solid-solid operations and processing -...

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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 FlowCharacterization, 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 undChemieingenierwesen 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 andDevelopment (Size Reduction)

Richard H. Snow, Ph.D. Engineering Advisor, IIT Research Institute (retired); Member,American Chemical Society, Sigma Xi; Fellow, American Institute of Chemical Engineers (SizeReduction)

Terry Allen, Ph.D. Senior Research Associate (retired), DuPont Central Research andDevelopment (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 ofQueensland; Member, Institution of Chemical Engineers (Australia) (Size Enlargement)

PARTICLE-SIZE ANALYSISParticle Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8

Specification for Particulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8Particle Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8Particle-Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9Model Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9Average Particle Sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9Specific Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-9

Particle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10Equivalent Projection Area of a Circle . . . . . . . . . . . . . . . . . . . . . . . . 21-10Feret’s Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10Sphericity, Aspect Ratio, and Convexity . . . . . . . . . . . . . . . . . . . . . . . 21-10

Fractal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10Sampling and Sample Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-10Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-11

Wet Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12Dry Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12

Particle-Size Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12Laser Diffraction Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12Image Analysis Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-13Dynamic Light Scattering Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . 21-14Acoustic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-14Single-Particle Light Interaction Methods . . . . . . . . . . . . . . . . . . . . . 21-15Small-Angle X-Ray Scattering Method . . . . . . . . . . . . . . . . . . . . . . . . 21-15Focused-Beam Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-15Electrical Sensing Zone Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-16Gravitational Sedimentation Methods . . . . . . . . . . . . . . . . . . . . . . . . . 21-16

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Sedimentation Balance Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-17Centrifugal Sedimentation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 21-17Sieving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-18Elutriation Methods and Classification . . . . . . . . . . . . . . . . . . . . . . . . 21-18Differential Electrical Mobility Analysis (DMA) . . . . . . . . . . . . . . . . 21-18Surface Area Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-18

Particle-Size Analysis in the Process Environment . . . . . . . . . . . . . . . . 21-18At-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19On-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19In-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19

Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19Reference Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-19

SOLIDS HANDLING: BULK SOLIDS FLOW CHARACTERIZATIONAn Introduction to Bulk Powder Behavior . . . . . . . . . . . . . . . . . . . . . . . 21-20Permeability and Aeration Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-20

Permeability and Deaeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-20Classifications of Fluidization Behavior. . . . . . . . . . . . . . . . . . . . . . . . 21-22Classifications of Conveying Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 21-22

Bulk Flow Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-23Shear Cell Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-23Yield Behavior of Powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-25Powder Yield Loci. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-27Flow Functions and Flowability Indices . . . . . . . . . . . . . . . . . . . . . . . 21-28Shear Cell Standards and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 21-29Stresses in Cylinders Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-29Mass Discharge Rates for Coarse Solids . . . . . . . . . . . . . . . . . . . . . . . 21-30Extensions to Mass Discharge Relations . . . . . . . . . . . . . . . . . . . . . . . 21-31Other Methods of Flow Characterization . . . . . . . . . . . . . . . . . . . . . . 21-31

SOLIDS HANDLING: CONVEYING OF BULK SOLIDSConveyor Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-33

Conveyor Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-34Conveyor Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-34Auxiliary Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-34Control of Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-34

Screw Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-34Belt Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-37Bucket Elevators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-39

Spaced-Bucket Centrifugal-Discharge Elevators . . . . . . . . . . . . . . . . 21-42Spaced-Bucket Positive-Discharge Elevators . . . . . . . . . . . . . . . . . . . 21-42Continuous-Bucket Elevators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-43Supercapacity Continuous-Bucket Elevators . . . . . . . . . . . . . . . . . . . 21-43V-Bucket Elevator-Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-43Skip Hoists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-43

Vibrating or Oscillating Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-43Continuous-Flow Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-46

Closed-Belt Conveyor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-47Flight Conveyors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-48Apron Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-48

Pneumatic Conveyors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-48Types of Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-49Nomographs for Preliminary Design. . . . . . . . . . . . . . . . . . . . . . . . . . 21-56

SOLIDS HANDLING: STORAGE, FEEDING, AND WEIGHING OFBULK SOLIDS

Storage Piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-56Discharge Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-56Reclaiming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-56

Storage Bins, Silos, and Hoppers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-56Material-Flow Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-57Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-58Specifying Bulk Materials for Best Flow . . . . . . . . . . . . . . . . . . . . . . . 21-60

Flow-Assisted Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-61Weighing of Bulk Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-62

Batch Weighing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-62Continuous Weighing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-63Weight Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-64

SOLIDS MIXINGPrinciples of Solids Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-65

Industrial Relevance of Solids Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 21-65Mixing Mechanisms: Dispersive and Convective Mixing . . . . . . . . . . 21-65Segregation in Solids and Demixing . . . . . . . . . . . . . . . . . . . . . . . . . . 21-66Transport Segregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-66Mixture Quality: The Statistical Definition of Homogeneity . . . . . . . 21-66

Ideal Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-68Measuring the Degree of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-69On-line Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70Sampling Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70

Equipment for Mixing of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70Mixed Stockpiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70Bunker and Silo Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-70Rotating Mixers or Mixers with Rotating Component . . . . . . . . . . . . 21-71Mixing by Feeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-72

Designing Solids Mixing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-74Goal and Task Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-74The Choice: Mixing with Batch or Continuous Mixers. . . . . . . . . . . . 21-74Batch Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-75Feeding and Weighing Equipment for a Batch Mixing Process. . . . . 21-76Continuous Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-77

PRINCIPLES OF SIZE REDUCTIONIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-78

Industrial Uses of Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-78Types of Grinding: Particle Fracture vs. Deagglomeration . . . . . . . . 21-78Wet vs. Dry Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-78Typical Grinding Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-78

Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-79Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-79Single-Particle Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-79

Energy Required and Scale-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-80Energy Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-80Fine Size Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-81Breakage Modes and Grindability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-81Grindability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82

Operational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82Mill Wear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-82Temperature Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-83Hygroscopicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-83Dispersing Agents and Grinding Aids . . . . . . . . . . . . . . . . . . . . . . . . . 21-83Cryogenic Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-83

Size Reduction Combined with Other Operations . . . . . . . . . . . . . . . . 21-84Size Reduction Combined with Size Classification. . . . . . . . . . . . . . . 21-84Size Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-84Other Systems Involving Size Reduction. . . . . . . . . . . . . . . . . . . . . . . 21-84Liberation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-85

MODELING AND SIMULATION OF GRINDING PROCESSESModeling of Milling Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-85Batch Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-85

Grinding Rate Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-85Breakage Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-85Solution of Batch-Mill Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-85

Continuous-Mill Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-86Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-86Solution for Continuous Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-87

Closed-Circuit Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-87Data on Behavior of Grinding Functions . . . . . . . . . . . . . . . . . . . . . . . 21-87

Grinding Rate Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-88Scale-Up and Control of Grinding Circuits . . . . . . . . . . . . . . . . . . . . . . 21-88

Scale-up Based on Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-88Parameters for Scale-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-88

CRUSHING AND GRINDING EQUIPMENT: DRY GRINDING—IMPACT AND ROLLER MILLS

Jaw Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89Design and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89Comparison of Crushers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89

Gyratory Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89Design and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-89Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-90Control of Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-90

Impact Breakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-91Hammer Crusher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-91Cage Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-91Prebreakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-91

Hammer Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-92Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-92

Roll Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-92

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Roll Press . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-92Roll Ring-Roller Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-93

Raymond Ring-Roller Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-93Pan Crushers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-93

Design and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-93Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-93

CRUSHING AND GRINDING EQUIPMENT: FLUID-ENERGY OR JET MILLS

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-93Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-94

Spiral Jet Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-94Opposed Jet Mill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-94Other Jet Mill Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-94

CRUSHING AND GRINDING EQUIPMENT: WET/DRY GRINDING—MEDIA MILLS

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-94Media selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-94Tumbling Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-95

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-95Multicompartmented Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-96Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-96Material and Ball Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-96Dry vs. Wet Grinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-97Dry Ball Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-97Wet Ball Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-97

Mill Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-97Capacity and Power Consumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-97

Stirred Media Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98Attritors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98Vertical Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98Horizontal Media Mills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98Annular Gap Mills. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98Manufacturers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98

Performance of Bead Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-98Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-99

Vibratory Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-99Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100

Hicom Mill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100Planetary Ball Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100Disk Attrition Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100Dispersers and Emulsifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100

Media Mills and Roll Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100Dispersion and Colloid Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-100Pressure Homogenizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101Microfluidizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101

CRUSHING AND GRINDING PRACTICECereals and Other Vegetable Products . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101

Flour and Feed Meal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101Soybeans, Soybean Cake, and Other Pressed Cakes . . . . . . . . . . . . . 21-101Starch and Other Flours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101

Ores and Minerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101Metalliferous Ores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-101Types of Milling Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Nonmetallic Minerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Clays and Kaolins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Talc and Soapstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Carbonates and Sulfates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Silica and Feldspar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Asbestos and Mica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Refractories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-102Crushed Stone and Aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103

Fertilizers and Phosphates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103Fertilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103Phosphates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103

Cement, Lime, and Gypsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103Portland Cement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103Dry-Process Cement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103Wet-Process Cement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-103Finish-Grinding of Cement Clinker . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104Lime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104Gypsum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104

Coal, Coke, and Other Carbon Products . . . . . . . . . . . . . . . . . . . . . . . . 21-104Bituminous Coal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104Anthracite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104Coke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104Other Carbon Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104

Chemicals, Pigments, and Soaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104Colors and Pigments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-104Chemicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105Soaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105

Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105Gums and Resins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105Rubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105Molding Powders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105Powder Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105

Processing Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105Pharmaceutical Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-105Biological Materials—Cell Disruption . . . . . . . . . . . . . . . . . . . . . . . . . . 21-106

PRINCIPLES OF SIZE ENLARGEMENTScope and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-106Mechanics of Size-Enlargement Processes . . . . . . . . . . . . . . . . . . . . . . 21-109

Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-109Compaction Microlevel Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-109Process vs. Formulation Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-110Key Historical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-111

Product Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-114Size and Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-114Porosity and Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-114Strength Testing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-114Flow Property Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-115Redispersion Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-115Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-115Physiochemical Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-115

AGGLOMERATION RATE PROCESSES AND MECHANICSWetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-115

Mechanics of the Wetting Rate Process . . . . . . . . . . . . . . . . . . . . . . . 21-115Methods of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-116Examples of the Impact of Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-118Regimes of Nucleation and Wetting . . . . . . . . . . . . . . . . . . . . . . . . . . 21-119

