Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UFE0044104/00001
## Material Information- Title:
- Combined Fundamental and Experimental Approach to Milling
- Creator:
- Starkey, Derek R
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2012
- Language:
- english
- Physical Description:
- 1 online resource (90 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Chemical Engineering
- Committee Chair:
- Svoronos, Spyros
- Committee Co-Chair:
- Powers, Kevin W
- Committee Members:
- Curtis, Jennifer S
Mecholsky, John J Iacocca, Ronald - Graduation Date:
- 5/5/2012
## Subjects- Subjects / Keywords:
- Bicarbonates ( jstor )
Grinding ( jstor ) Grinding mills ( jstor ) Hardness ( jstor ) Material properties ( jstor ) Mathematical independent variables ( jstor ) Modeling ( jstor ) Parametric models ( jstor ) Powder mills ( jstor ) Sodium ( jstor ) Chemical Engineering -- Dissertations, Academic -- UF air -- balance -- breakage -- characterization -- dependent -- function -- hardness -- jet -- measure -- mill -- milling -- model -- particle -- population -- powder - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Chemical Engineering thesis, Ph.D.
## Notes- Abstract:
- Air jet mills are important tools in the size reduction processing of pharmaceutical powders. The benefits of the air jet mill to the pharmaceutical industry are its sanitary design (no moving parts or media) and ability to produce narrow particle size distributions. Due to the high-value nature of active pharmaceutical ingredients, a trial-and-error approach to obtain optimal milling conditions for size reduction would lead to a needless expenditure of time and valuable resources. This work uses population balances for modeling the continuous milling of a spiral jet mill with inexpensive, readily available excipient powders, and predicts milling model parameters for high value powders with only small quantities being consumed. We have developed a multilevel model that describes the effect of material characteristics and mill operating variables on particle breakage in a specific air jet mill. This model allows us to predict product size distributions of brittle crystalline materials. The method used to develop this model can be utilized for many self-classifying mills. For a new mill, extensive initial milling with inexpensive excipient powders is required to determine mill-dependent model parameters. Subsequently, quick material characterization experiments can be made with limited powder consumption of expensive powders to determine powder-dependent mill parameters. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2012.
- Local:
- Adviser: Svoronos, Spyros.
- Local:
- Co-adviser: Powers, Kevin W.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-05-31
- Statement of Responsibility:
- by Derek R Starkey.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Starkey, Derek R. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 5/31/2014
- Resource Identifier:
- 864880665 ( OCLC )
- Classification:
- LD1780 2012 ( lcc )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

PAGE 1 1 COMBINED FUNDAMENTAL AND EXPERIMENTAL APPROACH TO MILLING By DEREK STARKEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 PAGE 2 2 2012 Derek Starkey PAGE 3 3 To my family and friends PAGE 4 4 ACKNOWLEDGMENTS I would like to acknowledge Eli Lilly and Company for sponsoring my research project I thank all my advisors and research team memb ers, who have provided me with invaluable help Thanks to all my friends and family who have supplied me with continuous support PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF ABBREVIATIONS ................................ ................................ ............................. 9 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 ORGANIZATION ................................ ................................ ................................ .... 13 2 REVIEW OF SIZE REDUCTION ................................ ................................ ............ 16 Comminution Equipment ................................ ................................ ......................... 16 Modeling Breakage ................................ ................................ ................................ 18 Modeling Air Jet Mills ................................ ................................ .............................. 19 Material Dependence in Breakage Functions ................................ ......................... 20 Measuring Material Properties ................................ ................................ .......... 21 Single Impact Milling ................................ ................................ ........................ 24 The Contribution of this Work ................................ ................................ ................. 26 3 MODEL STRUCTURE AND MODEL VALIDATION USING MILL EXPERIMENTS ................................ ................................ ................................ ...... 27 Introduction ................................ ................................ ................................ ............. 27 Background ................................ ................................ ................................ ............. 29 Modeling ................................ ................................ ................................ ................. 30 Population Balance Model ................................ ................................ ................ 30 Chipping Conditional Probability Simplification ................................ ................. 32 The Sigma Function: Probability of Breakage in the Mill ................................ .. 34 The k Function: Con ditional Probability of Chipping upon Breakage ................ 36 Model Summary ................................ ................................ ............................... 37 Materials and Methods ................................ ................................ ............................ 38 Results and Discussion ................................ ................................ ........................... 43 Conclusions ................................ ................................ ................................ ............ 47 4 POWDER DEPENDENT PARAMETERS FROM CHARACTERIZATION EXPERIMENTS ................................ ................................ ................................ ...... 51 Introduction ................................ ................................ ................................ ............. 51 Expanded Model Structure ................................ ................................ ..................... 53 Materials and Method s ................................ ................................ ............................ 55 Hardness: Microindentation ................................ ................................ .............. 56 PAGE 6 6 Microindentation procedure ................................ ................................ ........ 57 Mi croindentation analysis: hardness ................................ .......................... 58 Breakage Measurement: Single impact Micromilling ................................ ........ 58 Micromilling procedure ................................ ................................ ............... 59 Micromilling analysis: breakage measure ................................ .................. 60 Powder dependent Parameter Function Models ................................ ..................... 61 Results and Discussion ................................ ................................ ........................... 64 Conclusions ................................ ................................ ................................ ............ 66 5 FURTHER POWDER CHARACTERIZATION AND FUTURE WORK .................... 71 Further Characterization ................................ ................................ ......................... 71 Future Work ................................ ................................ ................................ ............ 75 Developing Milling Model Software ................................ ................................ .. 75 Applying the Mill Modeling Approach to New Mills ................................ ........... 76 Relating Breakage Measure to Material Properties ................................ .......... 78 APPENDIX A JET MILL RUN FORMS ................................ ................................ .......................... 79 B MILLING MODEL SOFTWARE USING MICROMILL BREAKAGE MEASURE ...... 81 C MILLING MODE L SOFTWARE USING MICROINDENTATION HARDNESS ........ 83 LIST OF REFERENCES ................................ ................................ ............................... 85 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 90 PAGE 7 7 LIST OF TABLES Table page 2 1 Mill types ................................ ................................ ................................ ............ 16 3 1 Material characteristics of the three test powders used in this study .................. 38 3 2 Jet mill operating conditions from fractional factorial design ............................... 39 3 3 Jet mill feed compositions according to sieving ................................ .................. 40 3 4 Bin definitions ................................ ................................ ................................ ..... 43 3 5 Parameters of the sigma and k functions for sodium bicarbonate ...................... 44 3 6 Sodium bicarbonate experimental and modeled all jet mill runs ......................... 48 3 7 Lactose monohydrate experimental and modeled all jet mill runs ...................... 49 3 8 Sucrose experimental and modeled a ll jet mill runs ................................ ............ 50 4 1 Powder dependent parameters of all test powders ................................ ............ 61 4 2 Hardness and breakage measure of all test powders ................................ ......... 61 4 3 Sodium bicarbonate experimental and modeled all jet mill runs using (Mod BM) and (Mod H) ................................ ................................ ................................ 68 4 4 Lactose monohydrate experimental and modeled all jet mill runs using (Mod BM) and (Mod H) ................................ ................................ ................................ 69 4 5 Sucrose experimental and modeled for all jet mill runs using (Mod BM) and (Mod H) ................................ ................................ ................................ .............. 70 PAGE 8 8 LIST OF FIGURES Figure page 2 1 Overloaded Vickers indentation used to measure fracture toughness. ............... 22 3 1 Air jet mill schematic showing gas and particle flows ................................ ......... 29 3 2 Logistic function used to model the probability of breakage function .................. 34 3 3 Two level model architecture ................................ ................................ .............. 37 3 4 6 bin population balance model equations ................................ ......................... 44 3 5 Sodium bicarbonate product from jet mill ................................ ........................... 45 3 6 Lactose monohydrate product from jet mill ................................ ......................... 45 3 7 Sucrose product from jet mil l ................................ ................................ .............. 46 4 1 Three level model architecture ................................ ................................ ........... 53 4 2 Optical image of 50 gram load Vickers indent of sodium bicarbonate ................ 58 4 3 Single impact micromill design ................................ ................................ ........... 59 4 4 Visualization of breakage measure with sodium bicarbonate micromill results .. 61 4 5 Breakage measure versus w size powder dependent parameter .......................... 62 4 6 Breakage measure versus a 0 powder dependent parameter ............................. 62 4 7 Hardness versus w size powder dependent parameter ................................ ......... 63 4 8 Hardness versus a 0 powder dependent parameter ................................ ............ 63 4 9 Three level model: sodium bicarbonate product from jet mill ............................. 65 4 10 Three level model: lactose monohydrate product from jet mill ........................... 65 4 11 Three level model: sucrose product from jet mill ................................ ................ 66 5 1 50 gram Knoop indentation on sodium bicarbonate ................................ ........... 71 5 2 50 gram Vickers indent with crack propagation on sucrose ............................... 73 5 3 50 g Vickers indent with cracks propagated to particle edge on lactose ............ 74 PAGE 9 9 LIST OF ABBREV IATIONS volume weighted arithmetic mean diameter of the feed volume weighted arithmetic mean diameter of the product volume weighted mean diameter of bin i (microns) a 0 powder dependent constant in k function AFM atomic force mi croscopy a FR mill dependent weight of feed rate in k function a GP mill dependent weight of grinding pressure in k function API active pharmaceutical ingredient a PP mill dependent weight of pusher pressure in k function b 1 mill dependent constant in sigma f unction BI brittlen e ss index b ij breakage distribution function, fraction of particles breaking from size j to size i BM breakage measure from single impact micromilling d average diagonal of Vickers indent E f mat resistan ce against fracture in impact comminution FR feed rate (g/s) GP grinding pressure (psig) H microindentation hardness IPA isopropanol k conditional probability of chipping upon breakage K c fracture toughness n number of bins PAGE 10 10 n mill energy per particle P in dentation load applied PP pusher pressure (psig) S j selection function, probability of a particle of size j breaking in some unit time w FR mill dependent weight of feed rate in sigma function w GP mill dependent weight of grinding pressure in sigma function w GPPP mill dependent weight in sigma function w m,min specific energy a particle can take up without comminution w PP mill dependent weight of pusher pressure in sigma function w size powder dependent weight of size i in sigma function y i mass fraction of bi n i in product y if mass fraction of bin i in feed ij probability size j breaks into size i j j j mean residence time of size j particle in mill sum of error squared or bin number feed composition number average productive impact angle PAGE 11 11 Abstract of Disse rtation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMBINED FUNDAMENTAL AND EXPERIMENTAL APPROACH TO MILLING By Derek Starkey May 2012 C hair: Spyros Svoronos Co chair: Kevin Powers Major: Chemical Engineering Air jet mills are important tools in the size reduction processing of pharmaceutical powders. The benefits of the air jet mill to the pharmaceutical industry are its sanitary design (no moving parts or media) and ability to produce narrow particle size distributions. Due to the high value nature of active pharmaceutical ingredients, a trial and error approach to obtain optimal milling conditions for size reduction would lead to a ne edless expenditure of time and valuable resources. This work uses population balances for modeling the continuous milling of a spiral jet mill with inexpensive, readily available excipient powders, and