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' 5'.‘ (m. =32 :2I 22““. ‘S‘I‘H‘Ri‘finz.II‘iIL.I‘l“W‘}ItI.II. .fiIII‘aJIIIIJi IIHWHWHIUIWWWHW 1293 00993 0979 lIBRARYj Michigan Stow . University i This is to certify that the thesis entitled NON-NEWTONIAN SHEAR RATE APPROXIMATIONS AND OPTIMAL EXPERIMENTAL CONDITIONS IN BACK EXTRUSION TESTING presented by Nelly J. Marte Guzman has been accepted towards fulfillment of the requirements for M.S. , Food En ineerin degree in g g 2 7)1 Major pr7ofessor Date 7J# 87 0-7639 MS U i: an Affirmative Action/Equal Opportunity Institution MSU RETURNING MATERIALS: Place in book drop to uaaAmss remove this checkout from ”- your record. FINES will be charged if book is returned after the date stamped below. NOV11 (39201111 NON-NEWTONIAN SHEAR RATE APPROXIMATIONS AND OPTIMAL EXPERIMENTAL CONDITIONS IN BACK EXTRUSION TESTING By Nelly J. Marte Guzman A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Food Science and Human Nutrition 198? ABSTRACT NON-NEWTONIAN SHEAR RATE APPROXIMATIONS AND OPTIMAL EXPERIMENTAL CONDITIONS IN BACK EXTRUSION TESTING By Nelly J. Marte Guzman Newtonian approximation for power law fluids and power law approximation for Herschel-Bulkley fluids, as well as optimal experimental conditions in back extrusion or annular pumping testing were considered in this study. The results obtained with the back extrusion device were in good agreement with those obtained with the Haake viscometer. Best results were obtained with the back extruder using a plunger to cup radius ratio between 0.77 and 0.85 and a plunger velocity combination in the ratio of 4.0 to 5.0. Under those optimal experimental conditions the power law approximation for Herschel- Bulkley fluids works in excellent agreement with actual values. The Newtonian approximation for power law fluids also works well but may vary with the rheological properties. To avoid and effects in back extrusion testing, the plunger bottom should be separated from the container bottom a minimum distance equivalent to the inner container diameter at the end of the experiment. To: my parents David and Lindin, my daughter Nellie Grace. ' my sister Nana and my husband Miguel Angel. fi ACKNOWLEDGMENTS The author wishes to thank Dr. Ronnie Morgan, Dr. Robert Ofoli and Dr. Mark Uebersax, members of the committee, fin~their helpful comments. Special recognition to Dr. James Steffe who serves as Major Advisor for his guidance and confidence. Sincere thanks also to The Partners of the Americas for its financial contribution in support of the program. Finally, thanks to all those friends who offered their time and expertise, especially Melania Molina, Fernando Osorio and Fernando Hirujo. iii TABLE OF CONTENTS Bass List of Tables ........................................ vii List of Figures ........................................ ix Nomenclature ........................................... xii CHARTERS 1. Introduction ....................................... 1 2. Literature Review .................................. 3 2.1 Newtonian Model ............................... 3 2.2 Non-Newtonian Models .......................... 4 2.2.1 The Power Law Fluid Model .............. 5 2.2.2 The Bingham Plastic Model .............. 6 2.2.3 The Herschel-Bulkley Model ............. 7 2.3 Viscometers and Rheometers .................... 7 2.3.1 Back Extrusion Technique ............... 8 3. Theoretical Development ............................ 10 3.1 Basic Theories ................................ 10 3.1.1 Back Extrusion of Newtonian Fluids ..... 10 3.1.2 Back Extrusion of Power Law Fluids ..... 13 3.1.3 Back Extrusion of Herschel-Bulkley Fluids 17 3.1.4 Procedure for the Determination of Rheological Properties of a Herschel- Bulkley Fluid .......................... 23 3.2 Computer Programs for Non-Newtonian Fluids .... 24 4. Experimental Procedure Applications ................ 26 4.1 Equipment, Materials and Methods .............. 26 iv 4.2 Effect of Plunger Velocity and Plunger Radius .. 4.3 Newtonian Approximation for Power Law Fluids Based on Simulated Data ....................... 4.4 Power Law Approximation for Herschel- Bulkley Fluids Based on Simulated Data ........ 4.5 Newtonian Approximation for Power Law Fluids Based on Experimental Data .................... 4.6 Power Law Approximation for Herschel- Bulkley Fluids Based on Experimental Data 4.7 End Effects .................................... 5.1 Shear Rate Approximations Based on Theoretical Data .............................. 5.1.1 Newtonian Approximation for Power Law Fluids .................................. 5.1.2 Power Law Approximation for Herschel- Bulkley Fluids ......................... 5.2 Shear Rate Approximation Based on Experimental Data .......................................... 5.2.1 Newtonian Approximation for Power Law Fluids .................................. 5.2.2 Power Law Approximation for Herschel Bulkley Fluids ........................ 5.3 End Effect Evaluation ......................... 6. Conclusions ........................................ 7. Practical Guidelines for Conducting Back Extrusion Testing ............................................ 6. Suggestions for Future Research .................... 9. References ......................................... 10. Appendices A: RHEO - Programs for Non-Newtonian Fluids in Back Extrusion Testing ......................... 27 29 32 37 42 44 48 48 48 58 72 72 85 92 99 101 102 104 107 Calculation example for the power law shear rate at the wall as an approximation to the shear rate at the wall for Herschel-Bulkley fluids, based on simulated data ........................ 1. Calculation example for the determination of rheological properties of a power law fluid using the back extrusion device ................ 2. Calculation example for shear stress and shear rate at the wall of Herschel-Bulkley fluid and approximated to power law values ..... Table of Results ............................... vi 113 118 121 125 LIST OF TABLES Table 1 10 11 12 13 Rheological properties and experimental conditions for power law fluids used in the Newtonian approximation based on simulated data ............... Rheological properties and experimental conditions for Herschel-Bulkley fluids used in the power law approximation based on simulated data ............... Shear rate at the wall for Newtonian fluids from Morgan’s equation ................................... Shear rate at the wall ranges for different flow behavior indices corresponding to different power law fluids .......................................... Rheological properties of power law fluids determined using the Haake viscometer data over a shear rate range of 10 - 100 s“-1 .............................. Shear stress and shear rate values for 1.0 X Methocel obtained using the flasks viscometer ................. Shear stress and shear rate values for 2.0 X Methocel obtained using the Haake viscometer ................. Shear stress and shear rate values for 1.0 X guar gum obtained using the Haake viscometer ................. Shear stress and shear rate values for 1.5 X guar gum obtained using the Haake viscometer ................. Shear stress and shear rate values for high sucrose corn syrup obtained using the Haake viscometer ...... Values of the experimental parameters for 1.25 x, 1.5 X and 2.0 X Kelset samples using the back extruder ............................................ Shear stress and shear rate values for 1.25 X Kelset obtained using the flasks viscometer ................. Shear stress and shear rate values for 1.5 x Kelset obtained using the Eaake viscometer .... ..... f ........ vii Page 38 49 53 74 75 76 77 78 79 86 87 14 15 16 17 18 19 Shear stress and shear rate values for 2.0 X Kelset obtained using the Haake viscometer ................. Experimental data, collected from the back extrusion device for 1.0 X Methocel, used in the end effect evaluation .......................................... Experimental data collected from the back extrusion device of 1.0 X Methocel at three plunger velocities .......................................... Experimental rheological parameters for 1.0 X Methocel for Vpl = 45.00 x 10-4 m/s, Vp2 = 62.00 x 10-4 m/s and R1 = 1.2 cm ..................................... Experimental rheological parameters for 1.0 X Methocel for Vpl = 45.00 x 10-4 m/s, Vp2 = 16.67 x 10-4 m/s and Bi = 1.2 cm ..................................... Experimental rheological parameters for 1.0 X Methocel for Vpl = 45.00 x 10-4 m/s, Vp2 = 83.33 x 10-4 m/s and R1 = 1.2 cm ..................................... viii 89 93 94 95 96 97 Figure 1 Geometry of the annular back extrusion device ........ 2 Schematic representation of coordinates describing axial flow in a back extrusion device (Osorio, 1985) ................................................ 3 Newtonian shear rate approximation of power law fluids based on simulated data ....................... 4 Power law shear rate approximation of Herschel-Bulkley fluids based on simulated data ....................... 5 Newtonian shear rate approximation of power law fluids based on experimental data .................... 6 Diagram of force versus distance obtained from the Instron to recorder for 1.5 X Methocel with plunger velocity of 8.33 x 10‘-3 m/s and a chart speed of 5.0 x 10‘-4 m/s (Osorio, 1985) ...................... 7 Power law shear rate approximation of Herschel-Bulkley fluids based on experimental data .................... 8 End effect evaluation ................................ 9 Shear rate at the wall for Newtonian fluids, based on Morgan’s equation (Equation 6) .............. 