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PAGE 1 EFFECT OF PARTICL E SHAPE ON HOPPER DISCHARGE RATE By KULDEEP MAMTANI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORID A IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2011 1 PAGE 2 2011 Kuldeep Mamtani 2 PAGE 3 To my Guru Asaram Bapuji and my parents 3 PAGE 4 ACK NOWLEDGMENTS I take this opportunity to thank so many people without whom this thesis would not have got completed. Firstly, I would like to sincerely thank my research advisor Dr. Jennifer Curtis for her constant encour agement, support and guidance throughout my research work. Her valuable inputs brought in immense value to this work. Secondly, I thank my colleagues in our research gr oup Casey, Poom, Deepak, Sarah, Henna and Karen for their help and meaningful suggestions during the course of this research work. I would also like to sincerely thank t he librarian of the chemical engineering department Dr. Donna Wrublewski for her c onstant guidance and valuable tips on literature survey and citation management tools. I sincerely appreciate her efforts in getting me the full text of lot of citations in this thesis which were actually quite difficult to get hold of. Last but not the least, I thank my Guru Asaram Bapuji and my parents to whom I would like to dedicate this thesis. 4 PAGE 5 TABL E OF CONTENTS page ACKNOWLEDG MENTS..................................................................................................4 LIST OF TABLES............................................................................................................7 LIST OF FI GURES ........................................................................................................10 LIST OF ABBR EVIATIONS...........................................................................................11 ABSTRACT ...................................................................................................................13 CHAPTER 1 INTRODUC TION....................................................................................................15 Backgroun d.............................................................................................................15 Correlations in the Literature to Predict Disch arge Rate s.................................15 Effect of Key Parameters on the Disc harge Ra te.............................................21 Fill hei ght....................................................................................................22 Hopper diam eter........................................................................................23 Orifice di ameter..........................................................................................23 Orifice shape..............................................................................................23 Particle size................................................................................................24 Cohesion and coeffici ent of fr iction ............................................................24 Particle shape and angularit y.....................................................................24 Cone angle of the bas e of the hopper ........................................................25 Particle size distribut ion.............................................................................26 Problem Defi nition..................................................................................................27 Objectives of the Stud y...........................................................................................28 2 MATERIALS A ND METHOD S................................................................................32 Test Part icles ..........................................................................................................32 Particle Charac terizati on.........................................................................................32 Experimental Apparatus..........................................................................................35 Experimental Procedur e.........................................................................................35 3 RESULTS AND DISCUSSION...............................................................................44 Comparison of Measured and Predicted Dischar ge Rate s.....................................44 Spherical s.........................................................................................................44 Nonspheric als .................................................................................................46 Prediction of discharge rates by modified Beverloo s correlation...............46 Prediction of discharge rates by Fo wler and Glastonburys correlation......47 Prediction of discharge rates by Harmenss correlati on.............................48 5 PAGE 6 Suggested Modifications in Modified Beverl oos correlation for Nonsphericals .....48 Proposed Definition of d in Modifi ed Beverloos Correlation for Nonsphericals......................................................................................................48 Proposed Definition of k in Modifi ed Beverloos Correlation for Nonsphericals......................................................................................................50 Proposed Definition of in Fowler and Glastonburys Correlation for Nonsphericals............................................................................................................51 Proposed Corre lation ..............................................................................................53 Effect of Cone Angle on the Dischar ge Rate ..........................................................54 4 CONCLUSIONS AND FUTURE WORK.................................................................82 Key Find ings...........................................................................................................82 Avenues for Further Resear ch................................................................................85 APPENDIX: VARIOUS MODIFICAT IONS FOR NONSPH ERICALS...........................87 LIST OF REFE RENCES.............................................................................................109 BIOGRAPHICAL SKETCH ..........................................................................................112 6 PAGE 7 LIST OF TABLES Table page 21 Physical properties of t he spherical materials used in the flow experiments......37 22 Physical properties of the nonspherical materials used in the flow experim ents........................................................................................................38 23 Definition of various diameters and their measurem ent techni ques...................39 24 Values of various di ameters for nonspherical ma terials used in the flow experim ents........................................................................................................40 31 Comparison of measured and predicted discharge rates by modified Beverloos correlation (equation 110) for spherical materials............................56 32 Comparison of measured and pr edicted discharge rates by Fowler and Glastonburys correlation (equation 14) for spherical materials.........................57 33 Comparison of measured and predicted discharge rates by Harmenss correlation (equation 110) fo r spherical ma terials..............................................58 34 Comparison of measured and predicted discharge rates by Rose and Tanakas correlation (equation 16) for spherical materials................................59 35 Comparison of the measured values of the filled and flowing bulk densities for some of the te st materi als.............................................................................60 36 Comparison of measured and predicted discharge rates by modified Beverloos correlation (equation 110) for nonspherical materials.....................61 37 Comparison of measured and pr edicted discharge rates by Fowler and Glastonburys correlation (equation 14) for nonspherica l materi als..................62 38 Comparison of measured and predicted discharge rates by Harmenss correlation (equation 111) for nonspherical ma terials......................................63 39 Comparison of the % difference between measured and predicted discharge rates by equation 110 for nonsphericals with d as dsi and dp...........................64 310 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k=1.4 and k = (dp/da)2..............................65 311 Comparison of the measured and predicted disc harge rates by equation 14 for nonsphericals with = (dp/dsi) and d=dp.......................................................66 312 Comparison of measured and pr edicted discharge rates by the new proposed correlation (equation 36) for spherical materials................................67 7 PAGE 8 313 Comparison of the measured and predicted discharge rates by the new proposed correlation (equation 36) for nons phericals......................................68 314 Effect of cone angle on the discharge rates fo r sphericals.................................69 315 Effect of cone angle on the di scharge rates for nonspheric als..........................70 316 Predicted % increase in the discharge rates for sphericals in a 55 hopper compared to a flatbottomed case by static angle of repose crit eria...................71 A1 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k=1.4 and d as dsi, da, etc........................87 A2 Comparison of the % differenc e between measured an d predicted by equation 110 for nonsphericals with k = (dp/da)2 and d as dsi, da, etc................88 A3 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (dF/da)2 and d as dsi, da, etc...............89 A4 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (dp/da) and d as dsi, da, etc.................90 A5 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (dF/da) and d as dsi, da, etc.................91 A6 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (dp/dsi) and d as dsi, da, etc................92 A7 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (da/dsi) and d as dsi, da, etc................93 A8 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (dF/dsi) and d as dsi, da, etc................94 A9 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (dp/dsi)2and d as dsi, da, etc................95 A10 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (da/dsi)2and d as dsi, da, etc................96 A11 Comparison of the % difference bet ween measured and predicted rates by equation 110 for nonsphericals with k = (dF/dsi)2 and d as dsi, da, etc...............97 A12 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with =1.0 and d as dsi, da, etc.........................98 A13 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dp/da)2 and d as dsi, da, etc..................99 8 PAGE 9 A14 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dF/da)2 and d as dsi, da, etc...............100 A15 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dp/da) and d as dsi, da, etc.................101 A16 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dF/da) and d as dsi, da, etc.................102 A17 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dp/dsi) and d as dsi, da, etc................103 A18 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (da/dsi) and d as dsi, da, etc................104 A19 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dF/dsi) and d as dsi, da, etc................105 A20 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dp/dsi)2and d as dsi, da, etc................106 A21 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (da/dsi)2 and d as dsi, da, etc...............107 A22 Comparison of the % difference bet ween measured and predicted rates by equation 14 for nonsphericals with = (dF/dsi)2 and d as dsi, da, etc...............108 9 PAGE 10 LIST OF FIGURES Figure page 11 Types of flow patterns in a dischargi ng hopper ..................................................30 12 Effect of cone angle on the discharge ra te.........................................................31 21 Schematic of the rect angular hoppers used for discharge rate measurements..41 22 Crosssection of the rectangular hoppers used for discharge rate measurem ents....................................................................................................42 23 Complete experimental assembly used for discharge rate measurements........43 31 Effect of particle shape on the difference between the measured and the predicted discharge rates by equat ion 110 for nonsphericals..........................72 32 Microscopic pictures of some test materials.......................................................74 10 PAGE 11 LIST OF ABBREVIAT IONS A projected area of the particle, L2 Ao flow area of the orifice, L2 AR particle aspect ratio, dimensionless C discharge coefficient in Beverl oos correlation, dimensionless D orifice diameter, L Dh hydraulic diameter of the orifice, L d average screen size of the particles, L. Subscripts si, a, p, F, v, s, sv refer to sieve, projected area, perimeter, Feret, volume, surface and surface volume diameters respectively. F Cohesive force between the particles, MLT2 F( ,) multiplicative correction factor for hopper angle, dimensionless f( ) function which gives dependence of discharge rate on cone angle in Rose and Tanakas correlation, dimensionless g acceleration due to gravity, LT2 H fill height, L k shape coefficient in Beverloos correlation, dimensionless L length of the orifice, L P perimeter of the particle, L Wo width of the orifice, L r roundness, dimensionless t time required for a given mass of a granular solid to discharge, M1T W mass discharge rate, MT1. Subscripts measured and predicted refer to the measured and predict ed discharge rates respectively. Z particle shape factor in Rose and Tanakas correlation Greek letters static angle of repose of the gr anular material, dimensionless 11 PAGE 12 voidage, dimensionless particle shape factor in Fowle r and Glastonburys and proposed new correlations, dimensionless coefficient of fric tion, dimensionless density of the granular material, ML3. Subscripts s, b and f refer to particle, filled bulk and flowing bulk densities respectively. /2 half hopper angle with respect to the vertical, dimensionless inclination of the stagnant zone boundary to the horizontal, dimensionless 12 PAGE 13 Abstract of Thesis Pres ented to the Graduate School of the University of Fl orida in Partial Fulf illment of the Requirements for t he Degree of Master of Science EFFECT OF PARTICLE SHAPE ON HOPPER DISCHARGE RATE By Kuldeep Mamtani December 2011 Chair: Jennifer Curtis Major: Chemical Engineering The discharge rates for granular material s of various shapes and sizes from a rectangular flatbottomed hopper with a rectangular orifice at the base were measured and compared to those predicted by the corre lations in the literature. The predicted discharge rates by modified Beverloos correlation and Fowler and Glastonburys correlation differed from the m easured values for most spherical materials by less than 10%. Harmenss correlation predicted discharge rates in good agreement with the measured values for spherical materials larger than 1000 m. Modified Beverloos correlation predicted discharge rates deviating by a maximum of 10% from the measured ra tes for nonspherical material s with an aspect ratio around unity and/or high roundness. However, the co rrelation significantly overestimated the actual discharge rates for materials wit h a high aspect ratio and/or low roundness and/or angular surface. Harmenss correlation al so failed to accurately predict discharge rates for such materials, though it gave bette r predictions when compared to modified Beverloos correlation. 