‘1 -----—-_-_-_I THE FLOW OF NON-NEWTONEAN FLUIDS THROUGH POROUS MSW 2313333 far the Dagree of Ph; D. MICHIGAN S‘E‘ATE UNIVERSITY HEP. CHUNG PARK 1932 mumm1111111111111an 293 78755 LIBRARY Juar- 5 Michigan State University This is to certify that the thesis entitled THE FLOW 0F NON-NEWTONIAN FLUIds THROUGH POROUS MEDIA presented by HEE CHUNG PARK has been accepted towards fulfillment of the requirements for PH.D. . CHEMICAL ENGINEERING \degree 1n\____ 67, [4/2711-6; Major professor 7 DateZ7/62/‘7 /Z /7_7Z.— 0-7639 ...9- «35.1-— . m “an.“ itm '14". n' I SEP, 212004 b. .11, . I..lv,$."mpl’m wwfllfllJa'fli . t. . FF J-ILH l 17. is ‘11.!!! . ‘4 7‘14... ‘l 1 ABSTRACT THE FLOW OF NON-NEWTONIAN FLUIDS THROUGH POROUS MEDIA BY Hee Chung Park The Ergun equation is widely used to relate pres- sure drop to volumetric flow rate of Newtonian fluids in packed beds. In this study the Ergun equation was ex- tended to non-Newtonian fluids by using an effective viscosity in place of the Newtonian viscosity. The effective viscosity was calculated based on the result of a hydrodynamic analysis of the capillary model of the packed bed using the appropriate constitutive equation for each non-Newtonian fluid. Measurements of pressure drOps and corresponding flow rates were made for several concentrations of aqueous solutions of three polymers (polyacrylamide, polyvinyl- pyrrolidone, and polymethylcellulose) flowing through packed beds, and the rheological properties of these aqueous polymer solutions were determined with a Weissen- berg rheogoniometer. The Sprigg's four-parameter model was selected to characterize the rheological properties of aqueous Hee Chung Park solutions of polyacrylamide because these solutions ex- hibited viscoelastic behavior. The average percent devi- ation in the apparent viscosities between values predicted from the model and experiment for 103 data points was 3.4% for shear rates in the range 0.00675 to 851.0 sec-1 for concentrations of 0.50, 0.25, 0.10, and 0.05 weight percent. The pressure drOp-flow rate data for aqueous solutions of polyacrylamide were correlated very well by the Ergun equation whose effective viscosity was calcu- lated based on the result of a hydrodynamic analysis of the capillary model of the packed Inna using the Sprigg's model. The average percent deviation between the experi- mental values of friction factor and the corresponding values from the Ergun equation was 5.7% for 117 experi- mental points with the Sprigg's model. The corresponding deviations for the power-law model and the Ellis model were 19.1% and 9.3%, respectively. Meter's four-parameter model was selected to characterize the rheological properties of aqueous solu- tions of polyvinylpyrrolidone because these solutions were purely viscous having both upper and lower limiting vis- cosities. The average percent deviation in the apparent viscosities between values predicted from the model and experiment for 98 data points was 2.1% for shear rates in the range 0.02689 to 1076.0 sec.1 for concentrations of 4.0, 3.0, 1.0, and 0.5 weight percent. Large Hee Chung Park deviations between experimental values of friction factor and those from the Ergun equation occurred for effective Reynolds numbers greater than one. A new correction parameter (DpGo/M(1 - e)nJ«301/pr) was used to account for the deviation. Agreement was excellent after cor- rection. The average percent deviation between the experimental values of friction factor and the corres- ponding values from the Ergun equation was 10.8% for 87 experimental points; before correction the deviation was greater than 60%. The Herschel-Bulkley three-parameter model was used to characterize the rheological prOperties of aqueous solutions of polymethylcellulose because these solutions exhibited yield stresses with non-linear flow curves. The average percent deviation in the apparent viscosities between values predicted from the model and experiment for 72 data points was 1.7% for shear rates in the range 1.076 to 851.0 sec-l, for concentrations of 0.3 and 0.5 weight percent of each of two different molecular weights. The pressure drop-flow rate data for aqueous solutions of polymethylcellulose were correlated very well by the Ergun equation using the Herschel-Bulkley model for effective viscosity calculation. The average percent deviation between the experimental values of friction factor and the corresponding values from the Ergun equation was 8.2% for 139 experimental points. Hee Chung Park Polymer solutions are adequately represented by the capillary model of the packed bed for effective Reynolds numbers less than one. The pressure drop-flow rate data for non-Newtonian fluids is not correlated very well by the Ergun equation for effective Reynolds numbers greater than one. It appears as though an improve- ment in the model for packed beds is required in order to account for inertial effects and surface effects of polymer solutions. Past investigators have used the power-law and the Ellis model to calculate the effective viscosity for packed beds. However, this work has shown that considerable improvement in accuracy of the friction factor predictions can be accomplished by using the rheological model which characterizes the shear stress- shear rate relationship with minimum error for calculating the effective viscosity for packed beds. THE FLOW OF NON-NEWTONIAN FLUIDS THROUGH POROUS MEDIA BY Hee Chung Park A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1972 To my brother Hee Bock ii ACKNOWLEDGMENTS No man of science is likely to achieve anything great, unless he brings to his work a zeal comparable with that of religion, and unless he is prepared to follow truth wherever it leads him. But zeal without strict discipline of the intellect will get him nowhere. The author wishes to express his sincere appreci- ation to his major professor, Dr. M. C. Hawley, for pro- viding his thoughtful guidance, genuine wisdom, and en- couragement throughout the course of this investigation, and also for his painstaking review of this manuscript. He also wishes to extend his thanks to Dr. R. F. Blanks whose discussions and suggestions concerning the molecular aspects of polymer solution behavior have furthered the author's understanding of the physical phenomena observed in this investigation. Sincere appreciation is extended, also, to Dr. B. J. Meister of Dow and the Dow Chemical Company for the technical assistance and for furnishing the polymers and the Weissenberg Rheogoniometer used in this investigation. Other guidance committee members, Drs. M. H. Chetrick, C. M. Cooper, D. A. Anderson, and iii R. H. Wasserman, are also deeply appreciated for their concern and interest in this investigation. Finally, the author wishes to thank his wife, Inoak, for her patience, understanding, and encouragement in the entire period of his study. iv Chapter 1. 2. TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . RHEOLOGY O O O O O O O I O O O O Time-independent Non-Newtonian Fluids . Time-dependent Non-Newtonian Fluids. . Viscoelastic Fluids . . . . . . . Definitions of Material Functions . . Generalized Newtonian Fluids . . . Specific Rheological Equation of State. Linear Response of Viscoelastic Fluids. Non-linear Response of Viscoelastic Fluids . . . . . . . . . . . NNNNNNNN o o o o o o o o QOUIfiUNH POROUS MEDIA PARAMETERS FOR VARIOUS CONSTITUTIVE EQUATIONS. . . . . . . POLYACRYLAMIDE SYSTEM . . . . . . . . 4.1 Chemistry . . . . . . . . . . 4.2 Physical Properties . . . . . . . 4.3 Solution Preparation. . . . . . 4.4 Rheology. . . 4.5 The Flow of Non- -Newtonian Fluids Through Porous Media. . . . . . . 4.6 Conclusions. . . . . . . . . POLYVINYLPYRROLIDONE SYSTEM . . . . . . 5.1 Chemistry . . . . . . . . . . 5.2 Physical Properties . . . . . . . 5.3 Rheology. . . . . . . . . . 5.4 The Flow of Non-Newtonian Fluids Through Porous Media. . . . . . . 5.5 Conclusions. . . . . . . . . . Page 10 12 14 14 18 22 25 51 56 56 58 S9 61 100 118 120 120 122 125 133 142 Chapter 6. POLYMETHYLCELLULOSE SYSTEM . . . . . 6 O 1 Chemistry 0 O O I O I O O 6.2 Physical Properties. . . . . . 6.3 Solution Preparation . . . . . 6.4 Rheology . . . . . 6.5 The Flow of Non-Newtonian Fluids Through Porous Media . . . . . 6.6 Conclusions . . . . . . . . 7. CONCLUSIONS AND RECOMMENDATIONS . . . 7.1 Conclusions . . . . . . . . 7.2 Recommendations . . . . . . . NOMENCIATURE O O O O O O O O O O O O BIBLIOGRAPHY O O O O O O O O O O O O APPENDICES Appendix A. Development of Packed Bed Equations . . B. Viscometric Experiment: Detailed Description, Weissenberg Rheogoniometer. C. Flow Experiment: Detailed Description . D. Summary of Viscometric Data Steady State Shear Stress Measurement. . . . . . E. Summary of Viscometric Data Steady State Normal Stress Measurement . . . . . F. Summary of Viscometric Data H(T), Relax- ation Time Spectrum vs. t, Time . . . G. Model Independent Parameters . . . . H. Model Independent Stress Parameters . . I. Summary of Viscometric Data and Ellis Model Calculation . . . . . . . . vi Page 146 146 148 152 153 161 169 172 172 177 179 183 190 234 252 260 264 266 270 272 275 Appendix J. K. Summary of Viscometric Data and Power-Law Model Calculation . . . . . . . . Summary of Viscometric Data and Spriggs Model Calculation . . . . . . . . Summary of the Viscometric Data and Bird- Carreau Model Calculation . . . . . Summary of the Constant Flow Rate Experi- ment Data and Newtonian Fluid Calculation Summary of the Constant Flow Rate Experi- ment Data and Power-Law Model Calculation Summary of the Constant Flow Rate Experi- ment Data and Ellis Model Calculation . Summary of the Constant Flow Rate Experi- ment Data and Sprigg's Model Calculation Summary of Viscometric Data and Meter Model Calculation . . . . . . . . . . Summary of the Constant Flow Rate Experi- ment Data and Meter's Model Calculation. Summary of Viscometric Data and Herschel- Bulkley Model Calculation . . . . . Summary of the Constant Flow Rate Experi- ment Data and Herschel-Bulkley Model Calculation . . . . . . . . . . vii Page 279 283 291 295 296 300 304 308 312 316 320 LIST OF TABLES Summary of Definitions of Material Functions . . . . . . . . . . . Components of AV . . . . . . . . . Components of V'VT. . . . . . . . . Simple Shear Flow Notation . . . . . . Porous Media Parameters . . . . . . . Power-law Parameters for Polyacrylamide Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C. . . . . . . Ellis Parameters for Polyacrylamide Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C. . . . . . . Spriggs 4-Constant Model Parameters for Polyacrylamide Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C . . Bird-Carreau Model Parameters for Poly- acrylamide Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C. . . Bird-Carreau Model Parameters for Poly- acrylamide Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C. . . Summary of Experimental Data for Power-law Modification . . . . . . . . . . Summary of Experimental Data for Ellis Model Modification . . . . . . . Summary of Experimental Data for Spriggs Model Modification . . . . . . . viii Page 15 28 29 30 54 86 88 89 99 101 112 114 115 Table 405-4 5.3-1 6.2-1 6.4-1 6.5-1 Page Summary of Error Analysis . . . . . . . 117 Meter Model Parameters for PVP Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C . . . . . . . . . . . . 132 Summary of Experimental Data for PVP Solutions. . . . . . . . . . . . 144 Viscosities of Methylcellulose of Various Molecular Weights . . . . . . . . . 150 Herschel-Bulkley Model Parameters for PMC Solutions as Obtained from a Weissenberg Rheogoniometer at 21°C . . . . . . . 159 Summary of Experimental Data for Herschel- Bulkley Modification . . . . . . . . 170 Torsion Bar Calibration. . . . . . . . 242 Normal Force Spring Calibration . . . . . 246 Gap Setting. . . . . . . . . . . . 247 Rotation Shear Rates for Various Platens . . 250 Results for the Weissenberg Rheogoniometer for 0.50% Polyacrylamide Solution at 21°C . 260 Results for the Weissenberg Rheogoniometer for 0.25% Polyacrylamide Solution at 21°C . 261 Results for the Weissenberg Rheogoniometer for 0.10% Polyacrylamide Solution at 21°C . 262 Results for the Weissenberg Rheogoniometer for 0.05% Polyacrylamide Solution at 21°C . 263 Results for the Weissenberg Rheogoniometer for 0.50% Polyacrylamide Solution at 21°C . 264 Results for the Weissenberg Theogoniometer . 265 Results of the Viscometric Experiments for 0.5% Separan Solution at 21°C . . . . . 266 Results of the Viscometric Experiments for 0.25% Separan Solution at 21°C. . . . . 267 ix Table Page F-3 Results of the Viscometric Experiments for 0.10% Separan Solution at 21°C . . . . 268 F-4 Results of the Viscometric Experiments for 0.05% Separan Solution at 21°C . . . . 269 G-l Model Independent Parameters for 0.50% Separan Solution at 21°C . . . . . . 270 G-2 Model Independent Parameters for 0.25% Separan Solution at 21°C . . . . . . 270 G-3 Model Independent Parameters for 0.10% Separan Solution at 21°C . . . . . . 271 G-4 Model Independent Parameters for 0.05% Separan Solution at 21°C . . . . . . 271 H-l Model Independent Stress Parameters for 0.5% Polyacrylamide Solution at 21°C . . 272 H-2 Model Independent Stress Parameters for 0.25% Polyacrylamide Solution at 21°C. . 273 H-3 Model Independent Stress Parameters for 0.1% Polyacrylamide Solution at 21°C . . 274 H-4 Model Independent Stress Parameters for 0.05% Polyacrylamide Solution at 21°C. . 274 I-l Results for the Weissenberg Rheogoniometer for 0.50% Polyacrylamide Solution at 21°C. . . . . . . . . . . . . 275 I-2 Results for the Weissenberg Rheogoniometer for 0.25% Polyacrylamide Solution at 21°C. . . . . . . . . . . . . 276 I-3 Results for the Weissenberg Rheogoniometer for 0.10% Polyacrylamide Solution at 21°C. 0 O I O O O O O O O O O 277 I-4 Results for the Weissenberg Rheogoniometer for 0.05% Polyacrylamide Solution at 21°C. 0 O O O O O O O I O O O 278 J-l Results for the Weissenberg Rheogoniometer for the 0.5% Polyacrylamide Solution at 21°C. . . . . . . . . . . . 279 Table K-2 Results for for the 0. at 21°C . Results for for 0.10% at 21°C . Results for for 0.05% at 21°C . Results for for 0.50% 21°C . . Results for for 0.25% 21°C . . Results for for 0.10% 21°C 0 0 Results for for 0.05% 21°C . . Results for for 0.50% Results for for 0.25% Results for for 0.10% Results for for 0.05% Results for for 0.50% Results for for 0.25% Results for for 0.10% the Weissenberg Rheogoniometer 25% Polyacrylamide Solution the Weissenberg Rheogoniometer Polyacrylamide Solution the Weissenberg Rheogoniometer Polyacrylamide Solution the Weissenberg Rheogoniometer Polyacrylamide Solution at the Weissenberg Rheogoniometer Polyacrylamide Solution at the Weissenberg Rheogoniometer Polyacrylamide Solution at the Weissenberg Rheogoniometer Polyacrylamide Solution at the Weissenberg Rheogoniometer Polyacrylamide Solution at 21°C. the Weissenberg Rheogoniometer Polyacrylamide Solution at 21°C. the Weissenberg Rheogoniometer Polyacrylamide Solution at 21°C. the Weissenberg Rheogoniometer Polyacrylamide Solution at 21°C. the Weissenberg Rheogoniometer Polyacrylamide Solution at 21°C. the Weissenberg Rheogoniometer Polyacrylamide Solution at 21°C. the Weissenberg Rheogoniometer Polyacrylamide Solution at 21°C. xi Page 280 281 282 283 284 285 286 287 288 289 290 291 292 293 Table L-4 Results for the Weissenberg Rheogoniometer for 0.05% Polyacrylamide Solution at 21°C. Results for the Constant Flow Rate Experi- ments for Newtonian (Distilled Water) Fluid at 21°C . . . . . . . . . . Results for ments for 21°C 0 0 Results for ments for 21°C . . Results for ments for 21°C . . Results for ments for 21°C 0 0 Results for ments for 21°C . . Results for ments for 21°C . . Results for ments for 21°C 0 0 Results for ments for 21°C . . Results for ments for 21°C 0 0 Results for ments for 21°C . . the Constant Flow Rate Experi- 0.50% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.25% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.10% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.05% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.5% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.25% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.10% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.05% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.50% Polyacrylamide Solution at the Constant Flow Rate Experi- 0.25% Polyacrylamide Solution at xii Page 294 295 296 297 298 299 300 301 302 303 304 305 Table Results for ments for at 21°C . Results for ments for at 21°C . Results for the Constant Flow Rate Experi- 0.10% Polyacrylamide Solution the Constant Flow Rate Experi— 0.05% Polyacrylamide Solution the Weissenberg Rheogoniometer for 4.0% PVP Solution at 21°C. . . . Results for the Weissenberg Rheogoniometer for 3.0% PVP Solution at 21°C. . . . Results for the Weissenberg Rheogoniometer for 1.0% PVP Solution at 21°C. . . . Results for the Weissenberg Rheogoniometer for 0.5% PVP Solution at 21°C. . . . Results for ments for Results for ments for Results for ments for Results for ments for Results for the Constant Flow 4.0% PVP Solution the Constant Flow 3.0% PVP Solution the Constant Flow 1.0% PVP Solution the Constant Flow Rate Experi— at 21°C. . Rate Experi- at 21°C. . Rate Experi- at 21°C. . Rate Experi- 0.50% PVP Solution at 21°C . the Weissenberg Rheogoniometer for 0.05% PMC 400 Solution at 21°C . . Results for the Weissenberg Rheogoniometer for 0.3% PMC 400 Solution at 21°C . . Results for the Weissenberg Rheogoniometer for 0.5% PMC 25 Solution at 21°C. . . Results for the Weissenberg Rheogoniometer for 0.3% PMC 25 Solution at 21°C. . . Results for the Constant Flow Rate Experi- ments for 0.50% PMC 400 Solution at 21°C Results for the Constant Flow Rate Experi4 ments for 0.30% PMC 400 Solution at 21°C xiii Page 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 Table Page T-3 Results for the Constant Flow Rate Experi- ments for 0.50% PMC 25 Solution at 21°C . 322 T-4 Results for the Constant Flow Rate Experi- ments for 0.30% PMC 25 Solution at 21°C . 323 xiv Figure 2.1-1 4.4-1 4.4-2 4.4-3 4.4-9 4.4-10 LIST OF FIGURES Flow Curves for Various Types of Time Independent Fluid . . . . . . . Flow Curves for Thixotropic and TheOpectic Fluids in Single Continuous Experiments . . . . . . . . . Schematic Diagram of the Stock Solution Preparation . . . . . . . . . Reservoir Platen Arrangement . . . . Start-Up Experiment . . . . . . . Shear Stress-Shear Rate Data for Aqueous Separan Ap 30 Solutions . . . . . Non-Newtonian Viscosity of Separan Ap 30 Solutions. . . . . . . . . . Primary Normal Stress Difference Coefficient of Separan Ap 30 Solutions Relaxation Spectrum of Separan Ap 30 Solutions. . . . . . . . . . n , Zero-Shear Viscosity, vs. C, Concentration . . . . . . . . Sw, Recoverable Shear Strain, vs. 1, Shear Rate . . . . . . . . . G Shear Modulus, vs. §, Shear Rate . w! Model Independent Parameter e/n vs. i . XV Page 11 13 60 64 65 68 69 70 71 77 81 83 85 Figure 4.4-11 504-2 (r12(t)/112(o)), Unsteady State Shear Stress, vs. t, Real Time for Polyacrylamide Solutions . . . . . . Schematic Diagram of the Equipment . . . f*, Friction Factor vs. NRev Reynolds Number for Newtonian Fluid . . . . Pressure Drop-Flow Rate Correlation for Flow of 0.50% Aqueous Solutions of Separan Ap 30 Through Packed Beds . . Pressure DrOp-Flow Rate Correlation for Flow of 0.25% Aqueous Solutions of Separan Ap 30 Through Packed Beds . . . Pressure Drop-Flow Rate Correlation for Flow of 0.10% Aqueous Solutions of Separan Ap 30 Through Packed Beds . . Pressure Drop-Flow Rate Correlation for Flow of 0.05% Aqueous Solutions of Separan Ap 30 Through Packed Beds . . . The Effects of the Wall Correction Factor in Friction Factor—Reynolds Number Correlation for .50% Separan Ap 30 Solution . . . . . . . . . Shear Stress-Shear Rate Behavior for Four Aqueous Solutions of PVP . . . Non-Newtonian Viscosity for Four Aqueous Solutions of PVP . . . . . . Comparison of Experimental and Calculated Values of the Apparent Viscosity for Four Aqueous Solutions of PVP . . . Pressure Drop-Flow Rate Correlation for Flow of 4.0% PVP Solutions Through Packed Beds . . . . . . . . . . Pressure DrOp-Flow Rate Correlation for Flow of 3.0% PVP Solutions Through Packed Beds . . . . . . . . . . xvi Page 91 103 106 107 108 109 110 111 128 129 131 136 137 Figure 5.4-3 5.4-4 Pressure Drop-Flow Rate Correlation for Flow of 1.0% PVP Solutions Through Packed Beds . . . . . . . . . Pressure Drop-Flow Rate Correlation for Flow of 0.5% PVP Solutions Through Packed Beds . . . . . . . . . . Pressure Drop—Flow Rate Correlation for Flow of PVP Solutions Through Packed Beds. . . . . . . . . . . . Viscosity of Polymethylcellulose of Various Molecular Weights. . . . . . Typical Moisture Absorption of Methocel Products at Relative Humidity Shown at 20°C. 0 O O O O I O O O O O Shear Stress-Shear Rate Behavior for Four Aqueous Solutions of Polymethylcellulose. Non-Newtonian Viscosity for Aqueous Solutions of Polymethylcellulose . . . Comparison of Experimental and Calculated Values of the Apparent Viscosity for Aqueous Solutions of Polymethylcellulose. Pressure DrOp-Flow Rate Correlation for Flow of 0.5% PMC 400 Solutions Through Packed Beds . . . . . . . . . . Pressure Drop-Flow Rate Correlation for Flow of .3% PMC 400 Solutions Through Packed Beds . . . . . . . . . . Pressure Drop-Flow Rate Correlation for Flow of 0.5% PMC 25 Solutions Through Packed Beds . . . . . . . . . Pressure Drop-Flow Rate Correlation for Flow of 0.3% PMC 25 Solutions Through Packed Beds . . . . . . . . . Pressure DrOp-Flow Rate Correlation for Flow of Aqueous Solutions of Poly- methylcellulose Through Packed Beds . . xvii Page 138 139 143 149 151 157 158 160 164 165 166 167 168 Figure Page A.3-1 Cylindrical Shell of Fluid Over Which .Momentum Balance is Made to Get the Velocity Profile . . . . . . . . 199 A.3-2 Momentum Flux and Velocity Distributions in Flow in Cylindrical Tubes . . . . 202 A.4-l Stress Acting on a Cylindrical Element of Fluid of Radius R in Steady Flow. . 212 A.5-l Packing of Uniform Spheres . . . . . 233 A.5-1 Pore Space in Packing of Uniform Spheres . . . . . . . . . . . 233 A.5-2 Pendular Ring Between Two Spheres . . . 233 3.1 Typical Arrangement of the Weissenberg Rheogoniometer. . . . . . . . . 235 3.2 Arrangement of Normal Force Measurement . 243 3.3 Schematic Diagram of Normal Force Measurement System . . . . . . . 244 8.4 Correction Terms for Normal Stress Difference . . . . . . . . . . 251 C.1 Brooks-Mite Rotometer . . . . . . . 255 xviii CHAPTER 1 INTRODUCTION The aims of this work were to measure and correlate friction factor data for the flow of non-Newtonian fluids through porous media and to characterize non-Newtonian fluids by a rheological equation of state more generally applicable than the commonly used power-law model. Rheo- logical measurements were made on a number of rather different types of non—Newtonian fluids and several rheo- logical models were compared and criticized in view of experimental data. Non-Newtonian fluids were categorized as: (a) purely viscous, (b) with yield stresses and non- linear flow curves, and (c) viscoelastic. Many studies of the flow of fluids through porous media have been concerned with Newtonian fluids; i.e., fluids for which the relation between shear stress and shear rate is a simple proportionality. This includes gases and most homogeneous, non-polymeric liquids. In many branches of engineering one is faced with design problems for non—Newtonian fluids; i.e., fluids for which the relation between shear stress and shear rate is not a simple proportionality. Suspensions and solutions of polymers are examples of this latter class of fluids. The vast majority of these "non-linear" fluids show a decrease of apparent viscosity with increasing shear rates. The open literature was essentially devoid of any studies on the flow of non—Newtonian fluids through porous media until recently. During the past few years a number of papers in this area have appeared. Several different approaches were taken to treat the experimental flow data. Bird and Sadowski [l] and Sadowski [2, 3] correlated a friction factor with a modified Reynolds number. The correlation was based on Darcy's law and the Ellis equation for non-Newtonian fluids. Christopher and Middleman [4] and Marshall and Metzner [5] used a similar approach ex- cept that they used the power-law model for non-Newtonian fluids. Gregory and Griskey [6] developed a friction factor versus Reynolds number correlation based on the Mooney-Rabinowitch equation. Chemical engineers have traditionally been employed in industries involving the manufacture of products from raw materials by chemical reactions and physical changes. The activities of many chemical engineers in industry involve the development of processes and the design and Operation of production facilities. A major portion of the skills required by the engineer engaged in these activities is obtained through study of the transport processes of mass, heat, and momentum. In particular, an understanding of fluid flow and the behavior of fluids in process equipment is a basic element of modern engineering training. The flow of polymeric fluids is an important aspect of engineering science in the polymer industry. The conversion of high polymers to useful products by operations such as extrusion, molding, mixing, and calendering constitutes a new and challenging branch of technology of polymeric fluids called polymer processing. This development has led to a demand for scientific engineering research into the area of polymeric fluid behavior, including synthetic polymers as well as natural polymers such as proteins, cellulose, and natural rubber. Knowledge of flow of fluids through porous media is basic for many scientific and engineering applications. It is essential to the individual problems of such diversified fields as soil mechanics, ground water hydrology, industrial filtration of polymer solutions and slurries, ceramic engineering, and the movement of aqueous polymer solutions through sand in secondary oil operations. To the chemical engineer, such knowledge forms the basic background for the design of packed towers and reactors containing granular catalysts. The impetus for this investigation is both of a practical and of a fundamental nature. There is interest in the secondary recovery of oil from underground reser- voirs by displacement of the oil with non-Newtonian fluids. Non-Newtonian fluid flow in liquid—solid chromatographic separations :us common in the pharmaceutical industry. Very small concentrations of high molecular weight polymers will increase the solution viscosities to values greater 1 Models which have than those for the reservoir oils. been used to describe non-Newtonian fluid behavior have been tested in relatively simple flow geometries. A test of these models for a very complex flow geometry, such as that which exists in a packed bed, may lead to more confi- dence in the application of the models to any arbitrary flow geometry. These problems will be discussed in Chapter 2. In order to construct a tractable mathematical model of the complicated flow system involved, it is neces- sary to resort to a number of simplifications. If the viscosity of the fluid is properly characterized, tra- ditional treatments for the study of the fluid flow in porous media may be adequate. The following assumptions were made for the present investigation: a. The fluid was treated as an incompressible and a continuous medium in which each point had a flow- path. l . . . . . . Typ1ca1 reservo1r o11 Viscos1t1es are generally less than 100 c.p. (Muskat, 1949, p. 96). b. The porous medium was isotrOpic, homogeneous, and of regular geometry. c. The flow of the fluids was isothermal and single-phase. d. The external forces on the fluids were homogeneous and time independent. e. For simplicity the gravity term was neglected. f. The inertia terms are omitted from the equations of fluid motion. 