Sedimentation and Thickening

MATHEMATICAL MODELLING: Theory and Applications

VOLUME 8

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The titles published in this series are listed at the end of this volume.

Sedimentation and Thickening Phenomenological Foundation and Mathematical Theory

by

Maria Cristina Bustos Department of Mathematical Engineering, University of Concepcion, Chile

Fernando Concha Department of Metallurgical Engineering, University of Concepcion, Chile

Raimund BUrger Institute of Mathematics A, University of Stuttgart, Germany

and

Elmer M. Tory Department of Mathematics and Computer Science, Mount Allison University, Sackville, New Brunswick, Canada

Springer-Science+Business Media, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Printed on acid-free paper

All Rights Reserved

ISBN 978-90-481-5316-9 ISBN 978-94-015-9327-4 (eBook) DOl 10.1007/978-94-015-9327-4

© 1999 Springer Science+Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1999.

Softcover reprint of the hardcover 1 st edition 1999

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

This book is dedicated to

PROFESSOR DR.-ING. WOLFGANG L. WENDLAND,

Professor of Mathematics at the University of Stuttgart.

His constant support of research on sedimentation for nearly two decades has made this book possible.

Table of Contents

Contents

Preface

Introduction Sedimentation processes in history Modern thickening research

1 Theory of mixtures 1.1 Introduction... 1.2 Theory of mixtures

1.2.1 Kinematics 1.2.2 Mass balance 1.2.3 Linear momentum balance.

2 Sedimentation of ideal suspensions 2.1 Introduction .................. . 2.2 Kinematical sedimentation process ..... .

2.2.1 Kynch theory of batch sedimentation. 2.2.2 Kynch theory of continuous sedimentation.

3 Sedimentation with compression 3.1 Introduction ..... 3.2 Macroscopic balance .. . 3.3 Localbalances ..... . 3.4 Constitutive assumptions

3.4.1 Kinematical constraints 3.4.2 Dynamical constraints 3.4.3 Constitutive equations

3.5 Dimensional analysis .... . 3.6 Dynamical variables .... .

3.6.1 Solid effective stress and pore pressure 3.6.2 Interaction force at equilibrium 3.6.3 Excess pore pressure .......... .

vii

vii

xi

1 1 4

7 7 7 8

17 22

27 27 27 29 31

35 35 38 39 40 41 42 43 44 46 46 48 48

viii Table of Contents

3.6.4 Solid flux density function . . . . . . 3.6.5 Dynamical sedimentation processes.

3.7 Extension to several space dimensions . . .

49 50 50

4 The initial value problem for a scalar conservation law 52 4.1 Weak solutions for a scalar conservation law . 52 4.2 Method of characteristics .......... 53 4.3 Uniqueness of the solution. . . . . . . . . . . 54

4.3.1 Oleinik's condition E (Oleinik 1957) . 55 4.3.2 Lax's shock admissibility criterion (Lax 1957, 1971, 1973) 57 4.3.3 Entropy admissibility criterion (Lax 1971, 1973) .... 58 4.3.4 Viscosity admissibility criterion (Hopf 1969, Lax 1971) . 59 4.3.5 Kruzkov's formulation (Kruzkov 1970) 60 4.3.6 Uniqueness of the solution. . . . . . . . 61

4.4 Existence of the global weak solution ..... 63 4.4.1 Properties of the Lax-Friedrichs scheme 63 4.4.2 Convergence of the Lax-Friedrichs scheme 69

5 The Riemann problem for a scalar conservation law 5.1 Introduction ....................... . 5.2 The Riemann problem for a convex flux density function . 5.3 Flux density function with one inflection point .

5.3.1 Properties of the flux density function .. 5.3.2 Construction of the global weak solution.

5.4 Flux density function with two inflection points. 5.4.1 Geometrical properties of the flux density function 5.4.2 Construction of the global weak solution . . . . . .

