ABSTRACT 
 
Title of Dissertation: CRYSTAL WET MILLING IN ROTOR-STATOR MIXERS 
Author: Kanan Ghaderzadeh 
Doctor of Philosophy, 2018 
Directed by: Professor Richard V. Calabrese  
Department of Chemical and Biomolecular Engineering 
 
Wet milling in rotor-stator mixers to reduce crystal size is an emerging practice 
with significant potential in crystallization and milling of active pharmaceutical 
ingredients. The complexities of crystal breakage behavior, the turbulent nature of flow, 
and multiscale geometry in rotor-stator mixers have limited the understanding of the 
milling process in these devices. The purpose of this work is to investigate the effect of 
milling conditions, crystal physical properties, and mixer geometry on the milling 
behavior of crystals, and to introduce a mechanistic framework on which to base mill 
design, scale up, and operating strategies. 
A Silverson L4R rotor-stator mixer with a square-hole stator head was used to 
conduct systematic wet milling experiments. Three mill head geometries were studied. 
There are inline units with a standard shear gap (clearance between rotor and stator) and 
an enlarged shear gap, and a batch unit with a standard shear gap. Three different crystals 
with different elastic modulus, hardness, and fracture toughness were milled in an anti-
solvent. Rotor speed and volumetric throughput were adjusted independently to vary 
energy input and mill head residence time. The milling rate was found to increase with 
  
higher rotor speed and lower throughput, while the ultimate particle size (maximum 
particle size at the end of milling) was only dependent on rotor speed. The effects of fluid 
agitation, particle-particle collisions, and particle-wall collisions on crystal breakage were 
assessed by changing particle concentration, coating the stator surfaces, and reducing 
rotor diameter. It was found that the concentration of particles in the slurry has a limited 
effect on milling rate and negligible effect on ultimate particle size. The effect of higher 
power input and smaller dispersion zone volume (the volume of the shear gap and stator 
hole regions), through changes to the mixing head geometry, showed to be more 
successful in concentrating the energy input, and hence, leads to higher milling rates and 
smaller ultimate particle size. 
To quantitatively explain the experimental results, a class of mechanistic models 
for crystal breakage were developed that consider the influence of plastic deformation, 
elastic deformation, and fracture toughness on breakage resistance. These models are in 
agreement with classical grinding theories. Based on the particle size scale, two classes of 
disruptive forces were studied considering either macroscale velocity (proportional to 
rotor tip speed) or inertial subrange turbulent eddy velocity (given by Kolmogorov’s 
inertial subrange model). Four cohesive force definitions were studied, each with 
different dependence on physical properties and particle size. The disruptive and cohesive 
forces were employed to construct eight different correlations for ultimate crystal size 
and its rate of approach. Model discrimination is based on comparison to the wet milling 
experimental data. The best-fitting model was based on the combination of inertial 
subrange model as the disruptive force and elastic-plastic deformation model as the 
cohesive force. 
  
For geometrically similar devices, a dimensionless comminution number was 
developed (ratio of disruptive to cohesive forces) to aid physical interpretation and scale 
up/down efforts. For devices without geometric similarity, the concept of local energy 
dissipation rate, defined as the power draw of the mixer per mass of fluid in shear gap 
and stator slot regions (calculated through computational fluid dynamics simulations), 
was introduced and exploited to compare the data from different mixers with different 
geometries. This approach was successful in correlating the maximum stable particle size 
resulting from wet milling of different crystals in a Silverson L4R inline mixer with 
standard and enlarged shear gap, and in a Silverson L4R batch mixer at different rotor 
speeds.  
The mechanistic theory is further utilized to provide breakage kernels based on 
probability of collision and collision rate theories. Application of the breakage functions 
to predict the milling rate as well as their implementation within a population balance 
framework is discussed. 
 
  
 
 
  
  
 
 
 
CRYSTAL WET MILLING IN ROTOR-STATOR MIXERS 
 
By 
 
Kanan Ghaderzadeh 
 
 
 
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 
2018 
 
 
 
 
Advisory Committee: 
Professor Richard V. Calabrese, Chair 
Professor Akua A. Asa-Awuku 
Professor Panagiotis Dimitrakopoulos 
Professor Kenneth T. Kiger 
Professor Michael R. Zachariah 
 
 
 
  
 
 
 
 
 
 
 
 
 
© Copyright by 
 
Kanan Ghaderzadeh 
2018 
 
 
 
 
 
 
 
 
  
 
Table of Contents 
Table of Contents ............................................................................................................... ii 
List of Tables .................................................................................................................. vvi 
List of Figures ................................................................................................................. viii 
Nomenclature ................................................................................................................ xviii 
Chapter 1: Introduction ...................................................................................................... 1 
1.1 –Motivation and Purpose ......................................................................................... 1 
1.2 –Approach ................................................................................................................ 2 
1.3 –Organization of the Dissertation ............................................................................ 5 
Chapter 2: Particle Size and Physical Property Measurements ......................................... 8 
2.1 –Particle Size ........................................................................................................... 8 
2.2 –Particle size distribution ....................................................................................... 12 
2.3 –Distribution Statistics ........................................................................................... 13 
2.4 –Sampling .............................................................................................................. 16 
2.5 –Particle Size Measurement Techniques ............................................................... 18 
2.5.1 –Sieving .......................................................................................................... 19 
2.5.2 –Image Analysis.............................................................................................. 20 
2.5.3 –Laser Diffraction ........................................................................................... 22 
2.5.4 –Focused Beam Reflectance Measurement (FBRM) ..................................... 25 
2.5.5 –Particle Vision Measurement (PVM)............................................................ 26 
2.6 –Particle Physical Properties.................................................................................. 28 
2.6.1 –Young’s Modulus.......................................................................................... 29 
2.6.2 –Hardness ........................................................................................................ 30 
2.6.3 –Fracture Toughness ....................................................................................... 30 
2.7 –Indentation Test ................................................................................................... 31 
2.7.1 –Measurements of Particle Mechanical Properties ......................................... 31 
2.8 – Summary ............................................................................................................. 33 
Chapter 3: Comminution Theories and Mechanistic Models for Particle Breakage ....... 35 
3.1 – Classical Comminution Theories ........................................................................ 35 
3.1.1 –Rittinger’s Law ............................................................................................. 36 
3.1.2 –Kick’s Law .................................................................................................... 37 
3.1.3 –Bond’s Law ................................................................................................... 37 
3.1.4 –Size Range Applicability of Comminution Laws ......................................... 39 
3.2 –Fracture Mechanics .............................................................................................. 40 
3.3 –Mechanistic Breakage and Attrition Theories ..................................................... 44 
3.3.1 –Indentation Mechanism ................................................................................. 45 
3.3.2 –Elastic Deformation Mechanism................................................................... 46 
ii 
 
 
3.3.3 –Plastic Deformation Mechanism ................................................................... 49 
3.4 –Breakage Model Based on Elastic-Plastic Deformation Mechanism .................. 52 
3.5 –Similarities between Comminution Laws and Mechanistic Models.................... 56 
3.5.1 –Lawn and Marshal Approach Compared to Kick’s Law .............................. 56 
3.5.2 –Ghadiri and Zhang Model Compared to Rittinger’s Law ............................. 56 
3.5.3 –Gahn and Mersmann Model Compared to Bond’s and Rittinger’s Law ...... 57 
3.5.4 –Current Study’s Model Compared to Bond’s Law ....................................... 58 
3.6 –Mechanistic Models for Wet Milling in Rotor Stator Mixers.............................. 58 
3.6.1 –Disruptive Forces .......................................................................................... 59 
3.6.2 –Cohesive Forces ............................................................................................ 60 
3.6.3 –Breakage Model ............................................................................................ 62 
3.7–Dimensionless Number for Particle Breakage in Geometrically Similar Systems65 
3.8 –Summary .............................................................................................................. 66 
Chapter 4: Wet Milling Experiments in Rotor Stator Mixers.......................................... 69 
4.1 –Introduction .......................................................................................................... 69 
4.2 –Rotor-Stator Mixer Geometry .............................................................................. 72 
4.3 –Experiment Setup ................................................................................................. 74 
4.3.1 –Inline Mixer Setup ........................................................................................ 74 
4.3.2 –Batch Mixer Setup ........................................................................................ 76 
4.4 –Particle Size Measurement Procedure ................................................................. 77 
4.5 –Mechanical Properties Measurement Procedure .................................................. 78 
4.6 –Experiment Procedure .......................................................................................... 79 
4.7 –Size Distribution Considerations ......................................................................... 80 
4.8 –Experiment Results .............................................................................................. 82 
4.8.1 –Silverson L4R inline mixer with standard shear gap .................................... 82 
4.8.2 –Silverson L4R Inline Mixer with Enlarged Shear Gap ............................... 100 
4.8.3 –Silverson L4R Batch Mixer ........................................................................ 107 
4.9 –Summary ............................................................................................................ 111 
Chapter 5: Model Validation and Mechanistic Correlations ......................................... 114 
5.1 – Suitability of Different Mechanistic Models to Predict the Wet Milling Behavior
.................................................................................................................................... 114 
5.2 –Scale Up strategy based on Local Energy Dissipation Rate .............................. 119 
5.3 –Rate of Size Reduction and Mechanistic Correlations for Breakage Kernel ..... 126 
5.4 –Summary ............................................................................................................ 135 
Chapter 6: CFD Simulations of the Silverson L4R Inline Mixer .................................. 138 
6.1 –Power Draw and Power Number in Rotor-Stator Mixers .................................. 138 
6.2 –Geometry Details and Grid Generation Strategy ............................................... 144 
6.3 –Solution Methods ............................................................................................... 148 
6.4 –Simulation Results and Discussion .................................................................... 150 
iii 
 
 
6.4.1 –Velocities within the Rotor and Stator Zones ............................................. 151 
6.4.2 –Total Energy Dissipation Rate and Turbulent Kinetic Energy ................... 158 
6.4.3 –Power Draw and Power Number Calculations ........................................... 170 
6.5 –Summary ............................................................................................................ 174 
Chapter 7: Summary and Conclusion ............................................................................ 176 
7.1 –Particle Size and Physical Properties ................................................................. 177 
7.2 –Mechanistic Models for particle breakage ......................................................... 178 
7.3 –Wet Milling Experiments in Rotor-Stator Mixers ............................................. 179 
7.4 –Model Validation and Mechanistic Correlations of Wet Milling ...................... 180 
7.5 –CFD Simulations and Power Draw Calculations ............................................... 181 
7.6 –Future Work ....................................................................................................... 182 
Appendix A: Wet milling data for sucrose-IPA system in the Silverson L4R inline mixer 
with standard shear gap .................................................................................................. 184 
A.1 – Wet Milling of Sucrose/IPA at 5000 rpm and 2.1 lpm .................................... 185 
A.2 – Wet Milling of Sucrose/IPA at 6500 rpm and 2.1 lpm .................................... 189 
A.3 – Wet Milling of Sucrose/IPA at 8000 rpm and 2.1 lpm .................................... 193 
A.4 – Wet Milling of Sucrose/IPA at 5000 rpm and 3.0 lpm .................................... 197 
A.5 – Wet Milling of Sucrose/IPA at 6500 rpm and 3.0 lpm .................................... 201 
A.6 – Wet Milling of Sucrose/IPA at 8000 rpm and 3.0 lpm .................................... 205 
A.7 – Wet Milling of Sucrose/IPA at 5000 rpm and 4.2 lpm .................................... 209 
A.8 – Wet Milling of Sucrose/IPA at 6500 rpm and 4.2 lpm .................................... 213 
A.9 – Wet Milling of Sucrose/IPA at 8000 rpm and 4.2 lpm .................................... 217 
Appendix B: Wet milling data for glycine-IPA system in the Silverson L4R inline mixer 
with standard shear gap .................................................................................................. 221 
B.1 – Wet Milling of glycine/IPA at 5000 rpm and 2.1 lpm ..................................... 221 
B.2 – Wet Milling of glycine/IPA at 6500 rpm and 3.0 lpm ..................................... 225 
B.3 – Wet Milling of glycine/IPA at 8000 rpm and 4.2 lpm ..................................... 229 
Appendix C: Wet milling data for ascorbic-IPA system in the Silverson L4R inline mixer 
with standard shear gap .................................................................................................. 234 
C.1 – Wet Milling of ascorbic acid/IPA at 5000 rpm and 2.1 lpm ............................ 234 
C.2 – Wet Milling of ascorbic acid/IPA at 6500 rpm and 3.0 lpm ............................ 238 
C.3 – Wet Milling of ascorbic acid/IPA at 8000 rpm and 4.2 lpm ............................ 242 
Appendix D: Wet milling data measured with the FBRM probe for sucrose-IPA system in 
the Silverson L4R inline mixer with standard shear gap ............................................... 247 
D.1 – FBRM results of wet milling of sucrose/IPA at 5000 rpm and 3.0 lpm .......... 247 
D.2 – FBRM results of wet milling of sucrose/IPA at 6500 rpm and 3.0 lpm .......... 249 
D.3 – FBRM results of wet milling of sucrose/IPA at 8000 rpm and 3.0 lpm .......... 251 
Appendix E: Flow Field Animations ............................................................................. 253 
iv 
 
 
References.. .................................................................................................................... 254 
v 
 
 
List of Tables 
Table 2.1 - Definitions of the equivalent diameters based on measured properties of the 
particle ....................................................................................................................... 10 
Table 3.1 - Definition of stress intensity factor and critical stress intensity factor .......... 41 
Table 3.2 - Disruptive stress, 𝜏𝐷 ~ 𝜌𝑢
2 predicted by different models: macroscale 
turbulence and inertial subrange turbulence models ................................................. 60 
Table 3.3 - Cohesive stresses, 𝜏𝐶, resisting particle breakage predicted by different 
mechanisms: indentation, elastic, plastic, and elastic-plastic models ....................... 62 
Table 3.4 - Breakage models: mechanistic correlations for maximum stable particle size 
(𝑥𝑚𝑎𝑥 𝑜𝑟 𝐷100) obtained by a balance between the cohesive and disruptive stresses
 ................................................................................................................................... 63 
Table 3.5 - Breakage models based on energy dissipation rate (disruptive stresses are 
calculated based on inertial subrange turbulence) ..................................................... 64 
Table 3.6 - Comminution number for geometrically similar systems: definition and 
application in different mechanistic models .............................................................. 66 
Table 4.1 - Silverson L4R inline and batch mixer geometry details ................................ 74 
Table 4.2 - Mechanical Properties of Crystals and Brittleness Indices ............................ 79 
Table 4.3 - The matrix of different experiments performed in the Silverson L4R inline 
mixer with the standard shear gap ............................................................................. 82 
Table 5.1 - Goodness of fit of different models to collapse the experimental data for 
milling of different materials in the inline mixer with the standard shear gap (IST = 
Inertial subrange of turbulence) ............................................................................... 116 
Table 5.2 - Volume of dissipersion zone for inline mixer with standard and enlarged 
shear gaps and batch mixer ...................................................................................... 120 
Table 5.3 - Tip speed, Reynolds number, power, and power number of Silverson L4R 
inline mixer with standadr shear gap at different rotor speeds – CFD investigation 
(see Chapter 6 for details) ........................................................................................ 121 
Table 5.4 - Tip speed, Reynolds number, power, and power number of Silverson L4R 
inline mixer with enlarged shear gap at different rotor speeds – CFD investigation 
(see Chapter 6 for details) ........................................................................................ 121 
vi 
 
 
Table 5.5 - Tip speed, Reynolds number, power, and power number of Silverson LR 
batch mixer at different rotor speeds based on data of Padron [63] ........................ 122 
Table 5.6 - Goodness of fit of different models to collapse the experimental data for 
milling of different materials in the inline mixer with the standard and enlarged shear 
gap and in the batch mixer ....................................................................................... 124 
Table 5.7 - Power law and exponential breakage kernels with inertial subrange turbulence 
model as the disruptive stress .................................................................................. 128 
Table 5.8 - Power law and exponential breakage kernels with macroscale model as the 
disruptive stress ....................................................................................................... 128 
Table 6.1 - Power number discrepancy between measurement and simulation results of 
Silverson L4R batch mixer with different stator geometries ................................... 142 
Table 6.2 - Geometry details of Silverson L4R inline and batch mixers ........................ 145 
Table 6.3 - Reynolds number, power number, and pumping number for Silverson L4R 
inline mixer with SSG and ESG at different rotor speeds ....................................... 173 
vii 
 
 
List of Figures 
Figure 2.1 - Definition of statistical diameters representing different dimensions of the 
particle ....................................................................................................................... 11 
Figure 2.2 - Unimodal frequency and cumulative size distribution obtained during wet 
milling (L4R batch mixer at 8000 rpm) of sucrose in IPA and its corresponding D10, 
D50, and D90 values measured by laser diffraction method ..................................... 15 
Figure 2.3 - Bimodal frequency and cumulative size distribution obtained during wet 
milling (L4R batch mixer at 8000 rpm) of sucrose in IPA and its corresponding D10, 
D50, and D90 values measured by laser diffraction method ..................................... 16 
Figure 2.4 - Degrees of dispersion - partial suspension: some particles rest on the bottom 
of the tank for short periods; complete suspension: all particles are off the bottom of 
the vessel; uniform suspension: particle concentration and size distribution are 
practically homogeneous throughout the vessel [2]. ................................................. 18 
Figure 2.5 - A microscope image of sucrose particles and its processed version using 
ImageJ image processing software ............................................................................ 21 
Figure 2.6 - Horiba LA 950 laser diffraction instrument and its sampling cell ................ 24 
Figure 2.7 - Lasentec D600 FBRM probe and schematic of probe internal parts showing 
the laser crossing a particle surface ........................................................................... 25 
Figure 2.8 - Particle Vision Measurement (PVM) probe, Particle View V19 from Mettler-
Toledo Autochem and schematic of probe tip ........................................................... 27 
Figure 2.9 - PVM images of sucrose crystals in IPA captured inline during wet milling 
(L4R inline mixer with standard shear gap), RBI of images are calculated at different 
milling times .............................................................................................................. 28 
Figure 2.10 - Elastic modulus is related to the atomic bonding forces; the higher atomic 
bonds (sharper slope), higher the elastic modulus ..................................................... 30 
Figure 2.11 - Indentation area and crack propagation in an indentation test .................... 32 
Figure 2.12 - TI 950 Hysitron nanoindentation device used in this study to measure 
elastic modulus, hardness, and fracture toughness of different crystals .................... 33 
Figure 3.1 - Size range applicability of grinding laws in a log-log representation [21]: 
Kick’s theory is applicable for large particle size, Rittinger’s for very small particle 
size, and the Bond’s work index is suitable for intermediate particle size ................ 39 
viii 
 
 
Figure 3.2 - Stress concentration near the crack tip of a solid under stress ...................... 40 
Figure 3.3 - Fracture behavior of materials under different process conditions 
(confinement is the geometric constraints on the surface of the particle) [7] ........... 43 
Figure 3.4 - Illustration of cone shape particle under impact energy as shown in Gahn and 
Mersmann (1997) for derivation of mechanistic model based on elastic mechanism47 
Figure 3.5 - Illustration of cubic shape particle under impact energy as shown in Ghadiri 
and Zhang (2002) for derivation of mechanistic model based on plastic mechanism
 ................................................................................................................................... 50 
Figure 3.6 - Illustration of plastic and elastic deformation of a particle under impact at its 
corner, a is plastic deformation length scale, li is elastic deformation length scale in 
fractured volume, and x is initial particle size ........................................................... 53 
Figure 4.1 - Silverson L4R Rotor Stator Mixers with Square Hole Stator (a) Inline mixer, 
(b) Batch mixer .......................................................................................................... 73 
Figure 4.2 - Experimental setup of Silverson inline rotor-stator mixer feed through a 
recirculation loop in line with a stirred vessel ........................................................... 75 
Figure 4.3 - Experimental setup of Silverson batch mixer in a glass vessel immersed in a 
cooling bath ............................................................................................................... 77 
Figure 4.4 - Microscope images of sucrose, glycine, and ascorbic acid before milling ... 78 
Figure 4.5 - Relationship between D90 and other cumulative particle sizes – data are 
from wet milling of sucrose/IPA in the Silverson L4R inline mixer with the standard 
shear gap at different rotor speeds and flow rates ..................................................... 81 
Figure 4.6 - Relationship between D90 and mean particle sizes – data are from wet 
milling of sucrose/IPA in the Silverson L4R inline mixer with the standard shear gap 
at different rotor speeds and flow rates ..................................................................... 81 
Figure 4.7 - Initial size distribution (obtained by sieving), measured with the laser 
diffraction method - wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm – 
Silverson L4R inline mixer with standard shear gap ................................................. 83 
Figure 4.8 - Particle size distribution, measured with the laser diffraction method,  
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 1 tank 
turnover – Silverson L4R inline mixer with standard shear gap ............................... 83 
ix 
 
 
Figure 4.9 - Particle size distribution, measured with the laser diffraction method,  
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 3 tank 
turnovers – Silverson L4R inline mixer with standard shear gap.............................. 84 
Figure 4.10 - Particle size distribution, measured with the laser diffraction method,  
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 5 tank 
turnovers – Silverson L4R inline mixer with standard shear gap.............................. 84 
Figure 4.11 - Particle size distribution, measured with the laser diffraction method,  
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 7 tank 
turnovers – Silverson L4R inline mixer with standard shear gap.............................. 84 
Figure 4.12 - Final size distribution, measured with the laser diffraction method,  resulted 
from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm – Silverson L4R inline 
mixer with standard shear gap ................................................................................... 85 
Figure 4.13 - Online PVM images of sucrose in IPA at initial size distribution - wet 
milling at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear 
gap ............................................................................................................................. 85 
Figure 4.14 - Online PVM images of sucrose in IPA after 1 tank turnover - wet milling at 
8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap ........... 85 
Figure 4.15 - Online PVM images of sucrose in IPA after 3 tank turnovers - wet milling 
at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap ....... 86 
Figure 4.16 - Online PVM images of sucrose in IPA after 5 tank turnovers - wet milling 
at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap ....... 86 
Figure 4.17 - Online PVM images of sucrose in IPA after 11 tank turnovers - wet milling 
at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap ....... 86 
Figure 4.18 - Online PVM images of sucrose in IPA after 15 tank turnovers - wet milling 
at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap ....... 87 
Figure 4.19 - SEM images of sucrose before and after milling at 8000 rpm and 3.0 lpm – 
Silverson L4R inline mixer with standard shear gap ................................................. 87 
Figure 4.20 - Rotor speed effect on sucrose breakage (laser diffraction measurements) – 
The flow rate is fixed at 2.1 lpm – lower graph is log-normal representation of the 
data in upper graph – Silverson L4R inline mixer with standard shear gap .............. 89 
Figure 4.21 - Rotor speed effect on sucrose breakage (laser diffraction measurements) – 
The flow rate is fixed at 4.2 lpm – lower graph is log-normal representation of the 
data in upper graph – Silverson L4R inline mixer with standard shear gap .............. 90 
x 
 
 
Figure 4.22 - Flow rate effect on particle breakage (laser diffraction measurements) - The 
rotor speed is fixed at 5000 rpm – lower graph is log-normal representation of the 
data in upper graph - Silverson L4R inline mixer with standard shear gap .............. 91 
Figure 4.23 - Flow rate effect on particle breakage (laser diffraction measurements) - The 
rotor speed is fixed at 8000 rpm – lower graph is log-normal representation of the 
data in upper graph - Silverson L4R inline mixer with standard shear gap .............. 92 
Figure 4.24 - Rotor speed and flow rate effect on wet milling of sucrose in IPA; chord 
length measurement by FBRM - Silverson L4R inline mixer with standard shear gap
 ................................................................................................................................... 94 
Figure 4.25 - Rotor speed and flow rate effect on wet milling of sucrose in IPA; RBI 
measurement by PVM - Silverson L4R inline mixer with standard shear gap ......... 94 
Figure 4.26 - Wet milling of different crystals, ascorbic acid, glycine, and sucrose (laser 
diffraction measurements) - 8000 rpm and 4.2 lpm – lower graph is log-normal 
representation of the data in upper graph - Silverson L4R inline mixer with standard 
shear gap .................................................................................................................... 95 
Figure 4.27 - Wet milling of different crystals, ascorbic acid, glycine, and sucrose (laser 
diffraction measurements) - 6500 rpm and 3.0 lpm – lower graph is log-normal 
representation of the data in upper graph - Silverson L4R inline mixer with standard 
shear gap .................................................................................................................... 96 
Figure 4.28- A particle with higher brittleness index breaks down to smaller D90 and 
larger D10 .................................................................................................................. 97 
Figure 4.29 - Size of attrited small chips in milling of different crystals, ascorbic acid, 
glycine, and sucrose, at 6500 rpm and 3.0 lpm (D10 is 10% cumulative size 
measured by laser diffraction method); lower grapg is log-normal representation of 
the data in upper graph - Silverson L4R inline mixer with standard shear gap ........ 97 
Figure 4.30 - Size of attrited small chips in milling of different crystals at 6500 rpm and 
3.0 lpm (D10 is the arithmetic or number mean measured by laser diffraction 
method); lower graph is log-normal representation of the data in upper graph - 
Silverson L4R inline mixer with standard shear gap ................................................. 98 
Figure 4.31- Concentration effect on wet milling performance – 6500 rpm and 3.0 lpm - 
Silverson L4R inline mixer with standard shear gap (laser diffraction measurements)
 ................................................................................................................................... 99 
xi 
 
 
Figure 4.32 - Particle-wall collisions cause the removal of paint coatings on inside wall 
and approach edges of the stator slots - Silverson L4R inline mixer with the standard 
shear gap .................................................................................................................. 100 
Figure 4.33 - Geometry of rotor and stator for standard and enlarged shear gap in 
Silverson L4R inline mixer...................................................................................... 101 
Figure 4.34 - Initial size distribution (obtained by sieving) measured with the laser 
diffraction method - wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm – 
Silverson L4R inline mixer with enlarged shear gap .............................................. 102 
Figure 4.35 - Particle size distribution measured with laser diffraction method resulted 
from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 15 tank turnovers – 
Silverson L4R inline mixer with enlarged shear gap .............................................. 102 
Figure 4.36 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm and 3.0 lpm after 30 tank turnovers – Silverson L4R inline mixer with 
enlarged shear gap ................................................................................................... 103 
Figure 4.37 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm and 3.0 lpm after 45 tank turnovers – Silverson L4R inline mixer with 
enlarged shear gap ................................................................................................... 103 
Figure 4.38 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm and 3.0 lpm after 90 tank turnovers – Silverson L4R inline mixer with 
enlarged shear gap ................................................................................................... 104 
Figure 4.39 - Final size distribution resulted from wet milling of sucrose in IPA at 8000 
rpm and 3.0 lpm – Silverson L4R inline mixer with enlarged shear gap ................ 104 
Figure 4.40 - Silverson L4R inline mixer with enlarged shear gap; rotor speed effect on 
sucrose breakage (laser diffraction measurement) – Flow rate is fixed at 3.0 lpm . 105 
Figure 4.41 - Shear gap effect on milling performance of inline rotor-stator device – wet 
milling of sucrose in IPA at 5000 rpm and 3.0 lpm ................................................ 105 
Figure 4.42 - Shear gap effect on milling performance of inline rotor-stator device – wet 
milling of sucrose in IPA at 6500 rpm and 3.0 lpm ................................................ 106 
Figure 4.43 - Shear gap effect on milling performance of inline rotor-stator device – wet 
milling of sucrose in IPA at 8000 rpm and 3.0 lpm ................................................ 106 
Figure 4.44 - Different loading modes in a rotor-stator mixer, a) Friction or compression, 
b) Particle-wall collision, c) Particle-particle collision ........................................... 107 
xii 
 
 
Figure 4.45 - Initial size distribution (obtained by sieving) measured with the laser 
diffraction method - wet milling of sucrose in IPA at 8000 rpm – Silverson L4R 
batch mixer .............................................................................................................. 108 
Figure 4.46 - Particle size distribution measured with the laser diffraction method 
resulted from wet milling of sucrose in IPA at 8000 rpm after 1 min – Silverson L4R 
batch mixer .............................................................................................................. 109 
Figure 4.47 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm after 2 mins – Silverson L4R batch mixer .............................................. 109 
Figure 4.48 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm after 3 mins – Silverson L4R batch mixer .............................................. 110 
Figure 4.49 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm after 10 mins – Silverson L4R batch mixer ............................................ 110 
Figure 4.50 - Final (t= 90 mins) particle size distribution resulted from wet milling of 
sucrose in IPA at 8000 rpm – Silverson L4R batch mixer ...................................... 110 
Figure 4.51 - Silverson L4R batch mixer; rotor speed effect on sucrose breakage (laser 
diffraction measurement) – lower graph is log-normal representation of the data in 
upper graph .............................................................................................................. 111 
Figure 5.1- Suitability of different models to fit the wet milling equilibrium data for 
different crystals in Silverson L4R inline mixer with standard shear gap (trend line 
comparison) ............................................................................................................. 115 
Figure 5.2 - Suitability of the model based on indentation approach (Lawn and Marshal 
model) to predict the wet milling behavior of different crystals in Silverson L4R 
inline rotor-stator mixer with standard shear gap at different rotor speeds ............. 117 
Figure 5.3 - Suitability of the model based on plastic mechanism (Ghadiri and Zhang 
model) to predict the wet milling behavior of different crystals in Silverson L4R 
inline rotor-stator mixer with standard shear gap at different rotor speeds ............. 117 
Figure 5.4 - Suitability of the model based on elastic mechanism (Gahn and Mersmann 
model) to predict the wet milling behavior of different crystals in Silverson L4R 
inline rotor-stator mixer with standard shear gap at different rotor speeds ............. 118 
Figure 5.5 - Suitability of the model based on elastic-plastic mechanism (current study) to 
predict the wet milling behavior of different crystals in Silverson L4R inline rotor-
stator mixer with standard shear gap at different rotor speeds ................................ 118 
xiii 
 