9. Different parts of one sample were macrosc0pically identical. This means that a fluid particle pro- ceeding through the porous medium found the same total probability for displacement along all points of its path. The analysis for this investigation included the characterization of the non-Newtonian fluids with an apprOpriate rheological equation of state and the develop- ment of the Ergun equation, derived from the equations of motion and the rheological equation of state for an appropriate model of the packed bed. CHAPTER 2 RHEOLOGY The subject of non-Newtonian flow is a subdivision of rheology, "the science of flow and deformation of matter." In this chapter the equations of change and several rheological equations of state will be presented. Emphasis will be placed on the several "generalized Newtonian models" as well as the empirical viscoelastic rheological equations of state of Spriggs and Bird- Carreau. The word fluid does not have a precise meaning. It may be said that the essential property of fluids is the absence of preferred configurations. Therefore, a fluid may be defined as that substance whose configur- ation can not be distorted. Some dictionaries define fluid as a substance capable of flowing. Non-Newtonian fluids are defined as materials which do not conform to a direct proportionality between shear stress and shear rate. Because of negative defi- nition of non-Newtonian behavior, an infinite number of possible rheological relationships exist for this class of fluids and, as yet, no single equation has been proven which can describe exactly the shear rate-shear stress relationships of all such materials over all ranges of shear rates. As a result non-Newtonian fluids are classi- fied in the following manner [120]: l. Time-independent fluids . . . those for which the rate of shear at a given point is solely dependent upon the instantaneous shear stress at that point. 2. Time-dependent fluids . . . those for which the shear rate is a function of both magnitude and the time lapse between consecutive applications of shear stress. 3. Viscoelastic fluids . . . those that show partial elastic recovery upon removal of a deforming shear stress. Such materials possess prOperties of both fluid and elastic solids. §2.l Time-independent Non—Newtonian Fluids These purely viscous fluids are usually classified into two groups, fluids with yield stresses and fluids without yield stresses. §2.1.1 Fluids with Yield Stresses The physical behavior of fluids with yield stresses is usually explained in terms of an internal structure in three dimensions which is capable of pre— venting movement for values of shear stress less than the yield value, Ty. For Tyx > Ty, the internal structure collapses completely, allowing shearing movement to occur. When Tyx < Ty, the internal structure is considered to be reformed virtually instantaneously. Examples of fluids with yield stresses may be found in the following materials: certain plastic melts, oil drilling mud, ores, sand in water, coal, cement, rock and chalk slurries, grain water suspension, chocolate mixtures, tooth paste, peat slurries, margarine and shortenings, greases, aqueous thorium oxide slurries, soap and detergent slurries, and paper pulp. §2.1.2 Fluids Without Yield Stresses A. PseudOplastic Fluids The majority of non-Newtonian materials are found in this category. A logarithmic plot of I vs. dvx/dy yx for these materials is often found to be linear over a wide range of shear rates. Fluid dispersions of asym- metric molecules or particles are probably characterized by extensive entanglement of the particles when the fluid is at rest. Progressive disentanglement should occur under the influence of shearing forces, the particles tending to orient themselves in the direction of shear. This orienting influence is proportional to shear rate and is Opposed by the randomly disorienting effects of Brownian movement [7], the extent of which is determined only by the temperature of a given fluid. PseudOplastic behavior would also be consistent with the existence of highly solvated molecules or parti- cles in the dispersion. Progressive shearing away of solvated layers with increasing shear rate would result in decreasing interaction between the particles (because of their smaller effective size) and consequent reduction in apparent viscosity. Examples of pseudOplastic fluids may be found in the following materials: rubber solutions, adhesives, certain polymer solutions or melts, greases, starch sus- pensions, cellulose acetate solutions used in rayon manu- facturing, mayonnaise, soap, detergent slurries, paper pulp, napalm, paints and dispersion media in certain pharmaceutical fluids. B. Dilatant Fluids Two phenomena have been observed with dilatant materials [8, 9]. Volumetric dilatancy denotes an increase in total volume under shear, whereas rheological dilatancy refers to an increase in apparent viscosity with increas- ing shear rate. It is this latter prOperty which is usually associated with dilatant fluids, although these materials are far less common than pseudoplastic fluids. I..,I|'| I; ‘1 10 Examples of materials which have been found to exhibit both volumetric and rheological dilatancy include the following materials: aqueous suspensions of titanium dioxide, some gum arabic and borax solutions, some corn flour and sugar solutions, starch, potassium silicate, gum arabic in water, quicksand, wet beach sand, many defloculated pigment dispersions containing high suspension concentrations, solids such as mica, powdered quartz, and iron powder in low viscosity liquids. Flow curves for various types of time-independent fluids and models relating Tyx to dvx/dy are shown in Figure 2.1—1. 52.2 Time-dependent Non-Newtonian Fluids These materials are usually classified into two groups, thixotrOpic fluids and rheopectic fluids, depending upon whether the shear stress decreases with time at a given shear rate and constant temperature. §2.2.1 ThixotrOpic Fluids These substances exhibit a reversible decrease in shear stress with time at a certain rate of shear and fixed temperature. Flow curves for thixotrOpic fluids in continuous experiments in which the shear rate is steadily increased from zero to a maximum value and then immediately 11 Fluid with a yield stress and a non- linear flow curve Shear Stress Tyx Shear Rate -—£ dy Figure 2.1-1 Flow Curves for Various Types of Time- independent Fluid 12 decreased steadily towards zero form a hysteresis loop as shown in Figure 2.2-1. Examples of thixotropic properties have been found in the following materials: some solutions or melts of high polymers, oil well drilling muds, greases, margarine and shortening, printing inks, and many food materials. §2.2.2 Rheopectic Fluids These materials, occasionally referred to as antithixotrOpic fluids,are relatively rare in occurrence. They exhibit a reversible increase in shear under iso- thermal conditions. Rheopectic behavior is often explained in terms analogous to those used to account for dilatancy but in this case with more prolonged time periods for the struc- tural changes involved. Although for some of these examples rheOpexy is confined to moderate rates of shear, rheOpectic characteristics have been observed in the following materials: bentonite clay suspensions, vanadium pentoxide suspensions, gypsum suspensions, certain soils, and dilute suspensions of ammonium oleate. 52.3 Viscoelastic Fluids These materials exhibit both viscous and elastic properties. Oldroyd [10, 11] has shown that dispersions of one Newtonian fluid in another may lead to emulsions 13 Thixotropic Shear Stress T XY Rheopectic de Shear Rate '3'- Y Figure 2.2-1 Flow Curves for ThixotrOpic and Rheopectic Fluids in Single Continuous Experiments 14 possessing both viscous and elastic characteristics. More discussions about viscoelasticity will be given in sections 6, 7, and 8. §2.4 Definitions of Material Functions Some investigators have defined various measurable material functions based on their experiments [14, 15, l6, 17]. The tube-flow experiment suggests the definition of a nOn-Newtonian viscosity n(?). The study of Weissenberg effects has led to the definition of a primary normal stress coefficient 9(4) and a secondary normal stress coefficient 8(9). The oscillatory flow experiments have suggested the definition of the complex viscosity n*(w) with its real and imaginary contributions n'(w) and n"(w). It is convenient to summarize the definition of these material functions at this point in terms of flow between two infinite flat plates. Such a summary is given in Table 2.4-1. §2.5 Generalized Newtonian Fluids In order to characterize the nature of different fluids, one introduces a rheological equation of state, or constitutive equation, which identifies the basic properties of the fluids. In particular, one must specify a relation between the deviatoric part of the shear stress tensor, Tij' and the shear rate tensor eij' 15 TABLE 2.4-1 Summary of Definitions of Material Functions l. Steady Shear Flow- . dvl v1 = V1(X2) Y : dx2 x2 . ——*V ,/1' X1 X3 a. Viscosity: 0(1) 3 le/(-1) b. Normal Stress Functions: e(§) (T11 — T22)/(‘YZ) O _ .2 B(Y) - (T22 — T33l/(-Y ) 2. Small Amplitude Sinusoidal Oscillations ++ (\Vf\’”d’ EB‘V = V (X ,t) 1 E R {Voelwt} 1 1 2 e X2 0 iwt }—> Xl_a T12 Z Re{T12e } o Ziwt x3 Tii Re{di+Tiie } a. Complex Viscosity ”*(w) E ”(w)-in1w) E 132/(’10) b. Complex Normal Stress Coefficients: I U _ ' I! : O _ o - .0 2 6*(w) — 6 (w) 16 (w) - (T11 122)/ (Y ) : ' ' I! Z O - O - .O 2 8*(w) — 8 (w) - 18 (w) - (122 T33)/ (Y ) c. Normal Stress Displacement Functions: ed '0 2 Re{dl - d2}/-IY I d I2 '0 B Re{d2 - d3}/-Iy 16 Consider a fluid placed in the region between two parallel plates which extend infinitely. The upper plate is held fixed, whereas the lower one is made to move with a constant velocity (see Table 2.4-1). The force per unit area in the x-direction required to maintain this motion is proportional to the velocity gradient: TYX = -udvx/dy (2.5-1) and the proportionality constant u is called the viscosity. This relation is called "Newton's law of viscosity." All the gases and simple liquids whose behaviors follow this relationship are called Newtonian fluids. It is also known that, if one can assume that the fluid is incom- pressible, then for motions more complicated than that above, Eq. (2.5-1) can be generalized to give [12]: av. avi T1:] = -u 5xi + ij -2ueij (2.5—2) which serves to define the quantities ei in Eq. (2.5—2) j: it is understood that i and j can be x, y, or z. Reiner [18] suggested a most useful and practical constitutive relation which is so-called "generalized Newtonian fluid," Tij = -2neij (2.5-3) 17 This relation is a simplification of the Reiner-Rivlin- Prager relation [19, 20] = -n e.. - n X Tij 13 c k eik ekj (2'5'4) where n and the "cross viscosity" nc are, in general, functions of the scalar invariants of the e tensor: eI = (ezd) = Z'eii (2.5-5) eII = (eze) = Zizjeijeji (2.5-6) eIII = det e = ZizjzkeijkeliereBk (2.5-7) Eq. (2.5-4) is the most general relation between Tij and eij which does not involve space time derivative of either Tij or eij' The exclusion of time effects restricts the relation to inelastic fluids. The reduction of Eq. (2.5-4) to Eq. (2.5-3) depends upon several assumptions. First, in all cases, the fluid must be considered to be incompressible; there- fore the first invariant eI is identically equal to zero. Second, cross viscosity effects must be neglected, i.e., nC = 0. There is little known about cross viscosity effects. Some investigators [21, 22] have assumed NC to be constant. Leigh [23], however, has shown that thermodynamic principles require that nc not be a con- stant. Most investigators are content to ignore this 18 function. Third, it must be assumed that there exists no functional dependence of the viscosity n on the third in- variant e There is little known about the effect of III' eIII on fluid flow. In simple flow geometries, e.g., flow through a tube or a slit, the third invariant is identically zero. §2.6 Specific Rheological Equation of State Numerous empirical functions of the form of Eq. (2.5-3) have been presented in the literature. Any pro- posed rheological model should represent the actual be- havior of a fluid with accuracy, convenience, and simplicity. No known model describes the behavior of all non-Newtonian fluids with a reasonable number of constants. Different models may be necessary to describe different fluids or even the same fluid under different conditions. The best relationship for a given fluid is not necessarily known until an experiment is made on the fluid to relate Tij and eij' The power-law model is a rheological equation of state which is used widely [12], n-l _ _ e:e 2 _ where K and n are two positive fluid parameters determi- nable from Viscometric experiments. When n = l and K = u, this model becomes Newton's law of viscosity. This model 19 does not describe the limiting viscosity at zero shear rate no, the limiting viscosity at infinite shear rate n”, and the relaxation time 1. It is also subject to Reiner's [l9] "dimension objection" since the dimension of K will vary with the value of n. Nonetheless, the model generally gives an adequate description of fluid behavior over an intermediate range of shear rates. Because of its mathematical simplicity, the power-low model has been used extensively by several investigators. It was used by Lyche and Bird [24] to extend the Graetz-Nusselt problem in heat transfer theory to non-Newtonian fluids. It was also used for an exact analysis of laminar tube and annular flow by Fredrickson and Bird [25], in an approximate analysis of flow around a sphere by Tomita [26] and Slattery [27], in a variational analysis of flow by Bird [28] and Schechter [29], and in laminar nonisothermal flow by Hanks and Christiansen [30], and in correlation of a friction factor with a modified Reynolds number by Christopher and Middleman [4], and Marshall and Metzner [5]. Ellis [19] and, more recently, Gee and Lyon [31], for plastic melts, prOposed the following three-parameter model, a-l _ _ T:T _2— _ eij - $0 + ¢1[T] T1]. (2.6 2) 20 where ¢o’ ¢1, and a are the fluid parameters. The Ellis model does not describe the limiting viscosity at infinite shear rate nm, but more important is the fact that it describes the limiting viscosity at zero shear rate no = 1/¢o for polymer solutions. An alternate form for Eq. (2.6-2) has been sug- gested by Bird, viz., e..=-——l—l+ Mad T.. (2.6-3) 1] no T8 1] where no, 1%, and a are the fluid parameters. The shear stress 55 is that value of shear stress for which the corresponding non-Newtonian viscosity has drOpped off to one-half of the value of the limiting viscosity at zero shear rate no. The introduction of the fluid parameter 1% enabled the elimination of the dimensional objection to Eq. (2.6-2). Slattery and Bird [32] used the Ellis model to predict the drag coefficients for flow around a sphere. They found that the three-parameter model in Eq. (2.6-3) provided a better description of non-Newtonian behavior than the two-parameter power-law model. Ree and Eyring [33] discussed several successful applications of Eq. (2.6-3) to polymeric and colloidal systems. The application of this model to various flow geometries (McEachern [34], for flow through tube, and Ziegenhagen [35], for flow around sphere) led to very formidable mathematical expressions of the flow behavior. 21 A four-parameter generalization of the Ellis model, in which there is limiting viscosities at both zero shear rate and infinite shear rate, is that pro- posed by Meter [36] r. n " 0 ‘— r.. = - “m + ° ” e.. (2.6-4) 13 1 + [75(1:T)]a-1 13 L. Tm J where no, n , T , and a are the fluid parameters. The m 00 Meter's model in Eq. (2.6-4) may be thought of as an ex- tension of the Peek-Mclean model (a = 2) [37] and the Reiner-Phillippoff model (a = 3) [19, 38]. The parameter Tm is the value of the shear stress for which the corres- ponding non-Newtonian viscosity has drOpped off to the arithmetic mean of the limiting viscosities, i.e., to the value of %(00 + nm). The constant a indicates the abrupt- ness of the transition from nO to nm. Meter [39] has successfully applied this model to the description of the turbulent flow through tubes of seven hydroxyethyl- cellulose solutions. For many fluids, nco is at least an order of magnitude smaller than nO so that Eq. (2.6-4) can be rewritten as: e.. = _.3; 1 + {/ET???7]a—1 ; [_ flg[/§T¥??7]a-1]j 1.. 13 o T j=0 T 13 m m (2.6-5) 22 by rearranging and then expanding in powers of nm/no. In this form, the Meter's model may be considered to be the Ellis model with a small perturbation on it. Equations (2.6-4) and (2.6-5) reduce to Newton's law for the limit in which Tm approaches infinity. Herschel and Bulkley [40, 41] prOposed a three- parameter model eij (2.6-6) where Ty, no, and m are the fluid parameters. This model is the combination of the Bingham plastic model and the power-law model. In evaluating the parameters in this model, the yield stress Ty is first read from the flow curve. Parameters m and “o are then obtainable from the lepe and intercept of a logarithmic plot of -dvz/dr vs. Trz - Ty. There are not many applications of this model reported in the literature. 52.7 Linear Response of Viscoelastic Fluids Although the generalized Newtonian theory, in its various forms, describes non-Newtonian fluid behavior quite well, it fails to predict the normal stress effects and the viscoelastic effects associated with time- dependent phenomena. Normal stress effects were demonstrated by Weissen- berg's [42] experiments in which various types of fluids 23 were sheared in a gap between two concentric cylinders. Greensmith and Rivlin [43], for a modified parallel-plate viscometer, found that the fluid, in manometer tubes placed at various radii into the fixed plate of the viscometer, tended to rise in the center manometer tubes, while the second plate rotated at a constant angular velocity. Markovitz and Williamson [44] noted the same effect for a modified cone-and-plate viscometer. Transient and dynamic (oscillatory) experiments [45] exhibit behavior, for viscoelastic fluids, which are not predicted by the generalized Newtonian theory. Linear behavior is defined as that in which the measured value of a material property is unaffected by a change in magnitude of shear stress and shear rate. Within the framework of linear behavior and sinusoidal shearing, a significant achievement of molecular theory exists. Zimm [46], Rouse [47], and Bueche [48] have all proposed theories which relate the response of dilute polymer solutions to such parameters c (number-concentrations of solute molecules), T (absolute temperature), no (solution viscosity at zero shear rate), and n(s)(solvent viscosity, Newtonian). Theoretical similarities between all three involve the assumptions: 1. Dilution to the extent that solute molecules have no direct interaction with each other. 24 2. "Segmented necklace" structural model, highly flexible to the point that a Gaussian distribution describes relative positions of the segments along the molecular chain. 3. Responsiveness to the random restorative forces of entropy, i.e., Brownian motion. From these and other postulates, a differential equation can be written to describe a balance between dis- torting and restoring forces acting on the c00perating segments of the entire molecules. Rouse and Zimm obtained solutions expressible in terms of the complex viscosityl n*(w)= (S) (2.7-1) ll 0 w *3 III"! Z * - 1 + i A n n w p where AP is the relaxation time of the p'th mode of motion, and k is Boltzmann's constant. The Rouse assumption of a "free-draining" molecular coil led to: 1:[‘ 1 _ —§ _ LiR] E7 (2.7 2) 1Complex notation is a convenience for manipulat- ing simultaneously two separate equations: one describing the component of stress in-phase with the shear, the other describing the out-of- -phase component. Thus we assume T12 = R e{T$2e1wT} and e1= Re{ef ele} in which T92, e92 are complex amplitudes and w = Hf is the radian frequency. Then we define: 132/e32 = n'(w) - in"(w) Ill n*(w) 25 where AR is the longest relaxation time constant. Zimm's introduction of hydrodynamic interaction between segments gave the result: A = [1R] '1711'1' (2.7-3) where the Ap are numerical solutions to a complicated equation. Bueche's concepts, similar to those of Rouse, produced a slightly different expression with relaxation times twice the magnitude of Rouse's. 52.8 Non-linear Response of Viscoelastic Fluids In recent years several models have been proposed which describe non-Newtonian viscosity, normal stresses, and oscillatory phenomena of viscoelastic fluids [49, 50, 51, 52, 53]. Most of these theories have been partially successful in describing some of the observed features of viscoelastic phenomena. Naturally none of them seems to be able to simulate, without ambiguity, all features of viscoelastic phenomena and certain other effects observed in complex fluids. Most of these theories and special cases of the more general theories are naturally subject to certain assumptions. Typical assumptions include: 1. An absence of peculiarities within the fluid or between the fluid and the confining boundaries which would cause the no-slip criterion to be violated. 26 2. No preferred orientation by the fluid with respect to the confining boundaries. 3. Absence of a yield stress. 4. No time-dependent behavior of the type referred to as thixotropy, rheopexy, etc. 5. The fluid is a continuum rather than a collection of discrete particles. Molecular theories have also been developed which attempt to portray certain viscoelastic phenomena and non- Newtonian viscosity behavior in terms of inter- and intra- molecular forces, polydispersity, coil rigidity, network and entanglement effects, and other morphological parameters [54, 55, 56]. Models of great generality, and great complexity, have been proposed by Rivlin and Erickson [57], in terms of functions of time rather than time constants, and by Coleman and Noll [14, 15], in terms in functionals evalu- ated at different times. 52.8.1 Sprigg's Four-parameter Model Spriggs prOposed [58] a non-linear extension of the generalized Maxwell model as a constitutive equation for viscoelastic fluids. The non-linear operator which is a special case of the one used by Oldroyd in formulating his 8-constant model [11] is given by 27 Dr F r = —— - (1 + €)(€'T + T°e - 3 t (T°e)6) (2.8-1) 6 Dt 3 r where DT 8T __.= -— + v-VT + w-T - T-w (2.8-2) Dt 3t Here 2e : VV + (VV)T, 2w 5 VV - (VV)T and e is an adjust- able dimensionless parameter. The components of VV and v-VT in rectangular, cylindrical, and spherical coordi- nates are tabulated in Tables 2.8-1 and 2.8-2. Simple shear flow notation is also tabulated in Table 2.8-3. In terms of Operator Fe defined in Eq. (2.8-l), the extension of the generalized Maxwell model is given by P P = _ - T + APFET 2npe (2.8 3) r = 2 TP (2.8-4) p=l where 1p and np are constants which characterize the linear viscoelastic behavior of the fluid. The Tp are given an interpretation in the "spring and dashpot" theory of linear viscoelasticity [59]. The components of Tp are differentiated and integrated in the same manner as the component of T. If one lets ‘0. A E 1 2.8-5 p p ( ) 28 TABLE 2.8-1 Components of AV Rectangular Coordinate (x,y,z) "av av av x y 2 5x 3x 3x BVX 3V 8Vz W = 5y 3y 3y BVX 3V 3Vz 52 z 52 Cylindrical Coordinate (r,0,z) ["szr av Br Br W=£avr_v2 13V9+ZE r' 35 r r' 35 r 3Vr ave L_ 32 az Spherical Coordinate (r,6,¢) fivr ave Br Br W..1_""r-‘_’.9 131.11; r 56 r r 36 r 1 BVr - :3 1 3V6 Z$~cote if sin6’8¢ r' r sing 8¢ H 29 TABLE 2.8-2 Components of v~VT _ _n--_‘-_‘*“._‘AL _ For Symmetrical Rectangular Coordinate (x,y,z) 3 a + 3 . I . V°V I Vx 5; + Vy 5; V2 3;», (V thij (V V)Tij — V V-V (V°V) _ (V )txx ( )Txy sz V-VT - (V'VHyx (V°V)Iyy (V'Vltyz Liv V)sz (V°V)TZY (VoV)Tzz For Cylindrical Coordinate (r,9,2) V a 9 3 a V'V ' Vr s—r- + -r— 55 4‘ V2 'a—i' (v-V)T - 2 29 Ire (V°V)T + 32(1 -T ) (V-V 11 r r6 1' rr ee “:2 VB VB V'VT = (V'V)‘tr6 + 7(trr'1’66) (V'V)T66 + 2 .1:- 11:8 (V.v)rtz Ve Ve LW'VHIZ .- 7:- I62 (wvnez + T Tr: (vwhzz For Spherical Coordinate (r,e,¢) V V 3 6 3 3 vv'vrfi+75§+rsin5fi [A B c v-VT - B D E C E F V V 3 o — _e_ — A A (V Vh’rr: 2 r' Tre 2 r Tra V V . . .9. - - .9; B (V VHr6 + r (Trr 166) r (16¢+Tr¢ COte) V V . . .9; - - .9. C (V V)TIO + r[(Trr T¢¢) + Tre cote] r' 18¢ V V I O .9- I A D (V V)Tee + 2 r' Tr6 2 r' Te¢ cote :1 V E - (V°V)T6° + r ['rre + (Tee-T¢¢)cot6] + — 1:4) W W C ‘V n.— u..- I" (V )T¢¢ 4' 2 r 11:0 + 2 r 19¢ cote u --._-\.~ C film .—q 62 n|&< rz 'm--a--. 30 TABLE 2.8-3 Simple Shear Flow Notation Coordinate Notation Flow Geometry l 2 3 6 z r l. Poiseuille Flow 2 r 6 -‘h~ 6 2. Couette Flow 0 r z 3. Parallel Plate Torsion 0 z r 4. Cone and Plate Torsion ¢ 6 r 31 and n 5"" Z A =---- (2.8-6) p o p P=1 p PaZ(a) where Z(a) = Z p-a denotes the Riemann zeta-function p=l [60], then, expressions for the material functions n*, 6*, 6d, 8*, 8d, n, 8, and B can be derived in terms of four parameters: (1) a zero-shear viscosity no, (2) a characteristic time A, (3) a dimensionless parameter a which describes the slope of log n' vs. logcoand log n vs. log 9 plots in the "power-law" region, and (4) a dimensionless parameter c which accounts for deviations from the Weissenberg [61] hypothesis that T22 - T33 = 0. 52.8.1.1 Derivation of Material Functions For simple shearing motion, the velocity field has the form Vl = V1(X2,t), V2 = 0, V3 = 0. Then the only non-vanishing components of the e and w tensors are — -w12 = j/Z, where 1 E dvl/dXZ' For the T tensor one has T12 = T21 and T13 = T3l = T23 = T32 = 0. Then Eq. (2.8-3) becomes: (1 + 1 P P p T11 T12 0 '39 Tp TF 0 4-i-A at 12 22 2 p P _° ° T331 F) 1 <1 =— . 0 np Y 0 2 0 <1 From this matrix BTP p 12 _ p T12 + 1p 3t 1P(ET11 P 8T 9 11 _ 5 111 + Ap —§E_ Ap(3 + P 8T 9 22 i - T22 + Ap at + Ap(3 an p 33 i T33 + Ap at + 1 (3 + Since T11 + T22 +'53 are independent. 32 [.2 2 P (3 + §€)T12 P _ P 11 (2+€)T -€T + (2+€)T €)T COIN 542- 2 '3'“ p T12 N|-< . 5.91 3€)T12 2 0, only three 22 P )1 22 2 _ P _ p T ETll (2+€)T22 O 2._‘2 P (3 §£r%2 o .1 p O 3(l+e)Tlg‘ (2.8-7) = - ' 2.8-8 an ( ) = 0 (2.8-9) = 0 (2.8-10) = 0 (2.8-11) of these equations 33 A. Non-Newtonian Viscosity and Normal Stresses For steady simple shearing, BTij/Bt = 0, i,j = 1,2,3, then Equations (2.8-8) through (2.8-ll) become: P _ P P 1,: _ _ T12 1p(8Tll + (2+e)T2 )2 npy (2.8 12) p-1291 = _ T11 Ap(3 + 3e)T12 2 0 (2.8 13) p 1-2.91 = - T22 + 1p(3 3e)T12 2 0 (2.8 14) p 1.4.21 = _ T33 + Ap(3 + 3€)T12 2 0 (2.8 15) From Equations (2.8-12), (2.8-13), and (2.8-l4) n i TEZ = - .12, , 2 (2.8-16) l + (c1 7) P where c is a shift factor and defined as: 2 2 - 2s + :2 c = (2.8-l7) 3 Then from Equations (2.8-5), (2.8-6), and (2.8-l6) the non-Newtonian viscosity can be Obtained as: °° Tl le/(-§) = X p . 2 p=1 l + (ckpy) nti) a P . (2.8-l8) 34 From Equations (2.8-13), (2.8-14), and (2.8-16) = - li—aEk (2.8—19) Then from Equations (2.8-5), (2.8-6), and (2.8-l9) the primary normal stress coefficient can be obtained as: e<§) 2 (:11 - :22)/<-§2) 2 An m = 2(a? pil 2a 12 2-2 (2'8’20’ p + c A Y From Equations (2.8-l4), (2.8-15), and (2.8-16) e A n §2 p _ TP = - .B, P (2.8-21) - 2 1 + Cl ( pY) Then from Equations (2.8-5), (2.8-6), and (2.8-21) the secondary normal stress coefficient can be obtained as: o _ o2 EAn m o 2 l = _ . (208-22) 22a) p—l p2a + c212y2 B. Oscillatory Shear Stresses and Normal Stresses Assume that the amplitude of the shearing vibrations is small enough that the terms of order §3 35 can be neglected. Since Tij = 0(§2), from Eq. (2.8-8) predicted 112 varies linearly with §. Let: § = Re{§° elwt} (2.8-23) r = a {1° eiwt} (2 8-24) 12 e 12 ' Then: ° _ o iwt '0 iwt ley — Re{T12 e } Re{Y e } 2 e 12 12Y ° Substituting Equations (2.8—23), (2.8-24), and (2.8-25) into Eq. (2.8-8) and equating terms with a similar time- dependence by removing Re- operator gives '0 n Y p0 = _ - T12 ETE‘IET; (2'8 26) From Equations (2.8-5), (2.8-6), and (2.8-26) the complex viscosity n*(w) is given as: n*(w) n'(w) - in"(w) wiz/(-§°) : n° 1 - i p=1 paz(a) 1 + wz zp-Za 1 + mzxzp’ wk -2a m 2a (2.8-27) 36 Therefore, T] m a o 2 pp n'(w) = _ (2.8-28) 2(a) p—l p2a + wZAZ n 00 o 2 wk n"(w) = _ (2.8-29) 2(a) p—l p2a + NZAZ From Equations (2.8-4), (2.8-9), (2.8-10), (2.8-11), and (2.8-26) the predicted oscillatory normal stress must be of the form: 0 Ziwt e } 1.. = R {d. + T.. e 3 33 33 j = 1,2,3 (2.8-30) Substituting Equations (2.8-23), (2.8-24), (2.8~25), and (2.8-30) into Equations (2.8-9), (2.8-10), and (2.8-11) and equating terms with a similar time-dependence by removing Re- operator gives: 1?? (1 + iZmAp) = 4 g 8 1p fig §° (2.8-31) r33 (1 + izwxp) = - 2 E 5 APT§§ §° (2.8-32) r33 (1 + izwxp) = - l_§_2 Aprgg §° (2.8—33) d? = 4 z e 1p1§§ $0 (2.8-34) d3 - - £_%_E Apng 3° (2.8—35) 37 l + a po To P — - _____ - d3 - 3 Aple y (2.8 36) From Equations (2.8-26), (2.8-31), and (2.8-32) .0 2 1 19° - 19° = - P n9 (Y ) . (2.8-37) 11 22 (l + iwxp)(l + Ziwkp) Hence, from Equations (2.8-5), (2.8-6), and (2.8-37) the primary complex normal stress coefficient 6*(m) is given as: 2 9*(w) s e'(w) - ie" a (Til - r32)/(-(y°) ’ = noA 2: p2a - 2w212 - iBwlpa (2.8-38) Zia) p=1 (pza + w212)(p2a + 4w2A2) Therefore, 6'(w) = JIA' E1 2 :a2_ 2;:A2 2 2 (2.8—39) (a) p— ( a + w A )( + 4w A ) n A m a Z 3 wkp 6"(w) = ‘3 _ (2.8-40) 2(a) p—l (p2a + w212)(p2a + 4w2XZ) From Equations (2.8-16), (2.8-34), and (2.8-35) n A (1 - iwl )|§°I P _ P _ _ 1PHP 12 . _ d1 d2 _ 2 2 (2 8 41) l + w A 38 Hence, from Equations (2.8-5), (2.8-6), and (2.8-41) the primary normal stress displacement function 6d(w) is given as: d '0 2) 9 (w) E Re{dl - d2}/(-IY I oo 2 l o = _ (2.8—42) Zia) p-l pZa + w2A2 Similarly, from Equations (2.8-5), (2.8-6), (2.8-16), (2.8-32), (2.8-33), (2.8-35), and (2.8-36) one can obtain the secondary complex normal stress coefficient 8*(w) and the secondary normal displacement function Bd(m) as: 2 A n (Yo) po _ po = _ g 29 p . . _ T22 T33 (2)11 + iwxp)(1 + xzwxp) (2 8 43) m a pm) = 6 Mo 2 92a -_21w2*2 ' 13mm 2 (2.8-44) I Zia) p=1 (pza + w212)(p2a + 4oz) ) A °° 2a 2 2 B'(w) = E rk) E ‘7 pz 2- Zgak 2 2 (2°8-45) 2 2(a) p—l (p 0L + (A) A )(p + 4Q) A) An m a e o 2 Bpr B"(w) = - _ , A (2.8-46) 2 Z a) p-1 (p2a + w212)(p2a + 4w2A2) An m d _ 6 Z 1 2.8-47) B (m) ‘ ‘2' Z(a) p=1 2a 2 2 ( 39 C. Stress Relaxation The transient stresses of a viscoelastic fluid can be studied under simple unsteady flows: stress relaxation after cessation of steady simple shear which has the following velocity distributions: Vl(x2,t) = YoX2[l - h(t)] V = V = 0 (2.8-48) where h(t) is the unit step function. For stress re- laxation, the following material functions are given by model as: ~ ° °° n expl-t/Ap] 2:12:31 3 ll T12(t) = 3L. 2 E» - 2 no no §o no p=1 1 + (ckao) = 1- Z PanPI‘Patlxg (2.8-49) 2(a) p—l pZa + (CKYO) 5 ° (t) _ (t) w _ (Yoot) E J; T11 T22 = J; 2 ZAPnEexp[ t/Ap] no no Yo2 no p=1 1 + (CA 9 )2 p o = 2A 2 exgt-pat/Al _ 2(a) p=1 2a (2'8 50) P + (ck§o)2 40 52.8.1.2 Computation of Material Functions For general a, simple rearrangement leads to the rapidly converging series. From Eq. (2.8-18) n(§) = 1. 