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Modeling of continuous particle classification in a liquid medium Chen, Aihua 2000

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M O D E L I N G O F C O N T I N U O U S P A R T I C L E C L A S S I F I C A T I O N IN A LIQUID M E D I U M By AIHUA C H E N B.Eng., Tsinghua University, P.R.China, 1988 M.Eng., Chinese Academy of Sciences, P.R.China, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Chemical & Biological Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A October, 2000 © Aihua Chen, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract Abstract Previous studies of sedimentation, wherein particles fall under the influence of gravity through a fluid in which they are suspended, are first reviewed. This technique can be used to separate particles having different settling velocities. Various hydrodynamic models, including a batch, a differential and a continuous model, of the particle segregation and classification in liquid fluidized beds for binary and polydisperse systems are described. A stochastic model, the Markov Chain Model, recently written for the liquid classifier by Zhang (1998) of our group, is also introduced. The emptying phenomenon encountered in industrial classifiers can be explained in terms of a limiting voidage in the column of the classifier, below which the particles cannot move downwards and be removed from the bottom of the classifier. Continuous classification of particles by size is studied in a solid- liquid classifier of 191 mm diameter and 1540 mm height under steady and unsteady state conditions, similar in geometry to industrial units. Spherical glass beads of uniform density, and Rosin-Rammler particle size distributions, were employed in the tests. The mean particle diameter and the operating conditions were dynamically similar to those used in industry. During the particle classification operation, a dense suspension of particles in water enters the classifier through a radial feed port near the top, while water without particles is injected upwards from the bottom. A relatively dilute stream containing mostly small particles is taken off the top as an overflow stream, while an underflow stream enriched in coarser particles is removed near the base of the column. Differential hydrodynamic models are developed to describe the steady and the unsteady state motions of the particles and liquid, based on mass and momentum ii Abstract conservation laws and knowledge of the sedimentation behavior of individual species within the mixture. The boundary conditions require that the particle concentration in the overflow stream be equal to the concentration at the top of the classifier and the particle concentration at any vertical level in the discharge stream equal the horizontally adjacent concentration in the column of the classifier. The correlation of Di Felice (1994) is used to calculate the drag force on the particles, and the particle dispersion force is introduced according to the concept of Thelen and Ramirez (1999). Any turbulence in the flow is taken into account indirectly via an axial dispersion coefficient, assumed to be uniform throughout the classifier. This sole fitted parameter is correlated in terms of the relevant dimensionless parameters. The degree of classification becomes better with increasing feed voidage, feed flow rate and fluidizing liquid flow rate, but is worse at higher underflow discharge rates. The performance of the classifier is better for a broad than for a narrow particle size distribution of the feed stream. The classification can be improved by increasing the height of the cylindrical zone. Predictions of the model agree reasonably well with experimental results. ii i Table of Contents TABLE OF CONTENTS Abstract ii List of Tables vi List of Figures vii Acknowledgements xii Chapter 1 Introduction 1 Chapter 2 Settling of Particles in Liquids 6 2.1 Free Settling of a Single Particle 6 2.2 Hindered Settling in Monodisperse Systems 10 2.2.1 Settling velocity 10 2.2.2 Drag Force 12 2.3 Applications and Modifications for Polydisperse Systems 13 2.4 Particle Segregation and Classification in Liquid Fluidized Beds 17 2.4.1 Segregation and Mixing of Binary Particles 18 2.4.2 Sedimentation of Multi-Species Particles 21 2.4.3 Markov Chain Model of Liquid Classifier 28 2.5 Explanation of Emptying Phenomenon in Industrial Classifier 35 Chapter 3 Experiments 40 3.1 Equipment and Particles 40 3.1.1 Equipment 40 3.1.2. Particles: 46 3.2 Procedures 48 3.2.1 Operating Conditions 48 3.2.2 Measurement of Parameters and Methods 49 3.3 Conductivity Probe 51 Chapter 4 Steady State Hydrodynamic Model 57 4.1 Conceptual Basis 57 4.2 Governing Equations and Boundary Conditions 60 4.3 Fluid Particle Drag and Particle Dispersion Force 67 4.4 Equations for the Eight Zones of the Classifier 69 4.5 Calculation Procedure 77 4.6 Calculated Results and Comparison with Experimental Data 83 4.7 Correlation of Axial Dispersion Coefficient 94 4.8 Predictions of Classifier Performance and Efficiency and Comparison with Experimental Data: Multi-Size Systems 107 4.8.1 Local Voidage 107 4.8.2 Voidage in Lower Portion of Column, Discharge and Overflow 110 iv Table of Contents 4.8.3 Increments of Mean Particle Size and Breadth of PSD 115 4.8.4 Classifier Efficiency 121 4.9 Comparison with Markov Chain Model 127 Chapter 5 Unsteady State Hydrodynamic Model 132 5.1 Governing Equations, Initial and Boundary Conditions 132 5.2 Equations for the Eight Zones of the Classifier 136 5.3 Numerical Methods 143 5.4 Predictions and Comparison with Experimental Data 147 Chapter 6 Conclusions and Recommendations 156 6.1 Conclusions 156 6.2 Recommendations 159 Nomenclature 160 References 165 Appendix A: Dimensions of UBC Classifier for Different Configurations 172 Appendix B: Tabulation of Experimental Operating Conditions and Experimental Results in UBC Runs 173 Appendix C: Sample Calculation in Calibrating Conductivity Probe 186 Appendix D: Calibration Curves for the Nine Conductivity Probes 187 Appendix E: Cross-Sectional Area of Conical Column and its Gradient 192 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model 193 Appendix G: Explanation of the Subroutine IVPAG in Microsoft Fortran PowerStation 4.0 (1994-1995) 217 Appendix H: Explanation of the Subroutine NEQNF and NEQNJ in Microsoft Fortran PowerStation 4.0 (1994-1995) 224 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model 228 v List of Tables List of Tables Table 2.1.1. Recommended drag correlations from Gift et al. (1978) 9 Table 2.1.2. Terminal settling velocity correlations for spheres (Grace, 1986) 10 Table 2.2.1. Richardson-Zaki (1954) correlation for index n with d p « D 12 Table 2.2.2. Correlations for the index n in Richardson and Zaki equation 12 Table 3.2.1. Measured parameters and methods 49 Table 3.3.1. Values of constant coefficients in third-order polynomial conductivity probe calibration 56 Table 4.5.1. Input data required by model 81 Table 4.7.1. Axial dispersion coefficient in different systems 94 Table 4.7.2. Number of data from different classifiers at U B C and M C C 101 Table 4.7.3. Exponents of parameters with respect to the axial dispersion coefficient. 102 Table B . l . Experimental operating conditions at U B C 173 Table B.2. Experimental data for mono-disperse suspensions with T-2 configuration .181 Table B.3. Experimental data for binary systems with dp=0.9/0.55mm and T-2 configuration 182 Table B.4. Some experimental data for multi-size systems with C-0 configuration 183 Table B.5. Some experimental data for multi-size systems with T-2 configuration 184 vi List of Figures List of Figures Figure 1.1. Schematic of classifier cold model columns: 3 Figure 2.1.1. Forces acting on a particle in a stagnant fluid 7 Figure 2.4.1. Mixing and segregation patterns for liquid fluidization of binary particles 18 Figure 2.4.2. Bed inversion phenomenon (Moritomi et al., 1982) 21 Figure 2.4.3. Formation of zones in batch sedimentation of polydisperse suspension... 23 Figure 2.4.4. Schematic of vertical settler for the model of Nasr-El-Din (1988) 27 Figure 2.4.5. Schematic of Markov Chain model 29 Figure 2.4.6. Forces exerted on a particle in the Markov Chain model 31 Figure 2.4.7. Schematic of determination of the height of a cell 32 Figure 2.4.8. Flow chart of Markov Chain Model 34 Figure 2.5.1. Variation of particle velocity with respect to voidage in the column at different discharge voidages 37 Figure 2.5.2. Effect of liquid fluidizing velocity on particle velocity at different discharge voidages 38 Figure 2.5.3. Effect of particle size on particle velocity at different discharge voidages. 3 8 Figure 2.5.4. Variation of particle velocity with respect to the voidage in column at different discharge voidages under commercial conditions 39 Figure 3.1.1 Schematic diagram of U B C particle classification process 41 Figure 3.1.2. Dimensions of hydrocyclone. A l l units are in mm 42 Figure 3.1.3. Dimensions of the classifier for the C-0 configuration. A l l units are in mm. 43 Figure 3.1.4 Dimensions of conductivity probe. Al l units are in mm 45 Figure 3.1.5. Dimensions of tube samplers shown in plan view with slot open 46 Figure 3.3.1. Electronic circuit connected to the conductivity probe 52 Figure 3.3.2. Variation of the voidage with respect to the relative resistance of probe #1 56 Figure 4.1.1. Schematic of cold model classifier 58 Figure 4.1.2. Radial profile of voidage for a mono-disperse suspension 59 Figure 4.2.1. Differential element of the classifier column 60 Figure 4.2.2. Comparison of the voidage in the lower and upper portion with voidage in discharge and overflow for mono-disperse, binary and multi-size systems 65 Figure 4.2.3. Comparison of the fractions of large particles in the lower and upper portions with those in the discharge and overflow for binary system 65 Figure 4.2.4. Comparison of mean particle sizes in the lower and upper portions with those in discharge and overflow for multi-size system 66 Figure 4.2.5. Comparison of the PSD breadths in the lower and upper portions with those in the discharge and overflow for multi-size system 66 Figure 4.4.1. Schematic of top zone I 69 Figure 4.4.2. Schematic of feed zone 70 Figure 4.4.3. Schematic of top zone II 71 Figure 4.4.4. Schematic of diverging zone 72 Figure 4.4.5. Schematic of column zone 73 Figure 4.4.6. Schematic of discharge zone 74 Figure 4.4.7. Schematic of bottom zone I 76 vii List of Figures Figure 4.4.8. Schematic of bottom zone II 77 Figure 4.5.1. Computer flow chart to solve the steady state hydrodynamic model 82 Figure 4.6.1. Comparison of predicted voidage from steady state model with experimental data for mono-disperse suspension: 84 Figure 4.6.2. Comparison of predicted discharge voidage with experimental data for mono-disperse suspensions: 85 Figure 4.6.3. Comparison of calculated voidage by steady state model with experimental data for a binary system: 86 Figure 4.6.4. Comparison of calculated liquid-free volume fraction of particles by steady state model with experimental data for binary system 87 Figure 4.6.5. Comparison of discharge voidage between the experimental data and hydrodynamic model for binary system 88 Figure 4.6.6. Comparison of large particle fraction in discharge between experimental data and hydrodynamic model for binary system 88 Figure 4.6.7. Comparison of calculated voidage by steady state model with experimental data for multi-size system 90 Figure 4.6.8. Comparison of particle species fractions predicted by steady state model with experimental data for multi-size system 91 Figure 4.6.9. Comparison of calculated cumulative particle size distribution by mass with experimental data for multi-size system 92 Figure 4.6.10. Comparison of experimental discharge voidage with predictions of hydrodynamic model for multi-size system 92 Figure 4.6.11. Comparison between experimental mean particle size, dpso„d, in discharge and hydrodynamic model for multi-size system 93 Figure 4.6.12. Comparison between experimental breadth of PSD, d9o,d/dpio,d, in discharge and hydrodynamic model for multi-size system 93 Figure 4.7.1. Voidage profile along the column predicted by hydrodynamic model with different axial dispersion coefficients for multi-size system 95 Figure 4.7.2. Cumulative PSD by mass in discharge predicted by hydrodynamic model with different axial dispersion coefficients for multi-size system 95 Figure 4.7.3. Variation of fitted dispersion coefficient with feed voidage, Sf, for multi-size system 97 Figure 4.7.4. Variation of fitted axial dispersion coefficient with PSD breadth in feed, d9o,d/dpio,f, for multi-size system 97 Figure 4.7.5. Variation of fitted axial dispersion coefficient with height of cylindrical region of the classifier for multi-size system 98 Figure 4.7.6. Variation of fitted axial dispersion coefficient with fluidizing velocity, U a , for multi-size system 98 Figure 4.7.7. Variation of fitted axial dispersion coefficient with discharge velocity, Ud, for multi-size system 99 Figure 4.7.8. Variation of fitted axial dispersion coefficient with feed velocity, Uf, for multi-size system 99 Figure 4.7.9. Comparison between fitted and calculated dispersion coefficients 103 Figure 4.7.10. Comparison of discharge voidage between U B C experimental data and hydrodynamic model for multi-size system 104 viii List of Figures Figure 4.7.11. Comparison of mean particle size, dP5o,d, in discharge between U B C experimental data and hydrodynamic model for multi-size system 104 Figure 4.7.12. Comparison of breadth of PSD in discharge between U B C experimental data and hydrodynamic model for multi-size system 105 Figure 4.7.13. Comparison of predicted discharge voidage with pilot and commercial M C C data 105 Figure 4.7.14. Comparison of predicted mean particle size in discharge, dpso,d, with pilot and commercial M C C data 106 Figure 4.7.15. Comparison of predicted breadth of PSD in discharge, dP9o,d/dpio,d, with pilot and commercial M C C data 106 Figure 4.8.1. Local voidage along the height at different fluidizing velocity, U a , for multi-size system 108 Figure 4.8.2. Local voidage along the height at different feed voidage, Sf, for multi-size system 109 Figure 4.8.3. Local voidage along the height at different feed velocity, Uf, for multi-size system 109 Figure 4.8.4. Local voidage along the height with different discharge velocity, Ud, for multi-size system 110 Figure 4.8.5. Variation of voidages with respect to fluidizing velocity, U a , for multi-size system 112 Figure 4.8.6. Variation of voidages with respect to feed voidage, Sf, for multi-size system 112 Figure 4.8.7. Variation of voidages due to changes in feed velocity, Uf, for multi-size system 113 Figure 4.8.8. Variation of voidages with respect to discharge velocity, Ud, for multi-size system 113 Figure 4.8.9. Variation of voidages with respect to breadth of PSD in feed, dP9o,f/dPio,f, for multi-size system 114 Figure 4.8.10. Variation of voidage with respect to height of cylindrical region, F£c, for multi-size system 115 Figure 4.8.11. Variation of increments of dpso and dP9o/dpio with respect to fluidizing velocity, U a , for multi-size system 116 Figure 4.8.12. Variation of increments of dpso and dP9o/dpio with respect to feed voidage, Sf, for multi-size system 117 Figure 4.8.13. Variation of increments of dpso and dP9o//dpio with respect to feed velocity, Uf, for multi-size system 118 Figure 4.8.14. Variation of increments of dpso and dP9o/dpio with respect to discharge velocity, Ud, for multi-size system 119 Figure 4.8.15. Variation of increments of dpso and dP9o/dpio with respect to breadth of PSD in feed, dP9o,f/dpio,f, for multi-size system 120 Figure 4.8.16. Variation of increments of dP5o and dP9o/dpio with respect to height of cylindrical region for multi-size system 120 Figure 4.8.17. Variation of classifier efficiency with respect to fluidizing velocity, U a , for multi-size system 123 Figure 4.8.18. Variation of classifier efficiency with respect to feed voidage, Sf, for multi-size system 124 ix List of Figures F i g u r e 4.8.19. Variation of classifier efficiency with respect to feed velocity, Uf, for multi-size system 124 F i g u r e 4.8.20. Variation of classifier efficiency with respect to discharge velocity, Ud, for multi-size system 125 F i g u r e 4.8.21. Variation of classifier efficiency with respect to breadth of PSD in feed, dp9o7f/dpio,f, for multi-size system 126 F i g u r e 4.8.22. Variation of classifier efficiency with respect to height of cylindrical zone for multi-size system 127 F i g u r e 4.9.1. Calculated cumulative PSD by mass in discharge by Markov Chain model with different turbulence intensity for multi-size system 128 F i g u r e 4.9.2. Comparison of calculated voidage profile along the column for Markov Chain model and Hydrodynamic model 129 F i g u r e 4.9.3. Comparison of calculated cumulative PSD by mass in discharge between Markov Chain model and Hydrodynamic model 130 F i g u r e 5.3.1. Representation of two-step Lax-Wendroff differencing scheme 144 F i g u r e 5.4.1. Variation of voidage with time for a step change in liquid fluidizing velocity, U a , att=20 s for mono-disperse suspension 149 F i g u r e 5.4.2. Variation of voidage with time for a step change in liquid fluidizing velocity, U a , at t=25 s for a multisize system 150 F i g u r e 5.4.3. Variation of increment of dpso and dP9o/dpio with time for a step change in liquid fluidizing velocity, U a , at t=25 s for multisize system 150 F i g u r e 5.4.4. Variation of voidage with time for a step change in the discharge velocity, Ud, at t=15 s for a mono-disperse suspension 152 F i g u r e 5.4.5. Variation of voidage with time for a step change in discharge velocity, Ud, at t=20 s for a multisize system 153 F i g u r e 5.4.6. Variation of increment of dP5o and dP9o/dpio with time for a step change in discharge velocity, Ud, at t=20 s for a multisize system 153 F i g u r e 5.4.7. Variation of voidage with time for a step change in feed velocity, Uf, at t=25 s for mono-disperse suspension 154 F i g u r e 5.4.8. Variation of voidage with time for a step change in feed velocity, Uf, at t=20 s for a multisize system 154 F i g u r e 5.4.9. Variation of increment of dpso and dP9 0/dpio with time for a step change in feed velocity, Uf, at t =20 s for multisize system 155 F i g u r e A . 1 . Dimensions of UBC classifier for different configurations 172 F i g u r e D . l . Variation of the voidage with respect to the relative resistance of probe #1 187 F i g u r e D.2. Variation of the voidage with respect to the relative resistance of probe #2 187 F i g u r e D.3. Variation of the voidage with respect to the relative resistance of probe #3 188 F i g u r e D.4. Variation of the voidage with respect to the relative resistance of probe #4 188 F i g u r e D.5. Variation of the voidage with respect to the relative resistance of probe #5 189 F i g u r e D.6. Variation of the voidage with respect to the relative resistance of probe #6 189 List of Figures Figure D.7. Variation of the voidage with respect to the relative resistance of probe #7 190 Figure D.8. Variation of the voidage with respect to the relative resistance of probe #8 190 Figure D.9. Variation of the voidage with respect to the relative resistance of probe #9 191 Figure E . l . Schematic of the conical column 192 xi Acknowledgements Acknowledgements I would like to thank the faculty, staff and fellow students in the Department of Chemical & Biological Engineering at the University of British Columbia for their steadfast support.. I would like to render my sincere appreciation to my supervisors, Drs. J. R. Grace, N . Epstein and C. J. Lim, who provided much of advice and patient support throughout this project. My other thesis committee members, Drs. B. D. Bowen, M . Church and F. Watanabe, contributed their time and effort to the completion of this work, which is greatly appreciated. The help of Dr. J. Zhang, Dr. X . B i , Mr. H . Cao, Mr. K. Mitsutani, Miss. P. Mehrani, Miss N . Rezai and other students and visiting scholars in our fluidization group is greatly appreciated. The financial support of the 4PP group of Mitsubishi Chemical Corporation in Japan is gratefully acknowledged. Honour is due to my parents, S. Zhu and H. Chen, and my parents-in-law, X . L i and S. Ma, who gave me more assistance than I am able to express. Finally, this thesis is dedicated to my most loved ones, my husband, J. Ma, and my son, K. Ma, who missed all the time we could have spent together. xii Chapter 1 Introduction Chapter 1 Introduction In industrial applications of solid-fluid suspensions, particles seldom all possess the same size and density. Particle diameters tend to vary quite considerably, as in crystallization operations, while different particles also commonly have different densities, as in mineral processing. In many cases, it is important to group together particles of similar sizes or densities, a unit operation called "classification". The classification of particles in solid-liquid systems has been the subject of numerous theoretical and experimental investigations for several decades. Svarovsky (1990) catalogued the variety of processes and equipment used in solid-liquid separation systems. Rushton et al. (1996) classified these processes into two principal modes of separation — filtration and sedimentation. For sedimentation, a density difference between the solids and the liquid is necessary, but not for filtration, where particles are intercepted at one or more barriers. Rushton et al. (1996) recommended that sedimentation be considered first when selecting the appropriate technology for a separation process. Gravity sedimentation is especially attractive where large, continuous liquid flows are involved (Pierson, 1990). A number of theoretical and experimental investigations have been conducted in the past decades on gravity sedimentation of polydisperse mixtures. However, most studies have been conducted under batch fluidization conditions with closed bottom sections, i.e. with accumulation of those particles which settle at the bottom. Little theoretical analysis has been carried out on the behavior of classifiers with particles of different sizes and densities with continuous addition and efflux of particles and liquid, especially for unsteady state dynamics. 1 Chapter 1 Introduction The Mitsubishi Chemical Corporation (MCC) of Japan sponsored experimental and theoretical work at U B C on classification in solid-liquid systems from 1996 to 2000 because of its wish to better understand and to scale up classifiers used in the production of polypropylene (Watanabe et al., 1998). Two cold model classifiers of diameter 508 mm and 191 mm were set up with geometry as shown in Figure 1.1. These are similar in geometry to, but significantly smaller than, the industrial classifiers utilized by M C C . In the cold model, glass beads enter as a dense suspension conveyed by water through a radial feed port near the top, while water without suspended particles is injected upwards from the bottom. A relatively dilute stream containing mostly small particles is taken off the top as an Overflow stream, while a stream enriched in coarser particles is removed as a Discharge stream from the side near the base of the column. As indicated in Figure 1.1, the base is conical while there is an enlarged section at the top. Qualitative observation of the classifier indicates that there are three different dominant flow patterns. The top region is dominated by a vigorous vortex caused by the feed stream, entering as a horizontal jet. Most small particles leave with the overflow stream at the top, while most of the largest particles settle downward through the central cylindrical zone where the turbulence is approximately isotropic. The settling particles are removed with a discharge stream near the bottom. Liquid introduced into the classifier from the bottom causes a vertical jet in the conical region. Under certain circumstances, a small defluidization region can also be observed along the sloping walls of the expanded upper section, as shown in Figure 1.1. 2 Chapter 1 Introduction Overflow Liquid ( a) Figure 1.1. Schematic of classifier cold model columns: (a) Side view; (b) Plan view at the feed level; (c) Plan view at discharge level. 3 Chapter 1 Introduction The purpose of this research project was to develop a fundamental model which can predict the complex steady and unsteady state behaviours of particles and fluid in a continuous classifier of this general geometry. The method is based on differential hydrodynamic models and knowledge of the sedimentation behavior of the individual species within the mixture. In the study, attention is confined to fluid dynamic aspects, rather than heat and mass transfer. Al l particles are considered to be spherical in shape. Both the liquid and particles are assumed to be incompressible. Friction at the column wall is neglected. Interactions between the particles, other than hydrodynamic effects transmitted through the fluid, are also neglected. Hence, electrokinetic and surface phenomena leading to flocculation and aggregation are beyond the scope of the study. In Chapter 2, the behavior of single and of many monodisperse particles settling in a fluid are first described. Literature on the sedimentation of particles of different sizes or densities is next reviewed. A Markov Chain Model of the liquid classifier, used to compare with the hydrodynamic model in Chapter 4, is also introduced. An explanation of the emptying phenomenon encountered in industrial classifiers is presented. In Chapter 3, the equipment and particles used in the experimental work, involving a 191 mm diameter cold model classifier, are described. The experimental procedures for measuring the voidage and particle concentrations in the classifier and in the discharge stream are introduced. The data obtained from the experiment are later used to test the predictions of the hydrodynamic model. In Chapter 4, a mechanistic steady state hydrodynamic model with one fitted parameter, an axial dispersion coefficient, is developed, including detailed discussion of the equations and boundary conditions. The calculation procedure used to solve this 4 Chapter 1 Introduction model is also described. A correlation to estimate the dispersion coefficient is established. Predictions of the model for classifier performance and efficiency at steady state conditions are presented and compared with experimental data. In addition, the Markov Chain model is compared with the hydrodynamic model developed in this work. In Chapter 5, an unsteady state hydrodynamic model is presented. The equations, initial and boundary conditions are discussed in detail, and the numerical methods employed to solve the resulting set of non-linear partial differential equations are described. Predictions of the unsteady state model in comparison with experimental data are presented. Finally, conclusions and recommendations for future work are summarized in Chapter 6. 5 Chapter 2 Settling of Particles in Liquids Chapter 2 Settling of Particles in Liquids Sedimentation, wherein particles fall under the influence of gravity through a fluid in which they are suspended, is one technique which can be used to separate particles with different settling velocities from one another. In this chapter, the free settling of a single particle is first described, leading to calculation of the particle terminal settling velocity. The settling velocity and drag force correlations for concentrated monodisperse suspensions are presented next. The modifications for sedimentation of particles of different sizes and densities are then reviewed. Phenomenological models, such as a batch model, a differential hydrodynamic model and a continuous model, of particle segregation and classification in liquid fluidized beds for binary and polydisperse systems are described. A stochastic model, the Markov Chain Model, recently written for the liquid classifier by Zhang (1998) of our group, is also introduced. The emptying phenomenon encountered in industrial classifiers is explained in section 2.5. 2 . 1 Free Settling of a Single Particle Consider a spherical particle of diameter d p and density p p suspended in a Newtonian fluid of density pf and viscosity pf as shown in Figure 2.1.1. The particle is acted on by a downward force due to gravity (F g) , an upward force due to buoyancy ( F B ) and a drag force (FD), resulting from fluid friction, which acts in the direction opposite to the direction of motion of the particles relative to the fluid. 6 Chapter 2 Settling of Particles in Liquids The gravitational force acts downwards and its magnitude is given by Fg^rcdpPpg (2.1.1) where g is the acceleration of gravity, while the buoyancy force acts upwards with magnitude ^ 1 ,3 F B =-7T.d p p f g (2.1.2) The drag force opposes the motion of the particle relative to the fluid and has magnitude (2.1.3) F D = c D . ^ . ^ d ? 2 4 where V is the particle velocity relative to the fluid and C D is the drag coefficient which is a function of Reynolds number (Re= P f V d p ) V T VT (pP>Pf) (pp<Pf) (a) (b) Figure 2.1.1. Forces acting on a particle in a stagnant fluid (a) Falling particle (b) Rising particle 7 Chapter 2 Settling of Particles in Liquids If the particle has a density greater than that of the fluid, upon release from rest it begins to fall, and the direction of the forces is as shown in Figure 2.1.1(a). Otherwise it rises and the forces are directed as shown Figure 2.1.1(b). As it accelerates, the drag force on the particle increases. Ultimately the net force on the particle is zero, and there is no further acceleration. Under these conditions the particle falls or rises at a constant velocity called the terminal velocity, V T , with F D =|F g -F B | Making use of definition of C D from Equation (2.1.3), yields C D • ^ • ~A dp = \ *dp I Pp - Pf I g (2.1.4) Hence the terminal velocity can be written as v ^ J i t o i ^ ( „ 5 ) V 3 Pf C D However, the relationship is generally difficult to use because C D is a function of Reynolds number, i.e., of V T itself. For the special case of very low Reynolds number (Re<0.1), we have Stokes or creeping flow with CD=24/Re T (2.1.6) where R e x = ^ I (2.1.7) Uf Substitution of Equations (2.1.6) and (2.1.7) into Equation (2.1.5) leads to _ g I Pp - Pf I dp V T -18u.f Chapter 2 Settling of Particles in Liquids which is the familiar Stokes law relationship for the terminal settling or rising velocity. For 750<Re<3.5 xlO 5 , the so called Newton's law range, Co«constant =0.445, leading to VT»1.73^gdp | p p - p f I/pf Table 2.1.1. Recommended drag correlations from Clift et al. (1978). Range Correlation (A) Re < 0.01 (B) 0 . 0 K R e < 2 0 (C) 20 < Re < 260 (D) 260 < Re < 1500 (E) 1.5 x 10 3 <Re< 1.2 x 104 (F) 1.2 x 10 4 <Re<4.4x 104 (G) 4.4x 10 4 <Re<3.38x 105 (FT) 3.38 x 105 < R e < 4 x 105 (I) 4. x 105 < Re < 106 (J) 106 < Re CD=3/16+24/Re C D R e log to 24 = -0.881 + 0.82w-0.05w^ i.e. C D = — Re l + 0 .1315Re(° - 8 2 - ° 0 5 w > l o g l 0 i.e. C D C D Re 24 24 -1 = - 0.7133 + 0.6305 w Re 1 + 0.193 5 R e 0 6 3 0 5 L o g 1 0 C D = 1.6435 - 1.1242W + 0.1558W2 Logio C D = -2.4571 + 2.5558w - 0.9295w2 + 0.1049w3 Logio C D = -1.9181 + 0.6370w - 0.0636w2 Logio C D = -4.3390 + 1.5809w - 0.1546w2 C D = 29.78-5.3w C d = -0.49 + 0.1W C D = 0.19-8 x 104/Re where w=logio(Re) 9 Chapter 2 Settling of Particles in Liquids For other ranges of the Reynolds number, there are numerous empirical correlations for the drag coefficient as a function of Re, with small numerical differences. Published correlations for the drag forces on a sphere moving in a fluid were reviewed by Khan and Richardson (1987), who also gave their own expression for the whole range of conditions encountered in chemical engineering. Table 2.1.1 lists recommended drag correlations from Clift et al. (1978) for the Reynolds number range of 0 to oc. Convenient equations for calculating the terminal velocity for spheres can be obtained from the drag correlations. Grace (1986) gives a compact and simple form, listed in Table 2.1.2, for calculating terminal settling velocities over a broad range of fluid and particle properties. Table 2.1.2. Terminal settling velocity correlations for spheres (Grace, 1986). dp* Range V T Range Equation "<3~8 <0.624 vT*= (dp*)2/18 - 3.1234 x 10"4(dp*)5 + 1.6415 x 10"6(dp*)8 - 7.278 x 10-10(dp*)u 3.8 to 7.58 0.624 to 1.63 Logi0(vT*)=-1.5446 + 2.9162w-1.0432w2 7.58 to 227 1.63 to 28 Logi0(vT*)=-1.64758 + 2.94786w-1.09703w2 + 0.17129w3 227 to 3350 28 to 91.7 Logio(vT*)=5-1837- 4.51034w + 1.687w2 - 0.189135w3 Definitions: dp*= d p [ p f g [ p p - p f j / u . f ] 1 / 3 w = logiodp* vT* = v T[ p f 2 / u . f g | p P - p f j ] 1 / 3 2.2 H i n d e r e d Set t l ing i n Monod i spe r se Systems 2.2.1 Settling velocity Sedimentation in suspensions composed of identical particles has also been investigated extensively. Theoretical and experimental analyses relating the particle settling velocity 10 Chapter 2 Settling of Particles in Liquids to solids concentration have been obtained in the dilute limit (Batchelor, 1972, 1982; Davis and Birdsell, 1988). The intermediate and high concentration limits are more important in practice but less amenable to rigorous theoretical analysis. Previous studies have shown that the magnitude of the sedimentation rate of any particle in a concentrated suspension is always less than the settling rate of the same particle in isolation. This is partly because the downward movement of particles in a container with a closed bottom causes an equal volumetric flowrate of displaced fluid relative to which the particles must move. Furthermore, for a given relative velocity, the average velocity gradients, and hence shear stresses, are greater in a concentrated suspension. If all particles are of uniform size and density, they settle with equal velocities in a quiescent fluid, apart from small statistical variations, and therefore there are few interparticle collisions. Many previous workers have expressed the particle velocity relative to the fluid, V, often called slip velocity, by V=vTf(e) (2.2.1) where s is the voidage of the suspension. Expressed in this form, f(s) is the same for both sedimentation and fluidization. Various functional forms have been reviewed by Barnea and Mizrahi (1973), Garside and Al-Dibouni (1977), Patwardhan and Tien (1985) and Di Felice (1995). The most popular of these is the Richardson and Zaki (1954) correlation, V^vts"" 1 (2.2.2) where n is an empirical index which is a strong function of the Reynolds number and a weak function of the ratio of the particle diameter, d p, to column diameter, D. For d p « D , The Richardson and Zaki (1954) values of n are given in Table 2.2.1. 11 Chapter 2 Settling of Particles in Liquids Table 2.2.1. Richardson-Zaki (1954) correlation for index n with d p « D . Range of Reynolds number n ReT<0.2 n=4.65 0.2<ReT<l n=4.4*ReT"uu3 KRe T<500 n=4.4*ReT' u l 500<ReT n=2.39 As presented in the review paper of Di Felice (1995), three simple correlations, proposed by previous researchers, to evaluate the parameter n over the complete range of Re are listed in Table 2.2.2. Table 2.2.2. Correlations for the index n in Richardson and Zaki equation. Authors Correlation Khan and Richardson (1989) 4 - 8 _ n = 0.043Ar0-5 7 n-2 .4 Rowe (1987) 4 7 ~ " n n c B 0.75 = 0.175ReT n-2.35 Garside and Al-Dibouni (1977) 5 1 " n = 0 . 1 R e T 0 - 9 n-2 .7 where Ar is the Archimedes number, dpPL I Pp - P L I g A 2.2.2 Drag Force It is common practice in fluidization research to express the effect of particle concentration on the drag force on a sphere in a multiparticle system (FQ) in terms of g(e)=-^- (2.2.3) ^DO where FDO is the drag on a single partilce. 12 Chapter 2 Settling of Particles in Liquids Here FD and FDO are forces evaluated at the same fluid superficial velocity. Di Felice (1995) reviewed many previous functions for g(s) and pointed out that of the many voidage functions proposed, the most popular has the form g(s)=s"P (2.2.4) where fi is a nearly constant coefficient ranging from 3.6 to 3.8 at low and high Re, but weakly dependent on the Reynolds number at intermediate Re. Di Felice (1994) gave the following relationship based on a critical examination of data for fixed and fluidized beds for a broad range of Reynolds number: P = 3 . 7 - 0 . 6 5 e x p [ - ( 1 5 - 1 O 2 g l ° R e ) 2 ] (2.2.5) Equation (2.2.5) assumes that the buoyant density is that of the fluid-particle mixture rather than of the fluid alone. If the fluid alone is assumed to be the source of buoyancy, then the constant 3.7 in Equation (2.2.5) becomes 4.7, as first postulated by Wen and Yu (1966). 