University of Wollongong Research Online University of Wollongong Thesis Collection University of Wollongong Thesis Collections 1991 Gravity flowrate of bulk solids from mass flow bins Zhi Hong Gu University of Wollongong Recommended Citation Gu, Zhi Hong, Gravity flowrate of bulk solids from mass flow bins, Doctor of Philosophy thesis, Department of Mechanical Engineering, University of Wollongong, 1991. http://ro.uow.edu.au/theses/2095 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au GRAVITY FLOWRATE OF BULK SOLIDS FROM MASS FLOW BINS A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy from The University of Wollongong by Zhi Hong G u B.Sc. ( NEUT ), M.Sc. ( NEUT ) Department of Mechanical Engineering 1991 DECLARATION This is to certify that the work presented in this thesis was carried out by the author in the Department of Mechanical Engineering of the University of Wollongong and has not been submitted for a degree to any other university or instituition. Zhi H o n g G u i Acknowledgements I would like to acknowledge my supervisor Professor P.C.Arnold, Head of Department of Mechanical Engineering, The University of Wollongong, for his grateful supervision, assistance, encouragement and other help during the period of this study. I also wish to thank m y co-supervisor Dr. A. G. McLean, Senior Lecturer in the Department of Mechanical Engineering of the University of Wollongong for his helpful suggestions and supervision during m y last two years of P h D candidature. I am very grateful to the University of Wollongong for providing me a Postgraduate Research Award for this study. Acknowledgement is also made to all the staff of the department, especially to Mr M . Wall and Mrs R. Hamlet. M a n y thanks are extended to the technical staff in the Workshop and Bulk Solids Handling Laboratory in this department with whose help and expertise the experimental apparatuses were constructed, in particular to Mr. D. Cook, Mr. S. Rodd, Mr. S. Watkins, Mr. R.Young and Mr. K. Maywald. I a m grateful to Mr. D. Jamieson for his help in using the computer systems. Finally, special acknowledgement is made to my dear wife Jun Dan, my son Yu Fan and m y parents for their encouragement and considerable help. ii ABSTRACT The gravity flow of bulk solids from mass flow bins and hoppers is a subject of considerable practical and theoretical interest M a n y published papers have brought greater understanding of the flowrate of coarse particles. Research on finer materials was insufficient due to imprecise predictions of the air pressure gradient at the hopper outlet, which is a very important factor in predicting the flowrate of the fine particles. The work in this thesis develops theoretical models for predicting the interstitial air pressure gradient and for predicting the flowrate of bulk solids from conical mass flow bins. T h e theoretical models are based on the continuum mechanics theory. T h e boundary conditions follow air pressure and bulk density continuity in a vertical direction. A close agreement between theoretical results and experimental results was obtained for predicting both flowrate and air pressure distribution. Both theoretical and experimental results indicate that the flowrate of free flowing bulk solids increases rapidly at first and then more gradually as the particle permeability constant increases. The use of 'permeability' enables the theories developed to be applied in describing the flow behaviour of both coarse and fine particle mixtures as well as composite particle size-distributed bulk solids. The use of consolidation-related bulk density and permeability enables the theoretical models to be applied to both compressible and incompressible materials. iii Results of both theoretical and experimental work on the effect of material surcharge level on the flowrate indicate that for fine materials this effect is significant Based on the original theoretical model, simplified expressions for predicting the dynamic deaeration coefficient K J and a simplified flowrate model are presented. Finally, the study is extended to examine, theoretically and experimentally, the use of standpipes attached at the hopper outlet to increase the limiting gravitational flowrate. iv Table of Contents Acknowledgements i Abstract ii Table of Contents iv List of Figures ix List of Plates xvii List of Tables xviii Nomenclature xx Chapter 1 Introduction 1 2 Literature Survey 7 2.1 Research on Flowrate without Air Retardation 2.2 Research on Flowrate with Air Retardation 7 20 3 Measurements of Bulk Solids Flow Properties 35 3.1 Introduction 35 3.2 Particle Size and Particle Size Distribution 37 3.3 Effective Angle of Internal Friction and Angle of Wall Friction 39 3.4 Particle Density 41 3.5 Bulk Density 41 3.5.1 Bulk Density Measurement 44 3.5.2 Models Used to Fit Experimental Results 45 3.5.3 The Models Appropriate to the Bulk Solids 47 3.5.4 Application of the Bulk Density Equation 54 3.6 Permeability 61 3.6.1 Permeability Measurement 61 3.6.2 Models Used to Fit Experimental Results 63 The Prediction of Air Pressure Gradients in Mass Flow Bins 67 4.1 Introduction 67 4.2 The Bulk Density Distribution in Mass Flow Bins 69 4.3 Superficial Velocity of Air Relative to Particles 74 4.3.1 Flowrate of Air 74 4.3.2 Relative Velocity of Air to Powder 75 4.4 Air Pressure Gradients in Mass Flow Bins 76 4.5 Air Pressure Distributions in Mass Flow Bins 78 4.6 Boundary Conditions for Application of Air Pressure Model 80 4.7 Discussion 81 4.7.1 General Observations 81 4.7.2 Comparison of Theoretical Model with Experiments 85 Prediction of the Flowrate of Bulk Solids from Mass-Flow Bins 96 5.1 Introduction 5.2 Theoretical Model for Predicting the Flowrate 5.3 96 98 5.2.1 Equation of Motion 5.2.2 Equation of Continuity 100 5.2.3 Equation for Predicting the Flowrate 102 Application of Theoretical Model 98 104 5.3.1 The Effect of Surcharge Level on the Flowrate 104 5.3.2 The Effect of Outlet Size on the Flowrate 113 5.3.3 The Effect of Permeability Constant on the Flowrate 119 VI 5.4 Comparison of Theoretical Results with Experimental Results 5.4.1 121 Sensitivity Analysis of the Various Terms in Theoretical Model 121 5.4.2 Comparison of Equation (5.17) with Other Flowrate Models 6 Experimental Facilities and Test Bulk Materials 125 134 6.1 Introduction 134 6.2 Test Rig for Measuring Mass Flowrate 135 6.3 Measurement of the Flowrate 139 6.3.1 Calibration Procedure 139 6.3.2 Test Procedure 142 6.4 6.5 Processing the Flowrate Data Measured 6.4.1 Original Fitting 6.4.2 Problem of Initial Fitting and its Improvement 6.4.3 Typical Flowrate Measurements 143 144 149 Preparation of the Bulk Solids Mixtures 150 6.5.1 The Range of Different Sand Mixtures 150 6.5.2 Median Particle Size and Size Distribution of the Bulk Materials 6.5.3 6.6 143 154 The Internal Friction Angle and the Wall Friction Angle 157 6.5.4 Particle Density 159 6.5.5 Bulk Density and Permeability 160 Measurement of Air Pressure Distribution in Mass Flow Bins 161 7 The Effect of Permeability on the Flowrate 168 7.1 Introduction 168 7.2 Experiments Using Glass Beads 168 7.2.1 Preparation of Glass Bead Mixtures 170 7.2.2 Mixtures of Particles for Laboratory Measurements 173 7.2.3 Mixtures of Particles for Flowrate Measurements 175 7.2.4 The Observed Particle Flowrates from the Bins 177 7.3 The Effect of Particle Size and Size Distribution 180 7.4 Discussion 184 7.5 Criterion for Classifying Coarse and Fine Bulk Solids in Terms of the Effect of Interstitial Air Pressure Gradients 189 The Effect of the Surcharge Level on the Flowrate 194 8.1 Experimental Results 194 8.2 Discussion 199 8.3 Problems Which M a y B e Caused by Increasing the Surcharge 8.4 Level 202 Summary 212 Simplification of the Model for Predicting the Particle Flowrate 217 9.1 9.2 Original Model for Estimating the Dynamic Deaeration Coefficient K d e a 217 Simplification of the Original K ^ e a Model 219 9.2.1 Random Simulation of K ^ e a 219 9.2.2 Sensitivity Analysis 222 9.2.3 Simplification of K<j e a Equation by Optimization Technique 9.2.4 Comparison of K ^ e a Using the Original Model and the Simplified Model 9.3 224 The Simplification of the Flowrate Model 231 239 10 Strategies for Increasing Limiting Flowrates 243 10.1 General Possibilities for Increasing Limiting Flowrates 243 10.2 Experiments and the Observations on Standpipe 246 10.3 Theoretical Model for Bin - Standpipe Configuration 250 10.4 Theoretical Predictions and Discussion 256 11 Conclusions 262 12 Suggestions for Further Work 268 References 270 Appendices I Theoretical Analysis Section 1-1 The Young's Modulus Coefficient K y for Bulk Solids under Uniaxial Test 1-2 283 Equivalent Major Principal Stresses in Hopper Region of a Mass Flow Bin 1-3 283 287 Difference Caused by Considering Variable Bulk Density in Walters' Equation for the Vertical Stress in Cylindrical Section of the Bins 293 II Experimental Measurement Section 295 III II-1 The Results of Particle Size Analysis by Laser Particle Sizer II-2 Instantaneous Yield Loci Measured for All Test Materials 300 n-3 Wall Yield Loci Measured for All Test Materials 307 Publications While P h D Candidate 295 314 IX List of Figures Figure Title Page 3.1 Gravity Flow of Bulk Solids from a Mass-Flow Bin 36 3.2 Yield Locus and Internal Friction Angle 40 3.3 Wall Yield Locus and Wall Friction Angle 40 3.4 Jenike Compressibility Tester (Arnold et al. 1980^) 44 3.5 A Typical Bulk Density Curve of a Fine Sand 45 3.6 The Flowchart of the Program to Calculate Minimum Variances 48 3.7 Total M i n i m u m Variance of Different Bulk Density Models for 79 Tests 3.8 3.9 49 Comparison of Experimental Results with Those Fitted by Model 13 in L o w Stress Range 50 Replotting of Figure 3.7 b) with Modified Model 13 51 3.10 The Typical Fitting Results for Best Fitting Models with Zero Stress Excluded 51 3.11 The Typical Fitting Results for Best Fitting Models with Zero Stress Included 52 3.12 Worst Fitting of Models 9,14,16 and Modified Model 13 52 3.13 Best Fitting of Models 9,14,16 and Modified 13 53 3.14 Pressing of Bulk Material in T w o States 55 3.15 Under-Estimation of Case I for Bulk Density for Plane Flow Hopper and Conical Hopper Compared with the Results of Case II 60 3.16 Jenike Permeability Tester (Arnold et al. 1980[3]) 62 3.17 Comparison of Fitting Results for Three Permeability Models 65 3.18 Worst Fitting of Permeability Model 2 66 3.19 Best Fitting of Permeability Model 2 66 Figure Title Page 4.1 Regions Defined for a Mass Flow Bin 69 4.2 Stress Field in Hopper (McLean 1979 [58] ) 4.3 A Plot of the Relation between T|mp, rj0 and T| m a x 4.4 Typical Distributions of Air Pressure and Bulk Density in a Bin (with Surcharge) 4.5 91 92 Comparison of Theoretical Results with Head's Experimental Results (Sand No. 6) 4.9 90 Comparison of Theoretical Results with Head's Experimental Results (Sand No. 10) 4.8 83 Comparison of Theoretical Results with Crewdson et al.'s Experimental Results 4.7 81 Comparison of Theoretical Results with Experimental Results for Alumina 4.6 73 Predicted Stress Distributions of Alumina During Flow 4.10 Predicted Bulk Density Distributions of Alumina During R o w 93 94 95 5.1 Flow from Region IU of a Conical Mass-Flow Bin 98 5.2 Predicted Flowrate Varying with H/D Ratio for Alumina, P V C Powder and Sugar 106 5.3 Predicted Flowrate Varying with H/D Ratio for Sand M l to Sand M 6 107 5.4 Predicted Flowrate Varying with H/D Ratio for Sand M D 1 to M D 4 108 5.5 Predicted Flowrate vs. H/D Ratio for Sand M D 3 109 5.6 Predicted Flowrate vs. H/D Ratio for Sand M 5 109 5.7 Predicted Flowrate vs. H/D Ratio for Sand M D 1 110 5.8 Predicted Flowrate vs. H/D Ratio for Sand M l 110 XI Figure Title Page 5.9 Predicted Air Pressure Gradient at Hopper Outlet vs. H/D Ratio for SandMD3 111 5.10 Predicted Air Pressure Gradient at Hopper Outlet vs. H / D Ratio for SandM5 111 5.11 Predicted Air Pressure Gradient at Hopper Outlet vs. H / D Ratio for SandMDl 112 5.12 Predicted Air Pressure Gradient at Hopper Outlet vs. H / D Ratio for Sand M l 112 5.13 Periodic Flow Behaviour for Fine Material under Fluidisation Condition 114 5.14 Predicted Flowrate vs. the Hopper Outlet Diameter for Sand M D 3 115 5.15 Predicted Flowrate vs. the Hopper Outlet Diameter for Sand M 5 115 5.16 Predicted Flowrate vs. the Hopper Outlet Diameter for Sand M D 1 116 5.17 Predicted Flowrate vs. the Hopper Outlet Diameter for Sand M l 116 5.18 Predicted Air Pressure Gradient at the Hopper Outlet vs. the Hopper Outlet Diameter for Sand M D 3 117 5.19 Predicted Air Pressure Gradient at the Hopper Outlet vs. the Hopper Outlet Diameter for Sand M 5 117 5.20 Predicted Air Pressure Gradient at the Hopper Outlet vs. the Hopper Outlet Diameter for Sand M D 1 118 5.21 Predicted Air Pressure Gradient at the Hopper Outlet vs. the Hopper Outlet Diameter for Sand M l 5.22 Predicted Flowrate for Varying Permeability Constant 118 120 5.23 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at Hopper Outlet for Sand M D 3 123 Xll Figure Title Page 5.24 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at Hopper Outlet for Sand M 5 123 5.25 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at Hopper Outlet for Sand M D 1 124 5.26 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at Hopper Outlet for Sand M l 124 5.27 Comparison of Theoretical Results with Experimental Results (D 0 = 0.02 m & H/D = 0.07) 127 5.28 Comparison of Theoretical Results with Experimental Results (D 0 = 0.02 m & H/D = 1.5) 128 5.29 Comparison of Theoretical Results with Experimental Results (D 0 = 0.0445 m & H/D = 0.07) 129 5.30 Comparison of Theoretical Results with Experimental Results (D 0 = 0.0445 m & H/D = 1.5) 130 5.31 The Percentage Error Between the Flowrates Obtained by Current Model and Brown's Theory for Coarse Bulk Solids 133 6.1 Mass Flowrate Qp v. H/D Ratio (Smith 1978[53]) 134 6.2 Schematic of the Double - Bin Apparatus 136 6.3 The Schematic of Bin Hanger 137 6.4 Transducer Circuitry 138 6.5 Instrumentation Schematic 139 6.6 Calibration Arrangement 140 6.7 Typical Calibration Recorder Output 141 6.8 Comparison of Indicated Load with Actual Load 142 xin Figure Title Page 6.9 Flowrate Prediction for Four Test Runs at One Surcharge Level for Coarse Sand (Sand M l ) 145 6.10 The Flowrate of Every Run after Smoothing 147 6.11 Flowchart of Fitting Procedure 148 6.12 Comparison of Flowrates for Different Loads Measured on the Double Bin Apparatus with 0.0445 m Outlet 149 6.13 Schematic of Large Screen Siever 152 6.14 The Frequency Distribution of Selected Sand Mixtures 153 6.15 Cumulative Size Distribution for Sugar, Alumina and P V C Powder 156 6.16 Cumulative Size Distribution for S and M 1 to M 7 156 6.17 Cumulative Size Distribution for Sand M D 1 to M D 4 157 6.18 Flowability Characteristic in Free Flowing Zone (According to Jenike's Flowability Zones'- ••) 159 6.19 The Apparatus for Measuring Air Pressure Distribution 163 6.20 Large Test Bin Geometry and Pressure Tapping Locations 164 6.21 Schematic of the Surcharge Level Control System 167 7.1 Mixture Bulk Density Characteristics 173 7.2 Mixture Permeability Characteristics 174 7.3 Mixture Equivalent Surface Area Characteristics 174 7.4 The Flowrates vs. Permeability Constant 179 7.5 Flowrate vs. Median Particle Size at Surcharge Level H/D = 1.5 180 7.6 Measured Flowrate vs. Median Particle Size for Sand M l to M 5 and 7.7 S a n d M D l t o M D 4 ( H / D = 1.5) 183 Flowrate v. Permeability for Sand Mixtures 183 Figure Title Page 7.8 Predicted Negative Air Pressure Distribution for Bulk Solids (from 0.02 m outlet) 7.9 185 Predicted Negative Air Pressure Distribution for Bulk Solids (from 0.0445 m outlet) dP 7.10 -«- at the Bin Outlet vs. Permeability Constant of Particles 185 186 7.11 Predicted Air Pressure Distribution for Sand M l (at H/D = 1.5) 187 7.12 Predicted Air Pressure Distribution for Sand M D 3 (at H/D = 1.5) 187 7.13 Negative Air Pressure Distribution (Willis[54]) 188 7.14 The Critical Curve of Air Affected Flow in Terms of Particle Diameter (Flowrate Data from Crewdson et al.' -fy 193 8.1 Measured Flowrate vs. H/D Ratio for PVC Powder and Sugar 196 8.2 Measured Flowrate vs. H/D Ratio for Sand M l to Sand M 5 197 8.3 Measured Flowrate vs. H/D Ratio for Sand M D 1 to Sand M D 4 198 8.4 Comparison of Experimental Results with Theoretical Results for Sand Mixtures Flowing from 0.0445 m Outlet 8.5 Pseudo-Steady Flowrate Affected by Surcharge Level (from 0.02 m Hopper Outlet) 8.6 208 Weight Curves for Discharge of 55 Lim Calcite (adopted from Miles (1968)[44]) 8.9 207 Mass Variation Plots for Sand M D 3 , Sand M 6 and Sand M 7 from 0.0445 m Outlet 8.8 206 Mass Variation Plots for Sand M D 3 , Sand M 6 and Sand M 7 from 0.02 m Outlet 8.7 204 209 Flowrate Variation Plots for Sand M 7 Discharging from 0.0445 m Outlet 210 XV Figure Tide Page 8.10 Variable Speed Belt Feeder 211 8.11 Flowrate Variations with Different Belt Feeder Velocities for Sand M 7 from 0.0445 m Hopper Outlet 214-215 8.12 The General Description of Flowrate Affected by Surcharge Level 216 9.1 The flowchart of Uniform 1 -1 Generator 221 9.2 Total Variance for Every Component (5000 Samples) 223 9.3 Total Variance for Every Case (5000 Samples) b, b, Typical Variations of K ^ with the — ratio and — 4 n r ratio 224 9.4 Po Po e 226 bi 9.5 9.6 9.7 9.8 Typical Variations of K d e a with sin(8) and * n < ratio p0(sin 8 ) U 1 Program Flowchart for Optimization Processing Comparison of K d e a Predicted by Simplified Model 1 with by 230 Original Model 233 Comparison of K(j e a Predicted by Simplified Model 2 with by Original Model 9.9 226 233 Comparison of K<i ea Predicted by Simplified Model 3 with by Original Model 234 9.10 Comparison of Simplified Model 2 with Simplified Model 1 234 9.11 Comparison of Simplified Model 3 with Simplified Model 1 235 9.12 Comparison of Predicted Pressure Gradient at Outlet Based on Simplified K^ea Model 1 with that on Original K^ea Model 235 9.13 Comparison of Predicted Pressure Gradient at Outlet Based on Simplified K^ea Model 2 with that on Original K ^ a Model 236 9.14 Comparison of Predicted Pressure Gradient at Outlet Based on Simplified K ^ Model 3 with that on Original K d e a Model 236 Figure Title Page 9.15 Comparison of Predicted Particle Flowrate Based on Simplified K ^ Model 1 with that on Original K^ea Model 237 9.16 Comparison of Predicted Particle Flowrate Based on Simplified Kjjea Model 2 with that on Original K^ea Model 237 9.17 Comparison of Predicted Particle Flowrate Based on Simplified K^ea Model 3 with that on Original K^ea Model 238 9.18 Comparison of Simplified Flowrate Model with Original Flowrate Model For the Experimental Parameters Examined 242 10.1 Schematic of Double - Hopper Bin Arrangement 244 10.2 Standpipe Installed at the Outlet of the Test Bin 10.3 Flowrate Q p vs. the L / D 0 Ratio of Standpipe for Alumina 248 Discharging from 0.0445 m Outlet (Surcharge Level H = 0.31 m ) 249 10.4 Typical Discharge Modes from a Standpipe 249 10.5 Flowrate vs. Feeder Belt Velocity for Different Clearances between Feeder and Standpipe Outlet (standpipe L/D 0 = 5) 250 10.6 Four Regions for Bin - Standpipe Arrangement 251 10.7 Comparison of Theoretical with Experimental Results 10.8 Flowrate Enhancement Factor for Different Materials (X = 0.75) 257 10.9 Predicted Air Pressure at the Hopper Outlet (X = 0.75) 260 10.10 Predicted Air Pressure Gradient at the Hopper Outlet (X = 0.75) 256 260 10.11 Predicted Air Pressure Distribution in the Bin and the Standpipe for Alumina (X = 0.75) 261 10.12 Predicted Air Pressure Distribution in the Bin and the Standpipe for Sand M l (X = 0.75) 261 Figure Tide Page A-I-1.1 A n Element of Bulk Solids Compacted by Three Stresses 284 A-I-1.2 Compressibility Tester 284 A-I-2.1 The Cylindrical Coordinates for the Plane Flow Hopper 288 A-I-2.2 The Spherical Coordinates for the Conical Hopper 289 A-II-2.1 Instantaneous Yield Loci for Alumina 300 A-H-2.2 Instantaneous Yield Loci for P V C Powder 300 A-E-2.3 Instantaneous Yield Loci for Sugar 301 A-H-2.4 to 2.10 Instantaneous Yield Loci for Sand M l to Sand M 7 A-II-2.11 to 2.14 Instantaneous Yield Loci for Sand M D 1 to Sand M D 4 301-304 305-306 A-H-3.1 Wall Yield Loci for Alumina 307 A-H-3.2 Wall Yield Loci for P V C Powder 307 A-H-3.3 Wall Yield Loci for Sugar 308 A-n-3.4 to 3.10 Wall Yield Loci for Sand M l to Sand M 7 A-H-3.11 to 3.14 Wall Yield Loci for Sand M D 1 to Sand M D 4 308-311 312-313 List of Plates Plate Title Page 6.1 Test Rig for Measuring Particle Flowrate 10.1 Bin - Standpipe - Belt Feeder Arrangement 10.2 Standpipes Used in Experiments * (-) opposite page 136(-)* 248(-) 248(-) xviii List of Tables Table Title Page 3.1 Shear Test Methods and Their Comparison (based on Schwedes 1983 t92] and Schwedes et al.l990[93'94]) 3.2 The Models for Bulk Density - Stress Relation 42 46 4.1 The Pressure Maxima and the Ratio between Two Extreme Pressures for Different Sized Bulk Solids and Hopper Outlet Sizes 85 4.2 The Flow Properties, Bin/Hopper Geometry and Observed Flowrates for the Experiments of Crewdson et al. and Head 89 5.1 The Comparison of Eqn (5.17) and Nedderman et al.'s Method with Head and Crewdson et al.'s Experiments 126 6.1 Bin Details 137 6.2 The Calibration Mass Increments 140 6.3 The Range of Particle Size for River Sand Mixtures 151 6.4 The Median Particle Size of the Test Materials 155 6.5 The Shear Test Results for the Test Materials 158 6.6 Measured Particle Densities 159 6.7 Measured Bulk Density Properties 160 6.8 Measured Permeability Properties 161 7.1 Bulk Densities of Glass Bead Mixtures 170 bi v 7.2 The - ^ Ratio for the Bulk Solids Prepared in Section 6.5.5 171 Po 7.3 Comparison of the Three Mixtures Parameters 176 7.4 The Observations for Mixtures with Similar Bulk Density 177 Table Title Page 7.5 The Observations for Mixtures with Similar Permeability 177 7.6 The Observations for Mixtures with Similar Equivalent Surface Area 178 7.7 The Relation between Experimental Flowrate from 0.02 m Outlet and Median Particle Size or Permeability Constant 182 7.8 The Relation between Experimental Flowrate from 0.0445 m Outlet and Median Particle Size or Permeability Constant 182 8.1 Actual Surcharge Level of Solids in the Test Bin with 0.02 m Outlet 194 8.2 Actual Surcharge Level of Solids in the Test Bin with 0.0445 m Outlet 195 8.3 Comparison of the Results with Other Previous Works in Terms of the Effect of Surcharge Level on Flowrate 203 8.4 The Classification of the Flow for All Sand Mixtures Used in Experiments in Terms of the Effect of Material Surcharge Level 213 9.1 The X components in Simulation 220 9.2 Simplified K ^ Model Minimization Results 231 10.1 A Summary of the Results in Using Standpipes 258 A-I-3.1 The G Values for the Materials Used in Experiments 294 The Results of Particle Size Analysis by Laser Particle Sizer: A-H-l.l for Alumina 295 A-H-1.2 for P V C Powder 296 A-n-1.3 to 1.8 for Sand M l to Sand M 6 A-H-1.9 for Sand M D 1 296-299 299 XX Nomenclature a exponent used to relate permeability to consolidation stress a** equivalent surface area of particles (* 10 -12 m 2 ) a n , b ll' c ll coefficients for flowrate model, eqn (5.17) &22' b22 coefficients for simplified flowrate model, eqn (9.21) A cross sectional area of flow channel (m 2 ) Ay cross sectional area of the hopper outlet (m 2 ) Ap cross sectional area of permeability test cylinder (m 2 ) b exponent used to relate bulk density to stress in eqn (2.7) bi, b 2 , b 3 constants used to relate bulk density to stress in density equation models B optimum parameter array in Chapter 3 B0 initial value of B in Chapter 3 Bf constant in Beverloo's flowrate model, eqn (2.1) c, cj, C2 constants C permeability of bulk solids, * 10 -9 ( m 4 N ^ s e c 1 ) CQ permeability constant of bulk solids (permeability at lowest compaction), * 10 -9 (m 4 N 4 sec 4 ) CCTi critical permeability value classifying the coarse and fine material, *10- 9 ( m 4 N 4 s e c 4 ) dso median particle size (|im) dg equivalent particle diameter used by in eqns (2.9) to (2.11) (|im) d« particle size (Lim) in Beverloo's model, eqn (2.1), the unit of dp is metre) D diameter of vertical section of the bin (m) DQ outlet diameter of the hopper (m) bulk density ratio, as defined by f = — Po bulk density ratio in Regions I, II, III, respectively, defined by eqns (9.2) to (9.6) function f of x first derivative of f with respect to x second derivative of f with respect to x flow factor for a converging channel, in eqn (2.3) and eqn (2.7) 'actual' flow factor for a flow situation, in eqn (2.3) flowrate enhancement factor due to standpipe acceleration due to gravity, gravitational acceleration (m/sec2) 1 dP effective gravitational acceleration, g* = g + ( j-) __ (m/sec2) Pout ° variable defined by eqn (A-3.8) vertical distance from the vertex of the flow channel (m) vertical distance from the vertex of the flow channel to outlet of hopper (m) vertical distance from the vertex of the flow channel to the transition of a hopper (m) height where the stress is assumed to be zero (near the hopper outlet), h* = r* cos a (m) surcharge level used in eqn (2.7) (m) material surcharge level, as defined as the height of bulk solids in vertical section of the bin (m) dimensionless surcharge level, defined in eqn (4.1) and eqn (10.1) height of powder bed (m) a function of the ratio of the volume of the arch to the perimeter of the arch, in eqn (2.7) k constant in Beverloo s flowrate model, eqn (2.1) kh k 2 constants Kj , K2 , K3 constants K^ea • ^ • J • • • , ^ l+sin8 ratio of major and minor principal stresses K = 1-sin 8 dynamic deaeration coefficient of bulk solids in the mass-flow bins K$ea dynamic deaeration coefficient evaluated by original K ^ model K^Ja dynamic deaeration coefficient evaluated by simplified K(j e a model Ky coefficient related to Young's modulus in Chapter 3 and Appendix I-1 Ks constant, K s = ( g ° ) 1 / b 2 K^ variable used in Walters' stress theory L length of standpipe (m) M mass flowrate of particles (considering the sense of flow), M = - Q p K (kg/sec) P air pressure in voids between particles (Pa) P0 atmospheric pressure, P Q = 0 (Pa) Pj air pressure at transition level (Pa) Q volumetric flowrate of air (m3/sec) QQ volumetric flowrate of air at top level of material in a bin (m3/sec) Qp flowrate of a bulk solid (kg/sec) Qpujax flowrate of a bulk solid unaffected by pressure gradient (kg/sec) r radial distance from the vertex of the flow channel (m) r* the distance from the vertex of the channel to where the stress is assumed to be zero, r* = 0.95 TQ ~ 0.99 TQ (m) r0 radial distance from the vertex to hopper outlet (m) rj radial distance from the vertex to the transition of a hopper (m) u relative velocity of air to the particles (m/sec) U variable defined by eqn (9.29) V particle velocity (m/sec) V5 bulk solids volume after compaction (m 3 ) Vbo bulk solids volume before compaction (m 3 ) Vbelt velocity of feeder belt (m/sec) V0 particle velocity at the hopper outlet (m/sec) Vt Wt terminal velocity of a bulk solid, in Table 10.1 (m/sec) 4hiM coefficient defined by W n = * (Pa) ° C0p07tD02k net weight of particles (kg) y constant used in Walters' stress theory W n 0 Y=[yj, ..., y n ] variables used in optimization analysis in Chapter 9 Z constant, Z = kdp (m) AP air pressure drop across by powder bed (Pa) AP air pressure difference caused by resistance of particles (Pa) AP0 air pressure difference caused by resistance of distribution gauze of permeability tester (Pa) APt total air pressure drop (Pa) % S and % L the percentage of small and large particles, respectively Greek Symbols a hopper half angle (degree) Tl weighting factor of the influence of the pressure gradient on flowrate 8 effective angle of internal friction of bulk solids (degree) e voidage of bulk solids En voidage of bulk solids corresponding to P Q voidage of bulk solids at one third of the w a y along the hopper wall from the hopper outlet, used in eqn (2.8) strain dimensionless vertical distance from the vertex of hopper, defined in eqn (4.1) and eqn (10.1) standpipe effect coefficient absolute viscosity of air (Pa .sec) bulk density of particles (kg/m 3 ) bulk density of particles at lowest compaction (kg/m 3 ) bulk density of particles at transition section of a bin (kg/m 3 ) bulk density corresponding to an arbitrary stress o e , used in eqn (2.7) (kg/m 3 ) density of interstitial fluid (kg/m 3 ) solids density of particles fl^g/m3) principal stress (kPa) stress (kPa) major principal stress (kPa) an arbitrary stress for bulk density equation used in eqn (2.7) (kPa) mean stress in three dimensional field (kPa) m e a n stress for compressibility tester (kPa) equivalent major principal stress (kPa) computed m e a n stress in plane hopper (kPa) computed m e a n stress in conical hopper (kPa) Poisson's ratio of bulk solids Poisson's ratio of bulk solids at m i n i m u m voidage friction angle between bulk solids and bin / hopper wall (degree) Molerus internal friction angle, infigure3.2 (degree) variance for optimization analysis in Chapters 3 and 9 CO constant used in permeability model 1 (Section 3.6.2) (VV)V convective terms, e.g., x-component in rectangular coordinat dWx dVx dWx A V_-!C* x dx + V„y~3y~ 5- + V , -r* Subscripts r r component 6 B component <D O component z z component out at hopper outlet mp minimum pressure position c conical hopper w plane flow hopper (wedge hopper) V vertical section max maximum min minimum sp standpipe Co-ordinate Svstems r, 0, z cylindrical co-ordinate system for • r,e,o spherical co-ordinate system for cc 1 Chapter 1 Introduction Bins and hoppers are used in a wide range of industries, such as agriculture, mining, chemical engineering, power plants, cement and food processing, where most bulk solids storage, handling and transportation systems are applied. O n e of the most important requirements is that the material should discharge smoothly and continuously w h e n the outlet is opened. O n e practical problem involved in designing the handling systems for particulate materials is the attainment of an adequate flow of material and the control of the flow at some desired rate. When bulk solids are allowed to flow out of a bin or hopper under gravity alone, it flow pattern can be basically of two types: mass flow or funnel flow^1 \ With mass flow, the hopper is sufficiently steep and smooth to cause flow of all the solids in the bin without 'dead' regions occurring during discharge. B y contrast, funnel flow occurs w h e n the hopper is not sufficiently steep and smooth to force material to slide along the walls or when the outlet of a bin is not fully effective, due to poor feeder or gate design. The bulk solids flow toward the outlet through a vertical channel that forms within stagnant material powders. From the viewpoint of processing, mass flow is preferred in making the bulk solids processing system efficient, reliable, predictable and more easily controlled. The flow of solid particles from mass flow bins or mass flow hoppers is, therefore, a subject of considerable practical and theoretical interest. During the past three decades, considerable advances have been made in the general understanding of the flow behaviour of particulate solids^ ' ' . Most publications deal with theoretical and experimental work concerning the flow 2 patterns in the bins and the mass flowrates which depend on the bulk solid itself and the bin/hopper geometry. Generally speaking, previous research on the flowrate of bulk solids from m a s s flow bins or hoppers can be classified as follows: i) according to different bulk materials concerned: coarse - cohesionless - incompressible - no effect of air pressure gradient on the flowrate considered' " ^ fine - relatively cohesionless - compressible - retarding effect of air pressure gradient on the flowrate considered™ " ^ ii) according to different bin/hopper configuration: conical bnVhoppert9-13'15'23-25'27'28'32'36-38'49'50'55-58'60] plane flow bi^oppert 2 2 ' 3 0 ' 3 1 ' 3 3 ' 3 4 ' 3 6 - 3 9 ' 5 1 - 5 4 ' 5 8 ^ flat-bottomed bint 1 0 - 1 2 ' 1 4 - 1 8 ^ iii) according to the models of calculating the flowrate: empirical correlations™ ' theoretical analysis based on: - fluid analogies[20,21] - continuum mechanics theoryt 22 - 24 ' 27 - 28 ' 31 - 33 ' 36 " 39 ' 46 ' 49 ' 51 ' 52 ' 54-56,58] - m i n i m u m energy theory^ ^ - stochastic model 125 - 26 ' 1133 - kinematic model^ 29 ' 303 Finite Element Analysis (a few applications for coarse material; normally associated with the determination of the stress field)™' * 3 iv) in terms of the method of solution used for the theoretical models simplification by assumptions, boundary conditions'22'27'28'34'38' 46,47,49,50,52,55-57,60] perturbation technique^24,31'32' numerical analysis'3 , 7 , ^S4'*8*^] Almost all the empirical correlations, Finite Element Analyses and a number of theoretical equations at present available in the literature to predict the flowrate of particles discharging from mass flow bins are applied only to coarse particles. A m o n g these studies, the most successful are those obtained by Beverloo et alJ ^ B r o w n ^ 1 J, Johanson^ \ They are verified by subsequent research. However, w h e n particles becomefiner,but still substantially cohesionless, their motions are significantly different from those of coarse material. The equations from the studies on coarse material often overpredict the flowrates for fine materials' ' ' 1 , a situation which needs to be avoided when dealing with practical applications. An important difference between the flow behaviour of coarse and fine materials is that the flowrate of fine particles is affected significantly by the interstitial air pressure resistance. S o m e attempts have been m a d e to consider this effect on the flowrate^43"60^. O f these attempts, Nedderman and his co-workers have paid more attention to and m a d e great progress in examining the flowrate affected by interstitial air pressure gradients^ 50 ' 51 ' 56 ' 571 . Particularly, the research by Nedderman et al. in 1 9 8 3 ^ is considered the most successful of their studies. Starting from the Beverloo empirical correlation, which w a s obtained for the flowrate of coarse material from a flat-bottomed container, Nedderman et al. developed semi-empirical equations to establish the relationship between the 4 flowrate for fine material and the air pressure gradients encountered by the particles. However, the equations provided by Nedderman and his co-workers include one variable A P (air pressure difference at the hopper outlet), which cannot be predicted from their theories. Therefore, it is difficult to predict the flowrate completely from their equations. Ford and Davies in 1990^6 * concluded that' it is now clear that an air deficiency, at the hopper outlet, appears for particles smaller than 500 |im and this deficiency increases as the m e a n size decreases reducing the flow predicted by the Beverloo equation', where the constant in the Beverloo equation is determined from Nedderman et al.'s semi-empirical results concerning the effect of interstitial air pressure gradients on the particle flow. In practice, it is difficult to apply the Nedderman model to a range of actual materials. Firstly, a model for predicting the air pressure gradient at the hopper outlet needs to be established. Secondly, using a single particle diameter of 500 |im to delineate between coarse and fine particles is inadequate. T h e bulk solids processed in industry often include a range of particle sizes. M a n y researchers have found that the flow properties are affected by the particle size distributicHi^63"703, indicating that a theory needs to be developed which can be applied to materials with a distribution of particle sizes. Parallel to Nedderman's work, the effect of interstitial air pressure gradients on the flowrate of bulk solids has been also studied for several years by Arnold, M c L e a n and their co-workers, at the University of Wollongong^ 52 ' 54,58 " 603 . A s a factor in relating the particle flow to the air pressure gradient, the parameter 'permeability' was introduced into their models which are based on Carleton's force balance theoretical model' 49 ' or Johanson's actual flow factor model' 22 '. Since insufficient experiments have been done to compare with their theoretical results, the theories 5 have not yet been published. However, the concept of using the parameter 'permeability' is most valuable, although m a n y researchers evade it due to its sensitivity to voidage between particles. However, according to the Carman Kozeny equation, it is this voidage that relates the air pressure gradient under a certain superficial velocity condition. T o simulate the air pressure gradient, the variation in voidage needs to be faced, therefore, permeability can be used for adequately predicting the air pressure gradient. In addition, permeability is a characteristic of a particle mixture, not of one single particle. Since the effect of particle size distribution on the flowrate is of practical interest (Thomson 1986' 3 ) , it seems that permeability is a useful parameter to describe the relationship between pressure gradient and superficial velocity for size-distributed particles. Another difference between fine particle flow and coarse particle flow relates to effect that surcharge levels of bulk solids have on the flowrate from mass flow bins. This effect has long been considered to be slight™. However, it is noted that this conclusion was drawn from experiments on coarse materials. Considering a extremely troublesome phenomenon of fine material - flooding, which occurs when particles become veryfineor by the addition of very fine particles, say less than 40 \im (Lloyd and W e b b , 1987^ 653 ), the particles can suddenly discharge from a bin at a very high flowrate in comparison with the normal steady state flowrate. The rapid flowrate m a y approach that of an inviscid liquid (Rathbone, Nedderman and Davidson, 1987 t 7 2 3 ) which is proportional to V 2 g ( h m a x - h 0 ) , namely, the flowrate in this case varies with the surcharge level. It is believed that as particle size varies from coarse towards fine, the effect of surcharge level cannot suddenly jump from insignificant in the case of coarse material to significant in the flooding case. There m a y be some transition phenomenon in-between. Willis (1978)' \ for example, observed that the flowrate decreases as the surcharge level increases for 6 fine material ( d ^ = 1 4 0 ^im). It is necessary to find a appropriate description of the effect of surcharge level on flowrate. The work described in this thesis, aiming to predicting the flowrate of bulk solids from conical mass flow bins, focuses on the following aspects: i) modelling adequately such flow properties of bulk solids as bulk density and permeability under condition of consolidation (Chapter 3); ii) establishing a theoretical model to predict the interstitial air pressure distribution (gradient) generated by particles during discharge from mass flow bins together with some experimental verification (Chapter 4); iii) establishing a theoretical model to predict the flowrate of bulk solids from mass flow bins, which is based on continuum mechanics theory and appliabletoboth coarse and fine materials (Chapter 5); iv) exarnining the effects of permeability and surcharge level on the flowrate both theoretically and experimentally (Chapter 5,7 and 8). This thesis also includes two extension aspects: v) simplifying the calculation of the dynamic deaeration coefficient by the M o n t e Carlo simulation and optimization technique and providing a simplified flowrate model to predict the particle flowrate from mass flow bins (Chapter 9); vi) examining experimentally and theoretically the strategy for increasing the limiting flowrate offinepowders by using standpipes (Chapter 10). 7 Chapter 2 Literature Survey Much work has been done on the gravity flowrates of bulk solids from hoppers and bins since early this century. Initial work has been carried out empirically. With more understanding of the discharge of bulk materials from these early works, the attempts have been m a d e to predict discharge rates using theoretical models. However, m u c h of the work has neglected the effect of air pressure gradients for the convenience of modelling. These models, therefore, are only used for coarse material. The effect of air drag has been taken into account in theoretical modelling only after the work of Miles et al. (1968)^ 3 and has become more widely recognized. In addition, the Finite Element Method is also of interest for use in this field for some particular purposes, such as calculating the stress field in the bin or wall loads. The research reported in the literature on the flowrates of bulk solids from hoppers and bins is summarized in two different categories: one concerns flowrate without air retardation; the other includes that with air retardation. 