Critical Regimes of Two-Phase Flows with a Polydisperse Solid Phase (eBook)

FM 2
Critical Regimes of Two-Phase Flowswith a Polydisperse Solid Phase 2
Critical Regimes ofTwo-Phase Flows with aPolydisperse Solid Phase 4
Annotation 6
Introduction 8
Contents 14
Chapter 1: General Ideas of Mass Transfer Processes in Critical Regimes 18
Granulometric Characteristics of Bulk Material 18
Distribution of Different Fractions in the Process of Separation 22
Fractional Separation Curves and Their Properties 24
Initial Composition 27
Solid Phase Concentration in the Flow 28
Process Stability 30
Flow Velocity and Particle Size 30
Chapter 2: Principles of Modeling Processes in Moving Media 35
Correlation Between a Full-Scale Process and Its Model 35
Mathematical Models Construction 37
Similarity Criteria Determination 42
Chapter 3: System of Particles of the Same Size Class in a Critical Flow 48
Dynamics of Mass Motion of Particles in a Flow 48
Definition of a Statistical System 53
Estimation of the State of a Statistical System 59
Principal Statistical Characteristics of the Separation Factor 68
Chapter 4: System of Particles of Several Size Classes 73
Interaction of Particles in a Flow 73
Forces Caused by Interactions of Particles of Various Classes 78
Two-Phase Flow Entropy in Critical Flow Regimes 81
Main Features of Entropy in Critical Regimes 87
Mobility Factor 95
Statistical Identities 100
Chapter 5: Principal Statistical Relations of Mass Transfer in Critical Flow 106
Mass Exchange Between the Zone and the Apparatus 106
Determination of Average Values 109
Cell and Apparatus, Entropy 111
Separation at Low Concentrations 113
General Regularities for the Zone 117
Chapter 6: Correlation Between the Apparatus and the Cell 119
Coarse Particles Separation 119
Fine Particles Separation 120
Definition of Mass Transfer Parameters 121
Cellular Model of Separation 126
Physical Meaning of Separation Factors 130
Chaotizing Factor 130
Flow Mobility 130
Separation Factor 130
Concentration Effect 131
Potential Extraction 133
Extraction from a Cell Located in the Zone 134
Chapter 7: Structural Model of Mass Transfer in Critical Regimes of Two-Phase Flows 137
Validation of the Distribution Coefficient 137
Physical Meaning of the Distribution Coefficient 139
Turbulent Overflow of Particles and Turbulent Regime of the Medium Motion in the Apparatus 144
Laminar Overflow Regime 146
Intermediate Regime of Overflow 147
Analysis of Distribution Coefficient 148
Analysis of Experimental Dependencies from the Standpoint of Structural Models 153
Check of the Structural Model Adequacy 159
Correlation Between the Structural and Cellular Models of the Process 163
Chapter 8: Correlation Between Statistical and Empirical Results 165
Approximation of Universal Separation Curve 165
Principal Separation Parameters Depending on the Apparatus Height 168
Equal Extractability of Various Size Classes 172
Chapter 9: Entropy of Composition: Optimization Criterion 181
Entropy and Particles Stratification 181
Evaluation of Heterogeneity of Powder Composition 185
Binary Separation 187
Multi-product Separation 188
Algorithms of Optimization of Separation into n Components 189
Algorithm 1: Complete Sorting-Out 190
Algorithm 2: Greedy Algorithm 190
Optimization of Separation into Four Components 192
Mathematical Model of Separation into n Components 198
Optimum Conditions for Binary Separation 199
Optimum Conditions for Multi-Product Separation 201
Chapter 10: Stability and Kinetic Aspects of Mass Distribution in Critical Regimes 208
Entropy Stability 208
Particles Distribution over the Channel Height 215
Velocity Distribution of Particles of a Narrow Size Class 218
Kinetic Aspect of the Material Distribution 221
Chapter 11: Critical Regimes of Two-Phase Flows in Complicated Systems 226
Problem Setting 226
Mathematical Model of a Duplex Cascade 227
Mathematical Model of a Cascade Process Allowing Control of the Effect of the Material Feed Site on Separation Results 231
Cascade Model with Two or More Material Inputs into the Apparatus 234
Combined Cascade Classifiers 236
Combined Cascades of n(z)Type 236
Working Schemes for Combined Cascades of n(z) Type 238
Connection Functions for Combined Cascades 240
Experimental Verification of the Adequacy of Mathematical Models of Combined Cascades 245
Quality Criterion for Combined Cascades 248
Fractal Principle of the Construction of Schemes of Combined Classifiers 252
Fractal Principle of Combination 252
Progressive Nature of Multi-element Apparatuses 255
Combined Scheme with Successive Recirculation of Both Products 257
Combined Cascade with an Alternating Bypass of Both Products 258
On the Potential of Fractal Combined Schemes 263
Some Methods of Combined Schemes Optimization 266
Multi-row Classifier 266
Method of Estimating a Multi-row Classifier 269
Optimal Scheme of a Multi-row Industrial Classifier 271
Chapter 12: Stochastic Model of Critical Regimes of Two-Phase Flows 276
Principal Definitions 276
Statistical Description of Gravitational Separation in Turbulent Flows 278
Equations of Particles Motion Taking into Account Their Rotation Around the Center of Mass in a Turbulent Flow 282
Description of One-Dimensional Stationary Process of Gravitation Separation in a Turbulent Flow 285
One-Dimensional Model of a Non-stationary Process 289
Statistical Equations of a Random Process of Gravitational Separation 289
Computation of Fractional Separation of a Narrow Class 292
Approximate Computation Method 294
Chapter 13: Mass Transfer in Critical Regimes of Two-Phase Flows 298
Mathematical Model of a Separating Cascade 298
Discrete Stationary Model of Critical Regimes of Vertical Two-Phase Flows 316
Optimization of Principal Parameters of Multi-stage Separation 330
Chapter 14: Universal Curves Criteria 343
Substantiation of the Curves Universality 343
Generalizing Criteria 347
Turbulent Regimes of Particles Overflow 350
Laminar Regimes of Particles Overflow 352
Universal Curves 354
Bibliography 355