3 pa no Zia) p=1 p2a + CZAZQZ = 1 z 1 - z C2A2§2 z(a5 p=1 a p=1 pa(pza + 02A2Y2) — 1 ’ E;%§%i :1 2a 1 2 2°2 (2'8'51) p p (p + c A Y ) Similarly, em?) = 1 >: 1 inox 2(a) p=1 p20 + C2A2Y2 2 2-2 ” Z(2a) c A Z l ( — -—T—}— _ . 2.8-52) Zia) Z a p—l p2a(p2a + 021272) .8111: 1 2 1 enol Z(a) p=1 p2a + c2A2§2 = 2(2a) _ czkziz 2 2 1 (2.8-53) Z(a) Z(a) p-l pZa(p a + CZAZYZ) 41 mu»): 1 2 °‘ no Zia) p=1 p2a + A2 2 2 2 m A w 2 1 - 1 _ (2.8-54) 2(a) p-l a( 2a + A2w2) n"(w) = 1. z 1 wnoA 2(a) p=1 p2a + A2 2 2 2 m - 2‘2“) — A m X 1 (2.8—55) 2(a) Z(a) p=1 p2a(p2a + A2w2) For cA§>>l Gregory's formula in the form [62] 00 f f(x)dx = Z f(p) + £é2l + error terms (2.8-56) o p=1 can be used to give asymptotic expressions for n(§), 9(1), B(§), n'(w), and n"(w). From Eq. (2.8-18) mi) = 1 2 p“ no 2(a) p=1 p2a + C2A2§2 Hence, 0. _ P f(O) _ f(P) ‘ 2a 2 2-2 ' 2 " 0' and p + c A Y CD (X) a I f(x)dx = I x . dx 0 o x2a + czAzy2 . "(cAW 1T 2a cos 5; Therefore, . . é-l n(Y) ___ N (CH) _ no 2a cos 1L Z(a) (2.8 57) 2a Similarly, l-1 n'(w) = n (Aw)a (2 8-58) no 2a cos 1L 2(a) . 2a n"(w) = 1 TI (Aw) W”) (2 8-59) wAn Zia) . n 2a 2 ' O 5111 ‘2—- a . . iii-2 . -2 6(1) = #1 1r (cAy) _ (GAY) (2 8_60) no Zia) sin 1L 2a 2 ' 2a 0 o %-2 o -2 8(1) = 1 n (cAy) (GAY) ‘] _ eAn Zia) . n 42a 2 (2'8 61) 0 $111 5&- J From Equations (2.8-18) and (2.8-S7) the limiting slope s of a plot log n vs. log i is given by s = % - 1 (2.8-62) 43 Hence, for high shear rate i, the viscosity function given by Equations (2.8-18) and (2.8-57) exhibits power-law behavior with n = %. §2.8.2 Bird-Carreau Model Bird and Carreau [63, 64, 65] proposed a non- linear extension of the generalized Maxwell model for polymeric fluids as a constitutive equation which is given by: i' t w n e-(t-t )/A T 3 = - f 2 -E -m p=1 A§p[1 + %I£(t')A 29 2 1p T13 dt' (2.8-63) where A1p and A2p are two sets of time constants; A1p is associated with the rate of creation of network junctions, whereas the second set A2p is associated with the rate of loss of junctions. The term 1 + %E%(t')Aip accounts for the structural changes of the material when undergoing strain. The finite strain tensor fl] is defined by [66] T13 = [[1 +§-][613(t') - §1jj + :2: alrajsx [arsaw - arsun] (2.8-64) 44 If one lets n = n A / E A ; (2.8-6S) p o 19 p=1 1p a _ 2 l , _ Alp - Al(E:I) , (2.8 66) and 2 a2 Azp = A2(Sfi) (2.8-'67) then, expressions for the material functions n*, 6*, 6d, d 8*, B , n, 6, and B can be derived in terms of six parameters: (1) a zero-shear viscosity no, (2) two sets of the characteristic time constants A1 and A2, (3) two sets of dimensionless parameters a1 and a2 which describe the slopes of log n vs. log i and log 6 vs. log i plots, respectively, and (4) a dimensionless parameter a which accounts for deviations from the Weissenberg hypothesis that 122 - 133 = 0. The Bird-Carreau model is an improvement over Sprigg's four-parameter model which has several weak points: (1) the curve of the log n vs. log y incorrectly has the same slope in the power-law region as log n' vs. low w, (2) in the power-law region the $10pe of normal stress curve is too rigidly related to the slope of the viscosity curve, and (3) the ratio of the normal stress difference is required to be constant. 45 §2.8.2.1 Derivation of Material Functions For a steady simple shearing motion for which the velocity profile is given by Vl = yxz and then IIe(t') = §2. The finite strain tensor fl) is given by: [$62 ye 0' —ij _ 2 r ya ya 0 (2.8-68) o o oj where 9 is the time elapsed (t-t') since a given inter- action was formed. Substitution of Eq. (2.8-68) into Eq. (2.8-63) and integrating yields the following equations for the shear and normal stresses: 112 = - z .9 (2.8-69) w n A §2 :11 - 122 = -2 p21 2_2, 29 . 2 (2.8-70) l + (Ale) w n A §2 29 T - T = -E g 19 . (2.8—71) 22 33 p—l 1 + (*1p7)2 Hence, the material functions n, 6, B are given as: 46 “(§)ETY3= Z P .2 p=1 1 + (Alpy) m a = ”0 Z P 1 (2 8-72 Zia ) -1 p=2 2a a 2 ' ) 1 1 1 ° 9 + (2 AlY) ° _ T11 - T22 m Up A2p 9(Y) = ,2 = 2 Z 2 'Y p=1 1 + (A1 Y) a2+l a -a = 2 A2n° ; P 1 2 (2 8-73 Z(a1) - l p=2 2a1 al . 2 ' ) p + (2 Aly) . T - T w B(y) : 22 .2 33 = E 2 no 12p 'Y p=1 ' 2 1 + (Alpy) a E2 szn m “1 0‘2 _ 0 12 - Z a - l 2 2a a 2 (2'8-74) l p=2 1 Note that Bii) is proportional to 6(y) only if E is taken to be a constant, a. Note further that, whereas n con- tains only the Al, the functions 6 and 8 contain both A1 and A2. Hence one would expect that, in the power-law region, the s10pes of n and 6 would not be connected in the same way for all fluids. For oscillatory, small-amplitude motion, the components of the complex viscosity n*(w) are: 47 n*(w) E n' - in" = m n w w n A :1 <9» 7 - i 2 -P 29 2 (2.8-75) P 1 + (12 w) p=1 1 + (12 w) n'(w) = 2: ”p 9‘1 1 + (12 w)2 no m P-al+2a2 = ZZal) - 1 :2 2&2 a2 2 (2’8-76) p p + (2 1200) n 00 0.) n A n (w) = 2 up 2p p=1 2 l + (12pm) O. 2 zwlzn m ‘°1+“2 = 0 2 P (2.8—77) 2(aly - 1 =2 202 a2 2 P p + (2 12w) The transient stresses of a viscoelastic fluid can be studied under simple unsteady flows; stress relax- ation after cessation of steady simple shear which has the following velocity distributions: V1(x2.t) = Yolel - h(t)] V = V = 0 (2.8-78) where h(t) is the unit step function. 48 For stress relaxation, the following material functions are given by the model as: ._______ E —— ——?——— -'- X n n Y n = - 2 o o o o p 1 1 + (Aleo) a a a 1 w p 1expl-(P 2/2 212)t] = ZTal) - 1 piz Zal a1 . ’2 (2'8-79) p + (2 AlYo) 39.913 3 _-__1. [117-112.2%}. °§ ”£23 exp““2219 n n _ ' o o o o p-l l + (Ale0) a +1 a -a a a 2 2 2 12 m p l 2exp[-(p 2/2 A2)t] 80 = 2(“13 ' 1 p22 201 “1 - 2 (2.8- ) p + (2 Alyo) 52.8.2.2 Computation of Material Functions As discussed in section 2.8.1.2, for low shear rate of frequency, the series for the material functions given above can be rearranged into rapidly convergent series, in which usually only the first few terms are needed. Hence, (2“11 § 2 o ) oo n(Y) = 1 _ 1 z 1 (2.8-81) n Z a - . 2 ° 1 p (p + (2 lle) ) ‘e ‘- 49 r _ e(') 2a2+lx a1 m ’(“1+“2’ __1_.= 2 - - ' 2 P no ZTalf - 1 z(0‘14’0‘2) 1 (2 AlY) :2 2a1 a1 . 2 p p + (2 AY) L. 1 .1 (2.8-82) 2 2 (2“ A w) m n'(w) _ 2 1 _ no ‘ 1 2(a1) - 1 piz a1 2a2 a2 2 (2'8 83) P (p + (2 12w) ) a 2 -(a +0 ) 2 A w a w l 2 32122... 2 - - 2 2 up no - 2(al) - 1 Z(al+a2) 1 (2 12w) :2 2oz a2 2 P p + (2 12m) (2.8-84) For Alp§>>l and 12pw>>l Gregory's formula in the form [62] 00 f f(x)dx = Z f(p) + Eégl + error terms (2.8-85) o p=1 can be used to give asymptotic expressions for n(§), e(§). n'(w). and n"(w). — l-al a . 1 - ‘1 no 2(al) - 1 . 1+0:1 a1 . 2 20.1 Sln(-§-a——1T) ”1+“ (2 AIY) L l 1 + % a1 - 2 (2.8-86) 0.1 . 2(1 + (2 AlY)) ._ 50 l-a -a2 a +1 a a . 2 A 1 ° 1 6( ) = 2 12 "(2 AlY) n 2(a ) - 1 1+a -a o 1 . 1 2a131n(—-23—1——1T) T l l + —(a — a ) 1 6 l 2 ' 0.1 2 " 0’1 2 (2.8-87) (cl-a2)(2 AIY) 2(1 + (2 11y) ) T - 1 a1 a a 2 n'(w) 1 "(2 A2”) 2 n = Z(a 1 - 1 1+2a -a o 1 . 2 l, 2a251n( 2a ") 2 l + l-(20L - a ) 1 6 2 l 88 - a2 2 - a 2 (2'8- ) 2 (2a2 - a1 + 1)(2 12w) 2(1 + (2 12w) ) F" l-al-a2 a a a 2 2 2 n"(w) _ 2 12w fl(2 12w) n Z(a ) - 1 l+a -a o l . l 2&281n(-—-2-a-2-——TT) L— —'| l 1 1 + 3402 - 01) (2 8-89) ' a2 2 " a2 2 ' (a2 - a1 + 1)(2 12w) 2(1 + (2 12w)) CHAPTER 3 POROUS MEDIA PARAMETERS FOR VARIOUS CONSTITUTIVE EQUATIONS When a fluid flows through a porous medium, the velocity of its elements changes rapidly from point to point along its tortuous flow path. The forces which produce these changes in velocity vary rapidly from point to point. It is reasonable to suppose that the random variations in flow path for any particular fluid element are uniformly distributed. Also the variations in magni— tude of velocity can be expected to be distributed uni- formly with mean zero. Thus, for steady laminar flow the lateral forces associated with the microsc0pic random variations in velocity can be expected to average to zero over any macroscopic volume. The only non-zero macro- sc0pic force exerted on the fluid by the solid is that associated with the viscous resistance to flow. For steady laminar flow this force must be in equilibrium with the external and body forces on the fluid. The literature on this subject is voluminous and experimental investigations have been done by many 51 52 workers [67, 68, 69, 70, 71, 72, 73, 74] to determine the correlation between the pressure drOp and the flow rate of fluids through packed beds. An excellent reference is the monograph of Scheidegger [75], which contains a par- ticularly good discussion of permeability concept. Other general references are those of Collins [76], Carman [77], Muskat [78], Leva [79], and Richardson [80]. Some effort has been extended toward establishing methods for predicting non-Newtonian flow behavior in porous media and for correlating pressure drop versus flow rate data with Viscometric data for porous media experi- ments. At the present time there is no universally acceptable scale-up method for flow of rheologically com- plex fluids in porous media. The various methods cur- rently employed or suggested, can be arbitrarily divided into at least three major categories. The method which seems to have received the most attention is based on the coupling of a particular model for a porous medium-~i.e., the so-called hydraulic radius model, with an assumed functional relationship between shear rate and shear stress to describe the rheological behavior of a non- Newtonian fluid. This method involves correlations of experimental data from one-dimensional flow experiments in unconsolidated porous media--i.e., mostly bead packs, with the appropriate rheological parameters derived from Viscometric experiments on the fluid of interest. The 53 power-law and Ellis models have been used to describe the purely viscous behavior of the non-Newtonian fluids. Another category involves generalized scale-up methods which adapt Darcy's law to non-Newtonian fluids without invoking a particular rheological model of purely viscous behavior. The appropriate rheological description can, in principle, be derived from Viscometric and porous media flow experiments. A third approach, based on the concept of the simple fluid, involves the application of dimen- sional analysis to the scale-up of porous media flow data for an arbitrary viscoelastic fluid. This work was mainly directed at developing a generalized scale-up method based on the capillary model. Wall effects were accounted for using the hydraulic radius concept. Various empirical and derived models for non-Newtonian fluids were applied to the problem of flow through porous media. Deve10pment of packed bed equations for various constitutive equations are contained in Appendix A and the results of the modified Ergun friction factor versus appropriate modified Reynolds number corre- lation for each rheological model are listed in Table 3-1. It has been indicated that Darcy's law is funda- mental within the assumptions made. Two significant aspects of this law are the analysis of capillary flow and the assumption of fluid homogeneity. It is the 54 .z-z ucmuowuao » m . Duo .0 shunned a. la- ( A 26;... n6 ":6 To 2. a aha}; . o . _ . I lnl .aoflxaam -e m u E _ x .. Iweonflo I «Hi I ouwwuflawuflwflunbm “4 E a e-» n~U£Unua= . o I (IV .I 9 .uI. a 11:0- 3 -3: 2 f4. . _ L - 1 in: «a A o; .2. -3. .LIIJUIH. .79]. _..- xi... 2. . 3 7.: o W . ._... . .1 a 3. 3.. o . . J: . .1 I UHGU ML 1— r on“. 0: mr A O and In . no Wm hug] I on II I If . P fl_ ‘8 . b is g ‘o accenoa anode: e.. e Tun. En . F; [I 2|; . «E. 66 64 .26 a . a In». ‘fi Jyv v N y I _ JD HO“; 3 Ti. . 73,. C 3 f “3.: 6.23.3 5 .. 2. cc 3 a r oc0o«~o~»>d~>c«>>~oa c I: o I u I y a a o . a d O I 1 . ILL... .1...- L.F2:- .. c u n . . _ I. o a um...‘ 24 . on I Sawu o ucIonIA e «In c 2.32.. on... «.33 a. u I w 3.. val. .mn.o .o~.o .no.o a I . Lu I In .93....» noun: a u a 0 333.3 5 a I 5- I a. a 4 o a» is I H; o I oval-«Paczom o o a c a :28 c U Q I MI“ I ‘ Tap. :u 526: a .03... «an... 6 OFIA 5 IL I 0d.“ on" 2.332— r» r y o 0506.! 00.0 on. I. u: .26 .26 .36 J ) u n A uni. 1 o: c 2 14% Nu nun] I cu noun: a 3.. u h o a J! n I 3. 3:» n a voduuu-«o cu Fury 0... u u .1 0353.60.30.— o. no LJE... . _ FF. 2 1.1:...- ....... I . V . -1. f. u 6 «MI..- c 2.6 I III...- I Rama and o accouoa 0 an d.) on.o vfil 'le u: a... .36 .36 .36 i 4 E u a .6 3332 o a no .Mm am. I on no»... 31 n Tamil- I of a 323.3 5 a 7.. wins: ... - 3...: Ivan-1.332 . u 0‘ U I o n «0:4 m _lml# Hflflh o m I z rdlmp u; U a r C a o noun: 0 I J .I o: u H you. 0 n n 00 o I mp." o mwm I down . u u WW WNJW I In . I la v0~u«uo«a : «Can I I a» anacOvIDI . n o aneuoaauna nonunz noaoc>oa acauuuucoucou ouou-Eduoa cosy-:vu caauavo! cg. “couch c0«~u«uh flan c04u3~0m «ovox I>4a3uuuncou «.008 cgvoz aneuOL abouIICMIL oaalx .9050. ~.. an '22) 1000 dyne/cm2 this fluctuation is 5%. The error in kN is negligible. Thus, primary normal stress difference data are accurate to within 15% for low values (< 500 dyne/cmz) and 5% for high values (> 1000 dyne/cmz). For values of -(Tll - 122)< 100 dyne/cm2 the errors are too great to consider the values reliable. §4.4.2 Results The basic rheological data for Separan AP30 are summarized in: Appendix D: Steady Shear Stress, as n(§) vs. Y Appendix E: Steady Normal Stress, as '(Tll - 122) vs. Y Appendix F: Stress Relaxation Spectrum, as H(T) vs. t As discussed earlier, the Viscometric curves should have a precision of about 3%. Some typical data 67 are plotted in Figures 4.4-3,4,5, and 6 as 112 vs. 9, n(§) vs. Q, 8(y) vs. §, and H(T) vs. t, respectively. There are two important effects which may in- fluence the Viscometric data; these are viscous heating and aging of the fluid. Viscous heating is a problem at high cone speed. Readings are not valid when the heat resulting from viscous dissipation produces significant temperature gradients within the fluid. Turian and Bird [100] have derived a formula for the threshold shear rate at which viscous heating would influence the steady state torque readings. They did not provide a method by which the observed readings may be corrected. There was no tendency for the torque reading to decrease with time for a fixed cone speed because of heating effects. Viscous heating was not a problem and shear degradation over the time of testing was not usually measurable on this instrument. No attempt was made to measure shear degradation rigorously but the viscosity curve was essentially retraced from low speed to high. It is possible that shearing during preparation of the solution had already reduced the polymer average molecular weight to the point where the solution was no longer sensitive to the levels of shear encountered visco- metrically. The high-shear behavior is determined by the coordinated motion of closely-neighboring segments of the molecular chain. This response is not greatly 68 mcoflusaom on do cmuwmom moowzvt MOM mumo mumm ummnmummmuum unocm muv.v musmam 00m .> Iowa Huonm Hoa ooa auoa ‘1‘1 ‘ 1 d. (4 .1 ‘1“1 11* q ‘ T dd.‘ 1 C ‘ 1 1 11“ 4 0H 0H on ma 5.363 366- om 64 533$ 636 4 on ma 5033 366 O om ac 536% 686 O JAIIAAL L lllLll J. ,4 N lnAnll O H O H OH zmo/auxp 'ZIl ssezas Jesus 69 chwusHom on ad cuuuamm mo huwmoomw> cuficou3OZucoz vnv.v ouomwh Hloom .w ovum udonm «.3 NS H3 63 . 73 _ «-3 1.1141111 4 d. .411111 1 d —<1<<<14 [4‘ lALLA on ma c833 $66 I am 9.. 536$ 636 4 on ac 596% 636 O on 9.. 533$ 63.6 0 OH OH OH oes mo/Jb 'u KirsooIrA Normal Streu Difference Coefficient e, gr/cm 10° p... O p... O 10 -3 -l l m 70 v wfifi 0.10‘ Separan Ap 30 A: l O 0.25% Separan hp 30 O 0.50% Separan Ap 30 A. 0.05‘ Separan Ap 30 . . ...l . . ....| . . ......l 10° 101 102 10 Sheer Rate §, sec-1 Pigure 4.4-5 Primary Normal Stress Difference Coefficient of Separan hp 30 Solutions 71 HI. HI 0 HI HID 1&11411‘i . 0mm mh.oa 00m who.a mm mmw~.o mm mmmm.o cuummmm «no.0- cmummmm mode ‘ c333 «25 O c333 89° 0 mcofluzaom om Q< :mummwm mo Esuuommm Godummemm owm tu mafia . a CA #141444 1 0 OH ‘44114‘ ‘ a -3 d114‘ 1 wuv.v wusmfim 4 1 N -3 N- A lALLl A 1ALJLA .AAALA 0H OH OH OH :39 ’(l)H mnzqoads uotqexctaa I... - .vp‘ nth. .5 'f on. U NI" ‘cl ' «A- '5... .._ e..: " 'Un . . e..: \ . I ‘u 72 affected by the long-chain entanglements which are destroyed by degradation processes. For most of the polyacrylamide solutions tested here, the entire shear range of the viscometer would have to be classed as ”low"; limiting values of n for each solution were approached. One difficulty was noted with measurements on the polyacrylamide solutions: an apparent flow instability developed at the higher shear rates, which effectively placed an upper limit on shear rates attainable with a given sample and geometry. This problem is apparently a common one for viscoelastic fluids in cone-and-plate instruments [101, 102]. In the instability region the fluid somewhat erratically exuded from the gap between cone and plate at some locations on the rim and pulled back from the rim at others. For any sample the condition developed near or somewhat beyond the range of shear rates where the viscosity begins to decrease with shear rate. The instability could be deferred and measurements ex- tended by using the sample reservoir and filling it deep enough to cover the gap. Eventually, however, the same problem again developed and the run had to be terminated. The location of the instability depended on the polymer molecular weight and concentration of the solu- tion. As a general rule it was possible to penetrate further into the non-Newtonian region for higher molecular 73 weights of the polymer and for higher polymer concen- trations. All experimental tests indicated that vis- cosities measured in this work were outside of the instability region. One measure of the reliability of the data was the consistency of the points in Figures 4.4-3 through 6, which were highly reproducible. The data which might be suspect are for the measurements of the primary normal stress difference of 0.05% solutions of Separan AP30. The primary normal stress data for 0.05% solution was furnished by Meister [117]. A more general warning of fundamental errors in normal stress measurement should be noted, however. It seems that the presence of time-dependent normal stress fluctuations--not seen before because of the damping and averaging action of manometers--could greatly influence results. 54.4.3 Discussions and Evaluations of Data 1. Zero-shear Viscosity The zero-shear viscosity exhibits a functional dependence on temperature, on solute concentration, and molecular weight which is smooth and roughly similar to the behavior predicted from molecular theory. For example, it is well known that no of many fluids--Newtonian or polymeric--can be characterized by . .t (I, w up. _1 L... 'Vf‘l u., 0 :-~ n... U. D1 (D (n I 'l. N» O CID 74 an inverse-exponential dependence on temperature. This is often expressed as Q - 0 no = n exp (- fig (4.4—4) where Qo is an activation energy (kcal/gr. mole) and H is a frequency factor; both are nearly independent of temperature. An important study of the temperature-dependent prOperties of polymers is the work of Williams, Landel, and Ferry [103]. They showed the temperature dependence of the relaxation time associated with viscoelastic response, and measured a coefficient aT defined by aT = nTror/ano (4°4'5) where the subscript r means that the quantity to which it is appended is measured at the temperature Tr' aT is essentially the shift factor defined in Eq. (4.4-6) aT (no - nS)TTO/(nO - ns)TOT (4.4-6) where To would be some arbitrary reference temperature. In this testing the zero-shear limiting viscosity, no, is obtained by extrapolating viscosity-shear rate curve as follows: = lim 75 Experimental evidence [104, 105] indicates that this voscosity-molecular weight relationship exhibits two distinct regions. In the low molecular weight region, the viscosity increases with a low power of M, generally in the neighborhood of unity. In the high molecular weight region, the viscosity shows a very strong molecular weight dependence, no being proportional to M3'4. The transition between these two regions occurs over such a narrow range of molecular weight that one is led to define a "critical" molecular weight MC as the intersection of the lines representing the high and low molecular regions. Such a sharp change in the molecular weight de- pendence must be associated with a corresponding change in the mechanism, on a molecular level, responsible for viscosity. Bueche [106, 107] has offered a theory which is in general agreement with observed behavior. The essential idea of his theory is that, above a certain chain length, molecules are sufficiently long to become entangled at various points along their lengths with neighboring molecules. The force required to move an entangled molecule is much greater than that for a shorter, unentangled molecule, because the entangled molecule drags other molecules with it, each of which may itself drag other molecules. The number of primary, secondary, etc., entanglements increases sharply with chain length, and thus the viscosity shows a strong dependence on molecular weight. 76 The influence of solute content on no is depicted by the solid line in Figure 4.4-7 for polyacrylamide. The slope of this line, as well as its elevation, is dependent on shear rate to a great extent [94]. Porter and Johnson [94] have remarked that in many concentrated solutions of linear polymers, n(§) is proportional to the "s" power of concentration, with 5 dependent upon shear rate. A zero-shear rate, values of s between 3 and l4--often near 5--have been measured. With the data taken by Meister [117] zero-shear limiting viscosity, no, is extended up to concentrations of 2.0%. An insufficient range of concentration was studied here to check this, but the data of polyacrylamide appeared to violate such a relationship. Most of the commonly used non-Newtonian models incorporate no. It is thus vital that a reliable means be deve10ped to estimate no. 2. Cone-and-Plate Viscosities Since the viscometer is characterized by constant- strain conditions everywhere under the cone, interpretation of the data is usually unambiguous. Even though shear degradation is possible in polyacrylamide, it was not noticeable during the short time of Viscometric testing. As is indicated in several instances in Appendix D, viscosity curves for all fluids could be retraced as cone speed was decreased after high-speed shearing. 77 1000‘"4 no in 43.5213 - in 2.6026 in 0.5 - in 0.05 1.0‘ = 1.2185 .1 l .01 .1 To Figure 4.4-7 no, Zero-shear Viscosity, vs. C, Concentration 78 3. Normal Stress Coefficients Analogous to the viscosity for describing shear stress are the normal stress coefficients for describing normal stress. One set of coefficients, used by Williams and Bird [16, 17] is . T - T (T - T ) 6(y) E _ ll .2 22 = _ ¢¢ 2 66 (4.4-8) y 4e¢e . T - T (T - T ) 8(Y) E _ 22 .2 33 = _ 60 3 rr (4.4_9) Y 4e¢0 where e(§) and B(§) are allowed to vary with shear rate. This is the approximation employed in presenting the normal stress data from Section 4.4.2 as 6(y) vs. § in Figure 4.4-7. The vast change in the magnitude of e(§) makes it difficult for one to appreciate the physical influence of the parameter. A more convenient variable might be the product §0(§), which increases from a very small number (0.05% polyacrylamide) to a maximum, then decreases to a plateau and seems to increase again as § becomes very large (0.1% polyacrylamide). It has the advantage of relatively small changes in magnitude and the familiar unit of gr. cm.-l sec- ; however, it looses significance as a purely material parameter. 79 4. Combination of 6(y) and n(§) A feel for the magnitude of viscosity is acquired from values of e<§> and n(§). Therefore, it is often preferable to express normal stress phenomena in terms of comparisons with the magnitude of shear stress in the same fluid. Many such comparisons are possible. Weissenberg [42], in extending theories of rubber-like continua to liquids, initiated for introduction of the recoverable shear (strain) Sw and the shear modulus Gw: T " T 3w 3 11 T 22 (4.4-10) 12 T - 12 GW = -----8 (4.4-11) W Physically, Sw is conceived as the amount of deformation experienced by a flowing elastic element in dynamic equilibrium with the imposed shear stress. As such it is a measure of stored elastic energy. Mooney [54] has inde- pendently evolved a continuum theory employing strain and two shear moduli; in the limit of T22 - T33 = 0 which is identical to the Weissenberg hypothesis. Using the definitions of 0(y) and n(y) leads to: U) II a N = 9%. (4.4-12) € F] H 2 80 c; = n__ (4.4-13) It is interesting to note that Gw is the same quantity which was predicted by molecular theory [61, 108]. From the steady state Viscometric data presented in section 4.4.2 Sw and Gw were calculated and tabulated in Appendix H. Values of Sw versus shear rate were plotted in Figure 4.4-8. For the aqueous solutions of polyacrylamide studied here, Sw is a steadily increasing function of § over al— most the entire range of shear rate. Solutions of low concentration exhibit low elasticity as seen by the con- centration dependence of Sw in Figure 4.4-8. The impli- cation of a single curve which is almost independent of concentration is that very similar recoverable-shear behavior is exhibited at all concentrations and molecular weights of the same type of molecular chain, as long as the solutions are "sufficiently" concentrated and long- range of entanglement couplings between molecules are the dominant feature in resisting stress. From this expla- nation, all four sets of the aqueous solutions of poly- acrylamide are concluded to be effectively dilute solutions. So far, the discussion has considered only re- coverable shear; the earlier-mentioned concept of the modulus Gw will now be examined. Values of modulus Gw and reduced modulus EQr are defined as follows: 81 10.- S W a 0" J", C>0.50% Separan Ap 30 (30.25% Separan Ap 30 40.10% Separan Ap 30 I0.05% Separan Ap 30 lTll'Tzzl 'e(') = 112 =3??? 1000. Figure 4.4-8 SW, Recoverable Shear Strain, vs. y, Shear Rate 82 1.0 2 6 = G ( ) (4.4-14) WI W C In the above equation c represents polymer concentration. Figure 4.4-9 is a plot of log Gw vs. log §, and the modulus is seen to be a slightly-increasing function of shear rate which attains an asymptotic value at high rates. This plateau or constant value is reached experi- mentally at lower shear rates for higher concentration solutions. Since Gw is nearly constant it can be seen from the following relationship that /0 is nearly pro- portional to n. / 2 /0 E 1— = n/ g; = n X const. (4.4-15) w G w is proportional to §“ 2n. Further, it can be shown that 112 and that T11 - T22 is proportional to Y S : T11 ‘ T22 = T12 w T12 Gw Therefore, T 2 -n 2 _ _ 12 _ (k ) _ °2n _ III 122 — Gw —.