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Multiphase Reactor And Polymerization System Hydrodynamics ADVANCES IN ENGINEERING FLUID MECHANICS SERIES

Nicholas P. Cheremisinoff, Editor


Nicholas P. Cheremisinoff, Editor

Gulf Publishing Company Houston, London, Paris, Zurich, Tokyo


Nicholas P. Cheremisino£f, Editor

in collaboration with— M. Abid A. Afacan K. M. Irdriss Ali M. A. Ali N. R. Amundson R. Aris U. D. N. Bajpai F. Berruti J. Bertrand N. Brauner Y. A. Buyevich J. B. L, M. Campos J. R. F. Guedes de Carvalho

J. Chaouki R. P. Chhabra K. S. Chian M. Chidambaram L. Choplin S. K. Das E. B. de la Fuente U. K. Ghosh R. O. E. Greiner V. K. Gupta P. K. Haider M. H. Han M. A. Kahn S. K. Kapbasov

J. Kaschta Y. Kawase J. K. Kim S. M. Kresta J. K. Kun D. M. Maron J. H. Masliyah H. A. Nasr-El-Din V. Nassehi Nivedita H. Orbey G. S. Patience M. Ravindranathan S. RavinHr'inathan

S. I. Sandler C. W. Stewart P. A. Tanguy L. Tassi K. C. Taylor A. Tecantel J. A. S. Teixeira K. Toi C. P. Tsonis S. N. Upadhyay E. Valles M. A. Villar C. Xuereb M. Yue


Copyright © 1996 by Gulf Publishing Company, Houston, Texas. All rights reserved. Printed in the United States of America. This book, or parts thereof, may not be reproduced in any form without permission of the publisher. Gulf Publishing Company Book Division P.O. Box 2608 • Houston, Texas 77252-2608 10

9 8 7 6 5 4 3 2 1

Library of Congress Cata!oging-in-Publication Data Cheremisinoff, Nicholas P. Advances in engineering fluid mechanics : multiphase reactor and polymerization system hydrodynamics / Nicholas P. Cheremisinoff, editor : in collaboration with M. Abid . . . [et al.]. p. cm. Includes bibliographical references and index. ISBN 0-88415-497-1 1. Polymers—Rheology. 2. Hydrodynamics. I. Abid, M. (Mohammed) II. Title. TP1092.C47 1996 668.9—dc20 95-51777 CIP Series ISBN 0-87201-492-4 Printed on Acid-Free Paper (oo)

CONTENTS

CONTRIBUTORS TO THIS VOLUME (For a note about the editor^ please see page xi)

viii

PREFACE

xii

1. The Viscosity of Liquid Hydrocarbons and their Mixtures 5. /. Sandler and H, Orbey 2. Experimental Studies for Characterization of Mixing Mechanisms J, K, Kim

1

25

3. Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns 49 J. R, F. Guedes de CarvalhOy J. B. L. M. Campos, and J. A. S, Teixeira 4. Numerical Solution of the Permeation, Sorption, and Desorption Rate Curves Incorporating the Dual-Mode Sorption and Transport Model K. Toi 5. Kinematic Viscosity and Viscous Flow in Binary Mixtures Containing Ethane-1,2-Diol L. Tassi 6. Reaction of a Continuous Mixture in a Bubbling Fluidized Bed A^. /?. Amundson and R. Arts 7. Fluid Dynamics of Coarse Dispersions y. A. Buyevich and S. K, Kapbasov 8. Combustion of Single Coal Particles in Turbulent Fluidized Beds P. K. Haider

67

79

105 119

167

9. Flow of Solids and Slurries in Rotary Drums H. A. Nasr-El-Diriy A. Afacan, and J. H. Masliyah

193

10. Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers G. 5. Patience^ J. Chaouki, and F. Berruti

255

11. Boundary Conditions Required for the CFD Simulation of Flows in Stirred Tanks 5. M. Kresta

297

12. Role of Interfacial Shear Modeling in Predicting Stability of Stratified Two-Phase Flow N. Brauner and D. M. Maron

317

13. Water Flow through Helical Coils in Turbulent Condition 5. K. Das 14. Modeling Coalescence of Bubble Clusters Rising Freely in Low-Viscosity Liquids C. W. Stewart

379

405

15. Oxygen Transfer in Non-Newtonian Fluids Stirred with a Helical Ribbon Screw Impeller 431 A. Tecante, E. B, de la Fuente, L. Chopliny and P. A. Tanguy 16. Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids in Stirred Tanks Equipped with Two-Blade Impellers C. Xuereb, M. Abid, and J. Bertrand

455

17. Non-Newtonian Liquid Flow through Globe and Gate Valves S. K. Das

487

18. Comparison of Numerical and Experimental Rheological Data of Homogeneous Non-Newtonian Suspensions V. Nassehi

507

19. Concentration Forcing of Isothermal Plug-Flow Reactors for Autocatalytic Reactions M. Chidambaram

525

20. Non-Newtonian Effects in Bubble Columns 539 R. P. Chhabra, U. K. Ghoshs Y. Kawase, and S. H. Upadhyay 21. Studies in Supported Titanium Catalyst System using Magnesium Dichloride-Alcohol Adduct V. K. Gupta, S. Ravindranathan, and M. Ravindranathan

571

22. Plasticizing Polyesters of Dimer Acids and 1,4-Butanediol U. D, N. Bajpai and Nivedita

583

Ti. Viscoelastic Properties of Model Silicone Networks with Pendant Chains M. A. Villar and E. M. Valles

599

24. Rheology of Water-Soluble Polymers used for Improved Oil Recovery H. A. Nasr-El-Din and K. C. Taylor

615

25. Relation of Rheological Properties of UV-Cured Films with Glass Transition Temperatures based on Fox Equation M. A. All, M. A. Kahn, K. M, I. All

669

26. Prediction and Calculation of the Shear Creep Behavior of Amorphous Polymers under Progressive Physical Aging R. O, E. Greiner and J. Kaschta

683

27. Die Extrusion Behavior of Carbon Black-Filled Block Copolymer Thermoplastic Elastomers J. K. Kim and M. H. Han

711

28. Polysulfides C. P. Tsonis

737

29. Properties and Applications of Thermoplastic Polyurethane Blends M. Yue and K. S. Chian

747

INDEX

763

Vll

CONTRIBUTORS TO THIS VOLUME

Mohammed Abid, Laboratoire de Genie Chimiqe URA CNRS 192, ENSIGC, 19 chemin de la Loge, 31078 Toulouse Cedex, FRANCE A. Afacan, Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, CANADA T6G 2G6 K. M. Irdriss Ali, Radiation and Polymer Chemistry Laboratory, Institute of Nuclear Science and Technology, Bangladesh Atomic Energy Commission, P.O. Box 3787, Bhaka, BANGLADESH M. Azam Ali, Radiation and Polymer Chemistry Laboratory, Institute of Nuclear Science and Technology, Bangladesh Atomic Energy Commission, P.O. Box 3787, Bhaka, BANGLADESH Neal R. Amundson, Department of Mathematics, University of Houston, Houston, Texas 77204, USA Rutherford Aris, Department of Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall, 421 Washington Avenue SE, Minneapolis, Minnesota 55455-0132, USA U. D. N. Bajpai, Polymer Research Laboratory, Department of Post-Graduate Studes and Research in Chemistry, R.D. University, Jabalpur—482001, M.P., INDIA Franco Berruti, Univeristy of Calgary, Calgary, Alberta, CANADA, 2TN 1N4 Joel Bertrand, Laboratoire de Genie Chiniqe URA CNRS 192, ENSIGC, 18 chemin de la Loge, 31078 Toulouse Cedex, FRANCE Neima Brauner, Department of Fluid Mechanics & Heat Transfer, School of Engineering, Tel-Aviv University, Tel-Aviv, 69978, ISRAEL Y. A. Buyevich, NASA Ames Research Center, Mail Stop 239-15, Moffett Field, California 94035-1000 U.S.A. J. B. L. M. Campos, Department of Chemical Engineering, University of Oporto, Oporto, PORTUGAL Jamal Chaouki, Ecole Polytechnique de Montreal, Montreal, Quebec, CANADA, H3C 3A7 R. P. Chhabra, Department of Chemical Engineering, Indian Institute of Technology, Kanpur, INDIA, 208016 K. S. Chian, School of Applied Science, Nanyang Technological University, Nanyand Avenue, SINGAPORE 2263

M. Chidambaram, Department of Chemical Engineering, Indian Institute of Technology, Madras 600 036 INDIA L. Choplin, GEMICO-ENSIC, 1 rue Grandville, B. P. 451, Nancy, 54001, FRANCE Supid Kumar Das, Chemical Engineering Department, Calcutta University, 92 A.P.C. Road, Calcutta—700 009, INDIA E. B. de la Fuente, Departamento de Alimentos y Biotechnologia, Facultad de Quimica— UNAM Mexico, D.F. 04510, MEXICO U. K. Ghosh, Department of Chemical Engineering, Banaras Hindu University, Varanasi, INDIA 221005 R. O. E. Greiner, Siemens AG, Corporate Research and Technology, 91050 Erlangen, GERMANY J. R. F. Guedes de Carvalho, Department of Chemical Engineering, University of Oporto, Oporto, PORTUGAL V. K. Gupta, Research Centre, Indian Petrochemicals Corporation Ltd., Vadodara—391 346, INDIA Prabir Kumar Haider, Department of Power Plant Engineering, Jadavpur University, Calcutta—700091, INDIA Min Hyeon Han, R&D Center, Kumho & Co., Inc., Sochondong, Kwangsanku, Kwangju 506-040, KOREA M. A. Kahn, Radiation and Polymer Chemistry Laboratory, Institute of Nuclear Science and Technology, Bangladesh Atomic Energy Commission, P.O. Box 3787, Bhaka, BANGLADESH S. K. Kapbasov, Department of Mathematical Physics, Urals State University, 620083 Yekaterinburg, RUSSIA J. Kaschta, University of Erlangen-Nurnberg, Institute for Material Science, Chair for Polymers, Martensstr. 7, 91058 Erlangen, GERMANY Y. Kawase, Department of Applied Chemistry, Faculty of Engineering, Toyo University, Kujirai, Kawagoe-Shil Saitama, 350 JAPAN Jin Kuk Kim, Department of Polymer Science & Engineering, Gyeongsang National University, 900 Kajwa-Dong Chinju, Gyeongnam 660-701, Seoul, KOREA Suzanne M. Kresta, University of Alberta, Edmonton, Alberta, CANADA, T6G 2G6 J. K. Kun, Department of Polymer Science & Engineering, Gyeongsang National University, Chinju 660 701 KOREA D. Moalem Maron, Department of Fluid Mechanics & Heat Transfer, School of Engineering, Tel-Aviv University, Tel-Aviv, 69978, ISRAEL J. H. Masliyah, Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, CANADA, T6G 2G6 H. A. Nasr-El-Din, Laboratory Research & Development, Saudi Aramco, P.O. Box 62, Dhahran 31311, SAUDI ARABIA V. Nassehi, Chemical Engineering Department, Loughborough University of Technology, Loughborough, Leicester, LEll 3TU U.K. Nivedita, Polymer Research Laboratory, Department of Post-Graduate Studies and Research in Chemistry, R.D. University, Jabalpur, 482001 M.P., INDIA

Hasan Orbey, Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA Gregory S. Patience, E.I. du Pont de Nemours, Wilmington, DE 19880-0262, USA M. Ravindranathan, Research Centre, Indian Petrochemicals Corporation Ltd., Petrochemicals, Vadodara-391-346, INDIA Shashikant Ravindranathan, Research Centre, Indian Petrochemicals Corporation Ltd., Petrochemicals, Vadodara-391-346, INDIA Stanley I. Sandler, Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA C. W. Stewart, Pacific Northwest Laboratory, Richland, WA 99352, USA P. A. Tanguy, Department of Genie Chimique, Ecole Polythecnique de Montreal, P.O. Box 6079 Station Centre Ville, Montreal, H3C 3A7, CANADA Lorenzo Tassi, University of Modena, Department of Chemistry, 41100 Modena, ITALY K. C. Taylor, Petroleum Recovery Institute 100, 3512 33rd Street NW, Calgary, Alberta, CANADA T2L 2A6 A. Tecante, Departamento de Alimentos y Biotecnologfa, Facultad de Quimica—UNAM Mexico, D.F., 04510, MEXICO J. A. S. Teixeira, Escola Superior Agraria, Instituto Politecnico de Braganca, Braganca, PORTUGAL Keio Toi, Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Minamiosawa, Hachioji, Tokyo 192-03, JAPAN C. P. Tsonis, Chemistry Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, SAUDI ARABIA S. N. Upadhyay, Department of Chemical Engineering, Banaras, Hindu University, Varanasil, INDIA 221005 E. M. Valles, Planta Piloto de Ingenieria Quimica, UNS-CONICET, 8000 Bahia Blanca, ARGENTINA M. A. Viler, Planta Piloto de Ingenieria Quimica, UNS-CONICET, 8000 Bahia Blanca, ARGENTINA Catherine Xuereb, Laboratoire de Genie Chiniqe USA CHRS 192, ENSIGC, 18 chemin de la Loge, 31078 Toulouse Cedex, FRANCE M. Yue, School of Applied Science, Nanyang Technological University, Nanyang Avenue, SINGAPORE 2263

ABOUT THE EDITOR

Nicholas P. Cheremisinoff is a consultant to private industry, government, and academia. He is an internationally recognized expert in multiphase flow system designs and polymer science. He has nearly 20 years of industry and applied research experience in petrochemicals manufacturing, synthetic fuels, elastomers, and emerging technologies for environmental restoration programs in the U.S., the Soviet Union, and the Far East. Dr. Cheremisinoff is with K&M Engineering and Consulting Corporation in Washington, D.C., is Donetsk Resident Director in Kiev and Donetsk, is Resident Director in the Ukraine, and is affiliated with the Donetsk University. He is the author, co-author and editor of more than 100 engineering textbooks, numerous patents and research articles. He received his B.S., M.S., and Ph.D. degrees in chemical engineering from Clarkson College of Technology.

PREFACE

This volume of the Advances in Engineering Fluid Mechanics Series covers topics in hydrodynamics related to polymerization of elastomers and plastics. Emphasis is given to advanced concepts in multiphase reactor systems often used in the manufacturing of these products. This volume is comprised of 30 chapters that address key subject areas such as multiphase mixing concepts, multicomponent reactors and the hydrodynamics associated with their operation, and slurry flow behavior associated with non-Newtonian flows. The intent of this book is to provide new concepts and an understanding of rheologically complex systems that undergo both phase changes and are subject to high transport exchanges. The series intends to explore additional areas including the dynamics of polymer processing operations. As in preceding volumes in this series. Multiphase Reactor and Polymerization System Hydrodynamics is comprised of contributions by recognized researchers and industry members. The efforts of these authors should be commended. A special thanks is extended to Gulf Publishing Company for its fine production of this series. Nicholas P. Cheremisinoff, Ph.D. Editor

CHAPTER 1 THE VISCOSITY OF LIQUID HYDROCARBONS AND THEIR MIXTURES Stanley I. Sandler and Hasan Orbey Center for Molecular and Engineering Thermodynamics Department of Chemical Engineering University of Delaware Newark, DE 19716 CONTENTS SCOPE, 1 TERMS AND DEFINITIONS, 2 EXPERIMENTAL BEHAVIOR, 2 CORRELATIONS FOR THE VISCOSITY OF PURE AND MIXED HYDROCARBONS, 7 VISCOSITY-TEMPERATURE RELATIONS AT LOW PRESSURES FOR PURE LIQUID, 7 Empirical Andrade-Type Relations, 7 Corresponding States Methods for Pure Hydrocarbons, 9 Other Prediction and Correlation Methods for the Viscosity of Pure Hydrocarbon Liquids, 11 VISCOSITY OF LIQUID HYDROCARBON MIXTURES AT AMBIENT PRESSURE, 13 Extension of Andrade-Type Correlations to Mixtures, 14 Extension of Corresponding States Methods for Viscosity of Mixtures, 15 Extension of the Theoretically Based Methods to Mixtures, 15 Viscosity Models for Undefined Mixtures, 16 VISCOSITY OF LIQUID HYDROCARBONS AND THEIR MIXTURES AS A FUNCTION OF PRESSURE, 17 Models that Correct Ambient Pressure Viscosity for Pressure, 18 Models that Incorporate Pressure Implicitly, 18 CONCLUSIONS AND RECOMMENDATIONS, 19 NOTATION, 20 REFERENCES, 21 1

2

Advances in Engineering Fluid Mechanics

SCOPE This chapter deals with correlation and prediction methods for the viscosity of liquid hydrocarbons and their mixtures. In particular, the change of viscosity of such fluids with temperature, pressure, and composition is considered. We begin with a brief introduction of terms and definitions, and then discuss the experimentally observed behavior of the viscosity of liquid hydrocarbons as a function of temperature, pressure, and composition. Next, the main types of viscosity models applicable to liquid hydrocarbons and their mixtures are reviewed. We also indicate the accuracy of several recent viscosity correlation and prediction methods that represent the general types of models in current use. The emphasis in this review is on the recent viscosity models, especially those after 1987, as reviews exist of the earlier methods [1,2], and because the recent methods are usually more accurate. TERMS AND DEFINITIONS When a Newtonian liquid, such as a hydrocarbon mixture, is subjected to a shearing stress, a velocity gradient develops within the fluid. Viscosity (or dynamic viscosity) is defined as the shear stress per unit area at any point within the fluid divided by the velocity gradient at that point. Consequently, the viscosity is a dynamic property; nevertheless, for Newtonian liquids it is a state property, that is, it depends only on state properties such as temperature and pressure or density. The dimensions of viscosity are force x time/length^ or equivalently mass/length x time. Occasionally kinematic viscosity, which is the ratio of dynamic viscosity to fluid density, is used instead of dynamic viscosity. The dimensions of kinematic viscosity are lengths/time. In the SI system the units of viscosity are N-s/m^ or Pa»s, and the units of kinematic viscosity are m^/s. In scientific and engineering work, the unit Poise (abbreviated P) is also used, with 1 Poise equal to 0.1 N-s/m^. Similarly for kinematic viscosity the unit Stoke (St) is used with 1 Stoke equal to 10"^ m^/s. EXPERIMENTAL BEHAVIOR The general viscosity behavior of hydrocarbon liquids, with respect to temperature, pressure and composition is reasonably well documented [3-6]. Temperature has the greatest effect on viscosity, with the viscosity being extremely high at the melting point of a fluid and decreasing by orders of magnitude as temperature increases. At low pressures (from the saturation pressure to a few bars above atmospheric), the viscosity is a function of temperature and essentially independent of pressure. The viscosity-temperature behavior of several liquid alkanes is shown in Figure 1. Other hydrocarbon fluids and their mixtures follow a similar trend. In general, the viscosity of a hydrocarbon decreases monotonically as the temperature increases, and the logarithm of viscosity decreases almost linearly with increasing temperature. At temperatures near and above the normal boiling point this linearity disappears for most liquids. At a given temperature, the viscosity of hydrocarbons generally increases with their molecular weight, though the effect of molecular structure is also significant

The Viscosity of Liquid Hydrocarbons and Their Mixtures '

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temperature, K Figure 1. Viscosity vs. tennperature at atnnospheric pressure for various alkane hydrocarbons. Data are from Knapstad et al. [40].

among the hydrocarbons with similar molecular weights. In Figure 2, the viscosities of several C^ hydrocarbons are shown as a function of temperature. There we see that at comparable temperatures, the cyclic molecules cyclohexane (melting temperature T^ = 278.7°K) and benzene (T^ = 279.6°K) have much higher viscosities, especially at lower temperatures due to their higher melting temperatures than n-hexane (T^ = 177.8°K). However, since 2-methyl pentane (T^ = 119.5°K) has a lower viscosity than normal hexane, and 2,2-dimethyl butane (T^ = 173.3°K) has a higher viscosity, the only general statement that can be made is that the effect of chain branching on viscosity is important, but smaller than the effect of either melting temperature or ring formation. Note also that while there is a large variation in the melting temperatures of the noncyclic alkanes (in addition to the melting points given in Figure 2, we have that 3-methyl pentane T^ = 155.0°K, 2,3-dimethyl butane T^ = 144.6°K, and 1-hexene T^ = 133.3°K), all the €5 hydrocarbons have normal boiling points within 20°K of each other. Consequently, as liquids have very

Advances in Engineering Fluid M e dlanics

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270 280 290 300 310 320 330 340 350 360 Temperature, K Figure 2 . Effect of molecular structure o n the viscosity of Cg hydrocarbons. Data are f r o m t h e compilation of Viswanath a n d Natarajan [1].

high viscosities near their melting points, this results in marked differences in viscosities of the components over the temperature range of 120''K to 290°K, but more similar behavior at higher temperatures. These statements concerning the effects of melting point, branching, and ring formation are also true for other hydrocarbons. The effect of large changes in pressure at constant temperature on the viscosity of various hydrocarbons is shown in Figure 3. There we see that the logarithm of the viscosity of liquid hydrocarbons and hydrocarbon mixtures increases almost linearly with increasing pressure. Alternatively, viscosity can be considered to be a function of density rather than pressure, and this is used in several of the models discussed later. The kinematic viscosity shows similar trends with respect to these variables mentioned above, however its variation with temperature is significantly more linear than dynamic viscosity so that the former is somewhat easier to correlate than the latter. Consequently, some correlations have been developed exclusively for the kinematic viscosity, as will be discussed later.

The Viscosity of Liquid Hydrocarbons and Their Mixtures 1

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pressure, bar Figure 3. Viscosity vs. pressure at 373°K for various alkane hydrocarbons. Data are fronn Ducoulombier et al. [4] and from Gouel [5].

There are some more specific observations that can be made about liquid viscosities. For example, except for the first members of a homologous series, the viscosity of most pure liquids at their normal boiling point is, to within ±30%: ^(Tb) - 0.29 cP

(1)

where T^ is the normal boiling point [7]. Also, from transition state theory [8] the temperature dependence of the viscosity is: lnii(T) = A + B ^ -

(2)

Further, the following dependence of the viscosity of oils on pressure [9] has been suggested In |i(T) = a + bP

(3)

6


Later we will show how the ideas in these three equations can be improved upon and used as the basis for a very simple and useful model for the viscosity of hydrocarbons. Other general observations [9, pp. 113-114] are that: (a) The viscosity of a branched compound is generally less than that of a straightchain compound; (b) Ring closure increases the viscosity; and (c) In a homologous hydrocarbon series, each additional methylene group increases the viscosity, but by a diminishing amount. It has been found that the viscosities of many non-hydrocarbon liquid mixtures at a fixed temperature and pressure exhibit a maximum or a minimum as a function of composition [2]. However, this effect is small for hydrocarbons as shown in Figure 4, where the viscosities of binary hydrocarbon mixtures are seen to change almost linearly with mole fraction.

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The Viscosity of Liquid Hydrocarbons and Their Mixtures

7

The discrepancy among measured viscosity data of different laboratories is generally about 5%. Examples of these discrepancies are shown in Figure 3 for the viscosity of decane and dodecane; a detailed analysis of the accuracy of viscosity data of pure liquid hydrocarbons is given by Oliveira and Wakeham [10]. As a consequence, at present it is not realistic to expect models for the liquid viscosity of hydrocarbons to be accurate to better than 5% when compared to experimental data. CORRELATIONS FOR THE VISCOSITY OF PURE AND MIXED HYDROCARBONS There are a large number of models used for the correlation and/or prediction of the viscosity of liquid hydrocarbons and their mixtures. Since there is no exact statistical mechanical or molecular-level theory for liquid viscosity, all of the models available contain some degree of empiricism. Also, there is considerable variation in the structure of these models in that most have been formulated to address only a specific viscosity estimation problem. For example, some liquid hydrocarbon viscosity models have been proposed only for predicting the viscosity of an undefined petroleum mixture, and their input parameters have been selected accordingly. There are models that use some experimental viscosity data, while others are completely predictive, at least within a class of substances. Some viscosity models are suitable for incompletely defined petroleum cuts, whereas others can be used only for well-defined hydrocarbons and their mixtures. Further, some models include the effects of pressure and dissolved gases on liquid hydrocarbon viscosity, while others are for use only at atmospheric pressure. It is not possible to provide an exhaustive review of all available models here. Instead, correlation and prediction methods will be categorized according to the nature of the model, and some contemporary and reasonably accurate models in each category will be described. First, methods that describe the behavior of the dynamic viscosity of pure hydrocarbons as a function of temperature at ambient pressure (actually from saturation pressure to a few bars) will be considered. These models can be divided into three main groups: empirical "Andrade"-type relations, corresponding states methods, and other (mainly theoretically based) methods. Next methods for estimating the viscosity of hydrocarbon mixtures at ambient pressures will be considered. These methods are categorized into two broad groups: those that are extensions of mixtures of the pure component viscosity prediction methods, and those that are specific to the correlation and/or prediction of only mixtures. This second group contains a number of methods that are exclusively for the kinematic viscosity. Finally, the extension of these methods to high pressures and dissolved gases is outlined. VISCOSITY-TEMPERATURE RELATIONS AT LOW PRESSURES FOR PURE LIQUID HYDROCARBONS Empirical Andrade-Type Relations Most liquid viscosity data have been collected either at the saturation pressure or at atmospheric pressure, and since in this pressure range the viscosity is essentially independent of pressure, these data can be used to develop correlations of viscosity

8


only as a function of temperature. With the exception of temperatures near that at which a pure liquid freezes or boils, the logarithm of the dynamic viscosity is found to correlate with respect to the reciprocal of absolute temperature, suggesting a relation attributed to Andrade: ln^ = A + -

(4)

Reid et al. [2] provide a compilation of the parameters for Equation 4 for a large number of pure liquids, including hydrocarbons, and the temperature range in which these parameters may be used to essentially reproduce the experimental data. As those parameters were obtained by regressing experimental data, they should not be used outside the indicated temperature range, especially at the low tem-perature end, if accurate viscosity predictions are necessary. There are several other correlations that use Andrade-type equations to correlate and/or predict the dynamic viscosities of pure hydrocarbons. They can all be used in a predictive mode for paraffinic hydrocarbons, or for correlation with substance specific parameters. One example is the correlation of Mehrotra [11]: log[|Li + 0.8] = 100(0. OIT)' with: (5) b = -1.396-

1358

258800

where T is the temperature and T^ is the normal boiling point in Kelvin. In Equation 5, the constants of the relation of b as a function of T^, are specific for paraffinic hydrocarbons. Mehrotra also reported parameters for other families of hydrocarbons, correlated the b parameter with the critical temperature and the acentric factor—as well as reporting substance specific values for b, and adopted this method for use with pure heavy hydrocarbons. With substance specific-parameters, this method provides estimates of the viscosity of paraffinic hydrocarbons with an average absolute deviation (AAD) of 12.1%, and similar accuracy for other hydrocarbon families (8.1% for 1-olefins, 6.6% for aromatics, etc.). The correlation of Allan and Teja [12] is:

Inji = A

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1

+ •( T - h C )

with: A = 145.73 +99.Oln-HO.83n' - 0 . 1 2 5 n '

(6)

B = 30.48 + 34.01n - 1.23n' + O.OlVn' C = -3.07-1.99n where T is in Kelvin, and n is the carbon number for paraffinic hydrocarbons and is an effective carbon number that can be obtained from an experimental measurement


9

of the dynamic viscosity of other hydrocarbons. For alkanes, this method reproduces the viscosities well within experimental error, with an A AD less than 5% in most cases, and with even better accuracy for hydrocarbons other than alkanes. The Orbey and Sandler [13] correlation is slighty different in that it combines the idea of corresponding states with an Andrade-type relation by correlating reduced viscosity versus reduced temperature as follows:

,„.jt^

-1.6866 + 1.4010| ^

j + 0.2406[

^

.^refy

with:

(7)

k = 0.143 + 0.00463T, - 0.00000405T' with jiref = 0.225 cP (or, equivalently, mPa«s) for paraffinic hydrocarbons. In this form, this model is completely predictive; however, it also can be used in a two-parameter correlative manner with k and jo^gf being obtained from data regression for the fluid of interest. These parameters are available for more than 60 hydro-carbons covering a wide range of molecular weights and structures. As shown by Orbey and Sandler, the model is capable of correlating the viscosity of a wide variety of hydrocarbons from methane to very heavy oils, such as perhydrochrysene, well within the experimental accuracy of 5%. The predictive form of this model has been tested for paraffinic hydrocarbons from propane to n-eicosane. With the exception of three paraffins (AAD for n-pentane 9.64%; for n-hexane, 6.04%, and for tetradecane, 5.82%), the viscosity of this group of hydrocarbons can be estimated to within 3% of the reported data. The predictions (not correlations) resulting from Equations 5 to 7 are compared with the results obtained using Equation 4 with the substance-specific parameters reported by Reid et al. [2] and with experimental data in Figure 5. While all the models show similar behavior, the best results are obtained with the Orbey-Sandler model, especially at lower temperatures and with heavier hydrocarbons. The OrbeySandler model also has the advantage that it has been generalized to make predictions [13], and in a modified form can be used for heavy oils and bitumens, and with dissolved gases [14] as will be discussed later. All the models discussed here are simple in form, easy to use, and stable in the sense that they do not lead to singularities or fail to converge to a solution outside the temperature range in which they are meant to be applied. Consequently, these models may be especially useful in process and reservoir simulation. Corresponding States Methods for Pure Hydrocarbons: The principle of corresponding states for viscosity is based on the idea that there is universal relation between a suitably defined dimensionless (or reduced) viscosity and a set of dimensionless parameters, such as reduced temperature and pressure. Among the important corresponding states prediction methods for viscosity is that of Ely and Hanley [15], which evolved into the TRAP? method. Others have suggested modifications to the Ely-Hanley method in recent years [3,16,17,18,19]. The TRAPP method and its modifications are useful for wide ranges of fluids and from the dilute gas to the compressed liquid. However, the mathematical complexity

10

Advances in Engineering Fluid Mechanics 6

I — I — I — I — I — I — I — \ — \ — \ — I — r

T—^—T-^—r

5 h

C/) I

cd

E

•^

2

260

300

340

380

420

460

500

Temperature, K Figure 5. Comparison of viscosity predictions of various Andrade-type models for some alkane hydrocarbons at atmospheric pressure. Data are from the compilation of Viswanath and Natarajan [1], solid lines are predictions by the Orbey-Sandler Method, long-dashed lines are predictions by the Allan-Teja Method, short-dashed lines are predictions by the Reid et al. method, and dotted lines are predictions by the Mehrotra Method. of these models may cause problems in simulation packages because of the extensive calculations involved, and the method may fail to converge to a solution, especially for heavy hydrocarbons. Pedersen et al. [20] and Pedersen and Fredenslund [21] also proposed corresponding states approaches; though Petersen et al. [22] developed a simpler method based on corresponding states principle that outperforms the earlier methods of this group. This correlation is as follows: l^ =

^^cx^^.(T.,P,)

Here, K is given by:

l^2(T2,P2)K |Li,(T,,P,)|ie.

(8)


K=

TMW. - M W ,1^ I MW2 - MW I /

11

(9)

where MW is molecular weight and the subscripts 1, 2, and x refer to reference components 1, 2, and to the component of interest, x. In Equation 8, jj^ = CMW^^^ P^^^ T"'^^, C being a constant, and [i^ and 1X2 are evaluated at conditions corresponding to a critical point ratio corrected for temperature and pressure as follows: T T, = T x ^T: ^

for i = 1 or 2

(10)

for i = 1 or 2

(H)

and p. = P x - ^

The Petersen et al. model also allows extrapolations to high pressures and to mixtures as will be discussed later, but it is of limited accuracy for hydrocarbons heavier than CJQ and for cyclic compounds. For example, AAD in viscosity estimates for hydrocarbons up to Cj2 is about 10%, whereas it increases to 39% for C,8 and to 47% for cyclohexane. Thus, none of the available corresponding states methods are better than the simpler Andrade-type equations of the previous section. The Orbey and Sandler [13] method mentioned earlier also can be considered to be a corresponding states method since it uses reduced variables. This method will be discussed further in the sections dealing with mixture and high pressure applications. Other Prediction and Correlation Methods for the Viscosity of Pure Hydrocarbon Liquids Other semi-empirical models have been proposed for the viscosity of fluids that are based in theory. As these models are applicable to all fluids, they also can be used for liquid hydrocarbons. The first of these models that we consider is that of Lee and Thodos [24,25], who proposed a model for the dynamic viscosity of pure fluids that is applicable to all state conditions from the dilute gas to a highly compressed liquid without discontinuity. This model uses the following expression for the excess of the viscosity above that for a dilute gas: \0\\i - |i*)y = [exp(2.9328g8-3264 + AMlAg^-^'^^^)] - 1

(12)

where \i and |X* are the dynamic and dilute gas viscosities of the fluid, respectively, in Poise. This latter quantity is estimated from the relation: 10^(|Li*vf M-'/'P;'/^ = 0.576T^'' - 35.50exp

10 • nnl/2

(13)

12


where v^ is critical volume in cubic centimeters per mole, M is molecular weight, TR is reduced temperature (T/T^,), and P^ is the critical pressure in atmospheres. The grouping of terms on the left-hand side of Equation 13 has its origin in the kinetic theory of low-density gases. In Equation 12, y is the viscosity parameter Y = v f M-'/^T;'/^

(14)

where T^ is the triple point temperature in K, and v^ is liquid molar volume at triple point in cmVmol. The parameter g of Equation 12 contains the effect of pressure through an imbedded density dependence as follows:

^

(0.976e)^"^/^"""

('^^

where e = v,/v^j is the so-called volume expansion factor, with v^^ equal to the solid molar volume at the triple point in cmVmol, and x is density-temperature variable given by: CO "^ ~

^0.070)^"

(16)

where T = T/Tj is a reduced temperature and co = p/p^ is a reduced density. For alkanes from methane to n-eicosane, the Lee and Thodos method results in viscosity predictions accurate to within the experimental error limit of about 5% with a few exceptions, such as methane, propane, n-heptadecane, and others for which the AAD increases but remains less than 10%. The model of Cao et al. [26] is based on Eyring's rate theory, and has then been formulated [27] in such a way that one obtains a group contribution viscosity model for mixtures based on the UNIFAC prediction method for activity coefficients. Their expression for the dynamic viscosity of a pure liquid is: [i^ =i-(27cRT)'/^

^M^'^^

Wf exp

^ zq.n.Uii

y

where R is gas constant and T is absolute temperature. For each pure fluid, Mj is the molecular weight, Vj is the molar volume, z is the coordination number of the liquid lattice, qj and r; are the area and volume parameters of the molecule calculated as in the UNIFAC model [28], Uji is a characteristic interaction energy between the molecules of the pure fluid, and nj is a segment proportionality constant. The following expression was proposed for the molar volume: gl+d-T/C.)"^!

Vi=-^^^

(18) i

where Aj, Bj, Cj and Dj are fluid specific parameters in the DIPPR data bank [29] and z is calculated from:

The Viscosity of Liquid Hydrocarbons and Their Mixtures z = 35.2 - 0.1272T + O.OOOHT^

13 (19)

[30]. The quantity Ujj is calculated from the molar heat of vaporization of the pure liquid as follows: (zq72)U, = R T - A H : ^ P ^

(20)

In Equation 20, AHJJ^^J'j^ is the molar heat of vaporization of the pure liquid obtained from: ^^Z)

= Aj (1 - T,^ )«i^C;T,-^D>T?^

(21)

where the parameters Aj, Bj, Cj, Dj also are obtained from DIPPR data bank, and T^j (=T/Tj,j) is the reduced temperature using the critical temperature T^,j obtained from the DIPPR data bank. The temperature-dependent, fluid specific parameter Uj needed in the model is given: ln(n,)=£A.T^

(22)

where Aj(j = 0, 1, 2, . . .) have been fit to experimental viscosity data and are provided by the investigators for more than 400 fluids, including many hydrocarbon liquids. With the Cao et al. method and its large number of parameters, it is possible to fit the experimental viscosity data of most fluids, including hydrocarbons, within experimental error. Among the advantages claimed for this model are that it reduces to some well-known practical viscosity equations—for example, the empirical Andrade-like expression (Equation 4) with a particular choice of its parameters— and that it can be extended to mixtures without additional adjustable parameters. VISCOSITY OF LIQUID HYDROCARBON MIXTURES AT AMBIENT PRESSURE Extension of the viscosity models developed for pure fluids to mixtures requires suitable mixing or combining rules for defined mixtures, as well as an appropriate characterization for an undefined mixture such as an oil or petroleum cut. The problem of identifying the best mixing model is made difficult by the limited amount of viscosity data for well-defined hydrocarbon mixtures. There are two general approaches to the prediction of the viscosity of the mixtures by the methods considered here. The first approach involves estimating the pure component viscosity of each of the constituents by some method and then combining these values to obtain the viscosity of the mixture. We refer to this approach as the multi-fluid model. A second approach is the so-called one-fluid model, in which the mixture is treated as a pseudo-pure fluid, with mixing rules for obtaining the parameters of the mixture from those of the pure components.

14


Extension of Andrade-Type Correlations to Mixtures All of the Andrade-type relations discussed can be extended to well-defined mixtures using the multi-fluid model. Orbey and Sandler [13] considered various combining rules for such hydrocarbon mixtures including: H.i. = X x , ^ ,

(23a)

H..x=exp[Xx, ln(n,)]

(23b)

and: H.. = [ I x , n ; " f

(23c)

where Xj is the mole fraction. For alkane hydrocarbon mixtures they concluded that the best combining rule is the cubic rule of Equation 23c. This approach also may be useful for petroleum cuts if they can be modeled as a mixture of identifiable hydrocarbon fractions. Orbey and Sandler tested their method for some well-defined alkane mixtures in both multi-fluid and one-fluid modes: the two options agreed very well, and with one exception in 18 cases they were able to predict mixture viscosities within 5%. The one-fluid approach can be used with some of the models presented earlier. In the method of Mehrotra [11], the one-fluid model can be used to obtain the needed normal boiling temperature, T^, from T^j^j^ = [SxjT^y-^^]^, if the mixture is well defined. This proposal has not been tested in that method. However, Orbey and Sandler [13] found this boiling-point estimation method to be suitable for use in their viscosity model as discussed later. Allan and Teja [12] proposed a one-fluid approach for their model by obtaining the effective carbon number from n^j^ = Xxj n;. They tested their method for 10 binary mixtures of various hydrocarbons including cyclic, aromatic, and paraffinic substances. They were able to estimate mixture viscosity with an overall AAD of 5.6%. In the Orbey-Sandler method [13], one needs mixing rules for the three parameters of the model, jo^g^, k and T^. However, they tested their model for only alkane mixtures for which w^f and k are identical for all species, eliminating the need for a mixing rule for those parameters. They then found that for the normal boiling point the cubic average, T^ ^^j^ = [ZxjT^y^^]^ gave the best results. For undefined mixtures, the Mehrotra and Orbey-Sandler methods can be applied directly if an estimate of the normal boiling point of the mixture is available. Allan and Teja [12] suggest that as long as the mixture is characterized by fractions each having an average boiling point, their method is applicable by assigning an effective carbon number to each fraction using one viscosity data point for that fraction. The Orbey-Sandler and Allan-Teja models have been tested for some undefined mixtures by those investigators. Orbey and Sandler reported that their method was capable of predicting the viscosity of 15 petroleum mixtures from various sources with an overall AAD of 6.4%. Allan and Teja considered nine similar petroleum mixtures and found an overall AAD in mixture viscosity predictions of 10.9%.


15

Extension of Corresponding States Methods for Viscosity to Mixtures The multi-fluid approach can always be used with corresponding states methods for well-defined mixtures. In the one-fluid approach, however, a mixing rule must be proposed for each of the input parameters. For the Petersen et al. [22] corresponding states model discussed earlier, the following relations [31] are used to extend the model to mixtures:

y

_

•

J

'

J

8 X XXx,xJ(T yP^,)V3 + (TJP^p'/3[(T ,Tp'/^ P =

y y,x.x.[(T./?.)"' + (T ./p .)'/3

(25)

Pedersen et al. [21] observed that larger molecules contribute more to the mixture viscosity than smaller ones, and thus adopted the following relations for the molecular weight in their model: MWî, = MW„ + 0.00867358(MW;,''''' - MW;^''"'') with: MW^ = £ X. M W y X î^W^

(26)

and: MW„ = y x.MW. In this form, the Pedersen et al. model can be extended only to well-defined mixtures for which critical properties of the constituents are available. They tested their models for 419 data points of seven binary hydrocarbon mixtures and were able to predict mixture viscosities with an average error of 7.4%. Extension of the Theoretically Based Methods to Mixtures The method of Lee and Thodos discussed previously was not extended to mixtures by them, presumably due to the number and nature of the input parameters required [25]. Consequently, the use of this estimation method appears suitable only for well defined mixtures using the multi-fluid approach.

