.H'. «A This is to certify that the dissertation entitled Accelerated Settling of a Sphere in a Herschel- Bulkley Fluid due to Vibration. presented by S. Paul Singh has been accepted towards fulfillment of the requirements for Ph.D. ' Agricultural Engineering degree in Date 56 a}; / [7(10 7 Ilflliu 1-- Afi—M ‘ ‘ " ' A 1 " ',' ‘ 0-12771 MSU LIBRARIES RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ——-’-r"' _ ___ ACCELERATED SETTLING OF A SPHERE IN A HERSCHEL-BULKLEY FLUID DUE TO VIBRATION BY Sherinder Paul Singh A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1987 Approved By: MIMI M flMajor Professor QIQZéfIZ/ézgt ”(Z/[C(W Department Chairperson 61:3 (7 / Date 3/? M7 Date Ti 7 Copyright by Sherinder Paul Singh 1987 ABSTRACT ACCELERATED SETTLING OF A SPHERE IN A HERSCHEL-BULKLEY FLUID DUE TO VIBRATION BY Sherinder Paul Singh This study investigates the effect of vibration on the settling of a Spherical particle in a Herschel-Bulkley (H-B) fluid. Most fluid sys- tems like soups, sauces, jams and preserves contain food particles that are suspended in a fluid medium. These particles could settle under the influence of vibration forces encountered in shipping due to the shear thinning characteristics of the fluid. A mathematical model was devel- oped to determine the motion of a spherical particle in a vibrating H-B fluids Dimensionless quantities were obtained from the governing dif- ferential equation. Kelsat solutions of varying concentrations were used to model H-B fluids and steel spheres of different diameters simu- lated the suspended particles. The effect of both sinusoidal and random vibration was considered. An equation was developed using dimensionless terms to predict the settling of a particle in an oscillating H-B fluid. A high correlation (R2 - 0.93) was obtained between experimental data and predicted values of settling time. A 'separation criterion' was established based on particle size, yield shear stress and vibration input. Three different modes of transportation (truck, rail and air- craft) were simulated. Example food systems such as spaghetti sauce and salad dressings were presented. Dedicated to my parents, whose love and guidance has inspired me beyond words iii ACKNOWLEDGEMENTS I wish to express my sincere gratitude to Gary J. Burgess, School of Packaging, whose guidance and friendship throughout my graduate program will be revered for years to come. Thanks to members of my committee, Dr. A. K. Srivastava, Chairperson, Dr. J. F. Steffe, Dr. R. Morgan, Agricultural Engineering; Dr. G. E. Mase, Metallugry, Mechanics and Material Science; Dr. G. Burgess, Dr. R. Brandenburg, School of Packaging. I also extend my appreciation to Dennis E. Young, School of Packaging, and Mohammad Usmaan, Mechanical Engineering, for assistance in data analysis. I appreciate the support of Dr. C. J. Mackson, Direc- tor Emeritus, School of Packaging, and Frank Bresk, Chairman, Lansmont Corporation, for the use of lab facilities in East Lansing, Michigan, and Monterey, California. Sincere thanks to Mary Corp for typing this manuscript. In the end, thanks to my family and friends who have been with me all along, for which I am deeply indebted to. iv TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE 1.0 INTRODUCTION 2.0 OBJECTIVES 3.0 LITERATURE REVIEW n-Newtonian Fluids 1.1 The Power-Law Model 1.2 The Bingham Plastic Model 1.3 The Herschel—Bulkley Model 1.4 The Ellis Model 3.1.5 The Powell-Eyring Model 3.2 Velocity Profile of Fluid Flow Over A Sphere 3.3 Rheological Models of Fluid Food Products 4.0 MODEL DEVELOPMENT 1 Provisions of the Model 2 Assumptions of the Model .3 Mathematical Model of System .4 Dimensional Analysis .5 Dimensionless Governing Differential Equations .6 Effect of Increased Number of Particles on the Settling Rate 7 Effect of Random Vibration on Settling of Particles 8 Effect of Non-Spherical Particles 5.0 EXPERIMENTAL DESIGN 6.0 RESULTS AND DISCUSSION Rheological Properties of the Herschel-Bulkley Fluid 1 2 Prediction Equations for Settling of Sphere 6.2.1. Factor Analysis 6. 6. V Page vii ix xi U1 Ht—n NOOOCDNNU‘ y—n U‘l 15 l7 17 26 29 32 35 37 39 46 46 47 55 6 3 Cross Validation of Prediction Equations 6.4 Absolute Separation Criterion 6.5 Effect of Random Vibration (Shipping Environments) 6.6 Sources of Error - Experimental and Theoretical 6 7 Condition of Settling of Particle During Shipping — Example Problem 6.8 Condition of No-Settling of Particle During Shipping - Example Problem 7.0 CONCLUSIONS 8.0 FUTURE RESEARCH BIBLIOGRAPHY APPENDIX A. REHEOLOGICAL DATA FOR KELSAT SOLUTIONS APPENDIX B. SETTLING TIME DATA APPENDIX C. VALUES OF DIMENSIONLESS CONSTANTS EVALUATED FROM EXPERIMENTAL DATA APPENDIX D. VALUES OF DRAG CORRECTION FACTOR, €(n) (Dahzi, 1985) vi 64 66 66 85 86 86 88 89 90 93 98 114 116 Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8 Table 9 Table 10 Table 11 LIST OF TABLES Rheological Parameters of Various Fluid Food Products Rheological Properties of Aqueous Kelsat Solutions After 24 hours of Storage at 24°C Using the Herschel-Bulkley Model Experimental Design Table for Determining Settling Time of Sphere in Vibrating Herschel-Bulkley Model Settling Time Data for Steel Spheres in a Vibrating 1.25% Aqueous Solution of Kelsat After 24 Hours of Storage at 24°C Settling Time Data for Steel Spheres in a Vibrating 1.50% Aqueous Solution of Kelsat After 24 Hours of Storage at 24°C Settling Time Data for Steel Spheres in a Vibrating 1.75% Aqueous Solution of Kelsat After 24 Hours of Storage at 24°C Settling Time Data for Steel Spheres in a Vibrating 1.75% Aqueous Solution of Kelsat After 24 Hours of Storage at 24°C Settling Time Data for Steel Spheres in a Vibrating 1.25% Aqueous Solution of Kelsat After 24 Hours of Storage at 24°C and Subjected to a Vibration Dwell for 1 Hour at 20 Hz and 1.0 g's Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model Statistical Analysis of Settling Time Data to Determine Prediction Equation Regression Coefficients Determined for Settling Time Prediction Equation vii Page 13 43 45 48 49 50 51 52 53 59 62 Table Table Table Table Table 12 l3 14 15 16 Factor Analysis of Data of Spherical Particle Settling in a Vibrating Herschel-bulkley Fluid Absolute Separation Criterion of Spherical Particle Suspended in a Herschel-Bulkley Fluid Due to Sinusoidal Vibration Upper and Lower Bounds of Yield Number that Predict Settling of a Particle Due to a 30 minute Random Vibration Dwell for Different Shipping Environments Simulated on an Electra-Hydraulic Vibration Table Absolute Separation Criterion of Spherical Particle Suspended in a Herschel-Bulkley Fluid Due to Random Vibration Using a Combined Truck-Air PSD Spectrum Statistical Analysis of Cross Validation Data Using Prediction Equation for Settling Time viii 63 67 69 7O 84 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure l. 2. 7. 8. 9. 10. ll. 12. 13. 14. LIST OF FIGURES Displacement Conventions for the Particle in the Fluid Column. Free Body Diagram of the Spherical Particle Spherical Coordinate System Used in the Drag Force Calculation Force versus Distance Diagram of a H-B Fluid Using the Back Extrusion Method (Osorio, 1985). Experimental Setup of Vibration Table with Fluid Column and Suspended Sphere VARIMAX Rotated Factor Analysis with Kaiser Normalization EQUIMAX Rotated Factor Analysis with Kaiser Normalization Predicted Versus Observed Values of Dimensionless Time Absolute Separation Curve for Spherical Particle Suspended in an H-B Fluid Due to Sinusoidal Vibration Power Spectral Density Curve for Simulation of Truck Transport (Lansmont Corporation) Power Spectral Density Curve for Simulation of Rail Transport (Lansmont Corporation) Power Spectral Density Curve for Simulation of Combined Truck-Air Transport (Lansmont Corporation) Acceleration Versus Time Plots for Random Vibration Truck Transport Simulated on a Vibration Table Acceleration Versus Time Plots for Random Vibration Rail Transport Simulated on a Vibration Table ix Page 16 19 21 41 42 57 58 65 68 72 73 74 75 76 Figure Figure Figure Figure Figure Figure Figure I6. l7. 18. 19. 20. 21. Acceleration Versus Time Plots for Random Vibration of Combined Truck-Air Transport Simulated on a Vibration Table Absolute Separation Curve for Spherical Particle Suspended in an H-B Fluid Due to Random Vibration Response Curve of Dimensionless Acceleration Versus Dimensionless Time Response Curve of Dimensionless Yield Number Versus Dimensionless Time Response Curve of Dimensionless Buoyancy Number Versus Dimensionless Time Response Curve of Dimensionless Drag Coefficient Versus Dimensionless Time Response Curve of Dimensionless Height (Alpha) Versus Dimensionless Time 79 78 79 80 81 82 83 NOMENCLATURE a = longest intercept, m b = longest intercept normal to a, m c = longest intercept normal to a and b, m f - frequency of vibration table, Hz (8.1) g = acceleration due to gravity, m s-2 h - height of fluid column, m m - mass of sphere, kg mf a displaced fluid mass, kg n = flow behavior index, dimensionless p = pressure v a volume of sphere V(t) - velocity of sphere at instant t secs, ms-l z(t) = position of sphere at instant t secs, m A s zero-pk displacement amplitude of vibration table, m B - radius of fluid column, m cl - constant in Equation (3-4), Pa- 3 2 - constant in Equation (3-4), Pa 8 C3 - constant in Equation (3-5), Pa 3 4 - constant in Equation (3-5), Pa- c5 - constant in Equation (3-5). 8 xi dA At AS' drag force on sphere, Pa m2 (N) drag force on sphere in power law fluid, Pa m2 (N) force in vertical direction (z-axis), Pa m2 (N) consistency coefficient, Pa sn plastic viscosity, Pa 8 velocity correction for additional particle, dimensionless distance of sphere surface from side wall, m distance between two spheres in one layer, m number of spheres in one layer, dimensionless radius of sphere, m sphericity, dimensionless structural parameter, dimensionless settling time, s velocity of sphere in flow field, ms"l distance travelled by particle in fluid, m surface area of element, 1112 small incremental time, 8 small incremental distance, m shear rate, 3-1 fluid mass density, kg mm3 sphere mass density, kg m53 fluid weight density, kg m-2 3-2 sphere weight density, kg m-2 3'2 shear stress, Pa xii s(n) 1D yield shear stress, Pa critical yield shear stress, Pa angle between element and x-axis, radians angle between element and z-axis, radians small time increment, s 1 radial frequency of vibration table, rad s- drag correction factor, dimensionless 2 5L2-= acceleration, dimensionless «2th0 m a yield number, dimensionless 8 ms 57'. buoyancy number, dimensionless f 6'" K 62:2) R C%§)n_l = drag correction number, dimensionless m m S %= height, dimensionless wT - settling time, dimensionless xiii 1.0. INTRODUCTION The effect of external vibration on a falling sphere in a shear thinning fluid with yield shear stress is a problem of great practical importance. Most fluid food systems like soups, sauces, jams, etc., are not homogeneous fluids but consist of particles suspended in a fluid system. The presence of external vibration has been measured and studied in different modes of transportation used to ship these food products (Ostrem, 1979). The decrease in viscosity with increasing shear rate is referred to as "shear thinning" and fluids that show such behavior are termed, by Bird (1977), as "pseudoplastic". It has been experimentally observed that the viscosity of such fluids appears to be much lower in high shear rate flows when compared with the idealized Newtonian fluids. This effect can be very significant in certain condi- tions where the viscosity may end up decreasing by a factor as high as "103". The fact that most food systems such as soups, jams, and pre- serves do contain food particles in suspension under normal storage conditions leads one to believe that in the presence of external vibration they could be forced to separate due to the "shear thinning" characteristics of these fluids (Bird, 1977). This effect becomes even more critical when companies ship bulk tanks filled with these products to regional filling plants. Any particles settled in transit have to be remixed in the whole system before they can be filled in retail customer containers to achieve uniform product distribution and meet label claims. Non-Newtonian fluids play a very important role in areas where application depends on the specific nature of their viscous behavior. Different media where these Non-Newtonian fluids are encountered include biological fluids, food products, rubber, plastic, paper pulp, paint, petroleum products, printing and heavy chemicals (Ferry, 1980). It is important to understand the rheological properties of these fluids to investigate their mechanics and heat transfer characteristics. In general, fluids discussed in the classical theories of fluid dynamics involve the ideal or perfect fluid and the Newtonian fluid. The former is completely nonviscous, resulting in no internal shear stresses, whereas the latter exhibits a linear relationship between the shear stress and the shear rate. Real life fluids such as foods, high polymer solutions, suspensions of solids in liquids, and emulsions have a non-linear relationship between the shear stress and the shear rate. Since the 1950's, the study of these real fluids has become very impor- tant because of the limitations of Newtonian theories in real situa- tions. Also, the presence of suspended particles in these type of fluids will closely simulate the ideal fluid food systems. The objective of the research was to study settling of particles in a Herschel Bulkley (H-B) fluid medium subjected to vibrational input with the purpose of developing a "criterion for separation" in the ship- ping environment. 