 
Figure 5.6 - Maximum stable particle size plotted against local maximum energy 
dissipation rate for wet milling of sucrose, glycine, and ascorbic acid in Silverson 
L4R inline mixer with standard and enlarged shear gap, and Silverson L4R batch 
mixer at 5000, 6000, 6500, and 8000 rpm ............................................................... 123 
Figure 5.7 - Suitability of the model based on indentation approach to collapse the wet 
milling equilibrium data for different crystals in Silverson L4R inline rotor-stator 
mixer with standard and enlarged shear gap, and batch mixer at different rotor speeds
 ................................................................................................................................. 124 
Figure 5.8 - Suitability of the model based on plastic mechanism to collapse the wet 
milling equilibrium data for different crystals in Silverson L4R inline rotor-stator 
mixer with standard and enlarged shear gap, and batch mixer at different rotor speeds
 ................................................................................................................................. 125 
Figure 5.9 - Suitability of the model based on elastic mechanism to collapse the wet 
milling equilibrium data for different crystals in Silverson L4R inline rotor-stator 
mixer with standard and enlarged shear gap, and batch mixer at different rotor speeds
 ................................................................................................................................. 125 
Figure 5.10 - Suitability of the model based on elastic-plastic mechanism to collapse the 
wet milling equilibrium data for different crystals in Silverson L4R inline mixer with 
standard and enlarged shear gap, and batch mixer at different rotor speeds ........... 126 
Figure 5.11 - Suitability of breakage kernel based on elastic-plastic model in predicting 
the milling rate of sucrose in inline mixer with standard and enlarged shear gaps at 
different rotor speeds and constant flow rate of 3.0 lpm, dotted lines are the model 
fits; lower graph is log-normal representation of the data in upper graph .............. 131 
Figure 5.12 - Suitability of breakage kernel based on plastic model in predicting the 
milling rate of sucrose in inline mixer with standard and enlarged shear gaps at 
different rotor speeds and constant flow rate of 3.0 lpm, dotted lines are the model 
fits; lower graph is log-normal representation of the data in upper graph .............. 132 
Figure 5.13 - Suitability of breakage kernel based on elastic model in predicting the 
milling rate of sucrose in inline mixer with standard and enlarged shear gaps at 
different rotor speeds and constant flow rate of 3.0 lpm, dotted lines are the model 
fits; lower graph is log-normal representation of the data in upper graph .............. 133 
Figure 5.14 - Suitability of breakage kernel based on elastic-plastic model in predicting 
the milling rate of sucrose, glycine, and ascorbic acid in inline mixer with standard 
shear gap at 6500 rpm and 3.0 lpm, dotted lines are the model fits; lower graph is 
log-normal representation of the data on upper graph ............................................. 134 
xiv 
 
 
Figure 5.15 - Suitability of breakage kernel based on elastic-plastic model in predicting 
the milling rate of glycine in inline mixer with standard shear gap at different milling 
conditions, dotted lines are the model fits; lower graph is log-normal representation 
of the data on upper graph ....................................................................................... 135 
Figure 5.16 - Suitability of breakage kernel based on elastic-plastic model in predicting 
the milling rate of ascorbic acid in inline mixer with standard shear gap at different 
milling conditions, dotted lines are the model fits; lower graph is log-normal 
representation of the data on upper graph ............................................................... 136 
Figure 6.1 - (a) centered  and (b) off-centered  configurations of Silverson L4R batch 
mixer inside a holding tank ..................................................................................... 143 
Figure 6.2 - Silverson L4R inline mixer. The image at the left is the entire assembly with 
attached inlet and outlet pipes. The right image show the high-shear mill head with 
the rotor surrounded by the stator. ........................................................................... 144 
Figure 6.3 - Illustration of rotating and stationary zones – the stationary zone is further 
divided into stator and volute zones for better control of meshing strategy ............ 146 
Figure 6.4 - Grid refinement in standard shear gap, Silverson L4R inline mixer........... 147 
Figure 6.5 - Grid refinement in enlarged shear gap, Silverson L4R inline mixer .......... 148 
Figure 6.6 - The coordinate system shown on the mid-plane of Silverson L4R inline 
mixer ........................................................................................................................ 150 
Figure 6.7 - Velocity vector flow field for Silverson L4R inline mixer with standard and 
enlarged shear gap at 8000 rpm and rotor position of Ѳ = -23° .............................. 153 
Figure 6.8 - Velocity vector flow field for Silverson L4R inline mixer with standard and 
enlarged shear gap at 8000 rpm and rotor position of Ѳ = -13° .............................. 154 
Figure 6.9 - Velocity vector flow field for Silverson L4R inline mixer with standard and 
enlarged shear gap at 8000 rpm and rotor position of Ѳ = 0 ................................... 155 
Figure 6.10 - Velocity vector flow field for Silverson L4R inline mixer with standard and 
enlarged shear gap at 8000 rpm and rotor position of Ѳ = 13° ............................... 156 
Figure 6.11 - Velocity vector flow field for Silverson L4R inline mixer with standard and 
enlarged shear gap at 8000 rpm and rotor position of Ѳ = 23° ............................... 157 
Figure 6.12 - Contours of normalized turbulent energy dissipation rate for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -23° 
(normalized by N3D2 of SSG) ................................................................................. 159 
xv 
 
 
Figure 6.13 - Contours of normalized turbulent energy dissipation rate for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -13° 
(normalized by N3D2 of SSG) ................................................................................. 160 
Figure 6.14 - Contours of normalized turbulent energy dissipation rate for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 0 
(normalized by N3D2 of SSG) ................................................................................. 161 
Figure 6.15 - Contours of normalized turbulent energy dissipation rate for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 13 
(normalized by N3D2 of SSG) ................................................................................ 162 
Figure 6.16 - Contours of normalized turbulent energy dissipation rate for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 23 
(normalized by N3D2 of SSG) ................................................................................ 163 
Figure 6.17 - Contours of normalized turbulent kinetic energy for Silverson L4R inline 
mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -23° (normalized 
by N2D2 of SSG) ...................................................................................................... 165 
Figure 6.18 - Contours of normalized turbulent kinetic energy for Silverson L4R inline 
mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -13° (normalized 
by N2D2 of SSG) ...................................................................................................... 166 
Figure 6.19 - Contours of normalized turbulent kinetic energy for Silverson L4R inline 
mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 0 (normalized by 
N2D2 of SSG) ........................................................................................................... 167 
Figure 6.20 - Contours of normalized turbulent kinetic energy for Silverson L4R inline 
mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 13 (normalized by 
N2D2 of SSG) ......................................................................................................... 168 
Figure 6.21 - Contours of normalized turbulent kinetic energy for Silverson L4R inline 
mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 23 (normalized by 
N2D2 of SSG) ......................................................................................................... 169 
Figure 6.22 - Torque on front and back surfaces of one blade, torque on bottom, top, and 
outer edge surfaces of all 4 blades, and total torque on all 4 blades of Silverson L4R 
inline mixer with SSG at 8000 rpm ......................................................................... 172 
Figure 6.23 - Torque on front and back surfaces of the blades (4 blades) and total torque 
on rotor for Silverson L4R inline mixer with ESG at 8000 rpm ............................. 172 
Figure 6.24 - Power draw as a function of shear gap width- Silverson L4R inline mixer at 
8000 rpm .................................................................................................................. 174 
xvi 
 
 
Figure A.1.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 185 
Figure A.1.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 185 
Figure A.1.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 186 
Figure A.1.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 186 
Figure A.1.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 187 
Figure A.1.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 187 
Figure A.1.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 188 
Figure A.1.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling at 
5000 rpm and 2.1 lpm .............................................................................................. 188 
xvii 
 
 
Nomenclature 
𝜎 - stress [𝑁/𝑚2] 
𝜎𝑓 - stress at fracture [𝑁/𝑚
2] 
𝑎 - flaw length [𝑚] 
𝑎𝑓 - flaw length at fracture [𝑚] 
𝐸 - elastic or Young’s modulus [𝑁/𝑚2] 
Γ - surface energy [𝐽/𝑚2] 
𝐾 - stress intensity factor [𝑁/𝑚3/2] 
𝐾𝑐 - critical stress intensity factor or fracture toughness [𝑁/𝑚
3/2] 
G - shear modulus [𝑁/𝑚2] 
K - bulk modulus [𝑁/𝑚2] 
𝑣 - Poisson ratio [-] 
H - hardness [𝑃𝑎] 
𝐸𝑠𝑝 - specific energy [𝐽/𝑘𝑔] 
𝑥 - particle length [𝑚] 
𝑙𝑖 - fractured fragment length [𝑚] 
𝑚 - particle mass [𝑘𝑔] 
n - number of particles 
𝑢 - impact velocity [𝑚/𝑠] 
𝜌 3𝑃 - particle density [𝑘𝑔/𝑚 ] 
𝑢𝑡𝑖𝑝 - rotor tip speed [𝑚/𝑠] 
𝜌 3𝑓 - fluid density [𝑘𝑔/𝑚 ] 
𝜖 - energy dissipation rate [𝐽⁄𝑘𝑔. 𝑠] 
N - rotor speed [1/𝑠] 
𝐷 - rotor diameter [𝑚] 
∆𝑉 - fractured fragment volume [𝑚3] 
𝑉 - particle initial volume [𝑚3] 
𝜏𝑐 - cohesive stress [𝑁/𝑚
2] 
xviii 
 
 
𝜏𝐷 - disruptive stress [𝑁/𝑚
2] 
𝑊𝑒 - Weber number [-] 
𝐶𝑜 - comminution number [-] 
𝑇 - time constant [𝑠] 
D100 - largest particle size of the sample [𝑚] 
D95 - size at which 95% (by volume) of particles in the sample are smaller [𝑚] 
D90 - size at which 90% (by volume) of particles in the sample are smaller [𝑚] 
D80 - size at which 80% (by volume) of particles in the sample are smaller [𝑚] 
D50 - size at which 50% (by volume) of particles in the sample are smaller [𝑚] 
D10 - size at which 10% (by volume) of particles in the sample are smaller [𝑚] 
D43 - Debroukere mean diameter [𝑚] 
D32 - Sauter mean diameter [𝑚] 
D10 - arithmetic mean diameter [𝑚] 
API - active pharmaceutical ingredient 
IPA - isopropyl alcohol 
CFD - computational fluid dynamics 
RANS - Reynolds-average Navier-Stokes 
FBRM - focused beam reflectance method 
PVM - particle vision method 
ASTM - American society of testing materials 
SEM - scanning electron microscopy 
TEM - transmission electron microscopy 
RBI - relative backscatter index 
   
   
  
  
  
  
xix 
 
 
Chapter 1: Introduction 
1.1 –Motivation and Purpose 
Size reduction and comminution is a critical step in many particulate processes, 
e.g., dry milling, wet milling, mineral processing, ceramic processing, mining and 
crystallization. It is used to create particles with specific size distribution and 
morphology to enhance product design, stability, solubility, and so on. Milling 
processes can be categorized to two broad categories: dry milling and wet milling. 
Wet milling has specific advantages over dry milling such as reducing number of 
operational steps, better control of heat generation and crystal morphology, reduced 
exposure to milling dust. Furthermore, it can be utilized in series with unit operations 
such as crystallization to ensure robust product delivery. 
Despite the increasing industrial interest, there are currently few studies to 
describe the wet milling of active pharmaceutical ingredients (APIs) or other 
crystalline substances, particularly in high shear rotor-stator mixers. Furthermore, no 
mechanistic framework has been developed to address the effect of different milling 
variables, particle mechanical properties, or mixer geometry/scale on wet milling 
performance. Consequently, the design and scale up of wet milling processes in high 
shear rotor-stator mixers is still predominantly based on experience and trial and 
error. 
The purpose of this study is to investigate the effect of milling conditions, 
crystal physical properties, and mixer geometry on milling behaviour of crystals; and 
to introduce a class of mechanistic models for breakage and attrition of crystals in 
1 
 
 
rotor-stator mixers based on elastic-plastic deformation and fracture mechanics 
concepts. The ultimate goal is to develop mechanistic correlations that lead to general 
laws for scaling up particle size from the laboratory to the industrial scale. Three steps 
are required to achieve this goal: 1) Conduct definitive wet milling experiments in a 
rotor-stator mixer to investigate the effect of milling conditions, mixer geometry, and 
particle physical properties on ultimate particle size and particle breakage kinetics; 2) 
resolve the complex flow in the high shear mixer to identify the extend of the particle-
fluid, particle-particle, and particle-rotor interactions; and 3) identify the comminution 
mechanisms and important physical properties in particle breakage. 
1.2 –Approach 
In experimental part of this study, a set of systematic experiments were 
performed in a Silverson L4R inline rotor-stator mixer with a standard shear gap 
(clearance between rotor and stator) and an enlarged shear gap, and in a Silverson 
L4R batch rotor-stator mixer to investigate the effect of milling conditions, mixer 
geometry, and particle physical properties on ultimate particle size (maximum stable 
particle size at equilibrium) and particle breakage rate. Rotor speed and volumetric 
throughput were adjusted to independently vary energy input and mill head residence 
time. To vary crystal physical properties (i.e. elastic modulus, hardness, fracture 
toughness), three different crystalline materials, sucrose, glycine, and ascorbic acid, 
were milled in isopropyl alcohol (IPA) as an anti-solvent. Using an anti-solvent 
allows discrimination between mechanical size reduction and solubility effects. The 
effects of fluid shear, particle-particle collisions, and particle-wall collisions on 
crystal breakage were assessed by changing particle concentration; coating the stator 
surfaces, and small geometric changes. Since the initial crystal size (before milling) is 
2 
 
 
greater than the standard shear gap width, the inline mixer rotor was modified to an 
enlarged shear gap to study the effect of shear gap width on milling performance. In 
addition, wet milling experiments in a Silverson L4R batch mixer were performed for 
the sucrose-IPA system at different rotor speeds. The batch mixer has a smaller 
mixing head and different power draw, so different milling behaviour was observed 
compared to the inline mixer. 
In the modelling part of this study, a class of mechanistic models was developed 
to interpret and correlate the data obtained in wet milling experiments.  These models 
consider the influence of plastic deformation, elastic deformation, and fracture 
resistance of a crystal during breakage. The cohesive forces resisting particle breakage 
were found to be a function of particle elastic modulus, hardness, fracture toughness, 
and size. Each mechanistic model predicts different relative importance of the 
material properties and particle size in resisting breakage. Therefore, different 
expressions for cohesive force were developed dependent on the particle deformation 
mechanics and breakage behavior. The disruptive forces that contribute to particle 
breakage were found to be a function of particle velocity magnitude at impact with 
surfaces and other crystals. While previous studies have related the disruptive force to 
macroscale velocity, particle sizes are such that they should be influenced by eddy 
encounters in the inertial subrange of turbulence. Therefore, two expressions for 
disruptive force were developed dependent on macroscale velocity, based on rotor tip 
speed, and eddy velocity given by Kolmogorov inertial subrange model. These 
cohesive and disruptive forces were employed to construct different correlations for 
maximum stable crystal size and the rate of approach to ultimate size. Model 
discrimination was based on comparison to the wet milling experimental data. 
3 
 
 
According to the proposed mechanistic models, power draw or energy 
dissipation rate data are required to collapse the milling data for different mixer 
geometries. These data are available in the literature for the Silverson L4R batch 
mixer but not for inline mixer. This gives rise to the numerical part of this study, in 
which a three-dimensional computational fluid dynamics (CFD) simulation was 
performed to investigate the effect of shear gap width on power draw and local 
velocity field in the Silverson L4R inline rotor-stator mixer. The Reynolds-Average 
Navier-Stokes (RANS) equations, with the realizable k-ε turbulence model and 
enhanced wall functions, were used to model the complex turbulent flow pattern 
generated by this mixer. The integral of the total moment exerted by pressure and 
viscous forces on the moving surfaces of the rotor was calculated to determine the 
torque and power draw of the mixer at different rotor speeds. It was found that the 
mixer with wider shear gap has a slightly better pumping number compared to mixer 
with standard shear gap whereas the later results in higher energy dissipation rates and 
hence better milling performance. The simulation results showed that the 
configuration with narrow shear gap draw significantly more power and power draw 
decreases with increasing the shear gap width. 
Finally, the aforementioned wet milling experimental data, mechanistic models 
of breakage, and power draw data were utilized to develop a dimensionless 
comminution number for geometrically similar devices, to aid physical interpretation 
and scale up/down efforts. For devices without geometric similarity, the concept of 
local energy dissipation rate (defined as the power draw of the mixer per mass of fluid 
in shear gap and stator slot regions) was introduced and exploited to compare the data 
from the different mixer geometries. This approach was successful in correlating the 
maximum stable particle size resulting from wet milling of different crystalline 
4 
 
 
materials in a Silverson L4R inline mixer with standard and enlarged shear gap, and a 
Silverson L4R batch mixer at different rotor speeds. The mechanistic theory is further 
utilized to provide breakage rate kernels based on probability of collision and 
collision rate theories. Application of the breakage functions to predict the attrition 
rate as well as their implementation within a population balance framework is 
discussed.  
The results of this study are not restricted to rotor-stator mixers; the scaling 
laws, non-dimensional correlations, and breakage functions developed herein can be 
applied to other processes in which particle breakage occurs according to the same 
mechanism(s). 
1.3 –Organization of the Dissertation 
This dissertation is divided into 7 chapters. Chapter 1, the current chapter, 
comprises the general outlines of the study. Chapter 2 focuses on the characterization 
of size distribution and physical properties of the particles. Various size measurement 
methods that are relevant to this study are discussed, which include: sieving, 
microscope-based image analyzing, laser diffraction, focused beam reflectance 
method (FBRM), and particle vision method (PVM). The measurement of particle 
physical properties (i.e., elastic modulus, hardness, and fracture toughness) using a 
nanoindentation method is discussed. 
Chapter 3 provides an introduction to fracture mechanics and development of 
mechanistic models for particle breakage. Particle breakage behavior based on 
indentation, elastic, plastic, and elastic-plastic deformation mechanisms are discussed. 
The functional forms of these models and their similarities with classical 
5 
 
 
comminution theories are presented and discussed. Different cohesive and disruptive 
forces that are relevant in wet milling process are presented, and different scale-up 
strategies for rotor-stator mixers are provided. 
Chapter 4 provides the results of wet milling experiments in the Silverson L4R 
inline mixers with standard and enlarged shear gap, and in the Silverson L4R batch 
mixer. The experiment setups and procedures for batch and inline modes are shown. 
The size measurement obtained from laser diffraction method is compared to FBRM 
and PVM results. The effect of milling conditions including rotor speed and 
volumetric throughput on ultimate particle size and particle breakage rate are 
presented. The effect of particle physical properties is discussed by presenting the 
experiment results for wet milling of different crystal-antisolvent systems (sucrose, 
glycine, ascorbic acid in IPA). Furthermore, the effect of solid concentration and 
mixer geometry on wet milling of sucrose-IPA system is presented and discussed. 
Chapter 5 reports the suitability of different mechanistic models that are 
presented in Chapter 3 to predict the wet milling data provided in Chapter 4. A 
definition for the dimensionless comminution number is presented and the scale up 
strategy for geometrically similar mixers is discussed. Furthermore, the concept of 
local energy dissipation rate is presented and a general scale up strategy is provided. 
Chapter 6 is concerned with calculation of power draw for the Silverson inline 
mixer with different shear gaps. The CFD simulation of Silverson L4R inline mixer is 
discussed and the geometry details, grid generation strategy, and numerical solution 
method is provided. The simulation results including the velocity field within the 
mixer, turbulent kinetic energy distribution, and total energy dissipation rate are 
6 
 
 
presented. The fluctuation of power draw with respect to rotor position is presented 
and discussed. 
Chapter 7 presents the conclusions of each portion of this study, summarizes the 
findings, and proposes recommendations for future work that would further improve 
the understanding of the wet milling processes in rotor-stator mixers. 
7 
 
 
Chapter 2: Particle Size and Physical Property Measurements 
This chapter provides a brief overview on various particle size definitions, size 
distributions, sampling errors, and size measurement techniques that are used in this 
study to characterize the milling performance of rotor-stator mixers. In addition, 
particle physical properties that play an important role in particle breakage are 
presented and their measurement method using a nanoindentation technique is 
discussed. 
2.1 –Particle Size 
Determination of particle sizes and size distribution of suspensions, emulsions, 
slurries, powders, granules, sprays, and aerosols is a critical step in many 
manufacturing and processing operations. Particle size and size distribution influence 
various properties of particulate materials such as stability in suspension, packing 
density and porosity, dissolution rate, reactivity, flowability and so on. For example: 
• In the pharmaceutical industry, the size of API controls their dissolution and 
absorption rates. 
• In the paint industry, the size of pigments affects appearance of the coated 
surface (matt, semi-glass, and glass finishes), depth of color, and rheological 
properties of the paint. 
• In the food processing industry, particle size can change the product flavor 
and texture, e.g., the size of coffee grinds affects the flavors released in the 
brewing process. 
8 
 
 
• In polymer processing, particle size distribution of catalysts influences the 
shape and molecular weight distribution of the polymer. 
Accurate definition and measurement of particle size can lead to a better 
understanding of the process dynamic and better control of the product quality. 
In quiescent liquid-liquid and gas-liquid processes, the resultant droplets and 
bubbles are usually spherical and their size can be fully described by their radius or 
diameter; whereas, in solid processing applications, the resultant particles usually 
have irregular shapes, and therefore the proper definition of their size is more 
challenging. One would need to store many data points to fully describe the particle’s 
irregular shape and various dimensions. 
In order to avoid the complexity of reporting various sizes for a single particle, 
it is preferred to report one diameter as the representative of the particle size. This is 
usually done by measuring a related size-dependent property and reducing it to a 
single linear dimension, dictated by the measurement technique. For example, image 
analysing techniques characterize the particles based on their surface area (projection 
area), and therefore the diameter of a sphere having the same surface area as the 
particle can be used to characterize the particle size. Table 2.1 summarizes some of 
the popular size definitions based on equivalent sphere diameter. It is important to 
note that these equivalent diameters provide a simple and acceptable result for 
particles with low aspect ratios, but problems can arise for needle and plate shaped 
particles in which the aspect ratios are high. 
 
9 
 
 
Table 2.1 -Definitions of the equivalent diameters based on measured properties of 
the particle 
Equivalent diameter Definition 
Diameter of a circle having the same area as 
Projected area diameter 
the projected area of the particle 
Diameter of a sphere having the same surface 
Surface diameter 
area as the particle 
Diameter of a sphere having the same volume 
Volume diameter 
as the particle 
Diameter of a sphere having the same surface 
Surface volume diameter 
to volume ratio as the particle 
Diameter of the sphere that can pass the sieve 
Sieve diameter 
aperture 
Diameter of a sphere having the same settling 
Stokes’ diameter velocity as the particle under laminar flow 
condition 
There are other size measures which can be used to quantify particle size such 
as Martin diameter, Feret diameter (including the maximum and minimum Feret 
diameter), and chord length. These diameters can be obtained using image analyzing 
methods. The Martin diameter is the length of the chord dividing the projected 
particle area into two equal halves. A Feret diameter (or calliper diameter) is the 
perpendicular distance between parallel tangents touching opposite sides of the 
particle’s projected area. Chord length is the length of a line segment whose 
10 
 
 
endpoints lie on the circumference of the projected area of the particle. Figure 2.1 
shows a graphical representation of these statistical diameters. 
 
Figure 2.1-Definition of statistical diameters representing different dimensions of the 
particle 
As mentioned earlier, there are an infinite number of linear dimensions to 
describe the particle size for irregular shaped particles. Moreover, the orientation of 
irregular particles during measurement can have a significant effect on the 
measurement result. Therefore, to have a statistically meaningful size, a large number 
of particles (from couple of hundred to several thousand particles) need to be analysed 
depending on the extent of the particle irregularity and the type of measurement 
technique. Given the vast variety of particle size definitions, it is important to note 
that the conversion from one size representing a dimension of the particle to that 
representing another dimension should be conducted with a careful consideration of 
the particle shape. 
11 
 
 
In this study, different measures of particle size are used to characterize the data 
from different measurement techniques. The sieving data are characterized by sieve 
diameter, laser diffraction data are characterized by volume diameter, image analysis 
uses projected area diameter, and FBRM uses the chord length as defined in Table 2.1 
and Figure 2.1. 
2.2 –Particle size distribution 
Particle populations are usually poly-dispersed, consisting of particles of 
different size. It is common practice to specify the size of these poly-dispersed 
particle populations in the form of either a frequency distribution, or a cumulative 
(undersize) frequency distribution. Frequency distribution is a two-dimensional 
representation of the particle sizes and the number of reoccurrence of each size. The 
number of reoccurrence of each particle size can be represented in different ways with 
respect to the weighting of individual particles based on their number, mass, volume, 
surface area, or other properties that is usually dictated by the measurement technique, 
e.g., mass distribution can be determined by the sieving technique, volume 
distribution by the laser diffraction method, and number distribution by microscopy 
methods. Each of these different distributions is advantageous for certain application. 
For example, in dry and wet milling processes, a volume distribution is a better 
measure for product evaluation compared to a number distribution because of the 
extensive generation of fine particles during milling that obscure the effect of larger 
particles on the number-based size distribution. 
It is important to note that it is mathematically possible to convert a distribution 
from one base to another (e.g., number distribution to mass or volume distribution) 
12 
 
 
but it requires the assumption that the particles have a specific shape, and thus can 
introduce significant error for irregular shaped particles. 
2.3 –Distribution Statistics 
It is common practice to define different distribution statistics and measures in 
order to compare different size distributions. This is especially useful when 
monitoring particulate process dynamics or comparing the products of different 
processes. It is important to note that a single measure cannot adequately describe a 
size distribution and usually different statistics of a distribution are reported together 
to give a better understanding of the size distribution. The type of process and the 
intended application is considered when choosing the appropriate distribution 
measures. Some of the frequently used distribution statistics are: 
• Mode: one of the popular measures of a size distribution is the mode of the 
distribution. It is defined as the size at which the frequency distribution is 
maximum (i.e., it is a measure of the particle size with maximum probability 
of occurrence in a size distribution).  
• Median: the median is the 50% size on a cumulative distribution. It is a 
measure of the particle size at which half the distribution population has 
particle sizes smaller than this size, and half has particle sizes larger than 
this size. 
• Mean: the mean is usually defined as the arithmetic average of a number 
based size distribution as: 
13 
 
 
∑ 𝑛 𝑥
?̅? 𝑜𝑟 𝐷 𝑖 𝑖10 =       (2.1) ∑ 𝑛𝑖
where 𝑛𝑖 is the number of particles with size 𝑥𝑖, and ?̅? is the mean. This is 
also called number weighted average size and shown as 𝐷10. It is important 
to note that for a normal distribution, the mean, the median, and the mode 
are all equivalent. 
• Weighted average: different weighted average sizes are defined to 
accommodate different type of processes and applications. In chemical 
reaction or dissolution problems, for example, the available surface area of 
the particles is important and therefore a surface area weighted average size 
is more relevant and can be defined as: 
∑ 𝑛 𝑥 3
𝐷 =  𝑖 𝑖32 2      (2.2) ∑ 𝑛𝑖𝑥𝑖
where 𝐷32 is the surface area weighted average size. It is also called the 
Suater mean diameter. Another popular size measure is the volume 
weighted average size, 𝐷43. It is defined as: 
∑ 𝑛 𝑥 4
𝐷 =  𝑖 𝑖43       (2.3) ∑ 𝑛 𝑥 3𝑖 𝑖
It is also called the De Brouckere mean diameter. This mean value is 
heavily weighted toward the large particles of the distribution and is a 
good measure for the processes that are sensitive to the presence of large 
particles. 
14 
 
 
• Cumulative percentile: the cumulative percentile indicates the number of 
particles in a population whose size is smaller than a given size. For 
example, the D90 is the particle size at which 90% (by mass or volume) of 
the particles have sizes smaller than this value. They are also called D-
values. Different D-values are representative of different aspects of a size 
distribution, e.g., D10 is a representative of the size of fine particles, D50 is 
the median, and D90 is a representative of the size of large particles in the 
distribution. Figure 2.2 and Figure 2.3 show a unimodal distribution and a 
bimodal distribution that are obtained during the wet milling of sucrose in 
IPA and measured by a laser diffraction method. The volume frequency and 
cumulative volume percentage of the distributions are illustrated along with 
the corresponding D10, D50, and D90 values. 
 