Growth and Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-121Granule Deformability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-122Types of Granule Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-123Deformability and Interparticle Forces. . . . . . . . . . . . . . . . . . . . . . . . 21-124Deformability and Wet Mass Rheology . . . . . . . . . . . . . . . . . . . . . . . . 21-127Low Agitation Intensity—Low Deformability Growth. . . . . . . . . . . . 21-127High Agitation Intensity Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-130Determination of St* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-131Granule Consolidation and Densification . . . . . . . . . . . . . . . . . . . . . . 21-133

Breakage and Attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-134Fracture Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-135Fracture Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-135Mechanisms of Attrition and Breakage . . . . . . . . . . . . . . . . . . . . . . . . 21-136

Powder Compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-137Powder Feeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-137Compact Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-137Compact Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-138Compaction Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-139Stress Transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-139Example 6: Lubrication, Blending, and Compact Uniformity . . . . . . 21-141Hiestand Tableting Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-141Compaction Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-141Controlling Powder Compaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-141

Paste Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-142Compaction in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-143Drag-Induced Flow in Straight Channels . . . . . . . . . . . . . . . . . . . . . . 21-143Paste Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-143

CONTROL AND DESIGN OF GRANULATION PROCESSESEngineering Approaches To Design 21-144

Scales of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-144Scale: Granule Size and Primary Feed Particles . . . . . . . . . . . . . . . 21-145Scale: Granule Volume Element . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-146Scale: Granulator Vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-147

Controlling Processing in Practice 21-148Controlling Wetting in Practice 21-150Controlling Growth and Consolidation in Practice . . . . . . . . . . . . . . . 21-150Controlling Breakage in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-152

SOLID-SOLID OPERATIONS AND PROCESSING 21-3


SIZE ENLARGEMENT EQUIPMENT AND PRACTICETumbling Granulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-153

Disc Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-153Drum Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-154Controlling Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . 21-155Moisture Control in Tumbling Granulation . . . . . . . . . . . . . . . . . . . . 21-156Granulator-Driers for Layering and Coating. . . . . . . . . . . . . . . . . . . . 21-157Relative Merits of Disc vs. Drum Granulators . . . . . . . . . . . . . . . . . . 21-157Scale-up and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-157

Mixer Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-158Low-Speed Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-158High-Speed Mixers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-158Powder Flow Patterns and Scaling of Mixing . . . . . . . . . . . . . . . . . . . 21-160Controlling Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . 21-162Scale-up and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-163

Fluidized-Bed and Related Granulators . . . . . . . . . . . . . . . . . . . . . . . . . 21-165Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-165Mass and Energy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-166Controlling Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . 21-166Scale-up and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-167Draft Tube Designs and Spouted Beds . . . . . . . . . . . . . . . . . . . . . . . . 21-168

Centrifugal Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-168Centrifugal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-169Particle Motion and Scale-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-169Granulation Rate Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-169

Spray Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-169

Spray Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-169Prilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-170Flash Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-171

Pressure Compaction Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-171Piston and Molding Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-171Tableting Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-171Roll Presses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-173Pellet Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-174Screw and Other Paste Extruders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-174

Thermal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-176Sintering and Heat Hardening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-176Drying and Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-177

MODELING AND SIMULATION OF GRANULATION PROCESSESThe Population Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-178Modeling Individual Growth Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 21-179

Nucleation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-179Layering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-179Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-180Attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-181

Solution of the Population Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-181Effects of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-181Analytical Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-182Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-182

Simulation of Granulation Circuits with Recycle . . . . . . . . . . . . . . . . . . 21-182


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

detectoras Specific surface per mass unit m2/g ft2/sB Empirically determined constant — —C Empirically determined constant — —C BET number — —CPF Area concentration 1/cm2 1/in2

D Translational diffusion coefficient m2/s ft2/sDm Concentration undersize — —e Elementary charge C Cfi Frequency i Hz Hzg Acceleration by due to gravity m/s2 ft/s2

I0 Illuminating intensity W/m2 fci 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 fcIθ Primary sound intensity W/m2 W/ft2

I(θ) Total scattered intensity W/m2 W/ft2

K Related extinction cross sectionKn Knudsen number — —k Wave number — —kB Boltzmann constant J/K J/Kk1, k2 Incident illumination vectors 1/m 1/ftL Loschmidt number 1/mol 1/moll Mean path of gas molecules m ftMk,r kth moment of dimension r

U.S.customary


m Refractive index — —n Real part of the refractive index — —n Number of classes — —na Amount of absorbed gas mol/g mol/lbnm Monolayer capacity mol/g mol/lbP Settled weight g lbp Number of elementary charges — —p Pressure Pa psipo Starting pressure Pa psiQ0(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/ftq Scattering vector 1/m 1/ftqr* (z) Logarithmic normal distribution — —qr* Logarithmic density distribution of — —

dimension rq0(x) Number density distribution 1/m 1/inq1(x) Length density distribution 1/m 1/inq2(x) Area density distribution 1/m 1/inq3(x) Volume or mass density distribution 1/m 1/inq⎯3,i Volume density distribution of class i 1/m 1/inq⎯ ∗

3,i Logarithmic volume density distrib-ution of class i 1/m 1/in

r, ri Measurement radius m ins Dimensionless standard deviation — —s,si Surface radius of a centrifuge m in



Nomenclature and Units for Solids Mixing

Symbol Definition Units

d Mixer diameter mD Axial coefficient of dispersion m/s2

EMix Mixing energy Wg Gravitational acceleration m2/sH Height of fluidized bed mL Mixer length mmp, mq Average particle weight of two components p kg

and q in mixtureM Coefficient of mixing m2/sM Mass of a sample kgM Mass of a batch kgn Random sampling scope —n Rotational frequency HzNe 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 Wq 1− p —r Mixer radius mRSD Relative standard deviation —S Empirical standard deviation —S2 Random sample variance —t, t´ Time stv Mean residence times s

sSymbol Definition Units

tf, tm, te, ti Filling, mixing, discharging, and idle time st* Mixing time —Tp Feed fluctuation period sv Axial velocity m/sVRR 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 kgtwo ingredients in mix

σ2 Variance —σ2

z Variance of a random mix —Φ(χ2) Cumulative function ofχ2 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

Nomenclature and Units for Particle-Size Analysis (Concluded )

U.S.customary


SV Volume specific surface m2/m3

S1(θ), Dimensionless, complex functions — —S2(θ) describing the change and amplitude

in the perpendicular and the parallelpolarized light

T Absolute temperature K Kt Time s su Settling velocity of particles m/s ft/sv1,v2 Particle velocities m/s ft/sW Weight undersize g lbxEQPC Particle size of the equivalent m in

projection area of a circlex⎯F Average Feret diameter m inxF,max Maximum Feret diameter m inxF,max 90 Feret diameter measured 90° to the m in

maximum Feret diameter

Δl Thickness of the suspension layer m inΔQ3,i Normalized volume fraction in — —

size class iΔxi Width of size class i m inε Extension of a particle ensemble in m in

the direction of a cameraΓ Decay rate 1/s 1/sη Hydrodynamic viscosity of the Pa s Poise

dispersing liquidκ Imaginary part of refractive index — —

U.S.customary


xF,min Minimum Feret diameter m inxi Size of class i m inx⎯k,0 Arithmetic average particle size for a m in2

number distributionx⎯k,r Average particle size m inxmin Minimum particle size m inxst Stokes diameter m inx⎯1,r Weighted average particle size m inx⎯1,2 Sauter diameter m inx50,r Mean size of dimension r m inz Integration variable m inZ(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 degσ Dimensionless wave number — —ω Radial velocity of an agglomerate rad/s rad/sω Radial velocity of a centrifuge rad/s rad/sψS Sphericity — —ψA Aspect ratio — —ψC Convexity — —

C��

Pa�s�m

C�

Pa�s�m

Greek Symbols



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/sA Apparent area of indentor contact cm2 in2 K Agglomerate deformabilityA 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 inB Nucleation rate cm3/s in3/s L Roll loading dyn lbfBf Fragmentation rate g/s lb/s (ΔL/L)c Critical agglomerate deformation strainBf 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 inδc Effective increase in crack length due cm in n(v,t) Number frequency size distribution by 1/cm6 1/ft6

to process zone size volumec Unloaded shear strength of powder kg/cm2 psf Nc Critical drum or disc speed rev/s rev/sd Harmonic average granule diameter cm in P Applied load dyn lbfd Primary particle diameter cm in P Pressure in powder kg/cm2 psfd Impeller diameter cm in Q Maximum compressive force kg/cm2 psfd Roll press pocket depth cm in Q Granulator flow rate cm3/s ft3/sdi Indentor diameter cm in rp Process zone radius cm indp Average feed particle size cm in R Capillary radius cm inD Die diameter cm in S Volumetric spray rate cm3/s ft3/sD Disc or drum diameter cm in St Stokes number, Eq. (21-48)D Roll diameter cm in St* Critical Stokes number representingDc Critical limit of granule size cm in energy required for rebounder Coefficient of restitution St0 Stokes number based on initial nuclei E Strain energy stored in particle J J diameterE* Reduced elastic modulus kg/cm2 psf t Time s sfc 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/sGc Critical strain energy release rate J/m2 J/m2 U Fluidization gas velocity cm/s ft/sF Indentation force dyn lbf Umf Minimum fluidization gas velocity cm/s ft/sF Roll separating force dyn lbf Ui Spouted-bed inlet gas velocity cm/s ft/sG Layering rate cm3/s in3/s V Volumetric wear rate cm3/s in3/sh Height of liquid capillary rise cm in V̇R Mixer swept volume ratio of impeller cm3/s ft3/sh Roll press gap distance cm in V Volume of granulator cm3 ft3

h Binder liquid layer thickness cm in w Weight fraction liquidhb 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 inH Individual bond strength dyn lbf x Granule or particle size cm inH 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 Δρ Relative fluid density with respect to g/cm3

between granules of volumes displaced gas or liquidu 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

ε b Interagglomerate bed voidage ρb Bulk density g/cm3 lb/ft3

ε g 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 psfφe Effective angle of friction deg deg σz Resulting axial stress in powder kg/cm2 psfφw Wall angle of friction deg deg σ Powder normal stress during shear kg/cm2 psfφw Roll friction angle deg deg σc Powder compaction normal stress kg/cm2 psfϕ(η) Relative size distribution σf Fracture stress under three-point bend loading kg/cm2 psfγ lv Liquid-vapor interfacial energy dyn/cm dyn/cm σT Granule tensile strength kg/cm2 psfγ sl Solid-liquid interfacial energy dyn/cm dyn/cm σy Granule yield strength kg/cm2 psfγ sv Solid-vapor interfacial energy dyn/cm dyn/cm τ Powder shear stress kg/cm2 psfμ Binder or fluid viscosity poise θ Contact angle deg degμ Coefficient of internal friction ς Parameter in Eq. (21-108)ω Impeller rotational speed rad/s rad/s η Parameter in Eq. (21-108)



