurement, of the materials used in the dispersion experiments are discussed, then 
the main body of work concerning equilibrium dispersion experiments is reported, 
and finally the in-line power draw experiments are presented and compared to drop 
size data. 
Chapter 2 defines the physical properties of interest for the batch dispersion 
experiments, which include: index of refraction, viscosity (of both phases), interfacial 
tension, and density. After such definitions, various measurement methods are 
discussed and the measurements that are relevant for this study are presented. 
Particular attention is given to the measurement of interfacial tension via the pendant 
 
 7 
 
drop technique because of the wide scatter in literature values caused by the large 
error inherent in the all methods of the measurement of interfacial tension of which 
the pendant drop method is no exception. A refined in-house methodology for the use 
of the pendant drop method is presented, the validity of which was extensively 
investigated and established much more thoroughly than the author believes is 
generally the case for methods of interfacial tension measurement (as evidenced by 
the unacceptably large spread in the reported values of interfacial tension). 
Chapter 3 discusses the theory for drop size distributions in dispersion studies. 
Various models are covered and classified with respect to the drop size distributions 
and how they relate to which types of flow regimes, viscosity ratios, concentrations, 
and mixer types (dynamic vs. equilibrium). These models are presented as bases for 
comparison for the drop size data which are reported in Chapters 6-7, and 9. There 
are two ways in which comparisons are made in subsequent chapters: the model can 
simply be verified in a flow regime in which it has not yet been thoroughly tested or 
the model can provide a partial explanation for observed effects (this is the case if the 
model is for a non-identical, though similar, situation). 
Chapter 4 shows the experimental procedures which are used to perform the 
dispersion experiments. Detailed description is given, first of the methods of 
emulsification and of drop size measurement of the equilibrium dispersion 
experiments taking place within the batch Silverson L4R mixer, then also of the 
power draw experiments performed on the IKA 2000/4 in-line mixer. Brief mention 
is made of the power draw measurement technique of the batch Silverson L4R mixer 
and the phase Doppler anemometry (PDA) dispersion experiments performed in the 
 
 8 
 
IKA 2000/4 in-line mixer. However, since that work was performed by others, its 
description is significantly abbreviated. 
The remainder of the chapters concern the actual dispersion experiments. 
Chapters 5-7 pertain to equilibrium experiments carried out with water dispersed into 
a viscous continuous phase, Crystal Oil, in a batch Silverson L4R mixer. Chapter 5 
argues for the domains of the laminar and turbulent regime as a function of Reynolds 
number and viscosity ratio for the aforementioned viscous continuous phase. This is 
done by comparison with power draw data using experiments performed by Padron 
(2001) and by qualitatively examining the behavior of dilute drop size distributions at 
varying viscosities and agitation rates. Some of the data is in an unknown or 
transitional flow regime, but in Chapter 5 it is shown that the dispersion experiments 
of chapters 6 & 7 are all definitively within the stated flow regime. 
Chapter 6 reports the results of the dilute dispersion experiments. The laminar 
and turbulent results are reported separately and compared to the theories relevant to 
each flow regime which were introduced in Chapter 3. These ideas are used not only 
to correlate the drop size, but also to investigate the breakage mechanisms for each 
flow regime when a viscous fluid is used as the continuous phase. The dilute 
dispersion experiments include varying the high-shear surfaces? wettability by 
application of a hydrophobic coating, varying the interfacial tension by adding 
Tergitol NP-4, and varying the viscosity of both phases. 
Chapter 7 reports the results of the non-dilute experiments. These results are 
more difficult to interpret than the dilute ones in Chapter 6. The theory of Chapter 3 is 
used to compare the effect of phase fraction for a low-viscosity continuous phase to 
 
 9 
 
the case in Chapter 7 where the continuous phase is viscous. The non-dilute 
dispersion experiments include varying the high-shear surfaces? wettability by 
application of a hydrophobic coating and varying the interfacial tension by adding 
Tergitol NP-4. 
By contrast with Chapters 5-7, Chapters 8-9 are concerned with power draw 
experiments in an in-line IKA 2000/4 mixer as compared to drop size experiments 
carried out with Immersion Oil dispersed dilutely into a low viscosity dispersed 
phase, water, in the same mixer. By comparing power draw data to drop size data, the 
approach to equilibrium and maximum rate of energy dissipation are investigated as a 
function of Reynolds number, continuous phase flowrate, and mixer geometry.  
Chapter 10 presents the most important conclusions of each portion of this 
study and provides recommendations for future work that would further improve the 
understanding of the material presented in this dissertation. 
Appendix A catalogs the interfacial and surface tensions of some common 
fluids listed in the literature to highlight discrepancies therein. Appendix B provides 
the surface and interfacial tension measurement computer programs. Appendix C 
provides the ImageJ macro used to identify the pixels which form the drop outline in 
the pendant drop method.
 
 10 
 
Chapter 2: Relevant Physical Properties and their Measurement 
 Physical properties such as viscosity and interfacial tension determine the 
magnitudes of the various forces that interact with each other to determine the drop 
size. The relative strengths of these forces are quantified by dimensionless numbers 
such as the Reynolds and Weber numbers. It is desirable to have control over these 
physical properties so that their effect on the DSD can be investigated. 
 Since the goal is to develop dimensionless scaling laws for drop sizes as a 
function of such dimensionless numbers, it is important to obtain accurate 
measurements of these physical properties. Any inaccuracies in physical property 
data would be manifested in the DSDs. Therefore, considerable effort was expended 
in obtaining such data. 
The physical properties discussed in this chapter pertain only to the batch 
experiments which make up the first part of this study. This is because the physical 
properties of Immersion Oil, which is used in the in-line portion of this study, were 
available in the literature, and so it was not necessary to determine them 
experimentally. 
2.1 ? Index of Refraction 
2.1.1 ? Definition of Index of Refraction 
The index of refraction of a material is defined as the ratio of the speed of 
light in a vacuum, c, to the speed of light in that material, v (Young and Freedman, 
2004): 
   
 
 
     (2.1.1-1) 
 
 11 
 
Differences in the speed of light in materials cause light to bend, or refract, 
whenever light passes through an interface from one material to one with a different 
refractive index. This is because the portion of the wave that enters the new material 
first adjusts its speed before the other portion of the wave has entered the new 
material. This effect is shown in Figure 2.1.1-1 where the speed of light is slower in 
material a than in material b, or ?b < ?a. 
The alteration in the trajectory of the light is a function of the difference in 
refractive indices of the two materials. The nature of this relationship is given by 
Snell's law (Young and Freedman, 2004). 
                   (2.1.1-2) 
where a and b refer to the materials shown in Figure 2.1.1-1, and ni is the 
appropriate index of refraction and  i is the appropriate angle. 
 
Figure 2.1.1-1: Illustration of Snell's Law. The angle of incidence, a, is shallower 
than the angle of refraction because the index of refraction in material a is greater 
than that of material b. 
2.1.2 ? Relevance of Index of Refraction: Determination of Oil for Use in 
Dispersion Experiments 
Unlike the other physical properties that are discussed in this chapter, the 
refractive index has no effect on the drop sizes in the dispersion experiments. Its 
 
 12 
 
relevance for this research is in the measurement of the drop sizes. An emulsion of 
two fluids which are both transparent and have the same refractive index would not 
scatter light and would, therefore, be transparent. The dispersed drops of such an 
emulsion would not be measurable by any method involving light. Generally, a 
greater difference in the indices of refraction of two liquids in a dispersion will result 
in easier drop size measurement with more accurate results. 
When the type of oil to be used was being selected for this research, the index 
of refraction was a key consideration. Since the method of measurement (which will 
be discussed more completely in Chapter 3) was optical microscopy, the difference in 
the indices of refraction between the drop and the continuous phase causes light to 
bend around the edges of each drop. This causes darker regions corresponding to the 
material interface to be apparent on a microscope. 
 The index of refraction was measured with a refractometer as a function of 
temperature for various oils which were considered for use in the water-in-oil 
dispersion experiments with the results plotted in Figure 2.1-2. 
Four types of oil were considered for use in this research, Lubsoil ND-50, 
silicone oil, Immersion oil and food grade Crystal Oil. Because they are expensive, 
immersion oils are typically used specifically for their high index of refraction, 
1.5150, which is not only close to that of glass, 1.51872 (Lide 2010), but also has the 
highest difference with water. Figure 2.1.2-1 shows that Lubsoil ND-50 and Crystal 
oil have a significant advantage in index of refraction compared to silicone oils. In the 
end, Crystal Oil (food grade) was selected because there were other issues - 
translucency and the suspicion of unwanted amphiphilic additives - which made 
 
 13 
 
Lubsoil ND-50 a poor choice. Figure 2.1.2-1 also shows that temperature does not 
affect the index of refraction difference between oil and water. 
 
 
Figure 2.1.2-1: Refractive index of various oil's that were considered for use in the 
dispersion experiments. The difference between the oil's index of refraction and 
water's determines the phase contrast for imaging purposes. 
2.2 ? Viscosity 
2.2.1 ? Definition of Viscosity 
The viscosity is the material property of a fluid that relates the rate of 
deformation of a fluid to the applied stress. The relationship of stress to rate of strain 
differs according to the type of the fluid. The simplest form of this relationship is that 
of an incompressible Newtonian fluid, where each shear stress component, ?ij, is 
Refractive Index vs. Temperature
 1.3
 1.35
 1.4
 1.45
 1.5
 1.55
 20 30 40 50 60 70
 Temperature (C)
 Ref
 ract
 ive
  In
 de
 x
 Water
 Lubsoil
 ND-50
 Silicone
 Oils
 Immersion
 Oils
 Crystal Oil
 
 14 
 
directly proportional to rate of strain,  
   
   
 
   
   
 . For a Newtonian fluid, the 
proportionality constant defines the viscosity,   (Middleman 1998). 
       
   
   
 
   
   
    (2.2.1-1) 
where ?ij is the stress from the j direction acting in the plane in the I direction, 
  is the viscosity which is here being defined, vi is the velocity in the i direction, and 
xj is the spatial direction in which the velocity is being differentiated. 
There are other, more complex relationships between stress and strain rate for 
non-Newtonian fluids which have been quantified. These fluids include Bingham 
fluids which have a yield stress below which no deformation can occur, power-law 
fluids in which the strain rate has an exponent which is not unity, and others (Bird, 
Stewart, and Lightfoot 1960). Newtonian fluids are the most common type of fluids 
and their properties are the simplest. Therefore, one of the criteria for the oil that was 
chosen for the dispersion experiments is that it be Newtonian. 
2.2.2 ? Relevance of Viscosity 
The viscosities of the continuous phase and the dispersed phase fluids have a 
role in both (assuming inertial forces do not dominate the breakup event) drop 
breakup (Grace 1982) and coalescence (Chesters 1991). Using a high viscosity 
continuous phase in a rotor-stator mixer is, in fact, the main distinguishing feature of 
this study. The effect of both viscosities is investigated in detail throughout this study. 
 
 15 
 
2.2.3 ? Experimental Measurement of Oil Viscosity 
The viscosity of Crystal Oil was measured using an Advanced Rheometer AR 
2000 manufactured by TA Instruments. This device is a cone-and-plate rheometer. 
The fluid is placed between a shallow cone (2? angle) and a plate that are a few 
millimeters apart and it is held in place by surface tension. The instrument works by 
rotating the cone while holding the plate stationary, thereby shearing the fluid. The 
amount of torque on the shaft of the cone is measured and is used to calculate the 
viscosity of the fluid.  
There were four viscosity grades of Crystal Oil available for use in this work: 
500, 350, 200, and 70 FG, the viscosity of all of which, as well as some blends of 
these grades, were measured as a function of temperature. After verification that the 
oil was Newtonian, all experiments were performed at a constant, arbitrary shear rate 
of 100 s-1.  
2.2.4 ? Verification that Crystal Oil is Newtonian 
As discussed in Section 2.2.1, Newtonian fluids are the simplest type of fluids 
and are therefore used in this research. No complex rheology was expected for 
Crystal Oil, but it was thought prudent to verify that it is indeed a Newtonian fluid. 
Figure 2.2.4-1 shows a plot of viscosity vs. shear rate for Crystal Oil 500 FG. Since 
the viscosity, the ratio of the shear stress to the shear rate, is not a function of shear 
rate, it can be concluded that Crystal Oil is a Newtonian fluid.  
 
 16 
 
 
Figure 2.2.4-1: Viscosity vs. shear rate for Crystal Oil 500 FG at 20.0? C. This 
shows, as expected, that Crystal Oil is a Newtonian fluid - that shear stress is linearly 
related to shear rate. 
2.2.5 ? Viscosity as a Function of Temperature and Viscosity Grade 
Viscosity is a significant parameter influencing the drop size in the water-in-
 oil dispersion experiments. Having a high-viscosity continuous phase is the most 
important feature that distinguishes this project from previous works. Therefore, it 
was important to be able to predictably control the viscosity. This was done by 
changing both the temperature and the grade of the oil. 
The viscosity of oil decreases with increasing temperature. For example, the 
viscosities of both diesel fuel and biodiesel decrease exponentially over the range of 
0-100? C (Barab?s and Todoru? 2011). As shown on Figure 2.2.5-1, each of the 
Crystal Oil curves also decreases exponentially over the range of temperatures that 
was measured. This is fortunate because this strong dependence on temperature 
 
 17 
 
allows the viscosity to be controlled fairly precisely during the dispersion 
experiments. 
 
Figure 2.2.5-1: Viscosity as a function of temperature for the four grades of Crystal 
Oil as well as water.  
The viscosity grade of the oil, naturally, had a significant effect on the 
viscosity. On Figure 2.2.5-1, it can be seen that, while most of the viscosities can be 
accessed by varying the temperature between 30 and 60? C and using 500 FG Crystal 
Oil, in order to gain access to lower viscosities 200 FG or 70 FG is required.   
Doing dispersion experiments at lower oil viscosities is crucial to obtaining a 
full set of experimental data. As it turns out, not only the phase viscosity is important, 
but also the viscosity ratio of the continuous and the dispersed phases. Figure 2.2.5-1 
also includes a curve for water (Lide 2010) (multiplied by 100 to show the curvature). 
It shows that water's viscosity is also a function of temperature though not as strongly 
 
 18 
 
as Crystal Oil's. Using this difference in the strength of the temperature dependence 
along with the different grades of oil, a range of combinations of continuous phase 
viscosities are available at a range of viscosity ratios. It is important to be able to 
control those two variables independently of one another for the dispersion 
experiments.  
Not all of the desired viscosities for the continuous phase and the dispersed 
phase were available and so, for a few of the experiments, blends were created. 
Figure 2.2.5-2 shows all the viscosity data for the oil that was measured including the 
blends. It is the same as Figure 2.2.5-1 except that it is more crowded with this 
additional information. There are no unexpected results from the blending - each 
blend was somewhere between the two viscosity grades from which it was composed. 
Also shown on Figure 2.2.5-1 are multiple curves for the same viscosity grade 
of oil. This is because, for some of the viscosity grades, more oil was purchased after 
the first was depleted. When measured, it was found that there were small differences 
in the viscosity between batches. Since it was measured and accounted for, it caused 
no problems.  
 
 19 
 
 
Figure 2.2.5-2: Viscosity of all the grades, batches, and blends of Crystal Oil 
including all the temperatures at which they were used in the dispersion experiments 
as well as that of water. 
2.3 ? Interfacial Tension 
2.3.1 ? Definition of Interfacial Tension 
Isaac Newton suspected that the whole of small scale phenomena might all 
depend upon something like interfacial tension (Rowlinson 2002). While it is not true 
that interfacial tension accounts for everything that occurs at small scales, it often has 
a major role when the length scale is sufficiently small - on the order of millimeters or 
less - and there are two phases present (Middleman 1998). 
Interfacial tension is a material property that is specific to each pair of 
immiscible substances. When two immiscible substances are in contact with each 
other, the boundary is called an interface. There is an amount of energy per unit area 
 
 20 
 
associated with every interface (Adamson 1976). The surface tension is the excess of 
free energy per unit area of the interface. 
An easy way to visualize interfacial tension as the free energy associated with 
an interface is to think of spreading soap films in air (Katoh 2004). The thought 
experiment is to imagine a square wire with one movable side as depicted in Figure 
2.3.1-1. As more area is created, the free energy of the system is increased because of 
the thermodynamically unfavorable interaction of the soap with air. 
 
Figure 2.3.1-1: Spreading of a soap film in air. As the red wire is pulled to the right 
by an external force the free energy of the system rises as a result of the greater 
interfacial area. 
2.3.2 ? Relevance of Interfacial Tension 
Interfacial tension is relevant to dispersion experiments because it is the 
primary force that holds drops together. Interfacial forces act to minimize the surface 
area, causing drops to assume as spherical a shape as possible (Leng and Calabrese 
2004). Drop breakage only occurs if this force can be overcome. Therefore, for the 
dispersion experiments, it is desirable to know as much possible about the interfacial 
force. Furthermore, some control over the strength of this force is established with the 
addition of surfactants. 
 
 21 
 
2.3.3 ? Scatter in the Published Values of Interfacial Tension 
There is a significant amount of scatter in the published values of interfacial 
tension. A literature search was conducted to illustrate some of the discrepancies in 
the published values of surface and interfacial tension. The results of this search are 
given in Appendix A and can be viewed in Table (A-1). 
Surface and interfacial tension vary with temperature. However, this is not the 
explanation for the scatter in Table A-1. Most of the temperatures for which surface 
or interfacial tension was published are close to 20? or 25? C. Also, the dependence of 
surface and interfacial tension on temperature is approximately linear and it can be 
seen that the slope of this linear relationship results in only minor adjustments to the 
values in A-1 (Jasper 1972; Lide 2009; Speight 2005). 
The significant amount of scatter evidenced in Table A-1 is the result of the 
many sources of error inherent in measuring interfacial tension. Of the many methods 
for measuring interfacial tension, most of them require knowledge of the density of 
the fluids in question. Therefore, any error in the density of the fluids propagates into 
error for the measured interfacial tension. Fortunately, for most of the fluids used as 
standards of calibration for interfacial tension systems, the densities are known to a 
high degree of accuracy. The most significant source of error and what accounts for 
the high degree of scatter in the reported values of surface or interfacial tension is the 
high amount of error caused by contamination. Often, this is due to the fact that the 
contaminant is so insoluble in a liquid that all the available contaminant adsorbs at the 
surface and forms a monolayer (Padday 1969). The two issues just discussed are 
general; however, each method has additional sources of error. 
 
 22 
 
2.3.4 ? Alternate Methods to Measure Interfacial Tension 
In this work it was deemed best to use the pendant drop method (explained in 
the next subsection) to measure the interfacial tension of all combinations of water 
and Crystal Plus oil (clean and at varying levels of surfactant concentration) used in 
the dispersion experiments. However, in order to understand the reasoning of this 
choice and to provide reference for a few alternate methods that would be more 
appropriate in other situations, some other methods for measuring surface and/or 
interfacial tension are here mentioned. 
Capillary Rise Method 
The Capillary Rise method was the first method used for measuring surface 
tension. Capillary action, as being somehow different from ?astronomical? (large-
 scale gravitational) forces was first studied by Young as well as Laplace (Rowlinson 
2002). This phenomenon is based on the difference in pressure on opposing sides of 
curved fluid interfaces. The pressure on the inside of a curved interface is greater than 
that on the outside of the interface as expressed in the Young-Laplace equation. 
            
 
  
 
 
  
    (2.3.4-1) 
where pin is the pressure on the inside of the curved interface, pout is the pressure on 
the outside of the curved interface,   is the interfacial tension, and 1/R1 is the radius 
of curvature in one direction and 1/R2 is the radius of curvature in the other. Since 
there is this pressure difference whenever a fluid-fluid interface is curved and there is 
a difference in the adhesion of one fluid to a solid surface relative to another 
(resulting in a curved interface), capillary rise occurs. The Capillary Rise method 
measures the surface tension by quantifying this effect when a capillary tube is 
 
 23 
 
inserted into a fluid interface. It is easiest to use a solid for which the contact angle is 
zero so that both radii of curvature are equal to the capillary radius. Then, the liquid 
height, in equilibrium with the atmospheric pressure, can be related to the hydrostatic 
pressure of the liquid column above the surface of the liquid. 
Wilhelmy Plate Method 
The Wilhelmy Plate method (Wilhelmy 1863) is, perhaps, conceptually, the 
simplest measurement method. The basic idea of this method is that a small plate 
starts immersed in one fluid and is pulled through the interface into another fluid. 
(The plate is often made of glass though it could also be roughened platinum or some 
other material such that the contact angle is close to zero). The additional force 
associated with the meniscus which forms as the plate is pulled through the interface 
is used to obtain the interfacial tension between the two fluids. 
DuN?oy Ring Method 
The DuN?oy Ring method was previously performed by Timberg (1887) as 
well as Sondhauss (1878). However, DuN?oy (1925) described a significantly easier 
to operate version of this apparatus, and so it is DuN?oy?s name that has become 
associated with this method (Padday 1969). This method is analogous to the 
Wilhelmy plate method except that, in place of the glass plate, a solid ring with a 
circular cross section is used. 
Maximum Bubble Pressure Method 
The Maximum Bubble Pressure method is a way of obtaining the surface 
tension of a liquid by measuring the maximum pressure needed to vertically inject an 
air bubble into the liquid. It is unique in that it is the only technique capable of 
 
 24 
 
measuring dynamic surface tension over a very short time range (down to 
milliseconds) (Fainerman 2004). However, it is limited in that it can only measure 
surface tension and not the interfacial tension between two liquids. The theory starts 
with the Young-Laplace equation (2.3.4-1). Equation (2.3.4-1) can be rewritten to 
find surface tension (Fainerman 2004). 
   
      
 
    (2.3.4-2) 
where RCap is the capillary radius, ?P is the pressure difference between the measured 
maximum pressure and the hydrostatic pressure, ? is the surface tension, and f is a 
correction factor which is unity when RCap < 0.1 mm. No correction is needed when 
the capillary radius is sufficiently small because in that case surface tension effects 
dominate the drop shape. If that is not the case (which is true for all commercial 
instruments (Fainerman 2004)), then f is not unity and additional subtractive terms 
are necessary, the magnitude of which are apparatus-dependent, to account for 
aerodynamic resistance in the capillary and fluid viscous resistance in the liquid. 
Drop Weight Method 
The Drop Weight method is performed by slowly increasing the volume of a 
drop hanging off of a dripping tip until the gravitational force is strong enough to 
overcome the interfacial force. The weight of the drop is measured and used as a basis 
for comparison to find the interfacial force at the moment the drop broke off, the 
moment at which the two forces just balanced each other. In order to do this with any 
sort of accuracy many drops are used and counted with their collective weight being 
divided by the number of drops in order to obtain the interfacial tension. In the last 
 
 25 
 
few decades the accuracy of this method has significantly increased with the advent 
of micrometer syringes and hypodermic needles (Lee 2008). 
Sessile Drop Method 
The Sessile Drop method is fairly similar to the pendant drop method so it will 
not be discussed significantly here. The main difference is that the sessile drop 
involves the shape of a drop resting on a horizontal surface while the pendant drop 
method has the drop hanging from a syringe. The pendant drop method has the 
advantage of eliminating the relevance of the contact between the solid and drop 
surfaces. However, there are some situations which are especially well suited to the 
sessile drop method, such as molten metals (Padday 1969) or other high temperature 
applications in which it would be problematic to utilize a syringe. 
2.3.5 ? Description of the Pendant Drop Method 
Pendant Drop Method 
The basic idea behind the Pendant Drop procedure is to hang a drop of the 
heavier fluid in a gravitational field while surrounding it with the fluid of lesser 
density. Gravity pulls the drop down while the interfacial tension between the two 
phases pulls it up. Since the strength of gravity is known, if the drop is at a static 
equilibrium, one can find the strength of the interfacial tension force by equating it to 
the gravitational force. This system may also be inverted so that the less dense phase 
is the drop phase and it is pulled up by buoyancy. The measured value of interfacial 
tension does not depend on which configuration is selected because, this phase 
inversion only serves to reverse the direction of the forces; their magnitudes are 
unchanged. These two options are represented in Figure 2.3.5-1. The fluid of lesser 
 
 26 
 
density is yellow and the fluid of higher density is blue, representing the situation of 
oil (yellow) and water (blue).  
 
Figure 2.3.5-1: The two configurations of the pendant drop method. The yellow fluid 
is less dense than the blue fluid. The measured value of the surface tension is the 
same regardless of which configuration is used. 
The strength of the interfacial force is reflected in the shape of the drop. An 
empirical correlation for this expressed in terms of shape parameters was developed 
by Bashforth and Adams (1883) and converted to a more straightforward tabular form 
by Adamson (1976). Adamson?s tables can be made into an empirical correlation of 
the form of equation (2.3.5-1). 
  
     
 
      
  
  
 
               
       
      (2.3.5-1) 
where g is the acceleration due to earth?s gravity, ?? is the difference in 
density between the two phases, and the lengths of De and Ds are shown in Figure 
2.3.5-2. These "diameters" quantify the amount of deformation, or the departure from 
the preferred spherical shape, that the hanging drop is experiencing when the 
gravitational and surface forces are balanced.  
 
 27 
 
 
Figure 2.3.5-2: The shape factors used to calculate interfacial tension via the pendant 
drop method. De is the equatorial diameter and Ds is the diameter of the drop at a 
height (measured from the bottom of the drop) equal to the equatorial diameter. 
According to Padron (2005), who performed pendant drop experiments with 
silicone oil-water surfactant systems, this expression is valid for minimum values of 
the Bond number, as defined in equation (2.3.5-2).  
   
 ?   
 
?
     (2.3.5-1) 
If the bond number is too low then, the surface forces are too much greater 
than the gravitational forces and the drop?s shape is too spherical for the shape factors 
to be accurately determined. Padron found that the experiments provided consistent 
data for all Bond numbers greater than 0.48. 
For all experiments performed in this study, the Bond number (as defined in 
equation 2.3.5-2) was always greater than 0.48 so this was not an issue. However, 
there were many other experimental complications which required specific, creative 
solutions. These are discussed in the following sections. 
 
 28 
 
Having completed the discussion on alternate methods of measuring 
interfacial tension as well as the pendant drop method, it is now appropriate to 
mention why the pendant drop method was chosen as the method to measure 
interfacial tension. The reasons are several. First, for the Pendant Drop method ? 
unlike the Sessile Drop, DuN?oy Ring, and Willhelmy Plate methods ? fluid-solid 
contact at the point of measurement does not affect the outcome. In the Sessile Drop 
method, any surface roughness or contamination could result in inaccuracies in the 
drop shape including asymmetry and hysteresis effects. For the DuN?oy Ring and 
Willhelmy Plate methods the large length of the three-phase contact line amplifies the 
effects of any solid contamination or roughness on the final result. 
Secondly, the pendant drop method is particularly well-suited to measure 
dynamic interfacial tension in the presence of surfactants, provided the diffusivity of 
surfactants in the bulk phase is sufficiently low so that the equilibrium surfactant 
concentration at the interface is reached in minutes or hours rather than seconds. If 
the dynamic interfacial tension was very fast, then a mechanism of quantifying time-
 zero would need to be developed. As aforementioned, the Maximum Bubble Pressure 
method is well-suited to measure fast dynamic interfacial tensions, being able to 
resolve milliseconds (Fainerman 2004). However, the Maximum Bubble Pressure 
method can only be used if the bubble is a gas so it is not suitable for use in this 
study. Also, for the systems used in the dispersion experiments of this study, the 
continuous phase is a viscous oil and so the rate of diffusion of the surfactant 
molecules within the oil is quite slow (on the order of hours) so quick measurements 
of the dynamic interfacial tension are unnecessary. 
 
 29 
 
The Drop Weight method was disqualified because it cannot measure dynamic 
interfacial tension by its very nature. Since drops are continuously falling from the 
end of a needle there is no time for surfactants to diffuse and for the interface to reach 
its equilibrium interfacial tension. 
The Capillary Rise method was deemed unsuitable because of the hysteresis 
of the contact line that would occur when the interfacial tension decreases for the 
dynamic interfacial tension experiments. Also, the time of the experiments would be 
expected to increase drastically due to the depletion of the surfactant near the 
interface caused by the narrowness of the capillary. That is, as the surfactant 
molecules became adsorbed at the interface, the oil near the interface would be 
surfactant-poor and it would take time for surfactant molecules to diffuse through this 
region. As will be seen in the following subsections, by comparison, the pendant drop 
method, which did not have this issue, took hours per experiment for the substances 
used in this study. 
The Pendant Drop method is a good method for this study because it is 
independent of contact angle (Morita 2002), is experimentally simple, and is capable 
of measuring dynamic interfacial tension. 
2.3.6 ? Pendant Drop Method as Used in this Study 
In this section all steps of the experimental procedure are described starting 
with the physical experiments and finishing with the data processing. In order to 
measure liquid-liquid interfacial tensions it was necessary to construct a cuvette. This 
was done by gluing 2? x 3? microscope slides together into a box shape with an open 
top with silicone glue as pictured in Figure 2.3.6-1. The function of this cuvette was 
 
 30 
 
to contain the surrounding phase while the syringe needle was immersed in it. The 
cuvette?s being made of glass permitted pictures to be taken through its wall of the 
drop?s shape. 
 
Figure 2.3.6-1: Glass cuvette constructed out of 2? x 3? microscope slides. Used to 
contain the surrounding phase in liquid-liquid pendant drop experiments to measure 
interfacial tension. 
First, this cuvette was filled with the surrounding phase and placed on a 
vibration isolation table. The vibration isolation table was used in order to prevent 
drops from falling off of the needle due to external vibrations. A 500 or 250  L 
syringe was filled with the drop phase and the needle was submerged into the 
surrounding phase in the cuvette. The needles that were used were made of type 316 
stainless steel and ranged in size from 26 to 16 gage (0.457 to 1.651 mm). If the drop 
phase was denser than the surrounding phase, then a straight needle was used so that 
gravity/buoyancy pulled down and interfacial tension pulled up as in the left-hand 
 
 31 
 
panel of Figure 2.3.5-1. If the drop phase was less dense than the surrounding phase, 
the forces acted in the opposite directions and the J-shaped needle used as in the 
right-hand panel of Figure 2.3.5-1. For either needle type, the syringe was held in 
place at the desired height with a clamp on a ringstand. A lamp with an incandescent 
light bulb was places behind the cuvette to provide back lighting. The cuvette itself 
was place on a vibration isolation table. After the syringe/needle was placed in the 
correct location and any air was cleared out of the syringe/needle assembly, the 
syringe piston was depressed to the degree that the largest possible stable drop was 
formed on the end of the needle. 
For clean systems, pictures could be taken immediately. Special 
considerations were necessary for surfactant-laden systems which are discussed in 
section 2.3.9. Pictures were taken with a Pulnix TM-1405GE camera on a tripod 
attached to an optical bellows and a 135 mm lens. For static pendant drop 
experiments, 5 pictures were taken in quick succession (1 every few seconds) of 
every drop. Dynamic experiments were only performed for surfactant-laden systems 
(which are discussed in section 2.3.9). For static experiments not only were 5 drops 
taken per picture, but 5 drops were imaged for a total of 25 images for each 
measurement of interfacial tension. 
When capturing the images it was important to ensure that the bottom of the 
drop did not touch the edge of the image and that there was an appreciable amount of 
the needle in the image. It was necessary to have the needle in the image so that the 
precise magnification of the image could be determined by reference to the known 
needle diameter. It was also necessary to have a sharp contrast between the drop and 
 
 32 
 
the surrounding fluid so that the drop boundary could be determined precisely. A 
sample image that meets these criteria for acceptability is shown in Figure 2.3.6-2. 
 