predicts milling model parameters for high value powd ers with only small quantities being consumed. We have developed a multilevel model that describes the effect of material characteristics and mill operating variables on particle breakage in a specific air jet mill. This model allows us to predict produc t size distributions of brittle crystalline materials. The method used to develop this model can be utilized for many self classifying mills. For a new mill, extensive initial milling with inexpensive excipient powders is required to determine mill depen dent model parameters. Subsequently, quick material PAGE 12 12 characterization experiments can be made with limited powder consumption of expensive powders to determine powder dependent mill ing parameters. PAGE 13 13 CHAPTER 1 ORGANIZATION This PhD dissertation is divided i nto four main parts: Chapter 2 presents a review of the relevant literature, Chapter 3 outlines a two level milling model which describes the continuous milling in an air jet mill, Chapter 4 describes a three level milling model which incorporates powder p roperties, and Chapter 5 discusses further characterization and future work. In Chapter 2, an appropriate review of literature provides information on common comminution equipment with a focus on air jet mills. A thorough history of modeling breakage func tions is provided spanning nearly sixty years. The use of these developed breakage functions to model and simulate air jet mills is highlighted. Finally, explanations of material properties and characterization techniques are provided with an emphasis on indentation measurements and single impact milling and their relation to material dependent parameters in milling models. In Chapter 3 a milling model for a continuous self classifying spiral air jet mill has been developed. Its foundation is a popula tion balance model with selection and breakage distribution functions that have been related to a minimal number of separable mill dependent and powder dependent parameters. Initially, experimentation is required to determine the mill dependent parameters for a specific mill, by milling a powder at multiple operating conditions. Powder dependent parameters can be determined from either mill experiments or from material characterization measurements that require small amounts of powder (presented in Chapter 4 ). The milling model uses as inputs the feed particle size distribution and mill operating conditions and predicts the product particle size distribution. Three crystalline powders, PAGE 14 14 sodium bicarbonate, lactose monohydrate, and sucrose, have bee n used to test the milling model. T he milling model for a self classifying spiral jet mill developed in Chapter 3 can be used to predict the product size distribution for a given mill and powder. For the same mill, the mill dependent parameters can be fix ed to the values determined previously, however extensive milling experiments are still required to determine the powder dependent parameters with the previous two level model. It would be preferable if only a small amount of powder was required to determ ine these powder dependent parameters using simple powder characterization and material property measurement techniques. In Chapter 4 the milling model described in Chapter 3 has been expanded to a three level model with the addition of powder dependent parameter function models using simple material characterization measurements as inputs. This allows the determination of these parameters with minimal consumption of powder. Specifically, the powder dependent parameters are related to material hardness from microindentation or to a breakage measure from single particle impact milling. The crystalline powders, sodium bicarbonate, lactose monohydrate, and sucrose, have been used to test the material characterization techniques and expanded milling model. Chapter 5 outlines preliminary results and guidance for determin ing elastic modulus from Knoop indentation and fracture toughness from overloaded Vickers indents. Also, suggested future work is presented as three main expansion paths: relate micromill br eakage measure to more material properties, use the mill modeling approach on another mill size or type, and employ the current milling model equations PAGE 15 15 to develop useful industrial software. Each of these three directions has great potential in advancing the understanding of breakage behavior and applying this knowledge to assist and advance size reduction in industry. Appendix A includes the jet mill run forms that were used to document each experiment. Appendix B and Appendix C contain the MATLAB code o f the milling model which predicts the breakage functions using micromill breakage measure or microindentation hardness measurements, respectively. PAGE 16 16 CHAPTER 2 REVIEW OF SIZE REDUCTION Comminution Equipment Size reduction is a crucial step in solids proces sing and milling is used to create specific product size distributions There are many different types of mills which have various advantages and disadvantages for the many industries that utilize size reduction. These mills can be divided into five cla sses : impact mills, ball media mills, air jet mills, roller mills, and shearing attrition mills [1] They can operate on three main stress mechanism s which caus e breakage: stress be tween two surfaces, stress applied at a single surface, and stress applied by a carrier medium [2] Table 2 1 lis ts a select few of the common mills in four of the five classes which will be discussed here Table 2 1 Mill types Mill class Mill type Product size range [2] Impact mills Pin mill Hammer mill 50 3000 microns Ball media mills Tumbling ball mill Stirred media ball mill < 50 3000 microns Air jet mills Loop (oval chamber) mill Fluidized bed opposed jet mill Spiral (pancake) jet mill < 50 microns Roller mills Crushing roll > 3000 microns When choosing a mill type, the important thing to consider is what product size is desired. If the starting material is very large and product greater than 3 mm is suitable, a roller mill can be used. Roller mills crush particles by passing them through a small gap between two cylindrical rollers, which rotate in opposite directions [2] If product between 50 microns and 3 mm is required, an impact mill can be used. Two commonly used impact mills a re pin mills and hammer mills. A pin mill breaks particles by passing PAGE 17 17 them, using a carrier gas, betwe en one stationary and one high speed rotating disc. Numerous pins projecting from each disc interconnect such that they pass very close to one another and impact fed material [2] Hammer mills contain a high speed rotating shaft to which several hammers are attached. Material entering the mill is stressed by multiple impacts with these hammers [3] A large range of product sizes are obtainable using two types of ball media mill s: v essel driven and agitator driven. Vessel driving ball mills are cylindrical vessels containing hard media which rotate and impact particles by tumbling the contents or, at high rotational speeds, compressing them against the chamber walls [1] Agitator driven ball mills, such as a stirred media mill, are capable of producing particles down to the nanometer range [4] To produce particles less than 50 microns, com monly used air jet mills such as s piral jet mills, l oop mills, and fluidized bed opposed jet mills are used The Micronizer Company introduced the spiral jet mill, a tangential jet mill type in 1934, which injects particles using a venturi into a high s peed gas vortex created within a pancake shaped grinding chamber by multiple high speed gas jets [5] Particle breakage within the grinding chamber is caused by particle particle collisions and impacts with the chamber walls. Grinding is dependent on causing collision s between slower moving particles in the rotating vortex with high speed particles accelerated by the tangentially entering grinding jets [6] Inside the vortex, l arger particles are pulled toward the walls of the chamber by centrifugal forces providing further imp act opportunities. As particles break down, the drag force created by gas exiting through a central exit exceeds the centrifugal force and pulls the particles out of the mill. This ensures that only particles below a certain size, the cut size, are able to leave the mill chamber, which means PAGE 18 18 spiral air jet mills are self classifying. Closely related loop mills are vertical versions of pancake jet mills and have higher operating capacities [5] Fluidized bed opposed jet mill s create intense particle particle impacts by accelerating particles using gas jets directly at one another, which has proven to be useful for milling materials that are very difficult to break [7] Modeling Breakage In 1948, Epstein laid the foundations of population bal ance modeling used to describe milling by constructing a statistical model for breakage mechanisms involving two functions [8] These two functions would evolve into the selection function, S j and breakage distribution function, b ij The selection function determine s the percentage of particles to be broken and the b reakage distribution function describe s the product particle fractions of each size created by breakage [9] Breakage in various batch grinding machines has been described by these breakage functions. Wet and dry batch ball milling has been simulate d by a grinding function which combines: selection parameters, breakage parameters, and the feed particle size distribution [10] Ball mills, rod mills, hammer mills, and shredder cutter mills have be en studied using the batch grinding equation assuming first order breakage kinetics [11] Later, Bilgili and Scarlett obs erved deviations from the linear theory and developed a non linear breakage kinetics theory to explain the effect of fines generation in batch ball milling [12] In order to determine the param eters of the breakage functions, many experiments were required with narrow feed size distributions. In order to cut down on the number of experiments, a back calculation optimization procedure was developed by assuming the forms of the selection and brea kage distribution functions, based on previous experiments [13] Another approach to minimize experiment time was to develop an PAGE 19 19 approximation method. Berthiaux and Varinot developed an appro ximation technique to calculate breakage parameters for stirred media batch milling by assuming the breakage distribution function : ( 2 1) T his is only suitable for short grinding times. [14] For continuous milling, modeling of the residence time distribution (RTD) of partic les in the mill h as to be done. Several methods used to incorporate residen ce time in continuous mill modeling include: combining batch milling data with the RTD, assum ing perfect mixing, assum ing plug flow, and us ing N mixers in series. [15] Residence time functions can be complicated when used to describe particle exchanges in mills which include both classification and grinding zones. To remedy this, Gommeren et al. developed a dynamic modeling method for spiral jet mills which used particle size dependent probability functions to describe particle exchange between three zones: grinding, internal classification, and external classification [16] Bilgili and Scarlett incorporated nonlinear effects in modeling continuous milling using a continuous stirred tank mill and a plug flow tube mill [17] Discrete models of mill classifier systems were also develo ped by relating convective flow and axial dispersion on particle size and describing recirculation as plug flow [18] Modeling Air Jet Mills Using these breakage functions, many started correlating mill operating variables to the breakage parameters of several ty pes of mills Since a spiral jet mill was used in this work, previous work involving common air jet mills will be presented. Using a loop mill with a closed outle t ( batch mode ) Nair related breakage parameters to grinding PAGE 20 20 nozzle pressures, nozzle sizes, and material feed rate [19, 20] Likewise, the volumetric flow rate of grinding gas, feed rate, and classification tube height were determined to be the most significant variables in spiral jet mill grinding [21] There have been many studies using horizontal and vertical jet mills involving different mill sizes, designs, and operating conditions. Midoux et al. completed a great summary of the design and previous work with jet mills, and they expanded it by testing three different sized spiral jet mills with organic crystals [22] In addition to mill models, simulations of spiral jet mills have also been completed To simulate spiral jet mills, flow in a nozzle, in a jet, in the zone between a jet hitti ng a flat target, and in a vortex was numerically studied [23] There are several numeri cal simulation methods that have been used to study spiral jet mills. Gommeren et al. used direct simulation Monte Carlo method to determine the frequency and intensity of particle colli sions related to the mill operating conditions [24] Discrete element method and computational flu id dynamics has also been used to simulate particle comminution and c oncluded that feed rate, nozzle angles, and pusher pressure have more influence on breakage [25] Simulations have shown that eddies form at the feed entrance and that gri nding pressure is the most influential operating variable [26] All of these methods used simplified two dimensional simulations. Expansion to three dimensional models was done to de scribe classification only by eliminating breakage [27] Material Dependence in Breakage Functions In addition to mill design and operating variables, the material characteristics of the powder being milled ha ve a great effect on grinding performance. There are several ways to characterize material dependence in milling. The most common techniques are PAGE 21 21 measuring material properties, such as hardness, fracture toughness, and elastic modulus, directly and using single impact milling. Measuring Material Properties Hardness (H) is a measure of a materials resistance to various types of permanent shape changes when a force is applied T wo common micro hardness tests are the Vickers and Knoop test s in which square or elongated diamond pyramids respectively, are used to indent a surface [28] Indentation hardness is calculated as the applied load over the area of the resulting indent. A size effect has been observed and a model has been dev eloped to describe how hardness increases as indentation depth decreases. This increase in hardness in metals is due to geometrically necessary dislocations which contribute to material work hardening as they act as obstacles to the motion of dislocations [29] However, in brittle materials there are various mechanisms leading to permanent deformation; including fracture, viscoelastic deformation, and crushing. For determination of hardness indentations should b e done using the maximum possible load which does not cause crack propagation. When a particle is overloaded during indentation using a Vickers tip, cracks will grow from the corners of the indent. Depending on the length of the crack, it can be assumed t o be either a half penny or Palmq u ist crack, and the fracture toughness (K C ) can be d etermined [30] Fracture toughness is a materials resistance to crack propagation. As illustrated in Figure 2 1 t he cr ack length c