2.3 Applications and Modifications for Polydisperse Systems Several investigations have been conducted on the behavior of mixtures of solid particles of different densities and sizes in sedimentation and fluidization. The observed behavior of suspensions of identical particles is a convenient starting point for explaining the behaviour of suspensions of particles of different sizes and densities. This approach was first used by Lockett and Al-Habbooby (1973) and is applied in the present work with modifications. 13 Chapter 2 Settling of Particles in Liquids Lockett and Al-Habbooby (1973, 1974, 1979) found that the Richardson and Zaki (1954) correlation originally proposed for monodisperse systems could be applied to each particle fraction of a polydisperse mixture. In order to allow for the interactions between particles settling at different velocities, Mirza and Richardson (1979) applied a correction factor of s ° 4 to the predicted sedimentation velocities to obtain a better representation of the experimental data. Selim et al. (1983a, 1983b) extended the Mirza and Richardson (1979) model by considering the buoyancy effect induced by smaller particles on the terminal velocity of larger particles. Masliyah (1979) developed an expression for the settling velocities of particles in a polydisperse suspension at low Reynolds numbers: Vi=v T i s n ' " 2 ( p p i - p b ) / ( p p i - p L ) (2.3.3) where p P i , p b and PL are the densities of particle species i , bulk and pure liquid, respectively. This model, unlike the earlier ones, was as effective for particles of different densities as for particles of different sizes. Patwardhan and Tien (1985a, 1985b) modified the Masliyah (1979) model by introducing a new concept — the apparent porosity of the ith species of particles in a suspension. The apparent porosity accounts for the difference between the region surrounding a particle of a given type in a suspension of particles of different sizes and densities and that encountered by the same type of particle in a suspension of identical particles at the same total porosity. Based on data for both sedimentation and fluidization of binary particle mixtures of different sizes and densities, Patwardhan and Tien concluded that their model represented a significant improvement over previous models. 14 Chapter 2 Settling of Particles in Liquids Law et al. (1987) tested the same five models to predict the settling velocity of heavy and light (buoyant) particle species in bidisperse suspensions. They concluded that the Masliyah (1979) model did somewhat better than the other four for this special case. Al-Naafa and Selim (1989), however, compared the above five models with experimental data for a large variety of polydisperse systems and found that the model of Selim et al. (1983a, 1983b) fitted the data more accurately than the other four approaches. Davis and Gecol (1994) proposed a model for which the only parameters required are the dimensionless sedimentation coefficients of Batchelor and Wen (1982). The model agrees with the Batchelor and Wen (1982) theory in the dilute concentration limit and reduces to the Richardson and Zaki (1954) correlation for monodisperse suspensions. Settling predictions are in good agreement with the measurements for both dilute and concentrated suspensions. van der Wielen et al. (1996) derived a new expression for the settling velocity of particles in fluidized beds of particles of different sizes or densities based on a steady-state force balance. Although the drag term was originally derived for monocomponent fluidization, the experimental verification supports straightforward application to polydisperse fluidized beds. Doheim et al. (1997) extended the Mirza and Richardson (1979) model by introducing an exponent parameter, ea, which depends on the nature of the system, to evaluate the particle settling velocity. For particles of different sizes and equal densities, a=0.4 as in the Mirza and Richardson (1979) model, while for particles of different sizes and different densities, a=0.5; for particles of the same size and different densities, a=0.7. 15 Chapter 2 Settling of Particles in Liquids This model not only showed improved predictions for sedimenting particles of different sizes and densities, but is also applicable for heavy and light particles of uniform size. Abu-Ali (1997) introduced a correction factor of s 0 2 into the Garside and A l -Dibouni (1977) model at intermediate particle Reynolds numbers by considering the crowding of particles and the collisions between particles. Smith (1965, 1966, 1967, 1997, 1998) developed a cell-model to account for differences in the local volume fraction of each species. Using his model, the settling velocity in creeping flow of a suspension at substantial volume fraction can be calculated without resort to empirical factors. Asif (1998a) demonstrated the capability of the Richardson and Zaki (1954) correlation to represent the overall expansion and the inversion behavior of binary-solid fluidized beds. His approach consisted of generalizing the Richardson-Zaki correlation by using mean values for the correlation parameters, i.e., the particle terminal velocity and the correlation index n, to account for the presence of two particle species in binary solid fluidized beds. The mean values of the correlation parameters were evaluated using the surface-volume mean particle diameter and the mean density for the two particle species. Galvin et al. (1999) proposed that the slip velocity of any species in a multispecies suspension, involving particles of different sizes and densities, can be a function of pressure gradient, based on the Richardson and Zaki equation, i.e. V = V T • M V T , i m i - i j dP /dz (2.3.4) ( P P , i - p L ) g where -dP /dz is the frictional pressure gradient opposing the buoyed weight of particles in the liquid. Hence, once the pressure gradient is known, the monocomponent expression 16 Chapter 2 Settling of Particles in Liquids becomes immediately applicable for describing the slip velocity in any suspension. This model is simple, easy to use, and also useful in process control, given that the pressure gradient is readily measured. Mostoufi and Chaouki (1999) developed a new correlation for calculating the parameter (3 in the form of effective drag experienced by a single particle falling in a fluidized bed of different particles. The parameter P was found not only to be a function of the terminal Reynolds number of the falling particle, but also to be a function of the Archimedes number of the fluidized particles. 2.4 Particle Segregation and Classification in Liquid Fluidized Beds When a mixture of several types of solid particles of different sizes or different densities or both is fluidized by liquid, the axial distribution of the particles is, in general, non-uniform. Larger-diameter or denser particles tend to occupy the lower region of the bed, while smaller or less dense particles tend to be found primarily in the upper region of the bed. Thus the local composition of the solids varies axially, as does the voidage. Different species of solid particles, in general, have different settling velocities and tend to segregate according to their settling velocities. On the other hand, solid particles in a fluidized bed are non-stationary and move in a random manner. As a result, different species tend to mix with one another. The steady-state distribution of particles is determined then by the relative magnitudes of these two opposing factors: the tendency to segregate and the tendency to mix. 17 Chapter 2 Settling of Particles in Liquids 2.4.1 Segregation and Mixing of Binary Particles Di Felice (1995) reviewed various aspects of liquid fluidization of binary-solid mixtures. Figure 2.4.1 summarizes observed steady state patterns for binary systems. o o o o o o O ° 0 ° o o o o o o o o O ° o o -o o o 0 ° O o O o o o o o O o o o o o o o o o o O ° 0 o o o o o o o o o O ° o o o o o o«o<>o o O ° o o O o o ' o O o o o o o 0 O 0 O 0 o o o O o o O o O ° o o o o o o ° o ° o oO© o O o ° °o° o o o o o o O ° 0 o O o O o o o O O o O ooooo o ° o ° o 0 O 0 O O O o O o o ooooo o O o O ° O o O o O o oo°o ooooo O o O o O O O O O o O o O O o ooooo o O ° O o o O O O O o o o ° o O O O O O O o O o O O O O O O ooooo O O O O O o O ° 0 ° ooooo O O O o O O O O O O O o Q o O ooooo O o Q O O (a) (b) (c) (d) Complete Perfect Partial Partial segregation mixing segregation 1 segregation 2 Figure 2.4.1. Mixing and segregation patterns for liquid fluidization of binary particles Figure 2.4.1(a) indicates complete segregation with the two solid species forming quite independent beds one above the other, each assuming the voidage appropriate to the corresponding single component bed. Figure 2.4.1(b) represents the other extreme of complete mixing in which the concentration of each species remains constant with height throughout the system. Figure 2.4.1 (c) depicts a gradual transition between two single component regions, representing an example of partial segregation, while Figure 2.4.1(d) contains no single component zones but manifests a continuously decreasing concentration of the less easily fluidized component with height. 18 Chapter 2 Settling of Particles in Liquids Case 1 Binary particles differing only in size or density: If the two types of particles differ only in size or in density, a two-layer bed will be generated, the larger or denser particles settling to the bottom, while the smaller or less dense particles collect at the top, as shown for a batch system in Figure 2.4.1(a). If the particle sizes or densities differ sufficiently, then the two layers will be sharply separated; otherwise there will be a mixing zone between the layers. In the mixing zone, the rival mechanisms of segregation and dispersion counterbalance each other, causing the overall flux to be zero at steady state. The flux of solids due to mixing can be quantified through Fick's law with the introduction of an axial dispersion coefficient, i.e. J d i s P , i = - D a , i ^ - 0=1,2) . (2.4.1) while the segregation flux can be written as Jseg, i = CjVj (2.4.2) D a,i is the axial dispersion coefficient of the ith species of particles. Equating the two fluxes yields a pair of differential equations for binary systems: - D . , i X - + C iVi=0 0-1,2) (2.4.3) dz Many previous methods to integrate the above equation have been summarized by Di Felice (1995). They include the approaches of Kennedy and Bretton (1966), A l -Dibouni and Garside (1979), Juma and Richardson (1979, 1983), Patwardhan and Tien (1984), van der Meer et al. (1984), Gibilaro et al. (1985), Di Felice et al. (1987), Dutta et al. (1988) and Asif and Petersen (1993). Recently Epstein and Pruden (1999) examined segregation by size of particles which are close in density. Segregation is shown to depend on the bulk density difference between the two particle components when each is 19 Chapter 2 Settling of Particles in Liquids fluidized separately by the same liquid at the same superficial velocity, normalized with respect to the buoyed density of the particles. Di Felice (1995) also reviewed previous work and concluded that, in spite of much effort, reliable numerical estimates of the axial dispersion coefficient are not available at the present time. Case 2 Binary particles differing in both size and density: If two species of particles differ both in size and density, but in opposite directions, a phenomenon called solid layer inversion, first studied systematically by Moritomi et al. (1982), can occur. Figure 2.4.2 shows the bed inversion phenomenon observed by Moritomi et al. (1982). The bed was almost always made up of two segregated layers. At low liquid rates, the small dense particles were found at the bottom, with the large but light particles at the top of the bed. On increasing the fluid velocity, some of the large less dense particles mixed into the bottom layer. A further increase caused the bed to be completely mixed (the inversion point). For liquid velocities greater than the inversion velocity, the bed was observed to invert, with the bottom layer being a mixture of the two species of particles and the top consisting purely of small dense particles. For even greater fluid velocities, all the small and dense particles migrated to the upper part of the bed leaving a bottom layer composed solely of large light particles. This layer inversion phenomenon is generally accompanied by a bulk density change in the relative position of the particle species. If a region of fluidized particles is heavier overall than another (i.e. if it has higher bulk density, epL+(l-e)pp), it tends to settle to the bottom of the bed. 20 Chapter 2 Settling of Particles in Liquids fa?*.'.*. 0 0 0 0 ; oqhb n * » 0 ^ >:°*2 P**>°. • or* • • V P •oy.q ?VP. o.oP. • J .o o*o*. .o* .o • • oP.o .oo-O O . p . p . • • O . Q o.°.b o-o.: °.o. # • • •••• • • • • • • •• • •••• • • • • O O O o o 0 ° o ° o Icroo ooo °oQol Inversion point Increasing fluid velocity Figure 2.4.2. Bed inversion phenomenon (Moritomi et al., 1982) Some approaches used to study the inversion phenomenon were reviewed by Di Felice (1995). Papers included in the review were by Moritomi et al. (1982, 1986), van Duijn and Rietema (1982), Epstein amd Leclair (1985), Matsuura and Akehata (1985), Patwardhan and Tien (1985b), Gibilaro et al. (1986a), Jean and Fan (1986) and Di Felice (1993). More recent work has been reported by Funamizu and Takakuwa (1995, 1996) and Asif (1998b). 2.4.2 Sedimentation of Multi-Species Particles Gravity sedimentation is commonly used to classify particles in industrial processes. When a multi-species suspension is subjected to sedimentation, particles of different 21 Chapter 2 Settling of Particles in Liquids densities and sizes settle at different velocities. Consequently, the final sediment consists of distinct layers of particles of different composition. This form of segregation is known as differential sedimentation, with faster settling species forming the bottom-most layers. Batch Model: Most previous studies of differential sedimentation, e.g. Lockett and Bassoon (1979), Mirza and Richardson (1979), Selim et al. (1983b), Zimmels (1983), Patwardhan and Tien (1985b), Stamatakis and Tien (1988), Al-Naafa and Selim (1989), Cheung et al. (1996) and Smith (1997), have been conducted under batch fluidization conditions with closed-bottom columns and no continuous addition or efflux of particles and liquid. Most of the work has been on sedimentation of multisized particles of uniform density. For a system consisting of m different sizes of particles (l,2,3,...,m, m being the largest), m zones of settling suspension (1,2,3...,m starting at the top) are formed, with clear liquid above and a sediment layer at the bottom, as shown in Fig. 2.4.3. The lowest zone m, just above the sediment boundary, contains all particle species at their initial concentration, whereas the region immediately above it is devoid of the largest particles. Each successive zone contains one fewer species than the zone below, with the upper zone 1 containing only the smallest particles 1. Each zone disappears in turn during the sedimentation process as its upper boundary becomes co-incident with that of the sediment layer. In batch sedimentation of a polydisperse suspension, as shown in Fig.2.4.3, the upward flowrate of fluid in zone k (k=l,2,3,...m) must equal the total downward flowrate of particles in the same zone, i.e. ' 22 Chapter 2 Settling of Particles in Liquids UkSk+X vuCi,k = 0 (i=l,2,3,...,k) i=i (2.4.4) sedimentation velocity of interface v u V 2 ,2 V3,3 Vk-l,k-l Vk,k Vk+l,k+l Vm-1 ,m-l clear liquid zone 1 zone 2 zone ; k - l zone k -zonem-1 zone m sediment contains only small particles 1 contains particles 1 and 2 contains particles 1, , k - l - contains particles 1, ,k . contains particles 1, ,m-l . contains all particles 1, ,m Figure 2.4.3. Formation of zones in batch sedimentation of polydisperse suspension (dpi<dp2< <dPk-i<dpk< <dpm-i<dpm). 23 Chapter 2 Settling of Particles in Liquids where Ci,k is the volumetric concentration of the ith particle fraction in zone k, Uk and Vi,k are the absolute velocities of the fluid and the ith particle fraction, respectively, in zone k, and 8k is the voidage in zone k. From volumetric considerations, k e k = l - X c i * (2.4.5) i=i If Vj,k is the relative velocity of the ith particle fraction relative to the fluid in zone k, vj,k = u k+Vi, k (2.4.6) Substituting from equations (2.4.5) and (2.4.6) into equation (2.4.4) gives k k u k(l-X Cj, k ) + X (uk+Vi,k)ci,k = 0 (2.4.7) i=l i=l k that is, u k = - ^  Vj.kCj.k i=l Substituting for Uk from equation (2.4.6) leads to k V i , k = V i , k - X Vi ,kCi > k (2.4.8) i=l The continuity equation for species i across the boundary between k and (k-l) yields Ci,k-i = Ci,k(vk,k-Vi,k)/(vk,k-Vi,k-i) (2.4.9) Since the concentrations of all particles in the bottom zone m are equal to the initial concentrations in the suspension, equation (2.4.8) permits direct calculation of the sedimentation velocities of all particles in the bottom zone. Then, proceeding zone by zone, the sedimentation velocity and the concentration of all particles in each zone in a polydisperse suspension can be calculated by iteration using eqns (2.4.8) and (2.4.9). This 24 Chapter 2 Settling of Particles in Liquids information suffices to give a complete description of the separation process at any instant. Differential Hydrodynamic Model: The sedimentation system consists of a sufficient number of particles that discontinuities can be smoothed out; therefore, derivatives of various properties are continuous (Gidaspow, 1993). Based on the mass and momentum conservation laws, differential hydrodynamic models have been proposed by several researchers, e.g. Greenspan and Ungarish (1982) and Shih et al. (1987), to describe the flow of non-dilute suspensions of particles of different sizes and densities. The continuity equation is — c; pi + V - CiPiVi = 0 (i = 0,l,2,...,n) (2.4.10) oi while the momentum equation is — C i P i V j + V - C i P i V i V i = -Ci V P + CiPig+f D , i (i = 0,l,2,...,n) (2.4.11) at where P is the total pressure due to the weight of particles and liquid, while foj is the drag force per unit volume for the ith species of particles. The continuous fluid phase is designated by index i=0, while the n particle species are represented by i=l,2,...,n. Equation (2.4.10) replaces Equation (2.4.4). However, this differential equation can be solved using finite-difference techniques that yield a set of equations similar to Equation (2.4.4). 25 Chapter 2 Settling of Particles in Liquids Continuous Model: — Nasr-El-Din and Masliyah Model (1988,1990) In practice, many industrial sedimentation systems operate continuously with continuous feed addition and underflow removal. Unlike batch sedimentation, continuous sedimentation studies are sparse. Continuous separation of binary suspensions containing particle species lighter and heavier than the suspending fluid in a vertical gravity settler was first studied by Nasr-El-Din et al. (1988, 1990). A theoretical model, based on mass balances and slip velocities, was developed. Figure 2.4.4 shows a schematic of their vertical settler together with terms used in the mathematical model. The settler is assumed to consist of a source zone located at the feed point of the settler. This zone supplies the suspension to the underflow and overflow streams. The solids concentration within this zone is assumed to be uniform. If the downward fluid velocities are larger than the rise velocity of particles of the ith species, particles are carried across the lower boundary of the source zone and report to the underflow stream. Otherwise the particles can only cross the upper boundary of the source zone to report to the overflow stream. A volumetric balance over the source zone is given by Fluid QfEf=A c8(uu-Uo) (2.4.12) Particle species i QfCif=A cCj(v u ,i-v 0 ,i) (2.4.13) where Ac is the cross-sectional area of the column and Q denotes volumetric flowrates, with subscripts f, u, o denoting the feed, underflow and overflow, respectively. The particle velocities at the overflow and underflow boundaries of the source zone are given by v0>i=min (0, u 0+V 0 ) i) (2.4.14) 26 Chapter 2 Settling of Particles in Liquids vu,i=max (0, uu+Vu >i) The overflow and underflow stream flow rates are Q o = - A c ( £ c i V 0 > i +eu 0) Q u = A c ( X c i v u i + 6 u u ) (2.4.15) (2.4.16) (2.4.17) The volume fractions of the particles in the overflow and underflow streams can be written as Co,i - A c V 0 , i C j / Q 0 Cu, i — A c Vu, iCj /Q u (2.4.18) (2.4.19) Feed Qf, 6f, Cf,i Overflow stream Q o , So, C 0 , j L Overflow boundary Cross-sectional area of column Ac Underflow boundary Underflow stream Q u , Su, Cu,i + Figure 2.4.4. Schematic of vertical settler for the model of Nasr-El-Din (1988) Equations (2.4.12) to (2.4.19) provide a simple model for the continuous separation of particles in a vertical settler in the absence of lateral concentration gradients. This model predicted all the observed qualitative trends and agreed quite well 27 Chapter 2 Settling of Particles in Liquids with the quantitative measurements. However, the assumption that the solids concentration within the source zone is uniform is not valid in a real classifier. Some recent experimental investigation, e.g. Spannenberg et al. (1996) and Galvin et al. (1998), have been performed on differential sedimentation of binary suspensions under continuous conditions. However, little theoretical analysis has been carried out on the behavior of liquid classifiers with particles having continuous size or density distributions, especially for unsteady state dynamics. 2.4.3 Markov Chain Model of Liquid Classifier Stochastic approaches have been applied by several groups to study the behavior of sedimentation and fluidization, e.g. Tsujimoto and Yamamoto (1986), Hesse (1993), Fan et al. (1995) and Chebbo et al. (1995). During the experimental study of the U B C classifier, Zhang (1998) proposed that the random movement of particles in a particle-liquid classifier can be considered as a stochastic or Markov process. For an event transferring from status I to another, if the transfer probability depends only on the condition of status I, then such a stochastic process is called a Markov Process. The classifier introduced in Chapter 1 is divided into Ni compartments (or cells) in series, as shown in Figure 2.4.5. Each cell represents a status in a Markov chain. Within At, or one step of transfer, particles in a cell can only move to adjacent cells or stay in the same cell. For a particle in cell I, whether a given particle will move to 1+1, to I- l or stay in cell I in an interval of time At (i.e. one step of transfer), depends on the transfer probability, which has nothing to do with where this particle has come from. The transfer probability is a function of the hydrodynamics in the cell, and is independent of the other cells. The 28 Chapter 2 Settling of Particles in Liquids fraction of particles of a given species moving to adjacent cells equals the transfer probability of this particle species. For simplicity the turbulence in the entire classifier is considered to be isotropic, and the overall intensity is assumed to be uniform for all cells. Particles in each cell are completely mixed. The Richardson-Zaki equation is applied to calculate the voidage in each cell. Overflow Q 0 , s0, c0,j Bi... N i - 1 . Feed Qf, Sf, Cfi 1±1 I..... .1-1. Discharge Qa, sd, cd,i Fluidization Q, 'a Figure 2.4.5. Schematic of Markov Chain model 29 Chapter 2 Settling of Particles in Liquids Forces exerted on a particle: In addition to the usual forces on a single particle described in section 2.1, namely Gravitational force: 1 3 F g - ' - ^ p P p g (2.4.20) Buoyancy force: ^ 1 J 3 F B = - 7 c d p p L g (2.4.21) Drag force: F D - C D — - — - d p (2.4.22) Zhang (1998) assumed a fluctuation force due to turbulence which may be written as F - ^ C D . a ^ _ . ^ d J ; F ' . z = - C D . i ^ . V ^ 2 4 -2 2 4 F y - C D • z : V F . y — C D : :dn f (2.4. 2 4 2 4 (2.4.23) »2 i 2 F ' - r P L V Z ^ H 2 - F ' - r P L V - Z * 2 , F X - C D - d p , F . X - C D - d p ; where V is the fluctuation slip velocity, equal to the turbulence intensity times the average slip velocity, V, i.e. V = L V The turbulence intensity is defined as it=v7v The forces with their directions are indicated in Figure 2.4.6. (2.4.24) (2.4.25) 30 Chapter 2 Settling of Particles in Liquids * F D Figure 2.4.6. Forces exerted on a particle in the Markov Chain model Transfer probability: It is assumed that a particle can move in 6 possible directions in each cell as shown in Figure 2.4.6. If the forces exerted on the particle are balanced, the probability of moving in any of the directions is 1/6. If the forces on the particle are unbalanced, a weight is added to modify the probability, i.e. 1 Pz = - Y z o P y = - y y P-z = I _1_ 6 Y z 1 1 where < l . 1 1 P x = T Y x P-X=-X — 6 6 y x Yx forces in z direction forces in - z direction forces in y direction forces in - y direction forces in X direction forces in - x direction (2.4.26) Since the turbulence in the entire classifier is assumed isotropic, F Z=F . z, F y=F . y, and F X=F _x., so that Yx=Yy=YZ : =l The normalized transfer probabilities for particle to move upwards and downwards are: 31 Chapter 2 Settling of Particles in Liquids Pz=" Yz Y z + 4 Y z + 1 P-Z= 1 Y z + 4 Y z + 1 (2.4.27) For Yz=l, p z = P - z = l / 6 . For y z>l, it can be shown that pz>l/6 while p.z<l/6, which means that particles have a greater probability of moving upwards rather than downwards. Determination of height of a cell: Figure 2.4.7. Schematic of determination of the height of a cell During the calculation, the height of a cell is determined first. As shown in Figure 2.4.7, for a cell, within At (or one step of transfer) the maximum distance a particle can move is Az„=V'zAt To ensure that the particle will not move to 1+1 and I-l at the same time, Ah should be greater than 2Az m , i.e. Ah=XV' zAt (X>2) (2.4.28) 32 Chapter 2 Settling of Particles in Liquids Calculation of voidage in the cell: The Richardson-Zaki equation is applied to calculate the voidage in every cell. At steady state, the particle and liquid absolute velocities are ( l - s d ) Q d / A c v=-0 - 8 c ) u _ Q a / A c - s d Q d / A c e c Hence the particle slip velocity can be written as V=v-U= - r f l - E d M d / A c ] _ [ Q a / A c - s d Q d / A C ] ^ ^ ( l - e c ) s c In the Markov Chain model, it is assumed that the voidage in the cell equals the voidage in the discharge stream, i.e. Sd=Sc Then, from Equation (2.4.29), the voidage and the particle slip velocity can be written as 1 Sc=Sd= r Q a / A c ^ V v T J (2.4.30) V = - - ^ ^ = - v T s c n - 1 (2.4.31) Calculation Procedures: Figure 2.4.8 gives the flow chart for the computations. With a single fitting parameter, L, the Markov Chain model predicts the particle size distribution for both the discharge and overflow streams and gives encouraging agreement with U B C experimental results, as well as pilot and commercial data (Zhang, 1998). However, the voidage in the cylindrical 33 Chapter 2 Settling of Particles in Liquids Initially, assume particle size distribution, PSD, to be uniform along the column. Obtain the mean particle size dpso and calculate vj and n of particle with size dpso Calculate s c and V from Eq uations (2.4.30) and (2.4.31) Set I t Determine the height of each cell from Equations (2.4.24) and (2.4.28) Calculate the amount of particle input and output to/from the column during time interval At Determine the Markov transfer probability based on the forces exerted on a particle from Equations (2.4.20) - (2.4.27) Calculate the quantity of particles, as well as its PSD, transferred to adjacent cells Determine the new PSD in each cell no Figure 2.4.8. Flow chart of Markov Chain Model 34 Chapter 2 Settling of Particles in Liquids region is predicted to be uniform along the column by using Equation (2.4.30) based on the assumption that the voidage in the column equals the discharge voidage. This does not agree with the experimental observations, except near the bottom at the discharge position. A discussion of the predictions of this model is provided in Chapter 4, together with a comparison with the hydrodynamic model developed in that chapter and experimental data measured as part of this thesis. 2.5 Explanation of Emptying Phenomenon in Industrial Classifier An emptying phenomenon is sometimes encountered in industrial classifiers. It may be explained as follows. For a mono-disperse suspension in the classifier (see Figure 2.4.5), the absolute particle velocity can be expressed by means of the Richardson-Zaki (1954) correlation: ^ U a - U j S d n-i ( 2 5 1 ) £ c where U a = Q a / A c and U d = Q d / A c . If v<0, the particles move downwards and are carried out by the discharge stream at the bottom. However, if the particle velocity, v>0, the particles move upwards and cannot be discharged at the bottom, leading to emptying of the classifier. Figure 2.5.1 shows the variation of particle velocity with respect to column voidage at different discharge voidages. It can be seen that there exists a limiting voidage s w which is about 0.61-0.64 under the conditions indicated in this figure. If the voidage in the column is less than s w, the particle velocity is positive, i.e. the particles move 35 Chapter 2 Settling of Particles in Liquids upwards. It can also be seen that the limiting voidage, sw, does not change much with changes of the discharge voidage from 0.4 to 1.0. The effect of the liquid fluidizing velocity, U a , on the particle velocity is shown in Figure 2.5.2. It is seen that the limiting voidage, ew, increases with increasing fluidizing velocity, U a . The effect of particle size on the particle velocity is shown in Figure 2.5.3. It can be seen that the smaller the particle size, the higher the limiting voidage, ew. It can thus be concluded that, with increasing liquid fluidizing velocity, U a , and decreasing particle size, d p, the limiting voidage, sw, increases. As shown in Figure 1.1, there is an expanded freeboard or bulge section at the top in the classifier. Defluidization and accumulation of particles may occur in the diverging section, resulting in a lower voidage within that section. If the voidage is lower than the limiting voidage, the particles cannot move downwards, causing emptying of the classifier. In commercial, as opposed to laboratory units, feed voidages tend to be lower, about 0.55-0.65, and the particle size is relatively small, about 0.4 mm. Figure 2.5.4 shows the variation of particle velocity with respect to the voidage for those conditions. It can be seen that the limiting voidage, ew, is about 0.62. It may be quite possible to make the voidage in the diverging region and in the upper section of the cylindrical region lower than the limiting voidage (sw «0.62) during operation, resulting in no particles settling downwards, i.e. the emptying of the classifier. As described in section 2.3, although the correlation of Richardson and Zaki (1954) was originally derived for mono-disperse suspensions, it is also applicable to systems with multi-species particles. For classifiers with multi-species particles, each species has its own limiting voidage, which is different and can be independently obtained in the 36 Chapter 2 Settling of Particles in Liquids same manner as described above for mono-disperse suspensions. If the voidage in the classifier is lower than the smallest limiting voidage, all the particle species cannot settle downwards, leading to emptying of the classifier. 0.10 -0.08 -\ -0.10 -j 1 1 1 1 1 1 1 1 1 1 r—'-\ 1 1 1 1 1 1 1 1 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 Voidage in column, sc F i g u r e 2.5.1. Variation of particle velocity with respect to voidage in the column at different discharge voidages. Ua=5.5 cm/s, U d = l . l cm/s, d p =l.l mm, pp=4970 kg/m3, pL=1000kg/m3, ui=0.001 Pa.s 37 Chapter 2 Settling of Particles in Liquids 0.50 0.55 0.60 0.65 0.70 Voidage in column, s 0.85 Figure 2.5.2. Effect of liquid fluidizing velocity on particle velocity at different discharge voidages. U d =l . l cm/s, d p=l.l mm, pp=4970 kg/m3, pL=1000 kg/m3, p.L=0.001 Pa.s 0.12 0.10 0.08 0.06 0.04-0.02-0.00 > o o U -0.02 -| 0) o ro -0.04 -I -0.06 -0.08 H -0.10 -0.12 (mm) T" 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 Voidage in column, s T" T" 0.75 0.80 0.85 Figure 2.5.3. Effect of particle size on particle velocity at different discharge voidages. Ua=5.5 cm/s,Ud=l.l cm/s, pp=4970 kg/m3,pL=1000 kg/m3, ui=0.001Pa.s 38 Chapter 2 Settling of Particles in Liquids 0.06 Voidage in column, e Figure 2.5.4. Variation of particle velocity with respect to the voidage in column at different discharge voidages under commercial conditions. U a=3.0 cm/s, U d=0.4 cm/s, dp=0.4 mm, p p=900 kg/m 3 , pL=399.3 kg/m 3 , p L=7.34xlO" 5 Pa.s 39 Chapter 3 Experiments Chapter 3 Experiments The solid-liquid classification equipment is located in the A M P E L building at UBC. Two cold model classifiers of diameter 508 mm and 191 mm have similar geometric configurations and are broadly similar to two industrial-scale classifiers operated by the Mitsubishi Chemical Corporation (MCC). Variations in geometric configuration have also been tested at UBC. Appendix A gives the detailed dimensions of some of the classifiers tested. The particle size and operating conditions at U B C were predetermined to give similar Archimedes and Reynolds numbers as those used at M C C . The methods to measure the flow rates, voidages and particle size distributions are described in this chapter. The steady and unsteady state hydrodynamic behaviors were investigated for mono-disperse, binary and multi-size particulate suspensions at different operating conditions. The experimental results are presented and discussed in Chapters 4 and 5 in comparison with the predicted results of the steady and unsteady state hydrodynamic models, which are also developed in these chapters. The columns, conductivity probes and samplers were designed and primarily operated by other members of the group. The author participated alongside other members of the group in obtaining all the mono-disperse, binary, and some of the multi-disperse data presented in this thesis. 3.1 Equipment and Particles 3.1.1 Equipment Figure 3.1.1 shows a schematic diagram of the U B C experimental particle classification system. The components include: 40 CD o , o O O o o O "5 '•6 (X O s t-, bO 2 '•3 o ' is s o t/3 0) s 41 Chapter 3 Experiments 1. A polyethylene holding tank of 1.22 m diameter and 1.22 m height equipped with a stirrer to mix the discharge and overflow streams coming from the classifier. 2. A centrifugal pump used to transfer mixtures of particles and liquid from the holding tank to a hydrocyclone. 3. A hydrocyclone made of steel with a 610 mm inside diameter and 1380 mm height which separates the mixture of particles and liquid into two streams, a virtually particle-free liquid stream taken from the top of the cyclone and a dense particle suspension stream leaving from the bottom. The dimensions of the hydrocyclone are shown in Figure 3.1.2. (b) r—*\ (a) Figure 3.1.2. Dimensions of hydrocyclone. All units are in mm. (a) side view; (b) top view 42 Chapter 3 Experiments DP: Pressure transducer CP: Conductivity probe SP: Sampler Liquid fluidization Figure 3.1.3. Dimensions of the classifier for the C-0 configuration. A l l units are in mm. 43 Chapter 3 Experiments 4. Two O M E G A - M A G magnetic flow meters, with meter ranges from 0 to 19 liters/s, that control and measure the feed flow rate, Qf, and liquid fluidization flow rate, Q a . 5. A classifier made from Plexiglas mounted in a vertical position. Figure 3.1.3 gives the detailed dimensions of the classifier which has a diameter of 191 mm and a height of 1540 mm. The classifier has an expanded cylindrical section (the "bulge") at the top, while the bottom is conical. The liquid stream from the top of the hydrocyclone is used as the liquid fluidization stream and is injected upwards into the classifier from the bottom, while the dense particle suspension from the bottom of the hydrocyclone provides the feed stream entering the classifier through a radial port near the top. A relatively dilute stream containing mostly small particles is taken off the top of the classifier as an overflow stream, while a more concentrated suspension stream, enriched in coarser particles, is removed as a discharge stream from one side near the base of the column. Both the overflow and underflow streams are returned to the holding tank for remixing and recirculation to the classifier. 6. Four pressure transducers (model: Endress+Hauser Deltabar PMD 230) mounted below the bulge section to measure the pressure gradients along the classifier column in order to determine the axial voidage variation. The positions of the pressure transducers are shown in Figure 3.1.3, The procedure used to calculate the voidage is described in section 3.2. 