Figure 2.3.6-2: Acceptable pendant drop image. The criteria that are met are: sharp 
contrast between phases, needle in the image, drop not touching image edges, and 
lack of (non-drop) dark spots in the image. 
Another important experimental consideration was to ensure that there were 
no dark spots present on the image. The problem with dark spots is that they cause 
errors in the drop edge detection software. A spot on the image could arise from dust 
particles on the surface of the camera?s LCD chip, and so therefore, if such spots 
were present, the surface of the camera was wiped with lens paper. Spots could also 
arise from air bubbles floating within the surrounding phase if the surrounding phase 
was viscous. For viscous surrounding phases, this issue could be resolved by simply 
waiting until the air bubbles all settled to the surface of the surrounding phase.  
 
 33 
 
The next step in the procedure is to convert the captured images into a list of 
data points describing the boundaries of the drop which will ultimately be used to 
calculate the interfacial tension. This was done in ImageJ, a free image processing 
software available from the National Institute of Health (NIH). A purpose-built macro 
was developed to quantify the position of the edges and thereby determine the shape 
factors which are illustrated in Figure 2.3.5-1. This macro can be found in Appendix 
B. 
As mentioned in Section 2.3.3, the measurement of interfacial tension is 
highly sensitive and subject to error. In equation 2.3.5-1, the value of Ds has a 
particularly strong influence on the reported value not only because of its exponent, 
but also because of the way in which it is determined. (Refer to Figure 2.3.5-2) The 
precise location of the drop edge determines first De, then the height of the bottom of 
the drop, then Ds. Since each step uses input from the step before, the error associated 
with determining the location of the drop?s edge is compounded three times. 
Therefore, great care had to be exercised in developing a robust method of edge 
determination.  
The simplest method would have been to define a threshold grayscale value 
(in grayscale, pixel values range from 0 = black to 255 = white) at which an image is 
converted to a binary image, an image in which every pixel is either black or white. 
The pixels that are lighter than the threshold get turned white and those that are 
darker get turned black. The drop edges would then be described by the outermost 
black pixels in Figure 2.3.6-3. 
 
 34 
 
 
Figure 2.3.6-3: Thresholded pendant drop image using straightforward thresholding. 
However, since the pixels on the edge of the drop are lighter than the interior 
pixels, if this straightforward method of defining the boundaries is used, then the 
location of the boundaries will necessarily be a function of the threshold value that is 
selected, as well as the background lighting. A higher threshold value (closer to 
white, meaning that more pixels would be turned black) would be accompanied by an 
apparent swelling of the drop?s boundaries. Therefore, a more creative approach must 
be taken to ensure that the measured lengths of the shape factors are not a function of 
the threshold. 
Therefore, an image processing procedure was developed which reports a list 
of the pixels describing the drop shape and is not a function of threshold. The ImageJ 
macro required some user input; it was not fully automated. The first step was to open 
one of the images in ImageJ and run ImageJ?s ?Enhance Contrast? plug-in to lighten 
the background, and then run ImageJ?s ?Find Edges? plug-in. This plug-in works by 
 
 35 
 
quantifying the spatial gradients in pixel lightness and turns pixels whiter wherever 
the gradient is higher. For our pictures this gradient is highest at the drop boundary, 
so the ?Find Edges? plug-in creates an image which is all very dark except for the 
drop and needle interfaces with the surrounding fluid (see Figure 2.3.6-4). The benefit 
of using the ?Find Edges? plug-in is that the interface between the two phases is no 
longer located on the edge of the pixels which describe the drop, but is now located in 
the middle of the pixels which describe the edge in Figure 2.3.6-4. 
 
Figure 2.3.6-4: Pendant drop image after the ?Enhance Contrast? and ?Find Edges? 
plug-ins have been executed. 
After this sample image was opened, the ?Adjust Threshold? command was 
selected ? this displays which pixels would be turned black or white for any given 
threshold. By varying the threshold slider, it can be seen that the thickness of the 3-10 
pixel thick group of whiter pixels in Figure 2.3.6-4 varies with threshold. However, 
the center of this band of whiter pixels does not vary with threshold and provides a 
consistent measurement; therefore finding the center of this band is the goal. 
 
 36 
 
With any image analysis procedure, pixilation does result in error and some 
variation with thresholding, but with Figure 2.3.6-4 the variation with thresholding is 
random, not systematic. In order to reduce this random error a range of thresholds 
was used for each measurement. A random image was opened for each data set to 
determine an acceptable range of thresholds. This image was tested to see for what 
thresholds of Figure 2.3.6-4, a continuous interface band could be established without 
any of the background turning white. 
The threshold range that was actually used for each data set was reduced by 
10-15 on each side of the acceptable range to allow for any single picture?s possibly 
requiring a slightly tighter threshold range. The threshold range was split arbitrarily 
and evenly into 7 values which were used along with the auto-threshold value for 
each data set. These 8 copies of each data set were processed by using the ImageJ 
macro in Appendix B. Using this method the effect of threshold is shown in Figure 
2.3.6-5. To obtain the most accurate values of interfacial tension these 8 different 
thresholds were averaged for every data point. 
 
 37 
 
 
Figure 2.3.6-5: Pendant drop image after the ?Enhance Contrast? and ?Find Edges? 
plug-ins have been executed. 
There were three purpose-developed computer programs (four files) that were 
used in measuring surface and interfacial tensions. The first is the ImageJ macro 
described above which transforms an image of a pendant drop into a list of points 
which describe the edges of the needle and the drop. The second is two MATLAB m-
 files which receive the text files outputted by the ImageJ macro as its input. These 
MATLAB m-files then report the interfacial tension for each pendant drop picture. 
The fourth is a purpose-developed curve fitting program used to find the equilibrium 
interfacial tension from dynamic data. All three of these programs (four files) can be 
found in Appendix B. 
The two MATLAB m-files used to measure interfacial tension require no 
additional user input other than simply listing the parameters of the experiment in the 
command line used when running the m-file. The output of the ImageJ macro is one 
 
 38 
 
text file per image. The first m-file received input from the user as to which images 
are to be processed, calls the second m-file as a subroutine, and then after running all 
of the images through the second m-file and storing the results, reports the interfacial 
tensions thus calculated in a list and provides the average and standard deviation. 
The user inputs which are required are those things which may vary from 
experiment to experiment including: the text files to be included, the density 
difference between the two phases, and the caliber of the needle from which the drop 
is hanging. 
The m-file which calculates the interfacial tension first checks for any gaps in 
the borders of the drop, then sets y=1 as the height of the center of the band which 
describes the bottom of the drop and adjusts the y-values of all rows relative to that. 
After eliminating any outliers to the side, the m-file determines the thickness of the 
bands on both sides and calculates the center x-value of each band. With all of the 
borders thus described, the pixel to real length calibration is done by comparison of 
the width of the needle on the top of the image as well as the known physical needle 
width. After calibration, the shape factors may be obtained. 
Using this method, consistency has been achieved in the determination of the 
shape factors. However despite this reliability (precision), the accuracy of choosing 
the middle of the band determined by the ?Find Edges? plug-in is unknown. 
Specifically, it was not clear exactly where to place the bottom of the drop, and, as 
aforementioned, the choice of the location of the bottom of the drop has a very large 
impact on the reported interfacial tension. Therefore, an ?adjustment factor? (?cal? in 
the pendant drop m-file contained in Appendix B) was introduced as a factor by 
 
 39 
 
which De was multiplied, effectively determining the relationship of the actual 
location of the drop?s bottom with that determined by the image processing 
procedure. This adjustment factor was obtained through processing a group of images 
which should yield the same interfacial tension while treating the adjustment factor as 
a variable. The standard deviation between the measurements was plotted as a 
function of adjustment factor in Figure 2.3.6-6. 
 
Figure 2.3.6-6: Standard deviation between identical interfacial tensions as a function 
of adjustment factor. The common minimum verifies the validity of this approach. 
Figure 2.3.6-6 shows a distinct minimum in the error between measurements 
of different thresholds and/or different drops in identical situations. This plot is 
particularly compelling since the minimum occurs at the same point for two different 
substances having significantly different interfacial tensions (72 mN/m for water and 
?34 mN/m for toluene ? the actual value for toluene is not known precisely in the 
literature, see Appendix A; however, that is not important for Figure 2.3.6-6). Based 
on this common minimum, the adjustment factor was fixed at 1.0075. This number is 
 
 40 
 
close to unity, which is unsurprising, however, the difference from unity is significant 
(see Figure 2.3.6-6), and so this issue could not have been neglected. Also, 
introducing this adjustment factor means that the shape factors are consistently 
determined, but it is not known whether some systematic error(s) remain(s). 
Therefore, after establishing consistency, calibration is still necessary. 
2.3.7 ? Calibration of the Pendant Drop Method 
For calibration, fluids were chosen which were relatively close in values in 
Table A-1. Choosing fluids with low scatter in the literature was the only way to have 
a reasonable basis of calibration. Such considerations were the reason for the detailed 
procedure that had to be developed in this study despite the preponderance of 
available interfacial tension data and measurement methods. The other requirement 
for the construction of a calibration curve is that the chosen fluids cover a wide range 
of interfacial tension values. Using fluids that meet these two criteria, the calibration 
curve is plotted in Figure 2.3.7-1. 
 
 41 
 
 
Figure 2.3.7-1: Calibration curve for pendant drop method as developed in this study. 
Adjusted, uncalibrated interfacial tension vs. literature interfacial tension. 
As shown in Figure 2.3.7-1, the calibration curve is linear, and the equation is 
given in equation 2.3.7-1. This calibration is included in the pendant drop program in 
Appendix B. 
                         (2.3.7-1) 
One additional consideration is that there proved to be slightly less scatter in 
the reported values of interfacial tension when using the larger size needles. So, 
therefore, while the procedure was validated for three needle sizes, for Figure 2.3.7-1 
and all the subsequent data gathering experiments, a 16 gage (1.651 mm) needle was 
always used. 
2.3.8 ? Clean Interfacial Tension Results 
The purpose of the development of Sections 2.3.3 through 2.3.7 was to 
develop a reliable method by which interfacial tension could be measured so that the 
 
 42 
 
interfacial tension of water and Crystal Oil could be determined for the dispersion 
experiments. After such development it was straightforward to determine the 
interfacial tension of clean (devoid of any surfactant presence) Crystal Oil with water. 
Water was suspended by a 16 gage needle into Crystal Oil 500FG using the 
aforementioned procedure. 5 drops of water were measured with 10 images of each 
drop captured at intervals of several seconds between image captures. The results of 
these 50 data points were averaged to yield an interfacial tension of 54.8 mN/m, 
which was used as the value for clean interfacial tension in the dispersion 
experiments. 
2.3.9 ? Dynamic Interfacial Tension Results 
Special considerations were necessary for surfactant-laden systems because 
their interfacial tensions are dynamic, changing with time as the surfactant diffuses 
through the quiescent oil and adsorbs on the interface. All surfactant-laden pendant 
drop experiments were carried out with water as the drop phase and oil as the 
continuous phase. This is done so to minimize changes in the bulk surfactant 
concentration as surfactant goes to the interface. 
In order to investigate the dynamic nature of surfactant-laden systems of water 
in Crystal oil, pendant drop images were captured at regular time intervals showing 
the approach to equilibrium. Surfactants diffuse very slowly through the viscous oil 
phase, so it would have taken prohibitively long (on the order of 15 hours) to wait for 
equilibrium at each surfactant concentration. This turned out not to be a problem in 
the dispersion experiments because of the large amount of convection present in the 
 
 43 
 
continuous phase, however, it is not possible to replicate that situation while 
performing the pendant drop technique. 
 The solution that was developed was to capture an image once every minute 
for three to five hours for every surfactant concentration. Then, using non-linear 
curve fitting regression, the equilibrium interfacial tension was found by fitting the 
data to its known functional form. 
?   ?  ?   
     ?    (2.3.9-1) 
where ?0 is the interfacial tension at time zero, ?? is the equilibrium 
interfacial tension, t is the time, and ? is the time constant associated with the decay 
of the interfacial tension from its initial to its equilibrium value. 
In this procedure, the data that was taken for t is inputted into the equation 
2.3.9-1, and that value for interfacial tension is compared to the data values. The error 
between those two quantities is minimized by treating ?0, ??, and ? as adjustable 
parameters. Specifically, the partial derivative with respect to each adjustable 
parameter of a function which was the sum of the errors was set to zero. This yields 
three implicit equations with three unknowns. Therefore, these equations were solved 
iteratively using the simple root finding method of bracketing the solution. This 
procedure was automated using a purpose-developed MATLAB m-file which can be 
found as the last file in Appendix B. A graph of a sample drop at a particular 
interfacial tension is shown in Figure 2.3.9-1. 
 
 44 
 
 
Figure 2.3.9-1: Dynamic interfacial tension of water in Crystal Oil 500FG at a 
Tergitol NP-4 concentration of 1x10-4 M. Approach to equilibrium is illustrated for a 
single drop. 
This procedure was carried out over a range of interfacial tensions to find the 
critical micelle concentration. The critical micelle concentration is that concentration 
of surfactants at which the addition of more surfactant does not further decrease the 
equilibrium interfacial tension because all of the additional surfactant molecules form 
micelles. Each point on Figure 2.3.9-2 represents one experiment of the type 
illustrated in Figure 2.3.9-1.  
 
 45 
 
 
Figure 2.3.9-2: Equilibrium interfacial tension vs. Tergitol NP-4 concentration for 
water and Crystal Oil 500FG. 
Using Figure 2.3.9-2, it was concluded that the critical micelle concentration 
(CMC) was 1x10-3 M and so in the dispersion experiments that is what is meant by 
the term ?CMC?. Unfortunately, it was not possible to obtain data above a surfactant 
concentration of 1x10-3 M. When experiments were attempted above that 
concentration, no stable drop could be formed on the end of the needle. Instead, a thin 
strand (significantly thinner than the diameter of the needle) of water fell slowly from 
the needle. However, this inability to measure interfacial tensions beyond the CMC 
was not a problem because all of the dispersion experiments were carried out at or 
below the CMC and because it just so happens that the interfacial tension at the CMC 
is close to zero. Since a negative interfacial tension is not possible (it would lead to 
spontaneous phase mixing), it was concluded that 1x10-3 M is the CMC. 
 
 46 
 
2.4 ? Physical Property Ranges 
The previous subsections of this chapter have described the measurement 
methods and results of some of the physical properties that are relevant for the batch 
dispersion experiments. Table 2.4-1 is a summary of all of the physical properties that 
are relevant in determining the drop size distribution in the batch experiments. Such 
relevance is reflected in the dimensionless numbers that can be used to correlate the 
data using methods such as those described in Chapter 3. 
 Rotor 
speed 
 
Temp. 
Oil 
visc. 
Water 
visc. 
 
IFT Phase 
fraction 
Oil 
density 
Rotor 
diameter 
Sauter 
mean 
diameter 
Sym-
 bol 
N ??  c  d ? ?? ?c D d32 
Max 9000 73? 153 0.94 54.8 0.001 0.838 2.81 35.1 
Min 3900 23? 18 0.39 6.2 0.46 0.864 2.81 3.3 
Max/
 Min 
2.3 N/A 8.5 2.4 8.8 460 1.03 1 10.6 
Units rpm ? C cP cP mN/m - g/cm
 3 cm  m 
Table 2.4-1: Range of physical properties which are used in the dispersion 
experiments. Yellow = symbol, green = adjustable parameters, and grey = units. 
In Table 2.4-1 the oil and water viscosities can be modified separately by 
changing the oil?s viscosity grade and the temperature. The interfacial tension was 
varied through the surfactant concentration. The Sauter mean diameter is the 
dependent variable and so it is counted as an adjustable parameter in that 35.1 was the 
maximum d32 which was measured and 3.3 was the minimum. The oil viscosity and 
the rotor diameter were not varied and so their effect was not tested in this study. 
Surface treatments were also not included in Table 2.4-1. However, as 
covered in Chapter 3, there were two states which were considered for the mill head: 
 
 47 
 
treated with a particular hydrophobic silane and untreated. Most of the dispersion 
experiments employed an untreated mill head. Where the experiments used a treated 
mill head, that is specifically indicated in this study. Otherwise it should be 
understood that the mill head is made of simply type 316 stainless steel which has not 
been subjected to any unusual conditions, and has particularly not been exposed to 
any hydrophobic silanes. 
Characterizing the physical properties is significantly simpler for the in-line 
experiments. Since the temperature is constant at 20? C, the viscosities are constant: 
 c = 1 cP and  d = 163 cP. No surfactants were used so interfacial tension is constant: 
? = 30 mN/m. Finally, the phase fraction was always dilute. 
 
 
 48 
 
Chapter 3: Batch Dispersion Theory 
Sections 3.1-3.4 are concerned with equilibrium drop size distributions, 
typically meaning those situations in which enough time has passed in a batch mixer 
so that the drop size distribution is independent of time. Section 3.5 is concerned with 
non-equilibrium drop size distributions, meaning those produced in an in-line mixer 
which usually do not have enough time to reach equilibrium, and are therefore 
dynamic. However, if the residence time is long enough in an in-line mixer 
equilibrium conditions can be reached (Berkman and Calabrese 1988; Middleman 
1974). 
3.1 ?Dilute Theory 
Dilute emulsions (dispersed phase fraction, ? = 0.001) are the simplest to 
analyze because the structure of the flow field is essentially unchanged by the 
presence of the drop phase, and because of the absence of coalescence due to the 
rarity of drop-drop interactions. For more concentrated emulsions (discussed in 
section 3.4), the equilibrium drop size distribution is reached when there is a dynamic 
balance between the rates of breakage and coalescence (Coulaloglou and Tavlarides 
1977). The absence of coalescence in dilute systems allows for the isolated study of 
the effect of breakage so that an emulsion may be said to have reached "equilibrium" 
when all of the drops are below the maximum stable drop size, as determined by the 
maximum deformation rate in the flow field (Leng and Calabrese 2004). This study 
includes dilute data from both the laminar and turbulent regimes with the results 
analyzed separately. 
 
 49 
 
3.1.1 ? Dilute, Turbulent Theory 
The analysis of drop breakup in turbulent flow began with Kolmogorov's 
(1941a, b, c; 1949) theory of cascading turbulent eddies and small scale isotropic 
turbulence. Hinze (1955) applied this work to describe a critical Weber number based 
on drop diameter which determines whether or not a drop breaks in a given 
deformation field. When inertial stresses control drop breakup, Hinze and 
Kolmogorov also provided an expression to relate the maximum stable drop diameter 
to the deformation rate and the physical properties of the fluids. They did this by 
equating the cohesive and disruptive stresses acting on an individual drop. Equating 
these stresses allows predictions of the largest surviving drop or maximum stable 
drop size, dmax. It does not allow prediction of the size of resulting satellite drops or 
those formed through processes such as tip streaming. 
For drops of low viscosity, resistance to deformation due to its internal 
viscosity can be ignored, and the cohesive stress , ?s, acting to stabilize a drop of size 
d is due to interfacial tension, ?, as given by equation (3.1.1-1). 
?            (3.1.1-1) 
The disruptive stress, ?c, due to turbulent velocity fluctuations is found by 
integrating the energy spectral density function, E(k), for the continuous phase over 
all eddies of length scale less than d. E(k) describes the amount of energy per unit 
volume contained in eddies of wave number k = 1/d to k + dk. The turbulent mean-
 square velocity difference,       , across the drop surface can be obtained from the 
energy spectral density function which is related to the ?c by equation (3.1.1-2). 
 
 50 
 
?                    
 
   
  (3.1.1-2) 
The energy spectral density function can be related to the energy dissipation 
rate, ?, and the kinematic viscosity, ?c, by different expressions based on the relative 
strengths of inertial and viscous forces, which are, in turn, determined by the length 
scale of the turbulent eddies.  Clearly, as the length scale decreases more energy is 
lost to viscous dissipation. The length scale of the smallest turbulent eddies, or 
Kolmogorov length scale is given by equation (3.1.1-3).  
   
  
 
 
 
   
    (3.1.1-3) 
Expressions for the energy spectral density function have been measured and 
used to develop expressions for the disruptive stress which was then equated to the 
cohesive stress, equation (3.1.1-1). Using this method as well as the fact that for 
constant power number ? ? N3D2, correlations were theoretically derived for the 
maximum stable drop size in an emulsion for different eddy length scales relative to 
drop size. These were expressed in terms of the dimensionless Weber and Reynolds 
numbers, We and Re, which are defined in terms of the agitation rate and physical 
properties to facilitate their practical use. 
     
   
   
 
    (3.1.1-4) 
     
    
 
  
    (3.1.1-5) 
where ?c is the continuous phase density, N is the impeller or rotor speed, D is 
the impeller diameter for a stirred tank and the rotor diameter for a rotor-stator mixer, 
? is the interfacial tension and  c is the continuous phase viscosity. 
 
 51 
 
The drop size correlations are listed in Table 3.1.1-1 along with the 
expressions for the disruptive stress and the energy spectral density function which 
were used to derive the correlations. Table 3.1.1-1 is grouped according to the length 
scale of the drops relative to the Kolmogorov length scale. 
Model name Applicabl
 e drop 
size 
E(k)              Scaling law Source 
inertial 
subrange 
 
D >> dmax 
>>  ? 
 
      ?         
 
    ?  ?  
    
    
 
         
 
Hinze 
(1955) 
sub-
 Kolmogorov 
inertial stress 
model 
 
 
dmax < ? 
 
 
      ? 
  ?    
 
 
    ? ? 
  ?   
 
    
 
              
 
Shinnar 
(1961) 
 
sub-
 Kolmogorov 
viscous 
stress model 
 
 
 
dmax << ? 
 
      ? 
  ?     
 
 
*?           
   
 
 
    ? ? 
  ?    
 
 
    ? ? 
   
?    
    
 
               
 
or 
    
 
            
Chen and 
Middleman 
(1967) 
 
 
 
Shinnar 
(1961) 
 
Table 3.1.1-1: Derivation of Turbulent Scaling Laws. * Shinnar?s sub-Kolmogorov viscous 
scaling law was derived using an alternative method in which the characteristic shear rate, ?
  
  
was defined to be the ratio of the Kolmogorov velocity to the Kolmogorov length scale. The 
stress then follows as ?    ?   This approach does not yield a value of E(k). 
The first row, when d > ?, reflects the case when inertial stresses control drop 
breakup. Many authors such as Chen and Middleman (1967) have verified the 
mechanistic correlation that applies for this case, which will be referred to herein as 
the inertial subrange model. 
    
 
            (3.1.1-6) 
Equation 3.1.1-2 assumes that the viscosity of the drop is not significantly 
contributing to the resistance of the drop to breakup and that the resistance to breakup 
is entirely a result of interfacial tension. However, sometimes the viscosity of the 
 
 52 
 
dispersed phase does contribute an appreciable resistance to drop breakup. Calabrese 
et al. (1986a; 1986b; 1986c) performed experiments and developed a mechanistic 
correlation for viscous drops whose ultimate size is determined by inertial range 
eddies, which required an additional viscosity term. 
   
 
               
   
 
 
   
 
   
   (3.1.1-7) 
where C is a flow-dependent constant and Vi is a dimensionless viscosity group 
which represents the ratio of viscous to surface forces which are both contributing to 
drop stabilization. Vi is defined by 
    
? 
? 
 
   
    
?
     (3.1.1-8) 
where ?d is the dispersed phase density and  d is the dispersed phase viscosity. 
In the inviscid limit equation (3.1.1-7) reduces to equation (3.1.1-6). This 
complication caused by the contribution of the dispersed phase viscosity to the 
resistance of drop breakup is not relevant to much the present work since the 
continuous phase is much more viscous than the dispersed phase. However, in 
Chapter 8 the continuous phase is much less viscous than the dispersed phase and the 
flow is turbulent as well as being in the inertial subrange. Therefore, equation (3.1.1-
 7) models the equilibrium situation for which the experiments reported in Chapter 8 
are the dynamic situation. 
The second row of Table 3.1.1-1, when d < ? but not d << ?, reflects the case 
when the drops are smaller than the Kolmogorov microscale, but not so much smaller 
that inertial effects can be ignored. Here, both inertial and viscous forces play a role 
in drop breakup. Shinnar (1961) developed this model, but did not provide 
 
 53 
 
experimental validation. It will be referred to herein as the sub-Kolmogorov inertial 
stress model.  
    
 
                (3.1.1-9) 
 The third row of Table 3.1.1-1, when d << ?, reflects the case where the 
drops break up entirely by viscous stresses. This row shows two models that have 
appeared in the literature (Chen and Middleman 1967; Shinnar 1961). Currently, it 
has not been established which of these expressions is more accurate although there is 
some experimental evidence (Boxall et al. 2012) that supports the Shinnar model. 
    
 
                 (3.1.1-10) 
Alternately, Shinnar (1961) developed equation 3.1.1-11, also for the case of 
geometric similarity when dmax << ?. 
    
 
              (3.1.1-11) 
The various expressions for the 3 possible breakup mechanisms also hold for 
long pipes and static mixers when We and Re are appropriately defined. As 
previously discussed, there is little data available for high viscosity continuous 
phases. Therefore the inertial subrange correlation, first row in Table 3.1.1-1, has 
been experimentally well-established, but the other correlations have not. To the 
authors? knowledge these correlations have not been well verified experimentally. 
Recently, Boxall et al. (2012) have correlated stirred tank data using sub-Kolmogorov 
models. Baldyga and Bourne (1993) discuss the importance of selecting the 
appropriate scaling regime when correlating mean drop size data. This paper claims 
to verify the sub-Kolmogorov inertial stress model, at least for one system. 
 
 54 
 
3.1.2 ? Dilute, Laminar Theory 
There have been many studies of single drop breakup in well-defined, 
idealized laminar flow fields (Grace 1982; Stone 1994). The manner in which drops 
break up in laminar flow, by contrast to turbulent flow and the theory of local 
isotropy, have to do with the structure of the flow field and with the pathline that each 
drop travels. Theoretically, this could be calculated, simulated, or measured; but, 
practically this is quite challenging. Therefore, simple, idealized cases of laminar 
drop breakup may be examined to serve as limiting bases for comparison when 
analyzing drop size distributions that have been formed in complex, practical flows. 
Seminal work of this type is given by Grace (1982) who performed laminar 
breakup studies in pure simple extensional flow (SEF) and in simple shear flow 
(SSF). For each of these conditions, Grace constructed stability curves which relate 
the maximum stable drop size, in the form of a critical capillary number ? sometimes 
reported as a Weber number (Walstra 1993) ? to the viscosity ratio (dispersed over 
continuous phase viscosity). It was found that simple shear flow is significantly less 
effective at breaking up a drop than extensional flow and that the smallest maximum 
stable drop size occurs when the viscosity ratio is near unity for both SEF and SSF. 
Others have also performed experiments in this vein. For example, Khismatullin et al. 
(2003) verified that there is an upper limit to the viscosity ratio beyond which drops 
do not break regardless of the magnitude of the imposed stress. Also, Torza et al. 
(1972) found that there is some dependence on the rate at which the shear field is 
developed and its overall duration. This is unsurprising since a finite time is required 
to break up a drop; even if the shear field is strong enough so that the drop falls above 
 
 55 
 
the critical Capillary number, if the drop has insufficient time to deform, then breakup 
will not occur. 
For SSF, Marks (1998) performed a thorough study of the breakup of a single 
drop in a viscous continuous phase at viscosity ratios ranging from 0.01 to 1. Those 
results along with those of Grace (1982) and Torza et al. (1972), as compiled by 
Marks, are shown in Figure 3.1.2-1. 
 
Figure 3.1.2-1: From Marks (1998). Critical Capillary number for the breakup of an 
individual drop in SSF as a function of the ratio of the drop viscosity to the matrix 
viscosity. 
The purpose of Figure 3.1.2-1 in the current work is to show consensus 
amongst those who have performed this type of experiment. Of particular interest, is 
the agreement between all authors, as well as Karam and Bellinger (1968), that there 
is a minimum in the critical Capillary curve at a viscosity ratio slightly less than 
 
 56 
 
unity. This agreement, in addition to having intrinsic significance, also allows a great 
degree of confidence to be placed in these results. 
Because of its broader range of viscosity ratios, as well as the inclusion of 
SEF in addition to SSF, the work of Grace (1982) alone will be used as the basis of 
comparison when the results of the laminar flow experiments are discussed in Chapter 
6. Grace?s results are shown in Figure 3.1.2-2. 
 