13 PAGE 14 14 It is proposed to define the particle diameter as the perimeter diameter in modified Beverloos correlation to better predict disch arge rates for nonsphericals. With this new definition, the predicted discharge rates deviated by a maximum of 20% from the measured values for most nonsphericals tested. The use of perimeter diameter is also physically justified because it correctly in corporates the angular particle outline. We also propose to define the particle shape factor in Fowler and Glastonburys correlation as the ratio of t he perimeter diameter to the sieve diameter. With the proposed definition, the predicted rates for most nonspherical materials tested were within 20% from the measured rates. We propose a new correlation to predict di scharge rates for various spherical and nonspherical materials. The predicted dischar ge rates by this new correlation deviated by less than 20% for all the test materials. The discharge rates for the various test materials were also measured in a hopper of half cone angle 55 and compared to the fl atbottomed case. The difference in the discharge rates was less than 5% for s phericals and less than 15% for most nonsphericals. PAGE 15 CHA PTER 1 INTRODUCTION Background Hoppers serve the dual purpose of storing and delivering granular materials at a controlled rate. They find wide applications in various chemical, pharmaceutical, food, agricultural, manufacturing and powder industries. Thus accurate prediction of the discharge rate for a granular material from a hopper is critical. Correlations in the Literature to Predict Discharge Rates A number of correlations have been propos ed in the past to predict discharge rates of granular materials from hoppers. Some of the widely used ones are discussed in the following several paragraphs. Deming and Mehring (1929) studied the discharge of crystallized urea and ammonium phosphate, potassium nitrate and urea pellets, crushed phosphate rock, marbles, glass beads, lead shots and a variety of seeds from conical bins provided with circular orifices at the bottom and proposed t hat the time required for a given mass of a granular solid to flow t can be calculated by equation 11. t 161.0130.0 2 sin4444.676.345.2D d Db (11) where t is in minutes per 100 g, D in mm and b in g/cc. The particle size d ranged from 0.131 to 13.5 mm, the bulk density b from 0.4 to 6.5 g/cc, the orifice diameter D from 1to 73 mm, the coefficient of friction from 0.38 to 1.07 and the half cone angle /2 from 15 to 45 in their study. 15 PAGE 16 Newton et al. (1945) examined the flow of 0.1 to 0.2 inch cylindrical cracking catalyst pellets from a flatbottomed hopper with a circular orifice and suggested that the limiting maximum flow rate can be predicted by equation 12. 04.096.250.8 HDW (12) where the discharge rate W is in lb/min, the orifice diameter D in in. and the head H in ft. Equation (12) holds good only when orifice diameter exceeds six times the particle diameter because for orifices sma ller than this size, the flow tends to be irregular (Newton et al., 1945). Franklin and Johanson (1955) studied discharge of variety of spherical and nonspherical materials like glass beads, ionexchange resins, lead shots, cracking catalysts, puffed rice, crushed olivine rock and coal from a cylindrical hopper provided with a circular orifice. According to them, the discharge rate of a granular solid can be predicted by equation 13. 90.44889.116.23228.693.2 d D Ws (13) where W is in lb/min, D and d in in. and s in lb/ft3. The particle size d ranged from 0.03 to 0.2 in., the particle density s from 7.3 to 676 lb/ft3 and orifice diameter D from 0.236 to 2.28 in. in their study. They found t he predicted discharge rates by equation 13 to be within 7% from t he experimentally measured di scharge rates on average. Fowler and Glastonbury (1959) studied the fl ow of wheat, rice, rape seed, sugar and variety of sand fractions from flatbotto med hoppers provided with either circular or noncircular orifices. They s uggested that the discharge rate s for various spherical and nonspherical materials can be predicted by equation 14. 16 PAGE 17 185.02236.0 d D gDA Wh h ob (14) where g is the acceleration due to gravity. The particle shape factor in equation 14 is defined as the ratio of spheric al diameter to the average screen size of the particles whereas the hydraulic diam eter of the orifice Dh is defined by equation 15. o o hP A D 4 (15) where Ao is the area of the orifice and Po is the perimeter of the orifice. The particle size ranged from 0.0272 to 0.1924 cm, t he true density from 1.1 to 2. 6 g/cc, angle of repose from 19.5 to 37.0, shape factor from 1.00 to 1.54 and hydraulic diameter from 0.996 to 5.066 cm in their experiments. Rose and Tanaka (1959) studied the effect of various paramet ers on the discharge rate from cylindrical hoppers provided wi th circular or noncircular apertures and proposed equation 16 to predict di scharge rates for bulk solids. 5.0 3.0 5.25 3 16.0 Zf d D gDWs (16) where 2 tan35.0 f if 90 2 (17) if 35.090tan f 90 2 (18) where is the static angle of repose of the material involved. The term f( ) gives the dependence of the cone angle on the discharge rate. The particle shape factor Z in equation 16 takes a value of 6.0 for spherical materials (Heywood, 1933). The particle size d varied from 0.0112 to 0.318 cm, the orif ice diameter D from 0.36 to 1.59 cm, half 17 PAGE 18 cone angle /2 from 15 to 90, the coefficient of friction from 0.71 to 1.25 in their measurements of discharge rates for silica sand, ground glass, steel balls and steel discs. When the cohesive effects are impor tant, they suggest an inclusion of a multiplicative factor of exp(7.7x106 F/d3sg) in equation 16 where F is the cohesive force between the particles. One of the most widely used correlations to predict discharge rates of granular materials from flatbottomed hoppers with circ ular orifices is equat ion 19, originally proposed by Beverloo et al. (1961). 5.2kdDgCWb (19) Beverloo et al. (1961) used linseed, spinach, watercress, rapeseed, kale, swede, turnip and sand fractions as the test materials. T he particle size d varied from 0.045 to 0.30 cm, the bulk density b from 0.57 to 1.5 g/cc and the orif ice diameter from 0.26 to 3.02 cm in their experiments. The constant s C and k in equation 19 are empirically determined and are called the discharge coefficient and shape coefficient respectively. The value of C typically lies in the range of 0.55 to 0.65 and that of k typically in the range of 1 to 2. A commonly used value of k is 1.4 though a higher value is reported for sand for which Beverloo et al. (1961) obtain ed 2.9 whereas Williams (1977) found it as 2.6. The discharge coefficient may depend on fr iction coefficient whereas the discharge coefficient is solely dependent on particle shape. The origin of the term kd in equation 19 has an interesti ng history. Beverloo et al. (1961) plotted W2/5 as a function of D and found that zero flow rate occurred for nonzero outlet diameter. They corre lated the intercept to the particle size and found that this intercept is equal to kd. Brown and Richards (1970) suggested that no particle centre 18 PAGE 19 can approach within a distance d/2 of the orifice edge a nd thus all particles will pass through an orifice of reduced diameter equal to (Dd), though this does not explain the reason for k being greater than unity. Myers and Sellers (1978) modified equation 19 and pr oposed equation 110 to predict discharge rates for granular mate rials from a flatbottomed hopper with a rectangular orifice. 5.1 003.1 kdWkdLg Wf (110) where L is the length of the orifice, Wo the outlet width and f the flowing bulk density. The flowing bulk density is defined as the ratio of the mass flow rate to the volumetric flow rate. Though equation 19 invo lved the filled bulk density of the material, Huntington and Rooney (1971) su ggested the use of flowing bulk density at the hopper outlet rather than filled bulk density. Thei r suggestion is based on the argument that granular material in the hopper dilates to so me voidage characteri stic of the flowing material and the extent of dilation is more with highly compacted beds. When the flow field is fully established, the flowing bulk density bec omes constant and does not depend on the initial voidage. Harmens (1963) proposed equation 111 to calculate the discharge rates for granular materials from flatbottomed or conical hoppers provided with either circular or noncircular orifices at the bottom. 160.0505.02 1F D d F gDAWh h s (111) where 19 PAGE 20 5.1 5.1 1045.0 38.0 h h hD d D d D d F 35.0 22 tan., F if 1 2 tan. (112) 1,2 F if 1 2 tan. (113) They tested the validity of equation 111 on a variety of spherical and nonspherical granules like glass beads, lead shots, wheat, rice, catalyst spheres and cylinders, sand, rapeseed and watercress seed and found the predi cted rates to be within 6% of the measured ones when the particle size d ranged from 0.110 to 0.381 cm, density s from 0.118 to 10.83 g/cc, the coefficient of friction from 0.34 to 0.79 and the orifice diameter from 0.5 to 5.79 cm. Brown and Richards (1970) der ived equation 114 to calculate discharge rates from conical hoppers. 2 sin 2 cos15.2 5.1 flatbottomWW (114) where Wflatbottom is the discharge rate from a fl atbottomed hopper with the same width and the outlet width as the conical hopper. Their derivation is based on the concept of free fall arch whic h refers to a surface in a fl owing granular material below which the particles are allowed to fall freely under gravity. Equation 114 does not account for the frictional properties of the materials or the wall friction. 20 PAGE 21 The correlations discussed so far assume that the influence of the solidfluid interactions can be neglected an d therefore are valid only when particle diameters are atleast of the order of 100 m. Below this si ze, the predicted discharge rates differ from the measured ones by as much as fa ctor of 10 (Barletta et al., 2003). Crewdson et al. (1977) incorporated the effect of interstiti al pressure gradients and proposed equation 115 to predict discharge rates for fine particles. 5.21 kdD r p gCWb b (115) They postulated that there is an extra body force equal to the pressure gradient at the orifice in addition to the weight of the ma terial in the hopper. The pressure gradient can be determined from the Carm an Kozeny equation. Humby et al. (1998) studied the discharge of binary mixtures prepared from acronitrile butadiene styrene ganules, acrylic beads, turnip seeds and radish seeds from flatbottomed and conical bunkers provided with circular orifices and modified equation 19 to predict discharge rates for bidis perse systems. The particle size of the component materials varied from 0.605 to 3.25 mm, whereas their densities from 1.02 to 1.1 g/cc in their study. They used an or ifice diameter D of 24 mm and the cone angle of 60 in all their experiments. Effect of Key Parameters on the Discharge Rate The various parameters that may infl uence the discharge rate of a granular material from a hopper are: Fill height Hopper diameter (or hopper wid th for rectangular hoppers) Orifice diameter (or hydraulic di ameter for noncircular orifices) Orifice shape Particle size 21 PAGE 22 Particle density Bulk density Particle shape Particle angularity Cohesion Coefficient of friction Angle of repose of the material Cone angle of the base of the hopper Particle size distribution It is interesting to note that some of the parameters though listed separately are infact coupled to some other ones. For exam ple, the angle of repose and the dilatancy behavior of granular materials are reported to be influenced by particle size, size distribution and shape (Sukumar an and Ashmawy, 2003). It mu st be pointed out that effect of fill height, hopper diam eter, orifice diameter, orifice shape, particle size, particle density, bulk density has been well established in the past w hereas the effects of the other parameters on the disc harge rate are not yet fully understood. We now discuss some of the key parameters in the list. Fill height Unlike fluids, the pressure at the bottom of a hopper is c onstant as long as the fill height is above a critical value. The pr essure at the bottom saturates because the weight of the bed is supported by the wa lls of the hopper and the discharge rate remains constant if the fill height is greater than the hopper diameter and hopper diameter is greater than 2.5 times the orif ice diameter (Brown and Richards, 1960; Ketchum, 1919).This pressure saturation effect was first reported by Janssen in 1895 from experiments carried out in cornfilled silo s, although there is evidence that this was known even earlier (Sperl, 2006). 