Nomenclature and Units

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/sequation qo Feed mass flow rate g/s lb/s

a Constant qp Mass flow rate of classifier product g/s lb/sak,k Coefficient in mill equations qR Mass flow rate of classifier tailings g/s lb/sak,n Coefficient in mill equations qR Recycle mass flow rate to a mill g/s lb/sB Matrix of breakage function R RecycleΔBk,u Breakage function R Reid solutionb Constant r Dimensionless parameter in size-C Constant distribution equationsCs Impact-crushing resistance kWh/cm ft⋅lb/in S Rate function S−1 S−1

D Diffusivity m2/s ft2/s S� 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 functiond Differential s Parameter in size-distribution d Distance between rolls of crusher cm in equationsE Work done in size reduction kWh hp⋅h s Peripheral speed of rolls cm/min in/minE Energy input to mill kW hp t Time s sEi Bond work index kWh/Mg hp⋅h/ton u Settling velocity of particles cm/s ft/sEi Work index of mill feed W Vector of differential size distributionE2 Net power input to laboratory mill kW hp of a streamerf Normal probability function wk Weight fraction retained on eachF As subscript, referring to feed stream screenF Bonding force kg/kg lb/ lb wu Weight fraction of upper-size particlesg Acceleration due to gravity cm/s2 ft/s2 wt Material holdup in mill g lbI Unit matrix in mill equations X Particle size or sieve size cm ini Tensile strength of agglomerates kg/cm2 lb/in2 X′ Parameter in size-distribution cm inK Constant equationsk Parameter in size-distribution equations cm in ΔXi Particle-size interval cm ink As subscript, referring to size of Xi Midpoint of particle-size interval ΔXi cm in

particles in mill and classifier X0 Constant, for classifier designparameters Xf Feed-particle size cm in

L As subscript, referring to discharge Xm Mean size of increment in size- cm infrom a mill or classifier distribution equations

L Length of rolls cm in Xp Product-particle size cm inL Inside length of tumbling mill m ft Xp Size of coarser feed to mill cm inM Mill matrix in mill equations X25 Particle size corresponding to 25 percent cm inm Dimensionless parameter in size- classifier-selectivity value

distribution equations X50 Particle size corresponding to 50 percent cm inN Mean-coordination number classifier-selectivity valueNc Critical speed of mill r/min r/min X75 Particle size corresponding to 75 percent cm inΔN Incremental number of particles in size- classifier-selectivity value

distribution equation ΔXk Difference between opening of cm inn Dimensionless parameter in size- successive screens

distribution equations x Weight fraction of liquidn Constant, general Y Cumulative fraction by weight undersizenr Percent critical speed of mill in size-distribution equationsO As subscript, referring to inlet stream Y Cumulative fraction by weight undersizeP As subscript, referring to product or oversize in classifier equations

stream ΔY Fraction of particles between two sievePk Fraction of particles coarser than a given sizes

sieve opening ΔY Incremental weight of particles in size- g lbp Number of short-time intervals in mill distribution equations

equations ΔYci Cumulative size-distribution intervals cm inQ Capacity of roll crusher cm3/min ft3/min of coarse fractionsq Total mass throughput of a mill g/s lb/s ΔYfi Cumulative size-distribution intervals cm inqc Coarse-fraction mass flow rate g/s lb/s of fine fractionsqF Mass flow rate of fresh material to mill g/s lb/s Z Matrix of exponentials

Greek Symbols

β Sharpness index of a classifier ρ� 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 SummationΖ Residence time in the mill s s σ Standard deviationηx Size-selectivity parameter σ Surface tension N/cm dyn/cmμ Viscosity of fluid N⋅S/m2 P υ Volumetric abundance ratio of ρf Density of fluid g/cm3 lb/in3 gangue to mineral

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GENERAL REFERENCES: Nedderman, Statics and Kinematics of GranularMaterials, Cambridge University Press, 1992. Wood, Soil Behavior and CriticalState Soil Mechanics, Cambridge University Press, 1990. J. F. Carr and D. M.Walker, Powder Technology, 1, 369 (1967). Thompson, Storage of ParticulateSolids, in Handbook of Powder Science and Technology, Fayed and Otten (eds.),Van Nostrand Reinhold, 1984. Brown and Richards, Principles of PowderMechanics, Pergamon Press, 1970. Schofield and Wroth, Critical State SoilMechanics, McGraw-Hill, 1968. M. J. Hvorslev, On the Physical Properties ofDisturbed Cohesive Soils, Ingeniorvidenskabelige Skrifter A, no. 45, 1937.Janssen, Zeits. D. Vereins Deutsch Ing., 39(35), 1045 (1895). Jenike, Storage andFlow of Bulk Solids, Bull. 123, Utah Eng. Expt. Stn., 1964. O. Reynolds, On theDilatancy of Media Composed of Rigid Particles in Contact: With ExperimentalIllustrations, Phil. Mag., Series 5, 20, 269 (1885). K. H., Roscoe, A. N. Schofield,and C. P. Wroth, On the Yielding of Soils, Geotechnique, 8, 22 (1958). Dhodap-kar et al., Fluid-Solid Transport in Ducts, in Multiphase Flow Handbook, Crowe(ed.), Taylor and Francis, 2006. Sanchez et al., Powder Technology, 138, 93(2003). Geldart, Powder Technology, 7, 285 (1973). Kaye, Powder Technology,1, 11 (1967).

AN INTRODUCTION TO BULK POWDER BEHAVIOR

Bulk solids flow affects nearly all solids processing operations throughmaterial handling problems and mechanical behavior. Measurementsof powder flow properties date back to Reynolds (loc. cit. 1885),Gibbs, Prandlt, Coulomb, and Mohr. However, the term flowabilityis rarely defined in an engineering sense. This often leads to a numberof misleading analogies being made with fluid behavior. Unique fea-tures with regard to powder behavior are as follows:

1. Powders can withstand stress without flowing, in contrast tomost liquids. The strength or yield stress of this powder is a functionof previous compaction, and is not unique, but depends on stressapplication. Powders fail only under applied shear stress, and notisotropic load, although they do compress. For a given applied hori-zontal load, failure can occur by either raising or lowering the normalstress, and two possible values of failure shear stress are obtained(active versus passive failure).

2. When failure does occur, the flow is frictional in nature andoften is a weak function of strain rate, depending instead on shearstrain. Prior to failure, the powder behaves as an elastic solid. In thissense, bulk powders do not have a viscosity in the bulk state.

3. Powders do not readily transmit stress. In the case of columns,normal stress or weight of the bulk solid is held by wall friction. Inaddition, normal stress is not isotropic, with radial stress being only afraction of normal stress. In fact, the end result is that stress in silosscales with diameter rather than bed height, a most obvious manifes-tation of this being the narrow aspect ratio of a corn silo.

4. A powder will not necessarily maintain a shear stress–imposedstrain rate gradient in the fluid sense. Due to force instabilities, it willsearch for a characteristic slip plane, with one mass of powder flowingagainst the next, an example being rat-hole discharge from a silo.

5. Bulk solids are also capable of two-phase flow, with large gasinteractions in silo mass discharge, fluidization, pneumatic conveying,and rapid compression and mixing. Under fluidized conditions, thebulk solid may now obtain traditional fluid behavior, e.g., pressurescaling with bed height. But there are other cases where fluidlike rhe-ology is misinterpreted, and is actually due to time dependent com-pression of interstitial fluid. After characteristic time scales related topermeability, stresses are transmitted to the solid skeleton. It may notbe of utility to combine the rheology of the solid and interstitial fluid,but rather to treat them as separate, as is often done in soil mechanics.

PERMEABILITY AND AERATION PROPERTIES

Permeability and Deaeration Various states of fluidization andpneumatic conveying exist for bulk solid. Fluidization and aerationbehavior may be characterized by a fluidization test rig, as illustratedin Fig. 21-25. A loosely poured powder is supported by a porous orperforated distributor plate. The quality and uniformity of this plateare critical to the design. Various methods of filling have been exploredto include vibration and vacuum filling of related permeameters

[Kaye, Powder Technology, 1, 11 (1967); Juhasz, Powder Technology,42, 123 (1985)]. Two key types of measurements may be performed.In the first, air or gas is introduced through the distributor, and thepressure drop across the bed is measured as a function of flow rate orsuperficial gas velocity (Fig. 21-26). In the second, the gas flow isstopped to an aerated bed, and the pressure drop or bed height ismeasured as a function of time, as the bed collapses and deaerates(Fig. 21-27).

For the first fluidization measurement, pressure drop willincrease with gas velocity while powder remains in a fixed-bed stateuntil it reaches a maximum plateau, after which the pressure dropequals the weight of the bed, provided the bed becomes uniformlyfluidized. Bed expansion will also occur. The point of transition isreferred to as the minimum fluidization velocity Umf. Various statesof a fluidized bed occur. For fine materials of limited cohesion, thebed will initially undergo homogeneous fluidization (also referredto as particulate fluidization), where bed expansion occurs without theformation of bubbles, and with further increases in gas velocity, it willtransition to a bubbling bed, or heterogeneous fluidization (alsoreferred to as bubbling or aggregative fluidization). Coarse materialsdo not expect the initial state of homogeneous fluidization, and Umb =Umf. The point at which bubbles form in the bed is referred to as theminimum bubbling velocity Umb. The various stages of fluidizationare described in detail in Sec. 17. In addition, for fine, cohesive pow-ders, channeling may occur instead of uniform fluidization, resultingin lower, more erratic pressure drops. Various states of fluidization areindicated in Fig. 21-26. Lastly, mixing, bed expansion, heat and masstransport, and forces acting in fluidized beds scale with excess gasvelocity, or U − Umf.

Prior to reaching minimum bubbling, a homogeneous fluidizedpowder will undergo a peak in pressure prior to settling down to itsplateau. This peak represents a measure of aerated cohesion, and itranges from 10 percent for fine, low-cohesion powders capable ofhomogeneous fluidization, to 50 percent for fine, extremely cohesivematerial, which generally undergoes channeling when fluidized.