Figure 3.1.2-2: From Grace (1982). Critical Capillary number for the breakup of an 
individual drop in SSF (upper curve) and SEF (lower curve ? flat line) as a function 
of the ratio of the drop viscosity to the matrix viscosity. 
In Figure 3.1.2-2 the higher curve is for the case of SSF, which is the same as 
what is reported in Figure 3.1.2-1. The lower curve is for the case of SEF. Although 
these experiments used single drops in idealized flow fields, the change in critical 
Capillary number with viscosity ratio is a useful criterion with which to investigate 
the breakup mechanism even in a complex flow field, such as is present inside of a 
rotor-stator mixer. In Chapter 6 the observed dependence of the Critical Capillary 
 
 57 
 
number with respect to the viscosity ratio will be used to comment on the mechanism 
of drop breakup. 
3.2 ? Surfactant-Laden Systems 
Surfactants, or emulsifiers, are in widespread use industrially and are 
important because they significantly decrease the drop size of emulsions both by 
lowering the interfacial tension and by sometimes inhibiting coalescence. Surfactants 
often decrease the drop size by decreasing the interfacial tension. In Chapter 6 there is 
a dilute dispersed phase, so in its analysis, coalescence is insignificant and only the 
reduction in interfacial tension is relevant. They have also been reported to decrease 
the drop size in non-dilute systems by the adsorbed surfactant molecules providing an 
additional barrier to coalescence (Lobo and Svereika 2003). 
Even for dilute systems, however, there is a potentially complicating factor 
which, if present, would not allow surfactant systems to be treated simply as clean 
systems with a lowered interfacial tension. The issue is the possible presence of 
Marangoni stresses which arise from the spatial gradient in surfactant concentration 
that occurs upon the stretching of a drop whose surface was initially uniformly 
populated with surfactant molecules. Padron (2005) found that for dilute dispersions 
of silicone oil in water there was a correlation between a peak in the DSD as a 
function of the concentration of water-soluble surfactant with the maximum surface 
dilatational modulus (a measure of how quickly uniform interfacial tension is restored 
after stretching). This implies that for the low continuous phase viscosity, turbulent 
system studied by Padron, Marangoni stresses affect the ultimate drop size by 
providing an additional resistance to drop deformation and breakup. 
 
 58 
 
One additional complication that has been reported when a long polymer, for 
example, a protein, is used as a surfactant is that the area of the interface which is 
covered by the surfactant increases because the polymer is free to change its 
conformation to do so. This results in an even lower interfacial tension, an effect 
which can be significant (de Feijter and Benjamins, 1982). However, this effect is not 
relevant to the present study since the surfactant which is used, Tergitol NP-4, is a 
relatively short-chain polymer whose average molecular weight is given by Dow 
Chemical Company (2003) as 396 g/mol. Due to entropic considerations a longer 
polymer is more readily conformed. Since Tergitol NP-4 is significantly smaller than 
the molecular weight of proteins, whose smallest molecular weights are on the order 
of several thousands of grams per mole, the previous assertion that this effect does 
not play a role is justified. 
3.3 ? Hydrophobic Treatment Theory 
3.3.1 ? Surface Free Energy 
This study is primarily concerned with the dispersion process; however, there 
are two reasons that the surface free energy is relevant. The surface free energy of the 
mill head, which determines the attraction of the solid surfaces for water, is varied in 
order to determine its potential effect on the breakage rate. The other reason for 
knowing about the surface free energy is to control the affinity of droplets in the 
emulsion to various surfaces, some of which are made of stainless steel and others of 
which are made of glass. 
 
 59 
 
Surface free energy is the thermodynamic excess of energy that results from 
the contact of two distinct substances with each other. If this energy is negative, then 
spontaneous mixing will occur. This would be mixing in the true sense rather than 
dispersion which is the phenomenon of interest in this study. In this study all surface 
free energies are positive, and so it really makes sense to speak about the difference in 
free energy in one situation relative to another. For example, if the interfacial tension 
of the two fluids of an emulsion is very high, then the area of all of the droplets is 
added together to obtain the total area in which those two fluids are in contact. The 
difference between the free energy in that case and in the phase separated case is what 
forms part of the driving force for the separation of emulsions ? in addition to the 
gravitational force for the case of quiescent emulsions. 
The previous example involved two liquid phases; however, the interface 
between a solid surface and a liquid surface has a free energy associated with it as 
well. The free energy between two substances, regardless of their phase, is, in 
general, determined by their intermolecular interactions. The well-known maxim 
?like dissolves like? is the natural starting point for this discussion. This statement is 
a colloquial way of saying that polar solvents tend to dissolve in polar solutes and that 
non-polar solvents tend to dissolve in non-polar solutes (Brown et al. 2003). Although 
this only refers to dissolution, which occurs only when the interfacial tension is 
negative, it is generally true that the interfacial tension is lower when substances have 
similar polarities. 
 
 60 
 
3.3.2 ? Surface Modification 
The significance of this for the current study is that the functional groups on 
the outside of a solid surface may be modified so that surface experienced by the 
liquid phase may rendered more or less hydrophobic (water-hating) or oleophobic 
(oil-hating). Glass and stainless steel are the surfaces of interest. Untreated glass, in 
particular, is covered with hydroxyl groups on its exterior surface.  Such polar groups 
are very attractive to suspended water drops and are the reason why untreated glass is 
very hydrophilic (water-loving). The basic idea of surface treatment is to replace 
these hydroxyl groups with a non-polar functional group, in the present case, a silane. 
This is illustrated in Figure 3.3.2-1. 
 
Figure 3.3.2-1: Qualitative difference between an untreated solid surface and a 
treated solid surface. 
The wettability is quantified by the contact angle made by the liquid resting 
on the solid surface in the presence of the surrounding fluid. This is determined by the 
degree of attraction of the drop fluid with the solid surface relative to the degree of 
attraction of the drop fluid with the surrounding fluid and the surrounding fluid with 
the solid surface. If there is a low surface free energy between two of the phases, then 
 
 61 
 
their contact area will be high. This directly determines contact angle, which is 
illustrated in Figure 3.3.2-2. 
 
Figure 3.3.2-2: Contact angle: the angle an adhering drop makes with its solid 
surface in the presence of the particular surrounding fluid. 
The contact angle is certainly an interesting feature, especially since it is most 
commonly used in industry, but it does not provide all of the information that is 
necessary to describe adhering drops. The actual amount of energy (per unit area) 
needed to remove an adhered drop from a solid surface is called the work of adhesion. 
It is defined by the surface free energy after adherence less the surface free energy 
before. This is best illustrated by a simple example of solid bars which are initially 
stuck together and then removed (Adamson 1976) as shown in Figure 3.3.2-3. 
 
 62 
 
 
Figure 3.3.2-3: Illustration of work of adhesion: work of adhesion is the difference in 
surface free energy after specified areas are created relative to the initial state of the 
system. 
In Figure 3.3.2-3, initially there is an amount of surface free energy associated 
with the interface of A and B. After the separation, that free energy is no longer 
present, but there are new energies associated with the interfaces of A and B with the 
surrounding fluid, which will be called ?C.? In this case the work of adhesion per unit 
area is defined in terms of the surface free energy per unit area as 
    ?   ?   ?     (3.3.2-1) 
where ?ij is defined as the surface free energy per unit area associated with the contact 
of generic surfaces i and j. It is important to note in the definition of work of adhesion 
that the surrounding fluid is just as important as the drop fluid and the solid surface. 
The previous example used solid surfaces immersed in a single fluid. The 
geometry is more complex for the case of droplets adhered to a solid surface in the 
presence of a second fluid. Young (1855) and Dupr? (1869) developed a means of 
 
 63 
 
expressing the work of adhesion per unit length in terms of the fluid interfacial 
tension and the contact angle. 
     ?                     (3.3.2-2) 
where W123 is the work of adhesion per unit length defined by reference to all three 
substances, ?fluid is the interfacial tension between the two fluids, and   is the contact 
angle of the drop as illustrated in Figure 3.3.2-2. 
The interfacial tension can be measured by the previously described methods 
and the contact angle can be observed directly or more accurately with the use of a 
goniometer as described in Chapter 4. Therefore, the work of adhesion may be 
determined using equation 3.3.2-2 for an arbitrary solid with one fluid as a drop and 
another surrounding it. This provides the information of how much energy must be 
supplied to a drop to strip it off of a solid surface, which is the primary issue when 
considering the likihood of drops leaving the dispersion to adhere to the solid surface 
? one of the situations of interest in the dispersion experiments.  
It is worth mentioning that both the contact angle and the work of adhesion 
play a role in determining whether the drop will be removed by fluid forces. The 
work of adhesion is relevant because it determines the amount of energy that must be 
supplied, and the contact angle is relevant because it determines the drop?s profile. It 
is clear that a taller drop is more exposed to fluid forces, both normal and shear, that 
act on the surface of the drop to potentially strip it away. 
The procedure of chemically binding a hydrophobic silane monolayer to 
achieve surface modification has been performed by other researchers who have 
reported several significant issues which were accounted for in this study. It is 
 
 64 
 
important to thoroughly clean the solid surfaces (often using highly concentrated acid 
for glass) before treatment to ensure a high density of hydroxyl groups which serve as 
reaction sites for silanization reactions which make the surface hydrophobic (Wright 
et al. 2006). Also, the roughness of a solid surface increases the hysteresis in the 
contact angle (Tang et al. 2008) and, therefore, depending on the situation in question 
either increases or decreases the work of adhesion. 
An alternative procedure which can only be used for stainless steel (glass is 
not electrically conductive) is to chemically deposit an anlkanethiol monolayer 
monolayer through the use of electrochemistry (Zhang et al. 2005). This method was 
not seriously investigated for use in the current study because of its high complexity 
and because of the fact that the straightforward chemical method was found to 
provide a sufficient difference in the hydrophobicity. 
The exact procedure used to achieve surface modification in this study was 
determined by trial-and-error experiments, which is discussed in Chapter 4, the 
chapter on experimental methods. 
3.4 ? Non-dilute Theory 
For non-dilute dispersions, the DSD of an emulsion or dispersion is, in 
general, determined by a dynamic equilibrium between the rates of drop coalescence 
and breakage (Coulaloglou and Tavlarides 1977; Leng and Calabrese 2004). There is 
a much greater understanding of breakage than of coalescence phenomena because 
drop breakage in emulsions can be studied independently of other effects by using a 
dilute dispersed phase (usually ? < 0.01). The advantages of using a dilute system are 
that the continuous phase flow field is essentially unchanged from that of a pure fluid 
 
 65 
 
except on the drop scale, and that coalescence is negligible due to the rarity of drop-
 drop collisions. For a drop to break up in a given deformation field, the imposed 
disruptive stress must be greater than the cohesive stress(es) (Leng and Calabrese 
2004). The disruptive stress decreases with decreasing drop size and the cohesive 
stress increases due to the length scale of interfacial phenomena. Once the drops 
reach a certain size, they will no longer break up and the equilibrium drop size is 
reached. Since breakup is due to stresses that the continuous phase exerts on 
individual drops, different flow regimes have been analyzed separately as 
summarized in Section 3.1 (in Table 3.1.1-1 for turbulent flow and the discussion of 
Grace?s (1982) work for laminar flow). Having discussed breakup fully in Section 
3.1, in order to understand the behavior of non-dilute dispersions it is now necessary 
to discuss coalescence phenomena. 
Coalescence occurs via a sequential procedure involving at least two drops. 
First, drops must collide, forming a thin film of continuous phase between them. The 
film must drain and finally rupture (Coulaloglou and Tavlarides 1977). Based on this 
mechanism, the coalescence rate can be expressed as the product of the collision rate 
and a collision efficiency, which is the probability that a collision will result in 
coalescence. Chesters (1991) provided methods for a first-estimate of the collision 
rate per unit volume, C, and the collision efficiency, P, for a monodisperse system. 
The collision rate is given by 
            (3.4-1) 
k is a flow-dependent constant, v is a characteristic velocity between two 
points separated by a distance d in the flow field, and n is the number of drops per 
 
 66 
 
unit volume (n ??? /d3). For turbulent flow, v is the square root of the turbulent mean-
 square velocity difference,           (listed in Table 3.1.1-1). For sub-Kolmogorov 
inertial flow where d < ?, but not d << ? (it is called ?fine-scale turbulence? in 
Chesters? paper (1991)), v ? (?/?c)
 1/2d. Substituting this into equation (3.4-1), along 
with the fact that for constant power number ? ? N3D2, the collision rate for sub-
 Kolmogorov inertial flow can be scaled by equation (3.4-2).  
  ?     ?    ? 
      (3.4-2) 
For viscous simple shear flow v is the product of the characteristic shear rate 
and the drop diameter. For now it is assumed that the characteristic shear rate in 
laminar flow is ? ? ND/?. This assumption will be tested in Chapter 6. Substituting 
this into equation (3.4-1), the collision rate can be scaled by equation (3.4-3).  
  ?  ?    ?    (3.4-3) 
It is interesting to note that in these expressions, C ? ? 2/d3 for both viscous 
simple shear and sub-Kolmogorov inertial turbulent flow. These are the two types of 
flow present in this study (these two ideas will be further developed and established 
conclusively in Chapter 6). 
Chesters models the collision efficiency is given by equation (3.4-4) 
                 (3.4-4) 
tc is the time required to drain the film so that it is thin enough for rupture to occur 
and ti is the interaction time brought about by each collision event. 
 
 67 
 
Equations 3.4-1 & 3.4-2 are general and apply to any flow regime. A 
significant amount of work has been done with turbulent flow in impeller stirred 
vessels with L >> d >> ?. This is for drops that are larger than the Kolmogorov 
microscale, ?, but small compared to the turbulent macro scale, L. Coulaloglou and 
Tavlarides (1977) developed expressions for the coalescence and breakage rates for 
this case. 
A number of authors (Delichatsios and Probstein 1976; Doulah 1975; Zerfa 
and Brooks 1996) have published relationships (which at dilute phase fractions 
reduce to the inertial subrange scaling expression represented by equation 3.1.1-6) for 
the maximum stable drop size which can be approximated by equation 3.4-5. 
              ?   
                          (3.4-5) 
C1 is a system-dependent constant and b ranges from about 1 to 10. This 
functional dependence of the drop size on phase fraction for inertial subrange scaling 
has been attributed to turbulence damping, coalescence, or both depending on the 
study (see Zhao and Kresta 1998; and Leng and Calabrese 2004). Doulah (1975) 
argued that b=3 in the absence of coalescence (due to turbulence suppression), and 
this may represent a lower limiting value. 
Equation 3.4-5 is an accepted way to quantify the effect of phase fraction on 
the equilibrium drop size. However, this relation should not be expected to be 
generally valid since it was developed and validated for inertial subrange scaling in 
turbulent flow. The mechanism of coalescence can be different if the flow is laminar 
or if the drops are smaller than the Kolmogorov microscale, making the flow locally 
laminar. In the inertial subrange the coalescence mechanism originates from the fact 
 
 68 
 
that drops are brought into contact and moved apart by eddies of the size of the drops 
(Tjaberinga et al. 1993). The coalescence mechanism is different if the flow field 
around the drop is locally laminar - a criterion which covers both sub-Kolmogorov 
turbulent systems and laminar systems. 
Using equation 3.4-4 and arguments for the interaction time and film drainage 
time, the collision efficiency for monodisperse, deformable drops in viscous simple 
shear flow can be approximated by equation 3.4-6 (Chesters 1991). 
                   
                  (3.4-6) 
                        (3.4-7) 
 
C2 is an unknown constant of order unity,  d is the dispersed phase viscosity,  c is the 
continuous phase viscosity, Ca is the Capillary number based on the continuous phase 
viscosity, ? is the interfacial tension, A is the Hamaker constant (material-dependent 
and related to the van der Waals force, but typically about 10-20 J), and    is the 
characteristic shear rate. 
The predicted inverse dependence on viscosity ratio, ?= d / c, in Equation 
3.4-6 has been qualitatively verified experimentally (Caserta et al. 2006; Lyu et al. 
2000; Perilla and Jana 2005; Priore and Walker 2001) with studies of coalescence in 
shear fields which were too weak to cause any breakage events. These authors present 
their results in terms of ?, yet it should be noted that dependency on ? alone is too 
simplistic due to the presence of Ca in equation. In equation 3.4-6, collision 
efficiency decays exponentially with - d c
 1/2, revealing that increasing either  d or  c 
individually decreases the collision efficiency. 
 
 69 
 
Film drainage is primarily dependent on interface mobility (Chesters 1991); 
that is, the ability of the drop surface to move with the film as it is squeezed out 
between colliding drops. Increasing  d decreases interface mobility, thereby 
increasing drainage time. Meanwhile, collision force increases with increasing  c, 
promoting coalescence. However, film drainage rate decreases with increasing  c, 
hindering coalescence, yet only in proportion to the steepness of the velocity profile 
within the film. If the interface was completely mobile, as can occur when  d <<  c or 
??0, then there would be no decrease of film drainage rate with increasing  c. One 
additional issue related to interface mobility is that surfactants are believed to hinder 
the interface mobility (Walstra 1993). 
 
 70 
 
Chapter 4: Experimental Methods for Batch Dispersion 
Experiments 
This chapter contains all of the experimental methods in this study which were 
relevant for the dispersion experiments. This includes all of the experimental methods 
except for those pertaining to the measurement of interfacial tension which were 
covered in Chapter 2 and those pertaining to the in-line experiments which are 
covered in Chapter 8. 
4.1 ? Batch Dispersion Process 
4.1.1 ? Experimental Setup 
The main piece of equipment used is a batch Silverson L4R rotor-stator mixer 
with a slotted stator. This mixer, along with the accompanying glass mixing vessel is 
pictured in Figure 4.1.1. 
 
 71 
 
 
Figure 4.1.1-1: Silverson L4R Batch rotor-stator mixer used in this study. a) mixer 
and stand, b) sideview of the mixing head, and c) bottom view of the mixing head. 
The main feature of rotor-stator mixers such as this is the very high shear rate 
caused by the close tolerance between the rotor and the stator. This mixer has a 
variable speed controller with a digital display so that the agitation rate may be 
controlled precisely. 
One important experimental consideration was the large amount of heat that 
was generated by shearing the viscous continuous phase. Sophisticated temperature 
 
 72 
 
control was required because the cooling requirements increase at higher rotor speeds 
due to increased viscous dissipation. This necessitated the use of a robust 
refrigeration unit. The unit selected was a HAAKE Pheonix II P2 heater/chiller 
recirculator made by Thermo Scientific. This has the capability to heat or chill a 
variety of working fluids and to pump them through whatever piping network is 
attached to its inlet and outlet nozzles. The piping network that was selected for these 
experiments was flexible rubber tubing attached with zip ties to a copper coil. The 
copper coil was manually bent into the shape shown in Figure 4.1.1-2 by first bending 
the copper tubing around a long cardboard cylinder, and then bending the linear coil 
into a semicircle. The copper coil?s purpose was to facilitate heat transfer from the 
temperature control unit to a water bath surrounding the mixing vessel. 
 
 73 
 
 
Figure 4.1.1-2: Experimental setup. Copper coil is used to transfer heat between the 
water bath and the refrigeration unit?s working fluid. Small unit on the right is a 
stirrer to keep the temperature in the water bath uniform. 
Due to the fact that this system has three different fluids (the refrigeration 
unit?s working fluid, the water bath surrounding the mixing vessel, and the emulsion 
itself), the refrigeration unit was capable of precise temperature control, which was a 
 
 74 
 
required feature in selecting the temperature control unit. The simplest method was to 
set the temperature of the working fluid as a constant. However, the more useful 
method was to use the internal controller in the refrigeration unit to keep the emulsion 
temperature constant. This was done using a Pt100 thermocouple which reported the 
emulsion temperature to the refrigeration unit. If the mixer speed is changed, then the 
viscous energy dissipation, and therefore the rate at which heat is to be removed, 
changes. The refrigeration unit?s controller can account for this with some time lag 
using basic control principles. 
The other two pieces of equipment are the stirrer and the overflow funnel. The 
stirrer was a VWR Scientific Polyscience 1112 Immersion Circulator and has the 
capability to heat as well as stir, but that capability was unused. The overflow 
funnel?s purpose was to provide a place to accommodate the oil?s thermal expansion 
when it was raised above room temperature. This was necessary because the mixing 
vessel was usually sealed off from the outside except for one port by which the 
overflow funnel was connected to the mixing vessel via rubber tubing. A schematic of 
the experimental setup is drawn in Figure 4.1.1-3 which represents what is pictured in 
Figure 4.1.1-2. The yellow represents the emulsion, and the blue represents the 
cooling bath water.  
 
 75 
 
 
Figure 4.1.1-3: Experimental schematic. Yellow represents the emulsion, and blue 
represents the temperature bath water. 
4.1.2 ? Experimental Procedure 
First, the glass mixing vessel was mostly filled with pure Food Grade Crystal 
Oil. The mixing lid, which has the mill head attached to it, was placed onto the 
mixing vessel and clamped in place with a ring clamp. Oil was poured through a 
closable NPT port on the top of the mixing lid, whose cap was subsequently screwed 
in place. With all ports closed so that the mixer was sealed, it was turned upside 
down. This caused air which was trapped adjacent to the rotor shaft and above the 
NPT port to rise to the floor of the mixing vessel. The mixer was then re-inverted and 
kept at such an angle that oil could be poured through the NPT port to replace the air 
bubble. The end of this is that the mixing vessel contained only pure oil with no air. If 
this procedure was not to be performed, then, upon spinning of the rotor shaft, air 
 
 76 
 
would be pulled down into the mill head and dispersed into the emulsion, an 
undesirable complication. 
After the mixing vessel was attached to the lid and fully charged with Crystal 
Oil as described, it was attached to the mixer as pictured in Figure 4.1.1-2 and Figure 
4.1.1-3. The water bath was filled up to the line of the closable NPT port, the 
overflow funnel was attached in place with its port open, the stirrer was turned on, the 
copper coil was put in place, and the refrigeration unit was turned on. The cap on one 
of the NPT ports on the mixer lid was removed and replaced with the Pt100 
thermocouple which acted as a seal in addition to reporting the temperature inside the 
mixing vessel. In a separate experiment using flexible thermocouples at the end of 
long wires (unlike the Pt100, which is located at the end of a 1/8? stainless steel rod), 
it was demonstrated that the temperature inside the mixing vessel is always uniform, 
even directly before and directly after the viscous oil interacts with the mill head. 
With the completion of the above steps, the setup was complete and the mixer 
was turned on and left overnight to reach thermal equilibrium at a rotor speed of 3900 
rpm. It should be remembered that the heat duty of the refrigerator is dependent on 
the rotor speed, and therefore the mixer must be set at its initial speed during the 
equilibration so that the refrigerator?s controller can accurately assess the required 
rate of heat removal. 
The next day, by which time thermal equilibrium was well-established, the 
main experiment was commenced. 2 mL of ultra-pure water (purified from deionized 
water using a Simplicity Millipore water purification system to obtain a resistivity of 
18.2 M??cm) was injected via a syringe through a port on the lid of the mixing 
 
 77 
 
vessel. It was visually evident that the first pass through the mixing head drastically 
reduced the drop size to microscopic sizes. Further passes served to bring the drops to 
the DSD. After 2.5 hours, when the drop size distribution had reached its ultimate or 
equilibrium value, a sample was withdrawn (at a depth of about 17 cm from the mixer 
lid ? this depth did not affect the drop size distribution as verified by one experiment 
at a different depth) using a wide-mouth pipette to avoid shearing the drops and about 
5 mL was deposited temporarily into a test tube. The test tube was carried to the 
microscope setup where the procedure of Section 4.1.3 was carried out on that sample 
to obtain images of the drops. 
After the first sample had been withdrawn at 2.5 hours, the rotor speed was 
increased from 3900 rpm to 5600 rpm and 2.5 hours were again allowed to establish 
the new equilibrium before withdrawing the sample for size analysis. This was 
repeated with the same time interval at increasing rotor speeds of 7300 rpm and 9000 
rpm so that these 4 rotor speeds were tested for each combination of viscosity and 
surfactant concentration. 
For emulsion concentrations with about ? > 0.2, it was observed that, once 
removed from the high shear fields of the mixing vessel, some amount of settling of 
the emulsion occurred in a relatively short time (on the order of tens of minutes) due 
to the drop density in the emulsion. Therefore, for all experiments with ? > 0.05, the 
~4 mL emulsion sample which was withdrawn via wide-mouth pipette was 
immediately deposited into a test tube already filled with ~10 mL of pure oil. The 
emulsion sample was stirred into this pure oil with a thin wooden dowel rod in order 
to decrease the drop density and prevent settling. This precaution was added to the 
 
 78 
 
procedure in all of the experiments with ? > 0.10 in addition to surfactant being 
present in all such experiments. Both the dilution and the presence of surfactant in all 
experiments in which ? > 0.05 allow for confidence that no coalescence occurs prior 
to the sample?s deposition on a microscope slide (described in Section 4.1.3; during 
those measurements it was directly observed that no coalescence occurred, also). 
One final consideration is that 2.5 hours may not strictly be necessary to 
achieve equilibrium. Based on preliminary experiments, where the drop size 
distribution was measured as a function of time, 1.5 to 1.75 hours seemed to be 
sufficient to reach the equilibrium drop size distribution. However, it was desirable to 
be absolutely certain that the drop size distributions had time to reach equilibrium. 
For turbulent flow, drop size correlates better with the maximum energy dissipation 
rate rather than the average energy dissipation rate (Zhou and Kresta 1998). The extra 
time was provided to ensure that all drops experienced the regions of maximum shear 
stress multiple times. For laminar flow in a high viscosity liquid-liquid system with a 
static mixer, Rao et al. (2007) found that shorter residence times resulted in a high-
 diameter tail which disappeared when drops were given more time in the mixer. Since 
the larger drops are generally the most influential feature of a drop size distribution, 
more time was added onto this study's experiments to ensure that no analogous tail 
would be formed in this data. 
4.1.3 ? Microscope Technique 
The preparation of the microscope slides was done on the same day as the 
main experiment. Microscope slides, which had been hydrophobically treated with 
the hydrophobic silane Glassclad 18 according to the procedure outlined in Section 
 
 79 
 
4.3, were washed using Contrex, a nonionic, slightly acidic detergent. After washing 
and drying the slides according to the procedure described in section 4.1.4, one layer 
of Matte Finish ScotchTM Tape was placed smoothly on their surface to support the 
cover slips at a distance of about 60  m from the slide (this distance varies with the 
brand and type of tape). This is so that the cover slips do not crush the drops causing 
them to deform and lose their spherical shape which is necessary to calculate the 
volume. The distance of 60  m was found to be the best distance because it was large 
enough to accommodate all of the drops that were present as well being small enough 
to just barely allow the 60x microscope objective to be sufficiently close to the 
sample to focus on water drops resting on the microscope slide at the bottom of the 
sample volume. If it had needed to be any closer, the cover slip would have been in 
the way. This is schematically illustrated in Figure 4.1.3-1. 
 
Figure 4.1.3-1: Schematic illustration of an emulsion sample set-up to be imaged by 
an optical microscope. The drops rest on the microscope slide at the bottom of the 
sample because they are more dense than the surrounding oil. 
The exactness of the depth made it important to place the tape smoothly on the 
surface with no wrinkles or bubbles in the tape. Each 1? x 3? microscope slide was 
able to accommodate 3 samples as shown in Figure 4.1.3-2. 
 
 80 
 
 
Figure 4.1.3-2: Prepared 1? x 3? microscope slide. Each slide was rendered 
hydrophobic, cleaned, taped carefully, and then the sample was deposited with a 
wooden dowel rod and the cover slip was placed atop it. 
Figure 4.1.3-2 shows two samples which have already been placed and the 
third sample being deposited. The sample deposition was done by dipping a thin 
dowel rod into a test tube containing the emulsion and then letting a single drop fall 
from the dowel rod onto the vacant space on the microscope slide. A cover slip was 
then placed on top. After about 20 minutes, at which point the water drops had had 
time to settle to the bottom of the sample so as to be all collected in the same focal 
plane, the image acquisition procedure was commenced. 
To acquire the drop size distributions, the samples which were withdrawn at 
each rotor speed were analyzed under a standard, backlit optical microscope using the 
samples prepared as shown in Figure 4.1.3-2. This treatment was necessary to prevent 
the water drops from spreading on the glass surface which would have made it 
inaccurate to determine the volume of a drop by simply measuring the projected area 
of a sphere. Using a "Watec America Corp. LCL-902K" low light camera, images 
were captured of individual drops (as many as 4 or 5 drops on a single frame) until 
 
 81 
 
the number of drops exceeded 1000. For moderately polydisperse distributions this is 
the number needed to ensure reasonably accurate statistical confidence (Paine 1993).  
The method of capturing this large number of drops was to scroll through the 
sample area by turning the stage knobs on the microscope. A grid pattern was made 
by starting near the upper left corner and scrolling down until a drop was found. An 
image was taken, and the scrolling continued down until the bottom of the slide was 
reached. Then, the sample was scrolled to the side a little further than the width of the 
camera?s view. The sample was scrolled up, taking pictures as drops became visible, 
until the top of the microscope slide was reached. This procedure was repeated until 
at least 1000 drops had been imaged. 
In preparing these samples there were several considerations which were 
necessary to ensure the successful capture of drops which truly represented the drop 
size. For every sample which was withdrawn, two microscope slide samples were 
prepared. The reasons for making this backup were so that if there was a problem 
with the first sample, the second one could be photographed (it would not have been 
possible to obtain a new 5 mL sample since the mixing speed would already have 
been increased). The four problems which sometimes occurred were: inadequate 
cleaning of the microscope slide resulting in non-spherical drops, tape which was not 
flat, accidentally crushing the cover slip into the sample and thereby deforming the 
drops, and canvassing the entire sample without reaching a drop count of 1000. 
The drop images were analyzed using the ImageJ image analysis software 
package. A purpose-built ImageJ macro was developed which works by subtracting 
the current image from a background image to find the outlines of the drops. These 
 
 82 
 
outlines are filled with pixels so that all pixels may be counted to find the area of each 
drop. Figure 4.1.3-3 shows a typical microscope image of drops and Appendix C 
provides the code of the ImageJ macro which used pictures like Figure 4.1.3-3 to 
measure the sizes of all of the drops so that the DSD could be obtained. Appendix C 
also discusses the two preliminary issues that must be performed to use the ImageJ 
macro for drop size determination: The image magnification first had to be calibrated, 
and for every folder of images a background image had to be supplied as the first 
image in the folder. 
 