7. Nine conductivity probes to measure the local voidage at different radial and axial positions in the classifier. As described in Chapter 1, the feed stream enters the classifier as a horizontal jet, resulting in a vigorous vortex within the top region of the classifier, so that the pressure transducer cannot give accurate results for the top 44 Chapter 3 Experiments region. For this reason, but also to give localized measurements, conductivity probes were also applied. The positions of the conductivity probes along the column are shown in Figure 3.1.3. Another probe used as a reference is located in the holding tank. The geometry and dimensions of the conductivity probes are shown in Figure 3.1.4. The cylindrical body is made of a Plexiglas tube of length 384 mm and diameter 12 mm, enveloped by two stainless steel rings of thickness about 100 pm. During the measurement, the end of the tube with the two rings, which are flush with the tube surface, is injected into the classifier. The conductivity of the suspension between the two rings is recorded. The principle of the conductivity probe to measure the voidage is described in section 3.3. 384 380 - 376" - 372 368 " Wires Handle - : - H * i 2 Steel plates Figure 3.1.4 Dimensions of conductivity probe. Al l units are in mm. Mehrani (1999). 45 Chapter 3 Experiments 8. Five tube samplers made of steel installed along the column to trap and then extract samples from the classifier in order to determine the particle size distribution. The positions of the ports through which the samplers were inserted are shown in Figure 3.1.3. Figure 3.1.5 portrays the geometry and dimensions of the tube samplers, each of 405 mm length and 40 mm diameter. Figure 3.1.5. Dimensions of tube samplers shown in plan view with slot open. Al l units are in mm. For details of construction and calibration, see Mehrani (1999). 9. A PC computer used to record and analyze the signals from the magnetic flow meters, pressure transducers and conductivity probes. 10. Several pails used to collect samples from the discharge and overflow streams in order to measure their voidages and particle size distributions. The calculations are described in section 3.2. 3.1.2. Particles: The particles used by M C C are made of polypropylene (pp=900 kg/m3 and 0=0.75) and are suspended in liquid propylene (pi_=399 kg/m3 and U.L =7.34xl0"5 Pa.s under operating 46 Chapter 3 Experiments conditions). The cumulative undersize particle size distribution can be fitted by a Rosin-Rammler distribution: F=l-exp (A Y 1 ' v x r y with xr= (ln2)" r (3.1.1) where dpso is the median particle size, where the curves of cumulative distribution by mass pass through 50%, and n r indicates the breadth of the particle size distribution. Another expression for the breadth of particle size distribution, dpgo /d pio, can be derived as dP9o/dpio-ln(O.l) ln(0.9) = 21.85"' where d9o and dpio are the particle diameters where the cumulative distribution curves pass through 90% and 10% respectively. If n r is higher, or dP9o/dpio is lower, the breadth of the particle size distribution is narrower. In the U B C unit, the particles and liquid were spherical glass beads (pp=2740 kg/m3 and O ^ l ) and water (pL=997 kg/m3 and pL=9.979xlO"4 Pa.s at 21°C), making pp/pL nearly the same as in the industrial unit. The mean particle diameter in the feed stream was typically about 1.1 mm, determined by matching the Archimedes number of the industrial unit, i.e. C 3 ^ PL(Pp-pL)g dp50 ^PL(Pp-pL)gdp50^ UBC (3.1.2) / M C C In the U B C unit, the range of the Rosin Rammler index n r is from 2 to 6, giving dp9o/dpio=1.7 to 4.7. 47 Chapter 3 Experiments 3.2 Procedures 3.2.1 Operating Conditions The base operating conditions are first determined by matching the Reynolds numbers and velocity ratios, i.e. >p50PLua 'dpSOPlUa^ (dnsnPT.U A M-L Jmc v M-L M UBC l U a J MCC UBC MCC J M C C (3.2.1) (3.2.2) (3.2.3) For example, for Ua=2 cm/s, Uf=5.0 cm/s and Ud=0.5 cm/s, dP5o=0.4 mm, pi=399 kg/m , Pi=7.34xl0"5 Pa.s in the industrial unit, 'dp50PLUaA (Re)Mcc= V M-L =43.5 J M C C v U a y =2.5 MCC v U a y =0.25 MCC This leads to Ua=3.96 cm/s, Uf=9.9 cm/s and Ud=0.99 cm/s in the U B C unit. The operating conditions used in the U B C experiments are given in Appendix B. During the experiments, the temperature in the holding tank was kept constant at about 2i+rc. 48 Chapter 3 Experiments 3.2.2 Measurement of Parameters and Methods Table 3.2.1. Measured parameters and methods Parameter Determined by Feed stream flowrate, Qf Magnetic flowmeter Liquid fluidization stream flowrate, Q a Magnetic flowmeter Discharge stream flowrate, Qd Pails sampling technique Overflow stream flowrate, Q 0 Mass balance, Equation (3.2.5) Discharge and overflow voidages, £d, e 0 Pails sampling technique Feed voidage, Sf Mass balance Local radial average voidage in classifier, s (z) Pressure gradient Local voidage in classifier, s (r,z) Conductivity probe Discharge and overflow particle size distributions, PSDd and P S D 0 Sieving dried samples Feed particle size distribution, PSDf Mass balance Local particle size distribution, PSD(r,z) Tube sampling and sieving Table 3.2.1 lists the parameters measured and the measurement methods. Feed flowrate, Qf, and liquid fluidization flowrate, Q a , can be controlled and measured by two magnetic flowmeters. The discharge flow rate, Qd, is obtained by recording the time to collect one pail of discharge stream, i.e. _ Volume of the pail (2 liters) —. TT (3-2.4) Time (s) 49 Chapter 3 Experiments The flowrate of the discharge stream Q d = U d A c , where A c is the cross-sectional area of the column, i.e. Ac=(7t/4)xl9.1 2 =287 cm 2. Therefore, a collection time of 6.3 s corresponds to a discharge velocity of 1.1 cm/s. The overflow flow rate , Q 0 , is calculated from an overall mass balance by Q 0 = Q f + Q a - Q d (3.2.5) The discharge and overflow voidages, Sd and s0, were measured by sampling. In the experiments, the mixtures of particles and liquid in the discharge and overflow streams were first collected in pails. Sd and s 0 are obtained after measuring the weight and volume of the collected samples, using the relations weight of the mixture in discharge ,„ „ . . ( l - 8 d ) P p + e d p L = — ~ — : . , 5 (3.2.6a) volume of the mixture in discharge weight of the mixture in overflow (l-8 0 )pp+s o P l =— . : - — (3.2.6b) volume of the mixture in overflow while the particle size distributions in the discharge and overflow streams are determined by sieving the samples after pouring off the liquid and then drying them. The feed voidage, Sf, and particle size distribution of the feed stream are calculated from mass balances: Qa+8fQf=SdQd+SoQo ( 3 . 2 . 7 ) Cf , iQf=Cd , iQd+c 0 , iQo ( 3 . 2 . 8 ) The axial profile of cross-sectional average voidage, ec(z), along the cylindrical region is derived from pressure gradient measurements. In the cylindrical region, the classifier behaves as a liquid fluidized bed, so the pressure gradient in the bed is given by A P -—={Pp[l-Sc(z)]+pLSc(z)}g (3 .2 .9) Az 50 Chapter 3 Experiments and -AP=pLg(AH+Az) (3.2.10) where Az is the difference in height between the two pressure ports on the classifier and A H is the recorded additional pressure drop expressed as a head (length) of the fluidizing fluid as measured by the pressure transducer. Substituting Equation (3.2.10) into Equation (3.2.9), the axial voidage in the classifier can be calculated by A H ec(z)=\--^- (3.2.11) f P _ l PL The local voidage can also be measured by a tube sampling technique. Particle samples in the column of the classifier were first taken by a tube sampler as described in section 3.1 and then dried, weighed and sieved. The voidage was then obtained by. mass of particles/p„ Ec(r,z)=l (3.2.12) volume of the tube while the local size distribution was determined by sieve analysis. 3.3 Conductivity Probe The conductivity of a mixture of solid particles and liquid depends on the conductivities of the two individual phases and their volume fractions. Therefore, measurement of the electrical conductivity of the continuous phase and the dispersed phase at different positions in the classifier can be related to the local voidage at those positions. Figure 3.1.3 shows the conductivity probe designed for the experiment. The Wheatstone bridge method is used to find the conductivity. Figure 3.3.1 shows a schematic diagram of the 51 Chapter 3 Experiments electronic circuit connected to the conductivity probe. Components include a Wheatstone bridge, an amplifier and a rectifier. R i , R 2 and R 3 are three fixed resistors, and the resistance of the probe, R p r ob, is to be measured. Wheatstone Bridge Figure 3.3.1. Electronic circuit connected to the conductivity probe Mehrani (1999). The Wheatstone bridge may act as an antenna and pick up 60 Hz waves from other electronic devices such as pumps and fluorescent lamps around the bridge. Therefore an amplifier is used to eliminate interference. Vi„ represents the voltage source for the bridge. Alternating current is used to minimize polarization effects. The rectifier changes the A C output voltage, V o u t , to a DC voltage that can be read by a computer. After measuring V D C and knowing the resistances R i , R2 and R 3 , the resistance of the probe, Rprob, can be calculated from 52 Chapter 3 Experiments Rprob - R3 R ' out R1+R2 Y (3.3.1) in J In our experiment, R.=19.9 Q, R2=100.1 Q, R3=19.9 O and V in=9.93 volts, so that f \ Rprob=19.9 1 0.8442- V ° u t 9.93 (3.3.2) The probe conductivity, C , can be determined by C = K / R p r o b (3.3.3) where K is a constant of the probe which depends on the configuration of its two electrodes. The probe conductivity, C , or its resistance, Rp r 0 b, varies with the particle concentration in the suspension. The relative conductivity, C L - S / C L , or relative resistance, Rprob,L/Rprob,L-s - C L - S / C L , is used to determine the voidage in the suspension. Here the subscripts L and L-S denote that the probe is in a pure liquid and in a liquid-solid suspension respectively. The procedure for calibrating the probe was as follows: 1. The conductivity of a mixture of solid particles and liquid varies with the properties of the two individual phases, which are functions of temperature. The properties of the liquid phase also change slightly after particles are introduced into the liquid because the particles are not clean and bring some dirt into the liquid. Therefore, a reference probe is needed to measure the liquid conductivity for the solid-liquid mixture at the known temperature. The temperature recorded in the mixing tank is assumed to be the same as that in the classifier. The suspension in the upper section 53 Chapter 3 Experiments of the tank is very dilute, and hence is considered to be pure liquid, even though there may be a few particles there. Hence, probe #0 is installed in the upper section of the mixing tank as a reference for the solid-liquid mixture system in step 6 below. 2. As described above, the classifier behaves as a liquid fluidized bed in the cylindrical region of the classifier. Therefore, the axial voidage variation in the cylindrical region can be measured by a pressure transducer, which can be mounted at any level in this region. The conductivity probe #1 to be calibrated is then installed at the same position as the pressure transducer. 3. When no particles are added into the system, both probes are in pure liquid. The initial output voltages for the two probes are then recorded as V ' 0 , L and V ' I , L respectively, where subscripts 0 and 1 denote probe #0 and #1, respectively. 4. Particles are added to the mixing tank and the flowrate of the fluidizing liquid is increased to expand the bed and vary the voidage. After the system reaches steady state, for each flowrate, the output voltages of both probes are recorded as VO,L-S and VI,L-S In the meantime, the voidage in the column based on the differential pressure transducer is recorded. 5. The corresponding resistances, R ' 0 , L, R ' I , L , RO,L-S and RI,L-S , are calculated from Equation (3.3.2). Because probe #0 is installed in the upper section of the mixing tank where the suspension is very dilute, RO,L-S is considered to be the resistance of pure liquid measured by probe #0 for the solid-liquid mixture system. 6. As discussed above, after the particles are added and the bed is expanded, the presence of particles introduces some small changes in the liquid properties. 54 Chapter 3 Experiments Therefore, the resistance of pure liquid measured by probe #1 for the solid-liquid mixture system should be modified as RI,L=(PVO,L-S/R'O,L)R'I,L (3.3.4) Then, the relative resistance for probe #1, RI ,L/RI ,L-S , can be obtained. 7. The liquid fluidizing velocity, U a , is varied and the corresponding voidage, s, measured by the pressure transducer, and relative resistance, Ri , i /Ri , L-S, obtained from the conductivity probe, are recorded. The variation of the voidage with respect to the relative resistance, R I , L / R I , L-S, is then plotted to obtain the calibration curve. A sample calculation is presented in Appendix C. The voidage, s, measured by the differential pressure transducer, and relative resistance, RI,L/RI,L-S , measured by the conductivity probe, were recorded at different liquid fluidizing velocities. Plotting RI,L/RI,L-S versus s led to the calibration curve for probe # 1 shown in Figure 3.3.2. The curve given in this figure is a fitted third order polynomial. The other eight probes were calibrated in the same manner and their calibration curves are shown in Appendix D. The parameters of the correlations, represented by the third order polynomials, to calculate the voidages for the nine probes are listed in Table 3.3.1. In the experiments, the relative resistances were measured by the same steps 1 to 6 as above, and the corresponding voidages were then obtained from the calibration curves or the correlations listed in Table 3.3.1 55 Chapter 3 Experiments Table 3.3.1. Values of constant coefficients in third-order polynomial conductivity probe calibration f e = a R prob,L ^Rprob.L-S J R \2 prob,L l^Rprob.L-S J f R prob,L \^  Rprob, L-S J + d Conductivity probe* C P I CP2 CP3 CP4 CP5 CP6 CP7 CP8 CP9 a -8.674 -3.64 -6.438 -2.822 -6.621 -7.310 -6.729 -3.790 -7.604 b 18.798 7.715 12.917 5.014 14.216 15.972 13.573 7.249 15.876 c -11.934 -4.147 -6.926 -1.369 -8.5254 -10.089 -7.416 -3.035 -9.274 d 2.810 1.075 1.448 0.178 1.930 2.429 1.572 0.576 2.002 *For positions of probes CPI to CP9, see Figure 3.1.3. 1.1 1.0-0.9 u 0.8 a> cn ro •D 0.7 o > 0.6 0.5 0.4 E =-8 .67 (R 1 L /R 1 L J 3 + 18 .80 (R 1 L /R 1 L.3) 2 -11.94(R /R ^+2.81 - ' 1 1 1 1 r~ 0.4 0.5 0.6 0.7 i 1 r~ 0.8 0.9 Relative resistance, (R 1 L /R 1 L S ) —i— 1.0 1.1 Figure 3.3.2. Variation of the voidage with respect to the relative resistance of probe #1 56 Chapter 4 Steady State Hydrodynamic Model Chapter 4 Steady State Hydrodynamic Model In order to predict the steady state behavior of the classifier, a steady state hydrodynamic model with only one fitted parameter, an axial dispersion coefficient, is developed in this chapter. The governing equations are based on mass and momentum balances, with the boundary conditions that the particle concentration at the top of the classifier equals that in the overflow stream, while the particle concentration in the lower portion near the discharge is equal to that in the discharge stream. The correlation of D i Felice (1994) is used to calculate the drag force on the particles, while the particle dispersion force is introduced in the same manner as by Thelen and Ramirez (1999). The calculation procedures to solve the model are presented in section 4.5. The calculated results for a mono-disperse suspension, binary system and multi-size system are shown and compared with experimental data in section 4.6. A correlation to calculate the dispersion coefficient is then established based on experimental data from both UBC and M C C . The predictions of the classifier performance and efficiency for different operating conditions are next shown in section 4.8, and compared with experimental results. Finally, the hydrodynamic model is compared with the Markov Chain model of Zhang (1998) in section 4.9. 4.1 Conceptual Basis Figure 4.1.1 shows a schematic of the U B C cold model classifier. At the top, multi-species particles enter the classifier conveyed by the feed stream and then begin to settle downwards. Liquid is introduced upwards from the bottom as a fluidizing stream. Some particles are removed via a discharge stream near the bottom, while some particles are 57 Chapter 4 Steady State Hydrodynamic Model entrained by an overflow stream leaving at the top. The classifier is divided into eight zones as shown. From top to the bottom, they are: top zone I, feed zone, top zone II, diverging zone, column zone, discharge zone, bottom zone I and bottom zone II. Cross-sectional area at top, At Qf , Sf, CR Feed stream Cross-sectional area of column, Ac Overflow stream Q 0 , s0, c 0 Top zone I Feed zone Top zone II Diverging zone Column zone +4 Discharge stream Q d , Sd, Cdi Discharge zone Bottom zone I Bottom zone II Q a = Fluidization flow Figure 4.1.1. Schematic of cold model classifier 58 Chapter 4 Steady State Hydrodynamic Model From the experimental results, e.g. those shown in Figure 4.1.2, it is found that there is a relatively uniform voidage distribution over the central core of the column, with a slightly lower voidage near the wall. Visual observations indicated that there was some internal circulation of particles, with more downward motion near the wall and some upward movement in the central region. This pattern probably caused the radial voidage variation. In the hydrodynamic model, the classifier system is assumed uniform in the lateral direction. 1.0 0.00 0.02 0.04 0.06 0.08 0.10 Radius (m) Figure 4.1.2. Radial profile of voidage for a mono-disperse suspension Ua=6.29 cm/s, Uf=22.13 cm/s, Ud=1.31 cm/s, 8f=0.88, dp=0.9 mm, h= 1.36 m, T-2 configuration On a local level, individual particles may move randomly, but their average velocity tends to be large in comparison with their velocity of fluctuations. Therefore, the fluctuations are neglected. However, turbulence in the flow is taken into account indirectly via an axial dispersion coefficient, which is assumed to be uniform throughout the entire volume. In the model, downwards is set to be positive. The forces exerted on 59 Chapter 4 Steady State Hydrodynamic Model the particles are one external force — gravity, one interaction force — drag due to the relative motion of the liquid, and one particle dispersion force which causes spreading of particles due to local turbulence. 4.2 Governing Equations and Boundary Conditions The hydrodynamic model is based on mass and momentum balances. Figure 4.2.1. shows the differential element of the classifier column. In general, considering one fluid and n species of particles, the continuity and momentum equations are written for one-dimensional flow as follows: \ / Cross-sectional area A i A. I. I. . z Az \ ckvh_uj3 f— rh + z I | \ I I / ^ Figure 4.2.1. Differential element of the classifier column Continuity: Solids: - ( c i P i V i A ) z + A z + ( c i P i V i A ) z -fihjAAz = ^ i P ^ M ( 4 . 2 . 1 ) dt Liquid: - ( s p L u A ) z + A z + (sp L uA) z + rh L AAz = ( 1 ( s P L A A z ) (4.2.2) dt where A is cross-sectional area of the classifier column, which is different for different zones. For the top zone, A=A t, while in the column zone, A=Ac. For diverging zone or bottom zone II, A=Ab, the vertically varying cross-sectional area, which can be related to its vertical position as shown in Appendix E. rhj and m L represent mass rates of particles 60 Chapter 4 Steady State Hvdrodvnamic Model and liquid addition (positive) or subtraction (negative) from the side of the column per unit volume of the bed. Since Az is independent of time, t, Equations (4.2.1) and (4.2.2) can also be written as partial differential forms. a (c i P i A) t afCjPjVjA) _ A A ( 4 2 3 ) dt dz a ( s P L A ) a(e P L uA) A ( 4 2 4 ) dt dz Since the cross-sectional area, A, does not change with time, t, the above two equations can be written as 8(c i P i ) dfCiPjVjA) = ^ dt Adz a^P k) + a (e P L uA) = ( 4 2 6 ) a Adz At steady state condition, the continuity equations for solids and liquid are d Adz d ( C i P i V j A ) = rii; (4.2.7) (sp LuA) = m L (4.2.8) Adz Equations (4.2.7) and (4.2.8) are one-dimensional in the vertical direction. If there is no addition or subtraction of particles and liquid from the side of the classifier in the horizontal direction, then rhj and rhLare equal to zero. However, in the feed zone, particles and liquid enter the classifier continuously from a feed pipe, while in the discharge zone, particles and liquid are removed from the classifier continuously through a discharge pipe, resulting in positive riij and rh L in the feed zone and negative riij and m L in the discharge zone. More details are provided in section 4.4. 61 Chapter 4 Steady State Hydrodynamic Model Momentum balances: Based on the two-fluid model (Boure and Delhaye, 1982), augmented by the dispersion term, momentum equations may be written as Solids: dfCiPiAAzVi) . . . , dP c . . . . - ^ J - ^ = - ( C i P i V j A v O z + A z +(c i P i v i Av i ) z +(-Ci — + P i c i g + c i f d r a g i + C i f d i s p i ) A A z dt dz (4.2.9) Liquid: d(epLAAzu) = _ ( g p L u A u ) z + A z + ( S P L U A U ) z + ( - s ^ + p Lsg - ^cf^ - Z c i f d i s p i ) A A z (4.2.10) where terms (cjpiViAv;) and ( S P L U A U ) are momentum rates of particles and liquid pass through the element. fdrag,i and fdisp,i are drag and dispersion forces per unit volume of particle species i. See Equations (4.3.7) and (4.3.9). Similar to continuity equations, equations (4.2.9) and (4.2.10) can also be written as d(ep, u) 3(ep, u 2 A ) dP _ „ _ „ At steady state condition, the momentum equations for solids and liquid are: -(CiPiVi'A) = -Cj — + P i c i g + c T d r ^ +cT d i s p i (4.2.13) » j v lri 1 ' 1 j Adz dz d , dP ( e p L u 2 A ) = - 8 — + p L s g - I c i f d r a g i - Z c i f d i s p i (4.2.14) Adz dz From Equation (4.2.14) 62 Chapter 4 Steady State Hydrodynamic Model ~T- = [ - Z r ^ u 2 A ) - Pi*g + I c j f ^ + IC ifd i s p i ]/s (4.2.15) dz Adz Substituting Equation (4.2.15) into (4.2.13) leads to d / * \ d v H e[V; — - ( C i P i V ; A) + c i P i Vi — ] = Adz dz C i [ u - ^ - ( s p L u A ) + sp LU^-] + C is(p i - p L ) g + Ci(Xcjfdra^ + Z C i f d i s p i ) + 8C i(fdragi +fdispi) Adz dz (4.2.16) Substituting Equations (4.2.7) and (4.2.8) into (4.2.16), we obtain dv ; du. eCjCPiVj — - p L u — ) + eVjmi - c i U m L dz dz (4.2.17) = CiS(pi - p L ) g + C i ( Z C i f d r a g i +Icifdlspi) + 6Ci(fdragi + fdispi ) Assuming P ; and P L are independent of z, then from Equations (4.2.7) and (4.2.8), dv. V: d^ m. v. dA ,.. „ „ „x —- = — - — L + — ! - — (4.2.18) dz Cj dz P i C ; A dz du u ds m, u dA , A — = + — (4.2.19) dz s dz pLe A dz Substitution of Equations (4.2.18) and (4.2.19) into (4.2.17) yields 2dc; , . 2 ^ d c i . , „ w „ . . 2 , „ „ 2 > dA - s P i v i ^ - c i P L u 2 Z ^ + 28v1mi - 2 C i u m L + 8C l(-p ivj' +pLu ) ( 4 ^ = Cje(Pi - p J g + CiXcjfj +80^ where f i=fdragi+fdis P i The particle concentration is related to voidage by s+Eci=l (4.2.21) and from overall mass balances on the particles and on the two inlet and two outlet streams, Q f C f i - Q d C d i - Q o C o i = 0 (4.2.22) 63 Chapter 4 Steady State Hydrodynamic Model Q f + Q a - Q d - Q o = 0 (4.2.23) In the steady state hydrodynamic model, Equation (4.2.20), with Equations (4.2.21) to (4.2.23), are the governing equations to be integrated to predict the particle concentration. The pressure variation through the column could also be calculated using Equations (4.2.13) and (4.2.14). Parameters, rhj, r h L , u and vi in Equation (4.2.20) can be written as functions of particle concentration, Q . Section 4.4 describes the procedure for the different zones in the classifier. Boundary Conditions: From the experimental results shown in Figure 4.2.1 and other similar results, the voidages in the column at the lower portion near the discharge and in the upper portion near the overflow are almost equal to the voidage in the discharge and overflow streams for mono-disperse, binary and multi-size systems. For the binary system, the fractions of large particles in the discharge and overflow streams are close to those in the lower and upper portions, as shown in Figure 4.2.2. For the multi-size system, the mean particle size and the breadth of particle size distribution in the lower and upper portion are close to those in the discharge and overflow, respectively, as shown in Figures 4.2.3 and 4.2.4. These results indicate that the particle concentrations in the overflow and discharge streams are almost the same as those in the upper portion and lower portion (near the discharge) respectively, in the column of the classifier. Hence, with z as the distance below the overflow level, the boundary conditions for the hydrodynamic model are at z=0, Ci=coi (4.2.24) at z=zd, Ci=cdzi (4.2.25) 64 Chapter 4 Steady State Hvdrodvnamic Model 1.0 £ E ro 3 ? 8 E o eg w H. 9 tr o 0.9 H II N 1 1 0 E m C CN cn • ro i-•o <u 1 I 0.8 0.7 H 0.6 0.5 0.4 • discharge/lowerformono-disperse systems • overflow/upper for mo no-disperse systems • discharge/lower for binary systems o overflow/upperforbinary systems A discharge/lower for multi-size systems 0 A overflow/upper for multi-size s y s t e m s ' # • n ' + 10% y^ •10% 0.4 —r~ 0.8 —I— 0.9 1.0 0.5 0.6 0.7 Voidage in discharge and overflow Figure 4.2.2. Comparison of the voidage in the lower and upper portion with voidage in discharge and overflow for mono-disperse, binary and multi-size systems. Figure 4.2.3. Comparison of the fractions of large particles in the lower and upper portions with those in the discharge and overflow for binary system 65 Chapter 4 Steady State Hvdrodvnamic Model Figure 4.2.4. Comparison of mean particle sizes in the lower and upper portions with those in discharge and overflow for multi-size system 4.5 c 1.5 -j 1 1 1 1 1 1 1 1 ' 1 1 1 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Breadth of P S D , d p 9 0 / d p 1 0 , in discharge and overflow Figure 4.2.5. Comparison of the PSD breadths in the lower and upper portions with those in the discharge and overflow for multi-size system 66 Chapter 4 Steady State Hydrodynamic Model 4.3 Fluid Particle Drag and Particle Dispersion Force Fluid particle drag force: In the absence of other particles, the force on a single particle in a fluid is |2 F n „ = C n / i " - V J ( U - V ) ^ (4.3.1) 1 D O ~ " - D O 2 4 Table 2.1 in Chapter 2 gives the correlations used to calculate the drag coefficient, C D O -For a multiparticle liquid system, the particles can be considered to be individually supported by the fluid (Richardson and Zaki, 1954). The influence of neighbouring particles on the drag force may be assumed to be solely a function of the local volumetric particle concentration, or void fraction, e (Di Felice, 1994): F D i = F D 0 i f ( s ) (4.3.2) The drag force, Fooi, for isolated particles of the ith species of particles in the liquid-solid suspension of voidage s should be 2 F - r ffir N P L g I u ~ Vj | s ( u - V i ) r c d p i ^DOi - c D 0 i ( K e i ) — (4.3.3) where the Reynolds number for the ith species is d D; I u - v; I p L s R e i =_PJj L L ^ (434) "L Di Felice (1994) demonstrated that for a wide variety of suspended-particle systems, the voidage function f(s) may be expressed as f(e)=e-p (4.3.5) with p - 3 . 7 - 0 . 6 5 e x p [ - ( 1 5 - 1 0 g l o R e i ) 2 ] (4.3.6) 67 Chapter 4 Steady State Hydrodynamic Model In the momentum Equations (4.2.13) and (4.2.14), fdragi represents the drag force per unit volume of particle phase i. Let FDi=Vjfdragi, where Vi is the volume of a single particle of species i , i.e. V =7r.dpi3/6. Then the fluid particle drag force per unit volume can be written as f * „ g i = F D , / V i = F D 0 i f ( e ) / V i = C D 0 , ( R e , ) P ^ l " - V ; | s ( " - V ' ) ^ f ( s ) / A (4.3.7) 2 4 6 4 c D o i ( ^ i ) s 2 | u ; V i l p L f ( g ) ( " - v , ) Particle dispersion force: Truesdell (1962) argued that particle diffusion (or dispersion), being a change of motion, can be considered to arise from forces on the particles. Thelen and Ramirez (1999) introduced dispersion forces in modeling solid-liquid fluidization in the Stokes flow regime. In our model, dispersion forces, which cause spreading of particles due to local turbulence, are likewise introduced into the momentum Equations (4.2.13) and (4.2.14), but over the whole range of Reynolds numbers. A dispersion velocity can be written v d i , p l = - B ^ (4.3.8) Cj dz where D a is an axial dispersion coefficient for the particles. According to Thelen and Ramirez (1999), the dispersion force can be expressed in a form similar to the drag force: f a ^ C ^ R e / | V ^ i s p i l p L f ( e ) v d i s p i (4.3.9) 4 d p i with 68 Chapter 4 Steady State Hydrodynamic Model R e r d p i ' V d U p i ' (4-3.10) M-L 4.4 Equations for the Eight Zones of the Classifier As described in section 4.1, the classifier is divided into eight zones. For different zones, the expression for parameters, irq, m L , u and V i , which are functions of Cj, can differ. The equations for the different zones are derived below, with the downward direction chosen as positive. Top zone I: As shown in Figure 4.4.1, in top zone I, there is no additional source of particle and liquid entering or leaving aside from those crossing the lower surface from below and those exiting with the overflow stream at the top. The mass rates of particle and liquid production are zero, i.e. rhj = 0 (4.4.1a) m L = 0 (4.4.2a) section j Feed Q f, Cf,, s f .• .> .•.» Overflow Q 0 , c0j, s 0 Vj v N/ U Cj s + H top I Figure 4.4.1. Schematic of top zone I 69 Chapter 4 Steady State Hydrodynamic Model From mass balances, the total flowrates of particles and liquid into and out of the section j (control volume), should balance each other. That is Solids: 0 - C j V i A t - Q o C i o = 0 (4.4.3a) Liquid: 0 - euA - Q 0 e o = 0 (4.4.4a) Feed zone: Q a - Q d S d , Cdi Figure 4.4.2. Schematic of feed zone A schematic of the feed zone is shown in Figure 4.4.2. The particle-liquid suspension in the feed stream is assumed to enter the classifier uniformly at the entrance port. At height z, the fraction of the area of cell Aj with height dz and length I to the total area of feed pipe is Act= I dz/Af where I is the length of cell Aj intercepted by the feed pipe and a is the ratio of the area of cell j intercepting the feed pipe to the total area of the feed pipe. From geometry, ^=2{R f 2 -[RKz-z f )] 2 } 0 5 (4.4.5) 70 Chapter 4 Steady State Hydrodynamic Model o=(J*dz)/A f (4.4.6) where Rf is the radius of the feed pipe. As discussed in section 4.2, in the feed zone, particles and liquid enter the classifier continuously from a feed pipe in the horizontal direction. Then the mass rates of particle and liquid addition from the feed pipe into cell Aj of the classifier per unit volume of the column are ^.cfth^aaa ^ ( 4 4 1 b ) A t d z . L = ( < d z / A ) Q f s p L = Q f e p L < / ( A f A ) A t d z From mass balances for section j , Solids: aQf Cf, - qviAt - QoCoi =0 (4.4.3b) Liquid: aQf&f - e u A t - QoS o=0 (4.4.4b) Top Zone II: Feed Q f, c f i, e f section j H t o p II + Q d , Ed, Cdi Figure 4.4.3. Schematic of top zone TJ 71 Chapter 4 Steady State Hydrodynamic Model Figure 4.4.3 shows a schematic of top zone II. In this zone, the mass rates of addition of particles and liquid are everywhere: rhi=0 (4.4.1c) m L = 0 (4.4.2c) Hence mass balances for section j yield: Solids: CjVjAt - Q dCdi =0 (4.4.3c) Liquid: euAt +(Qa - Qded)= 0 (4.4.4c) Diverging Zone: Cross-sectional area Ab + section j Q a ! L ^ Q d , ed, c d i Figure 4.4.4. Schematic of diverging zone Figure 4.4.4 shows the diverging zone in the classifier. There is no addition of particles or liquid from the side also, so rn, = 0 m L = 0 From mass balances for section j , Solids: QViAb- Q d c d i = 0 Liquid: suAb + Q a - Q d s d = 0 (4.4. Id) (4.4.2d) (4.4.3d) (4.4.4d) 72 Chapter 4 Steady State Hvdrodvnamic Model Column zone: Figure 4.4.5 shows the schematic of the column zone. With no chemical reaction and no side ports on this section, the mass rate of production of particles and liquid are rhj =0 m L = 0 From mass balances for section j , Solids: CjVjAc - QdCdi =0 Liquid: suAc + Q a - Qd8d = 0 (4.4. le) (4.4.2e) (4.4.3e) (4.4.4e) Section j + > Discharge Q d , Sd, Cdi Figure 4.4.5. Schematic of column zone Discharge zone: A schematic of the discharge zone is shown in Figure 4.4.6. At distance z below the top of the column, the fraction of the area of cell Aj to the total area of the discharge pipe is Aa=£dz/Ad 73 Chapter 4 Steady State Hvdrodvnamic Model QdCdi "Qa+QdSd Q a 1 Figure 4.4.6. Schematic of discharge zone Again, as discussed section 4.2, in the discharge zone, particles and liquid are removed form the classifier continuously through a discharge pipe in the horizontal direction, resulting in negative values of rhj and riiL in Equations (4.2.7) and (4.2.8). From the boundary condition, Equation (4.2.25), the particle concentration at any vertical level in the discharge stream is assumed equal to the horizontally adjacent concentration in the discharge zone. The composition of the discharge stream taken out of the classifier is assumed to be uniform at the discharge site. Therefore, the mass rates of addition of particles and liquid per unit volume can be written . _ ( < d 2 / A d ) Q d c i P , = _ Q d C . p / / ( A c A d ) ( 4 4 l f ) A c d z A C f t b M 1 ^ p L = A c d z Mass balances for section j give 74 Chapter 4 Steady State Hydrodynamic Model z Solids: QdCdi - CiViAc - / ( Q d ' A ^ d z = Q ( 4 4 3 f ) z d Liquid: -Q a+QdS d - suAc - J(Qd / Ad)&ftfc = 0 (4.4.4f) z d The length of cell Aj, £, and the ratio of the area of cell j to the total area of the discharge pipe can be calculated by ^=2{Rd2-[Rd-(z-zd)]2}0-5 (4.4.7) a=( |^dz)/A d (4.4.8) Zd where Rd is the radius of the discharge pipe. As discussed above for boundary conditions, at the discharge position the particle concentration in the column of the classifier is assumed equal to the particle concentration in the discharge stream. From mass balance on the discharge stream, the flowrate of each species of particles, i , should be zd+2Rd c d iQd= J(Q d/Ad)c,*dz z d The particle discharge concentration, c-di, can be written zd+2Rd cdi= Jc^dz/A d (4.4.9) Zd and the discharge voidage can be written z d+2R d z d+2R d ed=l-Zc d i = l - £ J c - i * d z / A d = jsJ?dz/A d (4.4.10) Z d Z d where £ can be calculated from Equation (4.4.7). 