SOLIDS HANDLING: BULK SOLIDS FLOW CHARACTERIZATION

FIG. 21-25 iFluid™ fluidization permeameter, illustrating powder bed sup-ported by distributor plate fluidized at a gas velocity U, with associated pressuretaps for multiple pressure gradient measurements dP/dh. (Courtesy iPowderSystems, E&G Associates, Inc.)


The pressure drop across the initial fixed bed (or final previouslyaerated bed) is a measure of permeability kP as defined by Darcy(1856), given by

U = = kP (21-26)ΔP�Hb�

μg

Q�Ab

otherwise known as Darcy’s law, which is strictly only valid for lowReynolds number. Comparing to the Kozeny-Carman relation[Kozeny (1927); Carman (1937)], permeability may be predicted fromparticle size (surface-volume average) and packing voidage:

kPo = (21-27)d2

pε3

��CP1(1 − ε)2

SOLIDS HANDLING: BULK SOLIDS FLOW CHARACTERIZATION 21-21

Fixed Bed Fluidized Bed Conveying

ΔP H( )

U = Q AbUmf Uc

ε = εmf

ε ≤ εmf εmf < ε < εc

ε = εc

ε →

ΔP H( )mf= Wb Ab

Be

d P

ress

ure

Dro

p

Superficial Gas Velocity

Pac

ked

Bed

Loos

e B

ed

Geldart Type A Behavior

Geldart Type C Behavior

ΔP H( )cohesion

FIG. 21-26 Fluidization measurement of permeabililty and fluidization behavior. Bed pressure drop ΔP�H for fixed and flu-idized beds as a function of gas velocity U. (After Rumpf, Particle Technology, Kluwer Academic, 1990.)

FIG. 21-27 Deareation measurement of deareation time and constant. Bed pressure drop(ΔP/H) decay following fluidization as a function of time. [Dhodapkar et al., Fluid-Solid Trans-port in Ducts, in Multiphase Flow Handbook, Crowe (ed.), Taylor & Francis, 2006, with permission.]


which is valid for low Reynolds number and loose packing. CP1 equals180 from the Kozeny-Carman relation and 150 from the Ergun rela-tion. For a wider range of gas velocities, Ergun’s relation should beutilized instead, where the pressure drop is given by

= (1 + 1.75ReP) where ReP = (21-28)

which can be rewritten to give

= = = E1 + E2U (21-29)

where E1 and E2 may be determined from plotting the slope in thefixed-bed region divided by velocity [or (dP/dh)/U] versus gas velocity.Theoretically, these constants are given by

E1 = = and E2 = (21-30)

where CP1 = 150 and CP2 = 1.75 based on Ergun’s relation. The stan-dard value of permeability is then related to the intercept E1, but avelocity dependence can be determined as well for high velocityrelated to conveying. And pf is another common definition of perme-ability, or permeability factor, which incorporates gas viscosity.

As with bulk density, permeability is a function of packing voidageand its uniformity, and in practice, it is best measured. It can varysubstantially with previous compaction of the sample. An example isthe change in bulk density—and therefore interstitial voidage—thatoccurs with a material as moves through a hopper. By applying a loadto the upper surface of the bed, permeability may be also deter-mined as a function of solids consolidation pressure (see “Bulk FlowProperties” and “Solids Handling: Storage, Feeding, and Weighingof Bulk Solids”). Permeability is a decreasing function of appliedsolids pressure, and bulk density is often written in log form, or

kP = kPo� m

(21-31)

From the second deaeration measurement, pressure drop ismeasured as a decaying function of time, given by one of the forms(Fig. 21-27)

= ae−t�td or = (21-32)

where td and Ad are a characteristic deaeration time and deaerationfactor, respectively. Large deaeration time or factor implies that thepowder retains air for long times. Also an additional deaeration factorhas been defined to account for particle density, or

Xd = (21-33)

Permeability and deaeration control both fluidization and pneu-matic conveying. In addition, they impact the gas volume and pres-sure requirement for air-augmented flow in hoppers and feeders.Materials of low permeability have lower mass discharge rates fromhopper openings (see “Mass Discharge Rates”) and limit the rate ofproduction in roll pressing, extrusion, and tableting, requiring vacuumto speed deaeration (“Compaction Processes”). Lastly permeabilityimpacts wetting phenomena and the rate of drop uptake in granula-tion (“Wetting and Nucleation”).

Classifications of Fluidization Behavior Geldart [PowderTechnology, 7, 285 (1973)] and later Dixon [Pneumatic Conveying,Plastics Conveying and Bulk Storage, Butters (ed.), Applied SciencePublishers, 1981] developed a classification of fluidization/aeration

Ad�ρs��(ΔP�H)mf

Ad�

tΔP�dh

ΔP�dh

ρbo�ρb

CP2ρg(1 − ε)��

dpε3

CP1μg(1 − ε)2

��d2

pε3

μg�kPo

dP�dh�

Uμg�kP

1�pf

ρgUdp��μg(1 − ε)

μgU�

kPo

dP�dh

behavior from studies of fluidized beds and slugging in vertical tubes,respectively (Fig. 21-28). The classification is based on particle size(surface-volume average for wide size distributions) and relative par-ticle density. Particle size controls interparticle cohesive forces,whereas density controls the driving force to be overcome by drag. Asummary of aeration behavior is provided in Table 21-4, where fromGeldart’s classification powders are broken down into group A (aer-atable) for fine materials of low cohesion which can exhibit homoge-neous fluidization, group B (bubbling) for coarser material whichimmediately bubbles upon fluidization, group C (cohesive) which typ-ically channels and retains air for long periods, and group D(spoutable) which is coarse material of high permeability with no airretention capability.

Classifications of Conveying Behavior Aeration behavior alsoimpacts mode and ease of pneumatic conveying [Dhodapkar et al.,Fluid-Solid Transport in Ducts, in Multiphase Flow Handbook,Crowe (ed.), Taylor & Francis, 2006]. Figure 21-29 illustrates theimpact of decreasing conveying velocity on flow pattern. At high gasflow, ideal dilute, homogeneous solids flow may occur (1). As gasvelocity is reduced past some characteristic velocity, the solids can nolonger be uniformly suspended and increasing amounts of solid willform on the bottom of the pipe, forming a moving stand of solids (2,3).With further decrease of gas velocity and deposited solids, movingdunes (4,5) and later slugs (6,7,8) will form which completely fill thepipe. Finally, ripple flow (9) and pipe pluggage (10) will occur.Dilute-phase conveying encompassed patterns 1 to 3, wheredense-phase conveying includes the remainder of 4 to 8. Dhodap-kar et al. (loc. cit.) further classified conveying patterns according toparticle size. Fine materials (plastic powder, fly ash, cement, fine coal,carbon fines) may be transported in all patterns, with a smooth, pre-dictable transition between regimes. At intermediate gas velocities,two-phase strand flow (2,3) is observed followed by dune flow at lowervelocities (4–8), where the solid flow can appear as turbulent or fast-moving bed, wave, or fluidized-bed modes. Conveying might also beachieved in patterns 9 and 10 for materials that readily aerate andretain air, in which case they are conveyed as a fluidized plug. Coarsematerials (pellets, grains, beans, large granules), however, form slugswhen conveyed at low velocities, which form on a regular, periodicbasis. The transition from dilute- to dense-phase conveying for coarsematerial is unstable and occurs under dune flow. Some coarse materi-als with substantial fines exhibit fine conveying modes.

Figure 21-30 provides classifications of conveying ability, wherepermeability and deaeration factor are plotted against pressure dropat minimum fluidization for a variety of materials [Mainwaring andReed, Bulk Solids Handling, 7, 415 (1987)]. Lines of constant minimumfluidization Umf = 0.05 m/s and deaeraton factor Xd = 0.001 m3 ⋅ s�kg areshown. From Fig. 21-30a, materials which lie above the line of high


FIG. 21-28 Geldart’s classification of aeration behavior with Dixon and Gel-dart boundaries. (From Mason, Ph.D. thesis, Thomes Polytechnic, London,1991, with permission.)


permeability can be conveyed in plug or slug form as they do not read-ily retain air, whereas those below the line of low permeability can beconveyed by moving-bed flow, as they more readily retain air, or bydilute-phase flow. Similarly from Fig. 21-30b, materials which liebelow the line with small deaeration constant (or time) can be con-veyed in plug or slug form, whereas those above the line with largedeaeration constant (or time) can be conveyed by moving-bed flow ordilute-phase flow, as they retain air. Jones and Miller [Powder Han-dling and Processing, 2, 117 (1990)] combined deaeration behaviorand permeability in a single classification, as shown in Fig. 21-31.Group 1 includes Geldart type A powders of low permeability andlarge deaeration time which conveyed as moving-bed flow, whereas at

the other extreme, group 3 includes Geldart type D materials withhigh permeability and short deaeration time conveyed as plug-typeflow. Dilute- and dense-phase conveying is possible for group 2 or typ-ically type B powder with (1) intermediate permeability and deaera-tion time, (2) small deaeration time and permeability, or (3) largedeaeration time and permeability. Type C material exhibits all threeforms of conveying. Sanchez et al. [Powder Technology, 138, 93(2003)] and Dhodapkar et al. (loc. cit.) provide current summaries ofthese classifications.

BULK FLOW PROPERTIES

Shear Cell Measurements The yield or flow behavior of bulksolids may be measured by shear cells. Figure 21-32 illustrates theseprinciples for the case of a direct rotary split cell. For flow measure-ments, powder is contained within two sets of rings. Normal stress isapplied to the powder bed through a horizontal roughened or pat-terned lid. The upper ring containing approximately one-half of thepowder is sheared with respect to the lower ring, forming a shearplane or lens between the two halves of powder. This is accomplishedby rotating the lower half of the powder mounted to a motorized base,which in turn attempts to rotate the upper half of powder through rota-tional shear stress transmitted through the shear plane. The upper halfof powder is instead held fixed by the upper lid, which transmits thisshear stress through an air bearing to a force transducer. Through thisgeometry, the shear stress between the two halves of powder, mea-sured as a torque by the force transducer, is measured versus time ordisplacement as a function of applied normal stress. In addition, anycorresponding changes in powder density are measured by changes invertical displacement for a linear voltage displacement transducer.

For wall friction measurements, a wall coupon is insertedbetween the rings, and powder in the upper ring alone is shearedagainst a coupon of interest. Wall friction and adhesion, both staticand dynamic, may be assessed against different materials of construc-tion or surface finish.