10 Shear rates difference versus annular gap, at different plunger velocities, for 1.5% guar gum 11 Shear rates difference versus annular gap, at different plunger velocities, for 2.0 X Methocel 12 Shear rates difference versus annular gap, at different plunger velocities, for 1.5 X Methocel 13 Shear rates difference versus annular gap, LIST OF FIGURES at different plunger velocities, for corn syrup ...... ix Page 20 31 34 39 41 45 46 50 54 55 56 57 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Shear rate at the wall versus annular gap, at different plunger velocities, for 1.5 X guar gum Shear rate at the wall versus annular gap, at different plunger velocities, for 2.0 X Methocel Shear rates difference versus annular gap, at different plunger velocities, for 1.0 X Kelset Shear rates difference versus annular gap, at different plunger velocities, for 1.5 X Kelset Shear stresses difference versus annular gap, at different plunger velocities, for 1.0 X Kelset Shear stresses difference versus annular gap, at different plunger velocities, for 2.0 X Kelset Shear rate at the wall versus annular gap, at different plunger velocities, for 2.0 X Kelset Effect of the dimensionless annular gap (K) on the power law approximation of a Herschel-Bulkley fluid (2.0 X Kelsat) ................................. Effect of the plunger velocity (Vp) on the power law approximation of a Herschel-Bulkley fluid (2.0 X Kelset) ....................................... Effect of the dimensionless annular gap (K) on the power law approximation of a Herschel-Bulkley fluid (1.5 X Kelset) ........................................ Rheograms for 1.0 X Methocel, considering the power law data from the back extruder, the Newtonian approximation of the power law fluid, and Haake viscometer data Rheograms for high sucrose corn syrup, consideringthe power law data from the back extruder, the Newtonian approximation of the power law fluid, and Kaake viscometer data ...................................... Rheograms for 1.5 X guar gum. considering the power law data from the back extruder, the Newtonian approximation of the power law fluid, and Haake viscometer data ...................................... Rheograms for 1.0 X guar gum, considering the power law data from the back extruder, the Newtonian approximation of the power law fluid, and Haake viscometer data ....................................... 59 60 62 63 65 66 68 69 70 71 80 81 82 83 28 29 Rheograms for obtained from and the power Rheograms for obtained from and the power 1.25 X Kelset, considering data the Haake viscometer, the back extruder, law approximation 1. 5 X Kelset, considering data the Haake viscometer, the back extruder, law approximation xi Csp Fr; :2- FT FT- PL R1 R0 To NOMENCLATURE chart speed of the recorder. m/s force applied to the plunger, N buoyancy force. N force corrected for buoyancy. N recorded force while plunger is traveling down, N recorded force after the plunger is stopped, N acceleration due to gravity, m/s2 ratio of radius of plunger to that of outer cylinder, dimensionless vertical length of wetted plunger surface, m chart length obtained for recorder, m flow behavior index, dimensionless initial level of fluid when the plunger has not been forced down in the sample, m pressure drop per unit of length. Pa/m pressure in excess of hydrostatic pressure at the plunger base. Pa radial coordinate from common axis of cylinder forming annulus, m radius of the plunger. m radius of outer cylinder of annulus, m reciprocal of n, dimensionless dimensionless shear stress, defined in Equation (18) dimensionless yield stress. defined in Equation (19) ' xii Tw VP dimensionless shear stress at the plunger wall, defined in Equation (22) velocity, m/s plunger velocity. m/s xiii 1w New 1w pl Yw HB Eh“ GREEK SYMBOLS sample density, Kg /m“3 consistency coefficient for power law fluids, Pa s‘n plastic viscosity, Pa s Herschel-Bulkley consistency coefficient, Pa s‘n value of dimensionless radial coordinate for which shear stress is zero limits of plug region in Herschel-Bulkley flow Newtonian viscosity, Pa 3 dimensionless radial coordinate shear stress, Pa yield stress, Pa shear stress at the plunger wall, Pa shear stress at the wall as a Newtonian fluid, Pa shear-stress at the wall as a power law fluid, Pa shear stress at the wall as a Herschel-Bulkley fluid, Pa shear rate, sec“-1 shear rate at the plunger wall, sec“-1 shear rate at the wall as a Newtonian fluid at the wall as a power law fluid a Herschel-Bulkley fluid shear rate shear rate at the wall as difference between the shear rate at the wall for the power law fluid and the shear rate at the wall approximated to Newtonian, defined in Equation (29) xiv difference between the shear rate at the wall for the ‘ Herschel-Bulkley fluid and the shear rate at the wall approximated to power law, defined in Equation (30) difference between the shear stress at the wall for the Herschel-Bulkley fluid and the shear stress at the wall approximated to power law, defined in Equation (31) dimensionless flow rate defined in Equation (21) dimensionless velocity defined in Equation (14) dimensionless geometric coefficient defined in Equation (5) XV Chapter 1 INTRODUCTION Rheological properties are critical in proper process and equipment design. They are also useful as indicators for detecting physico-chemical changes in food materials. Hence, they are key parameters required to solve food industry problems in numerous areas. The back extrusion, or annular pumping geometry, has been used for characterizing relative flow properties of various food materials. These devices may vary in size and shape, but usually operate in a similar manner. The sample is first placed in a vertical container. Then a plunger, traveling at a constant velocity, is forced down into the material causing a positive displacement of the sample. The sample flows upward through an annular space between the plunger and inner wall of the container. Force on the plunger is usually the measured, parameter and is correlated as a relative rheological property (Morgan,1979). To date, analytical expressions to obtain rheological properties of Newtonian fluids (Morgan, 1979) and non-Newtonian fluids (Osorio, 1985) in a back extrusion device have been developed. However, those expressions based on non-Newtonian rheological models such as the Herschel-Bulkley case are very complicated and their application in resolving real problems is very time consuming. Simple analytical methods or approximations, as well as optimum experimental conditions must be considered. The objectives of this work were: 1) To determine the effect of plunger velocity (Vp) and plunger radius (Ri), in determining the rheological properties of the power law and Herschel-Bulkley fluids according with the expressions developed by Osorio, 1985. 2) To consider the use of the power law and the Newtonian approximations for determining shear rate when dealing with Herschel-Bulkley and power law fluids respectively. 3) To study the end effect error in the determination of the rheological properties from back extrusion data. 4) To experimentally validate the results. Chapter 2 LITERATURE REVIEW Many industries, government research establishments and university research groups are concerned with a wide range of fluids which can be broadly classified as non-Newtonian. Polymer melts and polymer solutions come readily to mind in this connection, but the complete list is seemingly endless. So, rheometry plays a major role in the experimentalists approach to non-Newtonian fluid mechanics because those who are practically concerned with the flow behavior of these fluids invariably require measurements of their mechanical properties (Walters, 1975). 2.1 mm For many fluids, measurements of shear stress and shear rate (at various magnitudes of both) indicate a direct proportionality between the variables: t = uY' . Such materials, for which the ratio of shear stress to shear rate, or viscosity, is constant (i.e., independent of the magnitude of shear stress or shear rate), are called Newtonian. Most fluids of simple structure, composed of relatively simple molecules in a single phase, behave as Newtonian fluids (Darby, 1976). Water, and most aqueous solutions, organic liquids, silicones and liquid metals behave, within experimental accuracy, as Newtonian fluids. The model may be adequate, although not exact, for the dilute suspensions, emulsions, and solutions of moderately long chain molecules (Whorlow, 1980). 2.2 Nonzfleflienian_fled§l§ All those fluids for which the flow curve (shear stress versus shear rate) is not linear through the origin at a given temperature and pressure are said to be non-Newtonian. These fluids are commonly divided into three broad groups, although in reality these classifications are often by no means distinct or sharply defined. 1. Time-independent fluids are those for which the rate of shear at a given point is solely dependent upon the instantaneous shear stress at that point. (These materials are sometimes referred to as ’non-Newtonian viscous fluids’ or alternatively as ’purely viscous fluids’). 2. Time-dependent fluids are those for which the shear rate is a function of both the magnitude and the duration of shear and possibly of the time lapse between consecutive applications of shear stress. 3. Viscoelastic fluids are those that show partial elastic recovery upon the removal of a deforming shear stress. Such materials possess properties of both fluids and elastic solids (Skelland, 1967). Because of the wide variation in their structure and composition, foods exhibit flow behavior ranging from simple Newtonian to time dependent non-Newtonian and Viscoelastic. Furthermore, a given food may exhibit combinations of classical behaviors, depending on its origin, concentration, and previous history. Flow properties of food are used for a number of purposes, such as quality control (Kramer and Twigg, 1970), insights to physicochemical structure (Sherman, 1966), process engineering applications (Boger and Tiu, 1974; Bee and Anantheswaran, 1982; Steffe and Morgan, 1986), and correlation with sensory evaluation (Szchesmiak and Farkas, 1962). Non-Newtonian fluids may be distinctly classified by the way in which the shear stress varies with the shear rate, or in terms of the variation of apparent viscosity with shear stress or shear rate (Darby, 1976). Some of the most common rheological models, which have been used in axial laminar flow in a concentric annulus are the power law, the Bingham plastic, and the Herschel-Bulkley fluid models. 2.2.1 W This model, usually attributed to 0stwa1d but proposed independently by de Waele and others, is used to represent the behavior of many polymer solutions and melts, and sometimes of other systems. The equation for the model may be written as 1:: ny (1) where n is a constant sometimes referred to as the consistency index, and n may have any positive value. When n is greater than unity the material becomes less fluid as the shear rate increases. This is known as shear- thickening, or sometimes as dilatant behavior. When n is less than unity it is known as shear thinning or sometimes ' pseudoplastic behavior. When n is 1, of course, the flow curve corresponds to a Newtonian liquid. This model has proved very useful for approximating fluid behavior over a fixed range of shear rate values. However, it has limitations for small and very large shear rates. 