75 Chapter 4 Steady State Hydrodynamic Model Bottom zone I: Section Vi U Cj s Q d , Sd, Cdi Hbottom I t + Q a Figure 4.4.7. Schematic of bottom zone I Figure 4.4.7. shows the schematic of the bottom zone I of the classifier. The mass rates of addition of particles and liquid are zero so that rhi=0 (4.4. lg) rh L = 0 (4.4.2g) From mass balances for section j , Solids: C i V i A c = 0, i.e. v ; = 0 (4.4.3g) Liquid: s u A c + Q a = 0 (4.4.4g) Bottom zone II: The diverging bottom zone is shown in Figure 4.4.8. The parameters rhj and rh L in Equations (4.2.1) and (4.2.2) are simply m i = 0 (4.4. lh) m L = 0 (4.4.2h) From mass balances for section j , Solids: CjVi=0, i.e. Vj=0 (4.4.4h) 76 Chapter 4 Steady State Hydrodynamic Model Liquid: suAb + Q a = 0 (4.4.3h) For the entire bottom zone, the particle velocity, v., must be zero for steady state conditions. Figure 4.4.8. Schematic of bottom zone II 4.5 Calculation Procedure The particle concentration, Ci, can be obtained by integrating Equation (4.2.20) with the boundary conditions Equations (4.2.24) and (4.2.25). Equations (4.2.21) and (4.4.1) to (4.4.4) are used to calculate parameters e, rhj, r h L , u and Vj for every zone. Overall mass balance Equations (4.2.22) and (4.2.23) are also utilized in the calculation together with Equations (4.3.7) and (4.3.9) to compute the drag and dispersion forces. A computer flow chart is shown in Figure 4.5.1. The software package, Microsoft Fortran 77 Chapter 4 Steady State Hydrodynamic Model PowerStation 4.0 (1994-1995), was employed in the calculation. Appendix F lists the Fortran program written to solve the steady state hydrodynamic model. One routine I V P A G is supplied in Microsoft Fortran Power Station 4.0 to solve a stiff system of first-order ordinary differential equations of the form / = f (t, y) with boundary conditions y(t = to), where N is the number of differential equations, f, to be solved and y(t) = |y 1(t),...,yN(t)] r, f(t,y) - [f 1(t,y),..,fN(t,y)] T Two classes of implicit linear multistep methods are available in routine IVPAG. The first is the implicit Adams-Moulton method, while the second uses the backward differentiation formulas, BDF. The BDF method is often called Gear's stiff method. Equation (4.2.20) in the hydrodynamic model can be converted to the form y' = j{t, y) (see the program in Appendix F), so that the routine I V P A G can be adopted to do the integration. Both methods, Adams-Moulton and Gear BDF in routine IVPAG, are applicable for this case. Appendix G gives the description of routine IVPAG. The routines NEQNF and NEQNJ using a modified Powell hybrid method to find the zero of a system of nonlinear algebraic equations are supplied in the Microsoft Fortran Power Station 4.0. Both routines, explained in Appendix H , are also adopted in the program listed in Appendix F to solve the hydrodynamic model. Input data required by the model are listed in Table 4.5.1. In the calculation, the height step, Az, required in the integration of every zone in the classifier, should be determined. In Table 4.5.1, M is the number of height increments of every zone, so the incremental height is Az=H/M. During the calculation, the particle concentrations in the 78 Chapter 4 Steady State Hydrodynamic Model overflow are first guessed. Equation (4.2.20) is then integrated downward from the top to the bottom of the classifier, and the particle concentration in the discharge stream is then calculated from Equation (4.4.9). The new particle concentration in the overflow is next obtained from the overall mass balance, Equation (4.2.22), and this value is checked to see whether it is equal to the previously assumed one. If not, the particle concentration in the overflow is changed and the integration from top to bottom is repeated. This procedure continues until the relative error between the new value and the previous one is within a preset small tolerance, 0.001. As described in section 4.2, in the proposed hydrodynamic model, one parameter, the axial dispersion coefficient, must be optimized to "best fit" the experimental data. The least squares objective function, which is minimized to optimize D a , is the sum of the squares of the deviations between calculated voidages along the column in the classifier, together with the voidage in the discharge stream, and the corresponding experimental values, i.e. 2 2 f=2> c -e C ) e x p ) + ( S d - S d , e x p ) ( 4 - 5 1 ) As described in Chapter 3, nine voidages were measured along the column by nine conductivity probes, while one discharge voidage was measured by a sampler. These ten voidages were used in the calculation of the objective function. In addition, for mono-disperse systems, four voidages derived from pressure transducers were also included in the optimization. For binary and multi-size systems, the sum of the squares of the deviations between the calculated particle volumetric concentrations in the discharge and the 79 Chapter 4 Steady State Hydrodynamic Model corresponding experimental values is found to be good enough to act as the least squares objective function to optimize the parameter, D a , i.e. f=L(c d i -c d i ,exp) 2 0=1,2,. .,n) (4.5.2) For n particle species, n points are included in the calculation, i.e. for binary, two points, and for six species of particles, six points are included in the optimization. For binary and multi-size system, only data from the discharge were used. It should be pointed out that, for binary and multi-size systems, i f the values of particle concentration along the column in the classifier are included in the least squares objective function, the optimized D a does not change much. In most of our experiments, only particle concentrations in the discharge stream were measured for multi-size systems. Therefore, Equation (4.5.2) was used as the least squares objective function in the calculation. 80 Chapter 4 Steady State Hydrodynamic Model Table 4.5.1. Input data required by model Symbol Quantities Co,initial Initial guess of particle concentration in overflow Da, initial Initial guess of dispersion coefficient Tol Tolerance Uf, U d , U a Velocity in feed, discharge and fluidization Sf Feed voidage cfi Particle distribution in feed (i=l,2,...,n) dpi Particle diameter (i=l,2,. .,n) n Number of particle phases Pp, P L Density of particles and liquid H-L Liquid viscosity g Acceleration of gravity D e Diameter of liquid fluidizing pipe for the calculation on bottom zone II D c , D t Diameter of column and top zone in classifier Htop I, Hfeed, Htop II, Hdiverge, He, Height of eight zones in the classifier Hdischarge, Hbottom I, Hbottom II M t o p I, Mfeed, M t o p II, Mdiverge, M c , Mdischarge, Mbottom I, Mbottom II Number of height increments in every zone of the classifier needed in the integration 81 Chapter 4 Steady State Hydrodynamic Model Input data Change D ; Initial guess of D a w • 3 Initial gi i jess of c 0i Integrate Equation (4.2.20) downward Calculate cdj from Equation (4.4.9) From overall mass balance, calculate c0j' from Equation (4.2.22) Compare c 0i' with initial guess of c 0 error=|c0i'-Coi|/c0 Change c 0 Calculate objective function 2 2 f = Z ( 8 c - s c,exp) + ( s d - 8 d ,exp) f o r mono-disperse system f= (c^ - Cdi > e xp) f ° r binary and multi-size system no Figure 4.5.1. Computer flow chart to solve the steady state hydrodynamic model 82 Chapter 4 Steady State Hydrodynamic Model 4.6 Calculated Results and Comparison with Experimental Data In this section, the calculated results with the single fitted axial dispersion coefficient are compared with the experimental data at steady state conditions for mono-disperse, binary and multi-size systems in this section. It should be noted that the "steady state" mentioned here was never rock steady. The system was considered to be at steady state when the mean values of variables recorded by the computer, i.e. flowrates (Q a and Qf) controlled by flowmeters, and the voidage along the column zone measured by pressure transducers, were nearly invariant with time; however, there were still some residual fluctuations, likely associated with pump pulsations, experimental error, etc. Mono-disperse suspension: The hydrodynamic model was first solved for mono-disperse particles of 0.9±0.05 mm diameter using a classifier with a T-2 configuration. The geometry and dimensions are given in Appendix A. Figure 4.6.1 compares the experimental voidage profiles along the column with predictions from the steady state hydrodynamic model with one fitted parameter, the axial dispersion coefficient. The height, h, is the distance from the bottom of the classifier. This figure also shows the standard deviations of the voidages based on repeated measurements. It can been seen that, above the discharge portion, the voidage is predicted to decrease downwards in a realistic manner for a feed voidage 8f=0.88. It can also be seen that in the conical section at the bottom, the voidage always increases downwards due to the increasing superficial liquid velocity as one moves downwards in the conical region. The local, discharge and overflow voidages, calculated using one fitted axial dispersion coefficient, match the experimental data well. 83 Chapter 4 Steady State Hydrodynamic Model Figure 4.6.2 compares the predicted discharge voidages with experimental data for a wide range of operating conditions. It is shown that the predictions of the model calculated with one fitted parameter agree reasonably well with the experimental results for mono-disperse suspensions, although the predicted slope of 8 vs. h in the upper part of the column is noticeably less than that of the experimental points. Voidage, E Figure 4.6.1. Comparison of predicted voidage from steady state model with experimental data for mono-disperse suspension: Ua=6.29cm/s,Uf=22.1 cm/s,Ud= 1.31 cm/s,Sf=0.88, dp=0.9mm,Da=257cm2/s,T-2 configuration, Run ID=M09e01 84 Chapter 4 Steady State Hydrodynamic Model Figure 4.6.2. Comparison of predicted discharge voidage with experimental data for mono-disperse suspensions: Ua=3.31 to 8.31 cm/s, Uf=15.8 to 25.4 cm/s, Ud=0.64 to 3.46 cm/s, dp=0.9 mm, Sf=0.72 to 0.88, Da=210 to 745 cm2/s, T-2 configuration. Binary System: Particle classification was also studied for a binary system using the steady state hydrodynamic model. Particles having two different sizes, 0.9±0.05 mm and 0.55±0.05 mm diameter, are used in the calculation for a classifier with a T-2 configuration. Figure 4.6.3 gives a comparison of the experimental data with the predicted profiles from the steady state hydrodynamic model using the fitted axial dispersion coefficient. As for the mono-disperse suspension, the voidage in the column decreases downwards above the discharge portion, while increasing in the conical region below the discharge portion. This figure also shows the standard deviations of the voidages based on repeated measurements. The calculated local, discharge and overflow voidages again agree reasonable well with the corresponding experimental data. 85 Chapter 4 Steady State Hydrodynamic Model 2.0 Vo idage , e Figure 4.6.3. Comparison of calculated voidage by steady state model with experimental data for a binary system: Ua=6.32 cm/s, Uf=20.8 cm/s, Ud=1.28 cm/s, Sf=0.87, dPL=0-9 mm, dps=0.55 mm, Da=280 cm2/s, Cf,L=0.52, T-2 configuration, Run ID=B0509e2. * the standard deviations of data from samplers are obtained from a binary system done by Mehrani (1999). Figure 4.6.4 shows the particle hold-up profile along the column for the two particle species in the binary system. The fraction of large particles increases downwards, while the fraction of small particles increases upwards, indicating that most large particles settle and leave through the discharge stream, while small particles are more likely to be carried out by the overflow stream at the top. The calculated particle fractions in the column, discharge and overflow using a single fitted parameter, the axial dispersion coefficient, agree quite well with the experimental data. 86 Chapter 4 Steady State Hydrndynamic Model 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Liquid-free volume fraction of particles, C L and C s Figure 4.6.4. Comparison of calculated liquid-free volume fraction of particles by steady state model with experimental data for binary system: Ua=6.34 cm/s, Uf=20.15 cm/s, Ud=1.28 cm/s, epO.87, dpL=0.9 mm, dpS=0.55 mm, Da=280 cm2/s, Cf,L=0.52, T-2 configuration, Run H>=B0509e2. Over a wide range of operating conditions, the voidage and large particle fraction in the discharge stream calculated with a single fitted parameter, the axial dispersion coefficient, agree well with the experimental data for this binary system, as shown in Figures 4.6.5 and 4.6.6. 87 Chapter 4 Steady State Hydrodynamic Model 0.5 0.6 0.7 0.8 0.9 1.0 Experimental discharge voidage, ed Figure 4.6.5. Comparison of discharge voidage between the experimental data and hydrodynamic model for binary system: U a=3.27 to 8.34 cm/s, Uf=15.5 to 29.4 cm/s, Ud=0.71 to 2.63 cm/s, 8^=0.70 to 0.87, dpL=0.9 mm, dpS=0.55 mm, Da=250 to 708 cm2/s, Cf,L=0.50 to 0.54, T-2 configuration. Experimental fraction of large particles in the discharge stream Figure 4.6.6. Comparison of large particle fraction in discharge between experimental data and hydrodynamic model for binary system: U a=3.27 to 8.34 cm/s, Uf=15.5 to 29.4 cm/s, Ud=0.71 to 2.63 cm/s, s^0.70 to 0.87, dpL=0.9 mm, dpS=0.55 mm, Da=250 to 708 cm2/s, Cf)L=0.50 to 0.54, T-2 configuration. 88 Chapter 4 Steady State Hydrodynamic Model Multi-size System: The classifier with the C-0 configuration (see Appendix A) was next used to test the steady state behavior for a system with a continuous particle size distribution. The Rosin-Rammler expression is also applied to fit the particle size distribution to the U B C experimental results. Figure 4.6.7 compares the experimental voidage profile along the column with calculated results of the hydrodynamic model with a single fitted parameter, the axial dispersion coefficient. The predictions are seen to agree reasonably well with the experimental results, although they often lie outside the confidence range of the data, especially near the feed port, where there was considerable turbulence and radial variation due to the entering jet. A measure of the error in D a is given by the pair of values where the objective function (i.e. sum of square of deviations) is 10% greater than at the fitted values. This gives limits of 519 and 577 cm2/s about the fitted value of 552 cm2/s. The voidage in the column again decreases downwards above the discharge port, while increasing in the conical region below the discharge port. This figure also shows the standard deviations of the voidages based on repeated measurements. Figure 4.6.8 shows concentration profiles along the column for six particle size fractions. The proportion of larger particles increases downwards, while the hold-up of small particles increases upwards, meaning that coarse particles settle more easily in the classifier. The calculated results are consistent with the experimental data. Figure 4.6.9 compares the predicted and experimental feed, overflow and discharge cumulative size distributions. It can been seen that both the mean particle size and the breadth of the discharge stream distribution are greater than those of the feed and overflow streams, proving that particle classification does take place in the classifier. The 89 Chapter 4 Steady State Hydrodynamic Model model provides a good description of the particle size distribution in the discharge stream as well as within the classifier. Experimental and predicted voidages, mean particle sizes and breadths of the particle size distribution in the discharge stream are shown in Figures 4.6.10, 4.6.11 and 4.6.12 for a wide range of operating conditions. The predictions of the hydrodynamic model with a single fitted parameter are seen to be in good agreement with the experimental data for the investigated multi-size system. Voidage, e Figure 4.6.7. Comparison of calculated voidage by steady state model with experimental data for multi-size system: Ua=6.27 cm/s, Uf=21.2 cm/s, Ud=1.46 cm/s, Sf=0.88, dp50,f=1.13 mm, dp9o,f/dpio,i=3.62, C-0 configuration, Run ID=C0ef04. Da=552 cm2/s: fitted value, which corresponds to minimum of objective function (f=3.17xl0"4) Da=577 cm2/s: objective function 10% above fitted value Da=519 cm2/s: objective function 10% below fitted value * the standard deviations of data from samplers are estimated from a binary system done by Mehrani (1999). 90 o c 91 Chapter 4 Steady State Hydrodynamic Model 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Particle size, dp (mm) Figure 4.6.9. Comparison of calculated cumulative particle size distribution by mass with experimental data for multi-size system: Ua=6.27 cm/s, Uf=21.2 cm/s, Ud=1.46 cm/s, Sf=0.88, Da=551 cm2/s, dp5o,f=1.13 mm, dp9o,f/dpio,f=3.62, C-0 configuration Run ID=C0ef04. 0.4 -| 1 1 1 1 1 1 1 1 1 1 0.4 0.5 0.6 0.7 0.8 0.9 Experimental discharge voidage, s d Figure 4.6.10. Comparison of experimental discharge voidage with predictions of hydrodynamic model for multi-size system: Ua=2.53 to 7.39 cm/s, Uf=9.17 to 23.8 cm/s, Ud=0.39 to 2.97 cm/s, Sf=0.55 to 0.94, Da=300 to 1930 cm2/s, dp5o=107 to 1.25, dpgoVdpio.f^l^e to 3.63, C-0 configuration. 92 Chapter 4 Steady State Hydrodynamic Model 1.0 1.2 1.4 1.6 Experimental mean particle size, d p 5 0 d (mm) Figure 4.6.11. Comparison between experimental mean particle size, dpso„d, in discharge and hydrodynamic model for multi-size system: Ua=2.53 to 7.39 cm/s, Uf=9.17 to 23.8 cm/s, Ud=0.39 to 2.97 cm/s, Sf=0.55 to 0.94, Da=300 to 1930 cm2/s, dp5o=1.07 to 1.25, dp9osf/dpio,f=1.66 to 3.63, C-0 configuration. 5 1 0 Figure 4.6.12. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Experimental breadth of PSD, d p e o d / d p 1 0 d Comparison between experimental breadth of1 PSD, d9o,d/dpio,d, discharge and hydrodynamic model for multi-size system: Ua=2.53 to 7.39 cm/s, Uf=9.17 to 23.8 cm/s, Ud=0.39 to 2.97 cm/s, Sf=0.55 to 0.94, Da=300 to 1930 cm2/s, dp50=1.07 to 1.25, dp9o,f/dpio,f=1.66 to 3.63, C-0 configuration. in 93 Chapter 4 Steady State Hydrodynamic Model 4.7 Correlation of Axial Dispersion Coefficient The axial dispersion coefficient is a fitted parameter which accounts indirectly for turbulence, non-uniform velocity profiles and other factors which promote longitudinal mixing of particles. Table 4.7.1 compares the range of fitted axial dispersion coefficients for our liquid-solid classifier with typical magnitudes of diffusion and dispersion coefficients for some other systems. Table 4.7.1. Axial dispersion coefficient in different systems System Value (cm2/s) Multi-size particle axial dispersion coefficients in continuous liquid-solid classifier from this work 250 ~2000 Mono particle axial dispersion coefficients in liquid-fluidized beds (Epstein, 2000) 0.5-33 Binary particle axial dispersion coefficients in liquid-fluidized beds (Epstein, 2000) 0.24-515 Liquid axial dispersion coefficients in liquid fluidized beds (Epstein, 2000) 1 - 100 Liquid radial dispersion coefficients in liquid fluidized beds (Epstein, 2000) 0.1 -4 .0 Molecular self-diffusion coefficient of water at 25°C (Sherwood et al., 1975) 0.0000244 Eddy radial diffusion coefficients in water for duct flow at Reynolds numbers of 8000 ~ 800,000 (Sherwood et al., 1975) 0 .2-20 It is seen that the dispersion coefficients obtained in our work are higher than all the other values obtained, probably because of the substantial turbulence generated by the feed stream and the bottom apex liquid jet. A larger dispersion coefficient, due for example to a higher turbulence intensity, causes more particle mixing in the classifier and poorer particle classification. With a higher axial dispersion coefficient, the predicted voidage profile in the classifier becomes more uniform, as shown in Figure 4.7.1, and the cumulative particle size distribution curve in the discharge approaches that in the feed, as shown in Figure 4.7.2, indicating poor classification. 94 Chapter 4 Steady State Hydrodynamic Model Voidage, e Figure 4.7.1. Voidage profile along the column predicted by hydrodynamic model with different axial dispersion coefficients for multi-size system: Ua=6.3 cm/s, Uf=22.0 cm/s, Ud=1.3 cm/s, Sf=0.8, dP5o,f=l mm, dp9o,f/dpio,f=2.43, C-0 configuration. Particle size, d p (mm) Figure 4.7.2. Cumulative PSD by mass in discharge predicted by hydrodynamic model with different axial dispersion coefficients for multi-size system: U a=6.3 cm/s, Uf=22.0 cm/s, U d=1.3 cm/s, Sf=0.8, dP5o,f=l mm, dP9o,f/dpio,f=2.43, C-0 configuration. 95 Chapter 4 Steady State Hydrodynamic Model Figure 4.7.3 shows the variation of the fitted axial dispersion coefficient with respect to the feed voidage. It is seen that the fitted dispersion coefficient appears to reach a maximum at about sp0.65. For 0.65<ef<0.92, the fitted D a decreases with increasing feed voidage, while it decreases with decreasing Sf for 6f<0.65. As shown in Figure 4.7.4, with a broader particle size distribution in the feed, i.e., a higher dP9o,f/dpio,f ratio, the fitted axial dispersion coefficient is lower due to reduced mixing among the different species of particles in the sedimentation. The fitted dispersion coefficient is higher for a shorter classifier as shown in Figure 4.7.5. In the classifier, the turbulence is mostly generated due to the feed stream entering the top and by the fluidizing stream injected from the bottom. The shorter the cylindrical region of the classifier, the more turbulence there is in the classifier. As shown in Figures 4.7.6.and 4.7.7, the fitted axial dispersion coefficient is not very sensitive to changes in fluidizing velocity, U a , or discharge velocity, Ud, while it decreases slightly with increasing feed velocity, as shown in Figure 4.7.8. It was also found that the fitted axial dispersion coefficient for a larger classifier is higher than for a smaller classifier when the two classifiers are geometrically similar. Using this information and dimensional analysis, a correlation was developed to estimate the particle axial dispersion coefficient in terms of relevant dimensionless parameters. The system variables which may influence the dispersion coefficient are the liquid fluidizing velocity, U a , feed velocity, Uf, discharge velocity, Ud, feed voidage, Sf, feed particle mean size, dP5o,f, and breadth, dP9o,f/dpio,f, height and diameter of the column zone of the classifier, H c and D c , physical properties of particles and liquid, p p, P L and U L , and acceleration of gravity. Hence 96 Chapter 4 Steady State Hydrodynamic Model 2000 -r «T 1800-O, 16004 ^ 200-I 1 , 1 , 1 . 1 . 1 • 1 • 1 i 1 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 Feed voidage, ef Figure 4.7.3. Variation of fitted dispersion coefficient with feed voidage, 6f, for multi-size system: Ua=5.42 to 5.67 cm/s, Uf=18.1 to 20.3 cm/s, Ud=1.07 cm/s, dp5o,f=1.08 to 1.09 mm, dp9o,f/dpio,f=1.67 to 1.78, C-0 configuration. 2000 w 1800 E O 1600 . 1400 c O 1200-| g 1000 c o 800 0) C L W T3 T3 600 400 4 200 —I— 2.0 —I— 2.5 —I— 1.5 3.0 3.5 4.0 Breadth of PSD in feed, d p 9 0 /d p 1 0 f Figure 4.7.4. Variation of fitted axial dispersion coefficient with PSD breadth in feed, d9o,d/dpio,f, for multi-size system: Ua=5.39to 5.48 cm/s, Uf=17.8 to 21.0 cm/s, Ud=1.04 to 1.27 cm/s, Sf=0.55 to 0.62, dP5o,f==1.09 to 1.16 mm, C-0 configuration. —i— 3.5 97 Chapter 4 Steady State Hydrodynamic Model 2000 ~w 1800-CN"""" Height of column zone of classifier, H c (m) Figure 4.7.5. Variation of fitted axial dispersion coefficient with height of cylindrical region of the classifier for multi-size system: Ua=5.33 to 5.53 cm/s, Uf=18.7 to 20.0 cm/s, Ud=1.02 to 1.16 cm/s, Sf=0.57 to 0.62, dP5o,f=1.12 to 1.22 mm, dp9o,f/dpio,f=3.29 to 3.62. 2000 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Fluidizing velocity, U a (cm/s) Figure 4.7.6. Variation of fitted axial dispersion coefficient with fluidizing velocity, U a , for multi-size system: Uf=18.8 to 19.3 cm/s, Ud=1.09 cm/s, epO.82 to 0.88, dp5o,f=108 mm, dP9oydpio,f=1.67 to 1.71, C-0 configuration. 98 Chapter 4 Steady State Hydrodynamic Model 2000 -r «T 1800-§ , 1600-Q - 1400-c (D o 1200-Li_ 200-I , 1 1 1 . 1 1 0.5 1.0 1.5 2.0 2.5 Discharge velocity, U d (cm/s) Figure 4.7.7. Variation of fitted axial dispersion coefficient with discharge velocity, Ud, for multi-size system: Ua=5.42 to 5.52 cm/s, Uf=18.3 to 19.5 cm/s, Sf=0.83 to 0.86, dP5o,f=1.08 mm, d9o,f/dpio,f=1.67 to 1.71, C-0 configuration. 2000 i 'ST CN E o 1800-1600-ro Q 1400 -coefficient 1200-1000-spersion' 800-600 -• - — ^ . • T J T J B 400-b_ 200- 1 i i 1 i -i 1 1 1 1 1 1 1 1 1 1 < 1 1 r 10 12 14 16 18 20 22 24 Feed velocity, U f (cm/s) Figure 4.7.8. Variation of fitted axial dispersion coefficient with feed velocity, Uf, for multi-size system: Ua=5.42 to 5.52 cm/s, Us=0.96 to 1.09 cm/s, Sf=0.74 to 0.89, dP5o,i=1.08 to 1.09 mm, dP9o,f/dpio,f=1.66 to 1.72, C-0 configuration. 99 Chapter 4 Steady State Hydrodynamic Model D a=f(U a, U f , U d , 6f, dp50,f, dp9o,f/dpio,f, H C , D c , (pp-p^g, PL, UL) There are twelve parameters with three dimensions (length, time and mass). From Buckingham's Pi theorem, nine independent dimensionless groups are needed, e.g. Pe U . H , „ U a P L D c p _ = U a P L d P 5 0 , f . ^ (Pp " P L)gPL dp50 > f _ a c D„ ;ReD= ; Re d p=- A g . < W H ^ U f . U ^ " d p l 0 , f ' D c ' U a ' U a Here, Pe is a Peclet number, while ReD and Re d p are Reynolds numbers based respectively on the diameter of the cylindrical region and the mean particle diameter in the feed, dP5o,f- Ar is the Archimedes number. A correlation to estimate the particle axial dispersion coefficient was developed in the exponential form: Pe = k 0 ( l - s f ) k i ( s f - 0 . 4 ) k 2 up90,f V d P i o , f ; k3 ru A r 4 R e k ' Re k* Ar k ? v U a y kg v U a y k 9 (4.7.1) The axial dispersion coefficient is likely to be relatively small i f the feed stream is very dilute, and also when the feed particle concentration is very high causing the classifier to approach packed bed conditions where 8=0.4. The (1-Sf) and (eT-0.4) terms introduced into the correlation are intended to force the dispersion coefficient to approach these limits at high and low 8f. The coefficient and exponents in Equation (4.7.1) were obtained by minimizing the objective function, X rD - D ^ a,cal a,exp D , applying multi-variable regression analysis a,exp J 100 Chapter 4 Steady State Hydrodynamic Model using Excel software. Here D a , c a i is the dispersion coefficient calculated from the correlation and D^expis the corresponding dispersion coefficient fitted to the experimental data. Data were included for a wide range of operating conditions from U B C (listed in Appendix B) together with some pilot plant and commercial data from M C C . Table 4.7.2 lists the number of data from different classifiers at UBC and M C C used in fitting the axial dispersion coefficient. T a b l e 4.7.2. Number of data from different classifiers at U B C and M C C C o n f i g u r a t i o n C y l i n d r i c a l d i a m e t e r ( m m ) C y l i n d r i c a l height ( m m ) N u m b e r o f da ta i n c l u d e d U B C C-0 191 597 83 U B C D-500 508 1638 4 U B C C - l 191 25 7 U B C C-2 191 254 7 U B C C-3 191 940 7 M C C Commercial 1998 * * 39 M C C Commercial 1997 * * 28 M C C Pilot 200 200 1440 27 M C C Pilot 126 126 1520 9 Total N A N A 211 *Protected information The optimized constants are k0=0.219, ki=-0.733, k2=-0.633, k3=0.149, L r l . 1 2 1 , k5=0.469, k6=0.492, k7=-0.909, k8=0.127, k9=-0.052, and the objective function equals 9.90. The data utilized cover the following ranges of dimensionless variables: 0.54<8f<0.94; 1 . 5 2 < - ^ - <4.34; 0.133<—^<12.0; d pl0 , f D c 5.38xl03<ReD<1.73xl05; 30.9<Redp<165; 1.70xl04<Ar<1.41xl05 From Equation (4.7.1) and the fitted parameters, the dispersion coefficient increases with increasing feed voidage for 8f>0.68, while it decreases for Sf<0.68. With a broader 101 Chapter 4 Steady State Hydrodynamic Model particle size distribution in the feed, the axial dispersion coefficient is lower, while the dispersion coefficient is larger for shorter or larger diameter classifiers. The discharge velocity, Ud, does not have a great effect on the dispersion coefficient, while the dispersion coefficient decreases slightly with increasing feed velocity, Uf. It can also be found from Equation (4.7.1) and the fitted parameters that the estimated dispersion coefficient, D a , is proportional to U a 0 1 1 4 , meaning that the dispersion coefficient is predicted to increase with increasing liquid fluidizing velocity, U a , but only to a small extent. These trends are all consistent with previous discussion based on Figures 4.7.3 to 4.7.8. The axial dispersion coefficient is also predicted to increase with increasing particle diameter, as in previous studies (Juma and Richardson, 1983; Asif and Petersen, 1993). Table 4.7.3 also gives net exponents of other related parameters with respect to the axial dispersion coefficient, D a . Note that several of the exponents, especially those on Ud and p L are so small that negligible dependence could be assumed. It should also be noted that the terms with the strongest dependencies are Apg, U.L, and D c , and that all of these are in the expected direction. Table 4.7.3. Exponents of parameters with respect to the axial dispersion coefficient, D a Parameter Exponent Parameter Exponent U A 0.114 H C -0.121 u f -0.127 D c 0.652 u d 0.052 (pD-Pr)g 0.909 dp50,f 0.259 PL -0.052 dP9o,f/dpio,f -0.149 U L -0.857 Figure 4.7.9 compares the dispersion coefficients fitted using the model and experimental data with those calculated from Equation (4.7.1). UBC 200 and UBC 500 denote the results using the UBC classifiers with cylindrical region diameters of 191 mm 102 Chapter 4 Steady State Hydrodynamic Model and 508 mm respectively. The key dimensions of the cylindrical regions for the M C C pilot classifiers are listed in Table 4.7.1. Although there is considerable scatter, the predictions of the hydrodynamic model using D a from Equation (4.7.1) agree quite well with the experimental results from U B C , as shown in Figures 4.7.10 to 4.7.12. Comparison o f Sa, dP5o,d and dP9o,d/dpio,d between the M C C experimental data and the hydrodynamic model with D a obtained from Equation (4.7.1) are shown in Figures 4.7.13 to 4.7.15. It can be seen that the results predicted by the hydrodynamic model coupled with the D a correlation also agree moderately well with the M C C commercial and pilot data. 2500-o Q 2000-1 c (B (J '% 1 5 0 ° -o o c o CD C L w TJ TJ CU -*—< ro 3 O co O 500 + UBC 200 -50 % / T UBC 500 • commercial 1998 / ' y • commercial 1997 .' / A pilot 200 A pilot 126 / • • & m' 500 1 1000 1500 2000 .2, 2500 Fitted dispersion coefficient, D a (cm /s) Figure 4.7.9. Comparison between fitted and calculated dispersion coefficients. 103 Chapter 4 Steady State Hydrodynamic Model 1.0 „0-9 to CD O) CO -g 0.8 o > CJ) O ) . CO 0.7 o ? o . 6 H CD T J 2 0.5 0_ 0.4 + 1 0 % / ' ' • UBC 200 A UBC 500 / / '' - 1 0 % i 1 i | i 1 i 1 i 0.4 1.0 0.5 0.6 0.7 0.8 0.9 Experimental discharge voidage, s d Figure 4.7.10. Comparison of discharge voidage between UBC experimental data and hydrodynamic model for multi-size system: Ua=4.0 to 8.13 cm/s, Uf=l 1.2 to 29.4 cm/s, Ud=0.43 to 4.86 cm/s, Sf=0.57 to 0.94, dp50,f=1.08 to 1.25 mm, dp9oydpio,f=1.67 to3.67. E E 2.0 1.9 1.8 -O* 1-7-CD -— 1.6 CO 0) o CO CL c CO CO E T J & O T I CL 1.5 1.4 1.3 H 1.2 1.1 1.0 + 10% • UBC200 A UBC500 -10% 1.0 1.1 1.2 1.3 1.9 2.0 1.4 1.5 1.6 1.7 1.8 Experimental mean particle size, d p 5 0 d , (mm) Figure 4.7.11. Comparison of mean particle size, dP5o,d, in discharge between UBC experimental data and hydrodynamic model for multi-size system: Ua=4.0 to 8.13 cm/s, Uf=11.2 to 29.4 cm/s, Ud=0.43 to 4.86 cm/s, Sf=0.57 to 0.94, dP5o,f=1.08 to 1.25 mm, dP9o,f/dpio,f=1.67 to3.67. 104 Chapter 4 Steady State Hydrodynamic Model 4.0 1.0 -\ , 1 , 1 . 1 1 1 1 1 . 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Calculated d p £ O d /d p 1 0 d Figure 4.7.12. Comparison of breadth of PSD in discharge between U B C experimental data and hydrodynamic model for multi-size system: Ua=4.0 to 8.13 cm/s, Uf=l 1.2 to 29.4 cm/s, Ud=0.43 to 4.86 cm/s, Sf=0.57 to 0.94, dp50,f=1.08 to 1.25 mm, dp9o,f/dpio,f=1.67 to3.67. 1.0 0.9H CD ' O) CO ? 0 . 7 -> CO 0.6 .G =5 0.5 X 3 CD ."3 0.4 TS CD £ 0.3 0.2 • commercial 1998 A commercial 1997 o pilot 200 A pilot 126 + 3 0 o /o 30% - i 1 1 1 1 1 1 1 1 1 ' 1 1 1 <— 0.2 0.3 0.4 0.5 . 0.6 0.7 0.8 0.9 1.0 Experimental voidage, sd Figure 4.7.13. Comparison of predicted discharge voidage with pilot and commercial M C C data. 105 Chapter 4 Steady State Hydrodynamic Model 1.1 • 1 .r> 0.9' | 0.8-% 0.7-o IT) a. CD £ 0.5-T J CO rx 0-4 0.3 0.2 +30 % • commercial 1998 . - ' A commercial 1997 0 pilot 200 A pilot 126 ^ P * ^ • -30 % 0.2 0.3 0.4 0.5 1 — 1 — i — 1 — i — 1 — i — ' — r 0.6 0.7 0.8 0.9 1.0 Experimental d p 5 0 d, (mm) 1.1 Figure 4.7.14. Comparison of predicted mean particle size in discharge, dP5o,d, with pilot and commercial MCC data. Figure 4.7.15. Comparison of predicted breadth of PSD in discharge, dP9o,d/dpio,d, with pilot and commercial MCC data. 106 Chapter 4 Steady State Hydrodynamic Model 4.8 Predictions of Classifier Performance and Efficiency and Comparison with Experimental Data: Multi-Size Systems The hydrodynamic cold model classifier was also used to study the classifier performance and efficiency for a wide range of operating conditions for multi-size systems. In this section, Equation (4.7.1) is used to estimate the axial dispersion coefficient for the calculation of voidage in the column and discharge stream, increment of mean particle size and breadth of PSD, as well as the classifier efficiency, using the steady state hydrodynamic model. 4.8.1 Local Voidage The variation of local voidage with respect to height for different operating conditions is shown in Figures 4.8.1 to 4.8.4 for experiments carried out with the C-0 configuration (Appendix A). The predicted trends are consistent with the experimental results, although there are significant quantitative deviations in some cases. Figure 4.8.1 shows the effect of superficial velocity, U a , on the local voidage. It can be seen that the local voidage increases with increasing height above the discharge position, h=0.52 m, and with increasing U a . The variation with height occurs simply because the process involves continuing classification, with larger particles moving to the lower portion of the column, resulting in lower voidages. The influence of superficial velocity reflects the greater bed expansion caused by increasing the liquid flowrate. 107 Chapter 4 Steady State Hydrodynamic Model 1.64 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Voidage ec Figure 4.8 .1. Local voidage along the height at different fluidizing velocity, U a , for multi-size system (experimental data were obtained by pressure gradient): Uf=18.8 to 19.5 cm/s, Ud=1.09 cm/s, 6f=0.82 to 0.88, 0^ 50,1=1 08 mm, dP9o,f/dpio>f=1.67 tol.70, C-0 configuration. The influence of feed stream voidage, 6 f , is shown in Figure 4.8.2. The voidage profile along the axial direction changed as Sf varied from 0.62 to 0.92. Near the discharge port, h=0.52 m, the local voidage did not change greatly with Sf. However, higher in the column, the local voidage increased significantly with increasing ef. The effect of feed rate expressed as a superficial velocity, Uf, on the local voidage is shown in Figure 4.8.3. In the experiments, the voidage in the feed always increased with increasing feed velocity. It is clear that a higher U f coupled with a higher feed voidage results in higher voidage in the upper section of the classifier. However, the voidages in the lower portion of the column are not greatly influenced by the flow entering the top of the classifier for the range of conditions covered. 108 Chapter 4 Steady State Hydrodynamic Model 1.6 1.4 1.2 1 1 ' ° sz +s 0.8 D) 0.6 0.4 0.2 0.0 increase s. — 0.62 model •-•0.73 model | 0.80 model --- 0.85 model | 0.92 model • 0.62 experiment O 0.73 experiment • 0.80 experiment V 0.85 experiment + 0.92 experiment — i — 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Voidage ec Figure 4.8.2. Local voidage along the height at different feed Voidage, ef, for multi-size system (experimental data were obtained by pressure gradient): Ua=5.52to 5.67 cm/s, Uf=17.8 to 20.3 cm/s, Ud=1.07 cm/s, dP5o,f=1.08 to 1.10 mm, dp90,f/dpio,f=1.67 to 1.78, C-0 configuration. — i — — I — 1.0 1.6-1.4-1.2 £ 1-0-1 SZ 0.8 CD cu 0.6-1 0.4 0.2 0.0 increase U f and ef l/cm/s) —11.2 I 14.8 --- 18.7 | 23.8 • 11.2 O 14.8 A 18.7 V 23.8 0.74 model 0.78 model 0.84 model 0.89 model 0.74 experiment 0.78 experiment 0.84 experiment 0.89 experiment 0.0 0.1 0.2 0.3 0.4 0.5 Voidage ec Figure 4.8.3. Local voidage along the height at different feed velocity, Uf, for multi-size system (experimental data were obtained by pressure gradient): Ua=5.43 to 5.52 cm/s, Us=0.96 to 1.09 cm/s, Sf=0.75 to 0.89, dP5o,f=108 to 1.09 mm, dP9o,f/dpio,f=1.66 to 1.72, C-0 configuration. 109 Chapter 4 Steady State Hydrodynamic Model Figure 4.8.4 shows the effect of the discharge velocity, Ud, on the local voidage. The local voidage in the column increases slightly with increasing Ud. In this continuous classification process, increasing the discharge velocity caused more particles to discharge from the lower portion of the column, leading to a smaller average particle size in the column and, hence, to a somewhat higher voidage. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Vo idage ec Figure 4.8.4. Local voidage along the height with different discharge velocity, Ud, for multi-size system (experimental data were obtained by pressure gradient): Ua=6.17 to 6.27 cm/s, Uf=21.4 to 21.5 cm/s, Sf=0.77, dpscf^llS to 1.18 mm, dP9o,f/dpio,f=3.36, C-0 configuration. 