Figure 4.1.3-3: Typical microscope image used to measure drop size. Crystal Oil 
500FG surrounded by water under a 60x microscope objective with a ?Watec 
America Corp. LCL-902K.? 
4.1.4 ? Cleaning Procedure 
Impurities in the experimental equipment obviously need to be avoided and 
so, especially because of the high viscosity of the oil emulsion, a thorough cleaning 
procedure was required. The oil?s high viscosity necessitates the complete 
disassembly of all of the mixer?s parts which are exposed to the emulsion. After 
completely disassembling the mixer, toluene was sprayed onto each mixer part as 
 
 83 
 
well as the glass vessel using a solvent wash bottle. Acetone was then sprayed over 
every part using another solvent wash bottle. The toluene and acetone sprayings were 
repeated to remove the remaining oil. These sequential washings were effective 
because the oil portion of the emulsion is soluble in toluene, while the water portion 
is soluble in acetone, and the acetone and toluene are soluble in each other. By 
sequentially washing the mixer parts and the glass vessel in these two solvents, most 
of the emulsion can be removed without using any detergents. 
Aqueous nonionic detergents were used to clean off the remaining residue. 
Liquinox and Contrex were found to be acceptable detergents for this purpose. 
Liquinox was subjectively more effective at cleaning off the emulsion residue 
quickly. However, Liquinox solution is basic, as measured with pH paper, and it was 
found that bases degenerate any hydrophobic silane coatings. Therefore, Liquinox 
solution was used to clean any hydrophobically treated surfaces since it is slightly 
acidic. 
The nonionic detergent solutions were used by dissolving about 1% of the 
concentrated solution into water which was as warm as could be comfortably 
touched, about 45? C. A thick paper towel (not previously recycled) was soaked in 
the cleaning solution and used to wipe down all surfaces. All parts were rinsed. This 
washing and rinsing was repeated to be sure that all oil and solvents were removed. 
Then the solid surfaces were rinsed with deionized water followed by ultra-pure water 
(purified from deionized water using a Simplicity Millipore water purification system 
to obtain a resistivity of 18.2 M??cm). Immediately after these rinsings, the water 
was removed by drying the surfaces with pure nitrogen blown out of a nozzle from a 
 
 84 
 
storage tank. For a few parts, particularly the inside of the mixer lid, ethanol was 
sprayed into a deep recess in order to get the water out. Then the drying with nitrogen 
was performed. 
After this cleaning, the mixer was reassembled and ready for use in a new 
dispersion experiment. 
4.2 ? Recycling of Oil 
Each dispersion experiment requires about 2.5 liters of oil. This amount is 
burdensome in terms of waste production and cost. Therefore a means of recycling 
most of the oil used in each experiment was devised. Before starting the cleaning 
process, most of the emulsion was poured into a 1-gallon glass jug. The remainder, 
that which was stuck to the walls and that which was removed as microscope 
samples, was not recycled and was disposed of when cleaning. 
A little less than 2 liters of the emulsion to be recycled into oil at a time were 
poured into a crystallizing dish. This type of glassware can be described as a short 
cylinder with an open top and closed bottom that has an aspect ratio of about 1/3 with 
a volume of about 2 liters. A magnetic stirrer bar was placed within the emulsion and 
the crystallizing dish was placed on a magnetic stirring plate. The magnetic stirrer 
was left stirring overnight after which time the emulsion had separated. This could be 
confirmed visually because the oil was perfectly transparent again while the water 
was stuck to the sides of the glass container. 
This method of separation was possible because of the dilute phase fraction (? 
= 0.001) at which many of the experiments were performed. For the higher phase 
fraction experiments, the emulsion was allowed to sit on a shelf for a month or two so 
 
 85 
 
that much of the separation was accomplished by the density difference between the 
water and the oil. The oil-rich phase on top was fairly dilute in water and could be 
separated according to the preceding procedure. The water-rich phase was considered 
to be waste and was discarded. 
It is important to note that this recycling of the oil was only carried out on the 
clean emulsions. No surfactant-laden oil was ever recycled. To do so would have 
necessitated quantifying the amount of surfactant depletion in the bulk due to the 
creation of the surface area of the drops, as well as any possible loss of surfactant at 
solid liquid boundaries. Even if such quantification had been a simple matter, it 
would have introduced a new source of error, which would decrease the reliability of 
the results of this study. 
4.3 ? Hydrophobic Treatment of Surfaces 
4.3.1 ? Experimental Method 
A variety of hydrophobic surface treatment methods were tested before 
settling on how to treat the mixing head, the microscope slides, and the interior 
surfaces for the dispersion experiments. The general procedure that was employed 
was to treat a small sample surface using whatever method was being tested and then 
to examine the resulting contact angle to determine the effectiveness of that 
procedure. 
The sample surfaces used were 1? x 3? microscope slides for glass and 
custom-built substrates of approximately 1? x 3? 1/16? type 316-SS stainless steel. 
These custom-built substrates were made by cutting up a large sheet into the 
 
 86 
 
appropriate size to imitate microscope slides. The exact size was unimportant so long 
as these surfaces were able to fit inside the type of cuvette pictured in Figure 2.3.6-1. 
It is important to note that a fresh (never before used) substrate was used for each 
treatment and that the substrates had also been freshly cleaned according to the latter 
portion of the procedure in Section 4.1.4 (the toluene and acetone washings were not 
done). 
These substrates were placed in one of the cuvettes pictured in Figure 2.3.6-1 
on an elevated, level surface. A convenient way to create such a surface was to use 
socket bits of equal height from a socket set to support the substrate either stainless 
steel or glass. The cuvette was then filled with fresh oil. After waiting a few minutes 
for all air bubbles to rise out of the oil, drops were injected onto the surfaces of the 
substrate which had a diameter around 1 cm. 
The side view of this drop was used to find the contact angle, as represented in 
Figure 3.3.2-2. Preliminarily, the measurement was done by holding a protractor up to 
a piece of white paper placed behind the cuvette and estimating the contact angle 
visually. This method was accurate to within 5-10 ?. However, to obtain a more 
precise measurement for those methods which were ultimately selected for use in the 
other experiments, a goniometer was used to measure both the left and right contact 
angles. A goniometer is a specialized instrument which uses a consistently placed 
camera on rails with consistent back-lighting on rails to measure the contact angles 
more precisely than simply using a camera on a tripod or using a visual estimation. 
The left and right contact angles were averaged for at least 5 drops for each sample in 
which the goniometer was used as the method of measurement. 
 
 87 
 
4.3.2 ? Results & Treatment Selection 
The results of following the procedure of Section 4.3.1 can be summarized by 
Figures 4.3.2-1 through 4.3.2-4. Figures 4.3.2-1 & 4.3.2-2 show the work of adhesion 
for water in oil on glass and on steel, respectively. Water-in-oil-on-solid is being 
tested is because that is the situation inside the mixing vessel and on the microscope 
slide. Work of adhesion (given by equation (3.3.2-3)) is displayed because that is the 
relevant parameter to consider for how much energy is required to strip a solid a drop 
off of a solid surface. This amount of energy is equivalent to the amount of energy 
associated with a water drop?s binding to the solid surface, a quantity to be minimized 
so that the fluid flow near the wall will prevent such drops from adhering in the first 
place. 
In Figures 4.3.2-1 through 4.3.2-4, all samples were soaked overnight except 
for those using Glassclad 18 which were soaked for at least 20 minutes. 
 
 
 88 
 
Figure 4.3.2-1: Work of adhesion for various solid substrate treatment methods for 
water in oil on glass. The arrow and the underline indicate which method was 
eventually selected. 
 
Figure 4.3.2-2: Work of adhesion for various solid substrate treatment methods for 
water in oil on steel. The arrow and the underline indicate which method was 
eventually selected. 
Figures (4.3.2-3) & (4.3.2-4) show contact angles for water in oil on glass and 
on steel, respectively. These figures are shown in addition to Figures 4.3.2-1 & 4.3.2-
 2, not because they provide any new information (work of adhesion and contact angle 
are related by equation (3.3.2-3)), but because contact angle was the quantity that was 
directly measured and is the conventionally reported quantity industrially. One other 
advantage of reporting the contact angle is the ease of interpretation; 0? describes 
complete hydrophilicty where the drop spreads infinitely and 180? describes complete 
hydrophobicity where the drop is a perfect sphere resting undeformed on top of the 
solid substrate. However, the work of adhesion is still a quantity of interest as well. 
 
 89 
 
 
Figure 4.3.2-3: Contact angle for various solid substrate treatment methods for water 
in oil on glass. The arrow and the underline indicate which method was eventually 
selected. 
 
Figure 4.3.2-4: Contact angle for various solid substrate treatment methods for water 
in oil on steel. The arrow and the underline indicate which method was eventually 
selected. 
 
 90 
 
All of the treatment methods tested involved either a cleaning method alone or 
soaking the solid surfaces in a low concentration (2-4%) hydrophobic silane 
chemical. Of those with a hydrophobic silane some were dissolved into various 
organic solvents before soaking and some were dissolved into a 95% ethanol ? 5% 
water slightly acidic solution. The idea for the latter blend was taken from the website 
of Gelest, Inc (2006). However, the times for curing were experimentally found to be 
much too short. 
It should be mentioned that the criteria for choosing the hydrophobic 
treatment methods was not necessarily to find those which resulted in the maximum 
difference in contact angle and work of adhesion. Other factors such as toxicity, cost, 
and ease of application were relevant in the choice. In terms of the effectiveness of 
the coating, the requirement was that the hydrophobicity be demonstrably different 
from the untreated case. This difference may be quantified by comparing the columns 
with red arrows in Figures (4.3.2-1) through (4.3.2-4) to the columns on the left-hand 
side of the graph which represent the untreated case. 
As indicated by the red arrows in Figures (4.3.2-1) through (4.3.2-4), a 
separate method was chosen for glass and for stainless steel. For glass, 2 vol% 
Glassclad 18 (a proprietary silane formulation) was dissolved into a 95% ethanol ? 
5% water solution adjusted to a pH of 5 by using acetic acid. The glass surfaces were 
immersed in this solution for at least 20 minutes with all surface area to be treated 
exposed to the liquid. Glass slides were placed within slide containers to keep them 
from sticking together, while the glass mixing vessel was simply filled with this fluid 
since only the inside needed to be treated. For stainless steel, 3 vol% 
 
 91 
 
dimethyldichlorosilane was dissolved in cyclohexane and poured into the 
crystallizing dish described in Section 4.2. The stainless steel parts to be treated, the 
mill head (for some experiments) and the underside of the mixer lid, were immersed 
in the liquid overnight. To stop the fluid from disappearing by evaporation a bell jar 
was placed over this dish.  
One final consideration in the treatment of these surfaces was that the mixer 
lid was actually polished before applying the hydrophobic treatment. As can be seen 
in Figures (4.3.2-1) through (4.3.2-4), the glass treatment resulted in a far greater 
difference in hydrophobicity than that for stainless steel. Part of the reason for this is 
that clean glass is initially quite hydrophilic. In the dispersion experiments the 
treatment of the glass mixing vessel was discover to be sufficient to prevent the issue 
of glass sticking; however, while the treatment of the underside of the mixing lid was 
found to reduce the amount of sticking, some sticking still remained. This is shown in 
Figure (4.3.2-5). 
 
 92 
 
 
Figure 4.3.2-5: Drops sticking to the underside of the mixer lid. This figure shows 
that the hydrophobic treatment alone was insufficient to solve the drop sticking 
problem. 
In order to further mitigate such the effect of such drops, the mixer was 
sanded with progressively finer grades of sandpaper up to 3000 grit (average particle 
size less than 1  m) until a mirror-like finish was achieved. This was done, not to 
reduce the work of adhesion of the drops, but to decrease the hysteresis of drops on 
the surface as mentioned in Section 3.3. After the sanding, the mixer lid was retreated 
according to the aforementioned procedure. This whole procedure did further 
decrease the volume and size of the drops sticking to the lid. The effect of doing this 
is quantified in Figure 4.3.2-6. 
 
 93 
 
 
Figure 4.3.2-6: Effect of Polishing the mixer lid. 2 experiments were performed 
without polishing and then 2 experiments with polishing. There appears to be a slight 
difference caused by polishing. 
It is debatable whether Figure 4.3.2-6 conclusively establishes that there is a 
significant difference caused by polishing the mixer lid, but it does seem to provide 
some evidence that the drops are a little smaller when the mixer lid is polished. That 
is the direction of the effect that would be expected if there is indeed some error 
caused by large drops (based on the theory of work of adhesion large drops are more 
subject to being stripped than smaller ones) being stripped off right before sample 
withdrawal and, thereby, skewing the equilibrium DSD. Therefore, in all experiments 
from which data was taken in the Chapters 5 through 7, the mixer lid was polished 
and hydrophobically treated. 
This set of experiments and discussion is only present to show that adhering 
drops are not becoming stripped off just before sample withdrawal and disturbing the 
equilibrium DSD. The discussion of whether the DSD gets skewed by drops leaving 
 
 94 
 
the emulsion and remaining adhered to the solid surfaces is deferred to Chapter 7 
where it fits more appropriately with the discussion of the effect of phase fraction. 
After all, such a disturbance of the equilibrium DSD would be due to a change in the 
phase fraction caused by drops leaving the sample. It is known that drops do not stick 
to the high shear mixing surfaces both by direct observation of these surfaces after 
mixing and by the theory of work of adhesion which states that there is a threshold 
energy input from the fluid near the surface above which drops are not permitted to 
adhere. Clearly, this energy input is highest near the walls which are exposed to high 
shear rates. 
 
 95 
 
Chapter 5:  Flow Regime Determination for Batch Experiments 
Before the drop size models for the various flow situations which may be 
encountered can be discussed, the preliminary issue of flow regime identification 
must be discussed. Without definitive knowledge of the flow regime in which one is 
operating, it is not possible to make comparisons to previous models as well as to 
develop new ones. 
This study is concerned with the flow regime determination in two separate 
situations / mixing configurations. The first is that for the batch Silverson L4R mixer 
with the slotted stator. Fortunately, this work has been previously performed by 
Padron (2001) and so it was not necessary to repeat in this study. The second is that 
for the IKA in-line 2000/4 pilot scale mixer. Padron?s work is briefly summarized 
here and used as one of the bases of flow regime determination for the batch mixer 
along with the data obtained in this study. Work with the IKA mixer is deferred until 
Chapter 8 as part of the discussion about in-line systems. 
5.1 ? Power Draw vs. Reynolds Number Relationship 
A form of the friction factor, the power number, NP, can correlated with the 
Reynolds number.  
   
 
      
    (5.1-1) 
The power number represents the ratio of the power input into a system to the 
amount of that power which is spent in energy dissipation as opposed to bulk 
 
 96 
 
convection. The qualitative difference in the where the inputted power was spent, 
convection versus viscous dissipation, can be used to determine the flow regime.  
Rushton (1950) applied this concept to stirred tanks with various impeller 
geometries and demonstrated that Power number decays exponentially with 
increasing Reynolds number for laminar flow independent from impeller geometry. 
Rushton (1950) also discovered that Reynolds number reaches a different constant in 
turbulent flow for each impeller geometry, and that this constant can be used to 
quantify the efficiency of mixing. Rushton?s (1950) original plot is shown in Figure 
5.1-1. 
 
Figure 5.1-1: Rushton (1950) plot of Power number vs. Reynolds number. This type 
of plot, specific to each mixer, can be used to coarsely identify the transition between 
flow regimes. 
The issue of interest to the present study is that the flow regime transitions can 
be coarsely identified by the region at which the Power number vs. Reynolds number 
curve changes its behavior. When the Power number clearly decays exponentially 
with Reynolds number, the flow is definitively in the laminar regime, and when the 
 
 97 
 
Power number is clearly independent of Reynolds number, the flow is definitively in 
the turbulent regime. When neither of these conditions is true, there is some 
ambiguity as to the flow regime, which could be laminar, turbulent, or transitional. 
As illustrated in Figure 5.1-1, where a separate curve must be constructed for 
each impeller geometry, every type of mixer has its own curve which is measured 
experimentally. Due to the complex nature of the flow fields within these mixers, 
numerical simulation or direct experimentation are the only possible methods to 
obtain such curves. 
5.2 ? Padron?s (2001) Power Draw Work 
Fortunately for the sake of the present study, the Power number vs. Reynolds 
number plot for the batch Silverson L4R mixer has been previously performed by 
Padron (2001). The results of that work will be used in Section 5.2.2 to coarsely 
identify the flow regime boundaries. 
5.2.1 ? Brief Summary of Methodology 
It is not the purpose of this work to provide an in-depth analysis of Padron?s 
(2001) work; for such analysis, the reader is referred directly to that work. However, 
the essence of that work is that the power is measured using a power cell for each 
configuration of the mixer to be measured. This measured power is multiplied by 
motor?s efficiency (< 1). Then the difference of that quantity and the energy lost to 
friction in the seals and bearings is reported as the total power inputted into the fluid. 
The motor?s efficiency was calculated by replacing the mixing head with a 
Rushton turbine or similarly simple impeller which transmits nearly all of its power to 
 
 98 
 
the tank wall, losing very little to viscous dissipation. The energy transmitted to the 
tank wall is equated with the energy input to report the efficiency of the motor for 
each specific mixer. 
5.2.2 ? Results & Application to the Present Study 
The Power number was plotted for three separate mixers (this study is only 
interested in the data of the batch Silverson L4R) as a function of Reynolds number. 
The data of the Silverson L4R is displayed in Figure 5.2.2-1. 
 
Figure 5.2.2-1: Padron?s (2001) plot of Power number vs. Reynolds number for the 
batch Silverson L4R mixer. The range of Reynolds numbers for the dispersion 
experiments is bounded by the vertical lines. 
On the left-hand side of Figure 5.2.2-1, the Power number exponentially 
decays with Reynolds number, and on the right-hand side the Power number is a 
 
 99 
 
constant equal to 2.1. These regions correspond to laminar flow and turbulent flow 
respectively. The region bounded by the vertical lines, which corresponds to the 
region used in the dispersion experiments, does not obviously fall within one flow 
regime. However, this is helpful to the study because at the minimum Reynolds 
numbers the flow is laminar and at the maximum, the flow is turbulent. This allows 
both flow regimes to be studied and separate correlations to be made for each one. 
The need for different arguments for drop size correlations in each flow regime is 
what makes the identification of the flow regime transitions so crucial. 
Padron?s experiments were, of course, single phase and so the continuous 
phase kinematic viscosity in the Reynolds number is that of the single phase. It is 
listed in Figure 5.2.2-1 as the continuous phase viscosity because that is the parameter 
which is used in the dispersion experiments. The continuous phase viscosity is used 
because there was no available method to quantify the emulsion viscosity. 
Another notable issue which arises in the discussion in Section 5.3 is that the 
data points on Figure 5.2.2-1 are not all matched up perfectly well with the lines 
which have been drawn on top of the plot. This is because different viscosities were 
used by Padron (2001) for the various groups of data points; it would be impossible to 
cover the whole range of Reynolds numbers with a single viscosity. This lack of 
perfect harmony in the pattern of the data suggests that the flow regime transitions are 
broadly determined by the Reynolds number, but are also influenced by the viscosity. 
This conclusion is reinforced in Section 5.3. Finally, it can be observed that, with this 
clarification in mind, the flow regime transition from laminar to turbulent may be 
coarsely determined to be in the neighborhood of Re = 1000 with some dependence 
 
 100 
 
on viscosity. This does not mean that there is no transitional flow, but rather that if 
there is, then it occurs near Re = 1000. Furthermore, this means that laminar flow 
should is limited to near or below that value of Re and that turbulent flow is limited to 
near or above that value of Re. 
5.3 ? Qualitative Differences in Drop Size Behavior 
The above coarse determination of the flow regime is clearly insufficient for 
the purpose of classifying the drop size experiments by their flow regimes. However, 
the preceding coarse determination does provide necessary guidance for the fine 
determination of the flow regime transitions which was performed on the basis of 
qualitative differences in the behavior of the drop size data. 
5.3.1 ? Data Range of Physical Properties 
The range of the Reynolds numbers used in the dispersion experiments is 
shown in Figure 5.2.2-1 by the vertical lines. However, it is desirable to provide the 
range of the individual physical properties. This is done by reference to Table 2.4.1-1. 
The range of  c is 18 to 153 cP. ?c is approximately constant; it ranges from 0.838 to 
0.864 kg/m3. The range of N is 3900 to 9000 rpm. The value of D is constant at 2.81 
cm. All this results in a range of Reynolds numbers of 330 to 5600. 
5.3.2 ? Evidence from Outlier Data 
Before getting into the rigorous analysis of the drop size behavior which is 
used as the main criteria for separating the flow regimes, the drop sizes are plotted 
straightforwardly to investigate any qualitative differences. Drop size is plotted in the 
form of a Sauter mean diameter, d32, (the definition and reason for using this form for 
 
 101 
 
the drop size are discussed in Section 6.1.1) with respect to rotor speed in Figure 
5.3.2-1.  
 
Figure 5.3.2-1: Drop size vs. rotor speed. The two notable outliers have been pointed 
out with arrows. This plot provides evidence that those points are in a transitional 
flow regime. 
In Figure 5.3.1-1, most of the data follows a somewhat consistent pattern with 
drop size generally decreasing as  c increases. The effect of rotor speed is also 
somewhat consistent. However, there are 2 data points in particular that break from 
the trend. It is postulated that this is because these points lie within the transitional 
flow regime and, therefore, suffer from an increased amount of scatter and 
unpredictability compared to the other data. 
This evidence is not very compelling on its own, but it does have 
supplementary evidential value when taken together with Padron?s (2001) Power vs. 
Reynolds number plots and the more conclusive, systematic analysis of drop size 
behavior of the following section. 
 
 102 
 
5.3.3 ? Capillary Number Behavior 
More conclusive evidence that the transition from laminar to turbulent flow 
occurs at a Reynolds number of approximately 1000 was found from analyzing the 
drop size data acquired over the range of continuous phase viscosities. Trends in the 
data can be most easily seen by plotting d32 as part of a Capillary number, Ca, with 
respect to the rotor speed. 
   
      
 
           
   
 
   (3.4-1) 
where    is the nominal shear rate in the rotor stator gap and ? is the distance between 
the rotor tip and the inner wall of the stator. It is important to note that the nominal 
shear rate is defined as the rotor's tip speed divided by the width of the ?shear gap.? 
This may or may not be the true characteristic shear rate. However, it can be assumed 
that the true characteristic shear rate (which is unknown) is proportional to the rotor 
speed (Metzner and Otto 1957) and due to fixed geometry is proportional to the shear 
rate used here. 
 
 103 
 
 
Figure 5.3.3-1: Drop Capillary number as a function of rotor speed as a function of 
viscosity ratio (?) and Reynolds number (Re) for clean (no surfactant) systems. 
Dotted = laminar, dashed = transitional/unknown, solid = turbulent. 
The aforementioned experimental data is plotted as a function of viscosity 
ratio, ?, and Re in Figure 5.3.3-1. The dotted curves are laminar, the solid curves are 
turbulent, and the dashed curves are transitional or unknown. There is a qualitative 
difference between the behavior of the laminar and turbulent curves; for laminar flow 
there is a generally gentle increase in Ca with rotor speed, and for turbulent flow Ca 
is constant with rotor speed. The dashed curves in Figure 5.3.3-1 fall between the 
laminar and turbulent regimes and are more variable, not exhibiting consistent 
behavior. It is interesting to note that although this occurs near Re=1000, which 
verifies Padron's power measurement analysis for the boundary of the flow regimes, 
the boundary is not exact. In Figure 5.3.3-1, the dashed filled data points have 
 
 104 
 
Reynolds number ranges which are out of order while the viscosity ratio, ? =  d/ c, 
increases monotonically. So, although it is true that the flow regime transition is 
around Re=1000 and Re seems to broadly determine the flow regime, it appears to be 
 c rather than Re which provides the exact boundaries. 
Qualitative differences in the flow regime can also be seen by plotting the 
Capillary number against the viscosity ratio as done in Figure 5.3.3-2. While the 
boundaries between flow regimes are less clear (the precise boundaries in the flow 
regimes can only be seen in Figure 5.3.3-1), the laminar data decreases with 
increasing viscosity ratio, the turbulent data levels off to a constant Capillary number, 
and the transitional/unknown data exhibits greater variability. 
A remarkable fact about Figure 5.3.3-2 is the continuity of the overall curve 
despite the changes in the flow regime. Such continuity is not known to be necessary 
according to any conventional theory. Figure 5.3.3-2 could rightly be viewed as the 
concatenation of two plots, one for the laminar regime and one for the turbulent 
regime. 
 
 105 
 
 
Figure 5.3.3-2: Drop Capillary number based on Sauter mean diameter as a function 
of oil viscosity ( c) and Reynolds number (Re) for clean (no surfactant) systems. 
Using Figures 5.3.3-1 & 5.3.3-2 the equilibrium Sauter mean drop size data 
are separated into those produced in laminar, transitional, and turbulent flow. While 
Figure 5.3.3-1 is the means of precise separation, its evidence alone would be 
insufficient without Padron?s power draw study and is certainly supplemented by 
Figure 5.3.2-1. The laminar and turbulent data are analyzed separately in Chapter 6 
using the most appropriate mechanistic arguments. In the remainder of the 
presentation the data which is called ?laminar? or ?turbulent? is well within its 
respective regime. The transitional data is not used at all. 
 
 
 
 106 
 
Chapter 6:  Dilute, Batch Experimental Results 
This chapter contains the results and analysis for the all of the dilute 
experiments.  It begins with general remarks on characterization of the DSDs. Then, 
making use of the demarcations between flow regimes which were established in 
Chapter 5, laminar and turbulent flow experiments are reported and analyzed 
separately. The analysis of both flow regimes includes the effects of changing 
viscosities, adding surfactants, and hydrophobically treating the high-shear surfaces 
of the mill head. The effect of changing the phase fraction is deferred to Chapter 7. 
6.1 ? Drop Size Characterization 
6.1.1 ? Use of d32 to Characterize DSDs 
Since the DSDs are composed of a range of drops and there are many different 
experiments, it is necessary to compile the information from one experiment into a 
single number to facilitate comparison with other experiments. This could be done 
using the average drop diameter, d10. 
    
     
   
     (6.1.1-1) 
where ni is the number of drops belonging to each size category, or bin, and di is the 
value of the diameter in the middle of each bin. 
An alternative basis of characterization of the DSDs is the Sauter mean 
diameter, d32. 
    
     
 
     
      (6.1.1-2) 
 
 107 
 
where ni and di are defined as above. This quantity was chosen because it is 
subject to less statistical variability and is proportional to dmax (Padron 2005). The 
bins were defined by using three interlaced Fibonnaci sequences as discussed by 
Chang et al. (1991). Three interlaced Fibonacci sequences ([2, 1], [1, 1], and [2, 2]) 
were used to define the bin edges. Other Fibonacci sequences could just as well have 
been used. These three sequences were multiplied by a constant to ensure that the 
drops fit into at least 20 different bins. A sample DSD illustrating the use of such bins 
is plotted in Figure 6.1.1-1. 
 
Figure 6.1.1-1: Sample DSD. This plot illustrates the usage of Fibonacci sequences 
to define discrete size categories for use in the definition of d32. 
 
 108 
 
It is also worth mentioning that since every sample in this study contained at 
least 1000 drops, it was not possible to have significant errors due to binning choices. 
A number of bin choices (using different Fibonnaci sequences and different numbers 
of bins) were tested on the data and the maximum difference in d32 due to bin choices 
was 2% as the number of bins approached infinity. In other words, the bin choices are 
not a significant issue in this study because the sample sizes were large enough. 
The reasons for using d32 rather than d10 have not yet been established; they 
are threefold. First, there is more volume contained in the larger drops, and it is 
obvious mathematically that equation (6.1.1-2) is more influenced by larger drops 
than equation (6.1.1-1) which is influenced by all drops equally regardless of size. 
Volume is proportional to d3, and it is desirable to have a method of quantifying the 
DSD which is correlated more with volume that with diameter. Secondly, there is 
more error associated with the imaging of smaller drops. By biasing the DSD 
characterization to the larger drops, more accurate characterization is achieved. 
Finally, d32 has the physically meaningful interpretation of being the volume to 
surface ratio. This is useful when analyzing mass or heat transfer into or out of drops 
in an emulsion. 
6.1.2 ? Relationship of d32 and dmax 
Having thus established the benefits of characterizing the DSDs using d32, that 
is what was done throughout this study. However, in order to make comparisons with 
mechanistic arguments for both laminar and turbulent flow it is necessary to use the 
maximum drop diameter, dmax. Due to the large amount of scatter present in the value 
 
 109 
 
of dmax, it should not be used directly. Fortunately, a linear relationship between d32 
and dmax was found for both laminar and turbulent flow. 
When drop size is compared to that of single drops breaking up in idealized 
flow fields (discussed in Section 3.1.2), the Capillary number should be based on dmax 
as opposed to d32. This facilitates comparison with Grace's (1982) data which was 
obtained by experimentation on individual drops which are comparable to the 
maximum stable drop size. In order to convert from d32 to dmax, the values of d32 for 
all of the laminar experiments were plotted against dmax. Linear regression (plotted in 
Figure 6.1.2-1) was performed to obtain an appropriate correlation. Values of dmax 
were not used directly because they contain much more scatter than d32.  
                (6.1.2-1) 
 
Figure 6.1.2-1: Relationship between d32 and dmax for laminar flow data. 
 
 110 
 
As discussed in Section 3.1.1, the appropriate mechanistic correlation for the 
turbulent data is selected based on the drop size relative to the Kolmogorov 
microscale. ? should be compared to the maximum stable drop size rather than d32. 
As previously noted, it is not prudent to use dmax directly because of the high amount 
of scatter. Figure 6.1.2-2 is constructed by analogy to Figure 6.1.2-2 except that it is 
for the turbulent flow data. 
                (6.1.2-2) 
 
Figure 6.1.2-2: Relationship between d32 and dmax for turbulent flow data. 
While these two relationships between d32 and dmax have been plotted 
separately, it should not be inferred that a distinctness is being claimed. A distinctness 
in the relationships is possible since there seems to be more scatter in Figure 6.1.2-2, 
but such a difference is not thereby established. For the purposes of this study, it is 
not necessary to come to a conclusion on that issue; it is enough to simply note that 
 
 111 
 
for both flow regimes there is a reliable relationship between d32 and dmax and that it 
may be used to mitigate the high scatter present in measured values of dmax. 
6.2 ? Reproducibility of Experiments 
Having established a consistent method of representing the DSD numerically, 
it is useful to have some idea of the reproducibility of the experiments. Figure 6.2-1 
displays the results of performing the same experiment 5 times. This was done with 
an intermediate (not dilute, but still not highly concentrated) phase fraction of ? = 
0.0046. There were no surfactants, the mill head was not hydrophobically treated, and 
the flow was laminar. The standard deviation of d32, shown on Figure 6.2-1, varied 
with rotor speed. 
 
Figure 6.2-1: Quantification of the reproducibility of experiments. T = 43?,  c = 93.3 
cP, ? = 0.0066, and Re: 470-1100. 
 
 112 
 
6.3 ? Laminar Flow Results & Analysis 
As previously mentioned, there is a scarcity of laminar flow drop size data in 
rotor-stator and other mixing devices so our approach is to compare our results to 
those for idealized drop breakup experiments. Grace (1982) found that the maximum 
stable drop size is dependent on the viscosity ratio. At values of ? less than unity the 
critical Capillary number, Cac as defined in the first equality of equation (6.3-1), is a 
constant for simple extensional flow (SEF) and decreases with increasing ? for simple 
shear flow (SSF). An expression was estimated from Grace?s (1982) work for drops 
in SSF at low viscosity ratios.  
    
