21 solid solid operations and processing

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Published on March 31, 2014

Author: nguyennha1211

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21-1 Section 21 Solid-Solid Operations and Processing Bryan J. Ennis, Ph.D. President, E&G Associates, Inc., and CEO, iPowder Systems, Inc.; Co-Founder and Member, Particle Technology Forum, American Institute of Chemical Engi- neers; Member, American Association of Pharmaceutical Scientists (Section Editor, Bulk Flow Characterization, Solids Handling, Size Enlargement) Wolfgang Witt, Dr. rer. nat. Technical Director, Sympatec GmbH–System Partikel Tech- nik; Member, ISO Committee TC24/SC4, DIN, VDI Gesellschaft für Verfahrenstechnik und Chemieingenierwesen Fachausschuss “Partikelmesstechnik” (Germany) (Particle-Size Analysis) Ralf Weinekötter, Dr. sc. techn. Managing Director, Gericke AG, Switzerland; Mem- ber, DECHEMA (Solids Mixing) Douglas Sphar, Ph.D. Research Associate, DuPont Central Research and Development (Size Reduction) Erik Gommeran, Dr. sc. techn. Research Associate, DuPont Central Research and Development (Size Reduction) Richard H. Snow, Ph.D. Engineering Advisor, IIT Research Institute (retired); Fellow, American Institute of Chemical Engineers; Member, American Chemical Society, Sigma Xi (Size Reduction) Terry Allen, Ph.D. Senior Research Associate (retired), DuPont Central Research and Development (Particle-Size Analysis) Grantges J. Raymus, M.E., M.S. President, Raymus Associates, Inc.; Manager of Pack- aging Engineering (retired), Union Carbide Corporation; Registered Professional Engineer (California); Member, Institute of Packaging Professionals, ASME (Solids Handling) James D. Litster, Ph.D. Professor, Department of Chemical Engineering, University of Queensland; Member, Institution of Chemical Engineers (Australia) (Size Enlargement) PARTICLE-SIZE ANALYSIS Particle Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8 Specification for Particulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8 Particle Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8 Particle-Size Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8 Model Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9 Average Particle Sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9 Specific Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9 Particle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10 Equivalent Projection Area of a Circle . . . . . . . . . . . . . . . . . . . . . . . . 21-10 Feret’s Diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10 Sphericity, Aspect Ratio, and Convexity . . . . . . . . . . . . . . . . . . . . . . . 21-10 Fractal Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10 Sampling and Sample Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-11 Wet Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12 Dry Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12 Particle-Size Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12 Laser Diffraction Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12 Image Analysis Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-13 Dynamic Light Scattering Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . 21-14 Acoustic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-14 Single-Particle Light Interaction Methods . . . . . . . . . . . . . . . . . . . . . 21-15 Small-Angle X-Ray Scattering Method . . . . . . . . . . . . . . . . . . . . . . . . 21-15 Focused-Beam Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-15 Electrical Sensing Zone Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-16 Gravitational Sedimentation Methods. . . . . . . . . . . . . . . . . . . . . . . . . 21-16 Sedimentation Balance Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-17 Centrifugal Sedimentation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 21-17 Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

Sieving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-18 Elutriation Methods and Classification . . . . . . . . . . . . . . . . . . . . . . . . 21-18 Differential Electrical Mobility Analysis (DMA) . . . . . . . . . . . . . . . . 21-18 Surface Area Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-18 Particle-Size Analysis in the Process Environment . . . . . . . . . . . . . . . . 21-18 At-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19 On-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19 In-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19 Reference Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19 SOLIDS HANDLING: BULK SOLIDS FLOW CHARACTERIZATION An Introduction to Bulk Powder Behavior . . . . . . . . . . . . . . . . . . . . . . . 21-20 Permeability and Aeration Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-20 Permeability and Deaeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-20 Classifications of Fluidization Behavior. . . . . . . . . . . . . . . . . . . . . . . . 21-22 Classifications of Conveying Behavior. . . . . . . . . . . . . . . . . . . . . . . . . 21-22 Bulk Flow Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-23 Shear Cell Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-23 Yield Behavior of Powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-25 Powder Yield Loci. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-27 Flow Functions and Flowability Indices . . . . . . . . . . . . . . . . . . . . . . . 21-28 Shear Cell Standards and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 21-29 Stresses in Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-29 Mass Discharge Rates for Coarse Solids . . . . . . . . . . . . . . . . . . . . . . . 21-30 Extensions to Mass Discharge Relations . . . . . . . . . . . . . . . . . . . . . . . 21-31 Other Methods of Flow Characterization . . . . . . . . . . . . . . . . . . . . . . 21-31 SOLIDS MIXING Principles of Solids Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-33 Industrial Relevance of Solids Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 21-33 Mixing Mechanisms: Dispersive and Convective Mixing . . . . . . . . . . 21-33 Segregation in Solids and Demixing . . . . . . . . . . . . . . . . . . . . . . . . . . 21-34 Transport Segregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-34 Mixture Quality: The Statistical Definition of Homogeneity . . . . . . . 