2.1 Research on Flowrate without Air Retardation The work in this category includes three groups: empirical work, theoretical analyses and Finite Element Analyses. a) Empirical Correlations The first work on conical hoppers was studied by Deming and Mehring in 1929™3. They assumed that the flowrate varied with such parameters as hopper angle, diameter of orifice, the repose angle of bulk solids, particle size and bulk density of 8 the material. Their empirical correlation was based on dimensional analysis and their experimental results. They claimed their correlation would predict the flowrate of free flowing material above particle sizes of 200 mesh (74 jxm) with any density and particle shape. However, it is noted that the smallest particle size found in their experiments was 125 mesh (120 \s.m) for A m m o n i u m Phosphate crystals. The flowrates of this material were measured only from a very small outlet (0.001 m ) , in which the effect of air pressure gradient on the flowrate of such fine particles was not observed. Newton, Dunham and Simpson in 1945^ 3 studying the flow of catalyst pellets (2.54 - 5.08 m m in diameter), proposed that the flowrate varied with the orifice diameter and the height of material level. The authors found orifice blocking when the orifice diameter was less than six times the mean particle diameter. This limit corresponds with that suggested by Langmaid and Rose in 1957' \ as an insurance to prevent mechanical arching at the oudet Franklin and Johanson in 1955^13 studied the flow of such granules as glass beads, lead shot and puffed rice with particle diameter of 0.787~5.207 m m discharging from a cylindrical bin with an outlet varying in size from 6.6cL to 34dp. They correlated discharge rate with orifice diameter, particle diameter, panicle density and material friction. They reported no influence of material level on the flowrates observed. Fowler and Glastonbury in 1959[123 scrutinised the effects of changing orifice shape for discharge from flat bottomed bins. Materials of particle size ranging from 270 \Lm to 3300 ^tm were tested. They found that flowrate was related to hydraulic diameter of the orifice, m e a n particle size and shape factor for the material; the 9 effect of material head was found to be negligible. Rose and Tanaka in 1959^ 3 investigated the effects of the variables of hopper angle, material head, cohesive force, particle density, particle size, orifice diameter, diameter of the cylindrical section, shape factor and the repose angle of bulk solids. The variables were arranged into dimensionless groups and the effects of each group on the discharge rate were found by experiment. They concluded that the effect of material head on flowrate was insignificant based on the experiments for steel balls of 1.32 m m diameter. Experimental discharge results for particle sizes ranging from 112 \un to 910 |im were conducted to examine the effect of particle size. Linear correlations between log (flowrate) and log (D/d - 3) were observed for the particle sizes above 200 n m . For particles having diameters less than about 200 Jim, the flowrate decreased rapidly as particle size decreased. This was the first reported observation of the retarding effects of air drag on flowrate. Unfortunately, this retardation effect was explained as the effect of cohesive forces and functions of cohesion were empirically determined; no adjustment for air drag was modelled. Beverloo, Leniger and Van de Velde in 1961[143 investigated the flow of granular materials through m a n y differently shaped orifices in flat bottomed bins. Based on dimensional analysis, they proposed that discharge rate was proportional to p V g D 0 2 - 5 . A correction factor Z = kcL for D 0 must be applied to ensure their empirical correlation fit their experimental data. They presented the following equation for flowrate Qp = BfpVg(D0-kdp)2'5 (2.1) where Bf and k are constants. Beverloo et al. found Bf=0.58 and suggested k=1.4. This correlation has been found successful by many subsequent workers 150,51,56 ' 573 . Nedderman et al. noted in their review in 1982 143 that, though normally attributed to Beverloo, similar ideas had been in use prior to Beverloo's paper and indeed a correlation of this type had been proposed by Hagen as early as 1852[4]; Weighardt in 1952 [ 1 2 9 ] also gave the flowrate proportional to ( D Q - Z ) 2 5 ; and B r o w n and Richards in 1960^43 presented the concept of the 'empty annulus', where flowrate was also proportional to (Dft - Z ) 2 5 . The constant B f is dependent on hopper geometry; Nedderman et al.'-43' recommended the range 0.55<Bf<0.65; the constant k is related to particle shape and is differently valued for various materials within the range l<k<3 t513 . Bosley, Schofield and Shook in 1969^ 3 examined, in a photographic study, the effects of hopper shape, particle size, particle density and hopper size on velocity profiles in hopper discharge. In their investigations, only coarse particles (1-2.5 m m in particle size) were used to eliminate the effect of air pressure gradients. The velocity profiles were found to depend primarily on hopper shape. They claimed that m a x i m u m velocities agreed reasonably well with Brown's theoretical values but a significant effect due to wall friction was observed. Van Zuilichem and Van Egmond in 1974^163 studied the density behaviour of flowing granular material using G a m m a - R a y Absorption techniques. They observed that flowrate was proportional to the term: {a constant * (hydraulic diameter of orifice) }. b) Theoretical Analyses The initial significant theoretical analysis of the flowrate of free flowing bulk s 11 was probably the work of B r o w n and Richards; the papers in 1959^173 and 1961^183 and a book in 1970^ 193 reported their work. They used a 'minimum energy theory', established initially by Brown, to describe the flowrate of coarse incompressible bulk solids from bins and hoppers. T o obtain better correlations with experimental results, they accounted for the effect of the 'statistically empty annulus' adjacent to the aperture (a concept proposed by Wieghardt in 1952^ 1 2 9 3 ), to reduce the predicted values of flowrate. In their work, the independence of material head was also observed. T h e following equation was presented by B r o w n (1961 )^ 1 8 3 for conical hoppers: Qp = P 7c(D 0 -Z) 2 5 - ^i f 5 1 - cos a ^°' V 2 sin a It was found by m a n y researchers^ ^ ' ' (2.2) ) 3 that the flowrates computed by the Brown's theory showed a good agreement with the experimental results for coarse particles. Meanwhile, a quite different theoretical approach was taken by Jenike and his coworkers in the late 1950's to early 1960's[1,2'74"763. In their analytical work, the principles of soil mechanics and plasticity were applied to study the steady flow of bulk solids. The bulk solid was treated as arigid-plasticmaterial obeying Coulomb type yield criteria. B y neglecting the convective terms [ ( V V ) V ], the stress and the velocity fields under steady state condition were uncoupled allowing the stress distribution to be obtained first and then the boundary problem for velocities solved. Johanson in 1965' 3 developed a model for predicting flowrate which included the effects of inertia in the equilibrium of a cohesive arch of uniform thickness. This model is one of the most successful for predicting the flowrate of coarse cohesive bulk solids. Johanson derived expressions for the steady flowrate for both axisymmetric and plane flow hoppers. T h e following equation is the form for conical hoppers: -Mi--1 tan a ^ (2.3) ffa J Johanson's method involves the determination of the critical flow factor for arching ff and the actual flow factor for the material under dynamic conditions ffa, which are related to the flow properties of bulk solids and hopper geometry. E q n (2.3) gives a good prediction of the flowrate for coarse cohesive material. Experiments carried out by Johanson using several different bulk solids in both laboratory and fieldtests,supported his theory. However, he found the experimental discrepancies to be larger for finer materials, where the effect of the negative air pressure gradients becomes more significant. Therefore, eqn (2.3) is not able to predict the flowrate for fine material. Jones and Davidson in 1965^ and McDougall and Evans in 1965^ 3 described the flow of particles from an orifice by using the fluid analogy. They assumed that Bernoulli's fluid energy equation could be employed as the particles being discharged were analogous to a discharging fluid. In particular, Jones et al. studied the flow of particles through orifice plates and shaped nozzles in the side of an air fluidised bed. In their experiments the particle sizes of the bulk solids used ranged from 85 to 350 urn. The nozzles were shaped so that there were no interparticle pressure at all points in the nozzles, that is, the radius of cross-section r along the axis of the nozzles x was of the form of r = CjA/ ^ - where Cj and c 2 are constants. Their results indicated that the solids flow under the incipient fluidisation condition can be predicted by treating the fluidised solids as an inviscid liquid, in which the flowrate w a s proportional to the square root of the material head. McDougall et al. presented a theoretical equation for the flow of bulk solids from a flat bottomed bin. They retained the air pressure difference across the outlet when applying Bernoulli's theorem in their model and predicted the flowrate and the air pressure as functions of such parameters as the material level in the bin and bulk density. D u e to the difficulty of evaluating the air pressure difference, they presented a simplified model which involved the parameters of the bulk solids (particle internal friction angle and bulk density) and outlet size. However, the simplified model did not indicate well the effect of the air pressure difference for practical applications since the particle parameters internal friction angle and bulk density can describe only the interaction of particles not the interaction between particles and air. For example, the effect of the air pressure difference across the outlet is significant for fine material and insignificant for coarse material; the internal friction angle and bulk density can be applied for any material. Savage investigated the flow of bulk solids from a hopper in 1965^ 3, based on continuum mechanics theory and the Mohr-Coulomb yield criterion, in which the velocity and the stress fields are coupled by considering the convective terms in the theoretical m o d e l Compared with the B r o w n equation and the experimental results of D e m i n g and Mehring^ 93 , Savage's model presented an overestimation of the flowrate. H e extended the work in 1967 t243 and the modified model was solved by using a perturbation analysis. H e claimed that the theory agreed reasonably well with the D e m i n g and Mehring's experiments except for small values of a, where Savage suggested that the theory be modified by incorporating a correction for wall friction. This theory showed that the wall friction is a important parameter in reducing the flowrate for narrow angled hoppers. Savage showed that the change in flowrate due to wall friction could be more than 100%, whereas Nedderman et alJ 3 concluded in the review paper that the change in flowrate was rarely more than 1 0 % . Mullins in 1972 - i974i25.26,H3] pr0p0Se(j an alternative approach, named stochastic theory, to predict the velocity distribution in a hopper by modelling the flow as the upward diffusion of voids by random processes. The basic concept was the representation of convergent particle flow toward an open orifice, under gravity, as equivalent to a counterflow of voids entering the orifice and migrating upwards through the bed by a biased random flight as they are repeatedlyfilledby particles moving d o w n from above. The appropriateness of the model was not obvious. However, a continuous velocity distribution was predicted without considering the stress field which indicated that it is not necessary for the particle flow from hoppers to be considered as being driven by the stress field. This idea was sustained by the kinematic model of Nedderman and Tiizun (1979)^ I Davidson and Nedderman in 1973^273 derived an expression, known as the 'Hour Glass Theory' based on continuum mechanics, to model the flowrate of cohesionless material from a smooth walled hopper. They found the predicted flowrate was about double the observed result and concluded that this difference was caused by the effect of wall friction. They observed that the flowrate and stress distribution were very insensitive to surcharge at the top of the hopper. Williams in 1977[28] extended the theory of Davidson and Nedderman1273 to include a solution for wall friction. The author produced two solutions to the equations of m o m e n t u m and continuity to provide upper and lower limits. Since these limits differed by only about 2 0 % , Williams claimed that it was not necessary, for the prediction of discharge rates for design purposes, to obtain complete solutions to the equations. Comparison of the theory with experiments showed good agreement for coarse materials, (within 8 % ) , but over-predictions for fine particles less than about 500 |im (for sand with mean particle size 250 Jim, the mean of the upper and lower limits of predicted flowrate was about 3 0 % higher than the measured flowrate). This discrepancy was attributed to the fact that appreciable air pressure differences which developed in the airfillingthe void space were experienced by these particles. These air pressure differences were not considered in his model. Nedderman and Tuziin in 1979™3 developed a kinematic model for the flow of granular materials which, similar to Mullins' stochastic model' ' , 3 , m a d e no reference to the stress distribution within the material. T h e experimental measurements were m a d e on velocity distributions in a two-dimensional hopper with Nedderman et al. claiming excellent agreement with the kinematic model. They concluded that free-flowing granular materials discharging from hoppers are not driven by the stress field, as has usually been assumed, but m o v e simply by the particles in one layer slipping into the spaces vacated by the layer beneath. They suggested this model solely for steady flow of a dilated material. Drescher, Cousens and Bransby in 1978[3°3 presented a kinematically admissible velocity field for the mass flow of granular material in a plane flow hopper. The proposed solution was based on assumptions suggested by the flow pattern observed during experimental work. In particular, they incorporated rupture surfaces observed during flow into the solution. The material filling the hopper was treated as plastic, though different flow rules were specified for different areas of the material. Stresses were not considered because the static counterpart of the kinematic solution did not contribute to the approach adopted. The main thrust of the work was to provide a realistic mathematical model for understanding the velocity fields observed in experiments, rather than a complete solution based on simplifying assumptions to allow flowrates to be predicted. The predicted velocity field for the mass flow of granular media through a plane converging hopper, obtained from rather simple assumptions, agreed reasonably well with that observed experimentally. A criterion was suggested to indicate the transition from mass to funnel flow as the head of material in the hopper is reduced. However, this criterion cannot be predicted theoretically and has no practical implication. Nguyen, Brennen and Sabersky in 1979^ 3 presented an approximate solution to the flow of a cohesionless granular material in a conical hopper. The bulk solid was treated as a perfectly plastic continuum which satisfied the Mohr-Coulomb type yield condition. They concluded that the method presented was appropriate for small hopper angles with discrepancies from the experiments occurring at larger hopper angles. They pointed out that the continuum model described the behaviour of granular materials fairly well. Kaza and Jackson in 1982t333 provided a power series solution for solving their mathematical model instead of using a perturbation technique or introducing extra assumptions to simplify the model. Their theoretical model was based on the equations of continuity and m o m e n t u m balance for a cohesionless Coulomb powder discharging from a wedge-shaped hopper. Compared to the perturbation analysis of Brennen and Pearce in 1978 [ 3 1 3 , the power series solution was expected, as claimed by Kaza et al., to solve the hopper discharge problem more accurately over the range of values of hopper angles since it provided smaller computed residues. Unfortunately, the flowrates calculated from this solution lay further from the experimental data than those obtained from the Brennen-Pearce model. The authors suggested some possibilities for the gap between theory and experiment, such as the form of the boundary condition at the wall used m a y have been incorrect; the experimental characterization of the materialtodetermine internal friction angle and wall friction angle m a y have been in error or there was no continuous solution which remained bounded at the lower traction-free curve and approached the radial stress and velocity field high in the hopper. Michalowski in 1984^343 described the flow of granular material in a plane flow hopper in two stages, initial stage and advanced stage. H e concluded that a theoretical description of the kinematics of advanced flow, based on the plane plastic flow theory of incompressible material and coupled with the radial stress field, described the real velocity field accurately, particularly in hoppers with smooth walls. The description in his work gave results similar to those obtained by Drescher, Cousens and Bransby in 1978 [303 . For the initial stage of flow, only part of the material in the hopper experienced plastic deformation, which indicated that the traditional plastic flow theory did not seem to provide a promising framework for analysis in this stage of flow. Pitman in 1986 and 1988[36,373 presented a mathematical analysis based on continuum mechanics for incompressible granular material flowing from two- and three-dimensional hoppers. A numerical analysis was presented to solve the equations governing the flow of the particles. It was found that there was usually some oscillation during the material flow. For mass flow hoppers, the oscillations were slight. Moreover, the oscillations were more apparent in two dimensions while stable three-dimensional flows showed little oscillation. However, for both cases the computed stress field converged to the radial stress field, towards the outlet of the hopper. Jenike in 1987* 3 presented a theory which extended his previous theory developed nearly thirty years ago. H e pointed out that the design method based on his original theory is adequate for the design of mass-flow hoppers, but it predicts incorrect channel flow angles in funnel flow. T h e correction presented was obtained by replacing the modified Tresca yield pyramids of the original theory with conical yield surfaces and relating the strain rates to the stresses by the Levi flow rule which is that the stress deviatorics are proportional to the strain rate deviatorics. The corrected theory provided a significantly larger channel flow angle in funnel flow than the initial theory. The bulk solids considered were elasticplastic, frictional, cohesive and compressible. Poldeman, Boom, de Hilster and Scott in 1987' 3 presented a mathematical model for the solids velocity distribution in mass flow hoppers. T h e residence time distribution measurement and flow visualisation studies were carried out in experiments. B y comparing with the experimental results, the theoretical model which w a s based on classical plasticity theory, was claimed by the authors to provide a good prediction for the velocity distribution of free-flowing material in mass flow hoppers. However, the model of Poldeman et al. cannot predict the velocity distribution in a mass flow hopper completely as a constant in the model must be determined by individual experiment Ravi Prakash and Kesava Rao in 1988t393 studied the steady compressible flow of cohesionless granular materials from a wedge-shaped hopper. Their continuum model used the critical state theory of soil mechanics to generate the stress and density profiles and allowed the discharge velocities to be obtained. A n approximate expression for the discharge velocity was also presented, which predicted discharge velocities to within 1 3 % of the numerical values from their original model. The difference between the results from incompressible flow and compressible flow was examined for two materials, glass beads and Sacremento River sand. They observed that the effects of compressibility m a y safely be ignored for glass beads, but not for Sacremento River sand. Incompressible models predicted in some cases significandy higher discharge velocities than those obtained by the incorporation of density variation. For instance, the incompressible model overestimated discharge velocities for the above two materials by 2 % and 4 5 % , respectively. They pointed out that the discharge velocity was more sensitive to density variations than to stress profiles. c) Finite Element Analyses With its development for modelling complex flow behaviour, the Finite Element Method ( F E M ) seems to be an alternative method for use in the bulk solids gravity flow area. Probably, the first application of the FEM in this area was the work of Fried, Carson and Park in 1976^783. This application focussed on simulating the nonlinear flow problem of gas in a contact bed reactor. A s the solid particles m o v e d downward by gravity, gases were forced to flow through the particles from several particular points. Because of the complexity of the reactor geometry and the nonlinearities involved, the F E M was used to predict the gas velocities and pressure gradients near the gas inlets. Recendy, several attempts to use F E M to compute the velocityfieldin flowing bulk solids have been made, such as those by Haussler and Eibl (1984)^ 403 , Runesson and Nilsson (1986) [41] , W u (1990) t79] . The F E M has several advantages including: • modelling the flow of a bulk solid flowing from a bin with complex geometry; • showing the stress and velocity profiles at any position in the bin; • modelling the initial flow and steady flow for bulk solids. O n c e the velocity profile is determined, the flowrate of bulk material from silos / bins can be evaluated according to the velocity of particles at the outlet. However, using F E M has some limitations such as: • time consuming calculations; • indicating only indirectly the relationship between particle velocity and particle flow properties. D u e to these limitations, most F E M applications have attempted only to predict the stressfieldsin the bins or the wall load distributions; the prediction of the flowrate of bulk solids from a regular mass flow bin does not warrant calculation by F E M . 2.2 Research on Flowrate with Air Retardation As mentioned in Section 2.1 a), it is believed that the first appearance of the effe air retardation on flowrate of fine material was in the work of Rose and Tanaka (1959)^ 133 , although it did not c o m m e n d their attention at thattime.Bulsara, Zenz and Eckert in 1964 [ 4 3 3 conducted some experimental work on improving the flow of fine cohesive particles. They particularly aimed at rendering 'sticky* powders free-flowing. They found that the flow of cohesive particles can be improved by adding sufficient coarse particle into thefinepowders (e.g., adding sand with mean particle size of 715 ^im into fly ash with size of 20 Jim or into Micro-Cel of 3 |Am) or applying air pressure differentials to suck the solids out of the bin. This result indicated that the effect of air pressure on the flow of fine cohesive particles was considerable. Johanson in 19651-223 stated that drag effects could be very important to the flow offinematerial but his model avoided including this effect by assuming the material to be coarse. McDougall and Evans (1965) 1213 attempted to describe the particle flow affected by air pressure at the outlet, however, their equation did not take account of the interaction between particles and air. The first major experimental work involved with air drag on flowrate was conducted by Miles, Schofield and Valentin in 19681-443. They pointed out that w h e n air drag forces cannot be neglected, the B r o w n and Johanson equations overpredict the flowrates, with the over-estimation becoming more significant as the particle size decreased. They presented experimental evidence to clarify some of the processes which take place w h e n discharging fine powders. Flowrates of coarse gravel, fine sand and calcite, from a hopperfittedwith replaceable cones with different outlet size and wall angle, were recorded. Negative air pressures were observed for the fine sand discharge but no noticeable change in pressure was observed for gravel. For 55 fim calcite, the cycling between fast and slow flow was observed (as shown in Figure 8.8). Their experimental observations also indicated that both air injection into the flowing mass and the extension to the hopper oudet by means of a standpipe had a significant effect on increasing the flowrate of fine sand but no significant effect for the coarse gravel. However, the increase in flowrate of particles by air injection was not proportional to the flowrate of air. S o m e optimal case exists for particular material and bin geometry. The significance of using a standpipe below the hopper outlet lies in creating a suction effect at the hopper outlet to increase the flowrate. In their work the standpipe was found only effective w h e n it was full of material. This phenomenon was also observed by later workers 1 * 4 - 863 . Holland, Miles, Schofield and Shook in 19691-463 attempted to explain theoretically the phenomena observed in the above paper. They stated that the flowrates computed by Brown's equation are remarkably accurate if the particles are coarse. A s the particle size diminishes, however, the flowrates obtained by Brown's theory invariably exceed the experimental values; the effects neglected m a y be described as air drag forces and interparticle forces. The authors discussed the forces involved in the flow of particles, inertia forces, gravitational forces, fluid drag forces, interparticle pressure forces and interparticle shear forces. B y simplifying these forces (neglecting the fluid drag force and the interactions between particles), the authors derived a velocity equation which agrees exactly with the B r o w n equation for coarse material. B y considering the air drag effects, Holland et al. presented a relation between the air pressure gradient at the hopper outlet and the flowrate. The agreement of the presented air pressure gradient at the hopper outlet with the experimental data was claimed to be reasonably good. However, due to the complexity of the solution and the lack of boundary conditions such as voidage distribution in the hopper, no equation was presented to predict the flowrate of bulk solids. Shook, Carleton and Flain in 1970^473 proposed two correlation equations for calculating the flowrate under the influence of air drag forces: one was based on the Richardson-Zaki expression to estimate the drag force for hindered settling of particles; an alternative approach was based on the Carman-Kozeny equation for fluid flow to model fluid drag forces. Both equations required the determination of porosity in the hopper and at the hopper oudet but no procedures were proposed for these determinations. McDougall and Knowles in 1969' 83 presented a systematic investigation of the effect of pressure difference across an orifice on the flow from flat bottomed bunkers. T h e experimental work w a s conducted with Mustard Seed, average particle size of 2.2 m m , flowing from orifices ranging from 12.7 to 76.2 m m in diameter. The experiments were carried out on four types of flow: gravity flow, vacuum flow, pressure flow and counter-current flow. A m o n g these four types of experiments, the gravity flow can be considered as no air pressure difference existing across the orifice since the particle size was coarse. The three others created air pressure differences: for the vacuum flow test a suction effect was built up at the outlet to increase the flowrate; for the pressure flow test, a positive air pressure gradient was created to force the particles to flow quickly. Both these flow tests had a similar effect on the flowrate and, hence, the distinction between 'vacuum flow' and 'pressure flow' w a s not maintained throughout the experiments. For the countercurrent flow test, an air current w a s forced countercurrent to the solids flow. The results showed that the effect of the air pressure difference was quite dramatic, even for the coarse material used in the experiments. W h e n the adverse air pressure gradient was sufficiendy large, no flow of bulk solids at all was observed. The work led to a general conclusion that air pressure differences can be an important factor in the flow of solids from orifices. With free flowing materials, it was apparent that useful increases of flowrate resulted from pressurizing the hopper. Carleton in 1972^493 carried out a force balance on the fluid solid system and used the fluid drag coefficient on a single particle as the air drag model to predict the flowrate of fine material (particle sizes below about 200 p m ) . H e developed the following nonlinear equation: Ill 4Vosina 15pJ\iJvJ + D0 T Psdp3 =g (2-4) A nomograph was also presented to simplify the solution procedure of eqn (2.4). The equation proposed for the velocity profile was claimedtogive good agreement with experiment for free flowing materials except for hoppers with small orifices (below about 2 c m in orifice size) where wall effects become predominant. Crewdson, Ormond and Nedderman in 1977'503 investigated air retarded discharge theoretically and experimentally for conical mass flow hoppers. The authors, in their theoretical derivation, considered the effects of interstitial air pressure gradients and compressibility on the flowrate offinematerials (particle size less than 500 (Jm). A n adjustment was made to the Beverloo flowrate correlation to account for interstitial air pressure gradients. However, due to the lack of a reliable value of the compressibility index at low stress levels, only a qualitative comparison of the theoretical analysis with the experimental observations was reported. Spink and Nedderman in 1978^ 3 reported a theoretical and experimental study, based on the work of Spink 1976^803, observing the flowrate offineparticles (50 p m < d p < 500 p.m) from plane flow hoppers. The theoretical model yielded interrelated equations for particle flowrate and for the distributions of particle stress, bed voidage and interstitial air pressure in the hopper. G o o d agreement between experimental and theoretical voidage and air pressure distributions was found. The experimental study on the air pressure distribution showed that a pressure minima (negative) occurred near the hopper oudet and a visible pressure maxima (positive) near the top surface of material was observed for relatively large outlet cases. Furthermore, it was observed that the negative air pressure gradient increased with increase in the hopper outlet and with decrease in particle size. For very fine particles such as fine sand (with a particle size range from 63 to 90 p.m), unsteady flow w a s observed. The theoretical and experimental studies on the flowrate showed that the flowrate offinematerial (particle size range from about 100 to 500 |im) increased as its particle size increased. However, although they showed the correct dependency on the particle size, the theoretical discharge rates were about twice the measured values. For coarse material, from the concept of the 'empty annulus*' 3, Spink et al. (1978)^ 3 believed that the particle flowrate decreases as particle size increases. However, this variation, as pointed out by Head (1979)^ 3, is caused by orifice blocking and, therefore, is valid only for coarse material flowing from a small outlet. Smith in 1978^533 investigated the interactions of feeders with mass flow hoppers. O n e interesting finding was the variation of the free fall flowrate at various H / D ratios of the bulk solid (Shirley Phosphate with median particle size 140 p.m), as shown in Figure 6.1. In particular, he found that for a hopper with surcharge (H/D>0) the flowrate decreased as the surcharge level increased; for a hopper without surcharge (H/D<0) the flowrate increased with an increase in surcharge level. T h e decrease in flowrate for H / D > 0 was explained as the effect of the counterflow of air through the hopper oudet For the cases H/D<0, there was no adequate reason to explain the increase in flowrate. Nevertheless, this is the only work found which reported the variation of the flowrate with the surcharge level. Since Smith's results were observed under a variable material level during discharging (i.e., the readings of the material height reported were taken after filling the material in the test bin and before material was discharged), it seems that more work needs to be done to examine the effect of a constant material surcharge level on the flowrate. Willis in 1978^ 5 3 conducted a series of measurements on the interstitial gas pressures generated by flowing bulk solids in a plane flow variable geometry bin. It was found that varying thefillrate, hopper geometry and type of material had a significant effect on the interstitial gas pressures. It w a s claimed by Willis that the experimental values compared favourably with the values predicted by McLean's theory^583, which focused on the hopper section. However, Willis' work identified a major deficiency in the theoretical treatment. In particular, the experimental distributions indicated that the gas pressure at the transition is negative, suggesting the bin be analysed as a whole, since the air pressure distribution boundary conditions applying act at the oudet and top of the bin. McLean's analysis in 1979^5 3 considered the continuity of the bulk solid and the interstitial fluid, as well as the equation of motion including the effects of the interstitial gas pressure gradients (for fine material cases). In the derivation of the theory, consolidation-related bulk density and permeability equations were introduced for modelling the flow of compressible simple bulk solids. T h e predicted flowrates compared favourably with observed flowrates from an experimental plane flow variable geometry bin. Additionally, he concluded that this investigation provided initial estimates of the parameters for air injection techniques which require a suitable pressure, quantity and location to improve the flow. Head in 1979^553 investigated, theoretically and experimentally, the effects of the particle size of granular materials and the effect of interstitial air pressure on the flowrate as well as the effects of air injection technique on improving the flowrate. It w a s found that the flowrate was affected by its particle size in following way: for large sand particles (> 500 p m ) the flowrate decreased with increase in particle size due to particle mechanical interlocking at the orifice; for fine sand particles (< 500 p m ) the flowrate decreased with decrease in particle size due to particle air drag and compressibility. The study confirmed that for the sand less than 500 p m the flowrates are affected by negative air pressure gradients at the oudet; the flowrate can be varied by injecting air into the system and sensitive control of flowrate can be achieved. The experimental observation on the effect of air injection indicated that it should be located as near to the steady state negative pressure m i n i m u m position as possible to obtain best control of air pressure drop across the orifice. G o o d correlation for flowrate calculation, predicted using the models developed, was claimed. Nedderman, Tuziin and Thorpe in 1983' 3 presented theoretical and experimental work on the effect of interstitial air pressure on the flow of granular materials from hoppers. Significant effects were observed in air augmented discharge due to the compressibility of the air. Based on the work of Harrison and Mushin (1979)' 3, they proposed a theory for the case of low Reynold's number flow in a conical hopper. The following equations resulted 1 ~ ^ (* 2K-3 AP \2 Qp = Qpmax[l+2K-rTp!^J ^ where v5/2 _ rcpPp vpmax4 (1+K)g / yj 2(2K-3)sina The theory w a s also extended to higher Reynold's numbers and to cases where the compressibility of the air was important. However, their theoretical model overpredicted the measured flowrates. Some deficiencies in their theoretical model which may cause this discrepancy were suggested, e.g., the assumption of smooth hopper wall in theoretical analysis did not adequately represent the tested hopper wall; the assumption that the stresses drop to zero on the spherical surface r=r0 was indeed debatable. It was claimed that the agreement between the theory and the experiments was better if the discharge rate was correlated using the orifice pressure gradient as measured from the pressure profiles. To overcome the overprediction of the theoretical model for higher Reynold's numbers, a 'discharge coefficient' was introduced to their semi-empirical equation. According to their experiments this coefficient was valued as 0.92. They suggested further work be carried out to confirm this approach. McLean in 1984^603 modified the Johanson equation'23 by introducing an effective 1 dP gravitational acceleration g* = g + (-3-) instead of g. It can be seen that a Pout * r = r ° negative air pressure gradient at the hopper outlet retards the particle flow and a positive air pressure gradient assists the particle flow. His work enabled the Johanson's equation (2.3) to be used in both coarse and fine material cases, i.e., Qp = pDo j tan a ^ f fa j where for coarse material f g g*=i S + 1 ,dP, — <dF>r=ro Pout for fine material (2.6) In this modification, McLean provided the following equation for the prediction of the air pressure gradient at the hopper outlet. 1 (*£) = QP (JJLY ^ (( Wo) YRb" _ (1 ^dr^Urn AoCoutPeUpeJ l\DbffJ Uf b hJ , (2.7) provided the bulk density is described by P = Pe( —)b °e Recently, McLean[1153 developed an alternative approach to eqn (2.7) by introducing the voidage at one-third of the way along the hopper wall from the hopper oudet e ^ ; the following equation resulted fdPA Qp (eo-eifl) I dr J ^ ~ A 0 C o u t p s (l-eb)(l-cli3) (2 8) ' Ducker, Ducker and Nedderman in 1985t573 developed a theory to predict the flowrate of bulk solid from an unventilated hopper. In this case, air must enter the hopper through the orifice at a volumetric flowrate equaltothe volumetric outflow of solid. This counter-current flow will set up an adverse pressure gradient which will reduce the flow of solid even for such coarse material as Diakon, Kale Seed and Mustard Seed with particle size of 0.662,1.63 and 2.10 m m respectively. For sand with particle size of 285 p m , unsteady flow was observed from unventilated hoppers and this was attributed to fluidisation caused by a high counter-current air flow near the orifice. The theory was claimed to be in excellent agreement with experimental results. Experiments on the same bulk materials were carried out to obtain the flowrates from ventilated hoppers. The results showed a good agreement with the Beverloo correlation, by setting the constant B f in eqn (2.1) as 0.58,0.57 and 0.66 for Kale Seed, Mustard Seed and Diakon respectively. For the 285 p m Sand, flowrates less than the Beverloo prediction were observed due to the effect of interstitial pressure gradients. Arnold in 19861523 presented some attempts to predict the flowrate which were based on the Carleton method eqn (2.4), using equivalent particle size d e instead of particle size cL, i.e., \_ 2 4 4V?sina D0 15 + Pf 3 ji/ V 0 3 ~5 1 =S (2-9) .3 Psde In particular, two types of modifications were proposed: Drag force modification: The drag coefficient on a single sphere was replaced by that on a mass of particles by introducing the well known drag modification expression of Richardson and Zaki 1/e . Using the results obtained by W e n and Yu^ 1 3, n-1=4.7, the flowrate was calculated by eqn (2.9) and eqn (2.10). de - e 2 ' 8 2 ^ (2.10) Permeability modification: The equivalent particle size was estimated from permeability factor C in terms of the Carman-Kozeny equation. In this case, the flowrate was predicted by eqn (2.9) and eqn (2.11). 