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Handbook of Mathematical Relations in Particulate Materials Processing

ISBN: 978-0-470-36872-5
350 pages
January 2009
Handbook of Mathematical Relations in Particulate Materials Processing (0470368721) cover image


The only handbook of mathematical relations with a focus on particulate materials processing

The National Science Foundation estimates that over 35% of materials-related funding is now directed toward modeling. In part, this reflects the increased knowledge and the high cost of experimental work. However, currently there is no organized reference book to help the particulate materials community with sorting out various relations. This book fills that important need, providing readers with a quick-reference handbook for easy consultation.

This one-of-a-kind handbook gives readers the relevant mathematical relations needed to model behavior, generate computer simulations, analyze experiment data, and quantify physical and chemical phenomena commonly found in particulate materials processing. It goes beyond the traditional barriers of only one material class by covering the major areas in ceramics, cemented carbides, powder metallurgy, and particulate materials. In many cases, the governing equations are the same but the terms are material-specific. To rise above these differences, the authors have assembled the basic mathematics around the following topical structure:

  • Powder technology relations, such as those encountered in atomization, milling, powder production, powder characterization, mixing, particle packing, and powder testing

  • Powder processing, such as uniaxial compaction, injection molding, slurry and paste shaping techniques, polymer pyrolysis, sintering, hot isostatic pressing, and forging, with accompanying relations associated with microstructure development and microstructure coarsening

  • Finishing operations, such as surface treatments, heat treatments, microstructure analysis, material testing, data analysis, and structure-property relations

Handbook of Mathematical Relations in Particulate Materials Processing is suited for quick reference with stand-alone definitions, making it the perfect complement to existing resources used by academic researchers, corporate product and process developers, and various scientists, engineers, and technicians working in materials processing.

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Table of Contents



About the Authors.


Abnormal Grain Growth.

Abrasive Wearâ??See Friction and Wear Testing.

Acceleration of Free-settling Particles.

Activated Sintering, Early-stage Shrinkage.

Activation Energyâ??See Arrhenius Relation.

Adsorptionâ??See BET Specific Surface Area.

Agglomerate Strength.

Agglomeration Force.

Agglomeration of Nanoscale Particlesâ??See Nanoparticle Agglomeration.

Andreasen Size Distribution.


Ball Millingâ??See Jar Milling.

Bearing Strength.

Bell Curveâ??See Gaussian Distribution.

Bending-beam Viscosity.

Bending Test.

BET Equivalent Spherical-particle Diameter.

BET Specific Surface Area.

Bimodal Powder Packing.

Bimodal Powder Sintering.

Binder Burnoutâ??See Polymer Pyrolysis.


Cantilever-beam Testâ??See Bending-beam Viscosity.


Capillarity-induced Sinteringâ??See Surface Curvature-Driven Mass Flow in Sintering.

Capillary Pressure during Liquid-phase Sinteringâ??See Mean Capillary Pressure.

Capillary Riseâ??See Washburn Equation.

Capillary Stressâ??See Laplace Equation.

Case Carburization.

Casson Model.

Cemented-carbide Hardness.

Centrifugal Atomization Droplet Size.


Darcyâ??s Law.

Debindingâ??See Polymer Pyrolysis, Solvent Debinding Time, Thermal Debinding Time, Vacuum Thermal Debinding Time, and Wicking.

Debinding Master Curveâ??See Master Decomposition Curve.

Debinding Temperature.

Debinding Timeâ??See Solvent Debinding Time, Thermal Debinding Time, Vacuum Thermal Debinding Time, and Wicking.

Debinding by Solvent Immersionâ??See Solvent Debinding Time.

Debinding Weight Loss.

Delubricationâ??See Polymer Pyrolysis.


Densification in Liquid-phase Sinteringâ??See Dissolution-induced Densification.


Effective Pressure.

Ejection Stressâ??See Maximum Ejection Stress.

Elastic Behaviorâ??See Hookeâ??s Law.

Elastic deformation Neck-size Ratio.

Elastic-modulus Variation with Density.

Elastic-property Variation with Porosity.

Electrical-conductivity Variation with Porosity.

Electromigration Contributions to Spark Sintering.


Elongation Variation with Densityâ??See Sintered Ductility.


Feedstock Formulation.

Feedstock Viscosityâ??See Suspension Viscosity and Viscosity Model for Infection-molding Feedstock.

Feedstock Viscosity as a Function of Shear Rateâ??See Cross Model.

Feedstock Yield Strengthâ??See Yield Strength of Particle-Polymer Feedstock.

Fiber-fracture from Buckling.

Fiber-fracture Probability.

Fiber Packing Density.

Fickâ??s First Law.

Fickâ??s Second Law.

Field-activated Sintering.