22 PAGE 23 Hopper diameter The discharge rate is i ndependent of the container diameter if the container diameter is greater than 2.6 times the orifice diam eter (Rose and Tanaka, 1959). Ketchum (1919) and Brown and Richards (1960) suggest that discharge rate does not depend on container diameter if the container diameter is greater than 2.5 times the orifice diameter. Franklin and Johanson ( 1955) on the other hand suggest that the discharge rate does not vary with the hopper diameter if the difference between the hopper diameter and the orifice diameter exceeds thirty ti mes the particle diameter. Similarly for a rectangular hopper, as long as the hopper width is sufficiently large, the hopper walls do not affect the flow regime near the outlet and hence the discharge rate. (Anand et al., 2008). Orifice diameter The orifice diameter must be very large co mpared to the particle diameter to avoid intermittencies in the flow due to jamming. T he flow is intermittent and irreproducible when orifice diameter is less than about six particle diameter s (Nedderman et al., 1982). Brown and Richards (1959) suggest ed this critical value as 2.5 for slots and 4.0 for circular orifices whereas Deming and Mehrin g (1929) noticed irregularity in the flow when the orifice diameters r anged from 4 to 6.66 times the particle diameters depending on the particle shape. Orifice shape When apertures of same areas are consider ed, the granular materials are known to discharge at a higher rate from circ ular apertures than the noncircular ones. Kotchanova (1970) studied effect of various orifice shapes on the discharge rates of peas, corn, wheat, millet and mixed feed from flatbottomed bins and found highest 23 PAGE 24 discharge rates for circular orifices follo wed by square, semicircular and rectangular ones respectively. Brown and Ric hards (1959) also found higher discharge rates for glass beads and sand from cylindrical flatbottomed hoppers provided with circular orifices of the same areas as the noncirc ular apertures. The di scharge rate is more sensitive to the orifice shape when the or ifice area is small and the particles under consideration are large (Kotchanova, 1970). Particle size The discharge rate typically increases wit h a decrease in particle diameter for cohesionless materials (Rose and Tanaka, 1959). Cohesion and coefficient of friction The discharge rates of cohesionless materials are typically higher than those for materials with cohesion. Discharge rate is re ported to be independent of the coefficient of friction when the effects of cohesion ar e absent (Rose and Tanaka, 1959). Anand et al. (2008) reported that the par ticleparticle friction is much more important than the particlewall friction. They attributed this to the presenc e of a stagnant region between the central flowing core and the hopper walls. Further, because most of the interactions in a flowing hopper system are particleparticle interactions; particleparticle friction is expected to be more important t han the particlewall friction. Particle shape and angularity The particle shape and angularity significantly affect the flow behavior of bulk solids. Sukumaran and Ashmawy (2003) studied the flow behavior of a variety of sands and spherical glass beads in the size range of 0.30 to 0.50 mm and found that the flow rate decreases as the particle shape fa ctor and angularity increase. Brown and Richards (1959) explored the discharge of 0.027 and 0.110 cm spherical glass beads, 24 PAGE 25 0.055 cm rounded yellow sand and sharp grey sand fractions in the size range of 0.020 to 0.053 cm from cylindrical flatbotto med hoppers and obtained lower discharge rates for angular materials than those for spher ical ones. Cleary and Sawley (2002) also predicted discharge rates for elongated or angular particles to be about 30% lower than those for spherical ones on the basis of Discrete Element Method (DEM) simulations. This is not to say that spherical m aterial s always discharge faster than the nonspherical ones. For example, Jin et al., 2010 found di scharge rates for 6.5 mm hexahedron corn particles to be higher than 5.7 mm spherical soybean from a rectangular hopper. Li et al. (2004) reported flow rates for 19.22 mm di sc shaped chocolate beans to be higher than those for 14.74 mm spherical aniseed pa rticles by 2030%. Langston et al. (2004) also obtained 40% higher discharge rates fo r discshaped particles than the circular ones in DEM simulations. Cone angle of the base of the hopper The two possible flow patterns of a ma terial discharging from a hopper are represented in Figure 11. In ma ss flow, all the material is in motion whereas in core flow, there is a central flowing core surrounded by a stagnant zone (Rhodes, 2008). Mass flow typically occurs in narrow hoppers whereas core flow occurs in wideangled hoppers. The discharge rate depends on cone angle only in mass flow (Verghese and Nedderman, 1995). Rose and Tanaka (1959) sugges ted that whether the discharge rate is a function of the cone angle or not depends on the static angle of repose of the material. This static angle of repose criteria is represented by equations 17 and 18. According to this criteria, when 2180 where is the cone angle and is the static angle of repose of the material, discharge rate depends on the cone angle. 25 PAGE 26 Howev er, when ,2180 the discharge rate is i ndependent of the cone angle because in this case, when the material is discharging, a stagnant zone will occur and the walls will be buried within this stagnant mate rial and their inclination will not really matter. The two possible situations are represented in Figure 12. The static angle of repose criteria as suggested by Rose and Tanaka (1959) has been modified in terms of the flowing angle criteria represented by equations 116 and 117. 35.0tan 2 tan ),( F if )90( 2 (116) 1),( F if )90( 2 (117) where ),( F is the multiplicative factor and gives the dependence of discharge rate on the cone angle and is the inclination of t he stagnant zone boundary when the core flow occurs. Though the origin of the flo wing angle criteria is not known, it must be noted that is not the same as the angle of repose. For example, for glass ballotini, is 45 whereas the value of is 25 (Nedderman et al., 1982). It is difficult to measure directly and so is usually considered as a constant for a given particulate material. The value of can be taken as 45 for a granular material with an angle of repose of about 30(Ducker et al., 1985). Particle size distribution The discharge rate of a bidisperse granul ar system increases as the mass fraction of fines increases (Anand et al., 2008). The rate at which the discharge rate increases with increase in mass fraction of fines decreases as the fines mass fraction increases. 26 PAGE 27 This is attributed to th e fact that as the mass fraction of fines increases, the flowing density of the particulate system increases be cause the fines fill the voids between the coarser particles (Dias et al., 2004). When all the voids get filled, there is very little increase in the flowing densit y and hence the discharge rate. Problem Definition The last section discussed various correlations proposed in the literature to predict discharge rates for granular materials. Though, most of them can be considered to be applicable for both spherical and nonspherical materials, the effe ct of particle shape on the discharge rate is not yet fully underst ood. We say this because of the following reasons. Though it is agreed that the shape coefficien t k in Beverloos correlation (equation 19 or 110) is solely dependent on parti cle shape, there is not enough information on how exactly k depends on shape. The value of k is not entirely clear for nonspherical particles though there is some evidence that it is higher for angular materials than its usually reported value of 1.4 (Beverloo et al., 1961; Williams, 1977). Though Fowler and Glastonbury (1959) clearly defined the particle shape factor in equation 14 as the ratio of spherical di ameter to the sieve diameter, the term spherical diameter is not entirely obvious. They also did not clearly describe the approach they used to calculate the spher ical diameter or the shape factor. Further, Fowler and Glastonbury (1959) reported particle shape factors greater than unity whereas Beverloo et al. ( 1961) obtained values less than one even when both papers gave identical definitions for the particle shape factor. For example, for sand, the former obtained the value of as 0.85 whereas the later reported it in the range of 1.31 to 1.43. The particle diameter d in all the correlations is typi cally defined as the average screen size of the particles. Such a defin ition might not accurately represent particle size for highly nonspherical materi als like rice. This argument is based on the reasoning that it is ess ential to incorporate the outline of the particles as the particles get more and more nonspherical. Though some recent research publications considered the effect of particle shape on hopper discharge dynamics, so far, ther e has been no systematic experimental evaluation of the influence of particle shape on hopper discharge rate. Li et al. (2004) studied the flow behavior of disc shaped and spherical particles from a 27 PAGE 28 rectangular hopper experimentally and compared the results with DEM simulations. Zuriguel et al. (2005) conduc ted experiments with several spherical and nonspherical materials and reported that the arch formation is deeply influenced by the shape of the grains and does not depend on the material properties. Jin et al. (2010) experimentally investigat ed the flow behaviors of spherical, hexahedron and ellipsoid particles from a rectangular hopper. Several other researchers used DEM to model gr anular flow (Cleary and Sawley, 2002; Fraige et al., 2008; Langston et al., 2004). Investigating the effect of particle shape is relevant even from an industrial point of view. Nonspherical particles like discshaped tablets are fr equently encountered in various chemical, food and pharmaceutical industries. Polygon shapes are encountered in geotechnical engineering of soils and rocks, mining and mineral processing. The flow behavior of the polygon shapes can be significant ly different from that of spherical shaped particles due to the angular ity of the former. Further, nonspherical particles are more preferred in some applications and thus information on their discharge is critical. For example, cube shaped materials over spherical ones will be preferred in cases where minimizing voidage is im portant because the former offers much higher packing density. The surface to volume ratio of a cube is 24% higher than a sphere and thus cube shaped materials might be employed to enhance heat transfer, mass transfer and chemical reaction rates. The catalyst activity is also significantly influenced by catalystsupport contact area i.e. particle shape. Objectives of the Study As mentioned earlier, there are several param eters that affect discharge rate from a hopper and many of them need further research The goal of this work to address the effect of one of these namely particle shape. To uncouple the effect of particle shape from the effect of ot her parameters, all test materials considered were big enough so that no effects of cohesion are observed. Further, all materi als were monosized 28 PAGE 29 granules s o that the effect of particle size distribution can be kept at bay. The objectives of the study were as follows: To validate if the existing correlations in the literature accurately predict discharge rates for various nonspherical particles by comparing the predicted rates with the experimentally measured values To suggest modifications in the existing correlations or propose a new correlation if needed to better predict discharge rates for nonspherical materials To study effect of cone angle on the disc harge rates for the various spherical and nonspherical particles and examine the results in context of the existing corrections for wall incli nation in the literature 29 PAGE 30 A B Figure 11. Types of flow patterns in a disc harging hopper. A) Mass fl ow. B) Core flow 30 PAGE 31 31 A B Figure 12. Effect of cone angle on the dischar ge rate. A) Discharge rate is a function of cone angle. B) Discharge rate is independent of cone angle PAGE 32 CHA PTER 2 MATERIALS AND METHODS Test Particles The materials selected for the present study consisted of both spherical and nonspherical granules. Some of the nonspherical particles used were of welldefined shapes like cylindrical and cubical whereas some others were irregularly shaped. As far as the surface angularity is concerned, some of the materials were less angular than the others. Tables 21 and 22 list all the materials used along with their properties measured to characterize them. Material s like JSC1A and Beach sand showed an appreciably wide particle size distribution and so a relevant size fraction obtained from sieve analysis was selected for meas urement of the discharge rate. Particle Characterization The particle size was determined by siev ing the materials through standard sieves stacked on a laboratory scale sieve shaker The particle density was determined by using a Helium gas pycnometer (Quantachr ome Ultrapycnometer 1000, model number: UPY3). The bulk density was measured by filli ng a known mass of the material in a flatbottomed hopper and noting the volume of t he hopper occupied by the material. The relationship between the particle and the bul k densities was used to calculate the voidage. The static angle of repose was dete rmined from the material left in the flatbottomed hopper after the discharge was comple te. A set of two values was obtained for each material and they were averaged to determine the average angle of repose. The average value of angle of repose was used to calculate the friction coefficient for a given material where tan with as the static angle of repose of the granular material in question. 32 PAGE 33 To characterize particle shape and angularit y, its twodimensional projection was used. ImageJ software was employed to describe the outlines of particles. The software converts an optical image from a microscope or a camera into an electronic analogue signal, which is then converted to a digital signal suitable for processing. The choice between a microscope or a camera dependent on the si ze of the granular material involved. The software generated information on particles projected area, projected outline (i.e. perimeter), major and minor ax is. These quantities were then used to quantify particle shape and angularit y in terms of its aspect ra tio, AR and roundness, r. The aspect ratio was defined as the ratio of major to minor axis whereas roundness was calculated by its definition given in equation 22. 