  ?     
?
                                  ?   ?      (6.3-1) 
where    is the shear rate for SSF. We adjusted Grace?s original stability curve to 
replace radius by maximum stable drop size, dmax. 
The laminar portion of Figure 5.3.3-2 is qualitatively similar to Grace's curve 
for simple shear flow in that Cac exhibits a power law decrease with ?. When linear 
regression is performed on our laminar data the functional form of equation (6.3-2) is 
obtained.  
   
  
       
?
                             (6.3-2) 
where dmax is the maximum stable drop diameter as calculated from d32 using 
equation (6.1.2-1). 
Equation (6.3-1) is analogized to Equation (6.3-2). The most salient 
comparison between equations (6.3-1) and (6.3-2) is the magnitude of the exponent. 
In Grace's data the simple shear case has a higher dependence on viscosity ratio than 
 
 113 
 
the extensional case, and yet the data of this study's dependence on viscosity ratio is 
even stronger. This leads to the conclusion that the breakage mechanism for laminar 
flow is more similar to simple shear than to pure extensional flow. 
Another important consideration is the absolute value of the critical Capillary 
number. Our comparison is not very exact because the appropriate characteristic shear 
rate for laminar flow in a rotor-stator mixer is unknown. Following Metzner and Otto 
(1957) we assume that the correct shear rate is proportional to the agitation rate. Then 
the true characteristic shear rate,    , must be equal to the product of a coefficient, K, 
and the previously defined nominal shear rate for our mixer. 
      ?    
   
 
    (6.3-3) 
Assuming that the breakup mechanism is exactly that of simple shear and that 
the critical Capillary number curve for a rotor-stator mixer should correspond to that 
for SSF, the value of the coefficient, K, can be determined. If this reasoning is valid, 
then it can be concluded that the true shear rate is approximately 2.2 times greater 
than the nominal shear rate in the rotor-stator gap. This result is interesting but, at 
best, approximate because the dependency on ? is different. Equations (6.3-1) and 
(6.3-2) are plotted in Figure (6.3-1) along with equation (6.3-2) with    replaced by    . 
The value of K was determined by setting the middle data point of this study equal to 
its corresponding value on Grace's curve. 
 
 114 
 
 
Figure 6.3-1: Critical Capillary vs. viscosity ratio for this study compared to Grace's 
(1982) curve for simple shear flow with ?  <  ?0.05. 
Putting together equations (6.3-2) and (6.3-3), an expression for normalized 
drop size can be obtained.  
    
 
        
? ?
        
   
  
  
 
     
    or      
    
 
        
? ?
       
   
(6.3-3) 
The exponent of -0.96 on the viscosity ratio in equation (6.3-4) is close 
enough to unity that it is worth speculating that it ought to be -1, as approximated by 
the second expression in equation (6.3-4). The 95% confidence interval around the 
exponent is -1.16 < exponent < -0.76. 
 
 115 
 
If we define Cad as Cad= d N D/? as the Capillary number based on the 
dispersed phase viscosity and replace dmax by d32, equation (6.3-4) is obtained. 
   
 
        
 
 
    
      (6.3-4) 
where ?/D = 0.0066 in this study. Except for square root of density ratio, Cad is 
similar to the viscosity group used by Calabrese et al. (1986) to correlate data for 
viscous drops dispersed in turbulent stirred tanks. 
Unfortunately, there is no known previous study in laminar flow against 
which to compare our data. Unlike the turbulent flow analysis which relies on 
Kolmogorov's theory of local isotropy, laminar flows are locally directional. Because 
of the influence of geometry through the boundary conditions, derivation of 
mechanistic correlations is not straightforward. For these reasons this correlation is at 
best semi-empirical and should be used accordingly. 
An interesting feature of equation (6.3-4) is the independence of the drop size 
from continuous phase properties, a result which was not expected. The Grace (1982) 
data for SSF show that, for ? < 1, the drop size increases as ? decreases. He argued 
that, near ? = 1, the degree of deformation required for breakup reached a minimum, 
as did Cac. As ? ? 0, drops become highly elongated prior to breakup. The stability 
of such long threads should depend on their internal viscosity. However, as ? ? 0, 
the disruptive stress acting on the drop becomes unbounded, so the exact value of 
continuous phase viscosity may not be that important. However, the reason for  c 
independence is not obvious and requires further investigation. 
An important consideration with respect to equations (6.3-3) and (6.3-4) is 
that our experiments do not validate all of the variables. In particular, the choice to 
 
 116 
 
non-dimensionalize equation (6.3-3) with respect to D is arbitrary since only one 
value of D was tested; it would have been equally correct to non-dimensionalize 
equation (6.3-3) with respect to ?.  
The data presented previously only apply to clean (non-surfactant) systems for 
which ?=54.8 mN/m. Figure 6.3-2 compares equation (6.3-4) to both clean and 
surfactant systems. 
 
Figure 6.3-2: Empirical correlation for drop size scaling in rotor-stator mixers. This 
plot is comparable to equation (6.3-4) for clean and low surfactant concentration 
systems in laminar flow. 
In Figure (6.3-2) the data collapse to a single curve except for the higher 
concentrations of surfactant where the drop sizes are larger than would be predicted 
by equation (6.3-4). Since the low surfactant concentration systems follow the 
correlation, it can be concluded that bulk surfactant concentrations of 0.1xCMC or 
 
 117 
 
less are not sufficient to cause meaningful dynamic interfacial phenomena. At higher 
concentrations, surfactant transport from the bulk to the drop interface may be 
diffusion limited causing interfacial tension gradients along the interface during drop 
stretching. The resulting interfacial elasticity, or Marangoni stress, resists deformation 
and could possibly explain the larger drop size. 
6.4 ? Turbulent Flow Results & Analysis 
From equation (3.1.1-3), it is obvious that the Kolmogorov microscale is most 
strongly dependent on viscosity. Since a viscous continuous phase is used in this 
study, ? is larger than in most previous studies with a low viscosity continuous phase. 
There is some ambiguity in how to calculate the energy dissipation rate. Many 
authors, such as Zhou and Kresta (1996), have estimated the maximum energy 
dissipation rate experimentally from turbulence measurements, as a function of mixer 
dimensions and agitation rate, for various stirred tank impellers. However, there is no 
known comparable study for the particular mixer used in this study, and so the 
maximum energy dissipation rate is approximated here by analogy to the impeller 
swept volume approach (McManamey 1979). The impeller swept volume approach 
recognizes that the energy in a stirred tank is not spatially uniform. It approximates 
the region in which the energy from the impeller is dissipated as being equal to the 
region swept out by the impeller as it rotates. This gives a closer estimation of the 
energy dissipation rate as it relates to the breakage of drops. 
According to Figure (5.2.2-1), NP=2.1 for fully developed turbulent flow. The 
power draw, so obtained, is divided by the mass of fluid in the high shear region. The 
high-shear region is defined as being the cylinder that is swept out by the rotor at any 
 
 118 
 
point in its rotational path. This may not be precisely the correct volume; fortunately 
the effect of the choice for the definition of the high-shear region is not very 
significant, as is evident from the exponent of ? in equation (3.1.1-3). When the 
Kolmogorov scale is calculated in this fashion, it ranges from 40 to 139  m in the 
experiments of this study. The range of dmax was 11 to 49  m. In each experiment, the 
maximum drop size was always well below the Kolmogorov microscale. 
Since the drops were smaller than the Kolmogorov microscale, despite 
differences in the definition of the high-shear region, one of the sub-Kolmogorov 
models should be applicable to the turbulent data. In fact, as shown in Figure (6.4-1), 
these data are well-correlated by the sub-Kolmogorov inertial stress model presented 
generally in the second row of Table 3.1.1-1 and specifically for this system in 
equation (3.1.1-9). This is reasonable because for this data the Kolmogorov 
microscale is greater than d32, but not by orders of magnitude.  
   
 
                    (6.4-1) 
 
 119 
 
 
Figure 6.4-1: Empirical correlation for drop size scaling in rotor-stator mixers. This 
plot is comparable to equation (15) for clean and low surfactant concentration 
systems in laminar flow. 
In contrast to the results for laminar flow, the presence of surfactants in the 
turbulent experiments simply lowers the interfacial tension without manifesting any 
other effects. As a result they were included in the regression leading to equation 
(17).  Assuming that Marangoni stresses are, in fact, the cause of the departure of 
some of the laminar flow data from its correlation, a comparison of the laminar and 
the turbulent results implies that Marangoni stresses are less pronounced in turbulent 
flow. Perhaps transport of surfactant toward the drop surface by turbulent eddies is 
sufficient to prevent their occurrence. 
 
 120 
 
Finally, it is interesting that, by contrast to the empirical laminar correlation, 
equation (6.4-1) is inversely proportional to the continuous phase viscosity and 
uncorrelated to the dispersed phase viscosity. It is indeed peculiar to note that by 
comparing equations (6.3-4) and (6.4-1), it can be seen that in laminar flow the drop 
size is dependent on the dispersed phase viscosity and not the continuous phase 
viscosity, while in turbulent flow the drop size is dependent on the continuous phase 
viscosity and not the dispersed phase viscosity. A small part of the explanation for 
this is that in turbulent flow the square root of the turbulent mean-square velocity, 
also known as the Kolmogorov velocity scale, (see the second row of Table 3.1.1-1) 
is equal to ? 
    
?    , which depends on the continuous phase viscosity. This means 
that the structure of the turbulence eddies are themselves dependent on the continuous 
phase viscosity. The analogy of this is untrue for laminar flow whose flow field is not 
determined by the continuous phase viscosity. 
 
 121 
 
Chapter 7:  Non-dilute, Batch Experimental Results 
This chapter contains the results and analysis for the high concentration 
experiments. It begins with the observation that in both laminar and turbulent flow the 
drop size dependence on phase fraction is of the same form. The effects of adding one 
complication at a time are observed to change the strength, but not the form of the 
relationship. These complications are the hydrophobic treatment of the high-shear 
surfaces and the addition of surfactant to the continuous phase. All that work is 
performed with ? < 0.05. Then the high concentration experiments, when 0.05 < ? < 
0.5, are observed to change the form of the relationship. Finally the chapter is 
concluded with some experiments which investigate the possibility of the phase 
fraction changing due to adherence of the dispersed phase on low shear surfaces. 
7.1 ? Clean Systems 
Figures (7.1-1) & (7.1-2) show that for both laminar and turbulent flow with 
d32 < ?, the effect of phase fraction on the drop size can be quantified in log-linear 
form and is well-represented by equation (7.1-1).  
    ?)        ?       (7.1-1) 
 
 122 
 
 
Figure 7.1-1: Equilibrium Sauter mean diameter vs. water volume fraction for clean 
systems in laminar flow. All experiments use Crystal Oil 500FG. T = 41?,  c = 93.2 
cP, ? = 0.0069, and Re: 470-1100. 
 
 123 
 
 
Figure 7.1-2: Equilibrium Sauter mean diameter vs. water volume fraction for clean 
systems in turbulent flow. All experiments use Crystal Oil 70FG. T = 60?,  c = 18.1 
cP, ? = 0.026, and Re: 2400-5600. 
It does not make sense to compare the data of Figures (7.1-1) & (7.1-2) 
directly with previous data for which turbulent inertial subrange scaling is valid 
because the coalescence mechanisms may be fundamentally different (for example, 
bouncing occurs in inertial collisions, but not in viscous collisions (Chesters 1991)). 
Equation (7.1-1) is a clear departure from the prevailing notion that equation 
(3.4-5) describes the effect of phase fraction on drop size. Equation (3.4-5) may be 
normalized by the dilute case to investigate whether it characterizes the data. 
   
      
    ?    (7.1-2) 
If equation (3.4-5) were to be used to describe the DSD, then ?b? (which has 
been reported to be between 1 and 10) would not be constant and would initially be as 
 
 124 
 
high as 400. Since equation (3.4-5) has been applied to a turbulent coalescence 
mechanism, it should not be expected to describe the data in the current study. 
Equation (3.4-6), although derived for the specific case of viscous simple 
shear flow, is a reasonable first-approximation for the coalescence rate in flows which 
are locally viscous for both the laminar and turbulent data (d32 < ?) presented in 
Figures (7.1-1) & (7.1-2). The dilute drop size data for laminar flow are analyzed in 
Part 1 by reference to simple shear flow. 
Table 7.1-1 was constructed using the approximations of Chesters (1991) in 
equation (3.4-6). It is only intended to convey qualitative information about the 
collision efficiency, P, since there are many simplifying assumptions, such as the 
coalescing drops being of equal size and the flow field being viscous simple shear. 
Sample values were chosen near the middle of the ranges applicable to the data of 
Figures (7.1-1) & (7.1-2), and the shear rate used in the Capillary number is 
approximated by equation (7.1-3a) for laminar flow and equation (7.1-3b) for 
turbulent flow.  
             (7.1-3a) 
    ? ?  
       (7.1-3b) 
where ? is the distance between the rotor tip and the stator (0.185 mm) and ? is the 
energy dissipation rate as approximated by the ?impeller? (rotor) swept volume 
approach. 
 
 125 
 
Sample Values; d = 8  m, ? = 54.8 mN/m, rotor speed = 5600 rpm 
 Laminar,  c = 93.3 cP, T = 43? Turbulent,  c = 18.1 cP, T = 60? 
 d (cP)? 1.9 0.62 0.093 90.5 18.1 0.47 
? 0.02 0.0066 0.001 5 1 0.026 
P 0.01 0.26 0.81 0.01 0.36 0.97 
 
Table 7.1-1: Table of approximate values of collision efficiency. The table shows 
that the viscosities play a controlling role in determining the probability that a 
collision event will result in coalescence. Bolded columns indicate the results that 
correspond to Figures (7.1-1) & (7.1-2). 
The most striking feature of Table 7.1-1 is the extremely strong dependence of 
the collision efficiency on viscosity. To avoid confusion about the flow regime, only 
the dispersed phase viscosity was changed on either side of the table. The continuous 
phase viscosities are those for the actual experiments. With respect to  c, ?, and P, the 
bolded entries are for the actual experimental conditions. It should be remembered 
that equation (3.4-6) shows that collision efficiency decays exponentially with -
  d c
 1/2. It is seen that for the viscosity ratios of Figures (7.1-1) & (7.1-2) the collision 
efficiency is extremely large compared to that for ? = 1, for which correlation based 
on equation (3.4-5) has been reported. It is therefore possible that the data of Figures 
(7.1-1) & (7.1-2) are dominated by high degrees of coalescence. 
Even with this theoretical support for the likelihood of coalescence in these 
systems, it was decided to test for the presence of coalescence experimentally. A 
single experiment was performed in which an equilibrium drop size distribution was 
initially established for a phase fraction of 0.01 at a rotor speed of 3900 rpm with no 
surfactants. The mixing speed was increased to 9000 rpm for 2.5 hours, and then 
switched back to 3900 rpm. The drop size was measured just before each speed 
 
 126 
 
change and then at several times after the final speed adjustment. This experiment?s 
results are displayed in Figure (7.1-3). If there were no coalescence, the drop size 
would remain at the low value it reached when the rotor speed was 9000 rpm. 
However, it is clear that the drop size increases when the rotor speed is reduced, 
conclusively demonstrating the presence of coalescence in this system. 
 
Figure 7.1-3: Dynamic Sauter mean diameter for a clean (surfactant free) system in 
laminar flow. This experiment uses Crystal Oil 500FG. ? = 0.01, T = 41?,  c = 104.6 
cP, ? = 0.0062, and Re: 420 and 980. 
7.2 ? Clean Systems with Treated Mill Head 
For some laminar flow experiments the mixing head, consisting of the 
surfaces that were exposed to high shear rates, was treated with 
dimethyldichlorosilane in toluene, in order to modify the hydrophobicity so that the 
 
 127 
 
effect of the interaction between the dispersed phase and the metal surfaces could be 
examined. 
 
Figure 7.2-1: Equilibrium Sauter mean diameter vs. water volume fraction for clean 
systems in laminar flow with treated rotor-stator mill head. All experiments use 
Crystal Oil 500FG. T = 41?,  c = 93.2 cP, ? = 0.0069, and Re: 470-1100. 
The results of these experiments are plotted in Figure (7.2-1) and can be 
compared directly to the data of Figure (7.1-1). They can also be modeled by equation 
(7.1-1). In addition to the increase in the variability of the data, it appears that there is 
no effect of the treatment at dilute volume fractions (? < 0.001), but that at non-dilute 
concentrations the drop size is decreased relative to the untreated mill head data. This 
suggests the possibility of a heterogeneous coalescence mechanism whereby the 
dispersed phase's interaction with the high-shear untreated surfaces promotes 
coalescence. This is based on the idea that the treatment reduces the total amount of 
 
 128 
 
coalescence occurring via interaction with the metal surfaces, thus reducing the drop 
size. 
Figure (7.2-1) provides a quantitative method for comparison via the values of 
the coefficient ?a? displayed in a table within the figure. However, Figures (7.2-2) & 
(7.2-3) provide a graphical contrast of the data to show the effect of the hydrophobic 
treatment. It can be seen that at dilute phase fractions, the mill head treatment has no 
effect, but when the phase fraction increases its effect starts to become manifested. 
Figure (7.2-2) shows the effect of phase fraction with an untreated mill head, and 
Figure (7.2-3) shows the effect of phase fraction with a treated mill head. 
 
Figure 7.2-2: Equilibrium Sauter mean diameter as a function of phase fraction for 
clean systems in laminar flow with an untreated rotor-stator mill head. All 
experiments use Crystal Oil 500FG. T = 41?,  c = 93.2 cP, ? = 0.0069, and Re: 470-
 1100. 
 
 129 
 
 
Figure 7.2-3: Equilibrium Sauter mean diameter as a function of phase fraction for 
clean systems in laminar flow with a treated rotor-stator mill head. All experiments 
use Crystal Oil 500FG. T = 41?,  c = 93.2 cP, ? = 0.0069, and Re: 470-1100. 
7.3 ? Surfactant-Laden Systems 
All of the data was presented in order to understand the relationship of drop 
size with phase fraction in Sections 7.1 & 7.2. However, only a representative sample 
is necessary for understanding the general behavior of surfactant-laden systems. That 
sample is presented in Section 7.3.1, and the rest of the data is provided in Section 
7.3.2. The data presented in Section 7.3.2 does not contribute to the overall 
argumentation of the Chapter, but is useful as data. 
7.3.1 ? Behavior of Surfactant-Laden Systems 
For some experiments an oil-soluble surfactant, Tergitol NP-4, was added, the 
behavior of which in Crystal Oil - water systems having been described in Chapter 2. 
Experiments were performed at initial bulk surfactant concentrations of 0.01xCMC, 
 
 130 
 
0.1xCMC, and 1xCMC (CMC = 1x10-3 M). Since a large amount of interfacial area is 
produced (A = 6?/d32 where A = interfacial area per volume), the bulk surfactant 
concentration decreases as interfacial area increases. No attempt was made to 
quantify the degree of depletion of bulk surfactant.  
 
Figure 7.3.1-1: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 0.1xCMC (equilibrium interfacial tension = 23.8 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
Figure (7.3.1-1) shows the drop size behavior as a function of phase fraction 
for ? ? 0.05 and the initial bulk surfactant concentration is 0.1xCMC. It is seen that 
the drop size data in the presence of surfactant follow the same log-linear behavior as 
both the ?clean? laminar and turbulent flow experiments. Similar plots to Figure 
(7.3.1-1) can be made for the two other surfactant concentrations (0.01xCMC and 
1xCMC) and, while greater amounts of surfactant were found to significantly reduce 
the drop size and decrease the value of a in equation (7.1-1), they all have the same 
 
 131 
 
log-linear functionality. The effectiveness of the surfactant in decreasing the drop size 
is quantified by the values of a shown in Table 7.3.1-1. 
 0xCMC: 
? = 54.8 
mN/m 
0.01xCMC: 
? = 41.4 mN/m  
0.1xCMC: 
? = 23.8 mN/m 
1xCMC: 
? = 6.2 mN/m  
N (rpm) a R2 a R2 a R2 a R2 
3900 4.6 0.98 4 0.97 3 0.96 3.8 0.87 
5600 4.5 0.98 5.4 0.93 3.1 0.94 2.6 0.92 
7300 5.4 0.98 4.1 0.97 3.4 0.90 1.7 0.99 
9000 5.4 0.92 3.8 0.95 3.4 0.90 2.1 0.98 
Average 5.0 - 4.3 - 3.2 - 2.6 - 
St. Dev. 0.49 - 0.73 - 0.21 - 0.91 - 
 
Table 7.3.1-1: The effect of surfactant concentration on the value of a in laminar flow 
experiments. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
From Table 7.3.1-1, it is apparent that the chosen oil-soluble surfactant, 
Tergitol NP-4, is an effective surfactant for Crystal Oil and water emulsions. 
Furthermore, as the surfactant concentration increases, the value of a decreases and 
does so somewhat independently of rotor speed. The variations in the value of a with 
rotor speed are not systematic and arise from scatter in the data. The variations in the 
value of a with surfactant concentration are systematic; the systematic nature of this 
effect can be observed by noting the average value of a for each surfactant 
concentration. 
Further demonstration of the consistent effect of adding surfactant is shown in 
Figure (7.3.1-2). In Figure (7.3.1-2) the drop size as a function of phase fraction for a 
rotor speed of 7300 rpm is normalized with respect to the drop size at the most dilute 
phase fraction, which was ? = 0.001. The rotor speed of 7300 rpm is representative; 
 
 132 
 
similar plots may be made at the other 3 rotor speeds of 3900, 5600, and 9000 rpm. It 
is significant to note that the normalization was effective in Figure (7.3.1-2) for all 
surfactant concentrations, even the higher ones where dynamic interfacial effects 
were observed for dilute experiments in laminar flow (see Figure 6.4-1). 
 
Figure 7.3.1-2: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 7300 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 890. 
It can be concluded that surfactants reduce the interfacial tension, but they do 
not change the underlying phenomena. They facilitate breakage by reducing the free 
energy driving force for cohesion. It appears, for this system, that the surfactant 
lowers the drop size by lowering the interfacial tension in a consistent way, and does 
not necessarily provide a strong physical barrier to coalescence. Such a barrier has 
been reported by Lobo and Svereika (2003) for hexadecane oil-in-water emulsions. 
 
 133 
 
7.3.2 ?Surfactant-Laden Systems? Complete Data 
As aforementioned, Section 7.3.1 was provided sufficient data to illustrate the 
trends of interest for this study. However, in order to verify that these trends are 
representative of all the data more experiments were performed. The analogous 
figures to Figure (7.3.1-2) are Figures (7.3.2-1), (7.3.2-2), & (7.3.2-3). They are for 
the cases when N = 3900 rpm, 5600 rpm, and 9000 rpm, respectively. As Figure 
(7.3.1-2) does, they show that the presence of Tergitol NP-4 in the oil phase has the 
effect of simply lowering the equilibrium interfacial tension and that no other effect is 
manifested by the presence of the surfactant, regardless of its concentration. 
 
Figure 7.3.2-1: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 3900 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 470. 
Figure (7.3.2-1) is taken to illustrate the same principle as Figure (7.3.1-2) 
despite the obvious outlier data point when ? = 0.046 and ? = 1xCMC. This point is 
 
 134 
 
taken to be an outlier simply because it is the only point which breaks the pattern so 
significantly in all of Figures (7.3.1-2), (7.3.2-1), (7.3.2-2), & (7.3.2-3). 
 
Figure 7.3.2-2: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 5600 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 680. 
 
 135 
 
 
Figure 7.3.2-3: Normalized equilibrium Sauter mean diameter vs. water volume 
fraction for clean and surfactant systems in laminar flow. All experiments use Crystal 
Oil 500FG. N = 9000 rpm, T = 43?,  c = 93.3 cP, ? = 0.0066, and Re = 1100. 
The next three figures, Figures (7.3.2-4), (7.3.2-5), & (7.3.2-6), are 
supplementary evidence for the conclusions based on Figure (7.3.1-1) and serve as 
the basis for the construction of Table 7.3.1-1.  Each of them shows that at the 
particular surfactant concentration which it represents that the form of the relationship 
between d32 and ? can be described by Equation (7.1-1). A table is inset within each 
figure which repeats the information provided in Table 7.1-1. 
 
 136 
 
 
Figure 7.3.2-4: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 0xCMC (equilibrium interfacial tension = 54.8 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
Figure (7.3.2-4) is included in this section in order to have all of the 
surfactant-laden data reported within Section 7.3, however, it should be noted that 
0xCMC is the clean situation which was reported in Figure (7.1-1). Therefore, these 
two figures are identical. 
 
 137 
 
 
Figure 7.3.2-5: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 0.01xCMC (equilibrium interfacial tension = 41.4 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
 
 138 
 
 
Figure 7.3.2-6: Equilibrium Sauter mean diameter vs. water volume fraction for 
surfactant systems with 1xCMC (equilibrium interfacial tension = 41.4 mN/m) in 
laminar flow. All experiments use Crystal Oil 500FG. T = 43?,  c = 93.3 cP, ? = 
0.0066, and Re: 470-1100. 
By observing Table 7.1-1 or Figures (7.3.2-4), (7.3.2-5), (7.3.1-1), & (7.3.2-
 6), it can be seen that the value of a in Equation (7.1-1) steadily decreases with 
surfactant concentration. The rate of this decrease appears to follow a trend; this trend 
was plotted in Figure (7.3.2-7) and listed in equation (7.3.2-1).  
        ?        (7.3.2-1) 
 
 139 
 
 
Figure 7.3.2-7: Average slope in Equation (7.1-1) vs. equilibrium interfacial tension 
for surfactant systems in laminar flow. All experiments use Crystal Oil 500FG. T = 
43?,  c = 93.3 cP, ? = 0.0066, and Re: 470-1100. 
It is important to remember that Figure 7.3.2-7 is a sensible figure to construct 
only because there is no systematic variation in a with respect to rotor speed. The 
value of b is too sensitive to errors in the data to make any meaningful equations with 
it. However, even equation (7.3.2-1) is only useful at the particular viscosities at 
which it was constructed:  c = 93.3 cP and  d = 0.62 cP. Although it would require a 
massive experimental effort, it would be useful for future work to construct similar 
plots to Figure (7.3.2-7) at different viscosities. If this was done, then it may be 
possible to derive a mechanistic basis and a more complete formulation of the 
relationship of a with equilibrium interfacial tension. 
 
 140 
 
7.4 ? High Concentration Systems 
The previously presented data are restricted to ? < 0.05. It was not possible to 
achieve complete dispersion for clean systems above this volume fraction. However, 
the addition of a sufficient amount of surfactant allowed the acquisition of data in 
stable dispersions up to ? = 0.46. This was done for laminar flow with the untreated 
mill head at a surfactant concentration of 1xCMC. 
 
Figure 7.4-1: Equilibrium Sauter mean diameter vs. water volume fraction at high 
water phase concentrations for surfactant systems with 1xCMC (equilibrium 
interfacial tension = 6.16 mN/m) in laminar flow. All experiments use Crystal Oil 
500FG. T = 43?,  c = 93.3 cP, ? = 0.0066, and Re: 470-1100. 
As opposed to the preceding figures where ? < 0.05 and was plotted on a log 
scale, Figure (7.4-1) is plotted on a linear scale to give a clear representation of the 
effect of phase fraction over the entire range where ? ? 0.50. Figure (7.4-1) shows 
that d32 follows equation (7.1-1) initially, but then plateaus at higher phase fractions. 
 
 141 
 
This is attributed to the competing rates of breakage and coalescence. The breakage 
rate, as a single drop phenomenon, is approximately independent of phase fraction 
except for the effect of ? on emulsion viscosity. However, breakage rate does 
increase with drop diameter since: 1) drops break if they are too large to be stable in 
the deformation field and 2) drops that are much larger than the threshold of stability 
undergo catastrophic rather than binary breakage (Hinze 1955). 
 The rate of coalescence depends on both phase fraction and drop size. In 
equations (3.4-2) and (3.4-3), increasing the phase fraction significantly increases the 
collision rate. However, increasing the drop diameter decreases both the collision rate 
in equations (3.4-2) and (3.4-3) and the collision efficiency in equation (3.4-6). The 
interpretation of Figure (7.4-1) is that, at first, increasing ? only increases the 
collision rate, but as drop diameter starts to increase, both collision rate and collision 
efficiency decrease and the breakage rate increases. 
For water-in-oil dispersions produced in a Rushton turbine stirred tank, Boxall 
et al. (2012) reported that equilibrium mean drop size was independent of ? for water 
phase fractions of 0.1 < ? < 0.35, which is similar to the range of water phase 
fractions in this study for which there is no discernable systematic effect of phase 
fraction. These authors stated that their system was free of coalescence, but the data 
of Figure (9) call this assumption into question, since they did not investigate the 
lower phase fraction range. 
7.5 ? Tests for Drop Adherence on Low-Shear Surfaces 
One concern that was previously mentioned is the possible adherence of water 
drops to the low-shear surfaces that make up the interior of the mixing volume. This 
 
 142 
 
sticking phenomenon could potentially affect the DSD if drops are stripped off, 
causing a non-equilibrium DSD to be measured. It could also affect the measured 
DSD if it changed the phase fraction since the DSD for ? ? 0.1 is strongly dependent 
on the phase fraction. In order to investigate this, changes were made to the 
hydrophobic treatment, the amount, and the type of materials that were present as 
low-shear surfaces within the mixing volume. 
Several different configurations were tested. Some data sets used an untreated 
vessel, some used a vessel that is cylindrical, one used an untreated cylinder of 
stainless steel wrapped around the inside of the mixing vessel, and one involved the 
presence of a ?radiator? apparatus (pictured in Figure (7.5-1)). The purpose of this 
apparatus was to maximize any possible drop adsorption to low shear surfaces. It was 
created by gluing untreated (to maximize hydrophilicity) microscope slides with 
spacers together. 
 
 143 
 
 
Figure 7.5-1: ?Radiator? apparatus. The large amount of hydrophilic surface area 
provided by gluing microscope slides together in such a fashion should serve to 
maximize any possible drop adherence to low shear surfaces. 
 