2 de = 180p f (l-e) C 1 2 (2.11) 3 e W h e n studying the flow with air pressure retardation, a number of researchers have examined two main techniques for increasing the limiting flowrate. Air Injection - this technique has been reported in many publications^44,45'81" 83,116-121] ^ particular> Papazoglou and Pyle in 1970 [81] , Altiner in 1983 [82] , de Jong and Hoelen in 1975™ 3 investigated, both theoretically and experimentally, the flow of fine particles from aerated conical hoppers, plane flow hoppers and flat-bottomed bins. It was found that for low air flowrates the effect of air injection on improving the particle flow increased with increase in outlet size and with decrease in particle size. They also found that the effect of air injection w a s different for different aeration methods. A m o n g these researchers, Papazoglou et al. and Altiner studied the effect over relatively wider range of air flowrates in their aerating system. They discovered that there was an upper limittothe quantity of air that could usefully be injected. Beyond this, the excess air fluidised the bed above the injection point and simply escaped upwards without aiding the flow'43. Altiner reported that the limiting aeration rates were different for different size particles, being less forfinerparticle. H e also observed that thefinalm a x i m u m particle flowrate was the same for all sizes with the same aerating system. Furthermore, he found that different m a x i m u m particle flowrates were achieved with different aeration methods, air flow distributed along the hopper wall near the outlet giving the higher m a x i m u m particle flowrate than point air injection. Papazoglou et al.'s studies also involved the flow behaviour of particles at high air flowrates. It was noted that after the particle flowrate reached a m a x i m a the particle flowrate w a s proportionaltominus the air flowrate. Papazoglou et al. explained this decrease in particle flowrate with increasing air flow m a y be contributed to by significant m o m e n t u m flux in the air phase leaving the injector pipe and to compressibility effects. The experiments showed that the constant of proportionality was dependent on both the system geometry and the particle diameter. Standpipe - the use of a standpipe connected below the orifice of a container, as an alternate technique to improve particle flow, has been reported by m a n y researchers such as Miles, Schofield and Valentin (1968)^443; Yuasa and K u n o (1971)[122]; M c D o u g a i i a n d puiien (1973) [123] ; de Jong (1975) [124] ; Leung and his co-workers (1973 [125] , 1978 [ 1 2 6 ' 1 2 7 ] ); Ginestra, Rangachari and Jackson (1980) 1843 ; Chen, Rangachari and Jackson (1984) [85] ; Knowlton, Mountziaris and Jackson (1986)^ 3. O n e generalized result showed that the standpipe can increase the flowrate but it is only effective when full of the particles. A m o n g these researchers, Ginestra et al., Chen et al. and Knowlton et al. investigated theoretically the gravity flow of bulk solids through a configuration which included a hopper, a standpipe attached below the hopper outlet and a flow control device at standpipe outlet Ginestra et al. presented a one-dimensional theory which was based on a direct integration of the equations of continuity and m o m e n t u m balance for the two phases. The theory linked together equations describing the hopper, the standpipe and the flow control orifice at the lower end of the standpipe. The prediction indicated that the negative air pressure at the hopper outlet was considerably greater than the pressures above and below the system, inducing a vigorous flow of air downwards through the hopper thereby increasing the particle flowrate due to the concurrent drag force on the particles. However, it was claimed that the predictions of this theory did not agree with the observations. Ginestra et al. believed that this discrepancy was caused by the unrealistic simplifying assumptions such as the frictional forces between particles and standpipe walls being assumed to be negligible. Chen et al. extended the work of Ginestra et al. and presented a model by relaxing some of the simplifying assumptions in Ginestra et al.'s theory, such as considering frictional forces between particles and standpipe walls and treating the gas as a compressible fluid. T h e predictions of the improved theory were compared with measurements on a laboratory-scale standpipe and general agreement of theoretical prediction with experimental results was obtained. The research confirmed the fact that the use of a standpipe increases the particle flowrate more significantiy for fine materials than for coarse materials, because the suction effect built up at the hopper outlet is m u c h weaker in the case of the coarse bulk solids. Furthermore, they also noted the existence of two different particle flowrates under the same operating conditions in certain circumstances (such as applying some amount of air pressure at standpipe outlet which was greater than that at the top surface of material in the hopper). The upper value showed an increased particle flowrate, while the lower value indicated that the particle flowrate was reduced. In lower flowrate case, their predicted and measured air pressure profiles indicated that a positive air pressure occurred at the hopper oudet, so particle flow from the hopper was impeded rather than aided by the air pressure gradient. Physically, it is believed that in this lower flowrate case a relatively lean particle flow appeared in standpipe so that the positive air pressure applied at the standpipe outlet could reach up to the hopper outlet. This phenomenon supported the general result that the standpipe is only effective when it is full of particles. Based on the work of Chen et al, Knowlton et al. investigated the effect of standpipe length on the flow of particles for a standpipe restricted by a concentric orifice at its lower end. Comparing the theoretical results with the experimental results, it was found that general agreement between theory and observations was obtained. The research shown that the effect of standpipe length on particle flowrate depended on the percentage of the opened crosssectional area of the standpipe oudet W h e n the open area was small (e.g., 5 0 % or less), the particle flowrate was always smaller than that without the standpipe; the standpipe length had almost no effect on the particleflowrate.Otherwise, when the open area was large (e.g., 100%), the standpipe increased the flowrate and the standpipe length did affect the particle flow. The longer the standpipe length was, the more significant the effect became. However, Ginestra et al., Chen et al. and some other researchers such as Yuasa et al. studied a very long standpipe with the ratio between standpipe length and its diameter more than 100. This kind of long standpipe cannot be widely applied to the bulk solid handling system in industry. S o m e work needs to be done to study the extend to which the flowrate is increased by a standpipe with limited length. Chapter 3 M e a s u r e m e n t s of Bulk Solids Flow Properties 3.1 Introduction To model flow behaviour of bulk solids from mass flow bins, an essential step involves the determination and measurement of the flow properties of the bulk solids. The flow of the bulk solids from the bins depends on the geometry of the bin (e.g., hopper angle, outlet size), particle characteristics (e.g., particle size, particle size distribution, moisture content and particle shape) and the relationship between particles and the bin (e.g., wall friction). Since it is difficult, at the present time,toestablish a theoretical model which includes all these items, adequate flow properties need to be determined. Figure 3.1 summarised from studies in the literature how flowrate is related to different flow properties. Generally speaking, these flow properties can be collected into three groups of characteristics. The first group of characteristics is the physical and geometric characteristics for both the bulk solid and the bin, the second group of characteristics is the flow properties of the bulk solid which can be determined from laboratory measurements, while the third group of characteristics is usually predicted theoretically and/or measured in the laboratory from model experiments. Considerable research has been done to predict the flowrate of bulk materials from mass-flow bins and hoppers using a range of bulk solid properties. S o m e researchers have found that the effects of the angle of internal friction and the wall friction angle seem to be small (Nedderman 1982)™3. Therefore, the flowrate was calculated with knowledge of the particle size (normally median particle size), bulk density and the bin geometry parameters of oudet size and half hopper angle. First group Particle Characteristics Particle size dp Size distribution Particle shape Particle roughness Moisture content Particle density o r s Bin or Hopper Geometry Half hopper angle Outlet size Vertical section: - diameter - height Ct D0 D H - ^ ^ ^ ^ Bulk density p Second group Permeability C Internal friction angle Flow function Wall friction angle 1 Third group i ( empirical research ) ( theoretical research ) I Flow factor Stress / velocity field in bin or hopper Interstitial air pressure gradient Flow rate of particles from bin or hopper Figure 3.1 Qp Gravity Flow of Bulk Solids from a Mass-Flow Bin A s the bulk solids become finer, the flow phenomenon becomes more complicated. For example, the effects of particle size distribution, moisture content, cohesion of the particles and the interstitial air pressure gradients become significant. In this case, the median particle size is not sufficient to describe the size of solid mixtures; the bulk density is not a constant, but varies with the compressibility of bulk material; the permeability of bulk solids becomes an important parameter in predicting the flowrate. The solids used in this work are assumed to be fine and dry (or have a low and constant moisture content). Nedderman (1985)^53 suggested that if flow does occur the flowrate can be considered to be independent of the cohesion. Since this independence has not been tested in systematic w a y in Nedderman's work, it is necessary to assume which is that the effect of cohesion between particles concerned in this work is negligible for mass flow conditions. Hence, for this work the paramount flow properties of bulk solids are particle size, particle size distribution, effective internal friction angle, particle density, bulk density, permeability of particles and the friction angle between bulk material and bin/hopper wall. In view of the importance of these parameters, the methods used for their measurement are described in the following sections. 3.2 Particle Size and Particle Size Distribution The size of a spherical particle is uniquely defined by its diameter. For a nonspherical particle the particle size is defined by an equivalent diameter which depends on the method of measurement. There are several definitions of particle size, such as volume diameter, surface diameter, surface volume diameter, drag diameter, projected area diameter and sieve diameter 3 . Since the bulk solids handled in practice are mixtures of a large number of particles which commonly are not of uniform size but spread over a narrow or wide range of sizes, the properties such as flowability, segregation, compressibility and permeability will be affected by this particle size distribution, especially if there are fine particles present. There are a number of methods used to determine the particle size and size distribution of a particulate material. The methods can be briefly classified, mainly based on Ramanujan et al/ 3 and Jelinek™93, as mechanical methods (dry sieving, airjet sieving), gravitation methods (gravity sedimentation, centrifugal sedimentation), elutriation methods (gravity, centrifugal), optical methods (light microscope, ultramicroscope, electron microscope), translational methods (viscometry, permeametry), adsorption methods and laser methods. The methods used in this work are dry sieving and laser particle sizing. The Malvern Laser Diffraction Particle Sizer (2600 series) was used to measure the particle size distribution for materials finer than 565 p m . The median particle size used in this work was the volume percentile median diameter (volume diameter at 5 0 % point of the cumulative particle size distribution). The particle size in excess of 565 p m was measured by the dry sieving method. Since only a small amount of sample is needed in the Laser Particle Sizer extreme care must be taken to obtain a sample that represents the bulk material. Improper sampling often introduces errors that are larger than the errors produced by the instrument used for size analysis. In general, the Laser Particle Sizer can provide more reliable and accurate particle analysis for fine materials with a narrow size distribution. In some situations where bulk solids with wide size distributions were analysed and the effect of segregation could not be reduced satisfactorily, a combination of sieving and laser particle sizing was used. In particular the sieving method separated the whole particle mixture into several different size segments with narrower size distribution, then the L P S was used to analyse each narrow size segments. Finally the results were combined to obtain the results for the whole mixture. In addition to using the standard sieve series some large area screens were used for fine materials to achieve better results by increasing the sieving speed and avoiding crowding due to the sieve loading. It is necessary to mention that other particle characteristics such as particle shape and particle roughness can also affect such flow properties of particles as internal friction angle, bulk density and permeability. Since these flow properties will be measured directly the results will implicidy include these effects. 3.3 Effective Angle of Internal Friction and Angle of Wall Friction The state of stress at any point within the stress field can be described in the a, Tdiagram (o = compressive stress, x = shear stress) by a M o h r stress circle. The yield locus is the yield criterion of the material at a corresponding bulk density which depends cm the major consolidation stress aj. Figure 3.2 shows a diagram of a typical yield loci of a bulk material. The Effective Yield Locus, introduced by Jenike, is a straight line tangential to the stress circle at steady state flow and passing through the origin of the normal stress, shear stress axis. The angle between the E Y L and the a-axis is defined as the effective angle of internal friction. For cohesive material, such interactions as friction and adhesion between flowing particles have to be considered. Molerus™ ' 3 provided another steady state flow yield locus which can be used for cohesive materials. The intersection of the Molerus Yield Locus and the x - axis represents the cohesion in the non-consolidated state. For cohesionless material, the Molerus Yield Locus and die Jenike Effective Yield Locus coincide. Molerus Yield Locus ot Figure 3.2 o Yield Locus and Internal Friction Angle Considering the bulk materials used in this work and the assumption that the cohesion of the particles can be neglected (as discussed before), the effective angle of internal friction 6 can confidendy be used as a parameter in models developed to describe the flow of bulk solids from mass flow bins. The wall friction angle describes the yield criterion of bulk solids at the bin or hopper wall. It is determined by the material's Wall Yield Locus. A typical linearized Wall Yield Locus is plotted in Figure 3.3. Figure 3.3 Wall Yield Locus and Wall Friction Angle The Effective Yield Locus and Wall Yield Locus are determined by shear tests. There are a number of test methods available for measuring the shear properties of particles. Table 3.1 gives a survey of shear test methods and a comparison of shear testers. In this comparison the symbol " >/ " stands for " applicable ". The testers in bold font can measure both internal friction angle and wall friction angle. From Table 3.1, it is obvious that every method has its advantage. The use of a particular method depends on the field of application. In bin design and flowrate analysis, the Jenike shear tester offers a quick, reproducible and adequate method. It was used in this work to obtain the two angles stated above. The test procedures were conducted as detailed by Arnold et al.(1980)^ 3. 3.4 Particle Density Particle density is mass divided by the particle volume. The particle volume of a sample was obtained using a Beckman Air Comparison Pycnometer (Arnold et al. 1980^33). The mass of the sample was weighed on a Metder P C 4400 electronic balance. 3.5 Bulk Density Bulk density of a bulk solids is its mass divided by the volume of particles and voids it occupies. It is not only an important conversion parameter from a volumetric system to a mass system, but also a parameter to indicate the compressibility of bulk material and to predict the voidage of solids during processing. Table 3.1 Shear Test Methods and Their Comparison (based on Schwedes 1983[92] and Schwedes et al.l990[93,943) Shear Testers Direct Indirect - Shear Testers / Translational Rotational r-J—, . 1 . Comparison of \ Shear Testers N « 4> H & u s H U 04) 0) OS a> •8 I co E> II r u ** «5 t <n 4) H 4> h 09 0* « 41 H CO b03 03 0) A B CO O </> M B u © H OS 0) 1 on 4) H «- 8B9 CO M B JE 1s 1 OS PQ P OJ 0) *3 CO fc a 8 § co DC Angle of Internal Friction Angle of Wall Friction I £I Biaxial Triaxial 09 M B 0> U \ V V - V V V V - - - . . Shear Strength, Compressive Strength VV--VV-----V Stress Distribution VVV---VVVVVV Complete State of Stress - - V - -Time Consolidation Complete State of Deformation - V V V V - - - V - -VVVVVV- Suitable for Elastic Granular Materials Suitable for Coarse Material ( > 10 m m ) Suitable for Small Stress ( <lN/cm2 ) V V V V V V V V V In the prediction of flowrate of coarse bulk solids from a bin / hopper, the bulk density is generally considered as a constant parameter. Forfinematerials, the bulk density cannot be regarded as constant; the compressibility of the material must be taken into account in models to describe flow behaviour. In general, the value of bulk density depends not only on the composition of the particles, such as particle size, size distribution, moisture, shape and roughness, but also on its state of compaction. Therefore, the specified bulk density, such as loose packed density, dense packed density and tapped density, are used in an attempt to identify the bulk density at different compaction conditions. In practice, especially in theoretical analyses of solid handling, it is necessary to describe the bulk density of a certain powder as a function of the extent of consolidation. There are a number of relations to describe the variation between bulk density and applied stress, as shown in Table 3.2. S o m e models, theoretically, cannot be applied at zero or infinite stress. In fact, infinite stress in materials handling is impossible. However, low stress conditions are frequendy experienced. Hence, it is necessary to identify the best models which are both simple and sufficiently accurate to describe the bulk density variations for bulk solids in the range of stress from free external compaction to afiniteconsolidation. The bulk density - stress equation, normally, is obtained by applying uniaxial compaction in laboratory experiments. Here the stress is actually the major principal stress. However, in practice, the powder is handled in three dimensional stress systems. Furthermore present bulk density equations in practice utilize either the m e a n stress and or the major principal stress. Use of the m e a n stress in these equations, although convenient mathematically, m a y cause s o m e error. T h e magnitude of this error is evaluated in Section 3.5.4. 44 3.5.1 Bulk Density Measurement The Jenike Compressibility Tester (Arnold et al. 1980[33), as shown in Figure 3.4, was applied to record the relation between the bulk density and applied stress. Dial Indicator Indicator Holder ^ Sample Cover Weight Carrier & Figure 3.4 Jenike Compressibility Tester (Arnold et al. 1980^33) The tester cell (base of the tester) was 63.5mm in diameter and 19mm in depth. The ratio of thickness of the specimen and inner diameter of the ring typically ranges between 0.25 and 0.33. This measurement on this tester is a typical uniaxial compression test (Svarovsky 1987™ 3 ). A typical bulk density - stress curve is plotted in Figure 3.5. The bulk density is a function of the consolidation. It increases with increasing applied stress, varying rapidly at low stress and slowly at high stress. p = P ( o- ) (3.1) 1400 Major Principal Stress (kPa) Figure 3.5 A Typical Bulk Density Curve of a Fine Sand 3.5.2 Models Used to Fit Experimental Results There are many models, as listed in Table 3.2, which can be used to describe the density variation equation (3.1) for powders. An algorithm for least-squares estimation of nonlinear parameters (Marquardt 1963 [1013 ) was used to determine the coefficients in these models and further to find which models best fit the experimental results for the bulk solids examined. The optimization process of fitting is stated as follows: It is assumed that the experimental results are (o*j, pj) i = 1,2,..., n, one o the models in Table 3.2 is p = Dj ( o, B ) j = 1, 2,..., 16, the problem is to compute those estimates of the parameters B which will minimize the variance ^minJ = i I [ P i - P j ( ^ B ) ] 2 where B = (bj, b2,.... b m ) (j = 1, 2, ..., 16) (m<n) (3.2) Table 3.2 Model The Models for Bulk Density - Stress Relation Author a -->0 Expression 1 b1+b2lno a ->oo 1 Balshin* 2 Smith* 3 Ballhansen* 4 Jones* P= P 5 Athy* P = Ps"(Ps-Po)e"blCT Po Ps 6 Heckel** P = (Ps-p0)ebl-b2° (Ps-P 0 )e bl 0 7 Halldin** P = Ps-(Ps-Po)ebl-b2° Ps-(Ps " Po) e b l Ps 8 Rim**!96! P = (Ps - Po) e' bl ° Ps-P0 9 Nutting* Po 10 Cooper* P = P0eb^2 PsPO p = p0+b1o1/3 Pabie^a 1+ b i e b 2 C T S (—^T~) bj+b2lna m 11 Kawakita[97] 12 JenikeM Ps-(Ps-Po)We-b2/a l+bio P = PoPs Ps+PobiCT p = p 0 (1+20.89 o ) b l 13 Jenike^ 14 - oo 0 Po Psbi 1+ bl oo Ps - 0 4* P 0 0 oo PsPO b Ps"(Ps-Po) i Po Ps Po oo p = b2 ab2 0 oo Johansonf98] p = p 0 (1+ bl0-)b2 Po oo 15 Schofieldt"] p = p s (b 1 + b 2 hia) -oo oo 16 This work p = b 1 o b 2 + b3 b3 oo *<sited from Kawakita ** I97!. cited from German t10°]. After obtaining the optimum parameters B for all of the models in Table 3.2, the total m i n i m u m variances ^ m j n ; (jI = 1, 2, .... 16) were compared to identify the best fitting model. The flowchart of the program is plotted in Figure 3.6. 3.5.3 The Models Appropriate to the Bulk Solids 79 experimental results for 22 sand mixtures, sugar, 2 alumina mixtures, PVC powder, fine coal, Shirley Phosphate and glass beads were used to examine the appropriate models for the bulk solids. Considering that the models 1,4 and 15 are not suitable when the applied stress o approaches zero, the calculation for all of models was carried out in two cases. O n e case examined the no-zero stress situation, whereas the second examined the zero stress situation with the models 1, 4, and 15 excluded. The results of total m i n i m u m variance are plotted in Figure 3.7. From Figure 3.7, it can be seen that for the bulk materials handled in a compacted condition, the models 1, 4, 9, 13, 14, 15 and 16 provide the most accurate prediction of the bulk density variations. For situations involving very low compaction levels, Models 9, 14 and 16 give better density predictions. A s a special case, it is noticed that the very large variance for model 13, Figure 3.7 b) was caused by a near zero stress observation (Figure 3.8). Let O ( i ) = 0.0 i = 1, 2 16 min Input initial experimental data allowance error for calculation m a x i m u m iteration times Looping for every model i = 1,2,..., 16 Calculating the initial parameters B o Calculating the variance <J>. for current model I Calculating the gradients Finding out the B by the algorithm for nonlinear parameters and m i n i m u m variance 3P ; E = <D mm , I O (i)= min Output db T~ O (i)+E min I (i), B, O min j = 1, 2, ..., m p. < a , B ) i r Calculating the values of p by every model Yes Output thetotalvariance O (i) i=l,2,..., 16 min I Stop the Program Figure 3.6 The Flowchart of the Program to Calculate M i n i m u m Variances 1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 Model a) N o Zero Stress Involved 109 c B e o o § •fi > g 1 108 107 106 105 104 103 1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 Model b) With Zero Stress Figure 3.7 Total Minimum Variance of Different Models for 79 Tests 1500 i m IL 1 100 r ° M & --*-- 500 Measured Model 13 s PQ J l 2 Stress (kPa) Figure 3.8 Comparison of Experimental Results with Those Fitted by Model 13 in L o w Stress Range If it is assumed that piow is the bulk density at the lowest stress Oiow applied experiments (0.37172 kPa - weight of the cover in Figure 3.4), then model 13 can be modified to: G= 0 Po = < n _,_ (Plow-Pp)CT p0+ bl0-b2 « — CT low n^<r~ 0<a<alow (3.3) O^Olow Surprisingly, from the experimental data, the modified model 13 gave the same fitting results as those from model 14 at each test point. With the modified model 13, Figure 3.7 b) replots as Figure 3.9. The typical fitting results for the best models in each of the two cases are shown in Figure 3.10 and Figure 3.11 (for sand with particle size in the range 90-106 pm). • c E — e I "fi > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Model Figure 3.9 Replotting of Figure 3.7 b) with Modified Model 13 1400 1 1 1 1 1 r ' I -!- - 1380 A Measued - ° - Model l """" Model 4 -*- Model 9 "+• Model 13 - ° - Model 14 -*- Model 15 -*- Model 16 1360 1340^ iitn I 10 . I 20 . L 30 . — i 40 L • • i- 50 Stress (kPa) Figure 3.10 The Typical Fitting Results for Best Fitting Models with Zero Stress Excluded 1450 20 30 40 50 Stress ( k P a ) Figure 3.11 The Typical Fitting Results for Best Fitting Models with Zero Stress Included Models 9, 14 and 16 as well as the modified model 13 (equation (3.3)) are best fitting equations. As an indication of fitting accuracy, the worst fitting and best fitting of these models are displayed in Figure 3.12 and Figure 3.13, respectively. 1600 Minimum Variance ( 8 points ) & 1400 ffl 13C A O Model 9 -O- Model 13 (14) — - Model 16 Model 9 812.3555 Model 13 (14) 611.9850 Model 16 796.7709 10 20 Measured 30 40 50 Stress ( k P a ) Figure 3.12 Worst Fitting of Models 9,14,16 and Modified Model 13 1200 1 W> 1100 »9 s s Minimum Variance (8 points) 1000. 900 Model 9 Model 16 10 0~~ 1.963542 1.868106 20 Measured Model 9 Model 16 40 30 50 Stress (kPa) a) Best Fitting of Models 9 and 16 1500 09 c & 1300 1200 Minimum Variance (8 points) — O — — ° — 2.216909 10 20 30 Measured Model 13 (14) 40 50 Stress (kPa) b) Best Fitting of Model 14 and Modified Model 13 Figure 3.13 Best Fitting of Models 9,14,16 and Modified 13 From Table 3.2, models 9, 14 and 16 provide p 0 at o = 0 as expected. W h e n o tends to infinity, the bulk densities predicted also tend towards infinity. Theoretically, the bulk density varies rapidly at low stress, slowly at high stress and becomes almost constant at very high stress (assuming that the particles do not fracture under high stress). The equation^ \ named model 17 here, P =Pmax - (Pmax " Po> e '^^ provides a slighdy better fit to the results, in the range of the experimental stress compared to that obtained using models 9, 14 and 16. However, compared with these models, the necessity to use one more variable generates considerable inconvenience. Based on the above evaluation, models 9, 14 and 16 are the preferred bulk density expressions for practical applications. 3.5.4 Application of the Bulk Density Equation As mentioned previously, the bulk density equation is obtained by one-dimensional tests. However, bulk materials often are handled in a multi-dimension stress state. This suggests that the application of the bulk density equation based on uniaxial stresses m a y be of limited practical application. a) General Considerations It is assumed that the relation between strain { 6} and principal stress { O} for bulk solids should satisfy -D -X> -o 1 -D { o) _ -1> -1) 1 _ • {e} =ie 1 (3.4) Since the bulk solids are not linearly elastic, K y and Poisson's ratio o are not constants but vary with bulk material voidage. For the bulk solid in the uniaxial compressibility test, if density model 16 is used the coefficient related to Young's modulus K y becomes (referring to eqn (A-1.10) in Appendix 1-1) A n approximation, to estimate the Poisson's ratio, is possible by introducing two extreme pressing states, as shown in Figure 3.14. VJ 'A '/, /?!!!!!j!!!!!j!j!^ i ^liiiiiiiiiiiiii^ A—T^Y/. a). With High Voidage Figure 3.14 \VA b). With Low Voidage Pressing of Bulk Material in T w o States From Figure 3.14, it is considered that the compaction of bulk solids with high voidage can be regarded as a one dimensional problem, in this case, v = 0. O n the other hand, compaction of the material at very low voidage is considered a three dimensional problem, \) = o 0 . Obviously all bulk materials handling situations fall between these two states. Therefore, a simple way to estimate the Poisson's ratio is to assume v=\>0 60-e -JL—e 0" e min (3.5) or v.^-z^*. P m a x " PO Unfortunately, no publication on the Poisson's ratio for bulk solids has been found. Values of the Poisson's ratio provided by Lambe and Whitman^ ^ are \)0<0.3 for such materials as amphibolite, limestone, rock salt and steel; on^0.36 for ice and aluminum, for omax = (3 ~ 5) * Hr kPa. If the density model 16 is applied, the relation between Poisson's ratio and stress, eqn (3.5), can be rewritten as / t) = o0[ —— j * (3.6) v "max / For practical purposes, it is assumed that t) 0 = 0.3; o , ^ = 4 * 10 kPa. b) Application of the bulk density equation to mass flow bins The work presented here attempts to enable the uniaxial measured data from the laboratory to be applied to simulate the bulk density in different sections of a mass flow bin. Since the vertical movement of bulk material in the vertical section of the bin is considered similar to that in the uniaxial laboratory measurement, it is assumed that the vertical stress o*z in the vertical section can be direcdy usedtodetermine the bulk density in this region, despite the fact that a z is not the major principal stress in dynamic condition. That is: in vertical section p = p0 + blv (o^)^ (3.7) where blv = bx; b2v = b2 . However, for the hopper section, many theoretical predictions for stress deal w mean sn-ess[1»3»28»3l-33,39,58] j n ^ case> a n e q U i va i en t major principal stress needs to be estimated for predicting the bulk density. Based on the assumption that the bulk density can be evaluated by the mean stress acting, the considerations include the following steps: • assume the computed mean stresses for different hoppers are: <*r+ q 8 . plane flow hopper mean stress omw = conical hopper mean stress omc a. + o e + o 4 = —* j* *; convert the computed mean stressestoequivalent major principal stresses: (referring to eqn (A-2.3) and eqn (A-2.4) in Appendix 1-2) plane flow hopper ole = 2(1-D) o"mw; 3(1-D) conical hopper ale = a^; • insert the equivalent major principal stresses into bulk density model, e.g., using model 16: p = p0 + b! (o^)*2 (3.8) • rewrite the bulk density equations as a function of the computed mean stresses: (referring to eqn (A-2.5)toeqn (A-2.6) in Appendix 1-2) p = p0 + b l w ( o conical hopper P = Po + b l c ( a ^ ) ^ m w ) b 2 w plane flow hopper (3.9) (3.10) Once the experimental data are obtained on the Jenike Compressibility Tester (Figure 3.4) then by using eqn (3.6), the eqns (3.9) and (3.10) can be easily obtainedtogetherwith the eqn (3.7). c) Differences Resulting from Use of the Different Bulk Density Equations, In this section the differences resulting from applying the bulk density equation to two types of converging channel is made; it is assumed that the computed mean stress (cr m w or o ^ ) has been obtained. The cases are considered as: Case I: Simply using the computed mean stress (crmw or <*mc) in the bulk density model 16 instead of the major principal stress; Case II: Using the equivalent major principal stress, which is converted from the computed mean stress (<J m w or a m c ) , in the bulk density model 16 and calculating the bulk density using eqn (3.9) or eqn (3.10). It is assumed that the computed mean stress is in the range of 0 - 50 kPa. Comparison of Case I with Case II has been carried out for the bulk materials used in Section 3.5.3, the results indicate that the Case I gives a lower estimation of the bulk density. Compared to Case II the percentage under-estimation from Case I was 0.12% to 1.34% for a plane flow hopper and 0.17% to 2.05% for a conical hopper, at 50 kPa computed mean stress; the corresponding absolute underestimation was 1.5 to 26 (kg/m3) for the plane flow hopper and 2.2 to 40 (kg/m3) for the conical hopper. For the 79 bulk density measurements examined and in comparison to the results obtained in Case II, the extent of the under-estimation provided by Case I for the plane flow and the conical hoppers, in the range of computed mean stress 0 to 50 kPa, is between lower bound and upper bound plotted in Figure 3.15 in terms of absolute and percentage under-estimation, respectively. From Figure 3.15 a) and b), it can be seen that the under-estimation of bulk den in the conical hopper is larger than that in the plane hopper. This follows from the fact that the computed mean stress in plane flow hoppers is closer to the equivalent major principal stress. Namely, for the same stress condition, the inequality CT le > amw > cmc applies. m upper bound for conical hopper ^ 3 0 4> 2 IS upper bound for plane flow hopper t10 se lower bound for conical hopper lower bound for plane flow hopper ,e \ *=* 10 20 30 40 50 Computed mean stress (kPa) a) Absolute Under-Estimation 25 • 1 • 1 • r upper bound for conical hopper upper bound for plane flow hopper _ 05 • 0.0 ^ lower bound for conical hopper lower bound for plane flow hopper \ <-> r 1 i i i i i • 10 20 30 40 350 Computed Mean Stress (kPa) b) Relative Under-Estimation Figure 3.15 Under-Estimation of Case I for Bulk Density for Plane Flow Hopper and Conical Hopper Compared with the Results of Case II While the differences between Case I and Case II are small and either approach could be used in practical situations, the Case II approach (using an equivalent major principal stress) will be used in the theories developed in this work to predict the air pressure distribution and the particle flowrate. This approach has been taken as it is felt that the Case II approach is more realistic from a theoretical point of view. 3.6 Permeability The permeability C is a measure of the ability of a bulk material to allow air (or other gases) to flow through under certain circumstances. Permeability is defined, according to D'Arcy L a w , as a ratio of the superficial air velocity (relative to the particles) and the air pressure gradient. T h e permeability is complex property dependent on m a n y factors such as particle size, size distribution, shape, surface roughness, moisture content and as well as consolidation condition. 3.6.1 Permeability Measurement The permeability of solids can be measured on a packed bed or a fluidised bed of particles. Considering the handling condition for gravity flow of bulk solids from a mass flow bin, in which the particles are in contact with each other all thetimes,a Jenike Permeability Tester (Arnold et al. 1980 [31 , as shown in Figure 3.16) was used to find the permeability of a packed bed at different voidages (consolidationrelated permeability). o Air Pressure Gauge -p Air Pressure Gauge (P0) / Rotameter ^2' Si i o o •o o Y & M •P M V, •H O ^ Air supply 4 r iF2^ oo e o o •H£ o V • • u Test c linder y 63.5 mm H—=-:—H dia •r Pi PI E-i r-t •ri i ji8 ^ Sampl 2 0 sample 04 < Sample' in * i\ < o H Regulator -AP t •ri +> W 0 '" •ri Figure 3.16 Jenike Permeability Tester (Arnold et alS *) The bulk density was obtained from the weight and volume of the bulk solid in the test cylinder P " Ap Hp (3.11) The permeability was calculated from -Sa-a C = A P Ap (3.12) For a certain flowrate Q of air through the packed bed in Figure 3.16, the pressure drop APt includes the pressure drop due to the resistance to air flow of the particl AP and that caused by the resistance of the distribution gauze AP0, i.e., APt = A P + A P 0 (3.13) Substituting eqn (3.13) into eqn (3.12) gives Hn C = A-p£ APt - AP 0 (3.14) 3.6.2 Models UsedtoFit Experimental Results As mentioned at the beginning of Section 3.6, the permeability of bulk solid depends on particle size, particle size distribution, particle shape, particle roughness and voidage. For the bulk solids with fine particles, the permeability of bulk solids can be dramatically influenced by the appearance of the fine particles and the range of the particle size distribution^64,66,100^. Therefore, an adequate permeability model must be selectedtodescribe these factors. Under the conditions in which D'Arcy law is valid, viz., the flow of air bet packed particles consist of laminar, viscous flow without significant turbulence or inertial energy losses, the models commonly used for permeability are in three forms, i.e., model 1= C-K^l-e)^) model 2 : C = K, ri04i (Mahinda Samarasinhe et al. ) (Jenike and Johanson[98'105'1061) (1-e)' model 3 : C = K 3 (Carman-Kozeny equation[87'107"109]) (1-e)' where K j , K^, K 3 , a and co are constants. Generally, model 3 is applicable to experiments on coarse materials with a narrow particle size distribution (where the particle size d ^ can be readily identified) and in an unconsolidated condition^108,871. Therefore, some limitations on the application of model 3 are as follows: • it cannot describe the flow of air through a consolidated particle bed well^110^, namely, it cannot describe the consolidation-related permeability well; • it applies only over a limited range of porosities or to monosized particles and it is not suitable for fine powders^ ™. In comparison, Models 1 and 2 can be used for a greater range of particle mixtures because they include one more variable to fit the experimental data. F r o m the expressions, it can be seen that model 1 is identical to model 3 w h e n the index number co is equal to 3. For instance, Mahinda Samarasinhe et alJ •* observed in their research that CO = 3.2 for crushed glass and co = 5.2 for N e w Liskeard clay. In order to find the best fitting equation to the experimental data, a similar mathematical treatment to that used for the bulk density models was employed. It is necessary to mention that the 10 factor for permeability has been omitted from the following model comparison. The best permeability model was correlated to a total of 125 points of the 16 materials tested. T h e results for the total m i n i m u m variance for the 125 experimental data points are plotted in Figure 3.17. 65 Figure 3.17 Comparison of Fitting Results for Three Permeability Models From Figure 3.17, it is evident that model 2 provides the best fitting model. As an indication of fitting accuracy, the worst fitting and best fitting of model 2 are displayed in Figure 3.18 and Figure 3.19, respectively. In view of the accuracy of Model 2, this model will be used in this work. An alternate expression for Model 2 is -491 (3.15) where CQ is a constant standing for the permeability at lowest compaction (without external stress acting during permeability measurement) Q) = K2 ( 1 - e 0 )5 21000 1 1 1 1 ' | » 1 1 Bulk Material: Sugar S u m of Minimum Variance of 6 points: 79514.0 T—E1" 19000 • 17000 15000 •*~~ Measured Value "•""" Predicted Value x j. 13000 0.40 0.41 0.42 0.43 0.44 0.45 0.46 Voidage Figure 3.18 250 i I Worst Fitting of Permeability Model 2 i I i I i Bulk Material: Sand (0-425 p m ) S u m of Minimum Variance of 10 points : 157.43 200 150 100 "*— 50 0.40 0.42 0.44 0.46 Measured Value Predicted Value _L 0.48 Voidage Figure 3.19 Best Fitting of Permeability Model 2 0.50 Chapter 4 T h e Prediction of Air Pressure Gradients in M a s s Flow Bins 4.1 Introduction Analyses of the air pressure distribution in a mass flow bin have been conducted by Crewdson et al.(1977)[50], Spink et al.(1978)[51], Head(1979) t55] , Nedderman et al.(1983)t56' and McLean(1979)^ 5 -L In most cases, the pressure gradient is related to the superficial velocity by use of evaluated parameters consistent with the assumed actual stress field. The basis of these analyses is that the pressure gradient is m a d e proportional to the superficial velocity of air relative to the particles by employing either D'Arcy's law or the Carman-Kozeny equation. The majority of these air pressure distribution predictions do not include the effect of height of material in the vertical section of the bin. However, McLean (1979)^ ^ and McLean et al. (1980)^ ^ provided the prediction of pressure in the hoppers with or without surcharge. Head and Crewdson et al. used a log function to describe the bulk density variation in the hopper. Unfortunately such a representation of the density variation is not valid for very small stresses. In the adoption of this density model, Crewdson et al. deduced that the m i n i m u m air pressure in the hopper occurred constantly at r=2.718282r0. However, this predicted position of m i n i m u m air pressure was not confirmed by their experimental results. They observed that the position of m i n i m u m air pressure were in the range between 1.7r0 and 1.8r0 for all experiments. In an attempt to offset the above deficiencies, the following mathematical model attempts to include the effect of material head on the air pressure distributions in bins. Particle dynamic deaeration phenomena are also considered. This model for predicting the pressure distribution is based on the following considerations: i. The stress near the oudet of hopper is a linear function of the vertical distance measured from a point which is located between the apex of hopper and the oudet, as shown in Figure 4.2. ii. Bulk density and air pressure in the bin vary with the vertical distance continuously. iii. The flowrates of particles and air at any cross-section in a mass flow bin are constant. iv. The dynamic deaeration at the top surface of bulk material is proportional to the flowrate of particles. v. The mass flow bin is divided in three regions, as shown in Figure 4.1. The principal stresses are assumed to be vertical in region I, radial in region III while the region II is regarded as a transitional region. The dimensionless depth values used in Figure 4.1 are defined as follows: n h ^lrnax- hi n bo h* h max i mp (4.1) 69 Figure 4.1 Regions Defined for a Mass Flow Bin 4.2 The Bulk Density Distribution in Mass Flow Bins i) B u l k density distribution in region I A s presented in Section 3.5, the bulk density o f a bulk solid is a function of the extent of consolidation. T o evaluate the bulk density in the vertical section of the bin, the stress level m u s t b e estimated. O n e of the typical theoretical analyses o f stresses in the vertical section of the bin is the w o r k of Walters in 1 9 7 3 t l l l ] . H e extended Walker's theory to predict the stresses in the bulk solids under both static (initial fdling) a n d d y n a m i c (flow) conditions. Blight in 1 9 8 6 [ 1 1 2 ] presented a m e t h o d to determine the stresses of fine p o w d e r s in silo. H e considered the effect 70 of the pore air pressure during material filling. However, he recommended that this effect is very small especially for slow filling. It is acceptable that this effect on the vertical stress can be ignored for steady state flow where there is continuously filling while discharging material. Thus, once the effect of the pore air pressure on the vertical stress is neglected, Blight's theory becomes similar to Walters' theory but the lateral stress coefficient used in Blight's equations is not predictable. Therefore, Walters' equation for the material under dynamic condition is preferred to estimate the vertical stress distribution in current cases. The equations presented by Walters and Blight were derived assuming that the bulk density of the powder was a constant. However, it is believed in this thesis that the bulk density varies with the stress. If this variation is considered in Walters' equation, the derivation of an analytical equation for the vertical stress is not apparent F r o m Appendix I - 3, it is found that this variation can be neglected at this stage; the bulk density distribution in the vertical section can be approximated by using Walters' vertical stress distribution under dynamic conditions^ •• (referring to eqn (A-3.4) in Appendix I - 3), i.e., 4K ° (Tl)= where o QD w hl (» __x n ^Imax *\t 4K [1_e Kw = tan <|> cos 8 2— ( 1 + sin S ) - 2 y sin 8 y = 3V[1'(1"c)3/2] ( tan<|) V V tan8 ) ] (42) The bulk density distribution, using eqn (3.7), becomes 4K w h, P = Po + b l v [ 4 0 0 0 K ; ( l _ e b2> D ) In fact b 2 v « l»then ^v « p o b2 The above equation is rewritten as b P = Po + iv r 4K w h t Pogp (l-e D 4000 K w (nmax- 1 !) ) >2v (4.3) Hence, it follows that the bulk density at the transition level is Pl = Po + bi v ii) PogD 4000 K. (l-e" w 4K w h l „, -.b2 n D ) (4.4) Bulk density in region m From the consideration i) mentioned in Section 4.1, as shown in Figure 4.2, th stress in this region is described by: O m c = ki (*1 -11*) where rj* is suggested to be in the range of (0.95 ~ 0.99)rj0^58^, all theore results concerning this model in this work are obtained by using TI*=0.95TI0 From eqn (3.10), the bulk density in region in is (4.5) P = Po + bic K 3 (r) -Tj* ) 2c The bulk density at T| =ri m p, is P m p = Po + bi c K 3 ( limp - "H* ) iii) (4.6) 2c Bulk density in region II The stress in region II is a transitional stress field in the upper part and b equaltothe radial stress field in the lower part. For the hopper without surcharge, Figure 4.2 a), it is assumed that die flow stress field is approximated by a parabolic equation here, i.e., >mc = k 2 V ill - 11 In general, the stress distribution is assumed as ( °mc- < l/b2C Pi ~ Po " b ic ) + WTii-,n Therefore, the bulk density becomes p = Po + bic Pi-Po »ic l/b2c +WTii-,n .b2C a) Stress Field in Hopper Without Surcharge b) Stress Field in Hopper With Surcharge Figure 4.2 Stress Field in Hopper (McLean 1979 [58]) O n assuming the bulk density varies continuously with the depth, at T)=T| mp , P=Pmp» then 1 p=p0+ blc < fPi-po> °2C I bic J J_ JL • . 