24 P A r (22) where A is the projected area of the parti cle and P is the perimet er of the particle Thus, a perfect circle will have a roundness of unity. In other words, lower the roundness, higher is the deviation from a circ ular shape. It must be mentioned that equation 22 is only one of the many definitions for roundness given in Merkus (2009). It is worthwhile to note that such an analysis is orientation dependent because the particles in consideration are nonspherical. To address this issue, only particles resting in their most stable orientation were cons idered. Further, because the particle shape and angularity varied appreciabl y from particle to particl e for some materials, a statistical characterization was employed by taking images of atleast 150 particles of each material. It is believed that such a number is adequate to represent the average aspect ratio and roundness of even the most variable granular material considered. 33 PAGE 34 In addition to the aspect ratio and t he roundness, the particles were also characterized by their various other diameter s. A detailed account of all these diameters is given in Allen (1981). Tabl e 23 defines the diameters used to characterize all the test materials along with the method of measurement. It was possible to obtain the values of some of these diameters by using the so ftw are whereas some others had to be determined by using some other techniq ue. The method called manual counting in Table 23 refers to weighing a known number of particles (say 300) and then determining the weight per particle. The wei ght per particle in combination with particle density was used to obtain volume per par ticle which then can be used to determine volume diameter from the def inition. This method was used for particles big enough to be counted manually and more importantly quite uniform in their shape and size for a given material. The method called packedbed experiment refers to measurement of pressure drop per unit length of a packedbe d of a granular material involved as a function of gas flow rate. The packedbed experiment data in combination with CarmanKozeny equation was used to obtain the r equired diameter for a granular material involved. The values of the various diameters for all the granular materials used in the study are mentioned in Table 24. In case of some granular materials, it was not possible to determine all the diameters due to la ck of availability of a suitable technique and accordingly the corresponding values read as ** meaning they were not determined. The ratios of va rious diameters obtained were then used to define the discharge coefficient k in equation 19 (or 110) and shape factor in equation 14 such that the predicted and the measured discharge rates are in good agreement. 34 PAGE 35 Experimental Apparatus The experimental hoppers consisted of two rectangular hopper s made of clear acrylic and each fitted with a rectangular orif ice at the bottom. The two hoppers were identical in most respects. However, one of the hoppers was a flatbottomed one whereas the other was an angled one with a cone angle of 55 with the vertical. The two hoppers are schematically shown in Figure 21. Figure 22 shows their cross sections. The cross sections of the main parts of the hoppers were 15 cm x 12.5 cm and a height of 45 cm. The dimensions of the orifice were 15 cm x 2.5 cm. Experimental Procedure This section describes the experiment al procedure employed to measure the discharge rates of various granular materi als used in the study. The procedure is exactly the same in the two hoppers. So, we will describe the procedure in connection with only one of them. At the beginning of each experimental run, the hopper was thoroughly cleaned and any particl es from previous runs were removed to avoid any crosscontamination. The hopper was then pl aced on an Ohaus Champsquare balance which was connected to a computer using an Ohaus proprietary cable. Collectv6.1 software which was already installed on the computer was used to get the mass of the assembly on the balance. The orifice of the hopper was covered with a slide gate which can be removed when the discharge has to be started. Particles were charged into the hopper fr om the top and the bed was filled to a height of atleast 30 cm. The software was now started to record the mass of the assembly on the scale every 0.4 seconds di rectly onto an Excel sheet. The slide gate was removed and the discharge was started. The material leaving the hopper was collected in a plastic bucket which was not placed on the scale. After the discharge was 35 PAGE 36 complete, the software was stopped. The so ftware gave the mass of the material remaining in the hopper at any gi ven time. Using this data, mass of the granular material discharged from the hopper was plott ed as a function of time. The slope of the linear regime in the plot gave the discharge rate for a given ma terial. Five independent experimental runs were perfo rmed for a material and the di scharge rates from all the runs were averaged to obtain the average discharge rate. The deviation in the measured values of the discharge rates among various trials was less than 3% of the average reported values for all materials except polystyrene beads and polypropylene where the deviations were 6.5% and 4.7% re spectively. Figure 23 shows the complete experimental setup. 36 PAGE 37 Table 21. Physical properties of the spherical materials used in the flow experiments Material d ( m) s (g/cc) b (g/cc) () 2270 2.58 1.69 0.35 25.3 0.47 1530 2.50* 1.63 0.35 25.6 0.48 990 2.45* 1.57 0.36 21.4 0.39 Glass beads 546 2.49 1.59 0.36 22.9 0.42 1340 1.05 0.63 0.40 24.7 0.46 Polystyrene beads 301 1.06 0.66 0.38 25.8 0.48 Steel shots 317 7.64 4.68 0.39 26.7 0.50 assumed value 37 PAGE 38 Table 22. Physical properti es of the nonspherical materials used in the flow experiments Material d ( m) AR r s (g/cc) b (g/cc) () 2170 1.98 0.75 1.48 0.92 0.38 38.6 0.80 1830 2.76 0.63 1.49 0.83 0.44 43.8 0.96 Rice 1560 3.27 0.57 1.47 0.83 0.44 40.5 0.85 Snowwhite play sand 698 1. 36 0.77 2.64 1.44 0.46 35.9 0.72 Silicaquartz sand 223 1.52 0.69 2.64*1.43 0.46 35.9 0.72 Beach sand (212300 m) 226 1.47 0.69 2.64*1.36 0.48 39.6 0.83 Borosilicate crushed Glass 599 1.64 0.68 2.21 1.13 0.49 34.9 0.70 Polyamide nylon (cubical) 1100 1.12 0.83 1.13 0.67 0.41 41.7 0.89 1090 1.13 0.89 1.08*0.65 0.40 37.7 0.77 Polyamide nylon (cylindrical) 503 1.18 0.75 0.98*0.61 0.38 37.0 0.75 Polycarbonate polymer (cylindrical) 1120 1.16 0.86 1.21 0.73 0.40 37.3 0.76 Olivine 282 1.37 0.77 3. 26 1.65 0.50 37.3 0.76 Aluminum oxide 285 1.47 0. 75 3.90 1.88 0.52 37.0 0.75 JSC1A (150425 m) 202 1.38 0.7 2.86*1.43 0.50 39.8 0.83 Polypropylene 667 1.47 0.66 0.98*0.47 0.52 51.1 1.23 304 Stainless steel cut wires (cylindrical) 1100 1.14 0.75 7.77*4.76 0.39 34.6 0.69 assumed value 38 PAGE 39 Table 23. Definition of various diamet ers and their measurement techniques Sym bol Name Definition Formulae Technique dsi Sieve diameter Width of the minimum square aperture through which the particle will pass Sieving da Projected area diameter Diameter of a circle with the same area as the particles projected area resting in its most stable position A da4 ImageJ dp Perimeter diameter Diameter of a circle with the same perimeter as the particles projected outline P dp ImageJ dF Feret diameter Mean value of the distance between pairs of parallel tangents on the opposite sides of the projected outline of the particle ImageJ dv Volume diameter Diameter of a sphere with the same volume V as the particle 3/16 V dv ImageJ, manual counting ds Surface diameter Diameter of a sphere with the same surface area S as the particle S ds ImageJ dsv Surface volume diameter Diameter of a sphere with the same surface to volume ratio as the particle 2 3 s v svd d d ImageJ, packedbed experiment 39 PAGE 40 Table 24. Values of various diameters for nonspherical materials used in the flow experiments Material dsi ( m) da ( m) dp ( m) dF ( m) dv ( m) ds ( m) dsv ( m) 2170 3896 4520 5570 2872 3240 2257 1830 4084 5177 6805 2810 4120 1307 Rice 1560 4141 5502 7352 2747 4125 1218 Snowwhite play sand 698 959 1097 1172 ** ** 557 Silicaquartz sand 223 309 375 399 ** ** ** Beach sand (212300 m) 226 278 338 359 ** ** ** Borosilicate crushed glass 599 855 1045 1178 ** ** ** Polyamide nylon (cubical) 1100 1290 1420 1556 1602 1784 1291 1090 1662 1767 1925 1864 1997 1625 Polyamide nylon (cylindrical) 503 501 580 626 559 599 486 Polycarbonate polymer (cylindrical) 1120 1436 1553 1729 1604 1719 1397 Olivine 282 473 540 579 ** ** ** Aluminum oxide 285 407 471 518 ** ** ** JSC1A (150425 m) 202 341 412 428 ** ** ** Polypropylene 667 849 1059 1086 ** ** ** 304 Stainless steel cut wires (cylindrical) 1100 1201 1387 1457 1345 1440 1172 ** value was not determined 40 PAGE 41 A B Figure 21. Schematic of the rectangular hoppers used for discharge rate measurements. A) Flat bottomed hopper. B) Angl ed hopper making 55 with the vertical. 41 PAGE 42 42 Figure 22. Crosssection of the rectangular hoppers used for discharge rate measurements. PAGE 43 Figure 23. Complete experimental assembly used for discharge rate measurements 43 PAGE 44 CHA PTER 3 RESULTS AND DISCUSSION Comparison of Measured and Predicted Discharge Rates Sphericals The experimentally measured average di scharge rates for all the spherical particles used in the study were compared with those predicted by modified Beverloos correlation (equation 110), Fowler and Glastonburys correlation (equation 14), Harmenss correlation (equation 111) and Rose and Tanakas correlation (equation 16). The measured and the predicted discharge rates along with the % difference between the two are shown in Tables 31 to 34. Tables 31 and 32 clearly indicate that the predicted discharge rates by modified Beverloos correlation (equation 110) and those by Fowler and Glastonburys correlation (equation 14) are in good agree ment with the measured discharge rates. The predicted discharge rates by both the corre lations are within 10% for most of the spherical granules used in the study. Such a good agreement for particles as small as 301 m contradicts reports published in the past that the correlations give good predictions of discharge rates only when the particle size exceeds 500 m because otherwise the interstitial pressure gradients become significant (Barletta et al., 2003; Spink and Nedderman, 1978; Verghese and Nedderman, 1995). Thus, our findings indicate that the correlations hold good even when the particle size is much smaller, though the critical particle size, where the pressure gradients become important and should be accounted for, was no t determined. Hilton and Cleary (2011) suggest that this critical particle size to be of the order of 100 m. 44 PAGE 45 It must be pointed out that the predicted discharge rates by modified Beverloos correlation were obtained by using the filled bulk density and not the flowing bulk density in equation 110. There is not enough clarity to date on which density should be used. Huntington and Rooney (1971) suggested using flowing bulk density whereas Verghese and Nedderman (1995) used the filled bulk density to predict discharge rates for sand. We choose the filled bulk density over the flowing density owing to two reasons. First, the measurement of the flowing bul k density is difficult and subject to error. Second, the measurements indicated that the maximum difference between the flowing and the bulk densities for various ma terials was less than 5%. These results are shown in Table 35. Artega and Tzn (1990) found that the flowing bulk density differed from the filled bulk density by less th an 3% for almost all the binary mixtures they tested whereas Cleaver and Nedderman (1993) reported this difference to be less than 7% in most of their ex periments. It must be mentioned t hat the flowing bulk density can never be higher than the filled bulk densit y because a granular material dilates on flow and thus we attribute the values of fl owing bulk density higher than the filled one to the measurement error. The predicted discharge rates by Harmenss correlation (equation 111) correlate well with the measured ones only for larger particles as indicated by Table 33. However, the difference between the two increases as the particle size decreases. Table 34 shows that the predicted discharge rates by Rose and Tanakas correlation (equation 16) differ from the me asured ones by more than 10% for most of the materials. The deviation between the two increases further with the decrease in 45 PAGE 46 particle siz e decreases and for particles around 300 m, it is as high as 70%. The reason for such a high di fference remains unclear. Nonsphericals This section compares the measured and the predicted discharge rates for nonspherical materials by equations 110, 14 and 1.11. Equation 16 was not applied to predict discharge rates for nonsphericals because it predicted discharge rates significantly different than the m easured ones for some sphericals. Prediction of discharge rates by modified Beverloos correlation Table 36 compares the measured and the predicted discharge rates by modified Beverloos correlation (equation 110) for th e various nonspherical particles listed in Table 22. It must be noted that the predict ed discharge rates were obtained by using as the filled bulk density (and not the flowing bulk density), k as 1.4 and d as the average sieve diameter for all the materials. The reasons for choosing the filled bulk density over the flowing bulk dens ity are the same as the ones discussed in case of spherical materials. The value of 1.4 for k is the most commonly reported value in the literature and therefore was us ed while predicting discharge rates shown in Table 36. The particle size term in equation 110 was used in terms of the av erage sieve diameter because this is how d was originally defined in the correlation, though we believe that, for nonspherical particles, sieve diameter is not an appropriate way of expressing particle size. Figure 31 correlates the difference between the measured and the predicted discharge rates as a function of particle shape using the data from T able 36. It is found that this difference is less than 10% for a ll the nonsphericals with a low aspect ratio and/or high roundness. However, the difference increases as the particle gets more and 46 PAGE 47 more nonspherical, i.e. as the aspect ratio increases and/or as the roundnes s decreases. The predicted discharge rates for ma terials with a very high aspect ratio such as rice (AR=3.27) di ffered from the measured one by around 75% whereas those for materials with a low roundness such as polypropylene granules (r=0.66), this difference was around 40%. It is interesti ng to note that the predicted discharge rates were higher than the measured ones for all ma terials with a high aspect ratio and/or low roundness. As discussed in Chapter 2, angularity also affects the discharge rate in addition to the particle shape. The microscopic pictures of various nonspherical materials tested and Table 36 suggest that the predicted di scharge rates for materials with a highly jagged/angular surface deviates significantly from their measured ones. Example of such materials from Table 36 are JSC1A and polypropylene. The microscopic pictures of some of the nonspherical materials used in the study are shown in Figure 32. Prediction of discharge rates by Fowler and Glastonburys correlation Table 37 compares the measured and the predicted discharge rates by modified Fowler and Glastonburys correlation (equati on 14) for the various nonspherical particles listed in Table 22. It must be noted t hat the predicted discharge rates were obtained by using as 1.0 and d as the average sieve di ameter for all th e materials. It was found that predicted discharge rates agreed well with the experimentally measured values for materials with an aspect ratio around unity and/or high roundness. However, the correlation significantly overestimated the true discharge rates for high aspect ratio and/or less round and/or angular materials. 