4.8.2 Voidage in Lower Portion of Column, Discharge and Overflow Figures 4.8.5 to 4.8.10 show the variation of the predicted voidage in the lower portion of the column 0.07 m above the discharge port, in the discharge stream and in the overflow stream, for different operating conditions for some multisize systems. It is shown that the 110 Chapter 4 Steady State Hydrodynamic Model voidage predictions using the model and axial dispersion correlation agree well with the experimental data. Higher liquid flowrates cause greater bed expansion, as shown in Figure 4.8.5, resulting in higher voidage in the lower portion near the discharge port and in the discharge, but there is little influence on the overflow voidage. It is also seen in Figure 4.8.5 that the experimental voidage in the discharge stream is very close to the voidage in the lower portion of the column for Ua>5 cm/s. However, there is a substantial difference at lower U a , with Sd not varying significantly with U a while the voidage in the lower part of the column continues to drop as U a decreases below about 4 cm/s. This difference may result from instability of two-phase flow in the discharge pipe in the experiments. In the hydrodynamic model, one of the boundary conditions is that the voidage in the lower portion is equal to the voidage in the discharge stream, so this boundary condition is not applicable for low liquid fluidizing velocities. The influence of the feed stream voidage, Sf, on the voidage in the portion lower of the column, the discharge stream and the overflow stream is shown in Figure 4.8.6. Neither the voidage in the lower port of the column nor the discharge voidage changed significantly with Sf, while the voidage in the overflow increased significantly with increasing Sf. Similar results can be seen in Figure 4.8.7, which shows the effect of feed velocity, Uf, on the voidage in the lower portion of the column, in the discharge and the overflow. Again, with the feed stream entering near the top of the classifier, the feed conditions have little influence near the bottom of the classifier, while affecting the hydrodynamics near the top and hence the overflow stream. I l l Chapter 4 Steady State Hydrodynamic Model CD CD ra "a I 1.0-0.9-0.8 0.7 H 0.6 0.5 0.4 0.3 0.2 H 0.1 0.0 2.0 - - A -A- -experimental discharge experimental lower portion (0.07 m above discharge port) experimental overflow -predicted discharge • predicted overflow 2.5 3.0 —i— 3.5 —I— 4.0 —I— 4.5 —I— 5.0 —I— 5.5 —I— 6.0 6.5 7.0 Liquid Fluidization velocity, U (cm/s) Figure 4.8.5. Variation of voidages with respect to fluidizing velocity, U a , for multi-size system: Uf=18.8 to 19.3 cm/s, Ud=1.09 cm/s, Sf=0.82 to 0.88, dp5o,f=108 mm, dP9o,f/dpio,f=1.67 tol.71, C-0 configuration. cm cn ro I 1.0 0.9 0.8 -0.7 -0.6 0.5 4 0.4 0.3 -] 0.2 0.1 0.0 - A A-- - A ' • " " ' a n A - - A _A -a—5—•—fl-n-• experimental discharge • experimental lower portion (0.07 m above discharge port) A experimental overflow predicted discharge predicted overflow 0.6 0.7 I 0.8 0.9 1.0 F e e d voidage, e Figure 4.8.6. Variation of voidages with respect to feed voidage, Sf, for multi-size system: Ua=5.52 to 5.67 cm/s, Uf=17.8 to 20.3 cm/s, Ud=1.07 cm/s, 0^ 50,1=1-08 to 1.10 mm, dp9o,f/dpio,f=1.67 to 1.78, C-0 configuration. 112 Chapter 4 Steady State Hydrodynamic Model CD CO TJ O > 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10 12 • experimental discharge • experimental lower portion (0.07 m above discharge port) A experimental overflow predicted discharge predicted overflow 14 — r~ 16 I 18 20 22 24 26 Feed velocity, U f Figure 4.8.7. Variation of voidages due to changes in feed velocity, Uf, for multi-size system: Ua=5.43 to 5.52 cm/s, Us=0.96 to 1.09 cm/s, Ef=0.75 to 0.89, dpso^l-OS to 1.09 mm, dP9o,f/dpio,f=1.66 to 1.72, C-0 configuration. CD CD CO TJ O > 1.0 0.9-0.8-0.7-0.6 0.5 0.4 0.3 0.2 0.1 -| 0.0 —I ' 1— 0.5 1.0 • experimental discharge • experimental lower portion (0.07 m above discharge port) A experim ental overflow predicted discharge predicted overflow —I— 1.5 —I— 2.0 2.5 Discharge velocity, U d (cm/s) Figure 4.8.8. Variation of voidages with respect to discharge velocity, Ud, for multi-size system: Ua=5.42 to 5.52 cm/s, Uf=18.3 to 19.5 cm/s, Sf=0.83 to 0.86, dP5o,f=108 mm, d9o,f/dpio,f=1.67 to 1.71, C-0 configuration. 113 Chapter 4 Steady State Hydrodynamic Model Both the voidage in the lower portion and in the discharge increased slightly as the discharge flow rate was increased, as shown in Figure 4.8.8. This is probably due to the smaller average particle size in the lower portion of the column at higher values of Ud. The breadth of the particle size distribution in the feed and the height of classifier cylindrical zone do not have much influence on the voidage in the lower portion of the column, in the discharge or in the overflow stream, as shown in Figures 4.8.9.and 4.8.10. 1.0 -j 0.9-0.8 -to • CD 0.7-CD CO 0.6 -T3 O > E 0.5 -ro CD i_ 0.4-</> --*—' 0) 0.3 -o 0.2-0.1 -0.0 -tt- *V" • experimental discharge A experim ental overflow predicted discharge predicted overflow 1.5 2.0 2.5 3.0 3.5 Breadth of PSD in feed, d Q n /d , n f ' p90,r pio.f 4.0 Figure 4.8.9. Variation of voidages with respect to breadth of PSD in feed, dP9o,f/dPio,f, for multi-size system: Ua=5.39to 5.48 cm/s, Uf=17.8 to 21.0 cm/s, Ud=1.04 to 1.27 cm/s, Ef=0.55 to 0.62, dP5o,f=1.09 to 1.16 mm, C-0 configuration. 114 Chapter 4 Steady State Hydrodynamic Model 1.0 0.9 to 0)" 0.7 O) CO X J 0.6 o > g 0.5 CO £ 0.4 —• tn O 0.3 (5 0.2 0.1 0.0 A A i • - A A - -• • H c (mm) configuration • experimental discharge 25 C-1 A experimental overflow 254 C-2 predicted discharge 597 C-0 predicted overflow 940 C-3 0.0 0.2 0.4 0.6 0.8 1.0 Height of column zone, H c (m) Figure 4.8.10. Variation of voidage with respect to height of cylindrical region, H e , for multi-size system: Ua=5.46 to 5.50 cm/s, Uf=19.6 to 19.9 cm/s, Ud=2.46 to 2.78 cm/s, Sf=0.62 to 0.64, dp5o/=1.14 to 1.18 mm, dP9o,f/dpio,f=3.43 to 3.58. 4.8.3 Increments of Mean Particle Size and Breadth of PSD One way to evaluate the classification results is to compare the mean particle size, dP5o and the breadth of particle size distribution dP9o/dpio between the feed and discharge streams. Since both the feed PSD and discharge PSD could be described with reasonable accuracy as Rosin-Rammler distributions, the single parameter, dp9o/dpio, can be used to represent changes in the breadth of the PSD during classification. Increments of dpso and dP9o/dpio are defined as Increment of dpso= p 5 ° d -1 (4.8.1) d50f Increment of dp9o/dpio - p 9 0 d — E l ^ i -1 (4.8.2) d90f /dpiof 115 Chapter 4 Steady State Hydrodynamic Model It is noted that the increment of dpso has a positive value, while the increment of dP9o/dpio has a negative value because of the larger mean particle size and the narrower particle size distribution in the discharge after classification in the column. The higher the increment of dpso and the lower the increment of dP9o/dpio, the better the particle classification. Figures 4.8.11 to 4.8.16 show the variation of the increments of dpso and dP9o/dpio for different operating conditions. It is seen that the predictions of the hydrodynamic model with the axial dispersion coefficient fitted as described in section 4.7 agree well with the experimental increment data. 0.3 § 0.2 H T J c ro T J **-o <rt c CD E CD o c 0.1 H 0.0 <D -0.1 -0.2 2.0 —I— 2.5 • experimental increment of d ,^, • experimental increment of d^Jd^^ predicted increment of d^ ,, predicted increment of d^d,,,,, T T -Q-3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Fluidized liquid velocity, U (cm/s) 7.0 Figure 4.8.11. Variation of increments of dpso and dP9o/dpio with respect to fluidizing velocity, U a , for multi-size system: Uf=18.8 to 19.5 cm/s, Ud=1.09 cm/s, 8f=0.82 to 0.88, dP5o,f=108 mm, dP9o,f/dpio,f=1.67 tol.71, C-0 configuration. 116 Chapter 4 Steady State Hydrodynamic Model Figure 4.8.11 shows that the increment of dpso increases, while the increment of dp9o/dpio decreases, with increasing U a , indicating improved classification with increasing U a . This is because smaller particles are more likely to be entrained in the overflow stream at higher U a . Figure 4.8.12 shows the variation of the two increments with respect to the feed voidage, Sf. The increment of dP5o increases and the increment of dP9o/dpio decreases somewhat with increasing 8f. Very similar trends can be seen in Figure 4.8.13 regarding the influence of the feed velocity, Uf.. 0.30 0.25 t 0.20 0.15 £ 0.10-ro §. 0.05 -) T J O 0.00 tn §5 "0 05 E 2 -0.10 o — -0.15 -0.20 • — predicted increment of d p 9 (/d p 1 0 predicted increment of d p 5 0 • experimental increment of d p 9 ( /d p 1 0 • experimental increment of d p 5 0 0.6 0.7 0.8 Feed voidage, sf —r~ 0.9 1.0 Figure 4.8.12. Variation of increments of dP5o and dP9o/dpio with respect to feed voidage, Ef, for multi-size system: Ua=5.52 to 5.67 cm/s, Uf=17.8 to 20.3 cm/s, Ud=1.07 cm/s, dP5o,f=1.08 to 1.10 mm, dP9o,f/dpio,f=1.67 to 1.78, C-0 configuration. 117 Chapter 4 Steady State Hydrodynamic Model 0.30 -, 0 .25- • experimental increment of d p 5 0 • experimental increment of d p 9 0 /d p 1 0 | 0 .20-TJ w predicted increment of d p S 0 predicted increment of d p 9 0 /d p l 0 <& 0.15-TJ "g 0 .10-ro §. 0 .05-TJ , — ——~ • D • • o o.oo-1 ^ 0 5 -2 -0 .10-o — -0 .15-• -0 .20- i i i i ' I < I i I i I i i 1 I 1 I 8 10 12 14 16 18 20 22 24 26 F e e d velocity, U f (cm/s) Figure 4.8.13. Variation of increments of dpso and dP9o//dpio with respect to feed velocity, Uf, for multi-size system: Ua=5.43 to 5.52 cm/s, Us=0.96 to 1.09 cm/s, Sf=0.75 to 0.89, dP5o,f=1.08 to 1.09 mm, dP9o,f/dpio,f=1.66 to 1.72, C-0 configuration. Figure 4.8.14 shows the effect of the discharge velocity, Ud, on the increments of dP5o and dP9o/dpio. At higher Ud, more particles and liquid are withdrawn through the discharge port, drawing more small particles from the upper portion to the lower portion of the column. Also, since more liquid is withdrawn, the superficial liquid velocity inside the classifier is reduced, resulting in a small decrease in the increment of dpso and a small increase in the increment of dP9o/dpio. As shown in Figure 4.8.9, the feed stream particle size distribution has little influence on the voidage in the lower portion of the column and discharge stream. However, as shown in Figure 4.8.15, the breadth of the feed particle size distribution plays a critical role in the performance of the classifier. For a higher dp9o,f/dpio,f, i.e. a 118 Chapter 4 Steady State Hydrodynamic Model broader feed particle size distribution, the performance of the classifier is better than for a narrow feed particle size distribution. 0.3 * 0.2 XJ c CO o tn c co E cu b 0.1 o.o H -0.1 -0.2 • experimental increment of d ,^, • experimental increment of d „^/d 1 0 predicted increment of d ^ — predicted increment of d^Jd^ — I — 0.5 1.0 1.5 2.0 i 2.5 Discharge velocity, U (cm/s) Figure 4.8.14. Variation of increments of dpso and dP9o/dpio with respect to discharge velocity, Ud, for multi-size system: Ua=5.42 to 5.52 cm/s, Uf=18.3 to 19.5 cm/s, 6f=0.83 to 0.86, dP5o,f=1.08 mm, d9o,f/dpio,f=1.67 to 1.71, C-0 configuration. Although there is little change of voidage in the lower portion and in the discharge stream for different heights of the classifier cylindrical zone as shown in Figure 4.8.10, the classifier with the shortest cylindrical region performs relatively poorly as a classifier, yielding a lower increment of dpso and a higher increment of dP9o/dpio, as shown in Figure 4.8.16. The poorer separation almost certainly arises due to the greater turbulence coming from the feedstream at the top and liquid fluidizing stream at the bottom, leaving less quiescent space where turbulence is largely absent so that classification can proceed unimpeded. 119 Chapter 4 Steady State Hydrodynamic Model 0.5-1 0.4-• experimental increment of dpjo • experimental increment of d p 9 0 /d p , 0 n -J 0 .3 -1—>_, predicted increment of d p 5 0 predicted increment of d p S 0 /d p 1 0 • D O cn •o" 0 .2 -T 3 C CD 0.1 -o in C D -'S | * . 1 -CO ^ -0 2 - •*> b — - 0 . 3 --0 .4 - i • i • i • i 1 i 1 1.5 2.0 2.5 3.0 3.5 4.0 Breadth of P S D in feed, d p 9 0 / d p 1 0 f Figure 4.8.15. Variation of increments of dpso and dP9o/dpio with respect to breadth of PSD in feed, dP9o,f/dpio,f, for multi-size system: Ua=5.39to 5.48 cm/s, Uf=17.8 to 21.0 cm/s, Ud=1.04 to 1.27 cm/s, 8f=0.55 to 0.62, dP5o,f=1.09 to 1.16 mm, C-0 configuration. i 1 i 1 i 1 i 1 i ' i 0.0 0.2 0.4 0.6 0.8 1.0 Height of column zone , H. (m) Figure 4.8.16. Variation of increments of dpso and dP9o/dpio with respect to height of cylindrical region for multi-size system: Ua=5.46 to 5.50 cm/s, Uf=19.6 to 19.9 cm/s, Ud=2.46 to 2.78 cm/s, Sf=0.62 to 0.64, dP5o,f=1.14 to 1.18 mm, dp9o,f/dpio,f=3.43 to 3.58. 120 Chapter 4 Steady State Hydrodynamic Model 4.8.4 Classifier Efficiency One of the most important characteristics of a classifier is how sharply it is able in separating larger particles from smaller ones. There are a number of possible definitions of classification efficiency and selectivity (Snow and Allen, 1992). Let the desired "cut diameter" be d p c , An underflow discharge efficiency may be defined as the fraction of the feed particles larger than d p c that succeed in leaving with the discharge stream, i.e. = v d v an c i ^ p c A i Q f ( l - 8 f ) [ l - F f ( d p c ) ] Here F(d p c) is the cumulative mass fraction (or volume fraction if all the species of particle have the same density) of particles smaller than the cut diameter and subscripts f, d and o denote the feed, discharge and overflow streams, respectively. Similarly, an overflow efficiency can be defined as the fraction of particles smaller than the cut diameter in the feed stream which are carried out by the overflow stream, i.e. To = (4.8.4) Q f ( l - 8 f ) F f ( d p c ) The product of the above two efficiencies defines an overall classification efficiency, i.e. rj =r| dr| 0 (4.8.5) Perfect classification, i.e. all particles larger than d p c ending up in the underflow and all smaller than d p c in the overflow stream, would yield r) ci a s sif i er=l. Note that all three efficiencies defined above depend not only on the classifier geometry and operating conditions, including the flows, but also on the feed concentration, Cf, and the breadth of the feed particle size distribution. 121 Chapter 4 Steady State Hydrodynamic Model For example, i f we choose the cut diameter, dpc=dp5o,f, the variations of classifier efficiency, ndassifier, with respect to different operating conditions are plotted in Figures 4.8.17 to 4.8.22. The predictions from the model developed in section 4.2, with D a correlated as outlined in section 4.7, agree well with the experimental data. The effect of U a on the classifier efficiency is plotted in Figure 4.8.17. Increasing U a results in an increasing discharge voidage as shown above, i.e. in lower solids concentrations in the discharge stream. Hence, the particle discharge mass flow rate is decreased, which could reduce the efficiency according to its definition above. On the other hand, increasing U a increases the increment of dpso and decreases the increment of dp9o/dpio, again as shown above, and this is expected to increase the classifier efficiency. Experimental results shown in Figure 4.8.17 indicate that the combination of these two factors causes an initial small increase for Ua<3.5 cm/s and then a decrease in efficiency with increasing U a . The model does not predict a maximum, however. In the model, it is assumed that the voidage in the column near the discharge port equals that in the discharge stream. As shown in Figure 4.8.5 above, for the lower U a , there is a sizeable deviation of experimental voidages between the lower port of the column and the discharge stream, so that the hydrodynamic model is not applicable and fails to give good prediction of the classifier efficiency for the lower liquid fluidizing velocity. Figure 4.8.18 shows that the classifier efficiency increased dramatically with increasing feed stream voidage, 8f, for the range covered in the experimental program. This may be because the particle mass flow rate in the feed decreases with increasing Sf, causing the ratio of particle mass flow rate in the discharge stream to that in the feed stream to increase significantly. However there was only a small influence of feed 122 Chapter 4 Steady State Hydrodynamic Model velocity, Uf, on the classifier efficiency as indicated in Figure 4.8.19. Increasing Uf should decrease the ratio of particle mass flow rate in the discharge stream to that in the feed stream. However, in the experiments, feed voidage, Sf, increased simultaneously with increasing Uf, leading to a higher ratio of particle mass flow rate in the discharge stream to that in the feed stream. These two factors probably offset each other causing the classifier efficiency to vary little with Uf. 40 35 30 O 25 c CD O e CD 20 £ 15 'in in ™ 1 0 O 5^  2.0 experiment •predicted by model 2.5 — I — 3.0 —I— 3.5 — I — 4.0 — I — 4.5 5.0 5.5 — I — 6.0 —I— 6.5 7.0 Fluidizing liquid velocity, U (cm/s) Figure 4.8.17. Variation of classifier efficiency with respect to fluidizing velocity, U a , for multi-size system: Uf=18.8 to 19.5 cm/s, Ud=1.09 cm/s, Sf=0.82 to 0.88, dp5o,f=1.08 mm, dP9o,f/dpio,f=1.67 tol.70, C-0 configuration. 123 Chapter 4 Steady State Hydrodynamic Model 4 0 35 H o H , 1 , 1 1 1 1 1 1 1 0.5 0.6 0.7 0.8 0.9 1.0 F e e d vo idage, e f Figure 4.8.18. Variation of classifier efficiency with respect to feed voidage, Sf, for multi-size system: Ua=5.52 to 5.67 cm/s, Uf=17.8 to 20.3 cm/s, Ud=1.07 cm/s, dP5o/=l-08 to 1.1 mm, dP9o,f/dpio,f=1.67 to 1.78, C-0 configuration. 40-35 30 4 >, o 25-c CD O it 20-CD I— ifie 15-w tn CO 10-O experiment • predicted by model 10 12 14 16 I — ' — I 18 20 22 24 26 28 Feed velocity, U f (cm/s) Figure 4.8.19. Variation of classifier efficiency with respect to feed velocity, Uf, for multi-size system: Ua=5.43 to 5.52 cm/s, Us=0.96 to 1.09 cm/s, Ef=0.75 to 0.89, dP5o,f=108 to 1.09 mm, dP9o,f/dpio,f=1.66 to 1.72, C-0 configuration. 124 Chapter 4 Steady State Hydrodynamic Model The dependence of classifier efficiency on Ud is shown in Figure 4.8.20. The efficiency increases with increasing Ud. From a mass balance, Q0=Qf+Qa-Qd, i.e. the overflow decreases with increasing discharge flowrate. From Equation (4.8.3), the discharge efficiency increases with increasing discharge flowrate, while from Equation (4.8.4), the overflow efficiency decreases. The overall classifier efficiency is defined as the product of these two efficiencies. However, when the flowrate in the discharge stream is much lower than in the overflow stream, as in the work carried out both at U B C and M C C , increasing the discharge flowrate has little effect on the overflow rate, so that the net effect of increasing U d is an increase in overall classifier efficiency. 40 -| 35 -• experiment 3 0 _ predicted by model 5 H u-1 1 1 1 1 1 1 1 1 • 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Discharge velocity, U d (cm/s) Figure 4.8.20. Variation of classifier efficiency with respect to discharge velocity, Ud, for multi-size system: Ua=5.42 to 5.52 cm/s, Uf=18.3 to 19.5 cm/s, 8f=0.83 to 0.86, dP5o,f=108 mm, d9o,f/dpio,f=1.67 to 1.71, C-0 configuration. 125 Chapter 4 Steady State Hydrodynamic Model Figure 4.8.21 shows the influence of the feed PSD breadth parameter, dP9o/dpio, on the classifier efficiency. It can be seen that higher values of dP9o,f/dpio,f, i.e. broader particle size distributions in the feed, lead to a small improvement in classification efficiencies. The effect of the height of the cylindrical zone on the classifier efficiency appears in Figure 4.8.22. The longer this zone, the better the efficiency of the classifier, consistent with Figure 4.8.16. 40-j 35-30->i 25-o c CD -o 20-a> CD M— 15-in 10 CO 10-O 5-1.5 experiment predicted by model — i — 2.0 — i — 2.5 — i — 3.0 — i — 3.5 4.0 Breadth of PSD in feed, d x /d p90,f p10,f Figure 4.8.21. Variation of classifier efficiency with respect to breadth of PSD in feed, dP9oydpio,f, for multi-size system: Ua=5.39 to 5.48 cm/s, Uf=17.8 to 21.0 cm/s, Ud=1.04 to 1.27 cm/s, 6f=0.55 to 0.62, dpso^l-OQ to 1.16 mm, C-0 configuration. 126 Chapter 4 Steady State Hydrodynamic Model 4 0 -3 5 -3 0 ->i 2 5 -o c CD O U— 2 0 -t j ) <D 1 5 -"(fl cn CO 1 0 -O 5 -H c (mm) configuration • experiment 25 C-1 predicted by model 254 C-2 597 C-0 940 C-3 - i — — " — • 0.0 0.2 0.4 0.6 0.8 1.0 Height of co lumn z o n e , H (m) Figure 4.8.22. Variation of classifier efficiency with respect to height of cylindrical zone for multi-size system: Ua=5.46 to 5.50 cm/s, Uf=19.6 to 19.9 cm/s, Ud=2.46 to 2.78 cm/s, Sf=0.62 to 0.64, dP5o,f=1.14 to 1.18 mm, dp9o,f/dpio,£=3.43 to 3.58. 4 .9 Comparison with Markov Chain Model As described in Chapter 2, a stochastic Markov Chain model, developed by another member of our research group, can also be used to calculate the voidages and particle size distributions in the discharge and overflow streams, as well as within the classifier column. In this model, there is again a single fitted parameter, in this case the turbulence intensity, I t, to account for turbulence in the whole classifier. At a higher turbulence intensity, the cumulative particle size distribution in the discharge approaches that in the feed, leading to poor classification, as shown in Figure 4.9.1. Comparison of this figure with Figure 4.7.2, demonstrates that the effect of the turbulence intensity, I t, in the Markov Chain Model is quite similar to that of the dispersion coefficient, D a , in the hydrodynamic model. 127 Chapter 4 Steady State Hydrodynamic Model Particle size, d p (mm) Figure 4.9.1. Calculated cumulative PSD by mass in discharge by Markov Chain model with different turbulence intensity for multi-size system: U a = 5.8 cm/s, U d = 1.2 cm/s, U f = 20.9 cm/s, s f =0.57, dP5o,f=1.29 mm, dP9o,f/dpio,f=3.42, C-0 configuration. Figure 4.9.2 compares the calculated voidage profiles along the column derived from the Markov Chain Model and the hydrodynamic model with a single fitted parameter in each case, i.e. the turbulence intensity for the Markov Chain model and the axial dispersion coefficient for the hydrodynamic model. The calculation procedures are described in sections 2.4.3 and 4.5 respectively. Both models predict that the voidage in the cylindrical region does not change significantly with height. The voidage in the conical region increase downwards due to the change of superficial liquid velocity. In the top region, but below the feed position, h= 1.15 m, the predicted voidage drops below that in the cylindrical region because of the bulge (expanded top) section. Above the feed position, the voidage should increase with increasing height due to the higher overflow 128 Chapter 4 Steady State Hydrodynamic Model rate. This trend can be predicted by the hydrodynamic model. However, the voidage at the very top near the overflow is predicted to become smaller by the Markov Chain model, which is a weakness of this model. For the conditions simulated in Figure 4.9.2, the calculated discharge voidage predicted by the Markov Chain model is closer to the experimental data than that calculated by the hydrodynamic model. Voidage Figure 4.9.2. Comparison of calculated voidage profile along the column for Markov Chain model and Hydrodynamic model: U a = 5.8 cm/s, U d = 1.2 cm/s, Uf = 20.9 cm/s, s f =0.57, dp50,f=1.29 mm, dp9o,f/dpio,f=3.42, I t =8.26 for Markov Chain model, D a= 620 cm2/s for hydrodynamic model, C-0 configuration. Figure 4.9.3 shows cumulative particle size distributions in the discharge calculated from the Markov and hydrodynamic models. Both models agree well with the experimental data. However the mean particle size in the discharge, dpso,d, calculated by the hydrodynamic model is a little higher, and the particle size distribution in the 129 Chapter 4 Steady State Hydrodynamic Model discharge calculated by the hydrodynamic model a little broader, than calculated by the Markov Chain model for the specific operating conditions of Figure 4.9.3. For this particular case, the calculated results from the Markov Chain Model appear to provide better predictions than the hydrodynamic model. Particle size, d p (mm) Figure 4.9.3. Comparison of calculated cumulative PSD by mass in discharge between Markov Chain model and Hydrodynamic model: U a = 5.8 cm/s, U d = 1.2 cm/s, Uf = 20.9 cm/s, s f =0.57, dp5o,f=129 mm, dp9o,f/dpio,f=3.42, I t =8.26 for Markov Chain model, D a= 620 cm2/s for hydrodynamic model, C-0 configuration. It should be pointed out that the limited results reported by Zhang (1998) on the Markov Chain model are insufficient for general comparison with the hydrodynamic model. However, it can be concluded that both models show encouraging promise for 130 Chapter 4 Steady State Hydrodynamic Model predicting the discharge voidage and particle size distribution in a classifier with particles of different sizes but equal density. Comparing the computer programs for n particle species in the system, n initial guessed values of particle size distribution in the classifier for the Markov Chain model and of particle concentrations in the overflow stream for the hydrodynamic model must be provided first by the user and then compared with the final results, which are more easily and quickly reached if the initial guessed values are close to the final values. For the Markov Chain model, several hours must be spent on a Pentium II computer to obtain the results, while ten minutes at most are consumed for the hydrodynamic model. Furthermore, the hydrodynamic model has been tested over a much wider range of operating conditions and a correlation to estimate the axial dispersion coefficient has been developed based on experimental data from both UBC and M C C , while there is no similar correlation for the I t parameter needed by the Markov model. Therefore, the hydrodynamic model is recommended in preference to the Markov model at this time. 131 Chapter 5 Unsteady State Hydrodynamic Model Chapter 5 Unsteady State Hydrodynamic Model The industrial classifier sometimes shows unwanted transient excursions, and therefore an unsteady model was prepared as a possible tool for help in analyzing this problem. Dynamic experimental tests were carried out by making step changes of one operating variable at a time using the classifier with the T-2 configuration for both a mono-disperse suspension and a multi-size system. An unsteady state hydrodynamic model is developed in this chapter to study the unsteady state behavior of particles and liquid in the classifier. The governing equations are again based on mass and momentum balances, with the same boundary conditions as for the steady state hydrodynamic model. The drag and dispersion forces are also calculated using the same correlation as for the steady state hydrodynamic model described in Chapter 4. The two-step Lax-Wendroff scheme is applied to solve the partial differential equations which result. The predictions of the dynamic behavior for step changes of different operating conditions are discussed and compared with experimental results in section 5.4. 5.1 Governing Equations, Initial and Boundary Conditions The steady state hydrodynamic model described in Chapter 4 is extended to an unsteady state hydrodynamic model by adding transient terms as Equations (4.2.5), (4.2.6), (4.2.11) and (4.2.12). The governing equations for the unsteady state model are then: Continuity Equations: Solids: - ^ i ^ + - ^ - ( c i p i v i A ) = m i a A & V , F l 1 ' (5.1.1) 132 Chapter 5 Unsteady State Hydrodynamic Model Liquid: ^ ^ + - ^ ( s p L u A ) = m L (5.1.2) dt Adz Momentum equation: Solids: + j L ( C i P i V I 2 A ) = -c s £ + P i c i g + c ^ + C j f d i s p i (5.1.3) Liquid: a ( S ^ t U ) + ^ ( s p L u 2 A ) = - s ^ + pLeg - I c T ^ - IC j f d i s p i (5.1.4) Computations can be carried out using a mesh of finite difference cells. It is found that using the flux, rather than velocity term, helps to prevent cells from becoming drained completely and giving negative volume fractions. This also aids computational stability. Let F j = C i V j , FL=BU. The densities of particles and liquid are assumed to be constant with respect to time and position in the classifier. Therefore, the continuity and momentum equations can be rewritten as follows: dt dz Adz ^ 4 ( F L ) + F L - ^ - = m L / p L (5.1.6) at dz Adz Pi + Pi ^ F i V i ) + P i F i V i = " C i + P i C i S + C i f d « 9 + C i f d i s P i ( 5 - 1 7 > PL ^  + P L > | ^ U ) + P ^ U H" = + ^ " I C i f * * - 2>ifdispi ( 5 - 1 8 ) The second term on the left side of Equations (5.1.7) and (5.1.8) can be converted to A ( F i V , ) = V i § + F i ^ C5.1.9) oz dz dz i . ( H . u ) = u ^ + l t | (5.1.10) dz dz dz 133 Chapter 5 Unsteady State Hydrodynamic Model where Vi=Fi/ci u=Fi/e ( 5 . 1 . 1 1 ) ( 5 . 1 . 1 2 ) This leads to v c >y dz dz dz dz C J dz cf dz \_^L_^ds s dz e2 dz ( 5 . 1 . 1 3 ) ( 5 . 1 . 1 4 ) Substituting Equations ( 5 . 1 . 9 ) to ( 5 . 1 . 1 4 ) into an equation obtained by multiplying Equation ( 5 . 1 . 7 ) by 8 and Equation ( 5 . 1 . 8 ) by Q , then subtracting, we obtain d(F;) Fj SFj 1 dF: E dc-. ^ • fl + e p i F i ( - — 1 — L ) + 8 p i - L at C: oz C; dz cr cz F 2 5 A C; Adz 5(FL) F L oFL .1 dF, F, de. ^ - C ; P L F L ( - ^ ^ ^ ) - C i p L at s oz e dz e dz = c i e ( P i - p L ) g + £ C , f i + c i Z c i f i FldA_ 8 Adz ( 5 . 1 . 1 5 ) where fi=fdrag,,+fdisp,i and e=\-2Zci • Equations ( 5 . 1 . 5 ) and Equation ( 5 . 1 . 1 5 ) constitute the unsteady state hydrodynamic model for the liquid-solid classifier. In the equations, Cj and F; are the unknowns and 5F T 5F T other parameters, F L , ——, ——, m, and m L , can be written as functions of c; or Fj for dt dz different zones of the classifier, as described in section 5 .2 . The drag force and the dispersion force are calculated using Equations ( 4 . 3 . 7 ) and ( 4 . 3 . 9 ) in Chapter 4 . 1 3 4 Chapter 5 Unsteady State Hydrodynamic Model From an overall mass balance, the total flow rate into and out of the classifier for every particle species i must be equal to the accumulation of that particle species in the whole classifier. That is H total j QfCfi - QdCdi - Q o C o i = f - ^ - A d z (5.1.16) o d t where H t otai is the total height of the classifier. Initial conditions: At time 0, the particle concentration, Cj, particle velocity, V i , or particle flux, C M , have values from some previous steady state conditions. Hence at t=0, 0<Z<H t o t al: Ci=Ciniti al,i, Vi=Vinitial,i Or Fi=Finitial,i=Cinitial,iVinitial,i (5.1.17) Boundary Conditions: As in the steady-state model, it is assumed that the particle concentration in the overflow and discharge, c 0 i and Cd,i, are equal to those in the upper portion near the overflow and in the lower portion adjacent to the discharge port of the classifier, respectively, that is atz=0, Ci=c0i (5.1.18) atz=zd, Cj=c z di (5.1.19) From a mass balance, the particle flux in the upper zone near the overflow port is: at z=0, FrQoCoi/At (5.1.20) In the calculations, the downward direction is again chosen to be positive, so that at the upper portion near the overflow, z=0. The discharge height is at z=1.016 m for the classifier with the C-0 or T-2 configuration. 135 Chapter 5 Unsteady State Hydrodynamic Model 5.2 Equations for the Eight Zones of the Classifier CFT dFi To solve the unsteady state hydrodynamic model, the parameters, F L , ——,——, ihj and dt dz r h L , should first be written as functions of Fi and C,. As described in Chapter 4, the classifier is divided into eight zones. The expressions for the above parameters may be different for different zones. Summing Equation (5.1.1) for all species of particles and Equation (5.1.2), with £ Cj + s = 1, the following equation can be obtained: d Adz K E ^ +eu)A] = Zihj/pi +mL/pL (5.2.1) Top zone I: (see Figure 4.5.1) As for the steady state model, the mass rates of particle and liquid production are zero. That is: rhj = 0 (5.2.2a) m L = 0 (5.2.3a) From Equation (5.2.1), — (ZcjVi+su) = 0 dz From a mass balance for section j , 0 - ( I C i V i + e u ) A t - Q o = 0 and from Equation (4.2.14), Q0=Qf+Qa-Qd, so 136 Chapter 5 Unsteady State Hydrodynamic Model Hence: F L=su = ( - Q f + Q a - Q a _ Z C i V i ) = _ Q f + Q a - Q a _ Z F i ( 5 2 4 a ) A t A t ^ = ± d ( - Q f - Q a + Q d ) ( 5 2 5 a ) dt A t a ^ a ^ = - 1 ^ (5.2.6a) Sz dz Feed zone: (see Figure 4.4.2) As for the steady state hydrodynamic model in Chapter 4, the feed flowrate is assumed to be uniform across the entry port. Therefore, the mass rates of particle and liquid addition from the feed pipe into cell Aj of the classifier per unit volume of column are A t dz . L = ( M z / A ) Q f s p L A t dz where £ can be calculated using Equation (4.4.5) in Chapter 4. From Equation (5.2.1), we obtain ^ ( Z c i v i + s u ) = Q f ^ / ( A t A f ) dz From a mass balance for section j , a Q f - d C i V i + s u ^ - Q ^ O so ^ Q a - Q d + 0-oOQf XcjV; +eu = - — — — — i 137 Chapter 5 Unsteady State Hydrodynamic Model where a can be calculated using Equation (4.4.6) in Chapter 4. Since F L = su = - Q - - ^ + ( 1 - a ^ - l F , (5.2.4b) At then dFL_ 1 a ( - Q a + Q d - ( l - a ) Q f ) 8F{ b\ A t dt ^ dt c\=QLda_^dFL dz A t dz dz From Figure 4.4.2, for cell Aj, Aa=£ dz/Af, then 5 F L _ Q f £ dF{ - ^ = ——~L-z- (5.2.6b) dz A t A f dz Top zone II: (see Figure 4.4.3) The mass rate of production of particles and liquid are both zero, i.e. rhi=0 (5.2.2c) rh L = 0 (5.2.3c) From Equation (5.2.1), — ( 1 ^ + 8 ^ = 0 dz From a mass balance for section j , ( Z c i V i + 8 u ) A t + Q a - Q d = 0 that is, Q a - Q d XCjVj +8U = 138 Chapter 5 Unsteady State Hydrodynamic Model F L = e u = ( _ Q a ^ _ Z C i V i ) = _ C ^ _ Q d _ _ Z F j ( 5 2 A c ) SO 5 F L _ 1 d(-Q a +Qd) d¥{ a ^ a a dz ^ dz (5.2.5c) (5.2.6c) Diverging Zone: (see Figure 4.4.4) The terms rhj and m L are zero because no flow enters or leaves this section from the side, i.e. rhj = 0 (5.2.2d) m L = 0 (5.2.3d) From Equation 5.2.1, |-[(Ic iv i+8u)A b] = 0 dz From a mass balance for section j , (Zc i v i +8u)A b +Q a -Q d = 0 F l = 8 U = _ Qa_^d _ £ = _ Qa^Qd _ Z F . ( 5 2 4 d ) so that arj = I d(-QA+QD) y a F j a A b a ^ a ^ _ ^ + Q a - Q d d A j L ( 5 2 6 d ) dz dz A u dz 139 Chapter 5 Unsteady State Hydrodynamic Model dAu A b and — - can be calculated from the geometry as described in Appendix E. dz Column zone: (see Figure 4.4.5) As for steady state, the mass rates of addition of particles and liquid to the column zone are riii = 0 (5.2.2e) rh L = 0 (5.2.3e) From Equation (5.2.1), 0 — (Xc i v i +su) = 0 dz From a mass balance for section j , (2>iVi+su)Ac + Q a - Q d = 0 F L = SU = X C J V J = - Q & f Q i - X F j (5.2.4e) A c A c a t . i a t - Q . + Q d ) , ^ ( 5 2 5 e ) dt A c dt dt ^ = -1^- (5.2.6e) dz ^ dz J Discharge zone: (see Figure 4.4.6) As assumed in the steady state hydrodynamic model, the particle concentration near the discharge is equal to that in the discharge stream and the flow rate of the discharge stream leaving the classifier is assumed to be uniform across the discharge port. Therefore, the mass production rates of the particles and liquid per unit volume of the column are 140 Chapter 5 Unsteady State Hydrodynamic Model = _ P » « = - Q d c i P / / ( A c A d ) (5.2.2f) A c d z L = _ « d z / A d ) Q d s p L = _ Q d 8 M / ( A c A d ) ( 5 2 3 f ) m T = A c d z where I can be calculated by Equations (4.4.7) in Chapter 4. From Equation (5.2.1), — (Zcivi + 8 U ) = - Q d ^ ( A c A d ) cz From a mass balance for section j , Qd - ( Z C iVj + eu)A c - Q a - otQd = 0 Q a - ( l - a ) Q d XCjVi +su = Ac where a can be calculated by Equations (4.4.8) in Chapter 4. Then F L = e u = _ Q a - ( i - a ) Q d _ £ c i V i = _ Q a - a - g ) Q d _ £ F i ( 5 2 4 f ) A c A c so dFL _ 1 a [ - Q a + ( l - a ) Q d ] ^ o F , ( 5 2 5 f ) at A C a ^ a dF L Q d oa ^ dF, dz A c oz oz From Figure 4.4.6, in Aj cell, Aa=£ dz/Ad, so oK Q d £ ^ o F ; , oz A c A d oz Note that the discharged particle concentration, Cdi, can also be calculated as in the steady state model using Equation (4.4.9) in Chapter 4. 141 Chapter 5 Unsteady State Hydrodynamic Model Bottom zone I: (see Figure 4.4.7) As in the steady state hydrodynamic model, the mass rate of addition of particles and liquid is 0, i.e. mi=0 (5.2.2g) m L = 0 (5.2.3g) From Equation (5.2.1), From a mass balance, ^-(Zc i V i+eu) = 0 dz ( Z c i V i + s u ) A c + Q a =0 F L - e u ^ - ^ - Z c i V j = - ^ - Z F i (5.2.4g) A. c A c ^ = _ L 8 t f M _ v a ( 5. 2.5g) a A C a ^ a oz dz Bottom zone II: (see Figure 4.4.8) The terms m{ and m L i n Equations (5.1.1) and (5.1.2) are again zero: r h i = 0 (5.2.2h) rh L = 0 (5.2.3h) and ^ [ ( I c i v i + s u ) A b ] = 0 dz 142 Chapter 5 Unsteady State Hydrodynamic Model From a mass balance for section j , ( I C i V i + a O A b + Q ^ O Qa ^ . „ . _ Qa F L = su = - Ec jVi = - S F j (5.2.4h) ^ = _ _ L ^ _ y ^ L (5.2.5h) dt A b dt ^ dt ^ , _ Z ^ L + % ^ A b ( 5 2 6 h ) dz dz A b dz dAu Here Ab and —— can be calculated from Appendix E. dz 5.3 Numerical Methods Equations (5.1.1) and (5.1.15), together with the initial conditions, boundary conditions, Equations (5.1.17) to Equation (5.1.20), overall mass balance, Equation (5.1.16), and 5FT 6FI Equations (5.2.2)-(5.2.6) to calculate the parameters F L , — — , — — , rh, and rh L for dt dz different zones, are used to solve the unsteady state hydrodynamic model. Particle concentration and flux, c; and Fi, along the height of the classifier are the final outputs. Computations are carried out using a mesh of finite difference cells fixed in a two dimensional space. The two-step Lax-Wendroff scheme (Press et al., 1992), a method with second order accuracy in both time and space that avoids large numerical dissipation and mesh drifting, is adopted in the calculation. 