Shear cell testing of powders has its basis in the more comprehen-sive field of soil mechanics (Schofield and Wroth, Critical State SoilMechanics, McGraw-Hill, 1968), which may be further considered asubset of solid mechanics (Nadia, Theory of Flow and Fracture ofSolids, vols. 1 and 2, McGraw-Hill, 1950). The most comprehensivetesting of the shear and flow properties of soils is accomplished in tri-axial shear cells (Fig. 21-33). There are two such types of triaxial


TABLE 21-4 Characteristics of Geldart (1973) or Dixon (1981) Classification

Properties Group A Group B Group C Group D

Material Fine/medium powder Course powder Sand, Cohesive fine powder Granular Plastic pellets, Fly ash, pulverized salt, granules, mineral Cement, corn starch, tit- wheat, large glass beads,coal, plastic powders, powders, glass beads anium dioxide, carbon tablets, course sand,alumina, granular sugar, black powder, many seedspharma excipients pharma actives

Fluidization Good air retention, Poor air retention, low Cohesive and difficult to. Highly permeable, neglicharacteristics small bubble size, bed expansion, large fluidize. Tends to channel. gible expansion and no

considerable bed bubble, asymmetric Retains gas for extended air retention, largeexpansion slugging at higher period once aerated, bubbles, spouts, or

velocity or small beds adhesion to walls and axisymmetric slugs can surfaces. form.

Conveying Can be conveyed in Unlikely to convey in Difficult but possible to convey in Natural slugging ability and highcharacteristics fluidized- or moving-bed conventional dense phase, dense phase, forms impermeable permeability aid in slug or plug

mode, easy to convey, unsteady and unpredictable plugs that breakup. Requires flow conveying. Operationallydoes not form slugs plug formation, large special conveying easiest to dense phase conveynaturally pipe vibrations

Pressure drop at Umf : <50 >80 50–130 5–150(ΔP/H)mf [mbar/m]

Permeability factor 0.1 0.01–0.1 to 1 0.1 to 1 >1(kP/μg) = [m2/(bar⋅s)]

Deaeration Collapses slowly, air retention Collapses rapidly Collapses slowly, long air retention Collapses very rapidly

[Adapted from Dhodapkar et al., Fluid-Solid Transport in Ducts, in Multiphase Flow Handbook, Crowe (ed.), Taylor & Francis, 2006; and Sanchez et al. [PowderTechnology, 138, 93 (2003)].

Gas flow direction

FIG. 21-29 Pattern of solids flow in pneumatic conveying. [From Wen, U.S.Dept. of Interior, Bureau of Mines, PA, IC 8314 (1959) with permission.]



Plug or slug flow

Plug or slug flow

Moving-bed or dilute phase flow

Moving-bed or dilute phase flow

FIG. 21-30 Classification of pneumatic conveying based on (a) permeability factor and (b) deareation factor. [From Mainwaring and Reed, Bulk SolidsHandling, 7, 415 (1987) with permission.]

FIG. 21-31 Classification of pneumatic conveying based on combined permeability and deareation factors, based on Jones and Miller. [Sanchez et al., [Pow-der Technology, 138, 93 (2003), with permission.]


shear cells. In the traditional cylindrical triaxial cell, the axial andradial pressures acting on the sample contained within a rubber mem-brane are directly controlled through applied axial force and radialhydraulic oil pressure. Deviatoric stress, i.e., shear stress due to dif-ference in axial and radial pressure, is applied to the sample until fail-ure. In a true triaxial cell, all three principal stresses may be varied;whereas only the major and minor principal stresses are controlled inthe traditional cylindrical triaxial cell. Lastly, shear displacements aremeasured through a variety of strain gauges, and both the drained andundrained tests are possible. Such tests refer to simultaneous mea-surement of pressure of any interstitial fluid or gas. Interstitial fluidcan have pronounced effects on mitigating powder friction and chang-ing flow properties. While triaxial cells are not typically employed forpowder characterization in industrial processing, they do provide themost comprehensive information as well as a knowledge base of appli-cation in such results for bulk solids flow, including detailed simula-tions of multiphase flow of such systems. Their disadvantage is theirdifficulty of use and time required to perform measurements. Futureadvances in employing these designs are likely.

Direct shear cells were introduced due to drastically reducedtesting times, although the exact nature of stresses in the failure zoneis not as precisely defined as with triaxial cells. Direct cells haveundergone substantial automation in the last two decades. All have asa common feature that only the applied axial force or axial stress iscontrolled (Fig. 21-33). The shear stress required to accomplish fail-ure is measured as a function of the applied axial stress, where trans-lational or rotational motion is employed. Both cup and split celldesigns are available. Rotational cells include both full annulus andring cells. For a properly designed direct shear cell, failure occurswithin a specific region, in which both the plane of failure and the act-ing stresses may be clearly defined. In addition, direct shear cells maybe validated against an independent vendor standard, or the BCR116limestone powder (see “Shear Cell Standards and Validation”). Rotarysplit cell designs have two possible advantages: (1) Unlimited dis-placement of the sample is possible, allowing ease of sample condi-tioning and repeated sample shear on a single sample. (2) The shearplane is induced in a defined region between the two cell halves,allowing unconfined expansion in the shear plane (Fig. 21-32).

Yield Behavior of Powders The yield behavior of a powderdepends on the existing state of consolidation within the powderbed when it is caused to flow or yield under a given state of stress,defined by the acting normal and shear stresses. The consolidationstate controls the current bed voidage or porosity. Figure 21-34 illus-trates a times series of shears occurring for the BCR116 limestonestandard for a rotary shear cell. For each shear step, torque is applied

to the sample by cell rotation until sample failure; the cell is thenreversed until the shear force acting on the sample is removed. Twostages of a typical experiment may be noted. The first is a consolida-tion stage wherein repeated shears take place on the sample until theshear stress τ reaches a steady state, defined by either the maximumvalue or the steady value occurring after an initial peak. This occurswith a constant normal consolidation stress σ = σc acting on the sam-ple. During this step, the sample reaches a characteristic or criticaldensity or critical porosity εc related to the consolidation normalstress. A set of shear steps is then performed during a shear stagewith progressively smaller normal loads. In all cases, each shear step ispreceded by a shear at the original consolidation normal stress.

Three characteristic displacement profiles may be observed duringshear for shear stress and density (Fig. 21-35), which are unique to thestate of consolidation:

1. Critically consolidated. If a powder is sheared sufficiently, itwill obtain a constant density or critical porosity εc for this consolida-tion normal stress σc. This is defined as the critical state of the powder,discussed below. If a powder in such a state is sheared, initially thematerial will deform elastically, with shear forces increasing linearlywith displacement or strain. Beyond a certain shear stress, the mate-rial will fail or flow, after which the shear stress will remain approxi-mately constant as the bulk powder deforms plastically. Depending onthe type of material, a small peak may be displayed originating fromdifferences between static and dynamic density. Little change in den-sity is observed during shear, as the powder has already reached thedesired density for the given applied normal consolidation stress σc.

2. Overconsolidated. If the same sample is sheared, but at alower normal stress of σ < σc, the shear stress will increase elasticallyto a peak and then fail, with this peak being less than that observed forthe critically consolidated state, as the applied normal stress is lower.After the failure peak, the shear stress will decrease as the powderexpands due to dilation and density decreases, eventually leveling offto a lower shear stress and lower density. Overconsolidated shears areobserved during the shear stage of a shear cell experiment.

3. Underconsolidated. If the same sample is sheared, but at ahigher normal stress of σ > σc, the shear stress will progressivelyincrease to some value, while the material simultaneously densifies.Such underconsolidated responses are observed in the consolidationstage of an experiment.

In practice, following the filling of a cell, the powder is in an under-consolidated state. A set of shear steps is performed under a chosenconsolidation stress in the consolidaton stage to increase its densityand bring it into a critical state. A set of shears is then preformed atsmall normal stresses in the shear stage to determine the strength of


FIG. 21-32 iShear™ rotary, full annulus split cell, illustrating normal load weight application, rotational base,and shear stress/torque measurement. Vertical displacement of lid is monitored by displacement transducer.(Courtesy E&G Associates, Inc.)



FIG. 21-33 Examples of powder shear cells. Triaxial cells: (a) Traditional triaxial cell; (b) true triaxial. Directshear cells: (c) Translational split, Collin (1846), Jenike™ (1964); (d) rotational annulus, Carr and Walker (1967),Schulze™ (2000); (e) rotational split, Peschl and Colijn (1976), iShear™ (2003). [From Measuring Powder Flowabil-ity and Its Applications, E&G Associates, 2006, with permission.)

Consolidation Stage

Str

ess

(g/c

m2 )

Den

sity

(g

/cm

3 )

Shear Stage

FIG. 21-34 Time-series shearing profile for the BCR116 limestone validation powder, in an iShear rotary split cell. (FromMeasuring Powder Flowability and Its Applications, E&G Associates, 2006, with permission.)

95%



the powder as a function of normal load, in the overconsolidated orovercompacted state, each time reconsolidating the powder beforeperforming the next shear step.

Powder Yield Loci For a given shear step, as the applied shearstress is increased, the powder will reach a maximum sustainableshear stress τ, at which point it yields or flows. The functional rela-tionship between this limit of shear stress τ and applied normal load σis referred to as a yield locus, i.e., a locus of yield stresses that mayresult in powder failure beyond its elastic limit. This functional rela-tionship can be quite complex for powders, as illustrated in both prin-cipal stress space and shear versus normal stress in Fig. 21-36. SeeNadia (loc. cit.), Stanley-Wood (loc. cit.), and Nedderman (loc. cit.)for details. Only the most basic features for isotropic hardening ofthe yield surface are mentioned here.

1. There exists a critical state line, also referred to as the effec-tive yield locus. The effective yield locus represents the relationshipbetween shear stress and applied normal stress for powders always ina critically consolidated state. That is, the powder is not over- orundercompacted but rather has obtained a steady-state density. Thisdensity increases along the line with increases in normal stress, andbed porosity decreases.

2. A given yield locus generally has an envelope shape; the initialdensity for all points forming this locus prior to shear is constant. Thatis, the locus represents a set of points all beginning at the porosity, thiscritical state porosity determined by the intersection with the effectiveyield locus.

3. Points to the left of the effective yield locus are in a state of over-consolidation, and they dilate upon shear. If sheared long enough, thedensity and shear stress will continue to drop until reaching the effec-tive yield locus. Points to the right are underconsolidated and compactwith shear.