2.0. OBJECTIVES following steps. I. 4. 5. Develop analytical expressions to describe the drag force on a spherical particle suspended in a H-B fluid. Apply the Herschel—Bulkley drag force to the analysis of the problem of a single spherical particle moving in l-Dimension in a vibrating fluid using Newton's laws of motion. Determine dimensionless parameters that affect the settling of a sphere in a vibrating H-B fluid. Apply statistical techniques using experimental data to establish an equation to predict the settling of a sphere in a vibrating H-B fluid. Experimentally ‘validate this equation. developed in Step 4. The objective was accomplished by completing the 6. 7. Determine the effect of random vibration observed in the shipping environment on the settling of spherical particles suspended in a H-B fluid. Determine the ”separation criterion" for a particle suspended in an H-B fluid due to vibration input. 3.1. 3.0. ‘LITERATUBE REVIEW Ian-Newtonian Fluids Non-Newtonian fluids have a non-linear relationship between viscometric shear stress and shear rate for given temperature and pressure conditions. Ferry (1980) divided these fluids into three main.groups: 1. Time-independent Non-Newtonian fluids 2. Time-dependent Non-Newtonian fluids 3. Viscolastic fluids A similar approach was used by Metzner (1965): 1. Purely viscous fluids 2. Time-dependent fluids 3. Viscoelastic fluids where the Newtonian fluids are considered to be a subcategory of purely viscous fluids. The time-independent fluids are those for which shear rate, at a point, is purely a function of instantan- eous shear stress. However, in the case of time-dependent fluids, shear rate is a function of both the instantaneous shear stress and history of shear stress. Such behavior, where the shear stress decreases with time, is also called thioxtropy. Rheopectic fluids on the other hand show an increase in shear stress with time. Viscoelastic fluids exhibit a partial elastic recovery upon removal of the shear stress. Hence, they depict an intermediate 5 behavior between an ideal fluid and an elastic solid. Most of these models for interpretation of deformation and flow behavior have been applied to solid products like potatoes and apples. Coleman (1961) used the term ”Viscoelastic" to refer to those materials which do not behave according to the fundamental laws of fluid mechanics or elasticity. Both the Skelland (1967) and Metzner (1965) classifications described above are very similar; however, these groups are not finitely defined in regards to their viscous behavior. Several different mathematical models have been developed to predict their flow behavior (shear stress versus shear rate) for certain temper- ature and pressure conditions. Whorlow (1980) described the choice of this analytical equation somewhat as a matter of taste of the user. The need to develop non-Newtonian rheological models for var- ious fluid food products has arisen due to several engineering applications like: - Design of continuous processes. - Predict flow rates in pipes of different diameter. - Design pump sizes for required flow. - Mixing data and rates of heating for processes such as concentration, dehydration, pasteurization, and steri- lization. - Quality control parameters for raw products (maturity index). - Quality control parameters for processed foods (con- sistency index) Some of the popular models are briefly discussed next. 3.1.1 The Power-Law Model This is one of the most commonly used models. It was first described by Ostwald (1925) and de Waele (1923). The rheometric equation describing the flow curve for the Ostwald-de—Waele model can be written as: dV n T K '21-; (3’1) where: T = shear stress, Pa K = consistency coefficient, Pa sn dV _ -1 dr - shear rate, 8 n = flow behavior index, dimensionless When n a 1, this model reduces to the case of the simple Newtonian fluid. For most polymer solutions n < 1, which means that the shear stress decreases as the shear’ rate increases. Such fluids are called shear thinning or pseudoplastic (Skelland, 1967). However, for cases when n > 1, the shear stress increases as the shear rate increases and such fluids are called shear thickening or dialatant (Skelland, 1967). 3.1.2 The Bingha-.Plastic Model The rheometric equation for this particular model is similar to that for a Newtonian fluid but also includes a yield stress 3.1.3 £11| r = To + KP ldr (3-2) where: T = shear stress, Pa To a yield shear stress, Pa KP a plastic viscosity, Pa 8 é!-= shear rate, 8"1 dr The use of this model is somewhat limited (Skelland, 1967) because this type of plastic behavior is rarely seen in the case of real fluids. However, this is still a good approximation when deviation from the experimental results is small. The Herschel-Bulkley Model A combination of the power law model and the Bingham Plastic model provides a very good approximation to a vast number of Non- Newtonian fluids. Also referred to as the yield-pseudoplastic model (Hanks, 1979), the Herschel—Bulkley model can be written as, dV n a + — _ T To K Idr (3 3) where: T a shear stress, Pa 1 - yield shear stress, Pa K a consistency coefficient, Pa 3 n a flow behavior index, dimensionless 22.. shear rate, 8-1 dr 3.1.4 3.1.5 This model reduces to the power law model if To = 0, to the Bingham Plastic model when n = 1 and to the Newtonian fluid when To = 0 and n = l. The Ellis Hedel Another three constant model that is a combination of the power law model and the Newtonian fluid can be written as (Reiner, 1960), 1 dV T = -—- (3-4) (n-l) dr| C1 + C2 1 where: T = shear stress, Pa 1 -1 C1 = constant, Pa- 8 C2 = constant, Pa.3 3"1 n = flow behavior index, dimensionless dV - a;-= shear rate, 8 1 This model shows Newtonian behavior as the shear rate becomes very small and power law behavior at high shear rate. Thus, it overcomes a major drawback in the power law fluid that predicts infinitely large values of viscosity when the shear rate goes to zero, which is not the case in real fluids (Cho, 1979). The Powell-Eyring Model A modification of the Eyring model (Kincaid, 1941) was obtained by adding another constant to get the Powell—Eyring model (1944) written as, 3.2 10 c33|— W sinh'ltélgl) (3-5) where: T = shear stress, Pa C3 - constant, Pa 8 C4 - constant, Pa-a C5 - constant, 3'1 dV -1 dr shear rate, 8 This model can, therefore, accomodate both the high and low shear rate regions and reproduce the general behavior of Non- Newtonian viscosity. Several other mathematical models, such as the Sutterby Model (Sutterby, 1964) and the Williamson Model (Skelland, 1967), have been devised to provide additional flexibility to fit exper- imental results. However, only the Bingham IPlastic and the Herschel—Bulkley models can be used for fluids that show a yield stress. Velocity Profile of Fluid Flow Over A Sphere A classical problem in fluid dynamics is the determination of the fluid velocity field for slow moving flow over a spherical particle. For Newtonian flow, the drag coefficients for creeping flow, assuming no slip boundary conditions, has been developed by Stokes (1851). Non-Newtonian flows present considerable difficul- ties in obtaining solutions for the momentum equation because of the non-linear nature of the governing differential equations and the presence of any time dependency or thixotropy. 11 Several experimental and theoretical studies have been done to determine the effect of different Non-Newtonian fluids flowing over spherical particles. Some of the common falling ball visco- meter techniques for Non-Newtonian fluids have been discussed by Cho (1979). Bird (1960) determined that variational techniques could be used to predict the total drag on a sphere for a power law fluid. Similar numerical techniques were used by Wasserman and Slattery (1964). Ultman and Denn (1970) did another study using the Oseen solution for a simplified Maxwell fluid model. This study accounted for the Viscoelastic properties of the fluid flowing around the sphere. A closed form solution to determine the drag force on a spherical particle for creeping flow in a (inelastic) power law fluid was determined by Dahzi (1985). The drag on an unbounded fluid was estimated numerically using the finite element method. Earlier methods of approximation used include perturbation techniques, variational principles, finite difference methods and Oseen linearization techniques. Results obtained using the closed form solution were compared to experimental evaluations by Cho (1983). It was also shown that the more shear thinning the fluid, less important are the wall effects. Wall effects were found to be negligible for flow beha- vior index greater than 0.5. Another observation was that the flow field around the sphere is not viscometric but Newtonian and power law fluids have similar flow patterns. Harris (1977) deter- mined the effect of an oscillating wall on shear wave propagation. It was shown that Non-Newtonian fluids acquire higher shear stres- 3.3 12 ses as compared to Newtonian fluids at the same location from the side wall. Rheological Models of Fluid Food Products Generally most fluid food products do not show a simple New- tonian flow behavior. Holdsworth (1971) emphasized a great need to accurately describe the rheological properties of these pro- ducts in connection with engineering design problems. Several different types of mathematical models, as described in the pre- vious sections, have been applied to predict the flow behavior of various fluid foods. Generally, the power law model is used to predict the flow of common fluid food products. Table 1 describes some of the rheological parameters determined for soups, concen- trated juices and vegetable and fruit purees. 13 TABLE 1 Rheological Parameters of Various Fluid Food Products Product Soup Tomato Tomato 5.8% 12.8% 16.0% 25.0% 30.0% Tomato Ketchup French Mustard Puree Puree Conc. Conc. Conc. Conc. Conc. Sweetened Condensed Milk Apple Sauce Apricot Puree Temp. (°C) 12.8 47.8 32.0 32.0 32.0 32.0 32.0 25.0 95.0 25.0 25.0 27.2 30.0 25.0 Rheological Constants To Pa K n Pa 3 20.4 320 105 410 36-56 10.80 2.23 20.00 31.60 129.00 187.00 187.00 74.50 334.00 36.00 108.50 68.00 14.17 n Technique Ferranti 0.50 Portable Viscometer Brookfield 0.55 Viscometer LV Harper 0.59 Concentric Cylinder 0.43 0.45 0.41 0.40 0.28 0.25 0.40 0.83 Brookfield 0.38 Viscometer LV 0.30 Brookfield Viscometer LV 0 o 38 Haake Viscometer RV-12 Reference Wood (1971) Charm (1963) Harper (1961 and 1965) Higgs and Norrington (1971) Charm (1962) Saravacos and Moyer (1967) Ford and Steffe (1985) 14 Table 1 (continued) Rheological Constants Concentrated Temp. To K Product (°C) Pa Pa 8n n Technique Reference Merril Charm and Banana Puree 24.0 - 65.00 0.46 Viscometer Meril (1959) Brookfield Saravacos Pear Puree 30.0 - 56.00 0.35 Viscometer and Moyer LV (1967) Brookfield Saravacos Peach Puree 30.0 - 72.00 0.28 Viscometer and Moyer RVT (1967) Brookfield Saravacos Plum Puree 30.0 - 22.00 0.34 Viscometer and Moyer er (1970) Orange Brookfield JUice 000 703 18090 0068 Viscometer Charm (1963) Concentrated LV Orange Juice Contraves Saravacos (60 Brix) 30.0 - 15.50 0.55 Rheomat 15 (1970) 4.1 4.0. MODEL DEVELOPMENT Provisions of the Model Consider a cylindrical body of fluid with a spherical particle of radius R suspended in it (Figure 1). The spherical geometry was considered as a first approximation. Irregular geometries will result in complex flow patterns involving multi-directional flow and possibly local flow separation rendering the problem too diffi- cult to handle. The Herschel-Bulkley (H—B) model described by Equation (3-3) was chosen to represent the fluid because it shows the presence of a yield stress. Also, it shows a non-linear beha- vior between the shear stress and the shear rate and, hence, best describes such a Non-Newtonian fluid in general form. The column of fluid is vibrated sinusoidally with an amplitude "A" and radial frequency "w", and the particle responds by either vibrating in step with the fluid or breaking loose and moving rela- tive to the surrounding field. The spherical particle should be of a higher density, such that its weight overcomes the buoyancy when submerged. The purpose of this analysis is to determine combina- tions of "A" and "w" and To, K, n which will cause the suspended particle of a given mass and size to move in a Herschel-Bulkley fluid. The particular combinations which cause this will be related in an expression hereafter called the "Absolute Separation Criterion”. 