 144 
 
 
Figure 7.5-2: Effect of modifying interior, low shear surfaces. Very little effect is 
observed ? this invalidates the idea that possible drop sticking on the walls plays a 
significant role in the determining the equilibrium drop size. All experiments use 
clean Crystal Oil 500FG. T = 36?,  c = 132.7 cP, ? = 0.0053, and Re: 330-770. 
Figure (7.5-2) shows the results of these tests for ? = 0.001. The open 
diamond corresponds to the configuration that was used in all of the previously 
reported experiments. The differences in the data sets on Figure (7.5-2) are similar to 
typical scatter for this type of measurement. There are significant changes made in the 
interior low-shear surfaces present in these experiments, yet there is no corresponding 
large change in the equilibrium d32. It could be argued that the ?radiator? experiment 
(open circles) shows a slightly smaller drop size, but even if that is true, such a 
difference is insufficient to be a major cause in the functionality of the drop size?s 
dependence on phase fraction manifested in equation (7.1-1).  
 
 145 
 
Chapter 8: In-line Mixer Power Draw Experiments 
There are two portions to the IKA Labor Pilot 2000/4 study which are 
discussed in this and the following chapter to provide insight into in-line rotor-stator 
mixers. This Chapter is concerned with the single-phase power draw study, which is 
analogous to Chapter 5 except that it is for an in-line mixer, and Chapter 9 discusses 
the drop size study which is similar to Chapters 3, 6 & 7. The drop size work was not 
performed by the author; rather, the results were received as a personal 
communication from Murthy (2010). The drop size data is not the focus of this 
dissertation and is only included because it can be combined with the power draw 
study which is the author?s original work. The power draw work is covered in 
complete detail and the measurements are used to interpret Muthy?s (2010) drop size 
data in Chapter 9. 
8.1 ? In-line Theory 
8.1.1 ? Power Draw 
Much of the theory of power draw and flow regime determination is provided 
in Chapter 5. Rushton?s (1950) power draw data are shown in Figure 5.1-1 for the 
case of various turbine geometries in conventional stirred tanks. Padron?s (2001) 
power draw data is shown in Figure 5.2.2-1 for the specific case of a batch Silverson 
L4R mixer. Although those were for batch processes, the same ideas are present in 
the theory of power draw for in-line mixers. 
 
 146 
 
The most significant difference between in-line and batch flow is that for in-
 line mixers the flowrate is controlled independently from the rotor speed (Zhang et 
al., 2012). This can be understood by considering that for an in-line mixer fluid is 
pumped into the mixer by a pump unit and that the pump can be set to provide the 
desired pumping head. By contrast, in a batch mixer there is no fluid movement at all 
without the action of the impeller or rotor. This is why the power draw can be related 
to the rotor speed (along with fluid properties) alone in a batch mixer whereas for an 
in-line mixer an additional term contributes to the power draw which arises from the 
flow work done on the liquid (Kowalski, 2009). 
                              (8.1.1-1) 
where Pmeas is the total power draw measured by the motor, Pdiss is the viscous power 
dissipation from shearing the fluid, Pflow work is the power imparted to the fluid as flow 
work, and Pbear is the power lost by the motor due to bearings, vibrations, and other 
inefficiencies. 
It is not possible to eliminate Pflow and Pbear completely, but it is generally 
desirable to minimize their contribution to the total power draw since only Pdiss 
contributes to drop breakup and local mixing. However, given a particular mixer it is 
not possible to alter Pbear and it is difficult to influence Pflow work given a desired set 
of operating parameters. It has been found (Kowalski et al., 2011) that the efficiency 
of a mixer as a pump is a function of flowrate, meaning that the magnitude of Pflow is 
a function of flowrate. Since Pdiss is the term of interest for emulsification processes, 
yet P is the quantity that is measured, this implies that Pflow must be measured 
 
 147 
 
separately at every flowrate for a particular mixer. The value of Pflow can then be 
subtracted from P to obtain Pdiss. 
Pdiss can be related to the power per unit mass, ?, by defining an appropriate 
region in which the energy is dissipated. Finding the volume of that region has been 
done in various experimental studies such as those of Kresta and Wood (1991) and 
Zhou and Kresta (1996) by measuring ? through Laser Doppler Anemometry (LDA) 
or other methods. Unfortunately, to obtain the values of ?, experiments need to be 
performed on each type of mixer at each flow condition individually. When this has 
not been done for the particular mixer of interest, the best option is to use an analogy 
to the swept-volume approach of McManamey (1979) to approximate the region in 
which the energy is dissipated. 
The above discussion is useful for finding the energy dissipation rate which is 
itself useful in emulsification studies, because it has been found to be correlated to the 
maximum stable drop size across a variety of mixing devices (Davies, 1987). Davies 
published a plot of maximum stable drop size versus local energy dissipation rate for 
the inertial subrange (d > ?) which is shown in Figure (8.1.1-1). The remarkable 
feature of Figure (8.1.1-1) is that regardless of the geometry, or even the mechanism 
of the mixer, maximum stable drop size may be scaled by the maximum local energy 
dissipation rate alone provided that the drops are larger than the Kolmogorov 
microscale (given by equation (3.1.1-3)). The nature of this relationship is described 
by equation (8.1.1-2).  
         ?
          (8.1.1-2) 
 
 148 
 
 
Figure 8.1.1-1: Maximum stable drop size vs. maximum local energy dissipation rate 
for a variety of types of mixers. From Leng and Calabrese (2004).  
One clarification concerning Figure (8.1.1-1) is that the interfacial tension is 
equal to 30 mN/m. For the data that did not have an interfacial tension of 30 mN/m, a 
correction was employed such that it was as though the interfacial tension was 30 
mN/m. 
It is also important to note that the x-axis of Figure (8.1.1-1) is the local ? 
rather than the average ?. This distinction is based on the volume over which the 
energy is considered to have been dissipated. The average energy dissipation rate is 
calculated by considering a volume over the whole space where fluid is present, 
whereas the maximum local energy dissipation rate only accounts for energy 
dissipated very close to the ?high-shear? region of interest. 
 
 149 
 
The correlation of drop size with energy dissipation rate is rightly performed 
based on the local energy dissipation rate rather than the average energy dissipation 
rate (Zhou and Kresta, 1998). This is due to the fact that drop breakup occurs when a 
threshold intensity of the shear stress is reached. This makes it desirable to 
concentrate all of the energy into as small a volume as possible if the goal is to 
achieve the smallest drops possible. High energy dissipation density is characteristic 
of rotor-stator mixers and explains their use in emulsification processes. Most of the 
energy that is dissipated in a rotor-stator mixer is dissipated near the mixing head 
with relatively small amounts of energy being consumed in mild recirculation far 
from the mixing head (Yang, 2011). Therefore, most of the energy inputted into a 
rotor-stator mixer is spent increasing the maximum shear stress which results in 
smaller drop size distributions per inputted power. 
8.1.2 ? Single Pass Drop Size 
It is not the purpose of this work to perform a detailed review of single pass 
drop size data. This section briefly addresses some of the most basic distinctions; 
however, since the drop size data was not acquired by the author and rather received 
as a personal communication from Murthy (2010), this is not intended to be a full 
review of the matter. 
In a batch mixer the energy dissipated in the disruptive stresses (the 
turbulence intensity for the case of turbulent flow) which cause drop breakup is 
unevenly spatially distributed throughout the mixer. This means that the equilibrium 
drop size is not reached until every drop which is larger than dmax has experienced the 
region of maximum local ? (Leng and Calabrese 2004). In a batch mixer this criterion 
 
 150 
 
may be satisfied if the mixer is allowed to operate for a sufficient amount of time. 
The time to reach equilibrium has been reported to be several hours for various 
systems (Chen and Middleman 1967; Arai et al. 1977; Wang and Calabrese 1986). 
By contrast, in-line rotor stator mixers have residence times often on the order 
of seconds or tens of seconds. Furthermore, depending on the flowrate and the flow 
field, when passing through an in-line mixer drops may only experience the high 
shear region once before exiting the mixer. Therefore, the drop size distributions 
processed by an in-line mixer have often not reached equilibrium. 
However, it is possible for drops to have a long enough residence time in an 
in-line mixer to reach an equilibrium state, or in dilute cases in which there is no 
coalescence, an ultimate state. One example of a system in which drops sometimes 
reach an equilibrium state and at other conditions do not have sufficient time is a 
transient drop size study performed by Chang (1990). In this study paraffin oil was 
dispersed in water in a conventional stirred tank system. Drop size was photographed 
through the walls as the experiment progressed. The normalized transient drop size 
distributions are plotted in Figure (8.1.2-1). Figure (8.1.2-1) shows that there is a 
kinetic regime in which drop size decreases quickly with time, but that if the 
experiment is allowed to progress a steady state ?equilibrium? condition is reached. 
 
 
 151 
 
 
Figure 8.1.2-1: Illustration of the kinetic vs. equilibrium region for DSDs in a 
conventional stirred tank. Source: Chang (1990).  
 
 
 
 152 
 
8.2 ? In-line Experimental Procedure 
8.2.1 ? Mixer Details 
The mixer used in all of Chapter 8 is the IKA Labor Pilot 2000/4 in-line 
mixer. This mixer is an in-line rotor stator mixer which has a 3 phase 3 hp motor 
mounted on a base with rubber feet. This motor is connected by a rubber belt to an 
axle which spins the rotors. The rotors are located within the stators which are held in 
place also on the base. The mixer is pictured in Figure (8.2.1-1). 
 
Figure 8.2.1-1: IKA Labor Pilot 2000/4 in-line mixer. Motor on the left, mixer 
volume on the right. 
 
 153 
 
A closer look at the mixer volume is shown in Figure (8.2.1-2).  
 
Figure 8.2.1-2: IKA Labor Pilot 2000/4 in-line mixer. Motor on the left, mixer 
volume on the right. 
When the mixer was run for the drop size experiments it was placed on its 
side and when it was run for the power draw experiments it was placed vertically as is 
the case in Figure (8.2.1-1) and (8.2.1-2). The inlet is on the top and the outlet is on 
the side. The fluid is pumped by a MD 05-6LT Seepex progressive cavity pump and 
 
 154 
 
forced to pass through 1 to 3 rotor-stator stages. Either 1 or 3 of these stages of 
varying geometries may be placed between the inlet and the outlet. In order to have 2 
stages a so-called blank stage is substituted in place of the 1st stage. A blank stage is 
one which has no rotor or stator and simply serves to take space in the frame of the 
mixer. The only reason for this is that there were only connecting posts available of 
lengths capable of supporting 1 or 3 stages. The individual stages, called generators, 
which are the rotors and stators, are shown in Figures (8.2.1-3), (8.2.1-4), and (8.2.1-
 5). These figures show the medium, fine, and ultrafine generators respectively. 
 
Figure 8.2.1-3: Medium generator. Two rows of teeth on both the rotor and stator. 
 
 155 
 
 
Figure 8.2.1-4: Fine generator. Three rows of teeth on both the rotor and stator. 
 
Figure 8.2.1-5: Ultrafine generator. Four rows of teeth on both the rotor and stator. 
Unfortunately, due to considerations in the ease of fabrication, the different 
types of generators do not share identical slot widths, spacing and other dimensions. 
This must be taken into account when performing the analysis of the results. The 
dimensions and parameters of the various geometries are provided in Table 8.2.1-1. 
 
 156 
 
Generator Rows Slots 
 Slot 
Width 
(mm) 
Slot 
Height 
(mm) 
Shear 
Gap 
(mm) 
1st 
Row  
Diam 
(cm) 
2nd 
Row  
Diam 
(cm) 
3rd 
Row  
Diam 
(cm) 
4th 
Row  
Diam 
(cm) 
Medium 
 
2 
 
9 
 
1.575 
 
3.5 
 
0.5 
 
47 
 
57 
 
- 
 
- 
Fine 
 
3 
 
13 
 
0.965 
 
3.5 
 
0.5 
 
37 
 
47 
 
57 
 
 
- 
Ultrafine 
 
4 
 
22 
 
0.72 
 
3.5 
 
0.5 
 
34.5 
 
43 
 
52 
 
 
60 
Table 8.2.1-1: Mixer geometries? physical dimensions.  
As aforementioned, when the effect of 2 stages is desired, a blank stage is 
substituted for the first stage. Clearly, none of the parameters of Table 8.2.1-1 apply 
to the situation of a blank stage. A blank stage is shown next to an ultrafine stage for 
comparison in Figure (8.2.1-6). For a blank stage there is no rotor and no stator; there 
is only the outer metal which acts as a spacer. 
 
Figure 8.2.1-6: Blank stage as compared to an ultrafine stage. 
 
 157 
 
8.2.2 ? Power Draw 
To measure the power draw of the IKA Labor Pilot 2000/4 in-line mixer a 
Futek TRS600 (#FSH01995, range 2 N?m) torque sensor was installed in-line with the 
rotor shaft. The principle of this device is that the torque is the same at all points 
along the rotor shaft between the fluid and the location where the rotor shaft touches 
the belt connecting it to the motor. This is schematically illustrated in Figure (8.2.2-
 1). 
 
Figure 8.2.2-1: Principle of the operation of the torque meter. The torque meter was 
installed in-line on the rotor shaft. 
The sensor reports a voltage which is linearly related to the torque by a 
constant provided by the manufacturer. The full range of the this model of the torque 
meter is 5.00 V and 2 N?m. Therefore, the conversion constant from torque to voltage 
is 0.4 N?m/V. A sample calculation of the toque based on the voltage is shown in 
equation (8.2.2-1). 
               
 ? 
 
        ?    (8.2.2-1) 
The power may then be obtained from the torque by according to equation 
(8.2.2-2). 
 
 158 
 
             (8.2.2-2) 
It was important to choose a sensor with a sufficiently low upper bound so 
that the sensitivity was great enough to distinguish at the range of interest. 2 N?m was 
the maximum range of the sensor. Most of the available torque sensors on the market 
had a much greater range and were not sufficiently sensitive at the levels of torque 
used in this study. The appropriate size for the torque sensor was determined by 
measuring the maximum level of torque for the conditions of this study using a torque 
sensor of size 20 N?m. The rough accuracy of these torque sensors was verified both 
by the heat balance and by hanging weights off of a lever arm which was fixed to the 
shaft when the meter was not installed and was held stationary. In the latter case the 
torque was known and could be calculated by equation (8.2.2-3). 
            (8.2.2-3) 
where ? is the torque, w is the weight, and r is the lever arm. 
The installed torque meter is shown in Figure (8.2.2-2). This may be 
compared with Figure (8.2.1-1) to see the difference in the overall configuration with 
and without the torque meter. 
 
 159 
 
 
Figure 8.2.2-2: IKA Labor Pilot 2000/4 in-line mixer with Futek TRS600 torque 
meter installed in-line with the rotor shaft. 
It may also be seen in Figure (8.2.2-2) that a differential pressure gage (Dwyer 
Series 629) and two thermocouples (one at the inlet and one at the outlet) were 
installed. The pressure gage was installed to facilitate the calculation of the flow work 
that was either performed on the fluid or extracted from it (this depends on flowrate 
and rotor speed). The thermocouples were installed so that the energy dissipation rate 
 
 160 
 
in the fluid could be calculated calorimetrically. Calorimetric calculation of the power 
draw serves as an alternative to torque measurement, though either one may be used 
to calculate the energy dissipation rate (Cooke et al. 2011). 
Unfortunately the method of finding the power draw through a heat balance 
was not very precise and so it only served to verify the results of the torque sensor. 
The calorimetric calculation was done via equation (8.2.2-4).  
      ?            (8.2.2-4) 
where ? is the fluid density (single phase), Q is the volumetric flowrate, CV is the heat 
capacity, and ?T is the temperature difference between the inlet and the outlet 
measured as measured by the thermocouples. In equation (8.2.2-4), it is possible that 
some contribution to ?T may be from energy dissipated in flow work. However, it 
will be shown that the contribution of flow work is small (compare the axes of Figure 
(8.3.1-1) and (8.3.2-2) through (8.3.2-4)). 
The parameters necessary to perform the heat balance were measured along 
with the torque meter at a variety of experimental conditions. The geometries which 
were measured were 1 blank stage, 1 medium stage, 2 medium stages, 3 medium 
stages, 1 fine stage, 2 fine stages, 3 fine stages, 1 ultrafine stage, 2 ultrafine stages, 
and 3 ultrafine stages. The flowrates at which power was measured were 1 lpm (liters 
per minute), 2 lpm, 3 lpm, 4 lpm, and 5 lpm. Every combination of flowrate and 
geometry was measured except for cases where the amount of pumping done by the 
mixer would have caused a flowrate greater than the targeted value and artificial 
pressure increases would have caused the overall pressure in the system to be too 
large. 
 
 161 
 
To obtain the data, the pump was turned on and then the mixer. The torque 
was simply read from the voltmeter and then the next data point was collected. All of 
the data for each type of geometry was collected, and then the geometry was changed 
to the next type. Within each geometry?s data set, the smallest rotor speed, 4000 rpm, 
was performed first at each flowrate, and then the rotor speed was increased. This was 
done because the variable frequency drive of the mixer motor allowed for the mixer?s 
speed to be specified independently from the power draw. By contrast, the flowrate 
driven by the progressive cavity pump was a function of rotor speed. Since it changed 
with every data point anyway, it was controlled with a downstream valve for each 
data point (the pump?s power acted as a coarse tuner, but the fine tuning was done 
with the downstream valve). 
The rotor speed was read from a digital display on the variable frequency 
drive with which the IKA Labor Pilot 2000/4 in-line mixer was fitted. After setting 
the approximate flowrate and the rotor speed, the flowrate was tuned via a pressure 
valve downstream and was measured by a Great Plains Industries, Inc. flow meter 
(model #A109GMN025NA1). All flowrates were within 0.04 lpm of their nominal 
rates. As previously mentioned, the temperature difference was measured by inlet and 
outlet thermocouples and the pressure drop was measured by a differential pressure 
gage. 
As in the ?Acknowledgements? section at the beginning of this document, the 
author would like to thank Mr. Scott Anderson of IKA? for his work in modifying the 
torque meter so that it could be installed in-line with the IKA Labor Pilot. 
 
 162 
 
8.3 ? Power Draw Results & Analysis 
In the analysis of power draw it is essential to accurately account for the 
various ways in which power is inputted or removed from the system. The power 
which is calculated from the measurements done by the torque meter is influenced by 
three separate terms as expressed in equation (8.1.1-1). 
8.3.1 ? Mixer Pumping 
One other consideration of practical interest is how much the mixer pumps or, 
alternatively, how much it causes pressure to decrease due to its presence in a flow 
line. One issue that is not reflected in equation (8.1.1-1) is that some viscous 
dissipation of energy takes place even if the mixer is off, (Q?P)0. This energy is still 
viscously dissipated when the mixer is on and it represents the energy that is lost due 
to the fluid being required to move through the tortuous pathways necessitated by the 
rotor-stator geometry. (Q?P)0 is negative by convention since the inlet pressure is 
always less than the inlet pressure when the mixer is off. (Q?P)0 is plotted as a 
function of flowrate for all 9 geometry configurations as well as for the case of a 
blank stage in Figure (8.3.1-1). 
 
 163 
 
 
Figure 8.3.1-1: Viscous power dissipation when the mixer is off (the rotor-stator 
geometry acts as a static mixer in this case). 
In Figure (8.3.1-1) it is seen that the viscous power dissipation increases with 
flowrate and with the number and of stages and rows of teeth per stage. 
The criterion for the mixer?s contribution to pumping, whether it is providing 
flow work to the fluid, is given by equation (8.3.1-1). 
  ?     ?         (8.3.1-1) 
If equation (8.3.1-1) > 0, then the mixer is providing flow work to the fluid 
equal to the difference from zero. If equation (8.3.1-1) < 0, then the mixer is 
removing flow work from the system equal to the difference from zero.  
Since (Q?P)0 is always negative or near zero by convention, it increases the 
value of Q?P - (Q ?P)0. However, (Q?P) can be negative and greater in magnitude 
than (Q ?P)0 which is the case when the mixer is extracting flow work from the 
 
 164 
 
system rather than providing it. Figures (8.3.1-2) through (8.3.1-10) are plotted 
below. These figures show how much pumping is being performed by the mixer at 
every flowrate and for every geometry. 
 
Figure 8.3.1-2: Contribution to flow work by the mixer for 1 medium stage as a 
function of flowrate and rotor speed. 
 
 165 
 
 
Figure 8.3.1-3: Contribution to flow work by the mixer for 1 fine stage as a function 
of flowrate and rotor speed. 
 
Figure 8.3.1-4: Contribution to flow work by the mixer for 1 ultrafine stage as a 
function of flowrate and rotor speed. 
 
 166 
 
 
Figure 8.3.1-5: Contribution to flow work by the mixer for 2 medium stages as a 
function of flowrate and rotor speed. 
 
Figure 8.3.1-6: Contribution to flow work by the mixer for 2 fine stages as a function 
of flowrate and rotor speed. 
 
 167 
 
 
Figure 8.3.1-7: Contribution to flow work by the mixer for 2 ultrafine stages as a 
function of flowrate and rotor speed. 
 
Figure 8.3.1-8: Contribution to flow work by the mixer for 3 medium stages as a 
function of flowrate and rotor speed. 
 
 168 
 
 
Figure 8.3.1-9: Contribution to flow work by the mixer for 3 fine stages as a function 
of flowrate and rotor speed. 
 
Figure 8.3.1-10: Contribution to flow work by the mixer for 3 ultrafine stages as a 
function of flowrate and rotor speed. 
 
 169 
 
Figures (8.3.1-2) through (8.3.1.10) show that with fewer rows of teeth the 
mixer always pumps, particularly at high rotor speeds. With more rows of teeth, the 
mixer pumps at higher rotor speeds, but extracts flow work at lower rotor speeds, 
particularly at higher flowrates. The local minima sometimes experienced by these 
curves indicate competing effects of rotor speed. The intersections between curves at 
different flowrates similarly indicate competing effects of flowrate. 
8.3.2 ?Dissipated Power 
To account for frictional losses in the bearings and seals of the mixer, tests 
were run with the blank stage pictured in Figure (8.2.1-6) put in the place of the 
generators. 
 
Figure 8.3.2-1: Power measured by the torque sensor with 1 blank stage. This power 
does not contribute to the rate of viscous energy dissipation and must be subtracted 
from the measured power in future experiments. 
 
 170 
 
In this configuration there was very little power dissipation in the fluid, so all 
of the power measured by the torque sensor was taken to be spent on mechanical 
work to overcome friction in the bearings and other sources of energy loss outside the 
fluid volume. This power cannot be included in the rate of energy dissipation in the 
fluid which is used to correlate the strength of the flow fields which cause drop 
breakup. 
The values of power obtained from the torque sensor with a blank stage were 
subtracted from the readings in the cases where generators were used to find the 
power dissipated in the fluid. In order to also account for the effect of flow work, the 
calculation of the energy dissipated viscously in the fluid volume was done according 
to equation (8.3.2-1) which is a modified version of equation (8.1.1-1) based on the 
same principles. 
                                                (8.3.2-1) 
where Pdiss is the viscous power dissipation from shearing the fluid, Pmeas is the total 
power draw measured by the sensor, Pblank is the power measured with a blank stage 
as shown in Figure (8.3.2-1), Pflow work is the power imparted to the fluid as flow work, 
and Pblank flow work is the power imparted to the fluid as flow work when there is a blank 
stage. 
In equation (8.3.2-1), the flow work terms are quite small compared to the 
other terms. This is a desirable result because it means that, except for pump sizing 
and pressure concerns, little regard needs to be given to the amount of pumping. The 
small effect of pumping may be seen generally by comparing the magnitudes of the y-
 axes on the figures in Section 8.3.2 with the magnitudes of the y-axes on the figures 
 
 171 
 
in Section 8.3.1. However, the application of equation (8.3.2-1) is shown in Figures 
(8.3.2-2) through (8.3.2-5) for 3 sample cases to make this point clear. It is not shown 
for every case because that would necessitate showing 45 plots here, which is 
unnecessary since the only major contributors to Pdiss are Pmeas and Pblank. 
 
Figure 8.3.2-2: Power dissipated by the fluid as calculated by the measured power, 
the blank stage power, the flow work, and the blank flow work. 1 medium stage at a 
flowrate of 1lpm. 
 
 172 
 
 
Figure 8.3.2-3: Power dissipated by the fluid as calculated by the measured power, 
the blank stage power, the flow work, and the blank flow work. 2 fine stages at a 
flowrate of 3 lpm. 
 
Figure 8.3.2-4: Power dissipated by the fluid as calculated by the measured power, 
the blank stage power, the flow work, and the blank flow work. 3 ultrafine stages at a 
flowrate of 5 lpm. 
 
 
 173 
 
The power dissipated in the fluid was calculated similarly for all 45 cases (9 
generator configurations and 5 flowrates at a range of rotor speeds) and the results are 
plotted in Figures (8.3.2-5) through (8.3.2-13). The calorimetric results obtained 
using the inlet and outlet thermocouples along with equation (8.2.2-4) are plotted 
alongside the power measurements obtained from the torque sensor. There is 
significant error associated with the calorimetric method, mostly arising from the 
small temperature differences across the mixer and the lack of truly adiabatic 
conditions. The error was higher at lower levels of energy dissipation because the 
temperature difference in those cases was often a few tenths of a degree. The 
calorimetric results are included only as verification of the order of magnitude of the 
torque sensor. The calorimetric data does generally correspond to the torque data. All 
negative values are errors and should be ignored. 
 
 174 
 
 
Figure 8.3.2-5: Power dissipated as a function of rotor speed for various flowrates 
using 1 medium stage. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
 
 175 
 
 
Figure 8.3.2-6: Power dissipated as a function of rotor speed for various flowrates 
using 1 fine stage. Filled data points indicate the torque measurements and hollow 
data points indicate calorimetric measurements. 
 
Figure 8.3.2-7: Power dissipated as a function of rotor speed for various flowrates 
using 1 ultrafine stage. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
 
 176 
 
 
Figure 8.3.2-8: Power dissipated as a function of rotor speed for various flowrates 
using 2 medium stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
 
Figure 8.3.2-9: Power dissipated as a function of rotor speed for various flowrates 
using 2 fine stages. Filled data points indicate the torque measurements and hollow 
data points indicate calorimetric measurements. 
 
 177 
 
 
Figure 8.3.2-10: Power dissipated as a function of rotor speed for various flowrates 
using 2 ultrafine stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
 
Figure 8.3.2-11: Power dissipated as a function of rotor speed for various flowrates 
using 3 medium stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
 
 178 
 
 
Figure 8.3.2-12: Power dissipated as a function of rotor speed for various flowrates 
using 3 fine stages. Filled data points indicate the torque measurements and hollow 
data points indicate calorimetric measurements. 
 
Figure 8.3.2-13: Power dissipated as a function of rotor speed for various flowrates 
using 3 ultrafine stages. Filled data points indicate the torque measurements and 
hollow data points indicate calorimetric measurements. 
 
 179 
 
Figures (8.3.2-5) through (8.3.2-13) show that increasing the number and rows 
of teeth increases the power dissipation. An interesting result is the approximate 
independence of power dissipation with respect to flowrate. This is a very useful 
observation when correlating the power draw data. It was also found that the data 
could be correlated by number of stages (e.g. 2 stages of the same type, the power 
was twice as great as when there was 1 stage of that type). This is demonstrated in 
Figure (8.3.2-14) which is the energy dissipation per number of stages averaged over 
all flowrates as a function of rotor speed (averaging the power draw over all flowrates 
is appropriate because power dissipation is approximately independent of flowrate; it 
would also have been appropriate to use a sample flowrate). 
 
Figure 8.3.2-14: Power dissipated per stage averaged over all flowrates. For each 
generator type, the power dissipated per number of stages is independent of the 
number of stages. 
 
 180 
 
Since the curves fall on top of each other for each generator type in Figure 
(8.3.2-14), it may be considered the design equation for this mixer. This is, in itself, a 
very significant result because it clearly demonstrates the effect of adding each 
additional stage. It is also interesting to note that there is a much greater marginal 
difference in power dissipation for the ultrafine generator as opposed to the medium 
and the fine. However, further correlation of the power draw data is possible. 
By analogy to Figures (5.1-1) and (5.2.2-1), the Power number is plotted 
against the Reynolds number to investigate flow regime in Figure (8.3.2-15). The 
appropriate characteristic diameter is unknown and so in Figure (8.3.2-15) the 
diameter of the outer row of teeth on the rotor is used (the diameter issue is explored 
further in Figure (8.3.2-16)). 
 
Figure 8.3.2-15: Power number vs. Reynolds number for the IKA 2000/4 in-line 
mixer with water as the fluid using outer rotor diameter for D. 
 
 181 
 
In Figure (8.3.2-15), all flowrates are included which accounts for the scatter 
observed at low flowrates for the same number and type of generator. It is difficult to 
comment extensively on the behavior of the Power number at lower Reynolds 
number, but at higher Reyolds numbers, starting around Re = 250000, there is a 
definitive trend of leveling off towards a constant for each generator configuration. It 
is also clear that the value of that constant is a stronger function of the number of 
stages than it is of the type of stage. 
Since it is known from Figure (8.3.2-14) that the power per number of stage is 
constant, Figure (8.3.2-15) should be modified to account for that method of scaling 
the power. This makes the value of the Power number in turbulent flow as a function 
of Reynolds number independent of the number of stages. Additionally, it is possible 
to empirically determine diameters which collapse the data to make them independent 
of generator type. This is shown in Figure (8.3.2-16). 
 
 182 
 
 
Figure 8.3.2-16: Power number vs. Reynolds number normalized by number of 
stages and using a modified ?equivalent? diameter. This yields a constant Power 
number in turbulent flow. 
In Figure (8.3.2-16), the ?equivalent? diameter is defined by reference to the 
outer diameter of the medium rotor. The fine and ultrafine rotor diameters have been 
modified by a constant, K, displayed on Figure (8.3.2-16). Because of the exponents 
on Deq, these small values of K change the Power numbers of the fine and ultrafine 
generators by around 22% and 36% respectively, and the Reynolds numbers by 8% 
and 13% respectively. 
It is emphasized that the definition of the ?equivalent? diameter as being the 
outer diameter of the medium generator (which happens to be the same as that of the 
fine generator) is completely arbitrary. An arbitrary definition was necessary since it 
was not possible to correlate the various generator types on a theoretically justifiable 
 
 183 
 
basis. This was because the teeth widths, number of teeth per row, and diameters vary 
inconsistently across varying generator types, as seen in Table 8.2.1-1. However, the 
values of K found in Figure (8.3.2-16) can be justified because of their consistency 
across all three numbers of generators. Therefore, the values of K are only used to 
collapse the data to a single curve in the turbulent regime. The actual value of the 
constant,        = 0.083, is only as valid as the arbitrary definition of D being the outer 
rotor diameter of the medium generator. In the opinion of the author it would be 
necessary to acquire a set of generators in which the dimensions, number of teeth, and 
diameters were consistent in order to determine the true characteristic diameter. 
 