143 Chapter 5 Unsteady State Hydrodynamic Model For the ith species of particles, let q j , Fj j denote C,(ZJ, t„), FJ(ZJ, tn) respectively, and define intermediate values cfjly2, Fjj+i/\ at the half timesteps t„+i/2 and the half mesh points Zj+i/2. t or n j , n + l j-1/2 , n+1/2. / \ j+1/2, n+1/2 h n j+1 , n z orj Figure 5.3.1. Representation of two-step Lax-Wendroff differencing scheme As shown in Figure 5.3.1, two halfstep points (<8>) are calculated first. These, plus one of the original points, produce the new point via a staggered mesh of computational cells. Halfstep points are used only temporarily and do not require storage allocation on the grid. The successive steps of calculation are described as follows: Step one: 144 Chapter 5 Unsteady State Hydrodynamic Model If the known original point is at Zj.i, Zj, Zj+i and time is tn, the concentration of the ith species particle at time tn and point Zj+m can be written C i - C i j + i / 2 - - ( C i j + i + C i j ) and -> ~ n n dCj _ Cjj+i - Cj j dz Az - „n+l/2 n dc{ _ C i j + i / 2 - q , j + i / 2 2 The corresponding particle flux is F 1 = F i V / 2 = ^ ( F i V l + F u ) and dz Az ft " l A t 2 The above terms are substituted into Equations (5.1.1), (5.1.15) and Equations (5.2.2.) to (5.2.6) for different zones. The concentration and flux of the ith particle species at the half time steps, t„+i/2, and the half mesh points Zj+i/2, cij+i/ 22 a n ^ Fi j+Y/2, can then be calculated. The particle concentration, c y^ / / 2 2 , and flux, F " J ! L Y / 2 , at the other half step point can be obtained by the same formula as above via the original point, c ^ c ^ F ^ a n d F i J - l 145 Chapter 5 Unsteady State Hydrodynamic Model Step two: Using the particle concentration and flux at the half time and half step point from step one, the particle concentration and its gradients at time tn and point Zj can be expressed as Cj= Cj j n+l /2 n+1/2 dcj _ c i , j + l / 2 - C j , j - l / 2 dz Az n+1 n dci _ c i J ~ c i J dt At The corresponding flux and gradient are F = F n cn+1/2 pn+1/2 dE; _ M , j + l / 2 - F i J - l / 2 dz Az dti _ M,J M J dt At The above terms are again substituted into Equations (5.1.1), (5.1.15) and Equations (5.2.2) to (5.2.6) for different zones. The updated valuescjj"1 and Fj'j"1 can be obtained from such properly centered expressions. The provisional values cy^// 2 2 and F i j + V / 2 are now discarded. The Fortran program written to solve this unsteady state hydrodynamic model is listed in Appendix I. 146 Chapter 5 Unsteady State Hydrodynamic Model 5.4 Predictions and Comparison with Experimental Data Dynamic tests were carried out by changing one operating condition at a time stepwise, using the classifier with the T-2 configuration for both a mono-disperse suspension and for a multi-size system. The predictions of the unsteady state hydrodynamic model are shown in this section in comparison with the experimental data. The correlation, Equation (4.7.1), developed in Chapter 4 to estimate the axial dispersion coefficient for a multi-size system is employed again and, for want of an alternative, assumed to apply at every instant, i.e., the axial dispersion coefficient is also assumed to undergo a step change to its new value. Trial calculations showed that the response would be almost identical if D a were maintained at its pre-step-change value throughout the transient period. In the experiment, after the system reaches steady state, one operating variable, U a , Uf or Ua, was changed to a new value as quickly as possible (usually within 3 s). Then the voidages in the classifier column and in the discharge stream were recorded as functions of time until the system reached another steady state under the new operating condition. For a multi-size system, the particle size distribution in the discharge stream was also measured as a function of time by obtaining repeated samples and subsequent sieve analysis. Figures 5.4.1 and 5.4.2 show the variation of voidage with respect to time after a step change in the liquid fluidizing velocity, U a , for a mono-disperse suspension and a multi-size system, respectively. It can be seen that the voidage in the discharge stream increased after the step increase in U a , due to the higher bed expansion caused by the higher U a . The predicted and measured time constants for the response appear to be of order 20 s. 147 Chapter 5 Unsteady State Hydrodynamic Model During the experiments, it was also found that the discharge velocity increased in practice after the liquid fluidizing velocity was increased. However, because the discharge flow rate could only be measured by bucket collection, requiring about 30 s per measurement, it is not certain whether the discharge velocity changed in a stepwise manner with U a , or changed less quickly, or even after some delay time. A family of discharge voidage curves predicted by the unsteady state hydrodynamic model with different ramping delay times for increasing Ud ( i.e. Ud was assumed to change in a linear manner with time over the ramping period) is plotted in Figure 5.4.2. As discussed in Chapter 4, a higher Ud draws more small particles to the lower portion of the classifier, resulting in a slightly higher voidage in the discharge stream. Therefore, in the dynamic test involving a step increase in U a , the discharge voidage increases faster with a shorter ramping delay time in increasing Ud, as shown in Figure 5.4.2. The predictions of the hydrodynamic model agree reasonably well with the experimental data. Figure 5.4.3 shows the variation of the increment of dpso and the increment of dp9o/dpio with respect to time after a step change in U a for the multi-size system. It is seen that, at the beginning, the increment of dpso increases, while the increment of d P 9o /d pi 0 decreases. After a short period, the reverse occurs, i.e. the increment of dpso decreases and the increment of dP9o/dpio increases. As discussed in Chapter 4, at higher U a , the particle classification improves when there is a higher increment of dpso and lower increment of « dP9o/dpio- On the other hand, after the liquid fluidizing velocity was increased, the discharge velocity also increased, nullifying the beneficial effect and resulting in poorer classification. These two factors cause the increment of dpso to increase and the increment of dP9o/dpio to decrease first and then to move in the opposite direction. A family of 148 Chapter 5 Unsteady State Hydrodynamic Model curves for increments of dP5o and dP9o/dpio predicted from the unsteady state hydrodynamic model, with different ramping delay times for Ud, are plotted in Figure 5.4.3. It is seen that if the ramping delay time for Ua is longer, the decrease in the increment of dpso and the increase in the increment of dP9o/dpio happen later due to the delayed effect of Ud. It can also seen that the predicted trends with time of the increment of dP5o and the increment of dP9o/dpio, after a step change in U a , are reasonably consistent with the experimental trends, though there are offsets due to errors in predicting the respective steady state values. step change in U experimental • predicted 60 80 100 120 140 Time (s) Figure 5.4.1. Variation of voidage with time for a step change in liquid fluidizing velocity, U a , at t=20 s for mono-disperse suspension (T-2 configuration): U a(cm/s) Uf(cm/s) U d (cm/s) Sf d p (mm) initial: 5^96 23.4 L61 0.85 1.3 final: 7.90 22.5 2.03 0.84 1.3 149 Chapter 5 Unsteady State Hydrodynamic Model CD cn •g 0 7 o > CD CJ) i ro _c ° i 0.5 -• expeimental e d • I i predicted e d for • U d ramping delay time of 8 0 s - 4 0 s 20 s - O s 1 1 step change in U a i 1 i | i | i | i | i | i | i | i i i | i | i i i | i 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Time (s) Figure 5.4.2. Variation of voidage with time for a step change in liquid fluidizing velocity, U a , at t=25 s for a multisize system (T-2 configuration): initial: final: U a (cm/s) 6.02 8.10 U f (cm/s) 22.0 21.7 Ud (cm/s) 1.47 2.95 Sf 0.86 0.84 dP5o,f(mm) 1.07 1.10 dp9o,f/dpio,f 3.31 3.29 c CD E CD b C 0.6 0.4 -0.4 4 experiment • d P 5 o A <Wd, p10 ""pSO prediction d p 9 c / d p 1 0 U d ramping delay time 80s 40 s 20 s 0s step change in U a -i—i—i—|—i—|—i—i—i—i—i—i—i—i—i—i—i—i—i—|—i—i—i—|—i—|— 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Time, t (s) Figure 5.4.3. Variation of increment of dpso and dP9o/dpio with time for a step change in liquid fluidizing velocity, U a , at t=25 s for multisize system (T-2 configuration): Ua(cm/s) Uf(cm/s) U d (cm/s) e f dp5o,f(mm) dP9o,f/dpio,f initial: 6,02 22.0 L47 0.86 1.07 3.31 final: 8.10 21.7 2.95 0.84 1.10 3.29 150 Chapter 5 Unsteady State Hydrodynamic Model Figure 5.4.4 shows results for the voidage in the discharge stream after a step increase in the discharge velocity for a mono-disperse suspension. It is seen that neither the predicted voidage nor the experimental discharge voidage changes appreciably due to the change Ud, though there is some scatter in the experimental data. The voidage results after a step change in the discharge velocity for the multi-size system are shown in Figure 5.4.5 in comparison with the predictions of the hydrodynamic model. The experimental discharge voidage is not influenced much by increasing the discharge velocity. However, the predicted voidage from the model increases slightly with time in this multi-size case. As discussed in Chapter 4, at higher Ud, more smaller particles from the upper portion are drawn to the lower region, resulting in a higher voidage and poorer classification, hence a lower increment of dpso and a higher increment of dP9o/dpio. This is shown in Figure 5.4.6. It is seen that the predicted increment trends are consistent with the experimental data. The variation of voidage in the discharge stream after a step increase in the feed velocity for a mono-disperse suspension is shown in Figure 5.4.7. Because of the closed nature of the experimental system, the feed voidage increased with increasing feed velocity (as in Chapter 4). Experimentally, the voidage in the discharge stream did not change much after the increase in feed velocity, while a small decrease was predicted. Similar tendencies in the variation of voidage with respect to time after a step change in the feed velocity can be found for a multi-size system, though here the experimental scatter was greater, as shown in Figure 5.4.8. The increments of dpso and dP9o/dpio for the experimental data and the predictions of the hydrodynamic model after a step change in feed velocity for a multisize system are compared in Figure 5.4.9. After 151 Chapter 5 Unsteady State Hydrodynamic Model increasing the feed velocity, the particle classification was predicted to improve somewhat with a slightly higher increment of dpso and a slightly lower increment of dp9o/dpio. The predictions appear to agree moderately well with the experimental data, though the small magnitude of the predicted changes and the experimental scatter make it impossible to see how accurate the model is. 0.9 0.8 CO cn CO o > CO CO l _ CO JZ o CO b 0.7 0.6 0.5 experimental • predicted step change in U. 20 40 60 80 100 120 Time (s) Figure 5.4.4. Variation of voidage with time for a step change in the discharge velocity, Ua, at t=T5 s for a mono-disperse suspension (T-2 configuration): initial: final: U a (cm/s) 5.99 6.03 Uf (cm/s) 22.8 22.9 Ud (cm/s) 0.88 2.36 Sf 0.85 0.84 dp (mm) 1.3 1.3 152 Chapter 5 Unsteady State Hydrodynamic Model 0.9 • 0.8 CO cu" CO •g g 0.7 cu O) l _ ro x: o CO 6 0 6 0.5 experimental - predicted step change in U —r~ 20 40 60 80 100 120 —I ' 1— 140 160 180 200 Time (s) Figure 5.4.5. Variation of voidage with time for a step change in discharge velocity, Ua, at t=20 s for a multisize system (T-2 configuration): Ua(cm/s) Uf(cm/s) U d (cm/s) Sf dp5o,f(mm) dp9o,f/dpi0,f initial: 5.95 22.6 L34 0.87 1.06 3.25 final: 5.96 22.8 3.21 0.87 1.07 3.37 0.6 H 0.4 A C 0.2 CU E 2 -0.2 o c -0.4 -0.6 • experimental d p 5 0 A experimental d p 9 0 /d p l 0 predicted d p S 0 predicted d p 8 0 /d p 1 0 A A step change in U d 20 40 60 i— 80 140 160 180 200 100 120 Time (s) Figure 5.4.6. Variation of increment of dpso and dP9o/dpio with time for a step change in discharge velocity, U d , at t=20 s for a multisize system (T-2 configuration): Ua(cm/s) Uf(cm/s) U d(cm/s) gf dp5o,f(mm) dp9o,f/dpi 0,f initial: 5.95 22.6 L34 0.87 1.06 3.25 final: 5.96 22.8 3.21 0.87 1.07 3.37 153 Chapter 5 Unsteady State Hydrodynamic Model experimental • predicted step change in U. — i — 40 60 80 100 120 140 Time (s) Figure 5.4.7. Variation of voidage with time for a step change in feed velocity, U f , at t=25 s for mono-disperse suspension (T-2 configuration): initial: final: 0.9-U a (cm/s) 6.11 5.77 Uf (cm/s) 17.8 29.9 U d (cm/s) 2.06 1.99 Sf 0.80 0.88 dp (mm) 1.3 1.3 0.8 0.7 4 to CD CO ro "a o > CD CD \ ro o g 0.6-0.5 experimental • predicted ^ Step change in U, "T T 20 40 60 80 I 1 i 1 i 1 i • i 1 i 1 i 1 i r 100 120 140 160 180 200 220 240 260 Time (s) Figure 5.4.8. Variation of voidage with time for a step change in feed velocity, U f , at t=20 s for a multisize system (T-2 configuration): Ua(cm/s) Uf(cm/s) U d (cm/s) Sf dp5o,f(mm) dp9o,f/dpio,f initial: 6.37 152 2.64 (L81 1.10 3.26 final: 6.01 2S3 2.70 0,88 1.09 3.31 154 Chapter 5 Unsteady State Hydrodynamic Model 0.6 0.4 •£ 0.2 H CD E CD t> -0.2 q c -0.4 O experimental d p 5 0 experimental dp 9 l/dp 1 0 • predicted d p 5 0 predicted dp9 l/dp,0 20 O O Step change in U — 1 —i 40 60 T -80 T" T" 100 120 140 160 180 200 Time, t (s) F i g u r e 5.4.9. Variation of increment of dP5o and dP9o/dpio with time for a step change in feed velocity, U f , at t =20 s for multisize system (T-2 configuration): Ua(cm/s) Uf(cm/s) U d (cm/s) s f dp5o,f(mm) dp9oydpio,f initial: 6.37 15,2 2.64 0,81 1.10 3.26 final: 6.01 28.9 2.70 0.88 1.09 3.31 155 Chapter 6 Conclusions and Recommendations Chapter 6 Conclusions and Recommendations 6.1 Conclusions 1. Previous studies on the settling of particles in liquids demonstrate that the most popular equation to calculate the particle settling velocity in a concentrated liquid-solid suspension is the correlation of Richardson and Zaki (1954). Even though this was derived for suspensions of nearly identical particles, it is commonly used to predict the behavior of suspensions of particles of different sizes and densities. 2. Most published models of particle classification have been for sedimentation with binary systems under batch conditions. Although some recent experimental and theoretical investigations have been carried out on differential sedimentation of binary suspensions under continuous conditions, little theoretical analysis has been carried out on liquid classifiers with particles of multiple sizes and/or densities under continuous conditions. There is a particular shortage of information on unsteady state dynamics. 3. In the sedimentation of particles in a concentrated suspension, there is a limiting voidage below which the particles cannot move downwards and be discharged by the underflow from the classifier. This may account for the "emptying phenomenon" sometimes encountered in the industrial classifier. In the industrial classifier with high feed particle concentration, particles may accumulate in the diverging region, resulting in higher particle concentration, causing the particles to move only upward. 4. An experimental continuous solid-liquid classification system was erected at U B C to study the hydrodynamic behavior of particles and liquid under steady and unsteady 156 Chapter 6 Conclusions and Recommendations state conditions. Spherical glass beads of different sizes and uniform density enter the classifier as a dense suspension conveyed by water through a radial feed port near the top, while water without suspended particles is injected upwards from the bottom. A relatively dilute stream containing mostly small particles is taken off the top as an overflow stream, while an underflow stream enriched in coarser particles is removed from the side near the base of the column. The geometric configuration employed at U B C is similar to that in the industrial unit. The particle size and operating conditions applied at U B C are determined by matching the Archimedes number and Reynolds number employed by M C C . 5. Based on mass and momentum balances, a steady state hydrodynamic model was developed with only one fitted parameter, an axial dispersion coefficient. Any turbulence in the flow is taken into account in the model indirectly via the axial dispersion coefficient, which is assumed to be uniform throughout the classifier. As boundary conditions in the model, the particle concentrations at the top and at the lower portion near the discharge port were set to be the same as in the overflow stream and discharge stream respectively. The correlation of Di Felice (1994) is used to calculate the drag force on the particles, while the particle dispersion force is introduced according to the concept of Thelen and Ramirez (1999). 6. A correlation to estimate the particle dispersion coefficient in terms of relevant dimensionless parameters has been developed. The dispersion coefficient is predicted to increase slightly with increasing liquid velocity, U a , and decreasing feed velocity, Uf, but to be independent of discharge velocity, Ud. The dispersion coefficient increases with increasing feed voidage for Sf>0.68, while it decreases for Sf<0.68. 157 Chapter 6 Conclusions and Recommendations With a broader particle size distribution in the feed, a taller column zone of the classifier or a smaller mean particle size in the feed, the dispersion coefficient is predicted to be lower. The predictions of voidage and particle size distribution in the discharge, using the hydrodynamic model together with the dispersion coefficient correlation, agree well with the UBC experimental data, as well as with available industrial data. 7. The degree of classification in the liquid-solids classifier depends on the extent of axial dispersion, which in turn is a function of many variables. The classification becomes better with increasing feed voidage, feed flowrate and fluidizing liquid flow rate, but is worse at higher underflow discharge rates. The performance of the classifier is better for a broad than for a narrow particle size distribution of the feed stream. Increasing the height of the cylindrical zone of the classifier improves the particle classification. 8. The steady state hydrodynamic model has been extended to an unsteady state hydrodynamic model. Dynamic tests were carried out by examining the response to step changes in liquid fluidizing velocity, U a , feed velocity, Uf, or discharge velocity, Ud. Increasing Uf improves the particle classification, while increasing Ud leads to poorer classification. In the experimental equipment, an increase in U a causes an increase in Ud, resulting in poorer classification eventually. A l l these predictions are consistent with the experimental results. 158 Chapter 6 Conclusions and Recommendations 6.2 Recommendations 1. The hydrodynamic model, while useful, has certain deficiencies. Several approximations were made, each taking the model a step away from a faithful description of the real process. In the hydrodynamic model, the classifier is assumed to be uniform in the lateral direction and one-dimensional in the axial direction. However, as described in Chapter 1, the top region of the classifier is dominated by a vigorous vortex, caused by the feed stream entering as a horizontal radial jet, as shown in Figure 1.1. The flow pattern in this region is highly three-dimensional. The present one-dimensional hydrodynamic model should be modified to a three-dimensional model by introducing centrifugal terms in the top region. 2. The hydrodynamic model can give good predictions of classifier performance and efficiency for suspensions of particles having different sizes but equal densities. This model should be applied also to systems of particles having different densities. 3. The hydrodynamic models in Chapter 4 and 5 apply to a liquid-solid classifier with the geometric configuration shown in Figure 4.1.1. The models and the programs could also be rewritten to apply to different system geometries. 159 Nomenclature Nomenclature a = exponent in the Doheim et al. (1997) model or parameter in Table 3.3.1 A = cross-sectional area [m2] Ar = Archimedes number [dp3pL|pP-pL|g/!h.2] [-] b = parameter in Table 3.3.1 c = volumetric concentration of particles [-] or the parameter in Table 3.3.1 C = liquid-free volume fraction of particles[-] or conductivity [1/mQ] C D = drag coefficient [-] dp = particle diameter [m] D = diameter [m] D a = axial dispersion coefficient [m2/s] fD = drag force per unit particle volume [N/m3] f = objective function expressed in Equation (4.5.1) F = liquid or particle flux [m/s] or cumulative PSD undersize by mass [-] F G = gravitational force [N] F B = buoyancy force [N] F D = drag force [N] F = fluctuation force [N] g = acceleration due to gravity [m/s2] h = vertical coordinate measured upward from bottom of classifier [m] H = height [m] I = ith cell in Markov Chain Model 160 Nomenclature I t = turbulence intensity defined by Equation (2.4.25) [-] J = volumetric flux [m/s] k = constant of conductivity probe [1/m] £ = length defined in Equations (4.4.5) and (4.4.17) [m] M = Number of height increments needed in the integration, for each zone of the classifier [-] rh = external mass flow added into differential element per unit volume of element [kg/m3s] n = Richardson and Zaki index [-] n r = index of Rosin-Rammler distribution [-] N = number of cells in Markov Chain Model [-] p = transfer probability in Markov Chain Model [-] p = normalized transfer probability in Markov Chain Model[-] P = pressure due to weight of particles in the liquid [Pa] P = total pressure due to weight of particles and liquid [Pa] Pe = Peclet number [U aHCoiumn/D a] [-] Q = volumetric flow rate [m3/s] R = electrical resistance [fl] or radius [m] Re = Reynolds number [pLdpV/pJ [-] Reo = Reynolds number, [pLD c U a /pL] [-] Reap = Reynolds number, [pLdP5o,fUa/pL][-] t = time [s] 161 Nomenclature u = absolute local velocity of liquid [m/s] U = superficial velocity base on column cross-section [m/s] v = absolute local velocity of particles [m/s] V = velocity of particles relative to fluid [m/s] or voltage [volts] or volume [m3] V = fluctuation slip velocity defined by Equation (2.4.24) [m/s] VT = terminal settling velocity of individual particle [m/s] z = vertical coordinate measured downward from top of column [m] Greek Letters a - ratio of areas defined in Equations (4.4.6) and (4.4.12) P = drag force index in Equation (2.2.4) [-] y = defined by Equation (2.4.26) [-] s = voidage [-] rj = classifier efficiency [-] A. = defined by Equation (2.4.28) [-] u. = fluid viscosity [Pa.s] p = density [kg/m3] <b = sphericity [-] Subscripts 0 = single particle a = fluidization stream (i.e. stream injected upwards from bottom of column) 162 Nomenclature b = bulk or diverging zone or bottom zone II bottom I = bottom zone I bottom II = bottom zone II c = cylindrical column above discharge pipe d = discharge stream or discharge pipe discharge = discharge zone disp = dispersion diverge = diverging zone drag = drag e = fluidizing pipe exp = experimental data f = feed stream or feed pipe or fluid feed = feed zone i = ith particle fraction k = kth zone L = liquid or large 0 = overflow P = particle S = solid or small seg = segregation t = top top I = top zone I top II = top zone II Nomenclature T = terminal u = underflow w = limiting x, y, z = x, y, z directions References References Abu-AH, M . 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Smith, T.N., "Differential Settling of a Binary Mixture", Powder Technol., 92 171-178 (1997). Smith, T. N . , "The Differential Sedimentation of Particles of Various Species", Trans. Instn. Chem. Engrs., 45 T311-T313 (1967). Smith, T. N . , "The Sedimentation of Particles Having a Dispersion of Sizes", Trans. Instn. Chem. Engrs., 44 T153-157 (1966). Smith, T. N . , "The Differential Sedimentation of Particles of Two Different Species", Trans. Instn. Chem. Engrs., 43 T69-T73 (1965). Snow, R H . and Allen, T., 'Effectively Measure Particle -Size Classifier", Chem. Engr. Progr., 29-33, May, (1992). Spannenberg, A.B. , Raven, J., Scarboro, M.J. and Galvin, K.P., "Continuous Differential Sedimentation of a Binary Suspension", Powder Tech., 88 45-50 (1996). Stamatakis, K. and Tien C , "Dynamics of Batch Sedimentation of Polydispersed Suspensions", Powder Technol., 56 105-117 (1988). Svarovsky, L. , Chapter 1 in "Solid-Liquid Separation", 3 r d edn, L . Svarovsky (Ed.), Butterworths, Oxford, pp. 1-11, (1990). Thelen, T.V. and Ramirez, W.F., " Modelling of Solid-Liquid Fluidization in the Stokes Flow Regime Using Two-Phase Flow Theory", AIChE J., 45 708-723 (1999). Truesdell, C , "Mechanical Basis of Diffusion", J. Chem. Phys., 37, 2336- 2342(1962). 170 References Tsujimoto, T. and Yamamoto, T., "Stochastic Model for Determining Suspended Sediment Concentration Distribution with Particular Reference to Comparison with Diffusion Theory", Mem-Fac-Technol-Kanazawa-Univ., 19 33-42 (1986). van der Meer, A.P., Blanchard, C.M.R.J.P. and Wesselingh, J.A., "Mixing of Particles in Liquid Fluidized Beds", Chem. Eng. Res. Des., 62 214-222 (1984). van der Wielen, L . A . M . , van Dam, M.H.H. and Luyben, K.Ch.A.M. , " On the Relative Motion of a Particle in a Swarm of Different Particles", Chem. Eng. Sci., 51 995-1008 (1996). Van Duijn, G. and Rietema, K., "Segregation of Liquid Fluidised Solids", Chem. Eng. Sci., 37 727-733 (1982). Watanabe, F., Mitsutani, K., Yukawa, H. , Shimizu, F., Tanaka, E., Natori, Y . and Ko, G., " Model Based Process and Product Design of High-Impact Polypropylene Copolymer", 1998 Spring AIChE Meeting, New Orleans, March 8-12 (1998). Wen, C. Y . and Yu, Y . H . "Mechanics of Fluidization", Chem. Eng. Prog. Symp. Series, 62(62) 100-111 (1966). Zhang, J. P., " A Markov Chain Model for Particle Mixing and Segregation in a Fluidized Bed Classifier", Internal Report, Dept. of Chemical and Bio-Resource Engineering, Univ. of British Columbia (1998). Zimmels, Y . , "Theory of Hindered Sedimentation of Polydisperse Mixtures", AIChE J., 29 669-676 (1983). 171 Appendix A: Dimensions of U B C Classifier for Different Configurations Appendix A: Dimensions of UBC Classifier for Different Configurations H , H 2 Ha H f <t>:Dt <J)Df (1)DC N • (|)Dd 1 H 4 Figure A.1 . Dimensions of U B C classifier for different configurations Unit: mm C-0 C - l C-2 C-3 T-2 D500 D t 292 292 292 292 191 762 D c 191 191 191 191 191 508 H i 292 292 292 292 292 762 H 2 102 102 102 102 102 254 H 3 686 102 349 1035 686 1829 H 4 457 457 457 457 794 1219 H f 127 127 127 127 127 381 H d 64 64 64 64 64 152 D e 51 51 51 51 51 76 D f 51 51 51 51 51 102 D d 51 51 51 51 51 76 D 0 76 76 76 76 76 102 172 Voidage CO 0.88 0.84 0.78 0.72 0.75 0.74 0.74 0.73 0.74 0.79 0.80 0.75 0.81 0.83 0.67 0.72 0.73 0.69 0.72 0.83 0.84 Velocity (cm/s) •a » ro 0 r o 00 cs r o r o cs ro r o uo cs cs r o r-cs r o r o cs 0.64 N O C S cs O N 2.21 3.46 2.03 1.61 Velocity (cm/s) 22.1 21.9 21.9 22.1 22.1 22.1 22.0 22.1 22.2 15.7 18.8 22.1 25.5 28.9 22.9 22.5 22.6 21.5 21.9 22.5 22.9 Velocity (cm/s) « p 6.29 6.31 6.30 6.21 3.32 4.44 6.22 7.34 8.31 6.20 6.39 6.16 6.22 6.20 6.18 6.15 6.14 6.17 6.17 7.95 5.97 Particles: Glass Particles: Glass U Particles: Glass a ? Particles: Glass Particles: Glass in C O N 0 O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O O N O m ro Particles: Glass cu S ° - W) 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 Liquid: Water 3. 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N Liquid: Water O N O N t-O N O N r» O N O N r-O N O N f-O N O N r-O N O N r-O N O N O O N O N O N O N O N O N O N O N t-» O N O N t-> O N O N r-O N O N r-O N O N r~-O N O N c-~ O N O N f-O N O N r-O N O N f-O N O N r-O N O N d H O C S O C S O C S O C S O C S O C S O C S O C S O C S O C S O C S O C S O C S O C S 20.5 20.5 20.5 21.0 21.0 19.5 19.5 Config. T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 System Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Run ID M09e01 M09e02 M09e03 M09r04 M09a04 M09a03 M09a01 M09a02 M09a05 M09f01 zmon ro e O N O O N O M09f05 M09d01 M09d02 M09d03 M09d04 M09d05 Dymo041 Dymo042 3 CD P-1 C C o -a c o U C4 CO >? W o c H x •3 Sl Voidage CO 0.85 0.84 0.80 0.88 0.88 0.83 0.83 0.85 0.85 0.84 0.91 0.87 0.83 0.79 0.76 0.78 0.80 0.77 0.70 0.80 0.83 0.89 0.87 -5 c -a P 1.61 2.03 2.06 1.99 1.99 2.08 2.35 0.88 0.88 2.36 1.31 1.28 1.28 1.28 0.71 2.63 1.28 1.28 1.28 1.28 2.39 2.41 2.39 city (ci P 23.4 22.5 17.8 29.9 30.2 18.2 22.7 22.8 22.8 22.9 21.9 20.8 21.8 21.6 21.6 21.6 21.9 21.4 15.5 29.4 12.7 13.1 12.5 Velo P 5.96 7.90 6.11 5.77 5.76 6.09 6.03 5.99 5.99 6.03 6.31 6.32 6.29 6.41 6.23 6.25 3.24 8.32 6.19 6.36 5.38 2.99 3.91 o ON & 1.70 1.69 1.66 Particles: Glass u 0.50 0.52 0.53 0.54 0.53 0.54 0.53 0.54 0.53 0.54 Particles: Glass 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.55 Particles: Glass •o E ON O ON o ON o ON O ON O ON o ON O ON O ON o ON O Particles: Glass c n CO CO CO CO CO r o CO CO r o 1.07 1.0.8 1.07 Particles: Glass m P . E a W) 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 Liquid: Water •y « PM 3 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON o o ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON 00 ON ON o o ON ON 00 ON ON 00 ON ON o o ON ON Liquid: Water r -ON ON r-ON ON r-ON ON r-ON ON r-ON ON t— ON ON r-ON ON r-ON ON ON ON C-ON ON r-ON ON r-ON ON C-ON ON r-ON ON r-ON ON r-ON ON C -ON ON r-ON ON r-ON ON r-ON ON r -ON ON r-ON ON t-ON ON d. <U <= H 19.5 19.5 19.5 19.5 19.5 19.5 20.0 20.0 20.0 20.0 O CN © CN O CN O CN O CN o CN O CN O CN © CN o CN 20.0 20.0 20.0 Config. T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 o 1 O o 1 O o 6 System Mono Mono Mono Mono Mono Mono Mono Mono Mono Mono Binary Binary Binary Binary Binary Binary Binary Binary Binary Binary Multi Multi Multi Run ID Dvmo051 Dvmo052 NO o o B -, > Q Dvmo062 Dvmo071 Dvmo072 Dvmo081 Dvmo082 Dvmo091 Dvmo092 B0509el B0509e2 B0509e3 B0509e4 B0509dl B0509d2 B0509a2 B0509al ON o O PQ C ON o o PQ SBOl SUA20 SUA40 174 c CJ cu (4 cu £ CU o , W 13 C C _o "•*-> 13 C o O OJ c '5 CU o c cu CU D X W <4H o c o 3 X ) rt 1 3 C CU o , W) eS "2 o u o "S > es 6 • —• s-CS PH S O" CO 73 p U a. E U H ob C s o U s GO c 3 PH & E •o E •o E c £ uo ^1 CN 0 0 CN 0 0 r o r o CN Ti-r o CN O N O O IT) WO C N r o CN r o 0 0 0 0 t-r o r o CN r o O N N O NO NO NO o r o T f 0 0 O N O tr-od O N T f 0 0 0 0 ON o r o O N NO 0 0 CN O N NO NO NO NO 0 0 0 0 O N 0 0 0 0 0 0 NO ON o 0 0 od CN 0 0 ON O O N I 0 0 r o 0 0 O N NO O T f NO O N CN r o T f CN NO ON 0 0 T f CN uo O N 0 0 NO. 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CU 0 O O O O O O O O O O O O O O O O O O O O 0 O H CN CN CN CN CN CN CN CN CN CN CN CN CN CN C N CN CN CN C N CN CN CN •fig-0 CN CN CN CN CN CN C N CN CN CN CN CN CN CN CN CN CN O 1 O O O O O U H H H H H H H H H H H H H H H H H 6 6 6 6 System 3 3 3 3 3 3 3 1 3 3 1 3 3 1 3 3 3 1 3 3 3 1 Run ED r o O CN H T f O CN H IT) O CN H NO O CN H f -O CN H CN H C N CN H T f CN H ir> CN H CO CN H NO CN H C -CN H CN C N H CN CN CN H r o CN C N | H T f CN CN | H m CN CN H CN 0 0 r o e (0 0 0 T f e co 0 O m e (0 0 0 NO e co 0 O Voidage <*-CO 0.74 0.77 0.86 0.84 0.84 0.86 0.88 0.81 0.81 0.88 0.87 0.86 0.86 0.87 0.93 0.91 0.92 0.91 Vi " f i p 1.35 2.97 1.47 2.95 2.95 1.56 1.55 2.64 2.64 2.70 1.34 3.21 3.29 1.45 0.39 1.02 0.36 0.62 tcity (ci P 15.8 21.4 22.0 21.7 21.7 22.1 29.4 15.5 15.2 28.9 22.6 22.8 23.1 23.2 10.0 cs O N 00 O N cs O N Velc es P 6.23 6.17 6.02 8.10 8.14 6.08 5.99 6.36 6.37 6.01 5.95 5.96 6.04 5.98 4.05 3.69 3.34 3.61 Particles: Glass > 5 ° n 3.67 13.45 3.31 3.29 3.29 3.32 3.27 3.26 3.26 3.31 3.25 3.37 3.31 3.23 2.40 2.70 2.56 2.72 Particles: Glass p-3 U Particles: Glass SL £ Particles: Glass Particles: Glass •o £ CN oo t--o o o t-» o 00 o o o O N O N O o t— o t-N o N O o O N CS O N o o Particles: Glass 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 2749 Liquid: Water cn I t oo O N O N 00 O N O N 00 O N O N oo O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N 00 O N O N oo O N O N oo O N O N 00 O N O N 00 O N O N 00 O N O N oo O N O N Liquid: Water - £ r~. O N O N t> O N O N O N O N t--O N O N r-O N O N r-~ O N O N r-O N O N r-O N O N r-O N O N f-O N O N r-. O N O N t-O N O N r--O N O N r-O N O N r-O N O N t-O N O N t-~ O N O N t-~ O N O N d <u ° H 121.0 21.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 38.0 14.2 29.3 25.7 Config. o i O o 1 O T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 T-2 D500 . D500 D500 D500 System Multi Multi IMulti Multi Multi Multi Multi Multi Multi Multi Multi Multi Multi Multi 1 1 1 Mulri Run ID COufOl COudOl DymuOll Dymu012 Dymu021 Dymu022 Dymu031 Dymu032 DYmu041 Dymu042 Dymu051 Dymu052 Dymu061 Dymu062 Run5001 Run5002 Run5003 Run5004 180 c CJ 3 co CU •*-> c cu CU o X W -o c rt c o d o T3 C o O bl rt c u o, c u CU o, X W < + H O c o 3 Xi rt H C Q X *3 c <D O , P., < c o 3> c o o CN i H c _o 'co G c u O H CO 3 c u CL, O c o 6 rt T3 cu a. x W C N CQ 3 es H Os 00 00 T f 00 Os ON Os r-O 00 o 00 00 NO 00 NO 00 l-H 00 00 00 T f r-Os Os r-NO Os CO 00 o o 00 OS CO ON ON NO T f o Os o CS 00 r~ ON CN ON CO ON © CO ©" o O ©' o © ' o ©' ©' o o ©' ©' o o o o o o o t-H o o o o o o o o so t-w i r--Os NO T f NO r-NO T f r-00 r- 00 T f r-w i vs t-T f r-T f f- r- t -CO CN t-H CO r-NO NO VO NO CO r~ 00 NO NO NO NO vo 00 NO NO ON NO Os v q ON NO o o O ©' o o o ©' o ©' o ©' o o o o o o o o o o o o o o o o o o" o idage CO Voi 00 00 T f 00 00 r- CN W l T f r-T f CO r-T f ON o 00 w> t-H 00 CO 00 r-NO <N CO Os NO <N r-CO 00 T f 00 W l 00 T f 00 o 00 00 00 00 00 CO 00 CO 00 W l 00 W l °°. T f 00 CO o ©' ©' o o" o o o o o o d o o o o o o o o o o o o o o o o o o" o i—H ON Os o t-H CO Os t-H CO NO Os Os T f ON T f T f 00 W l NO CO t-H CN o NO NO T f T f 00 W l 00 W l f-' <N NO CN NO CN CN T f CN CN NO* CN oo' CN Ov CN ©' CS co' CN NO CN ©' CO co' CO oo' CN (--' CS NO' CN wi CN T f CN o CO T f CS wi CS CS CS ON CS 00 CN r-' CS NO CS wi CS NO (S wi CS o p CO CO O CO 00 CN CO CO CN CO t-H CO i n CN r -CN CO r~ CN CO CO CN T f NO NO cs CN Os CS CN NO T f co' CO o CS NO vo CO o CS NO o CS Os Os_ Os Os 00 o CS W l CO CN 00 00 o" oo oo_ o NO CO CS G U >-+•» <-< Os Os <-< «-< r-H © © CN f - 00 -H WI ON ON W l NO W l OS W l ON T f W l 00 Os cs CS r~ 00 00 Ov loci CN <N CN CN CN CN CN (N CN CN CN CN CN CN cs CN wi 00 CS cs vi cs oo' CN CN CS CN CN CS* CS CS t-H CN CN CN CN CS CO CS cs' cs Os" CS o CO oo' --H CN CN CN cs cs' cs CS' CS Ve Os rs CO o CO CN CN CO T f T f CN CS T f CO CO o CN Os CO NO CS CS o CN 00 W l T f l > W l ON r-OS VO ON o ON r-H l > NO f-OS o ro o ON ON ON ON CO o NO NO NO NO CO T f NO r-' 00 NO so' NO" NO NO NO NO NO NO NO r-' wi wi r-' NO wi wi NO NO wi wi NO n i= Os Os ov Os Os Ov ov Os Os Os Ov Os ON Os Os Os Os ON ON CO CO CO CO CO CO CO CO CO CO CO CO (mm) O © ' o ©' o O o o O O o o" o o o' o o o o (mm) & I S Run ED O CJ Os O 2 CN o CJ Os o 2 CO o CJ Os o 2 T f o CJ ON O 2 T f o rt Os O 2 <o o rt ON O 2 o rt ON o 2 CN o ca Os o 2 >o o rt Os O 2 e Os O 2 CS e Os o 2 CO e Os o 2 T f e Os o 2 W l e Os o 2 t-H o •a Os o 2 CN o •a Os o 2 CO o •a Os o 2 T f o •a Os o 2 W l o •a ON o 2 T f o o E >. D CS T f o o 1 D W l o o 1 Q CS W l o o D NO o o Q CS NO o o 1 Q t-H r-o o E D CS r-o o E D 00 o o 1 Q cs 00 o o i Q i-H ON o o 1 Q CS ON o o f Q 181 182 idage 0.94 0.66 0.97 0.92 0.63 0.95 0.88 0.64 0.92 0.82 0.63 0.87 0.77 0.63 0.83 0.74 0.65 0.83 0.77 0.65 0.85| o > 2.36 1.00 0.99 1.00 1.00 0.99 1.00 1.00 0.99 1.00 1.00 0.99 1.00 1.001 0.99 1.00 1.00 1.00 1.00 1.00 0.99 | 1.00 © 0.92 0.86 0.95 0.93 0.82 0.98 0.95 0.83 0.98 0.94 0.81 0.96 0.94 0.83 0.95 0.94 0.86 0.95 0.93 0.85 0.95 0.78 0.66 0.85 0.83| 0.58 0.93 0.84| 0.62 0.89 I 0.841 0.58 0.88 | 0.83 0.62 0.85 I 0.84 0.68 0.86 I 0.81 0.65 0.85 0.61 0.43 0.71 0.68 0.32 0.83 0.69 0.37 0.77 0.70 0.34 0.75 0.68! 0.39 0.71 0.69 0.47 0.72 0.65 0.43 0.71 1.18 0.45 0.24 0.57 0.541 0.15 0.70 0.54| 0.21 0.62 I 0.55| 0.18 0.61 1 0.53! 0.241 0.56 I 0.55 0.32 0.58 1 0.50 0.26 0.57 1.00 0.32 0.13 0.44 0.40 0.06 0.55 0.41 0.11 0.48 0.42 0.09; 0.47 0.39 0.13 0.42 0.42 0.20 0.45 0.37 0.15 0.43 PSD,F 0.85 0.22 0.05 0.33 0.29| 0.02 0.41 0.30| 0.05 0.36 0.311 0.04 0.35 I 0.28| 0.07 0.31 I 0.31 0.12 0.34 1 0.27 0.08 0.32 0.71 0.15 0.01 0.23 0.19 0.01 0.27 r o.i9 0.02 0.24 1 0.201 0.01 0.23 0.18 0.03 0.20! I 0.21 0.06 0.23 0.17 0.03 0.21 0.59 0.11 0.00 0.17 0.131 0.00 0.19 0.131 100 0.16 0.141 0.01 0.16 1 0.12 100 0.14 1 0.14 0.03 0.16 1 0.12 0.02 0.14 i n d 0.06 0.00 0.10 0.071 0.00 0.11 0.071 0.00 0.09 I 0.081 0.00 0.09 1 0.07 0.01 0.08 1 0.08 100 0.09 1 0.07 100 0.08 0.43 0.02 0.00 0.04 0.021 0.00 0.04 0.021 0.00 0.03 0.03 0.00 0.03 1 0.02| 0.00 0.02 1 0.03 0.00 0.03 1 0.02 0.00 0.03 0.