2.2.2 Magnum This material behaves as an ideal rigid solid (no deformation) when subjected to a shear stress smaller than a certain (yield) value (yield stress). For stresses above the yield value, the Bingham plastic flows as a fluid, with the shear stress being a linear function of shear rate. Although it is doubtful that any real material actually behaves precisely in this manner, the behavior of many materials, notable pastes, suspensions, slurries, paints, etc. may often be adequately approximated by this model over a suitable range of conditions (Darby, 1976). This model is expressed as a two parameter model, 'l-':To+nBY (2) where To is the yield stress and I? is the plastic viscosity. 2.2.3 W One of the most popular modifications of the Bingham model is to insert the power law term for the viscous component: n I: To #1“? (3) where 1:o is the yield stress, ”H is the H-B consistency index, and n is the flow behavior index. The Herschel- Bulkley model has the merit that it describes a flow curve in a small number of parameters and often is found to give a reasonably good representation of an experimentally determined curve. It does, however, suffer from the disadvantage that, unlike the simple power law, it is less easy to fit to the observed data because it usually involves an extrapolation in order to arrive at the value of yield stress (Prentice, 1984). Since Bingham plastic, power law and Newtonian behavior may be regarded as special cases of the Herschel-Bulkley model, the model represents the behavior of a large number of materials without being unduly difficult to handle mathematically (Whorlow, 1980). 2.3 W Under any conditions of motion and stress, a constant viscosity coefficient is sufficient to determine the behavior of incompressible Newtonian liquids. The measurement of this viscosity coefficient involves the use of a Viscometer defined simply as ’an instrument for the measurement of viscosity’. The viscosity of non-Newtonian elastic liquids may be shear rate dependent, the viscometer is therefore inadequate to characterize the behavior of these materials and has to be replaced by a rheometer defined as ’an instrument for measuring rheological properties’ (Walters, 1974). There are two basic objectives in Rheometry. The first involves a straightforward attempt to characterize the behavior of non-Newtonian liquids in a number of simple (rheometrical) flow situations using suitable defined material functions. The second objective involves the construction of rheological equations of state for the liquids which can be later used in the solution of flow problems of practical importance (Walters,1974). 2.3.1 WW Back extrusion apparatuses have been used for characterizing relative flow properties of various food materials (Morgan, 1979). In the back extrusion technique the following two facts are involved: 1) a plunger is forced down in a fluid, and 2) the fluid flows upward through a concentric annular space (Osorio, 1985). Some of the unique characteristics of this rheological apparatus is that it allows quick and easy measurements of small volume samples contained in standard glass test tubes. This provides for a wide range of variations in sample preparation and pre-treatment processes. This instrument should prove useful in quality control of the manufacturing of fluid foods (Steffe and Osorio, 1987). Several works have been developed in the area of the back extrusion technique. Morgan et al. (1979) developed mathematical relations for calculating shear stress, shear rate and velocity profiles for annular counter flow of a Newtonian fluid in a simple back extrusion device. Viscosity standards were used for experimental validation of the equations. Osorio (1985) developed mathematical expressions to describe the behavior of non-Newtonian fluids in a back extrusion device using the Herschel-Bulkley fluid model. With the mathematical model developed, expressed in form of dimensionless terms, it is possible to determine the rheological preperties of Newtonian, power law, Bingham plastic, and Herschel-Bulkley fluids. In addition, shear stress and shear rate at the wall may also be calculated. Chapter 3 THEORETICAL DEVELOPMENT 3.1 3511211192112: 3.1.1 W Morgan (1979) developed mathematical relationships for describing the operation of an annular back extrusion device, including equations which can be used for determining viscosity at known shear rates and certain Viscoelastic properties. Newtonian viscosity standards were used to validate mathematical relationships. Geometry of the annular back extrusion device used for this study is shown in Figure 1. As the plunger is lowered into the sample, fluid will begin to flow upward through the annulus. If the plunger velocity is constant, total plunger force will be the sum of shear forces on the plunger wall and the static pressure pushing upward on the bottom surface of the plunger. Static force at the base of the plunger is composed of buoyancy force, and that responsible for fluid flow in the upward direction. Buoyancy effects can easily be accounted for by simple calculation of the hydrostatic head. Assuming Newtonian flow, from the plunger force balance and the conservation of momentun equation the following expressions were developed by Morgan (1979): 10 ‘ u————Ro———s Figure 1. Geometry of the annular back extrusion device. 11 a ch ' tw = ---- (4) 2 1 R1 L Shear stress on wall of plunger, Pa where: w = ch = Total plunger force corrected for buoyancy, N R1 = Radius of plunger rod, m L = Vertical length of wetted plunger surface, m a = Dimensionless geometric coefficient (which depends only on physical geometry of the device for Newtonian flow conditions). Also 1 - K? a = ---- (5) 1 + K? where: K = Ri/Ro R0 = Radius of outside cylinder or container, m and, Y" = I). .___ (6) In K + a Ri where: Yw = Shear rate at plunger wall, a-1 Vp = Plunger velocity, m/s 12 Equations (4) and (6) can be combined to yield an expression for calculating the Newtonian viscosity: [ 1 fix: .' 1 a T u = 1-- 1n - 1 + . (7) L2 :4 Vp L .. K ln K3 3.1.2 WW: Osorio (1985) developed analytical expressions, using the power law fluid model, to describe the fluid behavior in a back extrusion device. The complete theoretical development for the equations presented in this section may be found in Osorio (1985) or Osorio and Steffe (1987). In the development of the theory for non-Newtonian fluids (both power law and Herschel-Bulkley) the following assumptions were made: . The density is constant. . The fluid is homogeneous. . The fluid has achieved steady state flow. There is no elasticity or time-dependent behavior. . The flow is laminar and fully developed. QO'IIFODNH The cylinders are sufficiently long that end effects may be neglected. 7. The temperature is constant. In addition, the following boundary conditions are assumed for the analysis: 13 3. There is no slip at the annulus walls and the velocity at the outer wall (the cup) is zero. b. The definition of a Herschel-Bulkley fluid implies a region of "plug flow" where the shear stress must reduce to zero at the boundary and inside the plug. When a power law fluid is tested in a back extrusion device, at two different plunger velocities, with a plunger radius and outside cylinder combination which are the same in both experiments, the value of s (defined as the inverse of the flow behaviOr index n) is given by r Vp2 1n: —" g Vpl :5 = L: (8) Fcb2 L1 ln . __ L Fcbl L2 where: Vp = plunger velocity, m/s Fcb = force corrected for buoyancy, N L = length of annular region, m The consistency coefficient can be determined from the following expression: PR0 33° in 71- -———. (9) 2 «vaK-J where: n = flow behavior index, dimensionless P pressure drop per unit of length, Pa/m 14 .Ro = radius of outer cylinder of annulus, m K = ratio of radius of plunger to that of outer cylinder, dimensionless n = consistency coefficient, Pa s“n Q = dimensionless flow rate Osorio and Steffe (1987) presented a plot of the dimensionless plunger radius (K), the flow behavior index (n) and the dimensionless flow rate ( Q ). They also presented a table from which, with the knowledge of the flow behavior index (n) and the dimensionless plunger radius (K), it is possible to obtain the A value which is the value of the dimensionless radial coordinate for which the shear stress is zero. The shear stress at the plunger wall is IN = ---- (10) where the dimensionless shear stress at the plunger wall (Tw) is obtained as Twz)‘ -K (11) K By applying a force balance on the plunger, the following expression was determined: 15 Fcb = Tw + K (12) I L P Ro Ri force corrected for buoyancy, N where: Fob L = length of annular region, m P = pressure drop per unit of length, Pa/m R0 = radius of outer cylinder of annulus, m R1 = radius of the plunger, m K = Ri/Ro, dimensionless Tw = dimensionless shear stress at the plunger wall The shear rate at the plunger wall is given by av _PRoSg¢_ Sp EFIsa ' n (13) o=k shear rate, 3-1 where: av/ar P Ro pressure drop per unit of length, Pa/m radius of outer cylinder of annulus, m n = flow behavior index n = consistency coefficient, Pa s“n p = dimensionless radial coordinate = r/Ro and e , the dimensionless velocity, is defined as !— 2 n l/n 9 ¢ = LpRon+1 . [v] (14) 16 Also, the dimensionless shear rate at the plunger wall is A‘ s : -'K (15) D: K The buoyancy force is given by 5¢ 60 2 Fb = C g L 1 Bi (16) 5'3 where L = (17) 1 - K? and 0B corresponds to the position of the plunger bottom, at the end of the test, measured with respect to the initial level of fluid, in meters. For a power law fluid, the buoyancy force, Fb, is equal to the recorded force after the plunger is stopped, FTe. With the above equations, and the tables and plots developed by Osorio and Steffe (1987), it is possible to calculate the rheological properties of a power law fluid. In addition, the shear stress and shear rate at which measurements are taken may be determined. 