47 PAGE 48 Prediction of discharge rates by Harmenss correlation The measured and the predicted discharge rates by Harmenss correlation (equation 111) for the various nonspherical particles tested along with the % difference between the two are shown in in Table 38. The predicted discharge rates for most of the materials fall within 20% of their measured values. Compared to equations 110 and 14, equation 111 gave better predictions of the discharge rates for materials with a high aspect ratio like rice or with a angular surface like silicaquartz sand, beach sand, JSC1A and polypropylene. Suggested Modifications in Modified Beverloos correlation for Nonsphericals As concluded in the last section, modified Beverloos correlation (equation 110) does not give good prediction of the discharge rates for highly nonspherical particles. Such behavior is attributed to the fact that the correlati on does not take into account particle shape effects and in pr inciple, will give a good prediction of the discharge rates only for sphericals (or nonsphericals with an aspect ratio around unity and/or high roundness). In this section, we suggest few m odifications in equation 110 so that the correlation incorporates particle shape effect s. Such a modified correlation can then be used to give good predictions of the discharge rates even for high aspect ratio materials like rice. Proposed Definition of d in Modified Beverloos Corre lation for Nonsphericals According to Beverloo et al. (1961), the par ticle size term d in equation 110 is defined as the average screen size of the part icles. Though, such a definition will work for spherical materials, it will not give an accu rate representation of particle size for nonsphericals especially with a high aspect ratio and/or low roundness and/or high angularity. Thus, it is essentia l to redefine d for nonspherical materials such that the 48 PAGE 49 particle shape is correctly incorporated in the correlation. Equation 110 can then be used with this new proposed definition of d to accurately predict discharge rates for nonspherical particles. The discharge rates for all the nonspherical particles were predicted by using the value of d as t hat corresponding to the various diameters defined in Table 23. The value of k in all cases was used as 1.4 and density as the filled bulk density as befor e. The predicted values were compared with the measured ones and the % difference was noted. Results are summarized in Table A1. It was found that amongst all the di ameters explored, the perim eter diameter gave the predicted discharge rates in good agreement with the measured values. The percentage difference between the measured and the predict ed discharge rates for the various nonspherical materials tested with d as the sieve diameter and as the perimeter diameter are indicated in Table 39. Table 39 clearly suggests that when d in equation 110 is defined as the perimeter diameter (and not as the sieve diameter), the predicted discharge rates for all the nonsphericals improved appreciably. The difference between the measured and predicted discharge rate for a material wit h a high aspect ratio and low roundness like rice (AR=3.27, r=0.57) improved from 74.92% to 11.15%. With the suggested modification, it was possible to predict di scharge rates within 20% of their measured values for almost all nonsphericals tested. The improvement in the predicted discharge rates when d is defined as the perimeter diameter can be physically explai ned by believing that as the particle gets more and more nonspherical, it is essential to capture the outline of a particle. When 49 PAGE 50 d is defined as the perimeter diameter, the outline is co rrectly captured and the shape is appropriately incorporated and thus the predicted rates agree to the measured ones. It must be pointed out that though the suggested modi fication improved the predicted discharge rates for materials lik e JSC1A and polypropylene also, the difference between the measured and predicted discharge rates is still quite high. We attribute this high difference to two reasons. First, the two material s have an extremely jagged/angular surface and even t he perimeter diameter fails to take into account their angularity. The reason ing applies even to materials with a comparatively lower angularity (as indicated by the microscopic pictures) like borosilicate crushed glass, silicaquartz sand and beach sand, where the predicted rates also did not improve remarkably. The microscopic pict ures of the some of the mate rials used in the study are shown in Figure 3.2. Second, the predict ed discharge rates for the two materials depend very weakly on the value of the diamet er used in equation 110. This is because the particle size of the two materials is so small that the term k d in equation 110 is almost a constant and insignificant to cont ribute, irrespective of the value of the diameter used. Proposed Definition of k in Modified Beverloos Corre lation for Nonsphericals The shape coefficient k in equation 110 is known to be a function of particle shape; however, to the best of our knowledg e, relationship between particle shape and the value of k is not known. Thus, to accurately predict discharge rates for nonspherical materials, it is essential to define k such that particle shape is incorporated in its definition. The values of the various diameters def ined in Table 23 were used to obtain many such definitions for the shape coeffi cient k. Such definitions for the shape 50 PAGE 51 coefficient were then used in combination with equation 110 to pr edict discharge rates for various nonspherical used in the study. The predicted values in all cases were compared with the measured values and t he % difference between the two was recorded. The results are shown in Tables A2 through A11. It is clear from the Tables A2 through A11 that the predicted discharge rates for highly nonspherical materials can be apprec iably improved by using some of these definitions for k in equation 110. For exampl e, the predicted discharge rate for rice (AR=3.42) was found to be within 10% of it s measured value when k was defined by equation 31 and d as the perimeter diameter dp (Table 310). 2 a pd d k (31) However, we also note that the same order of improvement in the predicted discharge rates was attained when d wa s used as the perimeter diameter dp and k as its most widely used value of 1.4. Thus, we choose not to redefine k for the sake of simplicity. We therefore reco mmend a value of 1.4 for k in combination with d redefined as the perimeter diameter dp in equation 110 to predict discharge rates within 20% of the measured values. Proposed Definition of in Fowler and Glastonburys Correlation for Nonsphericals Fowler and Glastonburys correlation (equat ion 14) involves a particle shape factor and thus the correlation can be used to predict discharge rates for nonspherical granules too. However, there are two difficulties. First, they defined the particle shape factor as the ratio of the spher ical diameter to the sieve di ameter without clearly stating 51 PAGE 52 the meaning of the term spher ical diameter. Therefore, the calculation of particle shape factor can not be made unl ess the term spherical diameter is clearly defined. Second, though their definition for the shape factor is identical to that given by Beverloo et al. (1961), the latter obtained v alues fo r the shape factor to be less than unity whereas Fowler and Glastonbury (1959) report ed values higher than one, even when some of the materials tested in both studies were the same (e.g. sand). To overcome these difficulties, we propose def inition for the shape factor, in equation 14. With being clearly defined in equation 14, the correlation c an then be used to predict discharge rates of even the nonspherical granules. The various diameters defined in Tabl e 23 were used to obtain several definitions for These definitions were t hen used along with equation 14 and discharge rates of the various nonspherical materials used in the study were predicted. The predicted discharge rates were com pared with the measured values and the % difference between the two was recorded. T he results are summarized in Tables A12 through A22. On the basis of thes e results, we propose to define by equation 32 in combination with d defined as the perimeter diameter dp. si pd d (32) Tables 311 shows the % difference between the measured and the predicted discharge rates for various nonsphericals with defined by equations 32 and d as dp. It is found that the predicted discharge rates lie within 20% of the measured ones for all the nonspherical particles tested except JSC1A. 52 PAGE 53 Proposed Correlation We now propose a correlation to predict discharge rates of granular solids from flatbottomed hoppers We consider that t he discharge rate can be influenced by the following factors. Orifice diameter (or hydraulic diameter for noncircular shapes) Particle diameter Bulk density of the granul ar material involved Particle shape factor Coefficient of friction Acceleration due to gravity The relationship can be symbolically expressed as g dDfWbph,,,,, (33) where all symbols have their meanings as before. The applicati on of the method of dimensional analysis to equation 33 gives the following result. ,,5.0 5.2 p h hbd D f gD W (34) where the groups within round brackets () are dimensionless and independent of each other. We now assume power law to be valid and obtain equation 35. dc b p h hbd D a gD W 5.0 5.2 (35) where a, b, c and d are constants and can be determined from regression analysis. The correlation coefficient r was 0.99 and the values of a, b, c and d were 1.000, 0.094, 0.289 and 0.098 respectively. Thus, t he final equation can be written as. 098.0 289.0 094.0 5.0 5.211 p h hbd D gD W (36) 53 PAGE 54 Table 312 and 313 compare the measured and the predi cted disc harge rates by equation 4 for spherical and nonspherical mate rials respectively. As evident from Tables 312 and 313, the difference between the predicted and the measured discharge rates for all the test materials was within 20% wh ich is quite good considering the range of wide range of particle sizes and shapes used. Effect of Cone Angle on the Discharge Rate The discharge rates for all the test materi als were also measured in an angled hopper provided with a conical base of c one angle 110 and the results were compared to the discharge rates obtained in a flatbottomed hopper. Tables 314 and 315 show the results for spherical and nonspherical materials respectively. It is clear from Table 314 that t he cone angled had a minimal effect on the discharge rate for sphericals. Decreasing t he cone angle from 180 (i.e. flatbottomed case) to 110 decreased the discharge rates by less than 5% for almost all the spherical particles. The reason for a slightly higher difference in case of 2270 m glass beads could not be ascertained. As evident from Table 315, the discharge rates for all the nonspherical materials in the two hoppers differed by less than 15% fo r all materials except rice (AR=2.76). Though, the difference in the discharge rate s measured in the two hoppers can not be correlated to particle shape, it is interesting to note that the dischar ge rate for two of the three types of rice (AR=1.98 and 3.42) and bor osilicate crushed glass differ by less than 3%. It is also worthwhile to note that t he measured discharge rates in the 55 hopper were found to be less than those measured in the flatbottomed hopper for all materials. This behavior is in agreement with the suggesti on by equation 113 where it is predicted 54 PAGE 55 that the discharge rates obtained in an angle d hopper will be less than those in a flatbottomed hopper (of the same width and outle t width). Equation 113 predicts a decrease of about 7% for all mate rials in case of a 55 hopper ( =110). However, the Rose and Tanaka (1959) predict much higher discharge rate s for sphericals in a 55 hopper whereas no change in the discharge ra tes for nonsphericals as compared to the flatbottomed case. They suggested that the discharge rate will be a function of cone angle only when the angle of repose, is less than the quantity (902 ) where is the cone angle. The least value of the measured static angle of repose (among the nonsphericals) was 35 thereby suggesting i dentical discharge rates in 55 and flatbottomed hoppers. Table 316 shows the predict ed difference in the discharge rate for sphericals according to the static angle of repose criteria repres ented by equations 17 and 18. On the other hand, the flowing criteria represen ted by equations 116 and 117 was predicts no difference in the discharge rates in the two hoppers for both sphericals and nonsphericals when the value of is assumed as 45 as suggested by Ducker et al. (1985). 