21-34 Ideal Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-36 Measuring the Degree of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-37 On-line Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-38 Sampling Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-38 Equipment for Mixing of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-38 Mixed Stockpiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-38 Bunker and Silo Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-38 Rotating Mixers or Mixers with Rotating Component . . . . . . . . . . . . 21-39 Mixing by Feeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-40 Designing Solids Mixing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-42 Goal and Task Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-42 The Choice: Mixing with Batch or Continuous Mixers. . . . . . . . . . . . 21-42 Batch Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-43 Feeding and Weighing Equipment for a Batch Mixing Process. . . . . 21-44 Continuous Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-45 PRINCIPLES OF SIZE REDUCTION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-45 Industrial Uses of Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-45 Types of Grinding: Particle Fracture vs. Deagglomeration . . . . . . . . 21-45 Wet vs. Dry Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-46 Typical Grinding Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-46 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-46 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-46 Single-Particle Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-46 Energy Required and Scale-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-47 Energy Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-47 Fine Size Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-48 Breakage Modes and Grindability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-48 Grindability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-49 Operational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-50 Mill Wear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-50 Safety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-50 Temperature Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-51 Hygroscopicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-51 Dispersing Agents and Grinding Aids . . . . . . . . . . . . . . . . . . . . . . . . . 21-51 Cryogenic Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-51 Size Reduction Combined with Other Operations . . . . . . . . . . . . . . . . 21-51 Size Reduction Combined with Size Classification. . . . . . . . . . . . . . . 21-51 Size Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-52 Other Systems Involving Size Reduction. . . . . . . . . . . . . . . . . . . . . . . 21-52 Liberation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-52 MODELING AND SIMULATION OF GRINDING PROCESSES Modeling of Milling Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-52 Batch Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-53 Grinding Rate Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-53 Breakage Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-53 Solution of Batch-Mill Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-53 Continuous-Mill Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-53 Residence Time Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-53 Solution for Continuous Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-54 Closed-Circuit Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-54 Data on Behavior of Grinding Functions . . . . . . . . . . . . . . . . . . . . . . . 21-55 Grinding Rate Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-55 Scale-Up and Control of Grinding Circuits . . . . . . . . . . . . . . . . . . . . . . 21-55 Scale-up Based on Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-55 Parameters for Scale-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-55 CRUSHING AND GRINDING EQUIPMENT: DRY GRINDING—IMPACT AND ROLLER MILLS Jaw Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-56 Design and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-56 Comparison of Crushers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-57 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-57 Gyratory Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-57 Design and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-57 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-58 Control of Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-58 Impact Breakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-58 Hammer Crusher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-58 Cage Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-59 Prebreakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-59 Hammer Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-59 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-59 Roll Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-60 Roll Press . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-60 Roll Ring-Roller Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-60 Raymond Ring-Roller Mill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-60 Pan Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61 Design and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61 CRUSHING AND GRINDING EQUIPMENT: FLUID-ENERGY OR JET MILLS Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61 Spiral Jet Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61 Opposed Jet Mill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61 Other Jet Mill Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-62 CRUSHING AND GRINDING EQUIPMENT: WET/DRY GRINDING—MEDIA MILLS Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-62 Media Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-62 Tumbling Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-63 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-63 Multicompartmented Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-63 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-64 Material and Ball Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-64 Dry vs. Wet Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-64 Dry Ball Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-64 Wet Ball Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-64 Mill Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65 Capacity and Power Consumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65 Stirred Media Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65 Attritors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65 Vertical Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65 Horizontal Media Mills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65 Annular Gap Mills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-66 Manufacturers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-66 Performance of Bead Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-66 Residence Time Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-66 Vibratory Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-66 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-67 Residence Time Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-67 21-2 SOLID-SOLID OPERATIONS AND PROCESSING