2C b r P m p - P o ^ c /"Pi-Po"\b2cl / 11 i~il A bic J I bic J J V ^ " ^ m p 4.3 (4.7) Superficial Velocity of Air Relative to Particles 4.3.1 Flowrate of Air Neglecting the change of air density in the bin, the absolute flowrate of air through any section is constant, i.e. M Qi where + i£j = Pi Qi + Mia (4.8) pj Q ^ Q ; are the volumetric flowrates of air relative to the particles at location i and j, respectively. M ^ M j are the mass flowrate of particles at location i and j, respectively. Considering the sense of flow, M = - Q « , where Q p is the absolute mass flowrate of particles. ' -' comes from the different sense between the particle flow (downward) and height (upward is positive) The subscripts i and j stand for arbitrary levels i and j, respectively, in the bin. Setting the reference level at the top level of material, then Po p Since e = 1 - — — ; Ps Pi and for mass flow, M Q = M j = M , then The term Q0 stands for the dynamic deaeration condition at the top surface. Assuming that Qo = " Kdea — (4.10) Po where K<iea is a dimensionless coefficient (named dynamic deaeration coefficient) then the general expression for the relative air flowrate Q is Q=£[('-K<iea>-£] (4.11) 4.3.2 Relative Velocity of Air to Powder The superficial velocity of the air relative to the particles is the ratio betw relative flowrate of air and cross-section area through which the air is passing, i.e., u = A In region I u= M PoA> [(l-Kdea)-y] (4.12) where A v is the cross-section area of the vertical section of bin In region U and EI u = M PoAc [(1-Kdea)--^ (4.13) where A c is the local hopper area 4.4 Air Pressure Gradients in Mass Flow Bins The mathematical model for the pressure gradient is based on the D'Arcy law, viz., dP u dh " " C or dP dP.dn, _u , dp ~ dh'dh " " C h i i) Li region I Substituting eqn (4.12) into eqn (4.14) yields ^p.-_2lf-P_Yl_JL_rri-ir ^ p0" dri " ColpoJ p 0 A v L U ***** "p~_ (4.14) 4h M Let W 0 = ^ " "t ^ 2 a n d f = 7"» then CoPo^Do^ p0 S-w.5'"[»-«*.)4] or g.w.fij'^-u-^', (4.15) where f=l + p lv Po Pog° (l-e 4000 Kw 4 K w hl („ D lTlmax b 2v _n\ 1]) ) (4.16) A similar method is usedtoobtain the pressure gradient in region II and region m . ii) In region II S-w.^E^-O-Wf'l (4.17) where f = —P Po with p determined from eqn (4.7) Pmp is calculated by eqn (4.6) (4.18) iii) In region III ^ = W„(^)2[fa-1-(l-Kdea)fa] (4.19) where f -*Po = i + _lC-^3(T| _ T1 *) b 2c Po 4.5 (420) Air Pressure Distributions in Mass Flow Bins The air pressure distributions in each region is obtained by integrating the gradients over the respective height ranges. i) In region I (T|j < T\ < Tjmax) 2 T\ a 1 P-Pi = W0-| [ [f - -(l-Kdea)fa]di1 (4.21) when r| = r\max, P = 0, then ~2 ilmax -Pi = W 0 - | | [fa-1-(l-Kdea)fa]dTi where Pi is the pressure at the transition level and f is determined by eqn (4.16) (4.22) 79 ii) In region II (ri mp <, T) <, r^ ) ,TI [f^-d-Kdea)^] P-Pmp = W 0 | i dn •^mo (n/Tin) or ^-(l-Kd P = Pmp + W 0 I Imp (il/ilo)2 ^'a dri (4.23) where f is determined by eqn (4.18) Pnm is the pressure at rj = r|mp When 11=11!,? = ?! given by fill [f^-d-Kdea)^] Pi - Pmp = W0 I 5 d^ mp in) (4 24) ' (T|/T!O) Inregionni (ri0 <> x\ <, ri mp ) < t A (•> v. \ r a< t [f•a-1 " -(l-K d e a )f ] P = W0[ J" dr| (il/ilo)' ril (4.25) When rt = Timp, P = P m p given by filmp Pmp = W 0 J V Jl lo [fa"1-(l-Kdea)fa] ™ dti dl/ifo) where f is determined by eqn (4.20) (4.26) 4.6 Boundary Conditions for Application of Air Pressure Model The unknown variables in this mathematical model are K3, K d e a and r ) m p . Hence, three independent boundary conditions are required to determine these variables. 1) Pressure Continuty eqn (4.22) + eqn (4.24) + eqn (4.26) = 0 (4.27) dP 2)Atr|=rimp,— =0 ( 1 - Kdea )-£m2- 1 = 0 Po Kdea = 1 - ~2j- (4.28) Pmp 3) Estimation of TJmp The pressure minima point rj = rjmp is that separating the positive pressure gra from the negative pressure gradient; it is also the point where the direction of superficial velocity of the air relative to the particles switches. It is assumed that rjmp depends on the bin geometry, mainly r|o and i]max. Figure 4.3 shows the relation between T]mp, rj0 and T)max observed in experiments. Since there were insufficient data to identify T[mp more accurately, a simple relation estimating Tjmp is used to avoid the effect from any particular experiment at th stage, i.e., ( l+ilmax'l ilmp = ilo * V l+Tlo (4.29) > 4.0 3.0 o 1 • waiis • Head * Experiment (Chap. 6) • Crewdson et al. ^^^ H X JW^ • ex E ^ r 2.0 - B X X - ^S* .x'x y^ 1.0 1.0 • i 2.0 • (1+rj v 3.0 )/(1 + TI A ) 'max' v 4.0 '0' Figure 4.3 A Plot of the Relation between rj m p, T|0 and i l m ^ 4.7 Discussion 4.7.1 General Observations A s one significant basis of this air pressure gradient model, the eqn (4.8) describes the motion of air between particles based on one kinematic consideration which is that the absolute air motion in the bin equalstothe vector sum of relative air motion to the particles and the absolute air motion with the particles. According to eqns (4.8), (4.11) and (4.28), the assumed constant absolute air flowrate in the bin equals M |. Since the bulk density at the m i n i m u m pressure position \Pmp Ps ) Pmp is always less than particle density p s , i.e., p m p<p s > ^ absolute air flow is always in the same sense as the particle flow (downward). However, the variation in the relative air flowrate which relates the air pressure gradient depends on the magnitude of the local bulk density. Inserting eqn (4.28) into eqn (4.11), the relative air flowrate is given by Q = M(^i--I) (4.30) It can be interpreted that the relative air flowrate is upward and downward as P < P m p and p > p m p respectively. Consequently, the air pressure gradient either retards or assists the particle flow depending on whether p < p m p or p > p m p respectively. A typical air pressure distribution in a mass flow bin is plotted in Figure 4.4, having a pressure minima near the hopper oudet and a pressure maxima near the top surface of the material. This pressure distribution has also been reported by M c L e a n [ 5 8 ] , Spink [ 8 0 1 and Spink et al.[51]. The physical meaning of this distribution can be explained easily with the current model. From eqn (4.28), the extreme point of air pressure occurs at p = P m p - During material discharge, the highest stress level appears at the transition section (Figure 4.9); the bulk density increases as the particles flow towards the transition level from the top surface then decreases as the particles discharge towards the hopper oudet. Figure 4.4 and Figure 4.10 show that the bulk density increases rapidly near the top surface then only slighdy in the remainder of the vertical section. Since a low value of bulk density appears at both ends of bin and the m a x i m u m bulk density (Pmax) occurs at the transition level for a hopper with surcharge or in the converging section for a hopper without surcharge, p m p is located between low and m a x i m u m bulk densities. There should be two extreme points which are located on either side of the m a x i m u m bulk density position, as shown in Figure 4.4. Furthermore, since 83 the air pressure self-generated by the flowing particles is continuous, these two extreme pressure values should be the maxima and minima values. hopper outlet height above the hopper oudet pressure maxima top surface of material Figure 4.4 Typical Distributions of Air Pressure and Bulk Density in a Bin (with Surcharge) The absolute value of the pressure maxima is very small compared with the absolute value of the pressure minima; the magnitude of the pressure maxima depends on the hopper oudet size and permeability of bulk solids. Table 4.1 shows the examples in which the flows of Sand M D 3 (a low permeability bulk solid) and Sand M l (a high permeability material) from both 0.02 m and 0.0445 m hopper outlets were considered (the flow properties of two bulk materials and bin geometry are detailed in Chapter 6). F r o m Table 4.1, it is apparent that the larger the hopper oudet size or the lower the permeability of bulk material, the higher the pressure m a x i m a becomes; the larger the oudet size, the greater the absolute ratios between pressure m a x i m a P m a x and pressure minima Pmin- F r o m eqn (4.29), as the hopper oudet size increases, the pressure rninima point moves toward the m a x i m u m bulk density point; the pressure m a x i m a point, therefore, moves away from the top surface so that the measurement of the pressure maxima can be less affected by the atmosphere at the top surface of the material. Furthermore, the magnitude of air pressure increases with an increase in the hopper oudet or with a decrease in permeability of the particles (as discussed in Section 4.7.2). For these reasons the pressure maxima is more visible in the experiments with the large hopper outlet and/or with low permeability material. This fact can also evident in the experimental results presented by S p i n k ^ and Spink et al.t51^ using plane flow hoppers. From Figure 4.4, a negative air pressure gradient occurs in region III; a significant negative air pressure gradient is generated near the hopper oudet. This is because that the bulk density in this region p is less than p m p , from eqn (4.30), a negative air pressure gradient is generated by flowing particles. Physically, the bulk density in region HI decreases rapidly, due to dilation. The increase in dilation results in an increase in negative pressure, causing an air flow opposite to the bulk solids flow, to fill the increasing void volume. Table 4.1 The Pressure Maxima and the Ratio between T w o Extreme Pressures for Different Sized Bulk Solids and Hopper Oudet Sizes Bulk Material SandMD3 (with low permeability) Sand M l (with high rjermeabiHty) Oudet Diameter (m) P P 0.02 0.0055 max p . min 0.00047 0.0445 0.2000 0.00647 0.02 0.0005 0.00047 0.0445 0.0140 0.00539 max (mmH20) 4.7.2 Comparison of Theoretical Model with Experiments The theoretical results predicted by the current pressure model are now comp with the experimental results obtained by i) experimental work (on alumina) detailed in Section 6.6; ii) Crewdson et al. (on three sand mixtures)'5™; and iii) Head (on two of his sand mixtures flowing from hoppers with two oudet sizes)1551. For the bulk materials used by Crewdson et al. and Head, their flow properties such as bulk density and permeability were estimated by comparing them to those materials in Chapter 6 with similar range of particle size. The flow properties and bin / hopper geometries for their experiments are listed in Table 4.2. The flowrates Q p for Crewdson et al. were digitized from Fig.l in their papert501; the bulk materials shown in 'referring material' in Table 4.2 are further detailed in Chapter 6; the surcharge level for Crewdson et al. is assumed to be 0.01 (m) to simulate the flow without surcharge; furthermore, the bj, b2 are used instead of the b l v , b 2 v and b l c , b 2 c ; the estimation of the permeability constant C Q for those mixtures which have two referring materials were initially from the relation which is assumed to be proportional to square of d 5 0 particle size (following the Carman-Kozeny correlation but neglecting the effect of the difference in porosities of two compared materials), i.e., ^0 = cl dso +c2 where ^ and C2 are constants. e.g., to estimate the C0 for Crewdson et al.'s sand sized 212-300 pm, the referring materials are Sand M 2 and M 3 with about 4360 and 2330 *10" 9 ( M 4 N^Sec" 1 ) in permeability constant; 310 and 200 p m in median particle size respectively. Therefore, the constants Cj and c 2 are determined as 0.0362 and 881.18 respectively. The median particle size d 5 0 for sand 212-300 p m is taken as 255 p m . Thus, the estimated permeability constant CQ equals 3235 *10"9; take as 3250 *10" 9 ( M 4 N _ 1 Sec _ 1 ) in Table 4.2. The comparisons of the theoretical and experimental air pressure distributions are depicted in Figure 4.5 to Figure 4.8. The predicted pressure distributions closely approximate the observed variations for sand mixtures and for alumina. The theoretical results, which are similar to the experimental findings (the measurements are detailed in Chapter 6), highlight the following: i) Air pressure in a mass flow bin (negative air pressure gradient at the hopper oudet) increases with increase of hopper outlet dimension (as shown in Figures 4.7 and 4.8). Since the dynamic stress in the hopper is small, the difference between stresses in the two bins with different outlet sizes is insignificant. Therefore, the differences between the two bulk density variations and permeability distributions under equilibrium conditions do not have a significant effect on the magnitude of air pressure gradients. A s the outlet is increased, the particle velocity increases. Consequendy, the increase in particle velocity relate to the increase in superficial relative velocity between particles and air with a corresponding increase in air pressure gradient. Therefore, the larger the hopper size, the greater the negative air pressure in the bin, and further, the greater the adverse air pressure gradient at the hopper oudet. This phenomenon can been also found from the experiments by Head* 5 5 1 on conical hoppers; by Willis*541 and Spink* 801 on plane flow hoppers (Figure 8.13 shows the Willis' experimental results). ii) Air pressure increases with increase of surcharge level (as shown in Figure 4.5). Physically, for a typical slice element of material in the bin, there are two types of air flow in the element of particles: one is the flow due to air being expelled from the element (relative motion of air); another air flow component is that due to drag by the discharging particles (absolute air motion with the particles). A s discussed in Section 4.7.1, the relative air velocity to particles is d o w n w a r d in the portion between two pressure extreme points as p > p m p . T h e higher the surcharge level, the longer this portion becomes. Therefore, the air pressure gradients generated in this portion of the bin result in the air pressure being proportional to the distance between two extreme pressure points and hence, the higher the surcharge level, the greater the negative pressure produced. iii) Air pressure increases with decrease in material permeability (as shown in Figure 4.6 to Figure 4.8). F r o m Section 4.5, it can be seen that the product C 0 p 0 is an important term to indicate the negative air pressure. In particular, the negative air pressure increases with the decrease in product C Q P Q . Furthermore, since the permeability indicates the ability of the bulk solids to allow the air to flow through it, the permeability constant C ft has greater effect on the pressure gradient than the bulk density constant p 0 does. For a lower permeability powder flowing from a mass flow bin, a counter-current air flow to fill the increasing void volume caused by particles dilation in region D I is more difficult to generate requiring a higher negative air pressure gradient For a finer material with lower permeability, a higher pressure gradient generated by powders results in more resistance to the flow of material in region in and a decrease in flowrate. Furthermore, if the permeability of material becomes very low, the resistance of air pressure gradient can cause difficulty in controlling the flow of material due to the occurrence of flooding (more discussion on this topic will be presented in Chapter 5 and Chapter 8). iv) Effect of using blv and b2v instead of the blc and b^ in hopper on the predicted air pressure is insignificant A s mentioned in Chapter 3, the predicted difference in bulk density by using b l v and b ^ instead of the b l c and b ^ in the hopper is small (less than 2.05% for the tested material at 50kPa stress level). The error in the predicted bulk density and permeability produced by application of these parameters will be further reduced due to the low stress applying during flow. According to the calculation for alumina used above, the over-estimation of pressure by using b l v and b 2 v instead of b l c and b^c in pressure prediction model is less than 2 % . 1 a 00 8 3 o en SO ON ON oo -^ CO 8 O s NO d co oo ^^ CO •a I £ o d vo •Sa vo vo oo ,—< t3 «o »-H CO d 00 <s s «n i-H d CO rt- CO d 2 vo •o "* O o d d CO VO CO 00 !Q o oo d r—t 8 d 3 CO o o d 8 8 00 d CO CO 6 =L S'a 1 gi is co •O CO «-H =t o 1 • co i—1 8 VO <* ON i—• VO o d a 8 •a CO .2 * o o § ^~v *-> ON <S 1 r-t O oo d 1—t o 00 CO w> en o VO CO CO ON OO VO d 8 ON d co i—i o d CO © CO 1—( T-H o d i o d b X! GO £ ! 1 CO 8 2 VO VO d CO CO 6^ 35 VO 8 E CO CO a =t I 1 ~ ^—\ 1 2. x> * /"^ ^ Q. CO /Cjisuoa^H /CjmqB -auusd a >-• o PQ •™\ 1Q cd •©- ? a^ 8 sa /-S o •8 Q 8 sOUPOIOar) 1 -a «? S t-H 1 1 I 2 „ 3 8 aa * 38 3so co8 8m • j «o> -O vo o © © II II © \ H = 0. H = 0. i-3 mental © W5 & • ° \ so © ! \ t t \ ro \ •A a I SO i © \ •\ u •a I B<a \ «« ••\ d £00 s X • \ i 8 o CO © "r\k \ \ • ^>i <Ml t • fl •• • m 1 O 03 1 •TJ» ••• \ o coo d 1 &»coo A o oo jf ID mm d o w i—i o 1 . i 8 i . i . d i 8 3 8 . I q 55 i (J3JBM unu ) ainssay JI y . I o ri . I o oo i . 91 •J t-; a d 09 3 <s VO d T o s 111 a CO CO CO "1 d 8 CO i Tt i cs «o e* cs •n -3 111 1 CO CO CO CO w CO c /"•S •<* a ^ o t<i) * e» d C ? 8 u -a =CS6 3 en d CO -a 06 I cs o s CO VO i E q ©"• CS (jajBM UIUI ) amssay ity 92 s 09 /-s a 09 0) as (U 00 T3 d T3 (V t& = 0.02805 = 0.0304 ( Si 09 ss 11 1 I tt a09 CU 3. ^-v a "3 8 OS *•» C*i iH 00 i 00 ©• CO a o s 1 1 aua ©o 8dII ••* Q w 09 • o n a i •• CO •-* u o» 03 03 a a *> o >w/ s O b 5 J3 00 <s d © ©W-l • * "O (jaiBM U I U I ) amssaid jny r? 93 s 09 a 09 PCS T3 0) oo d a •3 II II ss H as 3 a 0> "0)C aa 09 T3 03 0) 5? 0.0304 ( 3 w a 0.02805 0.0304 ( 09 00 0.02805 0) 09 II II ss • • VO 00 iH -a a • 03• VO i 1 CO 0 03 8 K 3 % © W •a CO i§ & •fl CO *-» i—H u 3 CO s Pi •a •a 8 «-> •a I i cs d 00 i E © ©^ • * cs ,o **o ^r I •o I (jaiBM unn) amssay Jiy 94 co *n vo © II II xi (50 3 ( B ^ ) ssaus 95 cs cs a a co m vo © © © II II X X cs •a S 3 I I 1 O ( rin/ib[) suig sqi ui uonnqinsiQ yCjxsuaa ^ g Chapter 5 Prediction of the Flowrate of Bulk Solids from M a s s F l o w Bins 5.1 Introduction The flowrate of bulk solids from mass-flow bins depends very much on the flow conditions experienced by the particles near the outlet of the hopper. Considering the dynamic balance of bulk solids flowing in the outlet region of a hopper, the forces acting on the particles, for a cohesionless material, are as follows (based on Carleton's classification^ ™ ) : i) equivalent inertial force due to the acceleration of the particles ii) gravity force iii) resistance by interstitial air pressure gradients iv) interaction between particles. Under steady state conditions, the equivalent inertial force is related to the rate of change in bulk density and cross-section along the flow stream. Most publications treat the bulk solids as incompressible materials, whereas M c L e a n (1978) [ ], considered compressible material in his research. Almost all publications consider the resistance by interstitial air pressure gradients is negligible for coarse free flowing materials discharging from large orifices. However, an increasing number of researchers realise that the air pressure gradients do affect significantly the flowrate of fine bulk solids. Unfortunately, there are different criteria to classify bulk solids as coarse or fine material. For instance, Carleton (1972) [49] suggested that the distinguishing criterion for particle size is about 200 (im. However, most other researchers including Williams (1977) [28] , Crewdson et al. (1977) t 5 0 ] and Spink (1978) [51] prefer 500 p m . In comparison the present examination and mathematical model, for predicting the flowrate, attempts to cover the complete practical bulk solids particle size range by introducing permeability as a parameter to indicate the effect of adverse air pressure gradients and by using consolidation-related bulk density to consider both compressible and incompressible materials. The theoretical analyses indicate that for the bulk solids under dynamic conditions in a mass flow bin, no matter if it is compressible or incompressible material, the bulk density and stress gradients only have insignificant effect on the flowrate. However, permeability and hence the air pressure gradient generated exercises significant control on the flowrate, especially for fine bulk solids. Following assumptions have been considered in the derivation of the model: i) The flow of bulk solids in the outlet region, as region III in Chapter 4, is radial; however, it is assumed that the results obtained in Chapter 4 (based on the consideration of particle flow through a horizontal cross-sectional plane in region HI) for the air pressure gradient at the hopper outlet can be applied without causing any significant error. ii) The wall of the hopper is so smooth that the effect of wall friction can be disregarded. iii) The flow of bulk solids is continuous so that the model presented can be based on continuum mechanics theory. 5.2 Theoretical Model for Predicting the Flowrate 5.2.1 Equation of Motion The spherical coordinates (r, 6,0) are used for the axisymmetrical hopper shown in Figure 5.1. Figure 5.1 Flow from Region HI of a Conical Mass-Flow Bin The equation of motion can be expressed in the form, according to Bird et al. (1960)[128], dVr 1 d,2 , o-e + a* dP /C1, P V r i r = - - J ¥ ( r o - r )+ ^ ^ + pgr-^ (5.1) where gr is a gravitational acceleration in r direction (gr = - g cos 6) Since the flow is converging, a 'passive type1 of stress distribution will prevail. Assuming o r is the minor principal stress, o e is major principal stress and adopting the Haar and van Karman hypothesis which states that the circumferential stress o* is equal to the major principal stress^1,3,361, the relation between three principal stresses is given by oe = o0 =K or (52) ^ . .j L, .j [1,3,27,28,55,58,75] L t . e , There is considerable evidence to suggest that the ratio of the major and minor principal stresses is constant and can be expressed as a function of the effective angle of internal friction of the material by, K = -^i"L« 1 - sin 8 (5.3, Substituting eqns (5.2), (5.3) into (5.1), and considering the effect of 0 (0 < a) being small, the equation of motion (5.1) becomes T7 dV r do,- 2 (K-l ) dP ,.,. According to the assumption i) in Section 5.1, the mean stress in region m assumed in Section 4.2 ii), is rewritten as 'mc = — - — ^ where r = then h x cos a *- = — a — a r = Ki (r - r*) ; r* = -hj cos a (5.5) CT CTr 3 ( r - r» ) " 2K + 1 K l <5 and hence da r 3 "dT~ " 2 K + 1K l (5 Therefore, eqn (5.4) on substitution becomes dV r 3Kt f 2 ( K - l ) , 1 dP = ^-dT 2KTTL r ('"«*)-1J-P8--5" (5 5.2.2 Equation of Continuity For steady state flow in a mass flow bin, the mass flowrate of bulk solids is Qp = A V r p (5 and the flowrate gradient should be equal to zero, i.e., £- (5. Eqns (5.9) and (5.10) lead to dA dVr dp V r p f + A p - ^ + A V r f =0 or P ~dT r L dr A dr J (5 Since the cross-section area A is 2 TC r2 (1 - cos a ) for a conical hopper, then dA . ,, N -r- = 4rcr ( 1 - cos a ) or J_dA 2 r A dr (5.12) From eqns (3.10) and (5.5), the bulk density equation will be fMr-r*)]b2c P = Po + b l c [ IOOQ—J then dp dF = b^ ( r - r* ) ( p " P o ) (5.13) Hence, on substituting eqn (5.13) into eqn (5.11) yields dV r 2f 02c . 2pl or dV r b2c ~ ( f t pfJl L (r-r*) *] ( P - Po ) + (5. 5.2.3 Equation for Predicting the Flowrate Substituting eqn (5.14) into eqn (5.8) yields (M<^<>-^¥ 3K£i r 22(K-1) X f (K-1), I dP 2K TT[ 7 (r-**)-iJ-pg-5Since at the hopper oudet, r = r0, hence r Qp -( A y Pout bfc , . , 2poutl r ^ T ( P o «.-Po)+ 3Kt 2(K 2 2K+1 1) 1 1 — J p g - ( -S)- ]- - -(f)r = r0 (5 The air pressure gradient at the outlet is (dP\ _fdP) where cosa is obtained from eqn (4.19) Therefore ^ d W r = r0 "Q^jA 1 [ 1 -< 1 - K dca ) ^ l c o s a (5 Equations (5.15) and (5.16) yield on substitution the following mathematical equation for predicting the flowrate: anQp +b11Qp-c11=0 (5.17) where ail= ( AoT^ut) [ p o u t U - r * ) ( P o u t - P o )+ i bn = ^ut [l-(l-Kdea)fout]cosa CoPoAoPout cn = g + (2K ^ D P o u t L r ° 'J ^ where K = 1 + sin 5 1 - sin S bicKsfdo-r"')^©:"!1^ fout=l + K 3 Po [ 1000 cos a __ 1/hv. K i= 3 ^ h. Kdea b Po f 1 ^ " i-Kdea ic I Timp-Ti* J rn cos a Ti m p is determined by eqn (4.29) and r\* = 0 . 9 5 — ^ 5.3 Application of Theoretical Model The theoretical model (5.17) has been applied to predict the flowrates of the bulk materials detailed in Chapter 6. A complete comparison of theoretical results with the experimental results will be presented in Chapter 8. In this section, some phenomena highlighted by the theoretical model are discussed. 5.3.1 The Effect of Surcharge Level on the Flowrate The effect of surcharge level on the flowrate has been observed for the materials in the current experiments. The predicted results for the effect of the surcharge level examined in the experiments are depicted in Figures 5.2 to 5.4. The theoretical results indicate that for coarse materials flowing from a small outlet the effect of surcharge level is insignificant; whereas for fine material the effect can be significant, depending on the outlet size of the bin. M o r e of this dependency is discussed in Chapter 8 which provides more details of this interesting finding. Furthermore, Chapter 8 will also compare these theoretical results with experimental results. In order to show theoretically that the effect of surcharge level depends on the particle size of the mixture and the outlet size of the hopper, Sand M D 3 , Sand M 5 , Sand M D 1 and Sand M l (with permeability constants of 575.561, 1155.835, 2227.654 and 6517.522 *10" 9 M 4 N" 1 Sec"1, respectively) were used as different particle-sized materials to check the above results over a wide range of surcharge level and outlet size. In particular the range of outlet size varied between 0.010 ~ 0.050 metre. T h e m a x i m u m outlet diameter was taken as 0.05 m from the consideration D / D 0 » 3.0 (D=0.145 m ) . The highest surcharge level H / D = 12.0 was considered as the m a x i m u m extent of surcharge likely to be used in practice. Predicted flowrates are shown in Figures 5.5 to 5.8. From these results, it can be seen that for fine material (with low permeability constant), such as Sand M D 3 , the effect of material level in the bin is significant when it is flowing from the larger oudet In comparison for coarse material (Sand M l ) the effect is insignificant This flowrate variation dependency is considered to be due to the retarding influence of interstitial air pressure gradient, as expressed in eqn (5.17). The predicted air pressure gradients at the hopper oudet corresponding to these predicted flowrates are plotted in Figures 5.9 to 5.12. These figures show that the interstitial air pressure gradient generated by the flow of fine material at the outlet is greater than that for coarse material. For instance, Sand M D 3 experiences a negative air pressure gradient about 6 - 1 6 times higher than Sand M l does in the range of outlet sizes and surcharge levels considered. It is also shown, from figures 5.5 to 5.8, that the rate of decrease in flowrate with AO, respect to material level, —rf-, decreases as the level increases. It is believed that as A(g) Ap the surcharge level increases, the rate of increase in bulk density, ~ | j ~ , decreases A(g) AK^ea and the rate of increase in dynamic deaeration coefficient, — g — , decreases (refer A%) to eqn (9.19)). This leads to a reduction in the rate of increase experienced by air pressure gradient at the hopper outlet A((f)r=r0) p (refer to term b A(g) u in eqn (5.17)), as shown in Figures 5.9 to 5.12. This reduction of increasing rate in pressure gradient results direcdy in a reduction of rate at which the particle flowrate decreases, as the surcharge level increases (refer to Figure 5.5 to 5.8). 0.12 • = c = s=i t=r J — — 0.10 o^ g o E Legend 0.08 — ,™™r~ B Alumina • P V C Powder • Sugar 0.06 0.04 1 2 3 H/D Ratio a) Qp vs. H/D (from 0.02 m Outlet) 1.00 8 0.80 Q/O, Legend 0.60 0 o E 1 I Alumina • P V C Powder • Sugar 0.40 0.20 2 3 4 H/D Ratio b) Qp vs. H/D (from 0.0445 m Outlet) Figure 5.2 Predicted Flowrate Varying with H/D Ratio for Alumina, PVC Powder and Sugar 0.20 * Material and Median Particle Size 0.15 (Hm) Sand Ml d50=370 Sand M2 d50=310 • SandM3d50=200 X SandM4d50=200 4 SandM5d50=155 0.10 £ 0.05 Sand M 6 d 5 0 = 80 0.00 2 3 H/D Ratio a) C^ vs. H/D (from 0.02 m Outlet) 15 Material and Median Particle Size i—I 8 (Hm) 1.0 SandMld50=370 rr** i~—-"•• i Sand M 2 (150=310 • SandM3d50=200 x Sand M 4 d 5 0 =200 SandM5d 5 0 =155 0.5 9 0.0 ——e © 2 -o Sand M 6 d 5 0 = 80 3 H/D Ratio b) Qp vs. H/D (from 0.0445 m Outlet) Figure 5.3 Predicted Flowrate Varying with H/D Ratio for Sand M l to Sand M 6 0.16 Legend H SandMDl • SandMD2 • SandMD3 * SandMD4 1 2 3 H/D Ratio a) Qp vs. H/D (from 0.02 m Outlet) 1.20 Legend a SandMDl • SandMD2 • SandMD3 * SandMD4 H/D Ratio b) <^> vs. H/D (from 0.0445 m Outlet) Figure 5.4 Predicted Flowrate Varying with H/D Ratio for Sand M D 1 to M D 4 Oudet Size ( m ) Db=0.050 Do=0.0445 1^=0.040 %=0.035 Db=0.030 Do=0.025 Dfo=0.020 Db=0.015 D,j=0.010 6 H / D Ratio 8 Figure 5.5 Predicted Flowrate vs. H / D Ratio for Sand M D 3 Oudet Size ( m ) Eo=0.050 Do=0.0445 Eb=0.040 Eb=0.035 Efo=0.030 Do=0.025 Efr=0.020 Efr=0.015 D0=O.O1O 6 H / D Ratio 8 10 Figure 5.6 Predicted Flowrate vs. H / D Ratio for Sand M 5 Outlet Size ( m ) Eo=0.050 Do=0.0445 E^O.040 D^O.035 Db=0.030 D^O.025 Efo=0.020 1^=0.015 D(p0.010 H / D Ratio Figure 5.7 Predicted Flowrate vs. H / D Ratio for Sand M D 1 2.0 • r——' T— i 1— 1.81 —i 1 1.6 OutletSize ( m ) EQ=0.050 1.4 Do=0.0445 1? E^j=0.040 1.0' B 0.8 D(f0.025 06 E^f=0.020 0.4 k 0.0 * i C1 E^O.035 Efo=0.030 • ' —* * iJtLx2 *4 1 X 6 — a — l"K 1 8 10 ! l2 H/D Ratio Figure 5.8 Predicted Flowrate vs. H / D Ratio for Sand M l M Do=0.015 Dg=0.010 -9500 Outlet Diameter (m) i -10500 - § E^O.010 Do=0.0l5 Eu=0.020 %=0.025 E„=0.030 -11500 CA «3 -12500 D(F0.035 Dfo=O.040 Eu=0.0445 D<j=0.050 -13500 -14500 4 6 H / D Ratio 8 10 12 Figure 5.9 Predicted Air Pressure Gradient at Hopper Oudet vs. H / D Ratio for Sand M D 3 -3000 r — • 1 • i 4000 S • Outlet Diameter (m) Eb=0.010 DpO.015 E^O.020 E^O.025 Eb=0.030 D(j=0.035 Eb=0.040 Eb=0.0445 D(j=0.050 -5000 -6000 •go o CO -7000 • -8000 -9000 4 6 H / D Ratio 8 Figure 5.10 Predicted Air Pressure Gradient at Hopper Oudet vs. H / D Ratio for Sand M 5 -1000 . Oudet Diameter (m) -x4 6 Eb=0.010 Do=0.015 Efr=0.020 Dfc=0.025 1^=0.030 D(j=0.035 E^O.040 Efo=0.0445 D(j=0.050 8 H/D Ratio Figure 5.11 Predicted Air Pressure Gradient at Hopper Oudet vs. H / D Ratio for Sand M D 1 -500 Oudet Diameter (m) g -1000 Eb=0.0l0 Do=0.015 Eb=0.020 Dfo=0.025 1^=0.030 D(j=0.035 E^=0.040 1^=0.0445 D(j=0.050 36 it "gO -2000 - 4 6 H/D Ratio Figure 5.12 Predicted Air Pressure Gradient at Hopper Oudet vs. H / D Ratio for Sand M l The generation of the air pressure gradient requires the flow of particles. It can be seen, from eqn (4.19) and eqn (5.17), that the air pressure gradient at the hopper oudet is proportional to the particle flowrate. However, it is also noted, from eqn (4.19) and eqn (5.17), that the air pressure gradient increases with a decrease in particle permeability, as discussed in Section 4.7.2. For bulk solids with very low permeability, the very high pressure gradients generated near the oudet of a bin reduce the flowrate dramatically and m a y cause difficulties to control the particle flow, if, for example, fluidisation effects occur. For sufficiendy fine material (with low cohesion), the air pressure gradient generated at the hopper oudet by flowing particles m a y be great enough to fluidize the particles at the hopper oudet T h e fluidisation results in more voidage between the particles which releases the high pressure gradient and increases the flowrate quickly (flooding) until the high air pressure gradient builds up again; the resulting periodic flow behaviour is illustrated in Figure 5.13. This periodic flow behaviour was observed in the experiments and will be detailed in Section 8.3. However, the flowrate prediction model eqn (5.17) does not include this effect. Further work needs to be done on the model to include the effect of the fluidisation. 5.3.2 The Effect of Oudet Size on the Flowrate It is a clear that the mass flowrate of material increases with the increase in outlet size. This variation is described by theoretical predictions shown in Figures 5.2 to 5.12. In addition, these figures show the effect of oudet size on the sensitivity of the influence of surcharge level. For convenience of describing these effects, Figures 5.5 to 5.12 can be replotted as flowrate or pressure gradient at outlet against the oudet size, as shown in Figures 5.14 to 5.21. From Figures 5.14 to 5.17, it can be seen that for a bulk solid flowing from a bin the effect of the outlet size on flowrate is significant. They also show that •%?£• increases with oudet size. Figures 5.18 to 5.21 illustrate that the negative air pressure gradients at the hopper oudet increase with oudet size, indicating that the increase in the pressure gradient is influenced by the increase in particle flowrate. However, the influenced pressure gradient by particle flowrate will affect the particle flowrate until a dynamic balance between them occurs. The effect of air pressure gradient on flowrate is discussed in the sensitivity analysis of parameters to flowrate reported in Section 5.4.1. Build up of the High Pressure Grad lent at oudet Release of the High Pressure Gradient at Oudet • t Discharging at L o w Flowrate -• Flooding Fluidisation Figure 5.13 Periodic Flow Behaviour for Fine Material under Fluidisation Condition : 0.80 I 0.01 • 1 r 0.02 0.03 0.04 0.05 Diameter of Oudet D 0 ( m ) Figure 5.14 Predicted Flowrate vs. the Hopper Oudet Diameter for Sand M D 3 0.01 0.02 0.03 Diameter of Oudet D Q 0.04 0.05 (m) Figure 5.15 Predicted Flowrate vs. the Hopper Oudet Diameter for Sand M 5 0.00 0.01 0.02 0.03 0.04 0.05 Diameter of Oudet Dft (m) Figure 5.16 Predicted Flowrate vs. the Hopper Oudet Diameter for Sand M D 1 8 * o O E 1 0.02 0.03 0.04 0.05 Diameter of Oudet D0 (m) Figure 5.17 Predicted Flowrate vs. the Hopper Oudet Diameter for Sand M 1 -9500 Surcharge Level (H/D Ratio) H/D=0.07 H/D=1.5 H/D=3.0 H/D=4.5 H/D=6.0 H/D=7.5 H/D=9.0 H/D=10.5 H/D=12.0 -14500 0.01 0.02 0.03 0.04 0.05 Oudet Diameter D 0 ( m ) Figure 5.18 Predicted Air Pressure Gradient at the Hopper Oudet vs. the Hopper Oudet Diameter for Sand M D 3 -3000 8I -4000 Surcharge Level (H/D Ratio) -5000 H/D=0.07 H/D=1.5 H/D=3.0 H/D=4.5 H/D=6.0 H/D=7.5 H/D=9.0 H/D=10.5 H/D=12.0 o -6000 - I -7000 - Z -8000 -9000 0.01 0.02 0.03 0.04 0.05 Oudet Diameter D 0 ( m ) Figure 5.19 Predicted Air Pressure Gradient at the Hopper Oudet vs. the Hopper Oudet Diameter for Sand M 5 -1000 14 rn <? Surcharge Level (H/D Ratio) -2000 H/D=0.07 -3000 H/D=1.5 H/D=3.0 -4000 •» H/D=4.5 H/D=6.0 -5000 • H/D=7.5 H/D=9.0 2 -6000 cd -7000 0.01 H/D=10.5 H/D=12.0 0.02 0.03 0.04 0.05 Oudet Diameter Dft ( m ) Figure 5.20 Predicted Air Pressure Gradient at the Hopper Oudet vs. the Hopper Oudet Diameter for Sand M D 1 -500 Surcharge Level (H/D Ratio) c U -1000 1 H/D=0.07 a; H/D=1.5 M -1500 • H/D=3.0 H/EM.5 H/D=6.0 H/D=7.5 -2000 H/D=9.0 •8* H/D=10.5 -2500 0.01 H/D=12.0 0.02 0.03 0.04 0.05 Oudet Diameter D Q ( m ) Figure 5.21 Predicted Air Pressure Gradient at the Hopper Oudet vs. the Hopper Oudet Diameter for Sand M l 5.3.3 The Effect of Permeability Constant on the Flowrate Permeability can describe the relationship between interstitial air pressure gradient and particle characteristics such as particle size distribution, particle shape and particle roughness, as mentioned in Chapter 3. Since the negative air pressure gradient plays an important role in retarding the material flow, especially for fine material, the permeability can be used to display the influence of the particle characteristics on the particle flowrate. F r o m the theoretical model presented in Chapter 4 and Section 5.2, a zero stress was assumed to occur at r* = 0.95 r0, the permeability at the hopper oudet is very close to the permeability constant at the lowest compaction C n discussed by eqn (3.15). For convenience, a Q p - C 0 diagram is used to describe the variation of the particle flowrate for different bulk solids. For the sand mixtures used in experiments detailed in Chapter 6, the predicted flowrates are illustrated in Figure 5.22. T h e theoretical results indicate that the flowrate of material increases with an increase in permeability. T h e increasing rate of flowrate reduces as the permeability constant of material increases. After a particular value of permeability constant, the effect of negative air pressure gradient on the particle flowrate can be considered as insignificant; in this case, the flowrate of particles depends on its bulk density. T h e criterion for this particular value will be discussed in Section 7.5. These theoretical results were verified by experimental results, which are detailed in Chapter 7 and Chapter 8. 