is the distance from the center of the Vickers indent to the end of the crack. Let the diagonal of the indent be 2a. If the ratio c/a is greater than 2.5, it ca n be assumed to be a half penny crack. PAGE 22 22 Figure 2 1 Overloaded Vickers indentation used to measure fracture toughness. Another obtainable material property from indentation is the elastic modulus (E), which is described as a materials resistance to elastic deformation. Due to the geometry of the Knoop i ndenter, elastic relaxation can be assumed to be in only the shorter direction, which allows for the determination of a materials elastic modulus [31] While both Vickers and Knoop indentation can give hard n ess, Vickers tests are usually more reliable. Thus, the hardness (H) is measured using the Vickers indent and the Knoop indentation is used to find the elastic modulus (E) from the E/H ratio : ( 2 2 ) final short diagonal length, and = 0.45 for Knoop [32] While microindentation can be used, an improved technique to measure hardness and elastic modulus is depth sensi ng nanoindentation [33] Typically, nanoindentation is done with a Berkovich (triangular pyramid) indenter tip or a sphere. O ther indenter PAGE 23 23 tips, including Vickers, have be en tested and it has been shown that these tips suffer more from tip defects [34] Mechanical characterization, including fracture toughness, of several pharmaceutical powders, using micro or nano indentation, can be used to predict size reduction from single crystal properties [35] The ratio of hardness to fracture toughness is one definition of the brittleness index of a material and can be tho ught of as the machinability, or millability, of a substance [36] Taylor et al. found that the brittleness index, correlated to size reduction in a hammer mill [37] Likewise, Zugner et al. observed the strong impact of the elastic plastic properties of calcite, lactose, so dium ascorbate, and sodium chloride to their individual breakage behaviors in a modified spiral jet mill [38] Nanoindentation can be done using a nanoindenter or an atomic force microscope (AFM). Using AFM nanoindentation, the hardness of sucrose, lactose, ascorbic acid, and ibuprofen has been measured [39] Taking it a step further d e Vegt et al. determined that the distribution of flaws influences the measured mechanical properties and the rate of breakage in a jet mill, which increases with decreasing hardness [40] The biggest challenge with indentation is specimen pre paration. Indentation requires flat surfaces, so typically polishing is used to create suitable specimen s For large, hard materials like ceramics with hardness values close to 30 GPa this is not so difficult; however, most literature does not ade q uately describe the specific specimen preparation method used which can affect the hardness measurement [41] W hen small, soft particles are prepared for indentation, typically the material is altered to allow for easier sampling by creating large, unnatural crystals or compacting particles into tablets. These techniques often eliminate the need for polishing, whi ch is difficult for PAGE 24 24 particulate samples that tend to fracture and pull out. For example, to indent paracetamol l arge single crystals were grown to 5 10 mm from stock powder [42] To measur e the elasticity and fracture toughness of microcrystalline cellulose, Hancock et [43, 44] These techniques will obviously change the material properties measur ed from the true properties of the received par ticles. In this work, a clear specimen preparation method is outlined for sampling soft, brittle crystalline powders on the order of 200 microns in diameter Single Impact Milling In order to eliminate challe nging specimen preparation required for indentation a single impact milling method can be used t o characterize materials. While indentation methods require only a few crystals, single impact milling consumes the amount of powder required for particle siz ing. However, the speed and ease of this technique is very attractive and the influence of all material properties and characteristics can be measured at one time. Single impact mills are typically designed to accelerate particles with compressed gas at a fixed target and use filtration to collect the resulting fragments. Over time, single impact mills have evolved to be used with finer material and variable impact angles [45] There are two main breakage mechanisms at high velocity impacts: chipping and fragmentation. Chi pping is the breakage of small fragments from the surface near an edge of the mother particle, and fragmentation is the breakage of the mother particle into large pieces Papadopoulos and Ghadiri used a single impact device to observe the dependence of ch ipping and fragmentation breakage mechanisms on impact velocity [46] They developed a chipping model which relates the fractional loss per impact with the impact velocity. Later, Ghadiri and Zhang added indentation hardness PAGE 25 25 and fracture toughness into their model. The fractional loss by single impact mill ing was found to correlate to hardness over fracture toughness squared [47, 48] This model was tested at low particle velocities ( less than 10 m/s) with a fixed normal target. When the angle was varied, it was f ound that the normal force determine d the extent of damage while the tangential force influence d breakage pattern [49, 50] At high velocities, a greater normal force causes fragmentation while greater tangential forces near an edge lead to chipping. Peukert outlined an approach to model stirred ball milling that separates mill and material dependence by defining two material dependent parameters : f mat the resistance of a particle against fracture and w m,min the specific energy a particle can take up without fracture [51] This approach was tested on a single impact milling device using several materials, and the material parameters were related to the probability of breakage for these materials [52] Tested materials included large size ranges of polymers, crystalline materials, glass, and limestone. The separation approach, with material dependent parameters found using a single impact mill, was used to simulate breakage in a hammer mill and an air classifier mill ; proving that single particle breakage properties could be translated to multiple impact milling devices [53] Similarly, this separation of mill and material dependent parameters approach has been tested on the milling of sucrose in a single ball mill [5 4] Following the work of Peukert et al. [51 53], r elationships were then developed between compression tests and impact tests, which allowed for the development of a fatigue model [55, 56] Toneva and Peukert at tempted to a pply their separation PAGE 26 26 mat and w m,min are not the same for both impact and compression comminution tests; they are dependent on the stres s mechanism [57] When compared to indentation measurements, it was found that these two material parameters could be expressed in terms of a brittleness index [58] Meier et al. have determined a general breakage function dependent on particle size and material used A stressing parameter, which incorporates the breakage parameters, was determined for several pharmaceutical powders from single impact milling or from mech anical properties determined using indentation [59] The Contribution of this Work This work attempts to unite some of the great ideas and progress made in the history of mill modeling by developing a simple modeling approach to predict the product size distribution from a continuous spiral jet mill given only the feed composition, mill operating variables (grinding pressure, pusher pressure, and feed rate), and simple material characteristics from either microindentation or single impact milling. This has been done by developing a population balance model for self classifying spiral jet mills that does not require separation of grinding and classification zones or complicated residence time distribution models. This method does not assume breakage behavior or relat e the breakage distribution function to the probability of breakage. Furthermore, a clear sampling procedure for microindentation has been outlined for soft brittle particles on the order of 200 microns and a new breakage measure, which requires minimal consumption of material and time, has been defined to relate single impact breakage to material dependence in the spiral jet mill. PAGE 27 27 CHAPTER 3 MODEL STRUCTURE AND MODEL VALIDATION USING MILL EXPERIMENTS Introduction In the pharmaceutical industry, size r eduction techniques are used to improve powder processing and increase the bioavailability of an active pharmaceutical ingredient (API) in order to obtain the desired therapeutic effect or to aid in drug product formulation. One common size reduction tech nique used by the industry is air jet milling. Air jet mills are beneficial to the pharmaceutical industry because of their ability to create narrow product size distributions relying on particle particle impacts to break particles and a simple sanitary d esign containing no moving parts. Since particles are accelerated by gas jets to break upon each other, this mill exhibits minimal contamination compared to other mills which require foreign media or high speed mechanical parts. However, there are several challenges in milling. Each powder breaks differently and many pharmaceutical materials, such as APIs, can be very expensive to manufacture and require careful exposure control due to their biological activity Therefore, it is undesirable to develop mi lling model parameters using optimization methods requiring extensive experimentation with large quantities of an API. A model that requires minimal consumption of high valued powders to establish its parameter values would be advantageous. Such a model is developed in the present work. The standard batch milling population balance model has been modified to describe and predict the continuous milling of a self classifying spiral jet mill. The breakage functions of this population balance model include parameters that can be subdivided into two categories: (1) parameters that are independent of the powder being milled and only PAGE 28 28 dependent on the mill characteristics (mill dependent) and (2) mill independent parameters that depend only on powder characteris tics (powder dependent). Thus, the mill dependent parameter values can be determined for a specific mill through experiments conducted with inexpensive powders, and powder dependent parameters can be determined either with mill experiments or with small q uantities of high value powders using characterization experiments (described in Chapter 4 ). Population balance models are used to explain the grinding process using two functions: the specific breakage rate (selection function, S j ) and the breakage distri bution function, b ij [8, 9, 11] The selection function, S j is the probability of a particle of size j breaking in some unit time. The breakage distribution function, b ij is the fraction of particles breaking f rom size j to size i. Population balance modeling for batch milling is well defined and has been developed extensively in the literature [10] [14] Also, it has been uti lized in modeling mills with internal and external classifiers, but the mill is usually separated into grinding and classification areas [16] [18] Others have used data from spiral jet mills that were physically altered to significantly change its milling abilities [19] [20] This paper presents a population balance model designed for continuous milling. Of the two breakage functions, the selection function has been studied in greater detail in previous literature. The b reakage distribution function is often excluded or incorporated in the selection function. This approach is not suitable for every case [14] The model presented here is capable of predicting the entire product size distribution with only the feed size distribution, mill operating variables, and simple material PAGE 29 29 characteristics as inputs. The material characterization techniques and connection s to powder dependent parameters are des cribed in Chapter 4 Background A typical air jet mill, as shown i n Figure 3 1 has three operating variables: grinding pressure, pusher pressure, and feed rate [20] [26] [21] Figure 3 1 Air jet mill schematic showing gas and particle flows Particles are fed into the feed funnel. The pusher gas enters into the feed apparatus and, us ing a venturi, creates a partial vacuum at the bottom of the feed funnel which causes the particles to be drawn into the system. The particles are then accelerated by the pusher pressure into the grinding chamber. The grinding gas enters into an outer ma nifold surrounding the grinding chamber. Grinding nozzles allow the grinding gas to PAGE 30 30 enter tangentially into the grinding chamber at high velocities creating a vortex of gas. Teng et al. and Muller et al. have been able to illustrate this vortex via simul ation and experimentation [26] [60] P articles injected into this gas vortex are accelerated toward the grind ing chamber walls by centrifugal force. Particles experience multiple impacts with the wall and other particles. A drag force is created by the gases exiting the milling chamber through a central exit Since centrifugal force is proportional to the part icle size cubed and drag force is proportional to the particle size squared, a s particles break into smaller fragments the centrifugal force decreases faster than the drag force. Once the drag force overcome s the centrifugal force the particle exits the mill chamber with the airflow. Depending on the operating variables, there is some given particle size (the cut size) which will be able to leave the grinding chamber. Once a particle breaks down to this critical size the drag force on the particle will exceed the centrifugal force and the particle will be carried out of the chamber. Therefore, this mill design is said to be self classifying. Modeling Population Balance Model The model developed here is applicable to self classification mills such as th e spiral jet mill. It makes the following assumptions: Particles in a ny j in the mill. Size j breaks the same whether it entered as feed or was created by breakage from a larger particle within the mill. The larger the particle the greater the probability of breakage Pa rticles cannot increase in size during milling PAGE 31 31 These assumptions are reasonable for isotropic crystalline powders which break in a brittle manner and exhibit minimal effect from fatigue. For illustration purposes, consider a three bin system where bin 1 is the largest size and bin 3 is the smallest. T he bins are defined such that particles of size 1 are in bin 1, those of size 2 are in bin 2, and particles smaller than size 2 are in bin 3. For our purposes bin 3 includes all particles smaller than bin 2 and therefore particles cannot j j and ij is the probability that size j breaks into size i. These are related to the traditional breakage functions by: ( 3 1) ( 3 2) If the mass fractions of each bin in the feed and product are denoted as and respectively ma ss balances utilizing the functions and lead to the following equations. ( 3 3) ( 3 4) ( 3 5) Particles in bin 1 that exit the mill must have been fed in bin 1 and did not br eak, and since the survival probability is this leads to Eq uation 3 3 Product particles in