16


The method of Cao et ai, by its construction, is also suitable only for well-defined mixtures [27,32]. However, for such mixtures it offers the advantage of being completely predictive. In the spirit of the UNIFAC group contribution method for activity coefficents, Cao et al. used the following expression for the dynamic viscosity: In^^mix = S ^Jn\ -^^,

| + 2^jln

'x ^

U

(27) i y

all groups k

where Vj and V are the molar volumes of component i and of the mixture, respectively, jj^ is the dynamic viscosity of pure component i, ([){ is the segment fraction of molecule i in the mixture, Xj is mole fraction, H,^j is the group residual viscosity of group k in component i in the mixture, and Sj[j^ is the group residual viscosity of group k for component in pure liquid i. The evaluation of these group residual viscosities is described by Cao et al. [27,32]. The important observation is that the parameters, H,^} and Ej^-^ are dependent on the group interaction parameters of the UNIFAC model, thus a knowledge of group interaction parameters obtained from vapor-liquid equilibrium data leads to a reasonably successful method for the prediction of the viscosity of mixtures. The Cao et al. [27,32] model was tested for a large number of well-defined mixtures of various fluids. For alkane hydrocarbons they report an overall AAD of 3.4% for 51 systems. Viscosity Models for Undefined Mixtures The viscosity of a liquid hydrocarbon mixture that is not well characterized is required in many industrial applications, of which crude oils and crude cuts are typical examples. Many correlations have been proposed for the dynamic or kinematic viscosity of such mixtures based on some selected (usually specifically measured) characteristics of the mixture. Puttagunta and co-workers introduced several such correlations [33-36]. These correlations are all empirical in nature and require at least one viscosity measurement for the mixture; in addition parameters of the model(s) have been fit to certain hydrocarbon mixtures. One model they [33-35] proposed for the calculation of the dynamic viscosity as a function of temperature and pressure is:

InjLi = 2.30259

T-30 14303.15

+ C -hB,Pexp(dT)

(28)

In this expression, the viscosity jii is in Pa-s, T is temperature in °C, P is gauge pressure in MPa, b is a characteristic parameter computed from a viscosity measurement at 30°C (jLi3o°c i" ^he equations following), BQ, S, and d are parameters that depend on b, and C is a constant. For Canadian bitumens they proposed: C = -3.0020 b = log,o II300C - C


S = 0.006694b + 3.53641

17

(29)

Bo = 0.0047424b + 0.0081709 d = -0.0015646b + 0.0061814 As a variation of this model for Middle East crude oil mixtures, they [34,35] used the the same form of the equation, but now for the kinematic viscosity, with the following parameters: C =0 b = log,o V300C + 3.0020

S = 0.006694b + 3.5364

(30)

Bo = 0.002067b + 0.0060148 d = 0.004185b - 0.021356 where in this case b is obtained from the measured kinematic viscosity v in m^/s at 30°C and the kinematic viscosity of the mixture is obtained in the same units. A third correlation [36] for the kinematic viscosity has been proposed that, except for the pressure term, is identical to the expression given in Equation 28, but uses different constants for petroleum mixtures. Dutt [37] suggested the following simple prediction method for the kinematic viscosity of petroleum cuts at ambient pressures, based on an Andrade-type relation, that uses an average boiling point as the only input:

inv =-3.0171 +

442.78+ 1.6452T,

T +239-0.19Tb

^

nn ^^^^

where v is kinematic viscosity in cS, T is temperature, and T^ is the average normal boiling point of the mixture, both in °C. The accuracy of this method has been shown to be about the same as the more elaborate methods mentioned above that require a measured viscosity data point. The methods discussed here typically predict the viscosities of undefined mixtures with an accuracy of 5%-10% without tuning to experimental data, and to within 5% if tuned to an experimental data point, provided the mixture is not close to its pour point [23]. VISCOSITY OF LIQUID HYDROCARBONS AND THEIR JVIIXTURES AS A FUNCTION OF PRESSURE There have been two approaches to modeling the effect of pressure on the viscosity of mixtures. Either the effect of pressure is implicitly included in the model (for example, as in the case of Cao et al. or in the TRAPP method, in which the effect of pressure on density directly affects the viscosity), or viscosities are first calculated at low pressure (from saturation to a few atmospheres) and then corrected

18


for pressure [13,33-36]. The methods that correct viscosity for pressure usually lead to more stable and reliable predictions, but they require the evaluation of empirical parameters, such as the slope of change of the logarithm of viscosity with pressure. In contrast, models that include pressure implicitly are completely predicitive, but can lead to significant errors due to the very strong dependence of viscosity on density, especially near the freezing point of the mixture. Several models that include the effect of pressure on viscosity are outlined herein. For applications at high pressures, one may also require estimates of the viscosity of liquid hydrocarbons and their mixtures with dissolved gases (such as with CO2, N2, H2S, etc.) because, due to the high solubility of such gases in hydrocarbon mixtures at elevated pressures, there is a very large reduction in the mixture viscosity. Indeed, such behavior is part of the basis for enhanced oil recovery by miscible gas injection. Even though the effect of dissolved gases is beyond the scope of this chapter, some comments about this are included due to the importance of this subject. Models that Correct Ambient Pressure Viscosity for Pressure This category includes the recent models of Puttagunta et al. [33], of Orbey and Sandler [13] and of Al-Besharah and co-workers [38,39]. The Puttagunta et al. model incorporates the pressure directly in Equation 28. This model can be used for mixtures as for pure compounds as long as there is a low pressure viscosity data point for the mixture to fix the value of the parameter b of the model. A quite different procedure suggested by Orbey and Sandler [13] based on Equation 3 is as follows: |Li(P)/ii(P^^) = exp[m(P - P^Ô] - exp[mP]

(32)

In this equation, the second equality is written neglecting the low saturation pressure with respect to the pressure interest, P, and the viscosity at saturation pressure, |i(P^^^), is obtained from Equation 7. They correlated available high pressure data for paraffinic hydrocarbons and obtained the value m = 0.98 x 10~^ kPa~^ Using this formulation, they were able to estimate the high pressure viscosity of pure paraffinic hydrocarbons to within 5% of available experimental data. For other groups of hydrocarbons, they found that a slightly different values for m; for example, m = 1.05 x 10~^ kPa~' for alkylbenzenes. In the absence of experimental data, a value of m = 1 x 10~^ kPa~' appears to be a good approximation for all fluids. This model was extended by Orbey and Sandler [13] to include mixtures of liquid hydrocarbons and compressed gases, and then developed further for use with bitumens and other very viscous fluids [14]. Models that Incorporate Pressure Implicitly All corresponding states methods that incorporate pressure effects directly, usually through density, fall into this category. The method of Cao et al. [26,27,32] includes the effect of pressure through the molar volume, and the model of Lee and Thodos


19

[24,25] incorporates the effect of pressure through the density dependent co term in Equation 16. Also, the TRAPP method mentioned earlier uses the density as a primary variable, so that the changes in density with pressure account for the pressure dependence of the viscosity. Consequently, in these methods, estimation of viscosity at high pressures is very sensitive to the density and- is very dependent on the accuracy of the density estimates. As a result, in the TRAPP method, a very complicated, multiparameter equation of state is used to obtained accurate liquid densities. Partly because of the lack of available data, none of the models discussed here have been systematically tested or compared against each other. CONCLUSIONS AND RECOMMENDATIONS Much experimental viscosity data for liquid hydrocarbons is for pure liquids at ambient pressures, and the agreement among different data sets for the same compounds is about 5%. Therefore, at present, it is not realistic to expect any better accuracy from viscosity models that have been developed using these data. For mixtures, and also at high pressures, fewer viscosity data are available for developing correlation models, and both the data and the models are of lower accuracy. As there is no theoretically based model for liquid viscosity, there is considerable variation in the types of models that have been developed, and the limited amount of data and their accuracy makes it difficult to discriminate among the models that have been proposed. Most viscosity correlations are essentially empirical in nature and generally require at least one parameter fit to experimental data for accurate estimates of the viscosities of pure liquids. Some of these models have been generalized for certain types of liquid hydrocarbons and provide reasonably accurate predictions, but only for those groups of substances. For the correlation of the pure-component dynamic viscosity of hydrocarbons at ambient pressures, we recommend the use of the simple Andrade-type models. These models, including the more complex models such as those of Cao et al. [27,32] and of Lee and Thodos [24,25], require fitting at least one parameter to experimentally measured data. At present it is not evident that the more complex liquid viscosity models yield markedly better accuracy than the simpler Andrade-type models. For the prediction of viscosity of well-defined mixtures of liquids, the method of Cao et al. is very good in general [27,32]. However, since hydrocarbon mixtures form almost thermodynamically ideal solutions and their viscosities are simple functions of composition, simpler methods such as those of Allan and Teja [12] or of Orbey and Sandler [13] result in almost equal accuracy without the need for binary interaction parameters. These last two, simpler models can only be used for hydrocarbons, while the model of Cao et al is of more general applicability. The method of correlation of the viscosity of undefined liquid hydrocarbon mixtures depends upon the information available. If it is possible to measure at least one viscosity data point, then the methods of Puttagunta and co-workers [33-35] can be used. Other Andrade-type equations mentioned earlier also may be used for such cases with equal accuracy by fitting one parameter to a measured data point [13]. If information is available on the average boiling point of the mixture, methods such as those of Allan and Teja [12], Orbey and Sandler [13], or of Dutt [37] can

20

Advances in Engineering Fluid Meciianics

also be used. The more complicated methods, such as those of Cao et al. [27,32] or of Lee and Thodos [25], cannot easily be used for undefined mixtures. There are few alternatives for predicting or correlating the viscosities of pure fluids and of mixtures at high pressures, and none of the methods has been extensively tested because of the l^ck of available data. Only the method of Orbey and Sandler [13] has been partially tested for the effect of dissolved gases on the viscosity of hydrocarbon liquids under pressure [14]. This method has been found to be successful, and therefore is recommended here. ACKNOWLEDGMENT This research was supported, in part, by grant no. DOE-FG02-85ER13436 from the U. S. Department of Energy and grant no. CTS-9123434 from the U. S. National Science Foundation, both to the University of Delaware. NOTATION A, B, C, D Equation constants in various equations a,b,c,d Equation constants in various equations g Density dependence function in Equation 15 AH^^ Molar heat of vaporization in Equation 20 M, MW Molecular weight n Equation constant in Equation 6, and in Equation 17 P Absolute pressure

q UNIFAC area parameter in Equation 17 r UNIFAC area parameter in Equation 17 R Gas constant T Absolute temperature U Characteristic interaction energy in Equation 17 V, V Volume X Mole fraction, or densitytemperature variable in Equation 16 z Lattice coordination number in Equation 17

Greek Letters Y Viscosity parameter in Equation 12 £ Volume expansion factor (v,/Vst) in Equation 15 |i Absolute (dynamic) viscosity V Kinematic viscosity v^'^ Number of group k in molecule i in Equation 27 p Density

X A dimensionless temperature defined as (T/T^) in Equation 16 (|) Segment fraction in Equation 27 CO Reduced density (p/p,t) in Equation 16 5 Group residual viscosity in Equation 27

Subscripts b Boiling point c Critical

1 Liquid m Melting point


R, r Reduced s Solid

21

ref Reference t Triple point

Superscript sat Saturation

REFERENCES 1. Viswanath, D. S., and Natajaran, G., "Data Book on the Viscosity of Liquids," Hemisphere, 1989. 2. Reid, R. C , Prausnitz, J. M., and Poling, B. E., 1987, The Properties of Gases and Liquids, Fourth Edition, McGraw-Hill. 3. Kanti, M., Zhou, H., Ye, S., Boned, C, Lagourette, B., Saint-Gurions, H., Xans, P., and Montel, P., 1989. "Viscosity of Liquid Hydrocarbons, Mixtures and Petroleum Cuts as a Function of Pressure and Temperature," J. Phys. Chem. 95.-3,860-3,864. 4. Ducoulombier, D., Zhou, H., Boned, C , Peyrelasse, J., Saint-Guirions, H., and Xans, P. ,1986. "Pressure and Temperature Dependence of the Viscosity of Liquid Hydrocarbons," /. Phys. Chem., 90.-1,692-1,700. 5. Gouel, P., 1978. "Viscosite des Alcanes des Cycliques et des Alkybenzenes," Bulletin des Centres de Recherches Exploration—Production Elf-Aquitaine, Volume 2, No.2 , November 30. 6. Griest, E. D., Webb, W., and Schiessler, R. W., 1958. "Effect of Pressure on Viscosity of Higher Hydrocarbons and Their Mixtures," /. Chem. Phys. 29: 711-720. 7. Perry, R. H., Chilton, C. H., and Kirkpatrick, S. D., 1963. Chemical Engineers' Handbook, Fourth Edition, McGraw-Hill, New York, pp. 3-228. 8. Bird, R. B., Stewart, W. E., and Lightfood, E. N., 1960. Transport Phenomena, John Wiley & Sons, New York, p.29. 9. Partington, J. R., 1951. "An Advanced Treatise on Physical Chemistry, Vol. II, The Properties of Liquids," Longmans, Green and Co., London, p. 89. 10. Oliveira, C. M. B. P., and Wakeham, W. A., 1992. "The Viscosity of Five Liquid Hydrocarbons at Pressures up to 250 Mpa," Int. J. Thermophysics 13:113-190. 11. Mehrotra, A. K., 1991. "Generalized One Parameter Viscosity Equation for Light and Medium Liquid Hydrocarbons," Ind. Eng. Chem. Res. 30:\,361-1,312. 12. Allan, J. M., and Teja, A. S., 1991. "Correlation and Prediction of the Viscosity of Defined and Undefined Hydrocarbon Liquids," Can. J. Chem. Eng. 69: 986-991. 13. Orbey, H., and Sandler, S. I., 1993. "The Prediction of the Viscosity of Liquid Hydrocarbons and Their Mixtures as a Function of Temperature and Pressure," Can. J. Chem. Eng. 71:431-446. 14. Fong W. S., Emanuel, A. S., and Sandler, S. I., 1993. "A Simple Predictive Calculation for the Viscosity of Liquid Phase Reservoir Fluids, with High Accuracy for CO2 Mixtures," SPE 26645. 15. Ely, J. F., and Hanley, H. J. M., 1981. "Prediction of Transport Properties: Viscosity of Fluids and Mixtures," Ind. Eng. Chem. Fund. 20.323-332.

22


16. Wayne, D. M., Mehrotra, A. K., and Svrcek, W. Y., 1991. "Modified Shape Factors for Improved Viscosity Predictions Using Corresponding States," Can. J. Chem. Eng. 69:\,2X3-1,2X1 17. Johnson, S. E., Svrcek, W., and Mehrotra, A. K., 1987. "Viscosity Prediction of Athabasca Bitumen Using the Extended Principle of Corresponding States," Ind. Eng. Chem. Res. 26;2,290-2,298. 18. Mehrotra, A. K., and Svrcek, W. Y., 1987. "Corresponding States Method for Calculating Bitumen Viscosity," J. Can. Petroleum Tech. 26:60-66. 19. Hwang, M-J, and Whiting, W. B., 1987. "A Corresponding States Treatment for the Viscosity of Polar Fluids," Ind. Eng. Chem. Res. 26.-1,758-1,766. 20. Pedersen, K. S., Fredenslund, A., Christensen, P. L., and Thomassen, P., 1984. "Viscosity of Crude Oils," Chem. Eng. Sci. 39.1,011-1,016. 21. Pedersen, K. S., and Fredenslund, A., 1987. "An Improved Corresponding States Model for the Prediction of Oil and Gas Viscosities and Thermal Conductivities," Chem. Eng. Sci. 42; 182-186. 22. Petersen-Aasberg, K., Knudsen, K., and Fredenslund, A., 1991. "Prediction of Viscosities of Hydrocarbon Mixtures," Fluid Phase Equilibria 70.293-308. 23. Orbey, H., and Sandler, S. I., 1994, Letter to the Editor, Can. J. Chem. Eng. 72.-558-560 24. Lee, H., and Thodos, G., 1988. "Generalized Viscosity Behavior of Fluids over the Complete Gaseous and Liquid States," Ind. Eng. Chem. Res. 27.2,377-2,384. 25. Lee, H., and Thodos, G., 1990. "Generalized Viscosity Behavior of Fluids over the Complete Gaseous and Liquid States," Ind. Eng. Chem. Res. 29.-1,404-1,412. 26. Cao, W., Fredenslund, A., and Rasmussen, P., 1992. "Statistical Thermodynamic Model for Viscosity of Pure Liquids and Liquid Mixtures," Ind. Eng. Chem. Res. i7.-2,603-2,619. 27. Cao, W., Knudsen, K., Fredenslund, and A., Rasmussen, P., 1993. "Simultaneous Correlation of Viscosity and Vapor-Liquid Equilibrium Data," Ind. Eng. Chem. Res. 52.-2,077-2,087 28. Hansen, H. K., Rasmussen, P., and Fredenslund, A., 1991. "Vapor-Liquid Equilibria by UNIFAC Group Contribution: Revision and Extension," Ind. Eng. Chem. Res. J0.-2,352-2,355. 29. Daubert T. E., and Danner, R. P., 1989. "Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation:, Hemisphere, New York. 30. Skold-Jorgensen, S., Rasmussen, P., and Fredenslund, A., 1980. "On the Temperature Dependence of the UNIQUAC/UNIFAC Models," Chem. Eng. Sci. J5.-2,389-2,403. 31. Mo, K. C , and Gubbins, K. E., 1976. "Conformal Solution Theory for Viscosity and Thermal Conductivity of Mixtures," Mol. Phys. 57.825. 32. Cao, W., Knudsen, K., Fredenslund, A., and Rasmussen, P., 1993. "GroupContribution Viscosity Predictions of Liquid Mixtures Using UNIFAC-VLE Parameters," Ind. Eng. Chem. Res. J2.-2,088-2,092. 33. Puttagunta, V.R., Singh, B., and Miadonye, A., 1993. "Correlation of Bitumen Viscosity with Temperature and Pressure," Can. J. Chem. Eng. 77.-447-450. 34. Singh B., Miadonye, A., and Puttagunta, V. R., 1993. "Modeling the Viscosity of Middle-East Crude Oil Mixtures," Ind. Eng. Chem. Res. i2.'2,183-2,186.


23

35. Singh, B., Miadonye, A., and Puttagunta, V. R., 1993. "Heavy Oil Viscosity Range from One Test," Hydrocarbon Processing, August, 157-162. 36. Puttagunta, V. R., Miadonye, A., and Singh, B., 1992. "Viscosity-Temperature Correlation for Prediction of Kinematic Viscosity of Conventional Petroleum Liquid," Chem. Eng. Res. Dev., 70:627-631. 37. Dutt, N.V., 1990. "A Simple Method of Estimating the Viscosity Petroleum Crude Oil and Fractions," Chem. Eng. J. 45:83-86. 38. Al-Besharah, J. M., Salman, O. A., and Akashah, S. A., 1987. "Viscosity of Crude Oil Blends," Ind. Eng. Chem. Res. 26.2,445-2,449. 39. Al-Besharah, J. M., Akashah, S. A., and Mumford, C. J., 1989. 'The Effect of Temperature and Pressure on the Viscosities of Crude Oils and Their Mixtures," Ind. Eng. Chem. Res. 28:213-221. 40. Knapstad B., Skjolsvik, P. A., and Oye, H. A., 1989. "Viscosity of Pure Hydrocarbons," J. Chem. Eng. Data 34:31-43. 41. Chevalier, J. L. E., Petrino, P. J., and Gaston-Homme, Y, H., 1990. "Viscosity and Density of Some Aliphatic, Cyclic, and Aromatic Hydrocarbons Binary Liquid Mixtures," J. Chem. Eng. Data, 35.206-212.

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CHAPTER 2 EXPERIMENTAL STUDIES FOR CHARACTERIZATION OF MIXING MECHANISMS

Jin Kuk Kim Gyeongsang National University Department of Polymer Science & Engineering 900 Kajwa-Dong Chinju, Gyeongnam 660-701, Korea CONTENTS INTRODUCTION, 25 CHARACTERIZATION OF MIXING MECHANISMS, 26 Optical Microscopy, 26 Electron Microscopy, 26 Small Angle Scattering, 27 Surface Roughness of Samples Using Surface Profiler, 27 Electrical Conductivity, 27 Summary, 28 EXPERIMENTAL, 28 Procedure, 28 Apparatus, 30 RESULTS AND DISCUSSION, 31 Dispersive Mixing, 31 Distributive Mixing, 31 CONCLUSIONS, 46 REFERENCES, 46 INTRODUCTION Mixing plays an important role in processing in the modern polymer industry. Specifically, mixing of an elastomer with filler, stabilizer, and accelerator is used to enhance the physical properties and reduce the cost of products. Typical examples are i. dispersion of carbon black in elastomer in an internal mixer; ii. compounding of a rubber formulation on a mill; and iii. blending of two polymers in a twin screw extruder.

25

26


However, it is difficult to study this area because mixing mechanisms are very complicated. This article discusses mixing in internal mixers. Internal mixers play an important role in the rubber industry. Internal mixers that have developed consist of two rotors mounted on parallel shafts which rotate in opposite directions. However, designs involving corotating rotors also exist. Internal mixers generally are operated under starved conditions. However, the analysis of flow in starved processing machines, especially machines of complex design, is quite difficult. Therefore, studies were made with partially filled internal mixers, and the location of void regions was determined. Mixing may be generally divided into dispersive mixing and distributive mixing. Distributive mixing means randomizing to achieve homogeneity without changing the size of particle agglomerates. Dispersive mixing usually refers to the breakup of agglomerates in a polymer matrix. CHARACTERIZATION OF MIXING MECHANISMS One of the obstacles in studying the mixing process is judging the quality of mixing. These characterizing techniques have been developed as follows: 1. Intensity of color difference, color homogeneity, and color comparison using the eye or the spectrophotometer [1]; 2. Optical microscopy [2-5]; 3. Electron microscopy [2,6-8]; 4. Small angle scattering [9-12]; 5. Surface roughness of samples using surface profiler [7,13,14]; and 6. Electrical conductivity [7,15-17]. Optical Microscopy Optical microscopy has been widely used to characterize the state of mixing because this method is very simple and straight forward in characterizing the mixing. However, tremendous efforts are required to produce reliable results. Recently, the image analyzer helped to reduce the problem. This technique is usually used for characterizing the carbon black dispersed in rubber as are the following methods. Thin frozen strips of vulcanized rubber were cut by sharp knife and viewed under a microscope with an eyepiece ruled in square micrometers. The mixing characterizes the measurement of the percentage of black agglomerates on the sections for a total area. Other methods for characterization of carbon black dispersion involve using an optical rating index with a set of standard photographs rated 1 to 10 from a poor to an excellent dispersion (Phillipips dispersion rating and Cabot dispersion rating). Electron Microscopy Electron microscopy is playing a role in the determination of the microstructural state of polymer materials. Electron microscopy is concerned with the emission and energy analysis of low-energy electrons.

Experimental Studies for Characterization of Mixing Mechanisms

27

Generally, the transmission electron microscopy (TEM) and the scanning electron microscope (SEM) have been used for characterization of the mixing mechanism. While scanning electron microscopy surveys the surface of mixtures, transmission electron microscopy surveys the microstructure of mixtures at much higher magnification levels. Therefore, the optical microscope is used for characterizing the large agglomerates (greater than 20JLUII) and electron microscopy is used for small agglomerates (500A ~ 1,000A). The experimental difficulty of SEM survey is preventing of pollution of the objective materials. The widely used method for preventing of pollution is coating with conductive material. The TEM survey requires a very thin specimen. The replication and ultrasectioning have been used to prepare the thin (less than 0.2|am) sample. Small Angle Scattering As mentioned earlier, microscopy techniques are very easy and simple methods. However, microscopy covers a very narrow region. Small angle scattering characterizes scatterers from submicron range to roughly 20|am in diameter. The background of this technique is: When electromagnetic waves are incident on an object, a path difference occurs between scattered beams. By using this phenomena, one can characterize the mixing mechanisms. Calculation of the scattering of a system can be approached in two ways. One is the direct calculation of the radiation of each scattering particle and adding all the contributions, taking into account the phase differences. Another is treating the scattering as the result of statistical fluctuations in the density or concentration. The scattering intensity depends on the polarizability of the particles compared with that of the medium in which they are mixed. It is also depends on the size of the particles and on their concentrations. One of the advantages of this method is reasonably strong theoretical basis. The principal basis of scattering theory was given by Reyleigh [18]. Thompson [19], Debye [20], and Guinier [21] developed the scattering theory. Surface Roughness of Samples Using Surface Profiler The surface roughness analysis technique is widely used to evaluate the degree of dispersion in rubber compounds. The technique is based on diverting the rupture path using agglomerates in the mixture. Recently, a dark field reflecting a light image was used to analyze the roughness of the surface of mixtures. Oscilloscope traces from a line scan across the dark field reflecting light images were used to determine the amount of dispersion in a quantitative manner. However, this technique is not capable of evaluating details of the structure of the dispersed phase. Electrical Conductivity Measurements of electrical conductivity for polymer conductive filler mixture is one of the unique techniques of dispersion. Most polymers have electrical con-

28


ductivities on the order of 10"^ or lower. When the conductive material (fillers) are mixed in polymer, a structure formed by conductive filler in mixture leads to increased electrical conductivity. There are two distinguishable mechanisms of the electrical conduction in such heterogeneous mixtures. One is the percolation mechanism, and the other is the electron tunneling mechanism. This technique is very easy and simple to measure. However, a weakness of this method is the requirement of reasonable conductivity difference between dispersed phase and matrix phase. Summary Optical and electron microscopy give a direct picture of mixing conditions of agglomerates. However, the methods require tremendous labor to obtain reliable results. Electron microscopy gives more attention to observing microstructure with high magnification. The method of surface analysis determines the entire roughness of the surface with low stylus. Electrical conductivity is useful especially when electrical conductivity of filler is much higher than that of polymer, but the lack of quantitative theory is a weakness for reasonable theoretical background. The small angle scattering method has a reasonable theoretical background, but there are many limitations in this experimental system. It is only able to characterize behavior in a limited size range. Unfortunately, there exists no universally accepted standards of performance of the mixing process. EXPERIMENTAL Procedure A butadiene-styrene copolymer (Firestone Duradene 760) was used as a base polymer for experiments. Sulfur and accelerators were introduced as curatives into the rubber. The curatives were used to preserve the shape of substances in Table 1. Rubber containing curatives was added to the internal mixer with various volume fractions from 0.5 to 0.9. The observation of the flow behavior through the visualization technique is the first experiment (Figure 1). This method enables us to see the flow motions in the mixer directly. The torque was measured as a function of time. After two to three minutes, when the torque had achieved a steady state, the rotors were stopped and Table 1 Materials for the Experiments

SBR (Firestone Duradene 706) Stearic Acid Santocure HS Thiurad (mono) Sulfur

100.0 1.0 1.2 0.2 2.0


29

Heater K;J

Heater

HAAKE INTERNAL MIXER (Rheomlx 750)

NORMAL

VIDEO CAMERA

VIDEO RECORDER Figure 1. The schennatic view of trhe system for flow visualization in an internal mixer.

the temperatures increased to 150°C to vulcanize the rubber compounds. The apparatus was subsequently cooled down, the rotors removed, and the disposition of the rubber determined. Other experiments were carried out to determine the circulation time within the mixer. This was carried out in the following manner. First, markers were prepared by adding less than 0.5% weight red color pigment to the same elastomer in the mixer. Subsequently, the pigmented elastomer was compression-molded at 100°C for 10 minutes and cut to small rectangular-shaped pieces. These markers were introduced onto the front of the left side rotor and brought to thermal equilibrium. The circulation time is defined as the traveling time for reappearance of maker on the left rotor end. The carbon black used in this experiment was Huber N990. The dimension of Huber is 0.3|im, and the type is medium thermal.

30


Apparatus A laboratory mixer (Haake Buchler) with two nonintermeshing counter-rotating rotors and an electrical heater in the barrel wall was used. A specially designed front piece with a glass was attached to measure the circulation time, shown schematically in Figure 1 [22-26]. Three sets of rotors were used in this study: 1. a pair of right-hand screws; 2. a combination of right- and left-hand screws; 3. a pair of rotors with double-flighted rotors. The rotors used are shown in Figure 2. The chamber has a capacity of 80cm^ for cases 1 and 2 and 70cm^ for case 3.

(a) R-R screw rotor

(b) R-L screw rotor

(c) double flighted rotor

Figure 2. Rotors used in this experiment.


31

Scanning electron microscopy (ISI Model SX-40) and optical microscopy (Olympus Optical Ltd.) were used to characterize the mixing mechanisms. RESULTS AND DISCUSSION Dispersive Mixing As mentioned earlier, any experimental methods for characterizing mixing mechanisms have been developed. Unfortunately, no standard method exists because of the difficulty of characterizing particle agglomerates ranging from lOOjitm (easily visible in an optical microscopy) to submicron particulates, which cannot be seen. One must then combine different methods to determine the entire distribution of particulate size. In general, the high magnification size of agglomerates is measured by electron microscopy. Methods used to study morphology are: 1. measurement of the size distribution of agglomerates from optical microscopy; 2. the state of agglomerates from scanning electron microscopy. Dispersive mixing ability was evaluated by measuring the size distribution of carbon black agglomerates. The rubber was initially masticated at 100°C for 2 minutes prior to addition of carbon black (10 parts N990 carbon black per hundred rubber were introduced to the rubber through the hopper). After mixing for 10 minutes, rubber samples were taken out and characterized by optical microscopy and scanning electron microscopy. Photographs of optical microscopy are shown in Figure 3. The average size distribution of carbon black agglomerates with various fill factor obtained from an image analyzer are calculated from optical micrographs, shown in Figure 4. The figure indicates that the average size of carbon black agglomerates decreases with increasing fill factor. The microlevel observation was carried out using scanning electron microscopy, results of which are shown in Figure 5. The photographs show that the minimum size of agglomerates occurs at 0.9 of fill factor in the rotors. The torque exerted by the internal mixer was measured as a function of time and volume fraction for the different rotor pairs. Typical plots of torque as a function of time for a slow rotor speed are shown in Figure 6. Values of maximum and steady state torque are presented as a function of mixer chamber fill factor in Figure 7. The trend for the R-R (right-right) screw and doubleflighted rotor design are similar to the torque for each being an increasing function of fill factor. In this manner, torque is deeply related to dispersion; the higher torque value represents the better dispersive mixing ability. Distributive Mixing The distributive mixing ability directly relates the flow of the materials in the mixer. As mentioned, internal mixers operate under starved conditions. A question which needs to be answered is how the rubber is distributed in the mixer. Kim and (text continued on page 35)

32


^Wm

wm^

••'(!\'ifti:¥:'.•{•'hii',:

•• ''^''^'•''Ad'kiP'Pt'l^ji: '•'• •i'''l!^''''''-.:i'h

•î^

Figure 3a. Optical nnicrographis with various fill factors (R-R screw rotor mixer).


^jl!:ir,

•' /:••.•.

•::,•••"

'^• j^.^'îJ

33

•. • % ' » ' ^ ) ; . ; -

||0§75|'

o.s>

Figure 3b. Optical nnicrographis with various fill factors (R-L screw rotor nnixer).

34


Figure 3c. Optical micrographis with various fill factors (double-flighted rotor mixer).


35

100.0 O A A

E 80.0

O TWO WING A R-R SCREW A R~L SCREW

3

99 60.0 I

or:

v-

CO

^ 40.01 M CO

20.0 4-

0.0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FILL FACTOR Figure 4. Average size distribution of carbon black agglonnerates with various fill factors. (text continued from page 31) White [23,24] observed the circulatory flow behavior using markers with flow visualization technique in the R-R handed screw and double-flighted rotors mixer (Figure 8). This makes possible measurement of the circulation time as a criteria of mixing. Since the flow visualization experiments have limitations in observing the overall flow motion in the mixer, the following experiment was carried out to determine the distribution of elastomers. The accelerates and sulfur were added to the elastomer, and the mixer was operated at 100°C. When the torque had achieved a steady state and the temperature increased to 150°C to vulcanize the rubber, the rotors were removed and the disposition of the rubber determined. Typical distribution for various rotors and fill factors are shown in Figure 9. On the R-L screw rotors, voids open first on the back of the flights at the same back side of the mixer. As fill factor decreases, the void gradually increases to include the entire back parts of the mixer behind the screw flights. In the R-R screw rotors, the voids open up behind the screw flights at opposite ends of the mixer. The similarly configured rotor in the R-R system pair developed the void at the same position as that rotor in the R-R rotor pair. With decreasing fill factor in the R-R system, the voids open up at opposite ends of the two rotors behind the flights. In the case of the double-flighted rotor screws, four separate void regions develop. In each case, these are behind the flights. This distribution of rubber indicated that the associated void region created by removing material from (text continued on page 39)

36


iMi

$m

;W^

Figure 5a. SEM graphs with various fill factors (R-R screw rotor mixer).


37

0J

0J5

m

Figure 5b. SEM graphs with various fill factors (R-L screw rotor mixer).

38


iî

0.75

OS

Figure 5c. SEM graphs with various fill factors (double-flighted rotor mixer).


39

o

FILL

FACTOR

f

MIXING TIME

r

0.9

10

(min.) Figure 6. Torque tinne plot.

(text continued from page 35) the low-pressure positions is the fully filled model (Figure 10). The theoretical model using hydrodynamic lubrication approximation was developed by Kim in 1989 [25]. The circulation time was measured with makers by using a visualization technique described earlier. From these results, the circulatory motions that occur are R-R screw rotors and double-flighted rotor. But in the R-L screw rotors, the flow motion is pumping forward like twin screw extruders. The circulation time at a rotor speed of 5 rpm is plotted as a function of fill factor for the R-R handed screw and double-flighted rotors. This is shown in Figure 11, which indicates that a void region shortens the circulation time. If there are no voids, the flow in the rotor is inhibited, leading to poor mixing. The circulation time for screw rotors is shorter than that of double-flighted rotors. (text continued on page 46)

40

Advances in Engineering Fluid Mechanics T0R0UCVS.nu.rAC70« (R'-'R screw rotor)

(a)

0.4 nil

O.S FACTOR

TORGUe vs. nil. FACTOR ( R - L scrs'w rocor)

Cti!

o O

0.-;

0.5

0.3

l.O

FiLL "ACTOR

TCRCUE VS. nUL .-ACTOR (two înq rotor)

o

C* 0.5 n u . FACTOR

Figure 7, Maximum and steady state as a function of fill factor, (a) R-R screw rotors, (b) R-L screw rotors, and (c) Double-flighted rotors.

Experimental Studies for Characterization of Mixing l\/lechanlsms

Figure 8. Flow nnotion for different rotor designs.

41

42

Advances in Engineering Fluid Meclianics

=Sii^ 0 or, in other words, when the mixing process of two (or more) species provides an ordered liquid structure more pronounced than that observable in the pure components. Typical examples of structure-making effects have been described in previous works dealing with ME/W and DME/W binary mixtures, where positive deviations from linearity (ideal behavior) have been detected at all experimental conditions with a clear maximum in the investigated property centered at X^ = 0.75 for MEAV [8] and X^ = 0.85 for DME/W [41].

92


80° C

CO

a

-60 h

Figure 4, Plots of v^/cSt vs. \ for the ED (1)/DME (2) solvent system at various temperatures from -10 to 80°C [12].

The points of interest in studies of aquo-mixed solvent systems are well-illustrated by some relatively simple interpreting models, such as those of Frank and Wen [42], and the more recent "four-segment-model" [43]. Now, rather than listing a lot of results of ED/W studies which have been obtained to date [44], only a few examples will be considered here to rationalize the macro- and microscopic behavior of this binary system. Figures 5a and 5b show the trend of v and v^ vs X^ for the three ED/W, MEAV, DMEAV binary solvent systems. Although such a maxima observed for aqueous mixtures of different organic solvents has been attributed to the formation of an association complex [45], an appropriate explanation may be given as follows. The ascending part of viscosity-composition curves in the W-rich region (Figure 5a) represents structural promotion in the mixtures by gradual formation of supraclusters of associated species (primary clusters). These supra-clusters aggregation may be provided in three different possible ways such as the association between i) the same species, ii) different species, and iii) the same and different species simultaneously. A progressive aggregation of these different types of primary clusters obviously would lead to an increase in the mixture viscosity and approach

Kinematic Viscosity and Viscous Flow In Binary Mixtures

i//cSt

Figure 5a. Trend of v/cSt vs. mole fraction X^^ of binary systems ED/W (—), ME/W (---) and DME/W {•••) at 25°C.

+3 CO

KJ

0

0.2

0.4

0.6

0.8

1

Figure 5b- Trend of v^/cSt vs. mole fraction X^ of binary systems ED/W (—), ME/W (---) and DME/W (•••) at 2 5 X .

93

94


to a maximum value when this heteroaggregation is maximum. So viscosity maximum is an excellent indication of maximum structuredness in solution. However, the participation of different types of associated molecular species (primary clusters) in aggregation to provide supra-clusters organization at viscosity maximum apparently brings heterogeniety in the liquid system (mixtures). Therefore, all these investigations (among many others in literature) have indicated that the presence of these organic hydrophobic compounds is propitious to self-association of water molecules that result in separation of W-microphases. Furthermore, even if in presence of microheterogeniety, a different influence of the concentration of these organic species on the size of W-aggregates (geometric domains) allows us, however, to suppose that structures of W-microphases in aqueous solution of ED, ME, and DME are not the same. As a rule, it is observed that binary mixtures containing ED show a negative excess property (both v^ and r|^); hence, one can deduce that breaking-structure effects always prevail when this species is involved. Probably, in the pure state this solvent is so highly structured (total homopolymer) that no further structural promotion is possible when adding a cosolvent more or less structured by itself. In fact, a very constructive and conclusive comparison is possible by taking into account the findings about v^ quantity of EDAV solvent system, where negative deviations have been observed at all experimental conditions [11], with a broad minimum in the plots centred at X^ = 0.6 (Figure 5b). Therefore, structure-breaking and structure-making effects for ED/W, ME/W, DMEAV seem to be strictly related to the magnitude of hydrophobic hydration phenomena which are consistently different for the three nonaqueous species and which appear more evident as the molecular complexity increases on passing from ED < ME < DME. A comprehensive thermodynamic investigation of ED/W mixtures was made recently by Huot et al. [44]; but these authors performed measurements in a limited range of temperature (5 < t/°C < 45) and, unfortunately, they overlooked some simple and informative thermomechanic properties, such as static relative permittivity e [46] and kinematic viscosity [11]. It is well-recognized that water has a well-defined characteristic liquid geometric structure [47]. On the contrary, the hydrophobic species ED, in spite of the high degree of self-association through hydrogen bonding, does not possess a defined liquid geometric structure, as shown by some earlier studies employing very different experimental techniques [25,48,49]. Therefore, the molecular incompatibility which arises because of the dissimilarities in the basic geometric liquid structures, as well as the differences in hydrogen bonding energies (water seems to be the quite unique hydroxilated species which undergoes geometric relaxation phenomena [47]), results in the reciprocal breakage of structural integrities of ED and W when mixed. As a consequence, the individual W molecules will get themselves loosely associated with ED molecules through hydrogen bonding, resulting in an overall less structured solvent mixture than that expected from ideal behavior [44]. Turning now to literature suggested by Fialkov [20] and Fort and Moore [21], we can suppose that the real behavior of binary liquid mixtures of protic species showing Iv^l ?t 0 is due to the formation of a complex adduct between the components, probably via hydrogen bonding network, whose composition can be defined

Kinematic Viscosity and Viscous Flow in Binary Mixtures

95

by the mole ratio corresponding to the greatest (positive or negative) deviations. For example, in the case of Figure 4 we detect a complex moiety ED : DME = 2 : 1 at all the temperatures investigated in the range -10 up to 80°C. However, as mentioned earlier when cosolvent is W, careful attention must be given in interpreting excess thermophysical properties. In fact, the results obtained in some of our previous papers indicate the formation of approximately ED : W = 2 : 3 adduct that appears thermostable at all the temperatures investigated. These findings are perfectly consistent with other authors in taking into account the excess quantities relative to other properties [44,50]. Despite good agreement, some authors disagree with the hypothesis of simple complex formation because they suggest that ED appears to be unable to form a clathrate or other well-defined hydrate in aqueous solutions [51,52]. Therefore, it is more likely the suggested stoichiometric ratio 2ED • 3W should correspond to the composition mixture where the true intermolecular complex moieties mED • nW are packaged with a welldefined and structured W-microphases (tetrahedral tetramers or pentamers [47]). THERMODYNAMICS OF VISCOUS FLOW The thermomechanical property v can also be used in a calculating procedure similar to that applied for any other classical solution thermodynamic property by applying some fundamental concepts in a suitable treatment. In fact, starting from the fundamental approach of Andrade's theory of viscosity

in perfect analogy with the Arrhenius theory of reaction rate, Eyring [32] developed a very elegant theory making explicit the preexponential factor P (which is thermally dependent), and attributing to Q (which has the dimensions of work) the meaning of "free activation energy (or Gibbs energy) of viscous flow", although the structural interpretation of this quantity is nowadays not quite clear. Therefore, according to the Eyring approach. Equation 16 takes the form: hN AG* In V = In ^^s + ^X^M. RT

(17)

where the other symbols have their usual significance. After algebraic manipulation, this relationship may be rewritten as: v Y X,M, Rln—^ = ^£L_AS* hN T

(18)

where AH* and AS* represent enthalpy and entropy of activation of the viscous flow, respectively. The form of Equation 18 suggests a method to evaluate these

96


quantities, this function being a linear relationship of 1/T. Thus, plots of the term on the left-hand side against 1/T, at constant composition, will provide AH* from the slope and AS* from the intercept. For ED/DME mixtures, the results of Figure 6 are obtained. These plots generally are not linear with the exception of pure DME (curve M), indicating that AH* is not temperature-invariant. Actually, the linearity of the trend for pure DME (r = 0.9996) suggests that the enthalpy of the viscous flow is almost constant with temperature in this solvent, as AH* = 6.97 kJ mol' and AS* = -73.1 J mol' K', respectively. For other mixtures, and pure ED, Equation 18 cannot be employed to evaluate AH* and AS* in such an extensive temperature range, but these quantities can be calculated from the slopes (and intercepts) of the curves at each investigated temperature. Analogous trends (Figure 6) have been obtained in all other cases investigated [4-12], but linear correlations have been observed only in a few cases for slightly polar and not associated species like to DME, DX, and partially, despite its polarity, for DMF, too [4,53], being well-recognized that this solvent is quite destructured in the pure state. On the basis of the linearity of curve M in Figure 6 and following the literature suggestions [32], it may be assumed that the viscous flow mechanism in pure DME, and other solvents like this one, would be a single thermally activated process. These conjectures are probably supported by the fact that DME molecules would scarcely

140 W o

120 52;

100

2.8

M

3.0

3.2

3.4

3.6

3.8

IO^T-I/K-I Figure 6. Plots of Rln(vM/hN)/J mol-^ K"' vs. T-VK"^ for the ED (1)/DME (2) binary solvent system [12].