3.1.3 W: For a detailed discussion of the back extrusion theory for Herschel-Bulkley fluids presented in this section, refer to 17 Osorio (1985) or Osorio and Steffe (1985). The Herschel-Bulkley model is: ,n TZTO'i' nHY where: ‘f = shear stress, Pa 0 = yield stress, Pa rm = consistency coefficient, Pa s‘n -- ._ 40 « '3 3 0 .c 4.) a.) O 8 30 -‘ to L L «I 0) .c U) 20 4 10, Figure 20. O 0" X p—s x N 75 3 K4 Annular gap ( K ) Shear_rate at the wall versus annular gap, at different plunger veloc1t1es, for 2.0 x Kelset. Pa Shear stress at the wall ( Tw ), 240- 220' 200' 180. 160. 140- 130. 100.. MB mm: 80* PLa _;_ ‘_§1 , ALL Vp CONSIDERED 1 0.652 60 . 2 0.773 3 0.682 4 4 0.568 40 . HBi 8 REAL RHHXERAM FOR THE HERSCHEL-BUIKIEY 20 ‘ FLUID AT Ki PLi 3 mm OF THE POWER LAW APPIDXMTION AT l(i 7 fi 1 r l t T T r V 0 10 20 30 40 50 60 70 80 90 100 Shear rate at the wall (7w), 5‘1 Figure 21. Effect of the dimensionless annular gap ( K ) on the power law approximation of a Herschel-Bulkley fluid (2.0 % Kelset). 6S) Pa (TH). Shear stress at the wall 170‘ 160 150 140 130 120 110‘ 100 90 80 POI-1mm: _1_ v2 11 10‘ m/s . nu. K cousmcnm 1 16.67 7°‘ 2 45.00 3 62.00 60 4 63.33 "31 I NEIL NW FOR THE "B num AT Vpi 50 PLl-NWOF'I‘ITEWMH APPROXIMATIW, AT Vpi 4o 10 20‘ 10 10 20 30 40 SO 60 {7 -1 Shear rate at the wall ( w ), s Figure 22. Effect of the plunger velocity (Vp) on the power law approximation of a Herschel-Bulkley fluid (2.0 % Kelset). 70 Pa Shear stress at the wall, 130 1 120 . 110 . 100 1 90 l 80 4 Nomenclature: i Ki 1 0.568 2 0.682 3 0.773 4 0.852 All Vpi are considered. H81: Rheogram for the HB fluid at Ki PLiz Power law approximation f I I '0 1 I 50 60 70 80 90 100 Shear rate at the wall, 5'1 Figure 23. Effect of the dimensionless annular gap (K) on the power law approximation of a Herschel-Bulkley fluid. (1.5 % Kelset). 71 means lower error in the approximation. Therefore, the rheogram of the power law approximation is affected by the annular gap. The H-B rheogram is not affected by K. 3- The higher the shear rate value, the larger the difference between the H-B and the power law rheograms. Experimental results of Newtonian approximation of power law fluids, using the back extrusion technique, are shown in the following tables (Appendix D): Tables D13 to D16 summarize the results for 1.0 X Methocel solution at different Bi and Vp. Tables D17 to D20 summarise the results for 2.0 X Methocel solution at different Vp and Ri. Tables D21 to D24 summarize the results for 1.0 X guar gum solution at different Vp and Ri. Tables D25 to D28 summarize the results for 1.5 X guar gum solution at different Vp and Ri. Tables D29 to D32 summarize the results for 1.5 X Methocel solution at different Vp and Ri. Tables D33 to D36 summarize the results for high sucrose corn syrup, at different Vp and R1. All tables include the difference Ale. In addition, at the bottom of all tables, from D13 to D36, the best two Vp 72 combinations, according to the smaller Ale values, were established. Rheological properties, determined with Haake data, for the different power law fluids are included in Table 5. The values of shear stress versus shear rate obtained from the Haake are included in Tables 6, 7, 8, 9, and 10, for 1.0 X Methocel, 2.0 X Methocel, 1.0 X guar gum, 1.5 X guar gum, and for high sucrose corn syrup respectively. For all different power law fluids, the rheograms obtained from the back extrusion data were plotted along with the rheograms obtained from the Haake Viscometer, and with the rheograms of the Newtonian approximation. Figures 24 to 27 are examples for 1.0 X Methocel, high sucrose corn syrup, 1.5 X guar gum and 1.0 X guar gum, of the rheograms obtained from Tables D13 to D36, for different power law fluids. From all those figures it can be observed that: 1- For all power law fluids considered, only when the plunger radius used is bigger than 1.36 cm (K = 0.77 or K = 0.85), is the back extrusion data in good agreement with the Haake viscometer values. 2- In general the best Vp combinations are Vp1 = 83.33 x 10-4 m/s and Vp2 = 16.67 x 10-4 m/s or Vp1 = 62.00 x 10-4 m/s and Vp2 = 16.67 x 10-4 m/s which means that the best Vp combinations to work with are when a high velocity is used with a lower one, considering that Vp = 1.667 x 10-4 m/s can not be used experimentally with confidence, as was explained in Chapter 4. 73 Table 5. Rheological Properties of power law fluids determined using the Haake viscometer data over the shear rate range of 10 - 100 srl. Fluid Samples Equation R2 1.5% Guar Gum = 14.195 '70-225 0.993 1.0% Guar Gum = 4.880 {(0.319 0.996 2.0% Methocel = 5.379 {(0603 0.996 1.0% Methocel = 1.218 70-6-73 0.998 74 Table 6. Shear stress and shear rate values for 1.0% Methocel obtained using the Haake viscometer. Shear Rate (s-‘) Shear Stress (Pa) 1.81 1.41 4.77 3.07 6.93 4.16 9.16 5.18 11.54~ 6.22 16.33 8.02 23.57 10.54 30.82 12.66 35.62 13.94 47.72 17.01 52.85 18.09 57.84 19.06 62.66 19.98 67.46 20.89 72.44 21.76 77.17 22.60 84.91 23.87 91.95 24.94 96.50 25.67 109.00 27.43 75 Table 7. Shear stress and shear rate values for 2.0% Methocel obtained using the Haake viscometer, Shear Rate (8“) Shear Stress (Pa) 2.18 9.64 4.82 17.57 7.06 23.33 11.66 34.34 23.83 54.39 36.26 69.50 49.07 80.93 74.06 98.50 99.34 111.20 76 Table 8. Shear stress and shear rate values for 1.0% guar gum obtained using the Haake Viscometer- Shear Rate (s-l) Shear Stress (Pa) 2.10 4.79 7.19 9.47 12.07 19.83 27.87 38.66 52.26 65.35 77.64 104.00 3.10 6.71 8.12 9.39 10.50 12.62 14.37 15.87 17.28 18.39 19.41 21.23 77 Table 9. Shear stress and shear rate values for 1.5% guar gum obtained using the Haake Viscometer. Shear Rate (8“) Shear Stress (Pa) 3. 12. 25. 40. .56. 70. 87. 98. 11 .94 .72 60 86 25 94 80 69 17 130 27 18.69 20.61 24.55 29.91 33.81 35.59 37.13 38.32 39.29 78 Table 10. Shear stress and shear rate values for high sucrose corn syrup obtained using the Haake viscometer. Shear Rate (s‘l) Shear Stress (Pa) 2.45 0.22 4.71 0.48 6.87 0.68 9.10 0.92 11.49 1.13 16.10 1.60 23.17 2.41 35.00 3.54 46.66 4.71 58.34 5.93 70.28 7.20 82.10 8.31 92.97 9.47 104.90 10.71 79 (Tw), Pa Shear stress at the wall 50« 40“ 30‘ 20‘ 101 MATURE _3L_ _§_ , ALL Vp CONSIDERED 1 0.568 2 0.682 3 0.773 4 0.852 PLi = Rheogram for the power law fluid at Ki Newi = Newtonian approximation at Ki HAAKE "5:4 , I 1 30.67 N wz / = 1.22 PLZ // " PLJI s" 4L3 ////N€W i I I l T I f t r 10 20 30 40 50 60 70 80 90 100 110 Shear rate at the wall ( 1w ), s-1 Figure 24. Rheograms for 1.0 X Methocel, considering the power law data from the back extruder, the Newtonian approximation of the power law fluid, and Haake viscometer data. 80 ), Pa (Tw Shear stress at the wall 10 . 9 ‘ ,Haake: n = 1.0 n V n = 0.09 Pa.s 8 . PL4 7 New 6 . 5 . Nomenclature: 1 191 x 104 m/s K1 4 1 16.67 0.568 2 45.00 0.682 3 3 62.00 0.773 . Newz 4 63.33 0.652 New PL. = Rheogram for the power law . l 2 PL PL2 fluid at Ki New PL . Newtonian approximation 1 ' ’,,¢a¢¢sfifififi:3’f 3 1 r I T l T I I I l I 10 20 30 40 50 60 70 80 90 100 2 (D Z N Shear rate at the wall ( Yw ), 5‘1 Figure 25. Rheograms for high sucrose corn syrup, considering the power law data from the back extruder, the Newtonian approximation of the power law fluid, and Haake viscometer data. £31 .nuoo roomsoum_> oxen: use .owzpe zap Luzon on» so cowuespxorane newcopzmz or» . .rooauuxm some we» sort some zap Luzon we» mcprmormcou .szm seem a m.H roe mseemomgm .om mr=m_u Hum . A 3+ v ppm; mg» no more swarm ON on an ac om om oH o copoos_xonano newcouzmz u wzmz _x on u_=Pm sop roses on» roe smcmomzm u _md res «mm.o mm.mm e men.o oo.~m m ~mm.o oo.me N mmm.o mo.o~ a as m\e ees x eee a row "oraon_o=msoz .om cm.od o~.eH u c mm.o u : hmxoozxir 11. see ezmz H on N 3oz use so: ‘( Ml ) [low aul 16 556415 JESUS 82 9d .mpmu Lmumeoumw> wxum: use .u_:pw 3o. Luzon mzp we cowums_xoggam copcopzmz on» .Lmuzguxm xuma mzu sag; mumu zap gmzoa asp mcwgmuwmcou .Ezm swam x o.H Low msmgmomsx .NN wg:m_m Hum .A zmv Fpmz mg» um mum; smmcm as oo om cc pm pN pH 0 comume_xogaam cmpcouzmz “szz wx pm vwspm zap gmzoa mgu Low Emgmomcm "PA; ,oH m: :m.ma o.¢ u c mm.o u c ”mxmm: .oN e I1. 4; 3mz 3mz use Nmm.o mm.mm e ‘.om mNN.o oo.No m Nam.o oo.me N mom.o No.0H H _x mNE eoH x Pa> P ”assumpucmsoz . oq ed ‘( Ml.) [[PM an; 12 533415 Jeans 83 3- The Newtonian approximation of power law fluids does work. However, further work needs to be done to determine the relationship between the real power law rheograms and the approximation to Newtonian. In general, in all figures the real rheograms (power law) and the rheograms approximated to Newtonian have the same shape; however for the same shear rate at the wall value, the Newtonian approximation gives higher shear stress at the wall values, and the difference is almost constant for all shear rates. This difference between the two rheograms could be a function of the rheological properties of the power law fluids because it is smaller when the rheological properties are closer to the Newtonian case ( n = 1), and it works better when R = 0.77 or K: 0.85 and Vp combinations are 83.33 with 18.67 or 62.00 with 16.67 (x 10-4 m/s). 4- The shear rate ranges are affected in the same way by the plunger velocity and by the plunger radius: the higher Vp is, and the higher K values are, the higher shear rates at the wall can be achieved for the same fluid. 5- In general the shear rate ranges for all velocities, and for all K, vary from 10 to 100 sec“-1, which are the same as those achieved with the Haake. 6- When the plunger rod used was Ri = 1.36 cm, for all experiments run at the laboratory for both power law and Herschel-Bulkley fluids, the back extrusion data was affected by an error that could be explained as a function of the slip present at the wall of this plunger which was made of a different material. Future works will included the use of several plunger rods made of the same material to avoid this problem. 84 5.2.2 WWW Following the procedures described in Chapter 4, the experimental parameters obtained in the back extrusion device for the three Kelset solutions (1.25 X, 1.5 X, 2.0 X), are included in Table 11. Tables D37 to D39 (Appendix D) summarize the shear rate at the wall and the real shear stress at the wall for the H-B fluids ( §w HB, Tw HB ); the shear rate and the shear stress at the wall for the H-B fluids approximated to power law ( ‘;w pl, 1'p'w p1 ), and the difference Alwz for 1.25 X Kelset. Tables D40 to D43 and Tables D44 to D47 summarize the same results for 1.5 x and 2.0 % Kelset, respectively. Tables 12 to 14 include the Haake values for the shear stress and the shear rate corresponding to the three Kelset solutions. Figures 28 and 29 show examples of the rheograms for all Herschel-Bulkley fluids according with the data obtained from: 1) the Haake, 2) the back extrusion data, 3) the approximated to power law data. From those figures, it may be observed that: 1- The plunger velocity Vp and the plunger radius Ri affect the results of the rheograms and the approximation. The best combination is when K = 0.85 or K=0.77 and the Vp are 83.33 with 16.67 or 62.00 with 16.67 (x 10-4 m/s). Under those conditions the rheological properties of the Herschel-Bulkley fluids obtained from the Haake are in good agreement with those obtained from the back extrusion device. 