Gas-absorption Surface Areaâ??See BET Specific Surface Area.

Gas-atomization Cooling Rate.

Gas-atomization Melt Flow Rate.

Gas-atomization Particle Size.

Gas-generated Final Pores.

Gas Permeabilityâ??See Kozeny-Carman Equation.

Gate Strain Rate in Injection Molding.

Gaudin-Schuhmann distribution.

Gaussian Distribution.

Gel-densification Model.


Hall-Petch Relation.

Hardenability Factor.


Hardness Variation with Grain Size in Cemented Carbides.

Heating-rate Effect in Transient Liquid-phase Sintering.

Heat Transfer in Sintered Materials.

Heat-transfer Rate in Modelingâ??See Cooling Rate in Molding.

Herring Scaling Law.

Hertzian stressâ??See Elastic Deformation Neck-size Ratio.

Heterodiffusionâ??See Mixed-powder Sintering Shrinkage.


Impregnationâ??See Infiltration Pressure.

Inertial-flow Equation.

Infiltration Depth.

Infiltration Pressure.

Infiltration Rate.

Inhibited Grain Growthâ??See Zener Relation.

Initial-stage Liquid-phase Sintering Stressâ??See Sintering Stress in Initial-stage Liquid-phase Sintering.

Initial-stage Neck Growth.

Initial-stage Sinteringâ??See Surface Diffusion-Controlled Neck Growth.

Initial-stage Sintering Modelâ??See Kuczynski Neck-growth Model.


Jar Milling.

Jet Mixing Time.


Kawakita Equation.

Kelvin Equation.

Kelvin Modelâ??See Viscoelastic Model for Powder-Polymer Mixtures.


Kingery Intermediate-stage Liquid-phase Sintering Modelâ??See Intermediateâ??stage Liquid-phase Sintering Model.

Kingery Model for Pressure-assisted Liquid-phase Sinteringâ??See Pressure-assisted Liquid-phase Sintering.

Kingery Rearrangement Shrinkage Kineticsâ??See Rearrangement Kinetics in Initial-stage Liquid-phase Sintering.

Kissinger Method.

Knoop Hardness.

Knudsen Diffusionâ??See Vapor Mean Free Path.


Laminar Flow Settlingâ??See Stokesâ?? Law.

Laplace Equation.

Laplace Numberâ??See Suratman Number.

Laser Sintering.

Lattice Diffusionâ??See Vacancy Diffusion.

Lifschwiz, Slyozov, Wagner Model.

Ligament Pinchingâ??See Raleigh Instability.

Limiting Neck Size.

Limiting Size for Sedimentation Analysis.

Liquid and Solid Compositions in Prealloy Particle Melting.


Macroscopic Sintering Model Constitutive Equations.

Magnetic Coercivity Correlation in Cemented Carbides.

Mass Flow Rate in Atomizationâ??See Gas-atomization Melt Flow Rate.

Master Decomposition Curve.

Master Sintering Curve.

Master Sintering Curve for Grain Growthâ??See Grain-growth Master Curve.

Maximum Density in Pressure-assisted Sintering.

Maximum Ejection Stress.

Maximum Grain Size in Sintering.

Maximum Lubricant Content.


Nabarro-Herring Creep-controlled Pressure-assisted Densification.

Nanoparticle Agglomeration.

Nanoparticle Melting-point Depression.

Nanoscale Particle-Agglomerate Spheroidization.

Nanoscale Particle-size Effect on Surface Energyâ??See Surface-energy variation with Droplet Size.

Neck-curvature Stress.

Neck Growth Early in Liquid-phase Sintering.

Neck Growth-induced Shrinkageâ??See Shrinkage Relation to Neck Size.

Neck Growth Limited by Grain Growth.

Neck-growth Modelâ??See Kuczynski neck-growth Model.


Open-pore Content.

Optimal Packing Particle-size Distributionâ??See Andreasen Size Distribution.

Optimal Mixer Rotational Speed.

Ordered Packing.

Osprey Processâ??See Spray Deposition.

Ostwald Ripening.

Oxide Reduction.


Packing Density for Log-normal Particles.

Particle Cooling in Atomizationâ??See Newtonian Cooling Approximation.

Particle Coordination Numberâ??See Coordination Number and Density.

Particle Diffusion in Mixing.

Particle Fracture in Milling.

Particle Packing.

Particle-shape Index.

Particle Sizeâ??See Equivalent Spherical Diameter and Mean Particle Size.

Particle-size Analysisâ??See Sieve Progression.

Particle size and Apparent Density.


Quantitative-microscopy Determination of Surface Areaâ??See Surface Area by Quantitative Microscopy.

Quasi-3-dimensional Energy-governing Equation for Powder Injection Moldingâ??See Energy-governing Equation for Powder Injection Molding.