97

interact because of the fairly low value of the dipole moment (jii = 1.17 D) and, mostly, because of the absence of hydrogen bond donor sites. Consequently, only weak dipolar interactions can be observed in this pure solvent, and a sufficient number of monomers should be available in pure DME at each selected temperature to facilitate the viscous process via activated state of the monomeric molecular species. As a very common feature, generally plots like those of Figure 6 show a marked curvature, much more evident as an extensive temperature range is taken into account. Therefore, an alternative approach that is more effective and more expeditious in evaluating these quantities starts from an application of a polynomial fitting procedure of the type ^^ Rln—^ hN

J- d =y — VT^

(19)

which is applied for each binary mixture and where d are the adjustment coefficients to be evaluated. Successively, by differentiating and rearranging, it is possible to obtain

^* = tji 0

(20)

A

and AS* = t ( j - l ) ^ 0

(21)

1

Also by applying Equation 19 to pure selected solvents, AH* and AS* values have been obtained for the pure species and have been reported in our previous papers [10-12,53] for sake of completeness. On the whole, as regards the binary mixtures to which this review is devoted, the enthalpy of viscous flow is always positive at all experimental conditions, decreasing both with increasing temperature and with increasing cosolvent mole fraction. Obviously, as it can be deduced from the curves in Figure 6 for ED/DME binaries, the largest variations are detected in pure ED and in the ED-rich composition region. On the contrary, negative entropy values are generally obtained for our binaries, with only a few exceptions in pure ED and mixtures nearest to ED in the low temperature region. A brief examination of these values makes evident the very large difference (much more than one order of magnitude) encountered in passing from one pure solvent to the other (AS* = 13.2 J mol"^ K~' for ED at -10°C, and -74.3 J mol"' K* for DMF at 80°C). However, both AH* and AS* trends always show a negative temperature coefficients (3Y*/3T < 0) at all our experimental conditions. On the basis of these considerations, it appears unlikely that in pure ED and in the ED-rich region for binary mixtures, the flow can take place simply by jumping of individual molecular units. A satisfactory elucidation of these facts probably arises from the more realistic hypothesis of Eyring, which explains the flow

98


mechanism by movement of dislocations or discontinuities in the fluid layers [32]. In a dynamic steady state, and in an oversimplified picture, the movement of a dislocation by one layer position requires the cooperation of at least two moving elementary units: one is moving out the standard position and requires energy, and the other is moving into this cavity and gives up energy. Therefore, the enthalpy of activation of viscous flow could be taken as a measure of the cooperation degree between the species involved in the flow process. Actually, in the liquid state the opportunity of the formation of many discontinuities is warranted by statistical fluctuations of local density. In the low temperature range, as well as for highly structured components, one may expect a considerable degree of order so that transport phenomena take place cooperatively; as a consequence, a great heat of activation associated to a relatively high value of flow entropy is observed. When the breaking in the ordered and polymerized fluid structure becomes very quick, by increasing the temperature or by adding a component that breaks a homopolymer hydrogen-bonding network, the movement of the individual units becomes more disordered and the cooperation degree is reduced, facilitating the viscous flow via the activated state of molecular species. As a consequence, the overall molecular order in the system should be reduced, and positive or less negative AS* values should be expected. Unfortunately, all our findings only fairly agree with these expectations. In fact, the experimental evidence obtained in our previous works appears quite intriguing because at the highest temperatures, as in the cosolvent (different to ED) rich region, the availability of randomly scattered monomers should be sufficient to provide the activated molecular species, which then lead to comparatively increased order as a result of viscous flow, giving the more negative AS* values and being perfectly coherent with those reported in our previous papers. Excess molar free energy of activation of viscous flow, AG*^, can be evaluated from Equation 14 in the proper form. The calculation results for ED/DME and ED/ W binary mixtures are shown in Figures 7 and 8 respectively, where the excess thermodynamic function is plotted against X2. The curves in these figures have been obtained by fitting the AG*^ quantities using Equation 15. It is worth noting that, by comparing Figures 7 and 8, the property AG*^ is always negative for ED/DME mixtures, showing a broad minimum that becomes deeper as the temperature becomes lower, and always centered at X^ = 0.5 (ED : DME = 1 : 1) at all experimental conditions. On the other hand, even if in presence of v^ < 0 for both solvent systems, we observe that the contrary is true for ED/W binaries with a clear maximum centered at X^ = 0.6 at all temperatures investigated. In the literature it has been pointed out that AG*^ can be considered a reliable criterion for detecting or excluding the presence of interactions of any kind between dissimilar molecules [54,55]. According to these suggestions, the magnitude of the deviations from ideality (AG*^ = 0) of multicomponent systems also can be considered an excellent indication of the strength of specific interactions. The results indicate that the extent of these interactions increases as the temperature decreases, and these variations are more evident in ED/DME than ED/W mixtures, even if their magnitude is always comparable. In fact, as the temperature increases specific intermolecular interactions should slacken and weaken probably because of the increased internal vibrational motion


* = k,vT = aX^\

a, = 16k fî^l (^] m J Vm,

^'xW

(4.11)

The other constituent of the work done by fluctuations is connected with the action of the interphase interaction force fluctuation. It is described by the two terms in Equation 4.3 that contain (f^) and fj^. These terms give both the energy input into the pseudo-turbulent motion from the mean relative fluid flow and the dissipation of fluctuation energy by viscous forces, henceforth denoted by q^ and q , respectively. The sum of the two mentioned terms is equal to q^ - q Obviously,

132


this dissipation is due to hydrodynamic resistance to particle fluctuating motion. It can be singled out by substituting identity u'= v'- w'into Equation 3.4 and isolating that part of n(fj^w') that is quadratic in w'. As a result, we arrive at the equation q_ = nm{F, <w'^) + F,u[(^'') + (("o • w')' >]} = oc_T, a_ = ( # , / m ) ( 3 F , + 4 F , u )

(4.12)

Quantity q^ describes the energy supply to pseudo-turbulent fluctuations. By using Equation 3.4 and Equation 3.5, it can easily be expressed in terms of means having the form (] = (3A -h B)T = a j ' ^ ' (w;^> = < w f ) ^ ( w f ) = T/m

^^'^^

These equations are formulated for the simplified axisymmetric case in which Equation 3.1 holds true. The first coordinate axis is directed along mean relative fluid velocity u, and two other axes are arbitrarily chosen in the plane normal to u. The averaged squares that appear in Equation 7.7 must be calculated in accordance with the following general rule [35]: < « > = j_dcojdk4',,,^(co,k),

i,j = 1,2,3

(7.8)

where a component of the spectral density tensor for particle velocity fluctuations forms the integrand. If the pseudo-turbulence is not axially symmetric, Equation 3.1 must be modified, and the second line of Equation 7.7 gives, in fact, two independent equations. At given values of suspension concentration and of other mean flow variables. Equations 7.7 may or may not have a physically acceptable solution. If there exists an acceptable solution, the above hypothesis about particle fluctuations being isotopic is correct. When there is no acceptable solution to Equation 7.7, this hypothesis is wrong. This means that interparticle collisions are not efficient enough to provide for redistribution of kinetic energy over all translational degrees of freedom when that kinetic energy is first of all supplied from macroscopic flow to longitudinal fluctuations directed along u. In this case, the requirement of kinetic energy equipartition cannot be satisfied, and a more sophisticated theory is necessary to model coarse dispersion flow. PSEUDO-TURBULENCE IN A HOMOGENEOUS FLUIDIZED BED In the preceding section we have pointed out a workable method to calculate fluctuation temperature T* at any values for mean variables locally describing

Fluid Dynamics of Coarse Dispersions

141

macroscopic suspension flow. This suffices to finally close the set of governing Equations 6.1-6.3. To get a rough idea about the properties of the pseudo-turbulence, we are going to consider a particular case of the uniform and stationary states of a coarse dispersion in more detail. Such states can be attained in practice when fluidizing a particulate system by an upward fluid flow. To simplify subsequent calculation, we will additionally neglect energy dissipation at collisions. This means that the right-hand side of the first equation in Equation 7.7 turns to zero, and this equation immediately results in a very simple relation, B = -3A

(8.1)

For a uniform stationary state, Equation 4.2 yields (f^^) = -((t)/n)p,g which permits reformulating the two last terms on the right-hand side of the second equation in Equation 7.1. Using Fourier-Stieltjes representations for all random variables in Equation 7.1, we arrive at the following linear equations for the random measures of these variables: (CO + u • k)dZ^ = ek • dZ^ -ikdZp/p,(^ + (1 - K-' )gdZ, = (ico + A)dZ, + B(u„ • dZ^ )u, (5.2)

= (F, + F,u)dZ, + F,(u, • d Z J u + [ ( F ; + ¥',xx)n - 8g]dZ, dZ, ^ d Z , - d Z , As has already been pointed out, these equations allow random measures dZ^, dZ^ and dZ to be expressed as quantities proportional to random measure dZ . After that, spectral densities of all fluctuations can be expressed in terms of the spectral density identified in Equation 7.5 while using the definition of spectral densities. For definitiveness, we now present a formal definition for the joint spectral density of random processes wj and w' [35],

^-.w.(»,k) = Jim J(dZ„,dZ;)/da)dk]

(8.3)

where the asterisk signifies the complex conjugation operation. It is easily understood that all spectral densities calculated in conformity with Equation 8.3 are proportional to the concentrational fluctuation spectral density. The proportionality coefficients depend on both the fluctuation frequency and the wavenumber vector. Mean flow variables are related to each other by Equations 4.1-4.3 and 5.1, 5.2 closed with the help of rheological equations for: 1) the mean interphase interaction force, 2) the stresses acting in both suspension phases, and 3) the fluctuation energy flux. It is easy to see that only momentum conservation Equations 4.2 and 5.2 remain informative for the uniform steady state under consideration. If pseudoturbulent contributions to the interaction force are neglected in accordance with the discussion in Sections 4-6, these equations assume the form

142


# , [F, ((|)) + F2 ((t))u]u - # , (eK"' + (|))g + (|)p,g = 0 -Vpo - # , [F, () + F2 ()u]u + # , (EK"' + (t))g + ep„g = 0

(8.4)

These equations completely determine constant uniform vector u of fluid slip velocity and mean fluid pressure p = p^ within the interstices. Solution of Equation 8.1 and derivation of spectral density expressions are trivial. However, the resultant formulae for these densities are rather unwieldy and cumbersome. Even more unwieldy are expressions for different variances and other averages that are to be obtained by integrating these formulae over frequency and wavenumber vector. (As shown in reference [25], the integration may be somewhat simplified by taking into account the fact that characteristic fluctuation frequency is much smaller than quantity F,((|)) + F2((t))u). For this reason, we do not list these formulae but confine ourselves to a brief discussion of some main conclusions resulting from a detailed analysis of pseudo-turbulent statistical characteristics. First of all, excluding A from Equation 7.7 with the aid of Equation 8.1 we arrive at a transcendental algebraic equation for B, the solution to which would enable us to ensure final closure of the developed theory. This equation has a solution only at sufficiently large suspension concentrations that exceed a certain critical value. This critical value depends on the particle Reynolds number, and it monotonously increases within the interval (0.2, 0.3) as the Reynolds number grows from zero to infinity. Thus, collisions can provide for energy equipartition only in fluidized beds of high concentration where the collision frequency is sufficiently high. In fluidized beds of lower concentrations, particle pseudo-turbulent fluctuations can be essentially anisotropic. This effect is seemingly due to the presence of interphase interaction force constituent (fh)n', since otherwise an isotropic particle fluctuation state would be possible at any concentration [25]. Indeed, this constituent can be proven to contribute substantially to the energy input from suspension macroscopic flow into longitudinal particle fluctuations. The fluctuation temperature of fluidized beds represents a function of suspension concentration that has a maximum at some intermediate concentration value. This maximum shifts to larger concentrations as the particle Reynolds number grows. Figure 1 illustrates the concentrational dependence of dimensionless fluctuation temperature for limiting cases of small particles whose Reynolds number is close to zero and of large particles whose Reynolds number considerably exceeds unity. In these limiting cases, it is sufficient to retain only one of the two hydrodynamic drag force terms specified in Equation 3.2. In both the cases, quantity u° which is used to define the dimensionless fluctuation temperature represents the terminal fall velocity of a single particle in an unbounded fluid. As follows from the curves in Figure 1, the models by Enskog and by CarnahanStarling lead to somewhat different results, and the problem of a proper choice between these models arises. The theoretical curves for fluctuation temperature are compared with the experimental data of Carlos and Richardson [42] in Figure 2. These experiments were conducted with metallic balls 8.9 mm in diameter fluidized by dimetilphtalate. The maximal fluctuation temperature was experimentally observed at / = 0.32 - 0.34, which agrees well with our theoretical prediction. On the whole, the agreement between the presented theory and experiments looks quite satisfactory.


_

I /

\

143

1

/ /^ M / /

¥2 L 1

\ \

// //

\\A

1 u 1 // '

1

0.2

\

2

\

/y''

\

"^

y^

\

1 ^ ^

\

x^

1

^

0

0.4

0.6

Figure 1. Dinnensionless fluctuation temperature for fluidized beds of snnall (1) and large (2) spherical particles according to the Carnahan-Starling and Enskog nnodels (solid and dashed curves, respectively); u*^ is the terminal fall velocity of a single particle; ^. = 0.6.

Figure 2. Comparison of the mean square of particle fluctuation velocity (curve) with experiments in reference [42] (dots).

144


The concentrational dependence of dimensionless particulate pressure is demonstrated in Figure 3 for the same limiting cases of small and large particles. The curves of Figure 3 show that the Enskog and Carnahan-Starling models result in qualitatively different conclusions with respect to particulate pressure behavior at concentrations approaching that of the close-packed state. The Carnahan-Starling model predicts particulate pressure to be a monotonously increasing function of concentration. Moreover, even the derivative of this pressure over concentration is a monotonously increasing function. (This derivative determines the "bulk modulus of elasticity" for a fluidized bed. The bulk elasticity modulus describes resistance of the bed dispersed phase to compression and plays a major role in studies of bed hydrodynamic stability, as established in reference [15,32,34], and also in reference [43,44]. In contrast, the Enskog model results in nonmonotonous dependencies for fluidized bed particulate pressure and the bulk elasticity modulus. Furthermore, both quantities fall off to zero as the bed attains the state of close packing. The problem of choosing one of the utilized approximate statistical models therefore assumes a fundamental significance. As has been pointed out, this problem can be successfully resolved by considering the behavior of fluidized beds at concentrations differing little from that corresponding to the close-packed state [25]. If the bulk elasticity modulus for a fluidized bed is negative at large concentrations, as is required by the Enskog model, the particulate pseudo-gas would be absolutely unstable with respect to virtual concentrational perturbations. At a negative bulk elasticity modulus, any occasional perturbation will grow under action

Figure 3. Dimensionless particulate pressure as a function of fluidized bed concentration; u^ is the superficial fluid velocity; notation is the same as in Figure 1.


145

of the ensuing particulate pressure difference between the perturbation itself and the surrounding bulk of the fluidized bed. As a result, the initially uniform state will eventually transform into a strongly nonuniform state, and this nonuniform state will be characterized by randomly intermittent patterns of different concentrations. In essence, inequality dp/dcj) < 0 represents the condition of absolute thermodynamic instability for the pseudo-gas. Similarly, transformation of the pseudo-gas uniform state to the chaotic nonuniform state resembles, to all appearances, the wellknown process of spinodal decomposition of thermodynamically unstable molecular and colloidal systems. Nothing like such a nonuniform state has ever been observed under conditions of incipient fluidization. This obviously calls into question the adequacy of nonmonotonous concentrational dependencies for fluidized bed particulate pressure which are occasionally derived in the literature (see, for example, reference [21]). Furthermore, the condition of hydrodynamic stability in the homogeneous fluidization state requires dpj/d(|) to be larger than a certain positive threshold value [34,43,44]. If the bulk elasticity modulus is a monotonously increasing function of concentration, we can expect that an increase in concentration, at least within a concentrational range near the close-packed state concentration, will facilitate stability. There is a great deal of experimental and theoretical evidence that bears this expectation to be true. It seems there always exists a more or less narrow interval for fluid velocities that only slightly exceed the minimum fluidization velocity, and that in this interval any fluidized bed is hydrodynamically stable. This fact is in qualitative agreement with conclusions based on use of the CarnahanStarling model, but certainly contradicts expectations set up in the Enskog model. These inferences directly lead to the conclusion that in the context in question the Enskog model is inadequate and the Carnahan-Starling model ought to be preferred. This is hardly surprising considering the purely empirical nature of the Enskog model. Although it leads to a fairly good estimate for the particulate pressure (or Enskog factor), it is difficult to expect good results in relation to the more subtle properties of the particulate pseudo-gas, such as the chemical potential of the particles and the associated variance of the concentrational fluctuations. There seems to be no doubt that Equation 7.3 underestimates the variance in the region of high concentration. In particular, this variance turns to zero for the close-packed state, meaning that Equation 7.3 does not allow for the concentrational fluctuations specific to chaotic states of close packing. This is why Enskog model results in the incorrect conclusion concerning the thermodynamic stability of the pseudo-gas in that region. On the other hand, applicability of the Carnahan-Starling model (as well as applicability of other approximate statistical models of the same kind) to particulate systems of very high concentrations appears to be questionable, to say the least. In particular, this possible inapplicability of this model can be due to its failure to account for spontaneous origination of ordered crystalline phase patterns at (|) = 0.55 - 0.59. This model also fails to describe a sharp increase in pressure and dynamic viscosity for the dispersed phase when on the verge of the closed-packed state. In contast to this, the Enskog model leads, however empirically, to physically correct conclusions that both pressure and viscosity tend to infinity as (|) approaches the value attributed to the state of close packing.

146


If we use the Carnahan-Starling model to calculate concentrational fluctuation variance and the Enskog model to describe pseudo-gas osmotic pressure and kinetic coefficients, we shall most likely arrive at a better approximation in the certain narrow range of suspension concentrations that adjoins the close-packed state concentration. (In fact, it is precisely the Enskog model that has already been used to formulate Equation 4.4 describing kinetic coefficients in a concentrated pseudogas.) Within the framework of such a combined model, the osmotic pressure correction function and Enskog factor follow from Equation 4.8, whereas concentrational fluctuation variance results from Equation 7.4. The soundness of this combined model is implicitly confirmed below by comparison of theoretical and experimental conclusions concerning stability of fluidized beds near the state of incipient fluidization. With the help of these equations, we can easily calculate other averaged pseudoturbulence characteristics in fluidized beds and vertical suspension flows using the same methodology as employed in reference [25]. Instead of stepping through these unwieldy calculations, we are once again going to omit them and shall only briefly enumerate some qualitative conclusions that follow from these calculations. First of all, fluid pseudo-turbulent fluctuations are essentially anisotropic, even if particle fluctuations are isotropic. This anisotropy somewhat weakens as particle size grows. In addition, the intensity of fluid velocity fluctuations is much higher than that of particle velocity fluctuations. The total fluid and particle volume fluxes are expressible as Qo = (ev) = ev - ((t)V>,

Q, = (^w> = ^w + ( f w ' )

(8.5)

(It should be remembered that angular brackets are dropped out when notating mean flow variables.) The second term on the right-hand side of these equations are different from zero. Of course, both terms are proportional to constant vector u (or g). Thus, these equations prove that fluid and particle volume fluxes in an actual fluctuating suspension differ from the corresponding fluxes in the suspension of the same particles at the same concentration, but without fluctuations. Fluid volume flux always increases owing to fluctuations. However, particle volume flux increases for that reason only in suspensions of large particles and decreases in suspensions of small particles. Equations 8.5 permit averaged velocities of the suspension phases to be introduced as •^''-e"'^

e

'

'''-

^ -""^

(),

(8.6)

Hence, it immediately follows that the averaged phase velocities introduced through mean volume fluxes associated with flow of the dispersed and the continuous phases differ from the mean velocity for a single particle and from the mean velocity for fluid in the interstitial space, respectively. These velocities coincide only at the dilute limit ([) -^ 0. Among other things, the very existence of such a difference in mean velocities proves the hindered settling velocity for a suspension to be dependent on the


147

experimental technique used to measure this settling velocity. There are two main types of experiments designed to determine the hindered settling velocity. The most common finds this velocity in concentrated suspensions by measuring either the velocity of the subsiding interface at the top of a sedimenting suspension or by measuring the accumulation rate for a close-packed sediment at the bottom of a vessel containing the suspension [45]. (This is equivalent to measuring flux Q, while regarding quantity Ac = JCj - c^l as the hindered settling velocity.) An alternative approach consists in tracing a random trajectory of an individual particle in the suspension bulk [46]. In this case, evaluation of particle trajectory data directly gives velocity w, and quantity u = Iw - vl is considered the hindered settling velocity. The difference between these two settling velocities has been thoroughly discussed in connection with the sedimentation of coUisionless finely dispersed suspensions in reference [14]. The cited paper demonstrates that the settling velocity determined through mean dispersed phase volume flux may be considerably smaller than the velocity evaluated by means of tracing particle trajectories. Further, coefficients X. in expression 3.3 for mean drag force in a fluctuating suspension can be found by substituting local flow variables in expression 3.2 by the sums of their means and fluctuations with respect to these means. Consequently, we can expand the resultant expression into a Taylor series. Keeping in mind that fluctuations are assumed to be relatively weak, we retain only terms up to the second order in fluctuations inclusive. There is no doubt that a term of the first order in fluctuations is identically equal to the force fluctuation given by Equation 3.4. By averaging the mentioned Taylor series, we get the mean force as a sum of two terms. One of these terms is independent of fluctuations. It is expressed by Equation 3.2 in which u and (|) are understood as the mean fluid relative velocity and suspension concentration. The other term is of the second order of magnitude in fluctuations, and it contains the averages of various fluctuation products. After a manipulation, this second-order term can be transformed into a vector proportional to u so that summing up these two terms results in an expression of the type of Equation 3.3. Coefficients X. are usually close to unity. In fact, they describe a change in the hydraulic resistance that impedes relative fluid flow in a fluctuating particulate assemblage as compared with resistance of the same assemblage without fluctuations. It can be shown that fluctuations in suspensions of small and moderate particles induce a drag reduction at any suspension concentration. In suspensions of large particles (for which only the quadratic term of the two-term drag law, 3.2, is essential), fluctuations produce a drag reduction at low and moderate concentrations. If the suspension concentration is sufficiently high, fluctuations cause a certain increase in particle hydraulic resistance. As early as 1972, it had been concluded that hydraulic resistance for a fluidized bed is somewhat lower than that of a stationary granular bed under otherwise identical conditions [47]. This conclusion is also supported by extensive experimental evidence. APPLICATION EXAMPLES For the remainder of this paper, we shall illustrate the general capacity of the above fluid dynamic scheme to adequately describe coarse dispersion flow. With this purpose in mind, we shall briefly consider the results of applying this scheme

148


to a few concrete flow problems. All of these problems had earlier been proven a matter of considerable difficulty, and they were all the subject of many publications. A comprehensive analysis of each problem requires a great deal of reasoning, not to mention exaction of necessary and tedious calculations, and each problem might well be a topic of a separate lengthy paper. For this reason, we are compelled to confine ourselves only to a short discussion of the peculiarities of problem formulation and to a concise summary of the expected main conclusions, without paying attention to calculation. Homogeneous Fluidization Stability Hydrodynamic stability of uniform vertical suspension flow has been theoretically treated for more than 30 years (see reference [15,20,29,32-34,43,47], and also reference [48-56]). Much of this work has been undertaken when analyzing the important problem of reasons causing the transition from homogeneous (particulate) fluidization to nonhomogeneous (aggregative) fluidization, and subsequently, providing for the spontaneous origination in a fluidized bed of cavities (bubbles) almost devoid of particles. The fact that the total number of particles must be conserved during the development of occasional disturbances in a uniform vertical flow or in a homogeneous fluidized bed in itself results in the formation of kinematic waves of constant amplitude, as was first demonstrated by Kynch [48]. Both particle inertia and the nonlinear dependence of the interphase interaction force on the suspension concentration cause an increase in this amplitude. This amounts to the appearance of a resultant flow instability with respect to infinitesimal concentration disturbances and with respect to other mean flow variable disturbances. Various dissipative effects can slow the rate at which instability develops, but cannot actually prevent its development. Therefore, investigating the linear stability of a flow without allowing for interparticle interaction leads inevitably to the conclusion that the flow always is unstable irrespective of its concentration and the physical parameters of its phases. This conclusion contradicts experimental evidence that proves suspension flows of sufficiently small particles in liquids to be hydrodynamically stable in wide concentration intervals [57-59]. Moreover, even flows of large particles in gases may be stable if the concentration is either very low or very high. Beginning with the paper by Jackson [20], disturbance stabilization in a fluidized bed is usually associated with the action of specific normal stresses inherent to the dispersed phase. These stresses impede volume deformations of the dispersed phase. Despite this fact having been understood for a long time, comprehensive development of a stability theory is hindered by the almost total absence of reliable information concerning the dependence of dispersed phase stresses (or of the corresponding bulk moduli of dispersed phase elasticity) on the suspension concentration and on the physical parameters. This lack of information partly invalidates all theoretical inferences bearing upon hydrodynamic stability in suspension flow. Attempts to introduce dispersed phase normal stresses on a purely phenomenological basis [32,43] or by means of simple mechanical models [51,52,56] are heuristic by their very nature and in no way help to solve the problem. Therefore, the results of the corresponding stability studies, and especially the studies treating


149

nonlinear evolution of finite disturbances [53-55], remain essentially formal. The lack of a reliable rheological model for the suspension dispersed phase, even in one of the latest stability research studies in the field of the mechanics of disperse systems [34], necessitates the use of inadequately based empirical relations for its elastic properties. Another stumbling block inherent in conventional stability models for both vertical suspension flows and homogeneous fluidized beds consists in the fact that there exist quite different opinions regarding the additional constituents of the interphase interaction force which are supplementary to those due to drag and buoyancy. These additional constituents, which have been designated by 5f' and discussed in connection with Equation 3.2, substantially affect neutral stability curves since they are of the same order of magnitude as the inertial forces on the left-hand side of momentum conservation Equations 4.2 and 5.2. Moreover, there remain serious uncertainties with respect not only to the actual existence of possible stable states, but also with respect to the nature of the scale factors that determine the wavelength, velocity, and growth rate of the maximal growth waves associated with suspension flow instability [59]. Thus, in order to render the stability theory completely determinate, we need to specify in an unequivocal form both the conservation equations governing macroscopic suspension flow and all the rheological equations of state. This is easily seen to be possible for coarse dispersions of small particles. For such dispersions, normal stresses in the dispersed phase may be approximately described in terms of the particulate pressure as explained in Section 4, and this pressure can be evaluated for uniform dispersion states with the help of Sections 7 and 8. As a result, particulate pressure appears to be a single-valued function of mean variables characterizing the uniform dispersion state under study and of the physical properties of its phases. This single-valued function involves neither unknown quantities nor arbitrary parameters. On the other hand, if the particle Reynolds number is small, all interphase interaction force constituents also can be expressed in an explicit consummate form with help from the theory in reference [24]. This expression for the fluid-particle interaction force recently has been employed as well in stability studies for flows of collisionless finely dispersed suspensions [15,60]. Equations 8.4 hold true for unperturbed states of a homogeneously fluidized bed the stability of which is under question. When the bed consists of small particles, these equations yield u = -F-U(|))8(l-K-^)g,

po = const + ((t)p^ + e p o ) g * r

(9.1)

Mean dispersed phase velocity identically equals zero in the convective reference frame, and unperturbed fluctuation temperature coincides with quantity T* determined as explained in Section 8. Particulate pressure is then defined in accordance with the rules described in Section 4. Both T* and p^ are single-valued functions of (|) and u. In compliance with the discussion in Section 8, we choose the Carnahan-Starling model to define concentrational fluctuation variance according to Equation 7.4. With the help of Equation 4.9, this same model can be employed to express the osmotic pressure correction function and the Enskog factor for practically all suspension concentrations which lie beyond a narrow concentration range adjoining the closed-

150


packed state concentration. Within this range, Equation 4.8 resulting from the approximate Enskog model has to be used to calculate the osmotic pressure function and Enskog factor. For definitiveness, function F,((|)) is taken from Equation 7.2 in all subsequent calculations. A set of linear equations describing the infinitesimal disturbances in the mean flow variables follow from Equations 4.1-4.3 and 5.1, 5.2 with allowance made for Equation 9.1. Analysis of this linear set and of its characteristic equation should be accomplished along the well-known standard lines of the hydrodynamic stability theory which are exemplified for similar stability problems in reference [15,34]. In addition to this general formulation of the stability problem, different simplified versions of this problem can be considered, and in particular, those corresponding to the simplified fluid dynamic models discussed in Section 6. To greatly simplify the calculations, we might be tempted to obfuscate the necessity of reckoning with fluctuation energy conservation Equation 4.3. This can be accomplished by introducing two limiting extreme regimes of mean flow variables disturbance evolution. In these limiting cases, the fluctuation temperature is either taken to be totally insensitive to the occurrence of the hydrodynamic disturbances, or it is assumed to passively follow the changes in the fluidized bed local state induced by these disturbances. In the first regime, T is identically equal to its value T*, specific to the unperturbed state, and so does not vary at all. This regime is likely to become established if the fluctuation temperature relaxation time (which can be precisely evaluated from Equation 4.3 supplemented by Equation 4.15) greatly exceeds the disturbance time scale. On the contrary, in the second regime, the fluctuation temperature relaxation time is presumed to be negligibly small as compared with the time scale of the disturbances so that T is given by the same function as T*, except for the fact that perturbed local instantaneous values of mean flow variables serve as arguments to this function instead of the mean flow variables characteristic of the unperturbed state. These limiting regimes may be conventionally termed respectively as those of constant and varying temperature. In the general case, fluctuation temperature can be justifiably expected to vary somewhere between the extremes corresponding to these limiting regimes. In the remainder of this subsection, we shall briefly enumerate the main conclusions about these limiting regimes that obtain from a stability analysis of a homogeneously fluidized bed with respect to infinitesimal plane vertical waves. It is easily demonstrated that waves of infinite wavelength are always neutrally stable. However, actual fluidized beds are of finite dimensions, meaning that the wavelength cannot exceed some critical value. Consequently, the wave number cannot be smaller than some threshold value, k^.âeVg, which is inversely proportional to fluidized bed height, coefficient k^.^ being dimensionless. The onset of instability can be proven to occur for the first time with respect to waves of precisely this minimal wave number. In Figure 4, representative neutral stability curves are plotted in plane (((), K), the stability regions being situated above the curves. These curves are distinguished by two parameters, Vi = a/T^g, Sc = Vi^k^^.^K'*

(9.2)


151

Figure 4. Neutral stability curves for fluidized beds as follow fronn the Carnahan-Starling nnodel at Vi = 1 and different values of IgSc (figures at the curves) in the llnniting regimes of constant and varying fluctuation temperature (solid and dashed curves, respectively); dotted curves correspond to the osmotic pressure correction function calculated with the help of the Enskog model. The first parameter appears as a result of quasi-viscous stresses in the dispersed phase affecting the development of initial plane waves. In fact, this parameter characterizes an influence on fluidized bed stability caused by dispersed phase viscosity. The occurrence of the second parameter is due to the restriction imposed from below on permissible wave numbers for these plane waves. Actually, the second parameter descibes a so-called scaling effect of the bed dimensions on bed stability. The curves in Figure 4 correspond to the Carnahan-Starling model, save for the dotted ones which have been drawn when using Equation 4.8 to represent the osmotic pressure correction function and the Enskog factor. Comparison of the neutral stability curves shown in Figure 4 proves that the limiting regimes of constant and varying fluctuation temperatures yield essentially different results only at low and moderate fluidized bed concentrations. As the concentration increases, the difference in the neutral stability curves for these limiting regimes gradually disappears. Consequently, fluctuation temperature relaxation phenomena are unlikely to affect conditions for the onset of instability in highly

152


concentrated fluidized beds and vertical suspension flows. As can be seen from such a comparison, possible variations in fluctuation temperature favor a break of stability for flows of low and moderate concentrations. In contrast, no matter whether the Carnahan-Starling or the Enskog statistical model is used to evaluate particulate osmotic pressure and the Enskog factor, the difference between the corresponding conclusions pertaining to flow stability appears to be quite negligible at low and moderate flow concentrations. However, this difference becomes very significant in the range of high concentrations. As has already been discussed in Section 8, the Enskog model is apparently preferable in this range. If so, we immediately arrive at the conclusion that there always exists a narrow interval of concentrations near the close-packed state concentration for which homogeneously fluidized beds and vertical suspension flows are hydrodynamically stable with respect to small disturbances. This conclusion lends support to the inference made in a number of works with respect to fluidized beds being stable within a certain fluidization velocity interval immediately after their having passed over the state of incipient fluidization [49,52,59]. The curves in Figure 4 also show that homogeneous fluidized beds and vertical suspension flows are unstable if flow concentration lies within a certain range of relatively low concentrations, which is apparently independent of phase material density ratio K. When concentration (|) and parameter Sc are fixed, stability can be achieved by increasing K under otherwise identical conditions. At first glance, both these conclusions seem to contradict natural expectations that a decrease in either concentration or density ratio should favor flow stabilization. Actually, our expectations are not thwarted. As a matter of fact, flow stabilization occurs for the mere fact that the action of particulate pressure smooths away concentration disturbances. This pressure represents a rapidly increasing function of concentration. It is also proportional to the particle terminal velocity squared, and this velocity increases from zero as K grows from unity. Hence it becomes evident that particulate pressure may well be weak enough to ensure flow stabilization at low concentrations and density ratios close to unity. On the other hand, the growth increment of disturbances growing because of instability can be proven to decrease with both (j) and K. At low values of the indicated quantities, this growth increment is too small to provide for a noticeable increase of disturbances before they altogether leave a fluidized bed of finite dimensions. As a result, in spite of such a fluidized bed being hydrodynamically unstable, it may well appear stable from the visual perspective. And finally, the curves in Figure 4 help us to understand the underlying reason for the scaling effect so frequently encountered in industrial practice. The essence of this effect is as follows. Let a bed of given particles fluidized by a given fluid be quite stable when produced in a laboratory apparatus. However, the same bed becomes unstable at the same concentration when it is brought into being in geometrically similar industrial devices of a larger linear size. The increase in apparatus size obviously corresponds to a decrease in scaling parameter Sc. As the last parameter decreases, the stability region is reduced as evidenced by Figure 4. It may well be that point ((|), K) characterizing the bed state lies inside the stability region for a smaller laboratory apparatus, but falls outside this region for a larger industrial device.


153

Of special importance for loss of stability are the maximal growth waves which exhibit the greatest growth rate. The wave number values which determine the maximum for the growth increment and corresponding wave frequency are also functions of (j), K and Vi. The dimensional wave number and frequency as well as the propagation speed of maximal growth waves are illustrated in Figure 5 for some T

0.60

0.30

0.45

0.60

12.5 h

0.30

0.45

0.60

0.30

0.60

§hl3h

0.30

0.45

0.60

0.30

0.60

Figure 5. Dimensional wave number k^, frequency co^, and propagation velocity c^ for waves of maximal growth under conditions of the experiments in reference [57] in which glass balls 0.083 and 0.156 cm in diameter (a and b, respectively) were fluidized by water; curves—theory, dots—experimental data.

154


of the experimental conditions in reference [58] where glass balls were fluidized by water. The theoretical curves in Figure 5 are obtained with no assumptions made concerning fluctuation temperature disturbances, instead explicitly taking into account the equation for these disturbances that follows from Equation 4.3. The particulate pressure and the Enskog factor needed to calculate the viscosity and fluctuation energy transfer coefficients for the dispersed phase are evaluated with the help of the Enskog model of dense gases. The agreement between theory and experimental data seems satisfactory. Among other things, this agreement confirms the notion in Section 8 about the Enskog dense gas model being preferable for evaluating particulate stresses in states near to that of close packing. Voidage Distribution Ahead of a Bubble in a Fluidized Bed Experimental studies of voidage in the vicinity of bubbles rising in a fluidized bed have shown that these bubbles are surrounded by an expanded "shell" where the void fraction is considerably in excess of that in the remote bulk of the bed [61,62]. Such voidage variations had been reported in earlier experiments conducted using two-dimensional model beds [63,64]. Because these observations are at variance with the available theories of steady bubble motion in fluidized beds [65], they have been used by a number of researchers to reexamine these theories. Thus, Collins [66] fits an equation to the results of reference [63], (t) = (t)oexp [-k(R/r)"],

k « 1/15,

n« 3

(9.3)

where the zero subscript refers to the conditions far away from the bubble, R is the bubble radius, and r the radial dimension. He concluded that voidage variations are very slight, justifying the assumption of an incompressible, inviscid phase made in the theory developed in reference [65]. A similar conclusion also has been drawn in reference [67]. However, the findings of recent experiments conducted in reference [61,62] prove these variations to be much more significant than follows from the analysis in reference [66,67]. This fact has stimulated a new attempt to address the problem [68]. As a first approximation, the analysis in reference [68] uses the well-known model of Davidson and Harrison [65] in which the bubble is assumed to be a spherical cavity without particles and in which the dispersed phase is characterized by uniform concentration (^^ everywhere outside the bubble. Relative interstitial fluid velocity, u, and mean particle velocity, w, can then be found on the basis of: 1) a simple filtration flow model for a homogeneous porous body containing a spherical cavity, and 2) an ideal fluid model for flow around a sphere. In particular, the vertical components of these velocities along vertical axis z of the coordinate system having its origin at the bubble center are: (^

u

Uo,

w= -

î

I \4

where U is the bubble rise velocity.

^

u,

u = |vp:

(9.4)


155

For the sake of simplicity, we assume the fluid to be a gas and the particles to be small so that their hydraulic resistance is linear in u. The components of momentum conservation Equations 4.2 and 5.2 along the z axis can then be formulated as follows: 3p,/3z = (|)pi[F,((|))u-8g-w(3w/az)] apo/3z = -(t)pi [F, ((|))u - (|)g]

(9 5)

Particulate pressure can be expressed as G((t))nT in conformity with Section 4, and T can be approximately represented as T*, which is a function of (|) and u to be calculated according to the method of Sections 7 and 8. After that, the first equation of 9.5 is reduced with the aid of Equation 9.4 to a first-order differential equation for unknown function (|)(z). The last equation must be integrated at the boundary condition where (|) turns to (j)^ at large z. This gives a profile of fluidized bed concentration along the vertical axis ahead of the bubble. For definitiveness, Equation 4.9 is used for G((|)) and Equation 7.2 is used for F^((t)) in this calculation. The resulting theoretical profile is shown in Figure 6 together with the line (^ = ^ J 1 - exp[-1.5(r/R - 1)]}

(9.6)

that best fits experimental findings of reference [61,62,68]. In addition. Figure 6 contains profile (9.3) and the profile calculated in reference [68] on the basis of the same model used in this paper, but with no allowance for interphase interaction force fluctuation constituent {fj,)n' during evaluation of T*. The agreement between the theory and experimental data seems quite good. Moreover, it can be concluded that the mentioned force constituent causes a significant influence on the particulate pressure. Binary Fluidization Polydisperse fluidization is widely used in chemical engineering in connection with the various technologies employed in the thermochemical processing of powdered materials in fluidized beds of inert particles, such as drying, burning, and oxidation-reduction. Polydisperse fluidization is also relevant to low-temperature combustion of coal and other dispersed fuels. The segregation of a mixture of solid particles with different properties within either fluidized beds or in vertical suspension flows provides a basis for ore recovery and for diverse enrichment processes in mineral dressing. Such important applications have stimulated an extensive study of segregation phenomena in fluid-particle mixtures over the few past decades. A representative sampling of experiments conducted with liquid-fluidized binary particulate mixtures, together with some semi-empirical model considerations, are to be found in reference [69-75]. As a rule, such considerations are based on the presumption that the natural tendency of heavier or larger particles to accumulate at the bottom of a fluidized bed is counteracted by their axial dispersion. This dispersion is supposed to make for the occurrence of transitional regions in which different particulate species are

156

Advances in Engineering Fluid Mechanics u.o

1

2^^^^^ 0.4

3

/f

4

0.2

n

1

1

2

_i

1

3

Figure 6. Dependence of particle volume concentration ahead of a bubble rising in a fluidized bed at (t)^ = 0.5 on dimensionless distance ^ = z/R; 1—formula (9.3) derived in reference [66], 2—theory in reference [68], 3—present theory, and 4—the curve correlating experimental findings in reference [61,62]. simultaneously present. A weak point of those models lies in the fact that the necessary axial dispersion coefficients for different particles are commonly introduced into the analysis in a purely empirical way, and actually they play the role of adjustable parameters. There is altogether no precise indication in the literature as to how to theoretically determine these dispersion coefficients.


157

An alternative approach makes use of two-phase flow momentum conservation equations. Doubtless, such an approach forms a more sound foundation for the theoretical study of polydisperse and, in particular, binary fluidization. In principle, this approach may be based on the same notions and concepts as the fluid dynamic theory of monodisperse coarse suspensions developed in the present article. However, a number of new problems arise which are specific for polydisperse particulate mixtures. Among these new problems are: 1. How to evaluate the effective mobility of a particle of certain size and density in a concentrated mixture of suspended particles belonging to different species; 2. How to describe concentrational fluctuation variances for particles of different species in a polydisperse mixture; and 3. How fluctuation velocities and particulate pressures associated with particles of different species are related among themselves. These and some other problems have been successfully resolved in reference [76] for stationary fluidized beds in which the mean velocity of any type particles equals zero. A certain deficiency in the analysis of the cited paper consists in the fact that it entirely overlooks those contributions to fluctuation forces acting on different particles that are proportional to the number concentration fluctuations of these particles. Nevertheless, however important it may be in a quantitative sense, this deficiency does not affect the fundamentals of the theory of homogeneous fluidization of polydisperse mixtures. These fundamentals are briefly outlined below as set forth in reference [76]. Quantitative results, which are used to check the adequacy of the theory by comparing its predictions with experimental data, are obtained with an accounting for the mentioned force contributions missing from the analysis in reference [76]. We consider a mixture of spherical particles belonging to two discernible species which may differ in both radius and density. These particles are suspended by an upward flow of a fluidizing fluid, and there is no macroscopic flow of these particle species relative to each other. The local fluidized bed composition is assumed to depend only on the vertical coordinate z. It can be described with the help of partial volume concentrations (|)j and ^^ for particles of these two species. The hydrodynamic force exerted by the fluid on a single particle of the j-th type (j = 1, 2) can be taken in a form similar to that of Equation 3.2, fhj = ni.[F,.((t)) + ¥,.{ 1 mm) the CO burns to CO2 at or very close to the carbon surface; for finer carbon particles (d < 0.1 mm) the CO burns far away from the carbon surface. The char particles burn at a constant density in accordance with a shrinking sphere model. Burning rates of char particles in a turbulent fluidized bed are higher than those observed in bubbling beds. This is attributed to higher rates of oxygen transfer to the particle surface. The interaction between combustion and attrition has important consequences for system models; yet, despite an excellent start to this area of research, much more detailed research work is required. Detailed evaluation of heat transfer rates from burning particles also is required. The interaction between heat, mass, gas, and solid-phase chemical kinetics requires further research. Until now, experiments have been conducted with single particle or batch addition of carbon. Experiments with a continuous feed would provide useful additional information on size distribution and carbon fraction in the bed. NOTATION A a c C c

Arrhenius pre-exponential factor Surface area of the particle Exponent of d Gas phase oxygen concentration Specific heat of coal/char particles C^ Surface oxygen concentration d Diameter of bed particle d^^ Average diameter of coal/char particle during combustion d Diameter of coal/char particle d Q Initial diameter of char particle D Diffusion coefficient of O. in a

2

N2 D Diffusion coefficient of O. in g

2

N2 in the bed DQ Diameter of sphere having the same initial mass of the particle E Activation energy E Elutriation rate of carbon

E, E^ Energy of activation and mean energy activation of thermal decomposition AH Heat liberated from combustion of carbon k Thermal conductivity of gas k^ Attrition rate constant k Overall mass transfer coefficient k ^ Particle convective mass transfer coefficient k^^ Gas convective mass transfer coefficient k, k„ Rate constants for r

0

devolatilization reactions k Devolatilization constant V

m m^ m^ m^

Mass of the carbon particle Diffusion limit Burning rate Mass of char particle before combustion

Combustion of Single Coal Particles in Turbulent Fluidized Beds

M^ n Nu R R^ Re^ R^ Re Sc Sh T Tj^ T T t^ U U^

Molar mass of carbon Apparent order of reaction Nusselt number Universal gas constant Reaction rate coefficient Particle Reynolds number at the terminal velocity Surface reaction rate Reynolds number Schmidt number Sherwood number Absolute temperature Bed temperature Particle temperature Absolute temperature of gas Devolatilization time Superficial gas velocity Bubble Rise velocity

189

U^ Velocity at which transition to turbulent fluidization begins Velocity at which transition to u, turbulent fluidization becomes complete u. Transport velocity u^ Terminal velocity of the bed particles Uo Minimum fluidization velocity u Fractional burnoff V. Weight of volatile released up to time t per weight of original coal v« Asymptotic weight of volatile released at long times w Weight of carbon particle in the c bed X Fractional mass transfer control 1

1

Greek Symbols a Coefficient for particle size change P Coefficient for particle density change 8 Cluster fraction c

e, 8^ Bed and cluster voidage respectively (e^ - 0 . 5 ) e Emmissivity of the particle 0, 0 , 6 Contact time, gas convective contact time and particle convective contact time, respectively

PA' PAO Apparent char particle density, initial apparent char particle density p^ Density of bed particles ([), ^Q Shape factor and initial shape factor a Standard deviation of s

Gaussian distribution of activation energy a Stefan-Boltzman constant r| Fraction of energy reaching carbon surface

REFERENCES 1. Basu, P. and Subbarao, D., Comb. Flame, 66, 261-269 (1986). 2. Haider, P. K., Datta, A. and Chattopadhyay, R., The Canadian J. of Chem. Eng., 71, 3-9 (1993). 3. Grace, J. R., Handbook of Multiphase Systems, ed. G. Hestroni, Hemisphere Publication, 8-52 (1986). 4. Yerushalmi, J. and Cankurt, N. T., Powder Technology, 24, 187-205 (1979). 5. Anthony, D. B., Hottel, J. B., and Meissner, H. P., 15th Symp. on Combustion (Int.), The Combustion Inst., Pittsburgh, 1,303 (1974). 6. Borghi, G., Sarofim, A. F. and Beer, J. M., Annual AIChE meeting (1977). 7. By water, R. J., Proc. of the 6th Int. Conf. on Fluidized Bed Combustion, Atlanta, Vol. Ill, 1,092-1,102 (1980).

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8. Wells, J. W., Krishnan, R. P. and Ball, C. E., Proc. of the 6th. Int. Conf. on FBC, III, Atlanta, 773-783 (1980). 9. La Nauze, R. D., Fuel, 61, 771-774 (1982). 10. Pillai, K. K., J. Inst, of Energy, 54, 142 (1981). 11. Avedesian, M. M. and Davidson, J. F., Trans, of Inst, of Chem. Engineers, 51, 121-131 (1973). 12. Basu, P., Broughton, J. and Elliot, D. E., Inst, of Fuel Symp. Ser., Vol. 1, No. 1, A3-1-A3-10 (1975). 13. Ross, I. B., and Davidson, J. F., Trans, of Inst, of Chem. Engrs, 59, 108-112 (1981). 14. Basu, P., Ph.D. thesis, University of Aston in Birmingham, 1976. 15. Mon, E. and Amundson, N. R., Ind. Eng. Chem. Fundam., 17, No. 4, 313-321 (1978). 16. Bukur, D. B. and Amundson, N. R., Chem. Eng. ScL, 36, 1,239-1,256 (1981). 17. La Nauze, R. D., Chem. Eng. Res. Dev., 63, 3-33 (1985). 18. Basu, P., Transaction of CSME, 9, No. 3, 141-149 (1985). 19. La Nauze, R. D. and Jung, K., I9th. Symp. (Int.) on Comb., The Comb. Inst., 1,087-1,092 (1982). 20. Beer, J. M., and Massimilla, L. and Sarofim, A. F., Inst. Fuel Ser., 4, IV 5.1 (1980). 21. Donsi, G., Massimilla, L. and Miccio, M., Combustion and Flame, 41, 57-64 (1981). 22. Arena, U., D'Amore, M. and Massimilla, L., AIChE J., 29, 40 (1983). 23. Haider, P. K., Salatino, P. and Arena, U., The Canadian, J. of Chem. Engg., 66, 163-167 (1988). 24. Haider, P. K. and Basu, P., Chem. Engg. ScL, 47, No. 3, 527-532 (1992). 25. Chirone, R., D'Amore, D., Massimilla, L. and Mazza, A., AIChE J., 31, 812-820 (1985). 26. Massimilla, L., Chirone, R., D'Amore, M. and Salatino, P., Final Report, DOE Grant No. DE-FG22-81PC40796, Dipartimento di Ingegneria Chimica, Universita di Napoli (1985). 27. Cammarota, A., Chirone, R., D'Amore, M. and Massimilla, L., 8th Int. Conf. on FBC, Houston, TX, 1985. 28. Frossling, N., Gerlands Beitr Geophys, 52, 170-216 (1938). 29. Rowe, P. N., Claxton, K. T. and Lewis, J. B., Trans. IChemE. 43: T14-T31 (1965). 30. Chakraborty, R. K. and Howard, J. R., J. Inst. Energy, 54, 48-54 (1981). 31. Subbarao, D., Powder Technol., 46, 101-107 (1986). 32. La Nauze, R. D., Jung, K. and Kastle, J., Chem. Eng. Sci, 39, 1,623-1,633 (1984)., 33. Haider, P. K., Datta, A. and Chattopadhyay, R., I2th Int. Conf on FBC, San Francisco, 1,223-1,227 (1993). 34. Field, M. A., Gill, D. W., Morgan, B. B. and Hawksley, P. G. W., BCURA, Leatherhead, UK, 329-345 (1967). 35. Young, B. C. and Smith, I. W., I8th Symp. (Int.) on Combustion, The Comb. Inst., 1,249 (1981).