Hicom Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-67 Planetary Ball Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-67 Disk Attrition Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-67 Dispersers and Emulsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68 Media Mills and Roll Mills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68 Dispersion and Colloid Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68 Pressure Homogenizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68 Microfluidizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68 CRUSHING AND GRINDING PRACTICE Cereals and Other Vegetable Products . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68 Flour and Feed Meal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68 Soybeans, Soybean Cake, and Other Pressed Cakes . . . . . . . . . . . . . 21-68 Starch and Other Flours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-69 Ores and Minerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-69 Metalliferous Ores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-69 Types of Milling Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-69 Nonmetallic Minerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-69 Clays and Kaolins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-69 Talc and Soapstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Carbonates and Sulfates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Silica and Feldspar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Asbestos and Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Refractories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Crushed Stone and Aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Fertilizers and Phosphates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Fertilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Phosphates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70 Cement, Lime, and Gypsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Portland Cement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Dry-Process Cement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Wet-Process Cement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Finish-Grinding of Cement Clinker . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Lime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Gypsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Coal, Coke, and Other Carbon Products . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Bituminous Coal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Anthracite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-71 Coke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Other Carbon Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Chemicals, Pigments, and Soaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Colors and Pigments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Chemicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Soaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Gums and Resins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Rubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Molding Powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Powder Coatings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Processing Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72 Pharmaceutical Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-73 Biological Materials—Cell Disruption . . . . . . . . . . . . . . . . . . . . . . . . . . 21-73 PRINCIPLES OF SIZE ENLARGEMENT Scope and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-73 Mechanics of Size-Enlargement Processes . . . . . . . . . . . . . . . . . . . . . . 21-74 Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-74 Compaction Microlevel Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-76 Process vs. Formulation Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-77 Key Historical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-80 Product Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-80 Size and Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-80 Porosity and Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-81 Strength of Agglomerates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-81 Strength Testing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-81 Flow Property Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82 Redispersion Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82 Physiochemical Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82 AGGLOMERATION RATE PROCESSES AND MECHANICS Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82 Mechanics of the Wetting Rate Process . . . . . . . . . . . . . . . . . . . . . . . 21-83 Methods of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-83 Examples of the Impact of Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-86 Regimes of Nucleation and Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . 21-86 Growth and Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89 Granule Deformability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89 Types of Granule Growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-90 Deformability and Interparticle Forces. . . . . . . . . . . . . . . . . . . . . . . . 21-92 Deformability and Wet Mass Rheology. . . . . . . . . . . . . . . . . . . . . . . . 21-93 Low Agitation Intensity—Low Deformability Growth. . . . . . . . . . . . 21-95 High Agitation Intensity Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-96 Determination of St* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98 Granule Consolidation and Densification . . . . . . . . . . . . . . . . . . . . . . 