bin 2 can come from unbroken particles from the feed or can be created from particles of size 1 breaking into size 2 (Equation 3 4 ). Finally, particles exiting in bin 3 can be unbroken size 3 from the feed or particles created by the following mechanisms: bin 2 feed breaking into bin 3, bin 1 feed breaking directly into bin 3, or PAGE 32 32 breaking from bin 1 to bin 2 to bin 3 (Equation 3 5 ). Obviously, as the nu mber of bins increases the number of and factors increase. However, not all and are independent as Equation 3 2 implies that: ( 3 6) Thus, the three bin model has three independent functions ( and ) with and obtained from E quation 3 6. In general, if there are n bins, there will be independent function s. In order to determine and for a set of mill conditions, ( n 1) experimental runs with single bin feeds would theo retically suffice. For example, with 4 bins the run with feed in bin 1 will generate measurements of and and therefore three equations. The run with bin 2 feed will give and and a run with bin 3 feed gives only The resulting six equations can then be used to determine the independent functions and Chipping Conditional Probability Simplification It was observed that only breakage to the next bin size or small est bin size would suffice in describing the breakage of the test powders, used in this study, in the spiral jet mill. The rationale for this observation is that i n high force impacts, two main mechanisms of breakage typically exist: fragmentation and chi pping [61] Fracture occurs when the largest crack in the greatest stress region propagates to the edges of the particle and creates fragmen ts from impact. Typically, the mechanism of breakage depends on a combination of the kinetic energy and the angle at which a particle impacts. A greater normal force will lead to fragmentation while a greater tangential component will cause chipping. Du e to the nature of milling in an air jet mill, which PAGE 33 33 facilitates impacts both between particles and particle wall collisions, a single breakage mechanism cannot be assumed. However, for the materials used in this study (sodium bicarbonate, lactose monohyd rate, and sucrose) the simplifying assumption can be made that particles either break into the next size bin or small fractions chip off (break into the smallest bin). Naming the conditional probability of chipping upon breakage k leads to: ( 3 7) I f a particle breaks down utilizing a chipping mechanism, small particles will be removed from a mother particle which will, more than likely, remain in its original bin. Upon further chipping, the mother particle will eventually fall into the next size bin. Therefore, ij Note that if a particle were to that all fragments end up in the smallest bin, it would fall into the chipping regime. However, since the energy of impact is proportional to the surfaces created, the probability of shattering is much less than that of chipping. Using th e above simplification, for an n bin system, the number of independent functions is reduced from to n. One of the independent functions will always define k while the other functions will define through Theoretically, only two experiments (with feed containing the largest bin size) are required in order to determ ine all the breakage functions for a given set of mill operating conditions. PAGE 34 34 The Sigma Function: Probability of Breakage in the Mill The following function which relates the probability of size i breaking in the mill, to the grinding pressure (GP ), pusher pressure (PP), feed rate (FR), and size i ( ) was developed: ( 3 8) where ( 3 9) Here is the volume weighted mean diameter of bin i and can be thought of as a measure of mill energy per particle. The sigmoidal shape of the logistic function mimics typical milling curves and allows for a smooth transition between little to no breakage at low energies j = 1) at high mill energies. The constant b 1 was constrained to be less than or equal to 2.9 to ensure little to n o breakage at low mill energy (see Figure 3 2 ). Figure 3 2 Logistic function used to model the probability of breakage function PAGE 35 35 In the jet mill, an increase in grinding or pusher pressures will lead to an increase in particle velocity and higher kinetic energy per particle. This will lead to a greater breakag e probability, hence the parameters and are positive. As particle size increases the probability of breakage increases ( > 0) because a larger particle : has a greater probability of contain ing larger flaws has a larger mass hence a higher kinetic energy expe riences greater centrifugal force which will keep it in the mill longer than a smaller particle, leading to more opportuniti es for collisions and breakage If the feed rate of particles to the mill is increased, the probability of breakage will decrease ( < 0). More particles will cause the energy provided by the gas to be distributed over a larger number of particles thereby leadin g to decreased kinetic energy per particle. Finally, the interaction term between GP and PP has a negative effect on the breakage probability which can be explained by an increase in turbulence. When turbulence increases larger particles can be carried out of the mill without breaking. Note that all interactions between operating variables (GP, PP, and FR) were analyz ed, and it was determined that for the three materials studied, the interaction between GP and PP was the only significant interaction between the three possible operating variable interactions. Thus only the GP/PP interaction was included in this functi on ( E quation 3 9) in order to minimize the number of parameters. Of the six parameters in this model, only one w size is considered powder dependent The five mill dependent parameters are constant for the specific mill used. The constant b 1 and all t he operating variables and interaction terms (w GP w PP w FR and w GPPP ) are mill dependent, because mill operating variables generate energy in the mill independent of the material used. In order to model material dependence (the PAGE 36 36 breakage probability of d ifferent powders under the same mill conditions) only the parameter weighting particle size (w size ) is required to change. This parameter is powder dependent because the energy induced on each particle by the mill depends on the properties of the materia l used. The w size parameter can be determined from jet mill runs or material characterization (described in Chapter 4 ). The k Function: Conditional Probability of Chipping upon Breakage A function which relates the conditional probability of chipping ( giv en breakage ) k, to the grinding pressure, pusher pressure, and feed rate was developed. A modified cosine function was chosen to take advantage of the fact that productive collisions (collisions that cause breakage) at low impact angles will more likely cause chipping while high angle impacts will lead to fragmentation. where is the average productive impact angle ( 3 10) ( 3 11) The modified cosine function allows for a smooth transition between little to no chipping at high angles ( 90 degrees, using a frame of reference that results in negative angles) and maximum chipping (or a k = 1) at low angles (0 degrees). The average productive impact angle, can be linearly related to the energy in the mill. For example, at a given low ener gy only high angle impacts can actually break the particle, leading to a high angle ; however, as the energy of the particle increases low angle impacts start becoming productive and the average productive impact angle moves toward a lower angle. PAGE 37 37 Similar to the sigma function, in the jet mill, if either the grinding pressure or pusher pressure is increased, more kinetic energy is supplied to each particle, which will increase the number of small angle collisions between particles resulting in chipping ( and ). At low energies, or lower grinding and pusher pressure, the low angle collisions between particles do not generate enough energy to break the particle. If the feed rate increases, more particles are present in the mill which wil l lead to lower energy per particle. With less energy, only high angle impacts will be productive so will be higher and the probability of chipping will decrease leading to Of the four parameters in this model, only a 0 is considered to b e powder dependent because the impact angle at which a particle can break is dependent on the material used. The a 0 parameter can be determined from jet mill runs or material characterization (described in Chapter 4 ). Model Summary The model developed, i llustrated in Figure 3 3 predicts the product size distribution from a continuous self classifying air jet mill using only the feed size distribution and mill operating variables (GP, PP, FR) given as inputs. Figure 3 3 Two l evel m odel a rchitecture PAGE 38 38 The model can be subdivided into two parts: level 1 population balance model and level 2 breakage function models. For a specific powder and mill, mill and powder dependent parameters can be determined. These parameters and the mill operating variables can be used to predict the breakage functions using the breakage function models presented previously. The breakage functions can then be used in the population balance model to determi ne the product size distribution exiting the air jet mill from a given feed size distribution. Materials and Methods Three common excipient powders were milled to determine mill dependent parameters of the model and illuminate powder dependent parameters. Sodium bicarbonate was obtained from Arm and Hammer lactose monohydrate was from Foremost Farms (Product Code 310), and sucrose was provided by Michigan Sugar Company. Material characteristics for all three materials are given in Table 3 1 Table 3 1 Material characteristics of the three test powders used in this study Powder Sodium Bicarbonate Lactose Monohydrate Sucrose Molecular Formula NaHCO 3 C 12 H 22 O 11 H 2 0 C 12 H 22 O 11 Molar Mass (g/mol) 84.01 360.31 342.30 Appearance White crystals White crystals White crystals Crystal Structure Monoclinic Monoclinic Monoclinic Solubility, in IPA Insoluble Practically insoluble Slightly soluble, practically insoluble in dehydrated IPA D 10 as received (microns) 21.0 7.3 25.1 D 50 as received (microns) 82.1 6 6.2 179.0 D 90 as received (microns) 155.3 146.7 446.2 Bulk d ensity (g/c m 3 ) 1.3 6 0.75 1.11 Particle density (g/cm 3 ) 2.20 1.53 1.59 Solubility, in water (g/100ml) 9 22 200 Microindentation hardness (GPa ) 0.89 1.06 0.76 Breakage measure ( ) (Chapter 4) 0.39 0.28 0.55 Melting Point ( C) 50 (decomposes) 2 14 186 PAGE 39 39 For this study, a Sturtevant two inch Micronizer with no liner and stainless steel walls, was used for milling. A nitrogen Dewar was used to supply the carrier gas and the pressures were controlled using valves. The use of nitrogen help ed to reduce humidity effects. A model 102M Accurate volumetric screw feeder was used to control the feed rate to the mill. Experiments were completed within defined ranges of the three mill operating variables (grinding pressure, pusher pre ssure, and feed rate) using a fractional factorial design of experiments. Ten operating conditions were set at three levels of grinding and pusher pressures (30, 65, and 100 psig) and two levels of feed rate (0.050 and 0.100 g/s for lactose and sucrose; a nd 0.100 and 0.200 g/s for sodium bicarbonate) using the fractional factorial design shown in Table 3 2 Table 3 2 Jet mill operating conditions from fractional factorial design Condition set Grinding p ressure (psig) Pusher pressure (psig) Feed rate (g/s) Material All powders All powders Sodium bicarbonate Lactose monohydrate Sucrose A 30 30 0.100 0.100 0.050 B 100 100 0.100 0.100 0.050 C 30 100 0.100 0.100 0.050 D 100 30 0.100 0.100 0.050 E 65 65 0.100 0.100 0.050 F 65 30 0.200 0.050 0.100 G 65 100 0.200 0.050 0.100 H 30 65 0.200 0.050 0.100 I 100 65 0.200 0.050 0.100 J 65 65 0.200 0.050 0.100 For each of the operating conditions of Table 3 2 runs were completed wi th six different feed compositions, shown in Table 3 3 The f eed compositions in Table 3 3 were created by sieving original stock powder into corresponding bins, and then mixing fractions of each bin tog ether In the case of sucrose, the as received powder was pre PAGE 40 40 milled with the jet mill (operated at a grinding pressure of 30 psig, pusher pressure of 30 psig, and feed rate of 0.400 g/s) to create a new stock powder for experiments. Table 3 3 Jet mill feed compositions according to sieving Feed composition 32 53 microns (270 450 mesh) 53 75 microns (200 270 mesh) 75 106 microns (140 200 mesh) 106 150 microns (100 140 mesh) 1 100.00% 0.00% 0.00% 0.00% 2 50.00% 50.00% 0.00% 0.00% 3 33.33% 33.33% 33.33% 0.00% 4 25.00% 25.00% 25.00% 25.00% 5 0.00% 0.00% 100.00% 0.00% 6 0.00% 0.00% 0.00% 100.00% After the feed was prepared, the volumetric screw feeder was calibrated to feed the powder at the desired feed rate. Once the desired feed rate was achieved, the grinding pressure and pusher pressure were set at the proper levels. Then, the twenty minute jet mill procedure commence d For the first three minutes, the feed was collected into a beaker and weighed to confirm the feed rate During the next three minutes, six feed rate fractions were collected from the feeder in 30 second intervals to confirm the feed rate was consistent. If the feed rate was incorrect or inconsistent, the feeder would be recalibrated and the process would start again. If conditions were ideal, t he feed was allowed to enter the mill continuously for ten minutes using nitrogen as the carrier gas. After milling, another six feed rate fractions were collected at 30 second intervals to confirm uniformity of t he feed rate over the duration of the mill run. The feeder was then turned off while the mill pressures operated for one additional minute in order to mill any powder remaining in the mill chamber. Finally, at the twenty minute mark, the pressure valves were closed. Then, the mill was carefully and systematically disassembled in order to collect all the product powder. PAGE 41 41 After each mill run, powder was collected into three containers: feed, coarse product, and fine product. Material that remained in the feeder was collected as feed. Powder from the product collection chamber, cyclone separator, and grinding chamber was labeled as coarse product. Finally, ultrafine particles found in the piping leading toward and in the filter bag were collected as fine product. Subsequently, a representative sample of each container was sized using a Coulter LS13320 laser diffraction particle size analyzer using the small volume wet module with isopropanol (IPA) as a medium. In addition to using the wet module, sonica tion was used to ensure minimal agglomeration. Powders were analyzed with the Coulter in duplicate or triplicate. The weights and size distributions of the collected coarse product and fine product were used to calculate the size distribution of the comb ined product. All particle size distributions and analysis used for modeling were obtained from laser diffraction measurements. For each set of milling conditions of Table 3 2 the experimental breakage functions and or k were determined by minimizing the sum of the squared errors: ( 3 12) where denotes the mass fraction of size i in a run with feed composition r ( Table 3 3 ). The fit was constrained such that if This constraint is