120 o o o o o o o oo C/3 s in o o oo • * u 1 u c u CO 3 C/5 ^H 1 C •4-» a rm> o o o ^^\ >—i 55 *—/ 1<u o OH »-H * •a 3 § & 0-( wo '^ > ,o <4-l 8 ^>- o E 1 o o o cs K • (D3S/33[) °£) ajBiMOu pajorpay o 5.4 Comparison of Theoretical Results with Experimental Results 5.4.1 Sensitivity Analysis of the Various Terms in Theoretical Model In equation (5.17), the flowrate expression includes five terms: i) cross - section area gradient ii) gravity g iii) bulk density gradient (-p)r=r • x iv) •, A- • stress gradient (7T~ ) r=r ,d<yr 2(K- (-^ f l)qrx r ) T=TQ and v) ah* pressure gradient (-j-) r_r It is obvious that the first two terms are essential for material flow from a converging channel by gravity. The effect of terms iii), iv) and v) can be judged qualitatively from Equation (5.17) as follows: consideration of the bulk density gradient in eqn (5.17) increases the value of &n and then reduces the magnitude of predicted flowrate, namely, the bulk density gradient resists the flow; consideration of the stress gradient in eqn (5.17) increases the value of cn and then enlarges the value of predicted flowrate, i.e., the stress gradient makes the flow increase; the negative air pressure gradient plays a role in reducing the discharge rate. Most researchers consider the effects of the bulk density gradient and stress gradient to be insignificant whereas the effect of the air pressure gradient for fine material is considered significant A series of quantitative analyses were conducted to examine these effects. Sand M D 3 , Sand M 5 , Sand M D 1 and Sand M l were used as the bulk solids samples. In particular the effect of each individual term was examined by setting the relevant term in eqn (5.17) to zero. It is only necessary to mention that for the analysis of 'disregarding the air pressure gradient', eqn (5.17) becomes 2 , 1 ,2 ^ 2_ r o { An o u A ^m&x "g 0 Pout i.e., which yields Qpmax - j P„ Do2"5 ^-f^- (5.18) This equation is the same as that provided by Carleton (1972)[49J for coarse material. The results, as shown in Figures 5.23 to 5.26, verify that the effect of bulk density gradient and stress gradient on the flowrate of bulk solids from the mass flow bin can be disregarded, while the effect of the air pressure gradient m a y be significant. The effect of the air pressure gradient on the flowrate becomes more significant with decreasing material permeability. "5b M ex & Q 0.01 0.02 0.03 0.04 0.05 Oudet Diameter D Q ( m ) Figure 5.23 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at the Hopper Oudet for Sand MD3 3 M Ci. CT ft 0.02 0.03 0.04 0.05 Oudet Diameter D 0 ( m ) Figure 5.24 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at the Hopper Oudet for Sand M 5 I PH & 13 0.02 0.03 0.04 0.05 Oudet Diameter D 0 ( m ) Figure 5.25 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at the Hopper Oudet for Sand M D 1 2.00 o.oo' o.oi 0.02 0.03 0.04 0.05 Oudet Diameter D Q ( m ) Figure 5.26 Comparison of the Effects of Density Gradient, Stress Gradient and Air Pressure Gradient at the Hopper Oudet for Sand M l 5.4.2 Comparison of Equation (5.17) with Other Flowrate Models In this section, the theoretical results have been calculated to compare the experimental results and the results predicted by published models for the materials used in the experiments. In this comparison, the published theoretical models which are considered the best existing models for fine material are as follows: i) Nedderman, Ttiztin and Thorpe Method[56] [ eqn (2.5) ] ii) Original Carleton Method [ 4 9 ] [ eqn (2.4) ] iii) Carleton - Drag Force Modification^ [ eqn (2.9) and eqn (2.10) ] iv) Carleton - Permeability Modification^ [ eqn (2.9) and eqn (2.11) ] v) Modified Johanson Method^ 115 ^ [ eqn (2.6) and eqn (2.8) ] In iii) to v), the estimation of the major principal stress was evaluated using the procedure described by Arnold et al. (1980/ \ The bulk density and air pressure gradient at the oudet in model i) were predicted by the present model developed in Section 5.2. Figures 5.27 to 5.30 show the comparisons of the theoretical results with experimental results for sand mixtures from the 0.02 m and 0.0445 m oudet at H / D = 0.07 and H / D = 1.5 surcharge level, respectively. In Figures 5.27 to 5.30, the comparisons show that equation (5.17) provides the most accurate prediction relative to the other models examined. In particular the worst fit occurs by the original Carleton method, especially for low permeability materials. This is because eqn (2.4) only involves the median particle size and does not include the effect of particle size distribution. This indicates that the median particle size alone is not sufficient to describe fully the flow properties of the particle mixture. In comparison the Carleton - Drag Force Modification predicts a much lower flowrate. Hence it is considered that the introduction of the voidage function e" 4 7 over-estimates the drag force, although it is well known that the drag force acting on a particle in a dense solids-gas mixture is larger than that on a single particle. For example, if voidage e is 0.5, the drag force will increase 26timesby introducing e . On the other hand, the model of Nedderman et al.[561 provided an over-estimation of flowrate. More comparisons of eqn (5.17) and Nedderman et al.'s method are carried out with the experimental results obtained by Crewdson et alJ50J and Head^551. The estimated flow properties of the bulk solids and bin geometry details are presented in Table 4.2. The comparative results are listed in Table 5.1. Table 5.1 The Comparison of Eqn (5.17) and Nedderman et al.'s Method with Head and Crewdson et al.'s Experiments Experimental Flowrate (kg/sec) Predicted Flowrate by Eqn (5.17) (kg/sec) Predicted Flowrate by Eqn (2.5) (kg/sec) Crewdson et al. Sand 355 - 422 ^ m 0.015 0.20 0.029 Crewdson et al. Sand 212 - 300 ^ m 0.015 0.019 0.026 Crewdson et al. Sand 90 -106 ^ m 0.010 0.011 0.015 Head-0.02805 m Oudet Sand 186 - 268 ^ m 0.406 0.380 0.585 Head-0.0304 m Oudet Sand 186 - 268 ^ m 0.477 0.462 0.712 Head-0.02805 m Oudet Sand 86 -130 >im 0.230 0.186 0.283 Head-0.0304 m Oudet Sand 86 -130 ^ m 0.274 0.221 0.337 case 127 co oo .2 ^ ON J e ~ ft 1 12 5 B I 21 Sill OoQ» o o o c -*d o o o 3 8 05 ^^ *o Ja e24 VO 51m S. a« Ed 9 g4) •o •Vo !1 £ "3 o o o >n o o o ^*"* t* *^ o o o en o o o CM o o o Qi § 8 &5 CO 3 qT ft * I ft 9 Qi o ! O 1 •g ! cs »o o © cO d 8 d o d ( 03S/S5J) " £) 9JBJMOTJ SSBJ^ d d K 128 ( D9S/S3[ )d o aiM^ou SSBJ^ 129 d n S o il_ 3 Qi 8 1 co -c 8 • ^ a I •i o> o ft PH ft I—< 23 3 Qi I I 1 •css K (03S/331) £) ajBIAVOtl SSBJ^ 130 ft (03S/xb[) £) 3JBJM0TJ SSBJ^ The results in Table 5.1 confirm the over-prediction associated with Nedderman et al.'s method. It is noted that Nedderman et al.'s results are always greate those predicted by eqn (5.17), because of the following discrepancies betwee two models. Both models have one format, i.e., where Qpmax is the maximum flowrate, applied as (T~ ) ~ ^ g Pout ^ ™" h = r0 Tj is the weighting factor of the influence of the pressure gradient. For the model of Nedderman et al. n *-l0 D2.5|- 1 + K g 11/2 ^>max - 4 P o u t M ) [ 2 ( 2 K - 3 ) sina j and r » 1 2 K 5.in8-l 2KO " ! 1 + 3 sin 5 For Equation (5.17) ** Qpmax** = Op™3* in equation (5.18) and r/* = 1 piy; The ratio of the two maximum flowrates is Q Pmax ?A/ l QD ** ~ V ^Pmax 1 + K 2 ( 2 K - 3 ) 2 " / . = *y 5 sin 8 - 1 (5.21) For practical values of 8 the ratio expressed by eqn (5.21) will be greater than 1 and Tj will be less than 1. For instance: Q * Vvm if 8 = 40°, then ±± = 1.344 and Tx* = 0.756 * W = 1.633 and r x * = 0.600 ^Pmax Q if 8 = 30°, then Pina ^Pmax This means that Nedderman et al.'s model will over-estimate the m a x i m u m flowrate and under-estimate the retarding effect of the adverse pressure gradient, resulting the prediction of higher flowrate. Furthermore, for coarse bulk solids, the flowrate predicted by eqn (5.18) is very close to that predicted by Brown's theory, eqn (2.2), with the percentage error described in eqn (5.22). The percentage error between the flowrates obtained by two models for hopper half angle within the range from 5 to 30 is plotted in Figure 5.31. This comparison indicates that the eqn (5.18) is reliable, since the Brown equation is one of the most successful for coarse bulk solids (as mentioned in Section 2.1). the flowrate predicted by eqn (2.2) _ / 2 the flowrate predicted by eqn (5.18) ' V l + cos a ._ .. ft I I o 10 15 20 Hopper Half Angle a (degree) Figure 5.31 30 The Percentage Error Between the Flowrates Obtained by Current Model and Brown's Theory for Coarse Bulk Solids The main conclusions can be drawn from above comparisons as follows: • the flowrate prediction model, eqn (5.17), provides the most accurate prediction relative to the other models examined; • by using the particle permeability as a parameter to account for the effect of negative air pressure gradient at the hopper oudet on particle flow, the current model, eqn (5.17), is valid for both fine and coarse bulk solids; for the coarse bulk solids, the prediction of eqn (5.17) is almost same as Brown's prediction; • eqn (5.17) includes the effect of compressibility of bulk solids on the particle flow, hence, it is valid for both compressible and incompressible bulk materials; • eqn (5.17) is only one model which describes the effect of material surcharge level on the flowrate. However, the numerical methods have to be used to solve the eqn (5.17). A relatively simplified model will be proposed in Chapter 9 to predict the particle flowrate from mass flow bins in semi-empirical fashion. Chapter 6 Experimental Facilities and Test Bulk Materials 6.1 Introduction To compare theoretically predicted flowrates to actual flowrates accurately, the steady flowrates was necessary to be measured. These measurements should be made noting that the manner in which bulk solids flow from a mass flow bin is dependent on the geometric variables of the bin and on the flow properties of the discharging material. The first set of experiments in the study were carried out on a single hopper apparatus. In these experiments, it was found that the flowrate changed during discharging and it also changed with different levels. The same phenomenon was observed in the experiments carried out by Smith in 1978^ \ as shown in Figure 6.1. These results raised the problem that the level of material perhaps affects the flowrate, although m a n y researchers^4'11"14'17'19,24'13^ considered that the flowrate is independent of the 'head' of the material. 2.2 -1 1—1 1 1— — | — " T 1 • X u •5 1* • BX B ' —r— « % M « i 1.6 \A 1.2 X K B • X X - Test 1 Test 2 Test 3 1.0 • 0.8 •• -2 Figure 6.1 • • A 0 --•— _J L. 1 i i_ L_ • 2 H/D Ratio Mass Flowrate Q D v. H/D Ratio (Smith 1978[53]) T o avoid this effect further experiments were conducted with two hoppers, one positioned above the other, the steady flowrates were measured by keeping the level of material in the bottom hopper constant With this arrangement the effects of three factors were studied: i) the effect of the material surcharge level on the flowrates by measuring the flowrates at the different material levels; ii) the effect of oudet diameter of the bottom hopper by using two different oudet sizes; iii) the effect of median particle size and particle size distribution on the flowrate. 6.2 Test Rig for Measuring Mass Flowrate The apparatus consisted essentially of two bins suspended by three force transducers, as shown in Plate 6.1 and Figure 6.2. O f the two bins, one is called a test bin, as depicted, and the other a storage bin. W h e n solid material in the test bin flowed out, it was replaced by solids from the storage bin above. In other words, the purpose of using this double-bin apparatus was to keep the level of material in the test bin constant, allowing a steady flowrate to be measured with greater accuracy. Both bins had vertical sections made of perspex to allow visual monitoring of the surcharge level during tests, whereas the hoppers or converging sections were m a d e of polished galvanised steel. The bins were filled by manual material transport. The location of the apparatus and the method for recirculating the materials limited the capacity of the bins used. T w o test bins were m a d e with different outlet sizes. The fixed height of the cylindrical section of the test bins was 0.4 m . A removable cylindrical perspex section was used to extend the height in the test bin for higher level tests. With the help of a tapered standpipe (2 or 3 degree) for lowest level measurements, the m a x i m u m difference in level obtainable on the 136(-) 136(-) Plate 6.1 Test Rig for Measuring Particle Flowrate bin hanger was 0.66 m , which was about 4.2 times the diameter of the cylindrical section. <M45(I.D.) Filling funnel Transducer Storage bin Bin hanger (detailed in Figure 6.3) Test bin Figure 6.2 Schematic of the Double - Bin Apparatus The bin hanger consisted of three steel strips suspended vertically by steel wires above and below, six turnbuckles at the ends of the wires and two bin holders, as shown in Figure 6.3. T o vary the level of material in the test bin, the steel strips m a d e with holes allowed the storage bin to be set at different levels. B y adjusting the turnbuckles, the storage bin and test bin were aligned vertically using a spirit level to ensure that the flow of bulk solids would be vertical. S o m e principal dimensions of the bins used are given in Table 6.1. 137 To Transducers 4 Turnbuckles Steel Strips Steel Wire Bin Holders Turnbuckles Figure 6.3 The Schematic of Bin Hanger Table 6.1 Parameters Bin Details Oudet 1 Hopper half angle a (degree) Diameter of oudet Do ( m ) Oudet 2 15 0.0445 0.02 Diameter of cylinder D (m ) 0.145 Range of surcharge H (m) 0.0 ~ 0.66 138 In order to keep the level of bulk material in the test bin constant during discharge, the flowrate of material from the storage bin must be greater than or equal to that from the test bin. The three force transducers used were Interface SM-50 transducers, each having a m a x i m u m capacity of 50 lbs. The circuit arrangement for the three transducers is shown in Figure 6.4. The chart recorder was a Y E W type 3066 pen recorder. The instruments were connected in a simple w a y as shown in Figure 6.5. The gate of the oudet from the test bin was hinged. This hinge was installed 0.15 m away from the oudet, so that the gate had no effect on the flowrate during steady discharge. The gate was m a d e of galvanised steel and built as light as possible to minimize any inertia effects when opened. 0 I m I A.C. Figure 6.4 Transducer Circuitry •o Transducers 4TT \ ( /• A _n/ -c_ -rif -LZ h\1 ' I-J Transducer Conditioning Chart Recorder Unit Figure 6.5 Instrumentation Schematic T o minimize dust emission, skirts were mounted between the filling funnel and the storage bin, and also between the storage bin and the test bin. Attention was paid to ensuring that no tensile force existed in the skirt between the filling funnel and the storage bin. 6.3 Measurement of the Flowrate The chart recorder was used to record the mass of solids flowing from the test bin versus discharge time. To get an accurate flowrate and reduce the effect of experimental deviations, at least four runs were conducted for each mixture at each level. At the beginning and end of the test, the recorder system (including transducers and instruments) was calibrated to check the calibration constants. 6.3.1 Cahbration Procedure The calibration procedure was carried out under the test bin as shown in Figure 6.6. The total calibrating mass was 20 kg. The mass increments are listed in Table 6.2. 140 Table 6.2 The Calibration Mass Increments Weight No Mass (kg) 0 Weight Hanger 0.500 1 10 lbs Weight 4.535 2 10 lbs Weight 4.535 3 10 lbs Weight 4.535 4 2 kgs Weight 1.955 5 2 kgs Weight 2.000 6 2 kgs Weight 1.940 Total 20.000 Mass Bin Hanger Test Bin Bin Holder Weight Hanger Weights Figure 6.6 Calibration Arrangement At the beginning of the test, the calibration procedure was conducted as follows. Firstiy, the scale was established by putting on the whole 2 0 kg weight and adjusting the scale to give measurement pen deflection consistent with the scale divisions. The linearity of the system was checked, as at the end of the tests, by taking away the mass increments one by one. The typical diagram of the recorder output is displayed in Figure 6.7. Figure 6.8 shows a typical calibration result for the recorder system. Mass Increments to 20 No. 6 No. 5 No. 4 V) 93 2 No. 3 10 No. 2 No. 1 No.Or-1 0 t Recorder Movement a). At the Beginning of Test b). At the End of Test Figure 6.7 Typical Calibration Recorder Output Actual Load ( k g ) Figure 6.8 Comparison of Indicated Load with Actual Load 6.3.2 Test Procedure T o prevent the flowrate from being affected by different loading conditions, say different filling speed, before recording the discharge process, the initial loading conditions in the test hopper created by filling were destroyed by use of the following test procedure: a), bulk material was filled into the bins up to the required level via the filling funnel, b). the gate was opened to predischarge the material in the test bin until the head of material in the storage bin was near to the oudet of the storage bin or until the material first filled into the test bin flowed out, c). refill the material, d). for the higher levels, say H £ 0.45 m , steps b) and c) were repeated twice, e). start the chart recorder and set the datum line, f). open the oudet to start discharge, g). stop the chart recorder w h e n the bins were empty, and then prepare for the next run. After taking the chart results from the chart recorder, a Talos digitizer was utilized to read the data. 6.4 Processing the Flowrate Data Measured For steady state flow, the tensile force on transducers stood for weight of material in the bins, i.e., the curve plotted on the chart recorder was the weight curve of the bulk solids discharging. T h e orthogonal polynomial method'- \ which is a method based on the least square analysis to fit efficientiy the experimental data, was used to obtain the approximation of weight curve. With this method the degree of polynomial was created by the program itself. The flowrate of material was then evaluated by taking the time derivative of this polynomial. 6.4.1 Original Fitting The polynomial in t used to fit the experimental data is: n W(t) = ^ p i t i=0 i + E where n the degree of polynomial {Pi } E coefficients of every term error term (6.1) Neglecting the error term, the weight discharging is: n W(t) = X P i l i (6.2) i=0 Differentiating eqn (6.2) with respect to t, the flowrate of the bulk material from the test bin can be obtained, i.e., n-1 Q P = -dT- = X a i t i <6-3) i=0 where Oj = (i + 1 ) p i + 1 6.4.2 Problem of Initial Fitting and its Improvement The error term E in eqn (6.1) is a combination of errors due to digitising and numerical curve fitting. Since the fit processing is based on the least square analysis, this error can be ignored for weight curve fitting. However, for evaluating the flowrate, the differentiation m a y cause an increase in the magnitude of this error. T h e flowrate curve obtained often appeared as a w a v y line. In some cases, for example for coarse bulk material of which the discharging weight varied with time linearly, as shown in Figure 6.9 a), the wave became rougher, as shown in Figure 6.9 b). In particular the flowrate of the 4th run m o v e s very roughly. T o obtain more accurate flowrate value, the initial fitting method needs to be improved. It is noted that the error largely occurs at digitized points. It is acceptable to state that the initial curve fitting procedure only gave a general variation of the flowrate overtime.Furthermore the curve roughness was exasperated by the fact that if the flowrate at tj goes up slighdy, then at some subsequent point it should go down. 21 If 1 1 1 1 1 I 1 " 1 18 "" 15 c 'Eb 12 - Run 1 Run 2 Run 3 Run 4 Xi s ^y^ Xi 60 I < ^ 1 _1 _L 1 1 6 8 10 i 12 I 14 - I 16 18 Time Recorded (sec) Weight Curves Measured a) 1.8 .3* T — • — i — ' — i — • — i — • — i — • — i — • — i — * — r 1.6 bfc run 4 tfSTSm, 1.0 I . I . 4 j_ 6 8 10 12 14 16 18 Time Recorded (sec) b) Flowrate Curves Obtained Figure 6.9 Flowrate Prediction for Four Test Runs at O n e Surcharge Level for Coarse Sand (Sand M l ) T o smooth out the variation in flowrate over time, cubic splines were introduced. The basic concept is described as follows: It is assumed that there is a smooth function, f(x), which is subject to *n i) J [ f ( x ) ] x dx is minimized (6.4) l n ii) ] £ [ f ( Xi ) - ( Q p )j f i=l <. Constant (6.5) Eqn (6.4) presents the smoothness of function f(x) and eqn (6.5) has a concept of the least square method- in so far as finding the values of constants in the chosen equation f(x) that minimize the s u m of the squared deviations of the observed values from those predicted by the equation. Eqn (6.5) can be expressed in terms of the standard deviation. i=l where d ( Q p ) j are smoothing factors. Actually, eqn (6.4) and eqn (6.6) raise the problem of minimizing the functional v[f( x )]. Using Variational Analysis, the reasonably smooth function consists of several sectionalized cubic polynomials which have continuous f(x), f (x) and f (x) at points of intersection (Reinsch 1967 [132] ). In this program, the smoothing factors were taken as d( Q p )j = s (i=l, 2,..., n). If smooth factor s = 0 or s is small, say under 0.001, or 0.005, the results will be th same as before. If s is bigger, say 0.1, or 0.5 (depending on how rough the original data were), the results after smoothing will be on a straight line. Usually the flowrate curve was smoothed till the waves disappeared. Only for the runs where the weight variations were virtually straight lines, was a large s used. In su cases, the straight lines after smoothing were almost horizontal. The flowrates after smoothing the original flowrates shown in Figure 6.9 b), are plotted in Figure 6.10. The flowchart of the final fitting procedure is briefly show in Figure 6.11. » i • i • i—'—i • i •—i—' i • i « i run 1 run 2 "'" run 3 run 4 1 "0 » « 2 4 • • 6 L- 1 8 I — 1 _ - J — 1 — • — 1 — • — 1 — 1 — 10 12 14 16 Time Recorded (sec) Figure 6.10 The Flowrate of Every Run after Smoothing 18 148 Start Fitting Process) Q H Input Initial Data I Plot the Weight Curve for Every Run Calculate and Plot the M e a n Value of Weight Original Fitting Process and Plot the Results Select the Line 3 Input the Smooth Factor Smooth Process and Plot the Results Yes Yes Figure 6.11 Flowchart of Fitting Procedure 6.4.3 Typical Flowrate Measurements The typical flowrate observations with river sand (minus 350 |im in particle size with a median particle size of 2 0 0 u m ) are shown in Figure 6.12, for different loading conditions. For the loading condition called '8 kg loading' in Figure 6.12, the level of material in the test bin reached right up to the outlet of the storage bin, that is, the performance was the same as in the single bin apparatus. For the three other tests (12kg, 16kg and 20kg loading), different levels of sand were attained within the storage bin for each individual run. Initial Loading effect 8 I o. O i Figure 6.12 Time T (second) Comparison of Flowrates for Different Loads Measured on the Double Bin Apparatus with 0.0445 m Oudet The results of the experiments indicate that using a double bin apparatus enables a steady flowrate to be observed for a significant period oftime.Further advantages of theriginclude its ability to reduce the effect of filling velocity and storage time on the resulting flowrate from the bin oudet. The results of flowrate measurements will be displayed and discussed in Chapter 7 and Chapter 8. 6.5 Preparation of the Bulk Solids Mixtures The bulk materials used for the experiments were river sand, alumina, PVC powder, sugar and glass beads. Sampling techniques to obtain the most accurate representative samples of the bulk solids, presented by Allen ( 1 9 8 1 ) ^ , were employed. 6.5.1 The Range of Different Sand Mixtures For river sand, several mixtures, graded into different ranges of particle size, we produced by using a mechanical sieve shaker, as shown in Figure 6.13. The size fractions produced are listed in Table 6.3. The nominal aperture sizes 180, 212, 300, 350 and 425 \itn were achieved by standard test sieves. In comparison the sizes 98,154,223 and 328 |im were made from woven wire cloth. In Table 6.3, sand mixtures MD1, MD2, MD3 and MD4 were specially made to examine the effect of particle size distribution on the flowrate. The aim in preparing these samples was to obtain the mixtures with the same median particle size but different distribution. In particular Sand M D 1 had a narrow size distribution, while the distribution of Sand M D 2 was wider than that of Sand M D 1 . The difference between M D 2 and M D 4 was that Sand M D 2 had a uni-modal distribution and Sand M D 4 a bi-modal distribution. Sand M D 3 and Sand M 7 had uni-modal and bi-modal distributions, respectively (as shown in Figure 6.14). The samples were prepared as follows: Five initial sand mixtures were produced by grinding raw river sand and then sieving it to provide: Sand Mixture 1: 300-425 \un (same as Sand Ml) Sand Mixture 2:223-328 (im Sand Mixture 3 :154-223 Jim (same as Sand M D 1 ) Sand Mixture 4 : 98-154jim (same as Sand M 5 ) Sand Mixture 5 : 0 - 98 ^im (same as Sand M 6 ) Table 6.3 The Range of Particle Size for River Sand Mixtures Sand Mixture Number Range of Particle Size dp (nm) Sand M l 300 - 425 SandM2 212 - 350 SandM3 180-212 SandM4 0-350 SandM5 98 -154 SandM6 0- 98 SandM7 0-425 SandMDl 154 - 223 SandMD2 98 - 328 SandMD3 0-425 SandMD4 98 - 328 Frame Filling funnel suspended on the frame Sieve screen Sieve holder Electronically controlled vibrator Buckets for collecting material a) Basic Arrangement of Large Screen Siever ^+ ^r •I:::::-:: i .1 i 1 •l:_:_ T - 4 b) W o v e n Wire Sieve Screen Figure 6.13 c) i : ::i 4^-r Sieve Holder Schematic of Large Screen Siever 40 i E O 30 + • 1—I c_ CD c= cu crr c_ 0 20 H I i—i—i—i Sand Sand Sand Sand Sand i f -i 1—r—r- MD1 MD2 MD3 MD4 M7 10 °io- 10^~ 10 Particle Size ( micron ) -J-1-1-1 3 10 J Figure 6.14 The Frequency Distribution of Selected Sand Mixtures The sand mixtures with different distribution were produced by mixing some initial sand mixtures together: Sand MD1 was made of 100% Sand Mixture 3. Sand M D 2 was made of Sand Mixture 2, 3 and 4, in approximately equal proportions. Sand M D 3 was made of Sand Mixture 1, 2, 3, 4 and 5, in approximately equal proportions. Sand M D 4 was made of about two thirds of Sand Mixture 2 and one third of Mixture 4. Sand M 7 was made of 4 1 % of Sand Mixture 1 and 5 9 % of Sand Mixture 5. In order to minimize the error which was caused by particle segregation during sampling for particle size measuring, especially for sand mixtures with wide particle size distributions as Sand M D 2 , M D 3 , M D 4 and Sand M 7 , the material was first sampled by riffle box, then classified by a number of sieves which were those with 0,75, 90,106,125,150,180, 212, 250, 300, 355, 425 and 500 \im nominal aperture sizes. It is assumed that there was no particle segregation during sampling and measurement of the particle size of these sub-grouped materials. The median particle sizes and particle size distributions of the mixtures were calculated by using the following formula: it XMi(d|0)i d50 ~ i=l i=1 (6.7) 11 Ad) = IX f^) i=l it i=l where superscripts m and g stand for mixture and group respectively; dso for median particle size; f(d) for frequency of band; M for mass of groups; n for the number of groups. 6.5.2 Median Particle Size and Size Distribution of the Bulk Materials There were three particle size analysis methods used. i) Malven Laser Particle Sizer (for alumina, P V C powder, Sand M l ~ M 6 and Sand M D 1 . The results are attached in Appendix II-1); ii) Mechanical Sieve Analysis (for sugar); iii) Combination of i) and ii) (for Sand M 7 , Sand M D 2 ~ M D 4 ) . The median cumulative percentage diameter of the bulk materials measured in experiments are tabulated in Table 6.4. The particle size distributions are depicted in Figures 6.15 - 6.17. Table 6.4 The Median Particle Size of the Test Materials Bulk Materials Alumina P V C Powder Sugar Median Particle Size d50 (jim) 100.2 127.2 784 Sand M l 370.8 SandM2 310.2 SandM3 197.6 SandM4 202.9 SandM5 155.7 SandM6 79.6 SandM7 113.0 SandMDl 201.6 SandMD2 201.4 SandMD3 197.3 SandMD4 199.1 Particle Size (Micron) Sugar "• Alumina — * — P. V.C. Powder Figure 6.15 Cumulative Size Distribution for Sugar, Alumina and P V C Powder I s x> 1 •a 1 Particle Size ( Micron) "0— "•— •»— Sand-Mi Sand-M2 Sand-M3 Figure 6.16 •+— •«— Sand-M4 Sand-M5 Sand-M6 Sand-M7 Cumulative Size Distribution for Sand M l ~ M 7 100 ft^ST 80 v ^ I 1 I 1 60 L-T 40 20 c -jSf** 10 10' io- Particle Size (Micron) ••— -•— Figure 6.17 Sand-MDl Sand-MD2 ••— -°— Sand-MD3 Sand-MD4 Cumulative Size Distribution for Sand M D 1 ~ M D 4 6.5.3 The Internal Friction Angle and the Wall Friction Angle The internal friction angle and the wall friction angle for all materials were tested the Jenike Direct Shear Tester method described by Arnold et al. (1980) [3] . The wall materials tested were Perspex and Galvanised Steel. The shear testresultsare attached in Appendix U-2 and Appendix II-3. Table 6.5 gives the results for low consolidation conditions (al < 1 kPa). The shear test results show that all of materials listed above are free flowing materials according to Jenike's flowability zones'^, as plotted in Figure 6.18. Table 6.5 Material The Shear Test Results for the Test Materials Angle of Internal Angle of Wall Frictionty(degree) Friction 5 (degree) Perspex Galvanised Steel Alumina 36.5 22 21 P V C Powder 35.5 19 19 Sugar 43 26 23 Sand M l 37 25 25 SandM2 37 23 25 SandM3 37 25.5 25.5 SandM4 39 23.5 24.5 SandM5 38.5 29 28.5 SandM6 40 29 28.5 SandM7 40 27 26 SandMDl 36 26.5 27 SandMD2 41 27.5 27.5 SandMD3 39 27 27 SandMD4 40 28 27.5 --13-" ~ '-" -" — — 2 4 6 8 10 12 Alumina PVC Powder Sugar Sand Ml SandM2 SandM3 SandM4 SandM5 SandM6 SandM7 SandMDl Sand M D 2 SandMD3 SandMD4 14 Major Consolidation Stress (kPa) Figure 6.18 Flowability Characteristic in Free Flowing Zone (According to Jenike's Flowability Zones^) 6.5.4 Particle Density Particle densities of all materials measured by the Beckman Air Pycnometer method are given in Table 6.6. Table 6.6 Bulk Solids Material Measured Particle Densities Alumina Glass Beads PVC Powder River Sand Sugar 4000 2520 1500 2700 1600 Particle Density Ps (kg/m3) 6.5.5 Bulk Density and Permeability The bulk densities and permeabilities of all test materials measured by the me described in Section 3.6 are displayed in Table 6.7 and Table 6.8, respectively. Table 6.7 Measured Bulk Density Properties Bulk Solids Po blv 3 b 2v *lc b 2c Material (kg/m ) Alumina 972.82 96.18 0.08508 103.82 0.08338 P V C Powder 593.16 47.26 0.10711 52.31 0.10498 Sugar 816.59 53.96 0.11817 60.51 0.11588 Sand M l 1329.25 125.15 0.05120 130.29 0.05029 SandM2 1309.36 127.33 0.05201 132.67 0.05108 SandM3 1264.70 151.29 0.05217 157.66 0.05124 SandM4 1318.44 183.64 0.05376 191.68 0.05278 SandM5 1155.32 129.76 0.05329 135.38 0.05233 SandM6 1031.01 249.14 0.07091 264.76 0.06952 SandM7 1333.31 256.58 0.06277 270.31 0.06157 SandMDl 1255.53 112.82 0.08362 121.580 0.08194 SandMD2 1281.21 201.61 0.05438 210.56 0.05339 SandMD3 1342.56 211.78 0.05100 220.43 0.05010 SandMD4 1236.40 166.51 0.05798 174.55 0.05690 Table 6.8 Measured Permeability Properties 6.6 Material Permeability Constant C Q ( *10- 9 M 4 N- 1 Sec" 1 ) a Alumina 398.384 6.42186 P V C Powder 1566.698 9.06178 Sugar 21357.310 4.60027 Sand M l 6517.522 5.92424 SandM2 4353.567 5.48957 SandM3 2327.521 5.68797 SandM4 1535.968 5.85956 SandM5 1155.835 4.48150 SandM6 297.621 6.68156 SandM7 246.341 7.72660 SandMDl 2227.654 5.44753 SandMD2 1054.413 6.18770 SandMD3 575.561 7.44598 SandMD4 1335.804 4.50717 Measurement of Air Pressure Distribution in Mass Flow Bins For fine materials, it is believed that the interstitial air pressure gradients provide significant retarding influence on the flow from the bins. M a n y workers, such as Miles et al. (1968) [44] , Miles (1970) [45] , Crewdson et al. (1977) [50] , Spink et al. (1978) [ 5 1 ] , and Willis (1978) [ 5 4 ] and H e a d (1979) [55] , have reported that the observed negative air pressure distribution is dependent upon the particle size of material and the oudet size of the bin or hopper. According to the experimental results of the current research described in Sections 6.2 - 6.4, the effect of the surcharge level of fine material in a bin can be significant Arnold et al. (1989) [133] stated that the decreased flowrate is due to the presence of adverse interstitial air pressure gradients. In view of this importance it is worth examining the effect of material level cm the air pressure gradients. Since the test rig described in Section 6.2 was not big enough to measure the air pressure distribution in the bin, a testrigbased also on the double bin model but twice as big as that described in Section 6.2 was constructed. The structure of the rig, as shown in Figure 6.19, consisted principally of: i) Test Bin and Storage Bin Perspex was used as the material for the vertical sections of the bins. The converging section of the test bin was m a d e of steel which was machined then galvanised. Figure 6.20 shows the geometry of the test bin and the location of the air pressure tappings. The air pressure probes were set into the wall of the bin with 3 m m holes on the inner side of the test bin. G O R T E X cloth was used to cover the inlet of the probe hole to prevent blockage. It was considered that the smooth side of the G O R T E X , which was aligned on the inner wall surface, would not resist the flow since the probe holes were very small compared with the bin geometry. Care was taken to ensure that there was no leakage in the pressure measurement system. The storage bin, with an external diameter 15 m m smaller than the inner diameter of the test bin, was designed to be able to m o v e up and d o w n in the test bin. A n adjustable gate was designed for the oudet of the storage bin. A skirt was mounted between the two bins to reduce dust emission. 163 Ceiling Ceiling Ceiling (RL 5800 m m ) Skip Controlled by Compressed Air and Lifted by Fork Lift Truck (RL 4280 m m ) Flexible Discharge Duct Transducers Surcharge Level Control System (detailed in Figure 6.21) Skip for Collecting Material (RL 0.0) Figure 6.19 The Apparatus for Measuring Air Pressure Distribution 164 0295 (LP.) Figure 6.20 Large Test Bin Geometry and Pressure Tapping Locations ii) Surcharge Level Control System The surcharge level control system, as illustrated in Figure 6.21, was designed to lift the storage bin up and d o w n to any desired position in the test bin by use of a winch. The balance plate of the system could be m o v e d up and d o w n vertically guided by the rods of the bin hanger. A flexible discharge duct connected the fixed inlet of therigto the movable storage bin. iii) Flowrate and Air Pressure Measurement Systems The flowrate was measured by recording the weight curve. The bin hanger was constructed to ensure the total weight of the discharging material would be suspended on three transducers. The transducers used were R S 250 kg load cells. The load record system was identical to that described in Section 6.2. The manometers with coloured water and video camera were used to measure and record the air pressure profile at each probe point. The pressure readings were taken under the steady state condition. iv) The Facilities of Filling and Collecting the Bulk Material The material was filled by skip, stool and fork lift truck (KOMATSU FG15-12C). The geometry of the rig was limited by the height of the ceiling and the lifting capacity of the fork lift truck. The m a x i m u m lifting height of the fork lift truck was 3.7 meters. A stool of 0.61 m effective height was used to extend the reach of the skip. A belt conveyor and skip covered with a plastic sheet to reduce dust were used to collect the material on discharge. The experiments were conducted for alumina as the test bulk material. The load record system was calibrated by putting a number of weights (totalling 100 kg) on the bin hanger. A number of experiments have been carried out to measure the air pressure distribution at two different material levels (0.05 m and 0.63 m ) . The experimental results highlighted the following: • use of the double - bin apparatus enabled steady air pressure distribution in the test bin to be observed for a significant period of time; the developed surcharge level control system on this enlarged test rig easily adjusted the material level to anyrequiredposition in the test bin to allow the measurement of air pressure as affected by the surcharge levels; • the experimental results indicated that the negative air pressure generated by flowing particles increased with an increase in material surcharge level; the experimentalresultswere displayed in Figure 4.5 and discussed in Section 4.7.2 together with the theoretical predictions; • the measured minimum pressure position (hmp) agreed with the assumption eqn (4.29)reasonablywell, as plotted in Figure 4.3. However, due to time constraints, only limited experiments have been done to examine the eqn (4.29). M o r e work is required to give further evidence for eqn (4.29) by changing the test bin geometry, material surcharge level and the bulk solids. 167 String II II II II 7 2 Storage Bin Holder ' I Balance Plate J? II ^ I EZZE 9 Winch Rods of the Bin Hanger (three) Test Bin Holder Z Figure 6.21 Schematic of the Surcharge Level Control System Chapter 7 T h e Effect of Permeability on the Flowrate 7.1 Introduction As illustrated in Section 5.4.1, the negative air pressure gradients have a significant effect on reducing the flowrate of fine materials. M a n y researchers^44'45'50'51'54'55' have reported that the magnitude of the negative pressure gradient increases as the particle size decreases or the oudet size of the bin or hopper increases. T h e work described in Section 4.7.2 indicated that the negative pressure gradient increases not only with the decrease in particle size and with the increase in the outlet size but also with an increase in material surcharge level. Furthermore, the predictions of flowrate for Sand M D 1 to Sand M D 4 , as shown in Figure 5.4, indicated that the flowrate also depends on the particle size distribution. The flow offinebulk solids is so complicated that using one or two of material characteristics, say particle size or particle size and bulk density, are not sufficient to describe the effect of particle characteristics on flowrate. Current theoretical models for predicting the air pressure gradient and particle flowrate employed the idea of McLean 1 5 8 ' 6 0 1 , Arnold [52] and Arnold et al.[3] which related flowrate and pressure gradients through permeability according to D'Arcy law. In the work reported in this chapter, particular attention is paid to the relationship between flowrate and permeability of the flowing bulk solids. 7.2 Experiments Using Glass Beads This series of experiments was aimed at examining the effect of bulk density and permeability on the flowrate of a range of bulk solids flowing from a model mass- flow bins. Before analysing the experimental observations, it isfirstappropriate to analytically predict the effect of bulk density and permeability on flowrate. This analysis commences by examining the relationship between pressure gradients across powder beds to the basic properties of the bed, using the Carman-Kozeny equation: A P = ( 1 - e )2 j ^ H*p ^ = a^ c3 e where u (7.D e = 1- — Ps By comparison the D'Arcy law indicates AP 1 Hp = C u (7-2) It can be seen that the relationship between air pressure gradient and superficial relative fluid velocity depends on the bulk density and on equivalent surface area or on permeability. Hence, from a knowledge of the packing of particles, a particle mixture can be made with a certain value of bulk density, equivalent surface area or permeability by mixing different particle size components together in certain proportions. However, bulk density and permeability are very dependent on the state of compaction, as discussed in Sections 3.6 - 3.7 and shown in Section 6.5.5. In order to reduce and neglect the effect of consolidation, the approximately incompressible bulk solids, glass beads, were used in these experiments. A n example of the 'manufacture' of mixture properties is illustrated in Table 7.1 which declares the results of bulk density measured for two glass bead mixtures. Table 7.1 Bulk Densities of Glass Bead Mixtures Mixture Po b lv 2v Wv/Po b 3 (kg/m ) Smallest glass bead Mixture in experiments djQ = 159 n m 1401.09 79.83 0.15649 0.057 O n e glass bead Mixture in experiments djQ = 325 Jim 1527.05 53.59 0.12423 0.0351 F r o m Table 7.1, it is apparent that the glass beads mixtures are relatively incompressible. Normally, for glass beads which are typically particles with regular shape and smooth surface, the compressibility of a mixture is inversely proportional to the particle size. In this case, the ratio b jy/po of any mixture should be less than that of smallest mixture. In particular, for all the glass bead mixtures tested, b jy/Po < 0.057. It will be noted that b l v / p 0 is smaller for the glass bead mixtures than for the bulk solids prepared in Section 6.5.5, as presented in Table 7.2. Furthermore, since the magnitude of the consolidation stresses during mass flowing is small, the effect of consolidation for glass beads, therefore, can be neglected 7.2.1 Preparation of Glass Bead Mixtures The particulate mixtures used were specially prepared to exhibit particular values of bulk density and permeability. Basically, three different sized glass beads were used. The selected size ranges had the following nominal sieve apertures: 106-212 nm 212-355 urn 355-425 M-m were referred to as small, medium and large particles,respectively.The small and large particles were used to produce the binary mixtures to compare with the results for medium particles. Here, the different particle size distributions were not regarded as another parameter. Table 7.2 The blv/p0 Ratio for the Bulk Solids Prepared in Section 6.5.5 Bulk Material Wv/Po Alumina 0.0989 P V C Powder 0.0797 Sugar 0.0661 Sand M l 0.0942 SandM2 0.0972 SandM3 0.1196 SandM4 0.1393 SandM5 0.1123 SandM6 0.2416 SandM7 0.1924 SandMDl 0.0898 SandMD2 0.1574 SandMD3 0.1577 SandMD4 0.1347 The bulk density and permeability were measured using the Jenike Compressibility and Permeability Testersrespectively,as shown in Figure 3.5 and Figure 3.19. In generating the selected mixture, each component of the mixture was weighed on a Mettler P C 4400 electronic balance. The components were mixed together very carefully in a plastic vessel to produce a uniform mixture and keep segregation to a minimum. Also, attention was paid to filling carefully the test cylinder by spooning the mixtures layer after layer to minimise the mixture segregation. The bulk density and permeability were obtained from eqns (3.14) and (3.17): W and 0 C = SP. A p APt - AP 0 (3.17) The equivalent surface area was determined from eqns (7.1) and (7.2), that is a** = Mf , Psp2 ,3 C (7.3) ( Ps " P ) 3 or a** = \i{ ~ E 3 C Making up the mixtures of large and small particles required two steps. The first step involved determining the mix of glass beads to produce the desired bulk density, permeability and equivalent surface area characteristics of glass beads (detailed in Section 7.2.2). The second step involved preparing the required quantities of particle mixtures with a desired value of bulk density, permeability or equivalent surface area for flowrate measurement, based on the mixes obtained in the first step (detailed in Section 7.2.3). 