21-99 Breakage and Attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100 Fracture Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101 Fracture Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101 Mechanisms of Attrition and Breakage . . . . . . . . . . . . . . . . . . . . . . . . 21-102 Powder Compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103 Powder Feeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104 Compact Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105 Compact Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105 Compaction Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105 Stress Transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-106 Hiestand Tableting Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-107 Compaction Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-107 Controlling Powder Compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-108 Paste Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-108 Compaction in a Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-108 Drag-Induced Flow in Straight Channels . . . . . . . . . . . . . . . . . . . . . . 21-108 Paste Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-108 CONTROL AND DESIGN OF GRANULATION PROCESSES Engineering Approaches to Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-110 Scales of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-110 Scale: Granule Size and Primary Feed Particles . . . . . . . . . . . . . . . . 21-111 Scale: Granule Volume Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-112 Scale: Granulator Vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-113 Controlling Processing in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-113 Controlling Wetting in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-113 Controlling Growth and Consolidation in Practice. . . . . . . . . . . . . . . 21-117 Controlling Breakage in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-117 SIZE ENLARGEMENT EQUIPMENT AND PRACTICE Tumbling Granulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-118 Disc Granulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-118 Drum Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-119 Controlling Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . 21-120 Moisture Control in Tumbling Granulation . . . . . . . . . . . . . . . . . . . . 21-121 Granulator-Driers for Layering and Coating. . . . . . . . . . . . . . . . . . . . 21-122 Relative Merits of Disc vs. Drum Granulators . . . . . . . . . . . . . . . . . . 21-122 Scale-up and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-123 Mixer Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-123 Low-Speed Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-123 High-Speed Mixers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-123 Powder Flow Patterns and Scaling of Mixing . . . . . . . . . . . . . . . . . . . 21-125 Controlling Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . 21-126 Scale-up and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-128 Fluidized-Bed and Related Granulators . . . . . . . . . . . . . . . . . . . . . . . . . 21-130 Hydrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-130 Mass and Energy Balances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-130 Controlling Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . 21-130 Scale-up and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-133 Draft Tube Designs and Spouted Beds . . . . . . . . . . . . . . . . . . . . . . . . 21-133 Centrifugal Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-134 Centrifugal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-134 Particle Motion and Scale-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-134 Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-135 Spray Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-135 Spray Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-135 Prilling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-135 Flash Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-136 Pressure Compaction Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-136 Piston and Molding Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-137 Tableting Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-137 Roll Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-137 Pellet Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-139 Screw and Other Paste Extruders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-139 Thermal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-142 Sintering and Heat Hardening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-142 Drying and Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-143 SOLID-SOLID OPERATIONS AND PROCESSING 21-3

MODELING AND SIMULATION OF GRANULATION PROCESSES The Population Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-143 Modeling Individual Growth Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 21-144 Nucleation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-144 Layering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-144 Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-144 Attrition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-145 Solution of the Population Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-146 Effects of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-146 Analytical Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-146 Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-146 Simulation of Granulation Circuits with Recycle . . . . . . . . . . . . . . . . . . 