—{%;——._-y x const (4.4 16) The ratio 6(§)/n(y), with units of time, may be considered as an effective time-constant characteristic of the modes of molecular motion which are on the verge of being 83 0.50% Separan Ap 30 0.25% Separan Ap 30 0.10% Separan Ap 30 0.05% Separan Ap 30 1001 I ><>o 2 T12 n2(Y) Gw = lTll-T22] - _§T?T _ W dyne 10 cm W'— W I l. I I 10. 100. 1000. Y, sec- Figure 4.4-9 Gw, Shear Modulus, vs. §, Shear Rate 84 extended in shear. Values of 9(§)/n(§) are tabulated in Appendix G and plotted in Figure 4.4-10 for the four sets of aqueous solutions of polyacrylamide. This concept of a time-constant is entirely independent of the specific rheological model. 5. Evaluations of Material Parameters Various material parameters of several rheologi- cal models were determined using the data as previously described. A. Power-law Model (n,K) The value of n was obtained from the slope of the plot log n vs. log § and K was chosen based on best fit for data. The material parameters for the power-law model are summarized in Table 4.4-1 and percent deviations of values calculated from data and model are tabulated in Appendix J. The power-law model provided a good fit to the Viscometric data at high shear rate. At low shear rate deviations are large because the model does not predict a limiting viscosity at zero-shear. The power-law model does not describe normal stress difference. The devi- ation of viscosity calculated from this model with the data at sufficiently large values of §(§ > 10.76 sec-l) was 6.36 percent. 85 > .m> c\o HmumEmumm unmocmmmocH Hobo: oanv.v musmwm Hloom .> ovum umwnw 0.00H o.oa o.H f _ Ho. om ma cmumamm wmo.o I om ma cmummmm woa.o ¢ nw., om ma cmumamw mmm.o . . I 11.. cm mm cmummmm mom 0 o o O O O . ro.H O: ‘U/e JanemEJEd 08$ 86 camoc ooa x In ude: I oamoc oamo. "mm Umumaooamo mmz : CH Houum w mmmum> musHomnm on» mo mmmum>4m mm.m .Hmmnnm.ma mvoo.m mmmv.o mnmo.o mo.o mm.m .Hmmummm.m Hmmh.v mmm¢.o mnmm.o oa.o Hm.m .Hmmumno.a nmam.m voav.o mmmm.o m~.o mo.v .Hmmumnm.o mamm.va hvov.o mmmm.o om.o > no mooam> Aanommv Ac 0mm .Eo\.umv Ammmacoamcmeflov A00\.umv Aw ucmflm3v mmwmq on» now ovum ummnm IN a sowumuu came: ea mo mmcmm x c auwmcoo Icmocoo uouum m .m>< Hmucmsfiummxm panda sofiusHom ooam um umuo80acomomnm mumncmmmfimz m Scum omcamuno mm mcowusaom moflEmHmuommaom How mumqumumm 3m~num3om Huw.v mamma 87 B. Ellis Model (n p a, T ) 0 6 From the viscosity data the zero-shear limiting viscosity, no, was determined as n = %1m 0 yo+o n (4.4-17) Then, the value of 1% was obtained immediately as: T = [1 | at n = l n (4 4-18) 4 12 2 o ' Next, log Inc/n - 1| vs. log lTlZ/Tgl was plotted. The SIOpe is equal to (a - l). A summary of Ellis parameters for polyacrylamide solutions is contained in Table 4.4-2. Calculated viscosities for each polyacrylamide solution with corresponding material parameters are tabulated in Appendix I. As discussed in the previous chapter the Ellis model does not describe normal stress differences. The agreement of the model with the data was characterized by an average deviation of 7.16%. C. Sprigg's Four-parameter Model (no, a, A, e) The Sprigg's model parameters for polyacrylamide solutions are summarized in Table 4.4-3. The value of s was set to be zero according to the Weissenberg hypothesis. Considerable controversy exists regarding the relative magnitudes of the primary and secondary normal stress differences. No attempt was made to measure secondary 88 .H xflocwmmm CH owumasnmu mum uouum mo mwsHm> musHomnm may mo mmmum>¢m mm.m ammuonoa. omm.m moma.m mmom.m mnmm.o mo.o mm.m ammumwmvo. omm.m Hmvm.m osmo.m mnmm.o oa.o mv.o ammummmmo. oam.m hwmv.~ «www.ma mmmm.o mm.o on.¢ Hmmlmmmoo. mma.n manv.m mamm.mv mmmm.o om.o Aauoomv AmEo Ammma Aoom .Eo\.umv Aoo\.umv Am unmfimsv onoc ovum ummcm \mmcmov uc0flmcwswpv coflumuu as no mmcmm we 6 0C .wuflmcwo Icmocou uouum m .m>¢ AmocoEHummxm Ufloam coauoaom ooam um umumeoficomomsm mumncmmmwmz m Eoum cmcwmuno mm uncunugom mowemiauomhaom MOM muoumfimumm madam muvé mqmda .0 ma u can mmHm.o mm o ..o.fi .mmsam> m>onm co comma @mumsam>m mm3 .K .ucmumcoo weflu coaumxmamu mnao .mem.O wfl 09 UQUMHSUHMO mm3 0 m0 05Hm> $399 .0 mo mcflmn mm coxmu was u mo moam> ones 89 hm.a mmmm.o oomm.H mmom.~ mnmm.o mo.o mo.m mm>¢.H mamo.m onwo.n mnmm.o oa.o mn.m vmom.m ao>~.m ~mwn.ma mmmm.o m~.o oa.m mnmw.m onum.m mamm.mw mmmm.o om.o HMO: Aommv Ammmacowmcmefiov Aomm .Eo\.umv A00\.umv Aw ucmflmzv ca woman 0 a coflumuu m monum>4 « a c muflmcoa Icmocoo ofion cofiusaom ooam um umumsoflcomomnm mumncmmmwwz m Eoum omcflmuno .mm mcoflusHom mUHEmHmuommaom How mumumfimumm Howe: unnumGOUIv momflumm muv.v mqmée 90 normal stress difference in this study, but there is evidence that secondary normal stress, (T22 - 133), is considerably smaller than the primary normal stress difference. Both positive and negative values for the coefficient of the secondary normal stress difference have been reported [44, 49, 50, 51, 52, 53]. The Sprigg's four-parameter model provided an excellent fit to the Viscometric data at both low and high shear rates. At intermediate shear rates where viscosity starts to decrease as shear rate increases the fit is somewhat poorer. Calculated viscosities and normal stresses for each of the polyacrylamide solutions are tabulated in Appendix K. The deviation of viscosity calculated from this model with the data was 3.43 percent. Comparisons of model predictions with the data for shear stress relaxation are given in Figure 4.4-11. The agreement is fair because all the parameters were determined from the data on steady shear flow only. This model adequately predicts how the ratio 112(t)/ 112(0) relaxes rapidly as the initial shear rate §O is increased. D. Bird—Carreau Model (no, a1, a2, Al, 12, e) The material parameters for the Bird—Carreau model are summarized in Table 4.4-4. The value of e was assumed to be zero according to the Weissenberg hypothesis. 91 mcoHusHom 0 was moflamawuommaom How wEflu Hmwm .u .m> .mmmuum umonm mumum wommumsb .klplll. Hanv.v madman Auvmap now .u on mm om r _ _ 1? «0f! 0 9 H.03 msmqo. n w coauoanmua Hmnoz.lulwll. on ma mm. m. Mpma O o.a 92 o_m mm o_m o a o O IN. [QC [0. cofluofivmum Hmoozlloll anomm mmom. u > o lum- mumn s cm ad wm. 04 wow NHP ESP 93 ma 0mm .u coHuofiomum Hobo: loll MHMQ 0 pass. n W on as mmm. 94 coauoflomum amoozillYll mumo 0 0mm .u H loom mme. > on ma mmm. 7.m. o.H 95 GW ”ON p.- 0 333668 Hocoslbl fluflfl 0 00m - .u —M a noon who.a I m on ad aoa. — ruo.a 96 _ 00m coauoanmum Hmsosuuoll. mama o .u m p L-H mmm.m u 0» om ad wH. oé 97 --m ”'3‘ 00m Pm coauOflomum HmUOZIIQII muma o H loom mmo.m u 0» on mm mmo. ENS ENS 98 now .u T COfl#OHUmHm HMUOE IIOII mumo e uowm mn.oa u 0» a om md wmo. Fm. are; 99 .wcauuwm Hamnmmw mp pwcfimuno mum3 mumumEmumm Hmooz Hams mvn~.o momm.a maam.o oomm.a mmom.m mnmm.o mo.o mmnm.o whmm.a mamm.o Namo.m osmo.m mnmm.o oa.o mmom.o Hmmm.m Hmam.a Hon~.m momn.ma mmmm.o m~.o mmn~.m mamo.m mmmm.a onmm.m mamm.mv mmmm.o om.o Aommv Aummv Ammma Ammma Aomm .Eo\.umv Aoo\.umv Aw ucmwmzv Icowmsmeflov Icofimcmsflov Q coaumuu 0c auwmcmo Icmucoo m4 HA «5 Ha pagan cowusaow Ooam um MoumEoflcomomcm mumncmmmwmz m Eoum omcfimuno mm mcoflusaom moHEmHmuommaom now mumuoEmumm Homo: smouumuuouflm vlv.v mamds 100 Viscosity and primary normal stress difference were calcu- lated for each polyacrylamide solution. As indicated pre- viously, the Bird-Carreau model is supposed to be an im- provement over the Sprigg's model. The predictions of vis- cosity with the Bird-Carreau model were similar to those of the Sprigg's model; however, the Bird-Carreau model correlated primary normal stress differences much better than the Sprigg's model. Viscosity data were weighted three times as heavily as normal stress data for evaluation of material parameters using a least squares criterion for best fit. Details of the least squares procedure and computer pro- gram used for data analysis are reported elsewhere [109, 110]. The material parameters for the Bird-Carreau model are contained in Table 4.4-5. Calculated viscosities and normal stresses for each polyacrylamide solution with the corresponding material parameters are tabulated in Appen- dix L. The deviation of viscosity calculated from this model with the data was 2.77 percent. 54.5 The Flow of Non-Newtonian Fluids Through Porous Media Equations for the modified Reynolds number based on the result of a hydrodynamic analysis of the capillary model of the packed bed for the power-law, Ellis, and Sprigg's model were developed in Appendix A 101 .muouum mumswm ummma may now mHm>Hmcm nousmsoo mg omcwmuno mums mumuwsmumm Hamm mmaa.o m~m~.H mham.o maam.a mmom.~ mnmm.o mo.o wnma.o mnHm.m mnaw.o mmHH.N onmo.m mnmm.o oa.o mmmm.a mmoo.m HovN.H H>o~.~ «www.ma mmmm.o mN.o mmmm.v mhnq.v Nnmm.a wnnm.m mHNm.mv mmmm.o om.o Aommv Aommv Ammoa Ammma Aomm .Eo\.umv A.oo\.umv Aw unmflmsv ncowmcmeov Isoflmcmsflov a soflumuu 2 : N. l. 0: 3mm“ ammmm anm um umquOficomoosm mumncommflma m Eoum omcwmuno mm msowusaom mwfiEmHmuomaaom How mumumEmumm Hmooz smouumuuouwm mlv.v mqmde 102 with results listed in Table 3-1. The modified Ergun friction factor for these models is: f* = —l§9——+ 1.75 (4.5-1) calc NRe,eff Data from flow experiments for aqueous solutions of poly— acrylamide in packed beds were correlated with the Ergun friction factor. §4.5.1 Experiments Polyacrylamide solutions of four different con- centrations (0.05, 0.10, 0.25, and 0.50 weight percent) were made up and filtered to avoid gel formation, and then poured into the jacketed storage tank maintained at 21°C. Gentle stirring was maintained for about one day preceding the flow experiments. All electrical circuits were switched on for about 5 minutes before runs were made. The schematic diagram of the jacketed flow system which maintained con- stant temperature 21 i 0.5°C is shown in Figure 4.5-1. Nitrogen tank A was used to cause the flow of polymer solution from the storage tank to the packed bed, whereas nitrogen tank B was used as a controlling device to main- tain a constant flow rate through the bed. The pressure transducer was connected into two taps of the column which were 1.5 ft. apart. Various flow rates were obtained Nitrogen Pressure Tank A 103 Polymer Solution N gas x M 2 Polymer Solution Polymer Solution Storage Tank r W Packed Column 7// Pressure Transducer Nitrogen Pressure Tank 8 LJ Recorder To Rotomerer Figure 4.5-1 Schematic Diagram of the Equipment 104 by adjusting valves at the t0p and bottom of the column and measured with the apprOpriate rotometer. Pressure drop for each flow rate was recorded. Details of experi- mental procedures are contained in Appendix C. 54.5.2 Results and Discussion The analysis of the flow experiments was based upon the packed bed friction factor and modified Reynolds number f* = f*(N ) (4.5-2) Re,eff as summarized in the previous chapter for various rheo- logical models of fluids in porous media. Ergun [83] and Carman [69] reported the value of C in the permeability equation for Newtonian fluid to be 150 and 180, respectively. Sadowski [2, 3] used the value of 180 for C for non—Newtonian fluids. ChristOpher and Middleman [4], Gregory and Griskey [6] used the value of 150 for C for non-Newtonian fluids. The form of data with Newtonian fluid (distilled water) exhibited good agreement with Ergun's correlation, i.e., C = 150 for values of NRe < 1.0. The average percent deviation for the friction factor, based on this value, was 4.6% for 33 experimental points. Henceforth, the value of C used for this study was 150. 105 The result of friction factor versus Reynolds number for the Newtonian fluid (distilled water) is plotted in Figure 4.5-2 and tabulated in Appendix M. Results for non-Newtonian fluids are plotted in Figures 4.5-3,4,5, and 6 and are tabulated in Appendices N, O, and P for the power-law, Ellis, and Sprigg's model modifications, re- spectively. The modified Reynolds numbers ranged between 10'5 and 1.0. Wall effects correction was introduced in each calculation with M defined in Eq. (A.3-32). This is shown in Figure 4.5-7. Power-law Model The data for all concentrations showed larger deviations from the theoretical line as seen in Figures 4.5-3,4,5, and 6. The average percent deviation is listed in Table 4.5-1. The friction factor, f*, was consistently too low for the entire range of Reynolds numbers. No trends were observed with respect to Reynolds numbers or fluid concentrations. The departure from the relation f* = ——l§9——-+ 1.75 (4.5—3) NRe,eff was so great as to suggest that it was a result of a failure in the development of theory rather than a result of experimental error. That is, N decreased faster Re,eff with an increasing shear rate than the power-law model predicted. 106 ( D 3 f* = EAR. .2 G 2 I. 1 - e o D GO NRe = (1 - 6) Theoretical 1000~ line of f Experimental Points f* 100“ 10 l I l .01 .1 1.0 10 Figure 4.5-2 f*, Friction Factor vs. NRer Reynolds Number for Newtonian Fluid 107 moon cmxuum nmsounu. on at cmuamum no 33.033 msowsqt momé mo 30:. 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As seen in Table 4.5-1, the percent deviation of experi- mental and calculated friction factors was quite large, 5 but uniform over the Reynolds number range of 10- to 1.0 using the power-law model. Since the Reynolds number is less than 1.0, these deviations are not due to inertial effects. Ellis Model Some improvement over the power-law model on the friction factor correlation was obtained with the Ellis model. The average percent deviation for the Ellis model correlation is listed in Table 4.5-2. Spriggis Model The Sprigg's model modification of the Ergun equation (see Appendix A for the development) yielded the best results. As shown in Figures 4.5-3 through 6, the data scatter about the theoretical line with an average error of 5.7%. No trends were observed with respect to Reynolds numbers or fluid concentrations. These results appear to confirm the utility of the correlation develOped in this study. Sprigg's model predicts the Newtonian behavior at very low shear rates as is characteristic of most real polymer solutions. The average percent devi- ation for the Sprigg's model correlation is listed in Table 4.5-3. 114 .o xHocmmmd CH cm>Hm mum mosHm> musHOmnm mo mmmum>m UHqucuHum 0Hmo a OOH x ume «m I mm cmumHsono was no m 000 H «m Hum w mmwum>.mHu O.mm nm.m m~.O v.v + 0» m.OHu 0.0m Om.n om.O Houum w mucHom Ammum.vv .om scum Aw ucmHo3v mo mmcmm HmucmEHummxm coHumHsonu «m 2H cOHumuucmocoo EdEmez mo nonEsz uouum w momuo>4 coHuoHom M nonsmoamacoz Hmcoz maHHm you coma HmucmEHuomxm m o mumEESO mum.v mqmda .m xHonmm< CH Cm>Hm mum mosHm> ouCHomnm mo mmmum>m oHquCuHud 0Hmo m * OOH x umxm UHmo «m I «w mm UGUMHSUHMO ads) HOHHO w mmwumxfiwm 115 ~.m+ ou H.0H- o.hHH ma.m mcoausaom Had mm.a+ ou m.mH- o.mm ~m.m no.0 a.h+ on H.oan o.mm Hm.m oa.o ~.m+ ou m.ma- o.m~ wo.m mm.o a.m+ 0» ~.HHI o.o~ HH.m om.o uouum w mucaom Comum.qc .wm scum is unmamzv mo omCmm HmoCoEHuomxm CoHumHCono «m CH Cowumuucmocou ECEmez mo umnECZ uouum w ommuo>m COHuCHom M COHuCOHMHooz Hobos momwumm How mono HmuCoEHuomxm mo anmEECm mIm.v mamCB 116 Comparison of Models A complete error analysis was performed at this point to check out the deviations stated above. For each concentration and model the low and high values of shear rate at the wall in the bed were calculated. On the basis and f* for each model of these values error in ”calc calc was calculated. As expected, the Sprigg's model gave the best results followed by the Ellis model. The deviation in f*c 1 for the Ellis model may be due to the imperfect a c curve fitting process. Comparison of the Ellis model with the power-law model gave an interesting fact. The error in n is the same order of magnitude for both calc models but the power-law model has an error in f*calc twice as large as the error in the Ellis model. This shows clearly that there is something wrong with the power-law model. A comparison of the error for each model is listed in Table 4.5-4. Some objections to the use of the power-law model should be pointed out now. Most real polymer solutions approach Newtonian behavior at very low flow rates. The power-law does not predict this behavior. As shown in Appendix J, the power-law predicts an infinite viscosity in the limit of vanishing shear rate. This is a funda- mental failure, and provides for a valid objection to the general applicability of the power-law model. By using Eq. (A.3-32), the bed shear rate at the wall, §w, for TABLE 4.5-4 Summary of Error Analysis 117 Solution Ranges of % % Concentration Experimental Error Error (weight %) (sec’ ) Shear in n in f* Rates in the Beds calc cal A. Sprigg's Model 0.50 1.13- 51.22 1.95 5.11 0.25 2.43-188.0 1.36 6.06 0.10 1.92-297.2 1.69 5.61 0.05 6.16-221.97 2.23 5.83 B. Ellis Model 0.50 1.36- 61.64 3.65 7.96 0.25 2.96-229.88 6.10 6.57 0.10 2.39-37l.26 8.36 11.06 0.05 7.80-280.99 8.30 9.98 C. Power-law Model 0.50 1.15- 52.12 3.70 10.07 0.25 2.51-194.l6 5.99 14.32 0.10 2.02—312.78 7.52 22.87 .05 6.54—253.59 10.29 23.57 118 each experimental point was calculated and compared with Viscometric data. Shear rate at the wall of the hypo- thetical capillary is in the power-law region; but the shear rate at the center of the capillary is zero. The power-law model would be expected to characterize the fluid close to the wall rather well, but not toward the center because of the existence of a limiting viscosity at low shear rate which is not predicted by the power-law model. A more serious problem is that it is an inelastic model. Polyacrylamide does exhibit viscoelastic behavior which is predicted by the Sprigg's model but not by the power-law or Ellis model. An elemental volume of fluid actually experiences continual acceleration and deceler- ation as the fluid moves through irregular interstices between particles. Hence, for flow of non-Newtonian fluids in porous media, we might expect to observe viscoelastic effects which do not show up in the steady state spatially homogeneous flows usually used to establish the rheologi- cal parameters. §4.6 Conclusions Results of the present investigation may be summarized as follows: 1. Material parameters for the power-law,Ellis, Sprigg's, and Bird-Carreau model for aqueous solutions of polyacrylamide were evaluated. 119 The Ellis model adequately represented the fluid. The power-law model represented the fluid well only in the power-law region. Both the power-law and the Ellis models do not account for normal stress differences. The Sprigg's model corre— lated the steady shear experimental data very well, but did not correlate the normal stress difference data well. The Bird-Carreau model correlated the normal stress difference data considerably better than the Sprigg's model and described the steady shear data very well. When the flow rate of the 0.50 percent solution passing through the packed bed was less than 1 cc./min., the flow rate continously decreased. Adsorption of large polymer molecules on the surface of beads and consequent gel formation was observed at very low flow rate and long test run. Polymer adsorption and gel formation re- duced the bed permeability which automatically lowered the flow rate. The Sprigg's model gave the most successful results for values of effective Reynolds numbers less than one. These results appear to confirm the utility of the correlation developed in this study. The error analysis confirmed that there is some objection in the use of the power-law model for flow problems. CHAPTER 5 POLYVINYLPYRROLIDONE SYSTEM Polyvinylpyrrolidone (PVP) is an essentially linear polymerization product of Vinylpyrrolidone. The polymerization is usually carried out in aqueous solution at somewhat elevated temperatures. Hydrogen peroxide is used as catalyst and ammonia or an amine as activator. Of various parameters determining the chain length, the concentration of hydrogen peroxide has the greatest effect [92]. The polymer, when prepared in a certain range of molecular weight, is a valuable blood plasma extender. A plasma extender is a compound which, when infused into the flood stream, keeps a normal volume of liquid in circu- lation by its osmotic effect. §5.l Chemistry Vinylpyrrolidone CH2 —— CH2 is easily CH C = 0 NH2 CH CH NHCH CH NH + 2 2 2 121 block-polymerized in a simple manner with hydrogen peroxide as catalyst. Thirty-five Kg of Vinylpyrrolidone are added to 150 cc of hydrogen peroxide (30%) and heated to 110°C. The polymerization is exothermic and the tem- perature rises to 180 to 190°C. The molten polymer is poured from the kettle, cooled on a plate and milled on a ball mill to a fine, white, somewhat hygroscopic powder. This process gives a more or less strongly dis- colored product due to the high temperature; the product also has an unpleasant odor attributable to decomposition products caused by the rapid initiation of polymerization. Furthermore, the polymer contains up to 10% of monomeric Vinylpyrrolidone, which is probably responsible for the hygroscopicity of the material. Chemical Reactions of Pyrrolidones The introduction of a pyrrolidone ring into a molecule imparts a hydrOphilic character which in some cases may even extend as far as causing solubility in water. By using two moles of butyrolactone per mole of diamine, two pyrrolidone residues can be introduced. For example with diethylenetriamine, CH N - CH CH NHCH CH N 2 2 2 2 1H2 c=o 2 2 2 2 \ / L"u n—‘J O O O '2_+———\ /__. 122 With these compounds, the presence of the amino group offers a possibility of further reaction. Pyrrolidone reacts with concentrated sodium hydroxide solution in acetone to give N-sodium pyrrolidone which reacts with alkyl halides, i.e., epichlorohydrine in alcohol. /O\ :’ "1 CH2 CH2 ClCHZCH-CHZ + N.0H |——)-H o + I I C H OCH CHOHCH N 1 : 2 CH2 c=o cznson 2 5 2 2 . n \ / H I ' N I ' | O I ‘‘‘‘‘‘‘‘‘‘‘ l ‘ N I_ ... __________ I 55.2 Physical Properties Commercial N-vynyl pyrrolidone may contain up to 2 percent oily impurities which are insoluble in water. In small quantities these impurities often cause turbity when diluted with water during the subsequent polymeri- zation. In accordance with the preceding investigation, the commercial production of the Polyvinylpyrrolidone ("Kollidone") is polymerized in a 30 percent solution be- cause the product can still be dried in this concentration. In order to dry the high viscosity solutions of the higher polymeric Polyvinylpyrrolidone, or the more concentrated solutions, drum drying is contemplated. Pure polyvinylpyrrolidones, polymerized by using light as a catalyst, are glass-clear masses. Exposed to air, they gradually absorb water and form highly viscous 123 solutions in water. The softening point is over 100°C. In contrast to the high polymer produced by light polymeri- zation, the block polymer is discolored yellow to brown and is of low viscosity. Solubility Polyvinylpyrrolidone forms clear solutions with water. Highly concentrated warm solutions of alkalies and sodium chloride will cause salting out, but precipitation will not occur in strong acids. The aqueous solutions are completely neutral and very resistant to saponifying agents; however, when boiled with concentrated alkali, they form an insoluble product. Polyvinylpyrrolidones (hereafter called PVP) are soluble and compatible with water and organic solvents. They dissolve easily in alcohols, ketones, tetrahydro- furane, chlorinated hydrocarbons, pyridine, and lactones. They swell in esters and aromatic hydrocarbons and are insoluble in ether and aliphatic hydrocarbons. The addition of ligroin to monomeric Vinylpyrrolidone gives a turbity or precipitate if the monomer has started to Polymerize in storage. The monomeric Vinylpyrrolidone is 1miscible with all organic solvents. .A.Eelications of Polyvinylpyrrolidone Polyvinylpyrrolidones (PVP) may be used for various leproses depending in large part upon the degree of 124 polymerization. They are of interest as gums and glues, raw materials for adhesives, substitutes for animal glues, bonding agents in the film, reproduction, and coating industries, and as thickeners for emulsions, solutions, and for soaps and cosmetic preparations. They may be used as sizing agents for papers, fibers, and fabrics; and are reported to bring out a deeper color tone, par- ticularly in combination with basic dyes. PVP films can be rendered water-insoluble by reaction with diisocyanates. A unique and important application for the low viscosity PVP (molecular weight approximately 40,000) was as a blood plasma substitute, where it was called "Periston." To prepare the Periston solution, PVP was dissolved in water to give a 20 percent solution, filtered and sterilized at 120°C. The manufacturing formula is as follows [119]: Nacl : 800 gr. Hcl (1710 cc) : 1728.8 gr. Kcl : 42 gr. NaHCO3 : 168.0 gr. Cac12-6H20 : 50 gr. PVP (20% solution) : 12,500 cc. Mgc12-6H20 : 0.5 gr. Water (distilled) : 101.3 Kg. The solution prepared from the above recipe is filtered and sterilized in 100, 250, and 500 cc. ampoules at 105°C for one hour. The ampoules are stored at 30 to 35°C for three weeks to permit detection of any separation of material from solution. 125 "Periston" is used as a blood plasma substitute to replace loss of blood in the following cases: 1. Acute loss of blood (lesions, Operations, child birth, etc.). 2. Shock due to trauma, Operations of narcosis. 3. Thickening of the blood resulting from increased loss of liquid due to diarrhea, vomiting, or protOplasmic collapse. "Periston,' which can only be used intravenously, has no toxic effects, and may be used simultaneously with other water-soluble pharmaceuticals. §5.3 Rheology For most systems in steady shearing flow the viscosity depends on the shear rate in a rather character- istic manner. At sufficiently low shear rates the vis— cosity is independent of the shear rate (the Newtonian region); however, within the same critical range of shear rates the viscosity begins to decrease as the shear rate is increased to still higher values, and apparently if the shear rate can be increased sufficiently the vis- cosity becomes constant, a limiting viscosity at infinite- shear rate. The limiting viscosities both at zero- and infinite-shear rate and the critical shear rate region may change by many orders of magnitude from one system to 126 another depending on the nature of polymer, its molecular weight, the solvent, and the concentration. Meter's four-parameter model was selected to characterize the rheological properties of aqueous solu- tions of polyvinylpyrrolidone because these solutions were purely viscous and have both upper and lower limit- ing viscosities. Meter's model and material functions were fully discussed in Chapter 2. §5.3.l Experiment Fluids investigated were aqueous solutions of PVP of four different concentrations, 0.5, 1.0, 3.0, and 4.0 weight percent. PVP was manufactured by Badische Anilin und Soda Fabrik, Ludwigshafen, Germany. Concentrations were converted to the units of gr./100 cc. of solution by assuming additivity of volumes. Viscosity and primary normal stress difference measurements of the four PVP solutions were made over the shear rate range of 0.02689 to 1076.0 sec.l with a Weissen- berg rheogoniometer, a commercial type cone-and-plate viscometer manufactured by Farol Research Engineers Ltd., England. A platen diameter of 10 cm. and a cone of angle 2.0083° was used with a 1/16" torsion bar and a 1-/16" normal force spring. Room temperature was care- fT’ully adjusted to 21°C prior to taking measurements so 51:3 to maintain temperatures of samples and reservoir 127 platen arrangement which is shown in Figure 4.4—1 con- stant at 21 1 0.5°C. The Operation and run procedure of the Weissenberg rheogoniometer is contained in Appendix B. 55.3.2 Results and Discussion Four material parameters of Meter's model (no, n a, Tm) were determined from shear stress versus shear 00' rate data for PVP. Shear stress versus shear rate values for PVP solutions are tabulated in Appendix R and typical results are plotted in Figures 5.3-1 and 5.3-2 as T12 vs. y, n vs. 9, respectively. The zero-shear limiting viscosity no and infinite- shear limiting viscosity n0° were obtained directly from Figure 5.3-2 and listed in Table 5.3-1. The value of Tm was obtained as the value of the shear stress at n=§m uo mcoHuaHom msoosum usom new uoH>ocom ovum noOCmInnouum umosm HIn.m ousmwm .Hcoon .w oucm unocm .OOOH .OOH .OH O.H H. .44‘41 1 1‘ 1. ~41111411‘ 1 ‘111141 4 1 ~141<11 d1 1 .14444‘ H 1 OH m>m om.O m>m 0O.H m>m 50.n m>m 0O.v C) II 4| II OH HOH sz/euxp ’ZIl 880138 atoqs 129 m>m we mCoHusHom msoosvt noon new qumoomH> CMHCOustIcoz ~Im.m unamHm loom .