derived from the fact that larger particles are more likely to break, since i t is more probable for them to contain a larger flaw compared to a smaller particle. The larger the flaw the easier it is for that flaw to be propagated and grow into a crack to cause particle fracture. PAGE 42 42 The 50 sigmas found by fitting the sodium bicarbonat e jet mill data to the population balance model were fit to the sigma function E quation 3 8 and Equation 3 9, by minimizing the sum of the squared differences between experimental and model The parameters determined for sodium bicarbonate were used as an initial guess for fitting lactose and sucrose data. The only parameter that changed substantially was the parameter. T his p arameter is dependent on the material while the other five parameters can be considered mill dependent. Therefore, in order to limit the number of powder dependent parameters and still fit the data, all parameters other than were fixed to the sodium bicarbonate values and only was allowed to change to fit the of lactose and sucrose. Similarly, the 10 sodium bicarbonate k values found from the population balance model fits were used to determine the parameters for the k function described previously. However, not all experimental k values are equally reliable. In some cases where all product particles end up in the smallest bin (bin 6 for this study) there is no guarantee that the k value found is correct since there can be infinite solutions for k. If all parti cles of sizes 1, 2, 3, 4, and 5 completely break, then it makes no difference if size 1 breaks sequentially from 1 to 2 to 3 to 4 to 5 to 6 or directly into 6 as in both cases size 6 exits the mill. Therefore, a weighting variable is used to highlight the impact on the model by crucial k values (low pressure conditions with less breakage) and lessen the impact of high breakage k values. The weighting value used is for each condition, where is the probability of breakage of particles in the smallest bin size that can break to smaller sizes. Therefore, where is close to unity (high breakage) the k value has a lesser effect on the model than where is lower. The PAGE 43 43 parameters for sodium bicarbonate were again used as an initi al guess for lactose and sucrose. Here, the constant was considered a powder dependent parameter while the other parameters were fixed at the sodium bicarbonate value. This approach gave minimal degradation between the model and experimental value s. Results and Discussion The base test powder was sodium bicarbonate. Sixty jet mill runs were completed at the 10 different mill operating conditions (A J) and 6 feed (particle size) compositions (1 6) shown in Table 3 2 and Table 3 3 An additional 10 repeat runs were made; one per operating condition at feed composition 5. For this study, the bins were defined according to Table 3 4 Table 3 4 Bin definitions Bin Size range (microns) Volume weighted mean diameter Bin 1 150 212 186 Bin 2 106 150 132 Bin 3 75 106 93 Bin 4 53 75 66 Bin 5 38 53 47 Bin 6 < 38 30 Once all runs were completed and mass fractions wer e calculated, the probabilities of i ) and the conditional probability of chipping upon breakage (k) were found by fitting the experimental jet mill data to a six bin population balance model shown in Figure 3 4 The mill dependent and powder dependent parameters derived from the baking soda fits are shown in Table 3 5 T he signs on these parameters are as expected from the earlier discussion The experimental and modeled mass fractions in each bin of the product powder obtained from a single jet mill run operated with 106 150 micron feed powder a grinding pressure of 30 psig, a pusher pressure of 30 psig, and a feed rate of 0.100 g/s for PAGE 44 44 sodium bicarbonate and lactose and 0.050 g/s for sucrose are shown in Figure 3 5 Figure 3 6 and Figure 3 7 respectively. Figure 3 4 6 bin population balance model equations Table 3 5 Parameters of the sigma and k functions for sodium bicarbonate Sigma Function k Function 2.90 E+00 1.45 E+00 1.09 E 01 3.72 E 02 5.92 E 02 1.41 E 02 9.93 E+00 3.33 E+00 1.81 E 02 8.11 E 04 PAGE 45 45 Figure 3 5 Sodium bicarbonate product from jet mill operated at: grinding pressure of 30 psig, pusher pressure of 30 psig, feed rate of 0.100 g/s, and feed size of 106 150 microns Figure 3 6 Lactose monohydrate product from jet mill ope rated at: grinding pressure of 30 psig, pusher pressure of 30 psig, feed rate of 0.100 g/s, and feed size of 106 150 microns 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 5 6 Mass Fraction Bin # Experimental Data Level 1: Population balance model (with fitted j and ij ) Level 1: Population balance model (with fitted j and k) Level 2: Breakage functions modeled (predicted j and k) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 5 6 Mass Fraction Bin # Experimental Data Level 1: Population balance model (with fitted j and ij ) Level 1: Population balance model (with fitted j and k) Level 2: Breakage functions modeled: mill dependent parameters obtained from sodium bicarbonate (predicted j and k) PAGE 46 46 Figure 3 7 Sucrose product from jet mill operated at: grinding pressure of 30 psig, pusher pressure of 30 psig, feed rate of 0.050 g/s, and feed size of 106 150 microns These mill operating conditions produce the worst level 1 fits and have the richest data with product powder in multiple bins. The differences between the experimental bars and population balance model bars show how well populat ion balance model describes milling. Finally, t he differences between the population balance model bars and breakage function model bars show the ability to predict the breakage functions with models, related to mill and powder dependent parameters The graphs also show that the k simplification had low impact for each powder tested In order to condense the data obtained, Table 3 6 Table 3 7 and Table 3 8 show the experimental and modeled volume we ighted geometric mean diameter of the product distribution for all runs of sodium bicarbonate, lactose monohydrate, and sucrose, respectively. The 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 5 6 Mass Fraction Bin # Experimental Data Level 1: Population balance model (with fitted j and ij ) Level 1: Population balance model (with fitted j and k) Level 2: Breakage functions modeled: mill dependent parameters obtained from sodium bicarbonate (predicted j and k) PAGE 47 47 volume weighted geometric mean diameters were calculated for each particle size distribution using the volum e weighted mean diameters of each bin shown in Table 3 4 Conclusions The modeling of a self classifying spiral jet mill was studied using a population balance model and separation of mill and material dependence. Using three common pharmaceutic al excipient powders, extensive air jet mill experiments were completed in order to determine mill dependent and powder dependent parameters. A population balance model for self classifying mills was developed and used to measure the probability of breaka j ) and the conditional probability of chipping upon breakage (k). Models were then developed to relate these milling model functions to mill operating variables. This procedure can be applied to any mill type that is self class ifying or equipped with a particle size classifier. A different mill would require different mill dependent parameters to be found by running extensive experiments with inexpensive excipient powders. Moving forward, more work is needed to extend this pro cedure to other mills to further understand the mill dependent parameters. Also, more materials and a larger range of operating conditions could be used to extend the development and applicability of the milling model. PAGE 48 48 Table 3 6 Sodium bicarbonate experimental and mo deled volume weighted geometric mean diameter of the product distributions for all jet mill runs reported in microns Operating Conditions GP (psig) 30 100 30 100 65 PP (psig) 30 100 100 30 65 FR (g/s) 0.100 0.100 0.100 0.100 0.100 Feed Exp Model Exp Model Exp Model Exp Model Exp Model 1 32.3 33.1 30.0 30.0 30.3 30.4 30.0 30.0 30.1 30.2 2 34.6 35.5 30.0 30.0 30.8 30.6 30.0 30.0 30.1 30.2 3 35.3 36.5 30.0 30.0 30.3 30.7 30.0 30.0 30.2 30.3 4 37.0 36.8 30.0 30.0 30.6 30.7 30.0 30.0 30.2 30.3 5a 39.7 40.0 30.0 30.0 31.4 30.9 30.0 30.0 30.0 30.3 5b 38.2 40.0 30.0 30.0 31.5 30.9 30.0 30.0 30.0 30.3 6 42.6 42.4 30.0 30.0 31.4 31.1 30.0 30.0 30.0 30.3 Operating Conditions GP (psig) 65 65 30 100 65 PP (psig) 30 1 00 65 65 65 FR (g/s) 0.200 0.200 0.200 0.200 0.200 Feed Exp Model Exp Model Exp Model Exp Model Exp Model 1 30.7 30.7 30.3 30.3 32.7 32.5 30.2 30.0 30.7 30.5 2 30.8 30.9 30.2 30.4 32.9 33.1 30.1 30.1 30.7 30.6 3 31.4 31.0 30.4 30.4 33.6 34.2 30.3 30.1 30.2 30.6 4 31.5 31.1 30.6 30.4 35.1 35.7 30.1 30.1 30.3 30.7 5a 32.5 31.5 30.8 30.5 37.7 38.1 30.0 30.1 31.1 30.9 5b 32.5 31.3 30.4 30.5 37.1 38.4 30.1 30.1 31.3 31.0 6 34.5 31.8 30.7 30.5 41.1 39.7 30.1 30.1 31.0 31.0 PAGE 49 49 Table 3 7 Lactose monohydrate experimental and modeled volume weighted geometric mean diameter of the product distributions for all jet mill runs reported in microns Operating Conditions GP (psig) 30 100 30 100 65 PP (psig) 30 100 100 30 65 FR (g/s) 0.100 0.100 0.100 0.100 0.100 Feed Exp Model Exp Model Exp Model Exp Model Exp Model 1 34.4 34.3 30.0 30. 1 30. 3 30. 3 30.1 30.0 30. 5 30.3 2 37. 5 37.0 30.0 30. 1 30. 8 30. 9 30.0 30.0 3 1.0 30.5 3 42. 5 41.6 30.0 30.1 31.1 31. 6 30.0 30.0 30.7 30. 7 4 47.6 46. 1 30.0 30.1 31. 6 32.0 30.0 30.0 30.9 30. 9 5a 57. 7 53.5 30.0 30. 2 33.3 33. 1 30.0 30. 1 31.0 31.2 5b 53.6 52. 8 30.0 30. 2 32.8 3 3.0 30.0 30. 1 31.3 31.2 6 63. 8 61.3 30.0 30. 2 34. 3 34. 2 30.0 30. 1 31.0 31. 5 Operating Conditions GP (psig) 65 65 3 0 100 65 PP (psig) 30 100 65 65 65 FR (g/s) 0.050 0.050 0.050 0.050 0.050 Feed Exp Model Exp Model Exp Model Exp Model Exp Model 1 30.0 30. 1 30. 1 30.1 30. 8 30. 7 30.0 30.0 30. 1 30.1 2 30.0 30.3 30.4 30. 2 31.5 31. 5 30.0 30.0 30.0 30. 3 3 30. 0 30. 5 30.0 30.2 3 2.0 32. 8 30. 1 30.0 30.0 30.3 4 30.3 30.6 30.3 30. 3 32. 6 33.9 30. 1 30.0 30.0 30.4 5 31. 3 31.0 30.0 30.4 34.0 36. 5 30.0 30. 1 30.0 30. 7 6 30.0 31.3 30.0 30. 5 33. 5 38. 9 30.7 30. 1 30.0 30.8 PAGE 50 50 Table 3 8 Sucrose exp erimental and modeled volume weighted geometric mean diameter of the product distributions for all jet mill runs reported in microns Operating Conditions GP (psig) 30 100 30 100 65 PP (psig) 30 100 100 30 65 FR (g/s) 0.050 0.050 0.050 0.05 0 0.050 Feed Exp Model Exp Model Exp Model Exp Model Exp Model 1 30.0 30. 5 NA 30.0 30.0 30. 1 30.0 30.0 30.0 30.0 2 30.9 30. 8 NA 30. 0 30.1 30. 1 30. 2 30.0 30. 1 30.0 3 30. 7 31.1 NA 30.0 30. 1 30. 1 30.0 30.0 30.1 30.0 4 31. 3 31. 4 NA 30.0 30.4 30. 1 30.0 3 0.0 30.7 30.0 5 30. 4 31. 8 NA 30.0 30. 9 30.1 30.0 30.0 30.0 30.0 6 31.9 31.9 NA 30.0 30. 4 30. 1 30.4 30.0 30.1 30.0 Operating Conditions GP (psig) 65 65 30 100 65 PP (psig) 30 100 65 65 65 FR (g/s) 0.100 0.100 0.100 0.100 0.100 Feed Exp Model Exp Model Exp Model Exp Model Exp Model 1 30. 1 30.0 NA 30.0 30.3 30. 3 30. 1 30.0 30.0 30.0 2 30.6 30. 1 NA 30.0 30. 8 30.3 30. 1 30.0 30. 5 30.0 3 30.5 30. 1 NA 30.0 30.7 30. 3 30. 1 30.0 30.1 30.0 4 30.6 30. 1 NA 30.0 31. 1 30. 6 30. 4 30.0 30.6 30.0 5 30 .0 30. 1 NA 30.0 31. 6 30.7 30.0 30.0 30.0 30.0 6 30. 5 30. 1 NA 30.0 34.1 30. 8 30.0 30.0 30. 1 30.0 PAGE 51 51 CHAPTER 4 POWDER DEPENDENT PARAMETERS FROM CHARACTERIZATIO N EXPERIMENTS Introduction In Chapter 3 a population balance model was developed to describe and predict the continuous milling of a spiral jet mill. The breakage functions of this population balance model include parameters that can be subdivided into two categories: (1) mill dependent and (2) powder dependent For a specific mill, t he mill depende nt parameter values can be determined through experiments conducted with inexpensive powders P owder dependent parameters can be determined with small quantities of high value powders using characterization experiments In previous literature, several mat erial characterization techniques have been used to measure material properties and powder characteristics which relate to particle size reduction. The most common techniques include: indentation (both micro and nano ), single impact milling, flaw analys is, compression tests, and solution theory. Indentation is the most common method, and can be used to obtain several material properties: hardness, elastic modulus, and fracture toughness. Oliver and Pharr developed a technique to determine hardness and elastic modulus from nanoindentation [33] The Oliver and Pharr technique has been applied to determine hardness measurements of several pharmaceutical solids using AFM nan oindentation [38, 39] Meier et al. used nanoindentation not only to determine hardness and elastic modulus, but also to measure fracture toughness from cracks resulting from overloaded indentations [58] Microindentation techniques were used by Singh et al. and Marshall et al. to obtain hardness measurements from indentation with a Vickers tip and elastic modulus values from Knoop indentations [31, 32] Others have simply related breakage parameters to PAGE 52 52 [62] In this study, t he microindentation technique was used to measure hardness. A descriptive procedure was developed and used for creating suitable specimen s for microindentation of soft, brittle powders. Single impact micromilling has been studied extensively by Meier et a l. and Vogel and Peukert [3, 45, 52, 53, 58, 59] They define two material parameters using a single impact mill: the resistance against fracture in impact comminution, f mat and the specific energy a particle ca n take up without comminution, w m,min These parameters are determined by inverting the population balance of the single impact mill. Vogel and Peukert have shown that these parameters can be used to build a master curve to model the breakage probability for multiple powders and sizes. [52] Meier et al. have even related the material parameters to material properties from nanoindentation. Specifically, they show a relation between the f mat and w m,min parameters and the brittleness index (BI) defined as hardness (H) divided by the fracture toughness (K c ). [58] (4 1) In this stud y, a micromill was designed to prevent multiple impacts and reduce the required powder consumption. Also, a new material characterization measurement, the breakage measure (BM), has been defined from single impact milling. This measurement is easy to obt ain and can be used to predict powder dependent parameters in the breakage function s De Vegt et al. have developed a breakage probability function which includes many powder properties: fracture energy, crack propagation stress and velocity flaw PAGE 53 53 size a nalysis, hardness, stress intensity factor, and more [40, 63] Many of these properties are not simple to measure and were approximated using a solubility parameter. Therefore, these approaches and properties are not used in this study. However, it is noteworthy that the breakage measure from single impact micromilling theoretically includes all material properties. Expanded Model Structure Expanding on the milling model described in Chapter 3 the model presente d in Figure 4 1 takes material properties into account so that it can be applied to multiple powders. Figure 4 1 Three level model architecture This model can determine the product size distributi on produced by a self classifying air jet mill with only small