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36. Daw, C. S. and Krishnan, R. P., Oak Ridge National Laboratory, Report No. ORNL TM - 8604, Oak Ridge, TN (1983). 37. Haider, P. K. and Basu, P., The Canadian J. of Chem. Eng., 65, 696-699 (1987). 38. Chakraborty, R. K, and Howard, J. R., /. Inst of Fuel, 51, 220-224 (1978). 39. Roscoe, J. C , Witkowski, A. R. and Harrison, D., Trans. Inst. Chem. Eng., 58, 69-72 (1980). 40. Mitchell, R. E., Combust. Sci. and Tech., 53, 165 (1987). 41. Haider, P. K., Ph.D. thesis, The Technical University of Nova Scotia, 1988.


CHAPTER 9 FLOW OF SOLIDS AND SLURRIES IN ROTARY DRUMS

H. A. Nasr-El-Din Saudi Aramco P.O. Box 62 Dhahran 31311, Saudi Arabia and A. Afacan and J. H. Masliyah Department of Chemical Engineering University of Alberta Edmonton, Alberta, Canada T6G 2G6 CONTENTS INTRODUCTION, 194 EXPERIMENTAL STUDIES, 196 Flow of Dry Solids in Rotary Drums, 196 Flow of Highly Settling Slurries in Rotary Drums, 196 Flow of Slightly Settling Slurries in Rotary Drums, 198 FLOW OF DRY SOLIDS IN ROTARY DRUMS, 199 Prediction of Solids Hold-up, 199 Drum Without Lifters and Without Discharge End Constriction, 202 Drum Without Lifters and With Discharge End Constriction, 204 Drum With Lifters, 204 FLOW OF HIGHLY SETTLING SLURRIES IN ROTARY DRUMS, 207 Drum With an Open-End Dishcarge and No Lifters, 207 Drums With an End-Constriction, 220 Drum Without Lifters, 220 Drum With Lifters, 231 FLOW OF SLIGHTLY SETTLING SLURRIES IN HORIZONTAL ROTARY DRUMS, 235 Slurry Modes of Transportation, 237 Slurry Hold-up and Solid Concentration, 239 Solids and Fluid Mean Resistance Times, 243 Prediction of Hold-up Solids Concentration, 243 CONCLUSIONS, 248

193

194


ACKNOWLEDGMENTS, 249 NOTATION, 250 INTRODUCTION Rotary drums are widely used in the chemical and metallurgical industries for processing large volumes of granular solids. In the chemical process industries, drying of solids can be achieved easily in rotary kilns, where hot gases flowing axially along the kiln are contacted with the cascading solids [1,2,3]. In the environmental areas, rotary drums can be used to treat contaminated soil. In such a process, a microbial culture together with nutrients are added to the soil. A low rotational drum speed can provide excellent mixing action for oxygen transfer to allow the microorganisms to decontaminate the soil. Rotary drums also are employed for bitumen extraction in the hot-water process [4] and in simultaneous extraction and upgrading of bitumen in direct thermal processes [5]. Optimizing the operating conditions for these drums as feed flow rate, drum speed, and inclination is critical. Parameters such as feed flow rate, feed solids concentration, drum internal, and endplate design can be adjusted to obtain the desired effects. Most of the previous work dealing with rotary drums has been done using dry solids. Abouzeid et al. conducted a thorough study on dry solids transport in rotary drums with an end-constriction (overflow discharge) [6]. Effects of solids feed flow rate, drum speed, discharge opening, and particle size on the drum hold-up and solids mean residence time were investigated. Similar studies were conducted by Hogg et al. [7], Karra and Fuerstenau [8], and Hehl et al. [9]. Abouzeid et al. examined solids residence time distribution in rotary drums using an axial dispersion model [10]. Abouzeid and Fuerstenau developed semi-empirical equations to predict solids hold-up in rotary drums [11]. Fuerstenau et al. studied flow of dry solids in ball mills with and without end-constriction [12]. Recently, axial mixing in rotary drums was studied by Rao et al. [13] and Das Gupta et al. [14]. Axial particle velocity and solids hold up in drums with aspect ratios, L/D, > 40 were examined by Sai et al. [15]. There are two types of solids motion in a horizontal rotary drum: longitudinal (axial) and transverse (radial). For a horizontal drum without lifters, solids transport in the axial direction occurs as a result of the difference in the bed height at the inlet and outlet ends of the drum. Afacan and Masliyah discussed various models to predict dry solids transport in rotary drums as a result of the bed height axial gradient [16]. Radial movement of solids in rotary drums was studied by various investigators [1724]. Below the drum critical speed, several types of radial movement may be discemed. Slipping motion occurs at low rotational speeds. Slumping and rolling occur at intermediate drum speeds, whereas cataracting and centrifuging occur at a higher drum rotational speed. Solids residence time distribution and hold-up in the drum depend on the type of the radial motion of the solids. Slipping motion is characterized by poor radial mixing, whereas more transverse mixing occurs for rolling beds. Rotary drums are frequently equipped with longitudinal lifters to increase the degree of mixing in the radial and axial directions and to eliminate solids bed

Flow of Solids and Slurries in Rotary Drums

195

slipping motion [17]. Particle motion in rotary drums with lifters and in the presence of an air stream was examined by various investigators [3,25-28]. Abouzeid and Fuerstenau examined the variation of the solids hold-up with solids feed flow rate, drum rotational speed and particle size [29]. The effect of lifters shape on dry solids hold-up and solids residence time distribution in a drum with an end-constriction was examined by Venkataraman and Fuerstenau [30]. They found that the solids hold-up increased linearly with the solids feed rate. The degree of mixing in the drum was found to be a function of the lifters configuration. Unlike flow of dry solids in rotating drums, previous work on slurry flow in rotary drums is sparse. Davis examined slurry flow in an overflow ball mill (i.e., with an end-constriction) operating at a constant drum speed [31]. Davis found the mean slurry solids concentration in the ball mill (72 wt%) to be significantly higher than that in the feed (40 wt%). Hogg and Rogovin conducted a similar study using sand particles having a mean diameter of 0.355 mm [32]. The slurry feed solids concentration was varied from 24 to 46% by volume; however, the drum speed was kept constant at 70% of the critical drum speed. They observed two regions in the drum: a pool region and a ball charge region. The slurry was transported in the drum through the pool region whereas grinding occurred in the ball charge region. They also found the mean residence time of solids in the drum to be higher than that of the fluid. Consequently, the solids concentration in the drum was higher than that in the feed. Gupta et al [33] and Moys [34] examined slurry flow in ball mills with a grate discharge. They found that the slurry hold-up increased with the slurry feed rate. Myers and Lewis examined the contents of a continuous, wet, overflow, industrial rod mill with an inner diameter 182.4 cm and a length 273.6 cm [35]. They studied the particle size distribution along the mill, but did not report any measurements on hold-up of solids or fluid. Horst investigated a wet grate-discharge ball mill 40.6 cm in diameter and 40.6 cm long [36]. Although he was able to measure axial hold-up and size distribution of solids along the mill, he could not determine the fluid hold-up along the drum because of the sampling method used. The study in which a continuous wet mill was sampled axially in detail was carried out by Rogovin [37] on an open-circuit grate-discharge ball mill. He conducted the experiments in a 30.4-cm diameter and 60.8-cm long mill with a 7.62-cm discharge opening. Rogovin varied the slurry feed rate and feed solids concen-tration, but kept the drum rotational speed constant at about 53 rpm. He was able to determine particle size distribution, solids and fluid local hold-up along the drum, as well as the overall hold-up of solids and fluid. He also used a pulse tracing technique to examine the motion of the solid particles along the mill. It was found that the solids hold-up was essentially uniform along the mill, the overall solids hold-up increased with increasing feed solids concentration, and the mean residence time for solids was greater than that of the fluid. It was also found that the solids concentration in the mill hold-up was higher than that of the feed or discharge stream. This chapter is divided into four parts. In the first part, a description of experimental studies on flow of solids and slurries in rotary drums is given. In the second part, the flow of dry solids in rotary drums will be examined. In the third part, flow of highly settling slurries in rotary drums will be reviewed. In the last part, flow of slightly settling slurries will be discussed.

196

Advances In Engineering Fluid Mechanics

EXPERIMENTAL STUDIES Flow of Dry Solids in Rotary Drums Figure 1 shows a schematic diagram of the experimental set-up used by Afacan and Masliyah to study the flow of dry solids in a horizontal rotary drum [16]. A screw-type solids feeder was used. A single drum was employed. The drum had an inside diameter of 0.192 m and a length of 1.05 m (the drum aspect ratio, L/D, was 5.46 and the drum critical speed, N^, was 96.5 rpm). The drum had a central opening of 0.0265 m in diameter for the feed end and a discharge end-constriction having a diameter, D^, of 0.108 m (D^/D = 0.563). The drum was rotated using rubber rollers linked to a variable-speed drive. Coarse sand was used as the granular feed material. The properties of the sand are given in Table 1. Four solids feed flow rates were employed, namely 6.3, 12.5, 21.3, 30.3 x 10"^ kg/s. The measurement of the solids hold-up was straightforward. The sand was continuously fed to the rotating drum until the system achieved steady state, which was assumed to have been reached when the discharge and inlet sand flow rates were within 2% for three sampling intervals. After steady state was achieved, the sand feeder and the drum's drive were stopped simultaneously. The contents of the rotating drum were emptied and weighed. The fractional hold-up and the solids residence time were then calculated. This procedure was repeated for a given drum configuration, sand flow rate and drum speed. Flow of Highly Settling Slurries in Rotary Drums A schematic diagram of the experimental set-up used by Afacan et al. [38] and Nasr-El-Din et al, [39] for the investigation of the transport of slurries through a

C

b|^\y\>vXy\yvx"v>

=L

n75'

i= S ^

Figure 1. Experimental set-up for dry solids. 1, screw feeder; 2, rotary drum (luclte); 3, rubber-lined rollers; 4, 1/4-hp motor (variable rpm); 5, exit chute; 6, sand collecting tank.


197

Table 1 Properties of Coarse Sand Particles 2 mm 1,640 kg/m^ 2,630 kg/m^ 36°

Particle mean diameter Solids bulk density Solids density Angle of repose

horizontal rotating drum is shown in Figure 2. The experiments were performed in a rotating Incite drum similar to the one used for dry solids. The bulk solids (sand) was fed at a desired rate by using a screw feeder. The fluid (water) was introduced at the sand feed end using a positive displacement pump via an annulus surrounding the screw feeder discharge tube. The feed rates of the solids and the water were maintained within ± 3 % . The properties of the solids and water are listed in Table 2. The water mass flow rate was measured using a rotameter. The drum was rotated at a desired speed by four rubber-covered friction rollers driven by a

. 2 -fl/

ro

g

j^y

Figure 2. Experimental set-up for slurries. 1, screw feeder; 2, rotary drum (lucite); 3, rubber-lined rollers; 4, 1/4-hp motor (variable speed); 5, slurry collecting tank; 6, rotameter; 7, positive displacement pump; 8, fluid tank; V, valve.

198

Advances in Engineering Fluid Mechanics Table 2 Properties of Sand and Water

Mean particle size Average sand density Average sand bulk density Fluid density Fluid surface tension

0.08, 0.5, 2 mm 2,630 kg/m^ 1,644 kg/m^ 1,000 kg/m^ 35, 70 mN/m

variable-speed motor. The drum rotational speed was measured using a Minarik Electric digital tachometer. The following independent variables were investigated: feed solids concentration (7 to 46% by volume), slurry feed rate (5 to 10~^ kg/s), drum speed (7 to 65 rpm), particle size (0.08 to 2.0 mm), and fluid surface tension (35 and 70 mN/m). An experimental run consisted of rotating the drum at a given speed and maintaining predetermined flow rates of solids and water. Steady state conditions were assumed to have been achieved when the discharge slurry had the same flow rate and composition as those of the feed inlet slurry. Steady state was usually reached about 30 minutes after start-up. When steady state conditions were reached, the drum rotation was stopped simultaneously with the slurry feed. The slurry hold-up in the drum and slurry solids concentration were determined from the mass and volume of the drum contents. The drum slurry hold-up and its solids concentration were measured within ± 5% of the value. The percent slurry hold-up is defined as the percentage of total volume of slurry in the drum to the drum volume. The solids volumetric concentration, C, was determined from the slurry hold-up density, P,. using C = (p^ - p,)/(p^ - pp

(1)

where p^, and p^ are the fluid and solids densities, respectively. The slurry hold-up density was calculated from the weight and volume of the drum contents. The slurry hold-up volume was measured using a calibrated cylinder. Flow of Slightly Settling Slurries in Rotary Drums Masliyah et al. examined the flow of slightly settling slurries using the experimental set-up shown in Figure 2 [40]. The experimental tests were conducted with 80 jim silica sand particles in a water slurry. The feed slurry mass flow rate was varied from 0.01 to 0.04 kg/s, and the feed slurry volumetric concentration was varied from 10 to 40%. Some experiments were also conducted with 267 and 630 |Lun silica sand in a triethylene glycol solution. The terminal settling velocities of the three sand fractions in their respective liquids were 0.0052, 0.0097, and 0.041 m/s. Table 3 summarizes the properties and range of parameters examined. Measurements of the slurry hold-up volume and mean solids concentration in the drum were conducted at steady state employing the same procedure used with the coarse sand particles.


199

Table 3 Physical Properties and Parameters Used by Masliyah etaL [40]

dso 80 267 630

Ps (kg/m3)

Pf (kg/m3)

(Pa • s)

(m/s)

2,630 2,630 2,630

1,000 1,080 1,080

0.001 0.0057 0.0057

0.0052 0.0097 0.041

Ren

Fr Equation (35)

4^ Equation (36)

0.42 61.8 0.04-0.2 0.004-0.017 0.49 53.5 0.042-0.21 0.0014-0.009 4.9 7.1 0.042-0.21 0.0014-0.007

FLOW OF DRY SOLIDS IN ROTARY DRUMS Prediction of Solids Hold-up Bed motion in rotary drums with no internals is a strong function of the drum dimensionless speed, n* = n/n^ , where n is the drum speed, rps, and n^ is the drum critical speed, rps, given as = (1/2 n) ^g / R and R is the drum radius [22]. As the drum rotational speed is increased, the bed motion changes from slumping, to rolling, to cascading, to cataracting, and finally to centrifuging [21,41]. It is of interest to note that, although the drum geometry and its internals are critical for the solids motion and, hence, the solids residence time, no general correlation has yet been developed for the solids hold-up. This section considers previous theoretical developments for solids hold-up in rotating drums and makes use of one correlation to show its possible use as a general correlation for various drum geometries for n* < 0.4. As was mentioned earlier, the bed motion is a function of the drum dimensionless rotational speed, n*. Any theoretical development has to take into account the type of bed motion taking place in the drum. For the region 0.01 < n* < 0.1, where bed rolling occurs, Saeman [42], Vahl and Kingma [43], and Kramers and Croockewit [44] arrived at the general equation for the bulk volumetric flow of solids through any cross section of the rotating drum. The equation is 47inR3 f

^ dh 2h (2) tan a ~ cos 0 — R 3 sin e V ' dx where F^ is the volumetric flow rate of the material, 0^ is the angle of repose of the material, tan a is the slope of the drum to the horizontal and h is the depth of the solids bed at an axial location x . Equation 2 can be re-arranged to give: dh dx

tana cos 9

3F tanO^ 21i 47cnR3 R

-T RJ

(3)

200


Here, in deriving Equation 3, it is assumed that a bulk of material at a given radial position from the drum axis describes a circular path of a constant radius in a plane perpendicular to the drum axis. The bulk material is assumed to have the same angular velocity as the drum wall. Once the granular material reaches the bed surface, it rolls (cascades) downwards. Equation 3 is a first-order ordinary differential equation which requires a boundary condition (value of h either at the entrance, X = 0, or at the exit, x = L) for its solution. Equation 3 does not have a simple analytical solution. Various assumptions were subsequently made to obtain approximate solutions for Equation 3. Normally, the integrated form of Equation 3 is given as the fractional hold-up in the drum or the mean residence time of the solids in the drum. The fractional hold-up in the drum is given by

H = ljx,dx

(4)

Lo

where X _ Y - sm Y r ~ 2n = 2cos-'flY =

(5)

-1

(6)

and the mean residence time of the solids is X

_ 7CR2LH

(7)

where F F . - -

(8)

Solution of Equation 3 with the appropriate boundary condition provides the variation of h with respect to x; hence, it becomes possible to evaluate the integral in Equation 4. Most previous studies concentrated on the solution of Equation 3 by providing an approximate functional form for the term in the square brackets. Normally, the assumption is made that at the discharge end h^ ~ 0 for a drum with no end constriction, otherwise is the depth of the constriction lip, i.e., h^ = (R - R^), where R^ is the radius of the constriction opening. The assumption for the values of h^^ is particularly correct for lightly loaded drums, and it becomes less accurate as the drum loading is increased. Various approximate solutions to Equation 3 are given below for different authors using a unified nomenclature.


201

Saeman [42]:

.3„.„3rp 4jtnR sin I — (^cosG, + tana) (9) 3 sin 0, where C = tan-'(hyL)

(9a)

h, / R = 1 - cosj^l

(9b)

and

(10)

2K

For given F^, drum geometry and material properties. Equation 9 is used to evaluate P, which is then used to evaluate the fractional hold-up, H. Equation 9 is applicable to inclined drums with no end constriction and no lifters. Vahl and Kingma [43]:

F =2.86n

R4 cot e (11)

R

H = 0.32 - ^ R

(12)

For 0.3

which is nearly the same as that given by Equation 12. Equation 13 is valid for a horizontal drum with or without an end constriction. Fuerstenau and co-workers: Hogg et al. [7], Karra and Fuerstenau [8] and Abouzeid and Fuerstenau [11,29] carried out extensive experimental studies on solids flow in rotating drums with and without end constriction. Abouzeid and Fuerstenau gave empirical expressions for the drum fractional hold-up in terms of a dimensionless feed rate function. Their expression for the fractional hold-up was designed to cover a wide range of drum dimensionless rotational speed, 0 < n* < 0.9. Unfortunately, their expressions of H in of their papers differ, and it was not possible to reconcile the two expressions [11,29]. However, their experimental findings will be used to assess model predictions based on Equation 3. With the advent of digital computers, there is no need to solve Equation 3 with any approximation to the function term in the parentheses. Afacan and Masliyah used a fourth-order Runge-Kutta algorithm to solve Equation 3 starting with x = L [16]. At the discharge end, they assumed h^ = 2d , where d is the particle diameter. Numerical difficulties could be encountered if h^ is set to zero where a small grid size in the discretization of Equation 3 will be needed. Drum Without Lifters and Without Discharge End Constriction Figure 3 illustrates the effect of the drum speed on the solids fractional hold-up in the drum. At a low drum speed the solids hold-up is relatively high and decreases with the drum speed. According to Abouzeid and Fuerstenau, this trend is due to the variation of the thickness of the shear zone of the cascading bed with the drum speed [29]. A comparison between the experimentally measured solids residence time with the approximate solution of Vahl and Kingma [43] and Hehl et al. [9] is shown in Figure 4. The agreement is excellent for dimensionless drum speeds up to n* = 0.8. At higher drum speeds, centrifugal force becomes important, and the basis on which Equation 3 was derived becomes invalid.


203

n/nc 0.20

0.20

20

0.41

0.62

0.82

40

60

80

100

Drum Speed,(rpm) Figure 3. Variation of solids fractional hold-up with drum speed (no lifters and no end constriction). Solids feed rate, (kg/s): • , 6.3 x 10-=^; • , 12.5 x 10-3; A , 21.3 X 10-3; ^

^

/

05 »-:]

w

5

—^ r /

r/ iK^

L 0 / /

n

1 •J

-^

^ 40

,20

_J

QF.

1

60

g/s

Figure 15. Variation of slurry hold-up with slurry feed flow rate. constant at 21 vol.% and d^^ = 2.0 mm, respectively. Slurry hold-up was found to increase with increasing slurry feed rate, in a manner very similar to the flow of dry solids [29]. The relationship is linear except at low slurry feed rates (below 10 g/s). The slurry hold-up increases due to the increase in the depth of the slurry bed and the increase in the slope of the bed surface (necessary to overcome the increased energy dissipation due to particle friction) [29,43]. At low drum speeds, there is little transverse mixing of solid particles inside the slurry bed. As a result, the slurry hold-up vs. slurry feed rate curve may continue to increase rather sharply at low drum speeds.

214


Figure 16 shows the variation of the solids and water mean residence times with the slurry feed flow rate for the same conditions as in Figure 15. The residence times for both the solids and water decreased with increasing the slurry feed rate. The rate of decrease of the mean residence times with the slurry feed rate, Qp, was higher at small values of Qp. This was especially true for the case of the solids at 7 rpm. 600

~1

T \

2.0

\

%

N

200

N

a

Solid.

•7

CO

rpixi

35

» S

m m

21.6

4rOO

CO

»—

120

Water

•

rpm

A O

•

Xi^

\

ao N N

4 0

\

\

--^^

\^\ ^

O

X

-L

20

4-0 QF.

6 0

fi/s

Figure 16. Effect of slurry feed rate on solids and water mean residence times.


215

Figure 17 displays the variation of C/Cp with the slurry feed rate Qp for the same conditions as in Figures 15 and 16. The ratio of C/Cp is constant except at small values of Qp and for the 7 rpm case. This behavior is a direct consequence of the variation of the mean residence times of the solids and water. Effect of Particle Size on Slurry Hold-up Figure 18 depicts the effect of the drum speed on the slurry hold-up for various particle sizes at a constant slurry feed rate and feed solids concentration. It is clear that the slurry hold-up is a strong function of solids particle size as well as drum *t

1

dso = 2.0 mm Cr = 21 % 3.6

'

N

]

• 7 rpm A 35 rpm

J

3.2

-^

U4

O O

" " ^ ^ * *

.

m.-± \ —1

2.8 -

•*•

X

^

A

J

"^ 1 -J

2.4

o

1

1

1

1

20

40 QF.

g/s

Figure 17. Effect of slurry feed rate on C/Cp.

1

60

216

Advances in Engineering Fluid Meciianics 1

16 [-

&^ 12

1

CF

= 20.5 %

QF

= 20.0 g/s

I t

I

dso • 0.08 mm A 0.50 mm • 2.00 mm

\

Q

o

1

\

8 \

>1

\ \ \

4h

\

\\

\

\ ^ -

0

0

f.

20

•.

40 N, rpm

60

Figure 18. Effect of particle size on slurry hold-up.

speed. Increasing the drum speed, the slurry hold-up decreases for all particle sizes. For a given drum speed, feed solids concentration and feed slurry rate, the slurry drum hold-up decreases with particle size. For very fine particles, the slurry flow would be similar to a flow of a homogeneous fluid. Indeed, for the case of fine sand, djQ = 0.08 mm, at N > 25 rpm, the ratio of xjx^ was found to be nearly unity. For the coarse sand, the fluid drag on the solids becomes minimal, and the wet solids behave similar to the flow of dry solids. However, Figure 18 shows that the holdup of the medium size sand for d^^ = 0.5 mm is higher than that for d^^ = 2.0 mm. This result is a direct consequence of the medium size sand agglomeration mentioned earlier which tended to impede its rolling motion down the surface of the bed.


217

Ejfect of Fluid Surface Tension on Slurry Hold-up Experimental studies show that for medium particles {d^^ = 0.5 mm), the cohesive forces (agglomeration) are dominant and lead to higher slurry hold-up as compared with the coarse particles. To reduce these forces, the fluid surface tension, a, was reduced from 70 to 35 mN/m by the addition of Triton X-100 (a nonionic surfactant) to tap water at a concentration of 120 ppm. Figure 19 shows that the effect surface tension on the slurry hold-up was not significant for both particle sizes (d^^ = 0.5 and 2.0 mm).

15

T

T

dso = 2.0 mm

12h

n o- = 35 m N / m • (7 = 70 m N / m J

?

dso — 0-5 m m

I

o a = 35 m N / m • (7 = 70 m N / m

9h

Q ^

>^

\

o •\

\

^

-8~. •~-a.

r-* -~ih

Cf = 21.6 % QF = 21.3 g / s

0

J_

0

20

40 N, r p m

60

Figure 19. Effect of fluid surface tension on slurry hold-up

218


Prediction of Solids

Hold-up

The solids hold-up in the drum was found to depend to a large extent on the mutual interaction between the water and the solids. For the case of fine solids, d^Q = 0.08 mm, the fluid drag is significant. At the other extreme, d^^ = 2.0 mm, the fluid drag is not significant. For the case of the coarse solids, Afacan et al. [38] used Kramers and Croockewit's [44] equation to predict solids hold-up in the drum. A comparison between the experimentally measured solids hold-up and the prediction from Equations 2 to 5 for the case of the coarse sand is shown in Figures 20 and 21. The agreement is excellent at low solids hold-up and deviates as much as 20% at the highest hold-up, where the drum is no longer lightly loaded.

20

dso = 2.0 mm QF = 20.8 g / s 15

I

S

10

o CO Q

o CO

± 20

40 N, rpm

60

Figure 20. Predicted and measured solids hold-up at various drum speeds.


219

a, I

a o in Q

o

Figure 21. Predicted and measured solids hold-up with various solid feed rates at different feed slurry solid concentrations. The variation of the solids hold-up with the solids feed rate is illustrated in Figure 21 for two different values of the drum speed. The variation is very similar to that of dry solids. The data points were obtained from a slurry feed having a solids concentration of 21.1 vol.%, except for two data points, one for a Cp of 7.8 vol.% and the other for a Cp of 45.8 vol.%. For a given drum speed, both data points fall within the data for a Cp of 21.1 vol.%, indicating that the solids within the slurry behaved as if they were dry solids.

220


Drums with an End-Constriction In the previous section, the flow of a sand-water slurry in a horizontal drum with an open-end discharge was reviewed. Slurry hold-up was found to be a function of particle size, drum speed, feed flow rate, and composition. The experimental results also showed relatively low slurry hold-up and poor radial (transverse) mixing inside the drum. To overcome these problems for the case of dry solids, an end-constriction and lifters are normally used. The effects of such additions on the slurry hold-up and mean residence times of solids and fluid phases will be discussed in the next section. Drum Without Lifters Flow visualization experiments conducted by Nasr-El-Din et al. indicated that movement of the liquid and the solid phases in the drum depended on the drum speed [39]. At low rotational speeds, the two phases moved independently in two separate regions. The first region consisted mainly of the liquid while the second region consisted mainly of solid particles. The particles moved peripherally with the drum to a certain angle after which they slipped back to their original position and the process repeated itself. This kind of motion was characterized by poor radial mixing within the slurry bed and between the two regions. As the drum speed was increased, the oscillating motion of the slurry bed increased and a slumping motion was observed. Mixing within the slurry bed and between the liquid pool and slurry bed significantly improved. However, at higher drum speeds, the slurry bed stopped oscillating, the bed size significantly increased, and the two phases moved independently in the drum. The presence of a liquid pool indicated that the hydrodynamic forces (viscous-drag and lift forces) exerted by the liquid on the solids were not adequate to suspend the solid particles (sand). It is worth noting that the pool and bed regions were observed for slurry flow in a rod mill by Myers and Lewis [35]; in an overflow ball mill by Hogg and Rogovin [32]; and in a rotary drum with an open-end discharge by Afacan et al. [38]. Prediction of Bed Frequency of Oscillation. A simple model based on the slurry bed oscillating motion just described can be made. The concept of the model is: During the upswing part of the motion (Figure 22), the solids move with the same speed as the drum wall. During this motion, the solids-wall friction coefficient is taken as the static value. The solids bed reaches a maximum height characterized by an angle 0^ then it swings downwards. During the downswing motion, the friction coefficient is less than the static value. This is mainly due to water entrainment into the solids bed resulting from the drum rotational motion. At higher values of the drum speed, enough water penetrates the solids bed such that the friction coefficient becomes negligible. Based on these assumptions, the force balance equation on the solid particles at the end of the upswing motion (i.e., at (|) = (j)^) is: (g/R) cos (t)^ + ^J(g/R) sin ^^ + (7iN/30)2] = 0

(16)


221

downswing nnotion F-i = mg cos 6

F4 = mg Figure 22, Schematic of the drunn for the mathematical model. where N is the drum rotational speed, rpm; and |x^ is the solids-wall friction coefficient during the upswing motion, which was taken as |X^. Solution of Equation 16 gives the maximum rise angle, (j)^, for the solids before slipping occurs. During the downswing motion, the solids momentum equation is: e = (g/R) cos e - îJ(g/R) sin 6 + G']

(17)

where 9 + (|) = 7C, G is the angular acceleration, G is the angular velocity and [i^ is the friction coefficient during the downstring motion. Equation 17 is a second order

222

Advances in Engineering Fluid IVIechanics

ordinary differential equation which was solved numerically by Nasr-El-Din et al. [39] using the following initial conditions: 9(0) = n-

-•

•

o

•-—•

r Cy

u 45 h

•

-A

7.8 %

• 21.0 %

1

A

A 45.9 %

40

-J

0

A

20

...

,

J

40 N, rpm

1

L

60

Figure 31. Effect of drum speed on slurry mean solids concentration in a drum with lifters. Effect of Feed Rate on Slurry Hold-up. Figure 33 shows the effect of the slurry feed rate on the slurry hold-up at drum speeds of 7 and 35 rpm for Cp = 21.1%. The slurry hold-up increased with the slurry feed flow rate. This trend is similar to that observed with the drum without lifters (Figure 28). However, the presence of lifters tended to decrease the slurry hold-up at a drum speed of 7 rpm. At a drum speed of 35 rpm, the effect of the lifters on the slurry hold-up was not significant. To examine the effect of the fluid flow rate on the solids hold-up for the drum with lifters, the solids hold-up is plotted in Figure 34 as a function of the solids

234


1

1

1

1

"1

1 dso = 2.0 mm [ QF = 20.7 g/s

'

•

• 7.8 % • 21.0 % A 45.9 % m

•

m ^

^

w

-m

•

•——•

•—

-A

_A

A——4

A-

10

1

CF

15

""•^^

'

OQ

5 ~ ._•

0

20

40 N, rpm

60

Figure 32, Effect of drunn speed on the solids/fluid residence time for a drum with lifters.

feed flow rate, as explained for Figure 29. For the case of dry solids, the effect of the lifters on the solids hold-up was not significant. This result is due to the fact that the lifters height (0.017 m) was less than the lip height (0.042 m) for the drum examined by Nasr-El-Din et al. [39]. Figure 34 also shows that for the cases of slurry feed examined the solids hold-up was independent of the solids feed rate at Q^ greater than 10 g/s. The effects of the fluid flow rate on the solids hold-up were similar to those obtained with the drum without lifter.


235

40 dso — 2.0 nun Cy = 21.1 % 35

I o «

30

3

25

20

40 QF.

60

g/s

Figure 33. Effect of slurry feed flow rate on slurry hold-up In a drum with lifters. FLOW OF SLIGHTLY SETTLING SLURRIES IN HORIZONTAL ROTARY DRUMS In the case of highly settling slurries, there are two regions of flow within the drum: a pool region, which contains mainly water and acts as a slow-moving stream, and a slurry bed region, which is mostly solids and travels much slower than the pool region [38]. However, for the case of slightly settling slurries, the water pool region becomes more loaded with solids. At high drum speeds, the distinction between the two regions becomes less apparent, and the slurry travels in the drum as a pseudo-homogeneous fluid [40].

236


30

T

'O-

o

..cr" d. D

20

.-o"

I

o I

N = 35 rpm

CO

•D

O 10 CO

0

CF

A 7.5% • 21.1% • 45.4% O 100% (Dry Solids)

±

± 20

10

30

Q 2.62 s~' both the solids and the water moved in the drum with the same forward velocity. At low drum speeds, the average hold-up solids concentration approached 45%, irrespective of the feed solids concentration. Figure 36a is similar to Figure 35a, but for a slurry feed rate of 0.04 kg/s. The percent slurry hold-up behavior was similar to that for Qp = 0.02 kg/s. The variation of the hold-up solids concentration ratio, C/Cp, is shown in Figure 36b. It is clear that for drum speeds > 2.62 s~', all the curves for the various Cp values approached a limiting value of C/Cp = 1.0. This result is, once again, indicative that both the water and the solids move with the same axial velocity similar to a homogeneous slurry. The maximum deviation of C/Cp from unity occurs at low drum speeds, signifying a large relative velocity for the water and the solids. This case is that of a stratified slurry flow. A comparison of the variation for the hold-up solids concentration for the three sand fractions used by Masliyah et al. [40] is shown in Figure 37. The slurry flow rate was 0.02 kg/s, and the feed solids concentration was 20%. Figure 37 shows the variation of C with the drum speed. For a given drum speed, both Fr and 4^ (defined by Equations 35 and 36, respectively) are nearly the same for the three sand fractions. The only variable is the single particle drag coefficient, Cj^ A trial and error procedure was used to evaluate C^^ [56]. In all cases, the hold-up solids concentration decreased with the drum speed. For the cases of C^^ = 61.8 and 53.5, C decreased rapidly with N*, whereby a homogeneous slurry is formed for N* > 2.62 s~\ For smaller C^^ values, the rate of decrease of C with N* is small. For the three solids tested, their curves tend to converge to one point on the C vs N plot. For the purpose of comparison, data taken from Nasr-El-Din et al. [39] were plotted on Figure 37 for the flow of sand/water slurry containing 2 mm particles. The single particle C^ value of this coarse sand is 0.54. For the 2 mm particles, the curve of C vs. N* falls well above the others and does not converge together at low values of N*. For the coarse sand, a stratified slurry flow occurs for all drum speeds. This has an important implication as will be shown later. (text continued on page 242)

240


40

DRUM SPEED, N (s"*)

DRUM SPEED, N (s"') Figure 35. Effect of drum speed on (a) percent slurry hold-up; (b) average hold-up solids concentration d^ = 80 jim; \i, = 1 mPa • s; Qp = 0.02 kg/s; Cp (vol. %): • 10; 20; A 30; 0 40.


40

1

r-

,

I

'

1

'

241 1

!

(a)

S^ -

30 h

0

I Q

0

^ 0

O

A

9

L

!

>. 20

10

1

t

1

1

.

_. L

.

1 2 3 DRUM SPEED, N (s"*)

j_

1

4

1 2 3 DRUM SPEED, N (s'^) Figure 36. Effect of drum speed on (a) percent slurry hold-up (b) hold-up solids concentration ratio, C/Cp. 6^ = 80 j^m; |x, = 1 mPa • s; Qp = 0.04 kg/s; Cp (vol. %): • 10; 20; A 30; 0 40.

242


(text continued from page 239) Figure 38 shows the variation of the percent slurry hold-up with the slurry mass flow rate, Qp, for the case of Cp = 20% for the 80 [xm sand. For a given slurry feed rate the slurry hold-up decreases with increasing the drum speed. The slurry feed rate is a product of the slurry flow area (which is directly related to the slurry hold-up) and the axial slurry velocity. A decrease in the slurry hold-up indicates that increasing the drum speed enhances the slurry axial velocity. This result is due to the fact that increasing the drum speed renders a homogeneous slurry and, hence, a higher forward slurry velocity along the drum. Solids and Fluid Mean Resistance Times Figure 39 shows the mean residence time for the solids, T^, water, T^, and the ratio xji^. As would be deduced from the variation of the hold-up solids concentration shown in Figures 35b and 36b, the mean residence time of the solids in the drum is higher than that of the water at low values of the drum speed. However, for N* > 2.62 s"', the ratio of xjx^ approached unity, indicating complete suspension of the solids. Figure 39c depicts that xjx^ was not sensitive to variation in the slurry feed rate. However, as would be expected, the individual values of the x^ and x^ were affected by the slurry flow rate (Figures 39a and 39b).

80 = 0.02 k g / s Cy = 20 vol.% QF

60 -O

^ 40 o >

u

20 h • 61.8 (d5o=80/tm) D 53.5 (d5o=267/Lim) A 7.1 (d5o=630/tm) 0 L O 0.54 (d5o=2000^m; N a s r - E l - D i n et al.. 1992)

0

1

2 3 4 DRUM SPEED, N (s"^)

5

Figure 37. Effect of drum speed on the average hold-up solids concentration.


243

^25 I Q

O

ffi 15 >-

= 20 voL% d5o = 80 / i m CF

•

D 1.047 •

§. GO

0.01

0.02 0.03 QF ( k g / s )

0.733 3.665

0.04

0.05

Figure 38. Effect of slurry mass flow rate on percent slurry hold-up with ji^ = 1 mPa • s. Prediction of Hold-up Solids Concentration Figure 40 illustrates the variation of the hold-up solids concentration with that in the feed for the 80 jim silica sand at Qp = 0.04 kg/s. At a given drum speed, the average solids volumetric concentration varied linearly with the feed solids concentration. All the lines for the various N* values converged to a single point having C = C^ Plots for Qp = 0.01 and 0.03 kg/s also showed similar behavior. A value of 45% is assigned to C^, the critical volumetric solids concentration. Figure 40 suggests that: C = a + bC,

(32)

where a and b are functions of Qp and N* only. Equating C to Cp and C^ leads to a = (1 - b) q and Equation 32 becomes (C - C)/(C - Cp) = b(Qp, N*)

(33)

A plot of the normalized solids concentration, (C^ - C)/(C^ - Cp), against the drum speed is shown in Figure 41 for different values of the slurry feed flow rate. The effect of Qp on the normalized solids concentration is small. Therefore, it should

244


40

(a) 730

20 0]

10

0 • 12

8

h

4

^—*0 (^

H

1

,

1

H

(c)

B A

0

h

cn

1

0

1

2

3

4

DRUM SPEED, N (s~^) Figure 39. Effect of drum speed on (a) solids mean residence time (b) water mean residence time d^^ = 80 \xm; n, = 1 mPa • s; Cp = 20 vol. %; Qp (kg/s): • 0.01; 0.02; • 0.03; 0 0.04.


70 1

•

60 L 1

QF = 0.040 kg/s dgo = 80 iJ.m

\

'

I

•

I

'

1

1

-]

245

1

-J

50 ^

40 h

I;^^^^

r—l

O >30 U

20 \

•

.-^^'^^î^^

A • 0

^ .yy^^^^^ Q T ^

10

K \

0 0

10

.

1

1

20 Cp

N (s-i)i 0.733 i 1.047 1 1.571 1 2.094 \ 3.665 1

1

1

1

30 40 (voL%)

_i

50

1

1

60

Figure 40. Effect of feed solids concentration on average hold-up solids concentration, (p.^ = 1 mPa • s).

be possible to bring close together the data presented in Figures 35a and 36a. The normalized solids concentration is a strong function of the drum speed where it approached unity as N* was increased and zero at lower drum speeds. As pointed out earlier, the axial slurry velocity in the drum was fairly low and did not exceed 0.005 m/s. Obviously, the axial velocity was not sufficient to suspend the solids. However, the peripheral drum velocity was in the range of 0.07 to 0.35 m/s. The agitation offered by the lifters at these velocities tend to suspend the solids, which were carried along the drum. Making use of the previous section on the various dimensionless groups used to correlate slurry transport, the function b of Equation 33 was assumed to take the form: b(Qp, N*) = 1 - exp[-d Fr« C'^^\

(34)

where Fr is a modified Froude number for slurry transport in the drum, and is defined as: Fr = V/[2 g R(S^ - \)f'

(35)

^ is a dimensionless slurry flow rate, and is given by 4^ = Qp/7cR2 p^ V

(36)

246


1.0

?

^ 0.8 U I " 0.6

= 10 vol.% dso = 80 /xm

CF

U

U

0.4

QF (kg/s) • 0.01 A 0.02 • 0.03 O 0.04

0.2 0.0

0

DRUM SPEED, N (s"') Figure 41. Effect of drum speed on normalized solids concentration for four different slurry flow rates, (î^ = 1 mPa • s).

The single particle settling velocity is given by Vp = g d;(p^ -pp/[(l + 0.15 Re«^«^)18^,]

(37)

The particle Reynolds number. Re , is defined by Equation 30. Table 3 gives the properties of the particles, fluids, and the range of variables used by Masliyah et al. [40]. The best fit of 74 experimental data for the three sand fractions gave ( q - C)/(q - Cp = 1 - exp[-19.2 (FrC^02.26 y).333]

(38)

The range of applicability of Equation 38 is 7.1 < C^ < 61.8, 0.04 < Fr < 0.21 and 0.0014 < \\f < 0.017. Model prediction for the solids concentration in the hold-up for various feed solid concentrations is shown in Figures 35b and 36b. The model predicts the experimental data fairly well. Experimental data of the normalized concentration, (C^ - C)/(C^ - Cp), for the 80 mm (Vp = 0.0052 m/s) and 630 |im (v = 0.041 m/s) sands are shown in Figure 42 for a Cp = 20%. For the case of the fine sand, the variation of the normalized concentration is an S-shape type approaching unity for N* = 3.14 s~'. For the case of the coarse sand, the variation of the normalized concentration with N* is very


247

different from that of the fine sand. Such a difference is attributed mainly to different values of the single particle drag coefficient. For the case of C^^ = 61.8 (fine sand), the model prediction is in fairly good agreement with the experimental data. For the coarse sand (Cj^ = 7.1), the model prediction indicates that N* has to be greater than 80 rpm for the normalized concentration to reach unity, where the particles become fully suspended. Figure 42 shows that particles with C^^ values of about 7 represent the limit of the applicability of the model for the flow of slightly settling slurries in a rotating drum. It should be noted that the drum radius was used in the dimensionless groups Fr and \\f. The drum radius was not varied in the experimental runs of Masliyah et al. [40]. Thus, caution should be exercised in using Equation 38 for drum diameters much different than that employed by those authors. It also should be noted that

"T

1.0

r

'-&—i-

cy

0.8 ^

/

/

/

/

/

&4

o I

0.6

u O

o

0.4

a I

0.2

0.0

/

I I I I

•

/ I I /

/ ^/

/

/

/

/

/

/

/

/

/

/

/

r

= 0.02 k g / s Cp = 20 vol.% QF

-|

Model Prediction O CD = 61.8 (dso = 80 ixva) J • CD = 7.1 (dso = 630 /xm) I

I

I

I

0

:

8

DRUM SPEED. N (s~') Figure 42. Effect of drum speed on normalized solids concentration for two different slurries.

248


solids transport due to solids bed gradient is not important for the range of experimental parameters used in this study. This conclusion is based on the fact that the model does not take into account the term (dh/dx), which is important in the analysis of highly settling solids transport in rotary drums [38]. Figure 43 shows a plot of the experimental and predicted C values. The scatter tends to be equally distributed about the straight line having a slope of unity, indicating no bias in the proposed correlation for all the three sand fractions. CONCLUSIONS Flow of dry solids in horizontal rotary drums is governed by many factors. Solids hold-up in rotary drums is a function of drum speed, drum design, and solids properties. Solids hold-up for lightly loaded drums, at n* < 0.4, can be accurately predicted using the Kramers and Croockewit equation [45]. More experimental and theoretical work are needed to predict solids hold-up for highly loaded drums.

50

.

