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Pharmaceutical Blending and Mixing

Pharmaceutical Blending and Mixing

Edited by

P.J. CullEn

School of Chemical Engineering,University of New South Wales,

Australia

Rodolfo J. RomañaCh

Department of Chemistry,University of Puerto Rico – Mayagüez,

Puerto Rico

niColas abatzoglou

Chemical Engineering and Biotechnological Engineering,Université de Sherbrooke, Canada

ChRis d. RiElly

Department of Chemical Engineering,Loughborough University, UK

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Contributor List xv

Preface xvii

Part I Fundamentals of Mixing 1

1 Mixing Theory 3Chris D. Rielly

1.1 Introduction 31.2 Describing Mixtures 51.3 Scale of Scrutiny 61.4 Quantifying Mixedness for Coarse and Fine‐Grained Mixtures 8

1.4.1 Coarse and Fine‐Grained Mixtures 81.4.2 Scale and Intensity of Segregation 9

1.5 Determining the End‐Point of Mixing: Comparison of Mixing Indices 151.6 Continuous Flow Mixers 19

1.6.1 Idealized Mixing Patterns 191.6.2 Residence Time Distributions 211.6.3 Back‐Mixing and Filtering of Disturbances Using a CSTR 23

References 24

2 Turbulent Mixing Fundamentals 27Suzanne M. Kresta

2.1 Introduction 272.2 The Velocity Field and Turbulence 282.3 Circulation and Macro‐Mixing 292.4 Fully Turbulent Limits and the Scaling of Turbulence 322.5 The Spectrum of Turbulent Length Scales, Injection of a Scalar

(Either Reagent or Additive) and the Macro‐, Meso‐ and Micro‐Scales of Mixing 34

2.6 Turbulence and Mixing of Solids, Liquids, and Gases 372.7 Specifying Mixing Requirements for a Process 382.8 Conclusions 39Notation 39Roman Characters 39Greek Characters 40References 40

Contents

vi Contents

3 Laminar Mixing Fundamentals 43P.J. Cullen and N.N. Misra

3.1 Laminar Flows 433.2 Mixing in Laminar Flows 44

3.2.1 Chaos and Laminar Chaotic Mixing 453.2.2 Granular Chaotic Mixing 50

3.3 Recent Advances 53References 54

4 Sampling and Determination of Adequacy of Mixing 57Rodolfo J. Romañach

4.1 Introduction, Process Understanding, and Regulations 574.2 Theory of Sampling 594.3 Sampling of Pharmaceutical Powder Blends 634.4 Stratified Sampling Approach 654.5 Testing 674.6 Process Knowledge/Process Analytical Technology 684.7 Real Time Spectroscopic Monitoring of Powder Blending 704.8 Looking Forward, Recommendations 734.9 Conclusion 744.10 Acknowledgments 75References 75

Part II Applications 79

5 Particles and Blending 81Reuben D. Domike and Charles L. Cooney

5.1 Introduction 815.2 Particle Geometry 82

5.2.1 Particle Size and Size Distribution 825.2.2 Particle Shape and Shape Distribution 83

5.3 Particle Interactions 845.3.1 van der Waals Forces 845.3.2 Electrostatic Forces 855.3.3 Adsorbed Liquid Layers and Liquid Bridges 855.3.4 Solid Bridges 865.3.5 Use of AFM to Measure Interparticle Forces 875.3.6 Interparticle Friction 89

5.4 Empirical Investigations of Particles and Blending 905.4.1 Blending of Powders 905.4.2 Impact of Particle Geometry on Blending 925.4.3 Impact of Interparticle Forces on Blending 935.4.4 Impact of Blender Conditions on Blending 95

Contents vii

5.5 Simulation Techniques 955.5.1 Full Physics Models Using Discrete Element Modeling 965.5.2 Continuum Models 975.5.3 Cellular Automata 98

References 98

6 Continuous Powder Mixing 101Juan G. Osorio, Aditya U. Vanarase, Rodolfo J. Romañach, and Fernando J. Muzzio

6.1 Introduction 1016.2 Overview 1026.3 Theoretical Characterization 107

6.3.1 Residence Time Distribution (RTD) Modeling 1076.3.2 Variance Reduction Ratio 108

6.4 Experimental Characterization 1086.4.1 Hold‐Up 1096.4.2 Residence Time Distribution (RTD) Measurements 1096.4.3 Mean Strain 110