quantities of powder being consumed. As inputs, the model PAGE 54 54 requires the feed size distribution entering the mill, the mill operating conditions (grinding pressure, pusher pressure, and feed rat e), and the hardness of the particle (measured by microindentation) or breakage measure (measured with a single impact micromill). The model can be subdivided into three parts: a population balance model (level 1), breakage function models (level 2) descr ibed in Chapter 3 and powder dependent parameter function models (level 3). The powder dependent parameter function models take the material properties measured by microindentation or single impact milling: hardness and breakage me asure, respectively, an d relate them to the powder dependent parameters in the breakage function models. The probability of brea kage function (sigma function) is given by: ( 4 2 ) where ( 4 3 ) where all paramete rs are mill dependent except for w size The probability a particle of size j breaks into size i is given by: ( 4 4 ) where k is c onditional probability of chipping function : ( 4 5 ) where a 0 is the only material depende nt parameter. In Chapter 3, w size and a 0 were determined for the three powders tested by fitting experimental results from jet mill experiments. Here it is shown that for the crystalline PAGE 55 55 powders tested both parameters have very good linear correlations wi th both th e hardness and breakage measure: (4 6 ) (4 7 ) where x can be either the breakage measure from micromilling or the hardness from microindentation. All powder independent parameters of Equation 4 3 and Equation 4 5 can be determined from experiments on the m ill with an inexpensive primary base powder. These experiments will also determine w size and a 0 for the primary base powder. A second set of a 0 and w size can be obtained by running a small number of mill experiments (a single experiment could suffice) wi th a secondary base powder. Measuring the hardness or breakage measure of the base powders allows determination of the parameters of Equation 4 6 and Equation 4 7 To predict the performance of the mill for any other powder all that is needed is measurem ent of hardness and/or the breakage measure. Equation 4 6 and Equation 4 7 provide the powder dependent parameters, which can be used together with the mill dependent parameters and mill operating conditions to calculate the breakage functions and through which the population balance model will predict the milled powder size distribution for any initial feed composition. Materials and Methods Three common excipient powders were used in this study: s odium bicarbonate obtained from Arm and Hammer lactose monohydrate from Foremost Farms (Product Code 310), and sucrose provided by Michigan Sugar Company. Two material PAGE 56 56 characterization methods were tested in this study: microindentation and single impact milling. EpoThin epoxy was purch ased from Buehler Hardness: Microindentatio n Most previous studies involving indentation use supersized single crystals that are blocks of material. As these techniques m ay alter the measured mechanical properties a process was developed to successfully create specimens for the microindentation of soft brittle materials. Typically, polishing is done with a lubricant in order to enhance sliding between the specimen and polishing pad and thus prevent excessive scratching. Because all three powders used in this study are water soluble, the most common lubricant used in polishing, water, could not be used. Many different alcohols, oils, and polymers were tested as possibl e lubricants. However, all lubricants tested actually held the specimen closer to the polishing pads. This phenomenon caused deep scratches which led to particle fracture and eventual pull out. It was determined that dry polishing, with no lubricant, wa s a more suitable technique for soft brittle particles embedded in epoxy, because the specimen could be held slightly above the polishing pad to minimize or prevent excessive scratching. Each of the three materials used in this study were sampled in the sa me way. First, a thin layer of EpoThin epoxy was painted onto a glass slide with a foam brush. Then, particles were dispersed onto the slide using vacuum dispersion. The largest particles available were used for indentation. Typically, the particles u sed were on the order of 200 microns in diameter The specimen was allowed to sit for 24 hours to allow the epoxy to harden. Ideally, the thickness of the epoxy layer is thinner than the particle PAGE 57 57 diameter so that the tips of the particles will stick out of the epoxy. If this is the case, only the tips of the particles need to be removed by polishing. Once the epoxy was completely hardened, the specimen was dry polished using a sequence of polishing pads from 30 microns to 1 micron grit size. Polishing was done by hand with no lubricant, as described previously. In order to polish uniformly, a figure eight motion was used with specimen s held just above the polishing pad. Optical microscopy was used to determine the success of each polishing step and wh en to move to the next grit size. If the majority of surface scratches were on the order of the most recent grit size, the next polishing pad would be used. If not, more polishing would be done with the current polishing pad. If larger scratches were fo und, a larger grit polishing pad would be reused depending on the scratch sizes. Typically, use of a previous grit size failed to recreate a polished surface from one that was excessively scratched. This was especially true if particles had begun to pull out. More often than not, trying to recreate a surface from an excessively scratched specimen failed. For this reason, foreign material was removed by frequently cleaning the specimen surface and polishing pad with a low gas flow. Microindentation p roce dure Samples of each of the powders were indented with a Tukon 2100 microindenter. Suitable particles were identified using the built in microscope on the microindenter. Once a particle was located, the low load indenter tip was positioned and the inden t was made at the load desired. Approximately 50 total indents were made for each powder using a Vickers tip at 10, 15, 20, and 50 gram loads. After the indentations were made, the specimen was taken to a high power optical microscope to PAGE 58 58 be imaged. Digi tal photographs of each indent were taken, labeled by material and load, and saved for analysis. An example indent is shown in Figure 4 2 Figure 4 2 Optical image of 50 gram load Vickers indent of sodium bicarbonate Microindentation a nalysis: h ardness Each indentation photo was used to measure the hardness of the imaged particle. Using ImageJ software, both diagonals of each indent were measured. Then, the hardness (H) was obtained from: ( 4 8 ) where P is the load and d is the average diagonal length Breakage Measurement : Single impact Micromilling A single impact micromill was designed, constructed, and used to determine the breakage measure of e ach material. The micromill design is shown in Figure 4 3 The micromill used the same feed funnel and eductor that was feeding the jet mill. Particles we re fed into the feed funnel and accelerated at a fixed 45 degree target. The 45 PAGE 59 59 degree angle was chosen in order to minimize reacceleration for multiple impacts and impose both tangential and normal forces. With both forces occurring at impact, multiple breakage mechanisms can be tested. If multiple powders are tested in the micromill at the same energy, any difference in breakage would derive from material differences. Figure 4 3 Single impact micromill design Micromilling procedure In order for the energy to be constant in each micromill ex periment, the feed pressure, feed size distribution, and feed rate must be identical for each experiment. Each micromill experiment was operated with a feed pressure of 100 psig. An Accurate 102M volumetric screw feeder was used to control the feed rate to the micromill. The PAGE 60 60 feed rate was set to 0.010 g/s, the lowest achievable level, to ensure minimal particle particle interactions. Feed material was created by sieving powder stocks. Powder between 106 150 microns (100 140 mesh) was used as feed. The micromill was fed for a total of 5 minutes so that approximately 3 grams of material would be used. A filter bag was attached over the exit of the micromill to collect product powder. After the experiment was complete, the filter was removed and the pro duct powder was collected. A sample of unmilled material from the feeder was also set aside. After the experiments were completed, a representative sample of each (feed and product) was sized using the Coulter LS13320 laser diffraction particle sizer. P owders were analyzed with the Coulter in duplicate or triplicate. Micromilling analysis: breakage measure For each feed and product particle size distribution, the volume weighted arithmetic mean diameter was determined. Then, a breakage measure (BM) wa s defined by: ( 4 9 ) H ere and are the volume weighted arithmetic mean diameters of the feed and product distributions, respectively. The harder the material the sm aller the difference between the feed and product mean diameters will be, as illustrated in Figure 4 4 Thus the breakage measure varies inversely with the hardness of the material. However, other powder characteristics such as elastic modulus and fracture toughness will affect the breakage measure. PAGE 61 61 Figure 4 4 Visualization of breakage measure with sodium bicarbonate micromill results Powder dependent Parameter Function Models The powder dependent parameters for sodium bicarbonate, lactose monohydrate, and sucrose, determined by fitting milling data to the breakage function models, are shown in Table 4 1 Table 4 1 Powder dependent parameters of all test powders Sodium Bicarbonate Lactose Monohydrate Sucrose w size ( m 1 ) 1.81E 02 4.95E 03 3.54E 02 a 0 ( ) 1.45E+00 1.17E+00 2.06E+00 A ll three powders were tested with the microindenter and the micromill to obtain t he hardness (H), in GPa, and dimensionless breakage measure These values are displayed in Table 4 2 Table 4 2 Hardness and breakage measure of all test powders Sodium Bicarbonate Lactose Monohydrate Sucrose H (GPa) 0.89 1.06 0.76 BM ( ) 0.39 0.28 0.55 PAGE 62 62 When the breakage measure was plotted versus t he powder dependent parameters w size and a 0 linear relations with positive slope were observed as seen in Figure 4 5 and Figure 4 6 respectively. Figure 4 5 Breakage me asure versus w size powder dependent parameter Figure 4 6 Breakage measure versus a 0 powder dependent parameter Similarly, when hardness was plotted versus w size and a 0 linear tren ds with negative slope w ere found, as shown in Figure 4 7 and Figure 4 8 R = 0.999 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02 2.50E-02 3.00E-02 3.50E-02 4.00E-02 0.00 0.10 0.20 0.30 0.40 0.50 0.60 w size Breakage Measure (BM) R = 0.990 0 0.5 1 1.5 2 2.5 0.00 0.10 0.20 0.30 0.40 0.50 0.60 a 0 Breakage Measure (BM) PAGE 63 63 Figure 4 7 Hardness versus w size powder dependent parameter Figure 4 8 Hardness versus a 0 powder dependent parameter The breakage measure correlations imply that as the micromill breakage measure increases, the breakage probability in the jet mill will increase. In other words, the more a powder breaks in a single impact mill the more likely it will break in a multi impact jet mill as expected Likewise, the more a powder breaks in a single impact the higher the conditional probability of chipping in the jet mill. Inversely, the hardness correlations R = 0.978 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02 2.50E-02 3.00E-02 3.50E-02 4.00E-02 0.00 0.20 0.40 0.60 0.80 1.00 1.20 w_size Hardness (H) R = 0.923 0 0.5 1 1.5 2 2.5 0.00 0.20 0.40 0.60 0.80 1.00 1.20 a_0 Hardness (H) PAGE 64 64 say as the hardness increases, the breakage probability in the jet mill will decrease as will the conditional probability of chipping The linear fits are better for the breakage measure correlations ( R 2 = 0.999 and R 2 = 0.990 ) compared to the hardness correlations (R 2 = 0.978 and R 2 = 0.923 ) This is not unexpected as the hardness of a powder is just one material property whereas the breakage measure is influenced by several material properties The linear equations determined from the above correlations are for breakage measure (BM): (4 1 0 ) (4 1 1 ) and for microindentation hardness (H): (4 1 2 ) (4 1 3 ) Results and Discussion In Chapter 3 it was shown how well the model describes breaka ge using the w size and a 0 parameters fitted from mill experiments. Here w size and a 0 are instead calculated either from microindentation hardness or from micromill breakage measure using Equation 4 10 and Equation 4 11 or Equation 4 12 and Equation 4 13. Figure 4 9 Figure 4 10 and Figure 4 11 show the experimental and modeled product size distribution for sodium bicarbonate, lactose monohydrate, and sucrose jet milled with a gr inding pressure of 30 psig, pusher pressure of 30 psig, feed size of 106 150 microns, and feed rate of 0.100 or 0.050 g/s (depending on material). Both the hardness measurements and breakage measure measurements lead to reasonably good predictions of powd er breakage. PAGE 65 65 Figure 4 9 Three level model: sodium bicarbonate product from jet mill operated at: grinding pressure of 30 psig, pusher pressure of 30 psig, feed rate of 0.100 g/s, and feed size of 106 150 microns Figure 4 10 Three level model: lactose m onohydrate product from jet mill operated at: grinding pressure of 30 psig, pusher pressure of 30 psig, feed rate of 0.100 g/s, and feed size of 106 150 microns 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 5 6 Mass Fraction Bin # Experimental Data 3 level full model: mill dependent parameters from sodium bicarbonate and powder dependent parameters obtained from micromill breakage measure 3 level full model: mill dependent parameters from sodium bicarbonated and powder dependent parameters obtained from microindentation hardness 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 5 6 Mass Fraction Bin # Experimental Data 3 level full model: mill dependent parameters from sodium bicarbonate and powder dependent parameters obtained from micromill breakage measure 3 level full model: mill dependent parameters from sodium bicarbonated and powder dependent parameters obtained from microindentation hardness PAGE 66 66 Figure 4 11 Three level model: sucrose product from jet mill operated at: grinding pressure of 30 psig, pusher pressure of 30 psig, feed rate of 0.050 g/s, and feed size of 106 150 microns Table 4 3 Table 4 4 and Table 4 5 show the experimental and modeled volume weighted geometric mean diameter obtained from breakage measure (Mod BM) and microindentation hardness (Mod H) for every experiment described in Chapter 3. For almost all experiments the model prediction is in fairly close agreement with the experimental value. Conclusions A single impact micromill was constructed and used to determine the newly defined breakage measure for sodium bicarbonate, lactose monohydrate, and sucrose. These breakage measures were then integrated into the developed powder dependent parameter function models to determine the powder dependent parameters for each material in the air jet mill. T his pro cedure can be applied to m any powders. S mall 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1 2 3 4 5 6 Mass Fraction Bin # Experimental Data 3 level full model: mill dependent parameters from sodium bicarbonate and powder dependent parameters obtained from micromill breakage measure 3 level full model: mill dependent parameters from sodium bicarbonated and powder dependent parameters obtained from microindentation hardness PAGE 67 67 quantities of material can be used in the micromill to determine powder dependent paramete rs for high value products (we used 3 grams) Powder dependent parameters have also been related to microindentation har dness. The benefit of using indentation is that less material is required compared to micromilling; only a few particles are required. However, specimen preparation is much more difficult. Also, hardness is only one material property that can cause diff erences in breakage. Therefore, modeling powder dependent parameters with only microindentation hardness may not work for different materials as well as micromilling where all powder properties affect the resulting breakage measure. Moving forward, more e xperimentation is needed to obtain more material properties such as toughness and elastic modulus. Also, more study is needed to understand the relationships between material properties and the breakage measure determined with the micromill. Finally mor e materials with a larger range of powder properties could be tested to extend the development a nd applicability of the milling model. PAGE 68 68 Table 4 3 Sodium bicarbonate experimental and modeled volume weighted geometric mean diamet er of the product distributions for all jet mill runs reported in microns, using micromill breakage measure (Mod BM) and microindentation hardness (Mod H) Grinding pressure (GP) and pusher pressure (PP) are reported in psig and feed rate (FR) is given i n g/s. Operating Conditions GP 30 100 30 100 65 PP 30 100 100 30 65 FR 0.100 0.100 0.100 0.100 0.100 Feed Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H 1 32.3 33.1 32.8 30.0 30.0 30.0 30.3 30.4 30.3 30.0 30.0 30. 