Figure 2.2 -Unimodal frequency and cumulative size distribution obtained during wet 
milling (L4R batch mixer at 8000 rpm) of sucrose in IPA and its corresponding D10, 
D50, and D90 values measured by laser diffraction method 
15 
 
 
 
Figure 2.3 -Bimodal frequency and cumulative size distribution obtained during wet 
milling (L4R batch mixer at 8000 rpm) of sucrose in IPA and its corresponding D10, 
D50, and D90 values measured by laser diffraction method 
2.4 –Sampling 
Particulate systems consist of a large population of particles with different sizes. 
It is not feasible to characterize the size of each and every particle in the entire 
population, so sampling is performed to obtain a sample with smaller fraction of the 
population. Special care must be taken in sampling steps to obtain a sample whose 
size distribution would be representative of that of the entire population; an incorrect 
and non-representative sampling can result in poor process characterization. 
The accuracy of a size distribution measurement depends on sampling errors, 
errors in measurement procedures (proper dispersion in measurement cell), operator 
errors (poor calibration), and the instrument limitations. Sometimes, the sample 
acquired in one sampling step is too large and must be further sub-divided to give an 
appropriate sample size for measurement. Sampling itself is prone to two types of 
16 
 
 
errors: one arises from a non-representative sample due to inhomogeneous bulk 
materials and lack of proper mixing and dispersion, the other is due to statistical 
errors with random fluctuations around the mean value [1]. In order to establish 
homogeneity, it is necessary to measure the size distribution of the different samples 
taken from the different regions of the bulk; the homogeneity is established if the 
variability of the size distribution between different samples is sufficiently small. The 
random fluctuation of the size distribution data around a mean value is a characteristic 
of a random process and cannot be completely suppressed; however, it is possible to 
reduce these fluctuations by increasing the sample size or sampling time. 
Proper mixing and dispersion in solid-liquid suspensions requires some mode of 
agitation. The agitation creates a turbulent flow field in which solid particles are 
dispersed and distributed throughout the liquid. The properties of both the liquid and 
the solid particles, holding vessel geometry, and agitation parameters influence the 
homogeneity of the suspension. The degree of dispersion can be classified into three 
levels of partial suspension, complete suspension, and uniform suspension as shown 
in Figure 2.4 [2]. Uniform suspension is the desired result at which particle 
concentration and size distribution are practically homogeneous throughout the 
holding vessel. In this study, to establish a uniform crystal-antisolvent suspension, the 
size and aspect ratio of the holding tank, impeller blades, and baffles are carefully 
selected and the agitation rate is adjusted to enhance the top-bottom mixing without 
aeration. The homogeneity of the sampling was examined by inline and offline 
measurement of samples acquired from different regions of the crystal-antisolvent 
suspension in the sampling/holding tank (see Chapter 4 for details). 
17 
 
 
 
Figure 2.4 - Degrees of dispersion - partial suspension: some particles rest on the 
bottom of the tank for short periods; complete suspension: all particles are off the 
bottom of the vessel; uniform suspension: particle concentration and size distribution 
are practically homogeneous throughout the vessel [2]. 
2.5 –Particle Size Measurement Techniques 
Particle size and size distribution can be determined using different 
measurement techniques such as sieving, sedimentation, microscopy and image 
analysis, electron zone sensing, laser diffraction, dynamic light scattering, focused 
beam reflectance method (FBRM), particle video microscopy (PVM) and many other 
techniques. Some of these techniques can be used inline (integrated into the process) 
with a fast response rate and some has to be used in an offline mode with a batch 
analysis procedure. The number of sampling steps, the required concentrations of the 
samples, and the size range capabilities are different in different techniques. The 
general factors that should be considered before selecting a measurement technique 
are: 
• Size range 
• The technique needs to generate size data in a form that is relevant to the 
process in hand (i.e., number based measurement, volume based 
measurement, and so on) 
• Accuracy and reproducibility  
18 
 
 
• Amount of sample available for analysis 
• Response rate of the measurement including sample preparation, 
measurement, and reporting, e.g., online methods usually have a faster 
response rate compared to off-line techniques 
• Cost of the instruments 
It is important to note that the resulting size distribution should be based in the 
format of the primary result for a given technique, e.g., laser diffraction technique 
generates results based on volume distributions and any specification should be 
volume based; conversion to another basis such as number can introduce significant 
errors. 
2.5.1 –Sieving 
Sieving is widely used in different disciplines and industries due to its relative 
simplicity, low cost, and high reliability in determining particle size distribution over 
a wide size range. It can characterize the dry powders as well as the slurries by 
shaking the sample through a stacked series of sieves with decreasing mesh size. The 
mesh size is defined as the number of wires per linear inch of the sieve. A variety of 
sieves with different aperture sizes are available. For woven wire sieves, the mesh 
size ranges from 5 inch mesh with 125 mm of aperture size to No. 635 mesh with 20 
μm of aperture size based on the American Society of Testing Materials (ASTM) 
standards. The electroformed sieves can have smaller aperture sizes down to 1 μm. 
Particles having two dimensions smaller than the aperture opening will pass through, 
whereas larger particles will be retained. The resulting data can be reported as a mass 
based size distribution or cumulative mass percentage. The resolution is bounded by 
19 
 
 
the aperture opening of the sieves and varies from 10s of mm for coarse sizes to 10s 
of μm for fine particles. Some of the limitations of this technique include the long 
analysis times, the shaking action may cause agglomeration or fracture of particles 
during the analysis, and type and duration of the mechanical shaking and blockage of 
apertures during sieving can affect the reproducibility of the results. 
In this study, in order to obtain a consistent initial particle size distribution for 
milling experiments, batches of 100 grams of crystals were sieved using a vibratory 
sieve shaker (WS Tyler RX-29 Roto Tap) that provided motion in both horizontal and 
vertical axes with variable amplitude for 20 minutes (until less than 0.5% of the 
sample passes through in any 10 minutes sieving period based on the ASTM 
standard). A fraction of particles retained between No. 40 and No. 60 sieves (250-420 
μm) was collected and used as the feed for the wet milling experiments. 
2.5.2 –Image Analysis 
Image analysis provides a direct and powerful tool to characterize size, shape, 
surface texture, and degree of dispersion of the particles. The images of the particles 
are usually magnified by an optical lens, captured by a digital camera, and analyzed 
by image analysis techniques. The required magnification depends on the size of the 
particles being studied and the extent of the resolution that is desired. Optical light 
microscopes, scanning electron microscopes (SEM), and transmission electron 
microscopes (TEM) can be used to obtain higher resolution images. 
Optical light microscopes are used to analyze particle size in the range from 
3μm to 150 μm [1]. The lower limit is due to diffraction of light passing through the 
camera aperture and causes a fuzzy edge surrounded by circles of light for fine 
20 
 
 
particles. The upper limit is due to narrow depth of field so that, for a sample having a 
wide range of sizes, only a portion of particles are in focus in any field of view. The 
particles larger than 150 μm can be removed using a sieving method and their image 
can be captured using a simple magnifying glass. The size distribution of particles 
with sizes smaller than 1 μm can be characterized by using higher magnification 
microscopes such as SEM and TEM. 
In image analysis step, the captured images are usually converted to gray-scale 
images to enhance the contrast between the particles and the background. The images 
are then set to a specific threshold so that, all the pixels that have a grayscale value 
above the threshold are converted to black, and those with grayscale below the 
threshold are converted to white. The projected area of the particles in the image is 
then calculated by counting the number of pixels with black identifiers at each particle 
region. Figure 2.5 shows a sample of a raw image obtained using optical microscopy 
and the processed image identified via ImageJ software [3]. The border particles are 
removed because their size/shape cannot be meaningfully described. 
 
Figure 2.5 -A microscope image of sucrose particles and its processed version using 
ImageJ image processing software 
21 
 
 
In this study, image analysis is primarily used as an off-line method to monitor 
the particle size and shape in small batches to confirm the size data obtained by other 
devices. It was also used to monitor the samples for agglomerate formation. 
2.5.3 –Laser Diffraction 
Laser diffraction is one of the popular techniques for measuring particle size 
distribution because of its capability in analyzing broad range of materials with wide 
range of size distributions, ease of use, and high speed of measurement. The 
underlying principle of measurement is the difference in the scattering light from 
particles with different sizes and materials. The theory of light scattering by small 
particles has been developed by many authors and the fundamentals can be found in 
standard texts, e.g., Bohren and Huffman (1983). The theory is mathematically 
complex and computationally expensive to apply to non-spherical particles, therefore 
most of the laser diffraction methods assume that the particles are spherical and 
provide corresponding equivalent spherical diameters. This assumption can lead to 
biases and erroneous data when measuring the size distribution of particles with high 
aspect ratios.  Another limitation of this method is due to multiple scattering where 
light scattered from one particle encounters with another particle and gets scattered 
again. To overcome to this problem, the concentration of particles in the sample 
should be diluted to reduce the probability of multiple scattering phenomena. The 
concentration limit depends on the design of the instrument, the nature of the incident 
light source, optical geometry, and light detection system. 
To use laser diffraction method (based on Mie theory), the refractive indices of 
the particles and the dispersing medium need to be known. These indices are used to 
22 
 
 
predict the amount of absorption and refraction of the light passing through the 
particle. This is especially important for fine particles with sizes close to wavelength 
of the laser source. For fine particles, the refractive index is usually a complex 
quantity with a real and an imaginary part. The real part describes the refractive 
properties of the particle, while the imaginary part describes the light absorbing 
properties. For opaque, non-spherical particles with sizes much larger than the 
wavelength of the laser source, the refractive index can be adequately quantified by a 
real number (zero imaginary part). 
The Horiba, LA950, laser diffraction instrument was used in this study as 
shown in Figure 2.6. It can measure wet and dry samples with size range of 10 nm to 
3 mm [4]. This device consists of: 
• Two light sources: a 5 mW 650 nanometers red laser diode and a 3 mW 405 
nanometers blue light emitting diode. The smaller wavelength light is used to 
extend the lower detection limit of the technique.  
• A beam expander to produce a uniform parallel beam for illuminating the sample 
volume. 
• A sample cell with transparent flat windows and capacity of 10 mL to introduce 
the sample and control the interaction of particles and the incident light. The cell 
is mounted at a slight angle to the beam to eliminate the back reflections.  
• The cell is stirred with a magnetic stirrer to keep the particles suspended. The size 
and rotational speed of the magnetic bar is carefully adjusted to ensure a uniform 
suspension of crystal-antisolvent system in the sample cell. This is particularly 
23 
 
 
important for measuring the samples taken at the early stages of the milling where 
there are larger particles with higher settling velocities. 
• The scattered light from the sample passes through several optical lenses, and 
forms a scattering pattern onto 87 silicon photodiode array detectors at different 
light scattering angles. 
• The detected signals are digitalized and compiled with Horiba software that 
controls the system during the measurement process and analyzes the light 
scattering data. The software utilizes Mie theory to convert the light scattering 
data to volume-based size distribution data. 
One cycle of measurement including loading the sample cell, measurement, saving 
the data, and rinse typically takes less than 5 minutes to complete. 
 
Figure 2.6 -Horiba LA 950 laser diffraction instrument and its sampling cell 
24 
 
 
2.5.4 –Focused Beam Reflectance Measurement (FBRM) 
The FBRM device is an in-situ probe to characterize the chord length 
distributions of particulate samples. The probe projects a focused laser beam to its tip 
where a rotating optical lens deflects the laser. The beam is focused to a fine point 
close the probe’s window and traces a circular path as shown in Figure 2.7. When the 
probe is inserted into a system of particles, the laser is reflected as it scans across the 
surface of a particle. The probe measures the reflectance time and determines the 
chord length by the product of the reflectance time and the laser scan-speed. 
Depending on the concentration of the particles, thousands of particles are usually 
counted during one measurement and the results are presented as a chord length 
distribution of particles. 
 
Figure 2.7 -Lasentec D600 FBRM probe and schematic of probe internal parts 
showing the laser crossing a particle surface 
Considerable efforts have been made to find numerically robust solutions to 
reconstruct the size distribution from a chord length distribution, but for non-spherical 
particles, the results are unreliable due to ill-posed nature of this inverse problem. 
Despite this drawback, the chord length distribution data obtained by a FBRM probe 
25 
 
 
can be used to qualitatively monitor the dynamic changes in particulate processes. 
Moreover, process control strategies may be correlated directly with chord lengths 
and chord length distribution. 
In this study, FBRM probe, a Mettler-Toledo Lasentec D600 particle size 
analyzer, was used in situ for inline measurement of chord length distribution as well 
as evaluation of complete solid suspension in the holding vessel of inline setup (see 
Chapter 4 for details). It utilizes a laser with power of 3 mW, wavelength of 790 
nanometers, and rotation speed of 2 m/s. It is capable to measure the chord length of 
particles between 1 μm and 2 mm [5]. The instrument was calibrated using a standard 
sample provided by Mettler-Toledo Autochem. 
2.5.5 –Particle Vision Measurement (PVM) 
A PVM probe, Particle View V19 from Mettler-Toledo Autochem as shown in 
Figure 2.8, was also used in this study to monitor the particle shape during the 
milling. PVM is an in situ video microscope that can be integrated to the system as an 
inline measurement technique. It uses 6 near infrared laser sources to illuminate a 
small area in front of the probe window (within the slurry). An integrated CCD 
camera records digital images of the illuminated area with a 826 μm × 619 μm field 
of view and rate of up to 10 images per second [6]. Particles larger than 20 μm can be 
identified from the PVM images; below this size scale, it can be difficult to 
distinguish individual particles. 
26 
 
 
 
Figure 2.8 -Particle Vision Measurement (PVM) probe, Particle View V19 from 
Mettler-Toledo Autochem and schematic of probe tip 
Using image analysis techniques to obtain quantitative information from PVM 
images is challenging. The images have non-uniform intensity with poorly defined 
particle outlines. The image processing problem is further complicated by particle 
overlap and presence of large number of fine particles that scatter the input light and 
saturate the images. Despite this drawback, a relative backscatter index (RBI) is 
defined by Mettler-Toldeo that can be used to qualitatively monitor the dynamic 
changes of particle size and concentration in the process. The RBI is a function of 
particle size, shape, concentration, brightness, and refractive index. It is defined as: 
𝑖𝑚𝑎𝑔𝑒 𝑏𝑟𝑖𝑔ℎ𝑡𝑛𝑒𝑠𝑠
𝑅𝐵𝐼 =       (2.4) 
𝑖𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡 𝑖𝑛𝑝𝑢𝑡
RBI decreases as the particles get finer and as the concentration of particles increases, 
i.e., image brightness decreases. An example of images captured by PVM probe and 
corresponding RBI during a wet milling experiment are given in Figure 2.9. 
27 
 
 
 
Figure 2.9 -PVM images of sucrose crystals in IPA captured inline during wet milling 
(L4R inline mixer with standard shear gap), RBI of images are calculated at different 
milling times 
2.6 –Particle Physical Properties 
All solid materials deform under the application of force or stress. Depending 
on the characteristics of this deformation, the process is classified as elastic or plastic. 
The characteristics of this deformation primarily depend on physical properties of the 
material, temperature, and magnitude of the applied load. An elastic deformation is a 
type of process in which the original size and shape can be recovered once the load is 
removed. This type of deformation involves stretching of the atomic bonds of the 
material. The atoms keep their original bonds and restore their original positions when 
the load is removed. If the loads are increased beyond a certain limit (yield strength) 
then the particle goes through a plastic deformation. The plastic deformation involves 
the breaking of a limited number of atomic bonds (e.g. the movement of dislocations). 
The required force to break the bonds of all the atoms in a crystal plane is very great. 
However, the movement of dislocations allows atoms in crystal planes to slip past one 
28 
 
 
another at a much lower stress levels. Since the energy required to move is lowest 
along the densest planes of atoms, dislocations have a preferred direction of travel 
within a grain of the material. In this case the particle cannot recover its original shape 
when the loads are removed. 
2.6.1 –Young’s Modulus 
For combined stress states that have a combination of tensile, compression, and 
shear stresses, the elastic behavior of a material is defined through the Young’s 
modulus (is also known as elastic modulus, E), the shear modulus (G), the bulk 
modulus (B), and Poisson’s ratio (ν). For an isotropic material, these properties are 
related to each other according to: 
𝐸 = 2𝐺(1 − 𝑣)      (2.5) 
𝐸
𝐺 =        (2.6) 
2(1+𝑣)
𝐸
𝐵 =        (2.7) 
3(1−2𝑣)
𝐸−2𝐺
𝑣 =        (2.8) 
2𝐺
3
For many materials, B ~ E, 𝐺 ~  E, and ν ~ 0.33 [7]. Low modulus materials 
8
are pliable, and stretch a lot when they are pulled. High modulus materials are the 
opposite; they stretch very little when pulled. Elastic modulus is related to the atomic 
binding energy for the case of inorganic materials or molecular binding energy for 
organic solids. As shown in Figure 2.10, the stronger the bond of a material, the 
higher the resistance to bond stretching and hence a higher Young’s modulus of the 
29 
 
 
material. This material property can be obtained by several methods; a 
nanoindentation technique is used here. 
 
Figure 2.10- Elastic modulus is related to the atomic bonding forces; the higher 
atomic bonds (sharper slope), higher the elastic modulus 
2.6.2 –Hardness 
The resistance of a material to plastic deformation can be characterized using 
the concept of hardness. Mohs [8] introduced a qualitative hardness scale based on the 
scratch resistance of a material relative to diamond. Tabor [9] developed an 
indentation technique to quantitatively characterize the hardness of different 
materials. For fine particulate solids, which is the case in this study, the standard 
indentation method is not possible due to size constraint and the most appropriate 
technique is the nanoindentation.  
2.6.3 –Fracture Toughness 
The fracture toughness or critical stress intensity factor (𝐾𝑐) is a material 
property that represents the material resistance to crack growth. It is originally 
developed for isotropic, linearly elastic materials subjected to small deformations. 
30 
 
 
American engineer and physicist George R. Irwin [10] did a major work to predict 
crack growth under loading based on stress intensity factor. He showed that the 
fracture toughness can provide a proper characterization of crack tip stressing even 
when linear elastic theory is not applicable to the process.  (see Chapter 3 for more 
details). 
2.7 –Indentation Test 
Indentation testing is a mechanical testing process designed to determine the 
properties of materials by putting an indentation in the surface of a material sample. 
An indenter with a well-defined geometry is pressed on the surface of a sample using 
a known test load. The stress-strain history and the indent size and depth can be 
obtained as the indenter penetrates into and is withdrawn from the sample. 
Nanoindentation is an indentation test in which the length scale of the 
penetration is in nanometers rather than microns or millimeters. In nanoindentation 
tests, the size of the residual impression is of the order of microns and it can be 
obtained by measuring the depth of penetration of the indenter into the specimen 
surface. The measured depth of the indentation together with the known geometry of 
the indenter can be used to calculate the hardness of the material. 
2.7.1 –Measurements of Particle Mechanical Properties 
By placing a load F on the indenter tip, it penetrates into the specimen to a 
measurable depth. At this point, the load may be held constant for a period or 
removed. The Young’s modulus can be estimated from stress-strain graph during the 
loading and unloading process. The hardness (H) can be calculated by 
31 
 
 
𝐻 = 𝐹/(2𝑎2)      (2.9) 
where F is the applied load and a is the diagonal length of indentation as shown in 
Figure 2.11. 
 
Figure 2.11- Indentation area and crack propagation in an indentation test 
Stress intensity factor (𝐾𝑐) describes the ability of a material containing a crack 
to resist fracture, and is estimated as 
𝐾𝑐 =  ξ (E⁄H)
2 (F⁄c1.5)    (2.10) 
where ξ is a calibration factor (=0.016 ± 0.004), E is elastic modulus, H is hardness, F 
is applied load, and c is the radial crack [11]. 
The penetration depth can be in the order of μm or nm: micro-indentation and 
nano-indentation respectively. In this work, indentation tests were performed using a 
TI 950 Hysitron TriboIndenter (Figure 2.12) with load resolution of 1 nN and 
displacement resolution of 0.02 nm.  
32 
 
 
 
Figure 2.12 - TI 950 Hysitron nanoindentation device used in this study to measure 
elastic modulus, hardness, and fracture toughness of different crystals 
2.8 –Summary 
• Various size measurement methods that are used in this study were discussed 
including sieving, microscope-based image analyzing, laser diffraction, FBRM, 
and PVM.  
• It was shown that a number based measurement technique can be overwhelmed 
and saturated by the large number of fine particles that are generated via attrition 
during a milling process. Therefore, to consider the maximum stable particle size, 
a volume or surface based measurement technique is preferred in size distribution 
measurement of milling processes.  
33 
 
 
• Mechanical properties of crystals that are important during breakage were 
discussed including, Young’s/elastic modulus, hardness, and fracture toughness. A 
nanoindentation technique was introduced to measure these properties at the scale 
of the crystals. 
34 
 
 
Chapter 3: Comminution Theories and Mechanistic Models for 
Particle Breakage 
Comminution is a generic term for processes in which solid particles are 
mechanically broken into smaller particles and fragments (i.e., size reduction, milling, 
grinding, and crushing). It is used to create particles of certain size distribution and 
shape to enhance product design, stability, solubility, and so on. Despite widespread 
application, comminution is one the least energy efficient unit operation processes 
[12]; only small amount of the energy delivered to the process is used to reduce 
particle size and create new surfaces. Therefore, the power requirement is one of the 
major considerations in designing comminution processes. There have been 
considerable research efforts to explain the actual energy input necessary in size 
reduction processes. Comminution theories explain the energy consumption as a 
function of size reduction, but do not consider the physics of the particle breakage and 
physical properties of the materials. Mechanistic models, on the other hand, use the 
concepts of solid and fracture mechanics to predict the milling performance. 
3.1 –Classical Comminution Theories 
Classical theories attempt to model the milling process by empirically relating 
the energy requirements to size reduction. They have been the topic of extensive 
research in different fields of engineering; yet, they are empirical laws that do not take 
into account the physics of the breakage and physical properties of the materials. 
These so called laws can be generalized as [13]: 
𝑑𝑥
𝑑𝐸𝑠𝑝   =  − 𝑐 𝑛    (3.1) 𝑥
35 
 
 
where Esp is specific (per unit mass or volume) energy consumption, x is particle size, 
c is a dimensional constant and n is a dimensionless constant. 
3.1.1 –Rittinger’s Law 
Rittinger’s model (1867) suggests that the energy consumption is related to the 
amount of fresh surface produced; therefore n=2: 
𝑑𝑥
𝑑𝐸𝑠𝑝 =  − 𝑐𝑅  𝑥2     (3.2) 
where 𝑐𝑅 is the Rittinger’s constant. This equation can be integrated (with boundary 
condition of 𝐸𝑠𝑝 = 0 𝑎𝑡 𝑥 = 𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙) to give: 
1 1
𝐸𝑠𝑝 = 𝑐𝑅  ( −  )    (3.3) 𝑥𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙
In the grinding process, even if the initial size of particles is 𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙, i.e., 
monodispersed particles, the products will have a distribution of sizes. It is possible to 
use certain statistics of the distribution to overcome this problem. Since 𝐸𝑠𝑝  ∝ 1/𝑥, a 
mean diameter can be defined based on -1 moments of the distribution as: 
1 𝑥𝑚𝑎𝑥 𝑃(𝑥,𝑡)𝑑𝑥=  ∫       (3.4) 
?̅? 0 𝑥
where 𝑃(𝑥, 𝑡) is the cumulative fraction of the size distribution. Furthermore, it is 
possible to assume that the energy input rate to mill is constant, so that the fresh 
surface area production is proportional to milling time [13]. 
36 
 
 
3.1.2 –Kick’s Law 
Kick’s model (1885) states that for geometrically similar size reduction, the 
energy required for comminution is proportional to particle volume and therefore 
n=1: 
𝑑𝑥
𝑑𝐸𝑠𝑝 =  − 𝑐𝐾       (3.5) 𝑥
where 𝑐𝐾 is the Kick’s constant. This equation can be integrated (with boundary 
condition of 𝐸𝑠𝑝 = 0 𝑎𝑡 𝑥 = 𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙) to give: 
𝑥
𝐸𝑠𝑝 = 𝑐𝐾 log (
𝑝𝑟𝑜𝑑𝑢𝑐𝑡)    (3.6) 
𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙
 
Similar to previous section, in order to consider the distribution of particles with 
different sizes, it is possible to use a certain statistic of the distribution. Here, a mean 
particle size, ?̅?, can be defined as: 
?̅? 𝑥 𝑥
log( ) =  𝑚𝑎𝑥∫ log ( ) 𝑃(𝑥, 𝑡)𝑑𝑥  (3.7) 
𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙 0 𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙
where 𝑃(𝑥, 𝑡) is the cumulative fraction of the size distribution. 
3.1.3 –Bond’s Law 
Bond (1952) has argued that neither the Rittinger theory, which relates only to 
surface area, nor the Kick theory, which relates only to particle volume, can 
adequately describe the grinding results. Size reduction is related to both surface and 
volume; the amount of impact energy is related to particle volume, but breakage starts 
with a crack, created by concentration of stresses on the surface. 
37 
 
 
Bond’s third law of grinding suggested to use n=3/2 which falls between the 
two previous models [15–19]. 
𝑑𝑥
𝑑𝐸𝑠𝑝 =  − 𝑐𝐵  3     (3.8) 
𝑥2
where 𝑐𝐵 is the Bond’s constant. It integrates to give: 
1 1
𝐸𝑠𝑝 = 𝑐𝐵  ( −  )   (3.9) 
√𝑥𝑝𝑟𝑜𝑑𝑢𝑐𝑡 √𝑥𝑖𝑛𝑖𝑡𝑖𝑎𝑙
The third theory of comminution is further developed by Bond to introduce a 
work index for different materials and different milling operations. The commonly 
used form of the equation is: 
1 1
𝑊 = 10 𝑊𝑖 ( − )    (3.10) 
√𝑃 √𝐹
where 
W = Predicted mill energy consumption in kWh/tone 
Wi = Bond’s work index in kWh/tone 
P = D80 of the product (80% passing size in μm of the product)  
F = D80 of the feed (80% passing size in μm of the feed) 
This theory is a widely used method in milling processes and has become an 
accepted approach for designing comminution processes [13,20]. It is important to 
note that this equation lumps particle mechanical properties as well as the energy 
delivery capacity of the milling device in one parameter, which is called work index. 
38 
 
 
3.1.4 –Size Range Applicability of Comminution Laws 
Application of Kick’s and Rittinger’s theories has been met with varied success 
and is not realistic for designing many industrial size reduction processes. Instead, 
Bond’s model is the most widely used method in milling processes and has become a 
standard design approach for milling processes [20]. It has been suggested that the 
three approaches to prediction of energy requirements mentioned above are each 
applicable to a different range of product size. It is common practice to assume that 
Kick’s proposal is applicable for large particle size (coarse crushing and crushing), 
Rittinger’s for very small particle size (ultra-fine grinding) and the Bond formula 
being suitable for intermediate particle size (the most common range for many 
industrial milling processes). This is shown in Figure 3.1, in which specific energy 
requirement is plotted against particle size on logarithmic scales [21]. 
 
Figure 3.1- Size range applicability of grinding laws in a log-log representation [21]: 
Kick’s theory is applicable for large particle size, Rittinger’s for very small particle 
size, and the Bond’s work index is suitable for intermediate particle size 
39 
 
 
3.2 –Fracture Mechanics 
Prior to the research of Griffith (1921), the general assumption was that in order 
to break a solid material, the atomic bonds of the material had to be overcome. 
However, Griffith found that the fracture strength from known physical parameters of 
molecular structure led to values that are orders of magnitude larger than that one 
experience in laboratory measurements, i.e., the stress needed to fracture glass fibers, 
depending on their size, is 10 to 100 times smaller than the theoretical stress needed 
for breaking glass atomic bonds. A theory was needed to reconcile these apparently 
contradictory results. Griffith proposed that the low fracture resistance observed in 
experiments, as well as the size-dependence of strength, was due to the presence of 
microscopic cracks in the bulk material. The presence of cracks changes the stress 
field in the material causing high stress concentration. Depending on the crack length 
and crack tip radius, the stress at the crack tip can be orders of magnitude higher than 
the applied stress as shown in Figure 3.2. This concentration of stresses can exceed 
the atomic bonding energy, resulting in crack propagation and fracture. 
 
Figure 3.2- Stress concentration near the crack tip of a solid under stress 
Based on linear elastic theory, Inglis (1913) developed a mathematical 
framework for the stress field surrounding an ellipse to prove that the presence of 
40 
 
 
cracks in a solid under strain could lead to stress concentration at the crack tip. Based 
on this mathematical approach, Griffith showed that the product of the square root of 
the flaw length, 𝑎𝑓, and the stress at fracture, 𝜎𝑓, was nearly constant and that constant 
can be estimated from a thermodynamic energy approach (with plane stress 
assumption): 
𝜎𝑓 √𝜋 𝑎𝑓 =  √𝐸 Γ     (3.11) 
Equation 3.11 relates the stress at fracture to the size of crack. The left side of 
this equation states that fracture will occur when stress magnitude or/and crack size 
reaches their critical value. The term 𝜎 √𝜋 𝑎 appears frequently in fracture mechanics 
and it is usually abbreviated to a single symbol K, called the stress intensity factor. 
The right side depends on material properties only: E is the elastic modulus and Γ is 
the energy required to generate a unit area of crack (surface tension). Therefore, 
fracture occurs when 𝐾 = 𝐾𝑐 where 𝐾𝑐 (= √𝐸 Γ) is the critical stress intensity factor, 
more usually called the fracture toughness (Table 3.1). 
Table 3.1- Definition of stress intensity factor and critical stress intensity factor 
Γ Energy Required to Generate a 𝐽 𝑚−2 
Unit Area of Crack 
𝐾    (= √𝐸 Γ ) Fracture Toughness or Critical −3/2𝑐 𝑁 𝑚  
Stress Intensity Factor 
𝐾  (= 𝜎 √𝜋 𝑎) Stress Intensity Factor 𝑁 𝑚−3/2 
 
Griffith’s analysis introduced fracture toughness into fracture mechanics 
literature as an important property of the material which quantifies the resistance of 
the particle against fracture [24]. This property can be used with elastic theory to 
41 
 
 
quantify brittle breakage. The brittle fracture mode happens when a particle breaks 
before reaching the yield stress and therefore no plastic deformation is presented. 
However, most of the particulate solid materials, as well as the crystals used in this 
study, break in a semi-brittle (elastic-plastic) mode [25–28]. 
Semi-brittle materials experience a plastic deformation at the crack tip before 
fracture (stress at the crack tip exceeds the yield strength). In the vicinity of the crack 
front, a region is formed where weakening and rupture of cohesive bonds occur. This 
region is enclosed by a zone of plastically deformed material. The deformations in 
this zone govern the processes of fracture and define the modes of fracture of a body. 
Thus, studies of plastic deformations around cracks are of fundamental importance for 
describing the process of fracture [29]. 
Irwin’s (1957) studies extended Griffith’s energy theory to accommodate this 
inelastic yielding that precedes the crack propagation. He showed that the critical 
stress intensity factor (𝐾𝑐), that is defined based on elastic theory and experiments on 
brittle material i.e. glass, can characterize crack stressing for semi-brittle materials as 
well. For the semi-brittle failure mode, the prevailing stresses reach the yield stress, 
and hence plastic deformation precedes the crack propagation. The important particle 
physical properties in this failure mode are the elastic modulus, hardness (H), and 
fracture toughness (Kc). 
Depending on the loading condition, particles can experience a combination of 
tensile, compression, and shear stresses. The elastic behaviour of a material under 
these stresses is defined through Young’s modulus (is also known as elastic modulus, 
E), bulk modulus (B), shear modulus (G), and Poisson’s ratio (ν). For an isotropic 
42 
 
 
3
material, these properties are related to each other and for many materials B ~ E, 𝐺 ~  
8
E, and ν ~ 0.33 [7]. Hardness represents the resistance of a material to plastic 
deformation. The mechanistic basis for characterization of hardness has been 
developed by Tabor (1996), based on the indentation technique. In Chapter 2, the 
measurement of elastic modulus, hardness, and fracture toughness of different crystals 
using a nanoindentation technique is discussed. 
It is important to note that, in the other side of brittleness spectrum, ductile 
materials consume all the receiving energy in the form of plastic deformation. The 
size reduction in these materials can be achieved in very high strain rates or very low 
temperatures. Figure 3.3 shows how failure mode can change under different 
condition [30]. 
 