 184 
 
Chapter 9: In-line Mixer Power Draw Comparison with Drop 
Size Data 
Chapter 8 is concerned with single-phase power draw, which is analogous to 
Chapter 5 except that it is for an in-line mixer, and this chapter discusses the drop size 
study which is similar to Chapters 3, 6 & 7. The drop size work was not performed by 
the author; rather, the results were received as a personal communication from 
Murthy (2010). Murthy used a phase Doppler anemometer (PDA) system to measure 
in-line drop sizes of oil in water. Cursory information is provided in this chapter 
concerning the details of Murthy?s work; however, drop size measurements are not 
the focus of this dissertation and are only included because the drops size data can be 
combined with the power draw study which is the author?s original work. 
9.1 ? Single Pass Drop Size Measurement 
Drop size data was received as a personal communication from Murthy (2010) 
who employed a Dantec FiberPDA Phase Doppler Anemometer (PDA) system 
(consisting of: 57X40 receiving optics, 58N70 FiberPDA detector unit, and a 58N80 
MultiPDA signal processor) to measure the drop sizes of Cargille Immersion Oil 
Type A dispersed in water using this mixer. The method of particle size detection 
through PDA is reviewed generally by Tropea et al. (2007). The oil has a density of 
0.93 g/cm3 viscosity of 163 cP, and an interfacial tension of 30 mN/m. 
When the drop size measurements were being performed the mixer was set up 
sideways relative to the way it was when measuring power draw in Figure (8.2.2-2). 
Also, the torque meter was not installed. Finally, a glass tube was installed on the exit 
 
 185 
 
with squared walls providing a window into the interrogation region. The drop size 
experimental setup is pictured in Figure (9.1-1). 
 
Figure 9.1-1: IKA Labor Pilot 2000/4 in-line mixer with Dantec ?FiberPDA? PDA 
system used to measure drop sizes after they exit the mixer. 
The experimental procedure was to use the PDA device to measure the 
daughter drop sizes resulting from the breakup of a single large parent oil drop after it 
passed through the mixer (see Figure 9.1-1). About 15  L of the oil is injected by 
syringe into the plumbing system upstream of the mixer. The oil is carried along by 
the continuous phase water flow (which is being fed by progressive cavity pump as 
was also the case for the power draw measurements) until it enters the mixer where it 
is finely dispersed. Daughter drops exit the mixer at various times, with drop size 
often decreasing as residence time increases. 
 
 186 
 
After leaving the mixer, drops pass through the interrogation window of the 
glass exit tube where two laser beams intersect. These laser beams were split from a 
single beam and shined at nearly parallel angles through different points in the 
transmitter. There is a 40 MHz difference between these two beams? frequencies 
which intersect within the optical cell to create fringes of constructive interference. 
Two detectors were present in the receiver which record the frequency of the light 
which is reflected by passing drops, which has been Doppler shifted by an amount 
proportional to the particle?s velocity. Multiple detectors allow the drop?s diameter to 
be calculated from this information based on the offset of the time at which the 
different detectors record the presence of the drop and its velocity. 
After one 15  L parent drop has been allowed several minutes to ensure that 
all daughter drops of oil have left the mixer, another burst is performed. For each 
DSD at least 10,000 drops were acquired to ensure the accuracy of this method. 
The DSD was measured for the following geometries: 1 medium stage, 2 
medium stages, 3 medium stages, 1 fine stage, 2 fine stages, 3 fine stages, 1 ultrafine 
stage, 2 ultrafine stages, and 3 ultrafine stages. These geometries were combined in 
every possible way with the flowrates: 1 lpm, 3 lpm, and 5 lpm, and with the rotor 
speeds: 4000 rpm, 5000 rpm, 6000 rpm, 7000 rpm, and 8000 rpm. 
9.2  ? Drop Size Results & Correlation with Power Draw 
In Section 9.2.1, Murthy?s (2010) drop size data are reported on a time-
 averaged basis; the d32 values and the number and volume frequencies reported 
therein are based on drops measured at the outlet over all times. In Section 9.2.2, 
 
 187 
 
transient effects are considered; that is, the effect of residence time on the size of the 
size of the drops is considered. 
9.2.1 ? Time-Averaged Drop Size Data 
Figures (9.2.1-1) through (9.2.1-9) report the Sauter mean diameter as 
obtained by Murthy using the procedure outlined in Section 9.1. 
 
Figure 9.2.1-1: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 1 medium stage. 
 
 188 
 
 
Figure 9.2.1-2: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 1 fine stage. 
 
Figure 9.2.1-3: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 1 ultrafine stage. 
 
 189 
 
 
Figure 9.2.1-4: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 2 medium stages. 
 
Figure 9.2.1-5: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 2 fine stages. 
 
 190 
 
 
Figure 9.2.1-6: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 2 ultrafine stages. 
 
Figure 9.2.1-7: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 3 medium stages. 
 
 191 
 
 
Figure 9.2.1-8: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 3 fine stages. 
 
Figure 9.2.1-9: Time-averaged Sauter mean drop size of Immersion oil in water as a 
function of rotor speed at 1, 3, and 5 lpm for 3 ultrafine stages. 
 
 192 
 
The most obvious features of Figures (9.2.1-1) through (9.2.1-9) are that the 
drop size decreases with increasing number of teeth and with increasing rotor speed. 
This result is consistent with expectations; the effect of flowrate is more subtle. As 
the rotor speed increases, the effect of flowrate diminishes. If d32, at a given flowrate, 
becomes independent of flowrate, this implies that power per mass (energy 
dissipation rate) controls drop size. This allows the postulation that drops at the 
higher rotor speeds are more likely to experience shear fields of sufficient strength 
and duration to reach their equilibrium size. This is a mechanistically important idea 
which merits further exploration. 
Information may also be obtained through the number and volume 
distributions. Figures (9.2.1-10) & (9.2.1-11) show the number and volume 
distributions for a sample condition. 
 
Figure 9.2.1-10: Time-averaged number distribution of Immersion oil drops in water 
with 2 fine stages at a flowrate of 1 lpm. 
 
 193 
 
 
Figure 9.2.1-11: Time-averaged volume distribution of Immersion oil drops in water 
with 2 fine stages at a flowrate of 1 lpm. 
The data collapse with respect to rotor speed for the number distribution, but 
the peak shifts to higher values with respect to rotor speed for the volume 
distribution. The interpretation of this is that there are a few larger drops which skew 
the distribution and result in the dependence of volume (but not number) distribution 
on rotor speed. 
It is interesting to note that the effect of changing the number of stages is 
opposite the effect just described; when the number of stages is changed there is a 
disparity in number distribution, but the volume distributions collapse. This effect is 
displayed for a sample condition in Figures (9.2.1-12) and (9.2.1-13). 
 
 194 
 
 
Figure 9.2.1-12: Time-averaged number distribution of Immersion oil drops in water 
at a rotor speed of 7000 rpm with 1 and 2 ultrafine generators at a flowrate of 1 lpm. 
 
Figure 9.2.1-13: Time-averaged number distribution of Immersion oil drops in water 
at a rotor speed of 7000 rpm with 1 and 2 ultrafine generators at a flowrate of 1 lpm. 
 
 195 
 
9.2.2 ? Transient Drop Size Data 
It is also possible to plot the drop number frequency as a function of the time 
normalized by residence time. Recall that single large drops are injected at the 
entrance and the resulting daughter DSD is measured over residence time at the exit. 
These data show the time at which daughter drops leave the mixer as a function of 
flowrate. This quantity can only be defined for single parent drop experiments where 
all of the daughter drops leave the mixer before a new parent drop is injected into the 
flow line. 
Figure (9.2.2-1) is for the case of 1 ultrafine stage and Figure (9.2.2-2) is for 
the case of 2 fine stages. These two plots are sufficient to illustrate the behavior at 
other conditions. In Figure (9.2.2-1) and (9.2.2-2), the time each drop spent in the 
mixer is normalized by the mean residence time, ?, which is a function of mixer 
volume and flowrate. The mean residence time per stage was: 5.0 s, 1.67 s, and 1.0 s 
at flowrates of 1 lpm, 3 lpm, and 5 lpm, respectively. 
 
 196 
 
 
Figure 9.2.2-1: Residence time distribution of Immersion oil drops in water with 1 
ultrafine generator. 
 
Figure 9.2.2-2: Residence time distribution of Immersion oil drops in water with 2 
fine generators. 
 
 197 
 
Taken together, Figure (9.2.2-1) and (9.2.2-2) show that with enough stages, 
the residence time distributions are self-similar, but with only one stage the 
normalized drop residence times depend on flowrate. 
One last quantity worth consideration is the cumulative Sauter mean diameter. 
This is the average Sauter mean diameter of all daughter drops which have previously 
exited the mixer. The cumulative Sauter mean diameter is plotted for 1 fine stage in 
Figure (9.2.2-3) and for 2 ultrafine stages in Figure (9.2.2-4). These figures are 
normalized by the average Sauter mean diameter over all times ? such figures always 
approach unity at long times. 
 
Figure 9.2.2-3: Cumulative Sauter mean diameter of Immersion oil drops in water 
with 1 fine generator. 
 
 198 
 
 
Figure 9.2.2-4: Cumulative Sauter mean diameter of Immersion oil drops in water 
with 2 ultrafine generators. 
The way to interpret Figures (9.2.2-3) and (9.2.2-4) is first to note that they 
are self-similar, but also to note at which dimensionless time the value levels off to 
unity. This can be used as a measure of how effective the shear fields within the 
mixer are at establishing equilibrium conditions. 
From Figures (9.2.2-3) and (9.2.2-4) we see that the largest drops exit the 
mixer early, and from Figures (9.2.2-1) and (9.2.2-2) we see that more of the drops 
exit earlier than later. Minimizing the extent of this effect is generally desirable since 
such drops are the ones that are not likely to have reached the equilibrium size. 
The purpose of reporting the preceding drop size data of Section 9.2 is to 
determine the operating conditions that provide an equilibrium, or ultimate, drop size. 
Equilibrium appears to be achieved at higher power input and lower flowrate. If 
equilibrium is reached then the drop distributions tend to be self-similar and 
 
 199 
 
characterizing d32,avg is sufficient to characterize the entire drop size distribution. This 
motivates the prediction of d32,avg based on energy dissipation rate. 
9.3  ? Drop Size Correlation with Power Draw 
In Section 9.2, the drop size results received from Murthy (2010) are reported 
without reference to the power draw data. In this section the drop sizes are correlated 
with the power draw. 
The sensible place to start in correlating the drop size with the energy 
dissipation rate is to examine the flow regime of the mixer. This can be done by 
examining where on the plot of NP versus Re the drop size experiments were 
performed. This is displayed graphically in Figure (9.3-1). 
 
 200 
 
 
Figure 9.3-1: PN vs. Re showing the region in which the drop size experiments were 
performed. 
The drop size experiments were essentially restricted to the region where the 
Power number is constant, implying turbulent flow conditions in the mixer. After 
determining that the flow is turbulent, the next step in correlating the drop size is to 
consider the Kolmogorov microscale. 
To calculate the energy dissipation rate it is necessary to define a high-shear 
region. The definition of this region is not obvious, and so, as an initial method, it will 
be taken to be the fluid volume inside of the generators, or mixing head. This region 
is schematically illustrated in Figure (9.3-2) for the cases of 3 stages on the left and 2 
stages plus one blank stage on the right. The red volume was taken to be the high-
 shear region. The volume was measured carefully by water displacement. 
 
 201 
 
    
Figure 9.3-2: Illustration of the high-shear region used to calculate the energy 
dissipation rate. Red indicates the region of high shear. 
Using the same data as Figure (9.3-1), the Kolmogorov scale, defined in 
equation (3.1.1-3), is plotted as a function of rotor speed in Figure (9.3-3). 
 
Figure 9.3-3: Kolmogorov microscale as a function of rotor speed for the generator 
configurations for a sample flowrate of 3 lpm. 
Comparing Figure (9.3-3), which shows a range of Kolmogorov microscales 
of 3.8 to 8.3  m, with Figures (9.2.1-1) through (9.2.1-9), it is clear that the drops are 
 
 202 
 
always greater than the Kolmogorov microscale ? this is usually the case when water 
is the continuous phase. This implies that an inertial correlation (where d > ?) should 
be used to characterize the drop size. Since large drops are injected one at a time into 
the mixer the correlation should include the assumption of dilute dispersed phase 
fraction. 
These criteria lead to two obvious candidates for the drop size correlation 
from the literature. If the drop phase is sufficiently inviscid, the correlation is listed in 
the first entry in Table 3.1.1-1 and can be written in terms of energy dissipation rate 
according to equation (9.3-1) (Hinze 1955). 
       
?   
? 
   ?
        (9.3-1) 
the subscript on drop diameter indicates that the correlation applies to both d32 and 
dmax ? that is, d32 is proportional to dmax. 
The reason for writing this correlation in terms of ? is that the characteristic 
device diameter is unknown (or only known in a relative sense). By correlating the 
data with respect to ? this issue can be skirted and a meaningful correlation may still 
be obtained. The other candidate correlation applies to the case of a very viscous 
dispersed phase and follows equation (9.3-2) (Calabrese et al. 1986). 
       
 
 
   
 ? ?  
   ?
        (9.3-2) 
In the case of this study it is not obvious which correlation is more likely to be 
accurate. With a continuous phase of water and a dispersed phase of oil having a 
viscosity of 163 cP, the criterion of a very viscous dispersed phase may not apply. 
Therefore, both correlations are tested to see which one fits the data better. The 
 
 203 
 
inviscid correlation is tested in Figure (9.3-4) and the viscous correlation is tested in 
Figure (9.3-5). 
 
 
Figure 9.3-4: Test of the inertial, inviscid correlation?s capability of fitting the drop 
size data. A straight line with a positive slope represents good correlation. 
In Figure (9.3-4), a straight line with a positive slope would best fit the data 
according to the inertial, inviscid correlation of equation (9.3-1). There is a large 
amount of scatter in this data. However, the scatter is more pronounced on the right-
 hand side of Figure (9.3-4) than on the left. This is because the x-axis is the reciprocal 
of the energy dissipation rate raised to a positive power. The left-hand side 
 
 204 
 
corresponds to those flow conditions where there is a greater rate of energy 
dissipation and so equilibrium is more likely. A higher amount of scatter may also be 
observed for those points which are more likely to be further from equilibrium such 
as the curve of the blue circles connected by a dashed line, which corresponds to 1 
medium stage at a flowrate of 5 lpm (this is the case likely to be furthest from 
equilibrium). 
 
 
Figure 9.3-5: Test of the inertial, viscous correlation?s capability of fitting the drop 
size data. A straight line with a positive slope represents good correlation. 
 
 205 
 
Figure (9.3-4) & (9.3-5) both appear to be reasonable ways to correlate the 
data. The problem is that the exponents -2/5 and -1/4 are fairly close. Therefore, the 
axes of these plots are redone so that a flat line represents good correlation. For 
equilibrium conditions, the y-axis should be independent of ?. Figures (9.3-6) and 
(9.3-7) show this. 
 
 
Figure 9.3-6: Test of the inertial, inviscid correlation?s capability of fitting the drop 
size data. A flat line represents good correlation. 
 
 206 
 
 
 
Figure 9.3-7: Test of the inertial, viscous correlation?s capability of fitting the drop 
size data. A flat line represents good correlation. 
It appears that Figure (9.3-6) approaches ? independence better than Figure 
(9.3-7). Therefore it is concluded that the data fit better with the inviscid correlation 
than with the viscous one. Apparently, the oil is insufficiently viscous to cause an 
additional resistance to drop breakup. More likely, the data scatter is too great to 
discern the difference. 
It is important to remember that equation (9.3-1) is an equilibrium correlation. 
Therefore, if the drops have not reached equilibrium then they will deviate from the 
correlation. Therefore, it is reasonable to expect that conditions further from 
 
 207 
 
equilibrium correspond to data in Figure (9.3-7) which is further away from the flat 
line. This is clearly true for the case furthest from equilibrium ? the blue circles 
connected by a dashed line which corresponds to 1 medium stage with a flowrate of 5 
lpm. To investigate this effect more quantitatively the data points were removed in 
which there was a difference in d32 greater than 10% at 3 lpm vs. 5lpm (this was the 
median difference for all the data). When this is done, Figure (9.3-6) is replotted as 
Figure (9.3-8). 
 
Figure 9.3-8: Replotted version of Figure (8.4.2-7). Data with an effect of flowrate 
for 3 lpm vs. 5 lpm has been removed. A flat line represents good correlation. 
 
 208 
 
There is indeed a relationship between those data with less of a dependence on 
flowrate and those which form a straighter line on Figure (9.3-8). This provides some 
experimental assurance that the claim of equilibrium being established at some of the 
conditions is valid. Unfortunately, there is not complete consistency in which of the 
data survived the purge. As shown on the top of Figure (9.3-8), more of the 1 stage, 
fine rotor speeds survived the purge than the 1 stage, ultrafine. Despite that, however, 
there is general sense in the purge in that, for example, more of the ultrafine 
configurations survive than the medium ones. 
Since the data follow the inertial inviscid correlation, they can be compared to 
the Davies (1987) plot shown in Figure (8.1.1-1). First, however, d32 must be related 
to dmax to facilitate the comparison with Davies plot which was done in terms of dmax. 
Fortunately, there is a linear relationship in this case between d32 and dmax. This 
relationship (which is for all data, not just those of Figure (9.3-8)) is shown in 
equation (9.3-3) as well as Figure (9.3-9). 
                 (9.3-3) 
 
 209 
 
 
Figure 9.3-9: Relationship of d32 with dmax for all drop size data. 
The Davies plot comparison is done in Figure (9.3-9), which shows all of the 
data alongside the Davies plot, and in Figure (9.3-10), which is the same except that it 
only includes the data which survived the purge. 
 
Figure 9.3-10: Comparison of drop size data with Davies (1987) plot. 
 
 210 
 
 
 
Figure 9.3-11: Comparison of drop size data with Davies (1987) plot including only 
data which survived the purge. 
In Figures (9.3-10) and (9.3-11), a fitted expression for the Davies plot is 
listed which was obtained by measuring the slope and intercept of Figure (8.1.1-1); 
this expression is given in equation (8.4.2-4). 
         ?
           (8.4.2-4) 
Because the energy dissipation rate was calculated using an estimated volume 
for the high-shear region, Figure (9.3-11) may be used to reverse engineer the volume 
of that region and thereby find the true maximum local shear rate, ?max, which breaks 
up the drops. Assuming that equilibrium has been established for these data (an 
assumption for which evidence has just been provided), this may be used as a method 
 
 211 
 
to find the relationship between ?avg and ?max for this mixer. When this is done, the 
result is given by equation (9.3-5) and illustrated in Figure (9.3-12). 
?    ?   ?       (9.3-5) 
 
Figure 9.3-12: Calculation of the maximum local energy dissipation rate in terms of 
the average by means of comparison with the Davies (1987) plot. 
In this way oil drops were essentially used as tracer particles to detect the 
shear rate and provide information about the shear field, a useful technique. 
 
 
 212 
 
Chapter 10:  Summary & Conclusions 
This chapter summarizes the most important findings of this study. It is 
derived from and set apart from Chapters 1-9 to allow for felicitous use by the reader. 
Potential future work is also discussed for both dilute and non-dilute systems because 
there is much more that could be learned. Additional work for non-dilute systems 
would be particularly useful since that is most often the situation in industrial settings. 
10.1 ? Interfacial Tension 
A systematic method for calculating interfacial tension using the ?Pendant 
Drop Method? was developed which was independent of lighting and thresholding, 
and was a significant improvement over the previously published methodology. 
Using this method the interfacial tension of water in Crystal Oil was measured as a 
function of the concentration of the oil-soluble surfacatant, Tergitol NP-4.  The 
diffusivity of the surfactant is small because of the high oil viscosity. The CMC of 
Tergitol NP-4 in Crystal Oil is 10-3 M. 
10.2 ? Flow Regime Determination ? Silverson L4R Batch Rotor-Stator 
Mixer 
Padron's (2001) prediction of the flow regime transition from Power number 
data for rotor-stator mixers was verified via drop size data; it is in the neighborhood 
of Re = 1000. However, the precise location of the transition (and of any transitional 
flow behavior) is better characterized by consideration of viscosity in addition to the 
Reynolds number. 
 
 213 
 
10.3 ? Dilute Systems ? Silverson L4R Batch Rotor-Stator Mixer 
For batch rotor-stator mixers in laminar flow, drops break up via a mechanism 
of simple shear. Based on comparison with Grace's (1982) idealized drop breakup 
data for SSF, the shear rate in question is possibly about twice that of the nominal 
shear rate in the rotor-stator shear gap. Also, a semi-empirical correlation shows that 
d32/D scales with ? ?/N D
 2  d, or Cad. Because it lacks a mechanistic basis, this 
correlation should be used with caution. Also significant, this correlation is 
independent of continuous phase viscosity. Determining the reason for the 
independence from continuous phase viscosity and the dependence on dispersed 
phase viscosity is a good goal for future work. The first step in this could be 
extending the range of the viscosity ratios studied to include cases other than only 
very low values of ?. This extension of the range of viscosity ratios would also be 
helpful since it would provide further illumination of the breakage mechanism of the 
drops. Also, a full-scale CFD (computational fluid dynamics) study of the flow field 
around the mixing head would be useful in studying both the breakage mechanism 
and the surprising dispersed phase viscosity dependence. Such a CFD study would be 
relatively cheap computationally since the flow is laminar rather than turbulent. 
For turbulent flow, when the Kolmogorov microscale is greater than the 
maximum stable drop size, but not too much greater (d32 < ?, but not d32 << ?), the 
data are well correlated by the sub-Kolmogorov inertial stress model. Because these 
are fit by a previously derived mechanistic model, this correlation can be used to 
scale up from the laboratory to industrial scales. A useful avenue of future work 
 
 214 
 
would be to obtain drops much smaller than the Kolmogorov scale (d << ?, rather 
than d < ?) in an attempt to verify the sub-Kolmogorov viscous stress model. 
Despite the slow diffusivity of surfactant through the viscous oil, surfactants 
were found to mostly decrease drop size by lowering the interfacial tension. 
Convective mixing ensures that surfactant is always available at the surface under 
equilibrium conditions, so diffusion limiting issues are not present for laminar flow at 
low surfactant concentrations or for turbulent flow at surfactant concentrations up to 
the CMC. It is likely that Marangoni stresses exist for laminar flow at surfactant 
concentrations near the CMC, but quantification of the Marangoni effect was not 
attempted. 
Future work includes the further investigation of the Marangoni effect by 
reference to the diffusivity of the surfactant in the oil and the surface dilatational 
modulus. This would assist in understanding the effect of the presence of the 
surfactant on the breakage mechanism. Also, of interest would be the quantification 
of the rate of adsorption as distinct from the rate of diffusion to the surface. It is 
possible that differences in this final step, the surfactant going from just outside the 
interface to bridging the interface, could account for differences in the absence of 
Marangoni stresses in this system relative to some others such as that of Padron 
(2005).  
10.4 ? Non-Dilute Systems ? Silverson L4R Batch Rotor-Stator Mixer 
Sauter mean diameter increases according to Equation (7.1-1) when 0.001 < ? 
< 0.05 invalidating the widely held notion that ? < 0.01 is a firm limit for defining a 
dilute (non-coalescing) system, at least for the systems covered in this study. There is 
 
 215 
 
a much greater dependence of drop size on phase fraction than was previously 
reported for turbulent flow with L >> d >> ?. This is because the flow conditions in 
this study are locally laminar around the drops (either the flow was laminar or d32 < 
?) resulting in drop collisions governed by viscous rather than inertial forces. 
There is apparently a much higher rate of coalescence when the flow is locally 
laminar, at least at the continuous and dispersed phase viscosities of this study. Based 
on work reported by others (Caserta et al. 2006; Chesters 1991; Lyu et al. 2000; 
Perilla and Jana 2005; Priore and Walker 2001), the coalescence rate in these systems 
is believed to be the result of sufficiently low values of  c and  d in flows that were 
locally laminar around the drops. The low viscosity of  d causes the interface to be 
completely mobile during film drainage. 
Adding surfactants and treating the mill head surfaces with a hydrophobic 
silane decreased the drop size, but did not qualitatively change the functional 
dependence of d32 on ?. This implies that the same coalescence phenomena are 
occurring, but are reduced in their effectiveness. 
At large volume fractions, the drop size plateaus (at least in the presence of 
surfactants). Just as the initial increase of the drop size is attributed to an increase in 
collision frequency, the plateauing is attributed to the effect of increasing the drop 
size: decreased collision frequency, decreased collision efficiency, and increased drop 
breakage rate. 
Finally, the potential of drops adhering to low-shear solid surfaces within the 
mixing volume, and thereby affecting the DSD, is not significant in explaining the 
high functionality of the drop size with phase fraction. It is, however, recommended 
 
 216 
 
that low shear-surfaces be hydrophobically treated to mitigate any error associated 
with the aforementioned adherence. The chemical used to treatment these surface 
were Glassclad 18 for glass and dimethyldichlorosilane for stainless steel (the steel 
first having been polished to a mirror-like finish). 
There are three areas where future work could be performed to extend the 
reach of this study. Figures similar to Figure (7.3.2-7) and equations similar to 
equation (7.3.2-1) could be constructed at different viscosities. If this were done, it 
might be possible to correlate the functionality of ?a? from equation (7.1-1) with 
continuous and dispersed phase viscosity, instead of just the surfactant concentration. 
Having done that it may be possible to develop a mechanistic correlation for that. 
Even more useful, however, would be to construct figures such as Figure (7.4-
 1) by doing experiments with ? > 0.05 at various viscosities. Since d32 plateaus when 
? > 0.10, an expression for the plateau value of d32 as a function of continuous and 
dispersed phase viscosities could be obtained. Such an expression would have 
tremendous industrial value. This could be done in turbulent flow as well, for which 
no experiments were performed with ? > 0.05. 
Also, it would be extremely useful to discover the location of the coalescence 
events within the flow. Some evidence was presented for at least some of the 
coalescence taking place near the mixing head because of the effect of the 
hydrophobic coating of the mill head on d32. However, a more refined and specific 
understanding of the coalescence location and dynamics is desirable. To do this, a 
CFD study could be performed with population balances including breakage kernels 
and coalescence kernels. In this way the birth and the death (both occurring through 
 
 217 
 
coalescence or breakup) of the drops could be tracked as a function of location. This 
would provide immeasurable insight into the experience of individual drops within a 
non-dilute mixing system. 
10.5 ? Non-Equilibrium Systems ? IKA Labor Pilot 2000/4 In-line 
Rotor-Stator Mixer 
For an inviscid fluid in an in-line rotor-stator mixer, the power put into 
pumping is significantly less than the power dissipated into the fluid. However, a 
significant amount of power (of the same order of magnitude as that dissipated into 
the fluid) is lost to bearings and other sources of friction in the system. The amount of 
power dissipated per stage is independent of the number of stages for the same type 
of generator. 
A Power number versus Reynolds number curve was constructed in which the 
power number included the number of stages in its denominator and where the 
diameters were modified empirically. The modification of the diameters was justified 
by the fact that doing so caused the Power number curves for all mill head geometries 
to collapse to a single value for turbulent flow. The fact that the effective diameter 
was independent of the number of stages verified the choices. It was deemed not 
possible to define the appropriate characteristic diameter of the system because of the 
inconstant ways in which the dimensions of the generator types varied, so the value of 
the constant Power number in NP vs. Re plot should be seen as arbitrary. 
There is drop size evidence of various types (actual values, residence times, 
number and volume distributions, and cumulative number distributions) which 
 
 218 
 
provide evidence that the drop size distributions reach equilibrium for some of the 
conditions tested. 
The turbulent, inertial, inviscid correlation for drop size with energy 
dissipation rate scales the equilibrium data slightly better than the turbulent, inertial, 
viscous one, even though the oil has a viscosity of 163 cP. Equilibrium operating 
conditions were selected on the basis of insensitivity to flowrate. The inviscid 
correlation is better followed at conditions in which there is less of an effect of 
flowrate. Finally, with the idea established that these drops have reached equilibrium, 
they act as a sort of ?energy dissipation sensor? whose sizes reveal, by comparison 
with the ?Davies (1987) Plot?, that ?max ? 9 ?avg. 
For future work, it would be useful to develop a more refined understanding 
of the shear field within this mixer, and so it would be useful to conduct experimental 
PIV or computational CFD studies. With that done, the result here that ?max ? 9 ?avg 
could be verified and the drop size could be correlated instead of being used as an 
?energy dissipation sensor,? as just mentioned. It is recommended that such a study 
be performed before acquiring more drop size data. Also, as in the batch study, the 
most useful (and also the most difficult) future work would be a CFD study which 
accounts for density differences and includes a population balance with both 
coalescence and breakage kernels. In this way the experience of individual drops 
within a non-dilute emulsion may be tracked and quantified. 
 