7.2.2 Mixtures of Particles for Laboratory Measurements A small amount of each mixture which fully filled the test cylinder of the permeability tester (about 0.7 kg) was produced by carefully packing the particles to obtain the selected bulk density, equivalent surface area or permeability characteristics while noting the different components of small and large particles (referred to as particle packing stage in Table 7.3). This allowed the determination of the required percentage of small and large particle components with approximately the same bulk density, equivalent surface area or permeability as the m e d i u m particles. During these measurements, packing of the particles was repeated four times for each mixture, and six air flowrate readings at each different pressure drop for each packing were taken. The results are shown in Figures 7.1 to 7.3; the average value w a s taken from the 24 measured values, for each point plotted. 3%S/97%L 87%S / 13%L I 1503.9 0.0 Small Figure 7.1 0.8 1.0 Large Medium Large Solids Fraction Mixture Bulk Density Characteristics 17%S / 83%L r 8 CO o 3393.45 U I •a 0.0 0.2 0.4 0.6 0.8 Small 1.0 Large Medium Large Solids Fraction Figure 7.2 Mixture Permeability Characteristics 27%S/73%L r 6 CS --* 331.9 i o * * 0.0 Small 0.2 0.4 0.6 0.8 1.0 Large Medium Large Solids Fraction Figure 7.3 Mixture Equivalent Surface Area Characteristics From Figures 7.1 to 7.3 it is seen that the bulk density characteristic is very typical of that determined by other researchers^ 100 ' 134 -^]. Furthermore, the permeability characteristic is very similar to those of German^ 100 ^, Leitzelement et alJ135^ and Standish et alJ ^. The equivalent surface area can berelatedto the permeability as indicated in eqn (7.3). 7.2.3 Mixtures of Particles for Flowrate Measurements The large quantity of each mixture (about 10 kg) required for measuring the flowrate of particles from bins was produced according to the percentage of components determined from the experiments in the previous section (referred to as mixture blending stage in Table 7.3). The mixtures produced were as follows: i. T w o mixtures having the same bulk density as the medium particles, see Figure 7.1. a. 3 % small and 9 7 % large particles b. 8 7 % small and 1 3 % large particles ii. One mixture having the same permeability as the medium particles, Figure 7.2. 1 7 % small and 8 3 % large particles iii. One mixture having the same equivalent surface area as the medium particles, Figure 7.3. 2 7 % small and 7 3 % large particles Precautions were taken to reduce the effect of segregation during measurement of these parameters, including: • the use of a riffler to obtain sub-samples for laboratory measurements; • repeated packing of the particles at least six times for each mixture, and taking of six air flowrate readings at each different pressure drop together with the weight of particles and height of particles in the permeability tester (Figure 3.19) for each packing; the average permeability was then calculated from these 36 air flowrate observations and the average bulk density from the 6 packing tests; the equivalent surface area was obtained using the average bulk density and average permeability in eqn (7.3). Table 7.3 shows the comparison of the results for various parameters measure thefirststage (particle packing) and second stage (mixture blending). Table 7.3 Comparison of the Three Mixtures Parameters Particle Packing Mixture Blending a** P C a** 3% S / 97% L 1501.94 5342.54 518 1496.99 6023.71 572 17% S / 83% L 1536.43 3367.06 379 1534.00 3395.14 379 27% S / 73% L 1556.89 2715.34 334 1552.73 2784.62 336 87% S / 13% L 1505.16 1565.20 154 1509.99 1590.12 159 1503.93 3393.45 332 Mixture medium particles P C 7.2.4 The Observed Particle Flowrates from the Bins The oudet sizes of the test bins D 0 tested were 0.0445 m and 0.02 m , respectively (as described in Section 6.2). The experiment observations are listed in Tables 7.4 to 7.6. Table 7.4 The Observations for Mixtures with Similar Bulk Density Flowrate of Particles Q p ( kg / sec) Bulk Density Mixture P 0.0445 m Oudet 0.02 m Oudet 3% S / 97% L 1496.99 1.732 0.248 87% S / 13% L 1509.99 1.599 0.234 medium particles 1503.93 1.706 0.246 Table 7.5 The Observations for Mixtures with Similar Permeability Flowrate of Particles Q p (kg/sec) Mixture Permeability C (*10'9) 0.0445 m Outlet 0.02 m Oudet 17% S / 83% L 3395.14 1.733 0.248 medium particles 3393.45 1.706 0.246 Table 7.6 The Observations for Mixtures with Similar Equivalent Surface Area Mixture Equivalent Surface Area Flowrate of Particles Q p (kg/sec) a**(*10" 12 ) 0.0445 m Oudet 0.02 m Oudet 27% S / 73% L 336 1.732 0.249 medium particles 332 1.706 0.246 From these Tables, it is apparent that for two different mixtures with same permeability constant or equivalent surface area, the difference of their flowrates were less than 1.6%. However, in the case of the mixtures with the same bulk density the difference of their flowrates could exceed 7 % . This implies that for fine bulk solids the bulk density cannot uniquely describe the effect of particle size distribution on the flowrate of different mixtures of the same particulate materials. In comparison, the parameter permeability can provide a more accurate description of bulk solid flow behaviour. Figure 7.4 gives the variation in flowrate as a function of permeability for two hopper oudet sizes. The flowrate increases with the increase of permeability. The equivalent surface area, since it has a close relationship to permeability as described in eqn (7.3), can be used as an alternate parameter. However, in this thesis, the work is concentrated on investigating the relationship between permeability and the flowrate offinebulk solids, since the permeability directiy links the air pressure gradient with the superficial velocity between air and particles. 0.30 —i— . 0.28 - i • i i • 0.26 0.24 0.22 0.20 • t r ^~- • . . - 0.18 0.16 0.14 0.12 0.10 1000 • • • • • 2000 3000 4000 5000 6000 7000 Permeability C (* 10'9 M 4 N" 1 Sec'1 ) a) From 0.02 m Oudet 1000 2000 3000 4000 5000 6000 Permeability C (* 10"9 M 4 N' 1 Sec"1 ) b) From 0.0445 m Oudet Figure 7.4 The Flowrates vs. Permeability Constant 7000 Unfortunately, since there were insufficient glass beads available to allow mixtures to be m a d e for measuring the flowrate within the low permeability region. In view of this deficiency further experiments were conducted using sand mixtures. 7.3 The Effect of Particle Size and Size Distribution To examine the effect of particle size on flowrate, further experiments were conducted using sand mixtures Sand M l to Sand M 7 . The relationship between flowrate and median particle size at level H / D = 1.5 is plotted in Figure 7.5. These results indicate that for the mixtures which have quite different particle size flowing from a large D Q / C L ratio outlet, where the effect of 'empty annulus1^ ^ can be neglected, the flowrate increases with an increase in particle size. This variation of flowrate was also reported by Crewdson et al.[50], Spink et al.[51], Head [ 5 5 ] and Ducker etal.[57]. 1.4 8 •5 O^ !P 1.2 1.0 0.8 B 0.0200 m outlet • 0.0445 m oudet 0.6 i 4) 0.4 0.2 o.o 100 200 300 400 Median Particle Size d 5 0 ((im) Figure 7.5 Flowrate vs. Median Particle Size at Surcharge Level H / D = 1.5 It is also shown, in Figure 7.5, that the median particle size is an important parameter to compare the flowrates of bulk solids with narrow particle distributions and different median particle sizes. However, the effect of wider particle size distributions on the particle flowrate has not been reported by previous researches, although m a n y researchers163"701 found that the flow properties of bulk solids are affected by the particle size distribution. In this work, thisresearchwas investigated using the sand mixtures Sand M D 1 to Sand M D 4 as the particle size distributed materials. The Sand M D 1 to Sand M D 4 are the powders with the same median particle size (d 50 = 200 pm) but with different particle size distributions. The results are tabulated in Table 7.7 and Table 7.8. F r o m the Tables 7.7 and 7.8, it is seen that the difference between the flowrates for the bulk solids with same median particle size m a y be significant. Specifically, the flowrate of Sand M D 1 is about twice as large as that of Sand M D 3 for both 0.02 m and 0.0445 m oudet. Figure 7.6 shows the variation of the flowrate with median particle size for Sand M l to M 5 and Sand M D 1 to M D 4 at surcharge level H / D = 1.5. F r o m the Figure 7.6, it is clear that the median particle size cannot alone be used to predict accurately the flowrate. However, Tables 7.7 and 7.8 indicate that the permeability can provide a more accurate prediction of the flowrate than that possible using the median particle size. The significance of permeability is readily apparent in Figure 7.7 which depicts the same results as those plotted in Figure 7.6, in which the Q p - C 0 diagram, as illustrated in Section 5.3.3 and Section 5.4.2, is used. 182 3 oo 00 VO VI •a 0- CO o 13 B "5a o en oo r» oo oo en VO lO M >*^ CX C CS 1 •n cs 00 VO o E VO s ts en o S E vo cs IT) in en en O cs VO b ON CS 00 «—i d CS s m o 1 I a: in m in VO in Ov 00 8 CO 0> «n vo r^ cs cs ©CO S vq o •" O en t-H o VO »n in rm vo en «n oo VO CS in en en cs cs en i—i •n o VO »n m' i> in ts e w g d cs d d o o i oo o> m d o 1 I a b •n d d d E d ON vo •a-ao en 8 8 Si i-H ©v t& s VO Si** 8 § o -0 en o cs d s oo •n en en 1.4 1 , E1 1.2 - a * * • • 1.0 • 0.8 - B » 0.6 B \ • 0.4 SandMl-M5 SandMDl SandMD2 SandMD3 SandMD4 from 0.0445 m Outlet - - • from 0.02 m Outlet 0.2 Q B - i 0.0 100 200 300 400 Median Particle Size d 5Q (nm) Figure 7.6 Measured Flowrate vs. Median Particle Size for Sand M l - M 5 and Sand M D 1 - M D 4 (H/D = 1.5) 1.4 12 B A X • • 1.0 0.8 0.6 from 0.0445 m Outlet 04 I- from 0.02 m Outlet L 0.2 - 0.0 0 SandMl-M5 SandMDl Sand MD2 SandMD3 SandMD4 1000 2000 3000 4000 5000 6000 Permeability Constant Co *10' 9 ( M 4 N ^ S e c 4 ) Figure 7.7 Flowrate v. Permeability for Sand Mixtures 7000 7.4 Discussion Permeability is a useful parameter to describe the flow behaviour of particles as it provides a relationship between pressure gradient and air velocity which pass through the particle packing. Permeability includes the effects of particle size, particle size distribution, particle shape and roughness [64,66,87,95,100] j^ regard to the effect of particle size, in general, the permeability of the particles increases with an increase in the particle size (in particular it is proportional to the square of the particle size [87*10°]). Whereas in regard to the effect of particle size distribution, the permeability increases as the width of the particle size distribution decreases or as the particle size distribution is narrowed without changing the median particle sjze[66,ioo] F r o m Figure 7.2 and the results of previous workers! 64 * 100 ' 135 * 137 ], the permeability increases with the addition of large particles into a bed of small particles. O n the other hand, permeability rapidly decreases as the small particles are added into a bed of large particles. In comparison, permeability is fairly insensitive to particle shape differences fee randomly packed structures in practical uset100!. The flowrate of particles from a mass flow bin increases as its permeability increases and remains constant or almost constant after permeability is greater than a critical value Q n , as shown in Figures 7.4 and 7.7. This result w a s supported by Spink et al. and Crewdson et al. and by the theoretical prediction displayed in Section 5.3.3. For bulk solids with high permeability, the effect of interstitial air pressure gradients can be neglected. However, at low permeability the flowrate rapidly decreases as the permeability decreases. Figures 7.8 and 7.9 show some theoretical results for air pressure distribution predicted by model detailed in Sections 4.5 and 4.6. Figure 7.10 gives the general variation of negative air pressure gradients at the bin oudet versus the permeability constant: the lower the permeability, the higher the negative air pressure gradient at the hopper outlet generated is. Furthermore, the lower the permeability, the greater the effect the pressure gradients have in reducing flowrate. 0.500 0.400 - 0.300 0.200 1 •a ! - 0.100 -20 •a 0.000 -10 Negative Air Pressure ( m m water) Figure 7.8 Predicted Negative Air Pressure Distribution for Bulk Solids (from 0.02 m oudet) 0.500 — a — Sand M l —•— SandMDl — " — SandMD4 —•— SandMD2 --•-- SandMD3 0.400 rS 0.300 •a - 0.200 -40 Figure 7.9 C. -30 a - 0.100 -20 -10 0.000 Negative Air Pressure ( m m water) Predicted Negative Air Pressure Distribution for Bulk Solids ( from 0.0445 m outlet) i •a Permeability Constant C Q * 1 0 " 9 ( M 4 N^Sec"1) 2000 ' 4000 • —•• 8000 1 1 1 i •§B B -10000 - * a s O C • a • Surcharge Level H/D = 1.5 S •5 6000 1 i i £ 1 B 0.0200 m Oudet • 0.0445 m Oudet -20000 £ a -30000 Figure 7.10 i 1 1 1 j _ i dP -«- at the Bin Oudet vs. Permeability Constant of Particles A n examination of Figure 7.10 suggests that there is a critical value of the permeability constant C below which C has a marked influence on flowrate. Figures 7.4 and 7.7 also indicate that outlet size has an effect on the onset of the influence of permeability. Terming the value of permeability constant below which the permeability begins to influence flowrate as O n , then it can be seen from Figure 7.7 that the value of Q n for the sand mixtures is approximately 2000* 1 0 9 (M 4 N 1 sec1) for the 0.02 m oudet and 3500* 10*9 (M 4 N 1 sec1) for the 0.0445 m outlet. This can be also explained by the effect of adverse air pressure gradients. In particular, Figure 7.11 and Figure 7.12 show the predicted air pressure distributions for Sand M l and Sand M D 3 , respectively. Figure 7.13 shows the experimental results obtained by Willis^54-'; similar results were also observed by H e a d ^ w h o examined the flow of fine sand from a conical hopper fitted with variable orifice diameters. Applying their findings to the current experimental investigation indicates that for the same bulk solids and material level, flow will be retarded by a greater negative pressure gradient the larger the oudet size. This re supports the observations on the effect of surcharge level detailed in Chapter 8. 1 -1 I 1 ' 1 — — i | 0.500 • • — • •0.400 a - 0.300 3 e — 0.0200 m Oudet • — 0.0445 m Oudet ******/ ' - 0.200 - 0.100 - _* -5 i _J -4 i* •"• -3 I « ^ » -2 ri., -1 n ,, 0.000 o a i •a $ Negative Air Pressure (mm water) Figure 7.11 Predicted Air Pressure Distribution for Sand M l (at H/D = 1.5) 0.500 •*— 0.0200 m Oudet •*— 0.0445 m Oudet -40 -30 -20 -10 Negative Air Pressure (mm water) Figure 7.12 Predicted Air Pressure Distribution for Sand M D 3 (at H/D = 1.5) 6 •r>a 0.8 2 06 i 0.4 •5 *> ed Width of the Hopper Oudet 0.04 m 0.06 m 0.08 m 1.0 5 .s PQ vvvv 12 £bO 02 S3 0 • r* O 20 40 60 80 100 Air Pressure (negative m m water) Figure 7.13 Negative Air Pressure Distribution (Willis^) (Wedge hopper, O p = 15 ° , Slot length 0.61 m ) Generally, the steady state flowrate of a particle mixture decreases with the addition of smaller particles due to the decrease in the permeability and, further, the increase of the effect of air pressure gradients. This conclusion is supported by Bird et al.t63! and Arnold et al.t133]. B y contrast, M e m o n et alJ67l presented results which give a reduction in flowrate with the addition of large particles into small particles. It should be noted, however, that the conditions of their experiments were quite different from those in this investigation. In particular: • different materials (limestone for large particles and sand for small) were used in their experiments which had different solids densities. In this situation, the comparison of the effect of large particle mass fraction on mass flowrate could provide different results. their hopper outiets, in comparison to the size of large particles size, were too small to avoid wall effect on the flowing particles (Do/d ranged between 1.85 ~ 11.78). Hence, under these conditions, mechanical hindrance of the large particles during flow through the relatively small oudet dimension would have been experienced. On other hand, for bulk materials of very low permeability, for which no steady state flowrate could be measured, whether the addition of coarser material could improve the flow behaviour depends on the permeability of resultant mixture. In particular, the flowrate should increase if the permeability of the mixture increases. For example, the steady state flowrate of Sand M 6 with a permeability constant of 297.621*10"9 ( M 4 N" 1 Sec"1) could not be observed except at the lowest surcharge level. However, if some coarser material was added to produce another mixture, say Sand M D 3 with the higher permeability of 575.561 * 1 0 9 ( M 4 N^Sec" 1 ), steady state flowrates were then obtained. If, however, the resulting mixture had a similar or even lower permeability, e.g., the bimodal-distributed Sand M 7 for which the permeability constant was 246.341 * 1 0 9 ( M 4 N ^ S e c " 1 ) , a steady state flowrate could still not be attained. 7.5 Criterion for Classifying Coarse and Fine Bulk Solids in Terms of the Effect of Interstitial Air Pressure Gradients The flow behaviour of bulk solids in a mass flow bin is dependent not only on the particles themselves but also on the bin or hopper geometry. A s an indication of the relation between pressure gradient and superficial relative velocity, the parameter permeability has been shown to be significant. It was demonstrated in the experiments that the flowrate of the particles from a mass flow bin increases as the permeability increases and remains constant after the permeability is greater than a critical value. This fact indicates that from the point of view of the effect of interstitial air pressure gradients, a material can be defined asfineor coarse material depending on whether its permeability is smaller or greater than the critical permeability. The critical permeability value Ccri depends on the bin geometry and bulk solid itself. A s discussed in Section 7.4, the critical permeability value was about 2000* 1 0 9 ( M 4 N^Sec" 1 ) for the 0.02 m oudet and 3500* 1 0 9 ( M 4 N 4 S e c _ 1 ) for the 0.0445 m oudet This indicates that for bulk material with a relative large permeability the effect of negative air pressure gradients could become significant if the bulk solid flows out of a bin with a large oudet. For convenience, it is assumed that the critical permeability Ccn is proportional to the constant flowrate as predicted by Beverloo's equation!14!, that is Ccri=c1Qp + c2 (7.4) Restating Beverloo's equation Qp = Bfpvg (D0-kdp)5/2 (2.1) where Bf is a constant dependent on hopper geometry and k is a constant dependent on particle shape, usually 1 < k < 3 t51l it can be seen that when Do » kdp then Qp = BfPViD05/2 (7.5) According to the above assumption, Qri becomes Ccri= c1BfpVgD05/2+ C2 (7.6) Using Qri = 2000 * 10-9 C c n = 3500 *10- 9 ( M 4 N^Sec" 1 ) for D 0 = 0.02 m; (M^Sec'1) for D 0 = 0.0445 m , then eqn (7.6) can be approximated as Qn = [( 0.85 p Vg D05/2 + 1.8) * 103] *10-9 (M4 N^Sec"1) (7.7) Since the critical permeability varies with such hopper geometry as oudet size, the criterion classifying fine and coarse bulk solids must be related to the hopper geometry. Taking the sand mixtures where median particle size and permeability are listed in Table 6.4 and Table 6.8 respectively, as examples and assuming for convenience that p = 1300 kg/m 3 , then for D0 = 0.02 m, Ccri - 2000 *109 (M4 N^Sec-1); comparing with Sand M 3 and Sand M D 1 suggests that Carleton's criterion of 200 uml 49 l is reasonable to materials with a narrow size range; for D0 = 0.1 m, Qn * 13000 *109 (M4 N^Sec-1) indicating that the criterion of 500 umt28*50*51*55'61*62] becomes more acceptable. In summary any criterion delineating coarse and fine particles cannot be made independent of hopper geometry factors. Crewdson et alJ ^ measured the flowrate of eight materials with different particle sizes from four different sized hoppers. From their results, a critical curve 1-1 can be found to distinguish the air affected flow in terms of particle diameter, as shown in Figure 7.14. Fitting a parabolic function to curve 1-1 gives: where ^Crewis *e ^owrate in Crewdson et alJ50] Due to lack of information, it is difficult to predict the permeability parameter from only particle diameter. F r o m the Carman-Kozeny equation, the permeability is proportional to the square of the particle diametert87*100^ i.e., C - d* (7.9) Assuming that the effect of porosity variation of bulk mixtures on the permeability is insignificant compared with the square of the particle diameter, the combining equations (7.8) and (7.9) is Qri x wCrcw This indicates that the relationship assumed in eqn (7.4) is reasonable. It is necessary to mention that the coefficients in eqn (7.7) were obtained from the experimental results at low surcharge level and hence, the critical permeability predicted by eqn (7.7) is the criterion for classifying coarse and fine bulk solids at similar material surcharge level conditions. A critical permeability to classify significant and insignificant effects of surcharge level on flowrate will be suggested in Section 8.2. 3000 • 1 I * D Q = 0.003930 m A D Q = 0.005145 m * D Q = 0.006895 m • Do = 0.009155 m 1 2500 - • • / ft 2000 "- O "S* I o/ O - . / E o VI VI 2 0 A* 1500 - a y/ - 1000 X 500 - - A/ A . ~+ 1 ^ f •*. "-* ' 200 400 600 800 Particle Diameter d p ( H m ) Figure 7.14 The Critical Curve of Air Affected Flow in Terms of Particle Diameter (Flowrate Data from Crewdson et alJ ^) Chapter 8 The Effect of the Surcharge Level on the Flowrate 8.1 Experimental Results A number of experiments, described in Sections 6.2 ~ 6.4, were conducted to examine the effect of surcharge level on the flowrate of bulk solids from mass flow bins. The actual surcharge levels and the bulk solids used in the two test bins are listed in Tables 8.1 and 8.2. Table 8.1 The Actual Surcharge Level of Solids in the Test Bin with 0.020 m Oudet Bulk Solids Level 1 Level 2 Level 3 Level 4 (m) (m) (m) (m) P V C Powder 0.010 0.220 0.480 0.590 Sugar 0.010 0.220 0.480 0.610 Sand M l 0.010 0.220 0.480 0.610 0.250 0.530 0.250 0.530 0.250 0.530 SandM2 SandM3 (no test) SandM4 (no test) SandM5 0.010 0.220 0.480 0.610 SandM6 0.010 0.220 0.480 0.600 SandM7 0.010 0.230 0.500 0.600 SandMDl 0.010 0.230 0.500 0.600 SandMD2 0.010 0.230 0.500 0.600 SandMD3 0.010 0.230 0.500 0.600 Sand M D 4 0.010 0.230 0.500 0.600 Table 8.2 The Actual Surcharge Level of Solids in the Test Bin with 0.0445 m Oudet Level 1 Level 2 Level 3 Level 4 (m) (m) (m) (m) P V C Powder 0.010 0.310 (no test) 0.570 Sugar 0.010 0.220 0.480 0.600 Sand M l 0.010 0.235 0.535 0.610 0.235 0.535 0.190 0.510 0.215 0.525 Bulk Solids SandM2 SandM3 (no test) SandM4 (no test) SandM5 0.010 0.220 0.480 0.610 SandM6 0.010 0.220 (no test) 0.600 SandM7 0.010 0.270 0.480 0.630 SandMDl 0.010 0.270 0.530 0.660 SandMD2 0.010 0.270 0.530 0.660 SandMD3 0.010 0.270 0.530 0.660 SandMD4 0.010 0.270 0.530 0.660 In Table 8.1 and Table 8.2, level 1 to level 4 stand for the different material heights in vertical section of the test bins (measured from transition section of the bins). The steady state flowrates obtained, corresponding to the surcharge levels in 8.1 and Table 8.2, are shown in Figures 8.1 to 8.3. 0.12 8 V3 M s / 0.10 - oP22 S 0.08 • P V C Powder • Sugar £o 0.06 (H 3 Vi t E 0.04 2 3 H/D Ratio a) Q p vs. H/D (from 0.020 m Outlet) /—\ o » •ta M 0.80 0.70 0.60 (X at> 0.50 iQ 0.40 t3 e"S § </> • P V C Powder • Sugar 030 0.20 cd i2 S 0.10 H/D Ratio b) Q p vs. H/D (from 0.0445 m Outlet) Figure 8.1 Measured Flowrate vs. H/D Ratio for P V C Powder and Sugar 0.20 8 a § 1 0.18 n 0.16 Sand Ml • SandM2 A SandM3 H SandM4 * SandM5 0.14 0.12 E 1 0.10 2 3 H/D Ratio <u s a) Q p vs. H/D (from 0.020 m Outlet) 8 •a H Sand Ml • SandM2 A SandM3 B SandM4 * SandM5 o E •a 1 I 2 3 H/D Ratio b) Q p vs. H/D (from 0.0445 m Outlet) Figure 8.2 Measured Flowrate vs. H/D Ratio for Sand M l to Sand M 5 0.18 /—»N 8 • a M 0.16 v—«" oP 22 £ % 0.14 E 0.10 O 15 0.12 13 SandMDl • SandMD2 • SandMD3 • SandMD4 0.08 VI si 3£ s 0.06 2 3 H/D Ratio a) Q p vs. H/D (from 0.020 m Outlet) 8 CM "5ft H I SandMDl • SandMD2 • SandMD3 • SandMD4 b) Q p vs. H/D (from 0.0445 m Outlet) Figure 8.3 Measured Flowrate vs. H/D Ratio for Sand M D 1 to Sand M D 4 8.2 Discussion Figures 8.1 to 8.3 show the variation of mass flowrates with H / D ratio for different bulk solids. In some instances, the differences in flowrate are due to the variations in bulk density, e.g., the difference in the flowrates between Sand M l and Sugar with bulk densities p 0 of 1330 kg/m 3 and 816 kg/m 3 , respectively. The relationship between measured mass flowrate Qp and H/D ratio can be considered as a linear function over the range of H / D ratio examined in the experiments. For the fine materials tested, increasing the material level caused the flowrate to decrease. This decreased flowrate is due to the presence of adverse interstitial air pressure gradients which are k n o w n to be significant for fine materials, as pointed out by Nedderman et al. (1982)[4], Arnold et al. ( 1 9 8 0 ) ^ and also as predicted in Section 5.4.1. With increasing H / D ratio, the adverse air pressure gradients occurring at the hopper outlet increase in magnitude, as discussed in Section 4.7.2. Hence for fine materials subject to unhindered gravity discharge from a bin, the lowest material level is associated with the highest discharge rate. From the experiments reported, it can be seen that the extent to which the mass flowrate depends on the H / D ratio varies with the outlet size of the bin and the particle size of the bulk solids. For the coarse solids the effect of level is insignificant, while for the finer solids, the effect of level becomes significant, especially w h e n the solids are discharged from the hopper with the larger outlet. Specifically, for Sand M 5 discharged from the 0.0445 m outlet, the flowrate reduced 2 2 . 5 % as the H / D varied from 0.07 to 4.2. In addition, the flowrate of P V C powder from the 0.0445 m oudet reduced 52.5% as the H/D changed from 0.07 to 3.93. In comparison, w h e n Sand M 5 and P V C powder were discharged from the 0.020 m hopper outlet, the flowrate decreased 13.5% and 20.9% respectively, as the H/D varied from 0.07 to 3.93. The extent to which the mass flowrate depends on the H/D ratio is also influenced by the particle size distribution. Sand M D 1 to Sand M D 4 had the same median particle size but different size distributions, however, from Figure 8.3 it is seen that the effect of level on the flowrate is different for each of these solids. This confirms that the use of median particle size alone is not sufficient to describe the size of solids mixtures; permeability is a more suitable parameter to describe the flow behaviour of the bulk solids, as discussed in Section 7.3. From Figure 8.3 b) it is noted that the flowrate line of Sand M D 3 crosses the line of Sand M D 2 at high surcharge level (H/D » 4). W h e n H/D < 4 the flowrate of Sand M D 3 is less than that of Sand M D 2 while H / D > 4 the flowrate of Sand M D 3 is greater than that of Sand M D 2 . It is believed that since the permeability of Sand M D 3 is very low, w h e n Sand M D 3 is discharged from the hopper with 0.0445 m outlet at high surcharge level, the negative air pressure gradient generated at the hopper outlet is high enough to introduce the fluidisation effect at the hopper outlet, as discussed in Section 4.7.2 and Section 5.3.1, resulting in the higher flowrate. M o r e details will be presented in Section 8.3. As described in Section 4.7.2, both theoretical and experimental results on air pressures indicated that the negative air pressure increases with increasing surcharge level; Willis ( 1 9 7 8 ) ^ found that the negative pressures depend on the hopper oudet size; Crewdson et al. (1977)'-50-' provided evidence of the variation in the negative air pressures for different sand mixtures; Head (1979)'- ^ and Spink et al. (1978) L J examined the flow of fine sand mixtures from a conical hopper and plane flow hopper fitted with variable orifice diameters respectively. In agreement with the above findings, the present experimental observations indicate that the negative pressure gradient retarding the flow of a fine bulk solid will increase with increasing hopper oudet size or with increasing surcharge level. A comparison of present experimental results with others found in the literature is tabulated in Table 8.3. This comparison has been restricted to situations where the bins had either conical hoppers or flat bottoms. From Table 8.3, it is obvious that the effect of material level in the bin on the flowrate is insignificant for coarse solids. T h e result of Beverloo^ 4 * for fine sand (which roughly corresponds to the Sand M 5 ) is interesting. H e reports that the dependence of flowrate on material level is insignificant. While the present results for the Sand M 5 show a significant dependence on material level for both outlet sizes. It is felt that there are two principal reasons for the difference, namely i) Beverloo used outlet sizes of ranging from 2.5 to 10 mm compared with 20 m m to 44.5 m m in the present work. ii) Beverloo's bin had a flat bottom which caused funnel flow and prevented the establishment of significant negative pressures in the converging flow channel. Such a situation can be compared with a hopper having semi-permeable walls. A further comparison of experimental flowrates was made with theoretical predictions generated using eqn (5.17), resulting in the general variations plotted in the Q D - C 0 diagram, shown in Figure 8.4. The general variations m e a n the variation obtained from dispersed data as shown in Figure 5.22, and Figures 5.27 to 5.30. In Figure 8.4 both experimental and theoretical results demonstrate that the flowrate of fine material flowing from a mass flow bin decreases as the surcharge level increases. This effect of material level depends on the oudet size of the bin and particle size distribution. For material with higher permeability, the effect is negligible. A s the permeability diminishes, the effect becomes more significant. However, w h e n the permeability of particles becomes very small, no steady flow occurs, as discussed in Section 8.3. From Figure 8.4, it seems appropriate to delineate the conditions when material level has a significant effect on flowrate using a similar concept to the critical permeability discussed in Section 7.5. T h e value of C Q where material level is insignificant is approximately 2C cri . This is confirmed by referring to Table 6.8 where it is seen that the Sand M l , the only sand whose flowrate was not affected by material surcharge level in the experiments, had a permeability constant of about 6500 * 10' 9 ( M 4 N" 1 Sec"1). 8.3 Problems Which May Be Caused by Increasing the Surcharge Level As mentioned in Section 5.3.1, for the low cohesive fine material with very low permeability, the negative air pressure generated near the outlet of a mass flow bin could be high enough to effectively fluidise the particles. The fluidised particles contain m o r e voidage and provide s o m e relief to the high negative pressure gradient. T h e reduction of the air pressure gradient results in an increase in particle flowrate, and m a y even result in flooding. Provided the flooding is not too severe, 8 o i 5 E 8 > Xi cs O "O TJ Q & •e-S 4-C o Cfl s T9S o d I oo TJ o •S NO © CO 4-1 W u 3 i CO CM CO 8 d 13 T) 1 co B B cs CO O odd *r\ •*+ •*+ O O O £ CO I o VO CO CO CO !!§. ° 2 2 •-H O O N N CI VI I I I I t I W-J >/"> lO cs" cs" CS o r- >n cs § 00 CO ro CO o d d TJ TJ CO CO E E E E « « £ OH co in oq I-H cs vi tt> H !!> co < CH CO 3 cs m 8 CO CO 5 cs o <o H ~H O o o >n >n cs VI ON CS J I H O ON o r~ co t o d CO o o cs 3 cs cs A V O o 8 i i ) 22^ u-i ON cs d ! TJ TJ TJ vn a i-H -H odd f- o l l i m v© CS CO NO cs pn odd CO ON I <*3 o d CQ •* 00 ID Vfl no CS CS ON ITi 2 ON -H O -H ON — T — ro CS cs 00 o CO CO o «8 CO cs I •<fr .. CO t O NO CO «i.s « s s S §.1 «5 a .S E o« E o CQ E o« vx m oo .1 o d d23 d to « ^ co 2 c S S S S ISESE 00 sss o oo oo d ( oo c-» o C C. j- ao Q d I I ON CO i-H o ON d CM •g fa to P JS I 3 «S £ E © CQ CQ E I" CQ •a a 132 W ON 23 ON »5 w 13 p S2 B C* I 2 > i-H CQ o CO ON CO «o "S3 00 E 8; co r* I 204 •3 1 • 1 • 1 M 11 • ^1 o • • b d Hi • II ^"E|Q 1 111 I i 1 111 o o o >o m i-H 1 11 1 1 1 II d y^x\a o 1 \l\l 1 1 © o • +-> © en 4-i II • >n •*" OO a I-H 9 00 i-H 1 d £ WvW m o & © \\\w CO \\v^ ffijp > \ ^ Xi ccj II J 1 OH i-H * © © © /E|Q y/> / • * 1 <u o ""> II O en II o U cs ffi|Q • «o / !—• II ffilQ 1 © © © ^^W^^^^v^^V. 5 X dII ' • ffi|Q 00 cs © (0QS/SS[) © " £ ) 9JttIAV0fcI SSBJM © <s © p ft the high negative pressure gradient gradually builds up again until fluidisation occurs again; no steady state flow results. Increasing the surcharge level aggravates this periodic phenomenon indicated in Figure 5.13, since a higher material level results in a greater air pressure gradient being relieved by the fluidising effect. The results of Sand M D 3 flowing from a 0.0445 m outlet at higher surcharge level were slightly affected by this flooding phenomenon. For Sand M 6 and Sand M 7 with lower permeability, this effect occurred even for flow from the 0.020 m outlet; Figure 8.5 shows the effect of material level on the flowrate of Sand M 6 and Sand M 7 flowing from the 0.020 m outlet. In this figure, the pseudo-steady flowrate represents a m e a n flowrate for cases where periodic flow takes place regularly with high frequency (see Section 5.3.1). From Figure 8.5, it can be seen that for Sand M6 flowing from 0.020 m outlet the variation of flowrate with the material level H / D ratio is similar to that for Sand M D 3 flowing from 0.0445 m outlet, Figure 8.3; flowrate decreases at lower surcharge levels then tends to increase at higher surcharge levels due to the fluidisation effects. The only difference between these two cases is that when H/D is greater than about 2.7 (point a, Figure 8.5) the flowrate of Sand M 6 is greater than that at the lowest surcharge level, while the flowrate of Sand M D 3 at the lowest surcharge level is the highest flowrate for the range of surcharge levels tested. T h e difference in these trends can be explained as the fluidising effect at higher material levels being more significant for Sand M 6 case. However, from Figure 8.5, the variation of flowrate for Sand M 7 gives evidence of a typical flooding phenomenon, the flowrate increases as the H / D ratio increases. Rathbone et al. (1987) [72] suggested that the flowrate of fine powder during flooding can be estimated by an inviscid flowrate model where the flowrate is proportional to the square root of height of material in the bin (measured from the hopper outlet). T h e inviscid flowrate fitted from the experimental data for Sand M7 is also plotted in Figure 8.5 where it can be seen that the variation of flowrate with surcharge level for Sand M7 is close to the inviscid flowrate curve. More information observed on unsteady flow for Sand MD3, Sand M6 and Sand M7 has been illustrated in Figure 8.6 and Figure 8.7, showing the mass variation plots of Sand MD3, Sand M6 and Sand M7 flowing from both 0.020 m and 0.0445 m outlets. From these figures the flow of bulk solids is more unsteady when the particles discharge from a larger hopper outlet, at a higher surcharge level and/or as the permeability of the bulk solid decreases, confirming the considerable effect of these three parameters on the particle flow. 0.07 "SB 0.06 - ctt O E OH 0.05 - 0.04 >N * CO 0.03 - 3 VI 0.02 H / D Ratio Figure 8.5 Pseudo-Steady Flowrate Affected by Surcharge Level (from 0.020 m Hopper Outlet) 207 0 At Level 1 ' • ' • • • • • • • ' • • ( M 4 N" 1 Sec"1) At Level 4 At Level 3 Sand M D 3 = 575.561 * 10"9 • 32.1 21.1 -/*,'* •"W-l 17-1 jgr 1 tCU.I gll.l " " ••• m >•• S.I -i , At Level 4 X 4.1 H 3.1 *':i : 0 Sand M 6 = 297.621 * 10" 9 ( M 4 N" 1 Sec'1) : . Si.iee.isejeuS8 JIB J H ^ I I .4 SI d l l J H j N M J- W i TIHECSicondsl' I ' I ' I ' I ' I ' t ' I ' Sand M 7 : 0 = 246.341 » 10"» At Level 4 Figure 8.6 Mass Variation Plots for Sand M D 3 , Sand M 6 and Sand M 7 from 0.020 m Oudet (in these plots, 1,2,3 and 4 represent different runs in experiments) ( M 4 N" 1 Sec"1) 208 0 Sand M D 3 = 575.561 * 10"9 (M 4 N"1 Sec"1) Sand M 6 C 0 = 297.621 * 10-» ( M 4 N" 1 Sec"1) TIHElS.eond.l" Sand M 7 C 0 = 246.341 * 10"» ( M 4 N"1 Sec"1) Figure 8.7 Mass Variation Plots for Sand M D 3 , Sand M 6 and Sand M 7 from 0.0445 m Outlet (in these plots, 1,2,3 and 4 represent different runs in experiments) Further observations were conducted on the flow behaviour of Sand M 7 to present more details about the unsteady flow. In Figure 8.7, w h e n Sand M 7 discharged from 0.0445 m outlet at levels 2 to 4, the slow - fast periodic flows are observed. This phenomenon was also observed by Miles et al. (1968)^ \ as presented in Figure 8.8. For periodic flow, the particle flowrate varies between two values, Q l o w and Q^o^ Figure 8.9 shows the flowrate variations obtained from the mass variations of Sand M 7 in Figure 8.7. From Figure 8.9, it is confirmed that the flow of Sand M 7 w a s steady at the lowest level (at level 1, H / D = 0.07) with the flowrate of Q i o w (approx 0.2 kg/sec) but unsteady at higher surcharge levels with the flowrate varying between Q l o w and Qh}gh> where Q l o w is about 0.2 kg/sec and Qjjlgjj is about 0.6-0.7 kg/sec. This indicates that Q l o w is the flowrate unaffected by fluidisation and Qjugjj is the flooding flowrate. i i i i i i i i CONTEN BOO - cc UJ \7-b cm ORIFICE g; 400 WEIGHT o u_ X o ^ c ^ J ^ 2 a n ORF1CE o \ t 20 40 i 60 i 80 i IOO i 120 i— HO i ^-H 160 TIME(J) Figure 8.8 Weight Curves for Discharge of 55 |im Calcite (adopted from Miles et al. (1968)[44]) 210 • i • i • i • i • i • i • i • i • i • i • i • i' i • i • *n i/» CB—. "o *• <n <MUJ <u 3 oE m 5 O oB CM .3 t/» • w» i • i • i •! • i ' ( 3»./8n) 3 1 H u i i • i bO C • i-H two S3 Xi o vt ['••/8D 31HU M01J M01J oo CM 4—1 I ' I ' I ' I ' I ' I ' 1 '-l-'-v^Ji 1 I ' I 'IS | • | • | ' | • | ' | ' | ' | ' I ' I ' I ' I ' I ' I ' O 5 § •a cci "S > i 5 O o < ^ E , I • I • I • I • I . I . I . I • I • I • I • I ./I • I . n r j - o ^ o r . j ^ . ^ r g (3e»/6if) 3 i uy MOTj • I • I • I • I • I • I- CJ» (3»./8^|) o> r~ UD I .1 I I I I »/»«»ff**vJ 3ioy MOId ON 00 3 bfl E In order to diminish the fluctuations during flow and obtain a steady flowrate, a small variable speed belt feeder, as illustrated in Figure 8.10, w a s designed to control the attainable steady flow. A tachometer was used to measure the belt velocity. The distance between the belt and outlet of the test bin was maintained at 10 mm. a) Belt Geometry b) Belt Feeder Configuration Figure 8.10 Variable Speed Belt Feeder The observations were carried out to focus on the effect of feeder belt velocity on flow behaviour of Sand M 7 from 0.0445 m hopper outlet. The flowrate variations obtained for different belt velocities are plotted in Figure 8.11 a) to f). These results show that reducing the belt speed reduces the extent of unstable flow. In particular when the belt velocity was lower than a critical value, steady flow occurred. For the flow conditions observed, this critical value was 0.2 kg/sec which was the value of Q l o w in Figure 8.7. This suggests that Q l o w is the m a x i m u m attainable steady flowrate for this particular bin configuration and material. 8.4 Summary Summarising all the effects of surcharge level on the flowrate, Figure 8.12 gives a general description in which the bulk solids have been grouped into three regions A, B and C. The flowrate of a bulk solid with permeability constant in Region C is independent of the surcharge level, while the flowrate of bulk solids with permeability constants in Regions A and B will be affected by the material surcharge level. In particular the flow of a bulk solid in Region B is steady and the flowrate decreases with an increase in surcharge level; the flow of a bulk solids in Region A is unsteady and the flooding phenomenon m a y occur, i.e., the flowrate increases as the surcharge level increases. The b - b line is a criterion to decide whether the effect of surcharge level is significant or not. This criterion was assumed to be 2C c r i in Section 8.2. The a - a line is a criterion to distinguish whether the flow is steady or unsteady depending on the surcharge level. These two critical lines depend on bin geometry and particle characteristics as mentioned in Chapter 3. According to the observations discussed in this chapter, the classification of the flow for all sand mixtures in Figure 8.12 is listed in Table 8.4. However, to describe the flow in Region A or determine the criterion a-a line quantitatively, further work is required. Table 8.4 The Classification of the Flow for All Sand Mixtures Used in Experiments in Terms of the Effect of Material Surcharge Level Permeability Constant C Q *10-9 In Flowing from In Flowing from 0.020 m Outlet 0.0445 m Outlet (N^N^Sec" 1 ) Case Case SandM7 246.341 Region A SandM6 297.621 Region A Sand Mixture Region A close to a-a line SandMD3 Region A 575.561 close to a-a line SandMD2 1054.413 SandM5 1155.835 SandMD4 1335.804 SandM4 1535.968 SandMDl 2227.654 SandM3 2327.521 SandM2 4353.567 Sand M l 6517.522 Region B Region B Region C Region C I • I ' 1 r No Feeder Qi ow Z. 120 80 160 200 240 Time (second) a) Without Feeder 1.0 0.6 lt + 9 £ °-4 02 1 1 • 1 1 1 ' 1 r I I Vbelt = 0.638 m/sec &*> 0.8 8 1 T • 9 iffiQrf 1 0.0 0 SJ 40 80 120 ^low I 160 200 240 Time (second) b) With Belt Feeder Velocity of 0.638 m/sec 80 120 160 Time (second) c) With Belt Feeder Velocity of 0.367 m/sec 240 120 160 240 Time (second) d) With Belt Feeder Velocity of 0.247 m/sec 1.0 —1 1 1— 0.8 - v 1 1 belt = ° - 200 1 1 1 1 - "^sec 0.6 0.4 - ..J^l 02 0.0 - Qlow « 40 1 80 ... _!__.. 120 .J 1 160 1 200 240 200 240 Time (second) e) With Belt Feeder Velocity of 0.200 m/sec 1.0 0.8 Vbelt = 0.193 (m/sec) 0.6 Vbelt = 0.108 (m/sec) 0.4 40 80 120 160 Time (second) f) With Belt Feeder Velocities of 0.193 and 0.108 m/sec Figure 8.11 Flowrate Variations with Different Belt Feeder Velocities for Sand M 7 from 0.0445 m Hopper Outlet 216 g bfl 1 CO X) 1 .8 i u VI a 8 •xl 1 .