6.5 Continuous Mixing Efficiency 1106.5.1 Variance Reduction Ratio 1106.5.2 Blend Homogeneity 111

6.6 Effects of Process Parameters on Mixing Behavior and Performance 1126.6.1 Hold‐Up 1136.6.2 RTD Measurements 113

6.7 Mixing Performance 1186.7.1 Modeling 1206.7.2 PAT, QbD, and Control 122

6.8 Conclusions and Continuing Efforts 124References 125

7 Dispersion of Fine Powders in Liquids: Particle Incorporation and Size Reduction 129Gül N. Özcan-Taşkın

7.1 Particle Incorporation into Liquids 1297.1.1 Wetting 1307.1.2 Stirred Tanks for Particle Incorporation 1327.1.3 In‐Line Devices Used for Particle Incorporation 140

7.2 Break Up of Fine Powder Clusters in Liquids 1437.2.1 Mechanisms of Break Up 1467.2.2 Process Devices for Deagglomeration\Size Reduction

of Agglomerates 147References 150

viii Contents

8 Wet Granulation and Mixing 153Karen P. Hapgood and Rachel M. Smith

8.1 Introduction 1538.2 Nucleation 154

8.2.1 Drop Penetration Time 1568.2.2 Dimensionless Spray Flux 1588.2.3 Nucleation Regime Map 160

8.3 Consolidation and Growth 1628.3.1 Granule Consolidation 1628.3.2 Granule Growth Behaviour 1648.3.3 Granule Growth Regime Map 165

8.4 Breakage 1678.4.1 Single Granule Strength and Deformation 1678.4.2 In‐Granulator Breakage Studies 1708.4.3 Aiding Controlled Granulation via Breakage 172

8.5 Endpoint Control 1748.5.1 Granulation Time 1758.5.2 Impeller Power Consumption 1768.5.3 Online Measurement of Granule Size 1768.5.4 NIR and Other Spectral Methods 177

References 178

9 Emulsions 183Andrzej W. Pacek

9.1 Introduction 1839.2 Properties of Emulsions 185

9.2.1 Morphology 1859.2.2 Volumetric Composition 1859.2.3 Drop Size Distributions and Average Drop Sizes 1869.2.4 Rheology 191

9.3 Emulsion Stability and Surface Forces 1959.3.1 Surface Forces 1959.3.2 Emulsion Stability 199

9.4 Principles of Emulsion Formation 2039.4.1 Low Energy Emulsification 2049.4.2 High Energy Emulsification 205

9.5 Emulsification Equipment 2169.5.1 Stirred Vessels 2169.5.2 Static Mixers 2189.5.3 High Shear Mixers 2199.5.4 High‐Pressure Homogenizers 2239.5.5 Ultrasonic Homogenizers 225

9.6 Concluding Remarks 226

Contents ix

Nomenclature 226Greek symbols 228References 228

10 Mixing of Pharmaceutical Solid‐Liquid Suspensions 233Mostafa Barigou and Frans L. Muller

10.1 Introduction 23310.1.1 Linking Solid‐Liquid Processing to Critical Quality Attributes 23310.1.2 Material Properties and Composition 23410.1.3 Impact of Blending and Homogenization 23410.1.4 Impact of Turbulence 23710.1.5 Impact of Heat Transfer 237

10.2 Scale‐Up of Operations Involving Solid Suspensions 23710.2.1 The Nature of Suspensions 23710.2.2 Scale‐Up and Scale‐Down Rules 23910.2.3 Identification of Agitator Duties 24010.2.4 Solid‐Liquid Unit Operations 242

10.3 General Principles of Solid‐Liquid Suspensions 24310.3.1 Rheological Behaviour of the Continuous Phase 24310.3.2 Rheology of Suspensions 24610.3.3 Terminal Velocity of Particles 24910.3.4 Turbulence 254

10.4 Solids Charging 25710.4.1 Charging to Batch Vessels 25710.4.2 Charging Difficult Powders 261

10.5 Solid Suspension 26110.5.1 States of Solid Suspension 26110.5.2 Prediction of Minimum Speed for Complete Suspension 262

10.6 Solid Distribution 26910.6.1 Agitator Speed 26910.6.2 Homogeneity 27010.6.3 Geometry 27110.6.4 Practical Guidelines 272

10.7 Blending in Solid‐Liquid Systems 27210.7.1 Mixing Time 27210.7.2 Viscoplastic Slurries Yield Stress and Cavern Formation 272

10.8 Mass Transfer 27510.9 Size Reduction, Deagglomeration and Attrition 277

10.9.1 Breaking Particles through Turbulent Forces 27710.9.2 Breaking Particles through Impact 278

Nomenclature 281Greek symbols 281Abbreviations 282References 282

x Contents

Part III Equipment 287

11 Powder Blending Equipment 289David S. Dickey

11.1 Introduction 28911.2 Blending Mechanisms 29011.3 Blend Time 29011.4 Fill Level 29111.5 Segregation 29111.6 Powder Processing Difficulties 29211.7 Blender Classification 292