> mama umonm a MS «3 1: OS 73 «IS 3 1...... 1...... . _.... .. ... ... 11.1.1 «I I . . 1 .73 - 03 CE 3.6. mi Séd . E3 8;”. HE 840 m 3 sea ma/Jb 'u Aztecs-1A 130 would certainly be true in the case of a fluid which forms a gel structure when not in motion; then there is no zero- shear limiting viscosity and a Tm would be impossible to determine. In this study a and Tm were obtained by using a computer program for determining the best fit of the shear stress versus shear rate data for aqueous solutions of PVP. Values of the material parameters for Meter's model for PVP solutions are listed in Table 5.3-1. Meter's model provided an excellent fit to the Viscometric data as can be seen in Figure 5.3-3 contain- ing calculated viscosities as point for each PVP solution with the determined material parameters. The solid line represents the best line through the data. The main deviation between the data and correlation is in the instability region where viscosity starts to decrease with increasing shear rate and in the high shear rate region. The average absolute percentage deviation be- tween values predicted from the model and experiment for the 98 data points was 2.10 percent. The parameters listed in Table 5.3-1 show that both no and 710° decrease with decreasing polymer concen- tration, tending to the Newtonian viscosity of the sol- Vewrt at infinite dilution. The parameter (1 also de- crseases with decreasing concentration, whereas Tm 6appears to approach a definite value, characteristic of the polymer, as infinite dilution is approached. 131 gm MO mEOflUSHOM MDOOSU‘ HON >UHmoomH> acoummmm can no moch> counHCOHmo can HnuCOEHuwmxm mo COnHuumfioo mum.m ounuHm O noon .> ovum MumCm OH NI OH m>m am.o m>m wO.H m>m vo.m gm wo.‘ COHuquonm Hove: I004. H II... l..1 NIOH OH OH OH can mD/JS ’u Karsoosrn .0 vaCmmam CH OwumHsnmu nonno mo mosHm> ouCHOmnm mo mmmuo>¢m 132 mm.m O>OHIFOOH.O mmnm.o VHwO.m mmH0.0 mvmo.o Onmm.o m.O mm.H OhOHIOFOH.O mvmm.o mmOH.~ mmmo.o mmON.O ommm.o O.H mn.H OnOHInmvo.O vam.O mmHm.m omm0.0 vmmn.o FNOO.H O.m mh.m OnOHImONO.O vmvn.o man.~ vmho.o Omnm.H evoo.H O.v Hmo C AHIoomv Hoom EO\.umv AmOOH Room EO\.umv Auom EO\.umv AOO\.HOO Aw .u3v CH comm umonw E ICOHmCmEHOV CoHumnu uouum mo omcmm H o 8c 0: ICOOCOU a HmquEHuomxm CoHuCHOm oon um uouoEOHComomCm muonCommHoz m Eoum OOCHmunO mm mCOHuCHOm m>m Com mnoumemumm HwOoz umuoz HIm.m Manda 133 §5.4 The Flow of Non-Newtonian Fluids Through Porous Media The equation for the modified Reynolds number based on the result of a hydrodynamic analysis of the capillary model of the packed bed for Meter's model was developed in Appendix A with the following result: D G T * a-l NRe eff = M(1€e? 1 + d:3(f0) I no m nm T * a-l T * a-l o m + (::)2(Tw*)20-2 2 {+ 4 (T *)a-l n T c+l 3a+1 T O m D 3 * = 292. _E. 51.. - fexpt ( 2><1_€) (5.4 1) MGO The modified Ergun friction factor for Meter's model is: * = —1—§—°—— + 1.75 (5.4-2) calc NRe,eff The pressure drOp-flow rate data for PVP solutions in packed beds were correlated with the Ergun friction factor. 134 55.4.1 Experiment Polyvinylpyrrolidone solutions of four different concentrations (0.50, 1.0, 3.0, and 4.0 weight percent) were made up and filtered to avoid gel formation, and then poured into the jacketed storage tank maintained at 21°C. Gentle stirring was maintained for about one day preceding the flow experiments. All electrical circuits were switched on for about 5 minutes before runs were made. The schematic diagram Of the jacketed flow system which maintained constant temperature 21 : 0.5°C is shown in Figure 4.5-1. Nitrogen tank A was used to cause the flow of polymer solution from the storage tank to the packed beds, whereas nitrogen tank B was used as a controlling device to maintain a constant flow rate through the bed. The pressure trans- ducer was connected into two taps of the column which were 1.5 ft. apart. Various flow rates were Obtained by adjusting the valves at the top and bottom of the column and measured with the appropriate rotometer. Pressure drop for each flow rate was recorded. Details of experi- mental procedure are contained in Appendix C. §5.4.2 Results and Discussion The analysis of the flow experiment is based upon the friction factor-Reynolds number analysis as developed in the previous section. The flow data are presented in terms of 135 f* = f*(N ) (5.4-1) Re,eff where f* is the friction factor for flow through a packed bed and NRe,eff is an effective Reynolds number for the flow of non-Newtonian fluid through a packed bed. The modified friction factor versus modified Reynolds number for aqueous solutions of PVP are plotted in Figures 5.4-1,2,3,4 and are tabulated in Appendix R. The Reynolds number ranged 10"2 to 102. Wall effect correction was introduced in each calculation with M defined in Eq. (A.3-32). The data for all concentrations agreed with the Ergun equation at the value of effective Reynolds numbers less than one. But the friction factor, f*, was con- sistently too high for N > 1.0. The departure from Re,eff the relation f* = ——1-§9—— + 1.75 (5.4-2) NRe,eff was so great as to suggest that it was a result of a failure in the theoretical development rather than a result of experimental error. That is, either the effective coefficient of viscosity neff' did not de- crease with an increasing rate of shear as rapidly as Meter's model would predictcnrthere is a possibility that the inertial terms have not been properly accounted for. 136 nmsouca mCOHusHom m>m OO. O uo onm Maw COHucHouuoo ouum onhImouO unannoum 2.1.1. :1 P H ton v + H N 059.3: upon ooxuum HIv.m OHCOHm H i H fiumrl H+a+ «+8 8 o. 3:~ 4. + H on an .3 om: ~39: .3953. 8303mm rHla f. H15 C Hla C NOH H OOH HIOH NIOH —|1«<< 1 4 4 d21m aO.m uo 30Hm HON coHuunuuou mumm 30Hhtmoun ousauwum Nuv.m ouslo H84 P d+dn + H+d 3w v N rH'dr. P H H 4 av wk + 5w nHHw+n+a up Mm . Iv 9.8+ 028-3: 2 up a: 3p N v 3» 8c 3p v H am no .uu as: uonasz noHocaia obHuoouuu 72 . a for O 75. In. mOH NOH HOH OOH . HIOH NIOH 1111114 4 1 4<<1#4d 4 4 d<1qqqd d d du<<-W*4 4 .111414 * fi 0°“ .‘AALI HOH + L 4 u 4NOH mquom HuucoEHuomxm O A A mch HQUHuouoona .Ill 1MOH ] 0.; 1033.3 notacyzg [2%] [a] {5; 138 avom umxoum nosoune mcoHusHom m>m wO.H no 30Hm now coHunHounou ovum 30Hmlmoua musmmwum muv.m ousmHm aw H+dn + H+d 3p v w Hide MW Ma. + ME+WM U. mm - -alp nl+.w+ ocaéx o. 3p a: :p u v 3w a: a» v H ou ma .mu 0:: nonasz nuHoaaom o>Huuouuu NIUN a N Hld c . . Hid. HIUC MOH NOH HOH OOH H|OH NIOH 14<<+1 1 4 q<<1<<411 < 1.4114114 1 H441W.1< 1 4 ~144¢¢1 i d OOH L L L - HOH mucHom HmucofiHuomxw . ocHH HQOHumHomna IIII . O . 1 NOH A U h 3 n L A O 0.; 103393 uoxqoyxa 1.39 momm vmxowm monoune mcoHusHom m>m am.O uo onh now GOHuuHouuoo mumm 3onimouo unduaoum civ.m ousuHm 3P H+dm + H+d 3» v a Hid c a O a 0 - L. ..m + “E + B J ...; - ..p 3 05-2: .. 3p 3: 3p H v :p 8c» 39 v + H o m .uu «m2 ugnasz ncHocxum o>Huuauuu NuaN . N Hun . Hid . Hun . o o . OH OH mOH NOH HOH O Hi 111411 a 1 #1111111 1 1 H+11fi11 1 1 11‘111 1 1 fi41111 1 1 1 o J J mucHom HuucoEHuoaxu . 1 . . 23 103885 ..I L. H O O O L L h N A L OH OH OH OH ‘.3 zouava vatzoxza 140 The flow behavior of PVP solutions in packed beds depends upon the bed geometry since it is this geometry which conditions the flow patterns of the solution. A porous medium may be visualized as made up of innumerable flow constrictions and expansions with interconnecting curved pore channels. An elemental volume of solution actually involves continual accelerations and decelerations as it proceeds through irregular interstices between par- ticles in the bed. This alternating, or oscillating, behavior becomes more manifest as the average solution velocity increases. It is this locally unsteady-state phenomenon which leads to the observed deviation. Hence, for flow of non-Newtonian fluids in porous media, we might expect to observe inertial effects and surface effects which do not show up in the steady-state spatially homogeneous flows usually used to establish rheological parameters. The experiment would indicate that an addi- tional parameter may be necessary to correlate packed bed flow data. By method of dimensional analysis, a new dimensionless group, for flow in tubes, arises, viz., [116] DVp VA (-fi-)(Tr) Similarly, the new dimensionless group, for packed bed, would be DpGo Go A ‘M<1-e>no"oop’ 141 As pointed out by Meter [39], the relaxation time con- stant A can be obtained from model parameters as l = nO/Tm with unit of time. Since the observed modified Reynolds number for a given friction factor is too large, the modified Reynolds number (Eq. 5.4-l) may be expressed as: D Go 4 Tw* a-l N = ———E———— 1 +-——-v——4 Re,eff M(l--e)no a+3 'Hn D GO conO C2 ‘ C1 High—n; ‘ p—o—T— (5-4‘3’ pm where c1 and c2 are constants to be determined from the packed bed experimental data. For each PVP solution, c2 was 1.36, whereas cl was 0.43 for 4.0% solution, 1.3 for 3.0% solution, 3.5 for 1.0% solution; and 4.0 for 0.5% solution. The agreement was excellent. The average absolute percent deviation between the calculated value of the friction factor (Eq. 5.4-2) and experimental value (Eq. 5.4-1) was 10.8% for 87 experimental points repre- senting four fluids of different concentration. This was in contrast to the average percent deviation of 60.4% for those same data points without the correction. 142 This is shown in Figure 5.4-5 and the average percent deviation is listed in Table 5.4-1. 55.5 Conclusions Results of the present investigation may be summarized as follows: 1. Material parameters for Meter's model for aqueous solutions of PVP were evaluated. Aqueous solu- tions of PVP did not exhibit viscoelastic be- havior. Meter's model correlated the steady shear experimental data very well over the entire range of shear rate. The application of the Meter's model to the capillary model of a porous media led to a generalized Darcy's law. This development led to a successful correlation of the flow rate data for polymer solutions of low and medium molecular weights for the value of effective Reynolds numbers less than one. For the high flow rates where the inertial effects become significant, the modified Ergun equation correlated the friction factor versus the modified Reynolds number well. Exten- sive experimental works are desired for the generalization of this correlation thereby 1H3 mpwm cmxumm nmoouna 50H mcoHusHom m>m mo onm MOM coHumHmuuou mumm onmnaouo muommmum mav m on .h Hnomm .+ mumm ummnm NS H3 . o3 HuoH — 4 1 111111 1 1 1‘111 1 1 .—‘ 11 a 1 d1 1 1 q m>m mm.O m>m «O.H m>m «O.m m>m «O.v mch HooHuouoona. [001- 111.1 1 OH OH OH '.J 10333;] UO'FQOIIJ 144 .m xHUcmmmd CH chMHsnmu mum mmsHm> musHomnm mo mmmum>m oHumficuHud 0Hmo «w OOH x uQXm .1 000 m i H 1 m co comma cmumHsono mmz coHuMH>m© w mommm>am mv.HN+ 0“ HH.mmi hm bh.OH mCOHuDHOm HH< mw.HN+ OH HH.0MI NN Nm.mH . m.o mm.NH+ 0» mm.hNi mH Hm.HH O.H hm.mH+ 0p Hm.mHi mm mm.m o.m N¢.NH+ OH «m.hi mm Hm.m 0.1 cmwpwwmwm . Hmpmwmwmmmxm cwwmwwwwmuwmwmmwmmw cwmpwmwwwwwoo EdEmez mo umnEoz . . coHusHom mcoHusHom m>m now mumo HmucmEHummxm mo wumEEdm Hiq.m Manda 145 setting up the procedure for use of this correlation to confirm the utility of the correlation developed in this study. CHAPTER 6 POLYMETHYLCELLULOSE SYSTEM The use of natural products to thicken fluids is an ancient art. These products from nature, however, can often be limited in their properties and capabilities. Nature has also given man cellulose, a substance existing abundantly in cotton and trees, that can be modified in many ways to provide new areas of utility based on unique and unusual combinations of properties. The chemist has discovered that the reaction of cellulose materials with caustic soda and appropriate organic chemical produces substances that not only thicken but do more. They bind, suspend, emulsify, stabilize. They act as nonionic surfactants, and in water solutions they gel on the application of heat and liquefy on cool- ing. 56.1 Chemistry Polymethylcellulose is manufactured and provided under trademark Methocel by the Dow Chemical Company 146 147 (Midland, Michigan). Methocel products are derived from and have the polymeric backbone of cellulose, a natural carbohydrate that contains a basic repeating structure of anhydroglucose units. The basic structure for Methocel (methylcellulose) is as follows: H OH HO 3 H H CI'IZOCH3 Cellulose fibers, obtained from cotton linters or wood pulp, are swelled by a caustic soda solution to produce alkali cellulose which is treated with methyl chloride, yielding the methyl ether of cellulose. The fibrous re- action product is purified and ground to a fine, uniform powder or granule, a Methocel MC product. The number of substituent groups on the ring determines the properties of the various Methocel products. Methocel contains 27.5 to 31.5% methoxyl or a methoxyl D.S.l of 1.64 to 1.92, this range yielding maximum water solubility. A lower degree of substitution gives pro- ducts soluble only in alkali while higher degrees of substitution produce methlcellulose products soluble only in organic solvents. 10.8. means degree of substitution. 148 56.2 Physical Properties Appearance Methocel products are produced as white, odorless, tasteless powders or granules. It is available in several viscosity designations, specified as the viscosity of 2% aqueous solution of a particular molecular weight material at 20°C (see Figure 6.2—1 and Table 6.2-1). The apparent density is in the range of 0.4-0.6 gr./cc., the ignition temperature is 360°C (680°F) and the relative flammability in a furnace at 700°C is 90+. Moisture Absorption Methocel sealed in its original shipping container absorbs scarcely any atmospheric moisture; once the con- tainer is opened, however, Methocel picks up moisture from the air. Allowance must be made when exposed Methocel is weighed since a portion of the total weight may be water; this weight must be correlated for moisture content to assure using the proper weight of Methocel to give the viscosity desired. For this reason, Opened bags should be tightly resealed. Typical moisture absorption is shown in Figure 6.2-2. thsical Characteristics of Aqueous Solutions Methocel products are water-soluble polymers which form aqueous dispersions by swelling and by 20°C Viscosity, cps, lk9 1'00 1000‘ 00 100 2 10 100.1 101 1'0 r r I T T ‘T 0 l 2 3 4 5 6 Methocel, % Figure 6.2-1 Viscosity of Polymethylcellulose of Various Molecular Weights TABLE 6 . 2-1 150 Viscosities of Methylcellulose of Various Molecular Weights Viscosity Intrinsic Number gumber . . verage Grade, 2% Viscos1ty Average M l 1 20C DP w°.°§: 3r (cps) (dl/g.) n egg n 10. 1.40 70. 13,000. 40. 2.05 110. 20,000. 100. 2.65 140. 26,000. 400. 3.90 220. 41,000. 1,500. 5.70 340. 63,000. 4,000. 7.50 460. 86,000. 8,000. 9.30 580. 110,000. 15,000. 11.00 650. 120,000. 19,000. 12.00 750. 140,000. Doom um czocm muHcHfism m>Humem um muoovoum Hmoonumz mo coHuQMOmn< musumHoz HMUHmmB mlm.m musmHm mason .mEHB 151 4.4mm oem mmH omH mm we we «N we 11\I\||| 1H1 .2; \ luI|I|I||||||iIIIII|||I||III \‘l\ 1\\\\\\J1\\\\|\. mmn \rl‘ll.‘ . 1\\\\\\\ mOOH OH ON om ov om ' 311831103 GIHQS TOW % 152 successive hydration of their structural layers. Because of the swelling mechanism, there is no sharp solubility limit. Methocel products have the unique and valuable property of cold-water solubility but they are insoluble in hot water. The water solubility of all Methocel pro- ducts in cold water is limited only by the range of viscosity a manufacturer is mechanically equipped to handle. (1) PrOperties of Aqueous Solutions of Methocel Products Specific gravity, 20°C/4°C (all types): 1%: 1.0012 5%: 1.0117 10%: 1.0245 Refractive Index (2%, all types): 1.336 Partial Specific Volume (Methocel MC 4000 cps): 0.725 cc./gr. PH (all types): Neutral Freezing Point (2%, all types): 0.0°C Surface Tension (below 500 cps at 25°C): 47-53 dynes/cm. 56.3 Solution Preparation The principle underlying the methods of solu— bilization of Methocel is to disperse it in water before 153 attempting to dissolve it. Such dispersion prevents lumping caused by the formation of a gelatinous membrane on the particle. Methocel in powder form is blended into water by first mixing it thoroughly with 1/5 to 1/3 the required amount of water as hot water (BO-90°C). Mixing should continue to assure that all particles are thoroughly wetted. The remainder of the water should then be added as cold water or even ice, if desired. Stir the mixture until smooth. For maximum clarity and reproducible viscosities the solution of Methocel should be cooled to a range from 0 to 5°C, for 20 to 40 minutes. Since Methocel products disperse well in water above 80°C, the preliminary use of hot water assures wetting all portions of the particle prior to solution in cold water. If cold water is mixed directly with the powder, it creates a gelatinuous membrane on the outside of the particles which causes lumping and slow diffusion of water into the interior of the particles. It is very important, however, to have adequate cooling after wetting with hot water to insure complete dissolution of Methocel. §6.4 Rheology For most systems in steady shearing flow the viscosity depends on shear rate in a rather characteristic manner. Any prOposed rheological model should represent 154 the actual behavior of a fluid with accuracy, convenience, and simplicity. Different models may be necessary to describe different fluids, or even the same fluid under different conditions. The best relationship for a given fluid is not necessarily known until an experiment is made on the fluid to relate Tij and eij' The physical behavior of fluids with a yield stress is usually explained in terms of a three-dimensional internal structure which is capable of preventing movement for values of shear stress less than the yield value. For the shear stress greater than the yield value the internal structure collapses completely, allowing shearing movement to occur. The internal structure is considered to be re- formed virtually instantaneously when the shear stress becomes less than the yield value. The Herschel-Bulkley three-parameter model was selected to characterize the rheological prOperties of aqueous solutions of polymethylcellulose because these solutions had yield stresses with non-linear flow curves. Herschel-Bulkley model parameters and material functions were fully described in Chapter 2. [— .13 U1“ _ e:e 2 _ Tij - Ty — |:lo(-T) e.. (2.6 6) 155 §6.4.1 Experiment Aqueous solutions of polymethylcellulose of molecular weight of 18,000 (25 cps material called PMC 25) of concentrations 0.3 and 0.5 weight percent and molecular weight of 41,000 (400 cps material called PMC 400) of con- centrations 0.3 and 0.5 weight percent were investigated. Concentrations were converted to the units of gr./100 cc. of solution by assuming additivity of volumes. Viscosity and primary normal stress difference measurements of the four PMC solutions were made over the shear rate range of 1.076 to 851.0 sec-l with a Weissenberg rheogoniometer, a commercial type cone-and—plate visco- meter manufactured by Farol Research Engineers Ltd., England. A platen diameter of 10 cm. and a cone of angle 2.0083° was used with a 1/16" torsion bar and a 1/16" normal force spring. Room temperature was carefully adjusted to 21°C prior to taking measurements so as to maintain the temperatures of 21 : 0.3°C of samples and reservoir platen arrangement which was shown in Figure 4.4-1. The Operation and run procedure of the Weissenberg rheogoniometer is contained in Appendix B. 56.4.2 Results and Discussion Three material parameters of the Herschel-Bulkley model (no, m, Ty) were determined from shear stress versus 156 shear rate data for aqueous solutions of PMC. Shear stress versus shear rate values for PMC solutions are tabulated in Appendix S and typical results are plotted in Figures 6.4-1 and 6.4-2 as T12 vs. §, n vs. §, respectively. The yield stress, Ty, of each solution was ob- tained from the intercept of Figure 6.4-2. Parameters m and no characterize the slope and intercept of a logarith- mic plot of § vs. 112 - Ty. In this study parameters m and “0 were obtained by trial-and-error until the best fit of the experimental curve was obtained. The values of the parameters thus obtained are presented in Table 6.4-1. The Herschel-Bulkley model provided an excellent fit to the Viscometric data. Calculated viscosities for each PMC solution with these material parameters are shown in Figure 6.4-3. The solid line represents the best line through the data. The average absolute percentage devi— ation between the values predicted from the model and experiment for the 72 data points was 1.71 percent. All the parameters listed in Table 6.4-1 decreased with de- creasing polymer concentration and molecular weight. Values of primary normal stress differences were small and on the order of 150 dyne/cmz. The error of measurements of the normal stress difference determined with a Weissenberg rheogoniometer is about 20% or greater for normal stress differences of 500 dyne/cm2 or less. 157 om0asaamoaanumahaom mo mc0wusaom msoosv< usom new uofl>wnmm mama unmnmnmmwuum umonw Hsv.m wusmwm ~-uou .w oemm amorm MB NB _2 o2 TS 141144.< . 4 _..11... TJ 44.4J41 4 « _..q.14J 4 dq44fi:q 4 4 ~IO~ 3 83.35 I . mN cinema 4 A H w. 2:. 02.1.35 0 H n . J 2:. o§m$md 4 1 o2 m 0 i i i I. . ”I‘M L P (A A u 3 . / . D 1 m2 1.2 A U L S Viscosity, gr/cm sec 158 -— Model prediction r 0. 5% PMC 400 0. 3% PMC 400 0. 5% PMC 25 IDOD 0. 5% PMC ZS 10 A A A L A L441 A A L k A A AA] A L A A A A 101 10 10 -1 10 Shear rate 9. sec Figure 6.4-2 Non-Newtonian Viscosity for Aqueous Solutions of Polymethylcellulose 159 .m xflocmmmd cw boundsnmu uouum mo modam> wusHoQO mo monum>¢m m.a Hmmnmno.a mH.o vmom.a mmo.o vnmm.o m.o ~.N Hmmamno.a ~n.o mnmm.a Nmo.o mnmm.o m.o mm 02m m.H ammumno.a om.m mmmw.a mv.o vumm.o m.o v.a ammumno.a mm.m mmmn.a om.a ommm.o m.o oov 02m oamo Adlommvw “Noon Ammma AHIENowm Aoo\.umv Am .uzv c mumm nmmcm EO\umv ucofimcmeflov EEU\Ean coHumuu C“ no mmcmm we E o: wuwmcmo ucmocou uouum w Hmucmfiflummxm owsam coausaom m Eoum ooam um umumEOHcoqomcm mumncommwmz omcwmuno mm mcoausHom 02m How mumumemumm Hobo: moaxasmuawzomumm Huv.m mamda 10 _ Viscosity, gr/cm sec .01 1.0 160 _. Model prediction C) 0.5% PMC 400 0 0.3% PMC 400 A 0.5% PMC 25 I 0.5% PMC 25 A L AAAAAAI J_ A LALA‘LI I 10.0 100.0 Shear Rate 7. sec- Figure 6.4-3 Comparison of Experimental and Calculated Values of the Apparent Viscosity for Aqueous Solutions of Poly- methylcellulose. 161 The normal stress difference data for PMC solutions were not analyzed further because of inaccuracy in the measure- ments. The stress relaxed instantaneously up to the point where yield stresses were prevailing; therefore, a relax- ation time spectrum could not be obtained. Shear rates less than 1.076 sec.1 were not repro- ducible. An explanation for this nonreproducible was postulated by Meister [117] as follows. Two chains of polymer are tightly linked around each other by bending back on themselves in a short-range contour. This means polymer molecules are entangled. When shear is applied the entanglements start to disentangle and shift to another preferred state. After shear is removed the material will revert to the entangled state, only after a long time. 56.5 The Flow of Non-Newtonian Fluids Through Porous Media The equation for the modified Reynolds number based on the capillary model of the packed bed for the Herschel-Bulkley fluid model was deve10ped in Appendix A with the following result: 162 3 I"T * 2 worm+ r* m+1(T‘”-1) N = PO} 0) _1 y Re,eff M(l-€)“0(Tw*)4t Ty m + 3 T * w + 2(TY l) + l m + 2 m + 1 A2 I) e3 f* = p _2 (605-1) expt MG 2 I. l - e o The modified Ergun friction factor for the Herschel- Bulkley model is: * = __l§9__.+ 1,75 (6.5-2) calc NRe,eff Data from flow experiments for PMC solutions in packed beds were correlated with the Ergun friction factor. §6.5.l Experiment Polymethylcellulose solutions of 0.3 and 0.5 weight percent for each molecular weight of 18,000 and 41,000 were made up and filtered to avoid gel formation. Gentle stirring was maintained in the constant temperature (21°C) storage tank for about one day preceding the flow experiments. All electrical circuits were switched on for about 5 minutes before runs were made. The schematic diagram of the jacketed flow system was shown in Figure 4.5-1. Nitrogen tank A was used to cause the 163 flow of polymer solution from the storage tank to the packed beds, whereas nitrogen tank B was used as a con- trolling device to maintain a constant flow rate through the bed. The pressure transducer was connected into two taps of the column which were 1.5 ft. apart. Various flow rates were obtained by adjusting the valves at the top and the bottom of the column and measured with the apprOpriate rotometer. Pressure drop for each flow rate was recorded. Details of experimental procedure are contained in Appendix C. 56.5.2 Results and Discussion The data of the flow experiments were correlated with the previously developed friction factor-Reynolds number relationship for the Herschel-Bulkley model, Eq. 6.5-2. The modified friction factor versus modified Reynolds number for aqueous solutions of PMC was plotted in Figures 6.5-l,2,3,4,5 and tabulated in Appendix T. 3 to 10°. Wall The Reynolds numbers ranged from 10- effect correction was introduced in each calculation with M defined in Eq. A.3-32. As shown in Figures 6.5-1,2,3,4, and 5, the data scattered about the expected line with an average devi- ation of 8.53 percent. This is excellent. No effects were observed on the correlation due to various 161+ noon usxuam announe 25.330...“ ca... 98 310 we .5lo sou cauusaouuoo ovum 33.7.3.3 0.330.: Hnmé 0.393 A a .3303... is «f- a w . . . 4|. 1 a . Ts: .-..Llwwlojl.... a...§.§2§.§:=. _a-Jflu H-|W E .»A?apoov _ A. 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H+E n+5 n+6 A a». «A. we : v .uuo.om . a + p. .514. w, ww.b fig n.JMw.nllJfllAWAWIII z nonssz mcaocxum o>auowuum _H - -.N . - Inv_ .a . .. m.e P o a v . ., ...." 1 .: 0 ca HOH ooH H-o~ N- H m- ca 1141: d 4 #1141141 1 1 44141111 j 4 ququfi 4 d a . o . 80.33....a Evan-o0 A d o . 333.:0 334:9. A A used Hauwuouoona II. Inca L «ca A A A ad I 'uJ 1033'! 00133113 0 ZS" 3W5 167 30m 6333 533:9 unadusHom mm 02m an.o mo 30am How coauoaouuoo oumm 3odhudouo.ousnuoum vim.w ounuwm H+E N+E. M+§ a flu. VAO3PvOJ S NH..0¢ 1m. + . .«J . . s1. : - lyl jlu . z .35... 8355. 33...: a n u.~ a n .p. 1:” F «+5 p u o v . 3P." 3P» ... C r I .. N I HS c3 73 «-3 T2 3 .444d1 4 1 s .114111 4 1 441114111 1 1j1<11 4 J N . m . u . so 83 .. a .8 3. u u a o . 0 :0 name. I an 8 vm.« I n . 093 HQOHHOHOGQB .II on ILA-AA O.H A A: ' .3 10339.: "013311.! 168 momm owxcmm :msouce wmoHsHHmoaxnuoexaom uo mcoflu5Hom wsomsqc mo 30Hm new cOHumeuuou mumm 30kudouo ousmmmum mum.w musmflh > wimtoEVAVE :.E N+E n+5 . A. n + >5 + N... ~+E: 3A.. >boUmQ :.l3 3 : - Ill 4 TE v N 3 A. A. nobEsZ avgoc>om wort—52 ... no. _o~ com ~n0~ ~-o_ muo_ ‘ ‘1 1 1 4 4 u 4 u . No— ‘Jope; uoTiaug ) _Ll... M S 0 20%! Eva Main CI 3'1 3 E ( .2 m... 055 .3 .o I 3 8.3 .E .o O . A 2:. 02m .1: .o 1 . 2:. 02m e..... .o . < A 0:: 3030.393. ..Il. .m om 169 concentrations. The average percent deviation for each concentration are listed in Table 6.5-1. §6.6 Conclusions The results of the present investigation may be summarized as follows: 1. The material parameters for the Herschel—Bulkley model for aqueous solutions of PMC were evaluated. The Herschel—Bulkley model correlated the steady shear experimental data very well over the entire range of shear rate. The data of primary normal stress difference of aqueous solutions of PMC were not reliable because of experimental error. The presence of a yield stress prevented measurement of the relaxation time spectrum. The modified friction factor-modified Reynolds number correlation was developed for the Herschel- Bulkley rheological model. Packed bed flow data for aqueous solutions of PMC were well corre- lated for effective Reynolds numbers less than one. This result appears to confirm the utility of the correlation developed in this study. It appears that an improvement could be made on the capillary model for the packed bed. There seems to be an upward deviation from the Ergun 170 .9 xflbcmmmm CH cm>flm mum modam> ousHombm mo mmmum>m oaumenuflud oamo am ooa x ummm ¥ MI oamo «H mm Umumasoamo mm3 uouum m mmmum>¢m vm.ma+ ou mm.mmu mma mm.m mcofiusaom Haé mm.aa+ 0» m¢.mmn om mo.oa m.o vm.ma+ ou mm.mmu mm mm.m m.o mm 02m on.oH+ on ~v.m~| mm mm.m m.o om.oa+ 0» ma.omu av mn.m m.o oov 02m oamo.u uouum w mucfiom H-m magma scum Aw unmamgv mo mmcmm Hmucmaflummxm cmumasoamu «m CH cowumuucwocoo Edaflxmz mo umnEsz uouum m mmmum>¢ coHusHom M coflumowMHboz xmaxasmlamcomumm now mama Hmucmawummxm mo humEEdm Hum.m mqmda 171 correlation in the effective Reynolds number region of one (see Figure 6.5-5). CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS §7.1 Conclusions The Ergun equation, D 3 pA p e = (l - e) E—g- I. l _ e 150 5;5;7E_ + 1.75 O is widely used to relate pressure drop to flow rate of Newtonian fluids in packed beds. In this study the Ergun equation was extended to non-Newtonian fluids by using an effective viscosity in place of the Newtonian viscosity, D 3 p—‘LEZ-IElee—lsoD‘é/S)+1.75 MGO p 0 eff The effective viscosity was calculated based on the result of a hydrodynamic analysis of the capillary model of the packed bed using the appropriate constitutive equation for each non-Newtonian fluid. Therefore, the effective viscosity is not only a function of the material parameters but also a function of the porous medium. 