Figure 3.3 -Fracture behavior of materials under different process conditions 
(confinement is the geometric constraints on the surface of the particle) [7] 
43 
 
 
3.3 –Mechanistic Breakage and Attrition Theories 
Despite the popular energy-size theories for comminution, there are few 
publications on quantifying the milling behaviour of materials according to their 
mechanical properties. Lawn and Marshal (1979) used indentation mechanics to study 
the breakage mechanism. Gahn and Mersmann (1995, 1997, 1999) developed a 
mechanistic model for attrition loss for crystals undergoing elastic deformation in 
suspension crystallizers. Attrition occurs at lower stress levels compared to those 
necessary for comminution, although the mechanism of both processes is similar: the 
propagation of cracks under impact. Another mechanistic model was developed by 
Ghadiri and Zhang [26] to relate the attrition of materials undergoing plastic 
deformation in single particle impact experiments to their physical properties 
measured from indentation tests. These studies will be explained in next sections 
under the title of mechanistic models based on indentation, elastic, and plastic 
mechanisms.  
Rumpf  (1973) presented a scaling analysis to correlate the energy consumption 
to particle size. They showed that to obtain a similar fracture pattern the following 
conditions must meet: 
𝑒𝑛𝑒𝑟𝑔𝑦 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛
 × 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑠𝑖𝑧𝑒 = 𝑐𝑜𝑛𝑠𝑡. (3.12) 
𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑣𝑜𝑙𝑢𝑚𝑒
𝑐𝑟𝑎𝑐𝑘 𝑜𝑟 𝑓𝑙𝑎𝑤 𝑠𝑖𝑧𝑒
= 𝑐𝑜𝑛𝑠𝑡.    (3.13) 
𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑠𝑖𝑧𝑒
They observed that the constant in equation 3.13 depends on particle size and, 
based on experimental data, they showed that fractured surface area per utilized 
energy input increases (fracture energy decreases) monotonically with particle size. 
44 
 
 
This means more energy is required to break smaller particles because the initial flaw 
pattern is less favourable for fracture propagation. 
Vogel and Peukert (2003, 2005) developed a model based on Rumpf’s approach 
to predict the particle impact behaviour in impact mills. They divided the milling 
process into a machine function and a material function. The machine function 
describes the type of mill and the operational parameters, and the material function 
comprises the material properties. They proposed a form of breakage probability 
based on Weibull statistics as: 
𝑆 = 1 −  𝑒− 𝑓𝑀𝑎𝑡 𝑘 𝑥 (𝑊𝑘𝑖𝑛− 𝑊𝑀𝑎𝑡,𝑚𝑖𝑛)   (3.14) 
In this equation the number of impacts, k, and the specific impact energy, Wkin, 
are the machine functions. The material functions are fMat, a measure for the resistance 
of the material against fracture, and WMat,min, characterizes the specific energy which a 
particle can take up without comminution. They obtained these parameters 
experimentally by single particle impact tests. Meier and Peukert (2009) correlated 
these empirical parameters to the material properties, i.e. H and Kc, based on Rumpf 
dimensional analysis approach. 
3.3.1 –Indentation Mechanism 
Based on indentation measurements, Lawn and Marshal (1979) proposed an 
index of brittleness in terms of hardness and fracture toughness parameters. This 
index is defined as: 
𝐼𝑛𝑑𝑒𝑥 𝑜𝑓 𝐵𝑟𝑖𝑡𝑡𝑙𝑒𝑛𝑒𝑠𝑠 =  𝐻⁄𝐾    (3.15) 𝑐
45 
 
 
They measured the minimum crack size and minimum crack load of different 
materials and concluded that the materials with higher brittleness index are easier to 
break. It is reported that the brittleness index is successful in ranking the breakage 
behavior of several different pharmaceutical materials [27]. Taylor et al. (2004) found 
good correlation between brittleness index and the size reductions of five 
pharmaceutical materials in a pilot-plant scale Apex hammer mill. 
3.3.2 –Elastic Deformation Mechanism 
Gahn and Mersmann (1995, 1997, 1999) developed a mechanistic model for 
attrition lost occurring in suspension crystallizers. They assumed that attrition 
happens only at the contact of a crystal corner with flat impeller surface. The 
mechanical properties of the crystals are assumed to be independent of strain rate and 
are described by average isotropic values. For Poisson’s ratio, a constant value of 0.25 
is assumed. Furthermore, they assumed that formation and propagation of the cracks 
occurs instantaneously. 
They considered a cone shape particle colliding with impeller surface as shown 
in Figure 3.4. They assumed that the stresses vary inversely as the square of the 
distance from the point of application of the impact force as proposed by Michell 
(1900). By integrating the elastic stress over the volume with elastic deformation, they 
obtained the average elastic energy per unit volume as: 
 𝐻2 𝑎4
𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒  ~ 4    (3.16)  𝐺 𝑙𝑖
where 𝐻 is hardness, 𝑎 is plastic deformation length scale, 𝐺 is shear modulus, and 𝑙𝑖 
is elastic deformation length scale. 
46 
 
 
 
Figure 3.4 - Illustration of cone shape particle under impact energy as shown in Gahn 
and Mersmann (1997) for derivation of mechanistic model based on elastic 
mechanism 
They assumed that the elastic energy obtained in Equation 3.16 is utilized to 
break the particle (which equates the elastic deformation length scale and the fracture 
length scale). Considering the plastic deformation mechanics and definition of the 
hardness, they obtained an expression to relate plastic deformation length scale, 𝑎, to 
plastic deformation energy, 𝑊𝑝𝑙𝑎𝑠𝑡𝑖𝑐, and hardness, 𝐻, as: 
𝑊 3𝑝𝑙𝑎𝑠𝑡𝑖𝑐 ~ 𝐻 𝑎      (3.17) 
Considering the relationship between impact energy, elastic deformation 
energy, and plastic deformation energy, they reasoned that the elastic energy 
contributes by less than 15% to the total impact energy and therefore assumed that the 
plastic energy is approximately equal to impact energy. Based on this assumption, 
Equation 3.17 becomes: 
𝑊 3𝑖𝑚𝑝𝑎𝑐𝑡 ~ 𝐻 𝑎      (3.18) 
Combining Equations 3.16 and 3.18 gives: 
47 
 
 
 𝐻2/3 𝑊 4/3
𝑊 𝑖𝑚𝑝𝑎𝑐𝑡𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒  ~ 4   (3.19)  𝐺  𝑙𝑖
This elastic energy is assumed to be the source for the fracture and a 
thermodynamic surface energy approach is used to give: 
𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒  𝑙𝑖 ~ Γ    (3.20) 
where 𝑤𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 is the elastic energy stored in a fragment with size 𝑙𝑖, 
and Γ is the particle surface energy. Combining Equations 3.19 and 3.20 gives: 
 𝐻2/3 𝑊 4/3𝑖𝑚𝑝𝑎𝑐𝑡
 𝐺  𝑙 3
 ~ Γ     (3.21) 
𝑖
3
Finally, substituting the fractured fragment volume as ∆𝑉 = 𝑙𝑖  in Equation 
3.21, they proposed the following model for fractional loss under impact: 
 𝐻2/3 𝑊 4/3
∆𝑉 ~ 𝑖𝑚𝑝𝑎𝑐𝑡      (3.22) 
 𝐺 Γ 
It is important to note that this model employs a surface energy approach to 
quantify the resistance against the fracture and does not account for stress 
concentration due to presence of cracks in the material. Consequently, plastic 
deformation at the crack tip is not considered in this model. In the remaining of this 
subsection, Equation 3.22 will be further established to give a comparable expression 
to other models presented in the next sections. 
 Equation 3.22 can be written in terms of collision or impact velocity, 𝑢, by 
substituting 𝑊𝑖𝑚𝑝𝑎𝑐𝑡 as: 
48 
 
 
𝑊𝑖𝑚𝑝𝑎𝑐𝑡 =  𝑚𝑢
2     (3.23) 
where 𝑚 is particle mass. Based on Equation 2.6 and constant Poisson ratio 
assumption (v = 0.25), shear modulus, 𝐺, can be substituted by elastic modulus, 𝐸. 
Furthermore, using the definition of fracture toughness, 𝐾𝑐 =  √𝐸 Γ, Equation 3.22 
becomes: 
 𝐻2/3 𝑚4/3 𝑢8/3
∆𝑉 ~ 2      (3.24)  𝐾𝑐  
By dividing the both sides of Equation 3.24 by the parent particle volume (𝑉 =
𝑥3), and substituting particle mass as 𝑚 = 𝜌𝑝𝑉, the final expression can be written as: 
∆𝑉 𝜌 4/3 𝑢8/3𝑝  𝑥 𝐻
2/3
 ~ 2      (3.25) 𝑉 𝐾𝑐
Where ∆𝑉 is fractured volume, V is initial or parent volume, 𝜌𝑝 is particle 
density, u is impact velocity, x is particle size, H is hardness, and 𝐾𝑐 is critical stress 
intensity factor (fracture toughness). Several studies used this model to address the 
attrition behavior of crystals under impact condition similar to those found in stirred 
tank crystallizers [40–43]. 
3.3.3 –Plastic Deformation Mechanism 
Another mechanistic model was developed by Ghadiri and Zhang [26] to relate 
the attrition of materials in single particle impact experiments to their physical 
properties measured from indentation tests. They assume that the fracture time scale is 
the same as plastic deformation time scale. Furthermore, they assumed that the 
49 
 
 
breakage of the particle is due to the formation of the subsurface lateral cracks 
induced by plastic deformation. 
Considering a single fragment being detached from a corner of the particle, they 
assumed an ideal geometry as shown in Figure 3.5. The volume of the fragment lost 
per impact is: 
∆𝑉 𝑙 2𝑎 
= 𝑖 3       (3.26) 𝑉 𝑥
where 𝑙𝑖 is the lateral crack length and a is the plastic deformation length scale. 
 
Figure 3.5 - Illustration of cubic shape particle under impact energy as shown in 
Ghadiri and Zhang (2002) for derivation of mechanistic model based on plastic 
mechanism 
In order to estimate the crack length, 𝑙𝑖, they used the halfpenny crack model in 
3/2
the generic form of 𝐾𝑐 =  𝑘𝑃⁄𝑙𝑖  [11], where 𝐾𝑐 is the fracture toughness, k is a 
proportionality constant, and P is the applied load that can be calculated from 
hardness data as 𝑃 = 𝐻𝑎2. Combining these two equations to eliminate 𝑃 gives: 
50 
 
 
𝐻𝑎2
𝑙𝑖 ~ ( )
2/3        (3.27) 
𝐾𝑐
They assumed that the impact energy is completely dissipated in the form of 
plastic deformation (𝐻~𝜌 𝑢2𝑝 ) and estimated its time scale as: 
𝑡𝑝 ~ √𝜌𝑝⁄𝐻  𝑥     (3.28) 
Based on Newton’s law, the relationship between applied load and plastic 
deformation time scale is: 
𝑑𝑢 3 𝑢𝑃 = 𝑚  ~ 𝜌𝑝𝑥      (3.29) 𝑑𝑡 𝑡𝑝
Using the definition of the hardness as 𝐻 = 𝑃/𝑎2 the above equation can be 
written as: 
𝐻 𝑎2
𝑢
 ~ 𝜌 𝑥3𝑝      (3.30) 𝑡𝑝
Combining Equations 3.28 and 3.30 to eliminate 𝑡𝑝 gives: 
𝜌𝑝
𝑎 ~ 𝑥𝑢1/2 ( )1/4     (3.31) 
𝐻
Ultimately, combining Equations 3.26, 3.27, and 3.31, they proposed the 
following model for fractional loss under impact condition: 
∆𝑉 𝜌𝑝 𝑢
2 𝑥 𝐻
 ~ 2       (3.32) 𝑉 𝐾𝑐
where ∆𝑉 is the fractured volume, 𝑉 is the initial volume, 𝜌𝑝 is the particle density, 𝑢 
is the impact velocity, 𝑥 is the particle size, 𝐻 is the hardness, and 𝐾𝑐 is the critical 
51 
 
 
stress intensity factor (fracture toughness). As seen in the Gahn and Mersmann model, 
Equation 3.25, particle breakage is related to the same parameters, but with different 
dependencies.  
Zhang and Ghadiri (2002) showed that this model can predict the single particle 
impact experiment results accurately up to a certain impact velocity. Several 
researchers used this model to quantify the attrition in different milling processes 
(Taylor et al. 2004; Bentham et al. 2004; Yang et al. 2007; Olusanmi et al. 2011). 
3.4 –Breakage Model Based on Elastic-Plastic Deformation Mechanism 
The aforementioned mechanistic models consider either elastic or plastic 
deformation. In reality, both mechanisms contribute to particle breakage. Assume a 
cubic particle collides with a hard surface at its corner with velocity u as shown in 
Figure 3.6. It was considered that the impact surface has a high elastic modulus (hard 
surface) and therefore all the kinetic energy of the impact is being transferred to the 
particle. The impact energy dissipates in two ways: elastic deformation and plastic 
deformation of the particle (Equation 3.33). The plastic deformation length scale is a, 
and the elastic deformation length scale in the fractured volume is li. 
𝑊𝑖𝑚𝑝𝑎𝑐𝑡 ~  𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐 +  𝑊𝑝𝑙𝑎𝑠𝑡𝑖𝑐   (3.33) 
52 
 
 
 
Figure 3.6- Illustration of plastic and elastic deformation of a particle under impact at 
its corner, a is plastic deformation length scale, li is elastic deformation length scale in 
fractured volume, and x is initial particle size 
The impact energy can be estimated as 𝑊 2𝑖𝑚𝑝𝑎𝑐𝑡 ~ 𝑚𝑢  (Equation 3.23) where 
m is particle mass, and u is impact velocity. Plastic deformation energy can be 
calculated as [25]: 
𝑊𝑝𝑙𝑎𝑠𝑡𝑖𝑐  ~   𝐻 𝑎
3     (3.34) 
where a is plastic deformation length scale and H is hardness. And finally, the elastic 
deformation energy can be estimated as [32,49]: 
𝐻2
𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐 ~  𝑎
3     (3.35) 
𝐸
where E is elastic modulus. The Poisson ratio was considered that of an isotropic 
crystal with ν=0.33 and hence the elastic behaviour of the crystal under different 
strain conditions can be evaluated using only the elastic modulus (shear modulus and 
bulk modulus are linearly related to elastic modulus). Combining equations 3.23 and 
3.33-35 gives: 
53 
 
 
3 𝑚𝑢
2
𝑎   ~  2      (3.36) 𝐻
 𝐻+
𝐸
The key observation here is that the plastic energy is consumed and dissipated 
to deform the particle permanently with the length scale of a, but, elastic energy is 
propagated through the particle and is the available source of energy for the breakage. 
By combining equations 3.35 and 3.36, the available elastic energy in the particle 
under impact can be written as: 
𝐻2 𝑚𝑢2
𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐  ~   𝐸 𝐻2     (3.37)  𝐻+
𝐸
In order for breakage to happen, 𝑊𝑒𝑙𝑎𝑠𝑡𝑖𝑐 needs to be great enough to produce 
enough stress, 𝜎𝑓 that, based on Griffith’s theory, 𝜎𝑓 √𝜋 𝑎𝑓  reaches the limit of 
𝐾𝑐 (= √𝐸 𝛤 ) as: 
𝜎𝑓 √𝜋 𝑎𝑓 =  𝐾𝑐     (3.38) 
where 𝜎𝑓  is stress at fracture and 𝑎𝑓 is flaw length scale, 𝐾𝑐 is stress intensity factor 
(fracture toughness), E is elastic modulus, and 𝛤 is surface energy. The stress at 
fracture can be estimated as: 
𝑊 2 2
𝜎  ~ 𝑒𝑙𝑎𝑠𝑡𝑖𝑐
1 𝐻 𝑚𝑢
𝑓 3  ~ 3   2    (3.39) 𝑙𝑖 𝑙𝑖 𝐸 𝐻 𝐻+
𝐸
For brittle and semi-brittle particles, the pre-existing cracks and flaws are 
always presented in the particle. The larger the particle, the larger the flaws and 
smaller stress needed for crack propagation. Therefore, the flaw length scale can be 
written as a function of particle size, x, as: 
54 
 
 
𝑎𝑓 =  𝑓(𝑥)      (3.40) 
The simplest functionality would be a linear functionality. A linear relationship 
between flaw size and the range of particle size is assumed here as: 
𝑎𝑓 ∝ 𝑥       (3.41) 
Substituting 𝜎𝑓 and 𝑎𝑓 from Equations 3.39 and 3.41, Equation 3.39 gives: 
𝐻2 𝑚𝑢2 √𝑥
 2  3   ~ 𝐾𝑐     (3.42) 𝐸 𝐻 𝐻+ 𝑙𝑖
𝐸
Substituting particle mass as 𝑚 = 𝜌𝑝𝑉, dividing the both sides of Equation 3.42 
by the parent particle volume (𝑉 = 𝑥3), and substituting the fracture volume 
as ∆𝑉 ~ 𝑙 3𝑖 , an expression for fracture volume can be obtained as: 
∆𝑉 𝜌 2𝑝 𝑢   𝐻
2√𝑥
 ~ 2      (3.43) 𝑉 𝐻𝐸 𝐾𝑐 (𝐻+ )𝐸
𝐻2
Generally  < 0.1 𝐻 [32] and the final model can be approximated as: 
 𝐸
∆𝑉 𝜌𝑝 𝑢
2  𝐻√𝑥
 ~      (3.44) 
𝑉 𝐸 𝐾𝑐
Equation 3.44 relates the fractional loss during the impact to kinetic energy of 
the impact, square root of the particle size, and mechanical properties of the particle. 
The plastic, elastic, and fracture behavior of the particle are important in fracture 
process and they are represented in the model by hardness, elastic modulus, and 
fracture toughness respectively. It is important to note that the stress concentration 
due to presence of cracks in the material as well as the plastic deformation at the crack 
55 
 
 
tip is considered via adopting Griffith fracture theory and employing the concept of 
fracture toughness in deriving the Equation 3.44. 
3.5 –Similarities between Comminution Laws and Mechanistic Models 
By some manipulation, it can be shown that the classical theories of 
comminution can be translated to mechanistic models. This discovery is particularly 
important in bridging the existing studies on particle fracture and milling in different 
disciplines and therefore be able to interpret the data from different experiments. 
3.5.1 –Lawn and Marshal Approach Compared to Kick’s Law 
Lawn and Marshal classified the brittleness as the ratio of hardness to critical 
stress intensity factor. Classifying the brittleness index as 𝐻⁄𝐾𝑐 without taking into 
account the effect of particle size, is similar to Kick’s law. Kick assumes that the 
energy required for size reduction is proportional to volume ratio of feed and 
produced particle: 
𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑣𝑜𝑙𝑢𝑚𝑒
𝐸𝑠𝑝  ~ log  ( )    (3.45) 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
Based on this model the energy required to reduce size from 100 to 10 is equal 
to the energy required for size reduction from 10 to 1. In other word, the particle size 
effect is not seen in Kick’s law. 
3.5.2 –Ghadiri and Zhang Model Compared to Rittinger’s Law 
By setting the specific energy as 𝐸 = 𝑢2𝑠𝑝  and rearranging the Ghadiri and 
Zhang model, equation 3.32, we obtain 
56 
 
 
2 𝐾
2∆𝑉
𝐸𝑠𝑝 =  𝑢  ~ 
𝑐       (3.46) 
𝜌𝑝 𝐻 𝑥 𝑉
where ∆𝑉~𝑑𝑥3and 𝑉~𝑥3. Differentiating the both sides of Equation 3.46 and 
considering 𝑑𝑥3~𝑥2𝑑𝑥, we obtain 
𝐾 2 𝑑𝑥
𝑑𝐸𝑠𝑝 ~  
𝑐       (3.47) 
𝜌𝑝 𝐻 𝑥2
This matches to Rittinger’s law as: 
𝑑𝑥
𝑑𝐸𝑠𝑝 =  − 𝑐𝑅  2    ,    𝑐𝑅 ~ − 𝐾
2
𝑐 ⁄𝜌𝑝 𝐻  (3.48) 𝑥
It is found that Rittinger’s constant can be expressed in terms of mechanical 
properties of the material. 
3.5.3 –Gahn and Mersmann Model Compared to Bond’s and Rittinger’s Law 
Setting the specific energy as 𝐸𝑠𝑝 = 𝑢
2 and rearranging the Gahn and 
Mersmann model, equation 3.25, gives: 
3/4
𝐾 2∆𝑉
𝐸𝑠𝑝 =  𝑢
2 ~ ( 𝑐4 2 )     (3.49) 
𝜌𝑝3 𝐻3 𝑥 𝑉
Differentiating the both sides gives: 
𝐾 3/2 𝑑𝑥
𝑑𝐸 𝑐𝑠𝑝 ~ 𝜌  𝐻1/2  7/4     (3.50) 𝑝 𝑥
This is analogized to an energy model in between Rittinger’s (𝑛 = 2) and 
Bond’s law (𝑛 = 3/2) with exponent 𝑛 = 7/4 as: 
57 
 
 
𝑑𝑥
𝑑𝐸𝑠𝑝 =  − 𝑐 7     ,    𝑐 ~ −𝐾
3/2
𝑐 ⁄𝜌𝑝 𝐻
1/2  (3.51) 
𝑥4
3.5.4 –Current Study’s Model Compared to Bond’s Law 
Setting the specific energy as 𝐸𝑠𝑝 = 𝑢
2 and rearranging the current study’s 
model, equation 3.44, gives: 
𝐸 =  𝑢2
𝐸 𝐾𝑐 ∆𝑉
𝑠𝑝  ~      (3.52) 𝜌𝑝 𝐻√𝑥  𝑉
Differentiating the both sides gives, 
𝐸 𝐾 𝑑𝑥
𝑑𝐸𝑠𝑝 ~ 
𝑐  3/2     (3.53) 𝜌𝑝 𝐻 𝑥
This translates to the Bond’s law as: 
𝑑𝑥
𝑑𝐸𝑠𝑝 =  − 𝑐𝐵  3    ,   𝑐𝐵 ~ − 𝐸 𝐾𝑐⁄𝜌𝑝 𝐻  (3.54) 
𝑥2
Based on this result, Bond’s constant can be expressed in terms of mechanical 
properties of the material. 
3.6 –Mechanistic Models for Wet Milling in Rotor Stator Mixers 
The forces acting on the particle in a wet milling process can be divided into 
two opposing forces: a cohesive force and a disruptive force. The cohesive force is a 
function of particle size and mechanical properties and acts to oppose the particle 
fracture. The disruptive force acts on a particle to break it and depends on the type of 
mill as well as the operating conditions. It is possible to estimate theses opposing 
58 
 
 
forces based on the aforementioned mechanistic models. This procedure is described 
in the following sections. 
3.6.1 –Disruptive Forces 
In the experimental part of this study, Chapter 4, it is concluded that the impact 
energy imparted to the particles due to particle-particle and particle-wall collisions is 
primarily important in breakage of particles. Given the highly turbulent flow field 
Re>10,000 in rotor-stator mixers, both macroscale turbulence and inertial subrange 
turbulence models were used to determine the collision velocities as summarized in 
Table 3.2. For particle collisions governed by the macroscale turbulence velocity 
(macroscale turbulence with length scale ℓ), it is reasonable to consider that the 
largest disruptive force available for breakage is proportional to rotor tip velocity, 
𝑢𝑡𝑖𝑝. On the other hand, turbulent eddy fluctuations determine the velocity at which 
particle-particle and particle-wall collisions happen in the turbulent flow. It is 
reasonable to assume the velocity of a particle with size x is proportional to velocity 
of the eddy with the same length scale. In the rotor-stator mixers, the Kolmogorov 
length scale (𝜂 ~ 1 - 10 μm) is at least one order of magnitude smaller than the mean 
particle size. Therefore, the inertial subrange model can be used to calculate the 
2
average magnitude of eddy fluctuation velocity ( √̅̅𝑢(̅̅𝑥̅̅)̅′  ). The Inertial subrange 
model estimates that the kinetic energy of eddies with size x is equal to energy 
dissipation rate (𝜖) at size x with the power of 2/3. By substituting 𝜖 ~ 𝑁3𝐷2 
and 𝑁𝐷 ~ 𝑢𝑡𝑖𝑝, the expression in the third row of Table 3.2 is obtained. It is important 
to note that 𝑢𝑡𝑖𝑝 can be used to compare the data in the case of geometrically similar 
systems, although local energy dissipation rate, which depends on the power draw and 
59 
 
 
geometry of the mill, is a better measure to compare the data between different mills 
with different geometries. 
Table 3.2- Disruptive stress, 𝜏𝐷 ~ 𝜌
2
𝑝𝑢  predicted by different models: macroscale 
turbulence and inertial subrange turbulence models 
Macroscale 
Inertial subrange turbulence 
Model turbulence 
𝜂 < x < ℓ 
x ~ ℓ 
Collision 
~ Tip speed ~ Eddy velocity on length scale of x 
speed 
2
2
𝜌 𝑢 2 𝜌 𝑢̅̅ (̅̅𝑥̅̅
2 𝑥 3
)̅′𝑝 𝑡𝑖𝑝 𝑃   ~  𝜌𝑃(𝜖𝑥)3  ~  𝜌𝑝 𝑢
2
𝑡𝑖𝑝  ( )  Disruptive 𝐷
stress  (𝑢 = 𝑢𝑡𝑖𝑝) 
(u = √
2
𝑢̅̅ (̅̅𝑥̅̅)̅′  ) 
3.6.2 –Cohesive Forces 
The cohesive forces that oppose the particle’s breakage can be estimated using 
the mechanistic models presented above. It is reasonable to assume that the size of 
fractured volume is proportional to corresponding disruptive and cohesive forces. 
This can be expressed as: 
𝛥𝑉 𝐷𝑖𝑠𝑟𝑢𝑝𝑡𝑖𝑣𝑒 𝑆𝑡𝑟𝑒𝑠𝑠
=       (3.56) 
𝑉 𝐶𝑜ℎ𝑒𝑠𝑖𝑣𝑒 𝑆𝑡𝑟𝑒𝑠𝑠
Rearranging the model based on plastic mechanism, equation 3.32, gives: 
∆𝑉 𝜌  𝑢2𝑝  
 ~       (3.57) 
𝑉 𝐾 2𝑐 /𝐻𝑥
60 
 
 
By comparing equations 3.56 and 3.57, we can extract the cohesive stress as: 
𝐾 2
𝜏 = 𝑐𝑐       (3.58) 𝐻 𝑥
Similarly, a cohesive stress can be extracted from the elastic mechanism model 
by rearranging equation 3.25 as: 
∆𝑉 23/4 𝜌𝑝 𝑢  (  )   ~ 3/2     (3.59) 𝑉 𝐾 /𝐻1/2 𝑥3/4𝑐
And by comparing to equation 3.56, the cohesive stress can be written as: 
𝐾 3/2
𝜏 𝑐𝑐 = 𝐻1/2 𝑥3/4      (3.60) 
In the same way, a cohesive stress can be extracted from the elastic-plastic 
model, equation 3.44, by rearranging it as: 
∆𝑉 𝜌𝑝 𝑢
2 
 ~      (3.61) 
𝑉 𝐸 𝐾𝑐/𝐻 𝑥1/2 
And the cohesive stress can be determined as: 
𝐸 𝐾
𝜏 𝑐𝑐 = 𝐻 𝑥1/2      (3.62) 
As shown above, each of these different models predicts different relative 
importance of the material physical properties and particle size in breakage process. 
Table 3.3 summarizes these different cohesive stresses. 
 
61 
 
 
Table 3.3-Cohesive stresses, 𝜏𝐶, resisting particle breakage predicted by different 
mechanisms: indentation, elastic, plastic, and elastic-plastic models 
Model Cohesive Stress (𝝉𝑪) 
𝐾𝑐
Indentation Mechanism*  𝑐  
𝐻
𝐾 3/2
Elastic Mechanism  𝑐  
𝐻1/2 𝑥3/4
𝐾 2
Plastic Mechanism  𝑐  
𝐻 𝑥
𝐸𝐾𝑐
Elastic-Plastic Mechanism   
𝐻 𝑥1/2
*c is a dimensional constant (Pa/𝑚1/2) 
3.6.3 –Breakage Model 
Breakage occurs when the particles are exposed to disruptive stresses that 
exceed the cohesive stresses. Thus, by comparing the disruptive and cohesive stresses, 
a criterion can be specified for the occurrence of breakage and predicting the largest 
particle size that can survive. That is to say, the maximum stable particle size, x, 
exists when the disruptive stress balances the cohesive stress (𝜏𝐶 ∼ 𝜏𝐷). As has been 
noted, 2 different disruptive stresses (Table 3.2)  and 4 different cohesive stresses 
(Table 3.3) are present in the wet milling process, and as a result 2×4=8 different 
mechanistic models can be predicted Table 3.4 and Table 3.5 summarize the different 
breakage models based on rotor tip speed and energy dissipation rate respectively. 
 
 
62 
 
 
Table 3.4- Breakage models: mechanistic correlations for maximum stable particle 
size (𝑥𝑚𝑎𝑥 𝑜𝑟 𝐷100) obtained by a balance between the cohesive and disruptive 
stresses 
Cohesive Disruptive Correlation for Maximum Stable 
  
Stress Model Stress Model Particle Size 
𝑥 1 𝜌 𝐻
ϯ − 
𝑃  𝑢 2
 MT    ~  𝑒 𝑐𝐾 𝑡𝑖𝑝𝑐  
𝐷 𝐷
Indentation 
Mechanism 
𝑥 𝑐𝐾𝑐 1
 IST‡   ~  ( )3/2   
𝐷 𝜌  𝐻 𝑢 3𝑃 𝑡𝑖𝑝
𝑥 𝐾 2𝑐 1
 MT   ~    
𝐷 𝜌𝑃 𝐻𝐷 𝑢 2𝑡𝑖𝑝
Plastic 
Mechanism 
𝑥 𝐾 2𝑐  1
 IST   ~ ( )3/5   
𝐷 𝜌  𝐻𝐷 𝑢 6/5𝑃 𝑡𝑖𝑝
3
𝑥 𝐾𝑐2 1
 MT   ~  ( 1 3)
4/3   
𝐷 𝑢 8/3
𝜌𝑃 𝐻2 𝐷4
𝑡𝑖𝑝
Elastic 
Mechanism 3
𝑥 𝐾𝑐2 1
 IST   ~  ( )12/17
𝐷 1 3
  
𝑢 24/17
𝜌𝑃 𝐻2 𝐷4
𝑡𝑖𝑝
𝑥 𝐸𝐾𝑐 1
 MT   ~  ( 
2
1)   𝐷 𝑢 4
𝜌𝑃 𝐻𝐷2
𝑡𝑖𝑝
Elastic-Plastic 
Mechanism 
𝑥 𝐸𝐾𝑐 1
 IST   ~  ( )
6/7   
𝐷 1 𝑢 12/7
𝜌𝑃 𝐻𝐷2
𝑡𝑖𝑝
ϯ MT = Macroscale Turbulence     ‡ IST = Inertial Subrange Turbulence 
 
 
63 
 
 
Table 3.5- Breakage models based on energy dissipation rate (disruptive stresses are 
calculated based on inertial subrange turbulence) 
Cohesive Stress Correlation for Maximum 
 Disruptive Stress Model  
Model Stable Particle Size 
Indentation Inertial Subrange 𝑐𝐾𝑐 1
  𝑥 ~  ( )3/2  
mechanism Turbulence 𝜌𝑃 𝐻 ɛ
Plastic Inertial Subrange 
  𝐾
2
𝑐  13/5
mechanism Turbulence 𝑥 ~ ( )    𝜌  𝐻 ɛ2/5𝑃
3
Plastic Inertial Subrange 𝐾𝑐2 1
  𝑥 ~  ( )12/17  
mechanism Turbulence 1 ɛ8/17
𝜌𝑃 𝐻2 
Elastic-plastic Inertial Subrange 𝐸𝐾𝑐 1
  𝑥 ~  ( )6/7  
mechanism Turbulence 𝜌  𝐻 ɛ4/7𝑃
 
Here D is the rotor diameter; x is the maximum stable particle size (D100) that 
can survive under the disruptive force that is dictated by the milling operational 
conditions, i.e. rotor tip speed, 𝑢𝑡𝑖𝑝~𝑁𝐷 (𝑁 is rotor speed and 𝐷 is rotor diameter). In 
practice, D95, D90, or D80 of the size distribution are used instead of largest particle 
size, because of statistical considerations and the high scatter in D100 measurements. 
In this study, D90 is considered as the representative of the largest particle size and it 
is shown that D95 and D80 are linearly correlated to D90. 
Again, each of these different models predicts different relative importance of 
the material physical properties and energy input to the breakage process. The validity 
of these models needs to be assessed by comparison to experimental data. In Chapter 
4, the detailed description of the experiment’s setup and procedure are given and the 
64 
 
 
experimental data for wet milling of different materials in different rotor-stator mixers 
are presented. 
3.7 –Dimensionless Number for Particle Breakage in Geometrically 
Similar Systems 
Analogous to the Weber number in drop breakup, a new dimensionless number 
for solid fracture was developed. It is the ratio of inertial force to cohesive force of 
solid particles, and it is called the Comminution number (Co): 
𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 
𝐶𝑜 =      (3.63) 
𝑆𝑜𝑙𝑖𝑑 𝑐𝑜ℎ𝑒𝑠𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒
In section 3.6, the solid cohesive stresses are obtained using different 
mechanistic models. Using these different models, different Comminution numbers 
can be defined. Table 3.6 summarizes the Comminution numbers and corresponding 
breakage models. The use of Comminution number will allow the important task of 
scale up/down in milling processes as well as data reduction of similar problems. 
 