85 Table 11. Values of the experimental parameters for 1.25%, 1.5% and 2.0% Kelset samples using the back extruder. R1 Vp Used % Kelset an TIME To (cm) x 104 m/s Solution (Pa.s“> (Pa) 1.00 62.00 & 16.67 1.25 0.50 15.95 3.91 1.20 45.00 & 16.67 1.25 0.60 10.75 . 2.07 1.36 - 1.25 0.52 - 0.00 1.50 62.00 & 16.67 1.25 0.40 7.25 4.96 1.00 62.00 & 16.67 1.50 0.60 12.33 4.70 1.20 62.00 & 45.00 1.50 0.60 9.20 3.74 1.36 62.00 & 45.00 1.50 0.60 5.05 0.51 1.50 62.00 & 45.00 1.50 0.40 9.15 6.40 1.00 62.00 & 16.67 2.00 0.60 2.10 7.18 1.20 62.00 8 16.67 2.00 0.60 23.10 6.81 1.36 83.33 & 45.00 2.00 0.60 9.20 1.68 1.50 62.00 a 16.67 2.00 0.40 22.45 7.99 86 Table 12. Shear stress and shear rate values for 1.25% Kelset obtained using the Haake viscometer. Shear Rate (5“) Shear Stress (Pa) 10.01 13.92 12.39 14.98 17.12 17.21 24.54 20.41 32.23_ 22.78 36.96 24.16 49.24 27.70 54.63 28.90 59.50 29.98 64.10 31.09 68.68 32.19 73.15 32.92 73.64 33.34 78.60 34.34 86.16 35.94 91.31 36.86 95.42 37.70 97.66 38.11 103.00 38.94 87 Table 13. Shear stress and shear rate values for 1.5% Kelset obtained using the Haake viscometer. Shear Rate (5") Shear Stress (Pa) 3.57 7.62 4.64 13.22 7.20 16.88 9.83 19.02 12.24 20.28 16.68 24.21 24.16 30.45 37.04 35.88 48.89 41.05 60.86 46.14 73.85 50.86 86.29 54.21 99.29 57.95 112.10 60.37 88 Table 14. Shear stress and shear rate values for 2.0% Kelset obtained using Haake viscometer. Shear Rate (8“) Shear Stress (Pa) 3.96 17.17 4.64 23.44 7.28 29.09 10.03 32.25 11.98 34.17 16.82 43.90 24.81 49.33 34.91 ' 55.87 39.90 55.88 50.46 64.62 56.01 66.07 59.80 68.72 64.14 70.75 69.68 73.56 75.14 75.29 79.36 77.57 92.32 80.48 100.20 81.32 105.60 85.17 89 Shear stress at the wall, Pa _;_ 11 , ALL Vp ARE CONSIDERED 1 .568 2 0.682 3 0.773 4 0.852 eon K = 0.852 , Vp1 = 62.00 AND Vp2 = 16.67 (104 m/s) , To = 4.95 80 "NB = 0.40 "PL = 0.321 n = :- HB 7.25 ”PL 11.19 70 J 60. me 2 SO-i PL PL 40 . 2 30 i 20 10 . HB.: Rheogram for the HB fluid at Ki PLi: Power law approximation at Ki 10 20 30 40 50 6O 70 80 90 100 110 Shear rate at the wall, 5-1 Figure 28. Rheograms for 1.25 % Kelset, considering data obtained from the Haake viscometer, the back extruder, and the power law approximation. 90 (Tw), Pa Shear stress at the wall Nomenclature: i Vpi x 104 m/s Ki 1 16.67 0.568 2 45.00 0.682 80 3 62.00 0.773 ‘ 4 83.33 0.852 70 . HB HB PL ’1 2 60- 1 50~ 40 - 30 - 20 . PL2 HBi: Rheogram for the HB fluid at Ki 10 " PLi: Power law approximation at Ki I I l I I I I I I l 0 10 20 30 4O 50 60 70 80 90 100 Shear rate at the wall (Yw), s'1 Figure 29. Rheograms for1.5 % Kelset, considering data obtained from the Haake viscometer, the back extruder, and the power law approximation. 91 2- For K = 0.85 the power law approximation curve is the same as the real Herschel-Bulkley curve, which means that under those conditions (K = 0.85 and Vp = 62.00 and 18.67 x 10-4 m/s), the approximation works well. 3- Since the yield stresses of the fluids range from 4 to 10 Pa, further studies need to be done in order to check for the effect of high yield stress values in the approximation. 4- In general the shear rate ranges for all different velocities, but not for Vp = 1.667 x 10-4 m/s, and for all X, go from 10 to 100 sec‘-1, which are the same as those achieved with the Haake. 5.3 MW Table 15 shows the experimental data collected from the back extrusion device, for the different test runs, at different fluid lengths in the container cylinder, for a constant plunger traveling distance, as explained in Section 4.7. Because the theory of power law fluids in a back extruder considers the use of two different plunger velocities, Table 16 shows the experimental data collected for the 1.0% Methocel solution at three different Vp. Tables 17 to 19 illustrate all the experimental values of consistency coefficient, shear rate at the wall and shear stress at the wall, for 1.0 X Methocel solution, at different fluid lengths ( 0 ), and at different Vp combinations. From Tables 15 to 19 the following observations can be made: 1- At a constant plunger radius (Ri = 1.2 cm) (K = 0.68), for a power law fluid sample, the end effect is present. 92 Table 15. Experimental data, collected from the back extrusion device for 1.0% Methocel, used in the end effect evaluation. length of FT FT- L 1ch fluid (N) (N) (m) (cm) ”0” (cm) 17.00 0.90 0.71 0.18 12.92 16.00 0.88 0.71 0.18 13.06 15.00 0.88 0.71 0.18 13.06 14.00 0.95 0.73 0.16 12.19 13.00 0.96 0.73 0.16 12.08 12.00 0.92 0.73 0.16 12.22 11.00 0.90 0.73 0.16 12.22 10.00 0.92 0.71 0.16 12.08 VP: = 45.00 x 10“ m/s Scale range = 5.0% (2.5 N) Time set=1.00 min R1 = 1.20 cm 93 Table 16. Experimental data Collected from the back extrusion for 1.0% Methocel at four plunger velocities. Vp x 104 Time FT FT- lch (m/s) Set (min) (B) (11) (cms) 83.33 0.50 1.04 0.72 14.23 62.00 0.80 0.95 0.72 12.06 45.00 1.00 0.90 0.73 13.34 16.67 1.50 0.81 0.75 22.00 (R1 = 1.20 cm) 94 Table 17. Experimental rheological parameters for 1.0% Vp1 = 45.00 x 10-4 m/s, Vp2 = 16.67 x 10”“ m/s and R1 = 1.2 cm. 11 "0" P1, P3 "1, ”J Tw Yw (cm) (Pa/m) (Pa.s") (Pa) (sr‘) 0.43 17.00 1560.79 1.73 5.07 12.50 1788.98 1.73 5.82 17.23 0.76 16.00 1381.69 0.90 4.69 8.85 1762.13 0.90 5.98 12.19 0.84 15.00 1343.15 0.77 4.59 8.43 1757.98 0.77 6.00 11.61 0.19 12.00 1714.51 2.88 5.27 22.72 1824.18 1.33 5.60 31.30 0.56 11.00 1821.92 2.88 6.04 10.53 1776.29 1.33 5.89 14.51 95 Table 18. Experimental rheological parameter for 1.0% Methocel for Vp1 = 45.00 x 10“ m/s, Vp2 = 16.67 x 10-‘ m/s and R1 = 1.2 cm. n "0" Pi, PJ ”1, ”J r... 3}... (cm) (Pa/m) (Pa.s") (Pa) (s“) 0.97 17.00 1528.48 0.71 5.27 7.87 584.17 0.71 2.01 2.91 0.86 16.00 1343.15 0.76 4.71 8.30 585.76 0.76 2.00 3.08 0.86 15.00 1343.15 0.72 4.58 8.43 586.19 0.74 2.03 3.12 1.15 14.00 1828.66 0.64 6.37 7.28 581.85 0.65 2.03 2.70 1.22 13.00 1950.78 0.63 6.82 7.11 581.12 0.62 2.03 2.64 1.04 12.00 1643.64 0.69 5.69 7.59 583.12 0.69 2.02 2.81 0.93 11.00 1467.74 0.73 5.05 8.01 584.70 0.73 2.00 2.97 1.13 10.00 1790.48 0.65 6.23 7.34 582.06 0.65 2.03 2.72 96 Table 19. Experimental rheological parameter for 1.0% Methocel for Vp1 = 45.00 x 10*4 m/s Vp2 = 83.33 x 10“ m/s and R1 = 1.2 cm. n "0" Pi, PJ “1, "J 1“, {{w (cm) (Pa/m) (Pa.s“) (Pa) (s“> 0.79 17.00 1535.97 0.95 5.22 8.68 2499.47 0.95 8.50 16.07 0.96 16.00 1373.70 0.65 4.73 7.87 2487.27 0.65 8.57 14.57 1.01 15.00 1337.33 0.59 4.62 7.73 2485.02 0.59 8.59 14.31 0.49 14.00 1867.79 1.85 6.14 11.37 2530.13 1.85 8.31 21.06 0.38 13.00 2007.63 2.38 6.48 13.36 2546.37 2.38 8.22 24.74 0.67 12.00 1661.62 1.24 5.58 9.48 2509.93 1.24 1.24 17.55 0.86 11.00 1470.40 0.82 5.03 8.30 2494.04 0.82 8.53 15.37 0.53 10.00 1824.60 1.71 6.03 10.89 2525.52 1.71 8.34 20.17 97' 2- For every specific Vp combinations, the height of fluid ( 0 ) inside the cylinder container affects the rheological properties. 3- For the three different Vp combinations used the one which gives better results in the determination of the rheological properties was 45.00 with 83.33 x10-4 m/s, from the comparison of the back extrusion data with those values obtained using the Haake. 4- When Vp1 = 45.00 x 10-4 m/s and Vp2 = 83.33 x 10-4 m/s, for a fluid length ( 0 = 12.0 cm ), the flow behavior index (n) and the consistency coefficient values are approximately the same as the values obtained from the Haake: 1.21.79§v0.8733 From back extruder: :, 0.6699 From Haake viscometer: T 1.2370 Then, in this case the best position is when the plunger bottom is separated from the container bottom at the end of the experiment by at least 2.0 cm. To express this result in dimensionless terms, knowing that the ratio of this distance (2.0 cm) to the outside cylinder container radius (1.78 cm) is approximately equals to 1.0, the best position of the plunger at the end of the experiment is when it is separated from the container bottom a minimum distance equivalent to the container radius (R0). Thus, to be conservative, a distance equivalent to the cup diameter is recommended for practical applications. 98 Chapter 8 CONCLUSIONS Wham minimums. 1. The Newtonian approximation for power law fluids and the power law approximation for Herschel-Bulkley (H-B) fluids work independently of the plunger velocity (Vp) used, but work better at the experimentally highest possible K value which corresponds to the smaller annular gap, K: 0.85. 2. The rheological properties of the power law fluids (n.71 ) affect the Newtonian approximation results: the closer the rheological properties to the Newtonian case (n = 1), the smaller the difference between the shear rate at the wall for the power law fluid and the shear rate at the wall for the Newtonian approximation. 3. The rheological properties of the H-B fluids ( To, n and n ) affect the power law approximation results: the closer the rheological properties are to the power law model (To = 0), the smaller the difference between the shear rate at the wall of the H-B fluid and the shear rate at the wall for the power law approximation. 99 .H ,: ;,.; 9;, -7 .; _ . . e.‘ e ,2 ., .-_.,_ For all power law and B-B fluids used, only when the plunger radius used was bigger than 1.36 cm (K = 0.77 or K = 0.85) was the back extrusion data for the rheograms in good agreement with the values determined from the Haake viscometer data. The best Vp combinations to work with are when Vp1/Vp2 = 4 to 5. Those combinations are: Vp1 83.33 x 10-4 m/s with Vp2 = 18.87 x 10-4 m/s or Vp1 = 62.00 x 10-4 m/s with Vp2 = 16.87 x 10-4 m/s. For K = 0.85, the power law approximation rheogram is the same as the real H-B curve which means that under the best operational conditions ( K = 0.85 and Vp = 62.00 and 18.67 x 10-4 m/s ) the approximation working is in excellent agreement with actual values. Under the best operational conditions (K = 0.85 and Vp = 62.00 and 16.87 x 10-4 m/s) the Newtonian approximation of power law fluids works but may vary with the rheological properties. At a constant plunger radius for a power law fluid sample, the end effect is present. To avoid end effects in back extrusion testing, the plunger bottom should be separated from the container bottom a minimum distance equivalent to the inner container diameter at the end of the experiment. 