15 16 5 her' at pgrtic'fe\ A/Fluid column NA sin wt Figure 1. DISPLACEMENT CONVENTIONS FOR THE PARTICLE IN THE FLUID COLUMN 17 4.2 Assumptions of the Model 1. 3. 4. 5. 7. 8. 9. 10. Vibration motion is present only in the veritical (z-axis) direction. Creeping flow. Wall effects are negligible. Constant sinusoidal acceleration. Fluid column is homogeneous and isotropic. Fluid model inelastic and non-time dependent Steady state environmental conditions (temperature, pressure, relative humidity). Flow is laminar. Density of the fluid remains constant. A no slip boundary condition exists on the surface of the sphere. 4.3 Mathematical Modeling of System The absolute separation criterion will involve the following parameters: sphere fluid vibration input mass, ms; radius, R; yield shear stress, To; consistency coeffi- cient, K; flow behavior index, n; weight density, of; . displacement amplitude, A; frequency, w 18 The free body diagram for the particle is shown in Figure 2. The three forces acting on the particle are the weight of the par- ticle, msg; the buoyancy, mfg; and the drag force, D. The drag force for Herschel-Bulkley fluid on a sphere can be obtained from the solution to the Navier Stokes (N-S) equations using the rheo- metric equation for the fluid, _ dV n T - To + K 1dr (4 1) where: T = shear stress, Pa To a yield shear stress, Pa consistency coefficient, Pa an n . flow behavior index, dimensionless dV -1 dr 8 shear rate, 3 Dahzi (1985) has determined the drag force on a sphere in a power law fluid whose rheological equation of state is, dV n T K la? (4’2) In his analysis, Dahzi (1985) assumes a low Reynolds number flow and describes the flow of an incompressible fluid using the conservation of mass equation, + V v s 0 (4-3) 19 Velocny of sphere msg = Weight relative to fluid Drog = D Inf 9 = Buoyancy Figure 2. FREE BODY DIAGRAM OF THE SPHERICA PARTICLE " 20 where "v" is the velocity vector, and the conservation of linear momentum equation in which the inertial and body forces were neglected, I7- T-Vp=0 (4‘4) where T is the deviatoric stress tensor and p is the pressure. A three dimensional flow was considered with the generalization given by Slattery (1964). No slip boundary conditions were assumed. The normalized drag force on a sphere in a power law fluid Dkun is calculated from the solution of the stress field by numer- ical integration. The computations were in the range of the power law index between 0.1 and 1.0. The values of drag correction factor C(n), to be used in the drag force equation determined by Dahzi (1985), are listed in Appendix D. Fortunately, his results can be used to determine the drag force on the sphere for an HrB fluid without having to solve the equations of motion. When Equation 4-1 is substituted into the equations of motion, the following results: Consider an element of area "dA" on the surface of the spherical particle as shown in Figure 3. Using the spherical coordinates, the drag force on this element in the vertical direction is: D = f T dA cos (90 - ¢) a f:" A: T R2 Sinz ¢ d ¢ d 0 (4‘5) J since dA = Rd ¢ R sin ¢ d 0 21 b-N 79in¢ A} 4"“ Flow direction Figure 3. SPHERICAL COORDINATES SYSTEM USED IN THE DRAG FORCE CALCULATIONS. 22 For a generalized Newtonian flow, the shear stress can be expressed as (Darby, 1976): Tr¢ = n ( -II) Ar¢ (4-6) where ’-II is the scalar, and for spherical coordinates defined in Figure 3 is described by Jflsz J/;(1_ 5 VI) 2 (v¢ cit 9)2 (5V9- 9¥992 (4-7) T and the apparent viscosity n a (~2-+ .7 R Y 1.0 ) Assuming no slip boundary conditions at the surface of the sphere, 6 = Vr = 0, and that -!i'= 0 and substituting these into (4- i.e. V 50 O 6) we have, after simplification: T -< ° + k JOE-1’1) (4-8) -II ( -11)1°r r This becomes a complex equation to solve mathematically, but a Taylor series expansion could be used to approximate (4-8) in the general form described by (4-1). It is convenient to split T into two parts, To and k(%x)nas described by the H-B fluid model (Equation 4-1), so that the drag force D becomes the sum of two integrals: (4-9) 23 Since the integrand in the integral represented by DT involves the o constant term To , the integeration can be performed without requiring a knowledge of the velocity profile, D = I In T R2 sin2 O d ¢ d 0 = NZRZT (4-10) 0 o o The remaining integral represented by Dkun is 6n n dv n 2 2 Dkun - f0 f0 K(E;) R sin ¢ d o d 0 (4-11) But Dahzi (1985) has already performed this integration in his determination of the drag on a sphere in a power law fluid, 1 V n- Dkun - 6 TI K (g V R €(n) (4-12) where I! is the magnitude velocity and E(n) is the drag correction factor (Dahzi, 1985). The total drag force on a spherical particle in a H-B fluid is, therefore, 2 2 V n-I D ='n R To + 6 n K 02R) V R. C(n) (4-13) where the direction of D is opposite to that of the motion of the particle relative to the fluid (Figure 2). 24 Since all forces in the free body diagram have been examined, Newton's laws of motion can be applied to determine the trajectory of the particle, Z F+ 8 m8 '3? (4-14) d2 -m g + m g - D - m -——-(2 + A sin.w t) (4-15) 3 f s dt2 Carrying out the differentiation and rearranging gives, u 2_.-‘£_' - '2 - z + ms m3 (mf ms) + AID sin w t (4 16) Substituting the drag force (4-13) in (4-16) and noting that V s z , which gives an upward positive convention for V . v2.2 To I, 6,, K I115” |v| m.) v + { m } sign (V) ( 8 4-17) -§I:(mf —ms) -+-AII;2 sinLIt Equation 4-17 governs the motion of a spherical particle in the fluid. For a given particle (R, mp) in a H—B fluid with pro- perties k, n, C(n), To and mf - 6f g-r R3 , subjected to a known vibration input (A, w ), the velocity V of the particle relative to the fluid container can, in principle, be determined at any time. Now, if the effects of weight, buoyancy, and input vibration are not sufficient to cause the particle to break loose from the fluid, then V - 0 and 2 remains constant. From (4-15), the drag force on the particle must then be 25 2 D - (mf — ms)g + m8 AID sin w t (4-18) But D must not exceed DT a r2 R? To ; otherwise the particle will 0 break loose and move (V ¢ 0). Therefore, the criterion for motion relative to the fluid is that 2 sin.m tl > g R. T (4-19) |(mf - m8)g + m8 AI» 0 When condition (4-19) is met, the velocity is determined by the solution to (4-17) using V - 0 as an initial condition. Otherwise, V -= 0. It is conceivable then that the particle may undergo alternate periods of motion and rest if condition (4-19) is periodically satisfied as time goes on. In fact, it may be concluded from (4-19) that the particle will never break loose unless the following absolute motion criterion is satisfied, ,2 22 Imf mslg + m8 A.» > n R To (4-20) Since (4-20) applies in general for any A and w , it must also apply for the static case A - 0, w - 0 . This allows for a simple experimental determination of the yield stressTo . When the particle is just ready to break loose in a H-B fluid at rest because of difference in buoyancy and weight, the condition 2 2 lmf - mslg - n R To (4-21) 4.4 26 is satisfied. Therefore, if the particle can be made sufficiently large and massive so that motion is impending, then T . lmf-mSIS - g-n R3l5f-5si "é§'l5 —6 I (4_22) o 112 R2 1‘2 R2 3T f 3 Using several steel spheres ((58 fixed) with different radii, all suspended in a H—B fluid (5 f known) , some will remain suspended and others will fall. The smallest sphere which falls determines R in (4-22). In an experiment performed using the 1.252 Kelsat solution, an R - 7.9375 x 10-4m steel sphere was the smallest sphere to fall. ”Sing (4-22), 4 -4 To --§7: (7.9375 x 10 ) (7961 - 1020) T . 2033 Pa O This is actually an upper bound on To (determined to be 0.80 Pa using back extrusion, Appendix A) because of practical limitations involving the inability to find the absolute smallest sphere which falls. Dimensional Analysis Dimensional Analysis is a method for obtaining qualitative information about pertinent variables of a phenomenon based solely upon their dimensions (Langhaar, 1951). All pertinent variables must be considered in the analysis. The end result is a number of 27 dimensionless quantities involving groups of these variables. The relationships between these dimensionless quantities must be deter- mined experimentally. In this particular problem, a spherical particle is suspended in an oscillating column filled with a Herschel-Bulkley fluid. Variables which affect the setting time ”t” are: t g fl(A’ D), To: K: n: R: g: 2, T: 6 5 (4'23) 8’ f) where: K = consistency coefficient of H-B fluid model g - acceleration due to gravity 2 a displacement of particle T - temperature A a displacement amplitude of vibration table w - radial frequency of vibration table T - yield shear stress of H-B fluid model 0 5f - fluid mass density 68 - sphere mass density R - radius of sphere n - flow behavior index of H-B fluid model Since all the variables are explicit functions of temperature, T may be removed from the list and t . f2(A’ w! To, K, n, R, g! z, 68’ 6f) (4-24) 28 Using Buckingham's theorem (Langhaar, 1951), which states that "if an equation is dimensionally homogenous, it can be reduced to a relationship among a complete set of dimensionless products", we have f(t: A: “)9 TO: K: I1, R: g: z: 68’ 6f) = 0 (4-25) A dimensionless TI-number may be constructed from any product of these variables in Equation 4-25. n . up A9 Mr r03 Kt a” gv z" 68x sfy (4-26) The variables in (4-26) can be represented in terms of the primary quantities of mass M, length L and time T, i.e. t--[T] A = [L] w = [T-l] T a [ML-2] n = dimensionless K - [ML’ZT‘n] R - [L] -2] s = [LT z - [L] 29 In this study, the governing differential equation was used to obtain the specific dimensionless quantities that effect the settling of a spherical particle. 4.5 Dimensionless Governing Differential Equation We have determined in Equation (4-17) that for sinusoidal vibration V + {II R2 To + 61TK lg—Rln-‘l IV] R c (n)} sign (V) =g—(m "111)+Aw2 sinnt mS f 5 (4-27) Let the independent variable X = I.) t V u .—Awa and the dependent variable ThenVsEl-‘l d (Aw u) a a A.w2 du dt d (Xfiu) 'a'io Using these in (4—27) and dividing by A m2, NZRZT + 6IIK lfl—Jln-l [Aw ul Rc(n) du 0 2R 1 EEI+ { 2 , sign (u) AID m s “‘f a ——£3-(-—-— 1) + sin X (4-28) m AI» 3 UZRZT du o g 63 K.E R A n-l n Therefore, —dx + { m g 2 + —' 2-n (-2R) 'u' } sign (u) 8 Arm m II s “f = -g§-(--- l) + sin X Aw ms (4-29) 30 Since both "u" and "X" are dimensionless, all terms in the above differential equation are dimensionless. The different dimensionless terms are: 2 G - £%- Acceleration anzto Y a -————- Yield Factor msg (4-30) ms Y I —- Buoyancy Factor “f B . 6" K 5 £3: R (%E)n-l Drag Correction (0 Coefficient m 8 Rewriting the equation with these dimensionless terms we have: du 1 n -111. dX + {G + 3 lul } sign(u) G Y + sin X (4'31) Initial conditions are at X - 0, u 8 0. Time of Descent : From (4-31) if "h" is the height of fluid column, 2 - h + I: vdt (4-32) 31 h Let us assume another dimensionless term, a 2'2 {T Set 2 = 0 a h + ,0 vdt (4-33) Using v = Aw u and t - % from before, 0 = h + I“ (A w u)(—dx (4—34) 0 O) wT 0 = a +Ifo udX. (4'35) Once the differential equation (4-35) is solved, we will have u as a function of the dimensionless terms a, 8, Y, Y and G and x which represents time. Hence, we see that the dimensionless descent time is a func- tion of five dimensionless terms, i.e.: X = f ((1, 8: Y9 f: G) (4‘36) First Analytic Guess at Functional Fbrm Assuming that once the particle separates and reaches its terminal velocity rapidly, and maintains this during the entire descent, the velocity in steady state remains constant. Therefore, also sign(u) - -1 (descent) and sin X + 0 32 Therefore, substituting in the differential equation (4-31), f n 1 - Y 0 + {G + B ( u) } ( 1) G Y + 0 (4 32) f n_Y—l _ E-+ 8 (-u) — G Y (4 33) n _ Y - 1 _ £_ 1_ _ (u) -(GY 6),, (434) Therefore, the terminal velocity, __1 1.:_1.