Quasi-3-dimensional Pressure-governing Equation for Powder Injection Modelingâ??See Pressure-governing Equation in Powder Injection Molding.

Quasi-3-dimensional Pressure-governing Equation for Powder Injection Molding with Slip-layer Modelâ??See Pressure-governing Equation in Powder Injection Molding with Slip-layer Model.

Quasi-3-dimensional Pressure-governing Equation for Powder Injection Molding with Slip-velocity Modelâ??See Pressure-governing Equation in 2.5 Dimensions for Powder Injection Molding with Slip-velocity Model.


Radial Crush Strengthâ??See Bearing Strength.

Radiant Heating.

Raleigh Instability.

Random Packing Density.

Random Packing Radial-distribution Function.

Reaction-controlled Grain Growthâ??See Grain-growth Master Curve, Interfacial Reaction Control, and Interface-controlled Grain Growth.

Reaction-rate Equationâ??See Avrami Equation.

Reactive Synthesis.

Rearrangement Kinetics in Liquid-phase Sintering.

Recalescence Temperature.


Saddle-surface Stressâ??See Neck-curvature Stress.

Scherrer Formula.

Screen Sizesâ??See Sieve Progression.

Secondary Dendrite-Arm Spacing.

Secondary Recrystallizationâ??See Abnormal Grain Growth.

Second-stage Liquid-phase Sintering Modelâ??See Intermediate-stage Liquid-phase Sintering Model.

Second-stage Sintering Densificationâ??See Intermediate-stage Sintering-density Model.

Second-stage Sintering Pore Eliminationâ??See Intermediate-stage Pore Elimination.

Second-stage Sintering Surface-area Reductionâ??See Intermediate-stage Surface-area Reduction.

Sedimentation Particle-size Analysisâ??See Stokesâ?? Law Particle-size Analysis.


Tap Densityâ??See Vibration-induced Particle Packing.

Temperature Adjustments for Equivalent Sintering.

Temperature Dependenceâ??See Arrhenius Relation.

Terminal Densityâ??See Final-stage Sintering Limited Density.

Terminal Neck Sizeâ??See Neck Growth Limited by Grain Growth.

Terminal Neck Size in Sinteringâ??See Limiting Neck Size.

Terminal Pore Sizeâ??See Final-stage Pore Size.

Terminal Settling Velocityâ??See Stokesâ?? Law.

Terminal Sinteringâ??See Trapped-gas Pore Stabilization.

Terminal Velocityâ??See Acceleration of Free-settling Particles.


Ultrasonic Velocity.


Vacancy Concentration Dependence on Surface Curvature.

Vacancy Diffusion.

Vacuum Debindingâ??See Vacuum Thermal Debinding.

Vacuum Distillation Rate.

Vacuum Flux in Sintering.

Vacuum Thermal Debinding.

Vapor Mean Free Path.

Vapor Pressure.

Vibration-induced Particle Packing.

Vickers Hardness Number.


Washburn Equation.

Water-atomization Particle Size.

Water Immersion Densityâ??See Archimedes Density.

Weber Number.

Weibull Distribution.

Wetting Angle.


Work Hardeningâ??See Strain Hardening.

Work of Sinteringâ??See Master Sintering Curve.


X-ray Line Broadeningâ??See Scherrer Formula.


Yield Strength in Viscous Flowâ??See Bingham Viscous-flow Model.

Yield Strength of Particleâ??Polymer Feedstock

Youngâ??s Equationâ??See Contact Angle and Wetting Angle.

Youngâ??s Modulusâ??See Elastic Modulus.


Zener Relation.

Zeta Potential.



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Author Information

Randall M. German, PhD, is the CAVS Chair Professor of Mechanical Engineering and Director of the Center for Advanced Vehicular Systems at Mississippi State University. He holds an Honorary Doctorate from the Universidad Carlos III de Madrid in Spain, is a Fellow of APMI and ASM, holds the Tesla Medal, and is listed in various issues of Who's Who. His accomplishments comprise 850 published articles, twenty-three issued patents, nineteen edited proceedings, and fourteen books, including Sintering Theory and Practice (Wiley).

Seong Jin Park, PhD, is Associate Research Professor in the Center for Advanced Vehicular Systems at Mississippi State University. He is the recipient of numerous awards and honors, including Leading Scientists of the World and Outstanding Scientists Worldwide, both awarded by the International Biographical Centre in 2007. Dr. Park is the author of over 190 published articles and three books, holds four patents, and created four commercialized software programs. His areas of specialization and interest include materials processing technology, numerical technology, and physics.

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"Suited for quick reference with stand alone definitions. It is the perfect complement to existing
textbooks since it will simply cut to the key relations. " (Alamogordo Daily News, 16 March 2011)


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