£2 o § •0 f C« Q o cN (D3S/Sjf) <IQ ajBlMOJJ SSBp^ 217 Chapter 9 Simplification of the Model for Predicting the Particle Flowrate From Chapters 4 and 5, the solution of the model for predicting the flowrate of particles from a mass flow bin involves a series of integrations which have to be solved by numerical methods. It is necessary to simplify the solution procedure of the model for practical uses. This can be done by concentrating on the simplification of the model for the dynamic deaeration coefficient Kdea. 9.1 Original Model for Estimating the Dynamic Deaeration Coefficient K^a In Section 4.1, it is assumed that the pressure distribution is continuous in the whole of the bin and hence from equation (4.27) a-1 T W „_i 1o I tfl ~ d-Kdea) 9 f f J ili f\i [fiT - U - l W f r J 1J *l + f ~ ~2 dTl a •'Hmp (Tl/rjo) fnmp[f3 a " 1 -(l-K d e a )f3 a ] X dri = 0 (9.1) 2 (Tl/Tlo) where ^•^{^"-"^^"V- ™ >2v 218 bip K o bop (9.4) f3=l + ^ 2 ( T 1 - T l * ) 2 C Po From eqn (4.6) P m p = Po + b i c K 3 Olmp-Tl*) 2c then blc K 3 = P m p ~ Po ^ (Tlmp - "H*) also from eqn (4.28) 1 " Kdea - Po Pmp then Kd,ea Po P m p - Po - i . K d ea Hence, on substitution {2 an(* h m &¥& (9-3) and (9.4) become f2=1+ ££P°H 2 Po fa=l + where r ( Po ^ e a ^ 1 L IP1-P0 !-KdeaJ JV Kd, Niea f T|-Tl* — T|* ^>2 C 1-KdeaUmp-^J px is determined by eqn (4.4) rjmp is estimated by eqn (4.29) / T ^ ^\\-^\TL"Tlmp >2C (9.5) (9.6) 9.2 Simplification of the Original K^jea Model 9.2.1 Random Simulation of K^ea As illustrated in Section 9.1, the mathematical model for evaluating the dynamic deaeration coefficient K d e a . as expressed in eqns (9.1), (9.2), (9.5) and (9.6), is highly nonlinear. K^ea is an implicit function of 12 variables, i.e., K dea = f(a» blv> b2v, blc, b2c, 5, <|>, p0, D0, D, H, a) (9.7) where blv, b2v and blc, b2c are not absolutely independent of each other, cf, Section 4.7.2, where it was shown that the effect of using b l v and b 2 v instead of b l c and b 2 c in the hopper on the predicted air pressure is insignificant. In these calculations, the results also indicate an insignificant effect on the Kd e a . Therefore, eqn (9.7) becomes a function with 10 variables, i.e., K dea = f( *, W, b2, 5, <|>, p0, D0, D, H, a) (9.8) In order to simplify the expression for estimating K^ea* a specific stochastic simulation, called the Monte Carlo simulation, is used. The Monte Carlo method is a technique using random or pseudorandom numbers for solution of the model. R a n d o m numbers are essentially independent random variables uniformly distributed over the unit interval [ 0, 1 ] (Rubinstein 1981) [138J . Hence, eqn (9.8) can be in the form shown in equation (9.9) by introducing the independent variables X listed in Table 5.1. Kdea = f ( x ) (9.9) T o simulate the coefficient K^ea m o r e practically, wider ranges for all variables were selected. From the flow properties of the materials used in the experiments detailed in Chapter 6, the ranges of some variables for these materials were: a: 4.5-9.1 bx: 50-264 b2: 0.05-0.12 p0: 593 - 1342 The ranges for these variables in this simulation were selected to be wider than these ranges, as listed in Table 9.1. Table 9.1 The X components in Simulation X Components Corresponding Lower Bound Upper Bound Arbitrary Initial x x Random Parameter low up Number R N Q x l a 4 10 0.17182818 x 2 bi 50 300 0.27182818 x 3 b2 0.01 0.15 0.37182818 x4 5 20 70 0.47182818 x 5 <t>/8 0.4 0.9 0.57182818 x 6 Po 500 1700 0.67182818 x 7 D0 0.002 0.1 0.77182818 x 8 »/D0 2 50 0.87182818 Xg H/D 0 5 0.97182818 a 5 60 0.70732926 x 10 The samples were drawn from a uniform distribution on the unit interval [ 0,1 ]. (9.10) *i " x ilow+ (xiup " "ilow^Lj where RN - the random number i j - component of variable (i = 1, 2,..., 10) - jth sampling of the variables (j = 1,2,..., N s ) In this program take N s = 5000 A simple uniform 0-1 generator used to create RN- is illustrated in Figure 9. When R N 4 • i = 0.0, RN: • i was replaced by an arbitrary random number to avoid "no random number obtained". In fact, this case did not apply in the simulation processing. Yes RNjj.! =3.1415926 ~ » R N T = RN.j.1*RNiJ.1 -J*- R N T = 10.0 * R N T Yes RN.. <== Decimal Part of R N T T Figure 9.1 The Flowchart of Uniform 0 -1 Generator 9.2.2 Sensitivity Analysis In eqn (9.9), there are 10 variables to determine K d e a . The following sensitivity analysis was used to find the contribution of every variable to Kd e a - The first part of this sensitivity analysis included: • calculating the original coefficient K^ with all of variables varying in their o w n range listed in Table 9.1 using the K d e a model in Section 9.1, then, • setting one of the random numbers as constant (0.5 in this work) each time to obtain relative dynamic deaeration coefficient (Kd e a)i f° r every component of X, and • comparing (Kdea)i ^^ Kdea by calculating their variance, i.e., j=Ns (Total Variance^ = ^ [ (Kdea>i j - K$!a j ] 2 (9.11) During these calculations, the random numbers were created in the same way for calculating K ^ a a nd (Kd ea )i to ensure the results were comparable in eqn (9.11). The physical meaning of eqn (9.11) is that a lower total variance for component i indicates less error produced by setting component i as the constant value, or alternatively, a higher total variance corresponds with a more sensitive variable. The results of this analysis for every component of X are plotted in Figure 9.2. F r o m this figure, the first six most sensitive variables to K d e a ^ e x2> x 6' x 3' x 7' x 4 and Xg. Serial Number of Component with Constant Value Figure 9.2 Total Variance for Every Component (5000 Samples) The second step was to set the four relatively insensitive parameters as constants and then check the total variance caused by this change. The result is plotted in Figure 9.3 (named case 11). These four constant values were taken as follows: a = 7.0 a s R N t =0.5 $ = 0.65 5 (degree) a s R N 5 =0.5 D=26.0D0 a s R N 8 =0.5 a = 20.0 (degree) a s R N i n =0.27 From Figure 9.3, it is clear that keeping these four variables constant, the total variance is larger than that from their individual cases but the first six most sensitive variables to Kdea *xtstm< x2» x6» x3» x7» x 4 an£i x 9' w n ich means the parameters b,, p 0 , b 2 , D 0 , 5 and H/D are more important to Kdea ma *i the others. 224 1 2 3 4 5 6 7 8 9 10 11 Serial Number of Component with Constant Value (or Case Number) Figure 9.3 Total Variance for Every Case (5000 Samples) 9.2.3 Simplification of Kdea Equation by Optimization Technique Knowing the most important variables affecting Kdea makes it possible to simplify the complex model for Kdea- O n e °f the most effective methods for this purpose is the use of an optimization technique. a) Assumed Explicit Models for Kdea To assume a reasonably explicit model for Kdea> ^ is essential to judge how the important variables bj, p 0 , b 2 , D Q , 5 and H/D contribute to the coefficient Kd e a i) Outlet diameter D Q is an essential parameter to the particle flow. Theoretically, when D 0 = 0, there is no particle flow at all, then certainly Kdea = 0- Th* s requirement should be included in the assumed explicit model for Kdea- ii) H / D ratio affects the air pressure gradient and further the particle flowrate. From Figure 4.9, the bulk density at the minimum pressure position under higher surcharge conditions is greater than that under lower surcharge conditions. Therefore, from eqn (4.28), Kdea can be assumed to be proportional to the H/D ratio. Considering Kdea * 0 as H/D = 0, the assumed explicit model Kdea includes the term [l+c1(H/D)C2], where Cj and c2 are constants. iii) The bulk solids (sand mixtures) used in the experiments detailed in Chapter 6 were used to examine the contribution of bulk density constants b1? b2, p0 and internal friction angle 8. It was found that the VfO^.pn) ratio b, or — r r r ratio (examined for n= 1,2,3 and 4) provided an apparently Poe^ b i predictable relation to Kdea- Figure 9.4 shows variations of Kdea with the — Po bi ratio and e ^r— ratio at surcharge level H/D = 1.5. However, no apparently Po ^ predictable relation to K d e a for internal friction angle 8 or sin(8) was been bi found until considered together with the—ratio. Figure 9.5 demonstrates the Po b variations of ICjea with sin(8) and i — T T T ratio. P 0 (sin8r 1 Therefore, the following three explicit models have assumed to represent Kdea: Model 1 v e / i.e., ( 1 V7 y [ yq «=^Ka fef — (9.12) ve *J 0.3 p 0.2 - I ' ' — ~l —1 1 • -i 1 0.3 ' Ratio o •s 0,2 o Xie^5 o. - - 0.1 __l bi rr Xi • 0.10 i 0.0 0.06 - 0.1 cN 0.08 0.12 K Ratio 0.14 0.0 0.16 0.18 dea bi b, Figure 9.4 Typical Variations of Kd e a with the - - ratio and — j g - rat10 1 Po Poe 050 T > T r « 1 > 1 0.3 0.48 0.2 0.46 to 1 CO c • fH CO • iH 0.1 042 r CO Ratio 01 [p 0 (Sin8) ] 040 038 0.06 0.08 0.10 J 0.12 u 0.0 0.14 0.16 0.18 K dea Figure 9.5 Typical Variations of Kd e a with sin(8)and — Q - T ratio p0(sin 8) " i Model 2 K Y2 .sun d e a - Yi D o yen VPo e J i.e., x K d e a ~ Yi x 7 2 V3/ 2x 3 ^x6e J 1 \y 4 {•d^T^y^ (9.13) Model 3 .sim K d e a = yi ( bi lPoe V2 f \Y3 2b 2 J V sin 8 [••»(§n i.e., /- i — \Y3 sim Kdea = yi JC2_^y2 4*i 2x-x ^x 6 e J ; \ sin x 4 ) [1 + y 4 4 5 1 (9.14) where x 2 , x 3 , x 4 , x 6 , x 7 and Xp are simulation variables yi (i = 1, 2,..., n) are optimization variables n = 7, 6 and 5 for Model 1, Model 2 and Model 3 respectively b) Objective Function for Optimization Analysis The total variance of using explicit model Kdea instead of implicit . Ko^na is utilised as the objective function for the optimization analysis, viz., a set of optimal val are selected for variables Y=[ yh y2,..., yn ], to enable K^ to best simulate K^a for all of the N s samples. Basically the mathematical treatment is to minimize the objective function 0 ( Y ) , i.e., minO(Y) (9.15) where j=Ns c) Optimization Process and Results Eqn (9.15) is an unconstrained optimization problem with 5-7 variables. The objective function 0(Y) m a y not be continuous. T o solve this problem, the simplex method of unconstained minimization devised by Spendley, Hext and Himsworth (1962) [139] , and later improved by Nelder and M e a d (1965) [140] seems to be the appropriate method because of its robust properties such as: • simple and reliable • no requirement for continuous objective function • no calculation of the gradient needed for the implementation of the algorithm (Avriel in 1976[141] and Subrahmanyam in 1989[142]). The simplex method of unconstrained minimization, called 'Flexible Polyhendron Search* by Himelblau (1972) [143] , is briefly explained as follows: Consider the minimization of the real function <D(Y), let Y 1 , Y2,..., Y n + 1 be the points in R n that form a current simplex. YGR" Select the vertices at which the highest and lowest function values occur, i.e., points Y and Y1 be defined by <D(Yh)= max <D(Yi) (9.16) O ( Y1) (9.17) i= 1,2,..., n+1 and O (Y1) = min i= 1,2,..., n+1 Denote by Y the centroid of all the vertices of the simplex except Y , i.e., n+1 Y = IY1 n - Y1 (9.18) i=l The strategy of the algorithm is to replace the vertex of the current simplex Y which has the highest function value by a new and better point The replacement of this point involves three types of steps: reflection, expansion and contraction which are detailed by Nelder et al.[140], Avriel[141] and Himelblau[143]. Nelder et al. suggested that the calculation of the algorithm should be terminated ( n n where + \ 1 2 [ ( Yi) X ° V2 E is a predetermined positive number In this program, E = 10" The program flowchart is plotted in Figure 9.6. ( ?)]2 (9.19) - ° < E I Start of Program |- Createthe Random Number Nj ; and Calculate X; ; »> j *> J (i=l,2,...,6 andj = 1,2,...,NS) Calculate initial Y k I and <D(Y k ) (k = 1,2,.... n + 1) Determine h, 1 and Calculate Y by Eqn (9.16) to Eqn (9.18) Y* = Y + a ( Y Reflection Step: Calculate «D ( Y r ) Reflection Coefficient a = 1 -Yh) I Calculate Second Highest Function Value: 0(Y*)=max{0(Y i ),YUY n } Yes I Expansion Step: Y *e = Y- + Y ( Y r - Y ) Calculate 0 ( Y e ) Expansion Coefficient Y = 2 Select 0( Y**) = min { 0 ( Y h ) , 0>( Y r )) J Contraction Step: Y C = Y + p (Y**- Y ) Calculate <&(YC) Contraction Coefficient p = 0.5 Replace All Y 1 : i = i T •%- ( i = 1,2,.... n+1 ) N o ^--'inequality (9.19)^ is existing ? Yes End of Program Figure 9.6 Program Flowchart for Optimization Processing The results of the optimization applications for the three models (9.12), (9.13) and (9.14) are listed in Table 9.2. Table 9.2 Simplified Kdea Model Minimization Results Optimization Model 1 Model 2 Model 3 yi 0.6879528 0.690126 0.6631737 Y2 0.06456032 0.06396075 0.8408236 Y3 0.8421384 0.841538 0.1125652 Y4 0.08429366 0.08336415 0.01541624 Y5 0.01513928 0.01500192 0.883275 Y6 0.8753874 0.8770201 _ Y7 Minimum Total 1.610751 . - Variance 0.4070573 0.4074295 0.4182612 Variables (for 5000 Samples) 9.2.4 Comparison of Kdea Using the Original Model and the Simplified Model From the sections above, it is obvious that the three simplified models for Kdeaeqns (9.12) to (9.14), and Table 9.2 give a simulation and simplification of the original model presented in Section 9.1 under the wider parameter ranges listed in Table 9.1. Comparison of Kdea usin g the original model (KJjea) and the simplified models (Kdea) involves two aspects. O n e covers the wider ranges for every parameter in Table 9.1. This part of the work has been done by the optimization process. The values of m i n i m u m total variance caused by the three simplified models for all 5000 samples, as listed in Table 9.2, indicate the achievement of a quite satisfactory simplification for KdeaThis is evident from the fact that the m i n i m u m total variance implies that the absolute average differences between original model and three simplified model for every sample are 0.0001276,0.00012766 and 0.000129346 respectively. Another comparison of Kdea and K ^ a is carried out for the sand mixtures used in the experiments. The outlet diameters and H / D ratio considered are: D Q = {0.01, 0.015, 0.02, 0.025, 0.03, 0.035, 0.04, 0.0445, 0.05} and H / D = {0.07, 1.5, 3.0, 4.5, 6.0, 7.5, 9.0, 10.5, 12.0}. The range of H / D ratio used here is well in excess of the range stated in Table 9.1. The comparative results are shown in Figures 9.7 to 9.9. In these figures, the results obtained by the simplified models, eqns (9.12) to (9.14), show good agreement with those calculated by the original model. In addition, comparisons of the simplified models 2 and 3 with simplified model 1 are illustrated in Figure 9.10 and Figure 9.11. These figures indicate that the three models predict similar values for Kd e a- Further comparisons were carried out to examine the accuracy of the predicted negative air pressure gradient at the hopper outlet and predicted particle flowrate by using simplified K d e a models instead of using the original K d e a model for same materials, outlet size and surcharge level used in above comparisons. The air dP pressure gradients at the hopper outlet (-^ ) _ and flowrate Q p are predicted from eqns (5.16) and (5.17). The results are plotted in Figures 9.12 to 9.17. F r o m the small deviations evident in these figures, it is obvious that the simplified K ( j e a models can be used to predict both the air pressure gradient at the hopper outlet and flowrate with very high accuracy. 0.20 "•'•• — r 1 Total: 891 points 0.15 - K dea • \ 0.10 *AT9*° J jjpr* i 0.05 0.05 0.10 K dga 0.20 0.15 Predicted by Original Model in Section 9.1 Figure 9.7 Comparison of Kdea Predicted by Simplified Model 1 with that Predicted by the Original Model 0.20 ' 1— —• Total: 891 points 0.15 - K T 1 ~ >" j(^' dea - o.io jjjFf" — 0.05 0.05 . — 1 _ _ 0.10 __i i 1 0.15 0.20 Kdea Predicted by Original Model in Section 9.1 Figure 9.8 Comparison of Kdea Predicted by Simplified Model 2 with that Predicted by the Original Model 0.20 Total: 891 points a K 0.15 - S dea . x7 1 •3 %*^r ^Jtpc%" w o.io - I r I 0.05 0.05 jTjSBr . 1 0.10 _^ . _. • 0.20 0.15 K^ga Predicted by Original Model in Section 9.1 Figure 9.9 Comparison of Kdea Predicted by Simplified Model 3 with that Predicted by the Original Model 0.20 • 0.18 I 1 1 • 1 ' 1 - i — i — i i — • i — - • Total: 891 points • 0.16 K 0.14 dea 0.12 - ^ ^ 11 0.10 0.08 • • ^ • 0.06 • 0.08 . i 0.10 . i 0.12 • i 0.14 i 1 — i 0.16 " — 0.18 0.20 Kdea Predicted by Model 1 Figure 9.10 Comparison of Simplified Model 2 with Simplified Model 1 0.20 0.18 h en r •• 5 0.16 - £ 0.14 - T »" Total: 891 points K dea 1 tS 0.12 0.10 - .§ 0.08 - -4 0.06 0.06 0.08 0.10 0.12 0.14 0.16 K dea Predicted by Model 1 0.18 0.20 Figure 9.11 Comparison of Simplified Model 3 with Simplified Model 1 1 CU c •a Total: 891 points , ^ooo I I u - -8000 1 JKt • 12000 • x^ -16000 . 1 -12000 * 1 -8000 Prediction from Original K ^ • -4000 Model (Pa/m) Figure 9.12 Comparison of Predicted Pressure Gradient at Outlet Based on Simplified Kdea Model 1 with that on Original Kdea Model 1 ^ 1 I -4000 - -8000 • -12000 - § u 1 -16000 -16000 -12000 -8000 Prediction from Original K ^ -4000 Model (Pa/m) Figure 9.13 Comparison of Predicted Pressure Gradient at Outlet Based on Simplified Kdea Model 2 with that on Original Kdea Model I CM — CO -4000 - -8000 - i -12000 - 8 •a o -16000 -16000 -12000 -8000 Prediction from Original K ^ -4000 Model (Pa/m) Figure 9.14 Comparison of Predicted Pressure Gradient at Outlet Based on Simplified Kdea Model 3 with that on Original Kdea Model 2.0 1.8 1.6 1.4 • - " — 1 — ' — I - : • • — i — 'r i 1 1 1 1 1 1 —~r —•—i -'— Total: 891points : QP '' ^ 1.2 1.0 0.8 0.6 0.4 0.2 *r. 0.0 0.0 i 0.2 . i i 0.4 . 0.6 i . 0.8 I . 1.0 I . 1.2 I . 1.4 1 1 1.6 J L- - 1.8 2.0 Prediction from Original K ^ Model ( kg/sec) Figure 9.15 Comparison of Predicted Particle Flowrate Based on Simplified Kdea Model 1 with that on Original Kdea Model 2.0 — I — 1 — l — l — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — "I 1.8 m 1.2 1 - Total: 891 points 1.6 14 ' " - • QP I | • | I | r-|-i | I 1.0 0.8 0.6 0.4 0.2 >\ 0.0 0.0 i 0.2 . _i_ 0.4 • • 0.6 • ' 0.8 I 1 1.0 • 1 1.2 • 1 1.4 •— 1 1.6 1 J 1.8 2.0 Prediction from Original K dea Model (kg/sec) Figure 9.16 Comparison of Predicted Particle Flowrate Based on Simplified Kdea Model 2 with that on Original K d e a Model 2.0 3 CO — I 1.8 h 1—l 1 1 — ' 1 — • — I •—i r- - i — | — i — i — i - Total: 891 points 1.6 f- QD 1.4 1.2 I B S 1.0 0.8 0.6 & a 0.4 h •S u 02 a 0.0 0.2 I 0.4 i L 0.6 J 0.8 i I 1.0 u 1.2 1.4 1.6 1.8 2.0 Prediction from Original K jga Model (kg/sec) Figure 9.17 Comparison of Predicted Particle Flowrate Based on Simplified K d e a Model 3 with that on Original Kdea Model In summary, all three simplified K d ^ models have the following main advantages: • The simplified models provide accurate alternative predictions for 1^^. The analysis conducted indicates that they can be used to confidently predict the air pressure and flowrate of bulk solids from conical mass flow bins. • The simplified models are reliable. Since these models were simulated and simplified for a wider range of parameters presented in Section 9.2.1, the models are applicable not only to the cases studied in the experiments, but also to the cases with a wider range of the parameters. Furthermore, it is noted from Figure 9.11 and Table 9.2 that although the error predicted by Model 3 is a little larger than that predicted by the other two models, its 239 greater simplicity (only 5 constants) suggests this model be used, i.e., K^ga can be accurately and conveniently evaluated using the eqn (9.20). K^=0.6631737 bl n y>.8408236|' ^ 2b, \0.1125652 v sin 5 / 0.883275- l+0.01541624[-jij j (9.20) 9.3 The Simplification of the Flowrate Model Utilizing the simplifying assumptions justified previously, that is: • -sim the simplified dynamic deaeration coefficient Kdea (e<ln (9.20) is employed); • the bulk density and stress gradients are insignificant, as indicated in Section 5.4.1, from the sensitivity analysis in flowrate model; • blv and b^ can be used in the hopper section instead of blc and b2c (i.e., take bj = bjv ~ blc; b2 = b2v ~ b^). the flowrate model, eqn (5.17) can be simplified to a 2 2 Q p 2 + b22Qp-g = 0 (9.21) where 22 * f 1 f 2 I A Q p out ) r0 (9.22) L .a-1 out (9.23) fout=l + ^ ( T l 0 - T l * ) b 2 (9.24) ^^opXpout 11 " 0 "^^ 0111 ^ 080 ' where K 3 T,S1III K L d» Pof 1 Y^ 1-KSS ^V^mp-^J (9.25) Substituting eqn (9.25) into eqn (9.24) yields fout= 1+ Kctea f ^ o - n * b2 l-KgL^mp-^J (9.26) Inserting eqn (4.29) and TJ* = 0.95 T|0 into eqn (9.26), then fout=l + „sun ^dea I-K3S L 0.05(l+T]o) lb2 0.05+ Timax-0.95 ri0 (9.27) Rewritting fout in eqn (9.27) as a function of D, D 0 , H, a, results in „sim l out l + f„„,= Niea TT T-U I-KS (9.28) where 0.05(1 + U = ^ ) H D0 1.05+ 2^-tan a-0.95-jj (9.29) From eqn (9.28), eqn (9.23) is rewritten as „sim fa_1 b22 = r l-(l-Ita) out CoPo^oPout > cos a i.e., fa"2 x , b22 = out „sim ( . TT v (9.30) — KJea (1 - U ) cos a Q)PoAo D,0 into eqn (9.22), yields Inserting rQ = 2 sin a 2 a22 "lA o P o f o u t J 4 sin a D0 (9.31) Finally, the simplified flowrate model eqn (9.21) is solved using eqn (9.20) and eqns (9.28) to (9.31). A comparison of above simplified flowrate model with the original model expressed in eqn (5.17) is carried out for sand mixtures, Sand M l to Sand M 5 and Sand M D 1 to Sand M D 4 described in Chapter 6, discharging from 0.020 m and 0.0445 m hopper outlets at surcharge levels H/D = 0.07, 1.5, 3.0 and 4.5. The results, as depicted in Figure 9.18, indicate that the simplified flowrate model agrees well with the original flowrate model. The simplified model can also predict the flowrate of bulk solids (coarse,fine,compressible or incompressible) from mass flow bins (with or without surcharge) and provides a simpler solution procedure (without solving the integrations numerically). Therefore, for practical purposes the simplified flowrate model should be preferred over the original flowrate model. 0.5 1.0 1.5 Flowrate Predicted by Simplified Model (kg/sec) Figure 9.18 Comparison of Simplified Flowrate Model with Original Flowrate Model For the Experimental Parameters Examined Chapter 10 Strategies for Increasing Limiting Flowrates 10.1 General Possibilities for Increasing Limiting Flowrates It is recognised that the flowrate of fine powder from mass flow bins can be orders of magnitude less that the flowrate obtainable for coarse powders. A s analysis of current data shows, the low flowrate of fine material is mainly caused by the negative air pressure gradient at the outlet of a bin. Therefore, one of the most important strategies for increasing limiting flowrate is to reduce the interstitial pressure gradient, especially at the outlet. Various possibilities to achieve this improvement are n o w proposed and discussed. i) Low Surcharge Level (Double - Hopper Structure) As detailed in Chapters 5 and 8, a fine powder free flowing from a mass flow bin, whose permeability constant is located in region B in Figure 8.12, provides a higher flowrate at lower surcharge level than at higher level. W h e n a large H/D ratio is required for the storage capacity, a double hopper structure, as illustrated in Figure 10.1, can increase the flowrate attainable. The upper hopper can be designed with a smaller hopper angle and larger outlet size than those for the lower mass flow hopper, the ventilate channel is used to keep the air pressure above the top surface of material in lower hopper at atmospheric level, for any greater negative air pressure in this area will cause a greater negative air pressure gradient at outlet of the lower hopper. O n e of the limitations of this structure is the reduction 244 of storage capacity. A n increase in outlet diameter of the upper hopper can assist reduce this loss of storage capacity but m a y diminish the effect of the doublehopper arrangement in improving the flowrate. Hence an optimal arrangement must exist for a particular application. Another limitation is that this strategy cannot increase the flowrate remarkably, compared with such techniques as air injection or the use of a standpipe. For instance, for alumina discharging from 0.0445 m outlet at H/D«2.1, the use of a standpipe of 0.212 m in length (L/D Q =5), can increase the flowrate by about 2 0 0 % , as illustrated in Figure 10.3, while only about a 4 0 % increase in flowrate can be achieved by discharging the material at a lower surcharge level (H/D=0.07), as predicted in Figure 5.2. Bulk material surcharge level Ventilate channel Upper hopper Lower hopper Figure 10.1 Schematic of Double - Hopper Bin Arrangement 245 ii) Addition of Coarse Material From research in this work, as highlighted in Chapter 7, fine powder mixed with coarse material flows more easily as long as the resultant mixture has a larger permeability constant. Theoretically, the addition of coarse material m a y tend to increase the flowrate; consistent with the fact that the flowability can be improved by the addition of coarse material into a fine mixture (Prabhakara R a o et al. 1 9 7 1 [144] & B i r d et ^ 1 9 7 6 [63]) Q u a n t i t a t i v e analysis on the addition of an optimal amount of coarse material into a particular fine powder needs to be developed in this case. For a large scale solids handling system, some additional equipment would be needed to separate the coarse particles from the fine ones. This requirement implies that this method can be only used for small scale systems. In addition, this strategy cannot increase the flowrate substantially. If the cost of additional facilities cannot match the benefits, this technique remains theoretical and not of practical value. iii) Air Injection As summarised in Section 2.2, the use of this technique is well documented by m a n y researchers such as de Jong (1969) [116] , Papazoglou et al. (1970/1971) [81] , Sutton et al. (1973) [117 ' 118] , de Lazzari (1981) [119] , Altiner (1983) [82] , Johanson (1990) [ 1 2 0 ] and W e s et al. (1990) [121] . The negative air pressure gradient at the hopper outlet generated by fine material, causes an inflow of air from the outlet which retards the particle flow. T o overcome this inflow of air through the outlet, it is possible to inject air at certain positions of the hopper to ehrninate the interstitial air pressure gradient developed during flow. For successful air permeation it is necessary to control the adverse air pressure gradients at the hopper outlet by controlling both the pressure and rate of air permeated into the hopper. A s this 246 technique is well documented, the work presented in this chapter will not deal further with this strategy. iv) Use of Standpipe Another technique, which can assist in increasing the flowrate of fine powders without reducing the storage capacity in the bin and or requiring the addition of coarse material or compressed air, is the use of standpipes. The simple addition of a vertical non-converging pipe to the bottom of an existing mass-flow hopper produces a pressure gradient in the standpipe and creates a vacuum at the hopper outlet This suction effect will substantially increase the flowrate of bulk solids from the hopper, as found in m a n y research studies such as Yuasa et al. (1972) [ 1 2 2 ] , McDougall et al. (1973) [123] , de Jong (1975) [124] , Ginestra et al. (1980) [ 8 4 ] , Chen et al. (1984) [85] , Knowlton et al. (1986) [ 8 6 ] and Johanson (1990)t12°l. The effect of the standpipe is greater as the particle size becomes smaller and the length of the standpipe becomes longer. In this chapter, some work on the use of standpipes is reported. 10.2 Experiments and Observations on Standpipes i) Test Apparatus The experiments were carried out on a modified double-bin apparatus shown in Figure 10.2, in which the standpipe was installed at the outlet of the test bin of the double-bin apparatus detailed in Section 6.2. The belt feeder shown in Figure 8.10 was used for a long standpipe to obtain a steady flowrate, as shown in Plate 10.1. The standpipes used in the experiments ranged in length-to-diameter ratio from 1 to 5, as illustrated in Plate 10.2. The joint between the standpipe and the outlet of the 247 hopper was sealed to avoid leaking. The measurements made were similar to those without a standpipe as detailed previously (Chapter 6). ii) Experimental Observations The observations were made on the flowrate improvement attainable using alumina as the bulk solid. Figure 10.3 summarises the results: the use of a standpipe increased the flowrate remarkably; the flowrate increased with the length of the standpipe. For instance, a standpipe with L/D 0 = 5 provided an increase in flowrate of over 3 0 0 % for alumina. iii) Problems When Using the Standpipe As the higher flowrates were approached with the long standpipe, fluctuations in the flowrate were observed. These fluctuations were explainable by two typical discharge modes observed for material flowing through the standpipe: in one m o d e the standpipe was partially filled with the bulk solids, while in the other m o d e the standpipe was fullyfilledwith the solids, as shown in Figure 10.4. The higher flowrate occurred only as long as the standpipe was full of material. Once it was partially filled with the solids, the limiting flowrate was governed by the void in the standpipe and a slower flowrate occurred. The transition state between these two flow modes resulted in flow instability. This instability was suggested by Matsen (1976)^ 145 ^ to be caused by bubble held stationary in the standpipe. Instability of the flow is a c o m m o n phenomenon when using a standpipe. The same problems have been also reported by Leung and Wilson (1973) [125] , Leung and Jones (1978) [ 1 2 6 ] and Johanson (1990) tl20] . A jump phenomenon has been reported by Chen et al. (1984) [85 ^ between the upper and the lower bound flowrate in some situations, with the actual bounds differing by an order of magnitude. 248(-) Plate 10.1 Bin - Standpipe - Belt Feeder Arrangement Plate 10.2 Standpipes Used in Experiments (L/D 0 = 1,2,3,4 and 5 from left) 248 Storage Bin Test Bin Standpipe Feeder Q f) Figure 10.2 Standpipe Installed at the Outlet of the Test Bin 1 2 3 Ratio L / D Q of Standpipe Figure 10.3 Flowrate Q p vs. the L/D n Ratio of Standpipe for Alumina Discharging from 0.0445 m Outlet (Surcharge Level H = 0.31 m ) a) Partially Filled b) Completely Filled Figure 10.4 Typical Discharge Modes from a Standpipe 250 T o reduce the extent of these fluctuations and to ensure that the standpipe remained full at all times a belt feeder was installed under the standpipe. With the belt feeder installed, the m a x i m u m steady flowrate attained, using the long standpipe of L / D 0 = 5 , w a s 0.728 kg/sec (Figure 10.5), which represents 9 7 . 4 % of the m a x i m u m attainable flowrate for discharging from this test bin, as plotted in Figure 10.3. A s Johanson (1990)^ 120J pointed out, it is essential that the feeder controls the flow slightly below the limiting rate as the higher flowrate occurs only as long as the standpipe remains full of solids. 10.3 Theoretical Model for Bin - Standpipe Configuration A theoretical model based on the previous models for predicting the pressure gradients and the flowrate of bulk solids has been established to explain the experimental observations for the standpipe. i i i i i i i M a x i m u m attainable flowrate with full standpipe •*— h = 21 m m •o— h = 2 8 m m Flowrate without standpipe ! j "o.O • i 1 1 • 1.0 2.0 Feeder Belt Velocity (m/sec) 1 3.0 1 4.0 Figure 10.5 Flowrate vs. Feeder Belt Velocity for Different Clearances between Feeder and Standpipe Outlet (standpipe L / D 0 = 5) 251 The geometry of the mass flow bin with a standpipe installed is divided into four regions, where regions I, II and HI are the same as those described in Section 4.1, while the standpipe is considered as region IV, as shown in Figure 10.6. The same definitions of dimensionless depth values presented in eqn (4.1) are used in this model. In addition, the dimensionless depth values for ( h m p )Sp and h 0 S p in Figure 10.6 are defined as (Tlmp )sp - ( h m p )sp hi h osp _ TlOSp" ^ Figure 10.6 Four Regions for Bin - Standpipe Arrangement (10.1) 252 The assumptions about the use of the standpipe, in addition to those in Chapter 4 and Chapter 5, are as follows: • The standpipe remains full of bulk solid with the stress distribution in regi IV being linear with respect to height, i.e., o-! = kxri +k2 ( TioSp < rj < rio ) (10.2) where klt k 2 are constants. • The minimum pressure position for bin-standpipe configuration (rimp )Sp is located in the hopper section between T]mp and rig- The effect of suction by the standpipe is assumed to be dependent on the length of the standpipe in the form ("Hmp )sp ~ ^ m p " A ^ m p where Anmp= (1 -e-xL/D°) (rimp-T]0) i.e., / x - /1 (ilmp)sp =T1<>(1+ c -A.L/pn "Hmax ~ ^o x ,^ n > 1 + TJ 0 nn^ (103) where A, is standpipe effect coefficient. (0 < X < 1). This assumption means that the suction effect generated by the standpipe reduces the magnitude of the negative pressure gradient at the outlet and is based on following observations: 253 F r o m the theoretical flowrate model, eqn (5.17), the flowrate is decreased by a negative air pressure gradient but increased by a positive air pressure gradient. If the air pressure gradient at the hopper outlet was changed from negative to positive due to the installed standpipe, the flowrate should change suddenly from a low value to a high value for the flow without standpipe and with standpipe respectively. However, from Figure 10.3, the flowrate increased gradually as standpipe L / D Q ratio increases from zero. This indicated that the reduction of negative air pressure gradient at the hopper outlet depends on the standpipe length, which can be simulated by setting the m i n i m u m pressure position in the hopper section between Tl m p and rjo- E q n (10.3) describes that (n m p) S p equals T| m p for the flow without a standpipe approaches Tjg with increasing standpipe length. This assumption is acceptable w h e n studying the flow behaviour of bulk solids from a standpipe with a limited length. • The wall of the standpipe is so smooth that there is no resistance to the flowing bulk solids. Hence, as long as the air pressure gradient at the hopper outlet has been determined, the flowrate model, eqn (5.17), can be directly used to evaluate the flowrate of bulk solids from bin - standpipe configuration. The pressure distributions in the four regions, similar to those illustrated in Chapte 4, are expressed as follows: i) in region I (r^ < T| < Tlma*) 2 T] P_P1 = W 0 - ^ [ [f a - 1 -(l-K d ea)f a ]dTi Tlf -^l (4.21) 254 when J\ = ri m a x , P = 0, then 2 MH13JC "Pi = W 0 ^ J f [f a - 1 -(l-K d ea)f a ]dTl (4.22) T|i Tli where Pj is the pressure at the transition level and f is determined by eqn (4.16) ii) inregionH ((Timp)sp < T] < r^ ) / x f1! [f*-1-(l-Kdca)fa] P-(Pmp)sp = W 0 l P F ATlmp)sp 2 dr, (Ti/rio) or .11 [f^-d-Kdea)^] P = (Pmp)sp + W 0 l 5 dn •VimpJsp dl/rjo) (10.4) where f is determined by eqn (4.18) (pmp )sp is me pressure at r, = (Tjmp )sp when T\ = r\l,P = Plt the pressure will be Ml [fa^1-(l-Kdca)fa] Pl-(Pmp)sp = W 0 | J T ( lmp)sp iii) — — 2 (ri/rio) dn (10.5) in region in ( Tj0 ^ rj ^ (Tlmp)sp ) ? [f^-d-Kdea)^: p-p O ut=w 0 | a r~~2 dT1 (10 6) - 255 A s ri = (Ti m p ) s p , P = ( P m p ) s p , it follows that frlnnO f^^tf^-d-Kdea)^] (Pmp)sp-Pout = W 0 where iv) losp JT - S (Tl/Tio)2 1O dri (10.7) f is detennined by eqn (4.20) in region IV ( rioSp ^ rj <r\0 ) J1 [ [fa"1-(l-Kdea)fa]dTl •'Host) P = W0 (10.8) At the hopper outlet, Pout = W 0 ^o I [fa ^Hoso 1 -(l-Kdea)fa]dTl (10.9) Based on the assumption, eqn (10.2), and density continuity, f in eqn (10.9) is determined by f =i + ^ 1 K 2C -ri* J ' * Y ^^M - ^ P r v (T,o_T| - dealTlmp-Tl*J lTlo-TlospJ According to the pressure continuity assumption, the air pressure distribution in bin-standpipe structure will be subject to the requirement eqn (4.22) + eqn (10.5) + eqn (10.7) + eqn (10.9) = 0 (10.10) which simply states that the air pressure across the combined bin - standpipe configuration is zero. 256 E q n (10.10) and eqn (10.3) provide a model to determine the unknown variable K ^ . The flowrates are predicted by the flowrate model, eqn (5.17), as derived in Chapter 5. 10.4 Theoretical Predictions and Discussion To verify this theoretical model, the flowrates of alumina flowing from the test bin with 0.0445 m outlet have been predicted and compared with the observed flowrates. A comparison of the theoretical results with the experimental results, for a sequence of fixed values of the coefficient X, is depicted in Figure 10.7. The value of X = 0 implies no effect of the standpipe on the m i n i m u m pressure position. 1.2 i • i • 1 • 1 • i 1 r L / D 0 Ratio Figure 10.7 Comparison of Theoretical with Experimental Results The comparison indicates that the predicted flowrate increases with increase in standpipe length, although for the standpipe with a lower L / D 0 ratiothe theoretical results are over-predicted. The over-prediction of the flowrate at lower L / D 0 ratio m a y be caused by a different flow m o d e in the standpipe. In particular the theoretical results are based on the assumption of the standpipe being completely filled with solids. It is noticed that in experiments the belt feeder has only been 257 used for a long standpipe (L/D 0 = 5). It m a y be possible that the conditions for the lower L / D 0 ratio standpipes used in the experiments did not match the assumption. Hence a lower suction effect m a y have occurred in the experimental standpipe. F r o m Figure 10.7, it is considered that the case X = 0.75 yields the most confident prediction. The theoretical model also predicts that the standpipe provides a greater increase in flowrate for fine material than for coarse material. Defining an enhancement factor F c „ as the ratio of the flowrate with standpipe to that without standpipe, Figure 10.8 shows the variations of enhancement factor F s p with L / D 0 ratio for alumina, Sand M D 2 and Sand M l , which have the permeability constants 398.384, 1054.413 and 6517.522 * 10" 9 ( M 4 N" 1 Sec"1) respectively. The results indicate that a standpipe can create a higher flowrate forfinerbulk solids than for coarse materials. Specifically, the flowrate enhancement factors produced by a standpipe with L / D 0 = 7 are 4, 2 and 1.06 for alumina, Sand M D 2 and Sand M l respectively. These results coincide with other researchers' results, as illustrated in Table 10.1, indicating that the use of a standpipe increases the flowrate more effectively for fine material than for coarse material. 