1

1

1

1

1

1—

1

^

L O CD = 61.8 (dso = 80 /^m)

40

• CD = 53.5 (dso = 267 /im) r A CD = 7.1 (dso = 630 /tm)

.^ C )

O/^^

o

'_

'-H 30 h CD

o

•J

J^*

r

20 h

u

-

^^

1 1

-^

i? •

10 h H

0 \/ 0

•

1

10

1

1

20

1

1

1

30

1

40

L-

50

C, E x p e r i m e n t a l Figure 43. Comparison of experimental values of C with those predicted from Equation 38.


249

Flow of slurries in rotary drums depends on, among other factors, slurry settling characteristics. For the case of highly settling slurries flowing in a rotary drum with open-end and no lifters, these conclusions were obtained: 1. For all slurry feed solids concentrations and particle sizes, the slurry hold-up decreased as the drum speed was increased. 2. For constant feed solids concentration and drum speed, the slurry hold-up increased as the slurry feed rate was increased. 3. For constant slurry feed rate, the slurry hold-up increased as the slurry feed solids concentration was increased. 4. The effect of the water flow rate on the solids motion in the drum was found to be insignificant for the coarse particles. 5. Solids concentration in the drum was found to be higher than that in the feed. For the flow of highly settling slurries through horizontal rotary drums with an end-constriction (overflow discharge), the following conclusions were obtained: a. For the drum without lifters The slurry bed within the drum was found to oscillate. The bed frequency of oscillation was found to be a strong function of the drum rotational speed. A mathematical model was developed to predict the bed frequency of oscillation. The model predictions were in reasonable agreement with the experimental measurements. b. For the drum with lifters The slurry hold-up-drum speed relationship exhibited a minimum. The presence of the lifters eliminated the abrupt changes in the slurry hold-up observed with the drum without lifters. For a given feed concentration, the mean solids concentration in the drum and the mean residence time ratio, tjt^, were lower than the corresponding values obtained for the drum without lifters. Transport of slightly settling slurries in a horizontal rotary drum can be modeled using the drum Froude number, the single particle drag coefficient, and a dimensionless slurry flow rate. The solids volumetric concentration in the drum was found to be a strong function of the single particle settling velocity and the drum speed. ACKNOWLEDGMENTS The authors wish to thank Elisabeth Renga for typing this manuscript and the Natural Sciences and Engineering Research Council of Canada for its financial support. NOMENCLATURE C Slurry mean solids concentration in the drum hold-up, vol. % C Critical solids concentration, vol. % c

'

Cp Single particle drag coefficient, -

Cp Slurry feed solids concentration, vol. % C Solids maximum packing concenmax

^

tration, vol. %

C"

250


C ^ Delivered solids concentration, vd

'

volume fraction D Drum inside diameter, m Dj Pipe diameter, m D^ Discharge end-constriction diameter, m D Diameter of feed solids, m D^Q Mean particle size, mm F Bed frequency of oscillation, s ' F^ Solids mass flow rate, kg/s Fr Froude number for slurry flow in the drum, V/[2 g R (Ss - 1) f Frj Modified Froude number for pipe

flow, v3/[g D, (s^ -1) r Fr^ ' F^ G H

Modified Froude number for open channel flow V*/[g L* (S^ - l)f' Solids volumetric flow rate, mVs Gravitational acceleration, m/s^ Solids bed depth at axial distance X along the drum, m H^ Solids bed depth at feed end ° (x = 0), m \ Solids bed depth at discharge end (x = L), m H Solids fractional hold-up, H^.^ Minimum hold-up as a percentage of the drum volume, i Slurry pressure gradient, Pa/m if Fluid pressure gradient, Pa/m L* Characteristic length, m Greek Letters a Drum angle of inclination, rad 6^ Repose angle of the solids, rad 0 Angle during the downswing motion, rad 6^ Maximum angle reached during the downswing motion, rad 6 Solids angular velocity, rad/s 9 Solids angular acceleration, rad/s^ (j) Angle during the upswing motion, rad (|)^ Maximum angle reached during the upswing motion, rad a Fluid surface tension, mN/m

L Length of rotating drum, m m Solids mass in the drum, kg n Drum speed, rps n^ Drum critical speed, rps n* Drum dimensionless speed, n/n^ N Drum speed, rpm N^ Drum critical speed, ' (60/27i)Vg7R»rpm N* Drum speed, s"' Qp Slurry feed flow rate, kg/s Q^ Solids mass flow rate, kg/s R Radius of rotating drum, m R^ Radius of discharge endconstriction, m Re Single particle Reynolds number, S^ Density ratio, pjp^ T^ Time for solids to reach 6^ (during the down-swing motion), s T Time for solids to reach 9 u

m

(during the upswing motion), s Vg Bulk or average velocity for slurry pipeline, m/s V* Characteristic flow velocity, m/s V Rotary drum peripheral speed, m/s V Single particle settling velocity, m/s X Axial distance along the drum, m X^ Solids bed fractional flow area, (flow area/TcR^), -

|Li Carrier fluid viscosity, Pa • s |Xj Solids-wall friction coefficient during the downswing motion, )l^ Solids-wall static friction coefficient, X | ^ Solids-wall friction coefficient during the upswing motion, p^ Bulk density of dry solids, kg/m^ Pf Fluid density, kg/m^ p^ Slurry density, kg/m^ p^ Solids density, kg/m^ T Solids mean resistance time, s s

Xf Fluid mean residence time, s


y Central angle extended by solids bed, rad \\f Dimensionless slurry flow rate defined by Equation 36

251

\|/' Dimensionless solids flow rate defined by Equation 24

Subscripts d Downswing motion f Fluid

s Solids u Upswing motion

REFERENCES 1. Tscheng, S. H. and Watkinson, A. P., Can. J. Chem. Eng., 57 (1979) 433-443 2. Tackie, E. N., Watkinson, A. P. and Brimacombe, J. K., Can. J. Chem. Eng., 67 (1989) 806-817. 3. Matchett, A. J. and Sheikh, M. S., Trans. IChem E., Part A, 68 (1990a) 139. 4. Carrigy, M. A., Research Council of Alberta, Edmonton, Alberta (October, 1963). 5. Taciuk, W., J. Can. Pet. TechnoL, 23 (1984) 56. 6. Abouzeid, A. Z. M., Mika, T. S., Sastry, K. V. and Fuerstenau, D. W., Powder TechnoL, 10 (1974) 273. 7. Hogg, R., Shoji, K. and Austin, L.G., Powder Technol., 9 (1974) 99. 8. Karra,V. K., and Fuerstenau, D. W., Powder Technol, 16 (1977) 23. 9. Hehl, M., Kroger, H., Helmrich, H. and Schiigerl, K., Powder Technol., 20 (1978) 29. 10. Abouzeid, A. Z. M., Fuerstenau, D. W. and Sastry, K. V. S., Powder Technol, 27 (1980) 29. 11. Abouzeid, A.-Z. M. and Fuerstenau, D. W., Powder Technol, 25 (1980b) 65. 12. Fuerstenau, D. W., Abouzeid, A. Z. M. and Swaroop, S. H. R., Powder Technol, 46 (1986) 273. 13. Rao, S. J., Bhatia, S. K and Khakhar, D. V., Powder Technol 67 (1991) 153-162. 14. Das Gupta, S., Khahhar, D. V. and Bhatia, S. K, Powder Technol, 67 (1991) 145. 15. Sai, P. S. T., Surender, G. D., Damodaran, G. D., Can. J. Chem. Eng., 70 (1992) 438-443. 16. Afacan, A. and Masliyah, J. H., Powder Technol, 61 (1990) 179. 17. Rutgers, R., Chem. Eng. ScL, 20 (1965) 1079. 18. Wes, G. W. J. Drinkenburg, A. A. H. and Stemerding, S., Powder Technol, 13 (1976) 177. 19. Lehmberg, J., Hehl, M. and Schugerl, K., Powder Technol, 18 (1977) 149. 20. Cross, M., Powder Technol, 22 (1979) 187. 21. Henein, H., Brimacombe, J. K. and Watkinson, A. P., Metall Trans. B., 14B (1983) 191. 22. Pollard, B. L., and Henein, H., Can. Metall Quat., 28 (1989) 29. 23. Perron, J. and Bui, R. T., Can. J. Chem. Eng., 68 (1990) 61.

252

24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.


Woodle, G. R. and Munro, J. M., Powder Technol. 76 (1993) 241-245. Prutton, C.F., Miller, CO. and Schuette, W.H., Trans AIChE, 38 (1942) 123. Kelly, J. J. and O'Donnell, P., Trans. IChemE,, 55 (1977) 243. Glikin, P. G., Trans. IChemE., 56 (1978) 120. Kamke, F. A. and Wilson, J. B., AIChE, 32 (1986) 269. Abouzeid, A. M. and Fuerstenau, D. W., Powder Technol, 25 (1980a) 21. Venkataraman, K.S. and Fuerstenau, D. W., Powder Technol, 46 (1986) 23. Davis, E. W., Trans. AIME., 169 (1946) 155. Hogg, R. and Rogovin, Z., XIV Inter. Process. Congr., Toronto, Ont., Canada, Oct. 17-23 (1982), 1-7.1. Gupta, V. K., Hodouin, D. and Everell, M. D., Int. J. Miner. Process., 8 (1981) 345. Moys, M. H., Int. J. Miner. Process., 18 (1985) 85. Myers, J. F. and Lewis, F. M., Trans. Soc. Min. Eng. AIME, 169 (1946) 106. Horst, W. E., Ph.D. Thesis, Univ. of Arizona, Tucson (1967). Rogovin, Z., Ph.D. Thesis, Pennsylvania State University (1983). Afacan, A., Masliyah, J. H. and Nasr-El-Din, H. A., Powder Technol, 63 (1990) 179. Nasr-El-Din, H. A., Afacan, A., Foster, J. and Masliyah, J. H., Powder Technol, 71 (1992) 51-261. Masliyah, J. H., Afacan, A., Wong, A. K. M. and Nasr-El-Din, H. A., Can. J. Chem. Eng., 70 (1992) 1083. Niyanand, N., Manley, B., and Henein, H., MetallTrans. B., 17B (June ,1986) 247. Saeman, W. C, Chem. Eng. Prog., 47 (1951) 508. Vahl, L. and Kingma, W. G., Chem. Eng. Scl, 1 (1952) 253. Kramers, H. and Croockewit, P., Chem. Eng. Scl, 1 (1952) 259. Langrish, T .A. G., Powder Technol 75 (1993) 61-65. Matchett, A. J. and Sheikh, M. S., Chem. Eng. Res. Design 8 (1990b) 1. Marchand, J. C, Hodouin, D. and Everell, M. D., 3rd IFAC Symp., Montreal, Que., Canada (1980) p. 295. Weller, K. R., 3rd IFAC Symposium, Montreal, Que., Canada (1980) p. 303. Clift, R., Wilson, K. C, Addic, G. R. and Carstens, M. R., Hydrotransport, 8 (1982) 91-101. Nasr-El-Din, H. A., Shook, C. A. and Colwell, J., Presented at the 10th Int. Conf. of the Hydraulic Transport of Solids in Pipes, Innsbruck, Austria, Oct. 29-31 (1986). Shook, C. A., Daniel, S. M., Scott, J. A., and Holgat, J. P., Can. J. Chem. Eng., 46, (1968) 238-244. Nasr-El-Din, H. A., Shook, C. A. and Colwell, J., Int. J. Multiphase Flow, 13 (1987) 661-670. Wilson, K. C, Hydrotransport, 4-Al (1976) 1-16 . Shook, C. A., Hass, D. B., Husband, W. H. W.., Small, M., and Gillies, R. G., J. Pipelines, 1 (1981) 83-92. Gillies, R. G., Shook, C. A. and Wilson, K. C, Can. J. Chem. Eng., 69 (1991) 173-178.


253

56. Wallis, B. D., One Dimensional Two Phase Flow, McGraw-Hill, New York (1969). 57. Yalin, M. S., Mechanics of Sedimentation Transport, 2nd Ed. Pergamon Press, Oxford (1977), pp. 117-122. 58. Novak, P. and Nalhuri, C , Hydrotransport 2, D4 (1972) 33-51.


CHAPTER 10 GAS PHASE HYDRODYNAMICS IN CIRCULATING FLUIDIZED BED RISERS Gregory S. Patience E. I. du Pont de Nemours Wilmington, DE 19880-0262 Jamal Chaouki Ecole Polytechnique de Montreal Montreal, Que., Canada, H3C 3A7 Franco Berruti University of Calgary Calgary, Alta., Canada, T2N 1N4 CONTENTS INTRODUCTION, 256 INDUSTRIAL APPLICATIONS, 257 PHYSICAL CHARACTERIZATION OF CIRCULATING FLUIDIZED BEDS, 260 EXPERIMENTAL METHODS, 264 Intrusive Probes, 265 Steady State Tracers, 267 Impulse Tracers, 269 Optical Tracers, 274 Chemical Reaction, 276 HYDRODYNAMIC MODELING, 278 Radial Gas Velocity Profiles, 278 Gas Mixing, 280 Radial and Axial Dispersion, 281 Core-Annular Flow, 284 Contact Efficiency, 285 DESIGN CONSIDERATIONS, 286 RESEARCH NEEDS, 287 NOTATION, 288 REFERENCES, 289 255

256


INTRODUCTION Gas/solid reactors are critical to numerous processes in the chemical, petrochemical ,and metallurgical industries, in the manufacture of fine powders and ceramics, in combustion, and environmental remediation. One specific type of gas/ solid reactor, the Circulating Fluidized Bed (CFB), is finding significant applications industrially because of its many intrinsic properties, such as high rates of heat and mass transfer, high gas and solids velocities, and operational flexibility. Its principal component is a riser in which a high velocity gas stream carries powder vertically. The two-phase mixture is separated at the top of the riser (reactor), and solids are returned to the bottom via a standpipe and other ancillary equipment, such as strippers, regenerators, and heat exchangers. During the past five decades, gas-solids hydrodynamics studies principally have concentrated on solids phase measurements and characterization and have largely ignored the gas phase. Pneumatic conveying is an example; solids are the commodity of interest; the gas phase is only important in the sense that power requirements for blowers and compressors should be minimized. In studies of bubbling and turbulent fluidized beds, experimentalists study the spatial and temporal distribution of bubbles, but, typically, they employ solids measurement devices from which gas phase hydrodynamics are inferred. Circulating fluidized bed researchers also have devoted considerable attention to the solids phase, but since 1988 only 40 publications have appeared that deal with gas phase hydrodynamics. The behavior of the gas phase is, however, extremely important, not only for design and scale-up, but also for reactor optimization. The radial gas velocity profile affects gas-particle backmixing, influences the radial solids volume fraction distribution, and solids velocity profile, which, in turn, regulates the rates of chemical reaction, and heat and mass transfer. Our understanding of gas phase flow patterns and mixing is steadily increasing with the growing body of experimental data. Empirical models are being refined to make them more broadly applicable and suitable for scale-up purposes. However, because of the intrinsic complexity of the hydrodynamics and the difficulty with gas phase measurements and data reduction, no model has yet gained broad acceptance in the research community. In this chapter, we first introduce industrial applications of CFB technology and describe typical operating conditions of the two most common applications: fluid catalytic cracking (FCC) and coal combustion. We characterize CFB hydrodynamics broadly and categorize regime transitions and then describe various techniques used to measure gas phase hydrodynamics, including an extensive list of references. Based on data generated using these methods, a number of models have been proposed to characterize the hydrodynamics. We review these models and propose a correlation for the radial gas velocity profile. We discuss aspects of gas injection and design that impact overall hydrodynamics, which may have the greatest impact on the success of any given application, but are the least understood and studied. Finally, we recommend different areas for future research.

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

257

INDUSTRIAL APPLICATIONS Squires described the early development of CFB reactor technology [1]. It began in 1938 when eight companies—Standard Oil of New Jersey, Standard of Indiana, Texas Company, Shell, Anglo-Iranian, M. W. Kellog, Universal Oil Products and I. G. Farben—formed a consortium to develop a catalytic process for cracking oil. Standard Oil Development Company, a subsidiary of Standard of New Jersey, conducted experiments in essentially a horizontal coil of tubing. However, due to significant plugging problems when flow was interrupted, a vertical "snake reactor" design was adopted for development on a pilot plant scale in which catalyst was alternatively conveyed vertically upward then downward over several runs. Lewis and Gilliland had suggested that plugging problems would be greatly reduced in vertical upward flow and initiated experiments at MIT to study this concept. They studied a wide velocity range from the bubbling fluidization regime to pneumatic conveying. Snake reactor performance at the pilot scale was poor; it was improved by adding pipe runs, but it was eventually abandoned for vertical solids transport, as had been proposed by Lewis and Gilliland. Early commercial catalytic crackers were operated in the bubbling/ turbulent fluid bed regime because reaction rates of the powdered amorphous catalysts were so low that high gas velocities would have resulted in very tall process equipment. Fischer-Tropsch synthesis was the first successful commercial process employing high gas velocities in a riser. In the 1960s, high velocity risers were designed for catalytic cracking to take advantage of extremely active zeolite catalysts. Reh invented the Lurgi fast bed for calcining alumina [2]. This is essentially a device for burning a liquid fuel and using the heat of combustion to calcine aluminum hydroxide to alumina. Since then, other industrial processes commercialized include combustion, incineration, and gasification during the 1980s and catalytic partial oxidation in the 1990s. Table 1 highlights milestones and potential applications of CFB technology. Several hundred CFB combustors have been built in the past 15 years, and many other high temperature applications are being pursued. DuPont is currently constructing a plant to convert n-butane to maleic anhydride in a CFB. This catalytic reaction is one of many that have been studied and patented over the past 40 years. Its successful demonstration may lead to commercial development of other processes. Operational flexibility is often cited as an advantage of CFB reactors over conventional technology. Other cited advantages include good gas-solids contacting at relatively short and adjustable residence times, excellent heat and mass transfer characteristics, staged addition of gases (and solids), good turndown capability, and high throughput per cross-sectional area. Fast reactions are ideally suited for CFB risers. Processes in which CFB reactors are preferred include those in which high selectivities are essential, coke formation or some other poisoning mechanism is rapid, gas plug flow characteristics are desirable (i.e., where product inhibition or degradation is significant), or reactions where the solids are the primary reactant. (text continued on page 260)

258

Advances in Engineering Fluid Mechanics Table 1 Milestones and Potential Applications of CFB Technology References

Fluid Catalytic Cracking (FCC)

Review of early history and development of the FCC process Review of recent process and hardware developments US capacity reached 10 MMb/d in 1992 with over 350 FCC units FCC CFB catalyst regenerators described Resid FCC development FIscher-Tropsch Synthesis:

Gasoline production began in a Synthol CFB in 1955 Larger Synthol reactors came on stream in 1980 and 1982 History and operational experience of Synthol CFB reactors Catalyst developments and comparison of fixed fluid bed and CFB Methanol synthesis Methanol-to-olefins

Squires [1] King [4] Reichle [3] King [5] Avidan [6] Jewel and Johnson [7] Dry [8] Shingles and McDonald [9] Silverman et al, [10] Bartholomew [11] Steynberg et al [12] Chanchlani et al. [13] Schonfelder et al. [14]

Solids processing:

AIF3 from Al(OH)3 and 98% HF in a CFB reactor Calcination of Al(OH)3 to alumina (AI2O3) Conversion of ferric chloride to iron oxide and chlorine Metallurgical and inorganic chemical industries waste gas scrubbing Reduction of iron ore Reduction of gold roasting Biomass pyrolysis

Reh [15] Reh [15] Reeves et al. [16] Reh [2] Zhiqing [17] Peinemann et al. [18] Bohn and Benham [19]

Combustions and Applications, Environmental Remediation

Over 220 units in operation by 1991 NO^ emission requirements Performance testing of a large combustor Tests using different fuels Municipal waste incinerator

Engstrom and Lee [20] Tang et al. [21] Moe et al. [22] Anders et al. [23] Hallstrom and Rarlsson [24]


• Combined pressurized gasifier and atmospheric combustion • Ethane and propane cracking • dNo/dSO X

3.

4.

5.

Anders et al. [23] Koyama and Dranoff [25] Herrmann and Weisweiler [26]

X

Potential heteregeneous catalytic applications: 1. Paraffin Oxidation • Partial oxidation of n-butane to maleic anhydride • Oxidative coupling of methane to ethane and ethylene

2.

259

Contractor [27,28] Pugsley et al. [29] Baerns et al. [30] Dutta and Jazayeri [31] Santamaria et al. [32] Tjatjopoulos et al. [33]

Oxidation of alkane to alcohols and ketones: - isobutane to ter-butylalcohol - propane to acetone - isopropyl alcohol and butane to Lyons [34] methyl ethyl ketone Methane to CO and H^ Lewis et al. [35] Ammoxidation • Acrylonitrile from propylene Beuther et al. [36] • Benzonitrile from toluene and acrylonitrile Gianetto et al. [37] from propylene • Acrylonitrile and methacrylonitrile from Huibers [38] propylene and butene Huibers [38] • Tetraphthalonitrile from p-xylene • Acylonitrile and methacrylonitrile from Kahney and McMinn [39,40] propane and i-butane Ally lie Oxidation • Maleic anhydride from C4 to C6 fractions Rollman [41] • Acrolein from propylene process patent Johnson [42] • Acrolein from propylene Callahan et al. [43] Patience and Mills [44] Rollman [41] Wain Wright and Hoffman [45] • Phthalic anhydride from o-xylene and naphtalene Gelbein [46] • Phthalic anhydride from o-xylene Epoxidation Rollman [41] • Ethylene oxide from ethylene Park and Gau [47] Dehydrogenation • Anaerobic oxidation of butane to butadiene Tmenov et al. [48] Murchison et al. [49] • Alkane oxidative dehydrogenation Sanfilippo et al. [50]

260

Advances in Engineering Fluid Mechanics Table 1 (continued) References

• Ethyl benzene to styrene and ethyl toluene to vinyltoluene • Butadiene from butane and butenes • Ethylene from ethane 6. Other • Glyoxal from ethylene glycol • Acetaldehyde from ethanol • HCl oxidation • Cyclohexanone ammoximation • Methanol to formaldehyde

Debras et al. [51] Woskow [52,53] Coudurier et al. [54] Gallezot et al. [55] Filho and Domingues [56] Pan et al. [57] Fieri et al. [58] Zaza et al. [59]

(text continued from page 257) Ideally, CFB risers operate at a relatively uniform temperature, which is achieved by a high solids recycle rate renewing the inventory of the riser. Catalytic reactions are generally carried out at relatively low temperatures (250-650°C) compared to combustion processes (>800°C). Low-temperature operation permits the use of mechanical devices to control solids mass flux. In combustion processes, the rate is controlled by non-mechanical devices. Fluid mechanics of CFB catalytic reactors and combustors are significantly different, as shown in Table 2. PHYSICAL CHARACTERIZATION OF CIRCULATING, FLUIDIZED BEDS When gas is introduced through a suitable distributor into a vertical column containing solid particles, different hydrodynamic regimes are observed. These regimes depend on particle characteristics, gas superficial velocity, and geometry. At low gas velocities, the column of solids is in a packed bed hydrodynamic regime. With increasing gas velocity the hydrodynamic regime changes to bubbling bed followed by slugging flow (this regime is most prevalent in small diameter experimental columns) then turbulent fluidization, fast fluidization, and, finally, pneumatic transport. Many studies have been conducted to characterize flow regime transitions, as summarized by Kunii and Levenspiel [60]. A bed of solids becomes fluid when the gas velocity exceeds a minimum value U^^, the minimum fluidization velocity. As U^^ is exceeded (or U^^—minimum bubbling velocity—for Geldart Group A powders), gas bubbles appear throughout the bed. When the gas velocity reaches a critical value, U^, the bed becomes turbulent (Chehbouni et al. [61]). At this point, entrainment of the bed of solids into the freeboard region becomes significant, and the suspension density in the freeboard decays exponentially. Over the entire reactor length, the axial solids suspension density is sigmoidal, as shown in Figure 1. Particle carryover through the top of the column increases further with gas velocity and multiple cyclone stages


261

Table 2 Operating Characteristics of Fast Fluidized Bed Catalytic Reactors and Combustors Characteristic

Catalytic Reactors

Superficial velocity Solids circulation rate Particle diameters Geldart classification Temperature Riser diameter Pressure Solids reinjection system

4-10 m/s > 250 kg/m^s 50-150 iLtm A 250-650°C 0.5-2 m > 1 bar mechanical or non-mechanical valve Abrupt

Exit geometry

Combustors

2-6 m/s 50-100 kg/m^s 250-500 Mm

B >800°C 5-30 m 1 bar non-mechanical valves (L, J or V valve) Abrupt

Gas Out

Cyclone

/

Standpipe Riser

L-Valve

t

Suspension Density

Reactor Feed Figure 1. Fixed Inventory System (FIS) with sigmoidal density distribution.

262


are installed to return elutriated solids back to the bed. Conceptually, a turbulent fluidized bed with a cyclone can be considered a Circulating Fluidized Bed. Fast fluidization follows the turbulent regime and begins when the gas velocity exceeds the transport velocity, U^^. In a fast fluidized bed, gas is the continuous phase and solids are dispersed. Solids are carried upward by the gas; solids backmixing is less severe than in the turbulent regime, but solids may flow down in the vicinity of the wall. The time-averaged axial solids hold-up decays exponentially from the entrance region until a point at which the flow becomes fully developed (i.e., constant solids velocity and holdup), as shown in Figure 2. For risers with abrupt right angle exit configurations, solids hold-up increases at the exit. Another transition velocity, U^^, from fast fluidization to pneumatic transport is defined as the minimum gas velocity required to fully suspend a given flux of solid particles over the entire length without solids downflow along the wall. In pneumatic transport, gas also is the continuous phase, and solids hold-up is very low (typically, less than 1%). Figure 3 shows that operating regimes from the onset of turbulent fluidization (beyond the transition velocity, U^) follow different paths depending upon the two general CFB designs defined by Kobro and Brereton [62] and Kunii and Levenspiel [63]. In the first, "Fixed Inventory System" (FIS), solids inventory in the return leg or standpipe is not controlled. Setting gas velocity and system inventory establishes the riser suspension density profile and solids circulation rate. In the second, "Variable Inventory System" (VIS), a vessel external to the riser acts as a

Stripper Regenerator Off Gas I

Regeneration Gas

9^

Riser

T Reactor Feed

Suspension Density

Figure 2. Variable Inventory System (VIS) with exponentially decaying density distribution.

-

F.I.S. G, is a dependent

Turbulent Fluidized Bed F.I.S.

U,

Fast Fluidized Bed F.I.S.

U~~

Pneumatic + Transport F.I.S.

Fluidized V-LS. G, is an independent variable

Bubbling Fluid Bed continuousphase: solids dispersed phase: gas

Turbulent Fluidized Bed

Turbulent Fluid Bed bottom: continuous phase: solids dispersed phase: gas top: continuous phase: gas dispersed phase: solidr

I V.I.S.

ucA=f(~,)Pneumatic W T r a n s o o r t

I

Fast Fluid Bed continuous phase: gas dispersed phase: solids

I

Pneumatic Transport continuous phase: gas dispersed phase: solids

Figure 3. Regime transitions for Fixed lnventory Systems (FIS) and Variable Inventory Systems (VIS).

264

Advances in Engineering Fluid l\/lechanics

solids reservoir to accommodate riser inventory changes with modified operating conditions. Fixing gas velocity and circulation rate establishes riser solids inventory and suspension density profile. If the solids mass flux is an independent variable (i.e. VIS design), the operating regime of the CFB riser may change, as illustrated in Figure 3. When U^^ is reached, the fast fluidization regime appears. However, if the imposed solids mass flux exceeds the carrying capacity of the gas, a dense phase is formed at the riser entrance, and the riser is in the turbulent regime; a higher gas velocity is required to reach the fast fluidization regime. So, U^^ is a function of the imposed mass flux. As with the fixed inventory system, U^.^ characterizes the transition to pneumatic transport. It also is a function of mass flux: It is constant below the saturation carrying capacity of the gas and increases with increasing mass flux above this limit. Therefore, a CFB may be operated under turbulent fluidization, fast fluidization, or pneumatic transport regimes. This broad definition is consistent with experimental observations reported in the literature dealing with the general appearance of the bed. In this chapter, the discussion focuses mainly on systems where the gas velocity and solids mass flux are independent variables (VIS), and concentrates on the fast fluidization regime. EXPERIMENTAL METHODS A variety of experimental techniques are available to measure gas-solids hydrodynamics. Yates and Simons [64] recently reviewed methods particular to fluidization. They conclude that the techniques under development for laboratory-scale units are becoming more sophisticated and reliable and less intrusive. However, they maintain most of these techniques are not readily adaptable to industrial scale vessels operating under hostile conditions of atmosphere, pressure and temperature. Methods described for measuring gas phase hydrodynamics by Yates and Simons include capacitance probes, pressure sensors, optical probes and other imagining techniques, such as X-ray and y-ray attenuation [64]. Bachalo reviewed imaging methods relating to particle flow and included Doppler particle analysis, laser Doppler velocimetry (LDV), and near-forward light scatter deflection [65]. Together with in-particle imaging methods, tomography and particle tracking techniques have been improved. However, none of these methods measure gas flow. Under dilute conditions, gas velocity may be deduced assuming that the slip velocity—the difference between gas velocity and particle velocity—equals the single particle terminal velocity. In vessels operating at high suspension densities, the techniques reviewed measure the dilute phase distribution in time and space, which are bubbles in the case of low velocity fluid beds. Although most of the gas flows through the bed as bubbles, their distribution in time and space does not address mass transfer or emulsion gas hydrodynamics. Measurement techniques for the gas phase in CFB risers are becoming increasingly more sophisticated and accurate, and many find application in industrial-size equipment. In this section, we classify the different methods into five categories: intrusive probes, steady state tracers, impulse (non-steady state) tracers, chemical reaction, and optical tracers. No single tracer technique is capable of quantifying

Gas Phase Hydrodynamics In Circulating Fluidlzed Bed Risers

265

riser hydrodynamics entirely. The most successful studies combine two techniques to quantify temporal and spatial gas distribution. Intrusive Probes Iso-kinetic sampling generally is used to measure solids mass flux profiles. The sampling probe is a thin tube inserted through the vessel wall. The tip of the tube is bent at a 90° angle so that it is parallel to the flow stream. The tube velocity is adjusted to match the surrounding velocity so as not to disturb the main flow. Solids enter the tube then are separated from the gas with a cyclone. Van Breugel et al. used this technique to measure gas phase radial velocity profiles (see Table 3) [66]. To obtain iso-kinetic conditions, they equalized static pressure inside and outside the probe by adjusting suction rate, which then gives the volumetric flow rate and, thus riser velocity. Because solids retained in the collection vessel displace gas, they maintained that they measure interstitial gas velocity. However, Harris and Davidson pointed out that this assumption is valid only when there is no slip between the phases [67]. Despite this limitation, van Breugel et al. *s integrated gas radial profile matched the known throughput well, and centerline velocities agreed within 10% of an impulse tracer measurement using methane [66]. Their reported profiles are illustrated in Figure 4, and the single-phase turbulent velocity profile they measured is included for comparison. With solids present, centerline velocities approach three times superficial velocities. Wall velocities are shown to be zero, but they report having sampling problems in this region. Yang et al. developed Pitot-static tube probes to measure radial gas velocity profiles at different heights in their riser [68]. They encased 0.5 mm ID hypodermic needles in a 5 mm OD tube. The static tube was sealed at the tip, and two 0.5 mm holes were drilled perpendicular to the planes of the tube 10 mm from the tip. The tip of the impact tube was flush with the surface. Using a standard equation for Pitot static tubes gives the local velocity, V^ = c(2AP/pp»^2

(1)

where the constant c is a characteristic of geometry and AP is the differential pressure between the static and impact tubes. They examined a wide range of test conditions and proposed a number of correlations to characterize the effect of height, gas velocity, circulation rate, and particle diameter on the radial gas velocity profile. However, they only examined very dilute suspensions and ignored solids momentum. Azzi et al. correctly included the effect of solids in their study using a momentum probe; at high solids loading, they showed that gas momentum was negligible [70]. Their probe consisted of two tubes purged with nitrogen or air to avoid plugging with one pointing downward and the other upward. Bader et al. also reported data generated with a Pitot tube and assumed that the momentum flux of the riser gas was negligible under their operating conditions [69]. Intrusive probes are robust and have been used to measure velocity profiles in industrial scale equipment. However, data interpretation is not straightforward, and

-.

3

Table 3 Intrusive probes

D m

van Breugel et al. [66] Bader et al. [69] Azzi et al. [70]

Hams and Davidson [67] Yang et al. [68]

0.305 0.305 0.19 0.75 0.41 0.94 0.14 0.20

H m

12.2 11.7 16 5.1 5.5

Riser

",

m/s

6.3 4.6

Particle Properties Gs d~ PP kg/m2s pm kg/m3

380 147 152

6.2 2.6 3.5-7.3

Y

co. 3

12.3 k ^ at similar rCC

sand

conditions (i.e., k decreases with increasing particle size and density) [94]. Typical values of k ranged between 0.01 and 0.07 m/s; they increased with solids flux and decreased with gas velocity. Kagawa et al. [109] utilized the two-zone (core-annulus) model of Brereton et al. [96] to interpret their results of gas-solid contacting using the ozone decomposition reaction. A cross-flow coefficient of 0.001 m/s was used to fit their data, and ryR was assumed equal to 0.85. Werther et al. [77] further developed the core-annular model by including radial dispersion in the core region. They found that r^R = 0.85 and that this ratio was independent of gas velocity and solids flux. Best fit values of the radial dispersion coefficient in the core ranged between 20 and 60 cmVs, increasing with gas velocity but approximately constant with respect to solids flux between 0 and 70 kg/m^s. Amos et al. used sulfur hexafluoride as an absorbing tracer to assess riser gas mixing [85]. They also reviewed some earlier studies and pointed out discrepancies

- Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

285

between the predicted trends of the radial dispersion coefficient with operating conditions. In particular, they highlighted that Yang et al. [73] found that radial dispersion increased with solids suspension density and was independent of axial coordinate, in contrast to Adams [74], who reported that radial dispersion decreased with increasing suspension density, and Werther et al. [77], who suggested that radial dispersion in the core region was independent of solids flux. They attributed these contradictory findings to differences in equipment configuration, operating conditions, and the model chosen to characterize the data. To characterize radial gas phase mixing, Amos et al. [85] proposed that mass transfer in the annulus was different than that in the core: at sufficiently high solids mass flux, gas in the annular region is well mixed. In the core, radial dispersion characterizes the mixing and applies over the whole cross-section at sufficiently low solids mass flux (or suspension density). Core radial dispersion decreased with increasing mass flux, and they reported values between 30 and 40 cm Vs. Patience and Chaouki [98] adopted the two-phase model of Brereton et al. [96] to interpret their gas RTD data obtained with a radioactive tracer gas. The two model parameters, crossflow coefficient, k, and O (ratio between core and riser cross-sections), were evaluated by fitting the model to the experimental data. They found that the crossflow coefficients varied between 0.03 to 0.1 m/s, and O varied g

from 0.98, at high gas velocities, to 0.5, at low velocities. They attributed gas crossflow between core and annulus by supposing that solids drag gas to the annulus as they "condense" along the wall and then carry it downward for a certain distance. Solids are reintroduced into the core as they are stripped off the wall and re-entrained into the core gas flow. They developed a correlation for describing this gas mass transfer based on the analogy with wetted wall towers, as: kDOf ^^^^^P.U.D^ ^ = 0.25 D p | l O 1/2 PU DvPg

1/2

(9)

vKg

Based on a large pool of experimental data from various authors. Patience and Chaouki [98] developed an empirical correlation for predicting O as a function of operating conditions: o„ = l + l.lFr

^G. V P U

^'

(10) ,

O decreases with increasing solids circulation, more rapidly as the gas superficial velocity is decreased. Contact Efficiency Dry, using a heat pulse technique, showed the effect of the presence of solids on the gas flow both at the entrance region and along the riser [116]. Solids enhance

286


the radial gas velocity distribution, which results in the characteristic core-annular flow structure. Dry et al. extended the studies on gas-solids contact efficiency showing a decrease in efficiency with increasing gas velocity [89]. In a subsequent study [95], they examined the effect of inlet geometry, circulation rate (pressure drop) and gas velocity. Three inlet sections were tested: 76 mm ID, 102 mm ID, and 200 mm ID. They showed that gas-solid contacting was a strong function of inlet geometry. At constant pressure drop, the larger diameter unit gave the highest contact efficiency. However, it is not clear how they corrected for the increased gas-solids contact time. In summary, a single model has not been developed that can fully characterize riser gas phase hydrodynamics. The studies indicate that under dense phase conditions, typical of commercial FCC riser operation, a simple axial dispersion model may be adequate to characterize gas mixing. Under dilute conditions, a two-phase core-annular model is a good first approximation to the flow structure. However, both radial dispersion and radial gas velocity profiles must be accounted for to provide a realistic and reliable interpretation. The model suggested by Martin et al. should be further developed and applied to risers of different geometry operating with different powders [83]. However, contact efficiency may provide the simplest means from which scale-up criteria can be developed. DESIGN CONSIDERATIONS The axial suspension density profile depends on gas velocity, solids mass flux, riser geometry, and particle characteristics. For any given application, gas and soHds residence times are chosen to achieve a degree of conversion to maximize economics. High gas velocities are generally preferred to the extent that they result in low solids hold-up and, thus, compressor costs. However, for catalytic reactions, higher solids inventories are required to maximize specific activity per unit volume of reactor for which lower gas velocities are preferred with higher suspension densities. In the case of reactions in which the solids are the primary oxidant, as for butane oxidation to maleic anhydride [28], high solids circulation rates result in increased production. Matsen observed that for industrial FCC risers the ratio of gas velocity to particle velocity, which is referred to as the slip factor, was approximately equal to 2 [117]. Based on small scale experimental riser data. Patience et al. [118] developed an equation based on the slip factor concept that related solids suspension density and operating conditions in the fully developed region.

^ " l + U^Pp/(G,H^)

^^1)

where the slip factor, \|/, is related to particle terminal velocity and riser diameter, \|/ = 1 + 5.6/Fr + 0.41 Fr,"^'

(12)

Solids hold-up increases with mass flux and decreasing gas velocity. The correlation predicts that density increases with increasing riser diameter. This effect can


287

be very significant, especially as far as scale-up is concerned, but it has not yet been clearly shown with experimental data. Gas velocity is a key operating parameter, and together with CFB geometry, as discussed, determines regime transitions. Design criteria for gas injection and pressure drop are well known for low gas velocity reactors operating in the bubbling and turbulent fluid bed regime. Design criteria for the fast fluidization regime are not as readily available, although gas injection impact hydrodynamics and gas-solids contacting significantly in FCC risers. (As discussed earlier, secondary gas injection into the dilute phase of combustors does not impact performance). Johnson et al. suggest that poor feed injection into a FCC riser results in poor reactor performance: FCC profitability and product selectivity are largely determined by feedstock-catalyst contacting at the point at which they meet in the riser [119]. Optimum oil feed distribution minimizes regions of high and low catalyst-to-oil ratios and reduces catalyst backmixing. The atomizer they developed produces superior atomization at modest pressure drop and improved both gasoline selectivity and overall conversion in two different commercial risers. Fligner et al. compared circumferential gas injection with internal injection through two nozzles in a pilot scale riser [120]. Circumferential gas injection resulted in uniform radial densities at the entrance with a corresponding increase in conversion by 3%. Saxton and Worley [121] reported that the number of nozzles through which oil is introduced into industrial scale risers changes radial solids distribution at the entrance and overall solids hold-up over the entire length. However, Fligner et al. [120] did not report any difference in suspension density between circumfrential and internal injection in the fully developed section farther up the column. Bernard suggested this discrepancy may be due to the injection nozzles: first generation nozzles were open pipes that resulted in poor mixing, which affected hydrodynamics all the way up the riser. Johnson et al. [119] showed that the open-pipe nozzle produces little atomization and slugging develops because of the poor mixing upstream of the nozzle exit orifice. Results of Weinstein et al. [123] agree with previous studies that gas injection geometry impacts entrance region hydrodynamics. They proposed that an optimal design will accelerate solids rapidly and uniformly over the riser cross section, which will minimize entrance length and maximize reactor performance. RESEARCH NEEDS Commercialization of new CFB processes for the production of chemicals^ specialty or commodity—has been hindered due to scale-up concerns and operational complexity. In particular, the effect of diameter, high solids mass flux, and high pressure on gas hydrodynamics are undocumented in the open literature. Innovative research aimed at the design of new solids feeding devices and gas-solids separation techniques may reduce operational complexities. However, studies on small scale equipment must be performed prudently and documented concisely to be useful for scale-up. Scale effects at the entrance region are considerable, and sufficient attention has not been devoted to this region. Modelling gas mass transfer in the riser is evolving rapidly. Improved measuring techniques have led to a better understanding of the flow characteristics in the

288


developed region of the riser. However, hydrodynamic studies generally have focused on the gas and solids phase separately. More research is needed to evaluate interactions between the two phases to provide a basis for mass transfer modelling, which would lead to improved scale-up criteria. Entrance region mass transfer of solids and gas generally has been ignored to present. A fluidization regime map should be developed that identifies regime transitions as a function of operating conditions and CFB design—PIS vs. VIS. Techniques for determining transition velocities have been reported but implementation is not exact. Available predictive correlations for transition velocities need to be modified to include the effect of imposed solids mass flux. Transient hydrodynamic models are non-existent. In catalytic reactors, solids provide thermal inertia that stabilize highly exothermic processes. A potential hazard exists in these reactors if a loss in circulation results in the vessel emptying before the control system responds [124]. Furthermore, with regard to exothermic reactions, internal heat transfer surface area may be introduced to maintain temperature uniformity. Internals may affect flow patterns and solids hold-up, but further research is required. In conclusion, a fundamental understanding of CFB hydrodynamics continues to lag behind commercial operating experience. Although CFB provides considerable advantages of operational flexibility, concerns about operational complexity and scale-up hinder commercialization of heterogeneous catalytic reactions. Concentrated research in the areas of gas-solid contact in the entry region, effect of internals, riser diameter, and solid fines content is required. The continued interaction between academia and industry, as seen in the four international conferences of CFB technology, will be necessary to further optimize commercial operations but also to further develop revolutionary industrial processes. NOTATION c Constant defined in Equation 1 C Concentration C Concentration in annulus

AP Pressure drop r Radial co-ordinate, distance from riser centerline

a

C Concentration in core c

C Local concentration at r

r Core radius c

R Riser radius

r

d Particle diameter

t Time

p

D Riser diameter D^ Axial dispersion D^ Radial dispersion D^ Diffusion Fr Froude number, Ug/^/gD Fr^ Terminal Froude number, V,/^/gD g Gravitational acceleration G^ Solids mass flux H Riser height k Core-annular cross flow L Length or riser height m Exponent shown in Equation 3 n Exponent shown in Equation 2

U^ Transition velocity from bubbling fluidization to turbulent U^, Centerline velocity U^^ Transition velocity from fast fluidization to pneumatic U Superficial gas velocity U^^^ Minimum bubbling velocity U^^ Minimum fluidization velocity U Solids superficial velocity, G/p LT Local superficial velocity at r U^^ Transition velocity from turbulent fluidization to fast V Interstitial gas velocity


V ^ Core interstitial gas velocity V Interstitial solids velocity V ^ Core interstitial particle velocity

289

V^ Single particle terminal velocity z Axial coordinate

Greek Symbols a Exponent shown in Equation 4 e Void fraction e^ Annular void fraction e^ Core void fraction p Density

p Gas density p Particle density jl Viscosity Og (r^R)^ ^ Slip factor

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29. 30.

31. 32. 33. 34.