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 o.oo! 0.00 0.00 0.00 0.00 0.00 dp (mm) Feed Discharge Overflowe Feed Discharge Overflowe Feed Discharge Overflowe Feed Discharge Overflowe Feed Discharge Overflowe Feed Discharge Overflowe I Feed Discharge tu i* o «a tu 6 e P 27.1 25.8 26.0 26.9 26.3 20.6 24.6 E c > P VO <n -* VO T f VO T t m m 2.97 Velocit P 22.1 20.9 21.2 22.2 21.5 15.8 21.4 Velocit C P 6.29 6.22 6.27 6.23 6.32 6.23 6.17 Run ID C0ef02 C0ef03 C0ef04 m e tl) o U m e tu o O COufOl COudOl 183 CO C CJ CO PH G cu e c cu o, X W T 3 C 03 co C o c o O G c cu e c o o, X W C+H o c o 3 X> cd H CQ X °*3 c cu o, o, <: c o '•ii 3) C + H c o o CN i H X J CO CU N cS cc! TJ CU e cu O H X cu cu s o CZ) W CD es H Voidage 0.86 0.68 0.90 0.84 0.75 0.90 0.84 0.75 0.90 0.86 0.66 0.90 0.88 0.69 0.91 0.81 0.68 0.89 0.81 0.68 0.89 0.88 0.68 0.91 0.87 0.71 | 0.90| 2.36 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 o rs 0.95 0.84 0.97 0.95 0.86 0.97 0.95 0.86 0.97 0.95 0.85 0.98 0.96 0.85 0.97 0.95 0.89 0.98 0.95 0.89 0.98 0.95 0.88 0.97 0.96 0.84 0.98 r-0.88 0.65 0.92 0.86 0.67 0.92 0.86| 0.67 0.92 0.88| 0.63 0.93 0.88 0.64 0.92 0.87 0.71 0.93 0.87 0.71 0.93 0.87 0.72 0.92 0.89 0.61 0.93 r H 0.74 0.33 0.81 0.72 0.41 0.80 0.72 0.41 0.80 0.74 0.35 0.82 0.73 0.33 0.80 0.72 0.46 0.82 0.72 0.46 0.82 0.72 0.47 0.80 0.74 0.30 0.81 1.18 0.60 0.16 0.68 0.58| 0.24 0.67 0.58| 0.24 0.67 0.60| 0.19 0.69 0.60 0.16 0.66 0.58 0.28 0.70 0.58 0.28 0.70 0.59 0.29 0.68 0.61 0.13 0.68 1.00 0.49 0.07 0.56 0.46 0.13 0.55 0.46 0.13 0.55 0.48 0.09 0.56 0.47 0.07 0.53 0.45: 0.17 0.57 0.45 0.17 0.57 0.46 0.17 0.56 0.49 0.06 0.56 PSD, F 0.85 0.31 100 0.36 0.28| 0.04 0.35 0.28| 0.04 0.35 0.30| 0.02 0.36 0.30| 0.02 0.34 0.28 0.06 0.37 0.28 0.06 0.37 0.29 0.06 0.36 0.30 TOO 0.35 0.71 0.21 0.00 0.25 0.20J 0.01 0.25 0.20 0.01 0.25 0.21 0.00 0.25 0.21 0.00! 0.241 0.19 0.02 0.26 0.19 0.02 0.26 0.20 0.02 0.26 0.21 0.00 0.24 0.59 0.14 0.00 0.17 0.131 0.00 0.17 0.131 0.00 0.17 0.141 0.00 0.17 0.141 0.00 0.16 0.13 0.01 0.18 0.13 0.01 0.18 0.14 TOO 0.18 0.14 0.00 0.16 © 0.08 0.00 0.09 0.071 0.00 0.09 0.071 0.00 0.09 0.081 0.00 0.10 0.08 0.00 0.09 0.07 0.00 0.10 0.07 0.00 0.10 0.08 0.00 0.10 0.08 0.00 0.09 0.43 0.03 0.00 0.03 0.021 0.00 0.03 0.021 0.00 0.03 0.031 0.00 0.03 0.031 0.00 0.03 0.02 0.00 0.03 0.02 0.00 0.03 0.03 0.00 0.03 0.03 0.00 0.03 0.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 ? £ a •a Feed Discharge Overflowe Feed! Discharge Overflowe Feed Discharge Overflowe Feed Discharge y is o <£ <L> 6 Feed Discharge CJ is o TJ CJ Feed Discharge Overflowe Feed Discharge Overflowe Feed Discharge Overflowe Feed Discharge Overflowe e P 26.6 26.9 26.9 26.6 33.8 19.2 18.9 32.2 27.2 f (cm/s P 1.47 2.95 2.95 1.56 1.55 2.64 2.64 c-1.34 "elocto P 22.0 21.7 21.7 22.1 29.4 15.5 15.2 28.9 22.6 e P 6.02 r—1 00° 8.14 6.08 5.99 6.36 6.37 6.01 5.95 Run ID DymuOll Dymu012 Dymu021 Dymu022 Dymu031 Dymu032 T f o Q rs T f O 1 D Dymu051 184 rt CO rt c CO CO Oi rt c ccj CO C O '-3 c o O bl O ccS C CO CO o X w C M O c o p H CQ >< 3^ c CO o, o, < Voidage 0.86 0.69 0.86 0.69 0.92 0.90 0.87 0.67 1 0.91| 2.36 1.00 1.00 1.00 1.00 1.00 1.001 1.00 1.00 | 1.00 p 0.96 0.92 0.98 0.95 0.91 0.98 0.96| 0.85 | 0.98 t~-0.88| 0.77 0.94 0.88 0.78 0.93 1 0.89 0.65 I 0.93 T t 0.73 0.52 0.83 0.74 0.55 0.82 0.75 0.35 0.82 1.18 0.60| 0.35 0.72 0.60 0.38 0.71 1 0.61| 0.17 0.70 1.00 0.49 0.23 0.61 0.48 0.24 0.59 0.49 0.08 0.57 PSD,F 0.85 0.311 0.09 0.41 0.30 0.10 0.40 I 0.30| 0.02 0.36 0.71 0.221 0.03 0.30 0.21 0.04 0.29 r o.2i 0.00 0.25 0.59 0.151 0.01 0.21 0.14 0.02 0.20 1 0.141 0.00 0.16 V ; d 0.091 0.00 0.12 0.08 0.00 0.12 1 0.08 0.00 0.09 0.43 0.031 0.00 0.04 0.03 0.00 0.04 1 0.02 0.00 0.03 0.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 E E, c. T3 Feed Discharge Overflowe Feed Discharge Overflowe Feed Discharge o •a tu v (cm/s) 1 c P 25.6 25.9 27.7 v (cm/s) 1 X p 3.21 3.29 1.45 'elocit^ p 22.8 23.1 23.2 p 5.96 6.04 5.98 Run ID Dymu052 Dymu061 Dymu062 185 Appendix C: Sample Calculation in Calibrating Conductivity Probe Appendix C: Sample Calculation in Calibrating Conductivity Probe In step 3 of section 3.3 of the thesis, the recorded output voltages in the pure liquid without particles for the two probes were measured to be V' 0 )L=5.09 volts, V'i,L=4.05 volts In step 4. the recorded output voltages in the solid-liquid suspension after particles are added and the bed is expanded were V 0 , L - S =5.08 volts, V i ; L . s =5.50 volts At Ua=1.64 cm/s, the voidage measured by the pressure transducer is 0.465. In step 5. the corresponding resistances calculated from Equation (3.3.2) are Probe #0: R'0,L=41.98 Q, RO,L-S =41.78 Q Probe #1: R'i ; l=26.78 Q, ^ ,^=51.09 0 In step 6. the modified resistance of pure liquid measured by probe #1 for the solid-liquid mixture system, as determined from Equation (3.3.4), is Ri,L=26.65 Q and the relative resistance for probe #1 is then Ri,iyRi>L-s=0.522 186 Appendix D: Calculation Curves for the Nine Conductivity Probes Appendix D: Calibration Curves for the Nine Conductivity Probes Figure D . l . Variation of the voidage with respect to the relative resistance of probe #1 187 Appendix D: Calculation Curves for the Nine Conductivity Probes 1.1 Relative resistance, (R4L/R4 L.s) Figure D.4. Variation of the voidage with respect to the relative resistance of probe #4 Appendix D: Calculation Curves for the Nine Conductivity Probes 1.1 1.0 w 0.8 4 e = - 6 . 6 2 1 ( R 5 L / R 5 L J 3 + 1 4 . 2 1 6 ( R 5 L / R 5 L . 3 ) 2 - 8 . 5 2 5 4 ( R 5 L / R 5 , S ) + 1 . 9 3 1.1 0.8 0.9 1.0 Relative resistance, (R 5 L /R 5 L S ) Figure D.5. Variation of the voidage with respect to the relative resistance of probe #5 1.1 1.0-0.9 4 M 0.8-| CD O) CO T2 0.7 o > 0.6 0.5 0.4 £ = - 7 . 3 1 ( R 6 L / R 6 L / + 1 5 . 9 7 ( R 6 L / R 6 L . 3 ) 2 -10.09(R 6 ] L /R 6 ] L .3) + 2.429 0.4 — i — 0.5 0.6 0.7 0.8 0.9 Relative resistance, ( R 6 L / R 6 L . S ) i 1.0 1.1 Figure D.6. Variation of the voidage with respect to the relative resistance of probe #6 189 Appendix D: Calculation Curves for the Nine Conductivity Probes 1.1 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Relative resistance, (R7L/R7 L.s) Figure D . 7 . Variation of the voidage with respect to the relative resistance of probe #7 i 1 1 1 1 1 1 1 1 1 r -0.5 0.6 0.7 0.8 0.9 1.0 Relative resistance, (R8L/R8 L.s) Figure D .8. Variation of the voidage with respect to the relative resistance of probe #8 190 Appendix D: Calculation Curves for the Nine Conductivity Probes 0.5 0.6 0.7 0.8 0.9 1.0 Relative resistance, (R9L/Rg L.s) Figure D .9 . Variation of the voidage with respect to the relative resistance of probe Appendix E: Cross-Sectional Area of Conical Column and its Gradient Appendix E : Cross-Sectional Area of Conical Column and its Gradient Figure C l shows a schematic of the conical section of the column connecting the cylindrical section and the expanded freeboard. D T and D B are the diameters at the top and bottom of this section, while h is the height of the conical section. D Z is the diameter of any horizontal section between the top and bottom; z is the vertical distance from the top surface to the section under consideration. Horizontal section Bottom Figure E . l . Schematic of the conical column From similar triangles, so that D Z - D B h - z D T - D B D z = P r ^ B ( h _ z ) + D B The area of the intersection plan is given by 4 Z 4 ( h - z ) + D B while the gradient of the area of the intersection plan is obtained by differentiation, yielding dAz dz 1 •7d D T - D B D T - D B ( h - z ) + D B 192 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model c c M A I N P R O G R A M TO SOLVE THE STEADY STATE H Y D R O D Y N A M I C M O D E L c c A R G U M E N T S c A C : A R R A Y OF THE CYL INDRICAL REGION c AT: A R R A Y OF THE TOP REGION c A M : A R R A Y OF M A S S R A T E OF PARTICLE c A M F : M A S S R A T E OF LIQUID c C: A R R A Y OF PARTICLE CONCNETRATION IN THE C O L U M N c CD: A R R A Y OF DISCHARGE PARTICLE CONCENTRATION c CDE: A R R A Y OF E X P E R I M E N T A L DISCHARGE PARTICLE c CONCENTRATION c CF: A R R A Y OF FEED PARTICLE CONCENTRATION c C F I : A R R A Y OF FEED PARTICLE FRACTION c CO: A R R A Y OF PARTICLE CONCENTRATION IN O V E R F L O W c COO: A R R A Y OF INITIAL GUESS OF PARTICLE CONCENTRAION IN c OVERFLOW c D: D IAMETER OF CYL INDRICAL REGION OF THE CLASSIFIER c DA: D IAMETER OF THE FLUDIZING PIPE c DI: DISPERISON COEFFICIENT c DIA: M A X I M U M DISPERSION COEFFICIENT c DII: M I N I M U M DISPERSION COEFFICIENT c DP: A R R A Y OF PARTICLE D IAMETER c DP50F: M E A N PARTICLE SIZE OF FEED c D P M A X : THE M A X I M U M PARTICLE SIZE c DPP: A R R A Y OF PARTICLE D IAMETER c DT:DIAMETER OF TOP REGION OF THE CLASSIFIER c DZ: DIFFERENCING OF HEIGHT c E: A R R A Y OF VOIDAGE IN THE C O L U M N c EDE: E X P E R I M E N T A L DISCHARGE VOIDAGE c EF: FEED VOIDAGE c FI: A R R A Y OF LIQUID A N D PARTICLE D R A G FORCE c F ID :ARRAY OF PARTICLE DISPERSION FORCE c G: G R A V I T Y A C C E L E R A T I O N c H: A R R A Y OF HEIGHT OF THE EIGHT ZONES OF THE CLASSIFIER c HT: T O T A L HEIGHT OF THE CLASSIFIER c M : A R R A Y OF DIFFERENCING POINT OF HEIGHT c M H : N U M B E R OF ZONES OF CLASSIFIER c MT: N U M B E R OF M E S H POINT OF THE HEIGHT c N: N U M B E R OF PARTICLE SPECIES c QA: LIQUID F L O W R A T E c QD: D ISCHARGE F L O W R A T E c QF: FEED FLOW R A T E c QO: O V E R F L O W R A T E c R: RADIUS OF CYL INDRICAL REGION c RO: RADIUS OF LIQUID FLUIDIZING PIPE c RT: RADIUS OF TOP REGION c RF: LIQUID DENSITY c RNF: INDEX OF ROSIN R A M M L E R PARTICLE SIZE DISTRIBUTION IN c FEED c RP: PARTICLE DENSITY 193 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model C C C C C C C C C C C C C C C C C T l : F INAL TIME THE C A L C U L A T I O N R E A C H E D TOL: T O L E R A N C E U A : LUQID FLUIDIZING V E L O C I T Y UD: D ISCHARGE V E L O C I T Y UF: FEED VELOCITY U U : LIQUID VISCOCITY V D : A R R A Y OF DISPERSION V E L C O C I T Y V : A R R A Y OF PARTICLE V E L C O C r T Y X T O L : DIFFERENCING OF THE DISPERSION COEFFICIENT Z: A R R A Y OF HEIGHT OF THE CLASSIFIER STEAD YIN: INPUT D A T A FILE OUTCV.DAT: OUTPUT FILE OUTEC.DTA: OUTPUT FILE OUTED.DAT: OUTPUT FILE OUTWT.DAT: OUTPUT FILE INTEGER N P A R A M E T E R (MH=8) DIMENSION DP(20),CF1(20) DIMENSION H(MH),M(MH),DZ(MH),DPP(20) DIMENSION CO0(20), CDE(20) COMMON/BLK0 /MHH,DZ,M,MT,HT COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLKF/RF,RP,G,DP,UU,PI C O M M O N / B L K Y / C O 0 , C D E , E D E C C INPUT P A R A M E T E R TO C A L C U L A T E THE DISPERSION COEFFICIENT C D A T A AKHD,AKEF,AI<y}90,AK0,AKREDC,AKREDP,AKAR,AKOT + 1.120554992,-0.73261928,0.149283113,0.217814339,0.468551131, + 0.491871891,-0.908425733,0.127162111,-0.052299796,-0.6328947/ C INPUT D A T A C M H H = M H OPEN(l,FELE="STE ADYIN.DAT") RE AD( 1, * )LL,LL0 READ(1,*) READ( 1 ,*)N,D,DT,D A,TOL R=D/2 R0=DA/2 RT=DT/2 READ(1,*) READ(1,*)(M(I),I=1,MH) READ(1,*) READ( 1, *)(H(I),I= 1 ,MH) READ(1,*) READ(1,*) UA,UF,UD,EF READ(1,*) READ(1,*) RP,RF,UU,G READ(1,*) READ(l , * )DI I ,DIA,XTOL READ(1,*) DO 5 K=1,N C 194 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model READ(1,*)DPP(K),DP(K),CF1(K),CDE(K) 5 CONTINUE IF(N. GT. 3 )THEN READ(1,*) READ(1,*)DP50F,RNF END IF READ(1,*) READ(1,* )EDE READ(1,*) RE AD( 1, *)(CO0(K),K= 1 ,N) CLOSE( l ) CF=1-EF PI=4.*ATAN(1.) AC=PI*R*R UA=UA*0.01 UD=UD*0.01 UF=UF*0.01 Q A = U A * A C QD=UD*AC QF=UF*AC QO=QF+QA-QD C C C A L C U L A T E THE DISPERSION COEFFICIENT FOR MULTI-SIZE S Y S T E M C IF(N.GT.3. AND.LLO.EQ. 1)THEN HL=H(4) DP90F=(LOG(0.1)/LOG(0.9))**(1/RNF) DHD=HL/D DEF=1-EF DDP90=DP90F DREDC=(UA*D*RF/UU) DREDP=(UA*DP50F*0.001*RF/UU) DAR=((RF*(RP-RF)*(DP50F*0.001)**3*G)/(UU**2)) DUF=(UF/UA) DUD=(UD/UA) DEF4=(EF-0.4) PE1=AK0*DHD**AKHD PE2=PE1*DEF**AKEF PE3=PE2*DDP90**AKD90 PE4=PE3 * D R E H * * A K R E D C PE5=PE4*DRED**AKREDP PE6=PE5*DAR**AKAR PE7=PE6*DUF**AKUF PE8=PE7*DUD**AKUD PE=PE8*DEF4**AKEF4 DIC=UA*HL/PE DII=DIC DIA=DIC XTOL=0 E N D IF MT=0 HT=0 DO 31 I=1,MH MT=MT+M(I) HT=HT+H(I) 31 CONTINUE 195 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model DO 10I=1,MH DZ(I)=HT/MT 10 CONTINUE C C C A L L SUBROUTINE F C N TO DO THE C A L C U L A T I O N C C A L L FCN(LL,DII ,DIA,XTOL,N,TOL) STOP E N D C C SUBROUTINE F C N TO BEGIN THE INTEGRATION A N D OUTPUT RESULTS C SUBROUTINE FCN(LL,DII ,DIA,XTOL,N,TOL) INTEGER M H R E A L XGUESS(20),X(20) DIMENSION Z(0:200000),DP(20),CF 1 (20),DCDZ0(20),DCDZ 1 (20) DIMENSION CO(20),CD(20),E2(0:200000),C2(20,0:200000) DIMENSION C1(20,0:200000),E 1(0:200000) DIMENSION H(8),M(8),DZ(8),CDE(20),dcdz2(20) DIMENSION CO0(20),C0(20) ,C(20,0:200000) DIMENSION COl(20),CDl(20) DIMENSION ZC0(20), U(0:200000) DIMENSION DCDZ(20,0:200000) DIMENSION DC1(20,0:200000),DC2(20,0:200000),AM(20,0:200000) DIMENSION CV(20,0:200000) C O M M O N / P L K D C / D C D Z 0 ,DCDZ1 COMMON/BLK0 /MH,DZ,M,MT,HT COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKD/QF,QA,QD,QO,EF,CF 1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKY/CO0,CDE,ede C O M M O N / B L K WY/Z ,E 1 ,C 1 ,DC 1 C O M M O N / B L T O L / T O L L TOLL=TOL C C SET THE INITIAL GRADIENT OF PARTICLE CONCENTRATION TO B E ZERO C DO 88 1=1,N DCDZ1(I)=0 88 CONTINUE XX=(DIA-DII)/XTOL M L = X X X Y = X X - M L IF( ABS(XY) . GT.0.5)ML=ML+1 Z(0)=0 DO 901 I=0,MT Z(I)=DZ(1)*I 901 CONTINUE C C SET THE OBJECTIVE FUNCTION TO A L A R G E V A L U E C F =1000000000 DI1=DII C 196 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model C BEGINE THE INTEGRATION C DO 110IJ=0,ML DI=DII+IJ*XTOL ZZ=0 C C C A L L SUBROUTINE ROOT FIND THE DISCHARGE PARTICLE CONCENTRATION C C A L L ROOT (CO0,N,X,XGUESS) C C C A L C U L A T E THE OBJECTIVE FUNCTION C ZZ=0 D 0 32I=N,1,-1 ZZ=ZZ+(CD(I)-(CDE(I)*(1-EDE)))**2 32 CONTINUE F1=ZZ WRITE(6,*)IJ WRITE(6,10)DLF 1,DI 1,F 10 FORMAT(1X,10E12.4) C C C H E C K WHETHER THE OBJECTIVE IS LESS T H A N PREVIOUS ONE C EF Y E S STORE THE C A L C U L A T E D V A L U E C EF(F1.LT.F)THEN DI1=DI DO 66I=1,N C01(I)=CO(I) CD1(I)=CD(I) DCDZ2(I)=DCDZ0(I) 66 CONTINUE E01=EO ED1=ED DO 77 I=1,MT E2(I)=E1(I) DO 99K=1,N C2(K,I)=C1(K,I) DCDZ(K,I)-DC1(K,I) 99 CONTINUE 77 CONTINUE F=F1 ENDIF DO 79K=1,N CO0(K)=Cl(K,MT) 79 CONTINUE 110 CONTINUE C C EF LL= 1, CONTINUE THE INTEGRATION TO THE B O T T O M C EF(LL.EQ. 1)THEN DO 260I=1,N C0(I)=C2(I,M(1)+M(2)) DCDZ0(I)=DCDZ2(I) 260 CONTINUE 1 9 7 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model MM=M(1)+M(2) DI=DI1 DO 20 1=2,1,-1 CALLPDE(C0,DC2,DCDZ0,DCDZ1,H(I),N,C,M(I),I) M1=MM-1 MM=MM-M(I) DO 30 J=M1,MM,-1 E E 1=0 D 0 35K=1,N C2(K,J)=C(K,M1-J+1) DCDZ(K, J)=DC2(K,M 1-J+l) EE1=EE1+C2(K,J) 35 CONTINUE E2(J)=1-EE1 30 CONTINUE DO 25 K = 1,N C0(K)=C(K,M(i)) 25 CONTINUE 20 CONTINUE E N D IF C C C A L C U L A T E THE PARTICLE F L U X FOR DIFFERENT ZONES C C TOP ZONE I C AC=PI*R*R AT=PI*RT*RT DO 600 I=MT,MT-M(8),-1 T=HT-Z(I) U(I)=-QO*EO 1/E2(I)/AT DO 601K=1,N AM(K,I)=0 CV(K,I)=-QO*C01(K)/AT 601 CONTINUE 600 CONTINUE C C FEED Z O N E C RFE=H(7)/2 AF=PI*RFE*RFE ZF=HT-H(8) DO 610 I=MT-M(8)-1,MT-M(8)-M(7),-1 T=ZF-Z(I) IF(T.LT.0)T=0 IF(T.GT.H(7))T=H(7) BL=2*(T<FE**2-(RFE-T)**2)**0.5 ZZ=(RFE-T)/RFE Zl=2*ACOS(ZZ) Z2=Z l /2 /PI*PI*RFE*RFE Z3=2*RFE*SIN(Zl/2) ACC=Z2-(Z3 *(RFE-T))/2 APH=ACC/AF U(I)=(-QO*E01+APH*QF*EF)/E2(I)/AT D 0 611K=1,N AM(K,I)=QF*CF1(K)*(1-EF)*RP*BL/AT/AF 198 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model CV(K,I)=(-QO*CO 1 (K)+APH*QF*CF 1 (K)*( 1 -EF))/AT 611 CONTINUE 610 CONTINUE C C TOP Z O N E II C DO 620 I=MT-M(8)-M(7)-1,MT-M(8)-M(7)-M(6),-1 U(I)=(-QA+QD*ED 1)/E2(I)/AT DO 621K=1,N AM(K,I)=0 CV(K,I)=QD*CD1(K)/AT 621 CONTINUE 620 CONTINUE C C DIVERGING ZONE C ZF=HT-H(8)-H(7)-H(6) DO 630 I=MT-M(8)-M(7)-M(6)-1 + ,MT-M(8)-M(7)-M(6)-M(5),-1 RB=R DABDT=0 T=ZF-Z(I) 1P(TIT.NE.R)THEN H0=H(5)*R/(RT-R) RB=(H(5)+H0-T)/(H(5)+H0)*RT DABDT=2*PI*RB*(-RT/(H(5)+H0)) ENDIF AB=PI*RB*RB U(I)=(-QA+QD*ED 1)/E2(I)/AB DO 631 K=1,N AM(K, I )=-CDl( I ) *QD*RP/AB/AB*DABDT/2 CV(K,I)=QD*CD 1(K)/AB 631 CONTINUE 630 CONTINUE C C C O L U M N Z O N E C DO 640 I=MT-M(8)-M(7)-M(6)-M(5)-1 + ,MT-M(8)-M(7)-M(6)-M(5)-M(4),-1 U(I)=(-QA+QD*ED 1)/E2(I)/AC DO 641K=1,N AM(K,I)=0 C V(K,I)=QD* CD 1 (K)/AC 641 CONTINUE 640 CONTINUE C C D ISCHARGE ZONE C D O 5 0 0 K = l , N ZC0(K)=0 500 CONTINUE ZE0=0 RD=H(3)/2 AD=PI*RD*RD ZF= H(l)+H(2)+H(3) 199 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model DO 650 I=MT-M(8)-M(7)-M(6)-M(5)-M(4)-1 + ,MT-M(8)-M(7)-M(6)-M(5)-M(4)-M(3),-1 T=ZF-Z(I) IF(T.LT.0)T=0 IF(T.GT.H(3))T=H(3) AL=2*(TO**2-(T<D-T)**2)**0.5 ZE0=ZE0+E1(I)*AL*DZ(3) U(I)=(-QA+QD*ED 1 -QD/AD*ZE0)/E2(I)/AC DO 651 K=1,N AM(K,I)=-QD*C2(K,I )*RP*AL/AC/AD ZC0(K)=ZC0(K)+C1(K,I)*AL*DZ(3) CV(K,I)=(QD*CD1(K)-QD/AD*ZC0(K))/AC 651 CONTINUE 650 CONTINUE C C B O T T O M Z O N E I C IF(11.EQ.1)THEN DO 660 I=MT-M(8)-M(7)-M(6)-M(5)-M(4)-M(3)-1 + ,MT-M(8)-M(7)-M(6)-M(5)-M(4)-M(3)-M(2),-1 U(I)=-QA/E2(I)/AC DO 661K=1,N CV(K,I)=0 661 CONTINUE 660 CONTINUE C C B O T T O M ZONE II C ZF=H(1) DO 670 I=MT-M(8)-M(7)-M(6)-M(5)-M(4)-M(3)-M(2)-1 + ,MT-M(8)-M(7)-M(6)-M(5)-M(4)-M(3)-M(2)-M(1),-1 T=ZF-Z(I) IF(T.LT.0)T=0 IF(T.GT.H(1))T=H(1) H0=H(1)*R0/(R-R0) RB=(H(1)+H0-T)/(H(1)+H0)*R AB=PI*RB*RB/2 U(I)=-QA/E2(I)/AB D 0 671K=1,N CV(K,I)=0 671 CONTINUE 670 CONTINUE ENDIF C C OUTPUT THE RESULTS C OPEN(3,FILE-OUTED.DAT' ) DO 220K=1,N WRITE(3,60)DP(K),CD 1(K) 220 CONTINUE M M = M T OPEN(2,FILE='OUTWT.DAT') OPEN(21, FILE='OUTCV.DAT , ) OPEN(41, file='OUTEC.dat') 200 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model WPJTE(2,*)'DI,F',DI1 ,F WRITE(2,*)'Ua=',QA/AC, 'Uf=', QF /AC WRITE(2,*) ,Ud= ,,QD/AC,'ef= ,,EF WRITE(2, *)'ED, CD',ED 1, (CD 1 (K)/(l -ed 1 ),K= 1 ,N) WRITE(41,60)H(1)+H(2),ED 1,(CD 1(K),K= 1,N) M M = M T EF(LL.EQ. 1)11=0 EF(LL.EQ.0)II=M(1)+M(2) DO 40 I=MM,II,-100 IF(E2(I).NE. 1) WRITE(2,60)Z(I),E2(I),(C2(K,I)/(1-E2(I)),K=1,N) EF(E2(I).EQ. 1) WRITE(2,60)Z(I),E2(I),(0,K=1,N) WRITE(41,60)Z(I),E2(I),(C2(K,I),K= 1 ,N) WRITE(21,60)Z(I),U(I),(CV(K,I),K=1,N) 60 F 0 R M A T ( 1 X , F10.4,20(1X, E14.7)) 80 FORMAT(1X,20(1X,E12.5)) 40 CONTINUE 48 CLOSE(2) CLOSE(3) CLOSE(41) CLOSE(21) R E T U R N E N D C C SUBROUTINE ROOT TO SOLVE THE NONLINEAR EQUATIONS C USING SUBROUTINE NEQNF C SUBROUTINE ROOT(CO0,N,X,XGUESS) DIMENSION CO0(N) INTEGER ITMAX, N R E A L E R R R E L INTEGER K R E A L F N O R M , X(N), XGUESS(N) E X T E R N A L F C N N , NEQNF, U M A C H C O M M O N / B L T O L / T O L L DO 10K=1,N XGUESS(K)=CO0(k) 10 CONTINUE E R R R E L = toll I T M A X = 50000 C A L L NEQNF (FCNN, ERRREL , N, ITMAX, XGUESS, X , FNORM) R E T U R N END C C SUBROUTINE ROOT TO SOLVE THE NONLINEAR EQUATIONS C USING SUBROUTINE NEQNJ C SUBROUTINE ROOT2(DCDZ,N,X,XGUESS) DIMENSION DCDZ(N) INTEGER ITMAX, N R E A L E R R R E L INTEGER K R E A L F N O R M , X(N), XGUESS(N) E X T E R N A L FCNN1, LSJAC, NEQNJ, U M A C H C O M M O N / B L T O L / T O L L D O 1 0 K = l , N 201 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model XGUESS(K)=DCDZ(k) 10 CONTINUE E R R R E L = toll I T M A X = 50000 C A L L NEQNJ (FCNN1, LSJAC, ERRREL , N, ITMAX, XGUESS, X , FNORM) R E T U R N E N D C C SUBROUTINE FCNN1 TO PROVIDE THE NONLINEAR EQUATIONS C SUBROUTINE FCNN1 ( X , F,N) INTEGER N R E A L X(N), F(N) DIMENSION FI(20),FID(20) DIMENSION DP(20),CF1(20),V(20) ,VD(20) DIMENSION C(20),AM(20) ,CD(20),CO(20) COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKK7C,E,AM,AMF,UL,V ,F I ,SUMFI C O M M O N / B L L L / L DO 10 I=1,N VD(I)=0 IF(C(I).NE.0.)THEN DD=DI VD(I)=-(-DD/C(I)*X(I)) UD=0 ENDIF 10 CONTINUE C C C A L C U L A T E THE DISPERSION FORCE C C A L L FORCE(N,C,RF,DP,UU,UD,VD,FID,SUMFD) SUMX=0 DO 15 I=1,N SUMX=SUMX+X(I) 15 CONTINUE C C EQUATIONS TO B E SOLVED C DO 20I=1,N F(I)=C(I)*E*(RP-RF)*G+C(I)*SUMFI+C(I)*SUMFD+ + E*C(I)*FI(I)+E*C(I)*FID(I) + +2*C(I)*UL*AMF-2*E*V(I)*AM(I) + +E*RP*V(I )**2*Xa)+C(I)*RF*UL*UL*SUMX IF(C(I).EQ.0)F(I)=X(I) 20 CONTINUE R E T U R N E N D C C SUBROUTINE F C N N TO PROVIDE THE NONLINEAR EQUATION C SUBROUTINE F C N N (X, F, N) DIMENSION H(8),DZ(8),M(8),Cl(20,0:200000),CD(20),CO(20),C0(20) Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model DIMENSION C(20,0:200000),Z(0:200000),ZCD(20),CF1(20) DIMENSION DP(20),DCDZ0(20) ,DCDZ1(20),DCDZ(20,0:200000) DIMENSION DC1(20,0:200000) ,E1(0:200000) INTEGER N R E A L X(N), F(N) COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLK0 /MH,DZ,M,MT,HT COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLK3 /MD,ZED,ZCD COMMON/BLKC/ED,CD,EO,CO,DI C O M M O N / B L K W Y / Z , E 1 ,C 1 ,DC 1 COMMON/PLKDC/DCDZ0 ,DCDZ 1 C C C A L C U L A T E THE O V E R L O W A N D DISCHARGE CONCENTRATION C ZZ=0 ZZD=0 DO 15 K=1,N DCDZO(K)=DCDZ 1 (K) CO(K)=X(K) CD(K)=(QF*CF 1 (K)*( 1 -EF)-QO*CO(K))/QD IF(CD(K).LE.O)THEN CD(K)=0 CO(K)=QF*CFl(K)*( l -EF)/QO END IF C0(K)=CO(K) C1(K,MT)=C0(K) ZZ=ZZ+C1(K,MT) ZZD=ZZD+CD(K) 15 CONTINUE EO=l -ZZ ED=1-ZZD E1(MT)=1-ZZ C C C A L L SUBROUTINE PDE TO BEGIN INTEGRATION C M M = M T DO 20 I=MH,3,-1 C A L L PDE(C0,DCDZ,DCDZ0,DCDZ 1 ,H(I),N,C,M(I),1) M1=MM-1 MM=MM-M(I) DO 30 J=M1,MM,-1 EE=0 DO 35 K=1,N C1(K,J)=C(K,M1-J+1) D C 1 (K, J)=DCDZ(K,M 1-J+1) EE=EE+C1(K,J) 35 CONTINUE E1(J)=1-EE 30 CONTINUE DO 25 K=1,N C0(K)=C(K,M(i)) 25 CONTINUE 20 CONTINUE C 203 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model C EQUATION TO B E SOLVED C DO 55 I=1,N F(I)=CD(I)-ZCD(I)/(PI*(H(3)/2)**2) 55 CONTINUE R E T U R N E N D C C SUBROUTINE PDE TO SOLVE THE ORDINARY DIFFERENTIAL EQUALTON C SUBROUTINE PDE(C0,DCDZ,DCDZ0,DCDZ1,AL,N,C,M,L) INTEGER M X P A R M , N P A R A M E T E R (MXPARM=50) INTEGER M A B S E , M B D F , M S O L V E P A R A M E T E R (MABSE=1, MBDF=2, MSOLVE=2) INTEGER EDO, ISTEP R E A L A ( l , 1), P A R A M ( M X P A R M ) , T, TEND, TOL, Y(20) R E A L YPRIME(20) DIMENSION C0(20),C(20,0:200000),ZCD(20),YP(20),YP1(20) DIMENSION DCDZ0(N),DCDZ1(N),DCDZ(20,0:200000) E X T E R N A L IVPAG, S SET E X T E R N A L FCN1,FCN2, FCN3,FCN4,FCN5,FCN6,FCN7,FCN8,FCNJ COMMON/BLK3 /MD,ZED,ZCD COMMON/BLK31/ ISTEP,YP,YP1 C O M M O N / B L L L / L L C O M M O N / B L T O L / T O L L M D = M LL=L N1=N C C SET INITIAL CONDITIONS C T = 0.0 ZED=0 DO 20 1=1,N Y(I) =C0(I) ZCD(I)=0 YP(I)=DCDZ0(I) 20 CONTINUE ZCDD=0 TOL = TOLL C A L L SSET ( M X P A R M , 0.0, P A R A M , 1) C C SELECT A B S O L U T E ERROR CONTROL C PARAM(IO) = M A B S E C C SELECT BDF M E T H O D C PARAM(12) = M B D F C C Select chord method and C a divided difference Jacobian. C PARAM(13) = M S O L V E 204 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model JDO= 1 ISTEP = 0 10 CONTINUE ISTEP = ISTEP + 1 TEND = AL /M* ISTEP P A R A M ( 1 )=0.001 * AB S(TEND-O) PARAM(4)=500000 IP(L.EQ.3)THEN RD=AL/2. AAL=0 IP(TXT.AL)AAL=2*(RD**2-(RD-T)**2)**0.5 Z=0 DO 14I=1,N ZCD(I)=ZCD(I)+Y(I)*AAL*AL/M Z=Z+Ya) 14 CONTINUE EE=1-Z Z E D = Z E D + E E * A A L * A L / M END IF DO 16 I=1,N YP1(I)=YP(I) 16 CONTINUE C C C A L L I V P A G TO DO THE INTEGRATION FOR DIFFERENT ZONES C L=8: TOP ZONE I; L=7: FEED ZONE; L=6 TOP ZONE II C L=5: D IVERGING ZONE; L=4: C O L U M N ZONE; L=3: D ISCHARGE Z O N E C L=2: B O T T O M ZONE I; L=3: B O T T O M ZONE II C F ( L . E Q . l ) + C A L L I V P A G (IDO, N, FCN1, FCNJ, A , T, TEND, TOL, P A R A M , Y) IF (L.EQ.2) + C A L L I V P A G (IDO, N, FCN2,FCNJ, A, T, TEND, TOL, P A R A M , Y ) IF (L.EQ.3) + C A L L IVPAG (IDO, N, FCN3, FCNJ, A, T, TEND, TOL, P A R A M , Y ) IF (L.EQ.4) + C A L L I V P A G (IDO, N, FCN4, FCNJ, A, T, TEND, TOL, P A R A M , Y) IF (L.EQ.5) + C A L L I V P A G (IDO, N, FCN5, FCNJ, A , T, TEND, TOL, P A R A M , Y ) IF (L.EQ.6) + C A L L IVPAG (IDO, N, FCN6, FCNJ, A , T, TEND, TOL, P A R A M , Y ) IF (L.EQ.7) + C A L L IVPAG (IDO, N, FCN7, FCNJ, A, T, TEND, TOL, P A R A M , Y) IF (L.EQ.8) + C A L L I V P A G (IDO, N, FCN8, FCNJ, A , T, TEND, TOL, P A R A M , Y ) C C STORE THE RESULTS FOR E V E R Y STEP A N D RESET THE INITIAL CONDITION C IF (ISTEP .LE. M) T H E N N2=N DO 102 K=1,N IF(Y (K).LT.0)THEN Y(K)=0 YP(K)=0 N2=N2-1 ENDIF 205 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model IF(N.EQ.N1)THEN C(K,ISTEP)=Y(K) DCDZ(K,ISTEP)=YP(K) E L S E KK=N1-N DO 879K1=1,KK C(K1,ISTEP)=C(K1,ISTEP-1) • DCDZ(K1,ISTEP)=0 879 CONTINUE C(K+KK,ISTEP)=Y(K) DCDZ(K+KK,ISTEP)=YP(K) E N D IF IF(ISTEP.EQ. 1. AND.L.EQ. 8)DCDZ 1 (K)=YP(K) 102 CONTINUE KK=N-N2 N=N2 DO 888K1=1,N Y(K1)=Y(K1+KK) YP(K1)=YP(K1+KK) 888 CONTINUE IF(N.EQ.0)GOTO 220 IF (ISTEP .EQ. M) IDO = 3 G O T O 10 E N D IF 220 DO 210 I=1,N DCDZ0(I)=YP(I) 210 CONTINUE N=N1 R E T U R N END C C SUBROUTINE FCNJ TO COMPUTE THE JACOBIAN. C THIS SUBROUTINE IS NOT C A L L E D IN THIS P R O G R A M C SUBROUTINE FCNJ (N, T, Y , DYPDY) INTEGER N R E A L T, Y(N), DYPDY(N,* ) R E T U R N END C C SUBROUTINE F C N 1 TO FCN8 TO E V A L U A T I O N FUNCTION FOR DIFFERENT ZONES C C B O T T O M Z O N E II C SUBROUTINE FCN1 (N, T, Y , YPRIME) INTEGER N R E A L T, Y(N), YPPJME(N),X(20),XGUESS(20) DIMENSION DP(20),CF1(20),V(20) DIMENSION C(20),H(8) ,AM(20),FI(20),DCDZ(20) DIMENSION YP(20),YP1(20) COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLK31/ ISTEP ,YP,YP1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKX/C ,E ,AM,AMF,UL ,V ,F I ,SUMFI 206 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model C A L L CVU(E,C,N,Y) AC=PI*R*R H0=H(1)*R0/(R-R0) RB=(H(1)+H0-T)/(H(1)+H0)*R AB=PI*RB*RB DABDT=2*PI*RB*(-R/(H(1)+H0)) UL=(-QA)/E/PI/RB/RB DO 15 I=1,N V(I)=0 AM(I)=0 DCDZ(I)=YP1(I) EF(C(I).EQ.O)DCDZ(I)=0 15 CONTINUE AMF=-( -QA)*RF/AB/AB*DABDT/2 C C C A L C U L A T E D R A G FORCE C C A L L FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) C C C A L C U L A T E THE DISPERSION FORCE C C A L L ROOT2(DCDZ,N,X,XGUESS) DO 20I=1,N YPRIME(I)=X(I) YP(I)=X(I) 20 CONTINUE R E T U R N E N D C C B O T T O M ZONE I C SUBROUTINE FCN2 (N, T, Y , YPPJME) INTEGER N R E A L T, Y(N), YPPJME(N),X(20),XGUESS(20) D IMENSION DP(20),CF1(20),V(20),CO(20) DIMENSION C(20),H(8) ,AM(20),FI(20) ,CD(20),DCDZ(20) DIMENSION YP(20),YP1(20) COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLK31/ ISTEP ,YP,YP1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKK/C ,E ,AM,AMF,UL ,V ,F I ,SUMFI C A L L CVU(E,C,N,Y) AC=PI*R*R UL=-QA/E/PI/R/R DO 15 I=1,N V(I)=0 AM(I)=0 DCDZ(I)=YP1(I) 15 CONTINUE AMF=0 C C C A L C U L A T E D R A G FORCE C C A L L FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) 2 0 7 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model C C C A L C U L A T E THE DISPERSION FORCE C C A L L ROOT2(DCDZ,N,X,XGUESS) DO 20I=1,N YPRIME(I)=X(I) YP(I)=X(I) 20 CONTINUE R E T U R N E N D C C D ISCHARGE ZONE C SUBROUTINE FCN3 (N, T, Y , YPRIME) INTEGER N R E A L T, Y(N), YPRIME(N),X(20),XGUESS(20) DIMENSION YP(20),YP1(20) DIMENSION DP(20),CF1(20),V(20),CO(20) DIMENSION C(20),H(8) ,AM(20),FI(20) ,CD(20),ZCD(20),DCDZ(20) COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKD/QF,QA,QD,QO,EF,CF 1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKK/C ,E ,AM,AMF,UL ,V ,F I ,SUMFI COMMON/BLK3 /MD,ZED,ZCD COMMON/BLK31/ ISTEP,YP,YP1 L=ISTEP ZE=0 ZC=0 C A L L CVU(E,C,N,Y) AC=PI*R*R RD=H(3)/2. AL=0 IF (TXT.H(3))AL=2*(RD**2-(RD-T)**2)**0.5 AM1=-QD*RP*AL/(PI*R*R)/(PI*RD*RD) AMF= -QD*E*RF*AL/ (PI*R*R) / (PI*RD*RD) DO 15 I=1,N AM(I)=0 EF(C(I).NE.0)AM(I)=AM1*C(I) 15 CONTINUE DZZ=T-(L-1)*H(3)/MD ZE=ZED+E*AL*DZZ UL=(-QA+QD*(ED-ZE/PI/RD/RD))/AC/E DO 33 I=1,N ZC=ZCD(I)+C(I)*AL*DZZ V(I)=0 IF(C(I).NE.0)V(I)=(QD*(CD(I)-ZC/PI/RD/RD))/AC/C(1) DCDZ(I)=YP1(I) 33 CONTINUE C C C A L C U L A T E D R A G FORCE C C A L L FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) C C C A L C U L A T E THE DISPERSION FORCE 208 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model C DO 38 I=1,N X(I)=0 38 CONTINUE CALL ROOT2(DCDZ,N,X,XGUESS) DO 20I=LN YPRIME(I)=X(I) YP(I)=YPRTME(I) 20 CONTINUE RETURN END C C COLUMN ZONE C SUBROUTINE FCN4 (N, T, Y, YPRIME) INTEGER N REAL T, Y(N), YPRIME(N),X(20),XGUESS(20) DIMENSION DP(20),CF1(20),V(20),CO(20) DIMENSION C(20),H(8) ,AM(20),FI(20) ,CD(20),DCDZ(20) DIMENSION YP(20),YP1(20) COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKD/QF,QA,QD,QO,EF,CF 1 COMMON/BLK31/ISTEP ,YP,YP1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R,RT,R0,H COMMON/BLI<aC/C,E,AM,AMF,UL>V,FI,SUMFI CALL CVU(E,C,N,Y) AC=PI*R*R , UL=(-QA+QD*ED)/E/PI/R/R DO 15 I=1,N V(I)=0 IF(C(I).NE.O)V(I)=QD*CD(I)/C(I)/PI/R/R AM(I)=0 DCDZ(I)=YP1(I) 15 CONTINUE AMF=0 C C CALCULATE DRAG FORCE C CALL FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) C C CALCULATE THE DISPERSION FORCE C CALL ROOT2(DCDZ,N,X,XGUESS) DO20I=l,N YPRIME(I)=X(I) YP(I)=X(I) 20 CONTINUE RETURN END C C DIVERGING ZONE C SUBROUTINE FCN5 (N, T, Y, YPRIME) INTEGER N REAL T, Y(N), YPPJME(N),X(20),XGUESS(20) 209 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model DIMENSION DP(20),CF1(20),V(20),CO(20) DIMENSION C(20),H(8) ,AM(20),FI(20) ,CD(20),DCDZ(20) DIMENSION YP(20),YP1(20) COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLK31/ ISTEP ,YP ,YP1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKK/C ,E ,AM,AMF,UL ,V ,F I ,SUMFI C A L L CVU(E,C,N,Y) AC=PI*RT*RT RB=R DABDT=0 IF(RT.NE.R)THEN H0=H(5)*R/(RT-R) RB=(H(5)+H0-T)/(H(5)+H0)*RT DABDT=2*PI*RB*(-RT/(H(5)+H0)) END IF AB=PI*RB*RB UL=(-QA+QD*ED)/E/PI/RB/RB DO 15 I=1,N V(I)=0 IF(C(I).NE.O)V(I)=QD*CD(I)/C(I)/PI/RB/RB AM(I)=-CD(I)*QD*RP/AB/AB*DABDT/2 DCDZ(I)=YP1(I) 15 CONTINUE AMF=-( -QA+QD*ED)*RF/AB/AB*DABDT C C C A L C U L A T E D R A G FORCE C C A L L FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) C C C A L C U L A T E THE DISPERSION FORCE C C A L L ROOT2(DCDZ,N,X,XGUESS) DO 20 I=1,N YPRTME(I)=X(I) YP(I)=X(I) 20 CONTINUE R E T U R N END C C TOP Z O N E II C SUBROUTINE FCN6 (N, T, Y , YPRIME) INTEGER N R E A L T, Y(N), YPPJME(N),X(20),XGUESS(20) DIMENSION DP(20),CF 1 (20), V(20),CO(20) DIMENSION C(20),H(8) ,AM(20),FI(20) ,CD(20),DCDZ(20) DIMENSION YP(20),YP1(20) COMMON/BLKC/ED,CD,EO,CO,DI C O M M O N / B L K D / Q F , Q A,QD,QO,EF,CF 1 COMMON/BLK31/ ISTEP ,YP,YP1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKK/C ,E ,AM,AMF,UL ,V ,F I ,SUMFI 2 1 0 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model C A L L CVU(E,C,N,Y) AC=PI*RT*RT UL=(-QA+QD*ED)/E/PI/RT/RT DO 15 I=1,N V(I)=0 EF(C(I).NE.O)V(I)=QD*CD(I)/C(I)/PI/RT/RT AM(I)=0 DCDZ(I)=YP1(I) 15 CONTINUE AMF=0 C C C A L C U L A T E D R A G FORCE C C A L L FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) C C C A L C U L A T E THE DISPERSION FORCE C C A L L ROOT2(DCDZ,N,X,XGUESS) DO 20 I=1,N YPPJME(I)=X(I) YP(I)=X(D 20 CONTINUE R E T U R N E N D C C FEED Z O N E C SUBROUTINE FCN7 (N, T, Y , YPRIME) INTEGER N R E A L T, Y(N), YPRIME(N),X(20),XGUESS(20) DIMENSION DP(20),CF 1 (20), V(20),CO(20) DIMENSION C(20),H(8) ,AM(20),FI(20),CD(20),DCDZ(20) DIMENSION YP(20),YP1(20) COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLK31/ ISTEP ,YP,YP1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKX/C ,E ,AM,AMF,UL ,V ,F I ,SUMFI C A L L CVU(E,C,N,Y) AC=PI*RT*RT Rfe=H(7)/2. AL=0 IF(TXT.H(7))AL=2*(IAfe**2-(Rfe-T)**2)**0.5 AMl=QF*RP*AL/(PI*RT*RT)/(PI*Rfe*Rfe) AMF= QF*EF*RF*AL/(PI*RT*RT)/(PI*Rfe*Rfe) DO 15 I=1,N AM(I)=0 IF(C(I).NE.0)AM(I)=AM1*CF1(I)*(1-EF) 15 CONTINUE Z=(RFE-T)/RFE EF(Z.LT.-1.0)Z=-1 Zl=2*ACOS(Z) Z2=Zl /2/PI*PI*RFE*RFE Z3=2*RFE*SIN(Zl/2) ACC=Z2-(Z3 *(RFE-T))/2 211 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model APH=ACC/PI /RFE/RFE UL=(-QO*EO+APH*QF*EF)/E/PI/RT/RT DO 33 I=1,N V(I)=0 IF(C(I).NE.O)V(I)=(-QO*CO(I)+APH*QF*CF1(I)*(1-EF))/C(I)/PI/RT/RT DCDZ(I)=YP1(I) 33 CONTINUE C C C A L C U L A T E D R A G FORCE C C A L L FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) C C C A L C U L A T E THE DISPERSION FORCE C C A L L ROOT2(DCDZ,N,X,XGUESS) D O 2 0 I = l , N YPRJME(I)=X(I) YP(I)=X(I) 20 CONTINUE R E T U R N E N D C C TOP ZONE I C SUBROUTINE FCN8 (N, T, Y , YPRIME) INTEGER N R E A L T, Y(N), YPRIME(N),X(20),XGUESS(20) DIMENSION DP(20),CF1(20),V(20),CO(20) DIMENSION C(20),H(8) ,AM(20),FI(20),CD(20),DCDZ(20) DIMENSION YP(20),YP1(20) COMMON/BLKC/ED,CD,EO,CO,DI COMMON/BLKD/QF,QA,QD,QO,EF ,CF l COMMON/BLK31/ ISTEP ,YP,YP1 COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKA/R ,RT ,R0 ,H COMMON/BLKK/C ,E ,AM,AMF,UL ,V ,F I ,SUMFI C A L L CVU(E,C,N,Y) AC=PI*RT*RT UL=-QO*EO/E/AC DO 15 1=1 ,N V(I)=0 LF(C(I).NE.0)V(I)=-QO*CO(I)/C(I)/AC AM(I)=0 DCDZ(I)=YP1(I) 15 CONTINUE AMF=0 C C C A L C U L A T E D R A G FORCE C C A L L FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) C C C A L C U L A T E THE DISPERSION FORCE C C A L L ROOT2(DCDZ,N,X,XGUESS) DO 20I=1,N YPRIME(I)=X(I) 212 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model YP(I)=X(I) 20 CONTINUE R E T U R N END C C SUBROUTINE C V U TO TRANSFER Y TO C C SUBROUTINE CVU(E,C,N,Y) R E A L Y(N) DIMENSION C(20) Z=0. DO 5 I=1,N IF(Y(I).LE.0)Y(I)=0 Z=Z+Y(I) 5 CONTINUE E=l -Z DO501 I= l ,N IF(E.LT.0.4)THEN Ya)=Y(I)/(l-E)*0.6 E=0.4 END IF IF (E.GT. l )THEN Y(I)=0 E=l ENDIF 501 CONTINUE DO 10 I=1,N C(I)=Y(I) 10 CONTINUE R E T U R N E N D C C SUBROUTINE FORCE TO C A L C U L A T E THE D R A G OR DISPERSION F O R C E C SUBROUTINE FORCE(N,C,RF,DP,UU,UL,V,FI,SUMFI) DIMENSION C(N),V(N),DP(N),FI(N) Z=0 SUMFI=0 DO 10 I=1,N Z=Z+C(I) FI(I)=0 10 CONTINUE E=l -Z Z1=0 DO 20 I=1,N IF(C(I).EQ.0)THEN Fia)=0 GOTO 20 ENDIF RE=DP(I)*ABS(UL-V(I))*RF*EAJU IF(RE.LT.0.0001)THEN FI(I)=0 GOTO 20 ENDIF C A L L C A L C U L A T E C D ( R E , C D 1) Y=10 213 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model B=2.7-0.65*EXP(-(1.5-LOG(RE)/LOG(Y))**2/2) FE=E**(-B) na)=374.*051*E*ABS(UI^V(D)*Ca)*RF/DP(I)*FE*(lJL-V©) 22 Z1=Z1+FI(I) 20 CONTINUE SUMFI=Z1 R E T U R N E N D C C SUBROUTINE C A L C U L A T E D TO C A L C U L A T E THE D R A G COEFFICIENT C SUBROUTINE CALCULATECD(RE,CD1) Y=10 W=LOG(RE)/LOG(Y) IF (RE.LE.0.01)CDl=3./16.+24./RE IF(RE.GT.0.01.AND.RE.LE.20)CD1=24./RE*(1+0.1315*RE**(0.82-0.05*W)) IF(RE.GT.20.AND.RE.LE.260)CD1=24./RE*(1+0.1935*RE**0.6305) IF(RE.GT.260.AND.REXE.1.5E3)CD1=10.**(1.6435-1.1242*W+0.1558*W*W) IF(RE.GT.1.5E3.AND.RE.LE.1.2E4)CD1=10.**(-2.4571+2.5558*W + -0.9295*W*W+0.1049*W*W*W) IF(RE.GT.1.2E4.AND.RE.LE.4.4E4)CD1=10.**(-1.9181+0.6370*W + -0.0636*W*W) IF(RE.GT.4.4E4.AND.RE.LE.3.38E5)CDl=10.**(-4.3390+1.5809*W + -0.1546*W*W) IF(RE.GT.3.38E5.AND.RE.LE.4.E5)CD1=29.78-5.3*W IF(RE.GT.4.E5.AND.RE.LE.l.E6)CDl=-0.49+0.1*W IF(RE. GT. 1 .E6)CD 1=0.19-8.E4/RE R E T U R N END C C SUBROUTINE LSJAC TO E V A L U A T E THE JACOBIAN C SUBROUTINE LSJAC(N, X , FJAC) INTEGER N R E A L X(N), FJAC(N,N) DIMENSION DF(20),CD(20),CO(20),DP(20),C(20) DIMENSION V(20),FI(20),AM(20) COMMON/BLKF/RF,RP,G,DP,UU,PI COMMON/BLKI^C ,E ,AHANIF ,UL ,V ,F I ,SUMFI C O M M O N / B L L L / L COMMON/BLKC/ED,CD,EO,CO,DI DO 40 I=1,N IF(C(I).EQ.0.)THEN DF(I)=0 GOTO 40 ENDIF DD=DI VD=-DD/C(I)*X(I) RE=DP( l ) *ABS(VD)*RF*E/UU IF(RE.EQ.0)THEN DF(I)=0 GOTO 40 ENDIF C A L L CALCULATECD(RE,CDD) DVD=-DD/C(I) DRE=DP(I )*RF*E/UU*DVD 214 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model IF(VD.LT.O)DRE=-DRE C A L L CALCULATEDCD(RE,DRE,CDD,DCD) D C D V D = V D * V D * D C D + C D D * 2 * V D * D V D Y=10 B=2.7-0.65*EXP(-(1.5-LOG(RE)/LOG(Y))**2/2) FE=E**(-B) DF(I)=3./4.*E*RF*C(I)*FE/DP(I)*DCDVD IF(VD.LT.O)DF=-DF 40 CONTINUE DO 50 J=1,N DO 60 J=1,N FJAC(I,J)=C(J)*RF*UL*UL+C(J)*DF(J) IFa.EQ.J)FJAC(I,J)=FJAC(I,j^+E*RP*V(I)*V(I)+E*DF(I) IF(C(I).EQ.O)FJAC(I,J)=0 60 CONTINUE EF(C(I).EQ.0)FJAC(I,D= 1 50 CONTINUE R E T U R N E N D C C SUBROUTINE C A L C U L A T E D C D C TO C A L C U L A T E GRADIENT OF DISPERSION COEFFICIENT C SUBROUTINE CALCULATEDCD(RE,DRE,CD,DCD) Y=10 LN10=LOG(Y) W=LOG(RE)/LN10 DW=1./RE/LN10*DRE IF (RE.LE.O.Ol)THEN DCD=-24./RE/RE*DRE E L S E IF(T<E.GT.0.01.AND.RE.LE.20)THEN B=(0.82-2*0.05*W)*DW A=(CD*RE/24-l)*LN10*24 DCD=(B*A-CD*DRE) /RE E L S E IF(RE.GT.20.AND.RE.LE.260)THEN B=0.6305*DW A=(CD*RE/24-l)*LN10*24 DCD=(B*A-CD*DRE)/RE E L S E EF(RE.GT.260.AND.RE.LE.1.5E3) T H E N B=(-1.1241+2.*0.1558*W)*DW A=CD*LN10 DCD=B*A E L S E (RE. GT. 1. 5E3. AND.RE.LE. 1.2E4)THEN B=(2.5558-2.*0.9295*W+3.*0.1049*W*W)*DW A=CD*LN10 DCD=B*A E L S E IF(RE.GT. 1.2E4.AND.RE.LE.4.4E4)THEN B=(0.6370-2.*0.0636*W)*DW A=CD*LN10 DCD=B*A E L S E IF(RE.GT.4.4E4.AND.RE.LE.3.38E5)THEN B=(1.5809-2.