172 173 Measurements of pressure drops and corresponding flow rates were made for several concentrations of aqueous solutions of three different polymers (poly- acrylamide, polyvinylpyrrolidone, and polymethylcellulose) flowing through packed beds, and the rheological prOper- ties of these polymer fluids were determined. The following conclusions were based on the analysis of these experiments and comparisons with the literature: 1. The Ergun equation can be used to correlate pressure drop-flow rate data for non-Newtonian fluids flowing through packed beds by using the apprOpriate effective viscosity. The rheological model which characterizes the shear stress-shear rate relationship with minimum error should be used to calculate the effective viscosity for use in the Ergun equation. Previous investigators [1, 2, 3, 4, 5] have calculated the effective viscosity based on either the power-law or the Ellis model. Polyacrylamide The Sprigg's model, Tp + A F Tp = -2n e P E P 174 where material parameters no, a, A, and a were listed in Table 4.4-3, correlated the shear stress-shear rate data with the least square error between predicted and experimental values of apparent viscosities when compared to the power-law and the Ellis model. Correspondingly, the least square error between experimental and calculated values of the friction factor was obtained using the effective viscosity calculated from the Sprigg's model when compared with the use of the effective viscosity from the power-law and the Ellis model. The effective Reynolds 5 1 number for packed beds ranged from 10_ to 10- in our experiments with aqueous solutions of polyacrylamide. Polyvinylpyrrolidone Meter's model, { n° - n°° } To. = - n + e.. l] m r———7——a-l l] m where material functions no, nm, a, and Tm were listed in Table 5.3-1, was used to correlate the shear stress-shear rate data. The average percent deviation between values predicted from the model and experimental values of the apparent 175 viscosity was 2.1 percent. Aqueous solutions of polyvinylpyrrolidone were purely viscous having both upper and lower limiting viscosities. The pressure drop-flow rate data for aqueous solutions of polyvinylpyrrolidone were correlated well (error less than 6.1%) by the Ergun equation with the use of the effective viscosity calculated from Meter's model for effective Reynolds numbers less than one. The effective Reynolds number for 2 to 102 in our experi- packed beds ranged from 10' ments for aqueous solutions of polyvinyl- pyrrolidone. Polymethylcellulose The Herschel-Bulkley model, Tij ' Ty = “0“5‘) eij where material parameters no, m, and Ty were listed in Table 6.4-1, was used to correlate the shear stress-shear rate data for aqueous solutions of polymethylcellulose because these solutions exhibited yield stresses with non-linear flow curves. The average percent deviation between values predicted from the model and experimental values of the apparent viscosity was 1.7 percent. The Ergun equation with the use of the effective 176 viscosity calculated from the Herschel-Bulkley model correlated pressure drop-flow rate data within an error of 8.2 percent. The effective Reynolds number for packed beds ranged from 10'-3 to 1.0 in our experiments with aqueous solutions of polymethylcellulose. Neither the power—law nor the Ellis model will predict the proper shapes of the apparent viscosity versus shear rate for aqueous solutions of poly- vinylpyrrolidone and polymethylcellulose; and therefore, neither model would be very useful for predicting the effective viscosities for calcu- lations of friction factors in packed beds. Basic rheological data on polyvinylpyrrolidone and polymethylcellulose were obtained; this information has not been reported in literature previously. The wall effect correction factor as defined in Eq. A.3-32 and reported in the literature [90] was determined to be applicable for non-Newtonian fluids as well as Newtonian fluids. An improvement in the model for packed beds is required in order to account for the inertial and surface effects for polymer solutions. This conclusion is based on the friction factor-Reynolds §7.2 177 number data for polyvinylpyrrolidone solutions for effective Reynolds numbers greater than one (see Figure 5.4-7). An attempt was made to im- prove the correlation of the friction factor data for effective Reynolds numbers greater than one. A correction parameter (DpGo)/(M(l-e)no)- «fig/pr) was used to account for the deviations. However, this correction should not be considered general at this time because of limited infor- mation. Recommendations Of greater potential value would be an investi- gation of time-dependent normal stress. For such a measurement one needs the response of the pressure-sensing system. If an appreciable time lag was introduced, a calibration of the system response characteristics might still permit mean- ingful data analysis. In that way, the secondary normal stress difference, can also (T22 ’ T33” be measured. Continued attention to the quantity EQr and its utility for data-correlation purposes, is recom- mended. The implications of its behavior with temperature should be explored in an effort to clarify its fundamental significance. 178 It is of paramount importance that the polymer fluids chosen for study remain stable in storage and in shear. Months, if necessary, should be taken to determine these stability characteristics for every fluid to be tested. Special attention to chemical preservations and their subsidiary effects will be a great help. Many water-soluble polymers are frequently poorly characterized with regard to molecular weight, thereby confusing the interpretation of measure- ments. Behavior in a polar solvent may be anomalous, with ionic and association effects implying structures not characteristic of the polymer chain. Future experimental work should definitely move into the area of non-aqueous solutions. Appropriately designed apparatus is desired for extensive work of the quantitative correlation of the data in regards to the polymer adsorption, gel formation, shear degradation, and surface effects. Multiple flow loop system is recom- mended for these purposes. NOMENCLATURE f'k NOMENCLATURE cross-sectional area of a packed bed, cm2 constant in permeability expression, dimensionless diameter of tube, diameter of particle, cm partial time derivative, sec Jaumann time derivative, sec shear-rate tensor, sec complex amplitude of shear rate in oscillation friction factor for flow in a tube, dimen- sionless friction factor for flow in packed beds, dimensionless mass velocity, gr/cm2 sec gravitational acceleration, cm/sec2 fluid consistency ingex £power-law parameter), dyne sec /cm proportionality constant in Darcy's law, cm3 sec/gr machine constants for Weissenberg rheogoniometer packed bed permeability, cm2 . . -l relaxation time spectrum, sec 179 -”3X3 _ 4 L L If k-matrix is symmetric, then rotation of the axes to a particular orientation produces a diagonal k-matrix. Thus, if k.. = k.. ; i,j=l,2,3 (A.l-lO) I then for a particular set of rectangular axis, xi :i=l,2,3, the k-matrix takes the form (denoted as k'-matrix): [kl o o I k'-matrix = 0 k2 0 (A.l-ll) 0 0 k I L 3) For the coordinate axes oriented parallel to the principal axes of the porous medium having orthogonal principal axes, the postulated form of Darcy's law becomes = - _ai . = .- Vi ki(p/u) 3x1 , 1 1,2,3 (A.l 12) Since, in general, not one of the prime coordinates is parallel to the vertical axis which is the direction of the gravitational force, it must be written as: 195 6 ll tn ll MW p x.’ cos a. + f d i 1 1 1 po 51%)— (A.l 13) §A.2 Permeability Relations Darcy's law permits a description of flow through porous media in certain flow regions. It provides no understanding of the property of "permeability." This understanding is possible only if the concept of permea- bility is reduced to more fundamental physical principles. Many attempts, theoretical and empirical, have been made to connect permeability to the properties of the porous medium. Such properties are porosity, pore structure, and grain size distribution. Despite innumerable investi- gations, there is no universally accepted correlation. There are three theoretical approaches for study- ing flow through porous media. Common to each approach is the assumption that the flow may be analyzed on the basis of microsc0pic flow within pores. Theoretical models which describe the microscopic structure of pores are pro- posed and the microsc0pic flow properties are derived. By extension, macrosc0pic flow behavior is deduced. In the first method, the hydraulic radius theory, the porous medium is assumed to be equivalent to an assemblage of channels. The Navier-Stokes equations are solved for a single channel and the results are applied to the col- lection. The second method, the drag theory, considers 196 the particles to be obstacles to an otherwise straight flow of the fluid. The drag on each particle is calcu- lated from the Navier-Stokes equations and the sum of the resistances of all the particles is thought to equal the total resistance of the bed to the flow of the fluid. The third method, the statistical theory, suggests that the porous medium should be considered intrinsically dis- ordered. This is the Opposite view of the other methods which consider the porous medium to be intrinsically ordered. The Navier-Stokes equations may not be used for such disordered media; a statistical mechanics method has to be applied instead. In practice, the hydraulic radius theory has been most successful in correlating flow experiments for Newtonian fluids flowing through dense beds. The basic concept of the hydraulic radius theory is a consequence of dimensional considerations. It is observed that the permeability, k, in absolute units, has the dimension of length squared. It is reasonable that there exists a characteristic length which describes the permeability of the porous medium. This length is termed the "hydraulic radius" of the porous medium. A possible measure of this length, R , would be the ratio of volume h to surface of the pore space, Presumably, the permea- bility will depend upon dimensionless quantities, e.g., some function of porosity, e. 197 Therefore an expression for permeability may have the form (Rh)2 k = C W (A.2-1) Blake [67] introduced the hydraulic radius con- cept and was led to the permeability expression 3 E _.2. (A.2-2) S W [I Che where s is the surface area per unit volume. Blake's approach does not indicate what the value of C should be. Kozeny [82] assumed the pore space to be equivalent to a bundle of capillaries of varying cross-section but of a definite length. This approach gave a specific meaning to the constant C. For example, for a circular cross— section C=2.00 while C=3.00 for a parallel slit. Carman [69] further modified the expression to include the effect of tortuosity, T, which accounted for the fact that the actual length of a pore is greater than the length of the bed. Carman obtained (A.2-3) or adopting the value of /2 for the tortuosity and the value 2.5 for C, 198 k = -l- 63 (A 2-4) 5 302‘1 _ Q2 where so is the "specific surface," i.e., the surface area per unit volume of solid (not porous) material. Eq. 3.2-4 is the Balke-Kozeny-Carman equation. Further modifications have been prOposed by Ergun [83], but they differ only in the value of the constant. Fahien and Schriver [84] have a modified equation to account for the effects of large porosities and of flow in the transition region. There have been two approaches to the drag theory of permeability. Iberal [85] prOposed the model of a random distribution of widely separated (high porosity), long, circular cylindrical fibers. Brinkman [86] and Ranz [87] assumed that the particles in the fluid are spheres held in position by external contact forces. Iberall arrives at: 2_[eD 2 I 2 - £n(DpVOp/ue) k = 16 l - e 4 - £n(DpVop/u67‘ (A.2-5) whereas Brinkman gives: 2 DE 4 8 k— .72 [3+1-6-3 117-3:} (A.2-6) One observes that the drag theories lead to different accounts of permeability than do the hydraulic 199 radius theories. Happel and Epstein [88] have shown that Brinkman's approach has been the most successful in describing experimental results for flow through expanded, cubical assemblage of uniform spheres. These corre- lations have proved most useful for the study of the sedimentation of particle aggregates. §A.3 The Flow of Newtonian Fluids Through Porous Media Consider the flow of an incompressible fluid with density 0 through a pipe as shown in Figure A.3-1. Momentum in Pressure, PC by flow r & F; l I 1 ' , Momentum in R u//4 and out by w I viscous L .__+W_+ : transfer / I v ‘ i , fi+—Tube wall 2 ‘ Ar-—+ / 1 r L T Momentum out Pressure, PL by flow Figure A.3—1 Cylindrical Shell of Fluid Over Which Momentum Balance is Made to Get the Velocity Profile 200 Setting up a momentum balance on a shell of thickness Ar. 2 2 ZflrLTrz r - 2anIrz r+Ar + 2errsz 2:0 anArsz z=L + 2nrAerg + 21rrAr(PO - P ) = 0 (A.3-l) L Dividing Eq. A.3-l by ZNLAr and taking the limit as Ar goes to zero; this gives P - P d _ o L _ d? (rTrz) — [ L r (A.3 2) where P = P - pgz, Ap = P0 - PL. Equation 4.3-2 may be integrated to give: c = AP. __1. .. Trz [2L] r +- r (A.3 3) r = (371;: r (A.3-4) Newton's law of viscosity for this situation is de Trz = ' “ dr (A.3-5) Hence, from Equations A.3-4 and A.3-5 d" .42. - [ZuL] r (A.3-6) 201 :-./113.2 .. Vz [4HL] r + c2 (A.3 7) From boundary condition Vz = 0 at r = R =_A_E.Z c2 4uL R Hence, =_A_P_2_£2 - ‘Vz [4UL] R [1 (R) ] (A.3 8) The average velocity is calculated by summing up all velocities over the cross-section and then dividing by the cross-sectional area: 2n R f f Vz r dr d6 2 2n R f f r dr d6 = _P__.A R (A.3-9) Momentum flux and velocity distributions in flow in cylindrical tube were shown in Figure A.3-2. 202 T I” I Z _ 0 a I 7 7 V2 - j . Parabolic A Velocity ' Distribution 5 — _____ v V Z,maX : | _ 1 ] Trz — 0 , E Linear Momentum _ AB R 5 Flux Distribution rz,max 2L ‘ L Figure A.3-2 Momentum Flux and Velocity Distributions in Flow in Cylindrical Tubes As suggested before, the results for flow through a tube will be applied to that for flow through a porous medium by means of hydraulic radius concept. Hydraulic radius, Rh, is defined as: D = 4Rh or R = ZRh (A.3-10) where R = Cross-section available for flow h Wetted perimeter _ Volume available for flow Total wetted perimeter (volume of voids)/(volume of medium) (wetted surface)/(volume of medium) e/S (A.3-ll) 203 Since the "specific" surface area, So' is defined by S = 80(1 - E) (A.3-12) and a mean particle diameter DP = 6/50 (A.3-13) Hence, from Equations A.3-1l, A.3-12, and A.3-13 a DP 8 E— -- _ § — So(1 - e) - 6 - E) (A.3 l4) _ 1 _ Rh ‘ 7 R ’ Substituting Equation A.3-l4 into Equation A.3-9) 2 =é£-B_ z 8uL 4Ap ha Ap DP 22 = —_—é—_i-— = A 2 (A.3-15) “ 72uL(1 - e) The commonly accepted Dupuit-Forchheimer assumption [69]. is: V = e o z Ap DP 83 = 2 (A.3‘16) 72uL(l - 5) Therefore the permeability expression is: 9P2 83 k = C (A.3-l7) 204 where the constant C is left to be determined from experi- ments for the flow of Newtonian fluids in packed beds. The effect of tortuosity results in an actual pore length greater than the length of the bed. The experi- mental measurements indicate that the number 72 in the denominator in Equation A.3-l6 be replaced by 150 [83]. Hence, Equation A.3—16 becomes: 2 3 Ap DP 8 ° 150uL(1 - e)2 This equation is known as the Blake-Kozeny [12, 69, 83] equation and valid for low flow rates. For highly turbulent flow, the friction factor is only a function of roughness when the Reynolds number is high. The friction factor f, a dimensionless quantity is also called a drag coefficient. It is approximately a constant at higher Reynolds numbers. For the flow of a fluid through a bed of spheres, the pressure drop, Ap, is given as follows: Ap = F/A (A.3-19) where F is the force exerted on the solid surfaces and A is the cross-sectional area. Consider the fluid flowing through a cylindrical tube. The fluid will exert force F on the solid surfaces which is equal to: 205 F = A' K f (A.3-20) where A' is the surface area of the column or wetted sur- face, K is the kinetic energy per unit volume, and f is the friction factor; therefore, K = %- p 2 and A! = 27TR L (A.3-21) l 2 F = (ZNR L)(§ p )f (A.3-22) From Equations A.3-l9 and A.3-22 f = RAP (A.3-23) 1 2 2L(§p) Since the hydraulic radius, Rh' is: E 6 DP E Rh = S = 80(1 - e) = 6(1 - E) (A.3-14) Rh AB 1 f = -—— (A.3-24) L 1 >2 7 p > D (A.3-32) C P §A.4 The Flow of Non-Newtonian Fluids Through Porous Media Some effort has been extended toward establish- ing methods for predicting non—Newtonian flow behavior in porous media and for correlating pressure drOp versus flow rate data with Viscometric data for porous media experiments. At the present time there is no universally acceptable scale-up method for flow of rheologically com- plex fluids in porous media. The various methods cur- rently employed can be arbitrarily divided into at least three major categories. The method which seems to have. received the most attention is based on the coupling of a particular model for a porous medium--i.e., the so- called hydraulic radius model, with an assumed functional 209 relationship between shear rate and shear stress to describe the rheological behavior of a non-Newtonian fluid. This method involves correlations of experimental data from one-dimensional flow experiments in unconsoli- dated porous media, i.e., mostly bead packs, with the appropriate rheological parameters derived from visco- metric experiments on the fluid of interest. The power- law and Ellis models have been used to describe the purely viscous behavior of the non-Newtonian fluids. Another category involves generalized scale-up methods which adapt Darcy's law to non-Newtonian fluids without invoking a particular rheological model of purely viscous behavior. The appropriate rheological description can, in principle, be derived from Viscometric and porous media flow experi- ments. A third approach, based on the concept of the simple fluid, involves the application of dimensional analysis to the scale-up of porous media flow data for an arbitrary viscoelastic fluid. This work was mainly directed at developing a generalized scale-up method based on the capillary model. Wall effects were accounted for by using the hydraulic concept. Various empirical and derived models for non- Newtonian fluids were applied to the problem of flow through porous media. In this section application of Sprigg's model will be developed. Several other models applied in this study will be similar to that of Sprigg's 210 model application. The results are listed in Table 3-1. For steady upward cylindrical tube flow as shown in Figure A.4-l the sum of all forces acting on the fluid between section 1 and 2 must be zero. The forces involved are those due to static pressure, gravity, and shear, so that nRzPl - nRzP2 - nRszg - 2nRLTw = 0 (A.4-l) then: ZnRLI = WR2(P -P - Lpg) = NRZAP w l 2 _ _ RAP _ (Trz)r=R - Tw — 2L (A°4 2) Similarly, if the shear stress at any r is Trz, where r s R, then: T ___ _ (A.4-3) (A.4-4) Equation A.4-4 indicates a linear distribution of shear stress in the fluid, from zero at the centerline to Tw at the tube wall, regardless of flow regime. 211 It is assumed that: (l) the fluid is in steady laminar flow, (2) the fluid is time-independent under the prevailing conditions, and (3) there is no slip between the fluid and the tube wall. The volumetric flow rate through the differential annulus between r and r + dr is do = Vz 2nr dr (A.4-5) where V2 is the local velocity at r. Q R R 2 Q=f dQ=1Tf VZZrdr=Trf Vz d(r) o o 0 Integrating by parts R2 de] (A.4-6) o 2 Q = HIVzr - f r2 The term Vzr2 is eliminated by assumption (3), and from assumptions (1) and (2), z_— — -—— — f(Trz) (A.4 7) From Equation A.4-4) R2 T2 2 _ rz ,_ R r - 2 , dr - 1?— dTrz T W 212 Figure A.4-1 Stress Acting on a Cylindrical Element of Fluid of Radius R in Steady Flow Substituting these into Equation A.4-6, 2 2 T R T _ w rz 3L Q — “f T 2 f(Trz) w dTrz w or '49? = _£§ [TwTiz f(Trz)dTrz (A°4-8) TrR Tw 0 Since Q = nR , therefore, . R Tw 2 = :7;- Trz f(Trz)dTrz (A.4-9) 213 Sprigg's 4—Constant Model Sprigg's model gives n l l - —— = Z . for low Y no 2(a) p=1 pa + czkf 2 I é-l n = 11 (CM) . - n 2(a) for high y 0 2a cos 1L- 2a Therefore, at Newtonian range the relationship of friction factor f*, and effective Reynolds number N is the Re,eff' same as the Newtonian case except for the viscosity: AE D 63 ft: = .2 p (A.4‘10) 2 I. l - 6 MG 0 M D G NRe eff = (1 —Ee): ; n l = :— (A'4-ll) ' 0 eff M no For the high shear rate: 1 l «—-l --l n n a -a 0 (cl) 7 _ n — 2(a) (2.8 57) Then, rz In this case, Substituting = a+l 1 6Ap ‘1 3 DP a+3 25AL 3(1 - €)M e z 6 6A a. e Dp a+l E¥§[§5AL] 3(1 - €)M (A.4-l4) (A.4-15) 215 where is average pore velocity. The permeability expression is: k = 1— -——€——— (A.4-16) C (1 — e)2 where the constant C is left to be determined from experiments for the flow of Newtonian fluids in packed beds. Rearranging Equation A.4-15, gives v = 6 5A9 a €¥EP a+l O a+3 25AL 3(1 - €)M = c 6Ap aeD a-l -l—a+lk n+3 25AL 1 - 6 3M = [l_2_]a-l[l]a[l_]a+l .72 [8 DP Ja-IAE a-lk 92 S A M 2a(a+3)3a+l 1 - e L L (A.4-l7) From Equation A.4-l7 we may obtain the Sprigg's model analog of Darcy's law, i.e., v0 = n k 91.2 (A.4-18) eff where k is the same permeability defined previously for Newtonian fluids, and neff is an effective viscosity obtained from Equation A.4-18. 216 1 = [3.3-]0-1 _1_-_ 1 .72 [E 9P a-l[AE]d-l neff 25 Ad Ma+1 2a(a+3)3a+l l - e L (A.4-l9) If we combine Equation A.4-ll and A.4-l9, the effective viscosity for the entire range of shear becomes: ' n 1' = 1_ 1 + [l£]a-l 4 El a-l 1 La cos §E’Z(a)] . ”eff M200 25 a+3 no 3a-1Ma-l Tra e D a-l D a-l _Ll _ 8 _LE (A.4—20) Equations A.4-18 and A.4-20 show that for non-Newtonian fluids the superficial velocity is not directly pro- portional to the pressure gradient. Equation A.4-20 shows that the effective viscosity is not only a function of fluid parameters but a function of the porous medium too. The bed friction factor f*, proposed by Ergun [83], is given: D 3 (1 - €)n A _E 6 eff *= = _ f ESE? I, 1 - e p C{ M D G } (A'4 21) o p o where G0 = pVO. For laminar flow: 217 *= _ f C/NRe'eff (A.4 22) where: N :..- M(1 - €)no 5 a+3 no N a a-l 1 [2a cos 23 2(0)] [EDP ] . _ - a - 6a lMa 1 fl 1 e D a-l E? (A.4-23) The bed structure parameters may be combined with the pressure gradient as D *_..1__2._P. 8 £2 _ Tw ‘ 25 6M 1 - e I. (A'4 24) Substituting Equation A.4-24 into Equation A.4-23 gives: n a N = D Go 1 + 4 2a cos 5; 2(a) (T*)a-l Re,eff M(1 - e) a+3 lk-l w no “(cA)a V [ W a ...—'17 ___ 'k _ DP Go 4 2d cos 2a 2(ai] clrw - M(l — e)n l + a+3 A a n o n o B ..J (A.4-25) 218 Therefore, the friction factor f*, for laminar flow, depends upon three dimensionless quantities * D A M(lp-Gg) ' C T , and a. no no The frictional characteristics of non-Newtonian fluids flowing through packed beds depends upon the irregular geometry of the flow channel since it is this irregularity which causes the inertial terms to be im- portant at Reynolds numbersgreater than one. For high flow rates, the friction factor f, dimensionless quanti- ties which is also called as a drag coefficient, is approximately constant at high Reynolds numbers. For the flow of a fluid through a bed of spheres, the pres- sure drop, Ap, is given as follows: Ap = F/A (A.4-26) where F is the force exerted on the solid surfaces and A is the cross-sectional area. For the fluid flowing through a cylindrical tube the force F on the solid surfaces exerted by the fluid is equal to: F = A' K f (A.4-27) where A' is the surface area of column or wetted surface, K is kinetic energy per unit volume, and f is friction factor. Therefore, F = (2mm (% p 2)f (A.4-28) 219 Introducing hydraulic radius Rh such that: R e 6 9p 8 Rh = f = 5 = 50(1 - e) = 6M(1 - e) (A°4"29) and rearranging Equation A.4-28) yields: 2 3M(l - e)Go f0 92.: (A.4-30) L 3 prE Experimental data indicate that 6fo = 3.50 [12, 83]. Hence, A 1.75(1 - €)MG02 D e p p Then the Ergun equation becomes: A 1 D { 150 iLl§ i? = fi——-—-——-+ 1.75 (A.4-32) MGo J l Re,eff At very low flow rates, lSO/NRe factor is dominant ,eff in the right-hand side of Equation A.4-32. At very high flow rates where the inertial terms are significant, ISO/N eff becomes very negligible comparing 1.75; Re, therefore, friction factor remains constant at 1.75. Estimation of Shear Rate in the Bed From Equation 4.5-9 Q __ 80 l f ——— ——— = T f(T )dT (A.4-8) “R wD3 Tw3 o rz rz rz 220 where Equation A-4-8 is multiplied through by T: and both sides are then differentiated with respect to Tw' using Leibnitz rule for differentiating integral, i.e., h(t) h(t) §%. f f(x,t)dx = f %% dx + [f(b,t)%% - f(a,t)%%) a(t) a(t) (A.4-33) __82_=—l—fTw 1'2 f(T )d'r "D3 T 3 o rz rz rz w 8 d ___ 3 dV ND 8Q 2 T + 3 ——- = f(T ) = (- ———) (A.4-34) w d Tw ["133] w dr w If Tw is replaced by g3? , Equation A.4-34 becomes: 3 dV ng DAp d(8g/nD ) _ _ _ z _ 3[fl03] + 4L d(DAp/4L - f(Tw) — dr w (A°4 35) This is the Rabinowitsch-Mooney equation giving the rate of shear at the wall of a tube when the flow is steady, laminar, and there is no slip at the wall. Rearrangement of Equation A.4-35 gives: 221 1 - dvz _ _3- 8 + 8 d[I(8/D)]/(8/D) dr w ‘ 4 D D d[DAp/4L]/(DAp/4L) = 8 2 + l d£n(8/D) D 4 4 d£n(DAp/4L) - 3:5 _ 8 2 + .1. = 3n' + 1 8 (A 4-36) dr D 4 4n' 4nT D ' w L where , _ d£n(DAp/4L) _ n “ d£n(8/D) (A'4 37) For Newtonian fluid WR4AE 2 Q — 8uL = "R DAp 8 0 din 4L - din D + (512/? 11 Thus, , _ d2n(DAp/4L _ _ ‘ d£n(8/D) ' 1'0 (A'4 38) - de 3 + 1 8 = 8 (A 4_39) dr vv 4 D D ' For power-law fluid 222 n %?+ 1 A2 3? = §H‘I‘I D [ZKL] (A.4-40) .1 n 8 C: Constant o 8 RAE z din[4L]+ndy1/YC +ndin D Thus, din(DAp/4L) ' _ — - ‘ din(8/ET ‘ n (A.4 41) 1 ' de = 3n + 1 8 (A 4-42) dr 4n D . w For Ellis model fluid 2 Tl-OL a _ R A 5 R RAE _ (v2) .. BUOL + _]_._ a+ [21,] (A-4 43) nod 8 G. 2 _ 3; DA + C DAB D n 4L 4L 0 0 o 8 din z = d J; + din [9—2] + d n C + O‘din[2AE] D no 4L 4L Thus, ‘ (A.4-44) n. = din(DAp/4L) _ 1 din<8/D) 5:1 223 dvz _ a+4 8 For Sprigg's model fluid For low shear rate it will be the same as Newtonian and for high shear rate a _ R RA _ ‘ a+3 [ZAL] (A'4 46) 8 DAp = adin [4L] din [ D Thus, din[%fi?] l n = 8 z d+3 z [ 'a‘r—w T D (A-4‘48’ For an estimate of the shear rate in the bed, can be replaced by GO/pe, and D by four times the hydraulic radius radius. 28 D _ = p D ’ 4Rh 3M(1 - e) Then 224 Newtonian: dvz 12 M (l - 6) GO _ Tj}. = 2 (A.4-49) w p D e P Power-law: _ dvz = 3n+l 8 dr v7 4n D - 3n+1 12 M (l - 6) GO - (A.4-50) 4n 2 p D e P Ellis: _ dvz = a+4 8 dr 4 D 12 M (l - e) G = a24 2 ° (A.4-51) p D e P Spriggs: _ dvz = a+3 8 dr 4 D w 12 M (l - e) G = “:3 2 ° (A.4-52) p D e P 225 §A.5 Limitations to the Darcy's Law It has been indicated that Darcy's law is funda- mental to the assumptions made. Two significant aspects of this law are the analogies to capillary flow and the assumption of fluid homogeneity. It is the failure of these very same aspects which leads to the more obvious deviations of Darcy's law. By analogy to flow through tubes, at high flow rates, deviations from Darcy's law are expected due to the onset of "turbulent" flow. By the same analogy, we would expect deviations to occur in systems where the pore diameters become comparable with, or less than, the molecular mean free paths of the flow- ing gas, i.e., the so-called "slip" flow region and the region of "molecular streaming" or knudsen flow. Other anomalies (based on Darcy's law) are adsorption, capillary condensation, and molecular diffusion. These latter effects may be referred to as surface flow effects, i.e., the behavior of non-homogeneous flows. Finally, there are deviations due to chemical reactions and ionic effects as they are not included in Darcy's law. Scheidegger [75] has given an extensive discussion of general equations which account both for high flow rates and for molecular effects. Carman [77] has presented a thorough summary of surface flow effects. Both theoretical and empirical equations have been proposed for describing flow at high flow rates. 