 
 
 
 
65 
 
 
Table 3.6- Comminution number for geometrically similar systems: definition and 
application in different mechanistic models 
Correlation for Maximum 
Model      Co Number Stable Particle Size based on 
Co Number 
𝜌  𝑢 2 
Indentation 𝑃 𝑡𝑖𝑝
 𝐶𝑜𝐼 =   
𝑥
−3/2
Mechanism 𝛼 𝐾
   ~ (𝐶𝑜𝐼)  𝑐 𝐷
𝐻
𝜌  𝑢 2𝑃 𝑡𝑖𝑝  
𝐶𝑜𝐸 =  3  𝑥
Elastic Mechanism −12/17𝐾𝑐2    ~ (𝐶𝑜 )  𝐷 𝐸
1 3
𝐻2 D4
𝜌  𝑢 2𝑃 𝑡𝑖𝑝  𝑥
Plastic Mechanism 𝐶𝑜𝑃 =  2     ~ (𝐶𝑜 )−3/5𝐾𝑐 𝐷 𝑃
 
H 𝐷
𝜌  𝑢 2𝑃 𝑡𝑖𝑝  
Elastic-Plastic 𝐶𝑜𝐸𝑃 =   𝑥 𝐸 𝐾 −6/7
Mechanism 𝑐
   ~ (𝐶𝑜𝐸𝑃)  
1 𝐷
𝐻D2
The proposed mechanistic models and equations in this chapter were assessed in 
Chapter 5 using the experimental data. As described in Chapter 4, these data were 
obtained in wet milling of different crystal-anti-solvent systems in both the Silverson 
L4R inline and batch mixers. The inline mixer was further modified to study the 
effect of the shear gap width on milling performance. The detailed summary of the 
experiment results and procedure is given in Chapter 4. 
3.8 –Summary 
• A class of mechanistic models for breakage and attrition of crystals and particulate 
solids have been developed based on elastic-plastic deformation and fracture 
mechanics concepts. These mechanistic models were further analyzed to give four 
66 
 
 
different forms of cohesive forces resisting the particle breakage under different 
deformation stresses based on elastic deformation, plastic deformation, and 
elastic-plastic deformation mechanisms. Furthermore, two different scales of 
velocities were identified for collision intensity based on macroscale turbulence 
and inertial subrange turbulence theories. Based on these 2 different disruptive 
stresses and 4 different cohesive stresses, 2×4=8 different mechanistic 
correlations for maximum stable particle size were introduced. 
• The proposed mechanistic models show that the breakage of a particle is a 
function of its mechanical properties, size, and the impact energy. Most of the 
impact energy usually consumed to plastically deform the particle and only small 
portion propagate as elastic energy. 
• The functional form of these models was shown to agree with classical 
comminution theories where the size reduction is empirically related to power 
consumption.  
• Based on the mechanistic models, a dimensionless comminution number is 
developed (ratio of disruptive to cohesive forces) to aid physical interpretation and 
scale up/down efforts for geometrically similar devices 
• In this study, the characteristic velocity of rotor-stator mixers, 𝑢𝑡𝑖𝑝 = 𝑁𝐷, is used 
to describe the particle collision velocity and to construct the mechanistic models 
of milling. However, these mechanistic models can be utilized in other milling 
devices by substituting the appropriate characteristic velocity, e.g., jet velocity in 
67 
 
 
impinging jets, superficial velocity in valve homogenizers, and impact velocity of 
milling balls in ball mills. 
68 
 
 
Chapter 4: Wet Milling Experiments in Rotor Stator Mixers 
4.1 –Introduction 
Wet milling involving high shear rotor-stator mixers to reduce crystal size is an 
emerging practice with significant potential in crystallization and milling of active 
pharmaceutical ingredients (API). It has certain advantages such as smaller footprint, 
higher yield, lower cost, and better control of heat generation and crystal morphology 
when compared to dry milling. Despite the increasing industrial interest, there are few 
works to address wet milling of API or other solid particles in high shear mixers.  
Lee et al. (2004) conducted wet milling experiments in a lab scale rotor-stator 
inline mixer with different crystal-antisolvent systems. The effect of concentration of 
slurry (14 to 20% by weight), morphology of crystals, rotor speed, and flow rate were 
studied on milling performance. They reported that the wet milling performance is 
relatively independent of slurry concentration and flow rate when operating the mixer 
in a free pumping mode. They stated that the shear frequency, defined as the product 
of rotation rate, number of slots on rotor, and number of slots on stator, can be used to 
scale up rotor –stator wet mills from lab to pilot scale manufacturing. Kamahara et al. 
(2007) studied a semi-batch crystallizer using a rotor-stator device to generate fine 
particles as a source of secondary crystallization and they concluded that a constant 
rotor tip speed is a criterion for scale up. Harter et al. (2013) conducted an 
experimental study to evaluate the effect of flow rate, slurry solids concentration, and 
starting particle size distribution in wet milling performance and terminal particle 
size. They concluded that by maintaining constant tip speed comparable terminal 
particle size could be achieved. In addition, they developed an empirical model to 
69 
 
 
incorporate the critical geometric parameters by considering the frequency and the 
probability of a slot event to account for geometrical differences of the mill heads 
across the scales. Engstrum et al. (2013) studied particle size reduction in two 
different lab scales and one pilot scale rotor-stator mixers. They assumed that the 
dominant mechanism of size reduction is rotor-particle collisions and influence of 
particle-particle collisions and fluid shear forces have a comparatively negligible 
effect. They introduced a scaling approach to correlate API particle size reduction to 
energy input normalized for rotor-stator mixers by accounting for mill geometry, tip 
speed, and mill head residence time. Luciani et al. (2015) presented a modelling-aided 
approach to scale-up high-shear rotor−stator wet milling processes for pharmaceutical 
applications. A population balance equation was used to provide insight into the 
significance of different scale-up factors. They concluded that the specific breakage 
rate in population balance equations can be correlated to shear rate, particle-wall 
collisions, and frequency of rotor-stator events. 
Given the relatively small amount of the current research on wet milling of 
crystalline materials in rotor-stator mixers, there is clearly a need for more 
investigation to address the effect of different milling variables, particle mechanical 
properties, and mixer geometry/scale on wet milling performance. Apart from limited 
amount of literature, the major shortcoming of the existing research summarized 
above is that they do not consider the effects of particle size and physical properties 
on milling performance. Consequently, the proposed correlations and scale up 
strategies are not mechanistic and remain case dependent. 
Other than wet milling of crystalline materials, incorporation of nanoparticles in 
the liquid phase, deagglomeration of nanoparticle clusters and stabilization are also 
70 
 
 
carried out in rotor-stator mixers. Despite the similarities in particle attrition and 
erosion mechanisms, the cohesive forces resisting the deagglomeration of 
nanoparticle clusters are different than those resisting the particle fracture. Therefore, 
a different approach is required to study the deagglomeration processes. 
 Baldyga et al. (2008) utilized a two-stage rotor-stator inline mixer to study 
break up of fumed silica clusters. They reasoned that erosion is the dominant breakup 
mechanism and concluded that the breakup occurs when the hydrodynamic stresses in 
turbulent flow, based on Kolmogorov theory, are larger than the bonding forces 
forming the agglomerates. Padron et al. (2008) studied the effect of rotor speed, flow 
rate, particle properties, and rheology of dispersions on the breakup of hydrophobic 
and hydrophilic fumed silica nanoparticle clusters in an inline rotor-stator mixer. They 
showed that the breakup occurs through erosion and smaller aggregates are gradually 
eroded from large clusters over a period. They concluded that rotor speed has a 
stronger effect on cluster breakup when compared to flowrate effect. Furthermore, 
they concluded that at higher concentrations the rate of fine generation of 
hydrophobic particles are higher and it can be due to difference in hydrogen bond and 
rheology of dispersions. 
In this chapter, the results of wet milling experiments performed in the 
Silverson L4R rotor-stator inline mixer with the standard and enlarged shear gap, and 
in the batch mixer are presented. These experiments are conducted to investigate the 
effect of milling conditions, mixer geometry, and particle physical properties on 
ultimate particle size and particle breakage rate. 
71 
 
 
4.2 –Rotor-Stator Mixer Geometry 
In mixing processes, there is generally a trade-off between the required power 
input and the resultant particle size distribution. The maximum particle size obtained 
at equilibrium is a function of maximum local energy dissipation rate rather than the 
average energy dissipation rate in the mixing head. The advantage of rotor-stator 
mixers is their capacity to concentrate most of the supplied power in the mixing head 
creating a high local energy dissipation region which enhance the mixing performance 
[57–59]. Furthermore, an in-line rotor–stator mixer can be used in a continuous 
process by placing the mixing head in a recirculation loop of a stirred vessel. 
Silverson rotor-stator L4R inline and batch mixers were used in this study. They 
have a 250 W, 110 Volt, single phase, 60 Hz motor with nominal maximum speed of 
9000 rpm. The rotor is a radial 4-blade impeller that rotates inside a stator with square 
holes as shown in Figure 4.1. The rotor moves the slurry radially out of the mixer 
head through the 4 rows of holes in the stator. The mixing capacity of the batch unit 
depends on the slurry viscosity. In the inline assembly, the centrifugal action of the 
rotor generates a pumping action which gives a throughput up to 12 litres/minute 
(lpm), reducing as the viscosity increases. A valve, in the pipeline on the outlet side of 
the mixer, is used to control the throughput independent of the rotor speed. 
72 
 
 
 
Figure 4.1- Silverson L4R Rotor Stator Mixers with Square Hole Stator (a) Inline 
mixer, (b) Batch mixer 
Most of the experiments were carried out in the inline mixer with the standard 
shear gap of 0.2 mm (clearance between rotor and stator). Since the initial crystal size 
is greater than the shear gap width, the rotor was modified to have a larger shear gap 
of 1.7 mm, and to study the effect of shear gap width on milling performance. The 
milling experiments were also performed in the batch mixer to investigate the effect 
of scale/geometry on particle breakage in terms of power draw and the volume of the 
dispersion zone. The lab scale Silverson L4R batch mixer has a smaller mixing head 
compared with the lab scale Silverson L4R inline mixer. Geometry details for the 
different mixers are listed in Table 4.1. 
 
 
 
 
73 
 
 
Table 4.1- Silverson L4R inline and batch mixer geometry details 
Inline Mixer Batch Mixer 
Mixer Dimensions 
Standard Enlarged 
Standard Shear Gap 
Shear Gap Shear Gap 
Rotor Diameter 31.1 mm 28.2 mm 28.1 mm 
Shear Gap Width 0.2 mm 1.7 mm 0.2 mm 
Square Hole Stator 104 Square Hole Slots 92 Square Hole Slots 
Size of Stator Slots 2.4 mm×2.4 mm×2.0 mm 2.6 mm×2.4 mm×1.6 mm 
Thickness of Stator 
2.0 mm 2.0 mm 
Ring 
Height of Stator Ring 18.9 mm 18.9 mm 
Inner Diameter of Stator 
31.5 mm 28.5 mm 
Ring 
4.3 –Experiment Setup 
4.3.1 –Inline Mixer Setup 
Rotor-stator mixer is fed through a recirculation loop in line with a stirred 
vessel as shown in Figure 4.2. The experimental rig consists of a 3 litres glass vessel 
to hold the solid suspension. The holding vessel is fitted with a down pumping 
pitched blade turbine (D = 0.5T, blades inclined at 45°, C = T/4) rotated at 900 rpm 
(using Waverly OS20 overhead stirrer) to ensure complete solids dispersion [60,61]. 
Four baffles were mounted close to the wall of the vessel to promote better top to 
bottom mixing. The uniform dispersion state was verified by sampling different 
74 
 
 
vertical locations within the holding vessel in situ using the FBRM probe and offline 
by the laser diffraction method. 
 
Figure 4.2- Experimental setup of Silverson inline rotor-stator mixer feed through a 
recirculation loop in line with a stirred vessel 
The slurry from the holding vessel was pumped via a 1/2 inch PTFE tube to an 
inline L4R Silverson high shear rotor–stator mixer. The mixer head used in this work 
was a one-stage rotor-stator design with the square-hole stator and the four-blade 
rotor. The rotor is driven by a motor controlled by a frequency inverter and may be 
run to a maximum rotor speed of 10,000 rpm with 100 rpm intervals. The holding 
vessel was charged with 2 litres of antisolvent (isopropyl alcohol) and left for half an 
hour to reach thermal equilibrium at a given rotor speed. The slurry was maintained at 
room temperature by running water as a cooling fluid over the mixer head. The 
temperature change was monitored by a differential thermometer before and after the 
mixer head to ensure zero net heat generation. A diaphragm valve was installed in the 
outlet (recycle) tube to control the flow rate. The time zero is taken when the crystals 
were added to the holding vessel. Experiments were carried out over a range of 
independently controlled flow rates and rotor speeds. 
75 
 
 
The samples removed from the holding vessel at different times during the 
milling, were analysed offline by Horiba LA-950 laser diffraction and video 
microscopy techniques. Furthermore, particle chord length and shape were 
continuously monitored in the holding vessel by two particle size analysis probes, a 
FBRM and a PVM probe. They were positioned at an angle to the horizontal (as 
suggested by the manufacturer) at a height just above the axial centre of the impeller 
for the best performance. In addition, the FBRM measurement confirmed that the 
crystals were uniformly suspended in the holding vessel. 
4.3.2 –Batch Mixer Setup 
A schematic of the batch mixer setup is shown in Figure 4.3. A 2 litres glass 
vessel was charged with antisolvent (isopropyl alcohol). It was then attached to the 
mixer, and left for half an hour to reach thermal equilibrium at a given rotor speed. 
The time zero is taken when the crystals were added to the mixing vessel. The slurry 
temperature was measured and maintained at room temperature by using a cooling 
unit, HAAKE Phoenix II C35P, with a model predictive control strategy. The samples 
removed from the holding vessel at different times during the milling, were analysed 
offline with the Horiba LA-950 laser diffraction and video microscopy techniques. 
Furthermore, particle chord length was monitored online in the batch vessel with the 
FBRM probe. The milling is continued up to 90 minutes, at which point no further 
change in size distribution is observed. 
76 
 
 
 
Figure 4.3- Experimental setup of Silverson batch mixer in a glass vessel immersed in 
a cooling bath 
4.4 –Particle Size Measurement Procedure 
In order to perform different experiments with a consistent initial particle size 
distribution, the crystals were sieved using a vibratory sieve shaker. A fraction of 250-
420 μm was collected and used as the feed for the wet milling experiments. Different 
techniques were used to measure the size distribution and shape of the particles during 
the milling. An FBRM probe, a Mettler-Toledo Lasentec D600 particle size analyser, 
was used in situ for inline measurement of size as well as evaluation of complete solid 
suspension in the holding vessel of inline setup. In addition, a PVM probe is used in 
situ for online monitoring of particle shape. Also from Metller-Toledo, it records 
digital images with a field of view of 826 micron by 619 microns. 
A laser diffraction device is used off line to measure the particle size 
distribution of samples taken from the holding vessel at different times of the milling. 
Horiba Partica LA-950 provides a volume based particle size distribution in range of 
77 
 
 
0.01 micron to 3 mm. The 10 mL sample removed from holding vessel was added to 
30 mL antisolvent (1:4 dilution). A fraction of the diluted slurry sample was 
transferred to the laser diffraction measurement cuvette. A magnetic stirrer bar is used 
to stir the slurry and keep the particles dispersed in the cuvette. Dispersion of particles 
larger than 400 microns in the measurement cuvette was challenging and a larger 
stirrer bar is used with a higher stirring rate to obtain a uniform suspension of crystals 
in the test sample. No sonication was applied to prevent undesired particle breakage 
[4].  
4.5 –Mechanical Properties Measurement Procedure 
Indentation tests were performed using a TI 950 Hysitron TriboIndenter to 
obtain the mechanical properties for 3 different crystals, sucrose, glycine, and 
ascorbic acid that are used in wet milling experiments. These crystals are primarily 
selected for milling experiments because they have different elastic modulus, 
hardness, and fracture toughness but similar morphology as shown in the microscopy 
images in Figure 4.4. Crystals with size range of 500-700 μm are selected for the 
indentation test. For each material, 10 different crystals are tested. Different 
mechanical properties, elastic modulus, hardness, and fracture toughness are obtained 
based on the indentation data and corresponding equations given in Chapter 2. 
 
Figure 4.4 - Microscope images of sucrose, glycine, and ascorbic acid before milling 
78 
 
 
The mechanical properties as well as the brittleness indices (defined in chapter 
3) are provided in Table 4.2. The brittleness indices are not very different because 
these crystals have similar crystalline structure. Nevertheless, different brittleness 
indices are obtained for different materials, e.g., sucrose has a higher brittleness index 
compared to glycine, and therefore we expect more brittle behaviour from sucrose. 
Table 4.2- Mechanical Properties of Crystals and Brittleness Indices 
Young’s 
 Hardness 
Fracture Brittleness Index 
Modulus Toughness 
 E (GPa) H (MPa) Kc (MPa m
½) H/Kc (10
-3 m- ½) 
Sucrose 21.1 1105 0.097 11.3 
Glycine 28.6 887 0.089 9.9 
Ascorbic 32.2 1059 0.069 12.2 
Acid 
It is important to mention that the indentation data for ascorbic acid have large 
scatter (up to 35% of mean value) due to anisotropic nature of ascorbic acid crystals, 
i.e., the hardness and fracture toughness are different for the same sample measured in 
different orientations. 
4.6 –Experiment Procedure 
Crystals with 10% by weight, dispersed in 2 litres IPA, were fed to the 
Silverson mixer at a series of flow rates 2.1, 3, and 4.2 L/min in the inline setup and 
rotor speeds 5000, 6500, and 8000 rpm in both the inline and batch setups. The 
samples were taken from the sampling point in the glass vessels at predefined time 
intervals to investigate particle size reduction. Extra milling time was provided to 
79 
 
 
ensure that all particles experienced the regions with maximum local energy 
dissipation rate multiple times and the pseudo steady state [62] is achieved.  
Here tank turnover is used as the time scale of the process. The tank turnover is 
the number of times the slurry in the holding vessel is passed through the mixer at a 
certain time and can be calculated as: 
𝑡
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑎𝑛𝑘 𝑡𝑢𝑟𝑛𝑜𝑣𝑒𝑟 = 𝑉     (4.1) ⁄𝑄𝑣
Where t is the milling time, 𝑉 is the volume of the slurry in holding vessel, and 
𝑄𝑣 is the flow rate. The time scale can also be expressed as the mill head passes. The 
mill head passes can be calculated using the same equation by substituting 𝑉 as the 
volume of the slurry in mill head. Since volume of the slurry in holding vessel (2000 
ml) is 25.3 times larger than mill head (79 ml), 1 tank turnover is equal to 25.3 mill 
head passes. 
4.7 –Size Distribution Considerations 
The particle size used to compare the data from different milling conditions is 
D90, based on cumulative volume percentage. D90s, obtained from wet milling of 
sucrose-IPA system in the Silverson inline mixer with the standard shear gap at 
different rotor speeds and flowrates, are plotted against other relevant sizes to 
determine the relationship between D90 and other measures of particle size. As shown 
in Figure 4.5 and Figure 4.6, D95 and D80 are linearly correlated to D90 whereas 
D50, D10, 𝐷43, 𝐷32, and 𝐷10 do not show a linear relationship. These correlations are 
shown for sucrose-IPA system but assumed to apply more generally. It is important to 
80 
 
 
note that the values of maximum stable particle size, D100, were not used directly 
because of the statistical considerations and the high scatter in D100 measurement.  
 
Figure 4.5- Relationship between D90 and other cumulative particle sizes – data are 
from wet milling of sucrose/IPA in the Silverson L4R inline mixer with the standard 
shear gap at different rotor speeds and flow rates 
 
Figure 4.6 - Relationship between D90 and mean particle sizes – data are from wet 
milling of sucrose/IPA in the Silverson L4R inline mixer with the standard shear gap 
at different rotor speeds and flow rates 
81 
 
 
4.8 –Experiment Results 
4.8.1 –Silverson L4R inline mixer with standard shear gap 
Wet milling experiments were conducted in the Silverson inline mixer with the 
standard shear gap to investigate the effect of rotor speed, throughput, particle 
physical properties, slurry concentration (particle-particle collisions), and particle-
wall collisions on particle size reduction as summarized in Table 4.3. 
Table 4.3 - The matrix of different experiments performed in the Silverson L4R inline 
mixer with the standard shear gap 
 
Figure 4.7 to Figure 4.12 show the change in the particle size distribution from 
the initial distribution (obtained by sieving) to the final distribution for wet milling of 
sucrose-IPA system at 8000 rpm and 3.0 lpm measured with the laser diffraction 
method. It is important to note that the initial and the final distributions are unimodal; 
however, the intermediate distributions are bimodal. This is the characteristic of the 
wet milling processes in rotor-stator mixers where transition from initial to final 
distribution occurs through the bimodal or polymodal distributions. The PVM images 
82 
 
 
of the sucrose crystals at different milling times were taken in situ as shown in Figure 
4.13 to Figure 4.18. As the number of small particles increases during the milling, the 
laser input scatters significantly and does not reflect back to the sensor. As a result, 
the contrast and resolution of the PVM images decreases as the sensor becomes 
increasingly saturated. Despite this shortcoming, the images can be used to 
qualitatively monitor the particle size and shape changes during the milling. The 
scanning electron microscopy images of sucrose before and after milling are shown in 
Figure 4.19 for comparison with PVM images.   
 
Figure 4.7 - Initial size distribution (obtained by sieving), measured with the laser 
diffraction method - wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm – 
Silverson L4R inline mixer with standard shear gap 
 
Figure 4.8 – Particle size distribution, measured with the laser diffraction method, 
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 1 tank 
turnover – Silverson L4R inline mixer with standard shear gap 
83 
 
 
 
Figure 4.9 - Particle size distribution, measured with the laser diffraction method, 
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 3 tank 
turnovers – Silverson L4R inline mixer with standard shear gap 
 
Figure 4.10 - Particle size distribution, measured with the laser diffraction method, 
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 5 tank 
turnovers – Silverson L4R inline mixer with standard shear gap 
 
Figure 4.11 - Particle size distribution, measured with the laser diffraction method, 
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 7 tank 
turnovers – Silverson L4R inline mixer with standard shear gap 
84 
 
 
 
Figure 4.12 - Final size distribution, measured with the laser diffraction method, 
resulted from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm – Silverson L4R 
inline mixer with standard shear gap 
 
Figure 4.13 - Online PVM images of sucrose in IPA at initial size distribution - wet 
milling at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap 
 
Figure 4.14 - Online PVM images of sucrose in IPA after 1 tank turnover - wet 
milling at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap 
85 
 
 
 
Figure 4.15 - Online PVM images of sucrose in IPA after 3 tank turnovers - wet 
milling at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap 
 
Figure 4.16 - Online PVM images of sucrose in IPA after 5 tank turnovers - wet 
milling at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap 
 
Figure 4.17 - Online PVM images of sucrose in IPA after 11 tank turnovers - wet 
milling at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap 
86 
 
 
 
Figure 4.18 - Online PVM images of sucrose in IPA after 15 tank turnovers - wet 
milling at 8000rpm and 3.0 lpm - Silverson L4R inline mixer with standard shear gap 
 
Figure 4.19- SEM images of sucrose before and after milling at 8000 rpm and 3.0 lpm 
– Silverson L4R inline mixer with standard shear gap 
87 
 
 
4.8.1.1 –Rotor speed effect on particle size reduction in the Silverson L4R inline 
mixer with the standard shear gap 
Sucrose wet milling experiments were conducted to find the effect of rotor 
speed on particle size reduction at constant flow rate. All the variables held constant 
except for rotor speed; enough time is allowed to establish the equilibrium. As shown 
in Figure 4.20 and Figure 4.21, the rotor speed has a strong effect on the rate of size 
reduction as well as the ultimate equilibrium size. Furthermore, a sharp breakage rate 
at the beginning followed by slow changes is observed in the particle breakage data. 
The sharp breakage slope is due to fracture mode which refers to the breakage of a 
particle into fragments of comparable size. The slow breakage part is due to an 
attrition mechanism by which small fragments are chipped off from larger particles.  
This can be explained based on fracture mechanics: larger particles have lower 
cohesive stress (Table 3.3) and are easier to break but once they are broken down to a 
critical particle size, further breakage cannot occur for a given rotor speed. 
Furthermore, the shear gap geometry can be important since milling starts with 
particle sizes larger than shear gap and ends with particles smaller than shear gap. The 
wet milling experiment results for other measures of the size distribution as well as 
different rotor speed conditions for sucrose-IPA system are given in Appendix A. 
 
88 
 
 
 
 
Figure 4.20 - Rotor speed effect on sucrose breakage (laser diffraction measurements) 
– The flow rate is fixed at 2.1 lpm – lower graph is log-normal representation of the 
data in upper graph – Silverson L4R inline mixer with standard shear gap 
89 
 
 
 
 
 
Figure 4.21 - Rotor speed effect on sucrose breakage (laser diffraction measurements) 
– The flow rate is fixed at 4.2 lpm – lower graph is log-normal representation of the 
data in upper graph – Silverson L4R inline mixer with standard shear gap 
4.8.1.2 –Flowrate effect on particle size reduction in the Silverson L4R inline mixer 
with the standard shear gap 
The number of “high shear events” depends on the residence time distribution 
of the particles in the mill. The lower residence time means fewer impacts and 
therefore less breakage. Figure 4.22 and Figure 4.23 present the particle size as a 
function of flow rate at constant rotor speeds of 5000 and 8000 rpm respectively. The 
final attainable particle size is not different for different flowrates; however, the rate 
of approach to the final equilibrium size is faster with lower flow rates. This is 
because the mill head residence time per tank turnover for the lower flow rates is 
higher. 
90 
 
 
 
 
Figure 4.22 - Flow rate effect on particle breakage (laser diffraction measurements) - 
The rotor speed is fixed at 5000 rpm – lower graph is log-normal representation of the 
data in upper graph - Silverson L4R inline mixer with standard shear gap 
91 
 
 
 
 
Figure 4.23 - Flow rate effect on particle breakage (laser diffraction measurements) - 
The rotor speed is fixed at 8000 rpm – lower graph is log-normal representation of the 
data in upper graph - Silverson L4R inline mixer with standard shear gap 
The effect of rotor speed and flowrate on breakage rate is also exploited in situ 
in the holding vessel by the FBRM and the PVM probes. However, the difficulty is 
that the measurements are in terms of the chord length and not the size; one gets a 
distribution of chord length even for mono-sized spherical particles. Since FBRM is a 
number based measurement technique, Sauter mean diameter (D32) of the chord 
length distribution is used here to compare the milling performances. 
The digital images that are recorded using a PVM probe can be analysed and a 
relative backscatter index (RBI) can be defined for each image. As it is discussed in 
Chapter 2, RBI is the ratio of image brightness to intensity of light output. It is a 
function of particle size, particle shape, and concentration. A higher RBI is associated 
92 
 
 
to a larger particle size; an increase in number concentration of particles, as a result of 
reduction in particle size, decreases the RBI values. 
 In spite of different measurement mechanisms, both FBRM and PVM show the 
right trend for wet milling of sucrose as shown in Figure 4.24 and Figure 4.25 
respectively; by increasing rotor speed and decreasing flow rate the milling 
performance improves. However, the measurement of the ultimate particle sizes is not 
precise. As the number of small particles increases during the milling, the contrast and 
resolution of the measurement decreases and eventually the sensor becomes saturated. 
This saturation causes a false detection of the time required to reach the equilibrium. 
For example, the PVM probe shows that the wet milling at 8000 rpm and 2.1 lpm 
reaches the final size distribution after 4 tank turnovers whereas the data obtained 
with the laser diffraction method shows the equilibrium condition after 10 tank 
turnovers. In Figure 4.24 and Figure 4.25, the initial D32 for FBRM data is 180 ± 12 
μm and initial RBI for PVM data is 4.2±0.3. Several chord length distributions 
measured with the FBRM probe during wet milling of sucrose-IPA system are 
provided in Appendix D. 
93 
 
 
 
Figure 4.24 - Rotor speed and flow rate effect on wet milling of sucrose in IPA; chord 
length measurement by FBRM - Silverson L4R inline mixer with standard shear gap 
 
Figure 4.25 - Rotor speed and flow rate effect on wet milling of sucrose in IPA; RBI 
measurement by PVM - Silverson L4R inline mixer with standard shear gap 
4.8.1.3 –The effect of particle physical properties on particle size reduction in the 
Silverson L4R inline mixer with the standard shear gap 
Three crystals, glycine, sucrose, and ascorbic acid were milled to investigate the 
effect of mechanical properties on the milling behavior. The refractive indices for 
sucrose, glycine, and ascorbic acid are 1.51, 1.46, and 1.71 respectively. The laser 
diffraction device corrects the readings based on refractive indices. The rotor speed 
and the flow rate held constant; enough time is allowed to establish the equilibrium. 
94 
 
 
Figure 4.26 and Figure 4.27 show the extent of influence of physical properties on 
size reduction of these three materials (additional wet milling data for glycine-IPA 
and ascorbic acid-IPA systems are in Appendix B and C). Each figure shows different 
rotor speed and flow rate scenarios. The brittleness index defined by Lawn and 
Marshal’s model predicts a larger brittleness index for sucrose compared with glycine. 
This means that the sucrose crystals are more susceptible to breakage compared with 
glycine. This observation is in agreement with the experiment results in which the 
sucrose shows a higher rate of breakage and the ultimate size (D90) of sucrose 
crystals is smaller than the glycine at the end of milling process. 
 