72 72 73 76 76 79 83 83 87

6 The initial-boundary value problem for a scalar conservation law 95 6.1 Formulation of the problem . . . . . . . 95 6.2 Characterization of the entropy solution 96 6.3 Entropy conditions . . . . . . . . 100 6.4 Existence of the entropy solution . . . . 105 6.5 Uniqueness and admissible states. . . . 107

6.5.1 Admissible states at the boundaries 108 6.5.2 Geometrical interpretation of the sets of admissible states 109

7 Batch sedimentation of ideal suspensions 7.1 Initial value problem . . . . . 7.2 Modes of sedimentation . . . . . . . . . . 7.3 Construction of the solution ....... .

7.3.1 Flux density function with one inflection point 7.3.2 Flux density function with two inflection points.

7.4 Non-homogeneous initial concentration .... 7.5 Numerical computation of curved trajectories ..... .

111 111 112 113 113 119 133 137

Table of Contents

7.6 Dafermos' polygonal approximation method 7.6.1 Polygonal flux-density function ... 7.6.2 Continuous flux-density function .. 7.6.3 Application to batch sedimentation.

8 Continuous sedimentation of ideal suspensions 8.1 Mathematical model for continuous sedimentation

lX

139 139 142 143

149 149

8.2 Modes of continuous sedimentation. . . . . . . . . 150 8.3 Flux density function with one inflection point .. 151

8.3.1 Case I: Both f'(a) and f'(r.poo) are positive 151 8.3.2 Case II: f'(a) is positive and f'(r.poo) is negative 156 8.3.3 Case III: Both f'(a) and f'(r.poo) are negative 159

8.4 Flux density function with two inflection points . . . . . 162 8.4.1 Case I: f'(a), f'(b) and f'(r.poo) are positive . . . 163 8.4.2 Case II: f'(a) is positive and f'(b) and f'(r.poo) are negative 168 8.4.3 Case III: f'(a), f'(b) and f'(r.poo) are negative. 172

8.5 Control of continuous sedimentation 176 8.5.1 Model of the control problem . . . . . . . . . . 177 8.5.2 Construction of the entropy solution 178

9 Mathematical theory for sedimentation with compression 184 9.1 The initial-boundary value problem. . . . . . . . . . . 184

9.1.1 Initial and boundary conditions. . . . . . . . . 184 9.1.2 Type degeneracy and smoothness assumptions 185

9.2 Definition of generalized solutions. 186 9.2.1 The space BV(QT). . . . . . . . . . . . . . . . 186 9.2.2 Definition of generalized solutions 187

9.3 Jump condition . . . . . . . . . . . 188 9.4 Entropy boundary condition. . . . 191 9.5 Existence, uniqueness and stability 191

9.5.1 The regularized problem. . 191 9.5.2 Existence of the solution of the regularized problem 193 9.5.3 Existence of a generalized solution ....... . 9.5.4 Stability and uniqueness of generalized solutions

9.6 Properties of generalized solutions ......... . 9.6.1 Range of generalized solutions ....... . 9.6.2 Construction of the boundary value at z = L 9.6.3 Entropy boundary condition at z = L 9.6.4 Boundary condition at z = 0 ..... 9.6.5 Monotonicity of concentration profiles

9.7 Discontinuous solid effective stress function .

193 194 194 194 195 195 197 197 198

x Table of Contents

10 Numerical simulation of sedimentation with compression 200 10.1 Numerical algorithm 201 10.2 Parameters . . . . . . . . . . . . . . . . . . . . 204 10.3 Batch sedimentation . . . . . . . . . . . . . . . 205

10.3.1 Batch settling of a uniform suspension. 205 10.3.2 Repeated batch sedimentation ..... 205 10.3.3 A membrane problem . . . . . . . . . . 206 10.3.4 Expansion of overcompressed sediment. 207 10.3.5 Simultaneous expansion and batch sedimentation . 207

10.4 Continuous thickening . . . . . . . . . . . . . . 208 10.4.1 Filling and emptying of a thickener. . . 208 10.4.2 Transition between three steady states. 209

11 Thickener design 215 11.1 Introduction: definition, equipment and

operation . . . . . . . . . 215 11.2 Classcial methods. . . . . . . . . . . 216

11.2.1 Mishler's equation . . . . . . 216 11.2.2 Coe and Clevenger's method 217

11.3 Kinematical design methods. . . . . 219 11.3.1 Analysis of the batch sedimentation curve 220 11.3.2 Design methods based on a batch sedimentation process 221 11.3.3 Thickener design methods based on a continuous