11.7.1 Tumble Blenders 29311.7.2 Rotating Element Blenders 29811.7.3 Granulators 30311.7.4 Other Blenders – Mullers and Custom Blenders 304

11.8 Continuous Blenders 30511.9 Blender Selection 30611.10 Equipment Specifications 307

11.10.1 Materials of Construction 30911.10.2 Electrical Classification 30911.10.3 Drives and Seals 309

References 310

12 Fluid Mixing Equipment Design 311David S. Dickey

12.1 Introduction 31112.2 Equipment Description 312

12.2.1 Laboratory Mixers 31212.2.2 Development Mixers 31312.2.3 Portable Mixers 31312.2.4 Top-Entering Mixers 31512.2.5 High-Shear Dispersers 31812.2.6 High Viscosity Mixers 31912.2.7 Multi-Shaft Mixers 31912.2.8 Bottom-Entering Mixers 32012.2.9 Glass-Lined Mixers and Vessels 32112.2.10 Side-Entering Mixers 32212.2.11 Vessel Geometry 32212.2.12 Baffles 323

12.3 Measurements 32312.3.1 Power 32412.3.2 Torque 32612.3.3 Tip Speed 32712.3.4 Blend Time 327

12.4 Mixing Classifications 32812.4.1 Liquid Mixing 328

Contents xi

12.4.2 Solids Suspension 33012.4.3 Gas Dispersion 33212.4.4 Viscous Mixing 333

12.5 Mechanical Design 33412.5.1 Shaft Design 33412.5.2 Shaft Seals 33512.5.3 Materials of Construction 33612.5.4 Surface Finish 33712.5.5 Motors 33812.5.6 Drives 339

12.6 Static Mixers 33912.6.1 Twisted Element 33912.6.2 Structured Element 33912.6.3 Basic Design 340

12.7 Challenges and Troubleshooting 34112.7.1 Careful Observations 34112.7.2 Process Problems 341

Nomenclature 342Greek 343References 343

13 Scale‐Up 345David S. Dickey

13.1 Introduction 34513.2 Similarity and Scale‐Up Concepts 346

13.2.1 Dimensional Analysis 34613.2.2 Similarity 34713.2.3 Applied Scale‐Up 349

13.3 Testing Methods 35013.4 Observation and Measurement 35213.5 Scale‐Up Methods 354

13.5.1 Scale‐Up with Geometric Similarity 35413.5.2 Example of Geometric Similarity Scale‐Up 35813.5.3 Scale‐Up Without Geometric Similarity 35913.5.4 Example of Non‐Geometric Scale‐Up 36113.5.5 Scale‐Up for Powder Mixing 364

13.6 Summary 367Nomenclature 367Greek 368References 368

14 Equipment Qualification, Process and Cleaning Validation 369Ian Jones and Chris Smalley

14.1 Introduction 36914.2 Blending Equipment Commissioning and Qualification 370

14.2.1 Outline of the Verification Approach 370

xii Contents

14.2.2 Requirements Phase 37114.2.3 Specifications and Design Review Phase 37314.2.4 Verification Phase 375

14.3 Blending and Mixing Validation 38014.3.1 Why do You Need to Validate Pharmaceutical Blends/Mixes? 38214.3.2 When do You Need to Validate Blending/Mixing? 38414.3.3 Components of Blending/Mixing Validation 38514.3.4 What to Validate 386

14.4 Blending Cleaning Validation 38914.4.1 Cleaning Development Studies 38914.4.2 Cleaning Validation 395

14.5 Conclusion 39814.6 Acknowledgements 399References 399

Part IV Optimization and Control 401

15 Process Analytical Technology for Blending 403Nicolas Abatzoglou

15.1 Introduction 40315.1.1 The Role of PAT in Pharmaceutical Manufacturing:

Is PAT Really New? 40415.1.2 Why PAT is Feasible 40515.1.3 Where PAT can be Applied in Pharmaceutical Manufacturing 40615.1.4 The Regulatory Framework 406

15.2 Chemometrics and Data Management 40815.2.1 PAT Data Management and Interpretation 409

15.3 Near‐Infrared Spectroscopy (NIRS) 41215.4 Raman Spectroscopy (RS) 41915.5 Image Analysis 42215.6 LIF Spectroscopy 42415.7 Effusivity 42615.8 Other Potential Sensor Technologies 42615.9 Comments on PAT in Liquid Formulation Mixing 427References 427