21-147 21-4 SOLID-SOLID OPERATIONS AND PROCESSING Nomenclature and Units for Particle-Size Analysis U.S. customary Symbol Definition SI units units A Empirically determined constant — — a Distance from the scatterer to the m ft detector as Specific surface per mass unit m2 /g ft2 /s B Empirically determined constant — — C Empirically determined constant — — C BET number — — CPF Area concentration 1/cm2 1/in2 D Translational diffusion coefficient m2 /s ft2 /s Dm Concentration undersize — — e Elementary charge C C fi Frequency i Hz Hz g Acceleration due to gravity m/s2 ft/s2 I0 Illuminating intensity W/m2 fc i Index of size class — — I Measured sound intensity W/m2 W/ft2 I Measured sound intensity W/m2 W/ft2 I0 Illuminating intensity W/m2 fc Iθ Primary sound intensity W/m2 W/ft2 I(θ) Total scattered intensity W/m2 W/ft2 K Related extinction cross section Kn Knudsen number — — k Wave number — — kB Boltzmann constant J/K J/K k1, k2 Incident illumination vectors 1/m 1/ft L Loschmidt number 1/mol 1/mol l Mean path of gas molecules m ft Mk,r kth moment of dimension r m Refractive index — — n Real part of the refractive index — — n Number of classes — — na Amount of absorbed gas mol/g mol/lb nm Monolayer capacity mol/g mol/lb P Settled weight g lb p Number of elementary charges — — p Pressure Pa psi po Starting pressure Pa psi Q0(x) Cumulative number distribution — — Q1(x) Cumulative length distribution — — Q2(x) Cumulative area distribution — — Q3(x) Cumulative volume or mass distribution — — Q3,i Cumulative volume distribution till class i — — q Modulus of the scattering vector 1/m 1/ft q Scattering vector 1/m 1/ft ∆l Thickness of the suspension layer m in ∆Q3,i Normalized volume fraction in — — size class i ∆xi Width of size class i m in ε Extension of a particle ensemble in m in the direction of a camera Γ Decay rate 1/s 1/s η Hydrodynamic viscosity of the Pa s Poise dispersing liquid κ Imaginary part of refractive index — — U.S. customary Symbol Definition SI units units qr* (z) Logarithmic normal distribution — — qr* Logarithmic density distribution of — — dimension r q0(x) Number density distribution 1/m 1/in q1(x) Length density distribution 1/m 1/in q2(x) Area density distribution 1/m 1/in q3(x) Volume or mass density distribution 1/m 1/in q⎯ 3,i Volume density distribution of class i 1/m 1/in q⎯∗3,i Logarithmic volume density distrib- ution of class i 1/m 1/in r, ri Measurement radius m in s Dimensionless standard deviation — — s,si Surface radius of a centrifuge m in SV Volume specific surface m2 /m3 S1(θ), Dimensionless, complex functions — — S2(θ) describing the change and amplitude in the perpendicular and the parallel polarized light T Absolute temperature K K t Time s s u Settling velocity of particles m/s ft/s v1,v2 Particle velocities m/s ft/s W Weight undersize g lb xEQPC Particle size of the equivalent m in projection area of a circle x⎯ F Average Feret diameter m in xF,max Maximum Feret diameter m in xF,max 90 Feret diameter measured 90° to the m in maximum Feret diameter xF,min Minimum Feret diameter m in xi Size of class i m in x⎯ k,0 Arithmetic average particle size for a m in2 number distribution x⎯ k,r Average particle size m in xmin Minimum particle size m in xst Stokes diameter m in x⎯ 1,r Weighted average particle size m in x⎯ 1,2 Sauter diameter m in x50,r Mean size of dimension r m in z Integration variable m in Z(x) Electrical mobility of particle size x ρf Density of the liquid g/cm3 lb/in3 ρS Density of the particle g/cm3 lb/in3 θ Scattering angle rad deg σ Dimensionless wave number — — ω Radial velocity of an agglomerate rad/s rad/s ω Radial velocity of a centrifuge rad/s rad/s ψS Sphericity — — ψA Aspect ratio — — ψC Convexity — — C ᎏ Paиsиm C ᎏ Paиsиm Greek Symbols

SOLID-SOLID OPERATIONS AND PROCESSING 21-5 Nomenclature and Units for Solids Mixing Symbol Definition Units d Mixer diameter m D Axial coefficient of dispersion m/s2 EMix Mixing energy W g Gravitational acceleration m2 /s H Height of fluidized bed m L Mixer length m mp, mq Average particle weight of two components p kg and q in mixture M Coefficient of mixing m2 /s M Mass of a sample kg M Mass of a batch kg n Random sampling scope — n Rotational frequency Hz Ne Newton number — Ng Number of samples in basic whole — p Tracer component concentration in basic whole — pg Proportional mass volume of coarse ingredient — P Power W q 1− p — r Mixer radius m RSD Relative standard deviation — S Empirical standard deviation — S2 Random sample variance — t, t´ Time s tv Mean residence times s Symbol Definition Units tf, tm, te, ti Filling, mixing, discharging, and idle time s t* Mixing time — Tp Feed fluctuation period s v Axial velocity m/s VRR Variance reduction ratio — W{ } Probability — x Concentration of tracer component — xi Concentration in i sample Greek Symbols µ Mean concentration — ρ Density of solids kg/m3 ρbulk Bulk density kg/m3 ρs Density of solids kg/m3 σp, σq Standard deviation of particle weight for kg two ingredients in mix σ2 Variance — σ2 z Variance of a random mix — Φ(χ2 ) Cumulative function of χ2 Chi square distribution — χ2l , χ2u Chi square distribution variables. In a two-sided — confidence interval, l stands for lower and u for upper limit. ω Angular velocity l/s

21-6 SOLID-SOLID OPERATIONS AND PROCESSING Nomenclature and Units for Size Enlargement and Practice U.S. U.S. customary customary Symbol Definition SI units units Symbol Definition SI units units A Parameter in Eq. (21-108) k Coalescence rate constant 1/s 1/s A Apparent area of indentor contact cm2 in2 K Agglomerate deformability A Attrition rate cm3 /s in3 /s Kc Fracture toughness MPa·m1/2 MPa·m1/2 Ai Spouted-bed inlet orifice area cm2 in2 l Wear displacement of indentor cm in B Nucleation rate cm3 /s in3 /s L Roll loading dyn lbf Bf Fragmentation rate g/s lb/s (∆L/L)c Critical agglomerate deformation strain Bf Wear rate g/s lb/s Nt Granules per unit volume 1/cm3 1/ft3 c Crack length cm in n Feed droplet size cm in δc Effective increase in crack length due cm in n(v,t) Number frequency size distribution by 1/cm6 1/ft6 to process zone size volume c Unloaded shear strength of powder kg/cm2 psf Nc Critical drum or disc speed rev/s rev/s d Harmonic average granule diameter cm in P Applied load dyn lbf d Primary particle diameter cm in P Pressure in powder kg/cm2 psf d Impeller diameter cm in Q Maximum compressive force kg/cm2 psf d Roll press pocket depth cm in Q Granulator flow rate cm3 /s ft3 /s di Indentor diameter cm in rp Process zone radius cm in dp Average feed particle size cm in R Capillary radius cm in D Die diameter cm in S Volumetric spray rate cm3 /s ft3 /s D Disc or drum diameter cm in St Stokes number, Eq. (21-48) D Roll diameter cm in St* Critical Stokes number representing Dc Critical limit of granule size cm in energy required for rebound er Coefficient of restitution St0 Stokes number based on initial nuclei E Strain energy stored in particle J J diameter E* Reduced elastic modulus kg/cm2 psf t Time s s fc Unconfined yield stress of powder kg/cm2 psf u,v Granule volumes cm3 in3 g Acceleration due to gravity cm/s2 ft/s2 u0 Relative granule collisional velocity cm/s in/s Gc Critical strain energy release rate J/m2 J/m2 U Fluidization gas velocity cm/s ft/s F Indentation force dyn lbf Umf Minimum fluidization gas velocity cm/s ft/s F Roll separating force dyn lbf Ui Spouted-bed inlet gas velocity cm/s