55 PAGE 56 Table 31. Comparison of measured and predicted disc harge rates by modified Beverloos correlation (equation 110) for spherical materials Material d ( m) Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2270 160.58 171.94 7.07 1530 182.07 179.04 1.66 990 201.43 182.06 9.62 Glass beads 546 213.34 192.23 9.90 1340 71.75 70.35 1.94 Polystyrene beads 301 87.09 81.77 6.11 Steel shots 317 658.41 578.96 12.07 56 PAGE 57 Table 32. Comparison of measured and predicted disc harge rates by Fowler and Glastonburys correlation (equation 14) for spherical materials Material d ( m) Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2270 160.58 157.38 1.99 1530 182.07 163.28 10.32 990 201.43 170.51 15.35 Glass beads 546 213.34 192.50 9.77 1340 71.75 64.68 9.85 Polystyrene beads 301 87.09 89.32 2.56 Steel shots 317 658.41 627.29 4.73 57 PAGE 58 Table 33. Comparison of measured and predicted disc harge rates by Harmenss correlation (equation 110) for spherical materials Material d ( m) Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2270 160.58 145.82 9.19 1530 182.07 155.67 14.50 990 201.43 170.50 15.36 Glass beads 546 213.34 171.79 19.48 1340 71.75 66.57 7.22 Polystyrene beads 301 87.09 73.10 16.07 Steel shots 317 658.41 519.37 21.12 58 PAGE 59 Table 34. Comparison of measured and predicted disc harge rates by Rose and Tanakas correlation (equation 16) for spherical materials Material d ( m) Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2270 160.58 186.51 16.14 1530 182.07 204.22 12.16 990 201.43 216.73 7.59 Glass beads 546 213.34 275.20 28.99 1340 71.75 89.20 24.3 Polystyrene beads 301 87.09 147.40 69.25 Steel shots 317 658.41 1059.08 60.85 59 PAGE 60 Table 35. Comparison of t he measured values of the fill ed and flowing bulk densities for some o f the test materials Material d ( m) b (g/cc) f (g/cc) Difference (%) Glass beads (spherical) 2270 1.69 1.64 2.96 Polystyrene beads (spher ical) 301 0.66 0.63 4.55 Steel shots (spherical) 317 4.68 4.62 1.28 Rice 2170 0.92 0.93 1.09 Snowwhite play sand 698 1.44 1.45 0.69 Borosilicate crushed glass 599 1.13 1.15 1.76 Polyamide nylon (cubi cal) 1100 0.67 0.65 2.99 Polycarbonate polymer (cylindrical) 1120 0.73 0.70 4.11 60 PAGE 61 Table 36. Comparison of measured and predicted disc harge rates by modified Beverloos correlation (equation 110) for nonspherical materials Material d ( m) AR r Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2170 1.98 0.75 71.87 94.78 31.88 1830 2.76 0.63 64.82 88.41 36.40 Rice 1560 3.27 0.57 51.96 90.89 74.93 Snowwhite play sand 698 1.36 0.77 167.84 171.76 2.33 Silicaquartz sand 223 1.52 0.69 148.64 178.36 19.99 Beach sand (212300 m) 226 1.47 0.69 140.30 169.31 20.68 Borosilicate crushed glass 599 1.64 0.68 112.00 136.07 21.50 Polyamide nylon (cubical) 1100 1.12 0.83 69.33 76.83 10.81 1090 1.13 0.89 71.70 74.71 4.20 Polyamide nylon (cylindrical) 503 1.18 0.75 74.73 74.39 0.46 Polycarbonate polymer (cylindrical) 1120 1.16 0.86 76.61 82.98 8.31 Olivine 282 1.37 0.77 195.74 204.33 4.39 Aluminum oxide 285 1. 47 0.75 217.95 233.27 7.03 JSC1A (150425 m) 202 1.38 0.7 122.72 178.84 45.73 Polypropylene 667 1.47 0.66 40.68 56.65 39.25 304 Stainless steel cut wires (cylindrical) 1100 1.14 0.75 560.77 546.22 2.66 calculated by using as the filled bulk density, k as 1.4 and d as the average sieve diameter in equation 110 61 PAGE 62 Table 37. Comparison of measured and predicted disc harge rates by Fowler and Glastonburys correlation (equation 14) for nonspherical materials Material d ( m) AR r Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2170 1.98 0.75 71.87 86.39 20.20 1830 2.76 0.63 64.82 80.43 24.09 Rice 1560 3.27 0.57 51.96 82.84 59.44 Snowwhite play sand 698 1.36 0.77 167.84 166.79 0.63 Silicaquartz sand 223 1.52 0.69 148.64 204.56 37.62 Beach sand (212300 m) 226 1.47 0.69 140.30 194.06 38.32 Borosilicate crushed glass 599 1.64 0.68 112.00 134.64 20.21 Polyamide nylon (cubical) 1100 1.12 0.83 69.33 71.34 2.90 1090 1.13 0.89 71.70 69.33 3.31 Polyamide nylon (cylindrical) 503 1.18 0.75 74.73 75.07 0.45 Polycarbonate polymer (cylindrical) 1120 1.16 0.86 76.61 77.47 1.12 Olivine 282 1.37 0.77 195.74 226.03 15.48 Aluminum oxide 285 1. 47 0.75 217.95 256.93 17.88 JSC1A (150425 m) 202 1.38 0.7 122.72 208.33 69.76 Polypropylene 667 1.47 0.66 40.68 54.90 34.95 304 Stainless steel cut wires (cylindrical) 1100 1.14 0.75 560.77 506.83 9.62 calculated by using as 1.0 and d as the average si eve diameter in equation 14 62 PAGE 63 Table 38. Comparison of measured and predicted disc harge rates by Harmenss correlation (equation 111) for nonspherical materials Material d ( m) AR r Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2170 1.98 0.75 71.87 71.99 0.16 1830 2.76 0.63 64.82 69.82 7.81 Rice 1560 3.27 0.57 51.96 75.56 45.42 Snowwhite play sand 698 1.36 0.77 167.84 159.52 4.96 Silicaquartz sand 223 1.52 0.69 148.64 165.43 11.30 Beach sand (212300 m) 226 1.47 0.69 140.30 150.70 7.41 Borosilicate crushed Glass 599 1.64 0.68 112.00 135.83 21.28 Polyamide nylon (cubical) 1100 1.12 0.83 69.33 60.95 12.09 1090 1.13 0.89 71.70 62.35 13.04 Polyamide nylon (cylindrical) 503 1.18 0.75 74.73 66.59 10.89 Polycarbonate polymer (cylindrical) 1120 1.16 0.86 76.61 68.63 10.41 Olivine 282 1.37 0.77 195.74 200.53 2.45 Aluminum oxide 285 1. 47 0.75 217.95 240.29 10.25 JSC1A (150425 m) 202 1.38 0.70 122.72 158.08 28.81 Polypropylene 667 1.47 0.66 40.68 36.84 9.43 304 Stainless steel cut wires (cylindrical) 1100 1.14 0.75 560.77 456.66 18.57 63 PAGE 64 Table 39. Comparison of the % difference between measur ed and predicted discharge rates by equation 110 for nonsphericals with d as dsi and dp Material d ( m) AR r Difference (%) between measured and predicted rates with d defined as dsi dp 2170 1.98 0.75 31.88 0.87 1830 2.76 0.63 36.40 7.06 Rice 1560 3.27 0.57 74.93 11.15 Snowwhite play sand 698 1.36 0.77 2.33 1.59 Silicaquartz sand 223 1.52 0.69 19.99 18.36 Beach sand (212300 m) 226 1.47 0.69 20.68 19.68 Borosilicate crushed glass 599 1.64 0.68 21.50 16.33 Polyamide nylon (cubical) 1100 1.12 0.83 10.81 7.33 1090 1.13 0.89 4.20 2.80 Polyamide nylon (cylindrical) 503 1.18 0.75 0.46 1.52 Polycarbonate polymer (cylindrical) 1120 1.16 0.86 8.31 4.42 Olivine 282 1.37 0.77 4.39 2.09 Aluminum oxide 285 1.47 0.75 7.03 5.16 JSC1A (150425 m) 202 1.38 0.70 45.73 42.86 Polypropylene 667 1.47 0.66 39.25 33.03 304 Stainless steel cut wires (cylindrical) 1100 1.14 0.75 2.66 5.41 64 PAGE 65 Table 310. Comparison of the % differenc e between measured and predicted rates by equation 110 for nonsphericals with k=1.4 and k = (dp/da)2 Material d ( m) AR r Difference (%) between measured and predicted rates k = 1.4 k = (dp/da)2 and d = dsi and d = dp 2170 1.98 0.75 31.88 2.87 1830 2.76 0.63 36.40 16.11 Rice 1560 3.27 0.57 74.93 9.20 Snowwhite play sand 698 1.36 0.77 2.33 0.90 Silicaquartz sand 223 1.52 0.69 19.99 18.15 Beach sand (212300 m) 226 1.47 0.69 20.68 19.46 Borosilicate crushed glass 599 1.64 0.68 21.50 15.56 Polyamide nylon (cubical) 1100 1.12 0.83 10.81 9.43 1090 1.13 0.89 4.20 0.63 Polyamide nylon (cylindrical) 503 1.18 0.75 0.46 1.29 Polycarbonate polymer (cylindrical) 1120 1.16 0.86 8.31 7.14 Olivine 282 1.37 0.77 4.39 2.46 Aluminum oxide 285 1.47 0.75 7.03 5.36 JSC1A (150425 m) 202 1.38 0.70 45.73 42.62 Polypropylene 667 1.47 0.66 39.25 31.44 304 Stainless steel cut wires (cylindrical) 1100 1.14 0.75 2.66 4.75 65 PAGE 66 Table 311. Comparison of the measured and pr edicted discharge rates by equation 14 for nonsphericals with = (dp/dsi) and d=dp Material AR r Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 1.98 0.75 71.87 65.98 8.19 2.76 0.63 64.82 54.74 15.55 Rice 3.27 0.57 51.96 51.61 0.67 Snowwhite play sand 1.36 0.77 167.84 141.12 15.92 Silicaquartz sand 1.52 0.69 148.64 168.80 13.56 Beach sand (212300 m) 1.47 0.69 140.30 167.12 19.12 Borosilicate crushed glass 1.64 0.68 112.00 109.64 2.11 Polyamide nylon (cubical) 1.12 0.83 69.33 64.92 6.37 1.13 0.89 71.70 57.99 19.13 Polyamide nylon (cylindrical) 1.18 0.75 74.73 71.25 4.66 Polycarbonate polymer (cylindrical) 1.16 0.86 76.61 68.61 10.44 Olivine 1.37 0.77 195.74 177.79 9.17 Aluminum oxide 1. 47 0.75 217.95 213.47 2.06 JSC1A (150425 m) 1.38 0.70 122.72 160.03 30.41 Polypropylene 1.47 0.66 40.68 46.25 13.70 304 Stainless steel cut wires (cylindrical) 1.14 0.75 560.77 465.23 17.04 66 PAGE 67 Table 312. Comparison of measured and predicted discharge rates by the new proposed correlation (equation 36) for spherical materials Material d ( m) Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 2270 160.58 190.84 18.84 1530 182.07 190.62 4.69 990 201.43 195.20 3.08 Glass beads 546 213.34 207.55 2.71 1340 71.75 74.91 4.40 Polystyrene beads 301 87.09 89.93 3.26 Steel shots 317 658.41 632.06 4.00 67 PAGE 68 Table 313. Comparison of the measured and predicted discharge rates by the new proposed correlation (equation 36) for nonsphericals Material AR r Wmeasured (g/s/cm) Wpredicted (g/s/cm) Difference (%) 1.98 0.75 71.87 74.76 4.03 2.76 0.63 64.82 59.88 7.61 Rice 3.27 0.57 51.96 56.53 8.80 Snowwhite play sand 1.36 0.77 167.84 146.53 12.69 Silicaquartz sand 1.52 0.69 148.64 157.84 6.19 Beach sand (212300 m) 1.47 0.69 140.30 154.64 10.22 Borosilicate crushed glass 1.64 0.68 112.00 112.39 0.34 Polyamide nylon (cubical) 1.12 0.83 69.33 68.99 0.49 1.13 0.89 71.70 62.26 13.15 Polyamide nylon (cylindrical) 1.18 0.75 74.73 71.78 3.93 Polycarbonate polymer (cylindrical) 1.16 0.86 76.61 74.15 3.20 Olivine 1.37 0.77 195.74 168.61 13.87 Aluminum oxide 1. 47 0.75 217.95 203.32 6.70 JSC1A (150425 m) 1.38 0.70 122.72 145.92 18.90 Polypropylene 1.47 0.66 40.68 45.39 11.59 304 Stainless steel cut wires (cylindrical) 1.14 0.75 560.77 507.06 9.57 68 PAGE 69 Table 314. Effect of cone angle on the discharge rates for sphericals Material d ( m) Wmeasured (g/s/cm) Difference (%) Flatbottomed hopper Angled hopper with half cone angle of 55 2270 160.58 172.37 7.34 1530 182.07 177.23 2.65 990 201.43 198.96 1.22 Glass beads 546 213.34 211.42 0.89 1340 71.75 71.18 0.79 Polystyrene beads 301 87.09 86.92 0.19 Steel shots 317 658.41 627.85 4.64 69 PAGE 70 Table 315. Effect of cone angle on the discharge rates for nonsphericals Material d ( m) AR r Wmeasured (g/s/cm) Difference (%) Flatbottomed hopper Angled hopper with half cone angle of 55 2170 1.98 0.75 71.87 67.87 5.56 1830 2.76 0.63 64.82 52.63 18.80 Rice 1560 3.27 0.57 51.96 50.53 2.75 Snowwhite play sand 698 1.36 0.77 167.84 152.15 9.34 Silicaquartz sand 223 1.52 0.69 148.64 137.73 7.33 Beach sand (212300 m) 226 1.47 0.69 140.30 124.09 11.55 Borosilicate crushed Glass 599 1.64 0.68 112.00 112.25 0.22 Polyamide nylon (cubical) 1100 1.12 0.83 69.33 60.61 12.57 1090 1.13 0.89 71.70 66.84 6.77 Polyamide nylon (cylindrical) 503 1.18 0.75 74.73 66.92 10.45 Polycarbonate polymer (cylindrical) 1120 1.16 0.86 76.61 68.94 10.01 Olivine 282 1.37 0.77 195.74 176.63 9.76 Aluminum oxide 285 1. 47 0.75 217.95 200.34 8.08 JSC1A (150425 m) 202 1.38 0.70 122.72 109.40 10.86 Polypropylene 667 1.47 0.66 40.68 36.20 11.01 304 Stainless steel cut 1100 1.14 0.75 560.77 503.03 10.29 wires (cylindrical) 70 PAGE 71 Table 316. Predicted % increase in the discharge rates for sphericals in a 55 hopper compared to a flatbottomed case by static angle of repose criteria Material d ( m) () Predicted increase in the discharge rates in a 55 hopper with respect to those in a flatbottomed one (%) 2270 25.3 14.28 1530 25.6 14.28 990 21.4 22.22 Glass beads 546 22.9 18.92 1340 24.7 15.78 Polystyrene beads 301 25.8 12.82 Steel shots 317 26.7 11.38 71 PAGE 72 A Figure 31. Effect of particle shape on the difference between the measured and the predicted discharge rates by equation 110 for nonsphericals. A) Aspect ratio. B) Roundness 72 PAGE 73 B Figure 31. Continued 73 PAGE 74 A Figure 32. Microscopic pictures of some test materials. A) Snowwhite play sand. B) Silica quartz sand. C) Beach sand. D) Borosilicate crushed glass. E) Olivine. F) Aluminum oxide. G) JSC1A. H) Polypropylene 74 PAGE 75 B Figure 32. Continued 75 PAGE 76 C Figure 32. Continued 76 PAGE 77 D Figure 32. Continued 77 PAGE 78 E Figure 32. Continued 78 PAGE 79 F Figure 32. Continued 79 PAGE 80 G Figure 32. Continued 80 PAGE 81 H Figure 32. Continued 81 PAGE 82 CHA PTER 4 CONCLUSIONS AND FUTURE WORK This chapter discusses the key findings fr om the study followed by the avenues for further research. Key Findings The various correlations proposed in the literature were used to predict discharge rates for granular materials of various shap es and the predicted values were compared with the experimentally measured ones. It was found that the predicted discharge rates by modified Beverloos correlation (equation 110) and Fowler and Glastonburys correlation (equation 14) were within 10% from the measured values for most sphericals tested. A good agreement bet ween the measured and the predicted discharge rates for materials even around 300 m contradicts some reports published in the past arguing that the correlations break down when the particle size is around 500 m due to the dominance of the interstitial pressure gradients. Thus, our findings indicate that the correlations are valid for mu ch smaller particle sizes though the critical size where the effects of interstitial pre ssure gradients should be accounted for was not determined. It must be noted that the predicted discharge ra tes by equation 110 were calculated by using the filled bulk dens ity and not the flowing bulk density. Our measurements indicate that the two densities differed by less than 5%. Further, the flowing bulk density is difficult to measure accurately. Thus it may be preferred to use the filled bulk density over t he flowing bulk density. Harmenss correlation (equation 111) pr edicted discharge rates within 15% from the measured values for sphericals larger than 1000 m. The difference between the measured and the predicted discharge rates by equation 111 increased for smaller 82 PAGE 83 sphericals and for materials around 300 m, the deviation was around 20%. The predicted discharge rates by Rose and Tanakas correlation (equation 16) for spherical materials around 500 m or less did not agree well wit h the measured values with the difference between the two as high as 70% for 300 m polystyrene beads. Equation 16 when compared to equation 110 or (equation 14) did not give good predictions even for materials larger than 500 m. The measured discharge rates for nons phericals were compared with those predicted by modified Beverloos correla tion (equation 110), Fowler and Glastonburys correlation (equation 14) and Harmenss co rrelation (equation 111). Our results indicate that the predicted discharge rates by equation 110 differed by only around 10% from the measured values for nonspherical materials with an aspect ratio around unity and/or with high roundness. However, the difference between the measured and the predicted discharge rates increased as t he aspect ratio increased and/or roundness decreased. The predicted discharge rate fo r rice (AR=3.27 and r=0.57) differed by around 75% from its m easured value. It was also found that the predicted discharge rates for materials with a jagged/angular surface such as JSC1A and polypropylene also differed significantly from the meas ured values. Equations 14 and 110 also predicted discharge rates which differed widel y from the measured values for highly nonspherical materials. It was interesting to find that all the correlations overestimated the actual discharge rates for all the materials with a high aspect ratio and/or low roundness and/or jagged surface. It was also noted that equation 111 when compared to equation 110 or equation 14 gave better predi ctions of the discharge rates for such materials. 