16 Imaging Fluid Mixing 431Mi Wang

16.1 Introduction 43116.2 Point Measurement Techniques 43316.3 Photographic Imaging 43516.4 Digital Particle Image Velocimetry 43916.5 Magnetic Resonance Imaging 44316.6 Positron Emission Particle Tracking Imaging 44416.7 Electrical Process Tomography 446References 452

Contents xiii

17 Discrete Element Method (DEM) Simulation of Powder Mixing Process 459Ali Hassanpour and Mojtaba Ghadiri

17.1 Introduction to DEM and its Application in Pharmaceutical Powder Processing 459

17.2 DEM Simulation of Powder Mixing 46117.3 Validation and Comparison with the Experiments 46817.4 Concluding Remarks 474References 475

Index 479

Nicolas Abatzoglou Chemical Engineering and Biotechnological Engineering, Université de Sherbrooke, Canada

Mostafa Barigou School of Chemical Engineering, University of Birmingham, UK

Charles L. Cooney Department of Chemical Engineering, Massachusetts Institute of Technology, USA

P.J. Cullen School of Chemical Engineering, University of New South Wales, Australia

David S. Dickey MixTech, Inc., USA

Reuben D. Domike Center for Biomedical Innovation, Massachusetts Institute of Technology, USA; School of Business at the University of Prince Edward Island, Canada

Mojtaba Ghadiri Institute of Particle Science and Engineering, School of Chemical and Process Engineering, University of Leeds, UK

Karen P. Hapgood Monash Advanced Particle Engineering Laboratory, Department of Chemical Engineering, Monash University, Australia

Ali Hassanpour Institute of Particle Science and Engineering, School of Chemical and Process Engineering, University of Leeds, UK

Ian Jones Innopharmalabs, Ireland

Suzanne M. Kresta Department of Chemical and Materials Engineering, University of Alberta, Canada

N.N. Misra School of Food Science & Environmental Health, Dublin Institute of Technology, Ireland

Frans L. Muller AstraZeneca, Hursfield Industrial Estate, UK

Fernando J. Muzzio Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, USA

Juan G. Osorio Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, USA

Contributor List

xvi Contributor List

Gül N. Özcan‐Taşkın BHR Group (trading name for VirtualPiE Ltd), UK

Andrzej W. Pacek School of Chemical Engineering, University of Birmingham, UK

Chris D. Rielly Department of Chemical Engineering, Loughborough University, UK

Rodolfo J. Romañach Department of Chemistry, University of Puerto Rico, Puerto Rico

Chris Smalley Merck Sharp and Dohme, USA

Rachel M. Smith Chemical and Biological Engineering, Sheffield University, UK

Aditya U. Vanarase Bristol‐Myers Squibb Co., USA

Mi Wang Institute of Particle Science and Engineering, School of Chemical and Process Engineering, University of Leeds, UK

Pharmaceutical Blending and Mixing: Role, Challenges and Trends

Pharmaceutical Blending and Mixing provides a dedicated reference for one of the most critical and ubiquitous unit operations within the pharmaceutical industry. The text aims to cover the unique role and challenges of mixing associated with pharmaceutical manufac-ture. This book has been prepared as a source of scientific knowledge in support of international Quality by Design (QbD) initiatives which acknowledge that increased pro-cess understanding leads to a more reliable and risk free product for patients.

This book covers the underlying principles and mechanisms of mixing necessary for designing and optimising processing thereby assuring final product quality. Equipment design, control strategies and numerical techniques available to advance the scientific understanding of pharmaceutical mixing are detailed. Detailed discussions of the fundamentals of the field are completed by a discussion of several applications from powder blending to emulsions.

Opportunities for Process Analytical Technology (PAT) and imaging strategies to monitor and control the current state of the system are discussed in Chapters 15 and 16. However, this important topic is also discussed in Chapter 4 because (1) the advantages of continuous manufacturing include the opportunity for meaningful PAT and modelling tech-niques to be implemented for automated process control leading to finished products with improved quality and (2) PAT will be ineffective if process monitoring is biased or affects the process stream. Since PAT is the main tool of QbD, it is more than worth mentioning that there is a high activity in terms of both fundamental and applied research/endeavours and, consequently, there is a continuous flow of mixing/blending knowledge production. A large part of this knowledge becomes public through scientific papers, manuscripts and reports, but the production and actual existence of in‐house knowledge must not be ignored. Much of the latter is kept proprietary for at least some years while the rest, although not necessarily publicly available through peer‐reviewed material, could be uncovered in con-ferences specialized on PAT (i.e. IFPAC, ISPE, pan‐European PAT and QbD). This book’s ambition in this area is to present the development so far as well as the most commonly used and promising PAT tools in Pharmaceutical Blending/Mixing.