0 30.1 30.2 30.1 2 34.6 35.4 34.8 30.0 30.0 30.0 30.8 30.6 30.5 30.0 30.0 30.0 30.1 30.2 30.2 3 35.3 36.5 35.7 30.0 30.0 30.0 30.3 30.7 30.6 30.0 30.0 30.0 30.2 30.3 30.2 4 37.0 36.8 35.9 30.0 30.0 30.0 30.6 30.7 30.6 30.0 30.0 30.0 30.2 30.3 30.2 5 39 .7 40.0 38.6 30.0 30.0 30.0 31.4 30.9 30.7 30.0 30.0 30.0 30.0 30.3 30.2 6 38.2 40.0 38.6 30.0 30.0 30.0 31.5 30.9 30.7 30.0 30.0 30.0 30.0 30.3 30.2 7 42.6 42.4 40.5 30.0 30.0 30.0 31.4 31.1 30.8 30.0 30.0 30.0 30.0 30.3 30.2 Operating Conditions GP 6 5 65 30 100 65 PP 30 100 65 65 65 FR 0.200 0.200 0.200 0.200 0.200 Feed Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H 1 30.7 30.7 30.6 30.3 30.3 30.2 32.7 32.5 32.2 30.2 30.0 30.0 30.7 30.5 30.4 2 30.8 30.9 30.7 30.2 30.4 30.3 32.9 33.1 32.8 30.1 30.1 30.0 30.7 30.6 30.5 3 31.3 31.0 30.9 30.4 30.4 30.3 33.6 34.2 33.6 30.3 30.1 30.0 30.2 30.6 30.5 4 31.5 31.1 30.9 30.6 30.4 30.3 35.1 35.7 34.9 30.1 30.1 30.0 30.3 30.7 30.6 5 32.5 31.5 31.2 30.8 30.5 30.4 37.7 38 .1 36.9 30.0 30.1 30.0 31.1 30.9 30.7 6 32.5 31.3 31.1 30.4 30.5 30.4 37.0 38.4 37.1 30.1 30.1 30.0 31.3 31.0 30.7 7 34.5 31.8 31.4 30.7 30.5 30.4 41.1 39.7 38.1 30.1 30.0 30.0 31.0 31.0 30.7 PAGE 69 69 Table 4 4 Lactose monohydrate experimental and modeled volume weighted geometric mean diameter of the product distributions for all jet mill runs reported in microns, using micromill breakage measure (Mod BM) and microindentation hardness (Mod H) Grinding pressure (GP) and pusher p ressure (PP) are reported in psig and feed rate (FR) is given in g/s. Operating Conditions GP 30 100 30 100 65 PP 30 100 100 30 65 FR 0.100 0.100 0.100 0.100 0.100 Feed Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H 1 34.4 34.3 34.7 30.0 30.0 30.1 30.3 30.3 30.3 30.1 30.0 30.0 30.5 30.3 30.4 2 37.5 37.0 37.6 30.0 30.1 30.1 30.8 30.9 31.0 30.0 30.0 30.0 31.0 30.5 30.6 3 42.5 41.6 42.8 30.0 30.1 30.1 31.1 31.5 31.8 30.0 30.0 30.0 30.7 30.7 30.8 4 47.6 46.0 48.1 30 .0 30.1 30.1 31.5 32.0 32.4 30.0 30.0 30.0 30.9 30.9 31.1 5 57.7 53.5 56.7 30.0 30.2 30.2 33.3 33.1 33.6 30.0 30.1 30.1 31.0 31.2 31.5 6 53.6 52.8 55.8 30.0 30.2 30.2 32.8 33.0 33.5 30.0 30.1 30.1 31.3 31.2 31.5 7 63.8 61.3 66.9 30.0 30.2 30.2 34.2 34.2 35.1 30.0 30.1 30.1 31.0 31.5 31.9 Operating Conditions GP 65 65 30 100 65 PP 30 100 65 65 65 FR 0.050 0.050 0.050 0.050 0.050 Feed Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H 1 30.0 30.1 30.1 30.1 30.1 30.1 30.8 30.7 30.8 30.0 30.0 30.0 30.1 30.1 30.1 2 30.0 30.3 30.4 30.4 30.2 30.2 31.5 31.5 31.6 30.0 30.0 30.0 30.0 30.3 30.3 3 30.0 30.5 30.6 30.0 30.2 30.3 32.0 32.8 33.2 30.1 30.0 30.0 30.0 30.3 30.4 4 30.3 30.6 30.7 30.3 30.3 30.4 32.5 33.9 34.6 30.1 30 .0 30.0 30.0 30.4 30.5 5 31.3 31.0 31.2 30.0 30.4 30.5 34.0 36.5 37.6 30.0 30.1 30.1 30.0 30.7 30.8 6 30.0 31.3 31.7 30.0 30.5 30.6 33.5 38.9 40.8 30.7 30.1 30.1 30.0 30.8 31.0 PAGE 70 70 Table 4 5 Sucrose experimental and modeled v olume weighted geometric mean diameter of the product distributions for all jet mill runs reported in microns, using micromill breakage measure (Mod BM) and microindentation hardness (Mod H) Grinding pressure (GP) and pusher pressure (PP) are reported i n psig and feed rate (FR) is given in g/s. Operating Conditions GP 30 100 30 100 65 PP 30 100 100 30 65 FR 0.050 0.050 0.050 0.050 0.050 Feed Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H 1 30.0 30.5 30.5 NA 30.0 30.0 30.0 30.0 30.1 30.0 30.0 30.0 30.0 30.0 30.0 2 30.9 30.8 30.9 NA 30.0 30.0 30.1 30.1 30.1 30.1 30.0 30.0 30.1 30.0 30.0 3 30.6 31.1 31.3 NA 30.0 30.0 30.1 30.1 30.1 30.0 30.0 30.0 30.1 30.0 30.0 4 31.3 31.4 31.6 NA 30.0 30.0 30.4 30.1 30.1 30.0 30 .0 30.0 30.7 30.0 30.0 5 30.4 31.8 32.0 NA 30.0 30.0 30.9 30.1 30.1 30.0 30.0 30.0 30.0 30.0 30.0 6 31.9 31.9 32.3 NA 30.0 30.0 30.4 30.1 30.1 30.4 30.0 30.0 30.1 30.0 30.0 Operating Conditions GP 65 65 30 100 65 PP 30 100 65 65 65 FR 0.100 0.100 0.1 00 0.100 0.100 Feed Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H Exp Mod BM Mod H 1 30.1 30.0 30.0 NA 30.0 30.0 30.3 30.3 30.3 30.1 30.0 30.0 30.0 30.0 30.0 2 30.6 30.1 30.1 NA 30.0 30.0 30.8 30.3 30.3 30.0 30.0 30.0 30.5 30.0 30. 0 3 30.5 30.1 30.1 NA 30.0 30.0 30.7 30.3 30.3 30.1 30.0 30.0 30.1 30.0 30.0 4 30.6 30.1 30.1 NA 30.0 30.0 31.1 30.6 30.7 30.4 30.0 30.0 30.6 30.0 30.0 5 30.0 30.1 30.1 NA 30.0 30.0 31.6 30.7 30.8 30.0 30.0 30.0 30.0 30.0 30.0 6 30.5 30.1 30.1 NA 30.0 30.0 34.1 30.8 30.9 30.0 30.0 30.0 30.1 30.0 30.0 PAGE 71 71 CHAPTER 5 FURTHER POWDER CHARACTERIZATION AND FUTURE WORK Further Characterization Besides hardness measurements, some other characterization efforts have been made using microindentation to measure elas tic modulus (E) and fracture toughness (K C ) as described previously. In order to measure elas tic modulus, Knoop indentation h a s b een done. Figure 5 1 shows a typical Knoop indentation on sodium bicarbonate. Figure 5 1 50 gram Knoop indentation on sodium bicarbonate Due to its elongated diamond geometry, Knoop indents can only elastically relax in one direction. This assumption allows for the determination of the hardness over elastic modulus rat io using Equation 2 2 and using the hardness measurements from Vickers PAGE 72 72 indentation given in Table 4 2 the elastic modulus can be calculated Unfortunately, not enough indents have been done to determine the elastic modulus of the three test powders used in this study. However, some guidelines have been formulated to aid in future endeavors. Specimen preparation for Knoop indentation was the exact same as for Vickers. This procedure is described in Chapter 4. Selecting partic les to indent is not as simple as for the Vickers test, since the indent is much more elongated. Due to this fact, particles must be rotated so that the indent is kept as far away from the particle edges as possible to minimize edge effects on measured pr operties. This requires a lot more time compared to Vickers indents, which can be made almost in any location on the particle Another issue with the Knoop indentations that have been made is that some analyzed indents gave negative elastic modulus measu rements. This was most likely caused by the measured short diagonal being longer than the approximated initial short diagonal. The cause for this inaccuracy could be from poor measurement of the short diagonal, which is very small for low loads, or inapp ropriate measurement of the long diagonal, which defines the initial short diagonal. To remedy these issues, Knoop indentations should be done at larger loads in the future. The negative measurements tended to be biased to ward the 10, 15, and 20 gram loa ds. The challenge is to accomplish this without fracturing particles. Another material property which can be measured by microindentation is fracture toughness. To do this, particles are overloaded with a Vickers indenter tip in order to produce cracks f rom the corners of the indent as shown in Figure 5 2 PAGE 73 73 Figure 5 2 50 gram Vickers indent with crack propagation on sucrose An appropriate load size has to be determined such that the cracks grow t o a desired length (at least 2 3 times half the indent diagonal), but not long enough to reach the particle edge. This proves to be very difficult for particles on the order of 200 microns. If the crack length c, measured from the center of the indent to the end of the crack, is at least 2 3 times half the diagonal length 2a, the crack can be assumed to be a half penny crack and the following equation can be used to calculate the fracture toughness: (5 1) where P is the inde ntation load, c is the crack length, and is 7 for Vickers. PAGE 74 74 Future measurements may be made with larger particles, but this may be impossible if larger particles are not available and growing or compacting particles would change the material properti es. For the three test materials used in this study, fracture toughness measurements could probably be accomplished for sodium bicarbonate with loads above 50 grams, for lactose monohydrate with loads between 20 and 50 grams, and for sucrose with loads be tween 20 and 50 grams. In the case of lactose, a 50 gram load propagates the cracks to the edges of the particles already (seen in Figure 5 3 ), so the preferred load is most likely closer to 20 grams. Figure 5 3 50 g Vickers indent with cracks propagated to particle edge on lactose PAGE 75 75 For sucrose, 50 gram loads created decent crack lengths. Unfortunately, not eno ugh indents created suitable cracks and the lengths measured are highly variable. It is challenging to reproduce an indent, but it is nearly impossible to recreate crack propagation. Therefore, many indents within specified loads (which need to be determined for each material) must be done to obtain suitable results. Future Work T he contin uation of the research presented previously includes three main expansion paths: (1) employ the current milling model equations to develop useful industrial software (2) use the developed mill modeling approach on another mill size or type, and (3) relate micromill breakage measure to material properties Each of these three directions ha s great potential in advancing the understanding of breakage behavior and applying this knowledge would assist and advance size reduction in industry. Developing Millin g Model Software The three level milling model developed in previous chapters predicts the product size distribution from a given feed size distribution, mill operating conditions (grinding pressure, pusher pressure, and feed rate), and simple material cha racterization measurements (either microindentation hardness or micromill breakage measure). However, in most industrial applications involving size reduction the product size distribution desired is generally known. In this case, a milling model that pr edict s the mill operating conditions required to obtain a given product size distribution would be most beneficial. The milling model equations described could be used to obtain optimal mill operating conditions for a desired product. The optimal mill o perating conditions PAGE 76 76 (grinding pressure, pusher pressure, and feed rate) could be determined by minimizing the performance measure: ( 5 2 ) Here, and are the vol ume weighted mean diameter of the desired and modeled product distribution and are the desired and modeled mass fraction s of particles one bin away from bin i, which is the bin containing the mean diame ter of t he desired distribution, and are the desired and modeled mass fraction s of particles two bins away from bin i, and so on. a re weighting coefficients that could be set to reward or punish product particles in certain bins in order to create narrow or broad distributions. For example, if a narrow size distribution is desired, all the would be selected and the coefficients could be set such that material two bins away from the mean is more heavily penalized compared to material only one bin outside the desired bin ( ). The developed software could also be used to determine whether or not the mill is capable of producing a desired product distribution Applying the Mill Modeling Approach to New Mills In Chapter 3 milling models were developed to relate breakage functions for a self classifying spiral jet mill to mill operating variables. This procedure could be applied to any mill type that is self classifying or equip ped with a particle size classifier that returns oversize material If the system is not self classifying, a revised population balance model would need to be determined for the system used. A different mill would require different mill dependent paramet ers to be found by running extensive PAGE 77 77 experiments with inexpensive excipient powders. However, once this is completed, small quantities of material can be used to determine material dependent parameters for high value products (see Chapter 4 ). Two possib le research studies to test the developed approach on new mills include: using spiral jet mills of varying scales and testing the closely related loop mill. Studies with multiple scales of pancake spiral jet mills may lead to relationships between mill de pendent parameters and the mill geometry, gas nozzle sizes, and nozzle positions. If these relations are obtainable, the mill dependent parameters could potentially be determined for other spiral jet mills from simple mill design knowledge. Using the mi lling model approach developed in this work on different mill types would be more difficult, since different mill operating conditions affect the breakage functions in different mills. For example, consider a continuous milling system containing a ball mi ll outfitted with an external particle size classifier which reintroduces particles over a given cut size back to the ball mill This system has no gas flows or pressure controls. Here, the mill operating variables most likely include: the rotational spe ed of the mill, the media size and material, and material feed rate. However, the breakage function models could be developed by relating the mill energy per particle to the mill operating variables of this system similar to the approach used to model the spiral jet mill breakage functions. However, this may not be straight forward. For example, in the ball mill system at higher rotational speeds particles are acted on by a compressive force rather than an impact force. However, the micromill is an impa ct testing device; therefore, the same device may not accurately describe powder