Kynch sedimentation process . . . . . . . . . . . . . . .. 225 11.4 Design method based on a dynamical sedimentation process . .. 229

11.4.1 Sedimentation of a compressible suspension at steady state 229 11.4.2 Capacity of an ICT treating a flocculated susp~nsion . 230 11.4.3 Adorjan's method of thickener design ............ 234

12 Alternate treatments and open problems 236 12.1 Introduction. . . . . . . 236 12.2 Inertial and end effects . . . . . . . . . . . 237 12.3 Heterogeneity problems .......... 238

12.3.1 Spatial heterogeneity of homogeneous components 238 12.3.2 Heterogeneity of solid particles 240

12.4 Constitutive equations . 242 12.5 Channeling and collapse 249 12.6 Roberts' equation. . . . 251

Bibliography 253

Notation Guide 265

Subject Index 275

Author Index 283

Preface

The aim of this book is to present a rigorous phenomenological and mathematical formulation of sedimentation processes and to show how this theory can be applied to the design and control of continuous thickeners. The book is directed to stu­dents and researchers in applied mathematics and engineering sciences, especially in metallurgical, chemical, mechanical and civil engineering, and to practicing en­gineers in the process industries. Such a vast and diverse audience should read this book differently. For this reason we have organized the chapters in such a way that the book can be read in two ways. Engineers and engineering students will find a rigorous formulation of the mathematical model of sedimentation and the exact and approximate solutions for the most important problems encountered in the laboratory and in industry in Chapters 1 to 3, 7 and 8, and 10 to 12, which form a self-contained subject. They can skip Chapters 4 to 6 and 9, which are most important to applied mathematicians, without losing the main features of sedimentation processes. On the other hand, applied mathematicians will find special interest in Chapters 4 to 6 and 9 which show some known but many recent results in the field of conservation laws of quasilinear hyperbolic and degenerate parabolic equations of great interest today. These two approaches to the theory keep their own styles: the mathematical approach with theorems and proofs, and the phenomenological approach with its deductive technique.

A great part of the theory, model formulation and results shown in this book is original and based on the publications of the authors from the sixties to the present day. The most distinctive feature of the book is its combination of a rigorous axiomatic phenomenological approach with a concern for mathematical correctness and its application to the most important practical industrial problems.

In Chapter 1 we give a rigorous, but limited, presentation of the theory of mixtures. We feel that, since this is a book on mechanics of particle-fluid motion, there is no necessity to develop the thermodynamics of mixtures. This chapter gives the framework for developing the concepts of a sedimentation process and the tools to study the two models, the kinematical and the dynamical sedimentation processes, which are presented in Chapters 2 and 3. An introduction discusses the condition required for a multiphase particle system to be regarded as a continuum. Next, the concept of mixture, components and configuration are laid down and the ideas of mass and deformation function for each component and for the mixture are discussed. The measures of deformation and motion lead to the macroscopic mass

Xl

xii Preface

and momentum balances. For regions where the variables are continuous, these balances yield the local mass and momentum balances while at discontinuities mass and momentum jump balances are obtained. A dynamical process is defined and the necessity of formulating constitutive equations for the stresses and the interaction forces is established.

Chapter 2 analyses the mixture of fine particles and a fluid regarded as super­imposed continua. The concepts of an ideal suspension and of an ideal thickener are presented and the field equations for the mass of the solid component and of the mixture are established. The models for batch and for continuous sedimen­tation are formulated and the kinematical or Kynch sedimentation processes are defined.

Chapter 3 studies the sedimentation of flocculated suspensions. The assump­tions are established for those suspensions to be regarded as superimposed con­tinua. Macroscopic and local balances are established and constitutive equations are formulated. Dimensional analysis shows that the convective terms are sev­eral orders of magnitude smaller than the other terms in the equation and can be eliminated from the local balances. Experimental variables are replaced for the solid and liquid component pressures and the dynamical sedimentation process is defined.