Pharmaceutical Blending and Mixing provides 17 chapters describing the current state of the field. We hope that this huge effort will be useful to QbD initiatives, and help to catalyse the incorporation of scientific and technological knowledge in the design, control and validation of pharmaceutical blending processes.

It is envisaged that the reference will complement general references on mixing such as the Handbook of Industrial Mixing: Science and Practice (Wiley, 2004).

P.J. Cullen, Rodolfo J. Romañach, Nicolas Abatzoglou and Chris RiellyMarch 2015

Preface

Part IFundamentals of Mixing

Pharmaceutical Blending and Mixing, First Edition. Edited by P.J. Cullen, Rodolfo J. Romañach, Nicolas Abatzoglou and Chris D. Rielly. © 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.

1.1 Introduction

Mixing of ingredients, or dispersion of one phase in another, is an essential step in many pharmaceuticals processes. For example, the vast majority of manufacturing routes to form an active pharmaceutical ingredient (API) make use of crystallization, which involves a number of mixing steps in a liquid phase, such as: dispersion and dissolution of solid reagents into a solvent, blending of liquid reagents with the solvent phase, creation of super‐saturation through mixing, for example with an anti‐solvent addition, chemical reac-tion, or heat removal and suspension of the API crystals during subsequent growth (Kirwan & Orella, 2002; Paul et al., 2004). Each of these operations involves a mixing step, which is aimed at removing gradients of concentration, temperature or solids mass fraction within the crystallizer vessel, to give a more uniform environment for chemical reaction and/or crystal growth.

A second example may be taken from later in a pharmaceutical manufacturing process: during the formulation of solid dosage forms, dry‐powder mixing of an API with excipients (themselves mixtures of binders, diluents, flow modifiers and granulating agents) is required to produce suitable physical, flow and mechanical properties for tableting (for example Lee, 2002). Here, the objective is to remove concentration differences within the dry powder mix, so that each tablet contains a mixture with exactly the same properties and with a tightly‐controlled amount of the API. Other forms of oral dosage may involve the

Mixing Theory

Chris D. RiellyDepartment of Chemical Engineering, Loughborough University, UK

1

4 Pharmaceutical Blending and Mixing

blending of suspensions, emulsions and syrups to give a formulated liquid product; again the objective of mixing is to ensure that each dosage contains almost exactly the same amount of the active ingredient.

These examples demonstrate that in a mixing process the objective is to reduce inhomo-geneities in composition to an acceptable level, to provide a more uniform processing environment and/or a more uniform product. The examples also illustrate that there are dif-ferences between fluid mixtures of miscible phases and particle mixtures, which can, in principle, unmix; for example, by segregation effects (Sommier et al., 2001). Segregation often occurs in free‐flowing powders and is driven by differences in particle size and density. The phenomenon occurs when particulate mixtures are shaken (Rosato et al., 1987), or dur-ing flow within or between vessels (e.g. discharge from a vessel). During shaking or shear flow, there is relative motion between particles and small particles can fall into gaps beneath larger particles. Thus, the larger particles tend to rise to the surface, whereas small particles percolate downwards. Therefore, segregation can cause a previously well‐mixed material to undergo unmixing into a non‐uniform solid form; a way to counteract the tendency to seg-regate is to introduce a binder or adjust the moisture content to produce cohesion within the particulate mixture. In many processes a granulation operation follows the blending stage to prevent segregation in subsequent processing steps (Fung & Ng, 2003).

A distinction may also be drawn between batch and continuous flow mixing processes, although similar measures of mixing quality may be defined for both. Almost all current pharmaceutical processes operate by transferring batches of material between stages of the manufacturing process, rather than by continuous inflow and outflow to process equip-ment. Therefore this chapter will focus mainly on batch mixing processes, where the purpose is to use fluid mechanics, molecular diffusion and dispersion effects to produce spatially homogeneous mixtures; up to a point, an increase in the batch time will lead to an improvement in the mixture quality, that is a reduction in the level of spatial inhomogene-ities, but thereafter, the degree of mixedness will not improve. The chapter will address the question of what is an ‘acceptable’ measure of mixedness; the idea of a scale of scrutiny of the mixture will be introduced in Section 1.3 and various measures of the quality of a mixture will be discussed. The examples given here consider two rather different situations of mixing (1) between components in a liquid and (2) between different types of solid particles. In this context it is useful to differentiate between fine and coarse‐grained mixtures and this is discussed in Section 1.4. Selection of different definitions of the end‐point for a mixing process will be considered in Section 1.5, to consider their sensitivity at various stages of mixing and their sensitivity to sampling methods.