-£ l/n - - INC, G>1 (435) For the time of descent, mT 1 T - 1 f l/n = ' - — -'———— .- O a + ,0 [B ( G I G)] dX (4 36) x - a (4-37) or - Ii (Y ’1 331”“ B G'Y G wT=u[ ”G 11/“ (4-38) 1 1 - -- f Y Tell BC 11/“ (4-39) m 1 l--Y--f 4.6 Effect of Increased Number of Particles on the Settling Rate So far, If B is the "Af" of the described by the settling of only one sphere has been considered. radius of the fluid column, then the true flow area fluid past the cross section of one sphere can be 33 2 AfsnBZ-TRZ-nnzu-35) (4-40) B To reduce side wall effects, a value of TB!- > 20 was chosen, so that 2 True flow area _ B . 1-3- _ 1 _ 1 ((6-41) Assumed area " B2 B2 400 Now, consider ”N" spheres lying in a layer as in a dense suspen- sion. Consider this layer to move down a distance 2R in time "A t" . This will result in a volume of fluid displaced by the spheres equal to N in R3 . This volume must travel through an 3 effective flow area of ll B2 - N r Rz' Therefore, the distance the fluid travels upwards is _4_ 3 N 3'" R AS . 2 2 (4-42) h B - N‘n R The fluid velocity relative to container then is AS vf/C '3 E (lb-43) Usingg as the average velocity of a sphere over time interval A t , 4 NR B “d a 3 (—)Z - N 3 N (——") V _ R 3 dt _ i ( Z ) (4-44) f/c At (52 _ N 3 1 (8)2 _ l R N R Now the true fluid velocity relative to the sphere Vf/s which determines drag is not 2 alone but the sum of Z and Vf/c 34 ‘ 2 z vf/s + - z +3.1. (2)2 -1 (4-45) N R Letting 2 L = (4-46) 1 B 2 3 b“ 0R) r 1] Vf/s+ . z (1 + L) (4-47) The value of vf/s+ must now be used to determine the drag force in 2 400 ' N ' l] Equation 4-13. Using-% = 20 as before, L =: 3i To obtain a 10% increase in Vf/si’ that will result in a significant increase in the settling of particles, we need 2 400 3 b—fi- - l] 3 001 or N = 52.1 particles per layer In food systems like salad dressings, the value of N is less than 20 for creamy based emulsions, whereas it is much higher than 100 in the case of vinegar or oil based dressings. For this study, the effect of increased number of particles was not considered in the experimental design. It is suggested that for future research in this area, 'L' be considered as a separate term in the dimensionless prediction equation (6-2). 4.7 35 Effect of Random Vibration on Settling of Particle The vibrational forces experienced by products and packages in the shipping environment are not a result of a single frequency, sinusoidal vibration. The vibration of a truck bed or rail car, for example, is random in nature as shown in Figure 13. Random vibration may be thought of as a mixture or sum of several sinusoidal vibrations at any single instant and this may have a complete different effect from a single frequency vibra- tion. Packaging technologists, therefore, use random vibration input to investigate the fragility of product in the shipping environment. Using the Fourier Analysis it is possible to decompose a complex waveform representing the random vibration of a truck bed or rail car into simple sine wave components of known frequencies and amplitudes. This is accomplished by using a Band Pass Filter (BPF) which is an electronic device that splits a complex waveform into its sine wave components, referred to as the fourier transfor— mation. Normal or Gaussian statistical distribution is used to describe the instantaneous peak acceleration amplitude of any of the component frequencies composing the random vibration. The amplitude of any of the component frequencies determined earlier may be represented as the "Power Density" (P.D.), Z 2 i (RMS)i/Ni 36 where: P.D. - Power Density, gz/Hz RMSi - Root Mean Square acceleration value measured at any instant i, g's. N1 - Number of instants sampled BW - Band Width of the EFF used to obtain the component frequency, Hz. A plot of Power Density versus frequency is known as Power Density Spectrum (Figure 10). Statistically, the Power Density at any given frequency is the variance about a mean value of zero acceleration. Therefore, based on probabilities associated with Gaussian distributions, the acceleration levels associated with any component frequency of the complex waveform can be predicted as i l P.D. values occur 68.32 (1 O) of the time. i 2 P.D. values occur 95.42 (2 O) of the time. i 3 P.D. values occur 99.72 (3 O) of the time. The separation criterion described earlier for sinusoidal vibration in Equation (4-15) can be modified for random vibration by replacing the sinusoidal amplitude A sin wt by an instantaneous random amplitude y(t), . d2 . . -m8g + mfg - D = m8-—§-(z + V(t)) (4-49) dt Carrying out the differentiation and substituting the drag force (4-20) we have 4.8 37 2 2 V n-l v + {II R To + 6n Kl-Z—R-l IV] R e (n)} sign(V) (4-50) a fi_, - - M m (mf ms) y s and therefore the criterion for motion relative to fluid is " 2 2 (mf ms)g yl > n R To (4-51) Hence, whenever y in (4-51) is such that Equation (4-51) is satisfied, the particle moves. Effect of Non-Spherical Particles The mathematical model developed for this study (Section 4.3) assumes a spherical geometry. A non-spherical particle will result in complex flow patterns involving multi-directional flow and poss- ible local flow separation. This will result in a complex model. However, the size and shape of the particle will influence the relative drag force due to fluid flow. Curray (1951) introduced 'sphericity' by assuming the volume of solid is equal to the volume of a triaxial ellipsoid with inter— cepts a, b, c, and that the diameter of the circumscribed sphere is the longest intercept 'a' of the ellipsoid. The degree of spheri- city can then be expressed as volume of solid sphericity a volume of circumscribed sphere or 1/3 S = [(njb) :bcll/3 = [2511/3 g ab: (4_52) n b a a 38 where: a longest intercept 0‘ ll longest intercept normal to a 0 ll longest intercept normal to a and b Therefore, sphericity can also be expressed as the ratio of the geometric mean diameter to the major diameter. Blueberries and cherries have percent sphericity values of 90 - 95 (Moshenin, 1980). The effect of sphericity can be studied on the settling rate by considering S as a separate dimensionless term in the prediction equation (6-4). 5.0. EXPERIMENTAL DESIGN The validity of the mathematical model was demonstrated by design— ing an experiment that satisfied the model criteria. Osorio (1985) determined that Kelsat (sodium calcium alganate, Kelco Company) aqueous solutions could be modeled as Herschel-Bulkley fluid. A back extrusion device was designed by Osorio and Steffe (1985) to determine the mater- ial constants for the Herschel-Bulkley fluid. Five different Kelsat aqueous solutions, 1.002, 1.25%, 1.50%, 1.752 and 2.00%, obtained by mixing the Kelsat powder in distilled water were used as Herschel-Bulkley fluids. To aid mixing, the pH of the water was reduced by adding 52 of 2N hydrochloric acid, stirring continuously. The powder was introduced in the stirred solution very slowly to avoid clumping. The pH of the final solution was stabilized by adding 51 of 2N sodium hydroxide solution. Care was taken in the mixing process to avoid the introduction of air bubbles. The fluids were then allowed to stabilize for 24 hours at 72°F (22.2°C) and 502 RH. The rheological properties of the five different solutions were measured using the back extrusion method developed by Osorio and Steffe (1985). Force versus deflection curves were collected for four differ- ent plunger speeds for the back extrusion device on a Model 4202 Instron Universal Testing machine. Three different plunger sizes were used. 39 40 The plunger was mounted on the crosshead and the graduated cylinder con- taining a sample of the fluid was placed on the compression load cell. The force-deflection curves were collected on a strip chart recorder. Figure 4 shows a typical force-deflection curve for a Herschel—Bulkley fluid. The data from these curves was then used to compute the differ- ent constants (Appendix A) required for the evaluation of rheological properties with the help of a computer program (Osorio and Steffe, 1985). The different fluid constants evaluated are listed in Table 2. The spherical particles used in the model were approximated by using chrome plated steel balls (Detroit Ball Bearing Company) with sphericity less than 9.84 x 10"8 m/m. The size of the balls chosen for each fluid model must satisfy the absolute separation criterion. The mass density of the steel material used to make these balls was mea- sured. An electrohydraulic vibration table (MTS Corporation) was used for the controlled sinusoidal vibration to the fluid column. Three differ— ent frequencies of 10, 15 and 20 Hz were used and acceleration levels varied from 0.5 g's to 3.5 g's. The setup is shown in Figure 5. The fluid was contained in a cylindrical container clamped to the table bed. The spheres were immersed at a distance at least M away from the side wall sudh that M/R is greater than 20, thereby reducing any side wall effect. Also, any variation in the experimental results due to surface tension effects were avoided by completely immersing the sphere in the fluid field. The table was then forced to vibrate with the controlled input and the time required for the particle to travel 41 Plunger Force Plunger velocity Vp FT I is stopped I e ' lunger FT ____________ ' b/P Experimental [W ‘W‘ Ideal-u Ere ~ ' Buoyancy force -—I- Fluid F ----------- -—- l Ty i Experimental FT), 1 i - Distance Figure 4. FORCE VERSUS DISTANCE DIAGRAM OF A H-B FLUID USING THE BACK EXTRUSION METHOD (Osorio, 1985) 42 ‘3'" h I J IEJ l I I J I ’///7// ///// //7/fl, Figure 5. EXPERIMENTAL SETUP OF VIBRATION TABLE WITH FLUID COLUMN AND SUSPENDED SPHERE 43 TABLE 2 Rheological Properties of Aqueous Kelsat Solutions After 24 Hours of Storage at 24 C Using the Herschel-Bulkley Model. n Concentration, % Ty, Pa k, Pa 5 1.00 0.507 1.32 1.25 0.800 3.55 1.50 0.870 8.20 1.75 2.193 9.40 2.00 20.632 20.60 44 between two previously marked points on the container was measured. Five repetitions were obtained using different but similar spheres to avoid variation. The experimental design is shown in Table 3. The average values determined for five replicates of settling time, determined experimentally for different combinations of fluids and spheres, are listed in Tables 4-7 (see Chapter 6.0. Results and Dis— cussion). Table 9 (see Chapter 6.0. Results and Discussion) shows values of the dimensionless constants determined using Equation (4-30). 45 TABLE 3 Experimental Design Table for Determining Settling Time of Sphere in Vibrating Herschel-Bulkley Model. SPHERE Mass (Kg) Radius (m) -5 A 1.63 x 10 0.000793750 -4 B 1.30 x 10 0.001587500 -4 C 6.97 x 10 0.002778125 -3 D 1.04 x 10 0.003175000 FLUID Concentration (%) Height of Column (m) l 1.00 0.097 2 1.25 0.097 3 1.50 0.097 4 1.75 0.097 5 2.00 0.097 VIBRATION Frequency (Hz) Acceleration (G's) Sinusoidal 10 15 0 5 - 2 5 20 6.0. RESULTS AND DISCUSSION 6.1. Rheological Properties of the Herschel-Bulkley Fluid The force versus distance diagrams obtained for the 1.00%, 1.25, 1.50, 1.75 and the 2.002 Kelsat solutions were straight lines which, when extrapolated to zero displacement gave a non—zero value of force due to the presence of a yield shear stress in the fluid F (Figure 4). The values for F , and the chart length, 1, T’ Te’ F‘ry were determined for the five solutions at different plunger velocities for use in calculating the rheological constants using the back extrusion method outlined in Osorio and Steffe (1985). These values are listed in Appendix A. Table 2 lists the different rheological constants determined for the 1.00%, 1.252, 1.50%, 1.75 and the 2.00% Kelsat solutions respectively. An experimental approach was used to investigate the effect of the time dependent behavior of the fluid. Equation 4-1 describing the behavior of the H-B fluid can be modified as dV n SPITO + k (a?) 1 (6-1) H II where Sp a structural parameter (equal to unity at time zero and to an equilibrium value Se at equilibrium), dimensionless. 