ft/s G Layering rate cm3 /s in3 /s V Volumetric wear rate cm3 /s in3 /s h Height of liquid capillary rise cm in V˙R Mixer swept volume ratio of impeller cm3 /s ft3 /s h Roll press gap distance cm in V Volume of granulator cm3 ft3 h Binder liquid layer thickness cm in w Weight fraction liquid hb Fluid-bed height cm in w Granule volume cm3 in3 ha Height of surface asperities cm in w* Critical average granule volume cm3 in3 he Maximum height of liquid capillary rise cm in W Roll width cm in H Individual bond strength dyn lbf x Granule or particle size cm in H Hardness of agglomerate or compact kg/cm2 psf y Liquid loading Y Calibration factor Greek Symbols β(u, v) Coalescence rate constant for collisions 1/s 1/s ∆ρ Relative fluid density with respect to g/cm3 between granules of volumes displaced gas or liquid u and v ρ Apparent agglomerate or granule density g/cm3 lb/ft3 ε Porosity of packed powder ρa Apparent agglomerate or granule density g/cm3 lb/ft3 εb Interagglomerate bed voidage ρb Bulk density g/cm3 lb/ft3 εg Intraagglomerate granule porosity ρg Apparent agglomerate or granule density g/cm3 lb/ft3 κ Compressibility of powder ρl Liquid density g/cm3 lb/ft3 φ Disc angle to horizontal deg deg ρs True skeletal solids density g/cm3 lb/ft3 φ Internal angle of friction deg deg σ0 Applied axial stress kg/cm2 psf φe Effective angle of friction deg deg σz Resulting axial stress in powder kg/cm2 psf φw Wall angle of friction deg deg σ Powder normal stress during shear kg/cm2 psf φw Roll friction angle deg deg σc Powder compaction normal stress kg/cm2 psf ϕ(η) Relative size distribution σf Fracture stress under three-point bend loading kg/cm2 psf γlv Liquid-vapor interfacial energy dyn/cm dyn/cm σT Granule tensile strength kg/cm2 psf γsl Solid-liquid interfacial energy dyn/cm dyn/cm σy Granule yield strength kg/cm2 psf γsv Solid-vapor interfacial energy dyn/cm dyn/cm τ Powder shear stress kg/cm2 psf µ Binder or fluid viscosity poise θ Contact angle deg deg µ Coefficient of internal friction ς Parameter in Eq. (21-108) ω Impeller rotational speed rad/s rad/s η Parameter in Eq. (21-108)

SOLID-SOLID OPERATIONS AND PROCESSING 21-7 Nomenclature and Units for Size Reduction and Size Enlargement U.S. U.S. customary customary Symbol Definition SI units units Symbol Definition SI units units A Coefficient in double Schumann qf Fine-fractiom mass flow rate g/s lb/s equation qo Feed mass flow rate g/s lb/s a Constant qp Mass flow rate of classifier product g/s lb/s ak,k Coefficient in mill equations qR Mass flow rate of classifier tailings g/s lb/s ak,n Coefficient in mill equations qR Recycle mass flow rate to a mill g/s lb/s B Matrix of breakage function R Recycle ∆Bk,u Breakage function R Reid solution b Constant r Dimensionless parameter in size- C Constant distribution equations Cs Impact-crushing resistance kWh/cm ft⋅lb/in S Rate function S−1 S−1 D Diffusivity m2 /s ft2 /s Sෆ Corrected rate function S−1 S−1 D Mill diameter m ft S′ Matrix of rate function Mg/kWh ton/(hp⋅h) Db Ball or rod diameter cm in SG(X) Grindability function S−1 S−1 Dmill Diameter of mill m ft Su Grinding-rate function d Differential s Parameter in size-distribution d Distance between rolls of crusher cm in equations E Work done in size reduction kWh hp⋅h s Peripheral speed of rolls cm/min in/min E Energy input to mill kW hp t Time s s Ei Bond work index kWh/Mg hp⋅h/ton u Settling velocity of particles cm/s ft/s Ei Work index of mill feed W Vector of differential size distribution E2 Net power input to laboratory mill kW hp of a stream erf Normal probability function wk Weight fraction retained on each F As subscript, referring to feed stream screen F Bonding force kg/kg lb/lb wu Weight fraction of upper-size particles g Acceleration due to gravity cm/s2 ft/s2 wt Material holdup in mill g lb I Unit matrix in mill equations X Particle size or sieve size cm in i Tensile strength of agglomerates kg/cm2 lb/in2 X′ Parameter in size-distribution cm in K Constant equations k Parameter in size-distribution equations cm in ∆Xi Particle-size interval cm in k As subscript, referring to size of Xi Midpoint of particle-size interval ∆Xi cm in particles in mill and classifier X0 Constant, for classifier design parameters Xf Feed-particle size cm in L As subscript, referring to discharge Xm Mean size of increment in size- cm in from a mill or classifier distribution equations L Length of rolls cm in Xp Product-particle size cm in L Inside length of tumbling mill m ft Xp Size of coarser feed to mill cm in M Mill matrix in mill equations X25 Particle size corresponding to 25 percent cm in m Dimensionless parameter in size- classifier-selectivity value distribution equations X50 Particle size corresponding to 50 percent cm in N Mean-coordination number classifier-selectivity value Nc Critical speed of mill r/min r/min X75 Particle size corresponding to 75 percent cm in ∆N Incremental number of particles in size- classifier-selectivity value distribution equation ∆Xk Difference between opening of cm in n Dimensionless parameter in size- successive screens distribution equations x Weight fraction of liquid n Constant, general Y Cumulative fraction by weight undersize nr Percent critical speed of mill in size-distribution equations O As subscript, referring to inlet stream Y Cumulative fraction by weight undersize P As subscript, referring to product or oversize in classifier equations stream ∆Y Fraction of particles between two sieve Pk Fraction of particles coarser than a given sizes sieve opening ∆Y Incremental weight of particles in size- g lb p Number of short-time intervals in mill distribution equations equations ∆Yci Cumulative size-distribution intervals cm in Q Capacity of roll crusher cm3 /min ft3 /min of coarse fractions q Total mass throughput of a mill g/s lb/s ∆Yfi Cumulative size-distribution intervals cm in qc Coarse-fraction mass flow rate g/s lb/s of fine fractions qF Mass flow rate of fresh material to mill g/s lb/s Z Matrix of exponentials Greek Symbols β Sharpness index of a classifier ρᐉ Density of liquid g/cm3 lb/in3 δ Angle of contact rad 0 ρs Density of solid g/cm3 lb/in3 ε Volume fraction of void space Σ Summation Ζ Residence time in the mill s s σ Standard deviation ηx Size-selectivity parameter σ Surface tension N/cm dyn/cm µ Viscosity of fluid N⋅S/m2 P υ Volumetric abundance ratio of ρf Density of fluid g/cm3 lb/in3 gangue to mineral