Recently the pharmaceuticals industries have paid increasing attention to continuous manu-facturing operations, as potentially they could significantly reduce production costs and pro-vide more reliable manufacturing routes; see, for example, Schaber et al. (2011). Therefore, the final section (Section 1.6) of this chapter will consider continuous mixing of ingredients. In such operations the mixing objective is to obtain a product with a homogeneous distribution of ingredients in the correct proportions, which requires careful metering of the feed flow rates, as well as achieving a high degree of homogeneity. In continuous flow devices, the output product composition should not vary in time and the processing history of each element of the mixture should be the same. Variations in the feed composition to a continuous flow mixer can be compensated to an extent by allowing ‘mixing in time’, that is not all elements of fluid spend the same amount in the mixer, allowing materials that have arrived early, to mix with

Mixing Theory 5

materials that have arrived late. Thus the concept of a residence time distribution will be intro-duced in Section 1.6 to describe the process of back‐mixing, or mixing in time. Furthermore it will be shown that back‐mixing can effectively filter out higher frequency variations in feed composition and still give a uniform product. Thus, there are processing advantages and disad-vantages in having some width to the residence time distribution.

Throughout this chapter, the term concentration will be used quite generally to described the composition of a material within a mixture; for a single liquid phase the term can be inter-preted as mass (or mole) fraction, or mass (or moles) per unit volume of a specific compo-nent; for particulate mixtures it could represent mass fraction, number fraction or volume fraction of one type of solid; for a multi‐phase mixture it could be the volume or mass fraction of a specific phase. In general, the mixedness will be judged from a statistical measure of the distribution of concentrations of key components within samples drawn from a mixture.

1.2 Describing Mixtures

In practice, the whole of the composition of a mixture cannot be determined at a single time, so sampling is often used to assess the state of mixedness; sampling at an appropriate scale of scrutiny will be discussed in Section 1.3, but first the degree of uniformity between samples will be considered. The average concentration of a species in the whole mixture is determined by the amounts of all components added and can be calculated straightfor-wardly from a mass balance. The average species concentrations obtained from samples drawn from this mixture ought to have values distributed about the average for the whole mixture; it is the width of this distribution that provides information about the quality of the mixture, not the average value from the various samples.

Figure 1.1 shows an example of an idealized mixture comprising 50% white particles and 50% black particles. The whole mixture is divided into 36 samples, each containing 16 parti-cles. Figure 1.1(a) is a homogeneous, but non‐random mixture; each sample contains exactly eight white particles (or 50% white particles), which is exactly the same as the mean concentration of the mixture. Figure 1.1(b) shows the number of particles in each sample and indicates that there are no spatial differences in concentration; hence the mixture can be regarded as perfectly mixed. This mixture is ‘perfect’ in the sense that each sample contains exactly the same concentration as the whole mixture average; in other words there is no vari-ance between the samples. The probability of forming such a mixture by a stochastic process is rather small, so this situation is very unlikely to occur in a conventional mixing process.

In contrast, Figure 1.1(c) shows a mixture that has been generated entirely randomly by giving each particle an equal probability of being black or white; the overall composition of the whole mixture is still 50% white particles, but each sample now shows deviations from the whole mixture mean, as shown in Figure 1.1(d). Some samples contain as few as four particles, whereas others have 12 or 13, compared to the expected eight, which might lead to the conclusion that the material is not well mixed. However, further mixing, or randomization, of the particles will not lead to any significant improvement in the distri-bution of white particles between the samples. Figure 1.1(c) represents a more realistic picture of a perfectly mixed material, yet it is highly likely that a given sample concentration will show a large difference from the mean value, particularly when the number of parti-cles in the sample is small.

6 Pharmaceutical Blending and Mixing

A simple definition of ‘complete mixing’ could be defined as the state where there are equal concentrations of components in each sample, which is the same as in the mixture overall. However, this example shows that statistical variations between sam-ples in a fully random mixture leads to the conclusion that such a simple definition is of no practical use. Therefore, the principle applied later in this chapter to define the ‘well‐mixed’ state will make use of a comparison back to the best state that can be achieved by random distribution processes, for example a mixture of the sort shown in Figure 1.1(c). Essentially, this will be what is regarded as ‘well‐mixed’ since any further mixing would yield no statistical improvement in the mixture quality. Thus any descrip-tion of the quality of a mixture must be able to distinguish between the sample‐to‐sample variations that can occur for a fully randomized mixture and those that result from incomplete mixing.