46 6.2 47 To see the behavior of Sp for this particular Kelsat aqueous solution, a 1.252 Kelsat solution was made. This was subjected to a vibration dwell at 20 Hz and 1.0 g's for 1 hour; after being allowed to stabilize for one day. Steel spheres of mass 1.63 x 10'"5 kg and radius equal to 7.937 x 10"4 m were then allowed to settle a distance of 0.097im, while being vibrated at 1.0 g's. The values of settling time measured are listed in Table 8. These results are compared to those determined in Table B-1 (Appendix B) and it is evident that within experimental error the fluid prOper- ties do not change as a result of vibration histories. Tables 4-7 show the average values of settling time measured experimentally. Prediction Equation for Settling of Sphere Using the dimensionless constants evaluated by Equation (4-30) and listed in Tables C1-C8 (Appendix C), a prediction equation of the form x = z Carbycsdae (6—2) was considered. Taking the log transformation of Equation (6-2), we have log X - log 2 + a log G + b log Y + c log Y + d log 8 + e log a (6'3) A stepwise linear regression can be performed using the experimen- tal data to ewaluate the different regression coefficients. The average values of dimensionless constants determined are listed in Table 9. TABLE 5 Settling Time Data for Steel Spheres in a Vibrating 1.50 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. AVERAGE * SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/8 1.0 10 17.0 0.05 15 15.7 0.05 20 16.7 0.05 2.0 10 6.6 0.05 15 6.1 0.03 20 6.6 0.05 2.5 10 3.6 0.03 15 3.8 0.03 20 4.0 0.09 7/32 1.0 10 2.4 0.04 15 3.0 0.04 20 3.3 0.04 1.5 10 1.5 0.04 15 2.2 0.05 20 2.4 0.04 * Average values based on five replicates (see Appendix B for all data). 49 52 TABLE 8 Settling Time Data for Steel Spheres in a Vibrating 1.25 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C and Subjected to a Vibration Dwell for 1 Hour at 20 Hz and 1.0 G's. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/16 1.0 10 66.15 66.80 67.35 0.01 67.52 66.98 15 69.74 70.08 70.94 0.01 70.74 71.49 20 76.52 77.25 77.51 0.01 76.58 77.18 53 TABLE 9 Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. * ACCELERATION YIELD BUOYANCY DRAG ALPHA AVERAGE FACTOR FACTOR CORRECTION SETTLING FACTOR TIME 1.414427 0.031120 7.819548 6.244965 27.60765 4216 1.414427 0.031120 7.819548 5.098993 62.11721 6662 1.414427 0.031120 7.819548 4.415857 110.4306 9486 2.121640 0.031120 7.819548 5.098993 18.40510 2391 2.121640 0.031120 7.819548 4.163310 4.141147 3916 2.121640 0.031120 7.819548 3.605532 73.62040 5398 0.707213 0.015608 7.819548 3.132090 55.21530 1450 0.707213 0.015608 7.819548 2.557341 124.2344 2386 0.707213 0.015608 7.819548 2.214722 220.8612 3337 1.414427 0.015608 7.819548 2.214722 27.60765 561 1.414427 0.015608 7.819548 1.808313 62.11721 926 1.414427 0.015608 7.819548 1.566045 110.4306 1291 2.121640 0.015608 7.819548 1.808313 18.40510 309 2.121640 0.015608 7.819548 1.476481 41.41147 520 2.121640 0.015608 7.819548 1.278670 73.62040 737 1.414427 0.016973 7.805151 5.115697 27.60765 1068 1.414427 0.016973 7.805151 4.176949 62.11721 1481 1.414427 0.016973 7.805151 3.617344 110.4306 2098 2.828854 0.016973 7.805151 3.617344 13.80382 413 2.828854 0.016973 7.805151 2.953549 31.05860 577 2.828854 0.016973 7.805151 2.557848 55.21530 832 3.536067 0.016973 7.805151 3.325450 11.04306 225 3.536067 0.016973 7.805151 2.641734 24.84688 350 3.536067 0.016973 7.805151 2.287809 44.17224 507 1.414427 0.009695 7.805151 2.208881 27.60765 152 1.414427 0.009695 7.805151 1.803544 62.11721 275 1.414427 0.009695 7.805151 1.561915 110.4306 412 2.121640 0.009695 7.805151 1.803544 18.40510 94 2.121640 0.009695 7.805151 1.472587 41.41147 209 2.121640 0.009695 7.805151 1.275298 73.62040 304 1.414427 0.024439 7.805151 2.532132 27.60765 2993 1.414427 0.024439 7.805151 2.067477 62.11721 2196 1.414427 0.024439 7.805151 1.790488 110.4306 2943 2.121640 0.024439 7.805151 2.067477 18.40510 995 2.121640 0.024439 7.805151 1.688088 41.41147 1591 Average Values of Dimensionless Settling Time Based on Five Replicates (see Appendix C for all data) 54 TABLE 9 (Continued) Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. * ACCELERATION YIELD BUOYANCY DRAG ALPHA AVERAGE FACTOR FACTOR CORRECTION SETTLING FACTOR TIME 2.121640 0.024439 7.805151 1.461927 73.62040 2360 2.828854 0.024439 7.805151 1.790488 13.80382 586 2.828854 0.024439 7.805151 1.461927 31.05860 912 2.828854 0.024439 7.805151 1.266066 55.21530 1643 3.536067 0.024439 7.805151 1.601464 11.04306 511 3.536067 0.024439 7.805151 1.307587 24.84688 769 3.536067 0.024439 7.805151 1.132404 44.17224 1206 1.414427 0.021392 7.761904 2.073355 27.60765 1147 1.414427 0.021392 7.761904 1.692887 62.11721 1677 1.414427 0.021392 7.761904 1.466083 110.4306 2224 2.121640 0.021392 7.761904 1.692887 18.40510 623 2.121640 0.021392 7.761904 1.382227 41.41147 921 2.121640 0.021392 7.761904 1.197052 73.62040 1392 2.828854 0.021392 7.761904 1.466083 13.80382 328 2.828854 0.021392 7.761904 1.197052 31.05860 616 2.828854 0.021392 7.761904 1.036677 55.21530 904 3.536067 0.021392 7.761904 1.311305 11.04306 285 3.536067 0.021392 7.761904 1.070676 24.84688 447 3.536067 0.021392 7.761904 0.927232 44.17224 774 Average Values of Dimensionless Settling Time Based on Five Replicates (see Appendix C for all data) 6.2.1 55 Factor Analysis Kim (1975) described the single most; distinctive Charac- teristic of Factor Analysis is its data reduction capability. Given an array of correlation coefficients for a set of vari- ables, factor analytic techniques enables the user to see if some underlying pattern of relationship exists such that the data may be ”rearranged" or ”reduced" to a smaller set of factors or components that may then be taken as source variables accounting for the observed interrelations in data. Using the data from Tables Cl-C8 (Appendix C), Factor Analy- sis was performed using the SPSS Software (1975). The factoring method used was "Principal Factoring With Iteration". Two dif- ferent orthogonal rotation techniques were used. These were the VARIMAX and the EQUIMAX. The graphical representation of the rotated. factors that have been determined by the two rotations are shown in Figures 6 and 7. To interpret these graphs, the following observations have to be seen: 1. Relative distance of the variable from the two axes. 2. Direction of variable in relation to the axis. 3. Clustering of variables and their relative position to each other. This information is used to obtain the degree of correlation between the factors. Figures 6 and 7 indicate, however, that since all six vari- ables are spaced significantly apart from each other, they are 56 mutually orthogonal and independent. This indicates that all six variables must be considered together. This can also be attributed to the fact that initially eleven variables (4-24) have been reduced to six dimensionless terms (4-30) and these dimensionless terms tend to be orthogonal and independent. A stepwise linear regression was performed (Equation 6-3) using two out of five replicates chosen randomly from the data listed in Tables C1-C8. The results of the regression are listed in. Tables 10 .and 11. The values of regression coefficients obtained in this analysis were transformed to determine the coefficients needed for Equation (6-2). The prediction Equation (6-2) can, therefore, be written as 31 (Y)2.4581 (69.28575 (a) G0.72868 Y27.518 0.51152 X = 1013 x 10 (6-4) An Resquare of 0.946 was obtained. Since the variation in Buouancy Factor was very small in the experimental design, a very high coefficient was obtained in the regression analysis. If the average value of‘Y is substituted (Table 16), Equation (6-4) can be reduced to 6 ( )2.458l (6)0'28575 (“)0.5115 X a 3019 x 10 Goo72868 57 .eoMumNMHmsuoz humane spa: mMmMHmea uouomm eoumoom MasHm<> Mm mmeon ‘K-K'k-kfl‘kfi'kfifikfikflfifl fi¥¥¥¥¥¥¥¥¥¥¥¥fifikfifikflMfiflifikflMfii¥¥¥t v N mOBU H mOBodh AdBZONHmom HSHB (mmq< BzmHlohmOU w05m mmmzbz quHw ZOHB « H m0&0¢m AflBZONHmom M k s WSHB N m M éméflm s BZWHUHmhmOU 04mg N v em MMQZDZ MUZ¢NODQ N m « MWQZDZ QAMHM N N . onaemmgmooa u M 59 v.hmmmhm moufl>.ao>>am MOImN.omHm0H MOIMF.mNmmmN MOIMm.ommmvaeOImm.onmva MOImb.bhhmwN eclmm.meMNm moimv.mmmmma m0Iflh.NmommNIv0Iuw.HmNmbv VOImh.mmwhva voimm.m0thN «OINH.mmommmImOImm.mwv>vm veimm.mNHmmm meimm.mvmmmHIVOImm.mamme MOIWM.mNm0Nm vOImm.¢vmmva «OImv.NOVNmH AH 0H m m 5 NH HH 0H m m h NHmBm mhm.N bN.HNm mm.m vm.H NH wh.mmN Nmm.H Nm.HhH vm.N No.0 HH hh.va mom.o vm.mm om.o no.0I 0H Hm.mw wa.o mm.mm mm.o mm.o m mm.0Nm hH>.HI mm.meI Hm.HI Ho.NI w vN.ma mmN.o Nm.Nm mm.o mH.0I h moumsom mo 55m and: saw Essaxmz EDEHaHz UODOOMMODCD AMSHBV GOA NH A dofluooaom unooua3 moa on H .0: ammo sumo OWMHHADS “GOHHUGSM AAWQOS GOA. QHDAM mlm OZHB ZH mmmmmm m0 MZHB UZHABBmm "mauflfi aHngfimanJHmme OHHH mumn .GOMumnom doauoaoonm OGHEMODOQ 0» upon DEAR mafiauuom mo mHmMHMG¢ HMUfiumaumum 0H mqmdfi 60 h.omova belmm.mommmm FOIWV.OHthm mOImM.ththIBOIMm.mmeNe helmm.HmmvaI NH wolmm.NOOMHH welmo.vmmHNhIOHImh.vHNHNHImoImm.mwNHomIHOImh.mmmNMMI HH helmN.mNmbH¢ mOImH.HwawHImOImo.NHNvmm hOIMm.momOVHI OH HHIMN.mm>mVH CHIWm.mvmmmHIOHImN.mmmhmw m HH 0H m NOImm.wvommH NOImm.mmhmom m helmN.hvvmmm h w h XHmBOUIWUZ N.vmmmmH mOImm.mHmwNm mOImH.movoov helmN.mVMHmHImOImm.VMHomv mOImN.0NwmmMI NH QOIMm.NHm0NH melmm.mNVNbblmOIMF.mmmmNHImOIMm.mmmvwmImOImo.mwNNmmI HH mOImH.HmHth NOIwo.mowbhHIonmm.monNw mOImm.wmmomHI 0H mOImm.hHNomH melam.mHm50NImoImm.HmmNmm m HH 0H m meiwh.thmON meimm.vamvm w meimm.mmhmmm b w b meB m NoH echmwo.H Hmsprom m ombmmw.mH aOHmmoumom up moumsom mo sum 0 m H m M H 4 Z d ~oa.o n .umm no How enmeemum mem.o n m deMuasz mem.o u museum m eoumahea mqm.o nimumsum me cofiomcMEHODoa mo unmaofiuuooo NOImNmmo.m oo+mw NOImeN>.m oo+mN NOImvmmH.m HOIMm NOIMNth.N oo+Mm NOImmmmm.m oo+mN mommo.Hm u uaooumucH ooum OsHm> B Demesum «moo Hmauumm .mumoo no .uum .eum Hafiuumm oov.o NOImemm.m HOImNmHH.m HH mMH.o Neimmew.h HOIMmhmw.N OH vwh.I Ho+mwm0H.H Ho+mNHmh.NI m Now.o NOINNVFm.h oo+memv.N m 0Hm.I NOImmem.h HOImwmmN.hI h .Hmom Houum ucOHDHmmoou Honesz .oum nuancmum aonmouoOm OHQMHHm> .GOHumsom aoHuDHUOMm OEHB mcHHuuow How UOGHEMODOQ muaOHOHmuOOU COHmmouuom HH mqmda Factor Analysis of Data of Spherical Particle Settling in a Vibrating Herschel-Bulkley Fluid. O‘UlubWNH 63 TABLE 12 VARIMAX ROTATED FACTOR MATRIX AFTER ROTATION WITH KAISER NORMALIZATION ACC YLD BUO BETA ALPHA TIME ACC YLD BUO BETA ALPHA TIME FACTOR 1 -.19238 .55747 -.71262 .84175 -.03486 .76983 FACTOR 1 -.19238 .55747 -.71262 .84175 -.03486 .76983 FACTOR 2 -.70812 -.18685 -.19689 .09582 .83524 .37355 EQUIMAX ROTATED FACTOR MATRIX AFTER ROTATION WITH KAISER NORMALIZATION FACTOR 2 -.70812 -.18685 -.l9689 .09582 .83524 .37355 64 6.3 Gross Validation of Prediction Equation To determine the validity of the Equation (6-4), the remaining three of the five replicate data values from Tables C1-C8 were used and the predicted settling time was determined using Equation (6-4). If the validation error is sz, then the confidence level is equal to 1 - sz. Therefore, v - = u (xpredicted xobserved)2 v SSX (6-5) where ssx = 2x2 - (2x)2/N (6-6) From 162 sets of data considered (Table 16), XX = 248714.04 2x2 - 861256174.63 2 2:(X ) 3 33569955.77 predicted - xobserved 2 _ Therefore, 1 - Rv = 0.07 or R 3 0093 V Hence, this indicates a high degree of confidence in being able to predict the settling time using Equation (6-4). Figure 8 describes the plot between the observed versus predicted values of dimensionless settling time. 65 m5? NEH. 33:0,."chan mo 831$ @2538 mamum> @30ng mEc. 05.3mm mmmEchmEE nmimmno maxv mbfi 0N6 . . . b . P - 05mm mmfi "w ”mm—BE b mm. Ewe u «a awu Bugmag ssaluogsuawgg psiogpeud 6.4 6.5 66 Absolute Separation Criterion Equation (4-20) describes the separation criterion for the particle suspended in an H-B fluid to start moving as a result of vibration input. The critical vibration acceleration required for a certain spherical particle suspended in a Kelsat solution was determined (Table 13). Figure 9 describes the plot of critical acceleration versus the dimensionless. yield factor. The curve indicates the boundary between separation and no-separation con- dition for a suspended sphere. Effect of Random Vibration (Shipping Environ-ent) To investigate the effect of vibration forces seen during the shipping environment, three different Power Spectral Density Curves were considered. The three curves were developed by ‘Lansmont Corporation and are used to simulate the following environments: 1. Truck 2. Railcar 3. Combined Truck/Air A Solartron Model 1209 Random Vibration Controller was used to drive a Lansmont Model 6000-15 Electrohydraulic Vibration Table. A 30 minute vibration dwell was used and different combinations of sphere sizes and Kelsat solutions were used to determine the upper and lower bounds of the yield factor that will describe the settling of a suspended sphere (Table 14). In addition, the Combined Truck/Air Spectrum was used and the intensity of the spectrum increased in steps to determine an Abso- lute Separation case for random vibration input (Table 15). 