"Chapter 1 General Ideas of Mass Transfer Processes in Critical Regimes (p. 1-2)

Abstract Experimental studies have shown that despite a visual chaos in critical regimes of two-phase flows, there is a definite, almost deterministic order in the polyfractional solid phase distribution along the flow and counter the flow. The affinity of separation curves as a function of principal parameters of the flow in a turbulent regime is substantiated. Criteria of the affinization of separation curves are empirically established and experimentally substantiated.

Keywords Regularity - Granulometric composition - Size - Density - Concentration - Velocity - Process stability - Fractional extraction -Affinity -Separation curve -Productivity -Separation

1.1 Granulometric Characteristics of Bulk Material

First of all, we examine characteristics of a solid phase constituting a two-phase flow, because in critical regimes the process of particles separation according to their size grade or density can be organized most easily. Processing of ground materials is among the most widespread processes in today’s industry.

Many millions of tons of various materials are ground daily in mining, in various branches of chemical industry, in metallurgy, at the production of cement, ceramics, glass and other building materials, as well as in most novel branches of industry. Various natural and artificial materials become pourable when ground, and in this state they pass all the stages of technological processes, namely, extraction of useful components, production of powders with a specified particle size, compounding of necessary mixtures and compositions, treatment of particle surface and addition of various elements, drying, baking, etc. In this state it is convenient to granulate materials from particles of any composition or press products of any shape. As a rule, a solid phase is introduced into moving flows in this state only. At present, more and more low-quality raw materials are being processed because of growing production volumes.At the same time, the requirements for the quality of the final products constantly grow.

To meet these requirements, separation processes are becoming more and more important. Most often, the separation is performed by particle size or density, and more rarely – by shape, color or other parameters. While formerly, with a rather rough technology, it was sufficient to use various sieves for the separation by size, presentday operations with fine powders require separation carried out using moving media – air or water. As for separation by other parameters (density, particle shape), it can be realized only in moving media.

Before determining separation parameters, we examine principal characteristics of a bulk material. Such a material can be characterized by its specific density of particles, bulk density, humidity, porosity, etc. Usually a ground material contains, depending on its size grade, many millions of particles. These particles can differ in size, shape, surface state, etc. However, the material’s principal characteristic is connected with dispersity."