 
Figure 4.26 - Wet milling of different crystals, ascorbic acid, glycine, and sucrose 
(laser diffraction measurements) - 8000 rpm and 4.2 lpm – lower graph is log-normal 
representation of the data in upper graph - Silverson L4R inline mixer with standard 
shear gap 
95 
 
 
 
 
Figure 4.27 - Wet milling of different crystals, ascorbic acid, glycine, and sucrose 
(laser diffraction measurements) - 6500 rpm and 3.0 lpm – lower graph is log-normal 
representation of the data in upper graph - Silverson L4R inline mixer with standard 
shear gap 
Difference in physical properties of these crystals can also explain the 
difference in size of chips and attrition products. A simple explanation is sketched in 
Figure 4.28; a higher brittleness index results in a larger chip size as represented by 
D10 or D10. Figure 4.29 and Figure 4.30 show that the chip size D10 (10% cumulative 
size) and D10 (arithmetic or number based mean) of sucrose is larger than that of 
glycine. As it mentioned before, the indentation data for ascorbic acid shows its 
anisotropic nature. Therefore, the brittleness index of this material is not successful in 
predicting its milling behaviour. 
96 
 
 
 
Figure 4.28- A particle with higher brittleness index breaks down to smaller D90 and 
larger D10 
 
 
Figure 4.29 - Size of attrited small chips in milling of different crystals, ascorbic acid, 
glycine, and sucrose, at 6500 rpm and 3.0 lpm (D10 is 10% cumulative size measured 
by laser diffraction method); lower graph is log-normal representation of the data in 
upper graph - Silverson L4R inline mixer with standard shear gap 
97 
 
 
 
 
Figure 4.30 - Size of attrited small chips in milling of different crystals at 6500 rpm 
and 3.0 lpm (D10 is the arithmetic or number mean measured by laser diffraction 
method); lower graph is log-normal representation of the data in upper graph - 
Silverson L4R inline mixer with standard shear gap 
4.8.1.4 –The effect of slurry concentration (particle-particle collision) on particle size 
reduction in the Silverson L4R inline mixer with the standard shear gap 
The breakage can happen as a result of particle-particle collision, particle-wall 
collision, and/or fluid shear effects. Several experiments were carried out to explore 
the scope of each mechanism on the particle breakage. Extend of particle-particle 
collision was examined by milling different concentration of the sucrose in IPA. It 
was reasoned that by increasing the concentration, the probability of particle-particle 
collisions increases with a power of 2 whereas the particle-wall collisions increase 
98 
 
 
proportionally. So if the mechanical energy for breakage is coming from particle-
particle collisions, then it should significantly change the milling behavior.  
It is important to note that in all other experiments presented in this study, the 
concentration of slurry is set to 10% by weight. Here, different concentrations of 
sucrose in IPA were milled using inline mixer at 6500 rpm and 3.0 lpm. The 
experiment results, as shown in Figure 4.31, do not show a strong dependency toward 
concentration. This suggests that the particle-particle effects are not strong when 
compared to energy input (rotor speed) or residence time (flow rate) effects. This is in 
agreement with experimental evidence from literature that particle size reduction in 
rotor-stator wet mills is mostly independent of slurry concentration [53].  
 
Figure 4.31- Concentration effect on wet milling performance – 6500 rpm and 3.0 
lpm - Silverson L4R inline mixer with standard shear gap (laser diffraction 
measurements) 
Another experiment was designed to investigate the significance of particle-wall 
collisions. In this experiment, several coats of primer and paint were applied to the 
stator walls and enough time was allowed, ensuring that the paint is dry. The slotted 
head stator was used because of its larger openings which give better access to apply 
consistent layers of coating. It was then assembled to the mixer and milling 
99 
 
 
experiment was run with pure IPA for an hour. At this point no sign of paint removal 
was observed in stator. Subsequently, the experiment was repeated with 10 percent 
weight sucrose in IPA for another hour. If the paint is removed from stator walls 
during the milling, then it may be implied that particle-wall collisions are prevailing 
in breaking the particles. As it is shown in Figure 4.32, the paint on the inside wall of 
stator and the walls of the approach edges of the slots were removed. This observation 
suggests that the particle-wall collisions are important in wet milling processes and 
they might be the dominant mechanism causing the breakage. 
 
Figure 4.32- Particle-wall collisions cause the removal of paint coatings on inside 
wall and approach edges of the stator slots - Silverson L4R inline mixer with the 
standard shear gap 
4.8.2 –Silverson L4R Inline Mixer with Enlarged Shear Gap 
Power draws in rotor-stator mixers vary with the geometry, the rotor size, the 
rotor speed, flowrate, and properties of the process fluids. In the case of batch mixers, 
power draw is also a function of the vessel size and structure (baffles, obstacles). In 
this part, the effect of shear gap width on the milling performance of Silverson L4R 
inline mixer is investigated. Figure 4.33 shows the schematic geometries of the 
standard and enlarged shear gaps. The difference between these two geometries is the 
size of the rotor which in turn results in difference in width of the shear gap. 
100 
 
 
 
Figure 4.33- Geometry of rotor and stator for standard and enlarged shear gap in 
Silverson L4R inline mixer 
Widening the shear gap has two effects on milling performance. First, it reduces 
the power draw by the mixer due to reduced rotor and stator interactions and the 
smaller rotor diameter. As a result, particles experience lower intensity of disruptive 
energy in the mixing zone. Second, a wider shear gap means a larger volume of the 
dispersion zone and as a result the maximum local energy dissipation rate decreases 
as will be shown in Chapter 5. 
Wet milling experiments were conducted to find the effect of shear gap width 
on particle size reduction. Sucrose-IPA system was milled in Silverson L4R inline 
mixer with enlarged shear gap at different rotor speeds of 5000, 6500, and 8000 rpm. 
The flow rate is held constant at 3.0 lpm. Figure 4.34 to Figure 4.39 show the 
evolution of particle size distribution from initial size distribution to final distribution. 
It is important to note that the mixer with enlarged shear gap is not able to draw and 
concentrate enough power to break down particles to a unimodal and narrow size 
distribution, and therefore the final distribution (Figure 4.39) is bimodal. 
101 
 
 
 
Figure 4.34 - Initial size distribution (obtained by sieving) measured with the laser 
diffraction method - wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm – 
Silverson L4R inline mixer with enlarged shear gap 
 
Figure 4.35 - Particle size distribution measured with laser diffraction method resulted 
from wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm after 15 tank turnovers – 
Silverson L4R inline mixer with enlarged shear gap 
102 
 
 
 
Figure 4.36 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm and 3.0 lpm after 30 tank turnovers – Silverson L4R inline mixer with 
enlarged shear gap 
 
Figure 4.37 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm and 3.0 lpm after 45 tank turnovers – Silverson L4R inline mixer with 
enlarged shear gap 
103 
 
 
 
Figure 4.38 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm and 3.0 lpm after 90 tank turnovers – Silverson L4R inline mixer with 
enlarged shear gap 
 
Figure 4.39 - Final size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm and 3.0 lpm – Silverson L4R inline mixer with enlarged shear gap 
Figure 4.40 shows the effect of rotor speed on the rate of size reduction in the 
Silverson L4R inline mixer with the enlarged shear gap. Similar to wet milling results 
in the mixer with the standard shear gap, the rotor speed has a strong effect on the rate 
of size reduction as well as the ultimate equilibrium size.  The sharp size reduction 
rate at the beginning is due to fracture mechanism in which the particles break into 
fragments of comparable size.  The following slow breakage part is due to attrition 
mechanism by which small fragments are chipped off from larger particles. 
104 
 
 
 
 
Figure 4.40 - Silverson L4R inline mixer with enlarged shear gap; rotor speed effect 
on sucrose breakage (laser diffraction measurement) – Flow rate is fixed at 3.0 lpm 
Figure 4.41 to Figure 4.43 compare the wet milling performance of the enlarged 
shear gap, 1.7 mm, with the standard shear gap, 0.2 mm, at 5000, 6500, and 8000 rpm 
respectively. There is apparently a much higher initial breakage rate and smaller 
ultimate particle size associated with standard shear gap in all cases. 
 
Figure 4.41 - Shear gap effect on milling performance of inline rotor-stator device – 
wet milling of sucrose in IPA at 5000 rpm and 3.0 lpm 
105 
 
 
 
Figure 4.42 - Shear gap effect on milling performance of inline rotor-stator device – 
wet milling of sucrose in IPA at 6500 rpm and 3.0 lpm 
 
Figure 4.43 - Shear gap effect on milling performance of inline rotor-stator device – 
wet milling of sucrose in IPA at 8000 rpm and 3.0 lpm 
In the case of the standard shear gap (0.2 mm), the initial D90 (420 μm) is larger 
than the shear gap and it mills down to D90s smaller than the shear gap. Whereas, in 
the case of the enlarged shear gap (1.7 mm), the initial D90 is smaller than shear gap 
and it still mills down to much smaller D90s. This observation implies that the 
breakage of particles is not due to compression or tearing (Figure 4.44.a) and single 
particle-wall impacts (Figure 4.44.b) are the dominant breakage mode. The 
significance of particle-particle collision (Figure 4.44.c) is discussed in previous 
section.  
106 
 
 
 
Figure 4.44- Different loading modes in a rotor-stator mixer, a) Friction or 
compression, b) Particle-wall collision, c) Particle-particle collision 
4.8.3 –Silverson L4R Batch Mixer 
The performance of a rotor–stator mixer can be related to the Reynolds number 
and the power number with rotor diameter as a characteristic dimension. The power 
curves of rotor–stator mixers have similar shape to that of stirred tank with the power 
number inversely proportional to the Reynolds number in the laminar flow regime and 
practically constant in the turbulent flow regime. The laminar flow regime, indicated 
by the constant slope of 1/Re of the power curve, extends to Re = 100 instead of Re = 
10 as in the case of stirred tank [63, 64]. In the turbulent flow regime, which is the 
case in this study, (Re >10,000), the power number is independent of Re number and 
varies from 0.8 to 3 depending on the geometry of the rotor and stator mixer.  
In spite of smaller rotor and stator dimensions compared to inline mixer 
(dimensions are given in Table 4.1), L4R batch mixer draws more power and has a 
higher power number (power draw and power number calculations are given in 
Chapter 5). Furthermore, due to the flow condition inside the holding tank, and its 
smaller dispersion zone, the batch mixer develops a higher local energy dissipation 
rates compared with the inline mixer. Consequently, it achieves higher milling rates 
and delivers a smaller ultimate (equilibrium) particle size.  
107 
 
 
Figure 4.45 to Figure 4.50 show the evolution of particle size distribution from 
initial to final distribution for wet milling of sucrose-IPA in Silverson L4R batch 
mixer at 8000 rpm. The effect of rotor speed on milling performance of sucrose is 
shown in Figure 4.51. The rotor speed has a strong effect on the rate of size reduction 
as well as the ultimate equilibrium size. Furthermore, sharp breakage rate at the 
beginning following by slow changes is observed in particle breakage data. 
 
Figure 4.45 - Initial size distribution (obtained by sieving) measured with the laser 
diffraction method - wet milling of sucrose in IPA at 8000 rpm – Silverson L4R batch 
mixer 
 
108 
 
 
 
Figure 4.46 - Particle size distribution measured with the laser diffraction method 
resulted from wet milling of sucrose in IPA at 8000 rpm after 1 min – Silverson L4R 
batch mixer 
 
 
Figure 4.47 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm after 2 mins – Silverson L4R batch mixer 
109 
 
 
 
Figure 4.48 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm after 3 mins – Silverson L4R batch mixer 
 
Figure 4.49 - Particle size distribution resulted from wet milling of sucrose in IPA at 
8000 rpm after 10 mins – Silverson L4R batch mixer 
 
Figure 4.50 – Final (t= 90 mins) particle size distribution resulted from wet milling of 
sucrose in IPA at 8000 rpm – Silverson L4R batch mixer 
110 
 
 
 
 
Figure 4.51 - Silverson L4R batch mixer; rotor speed effect on sucrose breakage (laser 
diffraction measurement) – lower graph is log-normal representation of the data in 
upper graph 
4.9 –Summary 
A series of wet milling experiments with different crystal-antisolvent systems 
has been developed and performed in the Silverson L4R rotor-stator mixers, to 
investigate the effect of milling conditions, mixer geometry, and particle physical 
111 
 
 
properties on ultimate particle size and particle breakage kinetics. Three test 
materials, i.e., sucrose, glycine, and ascorbic acid, were used in the experiments to 
provide a range of material properties and explore their influence on the wet milling 
behavior. Different rotor speeds were used to explore the effect of energy input and 
operational conditions on milling performance. Three different mixer geometries were 
used, i.e. a Silverson L4R inline mixer with standard and enlarged shear gap, and a 
Silverson L4R batch mixer, to examine the effect of mixer geometry on particle 
breakage behavior. Long milling times were provided to ensure that all particles 
experienced the regions with maximum local energy dissipation rate multiple times 
and the size distribution no longer changes from one tank turnover to the next. It was 
shown that:  
• Increasing rotor speed/power input at constant throughput, intensify the milling 
process and leads to a smaller ultimate particle size. Furthermore, the rate of 
milling increases with increasing the rotor speed. 
• Increasing throughput (flowrate) at constant rotor speed, decreases the milling rate 
however it does not have significant effect on the ultimate particle size. Lower 
throughput gives more residence time for particles in the mixing head leading to 
more collision and the higher breakage rate.  
• Particle physical properties determine their resistance against the fracture and their 
mode of deformation. Generally, Particles with higher hardness and lower fracture 
toughness are more liable to break.  
• The Silverson L4R batch mixer is a better mixer compared to the inline unit 
because it delivers higher size reduction rates and smaller ultimate particle size. 
112 
 
 
• The clearance between rotor and stator (shear gap width) in inline mixer has a 
significant effect on the milling rate and ultimate particle size.  
113 
 
 
Chapter 5: Model Validation and Mechanistic Correlations 
In this chapter, the suitability of mechanistic models developed in Chapter 3 to 
predict the wet milling experimental data for the different rotor-stator mixer 
geometries presented in Chapter 4 investigated. 
5.1 –Suitability of Different Mechanistic Models to Predict the Wet 
Milling Behavior 
The mechanistic correlation in Table 3.4 are given in the form of maximum 
stable particle size over the rotor diameter (𝑥/𝐷) as a function of reciprocal of rotor 
tip speed (1/𝑢𝑡𝑖𝑝). Here, as discussed in Chapter 4, D90 is used as a representative of 
maximum stable particle size, i.e., x = D90.  
In Figure 5.1, D90/D for the wet milling equilibrium data for sucrose, glycine, 
and ascorbic acid at different rotor speeds (the ultimate particle size is independent of 
the mill head throughput rate) are plotted against the corresponding 1⁄𝑢𝑡𝑖𝑝 for the 
Silverson L4R inline mixer with standard shear gap. The wet milling data for the 
inline mixer with enlarged shear gap and the batch mixer were not used here because 
they do not have geometrical similarity with the inline mixer with the standard shear 
gap. They will be used in next section where the mechanistic correlations for the 
devices without geometric similarity are introduced. 
It is shown that the ultimate particle size data, for different crystal-antisolvent 
systems in the inline mixer with the standard shear gap, are well fit by a line with 
slope of 1.6 on a logarithmic plot. This slope suggests that the inertial subrange 
models better fit the experimental data than the macroscale models, which predicted 
114 
 
 
much larger slopes. Furthermore, this slope is in agreement with models based on 
24
elastic mechanism of Gahn and Mersmann [33] which predicts the slope of = 1.4 
17
12
and elastic-plastic mechanism of this study which predicts slope of  = 1.7 . 
7
8000 rpm 6500 rpm 5000 rpm 
6000 rpm 
 
Figure 5.1- Suitability of different models to fit the wet milling equilibrium data for 
different crystals in Silverson L4R inline mixer with standard shear gap (trend line 
comparison) 
It is important to note that the model based on the indentation mechanism 
(predicting a power law dependency of -3 with the 𝑢𝑡𝑖𝑝) underestimates the energy 
required for size reduction whereas, the plastic mechanism model (predicting slope of 
6
= 1.2) overestimates the energy requirement. This conclusion is in agreement with 
5
classical theories where: 1) kick’s law (similar to indentation mechanism) does not 
account for the particle size and consequently underestimates the energy requirement, 
2) Rittinger’s law (similar to plastic mechanism) overestimates the energy 
requirement by assuming that surface energy of the material controls the particle 
115 
 
 
fracture, 3) Bond’s law (similar to elastic-plastic mechanism) describes the energy 
requirements properly. 
In Figure 5.2 to Figure 5.5, wet milling data for sucrose, glycine, and ascorbic 
acid in the Silverson L4R inline mixer with the standard shear gap is plotted along 
with a best fit line for breakage models (based on turbulent inertial subrange theory 
for disruptive stress as given in Table 3.5). The degree of success of each model to 
collapse the data is assessed based on the goodness of the fit as summarized in Table 
5.1. Other than predicting the trend, the model based on elastic-plastic mechanism for 
cohesive stress and turbulent inertial subrange theory for disruptive stress is able to 
collapse the data of different materials on a single curve with reasonable goodness of 
fit (Figure 5.5). The final expression is given by: 
D90
= 2.6 × 10−5  Co −6/7EP      (5.1) D
Table 5.1 - Goodness of fit of different models to collapse the experimental data for 
milling of different materials in the inline mixer with the standard shear gap 
(IST=Inertial subrange of turbulence) 
Disruptive 
Cohesive stress Breakage model Goodness of fit 
stress 
Indentation 𝑥
 ~  𝐶𝑜 −3/2IST 𝐼  R-square= -8.5 approach 𝐷
Plastic 𝑥 −3/5 R-square= 0.63 
IST  ~  𝐶𝑜  
mechanism 𝐷 𝑃 RMSE= 4.6e-4 
Elastic 𝑥 −12/17 R-square= 0.60 
IST  ~  𝐶𝑜  
mechanism 𝐷 𝐸 RMSE= 5.0e-4 
Elastic-plastic 𝑥
 ~  𝐶𝑜 −6/7
R-square= 0.91 
IST  
mechanism 𝐷 𝐸𝑃 RMSE= 1.7e-7 
116 
 
 
 
 
Figure 5.2 - Suitability of the model based on indentation approach (Lawn and 
Marshal model) to predict the wet milling behavior of different crystals in Silverson 
L4R inline rotor-stator mixer with standard shear gap at different rotor speeds 
 
Figure 5.3 - Suitability of the model based on plastic mechanism (Ghadiri and Zhang 
model) to predict the wet milling behavior of different crystals in Silverson L4R 
inline rotor-stator mixer with standard shear gap at different rotor speeds 
117 
 
 
 
Figure 5.4 - Suitability of the model based on elastic mechanism (Gahn and 
Mersmann model) to predict the wet milling behavior of different crystals in Silverson 
L4R inline rotor-stator mixer with standard shear gap at different rotor speeds 
 
Figure 5.5 - Suitability of the model based on elastic-plastic mechanism (current 
study) to predict the wet milling behavior of different crystals in Silverson L4R inline 
rotor-stator mixer with standard shear gap at different rotor speeds 
Here, a geometrically similar analysis is performed to investigate the suitability 
of breakage models to predict the wet milling experimental data in the inline mixer 
with the standard shear gap. Although we don’t have enough experimental data for the 
inline mixer with the enlarged shear gap and in the batch mixer to perform the 
geometrically similar analysis, but these data along with the data from the inline 
118 
 
 
mixer with the standard shear gap are used in next section for analysis of devices 
without geometrical similarity. 
5.2 –Scale Up strategy based on Local Energy Dissipation Rate 
Several investigators have studied the importance of turbulent energy 
dissipation rate and power draw in predicting the drop size in liquid-liquid dispersions 
at high Reynolds number [65–67]. It is possible to facilitate the scale up/down efforts, 
and compare the data from different geometries using the concept of local energy 
dissipation rate. Average energy dissipation rate is simply the available power per 
fluid mass. However, the maximum particle size obtained at equilibrium is 
determined by the maximum or local energy dissipation rate rather than the average 
energy dissipation rate in the mixing head. For example, Davies [67] showed that the 
local energy dissipation rate for agitated tanks can be estimated by considering the 
power generated by the stirrer in the swept volume of the impeller. Here for rotor-
stator mixers, local energy dissipation rate is defined as: 
P
ɛ =         (5.2) 
𝜌𝑓 VDZ
where P is power draw of the mixer, 𝜌𝑓 is fluid density, and VDZ is volume of 
dispersion zone (volume with highest dissipation rates). The dispersion zone of 
different geometries and scales are different. Knowledge of velocity field in the 
mixer, based on experiments or simulations, aid the proper evaluation of this zone. It 
is reasonable to define the dispersion zone for rotor stator mixers as: 
VDZ = Volume of Stator Holes + Shear Gap Volume  (5.3) 
119 
 
 
Most of the energy that is supplied to a rotor-stator mixer is dissipated in this 
region, i.e., the density of energy dissipation rate is much higher in this region when 
compared with other regions in the mixer. Table 5.2 summarizes the volume of 
dispersion zone of the mixers that are used in this study. 
Table 5.2 - Volume of dissipersion zone for inline mixer with standard and enlarged 
shear gaps and batch mixer 
Stator Hole Shear Gap Dispersion Zone 
Mixer Type 
Volume Volume Volume V  DZ
Inline Mixer with 
1.2 ml 0.4 ml 1.6 ml 
Standard Shear Gap 
Inline Mixer with 
1.2 ml 2.8 ml 4.0 ml 
Enlarged Shear Gap 
Batch Mixer 0.9 ml 0.2 ml 1.1 ml 
The power draw of different mixers is obtained using experimental and 
computational methods. For inline mixers with standard and enlarged shear gap, a 
CFD simulation is conducted and the power draw of the mixer is calculated based on 
the torque on the shaft and rotor blades at different rotor speeds as shown in Table 5.3 
and Table 5.4. The simulation details are given in Chapter 6. For the Silverson L4R 
batch mixer, experimental data as well as CFD simulations for power draw were 
available in literature [63,68] and presented in Table 5.5. It is important to note that 
the experimental data supports the computational results for power calculations. The 
power numbers of different mixers were calculated as:  
𝑃𝑜𝑤𝑒𝑟
𝑁𝑃 = 𝜌  𝑁3𝐷5      (5.4) 𝑓
120 
 
 
Where 𝜌𝑓 is fluid density, N is rotational speed (rpm), and D is rotor diameter. 
It is important to note that despite the fact that the power number of inline mixer with 
standard shear gap is smaller than that of enlarged shear gap, it draws more power and 
as a result higher energy dissipation rates are present in this mixer. Since power 
number goes as 𝐷5, small changes in rotor diameter, 𝐷, have a large effect on the 
magnitude of 𝑁𝑃. 
Table 5.3- Tip speed, Reynolds number, power, and power number of Silverson L4R 
inline mixer with standadr shear gap at different rotor speeds – CFD investigation (see 
Chapter 6 for details) 
Inline Mixer  
Rotor Speed 
Standard Shear Gap, D = 31.1 mm 
N (RPM) utip (m/s) Re Power (W) NP Local ɛ 
5000 8.12 80,000 13.9 0.9 11,052 
6500 10.56 104,000 30.6 0.9 24,332 
8000 12.99 129,000 57.1 0.9 45,403 
Table 5.4- Tip speed, Reynolds number, power, and power number of Silverson L4R 
inline mixer with enlarged shear gap at different rotor speeds – CFD investigation 
(see Chapter 6 for details) 
Inline Mixer  
Rotor Speed 
Enlarged Shear Gap, D = 28.2 mm 
N (RPM) utip  (m/s) Re Power (W) NP Local ɛ 
5000 7.36 66,000 9.5 1.0 3,321 
6500 9.57 86,000 20.9 1.0 7,347 
8000 11.78 106,000 39.0 1.0 13,404 
121 
 
 
Table 5.5- Tip speed, Reynolds number, power, and power number of Silverson LR 
batch mixer at different rotor speeds based on data of Padron [63] 
Batch Mixer,  
Rotor Speed 
Standard Shear Gap, D = 28.1 mm 
N (RPM) utip  (m/s) Re Power (W) NP Local ɛ 
5000 7.31 66,000 23.0 2.3 26,601 
6500 9.51 86,000 50.5 2.3 58,408 
8000 11.70 106,000 94.2 2.3 108,952 
To this point it was reasoned that the concept of local energy dissipation rate 
provides a better measure of disruptive stresses present in the mixer and by taking into 
account the power draw and geometry of the mixer, it can be used to scale up/down 
the wet milling processes in different milling devices. Figure 5.6 shows the 
equilibrium data for wet milling of sucrose, glycine, and ascorbic acid in different 
mixers that are plotted against local energy dissipation rate in a logarithmic scale. 
This graph shows that maximum stable particle size is a function of dissipation rate 
raised to a power of 0.5. This relationship was predicted by the elastic-plastic model 
as given in Table 3.5. 
122 
 
 
 
Figure 5.6 - Maximum stable particle size plotted against local maximum energy 
dissipation rate for wet milling of sucrose, glycine, and ascorbic acid in Silverson 
L4R inline mixer with standard and enlarged shear gap, and Silverson L4R batch 
mixer at 5000, 6000, 6500, and 8000 rpm 
In Figure 5.7 to Figure 5.10, wet milling data of different crystals obtained in 
different mixers with different geometries is plotted along with breakage models 
based on energy dissipation rate from Table 3.5 in Chapter 3. The degree of success 
of each model to collapse the data is assessed based on the goodness of the fit 
summarized in Table 5.6. In addition to predicting the trend, the elastic-plastic model 
is able to collapse the data of different materials on a single curve with acceptable 
goodness of fit. The final expression is given by: 
𝐸𝐾 1
𝐷90 = 13.6 ( 𝑐 )6/7 
𝜌  𝐻 ɛ4/7
     (5.5) 
𝑃
 
 
 
123 
 
 
 
Table 5.6 - Goodness of fit of different models to collapse the experimental data for 
milling of different materials in the inline mixer with the standard and enlarged shear 
gap and in the batch mixer 
Mechanism Breakage model Goodness of fit 
Indentation 𝑐𝐾𝑐 1𝑥 ~  ( )3/2  R-square= -4.3 
approach 𝜌𝑃 𝐻 ɛ
Plastic 𝐾 2𝑐  1 R-square= 0.69 3/5
mechanism 𝑥 ~ ( )    𝜌  𝐻 ɛ2/5 RMSE= 2.1e-2 𝑃
3
Elastic 𝐾𝑐2 1 R-square= 0.78 
𝑥 ~  ( )12/171   mechanism ɛ8/17 RMSE= 3.7e-2 
𝜌𝑃 𝐻2 
Elastic-plastic 𝐸𝐾𝑐 1𝑥 ~  ( )6/7 R-square= 0.92  
mechanism 𝜌 4/7𝑃 𝐻 ɛ RMSE= 2.9e-3 
 
 
 
 
Figure 5.7 - Suitability of the model based on indentation approach to collapse the wet 
milling equilibrium data for different crystals in Silverson L4R inline rotor-stator 
mixer with standard and enlarged shear gap, and batch mixer at different rotor speeds 
124 
 
 
 
Figure 5.8 - Suitability of the model based on plastic mechanism to collapse the wet 
milling equilibrium data for different crystals in Silverson L4R inline rotor-stator 
mixer with standard and enlarged shear gap, and batch mixer at different rotor speeds 
 
Figure 5.9 - Suitability of the model based on elastic mechanism to collapse the wet 
milling equilibrium data for different crystals in Silverson L4R inline rotor-stator 
mixer with standard and enlarged shear gap, and batch mixer at different rotor speeds 
125 
 
 
 
Figure 5.10 - Suitability of the model based on elastic-plastic mechanism to collapse 
the wet milling equilibrium data for different crystals in Silverson L4R inline mixer 
with standard and enlarged shear gap, and batch mixer at different rotor speeds 
5.3 –The Rate of Size Reduction and Mechanistic Correlations for 
Breakage Kernel 
There are two common methods to define a breakage kernel: the power law 
kernel and the exponential kernel. They can be written in general form as: 
1
g(x) = ×probability of breakage (5.6) 
breakage time
 There are different methods to define the breakage time. It can be defined 
based on time constant of the process, i.e., rotational rate of rotor in rotor-stator 
mixers. In slurries with turbulent condition, particles in the scale of turbulent inertial 
subrange have a breakage time proportional to the lifetime of eddies and can be 
written as (Batchelor): 
𝑡 2/3𝑏  ~  𝑥   ɛ
− 1/3     (5.7) 
126 
 
 
where 𝑡𝑏 is breakage time, 𝑥 is particle size, and ɛ is energy dissipation rate. Different 
kernels have different breakage probabilities. The breakage probability in the power 
law kernel is simply defined based on the ratio of cohesive stress to disruptive stress 
raised to a power as: 
τ
probability of breakage =  𝑐1 ( 
c  )𝑐2   (5.8) 
τD
Whereas, the exponential kernel is based on the reaction rate analogy and is defined 
as [69]: 
τ
− c
probability of breakage  = e τD    (5.9) 
Using the cohesive stresses, τc, and disruptive stresses, τD, developed in 
Chapter 3 and given in Table 3.2 and Table 3.3 ,  each kernel will result in 8 different 
expressions for probability of breakage based on different cohesive and disruptive 
mechanisms. Here, the disruptive stress based on inertial subrange model of 
turbulence and macroscale velocity is used to construct the Table 5.7 and Table 5.8 
respectively. 
 