67 TABLE 13 Absolute Separation Criterion of Spherical Particle Suspended in a Herschel-Bulkley Fluid Due to Sinusoidal Vibration. YIELD NUMBER ACCELERATION 0.033779 0.5657700 0.033779 0.4243280 0.033779 0.5657700 0.085148 0.7072135 0.085148 0.7072135 0.085148 0.8486560 0.042758 0.5657700 0.042758 0.4243280 0.042758 0.5657700 0.402281 6.3649220 0.402281 7.0721350 0.402281 6.7185280 0.229923 2.4752470 0.229923 2.1216400 0.229923 2.1216400 0.201267 1.5558690 0.201267 1.4144260 0.201267 1.5558690 6'. D! MING I ONLECS “COILIRR‘I‘ I OH ( 0- PE“ )6) 68 "IMHO" SMIIM 0" mm In li-B MID 1_a I SMIION GM! / 1/ g 51 // / L5. ‘ a T 1 1 r 0 .1 .2 ' .3 .4 .Si DIMIOIILBS YIELD ill- FIGURE 9: Absolute Separation Curve for Spherical Particle Suspended in a H-B Fluid Due to Sinusoidal Vibration. 69 TABLE 14 Upper and Lower Bounds of Yield Number that Predict Settling of a Particle Due to a 30 minute Random Vibration Dwell for Different Shipping Environments Simulated on a Electro-Hydraulic Vibration Table. SHIPPING YIELD NUMBER ENVIRONMENT SIMULATED No Separation Separation TRUCK 0.033843 0.016973 RAIL 0.016973 0.009695 TRUCK/AIR 0.033843 0.016973 70 TABLE 15 Absolute Separation Criterion of Spherical Particle Suspended in a Herschel-Bulkley Fluid Due to Random Vibration Using a Combined Truck-Air PSD Spectrum. YIELD NUMBER ACCELERATION (Ga/Hz) 0.016973 1.40 0.033843 1.40 0.042785 1.45 0.229927 1.85 71 Figures 10, 11 and 12 describe the three different spectrums used. Figures 13, 14 and 15 describe the Acceleration vs. Time output from the vibration table using an Endevco Piezoelectric Accelerometer and a Kikusui Model D SS 6520 Digital Storage Oscilloscope. Figure 16 describes the Separation Criterion for the Random Vibration input using a Combined Truck/Air Spectrum. Response curves were plotted between dimensionless settling time and each of the dimensionless variables. These were developed using the prediction Equation (6-4). The response of each dimen- sionless variable effecting the settling time was observed by holding the other variables constant at their mean value and vary- ing the response variables by two standard deviations from their mean values thereby representing 957’. of the experimental limits used (Figures 17, l8, 19, 20 and 21). 72 FREQUENCY (Hz) PSD (CZ/Hz) PSD (dB/OCT) ABORT (dB) 1 0.000001 10.00 10 4 0.010 0.00 10 16 0.010 - 7.56 10 40 0.001 0.00 10 80 0.001 -15.10 10 200 0.000001 0.00 10 IIHEI- 0439039? "WE TEEN-3 DEN_9 fi’~'_9 o kf fi* 1 .vfi 4‘7 +47 V . . . v1“ TVA—.24 fl *4 Av - ---~+—1 I a“ 1Mw*' .H«—Q~“W .—g——.‘-¢ , 5+ A 4— “4.0%3-4 . n V . 1 o ' , . .1! L; ‘ c l . TEEY fiVCE ’ FREG!H:3 i OJEFLif {RIOFT LIE: A 1660 IEELECI ! 4.4 TE‘T swim ‘50 m FIGURE 10: Power Spectral Density Curve for Simulation of Truck Transport (Lansmont Corporation). 73 FREQUENCY (Hz) PSD (G’le) PSD (dB/OCT) ABORT (dB) 1 0.000001 20.00 10 2 0.010 0.00 10 50 0.010 -10.00 10 200 0.000001 23.80 10 rxntz- 04=£¢=2z Pagan ’Esrn LE'v‘ELH-3 M" v: memos“ E-ez ,/_ . ~5: €7‘_ ' '12: avast 77 7777777777777 77 7:7 7 PSD ,/ . j i *f g 3 1: ,:, g-..-.l,,...-n.,..- ”.1... ”,4“, g _,_ N“ H: . - , , . . , u ' 7 1 ' 7 ‘ r 4 : ' E-l ll;;,.1 L:'l'.+ '1 1 TEST ave: . stenu:y o ovsataw iRDQFT Lin» 1000 EELEGT H OREFLfii ' RLRKM I h" on Linn trnz' FIGURE 11: Power Spectral Density Curve for Simulation of Rail Transport (Lansmont Corporation). 74 FREQUENCY (Hz) PSD (Gz/Hz) PSD (dB/OCT) ABORT (dB) 2 0.001 7.56 20 5 0.010 0.00 20 110 0.010 -1l.60 20 200 0.001 0.00 20 TIHE:- 04:21:55 "if m 'ESIIE LUEL'H—s “Vi—a l : . : ~ -. c... . . . . . . . > I a , . o O O - . . . . . » . . . ,—--—......-Q—W¢~ # : A :i.‘ . . . g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . o . . . A .. . . . . . . .. . . . . . . . , . . . , . . . . . . . . . . . . . . . . . . . n . - E.“ 11 ~ - - . c a - R __g;‘ A— A A4 I A TEE7 095E . FRE9aH:- 4 OVERLR‘ 'RLRSH L1H) FIGURE 12: Power Spectral Density Curve for Simulation of Combined Truck-Air Transport (Lansmont Corporation). 75 0.5 G's I Major Dlvlslon 50 msec I Major DMslon FIGURE 13: Acceleration Versus Time Plots for Random Vibration of Truck Transport Simulated on a Vibration Table. 76 0.5 G's I Major Dlvlslon 20 msec I Major Dlvlslon FIGURE 14: Acceleration Versus Time Plots for Random Vibration of Rail Transport Simulated on a Vibration Table. 1.0 G's I Major Division FIGURE 15: 77 10 msec I Major Division Acceleration Versus Time Plots for Random Vibration of Combined Truck-Air Transport Simulated on a Vibration Table. 78 m ”IMHO! :mnnou GIIEIION A SWIM" curmon 1.75. / Q / \ N c A O z z 1.5. U z A O I a A C a: u .3 B o 0 ‘1 -- . d .241 mansions: mm mm FIGURE 16: Absolute Separation Curve for Spherical Particle Suspended in a H-B Fluid Due to Random Vibration. 79 .GEHB mmeCOHmCQEHQ MSWHO> COHHMHGHGOU‘ wwwHCOHwCQF—HQ HO 0>HSU GWGOQmmm "NH WNDMVH...“ mEfi mcztom mmmEBmcoEE x 002. Donn coon oonN Doom Don P 000. com PbbbhbbPPbBLhP-bhbb.bbbbbbblhfibrbbb Jo l N UOQDJ9|933V SSSIUOESUSUJEG O 80 .mEHH wmoacoamcmEHn mamuo> HonEdz vane» mmoacowmcmefln mo m>usu uncommon "ma mmeHm 0E? mczfimm mmmEBmcoEE x 81% ooou 82 com. com o9. _ p _ . . p — . . . _ . r . _ . . . NFO.O w 4 mu. . w 1290 u e. . o 1 W. s m .085 s A 1 VA. A: T M e N 1.86 n 1. w nfi . .6 J . muod 8] .meH mwoacoflmcmEHn mamuo> Mogadz woomhosm mmoacoflmcmefln mo o>uso omcomwmm "ma mmwam 6E2. mezfiom mmoEBmcmEE x on». omm. spa. o¢__ one, coo. _ .....Lk.[..r#...p.»gbr........... . was $3. long A: f .va quwnN Kouoflorlg sseluogsuewgg 82 .mEflB mmmHCOHmcmEflQ mswum> powwoammmou mmun mmmaoowmcmEHQ mo m>usu owcoawmm "om Hmome mEt. oc=tom mmoEBmcoEE x 00m P ooW— 0mm 00¢ o P P iuegogyaoo 60.13 SSSIUOiSUSLUjQ 83 .mEflB mwoacoflmcmEaQ wsmum> Annmadv pnmflmm mmoHGOAmcmEHQ mo m>uso oncommmm 6:5. mEEom mmoEBmcmEE x DOON 00% — DOW- omm cm; 5 b b "Hm MMDth iqfigsH ssajuogsuawgg D 84 ooo.chnc~en~o cam.cwocmsnvcn-—m_a cv—.wonc~nma nnu.vhdom~uoov cco.vo-mvnmv hn~.~¢ca ~w~ a wink ho m¢<30m ocm.mns~vmv mac.-hmv-mhccoa ava.-~so~ vhfi.mmmaomnno com.mowosnn awn.v mod 0 .AUz<>O=G occ.c ccc.c o~°.° vo~.n «no.6 O~0.0 Nod m cunts: Gam~> mn0.c ham.o ~o~.~ uho.cmn onm.n fich.o ~o_ a 20nhoo macaquc> coo: Esm Easquz Essazqz ummou amass: cumccmbw «o z m—nc.ac> .mEHB ocfiauuom new cofiumsom :ofiuoacmum moan: mumn coflumnfiam> mmOuU mo mam>amc< Hmofiumaumum m. mqmFH 85 6.6 Sources of Error - Experimental and Theoretical Some of the experimental problems encountered in this study were: 1. Sensitivity of the load cell and the recorder used to determine the rheological constants of the H-B fluids using the back extrusion method. 2. Inability to maintain a perfectly vertical plun- ger moving into the fluid column, whereas any small inclination will result for radial compo- nents of fluid flow which are not acounted for in Osorio's (1985) model. 3. Presence of small air bubbles in the fluid cylin- der add to surface tension effects. 4. Density variations in the fluid due to minute clumping of powdered Kelsat particles. 5. Inability to obtain a pure sinusoidal input from the vibration table. 6. Human error. Some of the assumptions in the theoretical approach respon- sible for disagreement with the experimental values are: 1. Modelling Kelsat solutions as ideal Herschel- Bulkley fluids. 2. Laminar flow assumption. 3. One dimensional flow assumption. 4. Neglecting wall effects. 6.7 6.8 86 Condition for Settling of Particle During Shipping - Example Problem Let us consider the case of particles suspended in oil based salad dressings. The average radii of these small vegetable particles is about 0.002 m with a mass of 0.000125 kg (Peleg, 1983). Also, the fluid system consisting of the oily emulsion has very small yield shear stresses of less than 0.5 Pa (Kraft, Inc.). Knowing these three values, the yield number, Y, can be determined using Equation (4-30). For this case, Y 3 000161 From Figure 9, we determine that the vibration acceleration level needed to break this particle loose is 0.5 g. Also, Table 14 indicates that these particles will settle in all the three shipping modes since the yield number calculated is less than the lower bound levels for separation/no separation criterion. .A higher yield number would have permitted the particles to remain suspended in transit as shown in Section 6.8. Condition for Nb-Settling of Particle During Shipping A typical fluid food system with particles that do not separate during shipping is spaghetti sauce with vegetable and meat particles. The average size of these particles is the same as that discussed in 6.7. However, in this case the fluid system is a viscous tomato puree. A typical yield 87 shear stress of such tomato pastes is around 15 - 20 Pa (Table 1). Using these as the known parameters, the dimen- sionless yield number can be calculated for this case (Equation 4-30). A value of Y - 0.3466 was determined. From Figure 9 we see that acceleration levels as high as 6 g's are required to break the particles loose. These high levels are rare in vibration. Also, from Table 14, since this value is higher than the upper bounds for the separa- tion/no separation criterion for all three modes of shipping, we expect the particles to remain suspended during shipping. 7.0 CONCLUSIONS The following are the conclusions of this study: 1. 2. 3. 4. 5. 6. An 'absolute separation criterion' was established which defined the motion of a suspended spherical particle in a vibrating H-B fluid. The trajectory of a sphere moving in a vibrating Herschel-Bulkley fluid can be predicted with reason- able confidence using the prediction equation (R2 - 0.93). Particles suspended in an H—B fluid will separate and settle due to vibration observed in the shipping envi- ronment. It is possible to predict product quality of various fluid food systems (prevention from settling) during transportation. The absolute separation criterion can be used to design stable suspensions. This technique can be used as an accelerated falling ball viscometer to determine rheological coefficients of H-B fluids. 88 l. 8.0. FUTURE RESEARCH Effect of sphericity, S, and increased number of particles, L, on the settling of particles in an H—B fluid due to vibration. Develop an analytical constitutive equation between shear stress and shear rate for pseudoplastic fluid with yield stress based on first principles, and evaluate the drag force around a sphere. Effect of large changes in Buoyancy Factor on the settling of particles. 89 snuocmnr Bird, R.B. 1960. Phys. Fluids, 3:539. Bird, R.B., Armstrong, R.C., Hassanger, 0., Dynamics of Polymeric Liquids, Vol. 1, John Wiley and Sons, Inc., 1977. Charm S.E. 1962. The Nature and Role of Fluid Consistencyin Food Engineering Applications, Adv. Food Res., 11:356. Charm, S.E. 1963. Effect of Yield Stress on the Power Law Constants of Fluid Food Material Determined in Low Shear Rate Viscometers, Ing. Eng. Chem. (Proc. Design and Dev.), 2:62. Charm, S.E. and Merrill, E.W. 1959. Heat Transfer Coefficients for Pseudoplastic Materials in Laminar Flow, Food Res., 24:319. Cheng, D.C., A Design Procedure for Pipeline Flow of Non-Newtonian Dispersed Systems. In "Proceedings of the First International Conference on the Hydraulic Transport of Solids in Pipes", British Hydromechanics Research Association, Cranfield, Bedford, England, 1970. Cho, Y.I., The Study of Non-Newtonian Flows in the Falling Ball Vis- cometer, Ph.D. Thesis, University of Illinois at Chicago Circle, 1979. Cho, Y.I. Hartnett, J. 1977. Non-Newtonian Fluid Mech., 2:1. C01€man, BsDo, N011, W. 1961. an“. NOYO Acado SCis, 89:672. Curray, J.K., Analysis of Sphericity and Roundness of Quartz Grains. M.S. Thesis in Mineralogy. The Pennsylvania State University, PA. 1951. Darby, R., Viscoelastic Fluids, An Introduction to Their Properties and Behavior, Marcel Decker Inc., 1976. Dazhi, G., Tanner, R.I. 1985. The Drag on a Sphere in a Power-Law Fluid. Journal of Non-Newtonian Fluid Mechanics, 17:1-12. de waele, A. 1923. Oil and Color Chem. Assoc. Journal, 6:33. Ferry, J.D. Viscoelastic Properties of Polymers. John Wiley and Sons, New York. 19800 90 91 Greenberg, N.D., Foundations of Applied Mathematics, Prentice Hall, 1978. Hanks, R.W. 1979. The Axial Laminar Flow of Yiled-Pseudoplastic Fluids in a Concentric Annulus. Ind. Eng. Chem. Proces des. Dev. 18(3):488. Hanks, R.W., Ricks, B.L. 1974. Laminar-Turbulent Transition in Flow of Pseudoplastic Fluids With Yield Stresses. J. Hydronautic 8(4):163. Harper, J.C. 1961. Coaxial-Cylinder Viscometer for Non-Newtonian Fluids, Rev. Sci. Instrum. 