0 L_—I_J 0 • 1 2 i • 1 3 • • 4 ' • 5 ' • 6 • ' 7 L / D Q Ratio Figure 10.8 Flowrate Enhancement Factor for Different Materials ( X = 0.75) Table 10.1 A Summary of the Results in Using Standpipes Material Used Diameter of Dimensionless Enhancement Standpipe Length of Factor Researcher (Particle Size or Standpipe D0(m) Permeability *sp IVD 0 Constant) Fine Sand Chen et al. 8 in 1984[85] d 50 = 154 p:m 0.0254 130 (Experimental Coarse Sand 2.5 Results) d 50 = 556 p.m Ginestra et al. Unnamed Material in 1980[84] 0.030 100 7-8 8 = 30 (Predicted V t = 0.1 m/sec Results) Sand Knowlton et al 53 -177 \sm 6.4 Sand in 1986t861 0.038 185 5.4 (Measured 177 - 420 *im Sand 420 - 840 pjn Glass Beads dso = 127 p:m Yuasa et al. Glass Beads in 1972 [122] dso = 254 Jim & Predicted) Glass Beads Results) d 50 = 505 urnGlass Beads d50=1015 urn Alumina Current Experimental d 50 = 100 p.m (C0=398.38) Results Alumina d 50 = 100 p:m Predicted (C 0 = 398.38) Results SandMD2 by d 50 = 200 (im (C 0 = 1054.4) Current Sand M l Model d 50 = 310p:m (C 0 = 6517.5) 3.6 5.9 0.0091 165 (Measured 3.25 2.11 1.5 0.0445 5 3.14 4 0.0445 7 2 1.06 259 The increase in flowrate using the standpipe is caused by the vacuum suction at the hopper outlet and a reduction of the negative air pressure gradient. This is evident from Figure 10.9 and Figure 10.10 which show the variations of predicted air pressure and air pressure gradients at the hopper outlet for fine material alumina and coarse particles Sand M l , respectively. Corresponding to these results, Figures 10.11 and 10.12 show the air pressure distributions in the bin and standpipe generated by alumina and Sand M l respectively. From Figure 10.11 and Figure 10.12, it can be seen that the predicted air pressure distribution in the standpipe is a linear function of the depth for short standpipes, while it is a nonlinear variation for long standpipes. In contrast, Yuasa et al. (1972)^ ^ and Chen et al. (1984)™ ^ observed an almost linear relationship between negative air pressure and the depth in the standpipe; Knowlton et al. (1986)^ ^ obtained similar observations in most of their experiments. It is possible that the nonlinear relation produced by the current model is caused by the constraint of the m i n i m u m air pressure position within the hopper section. Indeed, for a very long standpipe, the experimental results obtained by Yuasa et al., Chen et al. and Knowlton et al. (for standpipes of UDQ>100) show the occurrence of the m i n i m u m air pressure below the hopper outlet In this case, a higher flowrate can, theoretically, be produced by the standpipe since a positive air pressure gradient developed at the hopper outlet accelerates the particle flow. Within the time constraints imposed on this thesis, only limited experimental results been obtained to verify and/or to provide a basis to develop the current standpipe model quantitatively. Hence, further work in this area is needed 260 0 1 2 3 4 5 6 7 L/D 0 Ratio Figure 10.9 Predicted Air Pressure at the Hopper Outlet (X = 0.75) 0 1 2 3 4 5 6 7 L/D n Ratio Figure 10.10 Predicted Air Pressure Gradient at the Hopper Outlet (X = 0.75) 261 -o- L/Do=0 L/Do=l L/Do=2 I7Do=3 L/Do=4 L/Do=5 I7Do=6 L/Do=7 Transition level Hopper outlet -400 -300 -200 -100 -l.o Q Air Pressure Distribution (Pa) Figure 10.11 Predicted Air Pressure Distribution in the Bin and the Standpipe for Alumina (X = 0.75) Air Pressure Distribution Figure 10.12 Predicted Air Pressure Distribution in the Bin and the Standpipe for Sand M l (X = 0.75) Chapter 11 Conclusions Based on the continuum mechanics theory, a theoretical model for predicting the flowrate of bulk solids from mass flow bins, eqn (5.17), was developed. T o ensure this flowrate model predicts adequately the flowrate required the selection of relevant flow properties of the bulk solids (especially the bulk density and permeability) and an estimation of the air pressure gradient at the hopper outlet. Considerable work was carried out to simplify the original flowrate model, eqn (5.17), and to examine the strategies for increasing the limiting flowrate of fine bulk solids. A series of experiments were performed to verify the theoretical models developed in this thesis. F r o m above work, several conclusions can be drawn relating to the particle flow from mass flow bins. a) From the comparisons of consolidation-related bulk density models and permeability models with the experiments, the best bulk density models for practical applications are models 9, 14 and 16, as presented in Table 3.2; the best permeability expression is Jenike and Johanson1 s model, as shown in eqn (3.15). b) 'Permeability' is a useful parameter for use in models to describe the flowrate of coarse and fine as well as size-distributed particle mixtures in terms of the effect of air pressure gradient on particle flowrate from mass flow bins. The theoretical and experimental results indicate that the effect of the air pressure gradient on the flowrate is significant for lower permeability material (e.g.,fineparticles and a particle mixture with a size range which includesfineparticles) and insignificant for higher permeability material (e.g., coarser material). c) G o o d agreement between the theoretical and experimental results for predicting the air pressure distribution and the particle flowrate were obtained. This suggests that the pressure distribution model (as presented in Sections 4.5 and 4.6) and the flowrate model, eqn (5.17), be used for practical applications with the following distinguishing features: • they provide adequate predictions (the flowrate model provides the most accurate prediction relative to the other researchers' models examined in this thesis); • they can be used to study the effect of the material surcharge level on particle flow; • since the effect of consolidation on the particle flow was considered by using consolidation-related bulk density and permeability equations, both models can be used for compressible or incompressible materials; • since the permeability w a s used to relate the particle motion to the air resistance, the models can be used for size-distributed bulk solids. d) Both theoretical and experimental results indicate that the flowrate of bulk solids flowing steadily from a mass flow bin increases rapidly at first and then more gradually as the particle permeability constant increases. A b o v e a critical permeability value C c r i the flow of the particles can be considered to be unhindered by interstitial air effects and the flowrate is only determined by the bulk density and such bin geometry parameters as the hopper outlet diameter and the hopper half angle (for D 0 » dp cases), as expressed in eqn (5.18). e) The criterion C^ for distinguishing fine material from coarse material must be related to the hopper geometry. In addition, as the bulk solids handled often have a range of particle sizes, one particle size (say 500 Jim which is widely recognised nowadays^ 28 ' 50 ' 51 ' 55 ' 61 ' 62 ]), cannot describe the criterion very adequately. A hopper outlet-related criterion in terms of the permeability C ^ , as presented in eqn (7.7), provides a more reasonable criterion for classifying fine and coarse bulk solids in terms of the effect of air pressure gradients on the particle flowrate. f) Theoretical sensitivity analysis on the original flowrate model indicated that the bulk density gradient and stress gradient at the hopper outlet can be ignored. The analysis also verified the air pressure gradient at the hopper outlet is one of the most important factors in determining the magnitude of the particle flowrate. This conclusion is valid for unaided (gravity) flow or aided (e.g., air injected) flow and also valid for any low cohesion material. g) The theoretical and exrjerimental studies on the effect of material surcharge leve on the particle flow suggested that particular attention should be paid to the effect of material level on the flowrate when a fine material being considered. In this study, three regions of particle permeability, as indicated in Figure 8.12, have been proposed to generalize the effect of surcharge level. Region A: bulk material: very fine or with very low permeability particle flow: unsteady with flooding from the bins likely effect of surcharge level: significant; the flowrate increases with increasing material surcharge level. Region B: bulk material: fine or with low permeability particle flow: steady; flowrate increases with increasing permeability of the bulk solids. effect of surcharge level: significant; the flowrate decreases as the material surcharge level increases; the lower the permeability, the more significant the effect of surcharge level on flowrate becomes. Region C: bulk solids: coarse or with high permeability particle flow: steady; the flowrate depends on the bin geometry and bulk density of the particulate material and independent of the permeability of the bulk solid. effect of surcharge level: insignificant. The criterion for judging a significant effect of surcharge level (the boundary between Region B and Region C ) depends on the bin geometry. F r o m the experimental results this criterion is considered to be about 2 0 ^ . h) For fine powders the median particle diameter cannot alone be used to predict accurately the flowrate. Account must be taken of the particle size distribution. i) A low bulk density occurs at both ends of a bin and the maximum bulk density occurs at the transition level for a hopper with surcharge or in the converging section for a hopper without surcharge. j) Theoretically, the air pressure distribution in a bin has a pressure minima near the hopper outlet and a pressure maxima near the top surface of the material. The absolute value of the pressure maxima is very small compared with the absolute value of the pressure minima. Therefore, the pressure m a x i m a is visible in practice only w h e n the hopper outlet is large and/or the permeability of the bulk solid is low. k) Under steady flow conditions, the air pressure gradient at the hopper outlet increases with increasing the outlet size, material surcharge level and/or decreasing particle permeability, while the particle flowrate increases with increasing the outlet size, permeability of bulk solid and/or decreasing surcharge level. 1) Experiments indicated that using a double bin apparatus enables a steady air pressure distribution and a steady particle flowrate to be observed for a significant period oftime.The experimental results confirmed that the negative air pressure generated by flowing particles increased with an increase in material surcharge level. The measured m i n i m u m pressure position agreed with the assumption, eqn (4.29). m) An unsteady flow caused by a high surcharge level has two flowrate values, Q l o w and Q^gh, where the Q h i g h is the flooding flowrate and the Q l o w is the flowrate without being affected by fluidisation. The value of Q l o w is also the m a x i m u m attainable steadyflowratewhen a feeder is installed below the hopper outlet to diminish the fluidisation effect n) The dynamic deaeration coefficient Kdea mainly depends on bulk density constants (p 0 , b l f b 2 ), surcharge level H/D, internal friction angle 5 and the hopper outlet diameter D 0 . Three simplified models for the K d e a (which resulted from a combined method of M o n t e Carlo simulation and an optimization technique) provide a simple but accurate method to approximate the original Kdea m °del. o) Based on the simplified K^ model, a simplified flowrate model, as presented in Section 9.3, w a s obtained having very good agreement with the original flowrate model, eqn (5.17) but without requiring the numerical analysis used to solve the integrations in the original model. The simplified flowrate model has all the distinguishing features of the original model and therefore has more practical implications. p) Strategies for increasing the limiting flowrate include: • low surcharge level with double hopper arrangement; • addition of coarse material into fine particles; • air injection; • use of a standpipe. The use of standpipes to increase the limiting flowrate by gravity was examined both theoretically and experimentally. The results indicate that the use of standpipes can increase the flowrate significantly for fine particles (lower permeability materials) and insignificantly for coarse particles (higher permeability materials). The longer the standpipe, the more significant the suction effect induced by the standpipe and the higher the flowrate that can be obtained. However, a long standpipe is only efficient when it is kept full of bulk solid. This suggests that some precaution needs being taken to ensure that the standpipe remains full, for instance, by using a feeder under the outlet of the standpipe to control the flowrate. Chapter 12 Suggestions for Further W o r k The work completed has been concerned with axisymmetric bins. For general bulk solids flow, the work needs to be extended to the flow of bulk solids from the plane flow bins. More experimental work should be carried out to measure the air pressure distribution in the mass flow bins, to give further evidence for the m i n i m u m position expression eqn (4.29) and to provide a wider range of experimental data to validate and/or improve the theoretical model for predicting the air pressure gradient at the hopper outlet The critical permeability value C^, eqn (7.7), as a criterion of distinguishing the air retarded flow from non-air retarded flow is deduced from the experimental results for the bulk solids flowing out of the hopper outlet with the diameter less than 0.045 m . It seems that the eqn (7.7) is acceptable for the hopper outlet up to about 0.1 m in diameter. M o r e research needs to be carried out for larger outlet diameter to examine this critical permeability expression. The critical permeability value to decide the surcharge level-affected flow is suggested as 20^. M o r e research needs to be done for large hopper outlet cases to provide a greater degree of generality. The theoretical model for predicting the flowrate in this dissertation has not includ the effect of the fluidization on the flow behaviour. M o r e work should be carried out to better describe the flow of very fine materials. This work should relate to the studies on the flooding phenomenon of very low permeable materials. In this work, the dynamic deaeration coefficient Kdea is used to predict successfully the air pressure distribution in a mass flow bin and the particle flowrate from the bin. Both original and simplified models are presented to evaluate this coefficient. Further work is suggested to study the concept of dynamic deaeration, defined as the deaeration of a bulk solid in a mass flow bin under flowing condition, and the prediction of K ^ . The theoretical model presented for the standpipe predicts a reduction of the negative air pressure gradient at the hopper outlet. For longer standpipes, a positive air pressure gradient could be created by the standpipe, which is excluded in current model. Therefore, the further work should be done to improve the theoretical model to predict the performance. The main flow properties for predicting the flowrate of bulk solids from a mass flow bin include essentially the bulk density (p 0 , b l s b 2 ), permeability (C 0 , a) and internal friction angle (8). All of these parameters are related to the physical particle characteristics, say particle size, particle size distribution, particle density, as shown in Figure 3.1. F r o m the point of view of rheology and the packing of the particles, these variables could be predictable from the particles size and particle size distribution. 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Molerus, O., Effect of Interparticle Cohesive Forces on the R o w Behaviour of Powders, Powder Technol., 20, 1978, pp 161-175. 91. Molerus, O., Invited Review: Flow Behaviour of Cohesive Materials, Chem. Eng. Commun. Vol. 15, 1982, pp 257-289. 92. Schwedes, J., Evolution of Bulk Solids Technology since 1974, Bulk Solids Handling, Vol. 3, No. 1, March, 1983, pp 143-147. 93. Schwedes, J. and Schulze, D., Measurement of Flow Properties of Bulk Solids, Proc. of Second World Congress P A R T I C L E T E C H N O L O G Y , Kyoto, Japan, September 19 -22,1990, pp 61-70 (Part II). 94. Schwedes, J. and Schulze, D., Measurement of Flow Properties of Bulk Solids, Powder Technol., Vol. 61, 1990, pp 59-68. 95. Svarovsky, L, Powder Testing Guide : Methods of Measuring the Physical Properties of Bulk Powders. Elsevier Applied Science, London, 1987. 96. Kim, K., Hot Compaction Equations for Metal Powders and Porous Preforms, Int. J. Powder Metallurgy, Vol. 24, 1988, pp 31-37. 97. Kawakita, K. and Liidde, K. H., Some Considerations on Powder Compression Equations, Powder Technol., Vol.4, 1970, pp 61-68. 98. Johanson, J. R. and Cox, B. D., Practical Solutions to Fine Powder Handling, Powder Handling & Processing, Vol. 1, No. 1, 1989, pp 61-65. 99. Schofield, A. N. and Wroth, C. P., Critical State Soil Mechanics, McGraw - Hill, N e w York, 1968. . German, R. M., Particle Packing Characteristics, Metal Powder Industries Federation Princeton, 1989. 101. Marquardt, D. W., A n Algorithm for Least-Squares Estimation of Nonlinear Parameters, J. Soc. Indust Appl. Math., Vol. 11, No.2, 1963, pp 431-441. 102. Yu, A. B. and Gu, Z. H., Private Communication, 1990. 103. Lambe, T. W . and Whitman, R. V., Soil Mechanics, John Wiley & Sons, Inc., N e w York , 1969. 104. Mahinda Samarasinghe, A., Huang, Y. H. and Drnevich, V. P., Permeability and Consolidation of Normally Consolidated Soils, A S C E J. Geotech. Engng. D M . , Vol. 108, (GT6), 1982, pp 835-850. 105. Johanson, J. R. and Jenike, A. W., The Effect of the Gaseous Phase on Pressures in a Cylindrical Silo, Powder Technology, Vol. 5, 1971 / 1972, pp 133-145. 106. Johanson, J. R. and Jenike, A. W., Settlement of Powders in Vertical Channels Caused by Gas Escape, J. Appl. Mech., Dec, 1972, pp 863-868. 107. Ergun, S. and Orning, A. A., Ruid Flow through Randomly Packed Columns and Ruidized Beds, Ind. Eng. Chem., Vol. 41, No. 6, 1949, pp 1179-1184. 108. Carman, P. C , Ruid R o w Through Granular Beds, Trans. Instn. Chem. Engrs., Vol. 15, 1937, pp 150-166. 109. Levenspiel, O., Engineering R o w and Heat Exchange, Plenum Press, N.Y., 1984. 110. Cornell, D. and Katz, D. L., R o w of Gases through Consolidated Porous Media, Ind. Eng. Chem., Vol. 45, No. 10, 1953, pp 2145-2152. 111. Walters, J. K., A Theoretical Analysis of Stresses in Silos with Vertical Walls, Chem. Eng. Sci., Vol. 28, 1973, pp 13-21. 112. Blight, G. 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Univ. of Wollongong, 1981. 120. Johanson, J. R., Retrofitting Bins and Hoppers to Prevent Fine Powder Rushing. Binside Scoop, Vol. 4, No. 1, Winter 1990. 121. Wes, W . J., Stemerding, S. and van Zuilichem, D. J., Control of R o w of Cohesive Powders by Means of Simultaneous Aeration and Vibration, Powder Technol., Vol. 61, 1990, pp 39-49. 122. Yuasa, Y. and Kuno, H., Effects of an Efflux Tube on the Rate of R o w of Glass Beads from a Hopper, Powder Technol., Vol. 6, 1972, pp 97-102. 123. McDougall, I. R. and Pullen, R. J. F., The Effect on Solids Mass Rowrate of an Expansion Chamber between a Hopper Outlet and a Vertical Standpipe, Powder Technol., Vol. 8, 1973, pp 231-242. 124. de Jong, J. A. H., Aerated Solids R o w through a Vertical Standpipe Below a Pneumatically Discharged Bunker, Powder Technology, Vol. 12, 1975, pp 197-200. Leung, L. S. and Wilson, L. A., Downflow of Solids in Standpipes, Powder Technol., Vol. 7, 1973, pp 343-349. 126. Leung, L. S. and Jones, P. 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M., R o w of Ruidized Solids and Bubbles in Standpipes and Risers, Powder Technol., Vol. 7, 1973, pp 93-96. Appendix I : Theoretical Analysis Section Appendix I -1 The Young's Modulus Coefficient K y for Bulk Solids under Uniaxial Test Since the Young's modulus coefficient K y for a bulk solid varies with the voidage of the bulk material. The attempt in this section is to discover this variation under uniaxial condition. From equation (3.4), the relation between strain { E} and principal stress { (J) for bulk solids is in the form { e } = K, 1 -0) -1) -V 1 -1) -1) -V 1 (3.4) {o- } For a three dimensional compacted bulk material bo (A-l.l) P= Po v The relation between bulk density and voidage is (A-1.2) p = ps(l-e) For a element which is compacted by three stresses (a ,wa , a ) , Fig.A-I-1.1, x ' y ' vz z Figure A-I-l.l A n Element of Bulk Solids Compacted by Three Stresses v bO=(1+ex)dx(l+Ey)dy(l+ez)dz Vjj =dxdydz from eqn (A-l.l) p= p0(l+ £*) (l+ey)(l+ez) or on taking a first order approximation P = P o ( 1 + ex + ey +e (A-1.3) z) For the Compressibility Tester, using cylindrical coordinates (r, 9, z), Figure A-I1.2, er = ee = 0. <*Z V^yH111 llll IIIIII III 111IITTTTTTT ~r* y y IWMv^^WWx. Figure A-I-1.2 Compressibility Tester +r which, from eqn ( 3.4), suggests that CTr= e = ° T VC* and E 2\) 2 z= K y ^ ' l-i) TT"^ (A_L4) In neglecting the second term in the bracket of eqn (A-1.4), it follows that e, = Ky cz (A-1.5) where Ky is a function of strain From eqns (A-1.3) and (A-1.5), the measured bulk density ( since e^= e = 0 ) p= p0(l+ Kyoz) On comparison with model 16, the factor Kj can be evaluated as follows. Firstl noting p = b1azb2 + b3 Let b3~p0,then p = b1azb2+p0 (A-1.6) The relation between voidage and strain,fromeqns (A-1.2) and (A-1.3), is e = e0-(l-e0)e2 (A-1.7) It follows on substituting eqns (A-1.6) and (A-1.7) into (A-1.2), that D l bo (A-1.8) h = — °z Po Comparing eqn (A-1.8) with eqn (A-1.5), yields K = J- e (M/b2) (A-1.9) ^y K s - z where the constant K s = N — l/b2 or Ky = fe fe) ( A "U0) Inserting eqn (A-1.9) into eqn (A-1.5), yields J_ e d-l/b2) CT z K s Ez °z e = E Hence, finally this suggests the relation between stress and strain is a^K.E,1^ or (Po V * (A-l.ll) Appendix 1-2 Equivalent Major Principal Stresses in Honner Region of a Mass Row Bin A s presented in Section 3.5.4, it is assumed that the bulk density can be evaluated by the mean stress acting. Alternatively the equivalent major principal stress ale f different situations can be used in place of the mean stress om, as applies in the compressibility tester. From Appendix I -1, for the material in the Compressibility Tester (Figure A-I-1.2), Ej. = eQ = 0, then °r = a9 =~T~~ 1- "0 °z The mean stress in comressibility tester becomes °mt- 3 i.e., CT l+l) mt = mt °7 3 ( 1 -v ) z where the major principal stress az is the equivalent major principal stress ale in this case, i.e., omt = ale (A-2.1) mt i) 3 ( 1 -v ) le Tn a plane flow hopper The motion of bulk solids in a plane flow hopper (Figure A-I-2.1) is a two dimensional problem since ez = 0. 288 Figure A-I-2.1 The Cylindrical Coordinates for the Plane R o w Hopper or+CTfl As the computed mean stress o ~ m w = — h y — a , from eqn (3.4), then oz=\)( ar+CTe ) Therefore, the mean stress acting on the bulk solids in the plane flow hoppe becomes q a m = z+ g r + q e i.e., CT 1 +\) . . m = — 3 - ( °r+ CTe ) (A-2.2) According to the assumption, a m = o m t . Combining eqn (A-2.1) into eqn (A-2.2), the equivalent major principal stress in the plane flow hopper is CT1P= 'le 2(1-D) a mw (A-2.3) 289 ii) In conical hoppers Here the spherical coordinates (r, 6 , 0 ) are used (Figure A-I-2.2). Rgure A-I-2.2 The Spherical Coordinates for the Conical Hopper The computed m e a n stress is a m c = — — f * . The equivalent major principal stress for use in the density equation, therefore, is a le 3(1-1) ) 'mc 1 + v Hence with knowledge of the computed mean stress c (A-2.4) m w and a m c , use of eqns (3.6), (A-2.3) and (A-2.4) and the density models (9), (14) or (16) provide a means to evaluate theoretically the appropriate bulk density in the hoppers. For example, inserting eqn (A-2.3) and eqn (A-2.4) together with eqn (3.6) into the density model 16 and refitting the bulk density equation, the bulk density equations can be expressed as follows: in plane flow hopper p = p0 + b l w ( c r m w ) b 2 w (A-2.5) in conical hopper p = p0 + blc ( o ~ m c ) b 2 c (A-2.6) Once the experimental data are obtained on the Compressibility Tester, the bulk densities in both plane flow hopper eqn (A-2.5) and conical hopper eqn (A-2.6) can be obtained together with that in vertical section of the bin eqn (3.7) during the processing of the uniaxial measured data. For instance, the bulk density equations of Shirley Phosphate are obtained as follows: in the vertical section 0 09544 p = 1246.61 + 231.16 ( o z ) in plane flow hopper p = 1246.61 + 244.84 (omw) in conical hopper n no^^i p = 1246.61 + 252.48 ( a m c ) Appendix I - 3 Difference Caused bv Considering Variable Bulk Density in Walters' Equation for the Vertical Stress in Cylindrical Section of the Bins The differential equation describing the stress distribution in vertical section of the bin presented by Walters^1u ^ is in the form do z 4KW (A-3.1) -fr "D"°«=pg + where o*_ is mean vertical stress (Pa) Kw = tan <}> cos 8 (1 + sin 8) - 2 y sin 8 (under dynamic conditions) y = j7(i-(i-c)»l V tan8 ) From eqn (3.7), the bulk density equation is >2v p = p 0 + b lv (3.8) W000 ) Inserting eqn (3.7) into eqn (A-3.1) yields daz 4K„ '2v i p 0 + b lv ^ 1000 ) g (A-3.2) or daz 4KW ( az IT + ~rTa'-MToooJ ^2v g= P° g This equation is difficult to solve analytically. Considering the Walters' solution for constant bulk density pb, the mean vertical stress becomes CT 4K. De z PbgD * = Tir(1- > (A 33) - Using variable bulk density p instead of constant bulk density pb in eqn (A-3.3), the mean vertical stress becomes Following work attempts to judge if it is acceptable to use eqn (A-3.4) as the solution of eqn (A-3.1). Differentiating eqn (A-3.4) with respect to z, gives 4K„ ^ - D e e " " dz ~ p g e Z + fziL^ p daz dz (A-3.5) 293 Combining eqn (A-3.5) with eqn (A-3.4) provides do z ( , 4^1^^ daz "dr=pgl1-^P-F J +G"dT (A-3-6) or . da z (1_G) ~d7 + 4K,, z=pg "D"° (A"3-7) where Gz n G = d P 7 d^ <A-3-8) from eqn (3.7), eqn (A-3.8) can be rewritten as blv b2V f CTZ >|b2v b2 G= "p—ITOOOJ = —(P"PO) (A"3-9) or G =b 2 v (l-y) Since b 2 v « 1 and p > p n , G « 1. (A-3.10) Table A-I-3.1 shows the G values at the assumed stress levels (1 and 50 kPa) for the materials used in experiments described in Chapter 6. For the powders under dynamic conditions, 50 kPa in vertical stress may never be reached. This table corifirms that the G values are very small. Therefore, eqn (A-3.4) can be used with confidence as the approximate solution of eqn (A-3.1) in terms of variable bulk density. Table A-I-3.1 The G Values for the Materials Used in Experiments Bulk Material a z = 50kPa alumina 0.010311 0.007655 P V C powder 0.011574 0.007904 sugar 0.011220 0.007325 Sand M l 0.005282 0.004406 SandM2 0.005539 0.004610 SandM3 0.006675 0.005574 SandM4 0.007885 0.006572 SandM5 0.006476 0.005381 SandM6 0.017146 0.013800 SandM7 0.012393 0.010130 SandMDl 0.009267 0.006894 SandMD2 0.008861 0.007394 SandMD3 0.008235 0.006949 SandMD4 0.008380 0.006882 CTZ = 1 kPa Appendix II : Experimental Measurement Section Appendix I -1 The Results of Particle Size Analysis bv Laser Particle Sizer Following particle size analysis results are obtained by Malven Laser Particle Sizer (MLPS). The tables show the output of M L P S for alumina, P V C powder, Sand M l to Sand M 6 and Sand M D 1 . In these tables, the data of the cumulative particle size distribution are detailed; D (v, 0.5), D (v, 0.9) and D (v, 0.1) represent the volume diameters at 5 0 % , 9 0 % and 1 0 % point of the cumulative particle size distribution respectively. A-II-1.1 For Alumina Malvern Irifctrurnerits Size rn i ci-oriw 188.0 87. a 53. 5 37.6 28. 1 £1. 5 16.7 13. 0 10. 1 7. 9 s. £ 4. s 3. 3. 2. 1. 8 0 4 9 MttSTER Particle- S u e t - M3. O 1 'A wncier 1 lOO. 0 1 38. a 4.8 4.7 2. 5 2. 3 2.3 1. O 0. 7 0. 7 0. 6 0.2 O. 1 0. 1 O. 1 0. 1 1 S i ;: eb.ir.ci micv •Oi-l-r 188.0 87.2 53. 5 37.6 28. 1 21. 5 16. 7 13. 0 10. 1 7.9 6.2 4.8 3.8 3. 0 2. 4 S a m p l e detai 1 s :-al urnina 22/7/88 **Averaged Result** •/. 07. 2 5^« *-» 37.6 23. 1 21. 5 1.=,. 7 13. 0 10. 1 7.9 6.2 4.8 3.8 3. 0 a. 4 1. 9 61. 2 34.0 0. 0 2. 2 0. 2 0. 0 0. 5 1. 2 0. 0 0.0 0. 4 0. 1 0. 0 0.0 0. 0 i 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -•<.- .1.3. i bunrc^^Ssuip & i"-i:oi-d Nio. = 0 Fucal length = 100 IOil'1. Experiment t ype pi 1 Volume dist ri but ion Beam length = 14. 3 mm. Obscuration =0.£343 Volume Cone. = 0. 0475 "/= 4. 91 Log. Diff. Model indp D(v,0.5) 100. 2 149. 5 D(v, 0.9) D(v,0.1) 62. 4 D(4,3) 101. 0 D<3,2) 59.6 Span = 0. 9 Spec. surf. area 0.05 so. n. /cc. urn >.im urn urn pm A-n-1.2 Malvern For P V C Powder Instruments Size m1crons MPSTtR Particle Sizer M3.0 "/. under 188. 0 87.2 53. 5 37. 6 £8. 1 £1.5 16. 7 13. 0 10. 1 7.9 6. 2 4. 3 3. 8 3.0 2. 4 1.9 iOO. 0 5. S 2. O 1. 4 1. 3 1. 3 1. 3 1. 2 O. O 0. 0 0. 0 0. o 0. o 0. o 0. 0 0. o Si: 'O band microns 188. n 87. iz! 53. 5 37. 6 28. 1 21. 5 16. 7 13. 0 10. 1 7. 9 6. £ 4. 8 3. 8 3. 0 2. 4 87. 53. 37. 28. 21. 16. 13. 2 5 8 1 5 7 0 io. 1 7. 9 6. 2 4. 3 3. 8 3. 0 i_. 4 1. 9 X 94. 2 3. 8 0. 6 0. O O.O 0. 0 0. 1 1. 2 0.0 0. 0 0. O 0.0 0.0 0. 0 O.O Result source-Sample Record No. 0 Focal length = 100 mm. Experiment t ype pi) Volume di^t r i but i.• n Beam length = 14. 3 mm. Obscurat ion = 0 . 1 4 ? :4 V o l u m e Cone. = 0. 04 1 3 •/ Log. Diff. = 4. 74 Model indp 127. 2 D(v,0.5) 161. 5 D(v, 0.9) D(v, O. 1) 97.5 126. 5 D(4,3) . D(3,£> 113.7 Span = 0. 5 Spec. surf. Bt^ea 0.05 sq. m. /cc urn Urn Urn urn urn S a m p l e detai Is :-corvi r- pvc powder 22/7/88 **flveraged R e s u l t * * A-n-1.3 For Sand M l Malvern I n s t r u m e n t s Size microns 564. 0 261. 7 160. 4 112. a 84. 64. 6 SO. 2 39. 0 30. 3 £3. 7 18. 14. 5 11. 4 9. 0 7. £ 5. 8 MASTER Particle Sizer M3.0 Size band % under 100. 0 12. 1 2. o 1. 1 0. 7 0. 6 0. 6 0. 6 0. 6 0. 5 0. 2 0. £ 0. £ 0. 2 0. £ 0. 1 rn i c r o n s 564. 0 £61. 7 160. 4 112. 8 84. 3 64. 6 50. £ 39. 0 30. 3 £3. 7 18. 5 14. 5 11. 4 9. O 7. £ Sample d e t a i I s : - 3 0 0 - 4 2 5 u m river sand **0veraged Result** £61. 7 160. 4 112. 8 84. 3 64. 6 50. £ 39. 0 30. 3 23. 7 18. 5 14. 5 11. 4 9. 0 7. £ 5. 8 54 87.9 10. 1 0.9 0.5 0. 1 0. 0 0.0 0.0 0. 1 0. 3 0.0 0. 0 0.0 0. 0 0.0 Result source=Sample Record No. 0 Focal length = 300 mm. Experiment t ype pil Volume dist ri but ion Beam length 14. .i mm. Obscurat ion =0.2013 Volume Cone. = 0. 1557 •/ Log. Diff. =4. 38 Model indp D(v,0.5) 370. 8 D(v,0.9) 480. 0 D(v,0.1) 238. 3 D(4,3) 364. 5 D(3,£) 254. 9 Span = 0. 7 Spec. surf. area O.Ot: sq. m./cc. urn urn urn urn urn A-n-1.4 For Sand M 2 Malvern Instruments Size microns 564.0 £61.7 160. 4 112. 8 84.3 64.6 50.2 39.0 30. 3 23. 7 18.5 14. 5 11. 4 9. 0 7.2 5.8 % MASTER Particle Sizer M3.0 under lOO.O 35. 5 2. 1 2. 0 1. 1 O. 7 0.7 0.7 O. 7 0.7 0. 3 O. 3 O. 3 0. 3 O. 3 0. 2 Siz e band microns 564. 0 261. 7 160. 4 112. 8 84. 3 64. 6 50. 2 39. 0 30. 3 £3. 7 18. 5 14. 5 11. 4 9. 0 7. 2 £61. 7 160. 4 11£. 8 84. 3 64. 6 50. 2 39. 0 30. 3 23. 7 18. 5 14. 5 11. 4 9. 0 7. 2 5. 8 X 64. 5 33. 3 0. 2 0.9 0. 4 0.0 0.0 0.0 0.0 0.3 0. 0 0.0 0. 0 0. 0 0. 1 Result source=Sarnpl e Record No. 0 Focal length = 300 mm. Experiment t ype pi'. Volume dist ri but ic• n 14. 3 mm. Beam length Obscurat i on =0.1893 Volume Cone. = 0. 1161 -A Log. Diff. =4. 93 Model indp D (v, 0. 5) = 310.2 453. 0 D(v,0. 9) D(v,0. 1) 194. 6 212. 9 D(4,3) D(3,2) £31.4 Span = O. 8 Spec. surf. BTea 0. 0 2 sq. m./cc. pm pm pm pm pm Sample detai Is: -£l£-350urn river sand ••Averaged A-H-1.5 Result** For Sand M 3 Malvern Instruments Size microns 564. 0 261.7 160. 4 112.8 84.3 64.6 50.2 39.0 30. 3 £3. 7 18. 5 14. 5 11.4 9.0 7. 2 5. 8 MASTER Particle Sizer M3. 0 •A under 100. o 93.8 £1.5 5.8 1. 8 1.3 1. 1 1. 1 1. 1 1. 1 0.7 0.7 0. 7 0.7 0. 7 0. 6 Size band microns 564. 0 £61. 7 160. 4 11£. 8 84. 3 64. 6 50. 2 39. 0 30. 3 £3. 7 18. 5 14. 5 11. 4 9. 0 7. 2 S a m p l e detai Is :-180-£12urn river sand ••Averaged Result** 261. 7 160. 4 112. 8 84. vf> 64. 6 50. £ 39. 0 30. 3 23. 7 18. 5 14. 5 11. 4 9. 0 7. £ 5. 3 54 6. 2 72. 3 15. 7 4. 0 0. 6 0. 2 0. O 0. 0 0. 0 0. 4 0. 0 0. 0 0. 0 0. 0 0. 1 Result source=Samp! e Record No. 0 Focal length = 300 Experiment t ype pi! Volume dist ri but ion Beam length = 14. j. mm. Obscurat ion =0.2075 Volume Cone. = 0.0754 % Log. Diff. = 4.31 Model indp D(v,0. 5) 197. 6 246.9 D«v,0.9) D(v, 0. 1 ) 129. 0 D ( 4, 3 ) 197. 8 D<3, 2> 152. 0 Span = 0. 6 Spec. surf. area U.U3 sq. m./ee. pm pm pm pm pm A-n-i.6 For Sand M 4 M a l v e r n Instruments Size microns MASTER Particle Sizer M3.0 X under 564.0 £61.7 160.4 112. 8 84. 3 64. 6 50.2 39. 0 30. 3 23. 7 18. 5 14.5 11. 4 9. O 7.2 5.8 1 1 lOO.O 70.4 34.5 1 1 1 5. 1 2. £ 1.6 1.2 1. 1 0.8 0.6 0.6 0. 6 0.6 0. 5 0. 4 1 1 1 1 1 1 1 1 1 1 1 1 i&. a i Si ze band microns 564. 0 £61.7 160. 4 11£. 8 84. 3 64. 6 50. £ 39. 0 30. 3 £3.7 18. 5 14.5 11. 4 9.0 7. 2 •/ £9. 6 35. 9 17. 6 11. B 2. 9 0. 6 0. 4 0. 1 0. 2 0. 2 0. 0 0., 0 0., 0 0., 1 0., 1 £61. 7 160. 4 112. 8 84. 3 64. 6 50. £ 39. 0 30.3 23. 7 18. 5 14. 5 11. 4 9.0 7. £ 5. 8 1 Result source=Sampl e 1 Record No. 0 = 300 1 Experiment t;ype pil 1 Volume dist ri but icin 1 Beam length 14. 3 mm. 1 Obscuration =0.1898 1 Volume Core. = 0.0691 % =4. 61 1 Log. Diff. 1 Model indp 1 202. 9 pm 1 D(v,0.5) 362. 9 pm 1 D(v,0.9> 1 D(v,0.1) 97. 3 pm 216. 0 pm 1 D(4,3> 1 D(3,£) 123. 4 pm 1 Span 1. 3 1 Spec. surf. area 1 0.02 sq. m. /cc. Sninple detai Is:—0—350um river sand "Averaged Result** A-n-i.7 Malvern For Sand M 5 Instrument; Size 1 rn i c r o n s 1 under 564. O 487.0 420.0 36£.O 31E.O £70.O £33.0 201.0 173.0 14 9.O 129. O 111. O 95. 9 82. 7 71.4 61. 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 lOO. 0 99. 7 98. 9 97.8 36.6 95.7 92.3 8 2.3 65.6 42. 8 21.2 11.5 8.0 Urn 5 5. 0 4.4 MASTER Particle Sizer M3. 0 Size 1 1 •A i n band 1 microns 1 0.3 1 0.8 1 1. 1 1 1. 2 1 1. O 1 3. 4 1 10.0 1 16.7 -1 22. 8 1 £1.6 1 9.7 1 3. 5 1 2. 5 1 0. 5 1 0.6 1 1.7 1 S a m p l e d e t a i I s : — r i v e r sand ••Averaged Result*^ •/. 1 under in band 1 £.8 1. 5 1. 1 1. 1 1. 1 1. 0 1.0 1.0 0.9 0. 9 0.9 0. 9 1.3 1 0. 4 1 0. 0 1 0. 1 1 0. 0 1 0.0 1 0. 1 1 0. 0 1 0. 1 1 0. 0 1 0. 0 1 0. O 1 0.0 1 0.0 1 0. £ 1 0. 3 1 53. 1 1 45.8 1 33. 5 1 34. 1 1 £9. 4 1 25.4 1 £1.9 -1 18. 9 1 16.3 1 14. 1 1 1£. 1 1 10. 5 1 9. 0 1 7. 8 1 6. 7 1 5. 8 1 <98-154urn) 0. a 0.8 0.7 0.6 17-3-88 Result source=Samp le Record No. = 0 Focal length = 300 rn rn • Experiment t ype pi:i Volume dist r i but i<:m 14.. 3 mm. Beam length Dbscurat ion =0. £364 Volume Cone. = 0. 0 7 23 y. Log. Diff. =4. 34 Model indp D(v,0.5) = 155. 7 222. 8 D<v,0.9> D<v,0. 1) 104. 1 164. 0 D(4,3) 106.0 D(3,£) Span = 0. 8 Spec. surf, area 0. 03 sq. rn. /cc. pm pm pm pm pen A-n-i.8 Malvern For Sand M 6 Instruments MASTER Particle Sizer M3.0 Size microns V. under 564.0 261. 7 160.4 112.8 84.3 64. 6 SO. 2 39.0 30. 3 £3. 7 18.5 14. 5 11.4 9.0 7.2 5. 8 lOO.O 99.6 95.9 BO. £ 54.7 36. 4 £7. O 18. 3 11. 1 7.5 5.0 3.9 3. 3 2. 8 2. 3 1.4 S a m p l e d e t a i l s : - r i v e r sand Si ze band M icr ons 564.0 £61.7 160. 4 112.8 84. 3 64. 6 50. £ 39.0 30. 3 £3. 7 18. 5 14.5 11. 4 9.0 7. 2 <0-98um> £61. 7 160. 4 112. 8 84. 3 B4. 6 50. 2 39. 0 30. 3 £3. 7 18. 5 14. 5 11. 4 9. 0 7.£ 5. 8 •/. 0.4 3. 7 15. 7 £5.5 18. 3 9. 4 8. 7 7. 1 3.7 £. 4 1. 1 0.6 0. 5 0. 5 0. 8 Result source-Sample Record No. = 0 Focal length = 300,rnm. Experiment type pil Volume distribution Beam length = 14. 3 mm Obscurat ion =0. 19£4 Volume Cone. = 0.0££6 X Log. Diff. =3.73 Model indp D(v,0.5> 79. 6 131. 5 D(v,0. 9) D(v,0. 1) £8. 6 BI. 4 D<4,3) D<3,£> 45. 8 Span = 1. 3 Spec. surf. area. 0. 0 4 sq. m./cc. prn prn pm prn prn 17-3-83 ••Averaged Result** A-n-i.9 For Sand M D 1 MASTER F'article Sizer M3. O M a l v e r n Instruments Size microns 'A under 564.0 261. 7 160. 4 112.8 84. 3 64. 6 SO. 2 39. 0 30. 3 23. 7 18. 5 14.5 11.4 9. 0 7.2 5. 8 100. 0 97.8 13.2 2. 2 1. 3 1. 1 1. 1 1.1 0. a 0.2 0.2 0. 2 0.2 0. £ 0.2 0. 2 S a m p l e detaiIs:-sand ••Averaged Result** Size band rn l crons 564.0 261.7 160. 4 112.8 84. 3 64. 6 50.2 39.0 30. 3 23. 7 16.5 14.5 11. 4 9.0 7. 2 261. 7 160. 4 112. 8 84. 3 64. 6 50. £ 39.0 30. 3 23. 7 IS. 5 14. 5 11. 4 9. 0 7.2 5. 8 154 -223um 9/10/88 /. 2. £ 84.5 11. 0 0.9 0. £ 0.0 0. 0 0. 3 0. 5 0.0 0.0 0. 0 0. 0 0. 0 0.0 Result source=Sample Record No. = 0 Focal length = 3 0 0 mm. Experiment type pil distribution Volume Beam length = 14. 3 mm Obscurat ion =0.1637 Volume Cone. = 0. 070£ -A =4. 19 Log. Diff. Model indp 201.6 D(v,O.S> 241. 0 D<v,0.9) 147.3 D(v,0. 1) 139.5 D(4,3) 159. 1 D(3,£) Span = 0. 5 Spec. surf, area O. 03 sq. rn. /cc pm pm prn prn pm Appendix II - 2 Instantaneous Yield Loci Measured for All Test Materials A-II-2.1 For Alumina B. 1- 2- 6. 7. 8. NORMAL STRESS - kPa 9. 10. 11. 12. INSTflNTRNEOUS YIELD LOCI r MATERIAL:alumina MOISTURE CONTENT:as received ;GURE= A-H-2.2 TESTED:14/7/88 TEMPERRTURE:RMBIENT For P V C Powder 6- n-r i . i i . i i M M I I I M I I I I I I | I I I I | I I I I | I M I | I I I I | 1 I I I | I I I I | I I I in •n UJ cc co cr cr LiJ r co i i i I I i i i I I I I I I I I I U J I i i i 0. 1 . 2. 3. 4. 5. 6. 7. 8. NORMRL STRESS - kPa INSTflNTRNEOUS YIELD LOCI FIGURE: MRTERIRL:PVC Powder MOISTURE CONTENT:as received 9- 10. 11. TESTED:14/7/88 TEMPERRTURE:RMBIENT 12. 301 A-II-2.3 For Sugar i i . | i i I I | I : i i | i i i i | e-0. i i i | i i I i | i I i I | i I i I i i I I i i I i i i | i i i i i i I i i 4. 5. 6. 7. NORMAL STRESS - kPa 1 • 8. INSTflNTRNEOUS Y I E L D LOCI for Sand M l A-H-2.4 6 - 10. TESTED:14/7/88 TEMPERATURE:AMBIENT MATERIAL:sugar MOISTURE CONTENT:as received IGURE: 9- I I j i i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i | i i i i_ 4. 01 CO " 2 2. 3. 4. NORMAL STRESS - kPa 5. INSTANTANEOUS YIELD LOCI FIGURE: MATERIAL:sand-ml MOISTURE CONTENT:as received 6. 7. 8 9. 10. 11. TESTED:14/7/88 TEMPERATURE:AMBIENT 12. A-n-2.5 For Sand M 2 i i i i i i i i ' • • • i z. 1 1 1 3. 1. i i""iIMIi 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 i ' '' ' i 111 i I i i i i 1 i i ii i I 10. 4. 5. 6. 7. NORMAL STRESS - kPa i i i i i i 11. i i 12. INSTANTANEOUS YIELD LOCI TESTED:14/7/88 TEMPERATURE:AMBIENT MATERIAL:sand-m2 MOISTURE CONTENT:as received GURE: A-H-2.6 6. For Sand M 3 i i I i i i i i i i i i I i i i i I i i i i j i i i i | i i i i | i i i i | i i i i | i i i i i|i i i i i i i 5. 4. 0. 1 3. 4. 5. 6-. 7. I NORMAL STRESS - kPa I N S T A N T A N E O U S YIELD LOCI IGURE: MRTERIRL:sand-m3 MOISTURE CONTENT:as received 10. 11. TESTED:14/7/88 TEMPERRTURE:RMBIENT 12. A-H-2.7 For Sand M 4 : 1 i i I . i i i i i i i i . i i i || i< i i i | I I I I | 1 I I I | I I I I | l i | | | | | i i | ; i | l""l -• h CD en cr i— CO cr cr 4. NORMAL STRESS - kPa 5. 6. 7. * 8. 9- 10. 11- 12. INSTANTANEOUS YIELD LOCI FIGURE: A-H-2.8 MATERIAL: sand-n.4 MOISTURE CONTENT:as received TESTED: 1 4/7/88 TEMPERATURE:AMBIENT For Sand M 5 cr cz 3. 4. NORMAL STRESS - kPa 5. 6. 7. 8 10. 11. INSTANTANEOUS YIELD LOCI IGURE: MATERIAL:sand-m5 MOISTURE CONTENT:as received TESTED:14/7/88 TEMPERATURE:AMBIENT 12. A-n-2.9 For Sand M 6 4. 5. 6. 7. NORMAL STRESS - kPa INSTANTANEOUS YIELD LOCI TESTED:14/7/88 TEMPERATURE:AMBIENT MATERIAL:sand-m6 MOISTURE CONTENT:as received FIGURE: A-n-2.10 7 - M For Sand M 7 I I I | I I I I | I I I I | I I I I | I I I I |I I I I |I I I I | I I I I | I I I I | I I I I | I I I I |I ! I I |I I CO CO DC t— CO cr cr CO 4. NORMAL STRESS - kPa 5- 6. 7. INSTANTANEOUS YIELD LOCI FIGURE: MATERIAL:sand-m7 MOISTURE C0NTEN1:as received 8. 9- 10- 1 TESTED:23/9/89 TEMPERATURE:AMBIENT A-II-2.11 For SandMDl i l i ; i i | M i i | i i i i | i i M | : M i | i i I I | i i i i | I i i i | i i i i | CO CO cr i— CO cr cz DZ CO 11. 4. 5. 6. 7. NORMAL STRESS - kPa 12. INSTANTANEOUS YIELD LOCI FIGURE: A-H-2.12 MATERIAL:sand-mdl MOISTURE CONTENT:as received TESTED:23/9/89 TEMPERATURE:AMBIENT For Sand M D 2 CO CO UJ cr CO cr 9. 10. 11. NORMAL STRESS INSTANTANEOUS YIELD LOCI FIGURE: MATERIAL:sand-md2 MOISTURE CONTENT:as received TESTED:23/9/89 TEMPERATURE:AMBIENT 12. A-II-2.13 For Sand M D 3 ~i i ; i i i r j I I i I | I"! I ! I 1 1 ~1 1 T" T T 1 "I T~ I I 1 I I I I I I I I I I I I I I' ' ' ' I 11. 12. NORMAL STRESS - kPa INSTANTANEOUS YIELD LOCI FIGURE: A-H-2.14 MATERIAL:sand-md3 MOISTURE CONTENT:as received TESTED:23/9/89 TEMPERATURE:AMBIENT For Sand M D 4 1'''' 1 I'''' I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I •j~> NORMAL STRESS - kPa INSTANTANEOUS YIELD LOCI CURE: MATERIAL:sand-md4 MOISTURE CONTENT:as received TESTE0:23/9/89 TEMPERATURE:AMBIENT 307 Appendix II - 3 Wall Yield Loci Measured for All Test Materials For Alumina A-n-3.1 1 r T, 1'''' i' ~r-"-r r *r .EC-END C Pers:ei ^ Gelvoniied B. 1. 2. 3. Steel 4. 5. 6. 7. 8. 9. 10. 11. 12- 13. 14. 15. NORMAL STRESS - kPo WALL YIELD LOCI MflTERIRLrfllumino MOISTURE C0NTENT:os received TESTE0:lB/7/88 TEMPERATURE: ombi snt FIGURE: For P V C Powder A-H-3.2 1<!. -""1 " " I 1 " 1 ! " " ! " " ! " " ! " 1 11. 1Z- I •'" 1 " ' ' I " ' ' I " ' ' I ' ' • • I ' ' ' ' I ' ' ' ' I ' LEGEND 0 Perspex Colvsniled Steel 9-r E. :7. r. B. '- - { ^ ^ S ^ t t t t l T i r i Z . 13. ,4. 15. NORMRL STRESS - kPi WALL YIELD LOCITESTED:16/7/88 ICU^C M^TERIRLiPVC Powder MOISTURE CONTENT:at received TEMPERATURE: ombi ent 308 A-II-3.3 For Sugar • i n',_rn' NORMAL STRESS - kPa WALL YIELD LOCI r A-H-3.4 MqTERIflL:Sugar MOISTURE CONTENT:os received TESTED:16/7/88 TEMPERATURE:»»bient IGU=! for Sand M l 1 1 1 1 1 1 1 H 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 " 1 1 ' " | " " | " '' | " " I " " I " " I " "• LEGEND O Pcrspei A Golvenlied Steel NORMAL STRESS - »Po WALL YIELD LOCI M3TERIAL:5.nd-»l MOISTURE C0NTENT:ot 'iteinl TESTED: 16/7/88 T E M P E R A T U R E : i.bient 309 For Sand M 2 A-II-3.5 12 1 1 \l a e ? E 5 NORMAL STRESS - kPa WALL YIELD LOCI "ICL^E: MHTERIAL:S«nd-(t,2 MOISTURE C0NTENT:«s TESTED:16/7/88 TEMPERATURE: t»bient received For Sand M 3 A-H-3.6 | in I| i I I i |i•rI|I i i i | l l l n i i l l | l i i l | i l M | i l l i | l 5. 6- NORMAL 7. 8. 9- IB- H . ni|i 12- 13. 14. IS- STRESS - kPo WALL YIELD LOCI M=ITERinL:S.nd-»3 MOISTURE C0NTENT:oe FIGURE: received TESTED:16/7/88, TEMPERATURE:.«bii 310 For Sand M 4 A-II-3.7 12. 11. ie. "I""l""l""r"'l""l""l""l""l""l""l' LEGEND O Perspai & Galvanised > | " " I1 Steel ,,!.,, , l , , , , l , , . , l . . . i l n . . 1 . . t . l i i n l i i i i l i i i i l i i i i l i i n l l M l M , , , M ,1, n m m u i M i ^ ^ ^" ^ ^ i0. il. 12. 13- illllll 14. 15 NORMAL STRESS - kPo WALL YIELD LOCI FIGURE: MSTERIAL:S«nd-»4 MOISTURE CONTENT:os received TESTED:16/7/88 TEMPERATURE: ombi ent For Sand M 5 A-H-3.8 I M i|l I u | • • i ,. I I I | M I I | I I I I | I I I I | I I i i | i . •' | _l ,,,,,nil, M l , , .III. ll.M jin ^ NORMAL STRESS - kPo ^ ^ ''.'Jl ''.'i! ''.'ii ''.'it 15 WALL YIELD LOCI riGURC: M3TERIA t :Send--5 MOISTURE CONTEN":es received | • ' TESTED:16/7/88 TEMPERATURE:o*bi ent 311 A-II-3.9 For Sand M 6 i [ 11 i i ; ri-rr | i n i | i 1 1''" i",ri' M""l' rjTTTn _EGENQ C_ C D Pers Pers = A Galvanised 5 I e e I 8- 1- 2- 3- <• 5. B. 7. 8. S. 10. 11. 12. 13. 14. IS. NORMAL STRESS - kPo WALL YIELD LOCI MATERIHL:Sond-m6 MOISTURE CONTENT:as received iGURE: A-H-3.10 TESTEO:16/7/88 TEMPERATURE:anbient For Sand M 7 r" " I ' ' " I ' " ' I ' " ' I " ' ' I " ' "l""l""l""l' F f LEGEND : O Perspei Galvanised & S. ! 2. 3. 4. Steel 5. 6. 7. E. 9. 10- 11NORMAL STRESS - kPa 12- 13- 14. WALL YIELD LOCI "I CURE: MaTERIfiL:Sond-m1 M3TERIfiL:Sond-m7 MOISTURE C0NTENT:a. received TESTED:12/I0/8S TEMPERATURE:embi 312 For SandMDl A-II-3.11 12. rp-rrry. 11. ie. "FT LEGEND O PsrsDei & Galvanised 'l""l' Steel 9. 4 - 5. 6. 7. 6. 9. 10. 11. 12. 13. 14. NORMAL STRESS - kPo 15. WALL YIELD LOCI MBTERIRL:S«nd-mdl MOISTURE CONTENT:as r e c e i v e c FIGURE: A-H-3.12 TESTED:12/10/89 TEMPERATURE:ambient For Sand M D 2 12. i 1. !Z. "'I""!""!""!""!""!' 1 1 " " I " " I " " I " " I " " I " " I " "J LECENO O Perspex ^ Galvanised Steel c, E. 7- E- '0. 1. 2. 3. 4. h I i i I I II i 111 , i i I i 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. IS. NORMAL STRESS - kPo WALL YIELD LOCI FIGURE: MATERJBl:Sand-»d2 MOISTURE C0NTENT:a. received TESTED:12/10/89 TEMPERATURE:ambient 313 For Sand M D 3 A-II-3.13 12. i i i i i i i i i ' I • • ' ' I ' ' > •i Ii ' ' • ' I 11 . ie. LEGENO ^ O Perspei £> Galvanised |""|'" i i i i i i 'I1111!""!""!""!1 Steel c. 8. 7. 6. 5. 6. 7. 6. 9. 10. 11. 12- 13. 14. NORMAL STRESS - kPa WALL YIELD LOCI FIGURE: A-H-3.14 MATERIAL:Send-md3 MOISTURE CONTENT:os received TESTEO:12/10/89 TEMPERATURE:embi For Sand M D 4 i i I I | LEGEND O Perspe> & Galvanised 11 i i | i 1 1 i i i i Steel 15. 5. B. 7. 8. 9. Jf NORMAL STRESS - kPo WALL YIELD LOCI IGURE: M3T ERIAL:Sand-md4 KOI STURE CONTENT:as TESTED:12/10/89 TEMPERATURE:ombi Appendix III Publications While P h D Candidate Arnold, P. C. and Gu, Z. HM The Effect of the Material Level on the Flowrate of Bulk Solids from Mass Flow Bins. Proc. Third Inter. Conf. on Bulk Materials, Storage, Handling and Transportation. Newcastle, 1989, pp 196-199. Arnold, P. C, Gu, Z. H. and McLean, A. G., On the Flowrate of Bulk Solids from Mass-Flow Bins, Proc. Second World Congress PARTICLE T E C H N O L O G Y , Kyoto, Japan, September 19-22,1990, pp 2-9 (Part II). Arnold, P. C. and Gu, Z. H., The Effect of Permeability on the Flowr of Bulk Solids from Mass-Flow Bins, Powder Handling & Processing, Vol. 2, No. 3, September 1990, pp 229-233.