1.3 Scale of Scrutiny

The previous section described how sampling is required to assess the variability of the concentrations in a mixture, which begs the question, ‘What is an appropriate size for each  sample?’ The end use for a mixed product determines the quality of mixing that

(a)

(b)

(c)

(d)

Figure  1.1 Idealized mixtures of 50% white and 50% black particles (a) non‐random perfect  mixture, (b) number of white particles in each 4 × 4 sample of the non‐random mixture (c) random mixture and (d) number of white particles in each 4 × 4 sample of the random mixture

Mixing Theory 7

will be required and this can only be established by viewing samples of the mixture at an appropriate scale of scrutiny. Danckwerts (1953a) defined this scale of scrutiny to be the ‘minimum size of regions of segregation which would cause the mixture to be regarded as unmixed’, Scrutinizing a mixture on the scale of a whole batch of formulated product is essentially useless: if the correct proportions of ingredients have been charged in the first place, then the whole mixture must have the required average composition. At the other extreme, scrutiny at the scale of a single particle will show a completely segregated mixture. What is required is to scrutinize a sample of the mixture at an appropriate scale, determined by the end use of the product. For example, a pharmaceutical product is designed to deliver a fixed amount of a key component, usually the API, in each unit dose taken by the patient. Thus, the scale of scrutiny could be the mass contained in one tablet of the product, which could lie between 10 mg and 5 g in typical human patient dosages (Berthiaux et al., 2008). In‐situ sampling of particulate mixtures from within a blender at this scale of scrutiny is not straightforward to achieve and involves removal of a represen-tative mass from within a flowing bed, for example using a thief probe. Muzzio et al. (2003) provide a detailed discussion of the requirements of various designs of thief probe and highlighted the difficulties in obtaining accurate composition data for their use in determining mixture quality. Thief probes cause a disruption to the powder mixture and there can be uneven flow of the different powder species into the probe; segregation of different components can occur as the mixture is sampled. These problems are com-pounded in continuous systems and with cohesive materials that do not flow easily in the sample cavity. Thus the issues with sampling are to obtain (1) an appropriate mass of the mixture, corresponding to the scale of scrutiny and (2) a representative sample, with the same composition as within the mixer.

Figure 1.2 illustrates the effect of changing the scale of scrutiny on the mixture quality in an idealized mixture. The left‐hand sample appears to be homogeneous and on this scale of scrutiny the mixture is completely mixed; there are no visual signs of concentration dif-ferences. Increasing the magnification at which the first sample is viewed shows up differ-ences in concentration, until at the highest magnification, the mixture appears to be completely segregated, since individual particles can be clearly identified. In all cases the sample contains the same proportions of the key component, since it is the same mixture. It appears that this mixture quality would be acceptable at the left‐hand scale of scrutiny, but completely unacceptable at the right‐hand scale of scrutiny.

Increasing magni�cation/decreasing scale of scrutiny

Figure  1.2 The effect of decreasing the scale of scrutiny on the perceived quality of the mixture

8 Pharmaceutical Blending and Mixing

As described previously, once the scale of scrutiny has been determined, then samples at this scale can be obtained from the mixture and assessed for their homogeneity using the statistical measures described later in Section 1.5. Thus, the scale of scrutiny determines the sample mass required for any off‐line analysis of product quality. For example, the FDA (2003) make recommendations about the analysis of samples drawn from blenders or from intermediate bulk containers; the guidelines state that sample sizes between 1 and 10 times the dosage unit should be investigated. Thus, the scale of scrutiny (that is the sample mass) is often taken as three times the dose mass.

In‐line assessment of the homogeneity of a blend using a Process Analytical Technology (PAT) instrument would require an assessment of the effective sampling mass to justify that the FDA requirements are satisfied. Pernenkil and Cooney (2006) provided an example of such as assessment for a NIR PAT assessment of two‐component powder blending, by esti-mating the sample size from the probe diameter and the measurement penetration depth. The mass of powder scanned per sample was estimated and compared to the mass in a single dose; hence the number of samples to be scanned and averaged to equate to a scale of scrutiny of 3 times the dose size could be calculated.