32:557. Harper, J.C. and El Sahrigi, A.F. 1965. Viscometric Behavior of Tomato Concentrates, J. Food Sci. 30:470. Harris, J., Rheology and Non-Newtonian Flow, Longman Pubishers, 1977. Higgs, S.J. and Norrington, R.J. 1971. Rheological Properties of Selected Foodstuffs, Process Ciochem. 615:52. Holdsworth, S.D. 1971. Applicabiity of Rheological Models to the Interpretation of Flow and Processing Behavior of Fluid Food Products, J. of Texture Studies, 2:393-418. Kim, J.0., Statistical Package for the Social Sciences, McGraw Hill, 1975. Kincaid, J.F., Eyring, H., Stearn, A.F. 1941. Chem. Revs., 28:01. Langhaar, B.L., Dimensional Analysis and Theory of Model, Wiley & Sons, New York, NY. 1951. Metzner, A.B., Advances in Heat Transfer, Academic Press, New York, Vol. 2, p. 357, 1965. Moshenin, N.N. Physical Properties of Plant and Animal Material. Gordon and Breach Science Publishers, New York. 1970. Murphy, G., Similitude in Engineering, The Ronald Press, New York, 1970. Osorio, F.A., Back Extrusion of Power Law, Bingham Plastic and Herschel- Bulkley Fluids. M.S. Thesis, Dept. of Food Science and Human Nutrition, Michign State University, East Lansing, Michigan, USA. 1985. Osorio, F.A., Steffe, J.F. Computer program "rehoproph6b:, Michigan State University, East Lansing, Michigan, USA. 1985. Osorio, F.A., Steffe, J.F. Back Extrusion of Herschel-Bulkley Fluids - Example Problem, A.S.A.E. Paper No. 85-6004, 1985. 92 Ostrem, F.E., Godshall, W.D. An Assessment of the Common Carrier Shipping Environment, General Technical Report, FPL22, 1979. Ostwald, W. 1925. Kolloid-Z., 36:99. Peleg, M. and E.G. Bagley, Editors. Physical Properties of Foods. AVI Publishing Company, Inc., Westport, CT. 1983. Powell, R.E., Eyring, H. 1944. Nature, 154:427. Reiner, M. Deformatin, Strain and Flow, Interscience, New York, 1960. Saravacos, G.D. 1970. Effect of Temperature on Viscosity of Fruit Juices and Purees, J. Food Sci., 35:122. Saravacos, G.D. and Moyer, J.C. 1967. Heating Rates of Fruit Products in an Agitated Kettle, Food Technol., 21:372. Skelland, A.H.P., Non-Newtonian Flow and Heat Transfer, John Wiley and Sons, Inc., New York. 1967. Stokes, G.G. 1851. Camb. Phil. Trans., 9:8. Sutterby, J.L., Ph.D. Thesis, University of Wisconsin, Madison. 1964. Walters, K. Rheometry. Chapman and Hall, London. 1975. Wasaerman, MoLo, Slattery, J.C. 19640 AoIoChoEo Journal, 10:383. Whorlow, R.S. Rheological Techniques. Halsted Press, New York. 1980. Wood, F.W., Psychophysical Studies on the Consistency of Liquid Foods, (In Rheology and Texture of Foodstuffs, SCI Monograph No. 27), The Society of Chemical Industry, London, 1968. Ultman, J.S., Denn, M.M. 1970. Trans. Soc. Rheol., 14:307. APPENDIX A 93 TABLE A-l Experimental Values Obtained From Back Extrusion Test (Osorio and Steffe) for a 1.00 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. PLUNGER EXPERIMENT RADIUS (cm) 1 2 3 4 1.00 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 25.0 Lch, cm 13.4 17.6 24.0 21.4 F , N 0.4147 0.4047 0.4077 0.3997 F:e, N 0.5147 0.4897 0.4677 0.4377 1.20 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 25.0 Lch, cm 13.1 17.3 23.6 21.0 F , N 0.7195 0.7295 0.6995 0.7095 F:e, N 0.9434 0.9194 0.8994 0.7745 1.50 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 37.5 Lch, cm 6.2 7.4 10.2 13.5 F , N 0.9693 0.9114 0.9194 0.8994 FTe, N 1.7788 1.4990 1.4590 1.1942 T PLU] RADIU 1.0 94 TABLE A-Z Experimental Values Obtained From Back Extrusion Test (Osorio and Steffe) for a 1.25 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. PLUNGER EXPERIMENT RADIUS (cm) 1 2 3 4 1.00 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 46.9 37.5 25.0 Lch, cm 12.3 10.7 12.0 21.6 F , N 0.3817 0.3967 0.4018 0.4057 F:e, N 0.6077 0.5889 0.5607 0.4910 1.20 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 46.9 37.5 25.0 Lch, cm 31.2 10.3 11.6 21.7 F , N 0.6955 0.6875 0.7015 0.7255 F:e, N 1.2405 1.1142 1.0623 0.9269 1.50 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 46.9 Lch, cm 14.1 17.0 16.0 18.3 F , N 0.9294 0.8964 0.8774 1.0278 Te F , N 2.1385 2.2035 1.9896 1.7918 95 TABLE A-3 Experimental Values Obtained From Back Extrusion Test (Osorio and Steffe) for a 1.50 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. PLUNGER EXPERIMENT RADIUS (cm) 1 2 3 4 1.00 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 37.5 25.0 18.8 Lch, cm 11.3 8.9 8.4 17.7 F , N 0.3508 0.4117 0.4237 0.4397 F:e, N 0.7941 0.8692 0.8155 0.6466 1.20 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 37.5 25.0 18.8 Lch, cm 11.0 6.6 6.8 14.0 F , N 0.6096 0.5546 0.6226 0.6276 F:e, N 1.6019 1.3861 1.4000 1.0633 1.50 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 37.5 Lch, cm 15.6 20.8 13.6 18.3 F , N 1.0793 1.0993 1.2492 1.2591 FTe, N 4.7728 4.6878 5.4435 3.3169 96 TABLE A-4 Experimental Values Obtained From Back Extrusion Test (Osorio and Steffe) for a 1.75 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. PLUNGER EXPERIMENT RADIUS (cm) 1 2 3 4 1.00 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 62.5 37.5 Lch, cm 21.1 27.3 20.4 33.4 F , N 0.4177 0.3997 0.4257 0.4257 F:e, N 1.0613 0.9215 0.8826 0.6908 1.20 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 62.5 37.5 25.0 Lch, cm 18.0 13.1 10.6 19.4 F , N 0.6046 0.6499 0.6536 0.6665 F:e, N 1.7178 1.7818 1.5559 1.1802 1.50 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 62.5 25.0 Lch, cm 12.6 16.3 12.1 13.5 F , N 1.3042 1.2492 1.3191 1.3881 FTe, N 7.9866 6.0629 6.3877 4.0373 97 TABLE A-5 Kelsat After 24 Hours of Storage at 24 C. Experimental Values Obtained From Back Extrusion Test (Osorio and Steffe) for a 2.00 % Aqueous Solution Of PLUNGER EXPERIMENT RADIUS (cm) 1 2 3 4 1.00 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 37.5 Lch, cm 12.2 15.9 22.3 28.1 F , N 0.5196 0.4797 0.5946 0.4547 F:e, N 2.0536 1.7838 1.6489 1.0853 1.20 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 46.9 Lch, cm 21.2 13.9 19.5 27.4 F , N 0.6296 0.7245 0.7095 0.6895 F:e, N 2.7911 2.6482 2.6982 1.9687 1.50 Vp, cm/s 50.0 37.2 27.0 10.0 Csp,cm/s 75.0 75.0 75.0 46.9 Lch, cm 12.3 16.9 20.3 16.1 F , N 1.1992 1.3895 0.9044 1.0753 FTe, N ****** ****** 6.5476 5.9971 T APPENDIX B TABLE B-l 98 Settling Time Data for Steel Spheres in a Vibrating 1.25 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. INCHES ACCELERATION FREQUENCY (O-PEAK) G's HERTZ SETTLING TIME SECS COEFFICIENT OF VARIATION 1/16 1. 1. 5 10 15 20 10 15 20 67.45 66.10 68.14 66.43 67.37 69.39 70.75 71.04 70.59 71.70 75.61 77.16 73.19 74.56 76.93 39.50 37.95 38.62 36.70 37.55 40.45 42.81 41.28 42.59 40.64 42.32 43.53 42.15 44.26 42.52 0.02 99 TABLE B-2 Settling Time Data for Steel Spheres in a Vibrating 1.25 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/8 0.5 10 23.48 22.65 23.11 0.02 22.65 23.54 15 24.94 26.08 25.29 0.02 25.03 25.26 20 26.27 27.21 26.23 0.02 26.89 26.17 1.0 10 8.67 9.19 8.98 0.04 8.50 9.31 15 9.43 10.40 9.89 0.04 9.48 9.92 20 9.82 10.17 10.87 0.05 9.72 10.80 100 TABLE B-3 Settling Time Data for Steel Spheres in a Vibrating 1.25 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/8 1.5 10 4.87 4.97 5.18 0.04 4.62 4.98 15 5.31 5.11 5.58 0.07 6.15 5.41 20 5.52 5.09 6.48 0.10 6.51 5.73 101 TABLE B-4 Settling Time Data for Steel Spheres in a Vibrating 1.50 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/8 1.0 10 17.8 15.6 17.7 0.05 17.5 16.4 15 16.3 14.2 15.6 0.05 16.1 16.4 20 17.8 15.6 16.8 0.05 16.1 17.2 2.0 10 0.05 15 0.03 20 0.05 mmqmm mmmmm mmmmm mmeN COHU‘IO ooxoooom 102 TABLE B-5 Settling Time Data for Steel Spheres in a Vibrating 1.50 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/8 2.5 10 3.6 3.7 3.4 0.03 3.6 3.6 15 3.9 3.6 3.6 0.03 3.8 3.7 20 4.4 3.6 3.8 0.09 4.5 3.9 7/32 1.0 10 2.3 2.4 2.4 0.04 2.6 2.4 15 2.8 2.9 2.8 0.04 3.1 3.0 20 3.1 3.3 3.2 0.04 3.3 3.5 103 TABLE B-6 Settling Time Data for Steel Spheres in a Vibrating 1.50 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 7/32 1.5 10 1.5 1.4 1.5 0.04 1.5 1.6 15 2.2 2.2 2.1 0.05 2.4 2.2 20 2.4 2.6 2.3 0.04 2.4 2.4 104 TABLE B-7 Settling Time Data for Steel Spheres in a Vibrating 1.75 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 7/32 1.0 10 48.2 45.3 50.5 0.03 46.5 47.7 15 23.3 24.1 22.4 0.02 23.6 23.1 20 23.6 22.9 23.5 0.01 23.6 23.5 1.5 10 17.0 15.3 16.1 0.04 15.3 15.5 15 16.5 16.8 16.2 0.03 17.8 17.1 20 18.9 18.8 19.2 0.01 18. 18.5 105 TABLE B-8 Settling Time Data for Steel Spheres in a Vibrating 1.75 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 7/32 2.0 10 9.3 9.3 9.6 0.01 9.2 9.3 15 9.6 9.8 9.4 0.01 9.8 9.8 20 13.2 12.9 12.9 0.01 13.0 13.4 2.5 10 8.6 8.0 8.0 0.03 7.8 8.3 15 6.1 8.3 9.2 0.06 8.1 9.1 20 9.8 9.5 9.6 0.02 9.3 9.8 106 TABLE B-9 Settling Time Data for Steel Spheres in a Vibrating 1.75 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/4 1.0 10 18.0 18.7 18.2 0.03 18.9 17.5 15 16.5 17.9 18.9 0.06 16.6 19.1 20 17.2 16.4 17.4 0.05 FHA an» O \HD 1.5 10 end 0.05 15 0.01 H \omxouuo \oxomoo 20 thPJ hue O 0.01 OWOH «momma quantum H 3.: O 11.0 107 TABLE B-lO Settling Time Data for Steel Spheres in a Vibrating 1.75 % Aqueous Solution Of Kelsat After 24 Hours of Storage at 24 C. SPHERE DIA. ACCELERATION FREQUENCY SETTLING TIME COEFFICIENT INCHES (O-PEAK) G's HERTZ SECS OF VARIATION 1/4 2.0 10 5.2 5.2 5.6 0.04 5.0 5.1 15 6.9 6.2 6.6 0.05 6.8 6.2 20 7.8 8.4 7.2 0.05 7.6 7.9 2.5 10 4.5 4.8 4.3 0.05 4.3 4.8 15 4.6 4.4 4.5 0.07 5.3 4.9 20 5.9 6.1 5.8 0.05 6.4 6.6 APPENDIX C 108 TABLE C-l Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 1.414427 0.031120 7.819548 6.244965 27.60765 4237.88 1.414427 0.031120 7.819548 6.244965 27.60765 4153.06 1.414427 0.031120 7.819548 6.244965 27.60765 4281.23 1.414427 0.031120 7.819548 6.244965 27.60765 4173.79 1.414427 0.031120 7.819548 6.244965 27.60765 4232.85 1.414427 0.031120 7.819548 5.098993 62.11721 6540.00 1.414427 0.031120 7.819548 5.098993 62.11721 6668.18 1.414427 0.031120 7.819548 5.098993 62.11721 6695.52 1.414427 0.031120 7.819548 5.098993 62.11721 6653.10 1.414427 0.031120 7.819548 5.098993 62.11721 6757.72 1.414427 0.031120 7.819548 4.415857 110.4306 9501.15 1.414427 0.031120 7.819548 4.415857 110.4306 9695.92 1.414427 0.031120 7.819548 4.415857 110.4306 9197.05 1.414427 0.031120 7.819548 4.415857 110.4306 9369.20 1.414427 0.031120 7.819548 4.415857 110.4306 9667.02 2.121640 0.031120 7.819548 5.098993 18.40510 2481.78 2.121640 0.031120 7.819548 5.098993 18.40510 2384.39 2.121640 0.031120 7.819548 5.098993 18.40510 2426.49 2.121640 0.031120 7.819548 5.098993 18.40510 2305.86 2.121640 0.031120 7.819548 5.098993 18.40510 2359.26 2.121640 0.031120 7.819548 4.163310 4.141147 3812.41 2.121640 0.031120 7.819548 4.163310 4.141147 4034.84 2.121640 0.031120 7.819548 4.163310 4.141147 3890.64 2.121640 0.031120 7.819548 4.163310 4.141147 4014.10 2.121640 0.031120 7.819548 4.163310 4.141147 3830.32 2.121640 0.031120 7.819548 3.605532 73.62040 5317.93 2.121640 0.031120 7.819548 3.605532 73.62040 5469.97 2.121640 0.031120 7.819548 3.605532 73.62040 5296.69 2.121640 0.031120 7.819548 3.605532 73.62040 5561.71 2.121640 0.031120 7.819548 3.605532 73.62040 5343.06 0.707213 0.015608 7.819548 3.132090 55.21530 1475.24 0.707213 0.015608 7.819548 3.132090 55.21530 1423.09 0.707213 0.015608 7.819548 3.132090 55.21530 1452.00 0.707213 0.015608 7.819548 3.132090 55.21530 1423.09 0.707213 0.015608 7.819548 3.132090 55.21530 1479.01 109 TABLE C-Z Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 0.707213 0.015608 7.819548 2.557341 124.2344 2350.59 0.707213 0.015608 7.819548 2.557341 124.2344 2458.04 0.707213 0.015608 7.819548 2.557341 124.2344 2383.58 0.707213 0.015608 7.819548 2.557341 124.2344 2359.07 0.707213 0.015608 7.819548 2.557341 124.2344 2380.75 0.707213 0.015608 7.819548 2.214722 220.8612 3301.08 0.707213 0.015608 7.819548 2.214722 220.8612 3419.20 0.707213 0.015608 7.819548 2.214722 220.8612 3296.06 0.707213 0.015608 7.819548 2.214722 220.8612 3378.99 0.707213 0.015608 7.819548 2.214722 220.8612 3288.52 1.414427 0.015608 7.819548 2.214722 27.60765 544.73 1.414427 0.015608 7.819548 2.214722 27.60765 577.40 1.414427 