dependence in compressive mills PAGE 78 78 Relating Breakage Measure to Material Properties The constructed single impact micromill could be applied to m any powders to determine breakag e measures which correlate extremely well with powder dependent parameters, a 0 and w size However, this correlation is not as good with microindentation hardness. Since hardness is probably the most varied material property for the three powders tested in this study, hardness alone has a strong correlation to these powder dependent parameters However, for other materials with more extreme material property differences, such as tougher crystalline materials, different shaped particles, and non crystalli ne materials, this correlation may not be successful. Therefore, a 0 and w size could be related to more material properties: ( 5 3 ) where H is microindentation hardness, E is elastic modulus, K c is fracture toughness, AR is aspect ratio, and is material density. It is probably more reliable to use nanoindentation methods to determine hardness, elastic modulus, and fracture toughness in the future, as prefaced in Chapter 2. However, specime n preparation may need to be improved in order to use nanoindentation techniques. Of course, o ther material properties could a ffect the breakage measure, but these are expected to be the most important. If obtainable, these advanced correlations would ext end the development a nd applicability of the milling model. PAGE 79 79 APPENDIX A JET MILL RUN FORMS PAGE 80 80 PAGE 81 81 APPENDIX B MILLING MODEL SOFTWA RE USING MICROMILL BREA KAGE MEASURE %% FINAL MILLING MODEL %% clear all %% INPUTS %% %% OPERATING CONDITIONS %% GP=input( 'Grinding Pressure (psig)' ); PP=input( 'Pusher Pressure (psig)' ); FR=input( 'Feed Rate (g/s)' ); SIZE = [186 132 93 66 47]; %vector of volume weighted mean diameters of bins %% MICROMILL PROPERTIES %% %Baking Soda %xbarF = 124.5; %xbarP = 75.67; %x_50F = 13 0.5; %x_50P = 74.67; %AR = 0.69; %Lactose Monohydrate %xbarF = 135.7; %xbarP = 97.37; %x_50F = 140.1; %x_50P = 101.2; %AR = 0.73; %Sucrose %xbarF = 124.3; %xbarP = 56.17; %x_50F = 136.9; %x_50P = 48.95; %AR = 0.79; %Microcrystalline Cellulose %xbarF = 138.9; %xbarP = 134.3; %x_50F = 146.8; %x_50P = 140.7; %AR = %Moon Simulant %xbarF = 151.7; %xbarP = 145.0; %x_50F = 153.9; %x_50P = 147.3; %AR = %% W_SIZE FUNCTION MODEL %% BM = (xbarF xbarP)/xbarF; s_0 = 2.9; s_1 = 1.09E 1; s_2 = 5.92E 2; s_3 = 9. 93; PAGE 82 82 s_4 = 1.14E 1*BM 2.70E 2; s_5 = 8.11E 4; %% A_0 FUNCTION MODEL %% k_0 = 3.38*BM+1.89E 1; k_1 = 3.72E 2; k_2 = 1.41E 2; k_3 = 3.33; %% K FUNCTION MODEL %% %% K CALCULATIONS %% k = 1/2*cos(k_0+k_1*GP+k_2*PP+k_3*FR)+1/2; %% SIGMA FUNCTION MODEL %% %% SIGMA CALCULATIONS %% [m,n]=size(SIZE); for i = 1:n num=(s_0+s_1*GP+s_2*PP+s_3*FR+s_4*SIZE(i)+s_5*GP*PP); SIGMA(i)= logsig(num); end SIGMA(n+1)=0; SIGMA(n+2)=k; SIGMA=SIGMA' PAGE 83 83 APPENDIX C MILLING MODEL SOFTWA RE USING MICROINDENT ATION HARDNESS %% FINAL MILLING MODEL (with Hardness) %% clear all %% INPUTS %% %% OPERATING CONDITIONS %% GP=input( 'Grinding Pressure (psig)' ); PP=input( 'Pusher Pressure (psig)' ); FR=input( 'Feed Rate (g/s)' ); SIZE = [186 132 93 66 47]; %vector of volume weighted mean diameters of bins %% MICROMILL PROPERTIES %% %Baking Soda %H = 0. 89 ; %Lactose Monohydrate %H = 1.056 ; %Sucrose %H = 0. 76 ; %Microcrystalline Cellulose %H = %Moon Simulant %H = %% W_SIZE FUNCTION MODEL %% s_0 = 2.9; s_1 = 1.09E 1; s_2 = 5.92E 2; s_3 = 9.93; s_4 = 1.02E 1*H+1.11E 1; %linear trendline s_5 = 8.11E 4; %% A_0 FUNCTION MODEL %% k_0 = 2.93*H+4.21; k_1 = 3.72E 2; k_2 = 1.41E 2; k_3 = 3.33; %% K FUNCTION MODEL %% %% K CALCULATIONS %% k = 1/2*cos(k_0+k_1*GP+k_2*PP+k_3*FR)+1/2; %% SIGMA FUNCTION MODEL %% %% SIGMA CALCULATIONS %% [m,n]=size(SIZE); for i = 1:n num=(s_0+s_1*GP+s_2*PP+s_3*FR+s_4*SIZE(i)+s_5*GP*PP); SIGMA(i)= logsig(num); PAGE 84 84 end SIGMA(n+1)=0; SIGMA(n+2)=k; SIGMA=SIGMA' PAGE 85 85 LIST OF REFERENCES [1] T. Yokoyama Y. I noue, Selection of Fine Grinding Mills. In: A.D. Salman, M. Ghadiri, M.J. Hounslow, First ed. Particle Breakage. Amsterdam, The Netherlands: Elsevier B.V. ( 2007 ) 487 508. [2] M.J. Rhodes Introduction to particle technology. Second ed. Chichester, England ; Hoboken, NJ: Wiley ( 2008 ) [3] L. Vogel, W. Peukert, Characterisation of grinding relevant particle properties by inverting a population balance model, Particle & Particle Systems Characterization. 19 (3) (2002) 149 157. [4] C. Knieke, M. Sommer, W. Peu kert Identifying the apparent and true grinding limit Powd er Technology. 195 (1) (2009) 25 30. [5] J.A. CAaD. Air Jet Milling. In: Salman A.D. GM, and Hounslow M.J., ed. Particle Breakage First Edition ed. Amsterdam, The Netherlands: Elsevier B.V. ( 2007 ) 421 35. [6] K. Wozniak, A. Wallisch Air jet Milling of Grinding Materials. Washington D.C.: National Aeronautics and Space Administration ( 1983 ) [7] A. Bentham, C. Kwan, R. Boerefijn, A. Ghadir i, Fluidised bed jet milling of pharmaceutical powders Pow d er Technology. 141 (3) (2004) 233 8. [8] B. Epstein, Logarithmico normal distribution in breakage of solids, industrial and engineering chemistry. 40 (12) (1948) 2289 2291. [9] S. Broadbent, T. Callcott, A matrix analysis of processes involving particle assemblies, Philosophical Transactions of the Royal Society of London Series a Mathematical and Physical Sciences. 249 (960) (1956) 99 &. [10] P.C. Kapur, Kinetics of batch grinding part B: an approximate solution to the grinding equation, Society of Min ing Engineers. 247 (1970) 309 13. [11] L. Austin, K. Shoji, V. Bhatia, V. Jindal, K. Savage, Some results on the description of size reduction as a rate process in various mills. Ind Eng Chem, Process Des Dev. 15 (1) (1976) 187 196. [12] E. Bilgili, B. Sca rlett Population balance modeling of non linear effects in milling processes Powder Tec hnology. 153 (1) (2005) 59 71. [13] A. Devaswithin B. Pitchumani A. Desilva, Modified back calculation method to predict particle size distributions for batch grindi ng in a ball mill, Industrial & Engineering Ch emistry Research. 27 (4) (1988) 723 6. PAGE 86 86 [14] H. Berthiaux, C. Varinot, J. Dodds, Approximate calculation of breakage parameters from batch grinding tests, Chemical Engineering Science. 51 (19) (1996) 4509 4516. [15] P. Toneva, W. Peukert Modelling of Mills and Milling Circuits. In: A.D. Salman M. Ghadiri M.J. Hounslow, First ed. Particle Breakage Amsterdam, The Netherlands: Elsevier B.V. ( 2007 ) 873 911. [16] H. Gommeren, D. Heitzmann, H. Kramer, K. Heiskanen, B. Scarlett, Dynamic modeling of a closed loop jet mill, International Journal of Mineral Processing. 44 5 (1996) 497 506. [17] E. Bilgili, B. Scarlett Nonlinear effects in particulate processes Nonlinear Analysis Theory Methods & Applications. 63 (5 7) (2005) E1131 E41. [18] P. Kis, C. Mihalyko, B. Lakatos, Discrete model for analysis and design of grinding mill classifier systems, Chemical Engineering and Processing. 45 (5) (2006) 340 349. [19 ] P Nair, Breakage parameters and the operating variables of a circular fluid energy mill Part I. Breakage distribution parameter, Powder Technology. 106 (1 2) (1999) 45 53. [20 ] P. Nair, Breakage parameters and the operating variables of a circular fluid energy mill Part II. Breakage rate parameter, Powder Technol ogy. 106 (1 2) (1999) 54 61. [21 ] R. Tuunila, L. Nystrom, Technical note effects of grinding parameters on product fineness in jet mill grinding, Minerals Engineering. 11 (11) (1998) 1089 1094. [22 ] N. Midoux, P. Hosek, L. Pailleres, J. Authelin Microniza tion of pharmaceutical substances in a spiral jet mill Pow der Technology. 104 (2) (1999) 113 20. [23] D. Eskin, S. Voropayev, O. Vasilkov Simulation of jet milling Pow der Technology. 105 (1 3) (1999) 257 65. [24 ] H. Gommeren, D. Heitzmann, J. Moolenaar, B. Scarlett Modelling and control of a jet mill plant Powd er Technology. 108 (2 3) (2000) 147 54. [25 ] T. Han, H. Kalman, A. L evy DEM simulation of particle comminution in jet milling Particulate Science an d Technology. 20 (4) (2002) 325 40. [26 ] S. T eng, P. Wang, L. Zhu, M. Young, C. Gogos, Experimental and numerical analysis of a lab scale fluid energy mill, Powder Technology. 195 (1) (2009) 31 39. [27 ] A. Levy, H. Kalman Numerical study of particle motion in jet milling Particulate Sc ience and Tec hnology. 25 (2) (2007) 197 204. PAGE 87 87 [28] D.J. Green An introduction to the mechanical properties of ceramics. Cambridge ; New York: Cambridge University Press 1998. [29] Y. Huang, F. Zhang, K. Hwang, W. Nix, G. Pharr, G. Feng A model of size effects in nano indentation Journal of the Mechanics and Physics of Solids. 54 (8) (2006) 1668 86. [30] W.C. Duncan Hewitt, G.C. Weatherly Evaluating the Fracture Toughness of Sucrose Crystals Using Microindentation Techniques Pharmaceutical Research. 6 (5) (1989) 373 8. [31] D. Marshall, T. Noma, A. Evans, A simple method for determining elastic modulus to hardness ratios using knoop indentation measurements, Journal of the American Ceramic Society. 65 (10) (1982) C175 C176. [32] J.P. Singh, M. Sutaria, M. Ferber, Use of indentation technique to measure elastic modulus of plasma sprayed zirconia thermal barrier coating, 21st Annual Conference & Exposition on Composites, Advanced Ceramics, Material, and Structure; Cocoa Beach, Florida, USA; 1997; American Ceramic Socie ty. [33] W. Oliver, G. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, Journal of Materials Research. 7 (6) (1992) 1564 1583. [34] T.Y. Tsui, W.C. Oliver, G.M. Pharr In denter geometry effects on the measurement of mechanical properties by nanoin dentation with sharp indenters; 1996 [35] L. Taylor, D. Papadopoulos, P. Dunn, A. Bentham, J. Mitchell, M. Snowden Mechanical characterisation of powders using nanoindentation Powd er Technology. 143 (2004) 179 85. [36] E. Tsitrou, S. Northeast, R. van Noort Brittleness index of machinable dental materials and its relation to the marginal chipping factor Journ al of Dentistry. 35(12) (2007) 897 902. [37] L. Taylor, D. Papadopoul os, P. Dunn, A. Bentham, N. Dawson, J. Mitchell et al, Predictive milling of pharmaceutical materials using nan oindentation of single crystals, Organic P rocess Research & Development. 8 (4) (2004) 674 9. [38] S. Zugner, K. Marquardt, I. Zimmermann, Influe nce of nanomechanical crystal properties on the comminution process of particulate solids in spiral jet mills, European Journal of Pharmaceutics and Biopharmaceutics. 62 (2) (2006) 194 201. [39] V. Masterson, X. Cao, Evaluating particle hardness of pharmac eutical solids using AFM nanoindentation, International Journal of Pharmaceutics. 362 (1 2) (2008) 163 171. PAGE 88 88 [40] O. de Vegt, H. Vromans, J. den Toonder, K. Maarschalk, Influence of flaws and crystal properties on particle fracture in a jet mill, Powder Tec hnology. 191 (1 2) (2009) 72 77. [41] H.J. Yount Hardness and Fracture Toughness of Heat Treated Advanced Ceramic Material for use as Fuel Coating and Inert Matrix Materials in Advanced Reactors: University of Wisconsin Madison; 2006. [42] K. Prasad, D. S heen, J. Sherwood Fracture property studies of paracetamol single crystals using microindentation techniques Pharmac eutical Research. 18 (6) (2001) 867 72. [43] B. Hancock, S. Clas, K. Christensen Micro scale measurement of the mechanical properties of compressed pharmaceutical powders. 1: The elasticity and fracture behavior of microcrystalline cellulose International Journal of Pharmaceutics. 209 (1 2) (2000) 27 35. [44] B. Hancock, C. D alton, S. Clas Micro scale measurement of the mechanical propert ies of compressed pharmaceutical powders. 2: The dynamic moduli of microcrystalline cellulose International Journal of Pharmace utics. 228 (1 2) (2001) 139 45. [45] M. Meier, E. John, D. Wieckhusen, W. Wirth, W. Peuker, Characterization of the grinding beh aviour in a single particle impact device: Studies on pharmaceutical powders, European Journal of Pharmaceutical Sciences. 34 (1) (2008) 45 55. [46] D. Papadopoulos, M. Ghadiri Impact breakage of poly methylmethacrylate (PMMA) extrudates .1. Chipping mech anism Advanc ed Powder Technology. 7 (3) (1996) 183 97. [47] M. Ghadiri, Z. Zhang Impact attrition of particulate solids. Part 1: A theoretical model of chipping Chemical Eng ineering Science. 57 (17) (2002) 3659 69. [48] Z. Zhang, M. Ghadiri Impact attr ition of particulate so lids. Part 2: Experimental work, Chemical Eng ineering Science. 57 (17) (2002) 3671 86. [49] R. Moreno, M. Ghadiri, S. Antony Effect of the impact angle on the breakage of agglomerates: a numerical study using DEM Powd er Technology. 130 (1 3) (2003) 132 7. [50] A. Samimi, R. Moreno, M. Ghadiri Analysis of impact damage of agglomerates: effect of impact angle Powder Technology. 143 (2004) 97 109. [51] W. Peukert Material properties in fine grinding International Journal of Mineral Processing. 74 (2004) S3 S17. PAGE 89 89 [52] L. Vogel, W. Peukert, Breadage behaviour of different materials construction of mastercurve for the breakage probability, Powder Technology. 129 (1 3) (2003) 101 110. [53] L. Vogel, W. Peukert, From single particle imp act behaviour to modelling of impact mills, Chemical Engineering Science. 60 (18) (2005) 5164 5176. [54] W. Yang, C. Kwan, Y. Ding, M. Ghadiri, K. Roberts Milling of sucrose Powd er Technology. 174 (1 2) (2007) 14 7. [55] T. Han, Y. Petukhov, A. Levy, H. Kalman Theoretical and experimental study of multi impact breakage of particles Advanced Powder Technology. 17 (2) (2006) 135 57. [56] H. Kalman, M. Hubert, E. Grant, Y. Petukhov, M. Haim Fatigue behavior of impact comminution and attrition units Powd e r Technology. 146 (1 2) (2004) 1 9. [57] P. Toneva, W. Peukert A general approach for the characterization of fragmentation problems, Advanced Powder Technology 18 (1) (2007) 39 51. [58] M. Meier, E. John, D. Wieckhusen, W. Wirth, W. Peukert, Influence o f mechanical properties on impact fracture: Prediction of the milling behaviour of pharmaceutical powders by nanoindentation, Powder Technology. 188 (3) (2009) 301 313. [59] M. Meier, E. John, D. Wieckhusen, W. Wirth, W. Peukert, Generally applicable break age functions derived from single particle comminution data, Powder Technology. 194 (1 2) (2009) 33 41. [60] F. Muller, R. Polke, G. Schadel, Spiral jet mills: Hold up and scale up, International Journal of Mineral Processing. 44 5 (1996) 315 326. [61] G.M .H. Meesters, Particle strength in an industrial environment, Elsevier, 2007. [62] P. Nair, Material characteristics and the breakage parameters in a circular fluid energy mill, Advanced Powder Technology. 10 (1) (1999) 21 36. [63] O. de Vegt, H. Vromans, F. Faassen, K. Maarschalk, Milling of organic solids in a jet mill. Part 1: Determination of the selection function and related mechanical material properties, Particle & Particle Systems Characterization. 22 (2) (2005) 133 140. PAGE 90 90 BIOGRAPHICAL SKETCH Derek Robert Starkey was born in Mesa, Arizona in 1986. He graduated second in his class from Waterloo High School in Waterloo, IL in 2004. Subsequently, he joined the Department of Energy, Environmental, and Chemical Engineering at Washington University in S t. Louis (St. Louis, MO). He received his Bachelors of Science in Chemical Engineering in 2008. In fall of 2008, he joined the Department of Chemical Engineering PhD program at the University of Florida (Gainesville, FL), where he started working project. Derek received his Ph.D. from the University of Florida in the spring of 2012. |