GENERAL REFERENCES: Allen, Particle Size Measurement, 4th ed., Chapman and Hall, 1990. Bart and Sun, Particle Size Analysis Review, Anal. Chem., 57, 151R (1985). Miller and Lines, Critical Reviews in Analytical Chemistry, 20(2), 75–116 (1988). Herdan, Small Particles Statistics, Butterworths, London. Orr and DalleValle, Fine Particle Measurement, 2d ed., Macmillan, New York, 1960. Kaye, Direct Characterization of Fine Particles, Wiley, New York, 1981. Van de Hulst, Light Scattering by Small Particles, Wiley, New York, 1957. K. Leschon- ski, Representation and Evaluation of Particle Size Analysis Data, Part. Part. Syst. Charact., 1, 89–95 (1984). Terence Allen, Particle Size Measurement, 5th ed., Vol. 1, Springer, 1996. Karl Sommer, Sampling of Powders and Bulk Mate- rials, Springer, 1986. M. Alderliesten, Mean Particle Diameters, Part I: Evalua- tion of Definition Systems, Part. Part. Syst. Charact., 7, 233–241 (1990); Part II: Standardization of Nomenclature, Part. Part. Syst. Charact., 8, 237–241 (1991); Part III: An Empirical Evaluation of Integration and Summation Methods for Estimating Mean Particle Diameters from Histogram Data, Part. Part. Syst. Charact., 19, 373–386 (2002); Part IV: Empirical Selection of the Proper Type of Mean Particle Diameter Describing a Product or Material Property, Part. Part. Syst. Charact., 21, 179–196 (2004); Part V: Theoretical Derivation of the Proper Type of Mean Particle Diameter Describing a Product or Process Prop- erty, Part. Part. Syst. Charact., 22, 233–245 (2005). ISO 9276, Representation of Results of Particle Size Analysis. H. C. van de Hulst, Light Scattering by Small Particles, Structure of Matter Series, Dover, 1981. Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley-Interscience, new edition. Bruce J. Berne and Robert Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics, unabridged edition, Dover, 2000. J. R. Allegra and S. A. Hawley, Attenuation of Sound in Suspensions and Emulsions: Theory and Experiment, J. Acoust. Soc. America 51, 1545–1564 (1972). PARTICLE SIZE Specification for Particulates The behavior of dispersed mat- ter is generally described by a large number of parameters, e.g., the powder’s bulk density, flowability, and degree of aggregation or agglomeration. Each parameter might be important for a specific application. In solids processes such as comminution, classification, agglomeration, mixing, crystallization, or polymerization, or in related material handling steps, particle size plays an important role. Often it is the dominant quality factor for the suitability of a specific product in the desired application. Particle Size As particles are extended three-dimensional objects, only a perfect spherical particle allows for a simple definition of the particle size x, as the diameter of the sphere. In practice, spher- ical particles are very rare. So usually equivalent diameters are used, representing the diameter of a sphere that behaves as the real (nonspherical) particle in a specific sizing experiment. Unfortunately, the measured size now depends on the method used for sizing. So one can only expect identical results for the particle size if either the par- ticles are spherical or similar sizing methods are employed that mea- sure the same equivalent diameter. In most applications more than one particle is observed. As each individual may have its own particle size, methods for data reduction have been introduced. These include the particle-size distribution, a variety of model distributions, and moments (or averages) of the dis- tribution. One should also note that these methods can be extended to other particle attributes. Examples include pore size, porosity, surface area, color, and electrostatic charge distributions, to name but a few. Particle-Size Distribution A particle-size distribution (PSD) can be displayed as a table or a diagram. In the simplest case, one can divide the range of measured particle sizes into size intervals and sort the particles into the corresponding size class, as displayed in Table 21-1 (shown for the case of volume fractions). Typically the fractions ∆Qr,i in the different size classes i are summed and normalized to 100 percent, resulting in the cumulative distribution Q(x), also known as the percentage undersize. For a given particle size x, the Q value represents the percentage of the par- ticles finer than x. If the quantity measure is “number,” Q0(x) is called a cumulative number distribution. If it is length, area, volume, or mass, then the corresponding length [Q1(x)], area [Q2(x)], volume, or mass distributions are formed [Q3(x)]; mass and volume are related by the specific density ρ. The index r in this notation represents the quantity measure (ISO 9276, Representation of Results—Part 1 Graphical Representation). The choice of the quantity measured is of decisive importance for the appear- ance of the PSD, which changes significantly when the dimension r is changed. As, e.g., one 100-µm particle has the same volume as 1000 10- µm particles or 106 /1-µm particles, a number distribution is always dom- inated by and biased to the fine fractions of the sample while a volume distribution is dominated by and biased to the coarse. The normalization of the fraction ∆Qr,i to the size of the corre- sponding interval leads to the distribution density q⎯ r,i, or q⎯ r,i = and Α n i=1 ∆Qr,i = Α n i=1 q⎯ r,i ∆xi = 1 = 100% (21-1) If Qr(x) is differentiable, the distribution density function qr(x) can be calculated as the first derivative of Qr(x), or qr(x) = or Qr(xi) = ͵xi xmin qr(x) dx (21-2) It is helpful in the graphical representation to identify the distribu- tion type, as shown for the cumulative volume distribution Q3(x) and volume distribution density q3(x) in Fig. 21-1. If qr(x) displays one maximum only, the distribution is called a monomodal size distrib- ution. If the sample is composed of two or more different-size regimes, qr(x) shows two or more maxima and is called a bimodal or multimodal size distribution. PSDs are often plotted on a logarithmic abscissa (Fig. 21-2). While the Qr(x) values remain the same, care has to be taken for the transfor- mation of the distribution density qr(x), as the corresponding areas under the distribution density curve must remain constant (in particular dQr(x) ᎏ dx ∆Qr,i ᎏ ∆xi PARTICLE-SIZE ANALYSIS TABLE 21-1 Tabular Presentation of Particle-Size Data 1 2 3 4 5 6 7 xi, ∆xi, q– 3,i = ∆Q3,i/∆xi i µm ∆Q3,i µm 1/µm Q3,i q– * 3,i 0 0.063 0.0000 1 0.090 0.0010 0.027 0.0370 0.0010 0.0028 2 0.125 0.0009 0.035 0.0257 0.0019 0.0027 3 0.180 0.0016 0.055 0.0291 0.0035 0.0044 4 0.250 0.0025 0.070 0.0357 0.0060 0.0076 5 0.355 0.0050 0.105 0.0476 0.0110 0.0143 6 0.500 0.0110 0.145 0.0759 0.0220 0.0321 7 0.710 0.0180 0.210 0.0857 0.0400 0.0513 8 1.000 0.0370 0.290 0.1276 0.0770 0.1080 9 1.400 0.0610 0.400 0.1525 0.1380 0.1813 10 2.000 0.1020 0.600 0.1700 0.2400 0.2860 11 2.800 0.1600 0.800 0.2000 0.4000 0.4755 12 4.000 0.2100 1.200 0.1750 0.6100 0.5888 13 5.600 0.2400 1.600 0.1500 0.8500 0.7133 14 8.000 0.1250 2.400 0.0521 0.9750 0.3505 15 11.20 0.0240 3.200 0.0075 0.9990 0.0713 16 16.000 0.0010 4.800 0.0002 1.0000 0.0028 21-8