83 PAGE 84 A high difference between the measured and the predicted dischar ge rates for highly nonspherical materials necessitated proposing some modifications in the correlations. In particular, it is proposed to define the particle size d as the perimeter diameter dp and not as the average sieve diameter, the way it was originally defined. The use of the perimeter diameter is physica lly justified because it is essential to capture the outline of the particles for highly nonspherical materials. Our results indicate that the predicted discharge rates si gnificantly improved when d was defined as dp in equation 110. For example, the predict ed discharge rates for all the three rice particles differed from their measured values by less than 15%. When this new definition of d was used in equation 110, the difference between the predicted and the measured discharge rates for all nonspherical s except JSC1A and polypropylene was within 20%. Microscopic pictures indicated that JSC1A and polypropylene have an extremely angular surface and a high diffe rence between the measured and the predicted rates even when d was defined as dp is attributed to this high degree of surface angularity. Though it is also possible to improve the predicted discharge rates for highly nonspherical materials by redefin ing k also in equation 110, we choose not to do it for the sake of simplic ity. Thus, we propose to use the value of k as 1.4, its most widely used value and that of d as dp, the perimeter diameter in equation 110 to better predict discharge rates for all nonspherical materials. We also propose an alternative defin ition for the particle shape factor in Fowler and as the ratio of perimeter diameter dp to the average sieve diameter dsi. When this definition for was used in combination with equati on 14, the predicted discharge rates 84 PAGE 85 for all the nonspheric al materials except JSC1A were within 20% from their measured rates. The discharge rates for all the test materi als were also measured in a conical hopper of cone angle 110. Results indicate that the discharge rates for almost all the spherical materials in the 110 hopper differ ed from those in the flatbottomed one by less than 5%. On the other hand, discharge ra tes in the 110 hopper for most of the nonsphericals differed by around 10% when co mpared to those in the flatbottomed case. No correlation between particle shape and the effect of cone angle could be established. These results were also evaluated in terms of the available corrections in the literature for hopper angle effects. Br own and Richards correlation (equation 114) predicts discharge rates in the 110 hopper to be less than those in the flatbottomed one by 7% for all materials. The static angle of repose criteria (equations 17 and 18) suggested a much higher difference in the discharge rates in the two hoppers for sphericals whereas no difference for nonspheric als. However, the flowing angle criteria (equations 116 and 117) predict ed identical discharge rates in the two hoppers for all the materials. Avenues for Further Research The present work reached some important conclusions on the effect of particle shape on mass discharge rate from rectangul ar hoppers. However, there are several areas where further work could be undertaken. The effect of particle shape should be experimentally investigated more rigorously by manufacturing several shapes using the same starting material. The information on the flow patterns of various shapes in a hopper could provide useful insight into discharge rate behavior which can also be 85 PAGE 86 explored by DEM. Measurement of other particle characteri stics like volume diameter also should be undertaken. The effect of cone angle can be studied in a more detai led manner by considering hoppers of several other angles too. Material s of smaller sizes where cohesive effects dominate and of various size distributions should also be tested to explore the effect of cohesion and particle size distri bution on the hopper discharge rate. 86 PAGE 87 APPENDIX VARIOUS MODIFICATIONS FOR NONSP HERICALS Table A1. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k=1.4 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k=1.4 and d as dsi da dp dF dv ds dsv 31.88 8.69 0.87 11.72 22.06 17.18 30.42 36.40 6.34 7.06 25.58 22.93 5.88 43.84 Rice 74.93 31.76 11.15 14.44 54.41 32.01 81.04 Snowwhite play sand 2.33 0.24 1.59 2.32 ** ** 3.73 Silicaquartz sand 19.99 19.11 18.36 18.09 ** ** ** Beach sand (212300 m) 20.68 20.36 19.68 19.44 ** ** ** Borosilicate crushed glass 21.50 18.52 16.33 14.81 ** ** ** Polyamide nylon (cubical) 10.81 8.74 7.33 5.87 5.38 3.43 8.73 4.20 1.75 2.80 4.37 3.77 5.09 1.38 Polyamide nylon (cylindrical) 0.46 0.77 1.52 1.96 1.33 1.70 0.63 Polycarbonate polymer (cylindrical) 8.31 5.66 4.42 2.56 3.88 2.67 6.08 Olivine 4.39 2.74 2.09 1.70 ** ** ** Aluminum oxide 7.03 5.80 5.16 4.68 ** ** ** JSC1A (150425 m) 45.73 43.83 42.86 42.64 ** ** ** Polypropylene 39.25 35.80 33.03 32.67 ** ** ** 304 Stainless steel cut wires (cylindrical) 2.66 3.63 5.41 6.07 5.01 5.91 3.35 ** value was not determined 87 PAGE 88 Table A2. Comparison of the % diffe rence between measured and predicted by equation 110 for nonsphericals with k = (dp/da)2 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dp/da)2 and d as dsi da dp dF dv ds dsv 32.69 10.46 2.87 9.39 23.44 18.70 31.53 32.56 1.27 16.11 36.30 17.32 1.77 41.03 Rice 67.67 15.07 9.20 38.27 42.42 15.37 75.28 Snowwhite play sand 2.78 0.37 0.90 1.58 ** ** 4.09 Silicaquartz sand 19.95 18.93 18.15 17.87 ** ** ** Beach sand (212300 m) 20.81 20.18 19.46 19.21 ** ** ** Borosilicate crushed Glass 21.05 17.88 15.56 13.94 ** ** 22.77 Polyamide nylon (cubical) 12.46 10.66 9.43 8.15 7.72 6.02 10.65 6.23 1.49 0.63 0.66 0.17 1.25 1.79 Polyamide nylon (cylindrical) 0.59 0.57 1.29 1.71 1.10 1.46 0.43 Polycarbonate polymer (cylindrical) 11.05 8.19 7.14 5.57 6.68 5.66 8.54 Olivine 4.83 3.08 2.46 2.11 ** ** ** Aluminum oxide 7.15 5.98 5.36 4.91 ** ** ** JSC1A (150425 m) 45.61 43.63 42.62 42.39 ** ** ** Polypropylene 37.20 34.51 31.44 31.05 ** ** ** 304 Stainless steel cut wires (cylindrical) 2.13 3.06 4.75 5.38 4.37 5.23 2.79 ** value was not determined 88 PAGE 89 Table A3. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (dF/da)2 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dF/da)2 and d as dsi da dp dF dv ds dsv 18.20 12.96 23.22 39.25 5.04 1.60 16.54 12.08 39.16 59.22 82.89 11.81 39.87 25.83 Rice 41.82 38.52 69.41 96.56 1.44 38.11 54.52 Snowwhite play sand 1.89 0.84 2.27 3.04 ** ** 3.38 Silicaquartz sand 19.59 18.43 17.55 17.23 ** ** ** Beach sand (212300 m) 20.46 19.75 18.94 18.66 ** ** ** Borosilicate crushed glass 19.00 15.01 12.08 10.05 ** ** ** Polyamide nylon (cubical) 10.39 8.24 6.79 5.27 4.76 2.76 8.23 4.54 1.04 2.05 3.56 2.97 4.24 0.68 Polyamide nylon (cylindrical) 1.34 1.32 2.15 2.64 1.93 2.35 1.16 Polycarbonate polymer (cylindrical) 8.62 5.12 3.83 1.91 3.28 2.02 5.55 Olivine 4.43 2.41 1.71 1.30 ** ** ** Aluminum oxide 6.58 5.16 4.41 3.87 ** ** ** JSC1A (150425 m) 45.37 43.23 42.14 41.89 ** ** ** Polypropylene 36.69 33.87 30.65 30.24 ** ** ** 304 Stainless steel cut wires (cylindrical) 3.19 4.21 6.07 6.76 5.65 6.59 3.92 ** value was not determined 89 PAGE 90 Table A4. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (dp/da) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dp/da) and d as dsi da dp dF dv ds dsv 36.81 17.33 10.61 0.30 28.73 24.58 35.80 38.80 11.17 1.26 18.60 26.45 10.75 45.59 Rice 76.31 35.02 15.20 9.58 56.73 35.27 82.14 Snowwhite play sand 3.62 1.51 0.41 0.19 ** ** 4.77 Silicaquartz sand 20.42 19.58 18.94 18.70 ** ** ** Beach sand (212300 m) 21.29 20.77 20.17 19.97 ** ** ** Borosilicate crushed Glass 22.40 19.79 17.87 16.54 ** ** 23.81 Polyamide nylon (cubical) 13.42 11.77 10.65 9.48 9.08 7.53 11.76 6.80 2.33 1.52 0.31 0.78 0.24 2.62 Polyamide nylon (cylindrical) 0.03 0.04 0.58 0.94 0.42 0.73 0.16 Polycarbonate polymer (cylindrical) 11.83 9.19 8.21 6.75 7.79 6.84 9.51 Olivine 5.15 3.61 3.07 2.76 ** ** ** Aluminum oxide 7.53 6.51 5.97 5.58 ** ** ** JSC1A (150425 m) 46.11 44.46 43.62 43.43 ** ** ** Polypropylene 39.17 37.00 34.52 34.20 ** ** ** 304 Stainless steel cut wires (cylindrical) 0.76 1.57 3.04 3.59 2.71 3.46 1.34 ** value was not determined 90 PAGE 91 Table A5. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (dF/da) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dF/da) and d as dsi da dp dF dv ds dsv 30.97 7.63 0.32 13.10 21.24 16.26 29.75 31.48 3.40 18.63 39.24 15.74 3.92 40.23 Rice 67.48 14.64 9.72 38.87 42.10 14.94 75.13 Snowwhite play sand 3.22 0.97 0.21 0.85 ** ** 4.45 Silicaquartz sand 20.28 19.38 18.69 18.44 ** ** ** Beach sand (212300 m) 21.16 20.61 19.98 19.76 ** ** ** Borosilicate crushed Glass 21.60 18.66 16.50 15.00 ** ** 23.19 Polyamide nylon (cubical) 12.46 10.66 9.43 8.15 7.72 6.02 10.65 5.99 1.12 0.24 1.08 0.57 1.68 1.44 Polyamide nylon (cylindrical) 0.28 0.26 0.94 1.32 0.76 1.10 0.14 Polycarbonate polymer (cylindrical) 10.79 7.86 6.78 5.17 6.32 5.27 8.22 Olivine 4.99 3.34 2.77 2.43 ** ** ** Aluminum oxide 7.30 6.18 5.60 5.17 ** ** ** JSC1A (150425 m) 46.01 44.29 43.42 43.22 ** ** ** Polypropylene 38.98 36.76 34.22 33.90 ** ** ** 304 Stainless steel cut wires (cylindrical) 1.22 2.06 3.61 4.19 3.26 4.05 1.82 ** value was not determined 91 PAGE 92 Table A6. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (dp/dsi) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dp/dsi) and d as dsi da dp dF dv ds dsv 17.39 14.25 24.63 40.83 4.01 2.72 15.69 11.24 40.61 60.78 84.34 12.97 41.33 25.19 Rice 35.03 92.96 81.32 8.80 50.40 49.00 Snowwhite play sand 1.49 1.38 2.88 3.69 ** ** 3.06 Silicaquartz sand 19.57 18.41 17.52 17.20 ** ** ** Beach sand (212300 m) 20.77 20.14 19.41 19.15 ** ** ** Borosilicate crushed glass 19.80 16.12 13.43 11.56 ** ** 21.80 Polyamide nylon (cubical) 11.77 9.85 8.54 7.19 6.73 4.93 9.84 2.30 4.36 5.56 7.35 6.66 8.16 3.93 Polyamide nylon (cylindrical) 0.06 0.08 0.54 0.90 0.38 0.69 0.19 Polycarbonate polymer (cylindrical) 9.14 5.77 4.54 2.69 4.00 2.80 6.19 Olivine 3.62 1.06 0.17 0.35 ** ** ** Aluminum oxide 6.51 5.07 4.31 3.76 ** ** ** JSC1A (150425 m) 44.46 41.70 40.30 39.90 ** ** ** Polypropylene 37.01 34.27 31.14 30.74 ** ** ** 304 Stainless steel cut wires (cylindrical) 1.60 2.48 4.09 4.69 3.72 4.54 2.23 ** value was not determined, the predi cted discharge rates were imaginary 92 PAGE 93 Table A7. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (da/dsi) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (da/dsi) and d as dsi da dp dF dv ds dsv 23.16 5.08 14.51 29.42 11.29 5.28 21.66 21.52 22.30 40.52 63.86 1.44 22.93 32.88 Rice 50.97 20.70 50.54 81.63 15.60 20.31 61.91 Snowwhite play sand 2.48 0.04 1.36 2.07 ** ** 3.85 Silicaquartz sand 20.10 19.13 18.39 18.12 ** ** ** Beach sand (212300 m) 21.27 20.75 20.15 19.94 ** ** ** Borosilicate crushed glass 21.35 18.30 16.07 14.52 ** ** 23.00 Polyamide nylon (cubical) 12.81 11.06 9.87 8.63 8.21 6.57 11.05 3.10 3.18 4.31 6.00 5.35 6.77 2.78 Polyamide nylon (cylindrical) 0.58 0.59 0.05 0.26 0.19 0.08 0.69 Polycarbonate polymer (cylindrical) 10.09 6.98 5.84 4.13 5.34 4.22 7.36 Olivine 4.07 1.82 1.03 0.57 ** ** ** Aluminum oxide 6.97 5.71 5.05 4.57 ** ** ** JSC1A (150425 m) 45.16 42.86 41.70 41.43 ** ** ** Polypropylene 39.05 36.84 34.32 34.00 ** ** ** 304 Stainless steel cut wires (cylindrical) 0.30 1.07 2.47 2.99 2.15 2.87 0.85 ** value was not determined 93 PAGE 94 Table A8. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (dF/dsi) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dF/dsi) and d as dsi da dp dF dv ds dsv 7.58 29.32 41.04 58.76 8.22 16.07 5.56 3.29 63.96 84.34 32.61 64.73 14.16 Rice 14.88 81.74 37.16 81.36 32.44 Snowwhite play sand 0.95 2.11 3.71 4.58 ** ** 2.63 Silicaquartz sand 19.38 18.14 17.19 16.84 ** ** ** Beach sand (212300 m) 20.61 19.93 19.16 18.89 ** ** ** Borosilicate crushed glass 18.66 14.52 11.49 9.39 ** ** 20.91 Polyamide nylon (cubical) 10.73 8.64 7.23 5.75 5.25 3.30 8.63 1.11 6.11 7.42 9.36 8.61 10.24 5.65 Polyamide nylon (cylindrical) 0.25 0.23 0.90 1.28 0.72 1.06 0.10 Polycarbonate polymer (cylindrical) 7.85 4.14 2.78 0.75 2.19 0.87 4.59 Olivine 3.34 0.60 0.36 0.91 ** ** ** Aluminum oxide 6.16 4.57 3.74 3.13 ** ** ** JSC1A (150425 m) 44.31 41.44 39.98 39.66 ** ** ** Polypropylene 36.76 33.95 30.75 30.35 ** ** ** 304 Stainless steel cut wires (cylindrical) 2.06 2.97 4.65 5.28 4.28 5.13 2.71 ** value was not determined, the predi cted discharge rates were imaginary 94 PAGE 95 Table A9. Comparison of the % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (dp/dsi)2and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dp/dsi)2 and d as dsi da dp dF dv ds dsv 24.76 73.34 85.83 99.18 46.88 57.21 27.67 60.71 95.56 32.63 Rice 81.24 92.96 * 54.75 Snowwhite play sand 2.89 7.30 9.60 10.83 ** ** 0.43 Silicaquartz sand 17.51 15.57 14.08 13.55 ** ** ** Beach sand (212300 m) 19.41 18.47 17.38 17.01 ** ** ** Borosilicate crushed glass 13.41 7.18 2.66 0.46 ** ** 16.83 Polyamide nylon (cubical) 8.50 6.05 4.39 2.67 2.09 0.20 6.04 5.59 15.92 17.75 20.49 19.44 21.72 15.26 Polyamide nylon (cylindrical) 0.55 0.54 1.25 1.66 1.06 1.42 0.40 Polycarbonate polymer (cylindrical) 4.60 0.05 1.61 4.09 2.33 3.95 0.61 Olivine 0.15 4.67 6.33 7.30 ** ** ** Aluminum oxide 4.30 1.93 0.70 0.20 ** ** ** JSC1A (150425 m) 40.30 34.77 31.99 31.36 ** ** ** Polypropylene 31.17 26.93 22.11 21.49 ** ** ** 304 Stainless steel cut wires (cylindrical) 4.09 5.19 7.19 7.94 6.74 7.76 4.88 ** value was not determined, the predi cted discharge rates were imaginary 95 PAGE 96 Table A10. Comparison of the % differenc e between measured and predicted rates by equation 110 for nonsphericals with k = (da/dsi)2and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (da/dsi)2 and d as dsi da dp dF dv ds dsv 4.87 47.43 60.21 78.42 23.43 32.48 7.26 22.35 88.64 56.77 89.30 0.69 Rice 20.92 81.03 2.10 Snowwhite play sand 0.07 3.50 5.29 6.26 ** ** 1.80 Silicaquartz sand 19.14 17.81 16.80 16.43 ** ** ** Beach sand (212300 m) 20.75 20.11 19.38 19.12 ** ** ** Borosilicate crushed glass 18.31 14.03 10.90 8.73 ** ** 20.65 Polyamide nylon (cubical) 10.99 8.94 7.55 6.11 5.62 3.70 8.93 3.19 12.44 14.09 16.55 15.60 17.66 11.85 Polyamide nylon (cylindrical) 0.58 0.59 0.05 0.26 0.19 0.08 0.69 Polycarbonate polymer (cylindrical) 6.99 3.06 1.62 0.53 0.99 0.41 3.54 Olivine 1.84 1.89 3.18 3.93 ** ** ** Aluminum oxide 5.71 3.93 3.00 2.32 ** ** ** JSC1A (150425 m) 42.87 39.04 37.10 36.66 ** ** ** Polypropylene 36.82 34.03 30.85 30.44 ** ** ** 304 Stainless steel cut wires (cylindrical) 1.07 1.90 3.42 3.99 3.08 3.85 1.66 ** value was not determined, the predi cted discharge rates were imaginary 96 PAGE 97 Table A11.Comparison of t he % difference between measur ed and predicted rates by equation 110 for nonsphericals with k = (dF/dsi)2 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with k = (dF/dsi)2 and d as dsi da dp dF dv ds dsv 58.67 82.93 92.20 62.11 * 79.08 Rice * Snowwhite play sand 4.57 9.56 12.15 13.55 ** ** 1.83 Silicaquartz sand 16.85 14.66 12.99 12.38 ** ** ** Beach sand (212300 m) 18.90 17.84 16.62 16.20 ** ** ** Borosilicate crushed glass 9.41 1.64 3.97 7.81 ** ** 13.71 Polyamide nylon (cubical) 5.70 2.81 0.85 1.19 1.87 4.56 2.79 9.31 21.29 23.41 26.54 25.34 27.96 20.54 Polyamide nylon (cylindrical) 1.31 1.28 2.11 2.59 1.89 2.31 1.13 Polycarbonate polymer (cylindrical) 0.73 4.80 6.82 9.81 7.69 9.64 4.13 Olivine 0.92 6.43 8.32 9.42 ** ** ** Aluminum oxide 3.14 0.30 1.19 2.27 ** ** ** JSC1A (150425 m) 39.66 33.71 30.71 30.04 ** ** ** Polypropylene 30.37 25.92 20.87 20.22 ** ** ** 304 Stainless steel cut wires (cylindrical) 5.29 6.49 8.68 9.50 8.19 9.30 6.15 ** value was not determined, the predi cted discharge rates were imaginary 97 PAGE 98 Table A12. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with =1.0 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with =1.0 and d as dsi da dp dF dv ds dsv 20.20 7.87 4.94 0.97 14.13 11.61 19.33 24.09 6.96 2.37 2.68 14.62 6.79 32.06 Rice 59.44 33.09 26.28 19.69 43.59 33.19 66.91 Snowwhite play sand 0.63 6.30 8.60 9.71 ** ** 3.61 Silicaquartz sand 37.62 29.56 25.00 23.58 ** ** ** Beach sand (212300 m) 38.32 33.12 28.39 26.97 ** ** ** Borosilicate crushed glass 20.21 12.55 8.45 6.08 ** ** 26.18 Polyamide nylon (cubical) 2.90 0.09 1.85 3.50 4.01 5.91 0.10 3.31 10.57 11.58 12.97 12.45 13.55 10.19 Polyamide nylon (cylindrical) 0.45 0.53 2.16 3.53 1.49 2.74 1.09 Polycarbonate polymer (cylindrical) 1.12 3.42 4.81 6.68 5.38 6.58 2.93 Olivine 15.48 4.92 2.38 1.07 ** ** ** Aluminum oxide 17.88 10.39 7.45 5.58 ** ** ** JSC1A (150425 m) 69.76 54.09 48.79 47.75 ** ** ** Polypropylene 34.95 29.06 23.89 23.31 ** ** ** 304 Stainless steel cut wires (cylindrical) 9.62 11.08 13.41 14.20 12.92 14.01 10.67 ** value was not determined 98 PAGE 99 Table A13. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (dp/da)2 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dp/da)2 and d as dsi da dp dF dv ds dsv 13.56 1.90 0.86 4.62 7.82 5.44 12.73 13.62 2.06 6.26 10.89 4.96 2.22 20.92 Rice 42.72 19.14 13.04 7.13 28.54 19.22 49.41 Snowwhite play sand 5.47 10.86 13.05 14.11 ** ** 1.44 Silicaquartz sand 28.15 20.65 16.40 15.08 ** ** ** Beach sand (212300 m) 28.64 23.81 19.41 18.09 ** ** ** Borosilicate crushed Glass 11.66 4.55 0.74 1.47 ** ** 17.21 Polyamide nylon (cubical) 0.67 3.55 5.25 6.84 7.34 9.17 3.57 5.47 12.57 13.55 14.91 14.40 15.49 12.20 Polyamide nylon (cylindrical) 4.84 4.77 7.32 8.62 6.68 7.87 4.23 Polycarbonate polymer (cylindrical) 1.77 6.19 7.54 9.35 8.09 9.26 5.71 Olivine 10.01 0.05 2.47 3.72 ** ** ** Aluminum oxide 11.67 4.57 1.79 0.01 ** ** ** JSC1A (150425 m) 58.29 43.67 38.73 37.76 ** ** ** Polypropylene 24.29 18.87 14.10 13.57 ** ** ** 304 Stainless steel cut wires (cylindrical) 14.26 15.65 17.86 18.61 17.39 18.43 15.26 ** value was not determined 99 PAGE 100 Table A14. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (dF/da)2 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dF/da)2 and d as dsi da dp dF dv ds dsv 5.44 5.38 7.94 11.43 0.12 2.09 4.68 2.70 11.47 15.27 19.45 5.13 11.61 9.30 Rice 27.84 6.72 1.25 4.03 15.14 6.79 33.83 Snowwhite play sand 7.69 12.96 15.10 16.13 ** ** 3.76 Silicaquartz sand 25.16 17.83 13.69 12.39 ** ** ** Beach sand (212300 m) 25.80 21.07 16.77 15.48 ** ** ** Borosilicate crushed glass 6.75 0.05 3.69 5.80 ** ** 12.05 Polyamide nylon (cubical) 3.94 6.73 8.37 9.91 10.39 12.16 6.74 8.40 15.28 16.24 17.55 17.06 18.11 14.93 Polyamide nylon (cylindrical) 7.48 7.41 9.89 11.15 9.27 10.42 6.89 Polycarbonate polymer (cylindrical) 5.60 9.84 11.13 12.88 11.66 12.79 9.38 Olivine 7.13 2.66 5.02 6.23 ** ** ** Aluminum oxide 7.82 0.97 1.72 3.44 ** ** ** JSC1A (150425 m) 55.99 41.59 36.72 35.76 ** ** ** Polypropylene 23.15 17.77 13.05 12.53 ** ** ** 304 Stainless steel cut wires (cylindrical) 15.84 17.19 19.37 20.10 18.91 19.93 16.82 ** value was not determined 100 PAGE 101 Table A15. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (dp/da) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dp/da) and d as dsi da dp dF dv ds dsv 16.95 4.95 2.10 1.77 11.04 8.59 16.10 18.72 2.34 2.06 6.89 9.67 2.17 26.35 Rice 50.83 25.91 19.46 13.22 35.84 26.00 57.90 Snowwhite play sand 3.01 8.54 10.79 11.87 ** ** 1.13 Silicaquartz sand 32.85 25.07 20.67 19.29 ** ** ** Beach sand (212300 m) 33.32 28.31 23.76 22.38 ** ** ** Borosilicate crushed glass 15.87 8.49 4.54 2.24 ** ** 21.62 Polyamide nylon (cubical) 1.10 1.84 3.56 5.18 5.69 7.55 1.85 4.35 11.53 12.52 13.90 13.38 14.48 11.16 Polyamide nylon (cylindrical) 2.27 2.20 4.81 6.14 4.16 5.38 1.64 Polycarbonate polymer (cylindrical) 0.31 4.79 6.16 8.00 6.72 7.90 4.30 Olivine 12.71 2.41 0.07 1.35 ** ** ** Aluminum oxide 14.69 7.40 4.54 2.24 ** ** ** JSC1A (150425 m) 63.88 48.75 43.64 42.63 ** ** ** Polypropylene 29.49 23.84 18.88 18.32 ** ** ** 304 Stainless steel cut wires (cylindrical) 11.93 13.35 15.62 16.39 15.14 16.21 12.95 ** value was not determined 101 PAGE 102 Table A16. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (dF/da) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dF/da) and d as dsi da dp dF dv ds dsv 12.51 0.96 1.78 5.50 6.82 4.46 11.69 12.86 2.72 6.89 11.49 4.25 2.88 20.11 Rice 42.72 19.14 13.04 7.13 28.54 19.22 49.41 Snowwhite play sand 4.22 9.68 11.90 12.97 ** ** 0.13 Silicaquartz sand 31.29 23.60 19.25 17.89 ** ** ** Beach sand (212300 m) 31.96 27.00 22.49 21.13 ** ** ** Borosilicate crushed glass 13.26 6.04 2.18 0.06 ** ** 18.88 Polyamide nylon (cubical) 0.67 3.55 5.25 6.84 7.34 9.17 3.57 5.93 12.99 13.97 15.32 14.82 15.90 12.63 Polyamide nylon (cylindrical) 3.61 3.54 6.12 7.43 5.47 6.67 2.99 Polycarbonate polymer (cylindrical) 2.23 6.62 7.97 9.78 8.52 9.68 6.15 Olivine 11.31 1.13 1.32 2.58 ** ** ** Aluminum oxide 12.78 5.62 2.80 1.01 ** ** ** JSC1A (150425 m) 62.66 47.64 42.56 41.56 ** ** ** Polypropylene 28.92 23.30 18.36 17.81 ** ** ** 304 Stainless steel cut wires (cylindrical) 12.75 14.16 16.41 17.17 15.94 16.99 13.77 ** value was not determined 102 PAGE 103 Table A17. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (dp/dsi) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dp/dsi) and d as dsi da dp dF dv ds dsv 5.16 5.63 8.19 11.67 0.15 2.36 4.40 2.36 11.76 15.55 19.72 5.44 11.91 8.94 Rice 25.42 4.69 0.67 5. 86 12.95 4.77 31.29 Snowwhite play sand 8.58 13.80 15.92 16.94 ** ** 4.69 Silicaquartz sand 25.03 17.70 13.56 12.27 ** ** ** Beach sand (212300 m) 28.32 23.50 19.12 17.80 ** ** ** Borosilicate crushed Glass 8.51 1.59 2.11 4.26 ** ** 13.89 Polyamide nylon (cubical) 1.84 4.69 6.37 7.94 8.43 10.24 4.70 11.56 18.20 19.13 20.40 19.92 20.94 17.86 Polyamide nylon (cylindrical) 2.11 2.04 4.66 5.99 4.00 5.22 1.49 Polycarbonate polymer (cylindrical) 4.85 9.13 10.44 12.20 10.97 12.10 8.67 Olivine 2.45 6.91 9.17 10.33 ** ** ** Aluminum oxide 7.45 0.63 2.06 3.77 ** ** ** JSC1A (150425 m) 48.79 35.05 30.41 29.49 ** ** ** Polypropylene 23.85 18.45 13.70 13.17 ** ** ** 304 Stainless steel cut wires (cylindrical) 13.40 14.80 17.04 17.79 16.56 17.61 14.41 ** value was not determined 103 PAGE 104 Table A18. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (da/dsi) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 110 with = (da/dsi) and d as dsi da dp dF dv ds dsv 8.27 2.84 5.48 9.06 2.80 0.53 7.48 6.98 7.79 11.74 16.10 1.18 7.94 13.85 Rice 32.59 10.68 5.01 0.47 19.41 10.76 38.80 Snowwhite play sand 6.25 11.60 13.77 14.82 ** ** 2.25 Silicaquartz sand 29.49 21.90 17.62 16.27 ** ** ** Beach sand (212300 m) 33.12 28.12 23.57 22.20 ** ** ** Borosilicate crushed Glass 12.52 5.35 1.51 0.72 ** ** 18.10 Polyamide nylon (cubical) 0.05 2.95 4.66 6.26 6.76 8.60 2.96 10.52 17.23 18.17 19.45 18.97 20.00 16.89 Polyamide nylon (cylindrical) 0.45 0.53 2.16 3.53 1.49 2.74 1.09 Polycarbonate polymer (cylindrical) 3.39 7.73 9.06 10.85 9.60 10.75 7.26 Olivine 4.91 4.68 6.99 8.18 ** ** ** Aluminum oxide 10.33 3.32 0.57 1.18 ** ** ** JSC1A (150425 m) 54.06 39.84 35.03 34.08 ** ** ** Polypropylene 29.11 23.48 18.53 17.98 ** ** ** 304 Stainless steel cut wires (cylindrical) 11.05 12.48 14.78 15.55 14.30 15.37 12.09 ** value was not determined 104 PAGE 105 Table A19. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (dF/dsi) and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dF/dsi) and d as dsi da dp dF dv ds dsv 1.31 9.09 11.55 14.90 3.81 5.93 0.58 2.68 16.12 19.72 23.68 10.11 16.25 3.57 Rice 18.74 0.88 5.96 10.87 6.94 0.81 24.30 Snowwhite play sand 9.72 14.87 16.96 17.97 ** ** 5.87 Silicaquartz sand 23.57 16.33 12.24 10.96 ** ** ** Beach sand (212300 m) 26.95 22.18 17.84 16.53 ** ** ** Borosilicate crushed glass 6.04 0.71 4.33 6.43 ** ** 11.31 Polyamide nylon (cubical) 3.44 6.24 7.89 9.44 9.93 11.70 6.26 13.00 19.53 20.44 21.69 21.22 22.22 19.20 Polyamide nylon (cylindrical) 3.46 3.40 5.98 7.30 5.33 6.54 2.85 Polycarbonate polymer (cylindrical) 6.64 10.84 12.12 13.85 12.64 13.75 10.38 Olivine 1.12 8.13 10.35 11.50 ** ** ** Aluminum oxide 5.52 1.18 3.82 5.50 ** ** ** JSC1A (150425 m) 47.73 34.09 29.48 28.57 ** ** ** Polypropylene 23.29 17.90 13.18 12.65 ** ** ** 304 Stainless steel cut wires (cylindrical) 14.14 15.53 17.75 18.49 17.28 18.32 15.14 ** value was not determined 105 PAGE 106 Table A20. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (dp/dsi)2and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dp/dsi)2 and d as dsi da dp dF dv ds dsv 8.38 17.78 20.01 23.04 13.01 14.93 9.05 15.54 27.19 30.32 33.76 21.98 27.31 10.11 Rice 0.01 16.51 20.79 24.92 9.93 16.45 4.70 Snowwhite play sand 15.93 20.73 22.68 23.62 ** ** 12.35 Silicaquartz sand 13.53 6.88 3.12 1.94 ** ** ** Beach sand (212300 m) 19.15 14.67 10.60 9.37 ** ** ** Borosilicate crushed glass 2.14 8.37 11.71 13.65 ** ** 2.72 Polyamide nylon (cubical) 6.41 9.13 10.73 12.23 12.70 14.42 9.15 19.15 25.22 26.06 27.22 26.79 27.71 24.91 Polyamide nylon (cylindrical) 4.71 4.64 7.19 8.49 6.55 7.74 4.10 Polycarbonate polymer (cylindrical) 10.37 14.40 15.63 17.29 16.14 17.20 13.96 Olivine 9.21 17.51 19.51 20.54 ** ** ** Aluminum oxide 2.11 8.32 10.77 12.32 ** ** ** JSC1A (150425 m) 30.41 18.37 14.30 13.50 ** ** ** Polypropylene 13.74 8.77 4.42 3.93 ** ** ** 304 Stainless steel cut wires (cylindrical) 17.05 18.39 20.53 21.25 20.08 21.08 18.02 ** value was not determined 106 PAGE 107 Table A21. Comparison of the % differenc e between measured and predicted rates by equation 14 for nonsphericals with = (da/dsi)2 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (da/dsi)2 and d as dsi da dp dF dv ds dsv 3.18 13.12 15.47 18.68 8.07 10.10 3.88 7.80 20.52 23.93 27.69 14.83 20.65 1.87 Rice 11.09 7.26 12.01 16.61 0.05 7.20 16.30 Snowwhite play sand 11. 67 16.71 18.75 19.74 ** ** 7.90 Silicaquartz sand 21.97 14.83 10.79 9.53 ** ** ** Beach sand (212300 m) 28.17 23.35 18.97 17.65 ** ** ** Borosilicate crushed glass 5.36 1.35 4.97 7.03 ** ** 10.59 Polyamide nylon (cubical) 3.05 5.87 7.53 9.08 9.57 11.35 5.88 17.25 23.46 24.32 25.51 25.07 26.02 23.14 Polyamide nylon (cylindrical) 0.45 0.53 2.16 3.53 1.49 2.74 1.09 Polycarbonate polymer (cylindrical) 7.72 11.87 13.14 14.84 13.65 14.75 11.42 Olivine 4.61 13.33 15.43 16.51 ** ** ** Aluminum oxide 3.32 3.25 5.83 7.47 ** ** ** JSC1A (150425 m) 38.96 26.95 22.58 21.72 ** ** ** Polypropylene 23.43 18.04 13.31 12.78 ** ** ** 304 Stainless steel cut wires (cylindrical) 12.48 13.89 16.15 16.92 15.68 16.73 13.50 ** value was not determined 107 PAGE 108 Table A22.Comparison of t he % difference between measur ed and predicted rates by equation 14 for nonsphericals with = (dF/dsi)2 and d as dsi, da, etc. Material Difference (%) betwe en measured and predicted rates by equation 14 with = (dF/dsi)2and d as dsi da dp dF dv ds dsv 15.20 23.90 25.96 28.77 19.48 21.26 15.81 23.67 34.21 37.03 40.14 29.49 34.31 18.77 Rice 10.16 25.00 28.84 32.56 19.09 24.95 5.95 Snowwhite play sand 17.97 22.65 24.55 25.47 ** ** 14.47 Silicaquartz sand 10.98 4.48 0.80 0.35 ** ** ** Beach sand (212300 m) 16.58 12.20 8.22 7.02 ** ** ** Borosilicate crushed glass 6.41 12.37 15.57 17.42 ** ** 1.77 Polyamide nylon (cubical) 9.48 12.11 13.66 15.11 15.57 17.23 12.13 21.66 27.54 28.36 29.49 29.07 29.96 27.24 Polyamide nylon (cylindrical) 7.37 7.30 9. 78 11.04 9.16 10.32 6.78 Polycarbonate polymer (cylindrical) 13.86 17.74 18.92 20.51 19.40 20.43 17.32 Olivine 11.53 19.61 21.56 22.56 ** ** ** Aluminum oxide 5.48 11.48 13.84 15.35 ** ** ** JSC1A (150425 m) 28.58 16.71 12.70 11.91 ** ** ** Polypropylene 12.69 7.77 3.45 2.97 ** ** ** 304 Stainless steel cut wires (cylindrical) 18.51 19.82 21.93 22.64 21.48 22.47 19.46 ** value was not determined 108 PAGE 109 LIST OF REFERENCES Allen, T., 1981. 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PAGE 112 BIOGRAPHICAL SKETCH Kuldeep Mamtani completed his undergraduat e studies in chemical engineering from Laxminarayan Institute of Technology, Nagpur, India in 2008. He then joined Unilever Research India, Bangalore as a Research Associate. During his stay at Unilever, he was involved in projects aimed at improving the quality a ttributes of tea by altering the conventional tea processing. His research work at Unilever lead to the development of two patented tea manufacturing processe s (Patent numbers: F2075/V and G2043/V). He then enrolled at the University of Florida, Gainesville for the Master of Science program in chem ical engineering in August 2010. He received his M.S. degree in December 2011. 112 