226 Forchheimer [91] modified Darcy's law by the addition of a second term, i.e., 91$"- : aVO + bvo2 (A.5-l) where a and b are constants depending on the properties of both fluid and porous media. This was derived from a semi-theoretical analogy to turbulent flow in tubes. Many studies have attempted to show that the Forchheimer equation is the correct general flow equation. Most deviations have depended upon a consideration of kinetic energy losses at high flow rates. Rose and Rizk [89] suggested that viscous effects and turbulent effects are not additive as implied by Equation A.5-l. They proposed %? = aV + bV 1'5 + cV 2 (A.5-2) o o The additional term is an empirical correction for the non-additive nature of the phenomenon. Ergun [83] and Fahien and Schriver [84] indicated that the porosity function is different for each flow region. The molecular effects of Knudsen flow, slip-flow, adsorption, and capillary condensation are phenomena which primarily occur for the flow of gases in micro- porous media. As the present study is concerned only with liquid flow, it will suffice to mention that the 227 necessity of adding a surface-flow term to the Darcy and Knudsen terms in flow through absorbent porous media seems definitely established. That is, introducing Klinkenberg's [92] "superficial" gas permeability ka = a + b/Pm (A.5-3) where a and b are constants and Pm is the mean gas pres— sure, and assuming, as does Carman [77], that either adsorbate flow or capillary condensate flow are examples of diffusion along a concentration gradient, one obtains for the total flow: VO 2 3 p p p (l-e)SO MRT uSoRT(l-€) (A.6-4) where the D's are diffusion coefficients and c is the concentration. The two terms on the right-hand side of the equation are the surface flow and slip flow terms, respectively. For each of these effects, the apparent permeability is increased above that for purely viscous flows. For liquids, the only comparable experiment is that of Debye and Cleveland [93]. They found that in porous Vycor glass the flow of normal paraffins deviated from Darcy's law. The permeability of the glass, rather than being constant, was a function of viscosity. 228 There is considerable petroleum engineering literature related to adsorption or retension by a porous media of polymeric materials, and to the resultant permea— bility reduction effects which follows. It is not the intention of this study to consider these investigations in detail. Instead, we will focus attention on some of the frequently cited examples, noting that considerable controversy exists as to how these phenomena relate to flow behavior of non-Newtonian fluids through porous media. Both Marshall and Metzner [5] and Dauben and Menzie [113] considered the role of progressive plugging and adsorption of polymer aggregates in the flow phenomena they observed. They concluded that these aberrations were absent from their experiments. By contrast, Sadowski [3] concluded that polymer adsorption and gel formation was aggravated during constant pressure experiments be- cause, under these conditions, any tendency towards gel formation would lower the permeability and thus auto- matically lower the flow rate. He reasoned that gel formation would be aggravated because, at lower flow rate, there would be less tendency to remove polymer molecules from the particle surface. Burcik [114] is of the Opinion that the shear-thickening behavior observed in flow of dilute polymeric solutions through porous media is related to polymer molecules retained within the pore structure. He suggests that these bound 229 molecules are uncoiled under the imposed (high) velocity gradient, thereby increasing the resistance to flow. Pre- sumably polymer retention may occur either by adsorption of mechanical entrapment. At this time, at least qualitatively, the adsorption of polymers from solution by solids is well understood. Polymer molecules are adsorbed on the solid surface at multiple points of attachment, involving only a fraction of the chain segments of the molecules. The remainder of the molecules extended more or less freely into the sol- vent. A dynamic adsorption-desorption equilibrium may be attained, but in such cases individual segments of the molecule are involved. The extent of adsorption is such as to suggest that the amount of polymer adsorbed is at least one order greater than could be possible if all the chain segments of each molecule occupied the adsorption sites. As might be expected, the amount of polymer ad- sorbed by a solid surface in "static" experiments increases with (a) increasing molecular weight, (b) decreasing particle size, and (c) decreasing solvent power. Here the solvent power represents the ability of a solvent to dissolve polymer. In a "dynamic" or flow system it is doubtful if a simple layer of adsorbed polymer affects the permea- bility of the bed to any extent. First, the amount of adsorbed polymer will be dependent upon the equilibrium 230 condition for adsorption. There will exist a tendency for the polymer molecule to be adsorbed by the solid sur— face and a tendency for the polymer molecule to be swept away by the motion of the fluid in the bulk stream. Second, the residence time of a polymer molecule in a packed bed will be extremely short, on the order of seconds. Even in the case of complete coverage of the available surface area in a packed bed by the adsorbed polymer, there would be no apparent effect on the bed permeability. The addi- tional layer of polymer, 1000 to 5000 A in thickness, would have a negligible effect on the effective porosity of the bed and, therefore, the permeability of the bed. If there is to be an effective change in the bed permeability, another mechanism is responsible. An alternate mechanism may be visualized as follows. The geometry of the bed is extremely complex with much of its surface area in the vicinity of points of particle con— tact (4 to 12 points of contact per particle). As shown in Figure A.5-l, in such a geometry the magnitude of the distance between two solid surfaces, along any arbitrary line, varies from a maximum value equal to the largest particle diameter to zero on the assumption that there exists no bridging between the particles. In the region in which the distance between two adjacent surfaces converge rapidly to zero (point contacts), it is possible for the adsorption phenomenon to become important. Here, a polymer molecule exists in a velocity field which has 231 a value slightly greater than zero. Although the rate of shear may be high, the tendency for the polymer molecule to be swept away from the solid surface is markedly lowered. The tendency for adsorption will be favored. The adsorbed polymer molecules, extending into the bulk fluid, may serve as attachment sites for other molecules from the immediately adjacent particle surfaces and from the bulk fluid. In this way, a gel structure may be formed within various parts of the bed. It is generally accepted that a gel owes its character to the fact that one or more of the components forms a coherent structure, or network, throughout the system. The points of coherence may be due to chemical bonds, physical attractive forces (Van der Waals bonds), or regions of crystallinity [111]. An important conse- quence of the network structure of a gel is that a small amount of the component which makes up the network immobilizes a large amount of the second component. An almost negligible amount of polymer; therefore, may induce significant gel formation which would affect the bed permeability by the plugging of minute pores or the filling of the pendular regions or both. A pendular ring between two spheres is shown in Figure A.5-2. The tendency for gel formation will be enhanced by an increase in the molecular weight of the polymer, by a decrease in the size of the particles, by a decrease in the bulk flow 232 rate of the fluid, and by an increase in the time during which the fluid flows through the bed. It is obvious that the occurrence of gel formation invalidates the flow theory as develOped in Section 3.1. To test the analysis of that section it is necessary to discover the circumstances under which gel formation will occur and degree of its importance. There are two basic flow experiments: (a) the flow rate of the fluid through the bed may be held constant, and (b) the applied pressure to the fluid in the bed may be held constant. In the absence of the "gelling" effect, each of these experiments is equivalent. In the presence of this effect, each experiment may give different results. 233 Rhombohedral Figure A.5-l Packing of Uniform Spheres Figure A.5-l Pore Space in Packing of Uniform Spheres [After Graton and Fraser, 1935] Figure A.5-2 Pendular Ring Between Two Spheres [After Von Engelhardt, 1955] APPENDIX B VISCOMETRIC EXPERIMENT: DETAILED DESCRIPTION WEISSENBERG RHEOGONIOMETER APPENDIX B VISCOMETRIC EXPERIMENT: DETAILED DESCRIPTION WEISSENBERG RHEOGONIOMETER §B.l General Description Drive Unit On the left of the instrument is mounted the motor/gearbox drive unit, as shown in Figure B-l. This consists of an aluminum base standing on anti-vibration mountings and carrying either one or two motors and gear- boxes. One motor and gearbox only is necessary if rotation and oscillation are required individually but it may be desirable to have two if frequent changes from rotation to oscillation are to be made, since with a single motor and gearbox the unit has to be mechanically uncoupled and moved each time a change is to be made. The drive between the motor and gearbox is through a flexible coupling and the drive from the gearbox to the rheogoniometer is by a shaft connected by specially chosen universal couplings. These precautions limit the effects of motor and gear vibrations on the instrument when it is being used on the most sensitive ranges. 234 235 I .xoa wean :2: ...z géuflw /. uwumgucovounm muuncaunfios any «0 unusucuuut Hagan. ad 925.: ufl:zmmfl . . mmrdfliflwmd :3 was. .3233 3.9.0:. 236 Electric Motor These are three-phase synchronous motors rated at 1 horse power. This type of motor is used so that the drive speed is rigidly constant through the duration of a test and also has very high consistency even over a longer period of time. The standard motor supplied with the instrument is 220 volt, 1800 r.p.m., 60 c.p.s. The motor is carefully made, the rotors being dynamically balanced to fine limits so as to reduce vibrations to a minimum. The motor is controlled from a three-phase switch mounted on the control panel. In the case of continuous rotation drive, this is a reversing switch which enables the motor to be run in each direction and an average reading taken so as to test the symmetry of the sample. Gearbox Internally this gearbox unit is in two parts, the -0.0 first five steps of approximately 1:10 (1:1) to 1:10-'0"4 (1:2.5). The second part has twelve steps of 1:10-0.5 -0.0 -5.5 from 1:10 (1:1) to 1:10 (l:316,200 reduction) and is driven by the output of the first part. The gear change mechanism associated with each part of the gearbox shows a number indicating the speed reduction to which that side of the gearbox is set as a power of ten. By simply adding these two indexes a figures is obtained which expresses the overall speed reduction 237 of the gearbox as a power of ten. Anti-friction needle roller bearings are used throughout the gearbox, thus eliminating the tendency towards stop-go motion which is often troublesome where journal bearings are employed at very low speeds. Gear change may be made in the following way: on each end face of the gearbox there is a gear selection and locking knob, each controlling the associated part of the gearbox. It can be unlocked by rotating it counter- clockwise as far as possible. The knob will be seen to move outwards. The whole end plate of the gearbox can be moved around, so that the selected gear index shows at the top of the box. Selection of the precise position is assisted by using the locking knob in its intermediate position. This can be achieved by rotating the knob clockwise when the end plate is in any position. The knob is not rune in its fully locked position but is held in a detent and the locking mechanism is sprung against the inner face of the gearbox. When the end plate is now moved to the precise position required, the locking mechanism will spring into place and the end plate will now be accurately in position and the gears correctly meshed. The locking knob should now be turned once more in a clockwise direction pressing it inwards which will move it to the fully locked position. The end plate is now completely locked and cannot move in use. It is 238 essential that gear changes are not made while the motor is running, and it is also essential that the locking knob is fully locked before the motor is run. Rheogoniometer A. Base The base of the instrument provides a plane sur- face and has webs cast in the underside for increased rigidity. There is an adjustable leg at each corner, which enables the base to be adjusted level. This is set by means of the pair of sprit levels mounted at the front. The base also contains the clamps for holding the two transducers associated with normal force measure- ment and servo operation. B. Main Drive Box This is mounted roughly at the center of the base and contains the main vertical shaft of the instrument on which the lower platen adaptor and lower platen is carried. This vertical shaft is held in two precision taper roller bearings which are specially selected. They are fitted in opposition and are carefully adjusted to give the smoothest possible operation. This shaft carries a bronze worm wheel, having 48 teeth, which is driven by a four-start steel worm mounted in front of it on a hori- zontal shaft. This gives a speed reduction of 12:1. The 239 right-hand bearing is held in a carrier which also serves to clap a transducer mounted at the end of the shaft and is allowed to pivot a small amount. C. Two-Way Drive Box This is mounted on the left-hand side of the base, locking at the front of the instrument, and is bolted rigidly to the main drive box. It contains the step-up gear for the rotation drive as well as the brake/drive unit on the extreme left-hand side. D. The Column This is fitted at the back of the main drive box and is accurately set up so that the working faces are parallel with the axis of the vertical shaft in the main drive box. The column has a high rigidity to minimize any bending under heavy normal forces. It carries the torsion head, the gap setting transducer, and various ancillaries. At the base of the column at the rear there is a slot through which the normal force springs are fitted. Rotation Drive The drive from the gear box is taken into the electro-magnetic brake/drive unit at the left-hand side of the instrument. This consists of two discs of high permeability metal, each containing a coil winding and 240 friction pad, mounted on either side of a clutch disc of the same metal. This clutch disc is fixed onto the hori— zontal drive shaft which carries the rotation drive to the step up spur gears in the two-way drive box. The left-hand or outer coil disc is mounted on a shaft running in sealed ball races and is continuously driven by the motor gearbox unit. The right-hand or inner coil disc is fixed rigidly to the main casting of the two—way drive box. By energizing one or more of the coil windings, the clutch disc is attracted magnetically and forced against the friction pad, so that the instrument is accelerated rapidly up to the chosen speed or rapidly stopped, depend- ing whether "Drive" or "Brake" is chosen on the control switch mounted on the control panel. Torsion Head The torsion head is a complete assembly which is mounted on the column of the Weissenberg Rheogoniometer and carries the top platen. It owes its extreme sensi- tivity to the air bearing, which is an integral part of the construction, and its very wide range to the variety of torsion bars and platens which can be used. Air is supplied from the laboratory air supply at about 80 p.s.i. or may be fed from a separate air com- pressor. The air passes through a filter unit mounted at the back of the instrument at the base of the column. It then passes through one regulator at the base of the 241 column and another at the t0p. Two regulators are pro- vided so as to reduce fluctuations of pressure at the bearing to a minimum, which might otherwise cause small variations in the readings of tangential stress when work- ing on the most sensitive ranges. The top regulator is used to select the working pressure (between 20 and 50 p.s.i.) and the bottom one should be preset to a value in excess of the maximum working value (55-65 p.s.i.) to prevent possible interaction between two regulators. The torsion bar against which the tangential stress is measured has a length of 2.36 in. The top of the torsion bar is held rigidly in a clamp at the top of the air bearing casting and the bottom is clamped on the top of the air bearing rotor. The clamp at the lower end of the torsion bar carries the 10 cm. radius arm, at the other end of which is fitted the armature of the torsion head transducer. The displacement of this transducer noted on the appropriate transducer meter gives, with the calibration constant of the particular torsion bar, a direct reading of torque produced on the top platen, and hence the tangential stress in the specimen. All the calibration constants of the particular torsion bar are tabulated in Table B-1. 242 TABLE B-l. Torsion Bar Calibration Torsion Platen Size (Diameter) Bar 2.5 cm 5.0 cm 7.5 cm 10 cm 1/32" RT = 53.8 RT = 6.72 kT = 1.99 kT = 0.84 1/16" kT = 525 kT = 65.6 kT = 19.54 kT = 8.21 1/8" kT = 8,800 kT = 1,100 kT = 328 kT = 138 1/4" k = 142,500 k = 17,840 k = 5,400 k = 2,230 3/8" k = 561,000 k = 70,240 k = 20,800 k 8,780 Normal Force Measurement When normal force measurements are not being made, the platen is mounted directly on the vertical worm wheel shaft by means of the lower platen adaptor. If normal force measurements are to be made, the lower platen adaptor at the top of this vertical shaft is removed and replaced by a special normal force platen assembly which is free to move in a vertical direction, but is located rigidly and driven in rotation through a diaphragm. The general arrangement of normal force measurement and schematic layout of normal force measurement system are shown in Figures B—2 and B-3, respectively. The drive is taken from the tOp of the vertical shaft by an outer sleeve, which carries at its top a beryllium copper 243 aOhOE O>aum ucmEmHSmmmz mouom HmEuoz wo usmfimmcmuu< m.m wusmflm GUCJOI amunow24mk ...Zwimaamaflfi uUmOm 4(5102 ‘298 02.539 3.3 \/ $30: $38.23 .auuzaa Owo<04 Imhm>m O>mum 025% .\ QEawm _“ 1 o wOmOu 435.02 m. . . ___ \(7 (1 _!_.IIII.V._L o o . rll...“0 o a —fit 0 0 UN! 0 ...-hub @ . /.\ .1. III: - -Ih III-) H \n . . «9.1.3.24 02mm \ .>mw< th<1_n_ mwBOJ wUmOm 4<2m02 245 diaphragm and also a locking ring. The diaphragm is free to move in the vertical direction, but it is restrained by the lower platen shaft, without touching, to the normal force pivot hearing. The normal force spring itself is a leaf spring one-half inch in width and of different thickness, depending on the spring rate required. The cantilever leaf spring has, in its "free" end, a steel ball which provides a low friction pick-up point for the spring plunger and micrometer which position this "free" end. During measurements of normal force it is important to keep the gap between the two platens constant. As the force increases the normal force spring will bend and the lower platen will thus move downwards. This can be detected by the transducer which bears on the under side of the normal force pivot bearing and its deflection seen on the apprOpriate transducer meter. Steady state normal forces can be measured by adjusting the micrometer by hand to give zero deflection of the transducer mounted beneath the normal force pivot bearing. The normal force springs have constant spring rates over the full range of deflection used in the instrument. The product of the reading of the microm- eter, and the spring rate of the particular normal force spring gives a direct reading of normal force. Normal force spring calibrations are listed in Table B-2. 246 TABLE B-2. Normal Force Spring Calibration Normal Platen Size (Diameter) Forces Spring 2.5 cm 5.0 cm 7.5 cm 10 cm 1/8" kN = 56,600 kN = 14,150 kN = 6,300 kN = 3,540 1/16" kN = 8,330 kN = 2,082 kN = 930 kN = 520 T - T _ _ 11 22 Tll-T22_kNAN ' 6’ 1} §B.2 Experimental Procedure Gap,Setting It is important for cone and plate to be precisely positioned, so that there is an accurate known gap. The conical platen is truncated so that the cone and plate are divided at the center by the amount of this truncation. Gap settings according to the platen size and cone angle are listed in Table B-3. The gap can be set as follows: 1. Set the transducer selector switch to "gap setting." Loosen the clamping screw on the right-hand side of the carriage and wind the torsion head down until the platens are seen to be almost touching. Continue lowering the torsion head very slowly watching the servo transducer meter. As soon as TABLE B-3 Gap Setting 247 Platens Nominal Cone Gap Part Diameter Angle Setting Number (cma) (°) (in.) 2.5 4.00 .00713 2.5 Flat -- 2.5 Flat -- 5.0 3.9128 .00685 1153 5.0 Flat -- 947 5.0 Flat -- 955 7.5 1.5333 .00259 554 7.5 0.5502 .00100 993 7.5 Flat -- 427 7.5 Flat -- 584 7.5 Flat -- 1964 10.0 2.0083 .00350 1139 10.0 Flat -- 1167 10.0 Flat -- Loading 248 the platens touch this transducer meter will show a small deflection. Adjust the gap setting transducer using the "set zero" on the transducer meter panel for the final operation, so as to give a deflection in a posi- tive direction on the transducer meter equal to the gap required. Raise torsion head slowly until transducer meter reads "zero." The platens are now correctly set and can be reset to this zero position at any time after loading the sample, etc. the Sample 1. Wind the torsion head up until there is about 3 inches gap between the platens. Apply the sample in sufficient amount just to over-fill the gap. Wind the torsion head down slowly until the gap setting transducer meter is at "zero." A short period should be allowed before commencing the test, so that any stress in the sample can sub- side. The clamping screw on the carriage should be fully tightened--checking that gap setting does not alter when this is done. 249 The surplus material should be removed from around the platens. If the sample is not correctly trimmed it can, especially in the case of more viscous samples, give errors both in tangential and normal force measurements. Steady State Experiment 1. Carry out gap setting procedure. Carry out loading the sample procedure. Set the gearbox from low speed. Lock the gearbox firmly. Turn the rotation motor switch in the main control panel to "FORWARD." Turn the off/brake/drive switch in the main control panel to "DRIVE." Measure AT and AN for the given gearbox set. Re-set the gearbox and repeat (3) through (7). Rotation shear rates for various platens are tabulated in Table B-4 and the correction terms for pri- mary normal stress difference are shown in Figure B-4. 250 thv00.0 0.m mev0.0 m.v thv.0 0.m NBN.¢ 0.N Nb.Nv 0.H N.th 0.0 AHIUmmv mfimm .Hmwflm muwmnmm XOQHMQU quflmumCH HOW H0.0 0.m H.0 0.v O.H 0.m 0H 0.N 00H O.H 000A 0.0 vmmm.0 0v¢v.0 mMN.H OVBH.0 VOBH.0 0MHH.0 0.0 NFN¢.0 hmmm.0 00m.H vaN.0 m¢HN.0 0mva.0 0.0 thm.0 hMOh.0 000.H mth.0 hmmm.0 mth.0 5.0 m50.0 mvwm.0 N0¢.N m0wm.0 mmm.0 0NN.0 0.0 Hmm.0 NHH.H mmo.m mmv.0 0N¢.0 va.0 m.0 050.H mov.H NNm.m Nmm.0 0vm.0 00m.0 v.0 0mm.H mmh.a 0mm.v v00.0 050.0 va.0 m.0 hmm.H MNN.N mmH.0 Hhm.0 Nmm.0 mmm.0 N.0 mMH.N N00.N mom.h mmO.H vn0.H 0H5.0 H.0 mmm.N NNm.m Hm.m 00m.H 0mm.H 000.0 0.0 maoo .mmoo.~ mcoo .mmmm.a mcoo .Nomm.o maoo .mmam.m mcoo .oo.v 2mm ommmm omEU 0.0H .mgnv mom. omEU mom. omEU Com omEU moN XOQHMOU mcmumam mdoflum> How moumm Hmmcm cofiumuom vim wands 251 100001 for p = 1.0 Line drawn from this point will slope = 2 1000- N E o \\ w c >1 *0 N N P l H H P 100‘ 10 r 10 100 1000 - —1 Y sec Figure B.4 Correction Terms for Normal Stress Difference APPENDIX C FLOW EXPERIMENT: DETAILED DESCRIPTION APPENDIX C FLOW EXPERIMENT: DETAILED DESCRIPTION §C.l General Description The apparatus used for this experiment consisted of a one-half-inch glass column, a one-inch glass column, a pressurized flow system, packing particles (glass beads) for the column, differential pressure transducer (Model KP 15, Pace Engineering Company, North Hollywood, Cali- fornia), a recorder (Linear/Log Varicord 43, Photovolt Company, Broadway, New York), a cathetometer accurate to 0.005 cm. (Gaertner Scientific Company, Chicago, Illinois), and three rotometers (Brooks Instrument Division, Emerson Electric Company, Hatfield, Pennsylvania). The schematic diagram of the jacketed flow system which maintained con— stant temperature 21 : 0.5°C was shown in Figure 4.5-1 previously. Glass Beads The bed was packed with glass beads which were spherical and uniform in size. Glass beads were sized by "0.8. Standard" sieves. This was done by hand to 252 253 minimize damage to the particles. The glass beads used in the experiments were sieve size Nos. 12, 30, and 40. For those particles under 1 mm. in diameter, the particle diameter was measured by means of a microscope, the eye- piece of which had a movable hairline. The movement of the hairline was controlled by a micrometer spindle which was calibrated against an accurately measured 1 mm. line which was etched onto a slide. A random sample, 10 to 15 particles, was placed on a slide having raised sides and one diameter was taken which appeared in the field of the microsc0pe as the hairline was moved from left to right. One hundred diameters were measured in this way. The average diameter of each size of the particles was the arithmetic average of the individual diameters. For particles larger than 1 mm. in diameter, a machinist's micrometer was used to measure directly the diameters. Again, one hundred particles, randomly selected, were measured. Glass Column There were two glass columns A and B, with l-inch and l/2-inch internal diameters, respectively. The jacket around the test section was made of an epoxy glass tube. Epoxy glass was selected for the test section in order to allow the experimenter to observe the flow of fluids within the packed beds. This was particularly useful for checking that air bubbles were not trapped within the 254 bed. Each glass column was made in several segments. An assembled glass column was made to fit between two aluminum flanges and the entire assembly was bolted together. The pressure tap connections to the top section and to the bottom section were l/4-inch stainless steel tube which is connected into the pressure transducer. The bed particles were supported by 60-mesh stainless steel screens at both . ends of the test section. At the exit end of each glass column was placed a l/2-inch tee. There was then 1-1/2 inches to 2 inches of packed bed before and after the : ”5H5“? . t pressure taps. This arrangement served to eliminate entrance and exit effects for the pressure drop readings. Rotometer The Brooks-Mite is a small variable area type flow meter. The tapered metering tube is machined directly into a clear acrylic plastic block, and a standard direct reading scale engraved on the meter body. A metering float moves vertically in the tapered tube to indicate rate of flow. Figure C-l illustrates the assembled rotometer. Pressure Transducer The transducer circuit is valved to permit either of two variable reluctance differential pressure trans— ducers to be connected to the two taps of the test section or to the calibration system. The transducer signal is (l) (2) (3) (4) (5) 255 Meter Body Float Two End Plugs and O-Rings Two Spring Float Stops Two Adapters and O-Rings Figure C-l Brooks-Mite Rotometer 256 amplified (Model CD 25, Pace) and recorded. The cali- bration system consisted of two reservoirs containing distilled water connected to the valving assembly by copper tubing to avoid expansion during calibration, the end of which is connected under the water level. While the down stream reservoir is always open to the atmosphere, the air space over the upstream reservoir can be pres- surized by a nitrogen tank. The air space is also con- nected to one leg of a mercury manometer so that the pres- ' sure can be determined. The level of the manometer legs «‘34.. - - ’1' is read with a cathetometer accurate to 0.005 cm. The transducer circuit is also valved to allow any portion to be flushed with distilled water. Measurement of Void Fraction The void factor was determined by the following procedure. First, the graduated cylinder was filled with water. The height of the water level measured with the cathetometer was X cms. Then, dry glass beads were poured into the cylinder. The cylinder was vibrated until the beads settled down uniformly. The water level in the cylinder rose because the beads were added. The height of the beads in the cylinder was 2 cms., whereas the new height of the water level was Y cms. These heights are proportional to the corresponding volumes. 