100 Chapter 7 PRACTICAL GUIDELINES FOR CONDUCTING BACK EXTRUSION TESTS Prepare the samples and place them in the cylindrical container. Avoid the presence of air bubbles in the fluid, and keep sample at constant temperature. Select an annular gap (K) between 0.77 and 0.85. Select a plunger velocity combination in the ratio of Vp1/Vp2 = 4 to 5. Allow the plunger to travel enough distance inside the fluid to ensure that the flow is laminar, fully developed, and at steady state. Experimentally was found that the ratio of this distance (Li) to the container diameter (Do) should be in the range of (Li/Do) = 10 to 15. To avoid end effects in back extrusion testing, the plunger bottom should be separated from the container bottom a minimum distance equivalent to the inner container diameter at the end of the experiment. Use the power law approximation when dealing with Herschel- Bulkley fluids to reduce the calculation time. Under the best operational conditions (K = 0.85, Vp1 = 62.00 x 10-4 m/s and Vp2 = 18.67 x 10-4 m/s) this approximation is in excellent agreement with actual values. 101 Chapter 8 SUGGESTIONS FOR FUTURE RESEARCH 1. From this study it was found that the Newtonian approximation for the shear rate of power law fluids does not work as well as the power law approximation for Herschel-Bulkley fluids. For all power law fluids considered the real rheograms (power law) and the rheograms approximated by Newtonian equation have the same shape; however, for the same shear rate at the wall value, the Newtonian approximation gives a higher shear stress at the wall and the difference between the two shear rates is almost constant for all shear rates. This difference could be a function of the rheological properties of the fluids (flow behavior index (n) and consistency coefficient) and it works better when the optimal experimental conditions for back extrusion testing are maintained. As a result, further work needs to be done to determine the relationship between the power law rheogram and the Newtonian approximation as a function of the rheological properties. 2. In the power law approximation of H-B fluids, the different Kelset solutions used have yield stress values ranging from 4 to 10 Pa. To check for the larger effect of the yield stress in the_ approximation, high yield stress samples (above 10 Pa) should be used in future experiments. 102 3. End effects were found for a power law fluid (1.0 X Methocel) using a plunger radius (Ri) of 1.2 cm, at three plunger velocity (Vp) combinations. However, future experiments need to be run for different non-Newtonian fluids to determine if the rheological properties affect the results. On the other hand, different plunger velocity combinations and optimum plunger radius could be used to evaluate the influence of those experimental parameters on the and effect. Finally, the same fluid height inside the container could be used for each velocity of the two Vp combinations to guarantee more reliable conclusions from the experiments. 4. Statistical analysis of the data should be conducted to obtain more specific results for the shear rate approximations of non-Newtonian fluids and for the effect of the experimental conditions in back extrusion testing. 103 Chapter 9 REFERENCES Chee, K.K., and Rudin, A. 1970. A study of low shear viscosity of polymer melts. The Canadian Journal of Chemical Engineering 48 (8):362. Chee, K.K., and Rudin, A. 1976. Flow analysis and end corrections in falling coaxial cylinder viscometry. The Canadian Journal of Chemical Engineering 54 (6):129. Darby, R. 1976. Viscoelastic fluids. Marcel Dekker, Inc. New York. Ellis, T.M.R. 1982 A Structural Approach to Fortran 77 Programming. Addison-Wesley Publishing Company. London, Massachusetts, California. Ferry, J.D. 1970. Viscoelastic Properties of Polymers. John Wiley and Sons, Inc. New York, New York. Fredrickson, A.G. 1959. Flow of non-Newtonian fluids in annuli. Ph.D. Thesis, University of Wisconsin. Geankoplis, C.J. 1983. Transport Processes and Unit Operations. Allyn and Bacon, Inc. Boston, London, Toronto. Hanks, R.W. 1979. The axial laminar flow of yield-pseudoplastic fluids in concentric annuli. Ind. Eng. Chem. Fundam. 19(1):33. Hirujo, F. and Marta, N. 1987. Adaptation of RHEO programs for non-Newtonian fluids in a back extruder. INTEC, Dominican Republic. Morgan, R.G., Suter, D.A., and Sweat, V.E. 1979. Mathematical analysis of a simple back-extrusion device. Paper No. 79-6001. American Society of Agricultural Engineers. St. Joseph, MI. Osorio-Lira, F.A. 1985. Back extrusion of power law, Bingham plastic and Herschel-Bulkley fluids. M.S. Thesis, Department of Food Science and Human Nutrition, Michigan State University. East Lansing, MI Osorio, F.A., and Steffe, J.F. 1985. Back extrusion of Herschel-Bulkley fluids -Example problem. Paper No. 85-6004. American Society of Agricultural Engineers. St. Joseph, MI. 104 Osorio, F.A., and Steffe, J.F. 1985. Back extrusion of power law fluids .-Example problem. Paper No. 85-6003. American Society of Agricultural Engineers. St. Joseph, MI. Osorio, F.A., and Steffe, J.F. 1987. Back extrusion of power law fluids. Journal of Texture Studies 18(1987):43-63. Prentice, J.H. 1984. Measurements in the Rheology of Foodstuffs. Elsevier Applied Science Publishers. London and New York. Rao, M.A., and Rizvi, S.S.H. 1988. Engineering Properties of Food. Marcell Dekker, Inc. New York. Schummer, P., and Worthoff, R.H. 1978. An elementary method for the evaluation of a flow curve. Chemical Engineering Science 33(3):?59-763. Skelland, A.H.P. 1967. Non-Newtonian Flow and Heat Transfer. John Wiley and Sons, Inc, New York. Steffe, J.F., and Ford, E.W. 1985. Rheological techniques to evaluate the shelf-stability of starch-thickened, strained apricots. Journal of Texture Studies 16(1985):179-192. Steffe, J.F., and Osorio, F.A. 1987. Back extrusion of non-Newtonian fluids. Food Technology 41(3):?2-77. Steffe. J.F., and Morgan, R.G. 1986. Pipeline design and pump selection for non-Newtonian fluid foods. Food Technology 40(12):?8-85. Tiu, 0., and Bhattacharyya, S. 1974. Developing and fully developed velocity profiles for inelastic power law fluids in an annulus. AICHE Journal 20(6):1140. Walters, K. 1975; Rheometry. Halsted Press. John Wiley and Sons, Inc. New York. Worlow, R.W. 1980. Rheological Techniques. Halsted Press. John Wiley and Sons, Inc. New York. 105 APPENDICES Appendix A RHEO - PROGRAMS BHEQ..'_1 This option computes the dimensionless shear stress at the wall (Tw), and the dimensionless flow rate § defined in Equation (22) for different values of n, To and K. Tw and § will be used in the calculation of the consistency coefficient for power law and Herschel-Bulkley fluids according with the Equation (9). W= Tolerance Max Initial value of n, To and K Final value of n, To and K Increment rate of n, To and K W: For each corresponding combinations of n, To and K it will gives PL pressure in excess of hydrostatic pressure at the plunger base Tw = dimensionless shear stress at the plunger wall § = dimensionless flow rate defined in Equation (22) ¢ = dimensionless velocity defined in Equation (21) 107 BHEQ_:_2 This option computes the rheological properties of Herschel- Bulkley and power law fluids, knowing the experimental conditions and the collected data from the back extrusion device. Ineui_rariahlea_are= Number of experiments Plunger radius, Ri Outside cylinder radius, Ro Fluid density, C For each experiment: Plunger velocity, Vp Chart Speed, Csp Chart Length, lch Recorded force after the plunger is stopped, FTe Recorded force while plunger is traveling down, FT Output_rariahles_ars= Average yield stress for all experiments, Toave For each experiment: Force corrected for buoyancy, Fob Buoyancy force, Fb Yield stress, To Guess for the dimensionless yield stress, To guess The ratio Fob/(n L Ro Ri) Then, for each experiment, at a specific plunger velocity, for selected n and To guess intervals, this program will generate 108 data of P(Tw + K) versus n which will be plotted in order to obtain the n and :1 values according with the procedure explained in Chapter 4. If the fluid is power law, R3: 0, this program will gives immediately the n value without asking for the n and To guess intervals. BBQ—:1 This option computes the shear stress at the wall Tw, and the shear rate at the wall Yw for power law or Herschel-Bulkley fluids when their rheological properties are known. Wm: Tolerance Max Flow behavior index, n Yield stress,Tb Plunger radius, Ri Outside cylinder radius, Ro Plunger velocity, Vp Consistency coefficient,n Lower limit of pressure drop interval guess Upper limit of pressure drop interval guess W: Pressure in excess of hydrostatic pressure at the plunger base, PL Pressure drop per unit of length, P 109 T Shear stress at the wall, w Shear rate at Dimensionless Dimensionless Dimensionless Dimensionless BHEQ_;_1 the wall, Yw shear stress at the wall, Tw flow rate, § velocity, 8 yield stress, To This option is only used when dealing with power law fluids. With this program, Knowing the experimental conditions of the equipment, the flow behavior index n can be obtained according with Equation (8). RHEO - 4 also calculates the yield stress To which can be used as reference when the Herschel-Bulkley fluids are being approximated to power law models. W= Cell capacity Annulus gap, K = Ri/Ro Fluid density, C Plunger radius, Ri Outside cylinder radius, Ro Number of experiments For each experiment: Scale range Plunger velocity, Vp Recorded force while plunger is traveling down, FT (cm) Recorded force after the plunger is stopped, FTe (cm) 110 Time set axis Chart length, obtained from the recorder, lch QHIEHI xeriebles gcg: Yield stress, To Force corrected for buoyancy, Fcb Length of annular region, L Fch Li] 1n ' '-"l 2 Fcbi L_J Vpi 1n --- , Vpi 1n (F) Flow behavior index, n = in (V) RHEQ_:_§ called ln(F) called ln(V) This option was created as an alternative to reduce the calculation time when dealing with power law fluids. After determining the flow behavior index n with the