4. For negative normal stresses, a state of tension exists in the sam-ple along the yield locus. This area is generally not measured by directshear cells, but can be measured by triaxial shear and tensile split cells.

5. Multiple yield loci exist. As a powder is progressively compactedalong the effective yield locus, it gains strength as density rises, reach-ing progressively higher yield loci. Yield loci of progressively largerenvelope size have higher critical density and lower critical voidage, asshown in Fig. 21-36. Therefore, the shear strength of a powder τ is afunction of the current normal stress σ, as well as its consolidation his-tory or stress σc, which determined the starting density prior to shear.

Currently in industrial practice, we are most concerned with theovercompacted state of the powder, and applications of the under-compacted and tensile data are less common, although they are find-ing applications in compaction processes of size enlargement (see“Powder Compaction”).

Although the yield locus in the overcompacted state may possesssignificant curvature, especially for fine materials, a common Mohr-Coulomb linear approximation to the yield locus as shown in Fig. 21-37is given by

τ = c + μσ = c + σ tan φ (21-34)

Here, μ is the coefficient of internal friction, φ is the internalangle of friction, and c is the shear strength of the powder in theabsence of any applied normal load. Overcompacted powders dilatewhen sheared, and the ability of powders to change volume with shearresults in the powder’s shear strength τ being a strong function of pre-vious compaction. There are therefore a series of yield loci (YL), asillustrated in Fig. 21-37, for increasing previous consolidation stress.The individual yield loci terminate at a critical state line or effectiveyield locus (EYL) defined early, which typically passes through thestress-strain origin, or

FIG. 21-35 Examples of yield behavior. (From Measuring Powder Flowability and Its Applications, E&G Associates, 2006, withpermission.)

FIG. 21-36 Family of yield loci for a typical powder. (Rumpf, loc. cit., with permission.)


τ = μeσ = σ tan φe (21-35)

where μe is the effective coefficient of powder friction and φe isthe effective angle of powder friction of the powder. In practice,there be a small cohesion offset in the effective yield locus, in whichcase the effective angle is determined from a line intercepting an ori-gin and touching the effective yield locus. In this case, the effectiveangle of friction is an asymptotic function of normal stress.

When sheared powders also experience friction along a wall, thisrelationship is described by the wall yield locus, or

τ = μwσ = σ tan φw (21-36)

where μw is the effective coefficient of wall friction and φw is theeffective angle of wall friction, respectively. In practice, there is asmall wall adhesion offset, making the effective angle of wall friction anasymptotic function of normal stress, as with effective powder friction.

Lastly, both static (incipient powder failure) and dynamic (contin-ued-flow) yield loci may be measured, giving both static and dynamicvalues of wall and powder friction angles as well as wall adhesion.

Flow Functions and Flowability Indices Consider a powdercompacted in a mold at a compaction pressure σ1. When it isremoved from the mold, we may measure the powder’s strength, orunconfined uniaxial compressive yield stress fc (Fig. 21-38). Theunconfined yield and compaction stresses are determined directlyfrom Mohr circle constructions to yield loci measurements (Fig. 21-36).This strength increases with increasing previous compaction, with thisrelationship referred to as the powder’s flow function FF.

The flow function is the paramount characterization of powderstrength and powder flowability. Common examples are illustrated inFig. 21-38. Typically the flow function curves toward the normal stressaxis with increasing load (A). An upward shift in the flow function indi-cates an overall gain of strength (B). If one were comparing the flowa-bility of two lots of material, this would indicate a decrease in flowability.In other words, greater stresses would be required in processing for lotB than for lot A (e.g., hoppers, feeders, mixers) to overcome thestrength of the powder and to induce flow of the mass. An upward shiftalso occurs with time consolidation, where a specified time of consol-idation is allowed prior to each shear step of the yield locus. The result-ing flow function is a time flow function, and it indicates the effect ofprolonged storage on flow. Flow functions often cross (C vs. A), indicat-ing lot C is more flowable at low pressure than lot A, but less flowable athigh pressure. An upward curvature of the flow function is indicative ofpowder or granule degradation (C), with large gains of strength asbreakdown of the material occurs, raising powder density and interpar-ticle contacts.

Under the linear Mohr-Coloumb approximation, if parallel yieldloci are assumed with constant angle of internal friction, and with zerointercept of the effective yield locus, the flow function is a straight linethrough the origin D, given by

fc = fco + Aσ1 = � � (sin φe − sin φ)�σ1 where fco = 0 (21-37)

1 + sin φ��1 + sin φe

2�cos φ


FIG. 21-38 Common flow functions of powder. [From Measuring Powder Flowability and Its Applications, E&G Associates,2006, with permission.)

FIG. 21-37 The yield loci of a powder, reflecting the increased shear stressrequired for flow as a function of applied normal load. YL1 through YL3 repre-sent yield loci for increasing previous compaction stress. EYL and WYL are theeffective and wall yield loci, respectively.


Other workers assume a linear form with a nonzero intercept fco. Thisimplies a minimum powder strength in the absence of gravity or anyother applied consolidation stresses. As described above, the flow func-tion is often curved, likely due to the angles of friction being a functionof applied stress, and various fitting relations are extrapolated to zero todetermine fco. While this is a typical practice, it has questionable basisas the flow function may have pronounced curvature at low stress.

The flow function and powder strength have a large impact on min-imum discharge opening sizes of hoppers to prevent arching and ratholing (see also “Solids Handling: Storage, Feeding and Weighing”),mass discharge rates, mixing and segregation, and compact strength.

One may compare the flowability of powders at similar pressures bycomparing their unconfined yield stress fc at a single normal stress, orone point off a flow function. In this case one should clearly state thepressure of comparison. Flow indices have been defined to aid suchone-point comparisons, given by the ratio of normal stress to strength, or

RelP = or RelJ = (21-38)

The first is due to Peschl (Peschl and Colijn, New Rotational ShearTesting Technique, Bulk Solids Handling and Processing Conference,Chicago, May 1976). For powders in the absence of caking it has a min-imum value of 1 for a perfectly plastic, cohesive powder. The seconddefinition is due to Jenike (Jenike, Storage and Flow of Bulk Solids,Bull. 123, Utah Eng Expt. Stn., 1964). The reciprocal of these relativeflow indices represents a normalized yield strength of the powder,normalized by maximum consolidation shear in the case of Peschl andconsolidation stress in the case of Jenike. Flowability increases withdecreasing powder strength, or increasing flow index. Table 21-5 pro-vides typical ranges of behavior for varying flow index. For powders ofvarying bulk density, absolute flow indices should be used, or

AbsP or J = RelP or J × (ρb�ρH2O) (21-39)

Therefore, for powders of equal powder strength, flowabilityincreases with increasing bulk density for gravity-driven flow.

Shear Cell Standards and Validation While shear cells vary indesign, and may in some cases provide differing values of powderstrength, the testing does have an engineering basis in geotechnicalengineering, and engineering properties are measured, i.e., yieldstresses of a powder versus consolidation. As opposed to other phe-nomenological, or instrument-specific, characterizations of powderflowability, shear cells generally provide a common reliable ranking offlowability, and such data are directly used in design, as discussedbelow. (See also “Solids Handling: Storage, Feeding, and Weighing.”)Rotary split cells (ASTM D6682-01), translation Jenike cells (ASTMD6128-97), and rotary annular ring cells (ASTM D6682-01) all haveASTM test methods. In addition, units may be validated against anindependent, international powder standard, namely, the BCR116limestone validation powder for shear cell testing (Commission of theEuropean Communities: Community Bureau of Reference). Table21-6 provides an excerpt of shear values expected for the standard,and Fig. 21-39 provides a yield loci comparison between differing celldesigns, and a comparison to the standard values.

Stresses in Cylinders Bulk solids do not uniformly transmitstress. Consider the forces acting on a differential slice of material in,

σ1�fc

σ1 − σ3�

fcsay, a cylindrical bin (Fig. 21-40). Prior to failure or within the elasticlimit, the axial stresses σz and radial stresses σr, under the assumptionthey are principal stresses, are related by

σr = σz (21-40)

where ν is the Poisson ratio. Under active incipient failure, the axialand radial stresses are related by a lateral stress coefficient Ka given by

Ka = = (active) (21-41)

In the case of wall friction, the axial and radial stresses differ some-what from the true principal stresses, and the stress coefficientbecomes (Walker)

Ka = = where sin ω =

(21-42)

This may be contrasted to, e.g., the isotropic pressure developed in afluid under pressure, with only nonnewtonian fluids able to developand sustain a nonisotropic distribution of normal stress. In addition,the radial normal stress acting at the wall develops a wall shear stressthat opposes gravity and helps support the weight of the powder. Asoriginally developed by Janssen [Zeits. D. Vereins Deutsch Ing.,39(35), 1045 (1895)], from a balance of forces on a differential slice,the axial stress σz as a function of depth z is given by

σz = (1 − e−(4μwKa�D)z) (21-43)

where D is the diameter of the column. Several comments may bemade of industrial practicality:

1. Pressure initially scales with height as one would expect for afluid, which may be verified by expanding Eq. (21-35) for small z. Orσz ≈ ρbgz.

2. For sufficient depth (at least one diameter), the pressurereaches a maximum value given by σz = ρbgD�(4μwKa). Note that thispressure scales with cylinder diameter, and not height. This is a criti-cal property to keep in mind in processing; that diameter often con-trols pressure in a powder rather than depth. A commonplaceexample would be comparing the tall aspect ratio of a corn silo to thatof a liquid storage vessel. The maximum pressure in the base of sucha silo is controlled by diameter, which is kept small.

3. The exact transition to constant pressure occurs at roughly 2zc,where zc = D�(4μwKa).

Stress transmission in powders controls flow out of hoppers, feed-ers, filling of tubes, and compaction problems such as tableting androll pressing. (See “Solids Handling: Storage, Feeding, and Weighing”and “Powder Compaction.”)