the total area remains 1, or 100 percent) independent of the transfor- mation of the abscissa. So the transformation has to be performed by q⎯*r (lnxi−1,lnxi) = (21-3) This equation also holds if the natural logarithm is replaced by the log- arithm to base 10. Example 1: From Table 21-1 one can calculate, e.g., q⎯ 3,11 = = = 0.2 µm−1 q⎯∗ 3,11 = q⎯∗ 3 (ln x10, ln x11) = = = = 0.4755 Model Distribution While a PSD with n intervals is represented by 2n + 1 numbers, further data reduction can be performed by fitting the size distribution to a specific mathematical model. The logarith- mic normal distribution or the logarithmic normal probability func- tion is one common model distribution used for the distribution density, and it is given by q∗ r (z) = e−0.5z2 with z = ln ΄ ΅ (21-4) x ᎏ x50,r 1 ᎏ s 1 ᎏ ͙2πෆ 0.16 ᎏ ln1.4 0.16 ᎏᎏᎏ ln(2.8 µm/2.0 µm) ∆Q3,11 ᎏᎏ ln(x11/x10) 0.16 ᎏ 0.8 µm ∆Q3,11 ᎏ ∆x11 ∆Qr,i ᎏᎏ ln(xi/xi−1) The PSD can then be expressed by two parameters, namely, the mean size x50,r and, e.g., by the dimensionless standard deviation s (ISO 9276, Part 5: Methods of Calculations Relating to Particle Size Analysis Using Logarithmic Normal Probability Distribution). The data reduction can be performed by plotting Qr(x) on logarithmic probability graph paper or using the fitting methods described in upcoming ISO 9276-3, Adjustment of an Experimental Curve to a Ref- erence Model. This method is mainly used for the analysis of powders obtained by grinding and crushing and has the advantage that the transformation between PSDs of different dimensions is simple. The transformation is also log-normal with the same slope s. Other model distributions used are the normal distribution (Laplace-Gauss), for powders obtained by precipitation, condensa- tion, or natural products (e.g., pollens); the Gates-Gaudin-Schuh- mann distribution (bilogarithmic), for analysis of the extreme values of fine particle distributions (Schuhmann, Am. Inst. Min. Metall. Pet. Eng., Tech. Paper 1189 Min. Tech., 1940); or the Rosin-Rammler- Sperling-Bennet distribution for the analysis of the extreme values of coarse particle distributions, e.g., in monitoring grinding operations [Rosin and Rammler, J. Inst. Fuel, 7, 29–36 (1933); Bennett, ibid., 10, 22–29 (1936)]. Moments Moments represent a PSD by a single value. With the help of moments, the average particle sizes, volume specific surfaces, and other mean values of the PSD can be calculated. The general definition of a moment is given by (ISO 9276, Part 2: Calculation of Average Particle Sizes/Diameters and Moments from Particle Size Distributions) Mk,r = ͵ xmax xmin xk qr(x) dx (21-5) where Mk,r is the kth moment of a qr(x) distribution density and k is the power of x. Average Particle Sizes A PSD has many average particle sizes. The general equation is given by x⎯ k,r = k ͙Mk,rෆ (21-6) Two typically employed average particle sizes are the arithmetic average particle size x⎯ k,0 = Mk,0 [e.g., for a number distribution (r = 0) obtained by counting methods], and the weighted average particle size x⎯ 1,r = M1,r [e.g., for a volume distribution (r = 3) obtained by sieve analysis], where x⎯ 1,r represents the center of gravity on the abscissa of the qr(x) distribution. Specific Surface The specific surface area can be calculated from size distribution data. For spherical particles this can simply be calculated by using moments. The volume specific surface is given by SV = or SV = = = 6⋅M−1,3 (21-7) where x⎯ 1,2 is the weighted average diameter of the area distribution, also known as Sauter mean diameter. It represents a particle having the same ratio of surface area to volume as the distribution, and it is also referred to as a surface-volume average diameter. The Sauter mean is an important average diameter used in solids handling and other processing applications where aspects of two-phase flow become important, as it appropriately weights the contributions of the fine fractions to surface area. For nonspherical particles, a shape fac- tor has to be considered. Example 2: The Sauter mean diameter and the volume weighted particle size and distribution given in Table 21-1 can be calculated by using FDIS-ISO 9276-2, Representation of Results of Particle Size Analysis—Part 2: Calculation of Average Particle Sizes/Diameters and Moments from Particle Size Distribu- tions via Table 21-2. The Sauter mean diameter is x⎯1,2 = M1,2 = = with M−1,3 = Α n i=1 ∆Q3,i ln(xi/xi−1) ᎏᎏ xi − xi−1 1 ᎏ M−1,3 M3,0 ᎏ M2,0 M2,0 ᎏ M3,0 6 ᎏ M1,2 6 ᎏ x⎯ 1,2 PARTICLE-SIZE ANALYSIS 21-9 FIG. 21-1 Histogram q⎯ 3(x) and Q3(x) plotted with linear abscissa. FIG. 21-2 Histogram q⎯∗ 3(x) and Q3(x) plotted with a logarithmic abscissa.

which yields x⎯ 1,2 = = 2.110882 The volume weighted average particle size is x⎯ 1,3 = M1,3 = Α n i=1 ∆Q3,i (xi + xi−1) which yields x⎯1,3 = ⋅7.280590 = 3.640295 PARTICLE SHAPE For many applications not only the particle size but also the shape are of importance; e.g., toner powders should be spherical while polishing powders should have sharp edges. Traditionally in microscopic meth- ods of size analysis, direct measurements are made on enlarged images of the particles by using a calibrated scale. While such mea- surements are always encouraged to gather a direct sense of the parti- cle shape and size, care should be taken in terms of drawing general conclusions from limited particle images. Furthermore, with the strong progress in computing power, instruments have become avail- able that acquire the projected area of many particles in short times, with a significant reduction in data manipulation times. Although a standardization of shape parameters is still in preparation (upcoming ISO 9276, Part 6: Descriptive and Qualitative Representation of Par- ticle Shape and

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