experimental data in RHEO - 4, and Tw and § with RHEO - 1, the values of the consistency coefficient r], the shear rate at the wall 7w, the shear stress at the wall 1w, and the pressure drop per unit of length P , can be obtained for two different plunger velocities according with Equations (9), (10), 111 (12) and (13). W= Flow behavior index, n Plunger radius, Ri Outside cylinder radius, Ro Plunger velocity, Vp Length of annular region, L Force corrected for buoyancy, Fcb Dimensionless shear stress at the plunger wall, Tw Dimensionless flow rate, § Wm: Pressure drop per unit of length, P Consistency coefficient, n Shear rate at the wall, Yw Shear stress at the wall, Tw 112 Appendix B CALCULATION EXAMPLE FOR THE POWER LAW SHEAR RATE AT THE WALL AS AN APPROXIMATION TO THE SHEAR RATE AT THE WALL FOR HERSCHEL-BULKLEY FLUIDS, BASED ON SIMULATED DATA Wigs; 2.0 ’6 Kelset Walden; 1: = 8.8 + 27.0 {,0.5 1. Using RHEO-3, the following set of data was generated, using the known rheological properties and at different conditions of plunger velocity (Vp) and plunger radius (Ri): K = (0.010 m/0.0178 m) = 0.588 P Vpx10“4 Tw 7w Tw Fcb w) (Pa) (£4) (N)— 22182.99 83.33 108.552 13.108 0.5458 2.7328 19542.89 82.00 93.849 9.878 0.5445 2.4047 17087.91 45.00 81.850 7.280 0.5430 2.0997 11550.83 18.87 54.580, 2.872 0.5388 1.4114 5598.09 1.887 25.307 0.374 0.5139 0.8898 In all cases, to calculate Fcb (Equation 12) the value L = 0.20 was assumed. With this value it is ensured that the plunger is traveling enough distance inside the fluid so that the flow is laminar, fully developed and at steady state. 113 For the other plunger radii (Ri = 1.2, 1.38 and 1.5 cm ) the same sets of data were generated, as shown in the following tables. 5.3.1.612 P Vpx10‘4 Tw 7w Tw Fcb (£31m) giggle) (Pa) (sf-1) (N) 41593.23 83.33 135.773 22.115 0.3709 5.8107 36439.60 62.00 118.812 16.602 0.3705 5.0888 31649.10 45.00 103.045 12.184 0.3700 4.4174 20851.08 16.87 67.495 4.726 0.3878 2.9044 9289.19 1.667 29.361 0.580 0.3592 1.2832 K = 9.213 P Vpx10‘4 Tw iw Tw Fcb (Razml gals) (Bale (sf-1) (N) 82059.62 83.33 181.555. 40.938 0.2514 12.6393 71578.93 62.00 158.275 30.648 0.2513 11.0232 61834.18 45.00 136.637 22.417 0.2511 9.5211 39883.35 16.67 87.881 8.579 0.2504 6.1369 16452.20 1.67 35.800 1.000 0.2473 2.5238 [1.5.2.2852 P Vpx10‘4 1w 7w Tw Feb 1231!” , (14:1 1P1) (3"‘11 iflL. 194512.11 83.33’ 268.913 92.811 0.1571 32.5674 189010.82 82.00 233.806 69.325 0.1571 28.2967 145311.47 45.00 200.793 50.564 0.1570 24.3278 91931.06 16.67 126.883 19.127 0.1568 15.3881 35070.08 1.67 48.136 2.123 0.1560 5.8653 114 2. In the power law approximation of Herschel-Bulkley fluids, the flow behavior index (n pl) was calculated for each specific plunger radius and for all possible two plunger velocity combinations according with the following equation: Fch Li 7 3' ijl ln - n iln -— I Fcbi LJ j L Vpi; Then, n (pl) is obtained, from the linear regression, as the slope of the curve corresponding to all Vp combinations and Bi used. For example: W W11 1,2 0 8886 2.3026 1,3 1.3277 3.2956 1,4 1 4742 3.6161 1,5 1 6110 3.9118 2,3 0 4392 0.9931 2,4 0.5857 1.3135 2,5 0 7225 1.6092 3,4 0 1465 0.3205 4,5 0 1368 0.2957 From the linear regression of the data, slope n (pl) = 0.3977 r‘2 0.9985 115 3. Using RHEO-1, with the annular gap (K = 0.773), To = 0.0 (approximated to power law), and n = 0.3977, the following values were generated: Tw = 0.2495 and Q = 190.159 x 109-6 Therefore, with 0, P, Ri, Vp, K and n known, the consistency coefficient was calculated from Equation (9). At this point, the approximated to power law rheological properties ( n pl and n pl) have been obtained. 4. In the calculation of the shear stress and the shear rate at the wall approximated to power law for the Herschel-Bulkley fluids, RHEO-3 is used. Those values were compared to the original shear stress and shear rate at the wall corresponding to the Herschel-Bulkley fluids calculated in Step 1 of this procedure. The differences (Equations 30 and 31) reflect how the power law approximation works for Herschel- Bulkley fluids. For example: 2 : Vp x10“4 “pl P Tw pl 1?». pl ATwz AYw2 W (2;) Ls"-il x X— 83.33 27.576 57251.33 125.686 45.334 -3o.77 10.74 62.00 27.054 49937.77 109.630 33.730 -30.73 10.06 45.00 26.549 43140.49 94.707 24.461 -3o.69 9.21 16.67 26.417 27625.72 61.087 9.069 -30.49 5.71 1.67 26.197 11459.11 25.157 0.903 —29.73 -9.70 116 Conclusions about the power law approximation for Herschel- Bulkley fluids are presented in Chapters 4 and 5 of this study, at different Vp and Bi conditions for 2.0%, 1.5 X and 1.0 X Kelset solutions used as standards in the data simulation procedure. All tables of results are included in Appendix D. 117 Appendix C 1. CALCULATION EXAMPLE FOR THE DETERMINATION OF RHEOLOGICAL PROPERTIES OF A POWER LAW FLUID USING THE BACK EXTRUSION DEVICE Sample solution: 1.0 x Methocel Plunger radii used: Ri = 1.0, 1.2, 1.36, and 1.5 cm Plunger velocities used: Vp = 83.33, 62.00, 45.00 and 16.67 (x 109-4 m/s) Cell capacity: 50 Newtons (compression) Outside cylinder radius: R0 = 1.78 cm RHEO-Programs used: RHEO-4 to determine n with the experimental data collected. RHEO-i to determine Tw and i. RHEO-5.to determine P, Tl , Tw p1 and Yw pl. The following table shows an example of the data collected from the Instron for 1.0 X Methocel sample: 118 Data collected from the Instron for 1.0 X Methocel. Ri Vp Time set FT FTe lch isml___isn£ninl_____iminl 181 (Hi QatL. 1.0 500.0 0.5 0.4081 0.3215 11.250 372.0 0.8 3 0.4738 0.4030 11.844 270.0 1.0 0.5074 0.4220 12.371 100.0 1.5 0.4389 0.4150 24.000 1.2 500.0 0.5 1.0390 0.7235 14.295 372.0 0.8 0.9476 0.7245 12.058 270.0 1.0 0.8980 0.7250 13.344 100.0 1.5 0.8134 0.7450 22.000 1.36 500.0 0.5 1.4670 0.9300 11.350 372.0 0.8 1.4510 0.9500 9.544 270.0 1.0 1.3180 0.9400 10.640 100.0 1.5 1.1010 0.9350 18.996 1.50 500.0 0.5 4.8540 1.5235 8.700 372.0 0.8 3.9150 1.4580 7.308 270.0 1.0 3.0580 1.5000 8.340 100.0 1.5 2.3540 1.4000 13.250 With the collected data , for using REED-4, generated: RHEO-1 and RHEO-S, the following set of data was 119 Bi = 1.38 cm as an example, eeee.e seem.~ Hemm.e Nee.eMee eee.e eeme.e Ne.ee seem.es mums.m seem.e eeN.eeeN em.eeae mme~.e eem.e sees.e ee.ee eeNm.e emee.m seem.~ Nmee.e mme.eMes eee.e ewme.e Ne.ee ewes.N~ ewes.“ Neee.e see.emem ee.mees eemN.e Hem.e emeH.e ee.~e mmee.e emem.ee eeee.e emee.e NNH.HeeN mae.e some.e ee.ee emee.e~ emee.e Neae.e “Ne.~mee eee.ee- Nem~.e see.e ewes.e ee.~e neee.e eeee.e Neee.m eee.e eNe.emHs eee.e eeee.e ~e.es eNme.Ne Hemm.e eee.e sem.~eem see.eems eNe~.e 5mm.e ease.e mm.me eNmN.e eeem.ee emee.e mee.e ee.mee~ eem.e HeeH.e ee.me Hemm.em “Ne~.e mee.e see.~e~m eee.maa eee~.e eme.e wees.e ee.me eemm.e em.e Ae-ev keev nem.eev Ae\eev eee x sz flee e\e eee x Asec Fe 3» _e 3e e: .2: we .ee e 3e He.e. see we .25 ee> .ee> e ea .Pmoosuoz a o.H Low memuosegea —eu=osmemaxo we» so mouse) 120 2. CALCULATION EXAMPLE FOR SHEAR STRESS AND SHEAR RATE AT THE WALL OF HERSCHEL-BULKLEY FLUID AND APPROXIMATED TO POWER LAW VALUES W 2.0 X Kelset The Kelset solution was prepared at the laboratory, at a constant room temperature of 25 i 1 °C. The back extrusion device was balanced, calibrated, and prepared to be used. The fluid sample was placed in glass containers of constant radius (Ro = 1.76 cm) and tests were run at different plunger velocities (Vp) and plunger radius (Ri) and all data were collected. With the collected data, using RHEO-4, the following set of parameters was obtained from the P(Tw + K) versus n graphs generated as explained in Chapter 3. For 2.0 X Kelset, Ri Vpi, ij n n T0 Lem) (10“Lnfl) (Pu‘n) (P1).— 1.0 82.00, 18.87 0.8 29.1 7.180 1.2 82.00, 18.87 0.8 23.1 8.810 1.38 83.33, 18.87 0.8 9.2 1.878 1.5 82.00, 18.87 0.4 22.45 7.994 121 Then, with the rheological properties obtained above, for different Ri values, the shear rates and the shear stresses at the wall for the Herschel-Bulkley fluids were calculated using RHEO-3 at different plunger velocities. For example, for 2.0 X Kelset solution at R1 = 1.0 cm, n = 0.6, n = 29.1 Pa s‘n, and to = 7.18 Pa, Vp.10“4 P 1:w Yw Tw G x 1096 lfllil: (Balm) (P§l_ .LEA'll 18.87 11704.89 58.790 2.433 0.5513 3719.437 45.00 19214.48 94.258 8.214 0.5574 4395.588 82.00 22757.72 111.923 8.454 0.5589 4587.135 83.33 28888.51 131.518 11.251 0.5599 4708.958 or at R1 = 1.2 cm, n = 0.6, n = 23.1 Pa s‘n and To = 6.8 Pa, Vp.10“4 P ‘tw '1w* Tw 5 x 10‘6 fulfil (Efilnl, (Pal_____L§f-11 18.87 18533.32 80.928 4.133 0.3738 1894.815 45.00 30984.18 102.475 10.881 0.3785 1942.221 82.00 38880.89 122.078 14.572 0.3783 2003.875 83.33 43380.71 143.823 19.438 0.3787 2052.808 Therefore, Tw and Yw correspond to the Herschel-Bulkley (H-B) shear stress and shear rate at the wall values: Tw HB and Yw HB, respectively. Then, the procedure to obtain the 122 approximated to power law values was: a) With the same data collected from the back extrusion device for the H-B fluids, run RHEO-2 to generate the flow behavior index as power law (n pl), at every two possible plunger velocity combinations, as explained in Chapter 4. b) With the (n pl) values generated, To = 0 (power law), and K value known, the Tw and 4 values were obtained using RHEO-l. c) Using the data obtained for Q, Tw, Vp, Ri, L, Fcb, etc., the consistency coefficient, the shear rate at the wall, and the shear stress at the wall, approximated to power law, were calculated using RHEO-S. For example, for 2.0 X Kelset, at Ri = 1.0 cm, n Vp(i,.i) L Fcb P n In p1 w p1 1.0.41.5 L L EM“!!! Pa emf-1 0.572 62.00 0.131 1.787 21568.319 28.678 108.604 7.454 45.00 0.112 1.235 17441.118 28.679 87.822 5.410 0.519 62.00 0.134 1.787 21858.320 34.782 107.154 8.733 16.67 0.136 0.934 11046.256 34.764 54.151 2.348 0.473 45.00 0.112 1.235 17766.574 34.927 86.195 6.757 16.67 0.136 0.934 11103.091 34.907 53.867 2.503 123 Finally, the YO'w pl and the Tw pl are compared to the 1.(w HB and the rw HB according with their differences (Equations 30 and 31). From those differences, the effect of Vp and R1 in the determination of the rheological properties of a H-B fluid in back extrusion testing are considered, as well as the power law approximation for H-B fluids. All results are presented in Appendix D, and discussion and conclusions about them are included in Chapter 5. 124 Appendix D TABLE OF RESULTS 125 om.m~- NN.HN- NH.mo- mN.no- I Hom.om Nmm.mm emm.me mmH.wm eem.mm www.mm mmm.mm~ Nom.NN . mmm.oo~ mm.mm om.mm- NN.HN- Na.mm- mN.Nm- . mom.NN ~mm.