35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

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95. Dry, R. J., White, R. B., and Close, R. C , "The Effect of Gas Inlet Geometry on Gas-Solid Contact Efficiency in a Circulating Fluidized Bed" in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.), Engineering Foundation, New York, 1992 96. Brereton, C. M. H., Grace, J. R., and Yu, J., "Axial Gas Mixing in a Circulating Fluidized Bed", in Circulating Fluidized Bed Technology II (P. Basu and J. F. Large, eds.), Pergamon Press, Oxford, 1988. 97. Bai, D., Yi, J., Jin, Y. and Yu, Z., "Residence Time Distributions of Gas and Solids in a Circulating Fluidized Bed", in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992. 98. Patience, G. S. and Chaouki, J., "Gas Phase Hydrodynamics in the Riser of a Circulating Fluidized Bed", Chem. Eng. ScL, 48 (18), 3,195(1993). 99. Viitanen, P. I., "Tracer Studies on a Riser Reactor of a Fluidized Catalyst Cracking Plant", Ind. Eng. Chem. Res., 32, 577(1993). 100. Zhang, Y.-F., Arastoopour, H., Wegerer, D. A., Lomas, D. A., and Hemler, C. L., "Experimental and Theoretical Analysis of Gas and Particles Dispersion in Large Scale CFB", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 101. Yang, Y.-L., Jin, Y., Yu, Z.-Q., and Wang, Z.-W., "Investigation on Slip Velocity Distributions in the Riser of Dilute Circulating Fluidized Bed", Powder TechnoL, 73, 67(1992). 102. Horio, M., More, K., Takei, Y., and Ishii, H., "Simultaneous Gas and Solid Velocity Measurements in Turbulent and Fast Fluidized Beds", in Fluidization VII (O. E. Potter and D. J. Niclin, eds.). Engineering Foundation, New York, 1992. 103. Donsi, G. and Osseo, L. S., "Gas Solid Flow Pattern in a Circulating Fluid Bed Operated at High Gas Velocity", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 104. Hartke, W., Helmrich, H. and Droger, H., and Schugerl, K., "Non-Catalytic Decomposition of Sodium Bicarbonated in a Circulating Fluidized Bed Reactor", Ger. Chem. Eng. 4, 203(1981). 105. Fujima, Y., Fujioka, Y., Hino, H., and Takamoku, H.,"Experimental Study on Sulfur Retention in CFBC", in Circulating Fluidized Bed Technology (P. Basu, ed.), Pergamon Press, Toronto, 1986. 106. Colorado-Ute Electric Association, "NUCLA Circulating Atmospheric Fluidized Bed Demonstration Project", DOE Report MD/25137-3046, 1991. 107. Boyd, T. J. and Friedman, M. A., "Operations and Test Program Summary at the 110 MWe NUCLA CFB", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M.Hasatani, eds.), Pergamon Press, Oxford, 1991. 108. Basu, P., Wu, S., and Greenblatt, J., "Development of a Simplified Model for Sulphur Absorption in Circulating Fluidized Beds and Experimental Verification in Pilot Scale and Large Commercial CFB Combustors", J. Chem. Eng. Japan, 24 (3), 356(1991). 109. Kagawa, H., Mineo, H., Yamazaki, R., and Yoshida, K., "A Gas-Solid Contacting Model for Fast Fluidized Bed", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M. Hasatani, eds.), Pergamon Press, Oxford, 1991.

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110. Jiang, P., Inokuchi, K., Jean, R.-H., Bi, H., and Fan, L.-S., "Ozone Decomposition in a Catalytic Circulating Fluidized Bed Reactor", in Circulating Fluidized Bed Technology III (P. Basu, M.Horio and M. Hasatani, eds.), Pergamon Press, Oxford, 1991 111. Wei, F., Lin, S., and Yang, G., "Gas and Solids Mixing in a Commercial FCC Regenerator", Chem. Eng. TechnoL, 16, 109 (1993). 112. Ouyang, S., Lin, J., and Potter, O. E., "Ozone Decomposition in a 0.254 m Diameter Circulating Fluidized Bed Reactor", Powder TechnoL, 74, 73(1993). 113. Ouyang, S. and Potter, O. E., "Modelling Chemical Reaction in a 0.254 m I.D. Circulating Fluidized Bed", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 114. Vollert, J. and Werther, J., "Mass Transfer and Reaction Behaviour of a Circulating Fluidized Bed Reactor", Chem. Eng. TechnoL, 17, 201(1994). 115. Berruti, F. and Kalogerakis, N, "Modelling the Internal Flow Structure of Circulating Fluidized Beds", Can J. Chem. Eng., 67, 1,010(1989). 116. Dry, R. J. "Radial Concentration Profiles in a Fast Fluidized Bed", Powder TechnoL, 49, 37(1986). 117. Matsen, J. M., "Some Characteristics of Large Solids Circulation Systems", in Fluidization Technology VoL 2 (D.L. Keairns, ed.). Hemisphere, New York, 1976. 118. Patience, G. S., Chaouki, J., Berruti, F., and Wong, R., "Scaling Considerations for Circulating Fluidized Bed Risers", Powder TechnoL, 72, 31 (1992). 119. Johnson, D. L., Avidan, A. A., Schipper, P. H., and Miller, R. B., "New Nozzle Improves FCC Feed Atomization Unit Yield Patterns", Oil and Gas J., 92 (43), 80(1994). 120. Fligner, M., Schipper, P. H., Sapre, A. V., and Krambeck, F. J., "Two Phase Cluster Model in Riser Reactors: Impact of Radial Density Distribution on Yields", presented at ISCRE 13, Baltimore, September 1994. 121. Saxton, A. L. and Worley, A. C , "Modern Catalytic-Cracking Design" Oil and Gas Journal 68 (20), 82(1970). 122. Bernard, J. R., personal communication, 1995. 123. Weinstein, H., Feindt, H. J., Chen. L., Pell, M., Contractor, R. M., and Jordan, S. P., "Acceleration and Distribution of Solids Downstream of a Riser Gas Feed Nozzle", to be presented at Fluidization VIII, Tours, France, May, 1995. 124. Contractor, R. M., Pell, M., Weinstein, H., and Feindt, H. J., "The Rate of Solids Loss in a Circulating Fluid Bed Following a Loss of Circulation Accident", in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992. 125. Yang, Y., Jin, Y., Yu, Z., and Bai, D., "The Radial Distribution of Local Particle Velocity in a Dilute Circulating Fluidized Bed", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio, and M. Hasatani, eds.). Pergamon Press, Oxford, 1991. 126. Rhodes, M. J., Wang, X. S., Cheng, H., and Hirama, T., "Similar Profiles of Solids Flux in Circulating Fluidized-Bed Risers", Chem. Eng. Sci., 47 1,635(1992).

CHAPTER 11 BOUNDARY CONDITIONS REQUIRED FOR THE CFD SIMULATION OF FLOWS IN STIRRED TANKS

Suzanne M. Kresta University of Alberta, Edmonton, Alberta, Canada, T6G 2G6 CONTENTS INTRODUCTION, 297 Computational Fluid Dynamics, 298 Stirred Tanks, 298 Impeller Modeling, 299 PHYSICAL ISSUES, 300 Turbulence Model, 300 Swirl Number, 301 Definition of the Geometry and Boundary conditions at the Edges of the Domain, 301 Impeller, 303 NUMERICAL ISSUES, 310 Gridding, 310 Convergence Criteria, 313 Solution Algorithms and Differencing Schemes, 313 VALIDATION, 313 CONCLUSIONS, 314 ACKNOWLEDGMENTS, 314 NOTATION, 314 REFERENCES, 315 INTRODUCTION Accurate CFD (computational fluid dynamic) simulation of the flow in stirred tanks requires correct specification of both the geometry and the physical conditions of the flow. While specification of the geometry, the gridding, and the solution algorithm is relatively straightforward, some other issues remain difficult. The most challenging problem is definition of a physically accurate, computationally tractable impeller or impeller model which incorporates the effect of the tank geometry. This 297

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chapter is intended to serve as an introduction to the CFD simulation of flows in stirred tanks, and particularly to impeller modeling. The development and current state of knowledge is reviewed in terms of the underlying physical and numerical issues, some of which are now well-understood, others of which pose many unresolved issues. The chapter also should prove useful to engineers who are attempting CFD simulations of other flows, since it provides an overview of many of the issues which need to be addressed, and a framework for thinking about CFD issues. Computational Fluid Dynamics With the advent of reliable, commercially available CFD software has come an increasing interest in the simulation of the flows in large scale, geometrically complex, industrial vessels (Bakker and Fasano [1], Colenbrander [2] and Sharratt [3] for some examples). CFD simulations provide access to almost unlimited data on the flow field, infinite variations on the geometry, infinite scale-up possibilities, and visually appealing results which are easily digested by everyone involved with the process. In addition, the numerical experiments performed using CFD seem to require much less investment than "wet" experimental work, which may be difficult or impossible under hostile plant conditions. Like traditional experimental work, however, CFD has its limitations. Two major challenges confront the would-be user of CFD codes: The first and most important is correct specification of the physical conditions, primarily through the boundary conditions, turbulence model, rheological model, and other physical models such as reaction kinetics, heat transfer, and phase interactions (for multiphase flow); the second is proper attention to numerically based issues, such as grid definition and convergence criteria. If any of these issues is neglected, misleading results easily can be generated. In this chapter, some of these issues are discussed in the context of simulation of the three-dimensional (3D) turbulent flow in stirred tanks. Stirred Tanks Stirred tanks are widely used in the process industries as reactors and mixing vessels, for liquid-liquid dispersion, solids suspension, and crystal precipitation. Without a detailed understanding of the hydrodynamics within the vessel, not much progress can be made in predicting the scale-up of these processes. Work in this field began in 1972, when Fort et al. and Desouza and Pike reported the first attempts to model the flow patterns in a stirred tank [4,5]. Further attempts were made from 1982-1987, generally with 2D user written CFD codes and ad hoc models of various aspects of the flow (see a detailed review by Kresta and Wood [6]). Since 1988, research on this flow field has accelerated due to improvements in both CFD codes and experimental techniques. The CFD software packages have increased in power and flexibility while data collection and measurement techniques are more robust and exact, allowing detailed and reliable validation of the simulation results. It is now possible to simulate very complex geometries and simple non-Newtonian rheologies for laminar flow [7,8]. Successful multiphase simulations have been reported by Gosman et al, Rasteiro et «/., Myers et al, and Weetman [9-12]. Time varying aspects have been added by Luo et al. [13], and by Mathur and Murthy

Boundary Conditions Required for the CFD Simulation ,

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[14] (both of whom used a sliding mesh to model the impeller), and by Tanguy et al.y [8] (who illustrated the mixing of viscous fluids). The use of CFD results for the simulation of other processes has been reported by Ranade et ai, [15] (blend time); Roekaerts [16-17] (selectivity of reaction); and Smith et al.y [18] scale up. Finally, Fokema et al, [19], Ranade et al, [20,21], and Sahu and Joshi [22] have concentrated on the validation and refinement of simulations of the time-averaged, three-dimensional, single-phase, turbulent flow field. Even with the substantial progress represented by these contributions, reliable simulation methods for industrial conditions and varying geometries remain a goal for the future. Impeller Modeling At present, the simulation of flow in stirred tanks requires particular attention to accurate treatment of the impeller. The rotating impeller is difficult to simulate directly in the context of a stationary CFD domain. Even with the introduction of sliding mesh techniques which allow the impeller to rotate in a fixed tank, and thus reproduce the trailing vortices behind the impeller blades [13,23]), only 5 to 10 rotations of the impeller have been reported [13]). Laroche reported that 16 sliding mesh steps, for 90° of tank simulation, took over 10 hours on a Cray [23]. Since the (time varying) bulk flows of interest take of the order of 50 rotations to become established, and the process results of interest may span 10,000 rotations (60 rpm for 3 hours on an industrial scale), this approach is still impractical for the typical user. In order to replace the impeller with a physically equivalent numerical model, the experimental evidence must be considered. From the accumulated data about flows generated by impellers (see in particular Yianneskis et al, [24]), it is known that impellers generate a jetlike flow, which is dominated by the trailing vortices at the tip of the impeller blades. For radial impellers (e.g., the Rushton turbine [RT]), the trailing vortices, and the impeller discharge flow, are directed radially towards the tank wall. For axial impellers (e.g., the pitched blade turbine [PBT]) the flow is downwards. For the high efficiency designs (e.g. the A310 by Lightnin'), the vortices are much weaker; thus the discharge flow is less jetlike at the tip and more uniform over the extent of the impeller blades. Kresta and Wood have shown that the PBT interacts strongly with the flow field, producing different discharge flows and bulk flow patterns at different off-bottom clearances of the impeller [25]. In all cases, the impeller can be successfully modeled as a set of inlet cells, with defined axial, radial, and tangential velocities and, where the k-8 turbulence model is used, turbulence quantities k and e. There are two ways to obtain the required boundary conditions for the inlet cells: the first (and most common) is through direct experiment, the second is by developing a more general model of the impeller, based on experimental evidence, which will predict the discharge flow condition. The rest of this chapter is devoted to examining the physical and numerical issues underlying CFD simulations of flow in stirred tanks. Under physical issues, the general issues of turbulence modeling and degree of swirl are addressed first. The comments in these sections can be applied to any flow field. The specific stirred tank flow field depends on the tank geometry, the boundary conditions at the edges of the computational domain, and the impeller model. These issues are discussed in detail. Many of the numerical issues which were the focus of early CFD research

300


have been addressed successfully by the software vendors. The main issues which are visible to the user are gridding of the domain and selection of convergence criteria. While these two issues are discussed in detail, the reader is referred to other sources for more information on solution algorithms and other numerical issues [26]. PHYSICAL ISSUES Without adequate consideration of the underlying physical issues which define the flow field, accurate CFD simulations will occur only by chance. The challenge is always to define an approximation to the physics which is detailed enough to reproduce the essential features of the flow, without becoming so detailed that the computational load becomes unmanageable, and unnecessarily detailed information is generated. To illustrate this point, consider the CPU time that would be required to produce an exact simulation of the flow using a sliding mesh to model the impeller, and direct numerical simulation of turbulence to model the decay of the vortices . . . then consider how all of the information produced could be: 1) validated, and 2) processed and reduced to a useable form. In this section, each of the physical simplifications currently used to make the simulation possible is examined in turn, beginning with the turbulence model and the degree of swirl, proceeding through the less contentious issues of defining the domain of the simulation and the boundary conditions at the walls, and concluding with modeling of the impeller. Turbulence Model The standard two-equation k-e model has been used for almost all of the simulations referred to in this chapter because it is the most tested and reliable turbulence model available. Although it will not give the amount of information that a mean Reynolds stress or an algebraic stress model will give, it requires an order of magnitude less CPU time and gives predictions of the mean velocities that are of comparable accuracy to the higher order models. It is important for the user to understand that the standard model constants for the k-e model (C = 0.09, C , = 1.43, C , = 1.92, a =1.3, and a = 1.0) are not ^

\i

' e l

' e 2

' e

'

k

^

tuning parameters. These model constants were derived from fundamental turbulence experiments as a part of the development of the k-e model. The experiments were chosen to represent the characteristics of the simple shear flows, for which the model was designed. Extension of the k-e model to flows other than those for which it was designed carries with it definite risks, but it is unlikely that arbitrary tuning of the turbulence model constants to obtain agreement between simulations and experimental data will yield an improved general turbulence model applicable to all stirred tank flows, especially when this tuning is done using sensitivity analysis, without consideration of the physical meaning of the model constants. These physical arguments notwithstanding, several authors have devoted substantial effort to investigating the effect of the turbulence model constants on the simulated flow field [22,20]. They find that while the a constants have a negligible effect on the results, the three C constants increase or decrease the amount of circulation as they are adjusted, changing the balance between the mean and

Boundary Conditions Required for the CFD Simulation

301

turbulent forms of energy. Fokema et ai, in a sensitivity analysis of the impeller boundary conditions, showed that the circulation also is very sensitive to the accuracy of the velocity boundary condition at the impeller [19]. While the turbulence model constants can be tuned to fit a better solution for the mean flow field, the results of this approach offer little hope for general applicability. Improved models of the impeller offer a more rational way to proceed. As better and more general turbulence models are developed, in parallel with sliding mesh techniques and faster computers, it may become possible to simulate the generation and decay of the trailing vortices up to their entry into the locally isotropic range. This would allow more direct simulation of drop break-up and coalescence, mixing, and reaction kinetics, as well as a more satisfying approach to both turbulence modeling and impeller modeling. At the time of writing, however, the standard k-e model appears to be the most stable, readily available way to deal with the highly turbulent flow near the impeller. While it tends to underpredict the dissipation in the bulk of the tank, it gives good agreement close to the impeller where the values are most critical [19]. Additional experimental work is needed in the bulk of the tank, first to determine the regime of flow (transitional or turbulent), then to establish reliable data for validation of CFD simulations. Swirl Number Periodically, it is suggested that the flow in a stirred tank is highly rotational, requiring addition of the Coriolis force and modifications to the turbulence model. The number used to evaluate the degree of swirl in a flow field is the swirl number: f V, Ver'dr f VjrRdr Jo

^

If the swirl number is 3 or greater, the flow is considered highly swirling. Physically, this means that the tangential, or angular velocity, is significantly larger than the axial velocity over much of the region. If the integration is applied to the discharge flow at the edge of the PBT blades from r = 0 to D/2 (data taken from Kresta and Wood ([25], Figures 5 and 6), the resulting swirl number is approximately 1.0. Since this is the most highly rotational part of the flow field, it is clear that the modifications for highly swirling flow are not needed in a baffled tank. DeHnitioii of the Geometry and Boundary Conditions at the Edges of the Domain Geometry: For a stirred tank, the geometry is cylindrical, with a small aspect ratio (the height of fluid in the tank [H], is one to three times the tank diameter [T]). Although many industrial vessels have a dished bottom (especially if solids suspension is involved), simulations to date have used the simpler flat bottom geometry. The impeller, of diameter D = T/4 to T/2, is placed at the desired off-bottom clearance (C). Around the tank walls, two to four rectangular baffles are evenly

302


spaced. They have a width of T/10 to T/20 and may be placed at some distance from the tank wall. Industrial applications may have additional internals, such as feed pipes, heating coils, or baffle geometries specific to a particular application. Specification and gridding of these geometric variations poses an additional challenge to the user, and the exact implementation will depend on the code in use. Three Dimensionality: The flow field in a baffled tank is highly three-dimensional. Baffles both reduce the swirl and produce top to bottom circulation in the tank. Without baffles, any rotating impeller will produce a two-dimensional flow which is mainly rotational. The baffles induce drag and force the swirling fluid up the wall. This transforms a two-dimensional, swirling flow into a three-dimensional flow with low swirl, particularly in the outer third of the tank. A survey of the simulation literature shows that 2D simulations of baffled tanks were virtually abandoned as soon as computational speed would support a 3D approach. No adequate way to model the effect of the baffles in two dimensions was ever found. Symmetries: In spite of the fully three-dimensional character of the flow field, some symmetries can be defined. Simulation of on^ baffle sector (90° for four baffles, 180° for two baffles) with the baffle centered in the domain and cyclic (or periodic) pressure boundaries at the edges of the domain reduces the computational task dramatically, with no reduction in generality for the time averaged case (full tank simulations would be required to capture, for instance, a precessing vortex). This cyclic division of the domain implies an axisymmetry condition on the axis of the tank. For axial flow impellers this may cause a problem with some codes. The axisymmetry condition appears to suppress the entrainment of fluid under the hub of the impeller into the bulk circulation, holding the velocity close to the zero starting value which it has at the impeller. Examination of results will show a steep velocity gradient at the center of the tank (similar to that seen close to a wall) if this problem is present. A vertical symmetry condition at the centerline of the impeller was used in initial simulations of the Rushton turbine to reduce the computational domain to 1/8 of the tank volume (note that the impeller was placed at an off-bottom clearance of T/2, H = T) [20,6]. This simplification is not possible for axial flow impellers and unnecessarily restricts the generality of the simulation. All experimental evidence shows that the time-averaged impeller boundary condition is axisymmetric, in spite of the 3D nature of the flow field. This simplifies the requirements for impeller modeling, requiring specification of values for only one traverse at the impeller discharge. These considerations leave the user with a 3D, cylindrical geometry containing one baffle (centered in the domain), and an axisymmetric impeller. Boundary Conditions: With the geometry specified, the boundary conditions can be defined. On the tank walls, on the baffle, and along the top and bottom surfaces of the tank the no slip boundary condition is specified and wall functions are used to deal with the steep velocity gradients in these regions. If the shape of the free surface is important, the domain must extend well beyond the surface of the liquid, and the top surface must be specified as a free slip boundary. The details of this issue are not addressed here since the vortexing of the free surface may entrain gas into the flow field, resulting in a two phase flow with important rotational characteristics. The single remaining boundary condition is the impeller model, which merits a separate section.


303

Impeller Since it is the impeller which drives the flow, an accurate representation of the impeller discharge conditions (the three components of the mean velocity, and the turbulence quantities k and £) is a prerequisite for successful modeling and simulation efforts. Much of the early work on modeling the flow in a stirred tank focused on developing a general mathematical model of the impeller, based on the assumption that the discharge flow generated by the impeller was independent of the geometry of the tank (i.e. the impeller operated in an essentially infinite body of fluid). This hypothesis led to the development of many complex models of the discharge flow for the Rushton turbine, culminating in the swirling radial jet model by Kresta and Wood [6], which required only the rotational speed of the impeller and the angle of discharge flow as inputs. This final model was carefully validated against available experimental results, and successfully used in 3D simulations of the flow field generated by D = T/2 and D = T/3 Rushton turbines at an off-bottom clearance of T/2 in a fully baffled tank. More recent experimental work has shown that the flow field generated by a PBT, including the impeller discharge condition, undergoes a distinct transition at a C/D ratio of 0.6 [25] Instead of the classic circulation pattern of a single circulation loop which extends over the whole tank for all geometries, evidence from flow visualization and LDA experiments shows a primary circulation loop which reaches the bottom of the tank only for a low clearance. When the clearance is increased, a secondary circulation loop appears in the bottom of the tank. The change in circulation pattern affects the angle of impeller discharge for the T/2 impeller, deflecting the discharge angle toward the horizontal. For the T/3 impeller, the secondary circulation loop is not as strong and does not affect the angle of the impeller discharge. The axial component of mean velocity for the four characteristic discharge conditions found by Kresta and Wood [25] is shown in Figure 1. There are two things to note from this figure: First, the transition between profiles occurs at the same C/D ratio for both impellers; second, the low clearance discharge conditions (where the effects of the secondary circulation loop are removed) roughly coincide over the extent of the impeller blades. Both observations indicate that some generalization of impeller boundary conditions still may be possible. This work disproves the hypothesis that all impellers generate discharge flows which are the same in any container. The influence of the tank walls is fed back to the impeller and into the impeller discharge stream. Changes in circulation patterns have a substantial impact on the discharge stream, even very close to the edge of the impeller blades. Using the boundary conditions defined by Kresta and Wood [25,27], Fokema et al. [19] performed simulations with varying boundary conditions at two off bottom clearances, C/D = 0.93 and C/D = 0.5. The impeller was modeled using a thin disk (thickness 0.0005 m and radius 0.0375 m) with inlets on the top and bottom surfaces to simulate the fluid flowing through the impeller. The velocity and turbulence quantities as measured at the impeller discharge [27,25] were specified on both the upper and lower surface of the impeller. The disk was made as thin as possible to minimize the error introduced by using only the discharge boundary condition on both surfaces. At the tip of the impeller, a free-slip wall was defined. A mass flow

304


0.2 0.1

h

T/2 T/2 T/3 T/3

HDC, C/D= 1 to 0.667 LDC, C/D=0.58 to 0.33 HDC, C/D=1.5 to 0.75 LDC 2, C/D=0.59 to 0.5

V TTND

2r D Figure 1. Experimentally determined impeller discharge conditions for D = T/2 and D = T/3 pitched blade turbines. The transition between the high clearance discharge condition (HDC) and the low clearance discharge condition (LDC) occurs at the same C/D ratio for both impellers.

boundary was needed to eliminate a small numerical discrepancy in the overall mass balance between the upper and lower edges of the impeller. Since the experimentally measured discharge conditions differ significantly in several components (i.e. V^, V^, k, e), several combinations of boundary conditions and of bottom clearances were used in the simulations. First, the boundary conditions were applied to tank geometries equivalent to the experimental conditions (low clearance condition at C/D = 0.5; high clearance condition at C/D = 0.93). Then the low clearance boundary condition was used in a simulation with C/D = 0.93 to determine whether the high clearance flow pattern was primarily attributable to the impeller boundary condition or could be generated by changing the tank geometry alone. The test was repeated by using the high clearance boundary condition in a simulation with C/D = 0.5. The results of these simulations show that the use of incorrect boundary conditions for the impeller results in the prediction of a highly distorted flow field, as shown in Figure 2. In both cases the flow patterns have been distorted. The secondary circulation loop in the high clearance geometry has been greatly reduced in size and intensity; and a small secondary circulation loop appears in the low clearance case, even though none exists in reality.


305

The components of the boundary condition that have the largest effect on the overall flow pattern were isolated by altering the boundary condition one component at a time and noting the effect of these actions. Because the high clearance case and low clearance case boundary conditions produce such different flow fields in both geometries, the five components were exchanged one at a time to assess the sensitivity of the flow field to each variable. The only variables that significantly affected the circulation pattern were the axial and radial velocities. For the simulation to be completely successful, the turbulence field must be as accurately predicted as the velocity field. Where the simulations by Fokema et al. [19] used a tank geometry equivalent to the experimental conditions, the simulated and experimental values of the turbulent kinetic energy (k) showed reasonable agreement. The predicted energy dissipation rate (e) profiles immediately below the impeller were of the correct magnitude, but further away from the impeller, the predicted values of e decayed to only a fraction of the experimental values. This underprediction of the dissipation rate was also evident in mean values above the impeller. Figure 3 shows the impact of using incorrect turbulence boundary conditions on the turbulence field. Although the mean velocity field is not significantly affected by incorrect specification of the turbulence quantities at the impeller, the distribution of turbulent kinetic energy dissipation rates shows substantial variations. Figure 3a shows profiles of the turbulent kinetic energy dissipation rate below the impeller using the high clearance case boundary conditions in the high clearance case geometry. If the low clearance boundary conditions are used in the high clearance geometry (Figure 3b), the peak value of e is much greater than it should be and a decay of values does not occur as one progresses towards the bottom of the tank. To obtain a steady decay of £ below the impeller, only the correct turbulence boundary conditions were required. Figure 3c shows the decay of 8 when the axial velocity from the low clearance case is specified in the high clearance geometry. The profiles are nearly identical to the correct solution. The same result occurred when using the incorrect radial and tangential velocity boundary conditions. Figure 3d, however, shows the effect of setting the turbulent kinetic energy and dissipation rate to zero or not specifying them at all. The turbulent kinetic energy dissipation rate becomes highly unpredictable. If the turbulence components of the impeller boundary condition are not properly specified the distribution of turbulence energy in the tank will be incorrect. In this section, two methods of modeling the impeller have been discussed: general mathematical models and use of experimental data. The newer sliding mesh techniques were reviewed earlier in this chapter. A final alternative which has been suggested is to model the impeller as a source of momentum. It is not clear, however, how this method should be applied, and successful simulations have never, to the knowledge of the author, been published. It has been demonstrated that accurate representation of the impeller is central to obtaining accurate CFD simulations. Four alternatives are available for impeller modeling. The sliding mesh technique requires no measurements or assumptions but consumes large amounts of CPU time. Impeller modeling based on generalized mathematical models assumes that there is no interaction between the impeller and (text continued on page 304)

306


t /

/ / f /

f / ^ " -^ ^ ^^\ \ \ \ \ \ M I I " ^^\ \ \ \ \ U I I I f r

t

-:ih =

-

^j3^j^

b

for u < u^ a

(13.2)

b

for u =^ u^ a

(13.3)

b

In horizontal gas-liquid flows, the gas velocity is of higher order of magnitude and, therefore, the interface is considered as free surface with respect to the liquid and as a stationary surface with respect to the fast gas phase (Agrawal et al. [18]). In general two-fluid systems, however, the velocities of the two phases may be of comparable levels and depending on the fluids properties, system inclination and operational conditions, one phase velocity exceeds the other. Therefore, an adjustable definition of the equivalent hydraulic diameters D^, D^^ is to be adopted as part of the solution procedure [20]. The constants c^, c^, n^, n^ in Equations 12 are chosen according to the flow regime in each phase, (c = 16, n = 1 for laminar flow and c = 0.046, n = 0.2 for turbulent flow conditions.) Clearly, the two phases in stratified flow may result in laminar-laminar (L-L), laminar-turbulent (L-T), turbulent-laminar (T-L), or turbulent-turbulent (T-T) regimes. The interfacial shear is evaluated based on the relative velocity between the phases: ^^^^ P(U,-UJK-U,|

^J^^^

with P = Pa and /. = /^ for u^ > u^, p= p^ and /. = / ,

for u^, > u^

(14.2)

Equation 14.2 implies that the interfacial shear friction factor is evaluated as equal to that obtained between the faster phase and the pipe wall. For u^ ^ u^^, T. is identically zero and, thus, the interface is considered as free surface with respect to both phases, consistent with 13.3. When the Blasius models for the shear stresses. Equations 12-14 are introduced into Equation 11, solutions for H, U^, U^^ at steady conditions and the pressure drop can be obtained for a variety of two fluids, horizontal, or inclined two-phase stratified flows (Brauner and Moalem Maron [20]).

326


The legitimacy of employing Blasius type models for the shear stresses in stratified flows was checked in several studies. Kowalski made direct measurements of the Reynolds shear stress in the gas for horizontal stratified flow in pipes and found that the gas-wall friction factors are well approximated by the Blasius equation provided that the hydraulic diameter is utilized [64]. For the liquid phase, Andritsos and Hanratty [28] found that the use of the Blasius equation to calculate x^ introduces some error. However, improvements achieved by using a more complicated model for x^^, which is based on velocity profile and eddy viscosity concepts, were found to be of mild effect on the integral flow characteristics. Rovinsky et al. compared the solutions for laminar-laminar stratified flows obtained via the exact model with those predicted by the two-fluid model. Equations 11-14 [12] It was found that the error in H/D is bounded by 2.5%, and a typical error in the system pressure drop is about 10%. The comparison shows that for laminar-laminar flows, in the range of 0.1 < (|X^Q^)/(|Li^Q^) < 10 and 10"^ < \ij\i^ < 10^ the accuracy of the two-fluid model is improved when the interface is modeled as free with respect to both phases (D^, D^^ defined by Equation 13.3). Otherwise, the interface is to be considered as a "wall" for the less viscous (faster) phase and as a "free" surface with respect to the viscous (slower) phase (either Equation 13.1 or 13.2). The major controversy arises with respect to the appropriate modelling of the interfacial friction factor, /.. The problem arises because the presence of waves augments the rate of momentum transfer across the interface. The many attempts made in improving the modelling of the interfacial shear have been focused on improving the model for f to account for the role of the waves in increasing the interfacial drag (Cheremisinoff and Davis [14], Kowalski [64], Sinai [65,66], Andritsos and Hanratty [28], Bontozoglou and Hanratty [67], Kang and Kim [32], Strand [68]). There is evidence that the presence of interfacial waves also affects the evolution of secondary flows in the liquid and gas phases. Simultaneous measurements of the liquid film thickness, wall shear stress, and gas flow turbulence in the stratified wavy regime performed by Hagiwara et al. [69] suggest the existence of a separation bubble formed in the gas phase in front of the large wave. They also observed an increase of the liquid-wall shear stress caused by the passing of large waves. Suzanne observed strong transversal secondary velocities (with a magnitude of about 5% of the axial velocity in the liquid phase and about 10% of the axial velocity in the gas phase) which appear as a pair of rolls directed at the interface region from the wall towards the middle of the duct [70]. The source for the secondary motions is attributed to circumferential variation of the wall shear stresses (Jayanti et al. [71]) and to nonlinear interactions between the mean flow field and the wave-induced Reynolds stresses (Nordsveen and Bertelsen [72]). Observations and analysis indicate that significant secondary flows appear in the regular wavy stratified flow regime, but are of no effect in the smooth stratified regime. Practically, for the stratified smooth regime, only the axial component of the velocity is non-zero. Measurements of x. carried out by Andritsos and Hanratty [28] in 2.54cm and 9.53cm horizontal pipes (for air-liquid, liquid viscosities of 1-70 centipoise) and by Andreussi and Persen [73] in slightly inclined flows showed that the values of fjf^ increase dramatically when K-H waves are present. However, for smooth interface they found that f.lf^ = 1, thus substantiating the assumption that for smooth


327

stratified flows the interfacial friction factor can be evaluated as equal to that obtained between the faster phase and the tube wall. The above evidences suggest that in the framework of two-fluid models, Blasius type relations for the interfacial and wall shear stresses with an appropriate definition of the hydraulic diameters as given in Equations 13 provide a reasonably simple tool for predicting the integral characteristics of smooth-stratifiQd flow, namely: phases average velocities U^, U^^, layer thickness H (insitu holdup), and the system pressure drop for a variety of two-fluid systems. Obviously, in the presence of interfacial waviness the interfacial friction factor /. ought to be augmented to account for the interfacial shear enhancement. STABILITY ANALYSIS WITH QUASI-STEADY MODELLING OF SHEAR STRESSES The transient formulation of the two-fluid equations requires closure laws for the local and instantaneous shear stresses. The conventional way of modelling the wall and interfacial shear stresses is by assuming quasi-steady relations, whereby x^, x^ and X. are modelled in terms of the local phases insitu holdup and velocities: X = X (h, u , u j ; a

a ^

'

a'

b^'

x^ = X. (h, u , u j ; b

b ^

'

a'

b^'

x. = x (h, u , u.) i

i ^

'

a'

b^

(15) ^

^

The same relations used for modelling the undisturbed steady stratified configuration are adopted for the transient formulation by replacing the steady flow variables with the local instantaneous values of the insitu holdup and phases velocities. This approach has been taken by many investigators [24,37,40-47,61,74-76]. In searching for the necessary conditions under which the smooth-stratified flow configuration is stable, linear stability analysis is carried out on the transient twofluid continuity and momentum Equations 1, 2, 7. The equations are perturbed around the smooth fully developed^ stratified flow pattern. Following the route of temporal stability analysis h ' = he^^'^^'"^^; u^ = u^ e^^""^""*^; u^ = u^^ gUkx-cot) ^j.^ substituted for the perturbed liquid level and phases velocities. This yields a dispersion equation which relates the real wave number, k to the complex wave velocity C = co/k (for details, see Brauner and Moalem Maron [43]): aC^ - 2(bj + ib^) C + dj + id^ = 0; C = co/k

(16.1)

C^, = - ( b , + ib,) ± i[(bj + i b / - a(d^ + id^)] a a

(16.2)

K

A: =

fdA K dh

K

328

Advances in Engineering Fluid Meclianics A:

YbU,+p3 4^Y.U

PaUf

2 A,

b, =

Aj^_3

a.

Â, au,

au,,

aAF,, A, a u .

A;

2k

7a - PbU

2

^\J_

I^

.Aâu,

au,,

A', aAF^ A, 3U,

d, = p. 4^Y.U^ + p, 4 ^ Y,U^ - [(p, - p j g c o s p + ak^]

+ A:

+ PaU

d, =

^^ "Ubau,

A, au,

û^_a__i\_a ,A, au, A, au,

A^^j aAF.

û

au.

A; an

i_A^ Ya A; an

^^ au.

(17)

an

In Equation 17, aAF^y3(H, U^, U^) are dAfJdih, u^, u^), at steady conditions H, U^, U^. It includes the derivatives of the quasi-steady models for the wall and interfacial shear stresses with respect to flow variables. Based on the dispersion equation, the neutral stability conditions are derived by requiring a zero imaginary part for wave velocity C whereby the neutrally stable wave number, k^, and the corresponding wave velocity, C^^, are obtained. The resulting neutral stability criteria reads: J + J = 1+J

(18)

U Ap DgcosP (1 - e ) '

1 - ^

u„

u

- ^

' au.

U.

-1

C

+(Ya-l)

(l-e) Ub

U. J

U,a

ay,

au,

y

+ (l-e)

ay. a(l-e)

(18.1)


J ^ Pb UL £^ ^ Ap DgcosP e^ C

- ^ - 1 |

u, au,

329

+(Y.-1) 1- 2 ^

(l-e)

u.

u. l ^ + e

(18.2)

Ok

Apg cos P u , aAF,. aAF, ( l - e ) 3U, ae au. j_aAF, 1 aAF, ( l - e ) 3U,

(18.3)

U, aAF,

c.„ =

(19)

where 8 represents the lower phase holdup 8 = A^/A, 8' = (38 / 3h)_ . (superscript ~ denotes normalized values, length scales by D, areas by D^). The operational conditions are represented by the phases superficial phases velocities, U^^, U^j(^=4Q^JnD^). Equation 18 is essentially the K-H stability criterion. It is arranged in a form which clearly shows the relative contributions of the stabilizing and destabilizing effects (normalized with respect to gravity). The gravity and surface tension stabilizing terms constitute the rhs of Equation 18. The Ihs includes the destabilizing terms due to the two-phase inertia (J^, J^^). These vanish for particular combinations of wave celerity and (constant) shape factor related by C^JU = Y ± (Y^ - Y)^^^- For instance, the J^^ term in Equation 18.2 vanishes for the case of plug flow and stationary liquid phase (y^ = 1, Cj^^= U^^ as conventionally assumed in gas-liquid stability analyses) or for highly sheared thin liquid films with a typical linear velocity profile, y^ = 4/3 and C^^^ = 2U^. Note that the derivatives of the shape factors, dy^Jd(E, U^, Uj^), are usually ignored. Inspection of the K-H stability condition indicates that the structure of Equation 18 is invariant with the specific modelling of the wall and interfacial shear stresses and evolves essentially from the continuity equations and the left hand side of the momentum equations. On the other hand. Equation 19 for Cj^^ is directly related to the quasi-steady models adopted for the various shear stresses terms (the rhs of the two-fluid momentum Equation 8). In this sense, the form of 18 is general and is affected by the specific modelling of shear stresses only indirectly through the Cj^^ value. Thus, given different correlations for the shear stresses, the general form of 19 provides the corresponding values for C^^^. It is of particular interest to note that the wave velocity at neutral stability is in fact identical to the definition of kinematic wave velocity, C^ (Wallis, [74]);

(20.1) U,n,4Fb.

330


U = U + U = eU, + (1 - e)U

(20.2)

The identity of Equation 20 and 19 have been shown in Brauner and Moalem Maron [45]. Thus, the neutrally stable wave actually represents a continuity wave, and its characteristic velocity can be determined either by stability analysis or via the derivative of the liquid flux with respect to its insitu holdup (concentration). Clearly, both the neutrally stable and continuity waves are based on the steady momentum equation. The general structure of stability Equation 18 remains unchanged when different quasi-steady models are applied for the various shear stresses terms. Moreover, even when the viscous effects are completely ignored, resorting to an inviscid K-H stability type of analysis, the structure of the resulting stability condition, Equation 18, is still maintained while Equation 19 for C^^ attains different expression. For instance, the long wave K-H stability analysis on two inviscid layers (rectangular channel) yields: _p,U,H + p , U , ( l - H ) Rn

/i

TTX

T~T

(21.1)

p,(l-H)-Hp,H which for (p^H)/pb(l - H) — K, "^ ' e

Dg COS p

(22)

Pa

As the remaining gas-phase destabilizing contribution has been found insufficient to balance the gravity term along the experimental stratified/slug transition boundary, various empirical correction factors, K,, have been introduced to match Equation 22 with the data. For instance, Wallis and Dobson [35], Taitel and Dukler [19], Kordyban [34], Mishima and Ishii [36] proposed to enhance the gas term by introducing a correction factor, such as K, = 0.5, 1 - H, 0.49, and 0.74, respectively. The disagreement of the gas-destabilizing term with experimental transitional data


331

is, as a matter of fact, expected in view of the unjustified Cj^^ = U^^ assumption (which led to Equation 22). A more careful inspection of Equations 21 indicates that even for gas-liquid flows, where p^p^^ lOO O l-H D 10-1

Experimental data Mandhane et al [77] Shoham [78] Andritsos & Hanratty [39]| b) D=9.53cm H=0.5/^

H=0.5

HJ

< 10-2

^. 10-3 D 10-^ 10-1 ^''i « I'l

10^

10^

10-2 10-1

10^

IQi

10^

SUPERFICIAL AIR VELOCITY, Uas[m/s] Figure 2. Typical neutral stability boundary obtained with quasi-steady modelling of the interfacial shear. O A D stratifled-smooth/slug, • • • stratified-smooth/wavy thin water layers (below H = 0.5 line), even the lower bound for instability as represented by the ZNS line overpredicts the observed transitional conditions. This implies an additional destabilizing effect not accounted for by the K-H mechanism. As shown in Figure 2 the gap between the K-H stability boundary and the data of SS/SW transition increases with an increasing tube diameter. This gap cannot be bridged by choosing different quasi-steady relations for the shear stresses as closure laws for the two-fluid model; implementing other models suggested for the interfacial friction factor results in a minor variability of the stability boundary, in particular for horizontal flows (see also Crowley et al. [83]). Moreover, enhancing the interfacial friction factor, /. > /^, may stabilize the interface and extend the region of stable smooth stratified flow, hence further increasing the gap between the data and the predicted stability boundary. DYNAMIC MODEL FOR THE INTERFACIAL SHEAR The stability conditions in Equations 18-19 correspond to a quasi-steady modelling of the various shear stresses; hence, the effect of axial convection of the wave-


333

induced turbulence properties adjacent to the wavy-free interface is not accounted for. Theoretical as well as experimental studies of turbulent flow over solid wavy boundaries indicate that it is impossible to relate the local shear stresses over a wavy boundary to the local bulk flow characteristics (Benjamin [84], Thorsness et al [85], Zilker et al [86,87], Buckles and Hanratty [88], Abrams and Hanratty [89]). Although the mean velocities are virtually symmetric at corresponding converging and diverging locations, the wavy boundary gives rise to a phase shift of the fluid velocity gradients at the proximity of the wavy boundary. Consequently, the shear stress at the interface is not symmetrically disposed about the wavy boundary. It is seen to be both augmented and shifted upstream at the wind-side, whereas at the lee-side of the wave, the local shear stresses are reduced compared to the expected value based on the local flow cross section. This indicates that turbulent shear flow past a wavy boundary is basically incapable of immediate adjustment to the surface geometry since the local turbulence structure, as induced along the wavy boundary, memorizes its upstream properties. Clearly, the interaction between the surface waviness and the turbulent flow field is intensified on steeper slopes, or as the ratio of amplitude/wavelength ratio, ak, increases [87-89]. The stability of the interface in stratified configuration is closely related to the problem of wind-generated waves. Jeffreys observed that the critical wind velocity required to initiate waves as predicted by the K-H stability criteria significantly overpredicts the wind velocity at which waves are generated over a water layer surface [90]. Jeffreys hypothesized a "sheltering theory" whereby air flow over a deformed liquid surface is unable to follow the surface geometry and separates behind the crest. Therefore, the leeside region, which is sheltered from the main current, experiences a lower pressure force than the slope facing the wind. This phenomenon gives rise to a periodic component of wind stress in phase with the wave slope, which was described by a "sheltering coefficient." This stress component destabilizes the interface; as for wind velocity, which exceeds the wave velocity, the stress component affects a transfer of energy from the air stream to the wave. The value of the "sheltering coefficient" was empirically tuned to predict the critical wind velocity for generating water waves. Later works by Miles [91-94]and Benjamin [84] showed that the evolution of wind stress component in phase with the wave slope arises from interaction of the surface perturbations and the mean turbulent airflow characterized by boundarylayer type velocity profiles. Their models are considered quasi-laminar, as the perturbations in turbulent Reynolds stresses induced by the surface perturbations were ignored. Experimental studies of shear stress variations along a solid wavy boundary over which turbulent air is flowing [85,89,95-97] revealed that the quasi-laminar assumption is valid only for large dimensionless wave numbers (k+ = vjd U^, U^ = (T/PQ)° 0, when the wave-induced variations are confined to a very thin layer near the wall where turbulence has a negligible effect. These experiments showed that the phase shift of the shear stress fluctuation with respect to the wave elevation and the amplitude are dependent on the dimensionless wave number, (Figure 3) indicating that interaction between wave-induced perturbations and turbulent fluctuations in the airflow give rise to systematic surface stresses, which are not negligible compared with those predicted by the quasi-laminar assumption.