Chapters 4, 5 and 6 are formulated without appealing to sedimentation prob­lems. Chapter 4 collects a variety of well-known results for initial value problems or Cauchy problems of scalar hyperbolic conservation laws. Weak solutions are defined and the method of characteristics is used to obtain the solution. At the intersection point of the characteristic lines, the solution is no longer unique and entropy conditions are imposed. Riemann problems of scalar conservation laws are solved. Chapter 5 studies all possible forms of solutions of Riemann prob­lems with flux density functions with up to two inflection points. In Chapter 6 the theoretical framework for hyperbolic conservation laws is extended to initial­boundary value problems. Initial data are given only on a closed interval rather than on the whole real axis and, at the boundaries of the computational domain corresponding to the endpoints of that interval, solution values are prescribed as time-dependent functions. However, these boundary data are not always assumed in a pointwise sense by entropy weak solutions. Entropy boundary conditions are derived and characterized, and existence and uniqueness of entropy weak solu­tions of the initial-boundary value problem are shown by the vanishing viscosity method.

Chapters 7 and 8 apply the previous results to problems of batch and contin­uous sedimentation of ideal suspensions. Batch sedimentation is described by two adjacent Riemann problems in Chapter 7. Here these are solved for flux density functions with up to two inflection points giving seven types of solutions, the so­called modes of sedimentation. Finally Dafermos' polygonal approximate method is used to solve batch sedimentation problems, allowing an a priori error estimate. In Chapter 8, one Riemann problem located in the interior of the ideal continuous thickener is considered as an initial state and the upper and lower constant are extended to the boundaries. All possible weak solutions of this configuration for

Preface Xlll

flux density functions with up to two inflection points are determined, giving rise to five modes of continuous sedimentation.

Chapter 9 formulates batch and continuous sedimentation of flocculated sus­pensions as an initial-boundary value problem for the governing quasilinear de­generate parabolic equation. An appropriate definition of weak or generalized solutions of this initial-boundary problem is formulated, from which jump and en­tropy boundary conditions are derived. Existence and uniqueness results for this problem and special properties of its solutions are summarized.

Chapter 10 presents a numerical algorithm together with a selection of exam­ples of simulated batch and continuous sedimentation processes illustrating the predictive power of the mathematical model of sedimentation with compression.

In the light of the discussion in the previous chapters, the different methods of thickener design that have been proposed in the literature are described and analyzed in Chapter 11. Three types of methods are distinguished: those based on macroscopic balances, those based on kinematical models and those based on dy­namical models. All, from Mishler's and Coe and Clevenger's methods to Kynch's, Talmage and Fitch's, Oltman's, Yoshioka and Hassett's, Wilhelm and Naide's and Adorjan's methods fall into these categories. This classification permits the anal­ysis of thickener design procedures with a clear perspective of their applicability and limitations.

Finally in Chapter 12 alternative treatments to sedimentation than those based on continuum mechanics are looked at, in particular cases where Kynch's theory is a useful approximation although his assumptions do not hold. Open problems are also examined and we try to assess which can be treated within the theory of mixtures and which may require other treatments.

We wish to acknowledge financial support of Dr. Elmer Tory as Visiting Pro­fessor at the University of Concepcion by the Iberoamerican Program of Science and Technology for Development (CYTED) and of Dr. Raimund Burger as Post­doctoral Fellow by Fundacion Andes/ Alexander-von-Humboldt-Stiftung, project 0-13131, and by FONDEF project D97-I2042.

We thank the University of Concepcion, its Engineering Faculty, its Faculty of Physics and Mathematics, its Research Council and its Graduate School for the permanent support they have given us during the last 25 years of research in the area of sedimentation. Our thanks also go to FONDECYT through the many projects they have financed during this period. We cannot forget to thank our many graduate and undergraduate students who have contributed to our work.

The research cooperation between the University of Concepcion and first the Technical University of Darmstadt and then the University of Stuttgart was made possible by the German Academic Exchange Service (DAAD) and by the German Research Foundation (DFG), which continues to support research on sedimenta­tion within the Sonderforschungsbereich 404 at the University of Stuttgart. The preparation of the final version of this book has also benefitted from the research cooperation of the University of Stuttgart with the University of Bergen, Nor­way, supported by the recently initiated 'Applied Mathematics for Industrial Flow Problems (AMIF)' program of the European Science Foundation (ESF).