ρbgD�4μwKa

sin φw�sin φe

1 − sin φe cos(ω − φw)��1 + sin φe cos(ω − φw)

σr�σz

1 − sin φe��1 + sin φe

σr�σz

ν�1 − ν


Table 21-5 Typical ranges of flowability for varying flowindes, Modified after PeschI

Flow index Level of cohesion Example

Relp : < 1 Bonding, solid Cakedmaterial, time coonsolidated= 1 Plasic material Wet mass

1 to 2 Extermely cohesive Magstearate, starc (nongravity)2 to 4 Very Cohesive Coare organic

4 to 10 Cohesive Granules inorgaics10 to 15 Slightly cohesive Hard silica,sand15 to 25 Cohesionless If fine, floodable

From Measuring powder flowability Its applications, E&G Associates (2006),with permission

TABLE 21-6 BCR 116 Limestone

COMMISION OF THE EUROPEAN COMMUNITIESCOMMUNITY REFERENCE MATERIAL

CERTIFICATE OF MEASUREMENTCRM 116

LIMESTONE POWDER FOR JENIKER SHEAR TESTING

CONSOLIDATION SHEAR MEAN UNCERTAINTYNORMAL STRESS NORMAL STRESS SHEAR STRESS

kpa kpa kpa kpa

3.0 3.0 2.14 ± 0.313.0 2.0 1.75 ± 0.193.0 1.75 1.64 ± 0.173.0 1.5 1.54 ± 0.143.0 1.25 1.41 ± 0.133.0 1.0 1.27 ± 0.10

Stear stress values


Mass Discharge Rates for Coarse Solids The mass dis-charge rate from a flat-bottom bin with a circular opening of diame-ter B has been shown experimentally to be independent of bindiameter D and bed fill height H, for H > 2B. Dimensional analysisthen indicates that the mass discharge rate W must be of the formW = Cρ�g�B5�2, where C is a constant function of powder friction.Such a form was verified by Beverloo [Beverloo et al., Chem. Eng.Sci., 15, 260 (1961)] and Hagen (1856), leading to the Beverlooequation of mass discharge, or

Wo = Cρb�g�(B − kdp)2.5 ≈ 0.52 ρbA�2gB� for B >> dp

(21-44)

Here, ρb is loose poured bulk density, C ~ 0.58 and is nearly indepen-dent of friction, k = 1.5 for spherical particles and is somewhat largerfor angular powders, dp is particle size, and A is the area of the open-ing. The correction term of particle size represents an excluded annu-lus effective lowering the opening diameter. See Nedderman (Staticsand Kinematics of Granular Materials, Cambridge University Press,1992) and Brown and Richards (Principles of Powder Mechanics,Pergamon Press, 1970) for reviews.

The Beverloo relation for solids discharge may be contrasted withthe mass flow rate of an inviscid fluid from an opening of area A, or

W = 0.64ρlA�2gH� (21-45)


0

2

4

6

8

10

12S

hea

r S

tres

s (k

Pa)

0 2 4 6 8 10 12 14 16

Normal Stress (kPa)

15 kPa_J

9 kPa_J

6 kPa_J

3 kPa_J

15 kPa_P

9 kPa_P

6 kPa_P

3 kPa_P

BCR

(a) (b)

FIG. 21-39 Shear cell BCR 116 limestone validation yield loci. (a) Comparison of Jenike translational to Peschl rotary shear cell data (DuPont,1994, used with permission). (b) Typical validation set performed on an iShear™ rotary shear cell as compared to BCR standard (2005). (CourtesyE&G Associates, Inc.)

FIG. 21-40 Stresses in a vertical cylinder. [From Measuring Powder Flowability and Its Applications, E&G Associates,2006, with permission.)


where ρb is fluid density. Note that mass flow rate scales with height,which controls fluid pressure, compared to mass discharge rateswhich scale with orifice diameter.

For coarse materials of typical friction, discharge rates predicted bythe Beverloo relation are within 5 percent for experimental values fordischarge from flat-bottom bins or from hoppers emptying by funnelflow (“Solids Handling: Storage, Feeding, and Weighing”), and aremost reliable for material of low powder cohesion, in the range of 400μm < dp < B/6. However, for fine materials less than 100 μm or mate-rials large enough to give mechanical interlocking, the Beverloo rela-tion can substantially overpredict discharge.

Equation (21-35) may be generalized for noncircular openings byreplacing diameter by hydraulic diameter, given by 4 times the open-ing area divided by the perimeter. The excluded annulus effect can beincorporated by subtracting kdp from all dimensions. For slot openingof length L >> B with B as slot width, discharge rates have been pre-dicted to within 1 percent for coarse materials (Myers and Sellers,Final Year Project, Department of Chemical Engineering, Universityof Cambridge, UK, 1977) by

Wo = ρb�g�(L − kdp)(B − kdp)1.5 (21-46)

Through solutions of radial stress fields acting at the opening, the dis-charge rate for smooth, wedge-shaped hoppers emptying by mass flow isgiven by the hourglass theory of discharge [Savage, Br. J. Mech. Sci., 16,1885 (1967); Sullivan, Ph.D. thesis, California Institute of Technology,1972; Davidson and Nedderman, Trans. Inst. Chem. Eng., 51, 29 (1973)]:

W = where C(Kp) = � (21-47)

where α is the vertical hopper half-angle, C = fn(Kp), and Kp is the pas-sive Rankine stress coefficient given by

Kp = (21-48)

Here C is a decreasing friction of powder friction, ranging from 0.64to 0.47 for values of φe ranging from 30° to 50°. Equation (21-35)generally overpredicts wedge hopper rates by a factor of 2, primarilydue to neglection of wall friction. The impact of wall friction may beincorporated through the work of Kaza and Jackson [Powder Tech-nology, 39, 915 (1984)] by replacing Kp with a modified coefficient κgiven by

κ = Kp + (21-49)

From Eqs. (21-xx) to (21-xx), the mass flow discharge rate from wedgehopper increases with increasing orifice diameter B2.5, increasing bulkdensity, decreasing powder friction and wall friction, and decreasingvertical hopper half-angle, and is independent of bed height.

Extensions to Mass Discharge Relations Johanson (Trans.Soc. Min. Eng., March 1965) extended the Beverloo relations toinclude the effect of powder cohesion, with mass discharge rate givenby

Wsc = 1.354 Wo��1 −�� (21-50)

Here Wsc is the steady-state discharge rate for a cohesive powder forunconfined uniaxial compressive strength fc, and m = 1 or 2 for a slothopper or a conical hopper, respectively. σ1a is the major consolidationstress acting at the hopper opening (“Solids Handling: Storage, Feed-ing, and Weighing). Note that the discharge rate increases withincreasing stress at the opening and decreasing powder strength, andthat the major stress σ1a must exceed the powder’s strength fc for flowto occur. In addition, Johanson determined an intial dynamic mass dis-charge rate given by

fc�σ1a

1��2 m tan α

(ω + φw)sin φe��

α(1 − sin φe)

1 + sin φe��1 − sin φe

1 + Kp��2(2Kp − 3)

π�4

Wo�sin1�2 α

4�2�C�

π

Wdc = Wsc�1 − 1.39 where T = � (21-51)

where T is the period required to achieve steady-state state flow,which increases with the increases in the required steady dischargerate and increasing powder cohesion fc.

It is also especially critical to note that an applied surface pres-sure to the top of the powder bed will not increase the flow rate. Infact, it is more likely to decrease the flow rate by increasing powdercohesive strength fc. Similarly, vibration will increase flow rate only ifthe powder is in motion, primarily by lowering wall friction. If dis-charge is halted, vibration can lower or stop the discharge rate bycompacting and raising powder strength.

Stresses in powders are an increasing function of diameter [cf. Eq.(21-43)]. Therefore, as a powder moves toward the opening, the stressacting upon it decreases and the powder undergoes a decrease in bulkdensity. The displaced solids volume due to the correspondingincrease in powder voidage must be matched by an inflow of gas. Forcoarse solids governed by the Beverloo relation, this inflow of gasoccurs with little air pressure change with negligible effect on massdischarge. However, for fine powders of low permeability definedabove, large gas pressure gradients will be created at the openingwhich oppose solids discharge. There is therefore a decrease in massdischarge with decreasing powder permeability, or decreasing parti-cle size of the bulk solid. Verghese (Ph.D. thesis, University of Cam-bridge, UK, 1991) proposed an initial relation of the form

W = Wo�1 − 1�2

≈ 1.48 × 10−8 m2 (21-52)

The decrease in mass discharge rate from the Beverloo relation fordecreasing particle size is illustrated in Fig. 21-41. For fine enoughmaterials, bubbling and fluidization actually halt flow from the orifice,after which a gain in bulk density will again initialize flow. This may bewitnessed with fine sands discharging from hourglasses. A similar rela-tion based on venting required predicted from the Carman-Kozenyequation gives a fine powder mass discharge rate of

W ≈ (21-53)

where μg is gas viscosity and ε is the bed voidage.Gas venting may be used to increase discharge rate, either through

venting in the hopper wall or through imposed pressure gradients.The involved pressure drops or required air volumes my be calculatedfrom standard pressure drop correlations, based on, e.g., Darcy’s lawor the Ergun equation (cf. Sec. xx]. For air-augmented flow, dischargerates are given by

W = Wo�1 + � �1�2

for Reo < 10 (21-54)

W = Wo�1 + � �1�2

for Reo large (21-55)

W = Wo�1 + � � � �1�2

for intermediate Reo (21-56)

where ro is the radial distance from the hopper apex, ΔP�ro is the pres-sure drop imposed across the orifice, and Reo is the gas Reynoldsnumber acting at the orifice (see Nedderman, Statics and Kinematicsof Granular Materials, Cambridge University Press, 1992).

Other Methods of Flow Characterization A variety of othertest methods to characterize flowability of powders have been pro-posed, which include density ratios, flow from funnels and orifices,

ΔP�ρgro

2κ − 3�2κ − 1

150 + 5.25Reo��150 + 1.75Reo

3 ΔP�ρgro

2κ − 3�2κ − 1

ΔP�ρgro

2κ − 3�2κ − 1

2π(B�sin α)3 ρ2b d2

p g(1 − cos α)ε3

��180μg(1 − ε)3

λ�ρbg

λ�ρbgd2

p

1��1 − fc�σ1a

Wsc�2ρbgA

T�t


Au: plsinserteq. no.


angles of repose and sliding, simplified indicizer flow testing, and tum-bling, avalanche methods. These methods should be used with caution,as (1) they are often a strong function of the test method and instru-ment itself, (2) engineering properties useful for either scale-up or apriori design are not measured, (3) they are only a crude characteriza-tion of flowability, and often suffer from lack of reproducibility, (4) theylack a fundamental basis of use, and (5) they suffer from the absence ofvalidation powders and methods. This first two points are particularlycrucial, the end result of which is that the ranking of powders deter-mined by the apparatus cannot be truly linked to process performance,as the states of stress in the process may differ from the apparatus, andfurther, the ranking of powders may very well change with scale-up. Incontrast, shear cells and permeability properties may be used directlyfor design, with no need for arbitrary scales of behavior, and the effectof changing stress state with scale-up can be predicted. Having saidthis, many of these methods have found favor due to the misleadingease of use. In some defined cases they may be useful for quality con-trol, but should not be viewed as a replacement for more rigorous flowtesting offered by shear cell and permeability testing.