257 Apparent volume of bed = z cms. Volume of beads = (Y-X) cms. Void fraction = (l-(Y-X)/Z) As X, Y, and Z were known, 5 was calculated accordingly. §C.2 Experimental Procedure Packing procedure was important as there was a possibility of air entrapment in the bed during packing. It was absolutely required that there be no air trapped in the column when the experiments were performed. The glass beads were soaked for a day or more in a beaker filled with distilled water. The test section was first assembled and bolted together. With the glass column filled with the fluid to be tested in a vertical position, particles were dropped into the glass column so as to fill the column to a height of about 2 inches. The partially filled bed was then tamped with a plunger. This was repeated for each 2 inches of bed height until the bed was complete. The bed was just barely overfilled to the level at which the screen support was to be fixed. When the screen was tightened down in place, the compressive force of the screen on the particles served to fix the bed in place. In this way subsequent handling of the column did not disturb the bed structure. About 6 inches of glass tube was connected at both ends of the column so as to avoid end effects. 258 The column was then inserted into the flow loop. All air bubbles were flushed from the lines and from the bed completely. Upon the completion of the flushing pro- cedure, the test fluid was allowed to flow through the test section at rates greater than would be expected in the experimental run. Any entrapped air bubbles were usually removed by this step. When a visual examination of the bed did not reveal any air bubbles in the system, a run was made. All electrical circuits were switched on for about 5 minutes before a run was made. Then, the test fluid was forced to flow through the packed column. The constant liquid level above the bed was achieved by apply- ing the gas pressure to the top of the fluid level in the bed. Constant flow rate, constant fluid level at the top of the bed, and the constant reading of the pressure drop were indications of steady flow. The valve on gas tank B was adjusted until the above three observations remained constant. Sometimes, the valve on the connection at the bottom of the bed was also adjusted. The entire system was held at equilibrium for at least five minutes before taking a measurement. The flow rate was measured by a rotometer connected at the bottom of the bed. Pressure drop for the flow rate was transferred into the differen- tial pressure transducer, of which the signal was ampli- fied and recorded. The same procedure was applied for the different flow rates. 259 For any given flow rate the temperature of the reservoir fluid was adjusted such that the temperature of the fluid leaving the test section was 21°C. The maximum difference in these temperatures was not more than 1°C. After the experimental run was completed, the test section was removed from the flow loop, dismantled, and washed thoroughly with distilled water. The glass beads were cleaned in a beaker with several rinsings of dis- tilled water. Next, the glass beads were soaked in dis- tilled water and allowed to stand overnight. The next day they were rinsed again and dried with room air in a ventilation hood. A clean particle was distinguished by the lack of white polymer dried onto the surface of the particles. Then, they were soaked again for the follow- ing experiment. Finally, the pressure transducer system was flushed with distilled water to remove polymer solu- tions from the line. APPENDIX D SUMMARY OF VISCOMETRIC DATA STEADY STATE SHEAR STRESS MEASUREMENT With 10 cm. platen 2.0083° cone 1/16" torsion bar KT = 8.21 260 TABLE D-l. Results for the Weissenberg Rheogoniometer for 0.50% Polyacrylamide solution at 21°C Y T -l A 12 2 n G.B. (sec ) T (dyne/cm. ) (gr./cm. sec) 5.6 0.00675 0.036 0.2948 43.6745 5.4 0.01076 0.058 0.4748 44.1248 5.2 0.01697 0.090 0.7389 43.5415 5.0 0.02689 0.140 1.1494 42.7445 4.8 0.04272 0.205 1.6831 39.3972 4.6 0.0675 0.295 2.4220 35.8807 4.4 0.1076 0.415 3.4072 31.6650 4.2 0.1697 0.565 4.6387 27.3344 4.0 0.2689 0.765 6.2807 23.3563 3.8 0.4272 1.030 8.4563 19.7947 3.6 0.6750 1.32 10.8372 16.0551 3.4 1.076 1.63 13.3823 12.4371 3.2 1.697 2.05 16.8305 9.9178 3.0 2.689 2.60 21.346 7.9383 2.8 4.272 3.20 26.272 6.1498 2.6 6.750 3.85 31.6085 4.6827 2.4 10.76 4.68 38.4228 3.5709 2.2 16.97 5.60 45.976 2.7093 2.0 26.89 6.78 55.6638 2.0701 1.8 42.72 8.18 67.1578 1.5720 1.6 67.50 9.80 80.4580 1.1920 1.4 107.6 12.0 98.5200 0.9156 1.2 169.7 14.4 118.2240 0.6967 1.0 268.9 17.8 146.1380 0.5435 0.9 339.4 19.5 160.095 0.4717 0.8 427.2 21.7 178.157 0.4170 0.7 537.2 24.1 197.861 0.3683 0.6 675.0 26.5 217.565 0.3223 0.5 851.0 30.0 246.300 0.2894 261 TABLE D-2. Results for the Weissenberg Rheogoniometer for 0.25% Polyacrylamide solution at 21°C Y-1 A T12 2 ” G.B. (sec ) T (dyne/cm. ) (gr./cm. sec) 5.0 0.02689 0.062 0.50902 18.9297 4.8 0.04272 0.097 0.7964 18.6427 4.6 0.06750 0.147 1.2106 17.9356 4.4 0.1076 0.214 1.7577 16.3354 4.2 0.1697 0.300 2.4769 14.5956 4.0 0.2689 0.425 3.4879 12.9712 3.8 0.4272 0.545 4.4751 10.4755 3.6 0.6750 0.712 5.8480 8.6638 3.4 1.076 0.910 7.4598 6.9329 3.2 1.697 1.150 9.4120 5.5463 3.0 2.689 1.390 11.4131 4.2444 2.8 4.272 1.730 14.1992 3.3238 2.6 6.750 2.150 17.6161 2.6098 2.4 10.76 2.670 21.9579 2.0407 2.2 16.97 3.280 26.9195 1.5863 2.0 26.89 4.060 33.3032 1.2385 1.8 42.72 4.82 39.5928 0.9268 1.6 67.50 5.94 48.8025 0.7230 1.4 107.6 7.30 59.9547 0.5572 1.2 169.7 8.85 72.6316 0.4280 1.0 268.9 11.10 91.1571 0.3390 0.9 339.4 12.43 102.0236 0.3006 0.8 427.2 13.42 110.2176 0.2580 0.7 537.2 15.10 123.9857 0.2308 0.6 675.0 16.50 135.4650 0.2000 0.5 851.0 18.80 154.3480 0.1837 262 TABLE D-3. Results for the Weissenberg Rheogoniometer for 0.10% Polyacrylamide solution at 21°C Y_1 A I12 2 n G.B. (sec ) T (dyne/cm. ) (gr./cm. sec) 4.8 0.04272 0.032 0.2617 6.1248 4.6 0.0675 0.050 0.4122 6.1064 4.4 0.1076 0.079 0.6487 6.0286 4.2 0.1697 0.119 0.9783 5.7654 4.0 0.2689 0.167 1.3736 5.1084 3.8 0.4272 0.238 1.9557 4.5781 3.6 0.675 0.322 2.6451 3.9187 3.4 1.076 0.432 3.5480 3.2974 3.2 1.697 0.556 4.5657 2.6905 3.0 2.689 0.737 6.0505 2.2501 2.8 4.272 0.890 7.3187 1.7132 2.6 6.750 1.110 9.1266 1.3521 2.4 10.76 1.400 11.5670 1.0750 2.2 16.97 1.730 14.1750 0.8353 2.0 26.89 2.150 17.6586 0.6567 1.8 42.72 2.610 21.4539 0.5022 1.6 67.50 3.280 26.9190 0.3988 1.4 107.6 4.090 33.5712 0.3120 1.2 169.7 5.130 42.1195 0.2482 1.0 268.9 6.320 51.8977 0.1930 0.9 339.4 7.280 59.7344 0.1760 0.8 427.2 8.230 67.5830 0.1582 0.7 537.2 9.300 76.2824 0.1420 0.6 675.0 10.36 85.0500 0.1260 0.5 851.0 11.73 96.3332 0.1132 263 TABLE D-4. Results for the Weissenberg Rheogoniometer for 0.05% Polyacrylamide solution at 21°C Y T -1 A 12 G.B. (sec T (dyne/cm. (gr./cm. sec) 4.4 0.1076 0.034 0.2784 2.5872 4.2 0.1697 0.054 0.4434 2.6128 4.0 0.2689 0.085 0.7012 2.6078 3.8 0.4272 0.127 1.0433 2.4422 3.6 0.675 0.181 1.4868 2.2027 3.4 1.076 0.251 2.0605 1.9150 3.2 1.697 0.329 2.6987 1.5903 3.0 2.689 0.423 3.4688 1.2900 2.8 4.272 0.522 4.2848 1.0030 2.6 6.750 0.675 5.5377 0.8204 2.4 10.76 0.867 7.1145 0.6612 2.2 16.97 1.120 9.1688 0.5403 2.0 26.89 1.440 11.7966 0.4387 1.8 42.72 2.000 16.3916 0.3837 1.6 67.50 2.300 18.8865 0.2798 1.4 107.6 2.880 23.6827 0.2201 1.2 169.7 3.720 30.5120 0.1798 1.0 268.9 4.640 38.0762 0.1416 0.9 339.4 5.400 44.1220 0.1300 0.8 427.2 5.940 48.7862 0.1142 0.7 537.2 6.75 55.4390 0.1032 0.6 675.0 7.670 62.9775 0.09325 0.5 851.0 9.320 76.5049 0.08992 E” ‘- APPENDIX E SUMMARY OF VISCOMETRIC DATA STEADY STATE NORMAL STRESS MEASUREMENT With 10 cm. platen 2.0083° cone 1/16" normal force spring KN = 520.0 264 TABLE E-l. Results for the Weissenberg Rheogoniometer for 0.50% Polyacrylamide Solution at 21°C -(T11-T22) (dyne/cm?) Y- G.B. (sec 1) AN Measurement Correction Value 3.0 2.689 0.008 4.160 . . 4.160 2.8 4.272 0.020 10.40 . . 10.40 2.6 6.750 0.046 23.92 . . 23.92 2.4 10.76 0.110 57.21 . . 57.21 2.2 16.97 0.220 114.4 . . 114.4 2.0 26.89 0.390 202.8 . . 202.8 1.8 42.72 0.700 364.0 8.000 372.0 1.6 67.50 1.040 540.8 22.00 562.8 1.4 107.6 1.350 702.0 56.00 758.0 1.2 169.7 1.920 998.4 140.0 1138.4 1.0 268.9 2.420 1258.4 340.0 1598.4 0.9 339.4 2.520 1310.4 540.0 1850.4 0.8 427.2 2.650 1378.0 860.0 2238.0 0.7 537.2 2.440 1268.8 1330. 2598.8 0.6 675.0 1.840 956.80 2150. 3106.8 0.5 851.0 0.800 416.00 3300. 3716.0 265 TABLE E-2. Results for the weissenberg Rheogoniometer. -(T -T ) (dyne/cm?) , 11 22 Y-1 A G.B. (sec ) N Measurement Correction Value For 0.25 % Polyacrylamide Solution at 21°C 2.4 10.76 0.019 9.880 . . 9.880 2.2 16.97 0.048 24.96 . . 24.96 2.0 26.89 0.105 54.60 . . 54.60 1.8 42.72 0.191 99.20 8.000 107.2 1.6 67.50 0.340 176.8 22.00 198.8 1.4 107.6 0.585 304.2 56.00 360.2 1.2 169.7 0.790 410.8 140.0 550.8 1.0 268.9 0.925 481.0 340.0 821.0 0.9 339.4 0.805 418.6 540.0 958.6 0.8 427.2 0.750 390.0 860.0 1250.0 0.7 537.2 0.387 201.2 1330. 1531.2 0.6 675.0 -0.750 -390.0 2150. 1760.0 0.5 851.0 -1.800 -936.0 3300. 2364.0 For 0.10 % Polyacrylamide Solution at 21°C 2.0 26.80 0.020 10.40 . . 10.40 1.8 42.72 0.038 19.60 8.000 27.60 1.6 67.50 0.083 43.20 22.00 65.20 1.4 107.6 0.162 84.40 56.00 140.4 1.2 169.7 0.215 118.8 140.0 251.8 1.0 268.9 0.272 141.2 340.0 481.2 0.9 339.4 0.154 80.20 540.0 620.2 0.8 427.2 —0.070 -36.40 860.0 823.6 0.7 537.2 —O.660 -343.2 1330. 986.8 0.6 675.0 -1.600 -832.0 2150. 1318. 0.5 851.0 -3.250 -1690. 3300. 1610. For 0.05 % Polyacrylamide Solution at 21°C 1.8 42.72 . . . . 8.000 8.000 1.6 67.50 -0.005 -2.600 22.00 19.40 1.4 107.6 -0.006 -3.120 56.00 52.90 1.2 169.7 -0.007 -3.640 140.0 136.4 1.0 268.9 -0.225 -117.0 340.0 223.0 0.9 339.4 -0.280 -145.6 540.0 394.4 0.8 427.2 -0.580 -301.6 860.0 558.4 0.7 537.2 -1.330 -691.6 1330. 638.4 0.6 675.0 -2.580 -1341.6 2150. 808.4 0.5 851.0 -4.500 -2340.0 3300. 960.0 APPENDIX F SUMMARY OF VISCOMETRIC DATA H(T), RELAXATION TIME SPECTRUM VS. t, TIME 266 TABLE F-l. Results of the Viscometric Experiments for 0.5% Separan Solution at 21°C A 2 1 t(sec) T 112(t) dyne/cm d112(t)/d1n t H(T) m N H*(I) ?°‘= 0.04272 sec-1 .03 .180 1.4778 .0739 57.654 .55 .619 35.7455 .05 .175 1.4368 .0928 43.433 .59 .634 27.5365 .07 .172 1.4121 .0952 31.847 .61 .641 20.4139 .10 .168 1.3793 .1199 28.059 .65 .727 20.3988 .15 .160 1.3136 .1314 20.499 .68 .754 15.4562 .30 .148 1.2151 .1494 11.659 .70 .771 8.9890 .50 .140 1.1494 .1683 7.879 .72 .790 6.2244 1.0 .127 1.0427 .2069 4.843 .76 .826 4.0003 1.5 .112 .9195 .2611 4.074 .80 .859 3.4995 2.0 .100 .8210 .2997 3.507 .85 .901 3.1598 3.0 .086 .7061 .2956 2.306 .90 .935 2.1561 5.0 .064 .5254 .2759 1.291 .94 .963 1.2432 7.0 .055 .4516 .2742 .917 1.0 1.00 .9170 10.0 .040 .3284 .2726 .638 1.8 1.042 .6647 15.0 .0285 .2340 .2874 .448 1.17 1.078 .4829 20.0 .0180 .1478 .2627 .308 1.29 1.112 .3424 30.0 .0062 .0509 .1970 .154 1.62 1.116 .1718 50.0 0 -1 ?o a 0.2689 sec .03 .76 6.2396 .3284 40.709 .40 .564 22.9598 .05 .74 6.0754 .4269- 31.753 .46 .586 18.6072 .07 .72 5.9112 .5008 26.606 .49 .597 15.8837 .10 .70 5.7470 .6568 24.425 .51 .604 14.7527 .15 .660 5.4186 .8210 20.355 .62 .645 13.1289 .30 .572 4.6961 1.1822 14.655 .70 .771 11.2990 .50 .500 4.1050 1.2216 9.086 .80 .859 7.8048 1.0 .405 3.3251 1.1987 4.458 .95 .969 4.3198 1.5 .342 2.8078 1.1822 2.931 1.00 1.00 2.9310 2.0 .300 2.4630 1.1757 2.186 1.05 1.026 2.2428 3.0 .240 1.9704 1.1248 1.394 1.18 1.082 1.5083 5.0 .173 1.4203 .8210 .611 1.22 1.094 .6684 7.0 .131 1.0765 .8144 .433 1.30 1.114 .4823 10.0 .101 .8292 .7553 .281 1.40 1.127 .3166 15.0 .0692 .5681 .6584 .163 1.50 1.128 .1838 20.0 .0445 .3653 .5419 .101 1.60 1.119 .1130 30.0 .0225 .1847 .3941 .0489 1.68 1.104 .0539 50.0 .0040 .0328 .3153 .0235 1.80 1.074 .0252 Notes: H(T) is experimentally determined relaxation time spectrum. H*(T) is corrected relaxation time spectrum. 267 TABLE F-2. Results of the Viscometric Experiments for 0.25% Separan Solution at 21°C t(sec) A,r 112(t) dyne/cm2 d112(t)/d£n t 11(1)l m N H*(T) yo = 0.2689 sec-1 .03 .313 2.5697 .1642 20.355 .21 .494 10.0553 .05 .305 2.5041 .1888 14.045 .35 .545 7.6545 .07 .293 2.4055 .2135 11.340 .48 .593 6.7246 .10 .286 2.3481 .2742 10.198 .57 .626 6.3839 .15 .270 2.2167 .3612 8.956 .65 .727 6.5110 .30 .230 1.8883 .5254 6.513 .80 .859 5.5946 .50 .194 1.5927 .5419 4.030 .95 .969 3.9050 1.0 .151 1.2397 .5254 1.954 1.02 1.01 1.9735 1.5 .123 1.0098 .5090 1.262 1.15 1.071 1.3516 2.0 .109 .8949 .4762 .885 1.18 1.082 .9575 3.0 .084 .6896 .4105 .509 1.24 1.100 .5599 5.0 .0585 .4803 .3645 .271 1.45 1.128 .3056 7.0 .0455 .3736 .3415 .181 1.50 1.128 .2041 10.0 .0312 .2562 .2874 .107 1.56 1.124 .1202 15.0 .020 .1642 .1855 .046 1.60 1.119 .0514 20.0 .016 .1314 .1543 .0287 1.80 1.074 .0308 30.0 .009 .0739 .1149 .0142 2.50 .753 .0106 _ -1 9° — 0.1697 sec .03 .207 1.6995 .0903 17.739 .16 .476 8.4437 .05 .201 1.6502 .1289 15.191 .30 .527 8.0056 .07 .197 1.6174 .1560 13.132 .37 .553 7.2619 .10 .189 1.5517 .2036 11.998 .50 .600 7.1988 .15 .177 1.4532 .2808 11.031 .60 .672 7.4128 .30 .150 1.2315 .2734 5.370 .75 .816 4.3819 .50 .139 1.1412 .2504 2.951 .88 .922 2.7208 1.0 .110 .9031 .2750 1.621 1.0 1.00 1.6210 1.5 .095 .7800 .3268 1.284 1.11 1.053 1.3520 2.0 .080 .6568 .3777 1.113 1.15 1.071 1.1920 3.0 .065 .5337 .3284 .645 1.17 1.078 .6953 5.0 .044 .3612 .2915 .345 1.18 1.082 .3732 7.0 .0296 .2430 .2586 .218 1.18 1.082 .2358 10.0 .0208 .1708 .2463 .145 1.22 1.094 .1586 15.0 .0072 .0591 .2463 .0968 1.24 1.100 .1200 20.0 0 268 TABLE F«3. Results of the Viscometric Experiments for 0.10% Separan Solution at 21°C 2 . 7 1 t(sec) AT I12(t) dyne/cm dtl,(t)/din t H(r) m N H*(r) 1 7° = 1.076 sec- .03 .2275 1.8678 .2135 6.613 .41 .568 3.7561 .05 .216 1.17734 .2594 4.822 .50 .600 2.8932 .07 .206 1.6913 .3005 3.989 .60 .672 2.6806 .10 .191 1.5681 .3612 3.357 .70 .771 2.5882 .15 .1705 1.3998 .4220 2.615 .77 .834 2.1809 .30 .1328 1.0903 .4516 1.399 .96 .976 1.3654 .50 .107 .8785 .4433 .824 1.05 1.026 .8454 1.0 .0695 .5706 .3908 .363 1.26 1.104 .4007 1.5 .055 .4516 .3350 .208 1.33 1.119 .2327 2.0 .0394 .3235 .2791 .130 1.37 1.124 .1461 3.0 .0278 .2282 .2233 .069 1.44 1.128 .0778 5.0 .0158 .1297 .1855 .0345 1.50 1.128 .0389 7.0 .0098 .0805 .1642 .0218 1.53 1.126 .0245 10.0 0 _ —1 $0 — 2.689 sec .03 .392 3.2183 .5008 49.799 .30 .527 26.2440 .05 .370 3.0377 .5419 40.302 .37 .553 22.2870 .0/ .346 2.8407 .6272 33.323 .45 .582 19.3939 .10 .315 2.5862 .7832 29.127 .52 .608 17.7092 .15 .270 2.2167 1.0180 25.240 .67 .746 18.8290 .30 .172 1.4121 1.1166 13.841 .95 .969 13.4119 .50 .120 .9852 .5714 4.250 1.2 1.089 4.6282 1 0 .082 .6732 .4204 1.563 1.54 1.126 1.7599 1 5 .058 .4762 .3875 .961 1.70 1.100 1.0571 2.0 .048 .3941 .3875 .721 1.83 1.065 .7678 3.0 .026 .2135 .3514 .436 2.05 .978 .4264 5 0 .0067 .0550 .2085 .155 2.54 .732 .1134 7 0 .0025 .0205 .0657 .035 2.89 .552 .0193 10.0 0 269 TABLE F~4. Results of the Viscometric Experiments for 0.05% Separan Solution at 21°C t(sec) AT 112(t) dyne/cm2 d112(t)/d£n t H(I)1 m N H*(T) . _ -1 Yo — 2.689 sec .03 .140 1.1494 .2709 33.585 .63 .710 23.8453 .05 .126 1.0345 .2956 22.983 .72 .790 18.1565 .07 .115 .9442 .3243 17.229 .80 .859 14.7997 .10 .0975 .8005 .4064 15.113 .90 .935 14.1306 .15 .0735 .6034 .3612 8.956 1.10 1.050 9.4038 .3 .0510 .4187 .2463 3.053 1.26 1.104 3.3705 .5 .0350 .2874 .1970 1.466 1.36 1.123 1.6463 1.0 .0180 .1478 .1535 .571 1.47 1.128 .6440 1.5 .0134 .1100 .1297 .322 1.55 1.127 .3628 2.0 .0067 .0550 .1067 .198 1.55 1.127 .2231 3.0 .0045 .0369 .0961 .119 1.55 1.127 .1341 5.0 0 _ -1 9° — 10.76 sec .03 .330 2.7093 1.0180 3.154 .55 .619 1.9523 .05 .250 2.0525 1.3136 2.442 .74 .808 1.9731 .07 .200 1.6420 1.3300 1.766 .97 .982 1.7342 .10 .120 .9852 1.5927 1.480 1.05 1.026 1.5184 .15 .074 .6075 .4926 .305 1.27 1.107 .3376 .30 .050 .4105 .2463 .0763 1.52 1.127 .0859 .50 .0365 .2997 .2135 .0397 1.74 1.089 .0432 1.0 .023 .1888 .1396 .0130 2.05 .978 .0127 1.5 .019 .1560 .0985 .0061 2.20 .908 .0055 2.0 .016 .1314 .0821 .0038 2.24 .887 .0033 3.0 .014 .1149 .0640 .0020 2.28 .867 .0017 5.0 .0080 .0657 .0361 .00067 2.33 .842 .0005 7.0 .0035 .0287 10.0 0 APPENDIX G MODEL INDEPENDENT PARAMETERS 270 omno. mmoo. owmm. m.hmv mnmo. mmoo. moom. e.mmm mmmo. mHHo. ommm. m.mmm mvoo. Hmao. omme. n.mma mmmo. Hamo. momm. m.noa momo. mmeo. ommh. om.nm mmmo. mmmo. mmmm. mn.mv mmao. mmoo. hmama. o.Hmm oamo. mmno. mmmm.a mm.m~ mmao. mmoo. ooom. o.mhm memo. mmmo. mmmm.a hm.ma ommo. mmoo. momm. ~.nmm mavo. mmmo. hoeo.~ mn.oa Aummvc\o AEU\HmV® Aomw Euvc Aauommvv Aummvc\o AEU\umv® Acum Euvc Aanommvw anm um coausHom cmummmm mm~.o now mnoumEmnmm usmocwmoocH Homo: .muo mamma mnao. Hmoo. vmmm. o.Hmm mmoH. mmma. omma.a om.nm Hamo. mmoo. mmmm. o.m>m mama. mmom. omhm.a mo.mv oomo. omoo. mmmm. ~.nmm mmma. oomm. Hobo.m mm.m~ mono. mmao. onae. m.nmv mmva. whom. mmon.m mm.ma ammo. omao. have. ¢.mmm vmma. Home. monm.m m>.oa homo. ammo. mmvm. m.mm~ HNHH. memm. nmmm.v m>.m nmmo. mmmo. nmmm. n.mma sumo. momm. mmva.m mnm.o vano. vmmo. mmam. m.hoH mmno. mmnm. mmmm.n mmm.m Aommvc\o AEO\umvo Aom” Soy: “anommv¢ Aowmvc\o A80\umvo Acom Euvc Aauoomow Uon um GOAusaom cmummmm wom.o now mumuoEmumm usmonmmoocH H0002 .Huw mqmda 271 mamo. «moo. mmoa. m.nmm mmmo. omoo. Neda. N.nmv ammo. mmoo. ooma. v.mmm mmmo. ovoo. mama. m.mmm mvao. maoo. mmmmo. o.Hmm ammo. hooo. mama. n.mma NmHo. maoo. mmmmo. o.mhm oomo. mwoo. Homm. m.noa com c o Eu no com Eu omw w omm c 0 Eu no G own 80 com + 11\1\514217111\1\114:.:-1 oon um coHuSHom cmummmm wmo.o now mumumfimumm ucmpcwmmocH Homo: .ono mummy vmmo. mvoo. mmmm. ~.>mo Homo. mmoo. omwa. «.mmm memo. mmoo. ommH. o.mm~ ammo. smoo. «mom. h.mmm homo. omoo. «mad. o.Hmm mmmo. ammo. omHm. m.hoH mmmo. mmoo. omNH. o.mnm ammo. memo. momm. m.nm ammo. «moo. omvH. ~.nmm Homo. ammo. mmom. Nh.~v oomvc 9 Eu Home 0mm Eu A owmvm omm c o Eu no G omm Eu umw v 1 \ 1 \ 1 ... E T 1 o\ 1 \ o 1 u E 17 o ooam um conusaom canaamw woa.o you mumumsmumm “smegmawocH Hone: .muw mamas APPENDIX H MODEL INDEPENDENT STRESS PARAMETERS It - I I Sw = 11 22 , Recoverable 12 Shear Strain T12 Gw = ?T_ , Shear Modulus w Weissenberg Ratio 272 TABLE H-l. Model Independent Stress Parameters for 0.5% Polyacrylamide Solution at 21°C ( 4 _1 T12 2 T11‘T222 T11'T22 G sec ) (dyne/cm ) (dyne/cm ) 2le w w 2.689 21.346 4.16 0.0974 0.1948 109.5790 4.272 26.272 10.40 0.1979 0.3958 66.3769 6.750 31.6085 23.92 0.3783 0.7566 41.7768 10.76 38.4228 57.21 0.7444 1.4888 25.8087 16.97 45.9760 114.4 1.2441 2.4882 18.4776 26.89 55.6638 202.8 1.8216 3.6432 15.2788 42.72 67.1578 372.0 2.7700 5.5400 12.1223 67.50 80.4580 562.8 3.4974 6.9948 11.5025 107.6 98.5200 758.0 3.8469 7.6938 12.8051 169.7 118.224 1138.4 4.8145 9.6290 12.2779 268.9 146.138 1598.4 5.4688 10.9376 13.3610 339.4 160.095 1850.4 5.7790 11.5580 13.8514 427.2 178.157 2238.0 6.2809 12.5618 14.1824 537.2 197.861 2598.8 6.5672 13.1344 15.0643 675.0 217.565 3106.8 7.1399 14.2798 15.2358 851.0 246.300 3716.0 7.5436 15.0872 16.3250 273 TABLE H-2. Model Independent Stress Parameters for 0.25% Polyacrylamide Solution at 21°C 4 _1 T12 2 T11"‘22 T11‘T22 S G (sec ) (dyne/cm ) (dyne/cmz) 2le w w 10.76 21.9579 9.88 0.2249 0.4498 48.8170 16.97 26.9195 24.96 0.4636 0.9272 29.0331 26.89 33.3032 54.60 0.8197 1.6394 20.3142 42.72 39.5928 107.2 1.3537 2.7074 14.6239 67.50 48.8025 198.8 2.0367 4.0734 11.9807 107.6 59.9547 360.2 3.0039 6.0078 9.9794 169.7 72.6316 550.8 3.7917 7.5834 9.5777 268.9 95.1571 821.0 4.5032 9.0064 10.5654 339.4 102.0236 958.6 4.6979 9.3958 10.8584 427.2 110.2176 1250.0 5.6706 11.3412 9.7183 537.2 123.9857 1531.2 6.1749 12.3498 10.0394 675.0 135.4650 1760.0 6.4961 12.9922 10.4266 851.0 154.3480 2364.0 7.6580 15.3160 10.0775 274 TABLE H-3. Model Independent Stress Parameters for 0.1% Polyacrylamide Solution at 21°C _1 T12 T11’T22 T11"‘22 S G (sec ) (dyne/cmz) (dyne/cmz) 2le w w 26.89 17.6586 10.4 0.2944 0.5888 29.9908 42.72 21.4539 27.6 0.6432 1.2864 16.6774 67.50 26.9190 65.2 1.2110 2.4220 11.1143 107.6 33.5712 140.4 2.0910 4.1820 8.0275 169.7 42.1195 251.8 2.9891 5.9782 7.0455 268.9 51.8977 481.2 4.6360 9.2720 5.5972 339.4 59.7344 620.2 5.1913 10.3826 5.7533 427.2 67.5830 823.6 6.0932 12.1864 5.5457 537.2 76.2824 986.8 6.4680 12.9360 5.8969 675.0 85.0500 1318.0 7.7483 15.4966 5.4883 851.0 96.3332 1610.0 8.3564 16.7128 5.7640 TABLE H-4. Model Independent Stress Parameters for 0.05% Polyacrylamide Solution at 21°C y -1 T12 T11"‘22 11’ 22 S G (sec ) (dyne/cmz) (dyne/cmz) 2I12 w w 67.50 18.8865 19.4 0.5135 1.0270 18.3899 107.6 23.6827 52.9 1.1168 2.2336 10.6029 169.7 30.5120 136.4 2.2351 4.4702 6.8256 268.9 38.0762 223.0 2.9283 5.8566 6.5014 339.4 44.1220 394.4 4.4694 8.9388 4.9360 427.2 48.7862 558.4 5.7229 11.4458 4.2623 537.2 55.4390 638.4 5.7576 11.5152 4.8144 675.0 62.9775 808.4 6.4181 12.8362 4.9062 851.0 76.5049 960.0 6.2741 12.5482 6.0968 APPENDIX I SUMMARY OF VISCOMETRIC DATA AND ELLIS MODEL CALCULATION _.v 275 am.m| mmmm. omaw. mma.mna N.n~v mm.m| move. bane. mmo.oma «.mmm mm.m| maam. mmvm. omma.mva m.mmm ma.ou mmmm. bmmm. ov-.maa >.mma ma.a| amom. mmam. oomm.mm m.>oa mm.a moam.a omma.a ommv.om om.nm mm.o| mmmm.a omhm.a mhma.nm mn.~¢ m¢.a| movo.m aoho.~ mmmm.mm mm.m~ mh.an mmmm.m mmoh.m omnm.mo hm.ma mm.v| mvov.m mohm.m mmmv.mm mh.oa mm.mn a-v.v nmwm.v momm.am mh.m am.m| ommm.m mmea.m omnm.m~ ~h~.e mh.mn omm~.> mmmm.b omvm.am mmm.~ am.mn omnm.m mnam.m momm.ma 5mm.a no.o mmv¢.ma ahmv.~a mmmm.ma mho.a no.4: mmnm.ma ammo.ma mnmm.oa omnm. w~.mn ~mma.ma nvmn.ma mmmv.m Nome. om.m mmom.m~ mmmm.mm homm.m mmmm. o~.¢ mmmm.mm evmm.hm nmmm.v hmma. hm.m mmmm.~m ommm.am moov.m mooa. am.o Nmo~.mm nomm.mm om~¢.~ mnmo. mm.a| momm.mm Numm.mm ammm.a mhmeo. em.vu mahn.o¢ mvoh.mv omva.a mmmmo. nm.mu mavo.~v maem.mv moms. nomao. mm.m| Nmm>.~v moma.v¢ mono. mnoao. hm.an mmma.mv mvhm.me meow. mhmoo. .oamo owm Eu .oamo umm Eu .pmxm .Na 0mm ..umxm c ca Houum w Ilmmfll. c .Ilmmll c NEO\mc>o P an o anm um coausaom moafimamnummaom mom.o How umpofioacomomgm mnmncmmmamz on» How muaSmwm .aIH mamda 276 m>.vmu whoa. mmma. omvm.vma o.amm am.~a| mhba. ooom. ommv.mma o.mnm mm.oal aaom. momm. nmwm.m~a N.wmm mm.m| 55mm. ommm. mham.oaa N.nmv mm.ma| mmmm. moom. mmmo.~oa ¢.mmm mo.m| oaam. ommm. abma.am m.mm~ mo.o mmmo. ommv. mamm.mh o.mma am.o aomm. momm. nvmm.mm m.noa no.~ amen. ommo. mmom.mo m.hm am.m mmmm. mmmm. mmmm.mm m>.~v -.a mmmm.a mmmm.a mmom.mm mm.mm mm.o mmmm.a mmmm.a mmao.mm hm.ma em.m nmma.m movo.m momm.am mm.oa om.m omem.m mmom.m amam.ma m>.m am.m ~5>m.m mmmm.m mama.va «mm.o mm.o avom.v ovem.o amav.aa mmm.m mm.m mvmh.m mmvm.m omav.m nom.a mm.~ mmva.> mmmm.m mmmv.m mno.a om.o omvn.m mmmm.m omvw.m mom. mm.o mavm.oa mmnv.oa amnv.v Nome. mo.m| mmea.ma manm.ma mnme.m mmmm. hm.mu omho.va mmmm.va mmnv.m mama. am.vu mmmm.ma vmmm.ma nnmh.a mnoa. mm.m| mom>.ma mmmm.na moam.a mnmo. mh.mn vamm.na nmvm.ma ammo. 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I no «mmauummoum 00m I a NO HMO HO mmm uouuo w umxocw oamoy mum mmz own 800. a c 00m 500. a c oama.mc lowimeo. oo $8.0 o.aN um ceausaom a > a cc.a uou mucmsaumaxm Iona 30am unnumcoo on» Io“ muaaumm .NIm mam m wom.o uOu mucwEaummxm ovum 30am ucmumcou on» uOu muaommm .vum mamdk APPENDIX S SUMMARY OF VISCOMETRIC DATA AND HERSCHEL-BULKLEY MODEL CALCULATION 316 oamo ooa x umxmc Icoamoc "mm chMasoamo ma uouum w womam>¢m mm.a| eoso.o maso.o smmo.am vv.s o.amm m.o mo.o smso.o smso.o maoa.mm sc.m o.msm m.o mm.m ammo.o smmo.o samo.mv am.m ¢.smm s.o om.o| smmo.o mmmo.o mmmm.mv ma.m m.smv m.o mm.v moaa.o vmoa.o mmss.mm mm.o a.mmm m.o mo.o| omma.o amma.o smmm.mm oa.v m.mmm o.a mm.au mmma.o mama.o mvmv.sm vm.m s.mma m.a am.o mmom.o maom.o mamo.mm mm.m m.soa o.a mm.o: mmmm.o masm.o vmmm.ma vm.m m.sm m.a mm.a| mmmm.o mamm.o amcv.ma mm.a ms.mc m.a ma.a: mamv.o vmmc.o amma.ma om.a mm.mm o.m sm.o| mmmm.o vmmm.o msmm.aa sm.a sm.ma m.m oa.o| omam.o mmam.o ommm.m om.a ms.oa v.m mm.m: mmom.a mamm.a mmmo.m oa.a ms.m m.m ms.o omsm.a mamm.a mamm.s so.o msm.c m.m mm.o amvs.m smms.m ammm.s om.o mmm.m oam mm.a mmso.v maoo.v ooms.m mm.o smm.a m.m ov.m ssmo.m mamm.m mmov.m ms.o mso.a v.m mmma.m cmmv.m sm.o msm.o m.m omsm.ma moms.m mm.o msmc.o m.m .oamoc aomm EO\umv aomm EU\umv amomm EO\HmV ac Aa ommvm .m.w mca uouum m oamoc umxmc may 0 oam um cOaucaom com 0 z m wmo.o now umumEOacomomzm oumncmmmamz mcu How muaowmm .anm mamde 317 sv.a| mmao.o mmvo.o oomm.mm mm.o o.amm m.o mm.m somo.o omeo.o smvo.mm mo.c o.msm m.o mm.o ommo.o mmmo.o omms.mm om.m c.smm s.o ov.m ommo.o mmmo.o moom.mm ma.m m.smv m.o om.o| mmmo.o mmmo.o mmmm.mm mm.m v.mmm m.o am.m mmso.o maso.o maaa.om mm.m m.mmm o.a vm.an ammo.o mmmo.o oomv.ma oo.m s.mma m.a mm.m amaa.o mmaa.o ooam.ma mm.a m.soa v.a mm.o: mama.o mmma.o oamm.oa om.a m.sm m.a mm.o mmma.o mvma.o ooam.m mo.a ms.mv m.a sm.a| osmm.o ammm.o smvo.s mm.o mm.mm o.m mm.o: mmcm.o mmvm.o mmmm.m as.o sm.ma m.m mm.o: cmmv.o mmm¢.o maco.m am.o ms.oa m.m om.mn momm.o ommm.o mmam.c mm.o ms.m m.m sm.o| oomm.o mmmm.o movm.m mv.o msm.v m.m mm.o: momm.a cmcm.a mmam.m oo.o mmm.m o.m vm.on ammm.a mosm.a mmvm.m av.o smm.a m.m ma.ou mamm.m mmmm.m mmaa.m mm.o mso.a v.m moss.m amcm.m am.o msm.o m.m mmoa.m msmm.m mm.o msmc.o m.m .oam0c aomm EU\umv Aomm EU\ro amowm EO\umv Bo aanommvm .m.u ca uouuo w oamoc c.9.ch map UoHN #0 cOauSaom cow 0 z m mm.o now uwumfioacomomcm mumncmmmamz mcu How muaSmom .mum mamce 318 sm.a msao.o msao.o omoo.ma mm.a o.amm m.o ma.m mmao.o mmao.o ommm.ma mm.a o.msm m.o oa.a mamo.o mamo.o oomv.aa om.a ¢.smm s.o ma.o| smmo.o smmo.o oama.oa mm.a m.smv mao ms.a ammo.o mmmo.o omsm.m mo.a a.mmm m.o mm.o mmmo.o mmmo.o ommm.s mm.o m.mmm o.a mm.m mmmo.o mmmo.o oscs.m os.o s.mma m.a mm.a mmao.o mmco.o asmm.c mm.o m.soa m.a ms.o memo.o mmmo.o mmmo.m vv.o m.sm m.a mo.a vmmo.o ssmo.o mamm.m mm.o ms.mv m.a mm.m ommo.o cmmo.o smam.m sm.o mm.mm o.m mm.o: omaa.o amaa.o «osm.a mm.o sm.ma m.m mm.o: mmma.o moma.o avms.a am.o ms.oa a.m mm.m: smom.o mmam.o mssc.a ma.o ms.o m.m mm.o cmmm.o mmmm.o mamm.a ma.o msm.v m.m mm.ma maav.o csmv.o cmva.a va.o mmm.m o.m mm.m vomm.o momm.o mmmm.o ma.o smm.a m.m as.o mosm.o mvmm.o oomm.o maa.o mso.a a.m mmam.a oamm.o oa.o msm.o m.m mmam.m mmmm.o ma.o msmw.o m.m .oamoc Aomm EO\HmV Aomm EO\Hm0 amomw EU\an Bo Aalommvm .m.o ca uouum m oamoc umxmc may ooam um :Caudaom mm 0 2 m mm.o How umumEOacomowcm mumncmmmamz on» How muaSmmm .mum mamda 319 ma.a mmoo.o smoo.o ommm.m oo.a o.amm m.o am.m soao.o moao.o samm.m mm.o o.msm m.o ma.mn maao.o maao.o smcm.m ss.o c.smm s.o am.m mmao.o amao.o mmma.m mm.o m.smv m.o va.m mmao.o mmao.o msmv.v mm.o a.mmm m.o mv.o| mmao.o mvao.o mmmm.m mm.o m.mmm o.a mm.m msao.o msao.o aaam.m mm.o s.mma m.a mo.al oamo.o mamo.o svmm.m mm.o m.soa m.a sm.o| mmmo.o vmmo.o smas.a am.o m.sm m.a sm.o momo.o vomo.o cmmm.a ma.o ms.m¢ m.a mm.a msmo.o mmmo.o mamm.o ma.o mm.mm o.m om.a mmao.o omco.o mams.o mmo.o sm.ma m.m os.o smmo.o mmmo.o osmm.o mso.o ms.oa v.m mo.an mmso.o mmso.o omam.o mmo.o ms.o m.m om.a: mmmo.o aaoa.o samc.o mmo.o msm.v m.m mm.m: ocma.o vsma.o mmmm.o mvo.o mmm.m o.m am.o| vcma.o mmma.o mcam.o mmo.o smm.a m.m os.o mmmm.o mmmm.o mmsm.o cmo.o mso.a a.m vmmm.o mmmm.o mmo.o msm.o m.m ammm.o aomm.o mmo.o msmv.o m.m .oamoc Aomm EU\umv aomm EO\nmv amomm EU\umv B< Aalommom .m.u ca Houum w oamoc umxmc may “coam um acausaom mm 0 z m mm.o now umumEOacomomcm mumncmmmamz map How muaommm .cim aamma APPENDIX T SUMMARY OF THE CONSTANT FLOW RATE EXPERIMENT DATA AND HERSCHEL-BULKLEY MODEL CALCULATION D 3 f* = iEHL .2} expt MG 2 I.J l - 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