 219 
 
Appendix A: Catalog of Literature Data for Interfacial and 
Surface Tension for some Common Fluids 
The following table, Table A-1, catalogs some of the published values of 
interfacial and surface tension in the scientific literature. The purpose of this is to 
show that there are some significant discrepancies in these published values and that 
any claim to be able to measure surface or interfacial tension must have an 
explanation for why this scatter in the published values does not affect the confidence 
which should be placed in the results. That is, there must be a measurement technique 
which avoids the problem of the scatter apparent in Table A-1. 
Table A-1 also illustrates the fact that many published values for surface 
tension are unoriginal and are themselves simply a citation of previous work. This, 
combined with the aforementioned scatter, decreases the confidence which can be 
placed in a single reported value from the literature. Therefore, the method of 
calibration must not be based on calibrating against a single published value. As 
covered in Section 2.3.7, the calibration method was based on a multiplicity of 
judiciously chosen published surface/interfacial tension values. 
 Also, illustrated here is the fact that there is both less scatter and more data 
available for surface tension than interfacial tension. 
Finally, surface and interfacial tension are known to be a function of 
temperature which is approximately linear (Jasper 1972; Lide 2009; Speight 2005). 
Most of the temperatures listed here are near 20? C, and, based on the slopes reported 
 
 220 
 
by Jasper, Lide, and Speight, the effect of temperature is not the main cause of the 
scatter. 
Explanation of symbols and abbreviations used in Table A-1: 
?*? = Data unoriginal (taken from another reference in this table) 
?^? = Another source listed in this table took data from this source 
?IFT? = Interfacial Tension with Water (mN/m) 
?ST? = Surface Tension (mN/m) 
?T? = Temperature (? Celsius) 
In % spread, values over 5% have been highlighted  
  Pentane Hexane Heptane 
Author Year IFT ST T IFT ST T IFT ST T 
DuNo?y^ 1925          
Speight* 2005  15.4 25  17.9 25  19.7 25 
Middleman 1998          
Young^ 1928    51.1 18.43 20    
Janssen* 1987    51.1 18.43 20    
Bernstein 1997  15.49 19.3  19.2 11.6  21.53 5.44 
Kahl^ 2003        19.63 24.7 
Lide* 2009  15.49 25  17.89 25  19.66 25 
Jasper^ 1972  15.5 25  17.9 25  19.7 25 
Goebel 1997  15.5 25 51.4 18.3 22 51.9 20.55 22 
Zeppieri 2001 50.9 15.9 22 50.38  25 50.71  25 
Fu 1986          
Range  0.5  1.0 1.3  1.2 1.9  
% Spread  2.9  2.0 7.2  2.32 9.4  
St. Dev.  0.19  0.43 0.48  0.84 0.78  
  
 
 221 
 
  Cyclohexane Benzene Isoamyl Alcohol 
Author Year IFT ST T IFT ST T IFT ST T 
DuNo?y^ 1925    32.3 29 25 4.4 24.2 23.5 
Speight* 2005  24.7 25  28.9 20  23.7 25 
Middleman 1998  25 25 40 29 25 4.8  25 
Young^ 1928    35 28.86 20 5  18 
Janssen* 1987 50.3 26.54 20 35 28.86 20    
Bernstein 1997  25 20  28.89 20  23.7 24.3 
Kahl^ 2003  24.2 24.7       
Lide* 2009  24.16 25  28.22 25  23.71 25 
Jasper^ 1972  24.7 25  28.9 20  23.7 25 
Goebel 1997          
Zeppieri 2001          
Fu 1986          
Range  2.4  7.7 0.8  0.6 0.5  
% Spread  9.6  21.9 2.7  12.7 2.1  
St. Dev.  0.80  2.9 0.25  0.31 0.22  
  Oleic Acid Carbon Tetrachloride Carbon Disulfide 
Author Year IFT ST T IFT ST T IFT ST T 
DuNo?y^ 1925 12.8 34.2 25 40 28 23 33.8 34 20 
Speight* 2005  32.8 20  26.4 25  31.6 25 
Middleman 
n 
1998 16  25       
Young^ 1928 15.59 32.5 20 45 26.66 20 48.36 31.38 20 
Janssen* 1987    45 26.66 20    
Bernstein 1997  32.6 25  26.324 25  31.61 25 
Kahl^ 2003          
Lide* 2009     26.43 25  31.58 25 
Jasper^ 1972     26.4 25  31.6 25 
Goebel 1997          
Zeppieri 2001          
Fu 1986          
Range 3.2 1.7  5.0 1.7  14.6 2.6  
% Spread 21.6 5.1  11.5 6.3  35.4 8.2  
St. Dev. 1.74 0.79  2.89 0.58  10.29 1.01  
  
 
 222 
 
  Water Methyl Propyl Ketone Benzyl Alcohol 
Author Year IFT ST T IFT ST T IFT ST T 
DuNo?y^ 1925 -      4.8 43.3 25 
Speight* 2005 -    23.3 25  34.8 25 
Middleman 
n 
1998 - 72 25       
Young^ 1928 -   6.28 24.15 20  39.71 22.5 
Janssen* 1987 -   6.28 24.15 20  39.71 25 
Bernstein 1997 - 71.98 25  25.09 20  35.97 25 
Kahl^ 2003 -         
Lide* 2009 - 71.99 25  23.25 25    
Jasper^ 1972 - 72.1 25  23.3 25  34.8 25 
Goebel 1997 -         
Zeppieri 2001 -         
Fu 1986 -         
Range - 0.2  0 1.8  0.1 8.5  
% Spread - 0.2  0 7.7  1.0 22.3  
St. Dev. - 0.07  0 0.75  0.03 3.42  
  Octane Toluene Diethyl Ether 
Author Year IFT ST T IFT ST T IFT ST T 
DuNo?y^ 1925    32.5 28 25 10.9 18  
Speight* 2005  21.1 25  27.9 25  16.7  
Middleman 1998    32  25    
Young^ 1928 50.81 21.77 20 36.1  25 10.7 17.1  
Janssen* 1987 50.81 21.77 20 36.1 28.43 20 10.7 17.1  
Bernstein 1997  23.25 6.4  27.76 26.1    
Kahl^ 2003     27.76 24.7    
Lide* 2009  21.14 25  27.73 25  16.65 25 
Jasper^ 1972  21.1 25  27.9 25  16.7 25 
Goebel 1997 52.5 21.55 22       
Zeppieri 2001 51.16  25       
Fu 1986          
Range 1.7 2.1  4.1 0.7  0.2 1.4  
% Spread 3.3 9.7  12.0 2.5  1.9 7.9  
St. Dev. 0.80 0.75  2.23 0.24  0.12 0.53  
  
 
 223 
 
  Chloroform Ethanol Methanol 
Author Year IFT ST T IFT ST T IFT ST T 
DuNo?y^ 1925 27.9 27.5 25 -   -   
Speight* 2005  26.7 25 - 22.0 25 - 22.1 25 
Middleman 1998    -   - 23 25 
Young^ 1928 32.8 27.13 20 -   -   
Janssen* 1987 32.8 27.13 20 - 22.27 20 - 22.55 20 
Bernstein 1997  26.6 25 - 21.75 25 - 21.76 27.1 
Kahl^ 2003    -   -   
Lide* 2009  26.67 25 - 21.97 25 - 22.07 25 
Jasper^ 1972  26.7 25 - 22.0 25 - 22.1 25 
Goebel 1997    -   -   
Zeppieri 2001    -   -   
Fu 1986    -   -   
Range 4.9 0.9  - 0.5  - 1.2  
% Spread 15.7 3.3  - 2.4  - 5.8  
St. Dev. 2.83 0.34  - 0.19  - 0.45  
  Bromoform ?Parrafin?/?Mineral? Oil  
Author Year IFT ST T IFT ST T     
DuNo?y^ 1925 27.5 39.5 25 47 32 25    
Speight* 2005  44.9 25       
Middleman 1998     30 25    
Young^ 1928 40.85 41.53 20       
Janssen* 1987          
Bernstein 1997  45.1 25       
Kahl^ 2003          
Lide* 2009  44.87 25       
Jasper^ 1972  44.9 25       
Goebel 1997          
Zeppieri 2001          
Fu 1986          
Range 13.4 5.6   2.0     
% Spread 39.1 12.9   6.5     
St. Dev. 9.44 2.37   1.41     
Table A-1: Literature values of surface and interfacial tension for some relatively 
common substances. This table illustrates the significant amount of scatter present in 
the reporting of these values. 
 
 224 
 
Appendix B: Computer Programs Used in Measuring Interfacial 
and Surface Tension 
As discussed in Sections 2.3.6, 2.3.7, and 2.3.9, there are three purpose-
 developed computer programs (four files) that were used in measuring surface and 
interfacial tensions. The text of the programs follows. The first is the ImageJ pendant 
drop macro which was used to convert pendant drop images into a list of pixels which 
describe the drop?s boundaries to be used as input for the MATLAB interfacial 
tension program. The second file is used to organize the images for the third file, 
which actually calculates the interfacial tension using the inputs generated by the first 
file. The fourth file takes a list of dynamic interfacial tensions at a particular 
surfactant concentration and, using purpose-developed non-linear curve fitting, 
converts them into values of equilibrium interfacial tension. 
ImageJ Pendant Drop Program 
//?_PendantDrop_Series.txt? Used to transform pendant drop images into a 
text file //which is a list of the pixels which make up the drop and needle interface 
with the //surrounding phase 
macro "Series Pendant Drop Analysis" { 
    dir = getDirectory("Directory #1"); 
    //lower = getNumber("Lower Threshold Value #1", 1); 
    dir2 = getDirectory("Directory #2"); 
    lower2 = getNumber("Lower Threshold Value #2", 2); 
    dir3 = getDirectory("Directory #3"); 
    lower3 = getNumber("Lower Threshold Value #3", 3); 
    dir4 = getDirectory("Directory #4"); 
    lower4 = getNumber("Lower Threshold Value #4", 4); 
    dir5 = getDirectory("Directory #5"); 
    lower5 = getNumber("Lower Threshold Value #5", 5); 
    dir6 = getDirectory("Directory #6"); 
 
 225 
 
    lower6 = getNumber("Lower Threshold Value #6", 6); 
    dir7 = getDirectory("Directory #7"); 
    lower7 = getNumber("Lower Threshold Value #7", 7); 
    dir8 = getDirectory("Directory #8"); 
    lower8 = getNumber("Lower Threshold Value #8", 8); 
 
// ----------------------- 11111111111111 ---------------------------- 
    print('1st') 
    list = getFileList(dir); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir+list[i]; 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
       setAutoThreshold(); 
                  //setThreshold(lower, 255);      // SET TO DESIRED VALUE 
       run("Convert to Mask"); 
//       run("Invert"); 
// Uncomment the above line if image has a dark background and threshold 
// results in an reverse image.  
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
        } 
    } 
// ----------------------- 22222222222222 ---------------------------- 
print('2nd') 
list = getFileList(dir2); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir2+list[i]; 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
                  setThreshold(lower2, 255); 
       run("Convert to Mask");  
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
 
 226 
 
        } 
    } 
// ----------------------- 333333333333333 ---------------------------- 
print('3rd') 
list = getFileList(dir3); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir3+list[i]; 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
                  setThreshold(lower3, 255); 
       run("Convert to Mask"); 
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
        } 
    } 
// ----------------------- 44444444444444 ---------------------------- 
print('4th') 
list = getFileList(dir4); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir4+list[i]; 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
                  setThreshold(lower4, 255); 
       run("Convert to Mask"); 
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
        } 
    } 
// ----------------------- 55555555555555 ---------------------------- 
print('5th') 
list = getFileList(dir5); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir5+list[i]; 
 
 227 
 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
                  setThreshold(lower5, 255); 
       run("Convert to Mask"); 
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
        } 
    } 
// ----------------------- 66666666666666 ---------------------------- 
print('6th') 
list = getFileList(dir6); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir6+list[i]; 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
                  setThreshold(lower6, 255); 
       run("Convert to Mask"); 
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
        } 
    } 
// ----------------------- 77777777777777 ---------------------------- 
print('7th') 
list = getFileList(dir7); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir7+list[i]; 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
                  setThreshold(lower7, 255); 
       run("Convert to Mask"); 
 
 228 
 
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
        } 
    } 
// ----------------------- 88888888888888 ---------------------------- 
print('8th') 
list = getFileList(dir8); 
    setBatchMode(true); 
 for (i=0; i<list.length; i++) { 
 path = dir8+list[i]; 
            showProgress(i, list.length); 
            if (!endsWith(path,"/")) open(path); 
            if (nImages>=1) { 
       run("8-bit"); 
       run("Enhance Contrast", "saturated=.5 normalize"); 
       run("Find Edges"); 
                  setThreshold(lower8, 255); 
       run("Convert to Mask"); 
           txtPath = path+".txt"; 
           run("Save XY Coordinates...", "background=0 save=["+txtPath+"]"); 
     close(); 
        } 
    } 
} 
 
MATLAB m-file to Organize the Processing of Images  
function 
[sigma]=Pendant_Drop_Batch(datafile_part,ext,start,stop,delta_rho,Dn
 ,cal) 
  
%   Inputs: 
%     datafile_part - The portion of the filename  
%        common to all datafiles, prior to the number 
%        IE for datafiles 'myfile_002.tif.txt', the  
%        value of datafile_part would be 'myfile_' 
%        Include full path if not in working directory 
% 
%     ext - The file extension - strictly any part of 
%        the filename falling after the number common 
%        to all files - typically '.TIF.txt' 
% 
%     start - what datafile number to start at (ex 1) 
% 
%     stop - what datafile to stop at (ex, 100) 
% 
%     corr - side correction for swelling. range: 0.0-1.0 
 
 229 
 
%     frac - bottom correction for swelling. range: 0.0-1.0 
% 
%     cal - always use 1.0075 as obtained from 'PD15' 
%      
%Example: 
%Pendant_Drop_Batch('D:\Paul\Pendant Drop\PD22 - relearning Pendant 
Drop Method\PD22A\PD22A - 
100\PD22A_','.tif.txt',1,50,858.796,1.651,1.0075) 
disp(' ') 
disp('------------------------------------') 
%disp('Approximate Bond # - Gustavo lower bound cutoff is about 
.12') 
num=stop-start+1; 
sigma=zeros(num,1);  %initialize results vector 
tic;  % record start time 
h=waitbar(0,'Processing Images...'); %Spawn Progress Bar 
for i=start:stop 
    zeropad=''; %Zeropad adjustments for filenames like "myfile_002" 
    if i<10 
        zeropad='00'; 
    elseif (i>=10) && (i<100) 
        zeropad='0'; 
    end 
% EDIT THE ABOVE IF YOUR FILENUMBERS ARE MORE/LESS THAN 3 DIGITS 
    datafile=load([datafile_part,zeropad,num2str(i),ext]); %read 
Data files 
    rel_start=i-start+1; % Accounts for the fact that start may not 
be equal to 1 
    sigma(rel_start)=Pendant_Drop(datafile,delta_rho,Dn,cal); 
%Calculate IFT using pdrop function 
    waitbar(rel_start/num,h) %Update Progress Bar 
end 
close(h); %Kill Progress Bar 
time = toc; 
timeh=fix(time/3600); 
timem=fix((time-timeh*3600)/60); 
times=time-timeh*3600-timem*60; 
% Collect Statistics for run 
  
out1=['Run time was ' num2str(timeh) ' hours ' num2str(timem) ' 
minutes ' num2str(times) ' seconds.']; 
effc=num*60/time; 
out2=[num2str(num) ' Images were processed at a rate of ' 
num2str(effc) ' images per minute']; 
meanIFT=mean(sigma); 
varIFT=std(sigma); 
out3=['Calculated IFT value was ' num2str(meanIFT) '?' 
num2str(varIFT) ' mN/m']; 
%Output Run Statistics 
disp('------------------------------------') 
disp(out1) 
disp(out2) 
disp(out3) 
 
MATLAB m-file to calculate the Interfacial Tension of an Individual Image 
 
 230 
 
% ONLY USE IN CONJUNCTION WITH BATCH M-FILE - I.E. NOT FOR 
INDEPENDENT USE 
function [sigma]=Pendant_Drop(datafile,delta_rho,Dn,cal) 
% ------------------------------------------------------------- 
% Specify Values - edit as nessescary 
%rhoL = 684; % Droplet Phase Density - kg/m^3  
%rhoG = 996.7; % Continuous Phase Density - kg/m^3 - Water = 996.7 
or ~=996 
                % Density of Air = 1.204   %  Density of Silicon Oil 
~= 950 
                % Density of Lubsoil ~= 844 %  Density of Crystal 
Oil = 860 
                % Density of Heptane = 684 %  Density of Toluene = 
866.9 
                % Density of Chloroform = 1473.5 %  Density of 
Benzene = 878.6 
                % Density of Cyclohexane = 774 
    %delta_rho=abs(rhoL-rhoG); 
accg = 9.80665;  % Acceleration due to Gravity - m/s^2 
%Dn = 1.651;   % 21 gage = .8128, .82(.81 U-needle) % Outer Diameter 
of Needle in mm 
              % 16 gage = 1.651, 1.65 (1.64) 
              % 26 gage = .457, .46 
              % 18 gage = 1.27 
corr = .5;  % correction for swelling of drop sideways, range: 0-1 
frac = .5;  % correction for swelling of drop downwards: range 0-1 
%cal=1.0075;   % Calibration, that is, factor for Dep_px -> Ds_px 
from De_px 
                % "cal" is really the Adjustment Factor; actual 
calibration occurs at the end of this program 
% ------------------------------------------------------------- 
% Import Data 
    X_px = datafile(:,1); % uncomment if X values are used 
    Y_px = datafile(:,2); %read Y values from data file 
% ------------------------------------------------------------- 
cond1=0; 
for i=1:(length(Y_px)-1) 
    if Y_px(i+1) > Y_px(i) + 1; % checks whether there is a gap in 
the Y-values 
        Y_px_sc = Y_px(i+1:length(Y_px)) - Y_px(i+1) + 1; 
                       % Y_px(i+1) is the minimum legitimate Y-value 
        Y_sc = max(Y_px_sc); 
        X_px_sc = X_px(i+1:length(X_px)); 
        cond1=1; 
    end 
end 
if cond1==0 
    Y_min = min(Y_px); % scale Y values  
    Y_px_sc = Y_px - Y_min + 1; 
    Y_sc = max(Y_px_sc); % finds number of discrete Y values 
    X_px_sc = X_px; 
end 
% ------------------------------------------------------------- 
% Eliminate Side Outliers & Make Vector Containing widths of each 
row 
Q = zeros(1,Y_sc); % Alternate Code 
p=1; 
 
 231 
 
Y_px_sc(length(Y_px_sc)+1) = 0; % This line prevents an error in the 
"if" loop 
X_px_sc(length(X_px_sc)+1) = 0; % This line prevents an error in the 
"if" loop 
cond2=0; 
for i=1:Y_sc 
    k=1; 
    m=1; 
    while Y_px_sc(p) == i 
        T{k}(m)=X_px_sc(p); 
        if X_px_sc(p+1) ~= X_px_sc(p) + 1 && Y_px_sc(p+1) == 
Y_px_sc(p) 
             k = k + 1; 
             m=0; 
        end 
        m = m + 1; 
        p = p + 1; 
    end 
    leftpt=T{1}(1); 
    left_in=T{1}(length(T{1})); 
    right_in=T{k}(1); 
    rightpt = T{k}(length(T{k})); 
    Q(i) = (corr*right_in+(1-corr)*rightpt)-(corr*left_in+(1-
 corr)*leftpt); 
    if i < 40 && right_in < left_in 
        extra_up=i; 
        cond2=1; 
    end 
% THIS PART OF THE CODE DOESN'T WORK ---- FIX LATER 
%     % This next 'if' loop causes distant outliers to be ignored % 
%     if i > 2 % Stops 'Q(0)' from trying to be accessed 
%     if Q(i-1)-10 > Q(i) || Q(i) < Q(i-1)+10 
%         Q(i) = Q(i-1)+(Q(i-1)-Q(i-2)); % Attempts to guess true 
value 
%     end 
%     end 
    clear T; 
end 
if cond2 == 0 
    extra_up=3; 
end 
% ------------------------------------------------------------- 
% Account for swelling downward 
extra_down=0; 
extra = extra_down*(1-frac) + extra_up*frac; 
extra = round(extra); 
Q = Q(extra+1:length(Q)); 
Y_sc=length(Q); 
% ------------------------------------------------------------- 
% Find diameters necessary for calculating shape factor - With 
Averaging % 
if length(Q) > 50 
%if max(Q) > 50 && max(Q) < 3000 
sorted_Q = sort(Q); 
De_px = mean(sorted_Q(length(sorted_Q)-4:length(sorted_Q))); 
De_px = round(De_px); 
% De_px = max(Q); 
 
 232 
 
% Ds_px = mean(Q(De_px-2-cal:De_px+2-cal)); 
% Ds_px = Q(De_px-cal); 
Dep_px = De_px*cal; 
Dep_px = round(Dep_px); 
Ds_px = Q(Dep_px); 
% ------------------------------------------------------------- 
% Find scale_fac - i.e. # of pixels/mm 
j=Y_sc; 
while Q(j) < Q(Y_sc)+3 && Q(j) > Q(Y_sc)-3 
    j=j-1; 
end 
j=j+(Y_sc-j)*2/3; 
j = round(j); 
if j + 15 < Y_sc 
    R=Q(j:Y_sc); 
else 
    R=Q(Y_sc-5:Y_sc); 
end 
Dn_px=mean(R); 
scale_fac=Dn_px/Dn; 
% scale_fac=145;  % Uncomment to assign scale_fac instead of 
calculating 
% ------------------------------------------------------------- 
% Convert from pixels to mm & Calculate Shape Factor and Surface 
Tension 
De=De_px/scale_fac; 
Ds=Ds_px/scale_fac; 
S = Ds/De;  % Shape Factor 
H = 1/(0.315*S^(-2.608));  % From Padron, 2005 
% Bond_num = H/4; 
% disp(num2str(Bond_num))     % Uncomment to display "Bond number" 
uncalibrated_sigma = 1000*delta_rho*accg*(De/1000)^2/H;  % Surface 
Tension in mN/m 
%sigma = uncalibrated_sigma*1; 
%sigma = uncalibrated_sigma*1.0041-4.1208;  -- OLD CALIBRATION - 
LESS ACCURATE 
sigma = uncalibrated_sigma*.9669+.7326;  % first runs were done with 
"-" instead of "+"  
else 
    sigma = 0; 
end 
 
MATLAB m-file to convert Dynamic Interfacial Tension into an Equilibrium 
Interfacial Tension 
clear all 
t = xlsread('H:\Pendant 
Drop\PD23\PD23L\PD23L1','Matlab_Input','a:a'); 
sigma = xlsread('H:\Pendant 
Drop\PD23\PD23L\PD23L1','Matlab_Input','b:b'); 
N=length(sigma); 
  
eps = 1e-4; 
 
 233 
 
tau_interval = [1 200]; 
tau_mid = (tau_interval(2) + tau_interval(1))/2; 
tau_diff = tau_interval(2) - tau_interval(1); 
iter = 1; 
for j=1:3 
    if j < 3 
        tau = tau_interval(j); 
    else 
        tau = tau_mid; 
    end 
    a=0; 
    b=0; 
    c=0; 
    d=0; 
    e=0; 
    f=0; 
    g=0; 
    for i=1:N 
        a = a + sigma(i); 
        b = b + exp(-t(i)/tau); 
        c = c + sigma(i)*exp(-t(i)/tau); 
        d = d + exp(-2*t(i)/tau); 
        e = e + sigma(i)*t(i)*exp(-t(i)/tau); 
        f = f + t(i)*exp(-2*t(i)/tau); 
        g = g + t(i)*exp(-t(i)/tau); 
    end 
    sigma_c = (a*b-N*c)/(b^2-N*d); 
    sigma_inf = (c-sigma_c*d)/b; 
    dHdt(j) = sigma_c/tau^2*(e-sigma_c*f-sigma_inf*g); 
end 
if (dHdt(1) < 0 && dHdt(3) < 0)  || (dHdt(1) > 0 && dHdt(3) > 0) 
    tau_interval = [tau_mid tau_interval(2)]; 
    dHdt(1) = dHdt(3); 
else 
    tau_interval = [tau_interval(1) tau_mid]; 
    dHdt(2) = dHdt(3); 
end 
  
while tau_diff > eps 
    iter = iter + 1; 
    tau_diff = tau_interval(2) - tau_interval(1); 
    tau_mid = (tau_interval(2) + tau_interval(1))/2; 
    tau = tau_mid; 
    a=0; 
    b=0; 
    c=0; 
    d=0; 
    e=0; 
    f=0; 
    g=0; 
    for i=1:N 
        a = a + sigma(i); 
        b = b + exp(-t(i)/tau); 
        c = c + sigma(i)*exp(-t(i)/tau); 
        d = d + exp(-2*t(i)/tau); 
        e = e + sigma(i)*t(i)*exp(-t(i)/tau); 
        f = f + t(i)*exp(-2*t(i)/tau); 
 
 234 
 
        g = g + t(i)*exp(-t(i)/tau); 
    end 
    sigma_c = (a*b-N*c)/(b^2-N*d); 
    sigma_inf = (c-sigma_c*d)/b; 
    dHdt(3) = sigma_c/tau^2*(e-sigma_c*f-sigma_inf*g); 
     
    if (dHdt(1) < 0 && dHdt(3) < 0)  || (dHdt(1) > 0 && dHdt(3) > 0) 
    tau_interval = [tau_mid tau_interval(2)]; 
    dHdt(1) = dHdt(3); 
    else 
    tau_interval = [tau_interval(1) tau_mid]; 
    dHdt(2) = dHdt(3); 
    end 
end 
disp(['tau = ', num2str(tau)]) 
disp(['sigma_c = sigma_0 - sigma_inf = ' num2str(sigma_c)]) 
disp(['sigma_inf = ' num2str(sigma_inf)]) 
disp(['Number of iterations on tau using midpoint method = ' 
num2str(iter) ' iterations']) 
  
  
dHdt 
 
 
 235 
 
Appendix C: ImageJ Macro Used to Measure Drop Sizes from 
Drop Images 
In order to use microscopy to find the actual sizes of real drops there must be 
some form of calibration. Therefore a calibration standard was acquired which 
allowed the degree of magnification to be quantified precisely. Images of this 
standard are shown in Figure C-1. Since all of the drops were imaged with the same 
microscope configuration and camera the conversion factor from pixels to microns 
was always the same at a value of 6.73 pixels/ m. This configuration was the 
maximum magnification that could be obtained which accommodated all drops and 
had a sufficiently long depth of field. 
 
Figure C-1: Calibration standard. Spacing between the small increments is 10  m 
and between the large increments it is 50  m. This image yields a calibration factor of 
6.73 pixels/ m. 
 
 236 
 
 
In every folder of drops to be analyzed by this macro, the first image is a 
blank image which is used as a background to eliminate any dark pixels not due to 
actual drops. The two sources of such pixels were dust particles on the camera lens, 
which are shown on Figure 4.1.3-3 (or Figure C-2) as the smallest circles, and some 
form of smudging within the microscope assembly which appears as faded, out-of-
 focus circles, such as the one in the upper right-hand corner of Figure 4.1.3-3. These 
features were identified by moving the microscope slide and rotating the camera. The 
first step that the macro takes for each image is to subtract the background grayscale 
values from the image which results in the only non-zero pixels being those of the 
drops. It should be noted that this subtraction procedure necessitates that the lighting 
which is used for the background image also be used for all images in the folder. The 
impossibility of always having precisely the same lighting is the reason for having a 
new background image for each folder. 
 
Figure C-2 (Figure 4.1.3-3): Typical microscope image used to measure drop size. 
Crystal Oil 500FG surrounded by water under a 60x microscope objective with a 
?Watec America Corp. LCL-902K.? 
 
 237 
 
With the preliminary issues of calibration and background subtraction 
accounted for, the stacks of images may be processed by the ImageJ macro. After 
background subtraction, the macro coverts the image to a binary image using 
thresholding, the same type of procedure as for the interfacial tension measurements. 
This results in black rings forming on a white background. The ?Fill Holes? macro, 
which is native to ImageJ, was used to fill the rings and make them solid black 
circles. The solid black circles were counted using the ?Analyze Particles? macro, 
also native to ImageJ. Any rings cut off by the edges were not filled, but were also 
not counted in the next step. The number of pixels of each circle was counted as the 
area which was then converted, using the calibration factor, into the area in microns. 
Drop size was determined by calculating the diameter based on the projected area. 
The Sauter mean diameter was calculated using an excel file for each experimental 
run. 
ImageJ Macro Used to Calculate Drop Sizes 
macro "Tagged Batch Drop Size Measurement" { 
 
dir = getDirectory("Choose a Directory "); 
list = getFileList(dir); 
open(dir+list[0]); 
run("Deinterlace "); 
run("8-bit"); 
background = getImageID(); 
    setBatchMode(true); 
    for (i=0; i<list.length; i++) { 
    showProgress(i+1, list.length); 
    open(dir+list[i]); 
    run("Deinterlace "); 
    run("8-bit"); 
    current_image = getImageID(); 
    imageCalculator ("Subtract create", background, current_image); 
//     
    temporary_image = getImageID(); 
 
 238 
 
    selectImage(current_image); 
    run("Subtract...", "value=255"); 
    imageCalculator("Add", current_image, temporary_image); 
// 
    setThreshold(20, 255); 
    run("Convert to Mask"); 
    run("Fill Holes"); 
    run("Analyze Particles...", "size=75-Infinity circularity=0.88-1.00 show=Nothing 
display exclude"); 
close(); 
      } 
close(); 
 } 
 
 239 
 
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 249 
 
Archival Publications: 
a) Publications that are under review: 
? Dispersion of Water into Oil in a Rotor-Stator Mixer. Part 1: Drop Breakup in Dilute 
Systems.? Paul E. Rueger and Richard V. Calabrese, Chemical Engineering Research and 
Design. 
? Dispersion of Water into Oil in a Rotor-Stator Mixer. Part 2: Effect of Phase Fraction.? Paul 
E. Rueger and Richard V. Calabrese, Chemical Engineering Research and Design. 
b) Publications that will be submitted after the defense: 
? Invited submission as part of AIChE Journal?s best paper program. ?Power Draw 
Measurements and Correlation of Single Pass Drop Size Data in an In-Line Rotor-Stator 
Mixer.? Paul E. Rueger, B. N. Murthy, Kenneth T. Kiger, and Richard V. Calabrese, AIChE 
Journal. 
Conference Proceedings: 
a) Proceedings that have been published: 
? ?The Effect of Phase Fraction on Drop Size Distribution in a High Shear Mixer with a 
Viscous Continuous Phase.? Paul E. Rueger and Richard V. Calabrese, Proceedings of the 
14th European Conference on Mixing, pages 419-424. EFCE Event #711. Warsaw, Poland. 
September 10-13, 2012. 
Conference Presentations: 
a) Presentations that have been delivered: 
?  ?Dispersion of Water in Viscous Oils in a High Shear Mixer.? Paul E. Rueger and Richard 
V. Calabrese, 2010 AIChE Annual Meeting. Salt Lake City, UT, USA. November 2010. 
Speaker: Paul E. Rueger 
? NAMF (North American Mixing Forum) 2012 Student Paper Contest Winner: ?Dilute 
Dispersion of Water into Oil in a Batch Rotor-Stator Mixer.? Paul E. Rueger and Richard 
V. Calabrese, Mixing XXIII, Canc?n, Mexico. June 2012. Speaker: Paul E. Rueger 
? ?The Effect of Phase Fraction on Drop Size Distribution in a High Shear Mixer with a 
Viscous Continuous Phase.? Paul E. Rueger and Richard V. Calabrese, 14th Annual 
Conference on Mixing. Warsaw, Poland. September 2012. Speaker: Richard V. Calabrese 
?  ?Power Draw Measurements and Correlation of Single Pass Drop Size Data in an In-Line 
Rotor-Stator Mixer.? Paul E. Rueger, B. N. Murthy, Kenneth T. Kiger, and Richard V. 
Calabrese, 2012 AIChE Annual Meeting. Pittsburgh, PA, USA. October 2012. Speaker: 
Richard V. Calabrese