*0.1546*W)*DW A=CD*LN10 DCD=B*A E L S E IF(RE.GT.3.38E5.AND.RE.LE.4.E5)THEN DCD=-5.3*DW 215 Appendix F: Fortran Program Written to Solve Steady State Hydrodynamic Model ELSE IE(TJE.GT.4.E5AND.PvE.LE.l.E6)THEN DCD=0.1*DW ELSE IF(RE.GT.1.E6)THEN DCD=8.E4/RE/RE*DRE ENDIF RETURN END 216 Appendix G: Explanation of the Subroutine I V P A G Appendix G: Explanation of the Subroutine IVPAG in Microsoft Fortran PowerStation 4.0 (1994-1995) Solve an initial-value problem for ordinary differential equations using either Adams-Moulton's or Gear's BDF method. Usage CALL IVPAG (IDO, N, FCN, F C N J , A , T, TEND, TOL, P A R A M , Y) Arguments IDO — Flag indicating the state of the computation. (Input/Output) IDO State 1 Initial entry 2 Normal re-entry 3 Final call to release workspace 4 Return because of interrupt 1 5 Return because of interrupt 2 with step accepted 6 Return because of interrupt 2 with step rejected 7 Return for new value of matrix A. Normally, the initial call is made with IDO = 1. The routine then sets IDO = 2, and this value is then used for all but the last call that is made with IDO = 3. This final call is only used to release workspace, which was automatically allocated by the initial call with IDO = 1. See Comment 5 for a description of the interrupts. When IDO = 7, the matrix A at t must be recomputed and IVPAG/DIVPAG called again. No other argument (including IDO) should be changed. This value of IDO is returned only if PARAM(19) = 2. N — Number of differential equations. (Input) FCN — User-supplied SUBROUTINE to evaluate functions. The usage is CALL FCN (N, T, Y, YPRIME) , where N — Number of equations. (Input) T — Independent variable, t. (Input) Y — Array of size N containing the dependent variable values, y. (Input) YPRIME — Array of size N containing the values of the vector / evaluated at (f, y). (Output) 217 Appendix G: Explanation of the Subroutine I V P A G See Comment : 3. F C N must be declared E X T E R N A L in the calling program. FCNJ — User-supplied S U B R O U T I N E to compute the Jacobian. The usage is C A L L F C N J (N, T , Y , D Y P D Y ) where N — Number of equations. (Input) T — Independent variable, t. (Input) Y — Array of size N containing the dependent variable values, y(t). (Input) D Y P D Y — An array, with data structure and type determined by P A R A M ( 1 4 ) = M T Y P E , containing the required partial derivatives df\/dy\. (Output) These derivatives are to be evaluated at the current values of (t, y). When the Jacobian is dense, M T Y P E = 0 or = 2, the leading dimension of D Y P D Y has the value N . When the Jacobian matrix is banded, M T Y P E = 1, and the leading dimension of D Y P D Y has the value 2 * N L C + N U C + 1. If the matrix is banded positive definite symmetric, M T Y P E = 3, and the leading dimension of D Y P D Y has the value N U C + 1. F C N J must be declared E X T E R N A L in the calling program. If P A R A M ( 1 9 ) = I A T Y P E is nonzero, then F C N J should compute the Jacobian of the righthand side of the equation Ay1 = f{t, y). The subroutine F C N J is used only if P A R A M ( 1 3) = M I T E R = 1 . A — Matrix structure used when the system is implicit. (Input) The matrix A is referenced only if P A R A M ( 1 9 ) = I A T Y P E is nonzero. Its data structure is determined by P A R A M ( 1 4 ) = M T Y P E . The matrix A must be nonsingular and M I T E R must be 1 or 2. See Comment 3. T — Independent variable, t. (Input/Output) On input, T contains the initial independent variable value. On output, T is replaced by T E N D unless error or other normal conditions arise. See I D O for details. TEND — Value of t = tend where the solution is required. (Input) The value tend may be less than the initial value of t. TOL — Tolerance for error control. (Input) An attempt is made to control the norm of the local error such that the global error is proportional to T O L . PARAM — A floating-point array of size 50 containing optional parameters. (Input/Output) If a parameter is zero, then the default value is used. These default values are given below. Parameters that concern values of the step size are applied in the direction of integration. The following parameters must be set by the user: 218 Appendix G: Explanation of the Subroutine I V P A G PARAM Meaning 1 HINIT Initial value of the step size H. A lways nonnegative. Default: 0.001 \ tend - t0\. 2 HMIN Minimum value of the step size H. Default: 0.0. 3 HMAX Maximum value of the step size H. Default: No limit, beyond the machine scale, is imposed on the step size. 4 MXSTEP Maximum number of steps allowed. Default: 500. 5 MX FCN Maximum number of function evaluations allowed. Default: No enforced limit. 6 MAXORD Maximum order of the method. Default: If Adams-Moulton method is used, then 12. If Gear's or BDF method is used, then 5. The defaults are the maximum values allowed. 7 INTRP1 If this value is set nonzero, the subroutine will return before every step with IDO = 4. See Comment 5. Default: 0. 8 INTRP2 If this value is nonzero, the subroutine will return after every successful step with IDO = 5 and return with IDO = 6 after every unsuccessful step. See Comment 5. Default: 0 9 SCALE A measure of the scale of the problem, such as an approximation to the average value of a norm of the Jacobian along the solution. Default: 1.0 10 I NORM Switch determining error norm. In the following, ei is the absolute value of an estimate of the error in y\(t). Default: 0.0 — min(absolute error, relative error) = max(ei/wi); / = 1, N, where w\ - max( | yi(?) |, 1 .0 ) .1—abso lu te error = max(ei), / = 1 N.2 — max(ei / wi), / = 1 N where w\ = max(| / i ( f ) | , FLOOR), and FLOOR is the value PARAM(11 ).3 — Scaled Euclidean norm defined as where wi = max(|yi(f) | , 1.0). Other definitions of YMAX can be specified by the user, as explained in Comment 1. 11 FLOOR Used in the norm computation associated the parameter INORM. Default: 1.0. 12 METH Integration method indicator. 1 = METH selects the Adams-Moulton method.2 = METH selects Gear's BDF method.Default: 1. 13 MITER Nonlinear solver method indicator.Note: If the problem is stiff and a chord or modified Newton method is most efficient, use MITER = 1 or = 2.0 = MITER selects functional iteration. The value IATYPE must be set to zero with this option. 1 = MITER selects a chord method with a user-provided Jacobian.2 = MITER selects a chord method with a divided-difference Jacobian.3 = MITER selects a chord method with the Jacobian replaced by a diagonal matrix based on a directional derivative. The value IATYPE must be set to zero with this option.Default: 0. 14 MTYPE Matrix type for A (if used) and the Jacobian (if MITER = 1 or = 2). When both are used, A and the Jacobian must be of the same 219 Appendix G: Explanation of the Subroutine I V P A G type.O = MTYPE selects full matrices. 1 = MTYPE selects banded matrices.2 = MTYPE selects symmetric positive definite matrices.3 = MTYPE selects banded symmetric positive definite matrices.Default: 0. 15 NLC Number of lower codiagonals, used if MTYPE = 1.Default: 0. 16 NUC Number of upper codiagonals, used if MTYPE = 1 or MTYPE = 3.Default: 0. 17 Not used. 18 EPSJ Relative tolerance used in computing divided difference Jacobians.Default: SQRT(7AMACH(4)) . 19 IATYPE Type of the matrix A.O = IATYPE implies A is not used (the system is explicit). 1 = IATYPE if A is a constant matrix.2 = IATYPE if A depends on ^.Default: 0. 20 LDA Leading dimension of array A exactly as specified in the dimension statement in the calling program. Used if IATYPE is not zero.Default:N if MTYPE = 0 or = 2 NUC + NLC + 1 if MTYPE = 1 NUC + 1 if MTYPE = 3 2 1 - 3 0 Not used. The following entries in the array PARAM are set by the program: PARAM Meaning 31 HTRIAL Current trial step size. 32 HMINC Computed minimum step size. 33 HMAXC Computed maximum step size. 34 NSTEP Number of steps taken. 35 NFCN Number of function evaluations used. 36 NJE Number of Jacobian evaluations. 37--50 Not used. V — Array of size N of dependent variables, y(t). (Input/Output) On input, Y contains the initial values, y{t0). On output, Y contains the approximate solution, y(t). Comments 1. Automatic workspace usage is IVPAG 4-N + NMETH + NPW + NIPVT, or DIVPAG 8N + 2 * NMETH + 2 * NPW + NIPVT units. Here, NMETH = 1 3N if METH is 1, NMETH = 6N if METH is 2. NPW = 2N + NPWM + NPWA 220 Appendix G: Explanation of the Subroutine I V P A G where NPWM = 0 if MITER is 0 or 3, NPWM = N2 if MITER is 1 or 2, and if MTYPE is 0 or 2. NPWM = N(2 * NLC + NUC + 1) if MITER is 1 or 2 and MTYPE = 1. NPWM = N(NLC + 1) if MITER is 1 or 2 and if MTYPE = 3. NPWA = 0 if IATYPE is 0. NPWA = N2 if IATYPE is nonzero and MTYPE = 0, NPWA = N(2 * NLC + NUC + 1) if IATYPE is nonzero and MTYPE = 1 NIPVT = N if MITER is 1 or 2 and MTYPE is 0 or 1, NIPVT = 1, otherwise. Workspace and a user-supplied error norm subroutine may be explicitly provided, if desired, by use of 12PAG/D12PAG. The reference is CALL I2PAG (IDO, N, FCN, FCNJ, A, T, TEND, TOL, PARAM, Y, YTEMP, YMAX, ERROR, SAVE1, SAVE2, PW, IPVT, VNORM) None of the additional array arguments should be changed from the first call with IDO = 1 until after the final call with IDO = 3. The additional arguments are as fol lows: YTEMP — Array of size NMETH. (Workspace) YMAX — Array of size N containing the maximum Y-values computed so far. (Output) ERROR — Array of size N containing error estimates for each component of Y. (Output) SAVE1 — Array of size N. (Workspace) SAVE2 — Array of size N. (Workspace) PW — Array of size NPW. PW is used both to store the Jacobian and as workspace. (Workspace) IPVT — Array of size N. (Workspace) VNORM — A Fortran SUBROUTINE to compute the norm of the error. (Input) The routine may be provided by the user, or the IMSL routine I3PRK/DI3PRK may be used. In either case, the name must be declared in a Fortran ENTERNAL statement. If usage of the IMSL routine is intended, then the name I3PRK/DI3PRK should be specified. The usage 221 Appendix G: Explanation of the Subroutine I V P A G of the error norm routine is CALL WORM (N, V, Y, YMAX, ENORM), where Arg. Definition N Number of equations. (Input) V Array of size N containing the vector whose norm is to be computed. (Input) Y Array of size N containing the values of the dependent variable. (Input) YMAX Array of size N containing the maximum values of \y (t)\. (Input) ENORM Norm of the vector V. (Output) VNORM must be declared EXTERNAL in the calling program. 2. Informational errors Type Code 4 1 After some initial success, the integration was halted by repeated error-test failures. 4 2 The maximum number of function evaluations have been used. 4 3 The maximum number of steps allowed have been used. The problem may be stiff. 4 4 On the next step T + H will equal T. Either TOL is too small, or the problem is stiff. Note: If the Adams-Moulton method is the one used in the integration, then users can switch to the BDF methods. If the BDF methods are being used, then these comments are gratuitous and indicate that the problem is too stiff for this combination of method and value of TOL. 4 5 After some initial success, the integration was halted by a test on TOL. 4 6 Integration was halted after failing to pass the error test even after dividing the initial step size by a factor of 1 .OE + 10. The value TOL may be too small. 4 7 Integration was halted after failing to achieve corrector convergence even after dividing the initial step size by a factor of 1. OE + 10. The value TOL may be too small. 4 8 IATYPE is nonzero and the input matrix A multiplying / is singular. 222 Appendix G: Explanation of the Subroutine I V P A G 3. Both explicit systems, of the form y1 = f (t, y), and implicit systems, Ay1 = f (t, y), can be solved. If the system is explicit, then PARAM(19) = 0; and the matrix A is not referenced. If the system is implicit, then PARAM(14) determines the data structure of the array A. If PARAM(1 9) = 1, then A is assumed to be a constant matrix. The value of A used on the first call (with IDO = 1) is saved until after a call with IDO = 3. The value of A must not be changed between these calls. If PARAM(1 9) = 2, then the matrix is assumed to be a function of t. 4. If MTYPE is greater than zero, then MITER must equal 1 or 2. 5. If PARAM(7) is nonzero, the subroutine returns with I D 0 = 4 and will resume calculation at the point of interruption if re-entered with IDO = 4. If PARAM(8) is nonzero, the subroutine will interrupt immediately after decides to accept the result of the most recent trial step. The value IDO = 5 is returned if the routine plans to accept, or IDO = 6 if it plans to reject. The value IDO may be changed by the user (by changing IDO from 6 to 5) to force acceptance of a step that would otherwise be rejected. Relevant parameters to observe after return from an interrupt are IDO, HTRIAL, NSTEP, NFCN, NJE, T and Y. The array Y contains the newly computed trial value y(i). Algorithm The routine IVPAG solves a system of first-order ordinary differential equations of the form / = f (t, y) or Ay1 = f (t, y) with initial conditions where A is a square nonsingular matrix of order N. Two classes of implicit linear multistep methods are available. The first is the implicit Adams-Moulton method (up to order twelve); the second uses the backward differentiation formulas BDF (up to order five). The BDF method is often called Gear's stiff method. In both cases, because basic formulas are implicit, a system of nonlinear equations must be solved at each step. The deriviative matrix in this system has the form L = A + r\J where r| is a small number computed by IVPAG and J is the Jacobian. When it is used, this matrix is computed in the user-supplied routine FCNJ or else it is approximated by divided differences as a default. Using defaults, A is the identity matrix. The data structure for the matrix L may be identified to be real general, real banded, symmetric positive definite, or banded symmetric positive definite. The default structure for L is real general. 223 Appendix H : Explanation of the Subroutine NEQNF and NEQNJ Appendix H: Explanation of the Subroutine NEQNF and NEQNJ in Microsoft Fortran PowerStation 4.0 (1994-1995) E . l . N E Q N F : Solve a system of nonlinear equations using a modified Powell hybrid algorithm and a finite-difference approximation to the Jacobian. Usage CALL NEQNF (FCN, ERRREL, N, ITMAX, X G U E S S , X , FNORM) Arguments FCN — User-supplied SUBROUTINE to evaluate the system of equations to be solved. The usage is CALL FCN (X, F, N) , where X — The point at which the functions are evaluated. (Input) x should not be changed by FCN. F — The computed function values at the point X. (Output) N — Length of X and F. (Input) FCN must be declared EXTERNAL in the calling program. ERRREL — Stopping criterion. (Input) The root is accepted if the relative error between two successive approximations to this root is less than ERRREL. N — The number of equations to be solved and the number of unknowns. (Input) ITMAX — The maximum allowable number of iterations. (Input) The maximum number of calls to FCN is ITMAX * (N + 1). Suggested value ITMAX = 200. XGUESS - A vector of length N. (Input) XGUESS contains the initial estimate of the root. X — A vector of length N. (Output) X contains the best estimate of the root found by NEQNF. FNORM — A scalar that has the value F(1)2 + ... + F(N)2 at the point X. (Output) Comments 1. Automatic workspace usage is 224 Appendix H : Explanation of the Subroutine NEONF and NEONJ N E Q N F 1.5 * N2 + 7.5 * N units, or D N E Q N F 3 * N2 + 1 5 * N units. Workspace may be explicitly provided, if desired, by use of N2QNF/DN2QNF. The reference is C A L L N2QNF ( F C N , E R R R E L , N, I T M A X , X G U E S S , X , F N O R M , F V E C , F J A C , R , Q T F , W K ) The additional arguments are as fol lows: FVEC — A vector of length N . F V E C contains the functions evaluated at the point X . FJAC — An N by N matrix. F J A C contains the orthogonal matrix Q produced by the Q R factorization of the final approximate Jacobian. R — A vector of length N * ( N + 1)/2. R contains the upper triangular matrix produced by the Q R factorization of the final approximate Jacobian. R is stored row-wise. QTF — A vector of length N . Q T F contains the vector T R A N S ( Q ) * F V E C . WK — A work vector of length 5 * N . 2. Informational errors Type Code 4 1 The number of calls to F C N has exceeded I T M A X * ( N + 1). A new initial guess may be tried. 4 2 E R R R E L is too small. No further improvement in the approximate solution is possible. 4 3 The iteration has not made good progress. A new initial guess may be tried. Algorithm Routine N E Q N F is based on the MINPACK subroutine HYBRD1, which uses a modification of M.J .D. Powell 's hybrid algorithm. This algorithm is a variation of Newton's method, which uses a finite-difference approximation to the Jacobian and takes precautions to avoid large step sizes or increasing residuals. For further description, see More et al. (1980). Since a finite-difference method is used to estimate the Jacobian, for single precision calculation, the Jacobian may be so incorrect that the algorithm terminates far from a root. In such cases, high precision arithmetic is recommended. Also, whenever the exact Jacobian can be easily provided, IMSL routine NEQNJ should be used instead. 225 Appendix H : Explanation of the Subroutine NEQNF and NEONJ E . l . N E Q N J : Solve a system of nonlinear equations using a modified Powell hybrid algorithm with a user-supplied Jacobian. Usage CALL NEQNJ (FCN, L S J A C , ERRREL, N, U M A X , X G U E S S , X , FNORM) Arguments FCN — User-supplied SUBROUTINE to evaluate the system of equations to be solved. The usage is CALL FCN (X, F, N), where x — The point at which the functions are evaluated. (Input) X should not be changed by FCN. F — The computed function values at the point X. (Output) N — Length of X, F. (Input) FCN must be declared EXTERNAL in the calling program. LSJAC — User-supplied SUBROUTINE to evaluate the Jacobian at a point X. The usage is CALL LSJAC (N, X, FJAC), where N — Length of X. (Input) X — The point at which the function is evaluated. (Input) x should not be changed by LSJAC. FJAC — The computed N by N Jacobian at the point X . (Output) LSJAC must be declared EXTERNAL in the calling program. ERRREL — Stopping criterion. (Input) The root is accepted if the relative error between two successive approximations to this root is less than ERRREL. N — The number of equations to be solved and the number of unknowns. (Input) /TMAX — The maximum allowable number of iterations. (Input) Suggested value = 200. XGUESS - A vector of length N. (Input) XGUESS contains the initial estimate of the root. X — A vector of length N. (Output) x contains the best estimate of the root found by NEQNJ. FNORM — A scalar that has the value F(1)2 + . . . + F(N)2 at the point X. (Output) 226 Appendix H : Explanation of the Subroutine NEQNF and NEQNJ Comments 1. Automatic workspace usage is NEQNJ 1.5 * N * *2 + 7.5 * N units, or DNEQNJ 3 * N * * 2 + 15 * N units. Workspace may be explicitly provided, if desired, by use of N2QNJ/DN2QNJ. The reference is CALL N2QNJ (FCN, LSJAC, ERRREL, N, ITMAX, XGUESS, X, FNORM, FVEC, FJAC, R, QTF, WK) The additional arguments are as fol lows: FVEC — A vector of length N. FVEC contains the functions evaluated at the point x. FJAC — An N by N matrix. FJAC contains the orthogonal matrix Q produced by the QR factorization of the final approximate Jacobian. R — A vector of length N * (N + 1)/2. R contains the upper triangular matrix produced by the QR factorization of the final approximate Jacobian. R is stored row-wise. QTF — A vector of length N. QTF contains the vector TRANS(Q) * FVEC. WK — A work vector of length 5 * N. 2. Informational errors Type Code 4 1 The number of calls to FCN has exceeded ITMAX. A new initial guess may be tried. 4 2 ERRREL is too small. No further improvement in the approximate solution is possible. 4 3 The iteration has not made good progress. A new initial guess may be tried. Algorithm Routine NEQNJ is based on the MINPACK subroutine HYBRDJ, which uses a modification of M.J .D. Powell 's hybrid algorithm. This algorithm is a variation of Newton's method, which takes precautions to avoid large step sizes or increasing residuals. For further description, see More et al. (1980). 227 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model C M A I N P R O G R A M TO SOLVE THE U N S T E A D Y STATE H Y D R O D Y N A M I C M O D E L C C C A R G U M E N T S C A C : A R R A Y OF THE CYL INDRICAL REGION C AT: A R R A Y OF THE TOP REGION C A M : A R R A Y OF M A S S R A T E OF PARTICLE C A M F : M A S S R A T E OF LIQUID C C: A R R A Y OF PARTICLE CONCNETRATION IN THE C O L U M N C CD: A R R A Y OF DISCHARGE PARTICLE CONCENTRATION C CF: A R R A Y OF FEED PARTICLE CONCENTRATION C CFO: A R R A Y OF FEED PARTICLE CONCNETRATION AT INITIAL C CONDISITON C C F L A R R A Y OF FEED PARTICLE CONCNETRATION AT F INAL C CONDISITON C CO: A R R A Y OF PARTICLE CONCENTRATION IN O V E R F L O W C D: D IAMETER OF CYL INDRICAL REGION OF THE CLASSIFIER C DA: D IAMETER OF THE FLUDIZING PIPE C DI: DISPERISON COEFFICIENT C DP: A R R A Y OF PARTICLE D IAMETER C DP500: M E A N PARTICLE SIZE OF FEED C D P M A X : THE M A X I M U M PARTICLE SIZE C D P P : A R R A Y OF PARTICLE D IAMETER C DT: TIME DIFFERENCE IN THE C A L C U L A T I O N C DTO:DIAMETER OF TOP REGION OF THE CLASSIFIER C DTEND: TIME W H E N THE RESULTS A R E WRITTEN IN THE OUT PUT C FILE C E: A R R A Y OF VOIDAGE IN THE C O L U M N C EF: FEED VOIDAGE C EFO: FEED VOIDAGE AT INITIAL CONDITION C EF 1: FEED VOIDAGE AT F INAL CONDITION C F: A R R A Y OF THE PARTICLE F L U X C FF: A R R A Y OF A C C U M U L A T I O N OF THE PARTICLE IN THE C O L U M N C FI: A R R A Y OF LIQUID A N D PARTICLE D R A G FORCE C FED: A R R A Y OF PARTICLE DISPERSION FORCE C G: G R A V I T Y A C C E L E R A T I O N C H: A R R A Y OF HEIGHT OF THE EIGHT ZONES OF THE CLASSIFIER C M H : N U M B E R OF THE ZONES OF CLASSIFIER C MT: N U M B E R OF M E S H POINT OF THE HEIGHT C NH: N U M B E R OF M E S H POINT OF HEIGHT C N M : N U M B E R OF SPECIES OF PARITCLE C N: N U M B E R OF PARTICLE SPECIES C QA: LIQUID F L O W R A T E C QD: DISCHARGE F L O W R A T E C QF: FEED FLOW R A T E C QO: O V E R F L O W R A T E C RF: LIQUID DENSITY C RNO: INDEX OF ROSIN-RAMMLER PARTICLE SIZE DISTRIBUTION C RP: PARTICLE DENSISTY C T1: F INAL TIME THE C A L C U L A T I O N R E A C H E D C TA: TIME OF CHANGING LIQUID FLUIDIZING V E L O C I T Y C TD: TIME OF CHANGING DISCHARGE V E L O C I T Y C TF: T IME OF CHANGING FEED VELOCITY 228 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model C UAO: LUQID FLUIDIZING VELOCITY AT INITIAL CONDITION C UA1 : LUQID FLUIDIZING VELOCITY AT F INAL CONDITION C UDO: D ISCHARGE V E L O C I T Y A T INITIAL CONDITION C UD 1: D ISCHARGE V E L O C I T Y AT FINAL CONDITION C UFO: FEED VELOCITY AT INITIAL CONDITION C UF1: FEED VELOCITY AT F INAL CONDITION C U L Q : A R R A Y OF THE ABSOLUTE LIQUID V E L C O C I T Y IN THE C C O L U M N C UL : ABSOLUT LIQUID VELOCITY C U U : LIQUID VISCOCITY C V D : A R R A Y OF DISPERISON V E L C O C I T Y C W : A R R A Y OF PARTICLE V E L C O C I T Y C Z: A R R A Y OF HEIGHT OF THE CLASSIFIER C C INITIALC.DAT: INPUT FILE TO INPUT INITIAL PARTICLE C CONCENTRATION A N D VOIDAGE C INITIALF.DAT: INPUT FILE TO INPUT INITIAL PARTICLE F L U X A N D C LIQUID V E L O C I T Y C TRANIN.DAT: INPUT D A T A FILE C OUTEC.DTA: OUTPUT FILE C OUTED.DAT: OUTPUT FILE C P A R A M E T E R (NM=6,NH=188) DIMENSION A(NM,NM+1),XT(NM) DIMENSION H(8),DP(NM),CF(NM), CD(NM) DIMENSION E(0:NIT;,Z(0:NH) )C(NM,0:NH),C1(NM,0:NH) DIMENSION ULQ(0:NH),CF0(NM),CFl(mi) DIMENSION F1(NM,0:NH) DIMENSION U(2*NM),UX(2*NM),W(TsIM), \TJ(NM),FI(T^ DIMENSION AM(NM) ,UT(2*NM),M(NM) >CO(NM),F(NM,0:NH) DIMENSION C0(>M,0:NH),C2(>IM,0:NH),F0(NM,0:NH) DIMENSION F2(NM,0:NH),FF(NM),Z12(0:NH),DPP(NM) C C INPUT THE P A R A M E T E R TO C A L C U L A T E THE DISPERSION COEFFICIENT C D A T A AKHD,AKEF,AJ<^90 ,AK0 ,AJCREDC,AKREDP ) AKAR,AKUF,AKUD,AKEF4/ + 1.120554992,-0.73261928,0.149283113,0.217814339,0.468551131, + 0.491871891,-0.908425733,0.127162111,-0.052299796,-0.6328947/ C C INPUT D A T A C N=NM MT=NH O P E N (1 ,FILE='TRANIN.D AT') READ(1,*) READ( 1, *)MH,D,DTO ,DA READ(1,*) RE AD( 1, *)T A,TF,TD,T 1 ,DT,DTEND R=D/2 RA=DA/2 RT=DTO/2 READ(1,*) READ(l,*)(h(I),I=l,MH) READ(1,*) READ(1,*) UA0,UF0,UD0 ,EF0 READ(1,*) UA1,UF1,UD1 ,EF1 229 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model READ(1,*) READ(1,*) RP,RF,UU,G,DI READ(1,*) DO 5 K=1,N READ(1,*)DPP(K),DP(K),CF0(K) 5 - CONTINUE READ(1,*) DO 51 K=1,N READ(1,*)DPP(K),DP(K),CF1(K) 51 CONTINUE READ(1,*) READ(1,*)DP500, RN0 ,DPMAX CLOSE (1) OPEN (2, F I L E - r N r n A L C . D A T ' ) READ(2,60)Z00,ED,(CD(K),K=1,N) DO 11 J=MT,0,-1 R E AD(2,60)Z(J),E(J),(C(K, J),K= 1 ,N) 11 CONTINUE CLOSE(2) HT=Z(MT) 60 F 0 R M A T ( 1 X , F10.4,20(1X, E14.7)) OPEN(3 ,FELE= ,INrTIAlF.D AT') DO 42 J=MT,0,-1 READ(3,60)Z(J),ULQ(J),(F(K,J),K=1,N) Z(J)=HT-Z(J) 42 CONTINUE CLOSE(3) DO 302 I=1,N COa)=C(I,MT) 302 CONTINUE PI=4.*ATAN(1.) AC=PI*R*R AT=PI*RT*RT UA1=UA 1*0.01 UA0=UA0*0.01 UD0=UD0*0.01 UD1=UD1*0.01 UF0=UF0*0.01 UF1=UF1*0.01 QA0=UA0*AC QA1=UA1*AC QD0=UD0*AC QD1=UD1*AC QF0=UF0*AC QF1=UF1*AC EO0=(QF0*EF0+QA-QD0*ED)/(QF0+QA-qD0) EO 1=(QF 1 *EF 1+QA-QD1 *ED)/(QF 1+Q A-qD 1) T=0 TEND 1 =DTEND DALL=10 OPEN( 1 ,FELE='OUTED.D AT') OPEN(2,FLLE= ,OUTEC.DAT') C C BEGIN LOOP C 230 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model DO WHILE (T.LE.T1) IF(T-DT.LT. 0)THEN DP50=DP500 RN=RN0 DO 3011=1,N F(I,MT)=-C(I,MT)*(QA0+QF0-QD0)/AT C(I,MT)=CO(I) 301 CONTINUE ENDIF T=T+DT IF(T.LT.TA)THEN QA=(QA 1 -QA0)/TA*T+QA0 DQA=(QA1-QA0)/TA ENDIF IF(T.LT.TF)THEN QF=(QF 1 -QF0)/TF*T+QF0 DQF=(QF 1 -QF0)/TF EF=(EF1-EF0)/TF*T+EF0 D 0 4K=1,N CF(K)=(CF 1 (K)-CF0(K))/TF*T+CF0(K) 4 CONTINUE ENDIF IF(T.LT.TD)THEN QD=(QD 1 -QD0)/TD*T+QD0 DQD=(QD 1 -QD0)/TD E N D I F IF(T.GE.TA)THEN DQA=0 QA=QA1 ENDIF IF(T.GE.TF)THEN QF=QF1 DQF=0 EF=EF1 DO 23K=1,N CF(K)=CF1(K) 23 CONTINUE ENDIF IF(T.GE.TD)THEN QD=QD1 DQD=0 ENDIF UF=QF/AC UA=QA/AC UD=QD/AC DP50E=DP50 RNE=RN C C C A L C U L A T E THE DISPERSION COEFFICIENT C EF(N. GT. 3 )THEN C C FIT THE PARTICLE SIZE DISTRIBUTION B Y ROSINf-RAMMLER DISTRIBUTION C C A L L NONLINE (DP50E, RNE, DP50F,RNF,DPMAX,DPP,CF ,N+1,N) HL=H(4) 231 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model DP90F=(LOG(0.1)/LOG(0.9))**(1/RNF) DHD=HL/D DEF=1-EF DDP9ODP90F DMDC=(TJA*D*RFAJU) DMDP=(UA*DP50F*0.001*I^/UTJ) DAR=((T<F*(RP-I^)*(DP50F*0.001)**3*G)/(TJU**2)) DW=(UF/UA) DUD=(UD/UA) DEF4=(EF-0.4) PE1=AK0*DHD**AKHD PE2=PE1*DEF**AKEF PE3=PE2*DDP90**AKD90 PE4=PE3 * D R E H * * A K R E D C PE5=PE4*DRED**AKREDP PE6=PE5*DAR**AKAR PE7=PE6*DUF**AKUF PE8=PE7*DUD**AKUD PE=PE8*DEF4**AKEF4 DIC=UA*HL/PE ENDIF DO 980K=1,N CD(K)=0 FF(K)=0 980 COOTINUE UTT=0 L=0 DO 201 J=MT-1,0,-1 DO 211 I=1,N C2(I,J)=(C(I,J+l)+Ca,J))/2 F2(I,J)=(F(I,J+l)+F(I,J))/2 211 CONTINUE 201 CONTINUE 300 DO 200 J=MT-1,0,-1 C C L=0: HALFSTEP C A L C U L A T I O N ; L= 1: UPDATED C A L C U L A T I O N C EF(L.EQ.0)THEN X=(Z(J+l)+Z(J))/2 Z12(J)=X DO210 I= l ,N U(I)=C2(I,J) U(I+N)=F2(I,J) UX(I)=(C(I,J+1)-C(I,J))/(Z(J+1)-Z(J)) UX(I+N)=(F(I, J+ 1)-F(L J))/(Z(J+ l)-Z(J)) 210 CONTINUE ENDIF EF(L.EQ.1)THEN X=Z(J) DO 220 1=1,N U(I)=C(I,J) U(I+N)=F(I,J) IF(J.GT.0)THEN UX(I)=(Cl(I,J)-Cia,M))/(Z12(J)-Z12(J-l)) UX(I+N)=(F1(I,J)-F1(I,J-1))/(Z12(J)-Z12(J-1)) Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model E L S E UX(I)=(C(I,J+1)-C(I,J))/(Z(J+1)-Z(J)) UX(I+N)=(F(I,J+1)-F(I,J))/(Z(J+1)-Z(J)) ENDIF 220 CONTINUE IF(J.EQ.0)THEN DO 89 I=1,N UT(l)=-UX(I+N) UT(I+N)=0 89 CONTINUE GOTO 222 ENDIF ENDIF YY=0 D 0 2I=1,N YY=YY+U(I) 2 CONTINUE E(J)=1-YY CV=0 DCVZ=0 DO 555 1=1,N CV=CV+U(I+N) DCVZ=DCVZ+UX(I+N) 555 CONTINUE C C C A L C U L A T E FOR DIFFERENT ZONES C C B O T T O M ZONE II C IF (X .GT. H(8)+H(7)+H(6)+H(5)+H(4)+H(3)+H(2) + . AND.X.LE.H(8)+H(7)+H(6)+H(5)+H(4)+H(3)+H(2)+H(1))THEN AB=AC DABDZ=0 IF(R. GT.RA)THEN ZH=H(8)+H(7)+H(6)+H(5)+H(4)+H(3)+H(2) XZ=X-ZH H0=RA/(R-RA)*H(1) RB=(H0+H(1)-XZ)/H0*RA AB=PI*RB*RB DABDZ=-2*PI*RB*RA/H0 ENDIF EU=-QA/AB-CV DO 12I=1,N AM(I)=-U(I+N)*DABDZ/AB 12 CONTINUE AMF=-EU*DABDZ/AB D E U Z = Q A / A B / A B * D A B D Z - D C V Z DEUT1=-DQA/AB ABC=AB C C B O T T O M Z O N E I C E L S E IF (X .GT. H(8)+H(7)+H(6)+H(5)+H(4)+H(3) + . AND.X.LE.H(8)+H(7)+H(6)+H(5)+H(4)+H(3)+H(2)) T H E N DO 4111=1,N AM(I)=0 233 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model 411 CONTINUE AMF=0 EU=-QA/AC-CV DEUZ=-DCVZ DEUT1=-DQA/AC ABC=AC C C D ISCHARGE ZONE C E L S E EF (X .GT. H(8)+H(7)+H(6)+H(5)+H(4) + AND.X.LE.H(8)+H(7)+H(6)+H(5)+H(4)+H(3)) T H E N ZH=H(8)+H(7)+H(6)+H(5)+H(4) XZ=X-ZH RD=H(3)/2 AD=PI*RD*RD BL=2*(RD**2-(RD-XZ)**2)**0.5 DO 10 1=1,N AM(I)=-QD*U(I)*BL/AC/AD 10 CONTINUE AMF=-QD*E(J)*BL/AC/AD ZZ=(RD-XZ)/RD Zl=2*ACOS(ZZ) Z2=Z1/2*RD*RD Z3=2*RD*SIN(Zl/2) ACC=Z2-(Z3*(RD-XZ))/2 APH=ACC/PI/RD/RD EU=-(QA-(1-APH)*QD)/AC-CV DEUZ=-QD*BL /AC/AD-DCVZ DEUT 1=( 1-APH)/AC*DQD-DQ A / A C ABC=AC C C C O L U M N Z O N E C E L S E IF(X.GT.H(8)+H(7)+H(6)+H(5) + .AND.X.LE.H(8)+H(7)+H(6)+H(5)+H(4))THEN D 0 111I=1,N AM(I)=0 111 CONTINUE AMF=0 EU=-(QA-QD)/AC-CV DEUZ=-DCVZ DEUT1=DQD/AC-DQA/AC ABC=AC C C DIVERGING ZONE C E L S E EF (X.GE.H(8)+H(7)+H(6) + .AND.X.LT.H(8)+H(7)+H(6)+H(5))THEN AB=AC DABDZ=0 IF(RT.GT.R)THEN ZH=H(8)+H(7)+H(6) XZ=X-ZH H0=R/(RT-R)*H(5) RB=(H0+H(5)-XZ)/H0*R AB=PI*RB*RB 234 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model DABDZ=-2*PI*RB*R/H0 ENDIF EU=-(QA-QD)/AB-CV DO 17I=1,N AM(I)=-U(I)*U(I+N)*DABDZ/AB 17 CONTINUE AMF=-E( J) *UL*D ABDZ/ AB DEUZ=(QA-QD)/AB/AB*DABDZ-DCVZ DEUT1=DQD/AB-DQA/AB ABC=AB C C TOP ZONE II C ELSE IF (X.GT.H(8)+H(7) + AND.X.LE.H(8)+H(7)+H(6))THEN DO 13 I=1,N AM(I)=0 13 CONTINUE AMF=0 EU=-(QA-QD)/AT-CV DEUZ=-DCVZ DEUT1 =DQD/AT-DQ A/AT ABC=AT C C FEED ZONE C ELSE IF (X.GT.H(8) + AND.X.LE.H(8)+H(7))THEN RFE=H(7)/2 AF=PI*RFE*RFE ZH=H(8) XZ=X-ZH BL=2*(WE**2-(RFE-XZ)**2)**0.5 DO 14 I=1,N AM(I)=QF*CF(I)*( 1 -EF)*BL/AT/AF 14 CONTINUE AMF=QF*EF*BL/AT/AF ZZ=(PvFE-XZ)/RFE Zl=2*ACOS(ZZ) Z2=Z1/2*RFE*RFE Z3=2*RFE*SIN(Zl/2) ACC=Z2-(Z3 *(RFE-XZ))/2 APH=ACC/AF EU=-(QA-QD+( 1 - APH)*QF)/AT-C V DEUZ=QF*BL/AT/AF-DCVZ DEUT 1 =DQD/AT-( 1. - APH)/AT*DQF-DQ A/AT ABC=AT C C TOP ZONE I C ELSE IF (X.GT.0 + AND.X.LE.H(8))THEN DO 15 1=1,N AM(I)=0 15 CONTINUE 235 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model AMF=0 EU=-(QA-QD+QF)/AT-CV DEUZ=-DCVZ DEUT1=(DQD-DQF-DQA)/AT ABC=AT ENDIF C C C A L C U L A T E D R A G FORCE C UL=EU/E(J) DO 151K=1,N W(K)=0 IF(U(K).NE.O)W(K)=U(K+N)/U(K) 151 CONTINUE C A L L FORCEfN,U,RF,DP,UU,UL,W,FI,SLnvIFI) C C C A L C U L A T E DISPERSION FORCE C DO 6011=1,N VD(I)=0 IF (U(I).NE.O)VD(I)=-(-DI/U(I)*UX(I)) 601 CONTINUE UD=0 C A L L FORCE(N,U,RF,DP,UU,UD,\TJ,FID,SUMFID) C C C A L C U L A T E DC/DT C DCZ=0 DO 700K=1,N DCZ=DCZ+UX(K) 700 CONTINUE DO 16I=1,N UT(I)=AM(I)-UX(I+N) EF(l.eq.l)THEN ff(ABS(UT(I)).GT.UTT)UTT=ABS(UT(I)) ENDIF 16 CONTINUE C C C H E C K HOW M A N Y SPECIES R E M A I N IN E A C H C E L L C MJ=0 DO 25 I=1,N EF(U(I).EQ.0)THEN UT(I+N)=0 ENDIF IF(U(I).NE.0)THEN MJ=MJ+1 M(MJ)=I ENDIF 25 CONTINUE NN=MJ IF(NN.EQ.0)GOTO 100 C C C A L C U L A T E DF/DT C 236 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model DO 251 I=1,NN II=M(I) DO 351 JI=1,NN A(I,JI)=RF*U(II) 351 CONTINUE A(I,I)=A(I,I)+E(J)*RP FW=U(II)*(E(J)*(I^-RF)*G-i-SUMFI+SUMFID)+ + E(J)*FI(Il)+E(J)*FID(II) FW1=2*E(J)*U(II+N)AJ(II)*RP*UX(II+N) + -RP*E(J)*U(II+N)**2/ rU(II)**2*UX(II) + -U(II)*RF*DEUT1 + -2*U(II)*RF*EU/E(J)*DEUZ + - I^*U(I I )*EU**2/E(J)**2*DCZ+AM(J)-AMF A(I,NN+1)=FW-FW1 251 CONTINUE IPATH=1 LDA=N C A L L GSXQF1(A,NN,XT) DO 40 I=1,NN II=M(I) UT(II+N)=XT(I) 40 CONTINUE 222 EE=0 C C C A L C U L A T E THE UPDATED C A N D F C 100 DO 102 1=1,N IF(L.EQ.0)THEN C1 (I, J)=UT(I) *DT/2+U(I) F1 (I, J)=UT(I+N)*DT/2+U(I+n) ENDIF IF (L .EQ. l )THEN C0(I,J)=UT(I)*DT+U(I) F0(I,J)=UT(I+N)*DT+U(I+n) IF(J.GT.0)FF(I)=FF(I)+UT(I)*ABC*ABS(Z12(J)-Z12(J-1)) IF(J.EQ.0)FF(I)=FF(I)+UT(I)*ABC*ABS(Z12(J)-Z(J)) ENDIF 102 CONTINUE 200 CONTINUE IF(L.EQ.0)THEN L=l GOTO 300 ENDIF C C C A L C U L A T E THE DISCHARGE CONCENTRATION A N D VOIDAGE C DO 320 J=0,MT-1 X=Z(J) EE=0 DO 310 I=1,N C(I,J)=C0(I,J) F(I,J)=F0(I,J) EE=C(I,J)+EE 310 CONTINUE E(J)=1-EE 237 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model IF (X .Gt. H(8)+H(7)+H(6)+H(5)+H(4) + .AND.X.LT.H(8)+H(7)+H(6)+H(5)+H(4)+H(3)) T H E N ZH=H(8)+H(7)+H(6)+H(5)+H(4) XZ=X-ZH RD=H(3)/2 AD=PI*RD*RD BL=2*(RD**2-(RD-XZ)**2)**0.5 EE=0 ] D 0 981K=1,N CD(K)=CD(X)+C(K,J)*BL*ABS(Z12(J)-Z12(J-1))/AD EE=EE+CD(K) 981 CONTINUE ED=1-EE ENDIF 320 CONTINUE C C C A L C U L A T E THE O V E R F L O W CONCENTRATION AND VOIDAGE C EE=0 DO 982K=1,N CO(K)=((QF*CF(K)*( 1 -EF)-QD*CD(K))-FF(K))/(QF+Q A-QD) IF(CO(K).LT.0)CO(K)=0 C(K,MT)=CO(K) F(K,MT)=-C(K,MT)*(QA+QF-QD)/AT EE=EE+CO(K) 982 CONTINUE EO=l -EE E(MT)=EO C C C H E C K THE O V E R A L L M A S S B A L A N C E C DAL=0 DO 983 K=1,N DA=ABS(QD*CD(K)+CO(K)*(QA+QF-QD)-QF*(l-EF)*CF(K)) + /(QF*(1-EF)*CF(K)) IF(DA.GT.DAL)DAL=DA 983 CONTINUE C C OUTPUT RESULTS C IF(ABS(T-TEND 1).LE.DT/2)THEN C A L L NONLINE (DP50E, RNE, DP50D,RND,DPMAX,DPP,CD/(1-ED),N+1,N) TEND 1=TEND 1+DTEND WRITE(6,*) 'SOLUTION AT T=',T,di,DAL,UTT WRITE(2,*) 'SOLUTION AT T=',T,di,DAL,UTT WRITE(l ,* ) 'SOLUTION AT T=',T,di,DAL,UTT WRITE(2,*)'discharge ,,ED,(CD(K),K=l,N) WRITE(2,*)'overflow',EO,(CO(K),K=l,N) DO 305 J=MT,0,-1 WRITE(2,60)HT-Z(J),E(J),(Ca,J),I= 1,N) 305 CONTINUE CCD=0 WRITE(l,*)'ed= ', e d , ' dp50d= \dp50d,' nd= ',rnd DO 402K=1,N CCD=CCD+CD(K)/(1-ED) WRITE(1,*)DP(K),CCD 238 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model 402 CONTINUE ENDIF END DO CLOSE(l) CLOSE(2) STOP END C C SUBROUTINE FORCE TO CALCULATE THE DRAG OR DISPERSION FORCE C SUBROUTINE FORCE(N, C,RF,DP,UU,UL, V,FI, SUMFI) DIMENSION C(N),V(N),DP(N),FI(N) Z=0 SUMFI=0 DO 10 I=1,N Z=Z+C(I) FI(I)=0 10 CONTINUE E=l-Z Z1=0 DO 20I=1,N IF(C(I).EQ.0)THEN FI(I)=0 GOTO 20 ENDIF RE=DP(I)*ABS(UL-V(I))*RF*E/UU IF(RE.LT.0.0001)THEN FI(I)=0 GOTO 20 ENDIF CALL CALCULATECD(RE,CD 1) Y=10 B=2.7-0.65*EXP(-(1.5-LOG(RE)/LOG(Y))**2/2) FE=E**(-B) FI(I)=374.*CD1*E*ABS(UL-V(I))*C(I)*RF/DP(I)*FE*(TJL-V(I)) 22 Z1=Z1+FI(I) 20 CONTINUE SUMFI=Z1 RETURN END C C SUBROUTINE CALCULATED TO CALCULATE THE DRAG COEFFICIENT C SUBROUTINE CALCULATECD(RE,CD1) Y=10 W=LOG(RE)/LOG(Y) IF (RE.LE.0.01)CDl=3./16.+24./RE IF(RE.GT.0.01.AND.RE.LE.20)CD1=24./RE*(1+0.1315*RE**(0.82-0.05*W)) EF(RE.GT.20.AND.RE.LE.260)CD1=24./RE*(1+0.1935*RE**0.6305) IF(RE.GT.260.AND.REXE.1.5E3)CDl=10.**(1.6435-1.1242*Wmi558*W*W) IF(RE.GT.1.5E3.AND.RE.LE.1.2E4)CD1=10.**(-2.4571+2.5558*W + -0.9295*W*W+0.1049*W*W*W) IF(1<JE.GT.1.2E4.AND.RE.LE.4.4E4)CD1=10.**(-1.9181+0.6370*W + -0.0636*W*W) LF(RE.GT.4.4E4.AND.RE.LE.3.38E5)CDl=10.**(-4.3390+1.5809*W 239 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model + -0.1546*W*W) IF(RE.GT.3.38E5.AND.RE.LE.4.E5)CD1=29.78-5.3*W IF(TlE.GT.4.E5.AND.RE.LE.l.E6)CDl=-0.49+0.1*W EF(RE.GT. 1 .E6)CD 1=0.19-8.E4/RE R E T U R N E N D C C SUBROUTINE NONLINE TO FIT PARTICLE SIZE DISTRIBUTION C SUBROUTINE NONLINE (DP50E, RNE, DP50M,RNM,DPMAX,DPP,CD,N,N1) P A R A M E T E R (M=2,L=1,K=1) DIMENSION AK(M),Y1(K,N),X(L,N),Y(X,N),CD(N1),DPP(N1) NSIG=4 AK(1)=DP50E AK(2)=RNE ZZ=0 X (1,N)=DPMAX Y1(1,N)=0 DO 20 I=N-1,1,-1 X(1,I)=DPP(I) ZZ=ZZ+CD(I) Y1(1,I)=ZZ 20 CONTINUE C A L L PARA(K,L ,M,N,X,Y1,Y,AK,NSIG) DP50M=AK(1) RNM=AK(2) R E T U R N E N D C C SUBROUTINE P A R A TO DO NONLINEAR REGRESSION C SUBROUTINE PARA(K,L ,M,N,X,Y1,Y,AK,NSIG) DIMENSION AK(M),Y1(X,N),X(L,N),G(X,M),A(>I,M+1),DK(M),Y(K,N) DIMENSION C(>LM) > YY(K),B(M),AA(M,M),AA1(M,M) D A T A GAMA,UU/0.2,1/ DO 5 I=1,M DK(I)=0.0 5 CONTINUE JN=0 100 D O 8 0 I = l , M AK(I)=AK(I)+UU*DK(I) 80 CONTINUE S1=0. DO 22 I=1,M DO 25 J=1,M+1 A(I,J)=0.0 IF(I.EQ.J)A(I,J)=GAMA*GAMA 25 CONTINUE 22 CONTINUE DO 40I=1,N C A L L GG(X,Y,AK, I ,K,L,M,N,G) C A L L JZXC2(G,K,M,C,1) D O 5 0 J = l , M D 0 55JJ=1,M 240 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model A(J,JJ)=A(J,JJ)+C(J,JJ) 55 CONTINUE 50 CONTINUE S2=0. DO 60 II=1,K YY(n)=YiaU)-Y(II,I) S2=S2+YY(n)*YY(II) 60 CONTINUE S1=S1+S2 C A L L JZXC1(G,K,M,YY,B,1) DO 70 J=1,M A(J,M+1)=A(J,M+1)+B(J) 70 CONTINUE 40 CONTINUE IF(JN.EQ.0)GOTO 110 IF(S1-S.GT.0.)THEN UU=UU*0.5 GOTO 100 ENDIF 110 DO200 I= l ,M DO 210 J=1,M AA(LJ)=A(I,J) 210 CONTINUE 200 CONTINUE C A L L GSXQF1 (A,M,DK) S=S1 Z=0. D 0 75I=1,M Z=Z+ABS(DK(I)/AK(I)) 75 CONTINUE JN=JN+1 IF(JN.GT. 100000)THEN WRITE(6,*) ' ITERATION » ' , JN R E T U R N ENDIF Zl=real(M)*l. /10**NSIG IF(Z-Zl.GT.0)GOTO 100 C A L L INVERS(AA1,AA,M) DE2=S/(N-M) R E T U R N E N D C C SUBROUTINE G G TO C A L C U L A T E JACOBIAN M A T R I X OF R O S I N - R A M M L E R EQUATION C SUBROUTINE GG(X,Y,AK,I ,K,L,M,N,G) DIMENSION AK(M),X(L,N),G(K,M),Y(K,N) XR=AK(l)/((LOG(2.))**(l/AK(2))) Y( 1,1)=EXP(-(X( 1,I)/XR)** AK(2)) YY=(X( 1,1)/AK( 1))** AK(2) YD=LOG(X(l , I ) /AK(l ) ) XX=LOG(10.) D Y D N = Y D * Y Y G(l, l)=y(l , i )*LOG(2.)*AK(2)*(X(l, I) /AK(l))**(AK(2)-l) + *X(1,I)/AK(1)**2 G(l,2)=-Y(l, I)*LOG(2.)*DYDN 241 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model R E T U R N E N D C C SUBROUTINE JZXC2 C SUBROUTINE JZXC2(A,L,M,C,LA) DIMENSION A(L,M), C(M,M) D O 1 0 K = l , M DO 20 I=1,M S=0.0 DO 30 J=1,L EF(LA.EQ. 1)S=S+A(J,I)*A(J,K) IF(LA.EQ.0)S=S+A(I,J)*A(J,K) 30 CONTINUE C(I,K)=S 20 CONTINUE 10 CONTINUE R E T U R N E N D C C SUBROUTINE JZXC1 C SUBROUTINE JZXC1(A,M,N,B,C,LA) DIMENSION A(M,N), B(M),C(N) DO 10 I=1,N S=0.0 DO 20 J=1,M IF(LA.EQ. 1)S=S+A(J,I)*B(J) IF(LA.EQ.0)S=S+A(I,J)*B(J) 20 CONTINUE C(I)=S 10 CONTINUE R E T U R N E N D C C SUBROUTINE INVERSE TO C A L C U L A T E INVERSE M A T R I X C SUBROUTINE INVERS(ZAF,ZA,N) DIMENSION ZA(N,N),ZAF(N,N),G(N,N+N) DO 10k=l,N DO 20 J=1,N G(K,J)=ZA(K,J) G(K,J+N)=0.0 EF(J.EQ.K)G(K,J+N)=1 20 CONTINUE 10 CONTINUE D O 3 0 K = l , N DO 40 I=K,N IF(Ga,K).NE.0.0)GOTO 50 40 CONTINUE R E T U R N 50 DO 60 J=K,2*N B=G(K,J) G(K,J)=G(I,J) Ga,J)=B 60 CONTINUE 242 Appendix I: Fortran Program to Solve the Unsteady State Hydrodynamic Model C=1.0/G(K,K) DO 70 J=K,2*N G(K,J)=C*G(K,J) 70 CONTINUE DO 80 I=1,N IF(I.NE.K) T H E N C=-Ga,K) DO 90 J=K,2*N G(I,J)=G(I,J>C*G(K,J) 90 CONTINUE ENDIF 80 CONTINUE 30 CONTINUE DO100I=l ,N DO 110 J=1,N ZAF(I,J)=G(I,J+N) 110 CONTINUE 100 CONTINUE E N D C C SUBROUTINE GXQF1 TO SOLVE A R E A L G E N E R A L S Y S T E M OF L INEAR EQUATIONS C SUBROUTINE GSXQF1(A,N,X) DIMENSION A(N,N+1), X(N) N1=N+1 DO 10K=1,N DO 20 I=K,N IF(A(I,K).NE.0.) GOTO30 20 CONTINUE R E T U R N 30 IF(I.NE.K) T H E N DO 40 J=K,N1 B=A(K,J) A(K,J)=A(I,J) A(I,J)=B 40 CONTINUE ENDIF C=1.0/A(K,K) DO 50 J=K,N1 A(K,J)=C*A(K,J) 50 CONTINUE DO60I=l ,N EF(I.NE.K) T H E N C=-A(I,K) DO 70 J=K,N1 A(I,J)=A(I,J)+C*A(K,J) 70 CONTINUE ENDIF 60 CONTINUE 10 CONTINUE DO 80 1=1,N X(I)=A(I,N1) 80 CONTINUE R E T U R N E N D 243 

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