334


Attempts to explore this complicated interaction and to model the response of the eddy viscosity and turbulent shear stresses to the time variation of pressure gradients in turbulent air flow over a solid wavy surface have been made by Thorsness et al. [85] and Abrams and Hanratty [89]. Large variations of the amplitude and phase angle of the surface shear stress with the dimensionless wave number were predicted (Figure 3). The analysis shows that the surface shear stress fluctuation is shifted upstream with respect to the wave elevation and the phase shift varies in the range of 0-80° in comparison to the constant phase shift predicted by Benjamin [84]. Recently, Hanratty presented a comprehensive review of the attempts to account for the interfacial waviness in modelling the interfacial shear stress for the stability analysis of gas-liquid two-phase flows [53]. Basically, the approach taken was to implement the models obtained for the surface stresses in air flow over a solid wavy boundary as a boundary condition for the momentum equation of the liquid layer over its it mobile wavy interface. Craik [98] adopted the interfacial stresses components which evolve from the quasi-laminar model by Benjamin [84]. Jurman and McCready [99], Jurman et al. [100], and Asali and Hanratty [101] used correlated experimental values of shear stress components (phase and amplitude) based on turbulent models which consider relaxation effects in the Van Driest mixing length. Since the characteristics of the predicted surface stresses are dependent on the wave number, Asali and Hanratty picked the phase and amplitude values which correspond to the wave lengths of the capillary ripples observed in their experiments of thin liquid layers sheared by high gas velocities [101]. It was shown that the growth of these ripples is controlled by the interfacial shear stress component in phase with the wave slope. Clearly, the disintegration of the two-phase stability problem, to treat separately the liquid film when subjugated to prescribed interfacial stresses, greatly simplifies the analysis. However, as the stability of the interface turns to be largely governed by the nature of the interaction between the phases at the free interface, the availability of fairly good estimates for these stresses is crucial. Estimates obtained from measurements of turbulent flow over a solid (rigid) boundary may be inappropriate for predicting the stresses which develop over a wavy mobile interface. The essential difference is the ability of a mobile interface to accept energy from the shear flow, which feeds the growing interfacial waves. The wavelengths and amplitude of these waves dynamically interact with the turbulent gas flow to yield the stresses at the phases interface. In view of this discussion, it is obvious that the closure law for the interfacial shear stress ought to reflect the micro-structure hydrodynamic phenomena at the vicinity of the mobile wavy boundary between the phases. Consequently, the quasisteady interfacial shear stress model for x. is to be replaced by a model which accounts for the dynamic interaction between the phases. A new form for interfacial shear, which incorporates an explicit functional dependence on the interfacial slope due to interfacial waviness, recently has been proposed by Brauner and Moalem Maron [102,103]:

-c. = P ( u ,

1 /- • / X ^ 3h - u , f -/.sign(u, - u , ) + C , —

2

ax

Ch = C , ( u , , u , , h ) > 0

(23)

Boundary Conditions Required for the CFD Simulation 100 1

CD

1 1— O Abrams & Hanratty [89] a Kendall [96] ^ JL / 80 L O Hsu & Kennedy [95] 1 A Sigal [97] J ^ ^ Quasi-laminar model 60 /

1

335

\

« . . . ?*'"'*{?" J^^"^ên " "

J H

^

00 40 h -

1

^1 Frozen turbulence model (/,« 0)

cd OH

20 rH

^^|7"*V^ ^ ^ ^ ^ \ .

o d

s.

A

vs.^

r" t i l l -20 1 0.00001

mil

^ Equilibrium turbulence model it, «-33.ilr|, » 0 1 M i l l ul 1 II iiml 1 1 1 mnl

0.0001

A

L_i 1II ml

0.1

0.01

0.001

1.0

k+=kVGAJS

100

o OKA

'^

1.0 k -

>_

o Abrams & Hanratty [89] n Kendall [96] OHsu& Kennedy [95] A Sigal [97]

(/i

0.1

3

I

0.01

Hquilibrium "turbulence model *, « - 3 3 . i t ^ « 0

Fiozen turbulence model ( / » 0 )

cd

Relaxation theory A , - - 3 3 . * ^ - 1650

0.001

-a o •l-H

0.0001

Quasi-laminar model

s 0.00001 I 0.00001

Miiil—I 1111 III—I

0.0001

0.001

I iiiiiil

0.01

I I nniil

0.1

I i miiJ

1.0

k+=kVG/UG Figure 3. Wave-induced variation of the surface shear stress (adopted from Abrams and Hanratty [89])

336


Equation 23 indicates that for mild interfacial slopes the interfacial shear reduces to the quasi-steady value in Equation 14, x^. However, as the interface becomes perturbed with (growing) unstable modes, the second term, Cj^8h/3x, dynamically interferes in determining the interface stability. Equation 23 implies a shear stress augmentation at the windward side of the wave and a relaxation at its leeward side. With reference to Figure 4a, for the case of u^ > u^, x^ > 0. In this case, dh/dx > 0 at the wind-side, and thus. Equation 23 yields augmentation of the interfacial shear stress, x. > x9 as expected at the windside. Clearly, shear stress relaxation occurs at the lee-side where dh/dx < 0. On the other hand, in the case of u^ < u^^. Figure 4b, corresponds to a negative quasi-steady interfacial shear, T|^ < 0 and thus the (negative) shear stress is augmented at dh/dx < 0, which is now the wind-side, and is relaxed at the lee-side, where dh/dx > 0. Hence, a positive dynamic coefficient in Equation 23, C^ > 0, consistently yields the expected augmentation of the interfacial shear at the wind-side, and its relaxation at the lee-side, for either u > u^ or u < a . a

b

a

b

The functional dependence of C^ on the local flow variables (u^, u^y h) is derived from the comparison of instability characteristics, predicted when the dynamic model for X. is used with available observations. STABILITY ANALYSIS WITH DYNAMIC MODEL FOR TJ Temporal stability analysis is carried out on the linearized set of Equations 1, 2 and 7, 8 with Equation 23 replacing the quasi-steady model for z.. The resulting dispersion equation obtained for the complex wave celerity, C = co/k, as a function of the real wave number, k, is of identical form to Equation 16, except that d, includes now an additional term, which evolves from the dynamic component of the interfacial shear: d, = P. ^

Y,UJ + p, ^

-C,p(u,-uJ^S;

A, au.

A: 3H

YÛf - [ ( p , - p J g cos p + ak'

1 1 — + —

Yb + P.Uf

+ A:

p.u^ A. au.

U3.J__u^_3 A, au. A, au,

LA) A : an

(24)

Based on the modified dispersion equation, the neutral stability condition is: (25.1)

J + J, + J = 1 + J a

b

h

a

with J^, J^, J^ remaining unchanged as in Equation 18 while the additional dynamic term I is: n

J^=^^T-^^' V Ap Dgcosp

J\,'

7C8(l-e)

S.=S,/D;

C, =C,(H,U„UJ (25.2)


wind-side

337

lee-side

Case (a).Faster Upper Phase,

UQXJJ)

UrUa-Ub 0 , TfUa Figure 4, Schematic representation of the "dynamic term" in the modelling of the interfacial shear.

338


Note also that the neutral wave celerity is independent of the dynamic term and is given by Equation 19. The AF^^ term in Equation 19 is the rhs of the combined momentum equation in steady conditions. Hence, it includes only the quasi-steady term of the model for the interfacial shear stress: (A

AF,^ = - I ^ + x%

1

1 A

1 + — A.

+ 'C,-^ + ( p , - p j g s i n p

(26)

However, the wave celerity of the amplified modes is affected by C^^ ^ 0. The Ihs of Equation 25.1 includes again the destabilizing terms of the two phases inertia (J^, J^^) and that due to the dynamic term J^^. The dynamic effect as embodied in the additional term, J^^, represents another source of instability in the two-fluid system. It does not degenerate even in the limit of long wave analysis (where 3h/3x —> 0) as the gravity stabilizing force is also proportional to the wave slope, 3h/ 3x. In view of Figure 4, the destabilizing nature of J^ is independent of the relative velocity between the two phases and the direction of the interfacial shear. Note that while for C^^ = 0, the structure of the stability condition Equation 18 is invariant with the specific modelling of the wall and interfacial shear stresses, with the inclusion of the dynamic interaction in the modelling of T., the stability condition, Equation 25 is directly affected by the interfacial shear stress through the J^^ term. CORRELATION FOR THE DYNAMIC COEFFICIENT C^ The destabilizing term J^^ in Equation 25 requires knowledge of the memory coefficient, C^^, as defined in Equation 23. The coefficient, C^^, is to be extracted from experimental findings which reflect the dynamic interfacial interactions. Observations of stratified-smooth/stratified-wavy transitional boundaries from various laboratories reported in the literature bear a potential of a data-base for correlating C^. These are summarized in Table 1. Stemming from the premise that the modified stability boundary as presented by Equation 25 with k„ —> 0 predicts the limits of smooth-stratified zone, Equation 25 is used to extract the required J^^—hence, the corresponding dynamic coefficient C^^, which yields agreement with the data (C^^ is calculated by Equation 19). The modified "zero neutral stability" line, which accounts for the "dynamic term," Cj^ i^ 0, is denoted by ZNS^. Note that in solving Equation 25 for J^ the shape factors Y^, Yb ^^^ required. These depend on the velocity distribution and, therefore, demonstrate a change mainly with laminar/turbulent flow regime transitions. As long as the flow regime is maintained, the shape factors variation 3Y/3H, 3Y/9U^, 3Y/9UJ^ may be practically neglected and, thus, constant Y^* Yb "^^Y ^^ "^^^ ^^^ evaluating the stability characteristics. For instance, for thin sheared laminar layer, Yb ~ 4/3 (linear velocity profile), while in the turbulent regime a thin sheared layer yields \ — 1.6. Obviously, for plug flows Y^* Yb ^ ^» ^^^ these values can reasonably represent the flow of relatively thick turbulent layers. Thus, experimental data of superficial phases velocities along the SS/SW transitional boundary can be used to extract the dynamic coefficient C^^ for a variety of two-fluid systems, tube diameters, and operational conditions.


339

Table 1 Data -base of Stratif ied-Smooth/Wavy Transition Reference

Fluids

Tube Diameter D [mm]

Inclination

Mandhane et al. [77]

air/water

25, 51

horizontal

Simpson et al [79]

air/water

127, 216

horizontal

Shoham [78] Barnea et al. [104]

air/water

25, 51

Luninski [80]

air/water

4, 6.15, 8.15, 9.85, 12.3

horizontal

25, 95

horizontal horizontal

Andritsos and Hanratty [39]

air/water air/glycerine solution |i = 16cp |i = 70cp |X = 12cp |i = 80cp jLi =

lOOcp

steam/water 3M Pa 5M Pa 7.5M Pa

Nalcamura et al. [82]

horizontal upwards and downwards inclined

25 25 95 95 95 horizontal 180 180 180

A relatively simple functional dependence of the dynamic coefficient on (ReyFrj^)"" has been recently found by Brauner and Moalem Maron [102]. Figure 5a shows that when data which corresponds to laminar regime in the liquid phase is used, the power m equals 1, while with turbulent regime in the liquid phase, (Figure 5b), a variable power m as function of the liquid Reynolds number is required to correlate the data. The correlation obtained for C,:

C, = 2 . 4 5 x 1 0 -

Re, Fr^

Y. = 1

(27)

where: m = 1 ; Re < 2500 b

m = 1.565 - 0.072 In (Re.) ; Re, > 2500

(27.1)

340


a) Laminar liquid layer, Ret3-

cr o

10

UJ

\

10

-J

.-2

L_J

I I IIII

LIQUID LAYER 1.50

10'

10^

10'

10

Ub FROUDE NUMBER, Frb= (gHbi 1/2

1.25

I.OOl1^—n-Q D

mo

tK

0.75!

\ eq. (27)

0.50

0.25

I

• ' ••"•I

10'' LIQUID

'

.1

10'

' •

I

10'-

• • • "•"'

I I I Mini

10^

LAYER REYNOLDS

10"

1 I 111 ml

1 i >'

10^

NUMBER, Reb =

10^ UbDb

Figure 5. Correlating the "dynanîic coefficient" C^ with plug flow-model in the liquid, 7^= 1. Notation in Figure 5a: D air/water D = 0.025 m [39,77,78] O D = 0.051 m [77,78] A D = 0.095 m [39] + XO air/glycerine D = 0.095 m, ii^^ = 100, 80, 12 cp [39] V IS solution D = 0.025 m, î^ = 16, 70 cp [39]


341

with Fr. =

U, (gcospDH)'/'

R e . = ^ ^

(27.2)

It is worth noting that in working out Figures 5 and Equation 27, a plug flow has been assumed for the two-phases, y^, Y^ = 1. For the relatively thick turbulent flow of the gas phase, y^ = 1 is a reasonable representative value. However, it is still of interest to evaluate the effect of y^^ > 1 for the sheared liquid layer. This is demonstrated in Figure 6, where y^^ = 3/4 and y^^ = 1.6 have been used for the laminar and turbulent regime, respectively. Comparison of Figures 5 and 6 points out that the main effect of introducing y^ ^t l is in producing what seems an asymptotic value for the power m in the C^ correlation. In the high Reynolds numbers region, the power of Re^^/Fr^ seems to attain a constant value, m = 2/3. However, this observation is still to be established in view of additional data in the highly turbulent region. Yet, the corresponding correlation for C^^, as derived in view of Figure 6 is

C, =3.0x10"^

Re, Fr?

Yb ^ 1

(28)

m=1

Re^ < 2500

m= 1.565 - 0.072 In (Re^,)

2,500 < Re, < 10^

(28.1)

D

Re, > 10'

m= 2/3

b

With the correlation obtained for C^^, the dynamic model for x. formulated in Equation 23 is completed and can be applied as a predictive tool for analyzing the stability characteristics of a variety of two-fluid systems. Amplitude and Phase of Shear Stress Fluctuation The dynamic model for the interfacial shear defined by Equation 23 with either Equation 27 or Equation 28 for Cj^ can be used to characterize the fluctuation of interfacial shear due to the evolution of interfacial waves. In the framework of linear analysis, the shear stress perturbation is obtained by linearizing Equation 23 to yield:

X = T,

3
U., U = U^, or U < U ) are detailed in Brauner and 1

^

a

b'

a

b'

a

Moalem Maron [43]. For harmonic perturbation:

b^

342

Advances in Engineering Fluid Mechanics 10'

,\

F a ) ^Laminar liquid layer, Rei3 c^ > c^ (since V^ > 0). Hence, it is the relation between kinematic and dynamic wave velocities which essentially determines the stability, as c^ > c^ corresponds to unstable modes, whereas modes with cl > cl are attenuated. '

d

k

In extracting the conditions for the stable or unstable modes from Equations 40, it should be emphasized that co^ and co, are expressed in terms of waves velocities relative to the weighted mean velocity and, therefore, c^, c^, and c^ may attain negative values. Thus, the condition for an unstable mode, for instance, c^ > c^, is equivalent to C^ > C^ when both c^, c^ are positive, whereas for negative c^, c^ the conditions becomes C^ < C^. In both cases, however, this means that in terms of absolute relative velocities, the kinematic wave exceeds the dynamic wave, •C^ - V^l > IC^ - V^l, for an unstable mode. For stable modes, the absolute relative k

0

d

0 '

'

velocity of the dynamic wave exceeds that of the kinematic wave.


349

ILL-POSEDNESS BOUNDARY—INVISCID STABILITY ANALYSIS It is commonly believed that a correct mathematical presentation of physical situations ought to result in properly posed problems. In two-phase flow problems, however, the existence of an assumed physical situation, e.g., stratified wavy flow configuration, is not certain under all operational conditions. Therefore, ill-posedness in some domains of the parameters space does not necessarily imply that the formulation is globally incorrect. Moreover, the boundary of the well-posed domain may have physical significance since it signals the existence of additional physical features which the original model neglects. When these features become consequential, one expects a different physical behavior, such as transition to a different flow pattern, and a different model is required to simulate this transition. It is shown by Brauner and Moalem Maron [40,45] that, indeed, the wavy-stratified regime is confined to a domain at whose boundaries the two-fluid formulation becomes ill-posed. The ill-posedness boundary is always located in the region of amplified waves since the stable smooth stratified zone, which is confined by the stability boundary, is always a sub-zone of the well-posed region [43-45]. The transient continuity equations and the combined momentum equation constitute a set of hyperbolic equations. The formulation is well-posed provided the equations possess real characteristics. The conditions of well-posedness of averaged two-fluid models were extensively discussed in the literature (e.g., Lyczkowski et al. [106], Ramshaw and Trapp [107], Banerjee and Chan [56], Drew [108], Jones and Prosperetti [109], Prosperetti and Jones [110], Moe [111]). The condition under which the characteristic roots of Equations 1, 2, 7 are real reads, (derived in 43 for q = 0):

+ — [ ( P , - Pa) g COS P + Ok' - q(U^ - U,) S^( A-i + A-')] > 0

(41)

The identity between condition (41) and condition (38) for stable dynamic wave indicates that the region of well-posedness coincides with that of stable dynamic waves, c2 > 0. The region of c^ < 0, corresponds to unstable waves and their evolution, as formulated by the initial value set of equations, is ill-posed. As the stability condition for inviscid flows (obtained with AF^^^ = 0) is equivalent to that of stable dynamic waves, the well-posedness condition (with Cj^ = 0, y^ = y^ = 1) is actually equivalent to the classical (inviscid) Kelvin-Hermholz stability condition. This identity explains the apparent relevance of the frequently applied inviscid analyses for predicting flow pattern transitions in two-phase viscid flows. It is of interest to note also that for horizontal gas-liquid plug flows (y^ = y^^ = 1) when typically U^ » U^ and Cj^yU^ -> 0, if Cj^yUj^ -> 1 is assumed (practically stationary liquid phase) the well-posedness condition. Equation 41, and the stability condition. Equation 33, become identical. Inspection of Equation 41 points out that for fluids of zero surface tension and equal densities (or zero gravity conditions) the formulation is ill-posed for plug flow (y^ = y^ =1). However, since the condition for well-posedness is independent of the

350


viscous shear terms (rhs of the combined momentum equation, AF^^^ in Equation 26), the assumption of inviscid fluids in the framework of two-fluid models is of no effect. The inclusion of either gravity or surface tension is, thus, necessary for obtaining real characteristics in two-fluid plug flows. However, for y^, Y^ > 1» wellposed formulation may still be obtained in the absence of gravity and surface tension forces, as in viscid flows velocity gradients may stabilize the system. Thus, while the neutral stability boundary may represent preliminary transition from smooth-stratified flow to a wavy interfacial structure, the well-posedness boundary, which is within the wavy unstable region, represents an upper bound for the existence of a stratified wavy configuration. Beyond the well-posedness boundary transition to a different flow pattern takes place. In the "ill-posed" region, the model is no longer capable of describing the physical phenomena involved; therefore, amplification rates predicted for ill-posed modes or numerical simulation of their growth is actually meaningless. It is to be noted that the criterion for ill-posedness is affected only by the terms which are proportional to the gradients of h, u^, u^^ (derivatives with respect to time and space) and, therefore, apparently unaffected by the quasi-steady modelling of the shear stresses. However, the test for well-posedness is carried out on a stratified wavy configuration, which is represented by the averaged values of H, U^ U^, (obtained from the solution of AF^^^ = 0, Equation 11). Obviously, their values depend on the models used for the wall and interfacial shear stresses. In particular, the modelling of x. deserves a special attention since in the wavy regime the augmentation of the interfacial friction factor, /., due to the interfacial waviness is to be considered. INTEGRATED STABILITY AND WELL-POSEDNESS CRITERIA Stability and Well-Posedness Map Given the fluid physical properties and system geometry (tube size and inclination) the stability and well-posedness boundaries can be mapped in the coordinate system of the two-fluids flow rates, U U^. The construction of the stability and well'

as

bs

•'

posedness map (SWP map) is demonstrated in Figure 8 for horizontal air-water flow in a 2.5 cm pipe. For each combination of (U^^, U^^^) the range of amplified wave modes 0 < k < k^ is obtained by solving Equation 33 with 19 for k^. In searching for all combinations of (U^^, Uj^^) for which k^ —> 0, the so-called it "zero neutral stability" boundaries is obtained. This boundary confines all possible smooth stratified flows. The locus of the curve itself represents the departure from smooth stratified structure. For any operational set (U^^, U^^^) outside the k^ = 0 boundary, the linear stability analysis predicts exponential growth with time for a finite range of wave numbers, 0 < k < k^. The wave growth in this region may either be damped (due to non-linear effects) and, thus, end up with "stable wavy" stratified flow, or may result in a different flow configuration (due to bridging, for instance). The stability boundary is constructed from three parts: For turbulent gas phase (right to the laminar-turbulent transition in the gas phase) the destabilizing effect of the dynamic shear stress term, J^, ought to be accounted for and the stratified


AIR-WATER, D« 2.54 cm, ZNS, ZNSm boundaries ZRC boundary

.E . JO

3

351

^=0"" 'buffer' zone K i M

^7

Ck>c|Ul,

2 2 2 / Ck U^ while along the right one U^ > U^. The existence of two branches points out the multiplicity of solutions, which become even more complicated along the ZNS, ZNS^ boundaries, also due to the discontinuities which evolve from laminar-turbulent flow regime transitions in either of the two phases (and the associated change in the shear stresses). Note, that for low gas holdup (H/D > 0.5), the destabilizing effect of J^ may be dampened even when the gas flow is turbulent due to the strong effect of the upper wall on the turbulent structures. Therefore, the ZNS line in Figure 8 is extended (beyond gas phase L-T transitional boundary) up to the H/D =0.5 line.

352


The structure of the SWP map in Figure 8 for horizontal air-water system (D = 2.5 cm) has been found typical to a variety of horizontal flows of air-viscous liquids over wide ranges of liquid viscosity and tube diameter. (Wide range of U^^ is included in Figures 8 for exposing some general features of SWP maps.) In Figure 8, the instability characteristics are further indicated with reference to the characteristics of kinematic and dynamic waves. In the smooth stratified region confined by the stability boundary, c^ > c^ and cj < c^ for all wave modes while along the stability boundary c^ = c^ for the long waves (k^ -^ 0). Clearly, for all other modes with finite wave lengths, c^ > c^ certainly prevails along the stability boundary (ZNS and ZNS^). In the ill-posed region beyond the ZRC line, there exists a range of unstable dynamic waves, 0 < k < k^^ for which c^ < 0 and c^ > c^. Along the ZRC boundary, c^ (k^^ -^ 0) = 0, and the dynamic (long) waves are marginally stable while all other dynamic waves are certainly stable. Between the limits of the stability and well-posedness boundaries (shaded areas), c^ >0 for all wave modes, including those unstable modes for which c^ > c^. (and c^ < c^ < c^). As a corollary, it can be stated that the condition of unstable dynamic waves or ill-posedness is sufficient to indicate instability, whereas the condition of c^ < 0, or well-posedness, is necessary but insufficient to ensure stability. The above ideas and interpretations as detailed above with regards to the horizontal system of Figure 8 also prevail basically in inclined flows, although limiting stability and wellposedness boundaries may demonstrate entirely different structures (Brauner and Moalem Maron [45]). As discussed, the test for well-posedness is carried out in the stratified wavy region. Therefore, the calculation of the stratified flow averaged values (H, U^, U^^) in this region should be, in principle, calculated while accounting for the interfacial shear stress augmentation due to the presence of waves (///^ > l).The effect of augmented f. on the location of the well-posedness (ZRC) boundary is demonstrated in Figure 9; increasing /. results in extension of the well-posed region towards higher gas and liquid flow rates. It is also shown in Figure 9 that along the ZRC boundary the stabilizing effect of the enhanced interfacial shear stress due to interfacial waviness is of the same order as the destabilizing effect due to the dynamic interaction obtained with the inclusion of C^ ^ 0 (Equation 27) in the wellposedness criterion. Equation 41. Thus, in the absence of well-established correlations for /. and C^ along the stratified-wavy boundaries (where transition to annular or intermittent pattern takes place), the well-posedness boundary predicted while ignoring the gas flow interaction with the wavy interface (///^ = 1 and C^^ = 0) may provide a reasonable estimation for the upper bound on the region of stable stratified-wavy configuration. Constructing the Stratified Flow Boundaries The general implication of the ''stability boundary" (ZNS or ZNS^) and the wellposedness boundary (ZRC) is in defining three zones; the area within the stability boundary is well-understood to be the stable smooth stratified zone. Beyond the ZRC boundary, the complex characteristics indicate that the governing equations of the stratified flow configuration are ill-posed with respect to long wave modes in the wave spectra. In this sense, the ZRC boundary represents an upper bound


353

AIR-WATER ^-Laminar - Turbulent transition in the air ZRC, fi/fa=IO, C h = 0 ZRC, fi/fa=IO. Ch=^0 ZRC, fi/fa=l > C h = 0 3

10' > lO'

io°L a) D = 2.53cm Q. 3

01

-I

10

10^

I Imill

lO'

I I iiiiiil

» " Imiiil

' « ' '•""

10^

10^

Superficial

10^

10*

10^

10^

lo""

Air Velocity. UQS [m/s]

Figure 9. Effect of f./f^ > 1 and C^ ^ 0 on the well-posedness boundary. for the existence of the wavy stratified configuration, beyond which another flow pattern develops. Indeed the stable smooth stratified zone as defined by the stability boundary is always a subzone, confined within the well-posed region (Figure 8). The ''buffer'' region between the stability and ill-posedness boundaries is characterized by the evolution of amplified interfacial waves (the growth of which as governed by the variation of (h, u^, u^) in space and time is still well-posed). As such, this buffer region bears a potential for flow pattern transition. Whether these disturbances trigger a departure from stratified configuration (due to blockage) depends on the insitu holdup. For thick liquid layer, H/D =^ 0.5—1, it is likely that the evolution of amplified interfacial disturbances will end up in tube blockage. Thus, when the entry from the smooth region into the "buffer" zone (along the stability boundary) is associated with a relatively thick liquid layer, the stability boundary also may predict the condition for the development of another flow pattern. On the other hand, when the entry to the "buffer" zone occurs with a relatively thin liquid layer H/D < H^^.^, the stability line will predict the development of stratified-wavy flow, but it plays no role in predicting the transition to another flow pattern. The transition from this rippled interface to another configuration may be "delayed" to the well-posedness boundary (ZRC) as long as the relative liquid layer thickness, H/D, remains small in the "buffer" zone. When the relative liquid layer is of the order of the conduit radius, H ~ H . , within the "buffer" region, the '

cnt

^

disturbed interface may trigger flow pattern transition, this time within the "buffer"

354


zone in the vicinity of H = H^^.^ line. Clearly, the choice of H ~ H^^.^ line as a criterion for transition in the "buffer" region comes instead of a complicated nonlinear stability analysis, which is required to determine whether the amplified wave in the "buffer" wavy region will stabilize with a finite amplitude or cause transition. The value adopted for the critical layer thickness by Brauner and Moalem Maron [40-45] was H^^.^ = 0.5 (calculated with ///^ = 1). The criterion of H == 0.5 has been substantiated by Barnea and Taitel [1121. crit

•'

*^

^

They carried out non-linear simulations of the wave growth due to a finite interfacial disturbance, using the method of characteristics proposed by Crowley et al. [47]. For thick liquid layers, H > 0.5, the characteristics at the wave crest become imaginary during the wave growth. It was argued that this locally ill-posedness during the nonlinear stage of the wave growth implies transition to slug flow. It is to be noted, however, that the simulations were carried out on a simplified model, assuming a quasi-steady condition for the gas phase. This simplification has been justified on the grounds of gas (upper phase) velocity much larger than the liquid velocity, which is not always the situation along the stratified/slug transition, in particular for downward inclined systems. In downward inclined systems the liquid velocity even exceeds the gas velocity along the transitional boundary to slug flow (Brauner and Moalem Maron [45]). Observations on the mechanisms involved in the stratified/slug transition have been reported by Lin and Hanratty [75] and Andritsos et al. [81,113]. At low gas rates and thick liquid layers, slugs indeed evolve from a wave growth process, taking place in the downstream direction; eventually one of the growing waves hits the top of the pipe to form a slug. The wave growth mechanism has been found to be characteristic of the transitional boundary, where increasing the transitional gas flowrate is associated with almost constant liquid flow rate and thinner liquid layer (as is the case along the ZNS boundary of thick liquid layers, H > 0.5 in Figure 8). For higher gas rates, however, the stratified/slug transition has been observed to take place along an almost constant critical layer thickness and increasing liquid flow rate with increasing the gas rate (consistent with the H = H^^.^ criterion just discussed). Along this section of the transitional boundary, slugs have been observed to be formed by coalescence and breaking of waves (rather than due to the growing process of a single wave). Experiments by Andritsos et al. [81] indicate a critical height of H . = 0.3 for air-water systems and H = 0.4 for air-viscous liquid systems (< H = 0.5 criterion just discussed). However, the calculation of the locus of the critical layer thickness in the buffer region (Figure 8) should be carried out while taking into account the enhanced interfacial shear stress due to the developed wave structure formed over the interface. In the absence of established correlation for the augmentation of /. in this region, the choice of a larger critical layer thickness as a transitional criterion, H^^.^ — 0.5 (calculated with /. = / J , may compensate for the neglect of the interfacial shear stress augmentation due to the waves. STRATIFIED-FLOW BOUNDARIES: COMPARISON WITH EXPERIMENTS The various boundaries which evolve from stability and well-posedness analyses and the associated physical interpretations form a basis for constructing a flow

Boundary Conditions Required for the CFD Sinnulation

355

pattern map for the stratified configuration and transitional boundaries to the other bounding flow patterns. These are tested in view of available experimental data. Figures 10-13 represent some typical comparisons between the analytical boundaries and experiments. The figures include both the "zero neutral stability line" (ZNS) obtained with quasi-steady modelling of the interfacial shear stress, C^^ = 0, and the corresponding modified ZNS^ line obtained with C^^, as evolved from Equation 27. Along the ZNS, ZNS^ lines, J^ -> 0, and, therefore, the destabilizing inertia terms J^, J^, and the memory term, J^^ (normalized with respect to gravity), should sumup to unity. Equation 33. The comparison with the experiments is elucidated in view of the interrelation between J^, J^^, and J^^ along the stability boundary. Stratified/Slug Transition, Along the ZNS, H > 0.5 For thick liquid layers, H :> 0.5, the dynamic term, is to be excluded due to the proximity of the wall and the relatively low Reynolds number of the upper phase (Re^ < 2,000 left to ^ Figures 10-12). In this region of thick liquid layers, the growing unstable modes tend to block the gas passage, affecting the formation of liquid slugs. Thus, the stratified-smooth/stratified-wavy boundary practically coincides with transition to the slug pattern and is governed by the ZNS line. Indeed, the data of stratified-smooth/slug transition is in fair agreement with the ZNS segment (in bold) in this region, and in the vicinity of H = 0.5 in the buffer region.

Superficial Air Velocity , Uas[ni/s] Figure 10. Effect of liquid viscosity on the stratified flow boundaries; comparison of theory with experiments (Andritsos and Hanratty [39], D = 9.53cm). O stratified-smooth/slug, • stratified-smooth/wavy, A stratified wavy/annular, ^ turbulent/laminar transition of upper phase.

356

Advances in Engineering Fluid Mechanics Theoretical: ZNS (Jh=0) ZNS^ (Jh:^0) ZRC H=0.5

AIR -WATER Experimental: • D Mandhane et al [77] A A Shoham [78] • o Andritsos & Hanratty [39] | \ rv

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Figure 11. Effect of tube diameter on the stratified-smooth boundary for air-water system; comparison of theory with experimental data. O A D stratified-smooth/slug, • A • stratified-smooth/wavy ^ turbulent/laminar transition of upper phase AIR-VISCOUS LIQUID , D=2.53cm THEORETICAL ZNS(Jh-O) ZNSm(Jh'*0) _^ lO'

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Superficial Air Velocity , Uas[m/s]

Figure 12. Effect of liquid viscosity on the stratified smooth boundary; comparison of theory with experiments (Andritsos and Hanratty [39], D = 2.53cm). • stratified-smooth/wavy C, turbulent/laminar transition of upper phase


357

Nondimensional Superficial Steam Velocity, 11^5= U^g

Figure 13. Effect of pressure on the stratified-flow boundaries In steam water systems; comparison of theory with experimental data (Nakamura et al. [82]). • stratifled/non-stratlfled, • stratified smooth/stratlfied-wavy. Figure 14 represents the liquid inertia destabilizing term, ]^ (normalized to gravity) along the stability boundary. Clearly, along the ZNS boundary, (k^, J^) -^ 0, and in the absence of a dynamic effect, J^^ =0, ]^ + ]^= 1. Figure 14c indicates that C,^^ almost always deviates significantly from the liquid velocity, and thus C^/^^, - 1 ^ 0 . Correspondingly, the liquid destabilizing effect as represented by J^ certainly cannot be ignored. Moreover, in view of Figures 10-12 in the range which corresponds to stratified/intermittent transition, the liquid destabilizing term, in fact, even dominates, J^ > 0.5. It also has to be emphasized that even when Cj^yU^^ « 1 (as in air-water systems, |ij^ « 1 c ), the neglect of the liquid relative contribution may be erroneous over most of the stratified/slug transition boundary. This is understandable (in view of Figure 14d), since in this range also Cj^^/U^ « 1. However, in view of Equation 18, (Cj^^/U^ - 1) and (Cj^^/U^ - 1) are not the only quantities which determine the relative contributions of the liquid and gas destabilizing effects. This is further demonstrated while following the effects of the liquid viscosity: It is shown in Figure 14 that as the liquid viscosity increases, Cj^yU^^ may significantly deviate from one and attain high values while simultaneously C^yU^ decreases. However, the liquid relative destabilizing contribution, J^, although proportional to (C^Jl^^ - 1), decreases with increasing viscosity and, thus, the range of liquid dominance is reduced. The statement given by Hanratty [114] that "the K-H in viscid theory becomes surprisingly more accurate as the liquid viscosity increases'," is indeed well-established in Figure 14. In view of Figures 14, the regions of thick liquid layers (and laminar gas phase) is associated with "liquid-controlled" transition. Consequently, the stratified/slug

358


10^

10'

Superficial

10^

10'

10"

10'

Air Velocity, UosCm/s]

Figure 14. Interfacial wave celerity and liquid destabilizing contribution along the ZNS boundary. J^^ along: H > 0,5 and laminar air flow H > 0.5 and turbulent air flow H < 0.5 and turbulent air flow


359

transition is almost always "liquid-controlled"—instability is governed by the destabilizing effect of the liquid inertia. As is shown in Figure 15 larger diameters affect a wider range for ''liquid-controlled" transition, and the effect of the tube size becomes more pronounced with increasing liquid viscosities. Note that the dashed and dotted sections in Figures (14-15) refer to turbulent gas flow, where the dynamic term J^^ ought to be included in the stability conditions. Therefore, only the solid sections in Figures 14-15 are relevant. At this point, it is relevant to emphasize that some previous studies related to stratified/slug flow pattern transition assumed in fact a "gas controlled" transition by unjustifiably ignoring totally the liquid contribution (see Equation 22). As their modelling is essentially based on the gas-contribution, which shares a minor part on the stratified/slug boundary, a reasonable comparison with experiments required the insertion of correction constants, which became greater for lower U ^ (greater H/D). The (1 - H) = K, correction suggested by Taitel and Dukler [19],''although it yields the larger correction required at lower U^^, is suitable only for adjusting stratified/slug transitional boundary in air-water systems, (D = 2.5, 5cm), but is not applicable to air viscous liquid or general two-fluid systems [37,38,41,81].

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360


Stratified-Smooth/Stratified-Wavy Transition, Along the ZNS„, H < 0.5 As the operational conditions move towards high gas rates with relatively low liquid rates, thin liquid layers are obtained, (H < 0.5) which are known to exhibit a stable rippled interfacial structure, denoted as a stratified-wavy pattern. Thus, the stratified-smooth/stratified-wavy transition is associated with turbulent gas flow over a relatively thin liquid layer and is largely controlled by the dynamic interaction between the phases as presented by the J^ term. The relatively large destabilizing contribution of J^^ along the ZNS^ lines in this region is manifested by a large gap between the ZNS and ZNS^ lines. The transitional data to stratified-wavy in the region of H < 0.5 do follow the ZNS^ line. Figures 16-17 demonstrate the relative contributions of the destabilizing terms (along that segment of the ZNS^ line relevant for transition) in various two-fluid systems and conduit sizes. The first indication of these figures is that the destabilizing effect of the dynamic term is in no way to be ignored, and its impact increases with the gas rate. On the other hand, the inertia destabilizing effects of the twophases may become insignificant in certain situations. For instance, for highly viscous liquids the liquid inertia contribution, J^^, becomes negligible (Figure 16b) and the remaining J^ and J^ terms balance the gravity stabilizing term (Figures 16a, 16c). It is worth noting that for viscous liquids the destabilizing contribution of the dynamic term does not degenerate; for X | ^ > lOcp it follows a uniform curve which is practically independent of the liquid viscosity. As the liquid viscosity is reduced, the gas destabilizing contribution decreases, and although the liquid contribution increases significantly, the role of J^ remains the dominant one. As is further seen in Figure 17 for larger tubes, D > 2.5cm, the dominancy of the dynamic term in controlling the initiation of interfacial instability becomes absolute (J^^ -^1). Thus, in spite of the fact that the dynamic interfacial shear stress term in Equation 23 vanishes for dh/dx —> 0, this term affects an additional destabilizing contribution, the impact of which on the neutral-stability boundary is dominant even in the limit of long wave analysis. Stratified Wavy/Annular Transition, Along the ZRC, H < 0.5 The well-posedness boundary (ZRC) (included in Figures 10, 11, 13) represents the limit of operational conditions (U^^, U^^^) for which the governing set of continuity and momentum equations is still well-posed with respect to all wave modes. Hence, it is considered as an upper bound for the stratified-wavy flow pattern. Indeed, the data of stratified-wavy/annular transition follows the ZRC curve in the region of H < 0.5. Effects of Physical Properties Further inspection of Figures 10-13 indicates the effects of physical properties. Increasing the liquid viscosity results in a reduced stratified-wavy zone due to opposite migrations of the two boundaries—the ZNS^ moves towards higher gas rates while the ZRC moves towards lower gas rates. Also, since the H < 0.5 line

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361

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363

is associated with higher gas rates as the liquid viscosity increases, the practical range of the stratified-wavy regime is further reduced. In high pressure steam-water systems Figure 13, the higher vapor density and lower liquid viscosity affect a higher impact of the dynamic interfacial shear stress term. As a result, in the region of thin liquid layers, the relative destabilizing contribution of J^ becomes dominant. It is also to be noted that due to the higher vapor density the upper phase flow stays turbulent in the region of H < 0.5, whereby the relevance of the ZNS^ line in predicting the transition to stratified-wavy flow may extend to the region of thick liquid layers (as compared to atmospheric airliquid systems. Figures (10-12). The other transitional data from stratified-wavy to slug, to wavy-dispersed or to annular flows, are confined by the well-posedness (ZRC) boundary. Note that the coordinates of Figure 13 are modified in order to follow the reported data in Nakamura et al. [82]. Inspection of Figure 13 shows that the main effects of increasing the gas phase density are higher liquid rates and lower gas rates for which stable stratification can be maintained. Consequently, the stratified/slug transition is delayed to higher liquid rates while the stratified/annular and stratified-smooth/wavy transitions occur at lower gas rates. It also should be noted that utilizing modified coordinates (pÂp)'^^ U^^ and (pj^/Ap)'^^ U^^ instead of U^^ and U^^^ significantly reduces the effect of the gas density on the location of the stratified-smooth/wavy and stratified-wavy/annular transitions. Therefore, such a presentation may be of advantage in scaling the stratified-smooth/ wavy and stratified-wavy/annular transitions observed in low pressure systems to high pressure systems. However, it provides no advantage for scaling the stratifiedsmooth/slug transition. Thus, no universal structure for the complete stratified/nonstratified transition boundary can be obtained. Table 2 indicates the various controlling destabilizing terms along the stratifiedsmooth/wavy transitional boundary in some limited physical situations. Effect of the Tube Size The effect of the tube diameter on the construction of the transitional boundaries deserves special discussion. In large tubes and high gas rates, (Figures 10-11) the Table 2 Destabilizing Terms in Limiting Physical Situations

Thick layers H»0.5

Large D or Small |Li

Thin layers

>

Ja + Jb = 1 Jb-1 Ja + Jb + Jh = 1

Turbulent upper phase

Large ji

H

n 10-^ 00

10Superficial air velocity, U^g [m / s ] Figure 18. Construction of stratified-smooth/stratified-wavy boundaries in small and large diameter tubes. Comparison with experimental data (Shoham [78], Luninski [80]).


365

coincides with stratified-smooth/wavy transitional data. However, as long as the gas phase is laminar, the flow pattern transition is governed by the ZNS boundary Equation 33.2. The bold lines in Figure 18 mark the complete stratified-smooth/wavy transitional lines as predicted by integrated considerations of stability and L-T flow regime transition. For the sake of clarity, the focus in Figure 18 is on the interplay and practical relevance of the stability boundaries (ZNS^ or ZNS) and the L-T laminar/turbulent flow regime transitional line in predicting the stratified-smooth/wavy flow pattern transition. Other transitional boundaries, which confine the stratified-smooth and stratified-wavy zones, are shown in Figure 19. For relatively low gas rates, the stratified-smooth zone is bounded by the slug or bubbly patterns while the stratifiedwavy zone, at high gas rates, is bounded by the transition to annular pattern. In small diameter tubes, surface tension effects play a significant role (Brauner and Moalem Maron [115], Brauner et al. [13]) and should be considered while analyzing the stratified flow boundaries. Since the characteristic length of the long wave is roughly of the order of the tube diameter (k = 27c/D), the stabilizing term due to surface tension forces (proportional to kô) ought to be accounted for (even in the framework of long wave stability analysis) when its contribution becomes comparable (or exceeds) the stabilizing term due to gravity. This yields a criterion in terms of the nondimensional Evots number [115]: 471 O

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366

Advances in Engineering Fluid l\/Iechanics

In two-phase systems of G ^ > 1, surface tension contributes a dominant stabilizing term in the well-posedness criterion. Hence, the region of stable wavy stratified pattern extends and the transition to annular flow is delayed to higher gas rates, compared to those predicted by (infinite) long wave analysis (k —> 0). For instance, for air-water flow in a 1 inch pipe, e^ < 0.01, while for D = 0.4cm, E^ ^ 20. Indeed, surface tension effects can be justifiably neglected for D > 2.5cm, as the well-posedness boundary practically coincides with the boundary predicted by the (infinite) long wave theory. In large diameter systems, reducing the tube size affects a migration of the stratified-wavy/annular transition towards lower gas rates (Figures 10a and 11). However, in systems of small diameters, e^ > 1, the dominance of the surface tension stabilization is reflected in an opposite trend, whereby with reducing the tube diameter, the stratified-wavy/annular transition is delayed to higher gas rates (Figure 19). On the other hand, surface tension forces also stabilize the dispersed pattern (Brauner and Moalem Maron [115]) and, therefore, with reducing the tube diameter the stratification takes place at higher gas rates (SD boundary in Figure 19). Thus, the stratified flow boundaries in two-phase systems with G^ > 1 are largely surface tension controlled [115]. Inclined Systems The importance of including the dynamic shear stress term while analyzing the stability of inclined systems is demonstrated in Figure 20 for an air-water system inclined upward at 0.5°. The observed sensitivity of the stratified flow boundaries to slight upwards inclination is well-known (see, for example, Brauner and Moalem Maron [45]). Compared to horizontal flows, a drastic thickening of the liquid layer and a dramatic change in the location of the stability boundary takes place already with a slight inclination. In view of Figure 20a (obtained with quasi-steady model for x., J^ = 0), the region within the ZNS boundary is expected to confine a stable stratified-smooth zone. In the buffer region, outside the ZNS boundary, the region of stable stratifiedwavy pattern is bounded either by H = 0.5 line (due to transition to slug) or by the ZRC line, which represents the upper limit for the existence of a stratified wavy configuration for thin liquid layers (H < 0.5). Thus, a bell shaped region of a stable stratified zone (smooth or wavy) is predicted by the area which is confined by the H = 0.5 and the ZRC lines [45]. As is demonstrated in Figure 20a, employing a quasi-steady model for the interfacial shear apparently predicts a smaller bell-shaped region of stratified-smooth pattern as a sub-zone of the stable stratified region. On the other hand, inspection of Figure 20b indicates that the observed data of stratified-wavy pattern spread over the entire area confined by the ZRC and H = 0.5 lines. Thus, the experimental findings reveal that no stratified-smooth pattern exists already with the slightest upward inclination. The appearance of the finite stable stratified-smooth zone in Figure 20a implies that an additional destabilizing term is missing in the stability criterion obtained with a quasi-steady model for the interfacial shear. Indeed, the implementation of a dynamic model for the interfacial shear as obtained with Equations 23 and 27 contributes the required J^ destabilizing term, and the otherwise predicted stratified-smooth zone vanishes (Figure 20b).