 
 
 
127 
 
 
Table 5.7- Power law and exponential breakage kernels with inertial subrange 
turbulence model as the disruptive stress 
 
Table 5.8-Power law and exponential breakage kernels with macroscale model as the 
disruptive stress 
 
128 
 
 
A population balance equation is used to model the milling kinetics which can be 
expressed as: 
𝜕𝑓(𝑥,𝑡) ∞
=  ∫ 𝑓(𝑥′, 𝑡) 𝑔(𝑥′) 𝛽(𝑥′, 𝑥) 𝑑𝑥′   − 𝑔(𝑥) 𝑓(𝑥, 𝑡)   (5.10) 
𝜕𝑡 𝑥
where 𝑓(𝑥, 𝑡) is cumulative fraction of particle with size x at time t, 𝑔(𝑥) is breakage 
kernel, and 𝛽(𝑥′, 𝑥) is breakage distribution function (accounts for the size 
distribution of daughter particles after the fracture of the parent particle). The left 
hand side is the accumulation term, and the right hand side includes birth (production 
due to breakage of larger particles) and death terms (due breakage). It is possible to 
apply this equation to the largest size interval where the birth term is zero, and assume 
first order kinetics as: 
𝜕𝑓(𝐷90(𝑡))
=  −  𝑔(𝐷90) 𝑓(𝐷90(𝑡))     (5.11) 
𝜕𝑡
By substituting the breakage function,  𝑔(𝐷90), with those given in Table 5.7 
and Table 5.8, Equation 5.11 can be solved analytically or numerically to obtain the 
D90 as a function of milling time. The suitability of the different breakage kernels to 
fit the wet milling experimental data investigated and it is found that the power law 
kernels cannot adequately fit data. Here, the suitability of exponential kernels based 
on inertial subrange turbulence and different deformation mechanisms are discussed.      
By substituting the breakage kernel based on elastic-plastic mechanism, plastic 
mechanism, and elastic mechanism (Table 5.7), the Equation 5.11 can be written as: 
𝑐
− 2
 𝐸 𝐾𝑐 
2 7
𝜕𝐷90 1
=  −  𝑐1 ɛ3   𝑒
𝜌𝑝 𝐻 ɛ3 𝐷906   𝐷901/3    (5.12) 
𝜕𝑡
129 
 
 
c  K  
𝜕D90 − 
2 c
2/3 5/3
=  −  c  ϵ1/3 e ρp H ϵ  D901 D90
1/3   (5.13) 
𝜕t
c2 K
3/2
− c𝜕D90
=  −  c  ϵ1/3 e ρ  H
1/2 ϵ2/3p  D90
17/12 1/3
1 D90   (5.14) 𝜕t
The degree of success of different breakage kernels given in Equations 5.12-14 
is assessed based on the milling rate data obtained in wet milling of sucrose in IPA at 
the Silverson L4R inline mixer with the standard and enlarged shear gap. Figure 5.11 
illustrates the result of applying elastic-plastic exponential breakage kernel to model 
the data from the inline mixer. Figure 5.12 and Figure 5.13 show the result of 
breakage kernels based on plastic deformation mechanism and elastic deformation 
mechanism respectively. Figure 5.14 to Figure 5.16 illustrates the data fit based on 
elastic-plastic kernel for different crystals, sucrose, glycine, and ascorbic acid in inline 
mixer with the standard shear gap at different milling conditions. It is found that the 
breakage kernel based on elastic-plastic model is more successful in predicting the 
rate of milling in the inline mixer with different shear gaps. 
 
 
130 
 
 
 
 
Figure 5.11 - Suitability of breakage kernel based on elastic-plastic model in 
predicting the milling rate of sucrose in inline mixer with standard and enlarged shear 
gaps at different rotor speeds and constant flow rate of 3.0 lpm, dotted lines are the 
model fits; lower graph is log-normal representation of the data in upper graph 
131 
 
 
 
 
Figure 5.12 - Suitability of breakage kernel based on plastic model in predicting the 
milling rate of sucrose in inline mixer with standard and enlarged shear gaps at 
different rotor speeds and constant flow rate of 3.0 lpm, dotted lines are the model 
fits; lower graph is log-normal representation of the data in upper graph 
132 
 
 
 
 
Figure 5.13 - Suitability of breakage kernel based on elastic model in predicting the 
milling rate of sucrose in inline mixer with standard and enlarged shear gaps at 
different rotor speeds and constant flow rate of 3.0 lpm, dotted lines are the model 
fits; lower graph is log-normal representation of the data in upper graph 
133 
 
 
 
 
Figure 5.14 - Suitability of breakage kernel based on elastic-plastic model in 
predicting the milling rate of sucrose, glycine, and ascorbic acid in inline mixer with 
standard shear gap at 6500 rpm and 3.0 lpm, dotted lines are the model fits; lower 
graph is log-normal representation of the data on upper graph 
134 
 
 
 
 
Figure 5.15 - Suitability of breakage kernel based on elastic-plastic model in 
predicting the milling rate of glycine in inline mixer with standard shear gap at 
different milling conditions, dotted lines are the model fits; lower graph is log-normal 
representation of the data on upper graph 
 
 
 
 
 
 
 
 
 
 
 
135 
 
 
 
 
Figure 5.16 - Suitability of breakage kernel based on elastic-plastic model in 
predicting the milling rate of ascorbic acid in inline mixer with standard shear gap at 
different milling conditions, dotted lines are the model fits; lower graph is log-normal 
representation of the data on upper graph 
5.4 –Summary 
The potential of mechanistic models to predict the crystal breakage behavior in 
rotor-stator mixers was investigated. Based upon the wet milling experimental data it 
was demonstrated that: 
• Because the size scale of the particles and the operating condition of the rotor-
stator mixers, an inertial subrange model for disruptive forces and an elastic-
136 
 
 
plastic deformation mechanism for cohesive forces adequately describes the 
crystal breakage behavior in these mixers.  
• It was shown that the Comminution number is useful to correlate the data for 
geometrically similar mixers. It provides a useful approach to predict the effect of 
different operating condition as well as crystal physical properties on size 
reduction.  
• The concept of local energy dissipation rate calculated as the power input to the 
mixer divided by the dispersion zone (volume between rotor and stator plus the 
volume of stator slots) is developed to correlate the data for devices without 
geometric similarity. This approach was successful in correlating the maximum 
stable particle size from mixers with different geometries. 
• The mechanistic theory is further utilized to provide breakage kernels based on 
probability of collision and collision rate theories. Different breakage kernels 
based on power law approach and exponential approach are developed. The 
suitability of different kernels to fit the wet milling experimental data are 
investigated. It was found that an exponential breakage kernel based on elastic-
plastic deformation mechanism and inertial subrange turbulence disruptive model 
is more successful to fit the experimental data. 
137 
 
 
Chapter 6: CFD Simulations of the Silverson L4R Inline Mixer 
In previous chapters, the Silverson L4R inline mixer with the standard shear gap 
(SSG) and the enlarged shear gap (ESG) was used to investigate the effect of mixer 
geometry on wet milling performance of different crystal-antisolvent systems. It was 
shown that the local energy dissipation rate can be utilized to predict the milling 
behavior of crystals in different mixers with different geometries. To obtain the local 
energy dissipation rates, knowledge of the power draw of different mixers is required. 
Here, CFD simulations are employed to find the power draw of the Silverson L4R 
inline mixer with the standard and the enlarged shear gap. 
6.1 –Power Draw and Power Number in Rotor-Stator Mixers 
High shear rotor-stator mixers are employed in many processes that include wet 
milling, homogenization, dispersion, dissolving, and chemical reaction [70]. High 
turbulence, high deformation rate, and high-speed jet impingement are believed to be 
the reasons that rotor-stator mixers are efficient as process intensification devices 
[58,71]. Owing to their widespread use, the body of knowledge for rotor-stator mixers 
is evolving rapidly. The major advancements are in the areas of flow characterization 
and fluid dynamics [68,71–77], power draw measurement [63,64,78–82], liquid-liquid 
dispersion [57–59,83–92], deagglomeration of nanoparticle clusters [55,56,93], and 
wet milling [50–54]. 
Power draw specification perhaps is the most crucial measure in a rotor-stator 
mixer. The costs associated with the power requirements are usually a significant part 
of the operational costs. Moreover, many scale-up rules and modeling efforts in rotor-
stator mixers depend heavily on the specific power input. Power draw is generally a 
138 
 
 
function of mixer geometry, rotor speed, viscosity, and density of the working fluid 
(i.e., a fluid with higher density will draw more power from the mixer motor). Based 
on an energy balance, it is possible to calculate the power input to the system by 
insulating the mixing system and measuring the rise in temperature. This method is 
called calorimetric method and requires high sensitivity thermistors and proper 
insulation. Another method to acquire the power draw of the mixer is to measure the 
electrical power input to the mixer motor. Based on motor efficiency charts provided 
by the manufacturer, it is possible to calculate the power output of the motor. In this 
case it is important to consider the energy loss due to bearings and friction which 
makes the power exerted to the working fluid lower than the motor’s power output. 
Another common practice is to measure the torque on the mixer shaft. A torque meter 
is used with a strain gauge transducer and a signal amplifier. When a torque is applied 
to the shaft, the strain derives a change in the electrical resistance of the gauge, 
producing an electrical signal that is proportional to the applied torque. This torque 
can be used to calculate the power draw of the mixer as: 
𝑃 = 2𝜋𝑁𝑇      (6.1) 
where P is the power draw, N is rotational speed, and T is the torque. 
 Padron [63] used a strain gauge load cell to measure the power draw of 
different mixers with different stator heads. He carefully examined the different 
available length scales in the rotor stator mixers, such as rotor diameter, shear gap 
width, and hydraulic diameter of stator holes. He reported that the definitions of the 
power number (𝑁𝑃) and Reynolds number (𝑅𝑒) based on rotor diameter, as shown in 
equations 6.2 and 6.3, can be used satisfactorily to correlate the power draw data for a 
139 
 
 
rotor-stator mixer. In the equations below, 𝜌 is fluid density, N is rotor speed, D is 
rotor diameter, μ fluid viscosity, and P is the power draw of the mixer.  
𝜌 𝑁 𝐷2
𝑅𝑒 =        (6.2) 
𝜇
𝑃
𝑁𝑃 =  3 5      (6.3) 𝜌 𝑁 𝐷
Padron demonstrated that the power number of rotor stator mixers is inversely 
proportional to the Reynolds number up to Re = 100 (laminar regime) and remains 
constant above Re = 10,000 (turbulent regime). For the Silverson L4R batch mixer in 
the turbulent regime, he reported a power number of 1.7 for the disintegrating head, 
2.1 for the slotted head, and 2.3 for the square hole stator head. It is important to note 
that the instantaneous power draw of the mixer varies as the rotor blades sweep the 
stator slots in the mixing head and the reported values are an average taken in one full 
revolution of the rotor. 
Recently, with the development of high speed computers and advanced 
numerical methods it is feasible to compute the flow field characteristics inside a 
rotor-stator mixer with both Reynolds-Averaged Navier-Stokes (RANS) and large-
eddy simulations (LES). This allows predicting the extent of the hydraulic forces and 
the amount of torque exerted by fluid to the rotor. Extensive research was conducted 
by Kevala (2001) on two-dimensional sliding-mesh simulations of an inline rotor 
stator mixer. In this study the flow fields for IKA prototype mixer with two different 
shear gap widths were simulated and the results were compared to experimental 
measurements. It was found that the RANS simulation results provided a good 
140 
 
 
qualitative estimate of the velocity field. However, quantitatively, the three-
dimensional flow field was not well-modelled with two-dimensional simulation. 
Barailler et al. [72] studied the behaviour of a viscous fluid in a VMI-Rayneri 
batch rotor-stator mixer with a shear gap of 1.04 mm. A three-dimentional CFD 
simulation using POLY3D was used to predict the torque, shear stress, flow rate, and 
velocity field for a number of low rotational speeds. They showed that the simulation 
result for torque analysis was in good agreement with experimental data. Furthermore, 
they reported that the maximum stress experienced in shear gap was consistent with 
the nominal shear stress defined as the tip speed divided by the shear gap width. 
Utomo et al. [73,74] studied the flow field in a Silverson L4R batch mixer with 
different stator heads, i.e. disintegrating, slotted, and square hole. For the CFD 
simulations, they used a transient sliding mesh technique, standard k–ε model, and 
enhanced wall function with 60 time steps per rotor revolution and 1.1 to 1.4 million 
computational cells. The mixing head is positioned at the center of the holding tank as 
shown in Figure 6.1.a. They calculated the power numbers for different stator 
geometries and reported that the predicted values are 10 to 20% lower than those 
measured experimentally by Padron (2001). Their simulation results for power 
number and Padron’s experimental measurements of power number for different 
stator geometries in the Silverson L4R batch mixer are given in Table 6.1. 
 
 
 
141 
 
 
Table 6.1- Power number discrepancy between measurement and simulation results of 
Silverson L4R batch mixer with different stator geometries 
NP           NP            
Stator geometry Discrepancy 
(measurement [63]) (simulation [74]) 
Disintegrating 1.7 1.53 10% 
head 
Slotted head 2.1 1.66 20.9% 
Square hole head 2.3 2.05 10.9% 
Yang [68] conducted RANS simulations for the Silverson L4R batch mixer (lab 
scale) at constant tip speed to study the scale up strategies. In particular, the Silverson 
L4R batch mixer with a slotted head stator was investigated. The mixing head was 
positioned in an off-center configuration (Figure 6.1.b) as suggested by the 
manufacturer to enable higher speed operations without vortexing to promote top to 
bottom mixing. A transient sliding mesh technique, realizable k–ε model, and 
enhanced wall functions with 720 time steps per rotor revolution was used to simulate 
the flow field. Several mesh with different number of computational cells, 1.6, 4.2, 
and 5.6 million, was used for simulation. A significant change in the flow field 
characteristics with the increase of the mesh size from 1.6 to 4.2 million cells was 
reported, whereas the differences in the mean and turbulent flow field between 4.2 
and 5.6 million cell simulations were not significant. The power number obtained 
from 1.6 million cell simulations was 1.7 whereas the power number obtained from 
4.2 and 5.6 million cells simulations was 2.1 which is in agreement with Padron’s 
experimental power draw measurements for the slotted head stator. 
142 
 
 
 
Figure 6.1- (a) centered  and (b) off-centered  configurations of Silverson L4R batch 
mixer inside a holding tank 
Ko [76] conducted a three-dimensional simulations of an IKA inline rotor stator 
mixer for different rotor speeds and flow conditions. He refined the number of 
computational cells in the shear gap to resolve the high shear stresses and the velocity 
field in this region. An increase in mesh size from 2.8 to 10 million cells (increasing 
the number of cells across the shear gap) was necessary for a mesh independent 
simulation of low-speed low-flow condition.  
The investigations summarized above indicate the necessity for a high density 
of computational cells to resolve the large velocity gradient in shear gap, stator slot, 
and slot jet regions of a rotor stator mixer. It is important to note that the size of the 
cells in the smallest scale of the geometry has a repercussion on the total number of 
cells because the size of adjacent cells must be in the same order to produce a 
consistent mesh. As a result, when a high density mesh is used in the shear gap 
region, this concentration propagates into the rest of the domain. 
143 
 
 
6.2 –Geometry Details and Grid Generation Strategy 
The Silverson L4R inline rotor-stator mixer assembly consists of an inlet pipe, 
outlet pipe, and the mixing head where the rotor and stator are located as shown in 
Figure 6.2. The rotor is a radial 4-blade impeller that rotates inside a stator with 
square-hole slots with clearance or shear gap of 0.2 mm. The stator head has 104 slots 
with the opening of 2.4 mm * 2.4 mm * 2 mm. The thickness of the stator wall is 2 
mm and the height and inner diameter are 18.9 mm and 31.5 mm, respectively. The 
geometry details are summarized in Table 6.2.  
 
Figure 6.2- Silverson L4R inline mixer. The image at the left is the entire assembly 
with attached inlet and outlet pipes. The right image show the high-shear mill head 
with the rotor surrounded by the stator. 
 
 
 
 
144 
 
 
Table 6.2- Geometry details of Silverson L4R inline and batch mixers 
Silverson L4R Inline Mixer 
Mixer Dimensions 
Standard Shear Gap Enlarged Shear Gap 
Rotor Diameter 31.1 mm 28.2 mm 
Shear Gap Width 0.2 mm 1.7 mm 
Square Hole Stator 104 Square Hole Slots 
Size of Stator Slots 2.4 mm × 2.4 mm × 2.0 mm 
Thickness of stator 2.0 mm 
Height of stator ring 18.9 mm 
Inner Diameter of stator 31.5 mm 
The standard rotor diameter is 31.1 mm. This diameter was modified to provide 
a range of shear gap widths. In this study, different shear gap widths of 0.2, 0.23, 
0.85, 1.65, and 1.70 mm were simulated. The goal was to investigate the effect of 
clearance width on power draw of the mixer. The SSG has 0.2 mm and the ESG has 
1.70 mm clearance between rotor and stator. Figure 4.33 shows a simplified 
schematic of the SSG and ESG labelled with characteristic dimensions. 
145 
 
 
ANSYS Workbench was used to develop the computational grid. The model 
volume was divided to two primary zones: rotating zone and stationary zone as shown 
in Figure 6.3. This is needed to model flow around the rotor and establish regions 
with different frames of reference. Different methods exist to solve the flow field 
around a rotating body such as the single reference frame (SRF), multiple reference 
frame (MRF), and sliding mesh techniques.  For rotor-stator mixers, the sliding mesh 
technique is most successful for evaluating the time-periodic features of the flow, and 
was thus applied to the Silverson inline mixer in this work. In this technique, the 
model volume must be separated into rotating and stationary zones. At each time step, 
the cells in the rotating zone rotate with a given velocity and hence, the relationship 
between cells at the interface of stationary and rotating zones must be re-evaluated 
similar to a non-conformal surface. 
 
Figure 6.3- Illustration of rotating and stationary zones – the stationary zone is further 
divided into stator and volute zones for better control of meshing strategy 
The stator slots have cubic geometry that is connected to a cylindrical wall. This 
geometry makes it difficult to obtain hexagonal cells in this region. Therefore, the 
stationary zone is divided to two zones: stator and volute. This strategy helped to get a 
symmetric stator zone that can be divided to 13 identical pieces. The grid generation 
146 
 
 
was performed on one piece; it was then rotated to give the complete stator volume. 
This in turn gave more control on grid generation process and resulted in consistent 
hexagonal cells in this region. The zone definitions are illustrated in Figure 6.3. The 
rotor zone includes the volume between the rotor blades and half of the shear gap. 
Hexahedral meshing was used exclusively in this zone. Using the sliding mesh 
technique, the rotor zone was set to rotate with the speed of 0.5° per time step. The 
volute volume is comprised of the inlet and outlet pipes as well as the volume 
between the face of the stator zone and the volute cover. The stator zone includes half 
the volume of the shear gap, stator slots, and the volume behind the stator. This fluid 
zone was also meshed using hexahedrons only and is in a stationary reference frame. 
The interface between each zone is non-conformal, meaning that a direct one-to-one 
relationship between interface cells of the adjacent zones does not exist. This requires 
Fluent to use an interpolation algorithm to calculate flow across the interface. Because 
of the rotation of the rotor zone, the relationship between rotor cells and stator cells at 
the interface must be re-evaluated at each time step. 
 
Figure 6.4 - Grid refinement in standard shear gap, Silverson L4R inline mixer 
147 
 
 
 
Figure 6.5 - Grid refinement in enlarged shear gap, Silverson L4R inline mixer 
The smallest length scale in rotor-stator mixers is the shear gap width, and 
therefore the grid was characterized primarily by the number of cells across the shear 
gap. This in turn has a significant influence on the total number of cells in the model 
because it is generally desired that the aspect ratio of cells (i.e., the ratio between the 
length of the longest side and the length of the shortest side) not exceed 10:1. Initially, 
a coarse mesh with 2 million cells was produced and simulated. The output from this 
preliminary mesh was read into the fine mesh to initialize the flow field. The fine grid 
consists of 8 million cells with 6 cells across the shear gap for the standard shear gap 
and 20 cells for the enlarged shear gap as shown in Figure 6.4 and Figure 6.5. 
6.3 –Solution Methods 
The RANS equations were used to model the flow field in full 3D geometry. 
The Boussinesq approximation was applied to approximate the Reynolds stresses in 
RANS equations, requiring additional equations to model the turbulent viscosity. The 
realizable k-ɛ turbulent model was used to calculate turbulent viscosity, where k is 
148 
 
 
turbulent kinetic energy and ɛ is the turbulence energy dissipation rate. To solve these 
transport equations, a pressure-based solver within the commercial software package, 
ANSYS Fluent was used. Fluent converts the continuous transport equations into 
discretized forms around each computational cell. Balance of the transport equations 
in each cell implies overall conservation within the model. A second order 
discretization for pressure, a second order upwind discretization for momentum, 
turbulent kinetic energy, and turbulence energy dissipation rate, and a second order 
implicit scheme for time stepping were used in this study. For boundary conditions, 
pressure inlet and pressure outlet with zero-gauge pressure were used. 
The finite volume equations at each time step essentially represent a boundary 
value problem, so the treatment of fluid boundary layers at model surfaces is critically 
important in CFD simulations. This is particularly important in the current work 
because the shear gap, one of the primary regions of interest, is strongly affected by 
boundary conditions. There are different models to approximate the flow near the 
walls. Kim [77] reported that both non-equilibrium wall functions (NEWF) and 
enhanced wall treatment (EWT) produce similar results in comparison between 
particle image velocimetry data and CFD simulations for an inline rotor-stator mixer, 
but EWT is computationally more efficient. Therefore, a EWT model was employed 
here to model the near wall velocities. 
In this study, the rotor was set to rotate in clockwise direction (from the top 
view), and the rotation rates were 4000 and 8000 rpm. Water was used as the working 
fluid with density and viscosity of 998.2 Kg/m3 and 0.001003 Pa·s respectively. The 
y-axis coincided with the vertical axis of the shaft. The origin of the coordinate 
system was located on the mid-plane of the rotor and the positive y direction was 
149 
 
 
upward. The convergence criterion on each time step was evaluated based on the 
residuals. They must reach a value smaller than 10-5 before proceeding to the next 
time step. Here, the residual was calculated as the imbalance in the algebraic 
equations summed overall the computational cells. 
6.4 –Simulation Results and Discussion 
The coordinate system used in the Silverson inline L4R mixer is shown in 
Figure 6.6. The XZ plane at y=-130 mm is shown where θ is the angle between the 
centerline of the rotor blade and the x axis. The rotor rotates in the clockwise 
direction. 
 
Figure 6.6 - The coordinate system shown on the mid-plane of Silverson L4R inline 
mixer 
150 
 
 
6.4.1 –Velocities within the Rotor and Stator Zones 
The high-speed rotation of blades in the clockwise direction generates a 
tangential flow in the rotor zone and shear gap region. This high speed tangential 
flow, in shear gap region collide with the downstream edge of the stator slot, and 
hence, the flow direction changes to tangential-radial and eventually to the radial 
direction and forms strong jets. These strong jets emerging from the downstream edge 
of the slots develop a counterclockwise flow in the volute region behind the stator, 
ultimately discharging the fluid through the outlet pipe. Figure 6.7 to Figure 6.11 
illustrate the velocity patterns on the XZ plane at y = -130 mm for both the standard 
and enlarged shear gap at blade positions of Ѳ = -23, -13, 0, 13, and 23 degrees 
respectively. The different positions of the blade as well as the magnification used in 
this figure are intended to give a clear picture of the flow field in shear gap and stator 
region. Different flow patterns are observed as the jets emerge through the stator 
holes. There is a significant amount of flow impinging on the downstream edge of the 
stator creating a recirculation region (vortex), where there is a strong jet on one side 
and low speed recirculation flow on the other side of the stator slots. 
For the case of the SSG, the jets dissipate close to the stator head and therefore 
the vortices are bounded inside the stator sloth. The center of these vortices moves in 
a circular path as a function of the rotor blade position. A different flow pattern is 
observed for the case of the ESG where two distinct regions of flow exist: a region of 
high speed tangential velocity that impinges on the downstream edge of stator and 
escapes further into the volute extending to the volute wall, and a low velocity region 
that re-enters the stator slot from the volute and gets entrained into the outward 
flowing stator jet. The resulting flow pattern has larger recirculation loops which 
151 
 
 
extend into the volute area. This can be explained by considering the narrower jet 
structure and lower intensity of counter clockwise tangential flow in the volute area 
for the ESG. Furthermore, for the case of the SSG with narrow clearance, the higher 
intensity of the impingement on the stator edge causes the jet pattern to change 
direction to radial and then tangential in the direction opposite to the rotation. This in 
turn creates a higher pressure resistance and hence shorter jet length. It is important to 
note that for both cases, when the blade moves towards the downstream edge, the jet 
speed decreases and the recirculation region widens as the blade blocks the fluid in 
the rotor region from passing through the stator square hole. The flow field 
animations are given in Appendix E. 
152 
 
 
 
 
Figure 6.7 - Velocity vector flow field for Silverson L4R inline mixer with standard 
and enlarged shear gap at 8000 rpm and rotor position of Ѳ = -23° 
153 
 
 
 
 
Figure 6.8 - Velocity vector flow field for Silverson L4R inline mixer with standard 
and enlarged shear gap at 8000 rpm and rotor position of Ѳ = -13° 
154 
 
 
 
 
Figure 6.9 - Velocity vector flow field for Silverson L4R inline mixer with standard 
and enlarged shear gap at 8000 rpm and rotor position of Ѳ = 0 
155 
 
 
 
 
Figure 6.10 - Velocity vector flow field for Silverson L4R inline mixer with standard 
and enlarged shear gap at 8000 rpm and rotor position of Ѳ = 13° 
156 
 
 
 
 
Figure 6.11 - Velocity vector flow field for Silverson L4R inline mixer with standard 
and enlarged shear gap at 8000 rpm and rotor position of Ѳ = 23° 
157 
 
 
6.4.2 –Total Energy Dissipation Rate and Turbulent Kinetic Energy 
The contours of normalized turbulent energy dissipation rate on XZ plane of the 
mixer with y = -130 mm are provided in Figure 6.12 to Figure 6.16 for both the SSG 
and ESG configurations at different rotor positions and at a rotational speed of 8000 
rpm. The dissipation rate values are normalized by 𝑁3𝐷2 where 𝐷 is always the 
standard rotor diameter to facilitate the comparison. The simulation results show a 
non-uniform distribution of energy dissipation rate in different regions of the mixer; 
the intensity of dissipation rate is at least one order of magnitude larger in the shear 
gap and jet region compared to other regions in the mixer. Furthermore, the mixer 
with SSG is more successful in focusing energy dissipation in the shear gap and jet 
region and creating larger volumes of high energy dissipation rates compared to the 
mixer with the ESG. It is important to note that for both cases, the blade position with 
respect to the stator sloth affects the intensity and volume of the high energy 
dissipation region in the jet zone. This effect is more pronounced for the case of the 
SSG. 
158 
 
 
 
 
Figure 6.12 - Contours of normalized turbulent energy dissipation rate for Silverson 
L4R inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -23° 
(normalized by N3D2 of SSG) 
159 
 
 
 
 
Figure 6.13 - Contours of normalized turbulent energy dissipation rate for Silverson 
L4R inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -13° 
(normalized by N3D2 of SSG) 
160 
 
 
 
 
Figure 6.14 - Contours of normalized turbulent energy dissipation rate for Silverson 
L4R inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 0 
(normalized by N3D2 of SSG) 
 
161 
 
 
 
 
Figure 6.15 - Contours of normalized turbulent energy dissipation rate for Silverson 
L4R inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 13 
(normalized by N3D2 of SSG) 
 
162 
 
 
 
 
Figure 6.16 - Contours of normalized turbulent energy dissipation rate for Silverson 
L4R inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 23 
(normalized by N3D2 of SSG) 
163 
 
 
The contours of turbulent kinetic energy (TKE) are provided in Figure 6.17 to 
Figure 6.21 for both the SSG and ESG configurations at different rotor positions. 
TKE is a measure of macroscale eddy fluctuating velocities in turbulent flow. This 
energy produced at the macroscale level with large eddies is transferred down to 
smaller and smaller eddies as an energy cascade and finally dissipates by viscous 
forces at or below the Kolmogorov scale. The simulation results show a higher 
intensity of TKE in the rotor swept volume and jet regions compared to other regions 
in the mixer. In the case of SSG, the velocity of impingement on the downstream edge 
of stator sloth is higher resulting in higher intensity of TKE compared to ESG 
configuration. 
164 
 
 
 
 
Figure 6.17 - Contours of normalized turbulent kinetic energy for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -23° 
(normalized by N2D2 of SSG) 
165 
 
 
 
 
Figure 6.18 - Contours of normalized turbulent kinetic energy for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = -13° 
(normalized by N2D2 of SSG) 
166 
 
 
 
 
Figure 6.19 - Contours of normalized turbulent kinetic energy for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 0 (normalized 
by N2D2 of SSG) 
 
167 
 
 
 
 
Figure 6.20 - Contours of normalized turbulent kinetic energy for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 13 
(normalized by N2D2 of SSG) 
 
168 
 
 
 
 
Figure 6.21 - Contours of normalized turbulent kinetic energy for Silverson L4R 
inline mixer with standard and enlarged shear gap at 8000 rpm and Ѳ = 23 
(normalized by N2D2 of SSG) 
169 
 
 
6.4.3 –Power Draw and Power Number Calculations 
The macroscale energy input to the mixer is produced by the drag force acting 
on the rotor blades. The resulting force is proportional to the rotor tip speed, which is 
proportional to the rotational rate, and rotor diameter as: 
𝐹 ∝  𝜌 𝑈2𝐷2      (6.4) 
where F is drag force, 𝜌 is fluid density, U is tip speed, and D is rotor diameter. The 
power can be defined as the intensity of force acting at certain velocity. By adopting 
Equation 6.4 and substituting the tip speed in terms of rotational velocity, 𝜋ND, and 
using the relationship between force and power, 𝐹 = 𝑃/𝑈, an expression for power 
scale is obtained as: 
𝑃 ∝  𝜌 𝑁3𝐷5      (6.5) 
This expression gives a power scale for turbulent flows that can be used in non-
dimensionalizing the power input and defining a power number as indicated in 
Equation 6.3. The above analysis shows that power draw is related to the drag 
experienced by the rotor and, thus, the power number behaves in the same way as a 
drag coefficient and reduces to a constant value under fully turbulent flow conditions 
(Padron 2001). In rotor stator mixers under turbulent condition, the power draw 
fluctuates with a certain frequency because of the interactions between rotor blades 
and stator slots, related to their relative position. Figure 6.22 and Figure 6.23 show the 
deviations of torque from its time averaged values as a function of rotor position, Ѳ, 
for SSG and ESG configurations respectively. The torque fluctuations due to rotor 
and slot interactions is very pronounced in the case of the SSG compared to ESG. The 
170 
 
 
frequency of these fluctuations can be determined by considering the arrangement of 
stator slots and the number of rotor blades. There are 26 square holes at each row of 
the stator. The holes on each row overlap with those on the row above and below. 
This overlap causes 26×2=52 identical geometry configurations on the stator head. On 
the other hand, the rotor consists of 4 blades positioned at right angles on the rotor 
shaft. The number of distinct interactions between the rotor and stator can be 
calculated by the least common multiple of 52 and 4. This calculation leads to 52 
value and is in agreement with simulation results shown in Figure 6.22; in one 
revolution (360°) of the rotor, there are 52 fluctuations in torque data.  
Moreover, there is a low frequency fluctuation with a larger period in the power 
draw data as shown in Figure 6.22 and Figure 6.23. This fluctuation can be explained 
by the asymmetry in flow field in the volute where the position of the outlet pipe 
creates local asymmetric velocity field. To further analyse the power draw data, the 
amount of torque on the front and back surfaces of rotor blades as well as on the top, 
bottom, and outer edge surfaces of the rotor were investigated. Figure 6.22 shows the 
total torque on the rotor of the Silverson L4R inline mixer with the SSG. The torque 
imparted to the front and back surfaces of one blade of the rotor as well as the torque 
on top, bottom, and outer edge surfaces are shown. Figure 6.23 shows the total torque 
on the rotor of the Silverson L4R inline mixer with the ESG. The torque on the front 
and back surfaces of all 4 blades are given as well. It can be seen that the fluid 
interactions with the front and back surfaces of the blades are the main contributors to 
total torque and the effect of drag on top, bottom, and outer edge surfaces of the rotor 
is not significant. 
171 
 
 
 
Figure 6.22- Torque on front and back surfaces of one blade, torque on bottom, top, 
and outer edge surfaces of all 4 blades, and total torque on all 4 blades of Silverson 
L4R inline mixer with SSG at 8000 rpm 
 
Figure 6.23- Torque on front and back surfaces of the blades (4 blades) and total 
torque on rotor for Silverson L4R inline mixer with ESG at 8000 rpm 
Table 6.3 gives the power draw of the mixers with SSG and ESG calculated 
using equation 6.1. The mean value of the fluctuations in torque data was used as the 
torque input. The results show a significant increase in power draw by decreasing the 
clearance width between rotor and stator (shear gap width). It is important to note that 
the shear gap width is increased 8.5 times and the rotor diameter is shortened by 10%; 
172 
 
 
this resulted in a 30% power draw reduction by the mixer with the ESG compared to 
the mixer with the SSG. Given the differences in energy dissipation rate and power 
draw data, the mixer with the narrow shear gap draws more power and produces large 
high-intensity energy dissipation regions which in turn enhances the mixing. Figure 
6.24 gives the power draw of the Silverson L4R inline mixer as a function of shear 
gap width. Deviations from the mean values as a result of asymmetry in flow field and 
rotor and stator interaction is given by the error bars. The simulation data show an 
inverse proportionality between power draw and the clearance width between rotor 
and stator. It is important to note that the power number is related to rotor diameter 
with a power of -5 and as a result even though power draw of mixer with standard 
shear gap is larger compared to the power draw of mixer with the enlarged shear gap, 
its power number is smaller. 
Table 6.3- Reynolds number, power number, and pumping number for Silverson L4R 
inline mixer with SSG and ESG at different rotor speeds 
 
Finally, the effect of shear gap width on the pumping capacity of the mixer was 
investigated. The volumetric flow rate, Q, was calculated by integration of the 
173 
 
 
velocity field at the inlet and outlet of the mixer. Pumping number can be calculated 
as: 
𝑄
𝑁𝑄 = 3      (6.6) 𝑁𝐷
As it is summarized in Table 6.3, the mixer with ESG has a higher pumping 
number compared with the SSG. The change in flow rate was 7% when the shear gap 
width is increased from 0.2 to 1.7 mm. 
 