1.4 Quantifying Mixedness for Coarse and Fine‐Grained Mixtures

1.4.1 Coarse and Fine‐Grained Mixtures

The mixtures discussed in the previous section and shown schematically in the right‐hand images of Figure 1.2 are examples of coarse‐grained mixtures. When examined at these scales of scrutiny, a sample contains a relatively small (countable) number of dis-crete particles, which can be individually identified. Figure  1.3(a) shows a coarse‐grained mixture comprising light and dark particles; the material appears highly segregated and the composition changes abruptly from point to point, when moving from a light particle to a dark particle. In contrast Figure 1.3(b) shows a fine‐grained mixture. At the same scale of scrutiny, a sample contains such a large number of parti-cles that the material can be treated as a continuum. In this case the concentration varies smoothly from point to point and finite concentration gradients exist within the mixture. Fluids behave as fine‐grained mixtures, since each sample will contain a very large number of molecules and for practically useful scales of scrutiny the material can be regarded as a continuum. Mixtures of very finely‐divided powders may also be consid-ered to be fine‐grained, since each sample will contain a very large number of individual particles.

Fluid and coarse‐grained particulate mixtures differ in other respects. With the former, random motion of the molecules leads to diffusion, which causes a flux of material down concentration gradients to produce a more uniform mixture. However, molecular diffu-sion is a slow process and so this mixing mechanism is only effective at removing very small scale concentration gradients. In contrast, there is no such random motion for coarse particles and moreover, in practice small differences in diameter can lead to unmixing or segregation effects (see Section  1.1); small particles tend to percolate through the interstices created between larger particles, creating an unmixing effect based on size.

Mixing Theory 9

1.4.2 Scale and Intensity of Segregation

Danckwerts (1953a) proposed that two measures are required to quantify the mixedness, namely a length scale of segregation and an intensity of segregation. The former indicates the physical size of the unmixed regions in an imperfect mixture, whereas the latter repre-sents the degree to which there are variations in concentration between samples. Both mea-sures are affected by the selection of the sample size or scale of scrutiny (see Section 1.3), as will be illustrated with some examples. The top left image in Figure 1.4 shows a highly segregated mixture (coarse‐grained), in which individual regions of distinctly different concentration are visible: in qualitative terms, the length scale of segregation is large, because the regions of unmixed material have a significant size compared to the scale of scrutiny. Moving to the right in Figure 1.4, the size of the unmixed regions becomes smaller (the length scale of segregation decreases), although the mixture remains highly segregated between black and white areas. Moving downwards in Figure 1.4, the regions of unmixed material remain of the same size, but the concentration gradients are increasingly smeared out, that is the intensity of segregation decreases. The effect is created here by blurring the interface regions between black and white regions, in a process analogous to diffusion. The combination of decreases in intensity and length scale of segregation results in the mixture becoming increasingly more fine‐grained as it approaches the perfectly mixed (uniform concentration) state, as shown in the bottom right image of Figure 1.4. Here, individual particles are not visible and neither are concentration gradients. In other words, the mixture is well‐mixed.

Danckwerts (1952) provided quantitative definitions for the intensity and length scale of segregation based on measurements of the concentration fields. Consider an imperfect binary mixture of components A (white) and B (black), containing mass fractions a and b, respectively, at any point in the mixture. A mass fraction a 1 would represent pure com-ponent A at that point in the mixture, whereas a 0 would represent pure component B and for a binary components a b 1. The average mass fraction of A in the mixture would be given by

a n ai

n

i

1

1

(1.1)

(a) (b)

Figure 1.3 (a) a coarse‐grained mixture and (b) a fine‐grained mixture

10 Pharmaceutical Blending and Mixing

where ai is the concentration of A in sample i drawn from the mixture and n is the number of samples. The sample size should be less than or equal to the required scale of scrutiny, as discussed in Section 1.3. The mean composition a provides almost no useful information about the quality of the mixture, only that it contains the correct proportions of A and B. In contrast, the variance of the sample concentrations, ai, provides a useful statistic to charac-terize differences from the mean, a , and is defined by

22

1

1

1na ai

i

n

(1.2)

A perfect mixture might be thought to have a variance of zero, but as Figure 1.1 illus-trates, this is unlikely to happen in practice. Lacey (1943) showed that for a fully randomized binary mixture of the same‐sized particles, the variance is given by

rp

a a

n2 1 (1.3)

where np is the number of particles in each sample (determined by the required scale of scrutiny). Returning to the random mixture shown in Figure 1.1(c), the samples shown each contain 16 particles and the variance calculated from equation (1.2) is 2 0 018. , which compares well with the value of r

2 0 017. from equation (1.3). Thus further randomization of the mixture in Figure 1.1(c) would not result in an improvement in its uniformity and hence r

2 represents the lowest variance that can practically be achieved.

Decreasing length scale of segregation

Decreasing intensity of segregation

Figure 1.4 The effects of changing scale and intensity of segregation on the quality of the mixture