0.015608 7.819548 2.214722 27.60765 564.21 1.414427 0.015608 7.819548 2.214722 27.60765 534.05 1.414427 0.015608 7.819548 2.214722 27.60765 584.94 1.414427 0.015608 7.819548 1.808313 62.11721 888.77 1.414427 0.015608 7.819548 1.808313 62.11721 980.20 1.414427 0.015608 7.819548 1.808313 62.11721 932.13 1.414427 0.015608 7.819548 1.808313 62.11721 893.49 1.414427 0.015608 7.819548 1.808313 62.11721 934.96 1.414427 0.015608 7.819548 1.566045 110.4306 1233.98 1.414427 0.015608 7.819548 1.566045 110.4306 1277.96 1.414427 0.015608 7.819548 1.566045 110.4306 1365.92 1.414427 0.015608 7.819548 1.566045 110.4306 1221.41 1.414427 0.015608 7.819548 1.566045 110.4306 1357.12 2.121640 0.015608 7.819548 1.808313 18.40510 305.98 2.121640 0.015608 7.819548 1.808313 18.40510 312.26 2.121640 0.015608 7.819548 1.808313 18.40510 325.45 2.121640 0.015608 7.819548 1.808313 18.40510 290.27 2.121640 0.015608 7.819548 1.808313 18.40510 312.89 2.121640 0.015608 7.819548 1.476481 41.41147 500.46 2.121640 0.015608 7.819548 1.476481 41.41147 481.61 2.121640 0.015608 7.819548 1.476481 41.41147 525.91 2.121640 0.015608 7.819548 1.476481 41.41147 579.63 2.121640 0.015608 7.819548 1.476481 41.41147 509.89 110 TABLE C-3 Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 2.121640 0.015608 7.819548 1.278670 73.62040 693.64 2.121640 0.015608 7.819548 1.278670 73.62040 639.60 2.121640 0.015608 7.819548 1.278670 73.62040 814.27 2.121640 0.015608 7.819548 1.278670 73.62040 818.04 2.121640 0.015608 7.819548 1.278670 73.62040 720.03 1.414427 0.016973 7.805151 5.115697 27.60765 1118.37 1.414427 0.016973 7.805151 5.115697 27.60765 980.14 1.414427 0.016973 7.805151 5.115697 27.60765 1112.09 1.414427 0.016973 7.805151 5.115697 27.60765 1099.52 1.414427 0.016973 7.805151 5.115697 27.60765 1030.41 1.414427 0.016973 7.805151 4.176949 62.11721 1536.27 1.414427 0.016973 7.805151 4.176949 62.11721 1338.35 1.414427 0.016973 7.805151 4.176949 62.11721 1470.30 1.414427 0.016973 7.805151 4.176949 62.11721 1517.42 1.414427 0.016973 7.805151 4.176949 62.11721 1545.70 1.414427 0.016973 7.805151 3.617344 110.4306 2236.74 1.414427 0.016973 7.805151 3.617344 110.4306 1960.29 1.414427 0.016973 7.805151 3.617344 110.4306 2111.08 1.414427 0.016973 7.805151 3.617344 110.4306 2023.12 1.414427 0.016973 7.805151 3.617344 110.4306 2161.35 2.828854 0.016973 7.805151 3.617344 13.80382 411.70 2.828854 0.016973 7.805151 3.617344 13.80382 424.18 2.828854 0.016973 7.805151 3.617344 13.80382 374.28 2.828854 0.016973 7.805151 3.617344 13.80382 430.42 2.828854 0.016973 7.805151 3.617344 13.80382 424.18 2.828854 0.016973 7.805151 2.953549 31.05860 565.50 2.828854 0.016973 7.805151 2.953549 31.05860 612.62 2.828854 0.016973 7.805151 2.953549 31.05860 574.92 2.828854 0.016973 7.805151 2.953549 31.05860 565.50 2.828854 0.016973 7.805151 2.953549 31.05860 565.50 2.828854 0.016973 7.805151 2.557848 55.21530 779.09 2.828854 0.016973 7.805151 2.557848 55.21530 854.48 2.828854 0.016973 7.805151 2.557848 55.21530 892.18 2.828854 0.016973 7.805151 2.557848 55.21530 804.22 2.828854 0.016973 7.805151 2.557848 55.21530 829.35 Hi TABLE C-4 Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 3.536067 0.016973 7.805151 3.325450 11.04306 226.18 3.536067 0.016973 7.805151 3.325450 11.04306 232.47 3.536067 0.016973 7.805151 3.325450 11.04306 213.62 3.536067 0.016973 7.805151 3.325450 11.04306 226.18 3.536067 0.016973 7.805151 3.325450 11.04306 226.18 3.536067 0.016973 7.805151 2.641734 24.84688 367.57 3.536067 0.016973 7.805151 2.641734 24.84688 339.30 3.536067 0.016973 7.805151 2.641734 24.84688 339.30 3.536067 0.016973 7.805151 2.641734 24.84688 358.15 3.536067 0.016973 7.805151 2.641734 24.84688 348.72 3.536067 0.016973 7.805151 2.287809 44.17224 552.90 3.536067 0.016973 7.805151 2.287809 44.17224 452.37 3.536067 0.016973 7.805151 2.287809 44.17224 477.50 3.536067 0.016973 7.805151 2.287809 44.17224 565.47 3.536067 0.016973 7.805151 2.287809 44.17224 490.07 1.414427 0.009695 7.805151 2.208881 27.60765 144.50 1.414427 0.009695 7.805151 2.208881 27.60765 150.79 1.414427 0.009695 7.805151 2.208881 27.60765 150.79 1.414427 0.009695 7.805151 2.208881 27.60765 163.35 1.414427 0.009695 7.805151 2.208881 27.60765 150.79 1.414427 0.009695 7.805151 1.803544 62.11721 263.90 1.414427 0.009695 7.805151 1.803544 62.11721 273.32 1.414427 0.009695 7.805151 1.803544 62.11721 263.90 1.414427 0.009695 7.805151 1.803544 62.11721 292.17 1.414427 0.009695 7.805151 1.803544 62.11721 282.75 1.414427 0.009695 7.805151 1.561915 110.4306 389.54 1.414427 0.009695 7.805151 1.561915 110.4306 414.67 1.414427 0.009695 7.805151 1.561915 110.4306 402.11 1.414427 0.009695 7.805151 1.561915 110.4306 414.67 1.414427 0.009695 7.805151 1.561915 110.4306 439.81 2.121640 0.009695 7.805151 1.803544 18.40510 94.24 2.121640 0.009695 7.805151 1.803544 18.40510 87.96 2.121640 0.009695 7.805151 1.803544 18.40510 94.24 2.121640 0.009695 7.805151 1.803544 18.40510 94.24 2.121640 0.009695 7.805151 1.803544 18.40510 100.52 112 TABLE C-5 Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 2.121640 0.009695 7.805151 1.472587 41.41147 207.35 2.121640 0.009695 7.805151 1.472587 41.41147 207.35 2.121640 0.009695 7.805151 1.472587 41.41147 197.92 2.121640 0.009695 7.805151 1.472587 41.41147 226.20 2.121640 0.009695 7.805151 1.472587 41.41147 207.35 2.121640 0.009695 7.805151 1.275298 73.62040 301.58 2.121640 0.009695 7.805151 1.275298 73.62040 326.71 2.121640 0.009695 7.805151 1.275298 73.62040 289.01 2.121640 0.009695 7.805151 1.275298 73.62040 301.58 2.121640 0.009695 7.805151 1.275298 73.62040 301.58 1.414427 0.024439 7.805151 2.532132 27.60765 3028.40 1.414427 0.024439 7.805151 2.532132 27.60765 2846.19 1.414427 0.024439 7.805151 2.532132 27.60765 3172.91 1.414427 0.024439 7.805151 2.532132 27.60765 2921.59 1.414427 0.024439 7.805151 2.532132 27.60765 2996.99 1.414427 0.024439 7.805151 2.067477 62.11721 2196.02 1.414427 0.024439 7.805151 2.067477 62.11721 2271.42 1.414427 0.024439 7.805151 2.067477 62.11721 2111.20 1.414427 0.024439 7.805151 2.067477 62.11721 2224.30 1.414427 0.024439 7.805151 2.067477 62.11721 2177.17 1.414427 0.024439 7.805151 1.790488 110.4306 2965.57 1.414427 0.024439 7.805151 1.790488 110.4306 2877.61 1.414427 0.024439 7.805151 1.790488 110.4306 2953.01 1.414427 0.024439 7.805151 1.790488 110.4306 2965.57 1.414427 0.024439 7.805151 1.790488 110.4306 2953.01 2.121640 0.024439 7.805151 2.067477 18.40510 1068.11 2.121640 0.024439 7.805151 2.067477 18.40510 961.29 2.121640 0.024439 7.805151 2.067477 18.40510 1011.56 2.121640 0.024439 7.805151 2.067477 18.40510 961.29 2.121640 0.024439 7.805151 2.067477 18.40510 973.86 2.121640 0.024439 7.805151 1.688088 41.41147 1555.12 2.121640 0.024439 7.805151 1.688088 41.41147 1583.40 2.121640 0.024439 7.805151 1.688088 41.41147 1526.85 2.121640 0.024439 7.805151 1.688088 41.41147 1677.65 2.121640 0.024439 7.805151 1.688088 41.41147 1611.67 113 TABLE C-6 Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 2.121640 0.024439 7.805151 1.461927 73.62040 2374.97 2.121640 0.024439 7.805151 1.461927 73.62040 2362.40 2.121640 0.024439 7.805151 1.461927 73.62040 2412.67 2.121640 0.024439 7.805151 1.461927 73.62040 2324.71 2.121640 0.024439 7.805151 1.461927 73.62040 2324.71 2.828854 0.024439 7.805151 1.790488 13.80382 584.31 2.828854 0.024439 7.805151 1.790488 13.80382 584.31 2.828854 0.024439 7.805151 1.790488 13.80382 603.16 2.828854 0.024439 7.805151 1.790488 13.80382 578.03 2.828854 0.024439 7.805151 1.790488 13.80382 584.31 2.828854 0.024439 7.805151 1.461927 31.05860 904.80 2.828854 0.024439 7.805151 1.461927 31.05860 923.65 2.828854 0.024439 7.805151 1.461927 31.05860 885.95 2.828854 0.024439 7.805151 1.461927 31.05860 923.65 2.828854 0.024439 7.805151 1.461927 31.05860 923.65 2.828854 0.024439 7.805151 1.266066 55.21530 1658.71 2.828854 0.024439 7.805151 1.266066 55.21530 1621.01 2.828854 0.024439 7.805151 1.266066 55.21530 1621.01 2.828854 0.024439 7.805151 1.266066 55.21530 1633.58 2.828854 0.024439 7.805151 1.266066 55.21530 1683.84 3.536067 0.024439 7.805151 1.601464 11.04306 540.33 3.536067 0.024439 7.805151 1.601464 11.04306 502.64 3.536067 0.024439 7.805151 1.601464 11.04306 502.64 3.536067 0.024439 7.805151 1.601464 11.04306 490.07 3.536067 0.024439 7.805151 1.601464 11.04306 521.48 3.536067 0.024439 7.805151 1.307587 24.84688 574.92 3.536067 0.024439 7.805151 1.307587 24.84688 782.27 3.536067 0.024439 7.805151 1.307587 24.84688 867.10 3.536067 0.024439 7.805151 1.307587 24.84688 763.42 3.536067 0.024439 7.805151 1.307587 24.84688 857.67 3.536067 0.024439 7.805151 1.132404 44.17224 1231.46 3.536067 0.024439 7.805151 1.132404 44.17224 1193.77 3.536067 0.024439 7.805151 1.132404 44.17224 1206.33 3.536067 0.024439 7.805151 1.132404 44.17224 1168.63 3.536067 0.024439 7.805151 1.132404 44.17224 1231.46 114 TABLE C-7 Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 1.414427 0.021392 7.761904 2.073355 27.60765 1130.94 1.414427 0.021392 7.761904 2.073355 27.60765 1174.92 1.414427 0.021392 7.761904 2.073355 27.60765 1143.50 1.414427 0.021392 7.761904 2.073355 27.60765 1187.48 1.414427 0.021392 7.761904 2.073355 27.60765 1099.52 1.414427 0.021392 7.761904 1.692887 62.11721 1555.12 1.414427 0.021392 7.761904 1.692887 62.11721 1687.07 1.414427 0.021392 7.761904 1.692887 62.11721 1781.32 1.414427 0.021392 7.761904 1.692887 62.11721 1564.55 1.414427 0.021392 7.761904 1.692887 62.11721 1800.17 1.414427 0.021392 7.761904 1.466083 110.4306 2161.35 1.414427 0.021392 7.761904 1.466083 110.4306 2060.82 1.414427 0.021392 7.761904 1.466083 110.4306 2186.48 1.414427 0.021392 7.761904 1.466083 110.4306 2362.40 1.414427 0.021392 7.761904 1.466083 110.4306 2349.84 2.121640 0.021392 7.761904 1.692887 18.40510 640.86 2.121640 0.021392 7.761904 1.692887 18.40510 665.99 2.121640 0.021392 7.761904 1.692887 18.40510 615.73 2.121640 0.021392 7.761904 1.692887 18.40510 584.31 2.121640 0.021392 7.761904 1.692887 18.40510 609.45 2.121640 0.021392 7.761904 1.382227 41.41147 914.22 2.121640 0.021392 7.761904 1.382227 41.41147 933.07 2.121640 0.021392 7.761904 1.382227 41.41147 904.80 2.121640 0.021392 7.761904 1.382227 41.41147 933.07 2.121640 0.021392 7.761904 1.382227 41.41147 923.65 2.121640 0.021392 7.761904 1.197052 73.62040 1394.82 2.121640 0.021392 7.761904 1.197052 73.62040 1382.26 2.121640 0.021392 7.761904 1.197052 73.62040 1419.95 2.121640 0.021392 7.761904 1.197052 73.62040 1382.26 2.121640 0.021392 7.761904 1.197052 73.62040 1382.26 2.828854 0.021392 7.761904 1.466083 13.80382 326.71 2.828854 0.021392 7.761904 1.466083 13.80382 326.71 2.828854 0.021392 7.761904 1.466083 13.80382 351.84 2.828854 0.021392 7.761904 1.466083 13.80382 314.15 2.828854 0.021392 7.761904 1.466083 13.80382 320.43 115 TABLE C-8 Dimensionless Values of Acceleration, Yield Factor, Buoyancy Factor, Drag Correction Coefficient, Alpha and Settling Time for Accelerated Settling of a Spherical Particle in a Vibrating Herschel-Bulkley Fluid Model. ACCELERATION YIELD BUOYANCY DRAG ALPHA SETTLING FACTOR FACTOR CORRECTION TIME FACTOR 2.828854 0.021392 7.761904 1.197052 31.05860 650.32 2.828854 0.021392 7.761904 1.197052 31.05860 584.35 2.828854 0.021392 7.761904 1.197052 31.05860 622.05 2.828854 0.021392 7.761904 1.197052 31.05860 640.90 2.828854 0.021392 7.761904 1.197052 31.05860 584.35 2.828854 0.021392 7.761904 1.036677 55.21530 980.14 2.828854 0.021392 7.761904 1.036677 55.21530 1055.54 2.828854 0.021392 7.761904 1.036677 55.21530 904.75 2.828854 0.021392 7.761904 1.036677 55.21530 955.01 2.828854 0.021392 7.761904 1.036677 55.21530 922.71 3.536067 0.021392 7.761904 1.311305 11.04306 282.73 3.536067 0.021392 7.761904 1.311305 11.04306 301.58 3.536067 0.021392 7.761904 1.311305 11.04306 270.16 3.536067 0.021392 7.761904 1.311305 11.04306 270.16 3.536067 0.021392 7.761904 1.311305 11.04306 301.58 3.536067 0.021392 7.761904 1.070676 24.84688 433.55 3.536067 0.021392 7.761904 1.070676 24.84688 414.70 3.536067 0.021392 7.761904 1.070676 24.84688 424.12 3.536067 0.021392 7.761904 1.070676 24.84688 499.52 3.536067 0.021392 7.761904 1.070676 24.84688 461.82 3.536067 0.021392 7.761904 0.927232 44.17224 741.39 3.536067 0.021392 7.761904 0.927232 44.17224 766.52 3.536067 0.021392 7.761904 0.927232 44.17224 728.82 3.536067 0.021392 7.761904 0.927232 44.17224 804.22 3.536067 0.021392 7.761904 0.927232 44.17224 829.35 APPENDIX D APPENDIX D VALUES or DRAG coanacrron FACTOR, €(n) (Dahzi, 1985) Power Law Index, n Drag Correction Factor, e(n) 1.0 1.002 0.9 1.140 0.8 1.240 0.7 1.320 0.6 1.382 0.5 1.420 0.4 1.442 0.3 1.458 0.2 1.413 0.1 1.354 116 ll 1 1 WI Iii/1111111111 10758 3076 1 my; j if!