Figure 6.24- Power draw as a function of shear gap width- Silverson L4R inline mixer 
at 8000 rpm 
6.5 –Summary 
RANS simulations for the Silverson L4R inline mixer with different shear gap 
widths were conducted. The effect of shear gap width on the velocity field, turbulent 
kinetic energy, energy dissipation rate, power draw, and flow number was 
investigated. It was shown that:  
174 
 
 
• The velocity field and flow pattern is different in different shear gap width 
configurations. In particular, the stator jets are wider in the case of standard shear 
gap which creates higher intensity tangential flow in the volute area. The mixer 
with enlarged shear gap creates larger recirculation loops behind the stator which 
extend into the volute area.  
• The mixer with narrow shear gap was more successful in focusing the power input 
in shear gap and stator slots regions. As a result, higher energy dissipation rates 
were observed in this mixer. The power draw of inline mixer with different shear 
gap widths is calculated via CFD simulations. The loss of the power draw due to 
enlarging the shear gap/shortening the rotor diameter is quantified. 
• The torque exerted from fluid on tangential surfaces of the rotor is negligible 
compared to front and back side of the rotor that have a direct interaction with the 
working fluid. 
 
175 
 
 
Chapter 7: Summary and Conclusion 
This chapter summarizes the important results and conclusions of this study. 
Potential future work is also discussed for size measurement considerations, 
mechanistic model development, population balance modelling, experimental studies, 
and CFD investigation of wet milling in rotor-stator devices. Additional work for wet 
milling of different crystal-antisolvent systems in industrial scale rotor-stator mixers 
and other higher power devices, e.g. valve homogenizers, would be particularly useful 
to further extend the scaling up/down strategies developed in this work. 
The most important result of this study is that there are mechanistic 
relationships between particle size reduction and power input in wet milling processes 
using rotor-stator mixers. The mechanistic models can describe and predict the 
milling behaviour of different crystals considering their size and mechanical 
properties, mill operating conditions, and mixer geometry. This finding is particularly 
useful in designing and scaling up wet milling processes in rotor-stator mixers. 
Furthermore, the scaling laws, non-dimensional correlations, and breakage functions 
developed here can be applied to other processes in which particle breakage occurs 
according to the same mechanism(s), e.g., by substituting the particle collision 
velocity with jet velocity in impinging jets or with superficial velocity in valve 
homogenizers, the equations presented in Table 3.4 can be used to predict the 
breakage and attrition behaviour of different crystals with different size ranges in 
these devices. 
176 
 
 
7.1 –Particle Size and Physical Properties 
Milling generates irregular-shaped particles. To avoid the complexity of 
reporting various sizes for a single irregular-shaped particle, the equivalent diameter 
of a particle was used as representative of the particle size, e.g. volume equivalent 
diameter is the diameter of a sphere having the same volume as the particle. 
Various size measurement methods that are used in this study were discussed 
including sieving, microscope-based image analyzing, laser diffraction, FBRM, and 
PVM. It was shown that a number based measurement technique can be overwhelmed 
and saturated by the large number of fine particles that are generated via attrition 
during a milling process. Therefore, to consider the maximum stable particle size, a 
volume or surface based measurement technique is preferred in size distribution 
measurement of milling processes. The sampling method was shown to influence the 
accuracy of a size distribution measurement. A non-representative sample due to 
inhomogeneous bulk materials and lack of proper dispersion can result in a large 
discrepancy in size measurement data. The homogeneity of the sampling was 
enhanced by proper mixing and confirmed by laser diffraction and FBRM 
measurements of size distributions acquired from different regions in the 
sampling/holding vessel.  
Mechanical properties of crystals that are important during breakage were 
discussed including, Young’s/elastic modulus, hardness, and fracture toughness. The 
characteristics of particle deformation depend on these physical properties, i.e., 
Young’s modulus is representative of material resistance to elastic deformation, 
hardness is representative of material resistance to plastic deformation, and fracture 
toughness characterizes the fracture behaviour of the material under stress. A 
177 
 
 
nanoindentation technique was used to measure these properties at the scale of the 
crystals. The small size scale and anisotropic behaviour of some crystals are the 
challenges of this measurement technique. 
7.2 –Mechanistic Models for particle breakage 
A class of mechanistic models for breakage and attrition of crystals and 
particulate solids have been developed based on elastic deformation, plastic 
deformation, and fracture mechanics concepts. The functional form of these models 
was shown to be in agreement with classical comminution theories where the size 
reduction is empirically related to power consumption. It was shown that the model 
constants describe the influence of particle physical properties on milling 
performance. 
The forces acting on the particle during milling can be divided into two 
opposing forces: cohesive forces that resist particle fracture, and disruptive forces to 
break it. It was shown that the cohesive forces are a function of elastic modulus, 
hardness, fracture toughness, and particle size. Each of the different mechanistic 
models predicts different relative importance of the material physical properties and 
particle size in breakage process. Therefore, 4 different classes of cohesive forces 
were developed dependent on particle deformation and breakage behaviour under the 
impacts. Disruptive forces, on the other hand, were found to be a function of particle 
velocity magnitude promoting particle-particle and particle-wall collisions. While 
previous studies have related the disruptive force to macroscale velocity, particle sizes 
are such that they should be influenced by eddy encounters in the inertial subrange of 
turbulence. Therefore, two classes of disruptive forces were developed dependent on 
178 
 
 
macroscale velocity, based on rotor tip speed, and eddy velocity given by 
Kolmogorov’s inertial subrange model. The later depends upon the local energy 
dissipation rate. The disruptive and cohesive forces were employed to construct eight 
different correlations for ultimate crystal size and its rate of approach. Based on these 
mechanistic models, a dimensionless comminution number is developed (ratio of 
disruptive to cohesive forces) to aid physical interpretation and scale up/down efforts 
for geometrically similar devices. Model discrimination was based on comparison to 
the wet milling experimental data and demonstrated in Chapter 5.  
7.3 –Wet Milling Experiments in Rotor-Stator Mixers 
An extensive series of wet milling experiments with different crystal-antisolvent 
systems were performed in three rotor stator mixers: Silverson L4R inline mixer with 
standard shear gap, Silverson L4R inline mixer with enlarged shear gap, and Silverson 
L4R batch mixer with standard shear gap. The effect of milling conditions, mixer 
geometry, and particle physical properties on ultimate particle size and particle 
breakage kinetics was investigated. Various breakage modes are examined by varying 
operating conditions, particle concentration, coatings, and small geometric changes to 
consider the importance of particle-particle, particle-wall, and fluid shear interactions. 
The data indicate that:  
• Increasing the rotor speed has two effects on milling behavior: first, it boosts the 
breakage rate; and second, it results in a smaller ultimate particle size at the end 
of the milling process. It was reasoned that a higher rotor speed draws more 
power from the motor, and thus higher energy intensity is available in the 
mixing head to enhance the breakage process.  
179 
 
 
• Decreasing the throughput through the inline mill head increases the breakage 
rate but otherwise does not have an effect on ultimate particle size. It was 
reasoned that a lower flow rate gives more residence time for particles in the 
mixing head leading to more collision events in one pass, and hence, the higher 
breakage rate.  
• Mechanical properties of particles play an important role in their breakage 
behavior. Particles with higher hardness and lower fracture toughness are more 
brittle.  
• The concentration of particles in the slurry has a limited effect on breakage rate 
and a negligible effect on the ultimate size distribution.  
• Particle-wall collisions are more important in crystal breakage when compared 
to particle-particle or fluid shear mechanisms. 
• At a given rotor speed, the mixer with higher power draw and smaller dispersion 
zone (shear gap volume plus stator hole volume) is more successful in 
concentring the dissipated energy, and hence, leads to higher breakage rate and 
smaller ultimate particle size. 
7.4 –Model Validation and Mechanistic Correlations of Wet Milling 
The suitability of mechanistic models to predict the crystal breakage behavior in 
different rotor-stator mixers was investigated. Based upon the experimental data it 
was shown that an inertial subrange model for crystals undergoing elastic-plastic 
deformation well describes the crystal breakage behavior in the different rotor-stator 
180 
 
 
mixer geometries. It was shown that the comminution number, Co, can be 
successfully apply to describe and predict the effect of operating condition (i.e., rotor 
speed), and crystal mechanical properties in geometrically similar devices. 
The shear gap and stator hole regions experience the highest densities of energy 
dissipation rate and most of the energy input to the mixer is dissipated in these 
regions. Therefore, for devices without geometric similarity, the concept of local 
energy dissipation rate, defined as the power draw of the mixer per mass of fluid in 
the shear gap and stator slot regions, was introduced and exploited to compare the 
data from different mixers with different geometries. This approach was successful in 
correlating the maximum stable particle size. 
The mechanistic theory is further utilized to provide breakage kernels based on 
probability of collision and collision rate theories. Application of the breakage 
functions to predict the milling rate as well as their implementation within a 
Population Balance framework is discussed. The mechanistic framework shows 
promise for application to a broad class of milling devices under different processing 
conditions. 
7.5 –CFD Simulations and Power Draw Calculations 
The effect of shear gap width on the velocity field, turbulent kinetic energy, 
energy dissipation rate, power draw, and flow number in the Silverson L4R inline 
rotor–stator mixers has been examined. It was found that the velocity field in the 
swept volume of the rotor was predominantly tangential and the intensity of the 
velocities was a function of tip speed. In the shear gap region, the velocity field close 
to rotor blade was tangential. The impingement of this tangential flow on the 
181 
 
 
downstream surface of the stator slots caused strong radial jets to emerge from the 
stator slots. The characteristics of the stator jets emerging from the enlarged shear gap 
were different than that of narrow shear gap. 
 In the case of enlarged shear gap, the stator jets extended up to the tank wall, 
while in the case of narrow shear gap, the jets dissipated close to the stator head due 
to wider jet structure and higher intensity of counter clockwise tangential flow in the 
volute area. The jets emerging from stator holes created recirculation loops (vortex) 
behind them. It was found that the structure and center location of these vortices were 
different for different shear gaps. The mixer with narrow shear gap was more 
successful in concentrating the power in shear gap and stator slots regions and higher 
energy dissipation rates were observed in this mixer. The power draw of the mixers 
was obtained by calculating the torque on the rotor. It was found that the power draw 
is a function of shear gap and rotor diameter and by increasing the shear gap, 
shortening the rotor diameter, the power draw decreases. The effect of shear gap 
width on the pumping capacity of the mixer was investigated. It was shown that the 
effect of shear gap width on the flow rate was not significant compared to change in 
power draw. 
7.6 –Future Work  
There are several areas where future work could be performed to extend the 
reach of this study such as:  
• The mechanistic correlation represented in Figures 5.3 and 5.10 could be further 
extended to cover larger range of energy input and particle sizes by acquiring 
milling data from industrial scale rotor-stator mixers and other higher power 
182 
 
 
devices such as valve homogenizers. This would be particularly useful to further 
extend the scaling up/down strategies developed in this work. 
• Wet milling experiments with different crystal-antisolvent systems can be 
performed to further evaluate the mechanistic correlations given in Tables 3.5 
and 3.6. 
• The crystals used in this study are cubic or very close to a cubic shape. It would 
be useful to study the wet milling behaviour of crystals with different 
morphology. Special care must be taken for size measurement of needle shape 
particles. 
• Further analysis and comparison with wet and dry milling literature will be 
useful to build a master curve based on the proposed models. The challenge 
would be to obtain the appropriate scales of the different devices and can 
calculate the power input and the highest available stress.  
• The transient behavior of the process can be explored in more detail by solving 
the population balance equations for the entire population. This will allow a 
more precise evaluation of the breakage kernels that are developed in this study.  
• Furthermore, a mathematical algorithm can be developed in order to map the 
results from different measurement techniques such as FBRM, PVM, and laser 
diffraction to help better comparison between different data sets. 
 
 
183 
 
 
Appendix A: Wet milling data for sucrose-IPA system in the 
Silverson L4R inline mixer with standard shear gap 
Sucrose crystals with 10% by weight, dispersed in 2 litres IPA, were fed to the 
Silverson L4R inline mixer with the standard shear gap at a series of flow rates 2.1, 
3.0, and 4.2 L/min and rotor speeds 5000, 6500, and 8000 rpm in the inline setup (see 
Chapter 4 for details). The samples were taken from the sampling point in the glass 
vessels at predefined time intervals to investigate particle size reduction. Extra milling 
time was provided to ensure that all particles experienced the regions with maximum 
local energy dissipation rate multiple times and the equilibrium is achieved. Here, the 
complete sets of experimental data acquired at different milling time are presented in 
terms of different cumulative diameters, D95, D90, D80, D50, and D10 as well as 
mean diameters, D43, D32, and D10 versus milling time in tank turnovers. A sharp 
breakage rate at the beginning followed by slow changes is observed in the all particle 
breakage data. The sharp breakage slope is due to fracture mode which refers to the 
breakage of a particle into fragments of comparable size. The slow breakage part is 
due to an attrition mechanism by which small fragments are chipped off from larger 
particles. 
184 
 
 
A.1 –Wet Milling of Sucrose/IPA at 5000 rpm and 2.1 lpm 
 
Figure A.1.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
 
Figure A.1.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
185 
 
 
 
Figure A.1.3 – D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
 
Figure A.1.4 – D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
186 
 
 
 
Figure A.1.5 – D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
 
Figure A.1.6 – D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
187 
 
 
 
Figure A.1.7 – D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
 
Figure A.1.8 – D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 2.1 lpm 
188 
 
 
A.2 –Wet Milling of Sucrose/IPA at 6500 rpm and 2.1 lpm 
 
Figure A.2.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
 
Figure A.2.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
189 
 
 
 
Figure A.2.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
 
Figure A.2.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
190 
 
 
 
Figure A.2.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
 
Figure A.2.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
191 
 
 
 
Figure A.2.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
 
Figure A.2.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 2.1 lpm 
192 
 
 
A.3 –Wet Milling of Sucrose/IPA at 8000 rpm and 2.1 lpm 
 
Figure A.3.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
 
Figure A.3.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
193 
 
 
 
Figure A.3.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
 
Figure A.3.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
194 
 
 
 
Figure A.3.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
 
Figure A.3.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
195 
 
 
 
Figure A.3.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
 
Figure A.3.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 2.1 lpm 
 
 
 
 
 
 
 
 
 
 
196 
 
 
A.4 –Wet Milling of Sucrose/IPA at 5000 rpm and 3.0 lpm 
 
Figure A.4.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
 
Figure A.4.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
 
197 
 
 
 
Figure A.4.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
 
Figure A.4.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
198 
 
 
 
Figure A.4.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
 
Figure A.4.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
199 
 
 
 
Figure A.4.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
 
Figure A.4.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 3.0 lpm 
200 
 
 
A.5 –Wet Milling of Sucrose/IPA at 6500 rpm and 3.0 lpm 
 
Figure A.5.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
 
Figure A.5.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
201 
 
 
 
Figure A.5.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
 
Figure A.5.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
202 
 
 
 
Figure A.5.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
 
Figure A.5.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
203 
 
 
 
Figure A.5.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
 
Figure A.5.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 3.0 lpm 
204 
 
 
A.6 –Wet Milling of Sucrose/IPA at 8000 rpm and 3.0 lpm 
 
Figure A.6.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
 
Figure A.6.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
205 
 
 
 
Figure A.6.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
 
Figure A.6.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
206 
 
 
 
Figure A.6.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
 
Figure A.6.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
207 
 
 
 
Figure A.6.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
 
Figure A.6.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 3.0 lpm 
208 
 
 
A.7 –Wet Milling of Sucrose/IPA at 5000 rpm and 4.2 lpm 
 
Figure A.7.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
 
Figure A.7.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
209 
 
 
 
Figure A.7.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
 
Figure A.7.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
210 
 
 
 
Figure A.7.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
 
Figure A.7.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
211 
 
 
 
Figure A.7.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
 
Figure A.7.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 5000 rpm and 4.2 lpm 
212 
 
 
A.8 –Wet Milling of Sucrose/IPA at 6500 rpm and 4.2 lpm 
 
Figure A.8.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
 
Figure A.8.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
213 
 
 
 
Figure A.8.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
 
Figure A.8.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
214 
 
 
 
Figure A.8.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
 
Figure A.8.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
215 
 
 
 
Figure A.8.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
 
Figure A.8.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 6500 rpm and 4.2 lpm 
216 
 
 
A.9 –Wet Milling of Sucrose/IPA at 8000 rpm and 4.2 lpm 
 
Figure A.9.1 - D95 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
 
Figure A.9.2 - D90 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
217 
 
 
 
Figure A.9.3 - D80 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
 
Figure A.9.4 - D50 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
218 
 
 
 
Figure A.9.5 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
 
Figure A.9.6 - D43 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
219 
 
 
 
Figure A.9.7 - D32 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
 
Figure A.9.8 - D10 versus milling time in tank turnovers for sucrose/IPA wet milling 
at 8000 rpm and 4.2 lpm 
  
220 
 
 
Appendix B: Wet milling data for glycine-IPA system in the Silverson 
L4R inline mixer with standard shear gap 
Glycine crystals with 10% by weight, dispersed in 2 litres IPA, were fed to the 
Silverson L4R inline mixer with the standard shear gap at 5000 rpm and 2.1 lpm, 
6500 rpm and 3.0 lpm, and 8000 rpm and 4.2 lpm. Extra milling time was provided to 
ensure that the equilibrium is achieved. Here, the experimental data acquired at 
different milling time are presented in terms of different cumulative diameters, D95, 
D90, D80, D50, and D10 versus milling time in tank turnovers. 
B.1 –Wet Milling of glycine/IPA at 5000 rpm and 2.1 lpm 
 
Figure B.1.1 - D95 versus milling time in tank turnovers for glycine/IPA wet milling 
at 5000 rpm and 2.1 lpm 
221 
 
 
 
Figure B.1.2 - D90 versus milling time in tank turnovers for glycine/IPA wet milling 
at 5000 rpm and 2.1 lpm 
 
Figure B.1.3 - D80 versus milling time in tank turnovers for glycine/IPA wet milling 
at 5000 rpm and 2.1 lpm 
222 
 
 
 
Figure B.1.4 - D50 versus milling time in tank turnovers for glycine/IPA wet milling 
at 5000 rpm and 2.1 lpm 
 
Figure B.1.5 - D10 versus milling time in tank turnovers for glycine/IPA wet milling 
at 5000 rpm and 2.1 lpm 
223 
 
 
 
Figure B.1.6 - D43 versus milling time in tank turnovers for glycine/IPA wet milling at 
5000 rpm and 2.1 lpm 
 
Figure B.1.7 - D32 versus milling time in tank turnovers for glycine/IPA wet milling at 
5000 rpm and 2.1 lpm 
224 
 
 
 
Figure B.1.8 - D10 versus milling time in tank turnovers for glycine/IPA wet milling at 
5000 rpm and 2.1 lpm 
B.2 –Wet Milling of glycine/IPA at 6500 rpm and 3.0 lpm 
 
Figure B.2.1 - D95 versus milling time in tank turnovers for glycine/IPA wet milling 
at 6500 rpm and 3.0 lpm 
225 
 
 
 
Figure B.2.2 - D90 versus milling time in tank turnovers for glycine/IPA wet milling 
at 6500 rpm and 3.0 lpm 
 
Figure B.2.3 - D80 versus milling time in tank turnovers for glycine/IPA wet milling 
at 6500 rpm and 3.0 lpm 
226 
 
 
 
Figure B.2.4 - D50 versus milling time in tank turnovers for glycine/IPA wet milling 
at 6500 rpm and 3.0 lpm 
 
Figure B.2.5 - D10 versus milling time in tank turnovers for glycine/IPA wet milling 
at 6500 rpm and 3.0 lpm 
227 
 
 
 
Figure B.2.6 - D43 versus milling time in tank turnovers for glycine/IPA wet milling at 
6500 rpm and 3.0 lpm 
 
Figure B.2.7 - D32 versus milling time in tank turnovers for glycine/IPA wet milling at 
6500 rpm and 3.0 lpm 
228 
 
 
 
Figure B.2.8 - D10 versus milling time in tank turnovers for glycine/IPA wet milling at 
6500 rpm and 3.0 lpm 
B.3 –Wet Milling of glycine/IPA at 8000 rpm and 4.2 lpm 
 
Figure B.3.1 - D95 versus milling time in tank turnovers for glycine/IPA wet milling 
at 8000 rpm and 4.2 lpm 
229 
 
 
 
Figure B.3.2 - D90 versus milling time in tank turnovers for glycine/IPA wet milling 
at 8000 rpm and 4.2 lpm 
 
Figure B.3.3 - D80 versus milling time in tank turnovers for glycine/IPA wet milling 
at 8000 rpm and 4.2 lpm 
230 
 
 
 
Figure B.3.4 - D50 versus milling time in tank turnovers for glycine/IPA wet milling 
at 8000 rpm and 4.2 lpm 
 
Figure B.3.5 - D10 versus milling time in tank turnovers for glycine/IPA wet milling 
at 8000 rpm and 4.2 lpm 
231 
 
 
 
Figure B.3.6 - D43 versus milling time in tank turnovers for glycine/IPA wet milling at 
8000 rpm and 4.2 lpm 
 
Figure B.3.7 - D32 versus milling time in tank turnovers for glycine/IPA wet milling at 
8000 rpm and 4.2 lpm 
232 
 
 
 
Figure B.3.8 - D10 versus milling time in tank turnovers for glycine/IPA wet milling at 
8000 rpm and 4.2 lpm 
  
233 
 
 
Appendix C: Wet milling data for ascorbic-IPA system in the 
Silverson L4R inline mixer with standard shear gap 
Ascorbic acid crystals with 10% by weight, dispersed in 2 litres IPA, were fed 
to the Silverson L4R inline mixer with the standard shear gap at 5000 rpm and 2.1 
lpm, 6500 rpm and 3.0 lpm, and 8000 rpm and 4.2 lpm. Extra milling time was 
provided to ensure that the equilibrium is achieved. Here, the experimental data 
acquired at different milling time are presented in terms of different cumulative 
diameters, D95, D90, D80, D50, and D10 versus milling time in tank turnovers. 
C.1 –Wet Milling of ascorbic acid/IPA at 5000 rpm and 2.1 lpm 
 
Figure C.1.1 - D95 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1 lpm 
234 
 
 
 
Figure C.1. 2 - D90 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1  lpm 
 
Figure C.1.3 - D80 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1  lpm 
235 
 
 
 
Figure C.1.4 - D50 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1  lpm 
 
Figure C.1.5 - D10 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1  lpm 
236 
 
 
 
Figure C.1.6 - D43 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1  lpm 
 
Figure C.1.7 - D32 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1  lpm 
237 
 
 
 
Figure C.1.8 - D10 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 5000 rpm and 2.1  lpm 
 
C.2 –Wet Milling of ascorbic acid/IPA at 6500 rpm and 3.0 lpm 
 
Figure C.2.1 - D95 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
238 
 
 
 
Figure C.2.2 - D90 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
 
Figure C.2.3 - D80 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
239 
 
 
 
Figure C.2.4 - D50 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
 
Figure C.2.5 - D10 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
240 
 
 
 
Figure C.2. 6 - D43 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
 
Figure C.2.7 - D32 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
241 
 
 
 
Figure C.2.8 - D10 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 6500 rpm and 3.0 lpm 
 
C.3 –Wet Milling of ascorbic acid/IPA at 8000 rpm and 4.2 lpm 
 
Figure C.3.1 - D95 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
242 
 
 
 
Figure C.3. 2 - D90 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
 
Figure C.3.3 – D80 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
243 
 
 
 
Figure C.3.4 – D50 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
 
Figure C.3.5 – D10 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
244 
 
 
 
Figure C.3.6 – D43 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
 
Figure C.3.7 – D32 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
245 
 
 
 
Figure C.3.8 – D10 versus milling time in tank turnovers for ascorbic acid/IPA wet 
milling at 8000 rpm and 4.2 lpm 
 
  
246 
 
 
 
Appendix D: Wet milling data measured with the FBRM probe for 
sucrose-IPA system in the Silverson L4R inline mixer with standard 
shear gap 
Glycine crystals with 10% by weight, dispersed in 2 litres IPA, were fed to the 
Silverson L4R inline mixer with the standard shear gap at 5000 rpm and 2.1 lpm, 
6500 rpm and 3.0 lpm, and 8000 rpm and 4.2 lpm. Extra milling time was provided to 
ensure that the equilibrium is achieved. Here, the experimental data acquired at 
different milling time are presented in terms of different cumulative diameters, D95, 
D90, D80, D50, and D10 versus milling time in tank turnovers. 
 
D.1 –FBRM results of wet milling of sucrose/IPA at 5000 rpm and 3.0 
lpm 
 
Figure D.1.1 - Initial chord length distribution of sucrose/IPA measured by FBRM 
probe 
247 
 
 
 
Figure D.1.2 - Wet milling of sucrose/IPA at 5000 rpm and 3.0 lpm – Chord length 
distribution at 2 tank turnovers measured by FBRM probe 
 
Figure D.1.3 - Wet milling of sucrose/IPA at 5000 rpm and 3.0 lpm – Chord length 
distribution at 4 tank turnovers measured by FBRM probe 
248 
 
 
 
Figure D.1.4 - Wet milling of sucrose/IPA at 5000 rpm and 3.0 lpm – Final chord 
length distribution measured by FBRM probe 
D.2 –FBRM results of wet milling of sucrose/IPA at 6500 rpm and 3.0 
lpm 
 
Figure D.2.1 – Initial chord length distribution of sucrose/IPA measured by FBRM 
probe 
249 
 
 
 
Figure D.2.2 – Wet milling of sucrose/IPA at 6500 rpm and 3.0 lpm – Chord length 
distribution at 2 tank turnovers measured by FBRM probe 
 
Figure D.2.3 - Wet milling of sucrose/IPA at 6500 rpm and 3.0 lpm – Chord length 
distribution at 4 tank turnovers measured by FBRM probe 
250 
 
 
 
Figure D.2.4 - Wet milling of sucrose/IPA at 6500 rpm and 3.0 lpm – Final chord 
length distribution measured by FBRM probe 
 
D.3 –FBRM results of wet milling of sucrose/IPA at 8000 rpm and 3.0 
lpm 
 
Figure D.3.1 - Initial chord length distribution of sucrose/IPA measured by FBRM 
probe 
251 
 
 
 
Figure D.3.2 - Wet milling of sucrose/IPA at 8000 rpm and 3.0 lpm – Chord length 
distribution at 2 tank turnovers measured by FBRM probe 
 
Figure D.3.3 - Wet milling of sucrose/IPA at 8000 rpm and 3.0 lpm – Chord length 
distribution at 4 tank turnovers measured by FBRM probe 
252 
 
 
 
Figure D.3.4 - Wet milling of sucrose/IPA at 8000 rpm and 3.0 lpm – Final chord 
length distribution measured by FBRM probe 
Appendix E: Flow Field Animations 
Two flow field animations are provided in a CD. The captions to these 
animations are included here. 
Animation 1 – The Silverson L4R inline mixer with the standard shear gap – 
Rotor speed of 8000 rpm - Mean velocity field colored by velocity magnitude in the 
XZ plane at y = 130 mm. Showing the center plane of rotor and stator. Pressure inlet, 
pressure outlet (free pumping), 7.8M cells, RANS simulations.  
Animation 2 – The Silverson L4R inline mixer with the enlarged shear gap – 
Rotor speed of 8000 rpm - Mean velocity field colored by velocity magnitude in the 
XZ plane at y = 130 mm. Showing the center plane of rotor and stator. Pressure inlet, 
pressure outlet (free pumping), 7.8M cells, RANS simulations. 
253 
 
 
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