~m wNo.Nm HNm.mm “Nm.oo mom.oo mNm.meH wNH.mm I HmN.mm oo.No om.mmu NN.HN- Nfi.mo- mN.Nmu No.mo- mNm.oH oem.me HHm.oN “an.Nm NNm.me mem.mm eem.NoH Ho~.ms mmH.oem Neo.~m oo.m¢ om.mN- NN.~N- NH.mm- mN.Nm- No.mc- NNH.o Nm~.~¢ mom.m omo.m¢ mmm.n~ mom.me dom.mm em~.mm NeN.H~m awn.mn No.0“ cm.mN- HN.~N- mo.mo- mN.No- No.mo- NHo.o Ham.wN mmm.o oom.~m mmN.H Hem.em smm.m eoo.mm mNH.~m Noo.em New.” .5 32 es 32 4% as.“ OH x > E 3» E :2. E z» 3 3e 3 3» 3 se 3 3» 3 3e 3 3+ 3 3P 9 mom.o Nmo.o mNN.o Nmm.o. omm.o x .mme.o u : ecu m~m.~m u c "sum Luna & m.fl so; .ups_m coweouzmz on umamEFxogqae ppm: one so was; scene we» use u_:Pm zap cases we spa: we» on was; semen use :mwzomn mucmcmmswu we» use _P~3 ago on one; geese use mmmcom semen zap gazes .Ho m_ee» 126 mm.mm- Hm.om- um.em- mm.Nm- - mem.m~ sm~.NH emN.mm mom.m~ HmH.Ho omm.NN mwo.NmH Nam.mN . www.me mm.mw ma.mm- Hm.om- um.¢m- mm.Nm- N~.om- Nem.¢a on.mfi mom.¢N moH.mH mm¢.m¢ ooN.HN Nee.No~ HmN.oN eom.o~m Nom.ee oo.No m¢.mm- Hm.mm- mm.em- mm.Nmn NH.om- “mm.ofl eem.efi enm.mfi mHN.o~ NHo.mm www.mH mmm.e~ meH.eN m~m.mwm emN.~e oo.me mm.mm- Hm.mm- hm.em- mm.Nm- NH.om- omm.m mNm.~H mmm.m omm.N~ mNN.NH mm~.m~ eem.NN Nmm.m~ 5mm.NHN Ham.~m so.cfl om.mm- Hm.mm- em.emu em.Nmu NH.om- mmm.o NeN.o moo.o mNH.~ mNN.H mmm.m emN.N mmN.oH emN.HN eum.u~ New.” 3.5 3.5 3.6 3.5 3.3 35 pa 3» pa 3P pa 3» _a 39 Fe 3» pa :9 pa 3» Fa 3P pa 3» Pa :9 cod x a> mom.o Nmm.o NNN.o Nmm.o omm.o x .mmmN.o u c use m~m.~ as "sea seam N o.~ sow .ceecouzmz o» umpmswxogaae Pym: we» on some scene we» use swaps zap noses we ppm; was as was; seven one awesome mocmeowewu one see ppm: was on one; eemem use mmogum semen zap Luzon .No m_aes 127 we.Nm- H~.om- ~m.mN- mm.oN- mm.¢N- omw.Hfi 9mm.9m mam.0N Noo.mN mem.wm mnm.mo9 NeN.mm meo.~m~ Hoe.eNN Noo.mne mm.mm we.Nm- H~.om- ”m.mNu mN.mN- mm.eN- Nom.w NmN.me mmN.mH Hom.mo mNm.mN mem.om moe.oo Nom.mmH ONo.mmm m~m.ooe oo.No me.Nm- HH.om- 9m.mNu mN.oN- mm.¢N- amm.o e~N.He owo.~H eme.mm NNo.oN mNm.oN mm9.me mmm.NHH eNN.~mm Nom.eem oo.me me.Nm- ~H.om- Hm.mN- mN.oNu mm.eN- mcm.N NNH.mN mo~.¢ on.mm omN.N mm~.oe emw.NH mNm.ON NNm.e¢~ ¢NN.NoN No.m9 me.Nm- oo.om- mN.mN- mN.mNu mm.eN- fimN.o m-.~ o~¢.o mMN.o~ mNN.o ma~.e~ mes.“ N9N.~N mme.¢~ Nem.mo Noo.H IIuW 45 I; z z . s 2 s we 2 ass _a 3» pm :9 Pa 3» _a 39 —a 2» Pa 39 pa 3» pa :9 Pa 3» Fe 39 so“ x a> mom.o Nmo.o men.o Nmm.o omm.o x .mNHm.o u : use mm~.m~ u c "Pmoogumz & m.N no» .cescouzmz o9 umpmspxogaae P—ez one as some gemcm 6:9 one 59:99 zap Loses me 9’s: one on mum; emcee we» cmmzamn mesmemeeae 6:9 use P’s: one an mmmeum Leone use some semen zap Luzon .ma open» 128 em.m9- mw.ofl- oN.mH- mN.eH- mme.m Nmn.es eeN.NH omN.HN meo.Nm Heo.em oee.o~ mmm.oe mm.mm em.ms- ew.es- on.ms- m~.eH- Ne.mH- mNN.N eHo.NH omm.NH NmN.N~ eem.eN HeN.NN mNo.~m mmo.me Nmo.moe oeN.NHN oo.No em.ms- em.m9- 9N.ms- mN.¢H- No.m9- mmN.m Nme.m Nsm.m oeN.eH mmN.NH omN.NN Nmm.He www.mm ce~.mmm NN~.oN~ oo.me em.ms- ew.ms- oN.mH- mN.¢9- Ne.m9- emm.s emm.e om¢.m mN9.N ~mm.o QNN.H~ Nmm.mH emo.oN www.mNs oma.mm No.69 Nm.ms- Hm.os- mo.mfi- ¢N.eH- me.ms- om~.o Nam.o mem.o use.” mme.o NmN.N mmm.~ 909.9 owm.N~ mNm.Ns New.” 93 a: 93 9: H3 m 9e 9e 9e 94 me A \sv pa 3» Fe 39 _a 3» pa :9 Pa 3» Fe 39 Fe 3+ Fe :9 Fe 3+ Fe :9 sea x a> wem.o Nmo.o m9~.o wa.o omm.o x .Neme.e e eee eeo.m u c "5855: 9 o.N tee .eeeeeezez e9 eeeeeexeeeee 99e3 9:9 9e sue: gem:m «:9 use zap sexes we spa: 9:9 9e m9ee :ee:» «:9 :69399: wozogowmwu 6:9 use ppm: 9:9 as was; Lem:m one mmogum Leo:m an. more; .ea o—ae9 129 mm.¢~n No.mH- NH.~Hu om.-- me.oH- m~m.m coo.» mw¢.o~ ~mm.oH mom.~m w-.- mom.mn ~m¢.mm “Hm.oom mNo.NmH mm.mm mm.e~- No.m~- N~.~H- om.HH- o¢.oH- mmm.o omm.m nom.- mH~.m e~m.m~ No~.¢H Hem.¢m mom.m~ eom.~m¢ ¢m~.omH oo.~m ¢N.¢H- mo.mfl- ~H.~Hn mm.HH- oe.o~- mmo.m coe.¢ eom.m mmm.m nmo.~H moH.H~ eom.mm omo.~m meu.n~m www.moH oo.me NN.¢H- Ho.mH- ~H.NH- om.-- oe.oH- mmm.~ HNH.~ mm~.m mm~.m mum.m Nom.m men.¢~ moo.o~ m~¢.-~ www.mc uc.m~ mm.m~- mo.m~- mH.~H- _Hm.HH- me.o~- om~.o umm.o omm.o omw.o mmm.o eem.o 55¢.“ Nm~.H H¢~.~H Hmm.m mwm.~ u 3.5 H 35 H 3»< a 3»< H 3»< Am >5 3 z E z E E E E E E E E 3 S x a> .» H. 3.» 3... 3.» 3» 3» J 3» » mom.o Nam.o mum o Nmm.o omm.o x .mn.o u : ucm mm.H up» "pmuo:uoz a m.H Low .cowcouzmz o» umunspxogqam ppm: m:a an wan: Lom:m m:u vcm zap gmzog mm p_m3 m:u um was; gum:m o:a :mmzuma wucmsmh—avwfi m...“ “Em 2.03 mzu Hm wn—ML Lumcm Ocm mmwsum mecm 3MP Lm3oa .MQwPDMH 130 oo.o oo.o oo.o oo.o oo.o mom.H om~.e mom.~ coo.» mmm.m mmw.e~ omo.mH mmm.em mon.mofl Nm~.om~ mo.m~ oo.o oo.o oo.o oo.o oo.o oo~.o -e.o nw~.o o-.o omm.o Mme.» “on.“ m¢.m Hnm.oH muo.mm som.fi A H .n .n .n 3»< 3z 3+< 3&4 3»< Am\sv pa 3» Pa 3P Pa 3» Fa 3P pa 3» pa 3P Pg 3+ Pa 3P _n 3+ pg 3P co“ x a> mom.o Nm0.o mn~.o mmm.c omm.o x co.“ n : van um.m u c Ha=:»m :sou gem .cmmcoa3mz op umumEPxogagm ppm3 m:p an was: 3am:m w:u new 3o— Lm3oa mm Fpm3 m:a um mum: me:m m:u :mm3um: mucmgwmwwc m:u can ppm3 m:» as man: gmm:m use mmmgum 3mm:m 3m» Lm3oa .oa mpnm» 131 oH.NN moo.m~ moH.mH mm.m~ nmm.m~ mHH.- ¢~.oH rmm.me wmm.o¢ HH.o~ mme.~HH Ham.um mm.mm mm.om mom.HH onm.m mm.¢~ mfio.ma om.oH mo.o~ mn.mm wem.om mo.mH www.mm ~m.mm oo.~o -.mfi mem.m omm.» m~.mH Now.m~ fm~.- -.m Hw¢.¢~ ~H¢.NN oo.mH Ho~.oo mm.om oo.m¢ o¢.HH Ho~.m Nmm.~ n~.m NHH.m c-.e Hm.m moo.m mum.m mm.o~ Hom.~N -.mfi mo.o~ HN.¢H- mam.o enm.o N.-- mom.o omm.o o~.mu mom.o ooo.~ No.¢ -~.~ m-.~ mom.fi 35 ~3»< _a 3» m: 3» ~3»< pa 3+ a: 3+ ~3+< —a 3+ m: 3+ N3+< pa 3+ m: 3+ ca x Q> mom.o wwo.o mun.o wa.o g .w.m u o: can m.o u c .o.- u c "ummpox u o.~ no; amps; 3mm:m o3» m:u :mm3pwn mucwgmmmwv m:u new man: me:m 3a» 3m3on op umume_xosaam .F_m3 m:u um mums 3mm:m »m—x_:mn—m:umsm= .mo mpnm» 132 mm.om mme.o~ Hmo.m~ m¢.mm www.mm mmm.- N~.m~ ~mm.m¢ mm~.mm me.- mmm.~o~ m¢¢.mm mm.mm He.Hm woN.mH e¢o3ofi oe.»m mmm.- ~a~.o~ -.o~ Hem.mm mm~.m~ Nm.ofi mom.m» cmm.em oo.~o vm.m¢ omo.HH mmm.~ om.mm mm~.o~ Hn~.- No.m~ mHm.m~ omm.- mm.¢~ mem.¢m mo~.»e oo.m¢ ~m.e~ mmo.e o»~.m m».m~ -o.m ono.m om.m~ Nmm.m oem.m em.m u~m.o~ mem.m~ No.0H mm.m~- Noe.o nmm.o mm.¢m- oom.o mm».o mm.H~- mmm.o 5mm.» om.o~- mmo.~ «me.~ som.H m a N3? 3 3» m: 3» ~35 a 3» m: 3» ~35 .3 J» m: 3» «as a J» m: 3» A \ V . . . . . . c S x :3 mom.o «mm.o m»n.o wa.o .2; u o» 2; 0.: u c .3 n: 33.3. 3 m4 :8 mmumg gam:m 03“ m:a :mm3uo: oucmgmmmwv,m:u vac .F—m3 o:u um mum: Lao:m . 3m» Lm3oa o» umums_xoganm .Fpm3 m:a pm mum: me:m »m»x—:mupm:umgm: .wo cyan» 133 Hm.¢~ mum.o~ wam.m~ r©.wfi cum.Hm mom.mm em.m~ Num.mm wom.m¢ ~m.m mm¢.m- mm.mo~ mm.mm mo.~m ¢~H.mH www.mfi c~.»~ me».m~ o»~.o~ Hw.- ¢~¢.He ku.om em.m mom.mw wdo.flm oo.~o mm.o~ mum.o~_¢mfi.m o¢.m~ ocm.»~ «mm.¢H mm.H~ emo.om mmm.o~ mo.m ~mm.¢c ¢o~.mm oo.m¢ mm.HH umo.e ~m0.m oH.m nwm.o emw.m m».o mm~.- mm¢.oH N».¢ Nom.m~ «mm.- um.ofi mo.m~- moe.o mo¢.o mo.-- omo.o mu».o mm.oH- moH.~ mm~.~ mm.m - omm.~ mam.~ som.H ~33 .3 3» m: 3» ~32 a 3» m: 3» ~3»< a 3» m: 3» ~33V a 3» m: 3» a»: co» x a> mom.o Nam.o m»».o Nmm.o x .moe.~ u 9» ucm ¢.o u : .~.m u c "ummpmx a o.“ 3o» mmam: Lam:m o3» m:u :mm3um: mocmgmwwwu m:u can .Ppm3 m:a a: mum: Loo:m 3o» 3m3oa ca umumswxogaam .ppm3 m:a an mum: 3mm:m »mpxp:m-_m:omgmz .mo mpao» 134 mm.¢m- mom.m¢ www.mofi me.fieu oNH.mN m»».mm~ N».om- www.mNH mmm.~mfi HN.0N- Hom.eHN mam.moN mm.mm mN.¢m- HNN.N¢ mem.mm No.H¢- mom.mo Nam.mfifi m».om- ome.mo~ mNN.me mH.oN- mm¢.omH oom.mmN oo.Nm mH.¢m- emv.mm omo.Hm mm.~¢- NcN.oo m¢o.mo~ mm.om- Nom.em Nmo.mm~ NH.¢N- mNN.omH mm».o0N oo.m¢ No.mm- mom.mN om.¢m m~.H¢- Ham.mm mm¢.»m m¢.om- swo.Hm Hmw.um mo.oN- Noe.~ofi mam.oNH mm.ma mm.~m- NeNnNH Nom.mN um.mm- mmo.»fl Hom.mN m».mN- NmHLmN oom.mm m».m~- mNo.mm mm~.m¢ New.” N3 _3 3 m: 3 N3 .3 3 m: 3 N3 pa 3 m: 3 N3 pa 3 m: 3 Am\sv »< P H »< p p »< a H 94 H H co” x Q> mom.o Nmo.o m»».o wa.o x mmwmmmgum gmm:m o3» m:u :mm3pm: mucmgm333v m:u vcm .—»m3 m:u an 3mm:m 3a» 3m3on op umumspxogaam .Fpm3 9:» an mmmgam 3mm:m »m_:»=m-—o:umgm= .m.m u o» new m.o u c .o.mN u : uuwmpmx a o.~ 30% mmwgum .oHo mpnm» 135 m».om- mmm.eN mmN.om mm.m¢u Ne».mm NNm.cm mm.om- ~H¢.mo woe.~m mm.0N- MNN.N~H dom.~¢fi mm.mm ma.mm- mom.HN omo.me om.o¢- o~¢.¢m mmN.mm mm.omn em».¢m mw~.mu mm.ON- www.mm mo.~NH oo.Nm em.Nm- ooo.mH Nme.oe mm.»m- NHN.Hm oom.om o¢.omu mom.m¢ mom.No n».oN- ¢H0.Hm mmN.NoH oo.m¢ mm.Hm- Ho».mH fimN.mN mm.mmu «mm.0N m¢w.mm mm.mN- mNN.om e»~.m¢ mm.cN- ~mm.¢¢ oo¢.Nm no.0» mm.»e- oo¢.m m¢H.o~ m~.Nm- moH.HH mNo.NH co.mN- mNm.¢H st.ON mm.m~u cfim.oN eoN.mN Noo.H N N N N 3»< Pa 3» m: 3» 3»< pa 3» m: 3» 3»< Pa 3» m: 3» 3»< _a 3» m: 3» Am\sv 303 x 3» mom.o Nmo.o m»~.o wa.o 3mm:m 3m» 303cm o» umpms_xogaam .__m3 m:» an mmmgum me:m »m—:F=mupo:omgmz .o¢~.m u p» can m.o u : .H.m u.» “pom—ox N m.H so» mmmmogum som:m 03» m:» :mm3um: mu:m3m»»»u m:» ucm .Ppm3 0:» »m mmmgpm .HHo mpnm» 136 mo.Nm- mNN.m mmN.mH mm.mm- mmH.mH mHm.HN mm.mN- NHN.mH mmo.»N em.m~- Hmm.mN com.om mm.mm mm.Hm- Nam.» c»o.o~ ¢¢.mm- Nmm.HH mm».m~ Nm.mN- cmm.N~ www.cN Nm.m~- omm.mN mmo.Nm oo.Nm m».~m- mmN.» fioo.m~ em.mm- mo».o~ me».»~_mm.mN- ”mm.mH omm.~N wN.mH- Nae.mN m»m.mN oo.m¢ HN.fim- mN¢.m ONH.~» cm.mmu mom.» mmm.NH Nm.mN- moN.HH emo.m~ ofl.mfiu mmm.o~ mum.0N no.m~ ¢¢.m¢- oofiqm omN.c NN.Nm- mem.¢ NNm.o mm.NN- me.m Huo.m om.mH- N¢H.m NNo.oH Nmm.H N3 —a 3 m: 3 N3 Pa 3 m: 3 N3 Pa 3 m: 3 N3 Pa 3 m: 3 Am\ev »< » » »< » » »< » » »< » » o» x q> mom.o Nmo.o m»».o Nmm.o .mo¢.N u 0» vac e.o u : .N.m u c u»mm~wx N o.» 3o» mmmmmgum gum:m o3» w:» :mm3um: mucmgm»»»u w:» ucw ,ppo3 m:» um mmmsum 3mm:m 3a» 3m3oa o» umums_xogaa~ .ppm3 m:» as mmmgam 3mm:m »w—:F:m-_o:umgm: .Nflo mFQm» 137 OHNNW .o3\- "m:o_»~:_:sou.:> 3303 m».m oo.¢m- mHm.¢ HmN.m Nom.m wmm.e mam.N NNN.H oo.m¢ m».m om.¢m- mam.» mom.o mam.»~ mom.m Nam.N mm».o mm.mm Nom.o m».o N¢.m- mam.H m»m.» mes.» Non.» Nmm.o mmm.o Nm.m~ m».o N¢.ma mem.m mmH.¢ om¢.o mNH.¢ mmm.o mmo.o oo.Nm mem.o HN.o Nm.N- mmm.N. mom.o w»~.m ¢mm.m ~mm.o mm».o mm.mm “N.o Nm.N- mam.» mum.» wno.» cum.» smm.o mmw.o No.9” mmm.o o.“ x N flum an Hum ma :m.mm m.ma mxs coax Eu H3 H3 3m 3 3w 3 a 3 a 3 a 3m an . a c » »< »<, z » z » p + F » p c z c . > » > .m .pmoo:»mz » o.H Lo» mgmumsmgma Pmpcms_gmaxm m:» we mmapo> .mHo mpam» 138 o~\~o .ofl\- .m¢\- "m:o_»~:»:sou 3» 3333 e».o mo.m mow.N oHo.N HN~.m Hoo.N mNN.o mo».o No.9» e».o mo.w o»m.o~ meo.o moc.H~ Hoo.m mNN.o ~mm.o oo.Nm omm.o Ne.o mo.¢ mom.N oHo.N moo.m moo.N ce».o mo».o Nm.o~ N¢.o mo.¢ “em.¢~ mam.m meo.m~ nem.m ¢¢».o mam.o mm.mm oom.o om.o N».m ecu.» Nmm.¢ mmm.w mNo.¢ m»w.o mom.o oo.me om.o NN.m o»o.o~ m¢o.m mNm.- Hmm.m mflm.o nom.o oo.No mow.o eo.o m¢.o ecu.» mmm.e Nm».» mmm.e «No.0 moo.o 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