Various angles of repose may be measured, referring to the hori-zontal angle formed along a powder surface. These include the angleof a heap, the angle of drain for material remaining in a flat-bottombin, the angle of sliding occurring when a dish of powder is inclined,rolling angles in cylinders, and dynamic and static discharge anglesonto vibrating feed chutes (Thompson, Storage of Particulate Solids,in Handbook of Powder Science and Technology, Fayed and Otten(eds.), Van Nostrand Reinhold Co., 1984). From Eq. (21-37) describ-ing the impact of the angles of friction—as measured by shear cell—on cohesive strength, the angle of repose may be demonstrated to lacka true connection to flowability. For cohesive powders, there will belarge differences between the internal and effective angles of friction,and the unconfined strength increases with an increase in the differ-ence in sine of the angles. When one is measuring the angle of reposein this case, wide variations in the angle of the heap will be observed,and it likely varies between the angles of friction, making the mea-surement of little utility in a practical, measurement sense. However,when the difference in the angles of friction approached zero, theangle of repose will be equal to both the internal and effective angle offriction. But at that point, the cohesive strength of the powder is zero[Eq. (21-37)], regardless of the angle of repose.

In is likely the above has formed the basis for the use of rotatingavalanche testers, where the size and frequency of avalanchesformed on the sliding, rotating bed are analyzed as a deviation of thetime between avalanches, as well as strange attractor diagrams. This

approach is more consistent with the variation in the angle of reposebeing related to powder strength [Eq. (21-37)].

The typical density ratios are the Carr and Hausner ratios, givenby (“Density, Porosity, and Surface Area.”)

FICarr[%] = 100 and FIH[ − ] =

(21-57)

where ρb(tapped) is the equilibrium packed bulk density achieved undertapping. It could equally be replaced with a bulk density achievedunder a given pressure. The Carr index is a measure of compress-ibility, or the gain in bulk density under stress, and is directly relatedto gain in powder strength. Large gains in density are connected todifferences in the state of packing in the over- and critically consoli-dated state defined above (“Yield Behavior of Powders”), which inturn results in differences between the internal and effective angles offriction, leading to a gain in unconfined yield strength [Eq. (21-37)].However, the results are a function of the method and may not be dis-criminating for free-flowing materials. Lastly, changes in density areonly one of many contributions to unconfined yield stress and powderflowability. Hence, Carr and Hausner indices may incorrectly rankflowability across ranges of material class that vary widely in particlemechanical and surface properties.

Two methods of hopper flow characterization are used. The firstis the Flowdex™ tester which consists of a cup with interchangeablebottoms of varying orifice size. The cup is filled from a funnel, and thecovering lid then drops from the opening. The minimum orifice inmillimeter required for flow to occur is determined as a ranking offlowability. This minimum orifice is analagous to the minimum orificediameter determined from shear cell data for hopper design (see“Solids Handling: Storage, Weighing, and Feeding”). Alternatively,the mass discharge rate out of the cup or from a funnel may be deter-mined. Various methods of vibration both before and after initiation offlow may be utilized. Mass discharge rates, as expected, rank with thecorrelations described above. The disadvantage of this characteriza-tion method is that it is a direct function of hopper/cup geometry andwall friction, and has a low state of stress that may differ from theactual process. If a process hopper differs in vertical half-angle, wallfriction, opening size, solids pressure, filling method, or a range ofother process parameters, the ranking of powder behavior in practicemay differ from the lab characterization, since scalable engineeringproperties are not measured.

ρb(tapped)�

ρb(loose)

ρb(tapped) − ρb(loose)��

ρb(tapped)


FIG. 21-41 The impact of decreasing particle size and bulk permeability on mass discharge rate.


The last set of tests includes solid indicizers pioneered by Johann-son. These include the Flow Rate and Hang-Up Indicizers™ [cf. Bellet al., Practical Evaluaton of the Johanson Hang-Up Indicizer, BulksSolids Handling, 14(1), 117 (Jan. 1994)]. They represent simplied ver-sions of permeability and shear cell tests. Assumptions are made withregard to typical pressures and wall frictions, and based on these, a

flow ranking is created. Their degree of success in an application willlargely rest on the validity of the property assumptions. For definedconditions, they can give similar ranking to shear cell and permeabil-ity tests. The choice of use is less warranted than in the past due to theprogress in automating shear cell and permeability tests, which hassimplified their ease of use.

SOLIDS HANDLING: CONVEYING OF BULK SOLIDS 21-33

SOLIDS HANDLING: CONVEYING OF BULK SOLIDS

CONVEYOR SELECTION

Selection of the correct conveyor for a specific bulk material in a spe-cific situation is complicated by the large number of interrelated fac-tors that must be considered. First, the alternatives among basic typesmust be weighed, and then the correct model and size must be cho-sen. Workability is the first criterion, but the degree of performanceperfection that can be afforded must be established.

Because standardized equipment designs and complete engi-neering data are available for many common types of conveyors, theirperformance can be accurately predicted when they are used withmaterials having well-known conveying characteristics. However,even the best conveyors can perform disappointingly if material char-acteristics are unfavorable. It is often true that conveyor engineeringis more of an art than a science; problems involving unusual materialsor equipment should be approached with caution.

Many preengineered conveyor components can be purchased offthe shelf; they are economical and easy to assemble, and they performwell on conventional applications (for which they are designed). How-ever, it is advisable to check with the manufacturer to be sure that theapplication is proper.

Capacity requirement is a prime factor in conveyor selection.Belt conveyors, which can be manufactured in relatively large sizes tooperate at high speeds, deliver large tonnages economically. On theother hand, screw conveyors become extremely cumbersome as theyget larger and cannot be operated at high speeds without creating seri-ous abrasion problems.

Length of travel is definitely limited for certain types of convey-ors. With high-tensile-strength belting, the length limit on belt con-veyors can be a matter of miles. Air conveyors are limited to 305 m(1000 ft); vibrating conveyors, to hundreds of meters or feet. In gen-eral, as length of travel increases, the choice among alternativesbecomes narrower.

Lift can usually be handled most economically by vertical orinclined bucket elevators, but when lift and horizontal travel are com-bined, other conveyors should be considered. Conveyors that com-bine several directions of travel in a single unit are generally moreexpensive, but since they require only a single drive, this feature oftencompensates for the added base cost.

Material characteristics, both chemical and physical, should beconsidered, especially flowability. Abrasiveness, friability, and lumpsize are also important. Chemical effects (e.g., the effect of oil on rub-ber or of acids on metal) may dictate the structural materials out ofwhich conveyor components are fabricated. Moisture or oxidationeffects from exposure to the atmosphere may be harmful to the mate-rial being conveyed and require total enclosure of the conveyor oreven an artificial atmosphere. Obviously, certain types of conveyorslend themselves to such special requirements better than others.

Processing requirements can be met by some conveyors with lit-tle or no change in design. For example, a continuous-flow conveyormay provide a desired cooling of the solids simply because it puts theconveyed material into direct contact with heat-conducting metals.Screen decks can be readily attached to vibrating conveyors for simplesizing and scalping operations, and special flights or casings on screwconveyors are available for a wide variety of processing operationssuch as mixing, dewatering, heating, and cooling.

Initial cost of a conveyor system is usually related to life expectancyas well as to the flow rate chosen. There is a great temptation tooverdesign, which should be resisted. The first really long-distance belt

conveyor was designed and fabricated to extremely high standards ofquality. After 35 years it was still in operation with almost all its originalcomponents. Had this operation been planned for only a 10-year life,the conveyor system would have represented a bad case of overdesign.While there is a market for used conveyor equipment, it is extremelylimited. Thus, it is important to choose conveyor quality for expectedlife of project.

Comparative costs for conveyor systems can be based only onstudies of specific problems. For example, belt-conveyor idlers areavailable in a range of qualities that may make the best unit cost threetimes as much as the cheapest. Bearing quality, steel thickness, anddiameter of rolls all affect cost, as does design for easy maintenanceand repair. Therefore, it is necessary to make cost comparisons on thebasis of a specific study for each conveyor application.

As a general guide to conveyor selection, Table 21-7 indicates con-veyor choices on the basis of some common functions. Table 21-8 isdesigned to aid in feeder selection on the basis of the physical charac-teristics of the material to be handled. Table 21-9 is a coded listing ofmaterial characteristics to be used with Table 21-10, which describesthe conveying qualities of some common materials. While these tablesmay serve as valuable guides, conveyor selection must be based on theas-conveyed characteristics of a material. For instance, if packing oraerating can occur in the conveyor, the machine’s performance will notmeet expectations if calculations are based on an average weight percubic meter. Storage conditions, variations in ambient temperature and

TABLE 21-8 Feeders for Bulk Materials*

Material characteristics Feeder type

Fine, free-flowing materials Bar flight, belt, oscillating or vibrating,rotary vane, screw

Nonabrasive and granular materials, Apron, bar flight, belt, oscillating ormaterials with some lumps vibrating, reciprocating, rotary

plate, screwMaterials difficult to handle because Apron, bar flight, belt, oscillating orof being hot, abrasive, lumpy, or vibrating, reciprocatingstringy

Heavy, lumpy, or abrasive materials Apron, oscillating or vibrating,similar to pit-run, stone, and ore reciprocating

*From FMC Corporation, Material Handling Systems Division.

TABLE 21-7 Conveyors for Bulk Materials*

Function Conveyor type

Conveying materials horizontally Apron, belt, continuous flow, dragflight, screw, vibrating, bucket,pivoted bucket, air

Conveying materials up or down an Apron, belt, continuous flow, flight,incline screw, skip hoist, air

Elevating materials Bucket elevator, continuous flow,skip hoist, air

Handling materials over a combination Continuous flow, gravity-dischargehorizontal and vertical path bucket, pivoted bucket, air

Distributing materials to or collecting Belt, flight, screw, continuous flow,materials from bins, bunkers, gravity-discharge bucket, pivotedetc. bucket, air

Removing materials from rail cars, Car dumper, grain-car unloader, cartrucks, etc. shaker, power shovel, air

*From FMC Corporation, Material Handling Systems Division.

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