367

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368


S

Pa

Pb

s = 16KRe{^'"-»)/Fr2'" = 16q/Rej,

(43.2)

The structure of Equation 43.1 is identical to the condition derived by Jeffreys [90] for the evolution of wind-generated waves, except that in Jeffreys' analysis, the coefficient s was introduced as a tunable constant and denoted as the "sheltering coefficient." The value of s was adjusted empirically, s = 0.2—0.3 to fit experimental data of the critical wind velocity required to initiate waves. Later experiments by Stanton et al. [116] indicate a "sheltering coefficient" of about one tenth of the value obtained by Jeffreys. However, the expression for s derived here in Equation 43.2 indicates that even in cases where the K-H mechanism of wave formation can be justifiably ruled out (J^, J^^ = 0), the "sheltering coefficient," s, varies with the flow conditions and the physical system under consideration. Consequently, even for a specified two-fluid system, the value of s varies along the transitional boundary. This is demonstrated in Figure 21, where the value of s along the transitional boundary is presented for various liquid viscosities (D = 2.5, 9.5cm). As is shown in Figure 21 for an air-water system, the "sheltering coefficient" obtained by Equation 43.2 varies around s =^ 0.02, but it significantly increases with increased liquid viscosity. Nevertheless, the resulting destabilizing contribution of the dynamic term J^^ shows a moderate reduction for viscous liquids and for \x^ > lOcp Jj^ remains practically unchanged (Figure 16). The observation that in the case of air flow over viscous liquids (IJ^ > 15cp) the K-H instability occurs at a lower gas velocity than the Jeffreys instability (Hanratty [53], Andritsos et al [113]) seems to evolve from the neglect of the increase of the "sheltering coefficient" with increased liquid viscosity as shown in Figure 21. Figure 16 implies that the K-H mechanism and the Jeffreys-Miles-Benjamin "sheltering" mechanism both share important roles in the evolution of waves over viscous liquid layers. Jeffreys' stability condition (Equation 43.1) with a constant value for s was applied by Taitel and Dukler [19] in attempting to predict the stratified-smooth/wavy transitional boundary for air-water flows in closed conduits. The value needed for the "sheltering coefficient," in order to fit transitional data in 2.5 and 5.1cm tubes, was s — 0.01, in reasonable agreement with Figure 21. However, the omission of the inertia destabilizing terms, Ja, Jb in Taitel and Dukler [19], while employing Equation 43.1 for small conduits, should be carefully considered. Note that for sufficiently thin liquid layers, where the stratified-wavy transition is associated with a laminar layer (m = 1), Equation 43.2 yields: s = ——\

Laminar liquid layer

(43.3)

Thus, the "sheltering coefficient" is determined by the liquid layer Froude number. However, for a given two-fluid system, the Froude number along the stratifiedwavy transition boundary demonstrates a relatively small variation. Therefore,


369

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CD

cn ii JO

1 reduces to a critical superficial gas phase Reynolds number, Re^^ > 1.113 x 10^ [103]. CONCLUSION The structure of the closure laws used for the shear stresses in two-fluid models bears crucial consequences on the capability of two-fluid models to predict the stability characteristics of the presumed flow configuration. Quasi-steady closure laws for the interfacial shear stresses, which are widely used in stability analyses of the stratified flow configuration, are insufficient for capturing the physical phenomena involved during the evolution of waves over a liquid interface sheared by a turbulent gas phase. Modification of the interfacial shear stress model to include a dynamic term is essential for rendering a closure law which is capable of bridging the gap between the micro-scale phenomena at the vicinity of the phases interface and the macro-averaged representation of the flow. The dynamic term introduces a dependence of the interfacial shear on the local and instantaneous interfacial slope, hence on the gradients associated with the evolution and growth of interfacial waves. The inclusion of the dynamic term, while analyzing the stability of the flow, is necessary even in the limit of long wave analysis, where the interfacial slope degenerates since the gravity stabilizing term also is proportional to the local slope. The stability condition obtained when a dynamic model is used for the interfacial shear stress unifies the K-H mechanism and the Jeffery's "sheltering" mechanism in a single stability criterion. At marginal stability, the gravity and surface tension stabilizing terms balance the destabilizing terms which are due to the two phases inertia (K-H mechanism) and also that due to the dynamic shear stress component C'sheltering" mechanism). Various two-fluid systems differ in the relative contributions of the three destabilizing terms in controlling the evolution of interfacial waves. The dynamic component of the closure law proposed includes a dynamic coefficient, C^. This coefficient has been found to depend on the liquid layer Froude and Reynolds numbers. With the proposed dynamic model for the interfacial shear stress the transient two-fluid equations are capable of predicting the conditions for the evolution of waves in a variety of two-fluid systems (without any further tuning). The data-base which has been found suitable for extracting the information on the dynamic interaction and correlating the dynamic interfacial shear stress component consists of the fluids flow rates along a stratified-smooth/wavy transitional boundary. More experimental data is needed to further substantiate the correlation for the dynamic coefficient. In particular, there is a need for additional data on this transition in systems of high liquid Reynolds numbers (e.g., high pressure steam/ water systems and large diameter tubes) and in two-fluid systems of either comparable phase velocities or faster lower turbulent layers (e.g., viscous-oil/water systems, downward inclined gas liquid systems).


371

NOMENCLATURE a A c c , c^ a'

Wave amplitude, m Cross-sectional flow area, m^ Wave celerity relative to V^, m/s Coefficient of friction factor

K Numerical coefficient of C^ n

correlation K, Coefficient in Equation 22 m Power in C^ correlation

b

C C^ D D^, D^ / Fr g h H J

k

correlation. Equation 12 Wave celerity, m/s Dynamic memory coefficient. Equations 23, 27 Pipe diameter, m Hydraulic diameters. Equation 13,m Friction factor Froude number. Equation 27.2 Gravity acceleration, mVs Instantaneous layer depth, m Layer depth at steady-state, m Non-dimensional terms in stability conditions (Equations 18, 25) Wave number, m*

n

n Power in friction factor correlation. Equation 12 Q Input volumetric flow rate, mVs P Pressure, N/m^ Re Reynolds number s Sheltering coefficient. Equation 43 S Perimeter, m t Time, sec u Instantaneous axial velocity, m/s U Axial velocity at steady state, m/s VQ Weighted mean velocity. Equations 34, 35, m/s X Coordinate in the downstream direction, m V , V Defined in Equations 34, 35

Greek Letters

p Inclination angle

ji Viscosity, kg/m - s V Kinematic viscosity, mVs p Density (faster phase), kg/m^ ^^ Weighted density. Equation 38.2, kg/nP Pa'P, Dimensionless densities, Equation 38.2 o Surface tension, N/m T Shear stress, N/m^ CO Wave frequency, rad/s

Y Shape factor. Equation 5 "^ ba'

RHS of Equation 7 at transient, steady-state, N/m^ Ap Density difference = p^^ - p^ e In situ holdup of phase b 8' 38/3 h Evots number, dimensionless. Equation 42 e Phase shift of t ' with ba

1

respect to h' Subscripts a as b bs d G i I

Lighter fluid Superficial, lighter fluid Heavier fluid Superficial, heavier fluid Dynamic wave Gas phase Interfacial Imaginary part

h k L n re R s

Dynamic Kinetic wave Liquid phase Neutral stable Real characteristics Real part Solid wavy surface

372


Superscripts ~ Dimensionless (length normalized by D, area by D^) ' Fluctuating, or derivative with respect to h

° Quasi-steady " Amplitude * Frictional velocity

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CHAPTER 13 WATER FLOW THROUGH HELICAL COILS IN TURBULENT CONDITION

Sudip Kumar Das Chemical Engineering Department Calcutta University 92 A. P. C. Road Calcutta - 700 009 India CONTENTS INTRODUCTION, 379 PREVIEW, 381 Fluid Flow Through Rough Horizontal Conduit, 381 Fluid Flow Through Curved Conduit, 384 EXPERIMENTAL STUDIES, 395 Measurement of Relative Roughness, 397 NOTATION, 397 REFERENCES, 398 INTRODUCTION Flow in curved pipes has attracted much attention from researchers because of its enormous engineering applications in the heat exchanger network, heating or cooling coils, piping systems, intake in aircraft, fluid amplifiers, and many others. Flow through curved pipe appears in technological situations as well as in biological systems like the mammalian circulatory system. It is well-known that in fluid flow the fluid flowing through a straight pipe attains a characteristic velocity profile which is independent of the distance along the pipe, i.e., the flow becomes fully developed. However, if the flow direction is changed with a curved pipe, the flow structure of the fluid is completely changed. The fluid is subjected to varying degrees of centrifugal forces from the neighborhood of the curved wall to the center of the pipe-the fluid near the axis of the duct having the highest velocity is subjected to a larger centrifugal force than the slow moving fluid in the neighborhood of the duct wall. Due to the interaction primarily between centrifugal and viscous forces in the curved portion of the flow, certain characteristic motion, known as secondary flow, is generated which causes shifting of the maximum velocity from the inner portion of the curved pipe to the outer portion 379

380


of the curved pipe, i.e., the fluid in the central region of the pipe moves away from the center of curvature and the fluid near the pipe wall flows towards the center of curvature, thus causing increased pressure drop. With respect to the plane of curvature the secondary flow consists of a pair of helical vortices as shown in Figure 1. The secondary flow effects occur in pipes of any cross section, and since they are generated and sustained by the walls in the plane of the curved pipe, the cross-sectional shape has an important bearing on the magnitude of the secondary flow processes. The strength of the secondary flow depends on the curvature of the surface. Helical coil curvature remains constant throughout the length of the coil, giving rise to fully developed downstream flow [1]. The curve flow has important implications for blood flow. Blood flow in the aorta occurs through curved geometries. Coil capillaries were used first by Grindley and Gibson for viscosity measurement of air because of their compactness and increased pressure drop that permits measurement with higher accuracy [2]. Dawe has also tested the curved pipe capillary viscometer [3]. Coil heat exchangers are widely used for heating and cooling of fluids in a wide variety of industries. Their main advantages over the straight tube exchangers are (1) high packing efficiency and (2) lower heat transfer area requirement. But the main disadvantage is the higher pumping power required as compared to the straight tube exchanger. Merkel showed that the average heat transfer coefficient in a coil exceeded that of a straight tube by a factor (1 + 3.54 D/D^); the straight tube heat transfer coefficient was to be obtained from the Dittus-Boelter correlation [4]. Dravid et al. reported that the ratio of the heat transfer coefficient in coil to straight tube varied as De'^^ near the inlet, and the factor increased progressively and reached De'^^ in the fully developed region [5]. There are also applications where curved flow passages can be used to improve mass transfer rates, such as in membrane blood oxygenator (Wiessman and Mockros [6], Richardson et al. [7] and Tanishita et al. [8]), in kidney dialysis devices (Dravid [9]) and in reverse osmosis units (Srinivasan and Tien [10], Nunge and Adams [11]). The increase in heat and mass transfer is caused by the fact that the transport in radial direction takes place not only by means of diffusion but also by convection. The presence of secondary flow gives a marked variation in local transfer coefficients around the periphery of the curved duct. Besides, the secondary

Figure 1. Secondary flow in the cross section of a helically coiled tube.

Water Flow Through Helical Coils in Turbulent Condition

381

flow greatly reduces the differences between the mean axial velocities for the various streamlines, which results in the decrease of the axial dispersion. Koutsky and Adler confirmed from their experiment that helical tubes were superior to straight tubes or packed beds in minimizing axial dispersion and approached plug flow [12]. Janssen reported that in helical coil reactor for De^Sc < 100, the axial dispersion is the same as with the straight tube, but for higher De^Sc it would decrease more than three times [13]. A helical coil distillation column operation was first reported by Atkeson [14]. He claimed that his unit realized up to 62 theoretical plates at relatively rapid rates of distillation. Morton et al reported experimental studies on a helical coil distillation column [15,16]. Smol'skiy and Chokol'skiy conducted experimental study on condensation of water vapor from moist air in curved channel [17]. They found that heat and mass transfer rates and pressure drop were significantly higher than those in the straight channel. Drag reduction was observed by Barnes and Walters [18], Mashelkar and Devarajan [19,20] for non-Newtonian liquid flow through curved pipe in laminar flow condition. The secondary flow causes a certain amount of drag reduction for shearthinning and visco-elastic fluids. The residence time distribution for helical coils is narrower than for straight circular pipe (Ruthven [21,22], Trivedi and Vasudeva [23], Saxena and Nigam [24]). The recent interest in curved pipe flow has been concentrated on unsteady flow with reference to physiological problems. A fair number of studies have been reported on oscillatory or pulsatory flow Lyne [25], Zalosh and Nelson [26], Munson [27], Smith [28], Chandran et al [29], Mullin and Created [30,31], Lin and Tarbell [32], Talbot and Gong [33], Berger et al [34], Chang and Tarbell [35], Nandakumar and Masliyah [36], Sudo et al [37], Swanson et al [38], Webster and Humphrey [39] etc. Thus, it is becoming increasingly apparent that the complex nature of the primary (axial) and secondary (radial and tangential) flow patterns in a curved geometry—due to centrifugal forces—gives some definite advantages over the straight tube despite the relative increase in pressure drop at higher flow rates. PREVIEW The flow through a helical coil is uniquely different from that through a straight pipe due to the secondary flow pattern induced by the imbalance in the radial direction between the outwards-directed centrifugal force and the inwards-directed pressure force acting on the fluid. Reviews by Berger et al [34], Nandakumar and Masliyah [36], and Ito [40] summarize recent studies of curve pipe flows. Ali reported the pressure drop performance of different types of coiled tubes [41]. The literature review attempted here is not broad or deep, but focuses mainly on the pressure drop of fluid flow through horizontal conduits and curved pipes. Fluid Flow Through Rough Horizontal Conduit For evaluation of pressure drop for flow through a pipe one needs to know the friction factor. In laminar flow regime the friction factor is a function of Reynolds number only, and in the case of turbulent flow the friction factor is a function of Reynolds number and also the relative roughness factor. Blasius showed analytically

382


that the friction factor for turbulent flow in a smooth pipe was a function of Reynolds number only [42]. He examined the experimental data on pressure loss and obtained the following empirical relation f ^ = 0.079 Re-«25

(1)

Prandtl, [43] using boundary layer theory, mixing length hypothesis and law of the wall, developed the following theoretical law of friction for smooth pipes in turbulent flow - ^

= - 4 1og(1.26/Re^)

(2)

This simple equation fits the reliable data with a good degree of accuracy in the range 3,000 < Re < 100,000. The best fit of the experimental data points of Nikuradse [44] for the full practical range of turbulent flow 3,000 < Re < 3,000,000 is given by 1

V^

= -4 1 o g ( R e ^ ) - 0 . 4

(3)

In the case of fully developed turbulent flow condition, von Karman [45] first established the following relation for friction in a fully rough pipe 1

=41ogr^^l

,4,

Colebrook developed a mathematical function which, as he claimed, gave a transition curve between smooth and rough pipes and agreed more closely with actual measurements [46]. He simply combined the expressions for the friction factor for smooth and rough pipes, i.e., Prandtl's smooth pipe law of friction and von Karman's fully rough pipe law of friction as = -4 log

e/D,

3.76

1.26

ReVf^J

(5)

In Equation 5, f^^ appears on both sides of the equation, and the solution can only be obtained by the use of iterative procedure. Friction factor can also be determined from a graph commonly known as Moody's diagram (Moody [47]). Barr [48] proposed the following equation 1 ^

,,

f 5A5 \RC''''

E/DA 3.7


383

Later [49] he modified the above equation as 1

,,

("5.1286

£/D, ^ (7)

Churchill [50] proposed the following

V?^

£/D,

1_ Re

= -4 log

(8)

3.7

Swamee and Jain [51] proposed a similar type of equation as

îK

6.97 Re

= -4 log

£/Dt

(9)

3.7

Chen Ning Hsing [52] gave the following equation 1

£/D, 3.7065

= -4 log

5.0452 1.1098 _j_ 5 . 8 5 0 6 ^ 1 log (e/Dt) Re 2.8257 Re"

(10)

Barr [53] gave 1 ^|^

= -4 log

r5.021og{Re/4.5181og(Re/7)} [ Re{l + R e ' ' ' ( 8 / D j ' V 2 9 }

e/D, ^ 3.7

(11)

While later [54] he recommended the less cumbersome form 1 = -4 log

V?::

4.5181og(Re/7) Re{l + Re'-'' (e/D, )'-729}

8/D,

(12)

3.7

Zigrang and Sylvester [55] suggested the following equations 1

4^ Vf^

= -4 log

= -4 log

3.7

Re

I 3.7

3.7

Re

3.7

(13)

Re

Re

I 3.7

Re

(14)

Equation 14 is a better approximation than Equation 13, obtained by substitution of (13) into the right hand side of Equation 5.

384


Haaland [56] gave the following equation 1

-3.6 log

6.9^ ^ re/D,

(15)

Re J I 3.7

Chen [57] recommended the following equation 1

-4 log

4.52 Re

Mf].|^

(16)

He also analyzed the applicability of all these equations by comparing this equation with experimental data by calculating average absolute deviation. Urbina [58], Paraqueima [59], Norum [60], von Bernuth and Wilson [61] and Venkatesan et al. [62] have published friction factor data for the extruded plastic pipes and showed that these values were higher than those obtained using the smooth pipe equation. They all correlated their experimental friction factor data and the Reynolds number in the form of the following power function for individual pipe. F^, = A Re«

(17)

The coefficients A and B were found by regression analysis and are shown in Table 1. Fluid Flow Through Curved Conduit Developing Flow Hawthorne first studied the development of flow in a straight tube to fully developed curved tube flow [63]. Development of velocity profile in the entry region Table 1 Coefficients of Power Factor in Equation 17

Investigator Paraqueima Paraqueima von Bernuth et al. von Bernuth et al. von Bernuth et al. Urbina Urbina Urbina Norum Venkatesan et al. Venkatesan et al.

Material

Inside Diameter cm

A

B

Polyethylene PVC PVC PVC Polyethylene Polyethylene Polyethylene Polyethylene Polyethylene PVC PVC

1.55 1.76 1.56 1.62 1.44 2.10 1.40 0.89 1.57 1.27 2.54

0.380 0.284 0.360 0.330 0.343 0.609 0.687 0.192 0.216 0.4063 0.4356

-0.282 -0.246 -0.257 -0.251 -0.249 -0.315 -0.330 -0.206 -0.214 -0.2558 -0.2733

Coefficients


385

was first measured by Austin and Seader using hot wire anemometer [64]. They also proposed the condition to achieve fully developed viscous flow in curved pipe as e = 49[De/(D/D)]'

(18)

Singh [65] and Yao and Berger [66] calculated the laminar flow in a curved pipe as it develops from a uniformly distributed velocity at the entrance to a fully developed profile using matched solution of an in viscid core and boundary layer flow. Numerical procedures for developing laminar flow were reported by Patankar et al [67], Humphrey [68], Roscoe [69], Rushmore and Tanlbee [70], Bara et al. [71]. Experiments to find the velocity profile were performed by Agrawal et al [72], Choi et al. [73], Kaczinosky et al [74], Shiragami and Inoue [75] and Bara et al [71]. Detailed studies on turbulent developing flow for non-Newtonian liquid was reported by Takami et al [76]. They found that the developed location as 0 = 810/(D/D)]0 5

(19)

Nandakumar and Masliyah [36] summarized the flow characteristics in the entry region. Laminar

Flow

Thomson [77,78] reported the open channel water flow through the curved surface and the effect of curvature. Williams et al. noted that the location of the maximum axial velocity was shifted towards the outer wall of a curved pipe [79]. Grindley and Gibson first observed the effect of curvature on the flow through coiled pipe with their measurements of the viscosity of air [2]. The first experimental work on water flow in laminar flow through coil was reported by Eustice [80]. He used thickwall flexible pressure tube made of grey and red rubber to avoid the variations in both roughness and inside diameter. He found that for a given pressure drop, the volumetric flow rate of water through coil was less than the rate for an equal length of straight pipe. Later, Eustice demonstrated the existence of a secondary flow by dye injection (in the upstream section) into water flow through coiled pipe [81]. Dean may be credited with the first major theoretical advancement due to his pioneering studies on fully developed single phase flow in curved pipes [82,83]. He showed mathematically the existence of one pair of counter rotating vortices for the fully developed viscous flow of a Newtonian fluid, i.e., secondary flow. He also showed that the flow in curved pipes primarily depended on the ratio of the square root of the product of the inertia and the centrifugal force to the viscous force called Dean number, which was a measure of secondary flow. He solved analytically the Navier-Stokes equation for fully developed laminar flow in a circular curved pipe for small values of Dean number by the perturbation method using a concentric toroidal co-ordinate system. He derived the following expression for De < 20, f.

1-0.03058

1,288 j

01195

2£_ 288

(20)

386


Taylor observed the effect predicted by Dean, a circulatory motion on each side of the center plane of the coil superimposed on the bulk flow, i.e., the existence of secondary flow by introducing color filament in the curved section [84]. White [85] correlated his experimental data on pressure drop and proposed relationship as f. _

•-*'-i^^"""

(21)

which is applicable for 11.6 < De < 2,000. Adler measured the velocity distribution in a coiled pipe for laminar flow [86]. He found that the velocity profile differed considerably from the parabolic one which was due to the existence of secondary flow field. Using the boundary layer theory he derived the following relationship for relatively high flow rate but laminar range, f - ^ = 0.1064 De^^

(22)

St

Prandtl [87] suggested the following empirical equation which was valid in the range of 40 < De < 2,000 f ^

= 0.37(De/2)«-^^

(23)

St

Hasson [88] proposed empirical correlation to correlate White's data in the range 30 < De < 2,000 as f - ^ = 0.556 + 0.0969 De^^^

(24)

St

Ito [89] proposed the following equation through his experimental studies on the 13.5 < De < 2,000 range f - ^ =21.5 De/(1.56 + log De)^^^

(25)

St

Kubair and Varrier [90] have recommended the following equation f = 0.7716[exp(3.553 D/D^)]Re-«5

(26)

for 2,000 < Re < 9,000 and 0.037 < D/D < 0.097. '

'

1

C

In 1963, Barua assumed that the flow through curved pipe consisted of an inviscid core plus a thin boundary layer, the flow in the core lay in planes parallel to the plane of symmetry; he used Pohlhausen momentum integral method to solve the


387

boundary layer [91]. He derived the following expression for a large Dean number, but did not give the range of Dean number for which his equation was valid. f - ^ = 0.509 + 0.0918 De0 5

(27)

St

Kubair and Kuloor generated experimental data on non-isothermal pressure drop for different aqueous solutions of glycerol flowing in helical coils of different geometries, placed in horizontal position [92]. They proposed the following equation for isothermal condition f^ = 1[2.8 + 12(D/D^)]Re-i '5

(28)

for 170 < Re < 9,000 and 0.037 < (D/D) < 0.097. For non-isothermal case they suggested multiplication of the above equation by 1.1(M,/HJ°^'. Mori and Nakayama published a comprehensive experimental and theoretical study for laminar flow in curved pipe [93]. They used approximation technique for a series solution and obtained the following equations for the first and second approximation respectively. f - ^ = 0.1080 De^^

(29)

St

f, ^ 0.1080De"^ f,, ~ (1-3.253 VDe)

^^^^

Topakoglu [94] extended Dean's series solution by developing a double power series in Dean number and curvature ratio as

l^ = l-0.03058h5£l

_ 0.1833 D£lV^ + -L D '

(31)

Ito [95] improved the boundary layer model of Barua [91] and used Pohlhausen integral method to solve it. He gave the following correlation. ^ = 0.1033De'' f.

j _ ^ 1.729Y' De ;

n.729^'' V De

(32)

Srinivasan et al. [96] conducted an experiment to generate the pressure drop data for flow through the helical coils. They suggested the following equation f^ = 5.22/[Re(DyD/^]«6 for 30 < Re < 300 and 0.0097 < D.D < 0.135.

(33)

388


Larrain and Bonilla conducted theoretical analysis of pressure drop in laminar flow of fluid in a coiled pipe [97]. They extended the series to 14th order and solved by means of computer. Austin and Seader came up with a comprehensive review of previous work and gave a detailed numerical solution in the whole laminar range [98]. Their solution, based on the vorticity field, gave excellent agreement with experiments; but it did not yield any understanding of the complex interactions between the different forces. Rao and Sadasivudu [99] studied the pressure drop through helical coil and suggested the following equation f^ = 1.55 exp (14.12 D/D^)Re-
6. II. Empirical resistance formula deduced from the logarithmic velocity distribution law /

\0.5

= 0.00807 + 0 . 4 Vêy

(57)

for Y2(D/D^)o^ < 12 where, Y^ e^ = R e ( D y D / ^ ^ ^ 0.2965

(58) (59)

for Y2(D/D^)«^ > 5.3. Spiers [114] proposed -^

= exp(27i D/D^)

(60)

Kubair and Varrier [109] suggested the following equation f^ = 0.003538 Re«o^ exp (1.887D/D^)

[61]

for 9,000 < Re < 25,000 and 0.037 < (D/D^) < 0.097. KoutsJcy and Adler [12] correlated the pressure drop in turbulent flow as f. = 0 . 0 0 7 2 5 + for

0.076(D,/DJ"^ (62)


393

Re (D/D^)2 < 300 and ^ 0.079(D,/D,f^

for Re (D/D^)2 > 300 7,000 < Re < 50,000 0.025 < D / D < 0.150 t

c

Schmidt [115] proposed the following correlation f^ = f J l + 0.0823(1 + D/D^)(D/D^)o^3Re0.25]

(54)

for 2 X 10^ < Re < 1.5 X 10^ 0.012 < D / D < 0.2 t

c

Mori and Nakayama [116] analyzed the turbulent flow through coils and proposed the following equation f^(DyD/5 = 0.075[Re(D/D^)2]-o2 {i + 0.112(Re(D/D^)V2]

(65)

Srinivasan et al. [96] proposed the following empirical equation f^ = 0.084/[Re(DyD/^]«2

(66)

for Re ., , < Re < 14,000 and 0.0097 < D / D < 0.135. critical

'

c

t

Stevens et al. [117] suggested the following equation f^ = f ^ [ 1 . 8 3 ( D / D / n

(67)

Anglesea et al. [118] correlation is as follows f^(DyD/5 = 0.00412 + 0.0788[Re(D/D^)2]-o227 for 0.5 < Re {T>pf < 85.

(68)

2 X lO'* < Re < 1.5 X 10^ Rao and Sadasivudu [99] developed the following correlation from their experimental data as f^ = 0.0382 exp (11.17 D/D^) Re^^

(69)

394


for

Re .,

< Re < 27,000

critical

'

and 0.0159 < D/D < 0.0556 t

c

Mishra and Gupta [102] conducted experiments through different types of helical coils to generate the pressure drop data for turbulent flow condition and proposed the following empirical correlation f^ - f ^ = 0.0075(D/D^)«^

(70)

where, f^^ is calculated from Equation 1 and it is valid for the Reynolds number range of'4,500 to 10^ D/D^ range of 0.289 x 10^ to 0.15 and p/D^ range of 0 to 25.4. Singh and Mishra [119] showed that Equation 70 also satisfies the pressure drop data for non-Newtonian pseudoplastic liquid flow through helical coils. Hart et al. [104] observed that their experimental data were well represented by the equation proposed by White [85]. They also plotted the friction factor as a function of Reynolds number with D/D^ as a parameter. All the above correlations deal with smooth tube data. But Ruffell conducted an experiment on water/steam flow through commercial helical tube after some period of commercial operation [120]. He correlated his experimental single phase pressure drop data as f^. = 0.00375 + 0.633 (D/D/275 Re-o.4

(71)

for 5 X 10^ < Re < 6 X 10^ 0.014 < D/D^. < 0.229 and f^ = 0.0475 + 193.5 (D/Df-^''

Re"'"

(72)

for 6 X lO'* < Re < 2 X 10^ 0.0054 < D/D^. < 0.0208 Gill et al. compared their experimental single phase pressure drop data with the existing correlations and confirmed that the roughness had a large effect [121]. But they did not give any correlation. Das conducted an experiment on water flow through helical coils in turbulent condition [122]. The coils were made of thick-walled, flexible, transparent PVC pipes with finite roughness. Detailed dimensions of the coils are shown in Table 2.


395

Table 2 Dimension of the IHelical Coil Tube Diameter Dt m

Roughness Height m X 10^

Ratio of Diameter of Tube to Helical Coil D/Dc

0.008 0.008 0.008 0.0127 0.0127 0.0127

33.2 33.2 33.2 31.8 31.8 31.8

0.0308 0.0374 0.0635 0.0478 0.0578 0.0964

£

Initially he conducted the experiment on water flow through horizontal pipe (same PVC pipe for both cases) to find the roughness values (Venkatesan et al. [62]. He compared the experimental friction factor for coil with the values obtained from smooth coil equation and observed that the experimental friction factor was higher than that of smooth coil. He found that as coil diameter increased the friction factor decreased (Figure 2). He presented a generalized correlation for predicting the frictional pressure drop across the rough helical coils as f^^ - f^ = 17.5782 Re-o-3137 (D/D^)0-362i (e/D/-^^^^

(73)

where f^ represents the Mishra and Gupta [102] Equation 70. EXPERIMENTAL STUDIES The literature on experimental studies of curved-pipe flows is extensive. Most of these have been mentioned. Most experimental work on the flow through curved geometry was on the measurement of friction factor of coil (f^) or the ratio of friction factor iiJ^J- Adler measured axial velocity distributions for fully laminar condition by pilot tube in curved circular pipes [86]. His results clearly indicate the existence of the maximum velocity towards the outer wall of the curved pipe. Similar results are obtained for turbulent flow. More recent works on developing flow are based on considering the inlet condition, either inviscid with flat velocity profile produced by a bell-mouth entry from a large chamber (Agrawal et al. [72] or a fully developed straight pipe velocity profile. Laser Doppler velocimeter is used to measure the axial and secondary velocity distributions (Agrawal et al. [72], Humphrey et al. [123], Taylor et al. [124], Enayet et al. [125], Hille et al. [126], Azzola et al. [127], Takami et al. [76], Bara et al. [71], Webster and Humphrey [39] etc.) in the curved geometry. Typical experimental setup used by Das [122] is shown in Figure 3. The experimental apparatus consisted of a water storage tank, a centrifugal pump, a test section, control and measuring systems for the flow rate, and pressure drop. The pipes were wound round a wooden cylindrical frame of known diameter to form a helical coil. The coil diameter could be varied by changing the diameter of the

396


0.06

O

PIPE DIAMETER = 1. 27 SYMBOL 0 ^ X — £S, —

0.02

cm Dt/pc 0 0478 0 .0578 0. 0 9 6 4

o

< u.

0.01

o cr 0.006 0.004h

±

0.4

J

J

LJ_

l__L

0 . 8 1.0 3.0 6.0 9.0 REYNOLDS NUMBER, Re x 10^

Figure 2. Variation of friction factor with Reynolds number.

V2 •tXH

H.C.

iiiiiiiiiiiiiiiiimm

rS TO FROM TANK TANK (underground)

MANOMETER TO TANK

Figure 3. Schematic diagram of the experimental setup: H. C. = helical coil, P = pump, V,, Vg = valves.


397

wooden frame. The tubes were wound in a closed packed fashion so that pitch equalled the outer diameter of the tube and was maintained constant for all cases. All the systems were more than 10 m long. Pressure drops were measured over 3 m long section by means of simple U-tube manometer, containing mercury beneath water. Sufficient upstream and downstream length were provided to achieve fully developed flow. Measurement of Relative Roughness The average velocity of the fluid flowing in a pipe is greatly affected by the pipe roughness. So the relative roughness is one of the most important factors for all pipe flow calculations. The relative roughness is defined as e/D^. There are several approaches that could be taken for calculating the relative roughness factors. 1. The Colebrook Equation 5 is to force e to be non-negative to calculate e/D^. 2. According to Urbina [58], Paraqueima [59], and Norum [60] the maximum value for the second coefficient in the Colebrook equation is 1.67 if the nonnegative assumption is retained. 3. The third approach is to allow both the roughness and the second coefficient to be determined by regression. Von Bernuth and Wilson [61] calculated the roughness factor by this approach. Venkatesan et at. [62] and Das [122] used the second method to calculate the relative roughness. NOTATION A, B D D De

Coefficient in Equation 17 Diameter, m Diffusivity, m^/s Dean number. Re (D/D f \ dimensionless f Fannings friction factor, dimensionless Kj, K^ Constants in Equation 52

p Pitch, m r Radius, m Re Reynolds number VDp/|x, dimensionless Sc Schmidt number jo/pD, dimensionless V Velocity, m/s

Greek Letters 8 Thickness of shedding layer, m e Roughness height, m 0 Angle measured in the axial direction, deg.

|X Viscosity, Ns/m^ p Density, kg/m^

Subscripts b c g 0)

Bulk Coil Generalized Wall

t cr MR St

Tube Coil with finite roughness Metzner-Reed Straight

398


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CHAPTER 14 MODELING COALESCENCE OF BUBBLE CLUSTERS RISING FREELY IN LOW-VISCOSITY LIQUIDS C. W. Stewart Pacific Northwest Laboratory Richland, WA 99352' CONTENTS INTRODUCTION, 405 BACKGROUND, 406 BUBBLE MOTION WITH RESPECT TO THE LIQUID, 407 EXPERIMENTAL OBSERVATIONS OF BUBBLE INTERACTION, 409 A STOCHASTIC MODEL FOR BINARY COALESCENCE, 413 A CLUSTER COALESCENCE MODEL, 418 COMPARISON WITH BUBBLE SIZE DATA, 420 RESULTS OF THE PULSED SWARM EXPERIMENT, 422 CONCLUSION, 426 NOTATION, 427 REFERENCES, 427 INTRODUCTION The mechanisms by which freely rising bubbles interact with each other in relatively low-viscosity liquids and, specifically, how they approach, contact, and coalesce or break up are important aspects of multi-phase flow. Coalescence and breakup can control the interfacial area and mass transfer rate in bubble columns and gas-sparged chemical and biological reactors. Bubble interaction is fundamental in two-phase flow instability that plagues boilers and oil and gas wells. But bubble interaction remains a relatively mysterious area. Good models for bubble swarm dynamics, coalescence and breakup rates, interfacial area transport, and bubble size distributions must be based on real physical phenomena. Bubbly flow instability, for example, has typically been treated as a * This research was supported by Conservation and Renewable Energy, Office of Industrial Technologies, U.S. Department of Energy, through the Northwest College and University Association for Science (Washington State University) under Grant DE-FG06-89ER-75522. Publisher asknowledges the U.S. Government's right to retain a non-exclusive, royalty-free license in and to any copyright covering this paper. 405

406


kinematic wave, described mathematically by the eigenvalues of a linearized system of mass and momentum equations for the gas and liquid fields. But stability conditions so derived have not yet been related to actual bubble motion. The mathematics could be interpreted more clearly if the interaction creating the kinematic wave was known. This chapter presents the results of a study of bubble interaction that provides new insights into the process. Visual studies of pulsed swarms of 8-25 bubbles revealed a basic interaction mechanism fundamental to bubble coalescence and breakup. Groups of bubbles interact by moving in and out of clusters. This may also account for bubble flow instability and the transition from bubbly to slug flow. This chapter gives an analysis of the effects of clusters and presents several models for predicting bubble size distributions. The following section gives some of the background on bubble dynamics and interaction that will be useful in this study.

BACKGROUND The density and viscosity of the gas can be neglected in favor of the liquid properties, and the dynamic behavior of a single bubble can be correlated with three independent, dimensionless groups commonly defined as the Reynolds number, Eotvos number, and Morton number (Grace et al., [1]). These are given, respectively, by

and Eo

gpD?

and M-

.

where the density and viscosity are those of the liquid. The equivalent diameter, Dg, is given in terms of bubble volume, V^^ by -,1/3

D. =

71

Bubble shape can be correlated with some precision on a map of Re vs. Eo with M as a parameter [1]. Changes in bubble behavior and shape transitions in liquids of differing viscosities lead to the classification of liquids as "high-M" or "low-M." The boundary between low-M and high-M liquids is approximately M = 4(10~^), where a local maximum in terminal velocity begins to appear as M increases [2].

Modeling Coalescence of Bubble Clusters

407

A bubble's dynamic behavior is related to its shape. Ellipsoidal bubbles in lowM liquids exhibit a wobbling rise path. Wobbling begins at a Reynolds number of about 200 [3,4]. The wobble period is constant once established, but its onset is highly dependent on Morton number. From photographs of the wake, Lindt and DeGroot [5] found that the period became constant when what they described as an attached helical vortex reached its maximum length. They further observed that the transition from ellipsoidal to cap shape was coupled with significant changes in wake structure. The turbulent wake structure consists of a chain of looping horseshoe vortices originating at the bubble base [6]. The bubble wake is the main driver of interaction. If a bubble enters the rising column of liquid in another's wake, it will usually overtake the leader in an inline collision that may result in coalescence or breakup. Successful coalescence usually follows in-line collisions of large cap bubbles in relatively viscous liquids, whereas pairs of smaller ellipsoidal or spherical bubbles tend to repel each other. Bubbles in less-viscous liquids tend to have turbulent wakes and do not coalesce readily. The turbulent wake, especially behind wobbling bubbles, has a weak downstream influence on coalescence because it is irregular and intermittent, and turbulence often causes trailing bubbles to break up in the wake. Collisions in lowviscosity liquids occur at high relative velocity. This traps a liquid barrier between them that prevents coalescence [7-12,2]. The transition between viscous "coalescing" liquids and less-viscous "non-coalescing" liquids is approximately at M = 4(10~^), the same as was noted between "high-M" and "low-M" liquids above. Individual coalescence and breakup events are essentially impossible to observe in a bubble swarm. They must be inferred indirectly from the bubble size distribution. Instrumentation has not yet been able to resolve the details of the coalescence process itself [13-15]. Because of this, models of bubble swarms often use simple exponential coalescence rates assuming a random binary process [16,17]. More complex computational models using Monte Carlo methods have attempted to predict bubble size distributions for a combination of breakup and coalescence. These models typically treat bubble coalescence by analogy with the kinetic theory where bubbles are assumed to act as solid particles [18,19]. They use a binary collision rate (probability) and a collision efficiency factor to account for collisions that do not lead to coalescence. Since collision is assumed to be a random process in these models, turbulence of the same scale as the bubbles or smaller would increase collisions and, therefore, also increase the coalescence rate. BUBBLE MOTION WITH RESPECT TO THE LIQUID Before studying multiple bubble interactions, we must understand the behavior of a single bubble in response to acceleration of the surrounding continuous liquid. Neglecting surface tension effects and0 turbulent fluctuations smaller than the scale of the bubble, and assuming incompressible adiabatic flow, Stewart and Crowe [20] derived the averaged momentum conservation equations for dispersed bubbles and the surrounding continuous liquid [20]. These can be expressed, respectively, by: DhUh

«bPb - ^ p + ttbPfCv

DbUb

Dt

Dt

= a.p^g + a,V • Of - a , p f K , f ( U , (1)

408


ML Dt

= a^p^g + ttfV • o, - a^pfK,, ( U , - U^)

Dt

(2) where a is the volume fraction, C,,^, is the virtual mass coefficient, a^ is the stress VM

'

I

field in the continuous liquid (consisting of both pressure and shear stress), and Kj^^ is the interphase drag coefficient. The total derivative following the bubbles is

Dt

at

'

with a similar expression for the derivative following the liquid. Combining (1) and (2) by eliminating the stress term yields a single equation for the relative velocity,

1 + —^ K.

M ^ + U, Dt

^

' VUf + •

a, J

Pf •UR

=

\ . ^ ' Pf

af

1 + —^ J

Dt

-g

Pf

(3) Particles develop relative motion only if their density differs from the surrounding liquid. The source term on the right side of (3) is identically zero if p^ = p^. Neutrally buoyant particles can have no relative motion unless it is imposed as an initial or boundary condition. The sign of the right side reverses for density ratios greater or less then 1.0 so that heavier particles must move slower and lighter ones faster than the liquid. This is why bubbles always lead and solid particles always lag the liquid motion. The same sign change affects the gravity term, making heavy particles fall and light ones, such as bubbles, rise. In the case of a few spherical bubbles, where a^ ~ 0, p^^ m ..

(13)

The probability that a bubble size will be larger than m after k coalescences is equal to the number larger than m, divided by the total number. This probability is the "exceedance," which is the complement of the cumulative distribution function. That is, F(m|k) = l - F ( m | k ) =

^

Then, dividing (13) by N^*"^ and using the above definition, we can write ciF(m|k) 1 -^ ,^^ — — ^ = --F(mk) dm |LI

(14) ^ ^

The exponential distribution Jfunction, Equation 12 is a solution of Equation 14 with the initial condition that F (Oik) = 1 and using the definition of the density

dm

dm

The expression (Equation 13) approximates the probabilities (Equation 6) and (Equation 7) by letting the coalescence probability of a range of bubble sizes depend on the number available in that range while ignoring all the detailed permutations of different size combinations. For sufficiently large N^"^ and k, this is a satisfactory approximation. A CLUSTER COALESCENCE MODEL The bubble size distribution from a completely random, binary coalescence process is modeled well by the geometric and exponential distributions. We now develop a simple model for non-binary, clusterwise coalescence. In the random binary model (13), that leads to the exponential distribution, the effect of coalescence is linear. But now assume that coalescence occurs not between pairs of bubbles, but simultaneously among clusters of N^ bubbles. Then the change in the number of bubbles with volume, m, is the product of the number in the cluster (dN^^dN^.) and the change in the number of clusters with volume. That is,


dm

dNc

dm

419

^^^^

Now assume further that both the cluster size and the cluster coalescence rate are linearly proportional to N . With proportionality constants, a and (J,

= aN>, dNc . = -pN, dm

Substituting these relations into (15) and letting X - a^ yields dN.

= -^N^

dm

(16)

Cluster coalescence changes the dependence on number from linear to quadratic. That is, we should expect to find the number of bubbles decreasing much more rapidly with size because more of the smaller bubbles combine into a few much larger ones. To find the distribution function corresponding to (16), we replace N^^ with the exceedance by dividing by N^^^ as before, and let the exponent on the right side be a parameter, r| > 0, rather than an exact quadratic, r| = 2.

%

^ dm

= -?lF(m|kr

(17)

Equation 17 can be solved directly by separating variables with the initial condition F(Olk) = 1 to yield f(m|k) = - p 1 + H |

(18)

0 where new parameters have been defined as

Tl-1

X(Tl-l)

It is surprising and encouraging that (18) is exactly the definition of the Pareto distribution function. This distribution was originally used in economics as a model for income during earlier times when a very few were wealthy. More recently, it has been used to characterize the length of wire between flaws and in biomedical

420


problems such as survival time following a heart transplant. The Pareto distribution also arises from compounding the exponential and gamma distributions modeling a collection of systems with varying failure rates [26]. In our application this would be analogous to modeling systems (clusters) of bubbles with different coalescence rates—exactly the idea we started with. COMPARISON WITH BUBBLE SIZE DATA Five bubble size distributions were selected from the literature to compare with the binary and cluster coalescence models. Four were measured in small-scale bubble column test sections and one in a sieve tray. Newtonian systems of air-water, oxygen-water, and oxygen-glycol, and a non-Newtonian oxygen-PAA solution were used. The data sets are described in Table 1. The mean bubble diameter is the equivalent diameter of a sphere of equal volume. Bubble shapes are generally ellipsoidal, though larger cap bubbles of ~2 cm in diameter appear in several tests. Comparisons of continuous distributions to discrete experimental data must accommodate the size ranges or "bins" selected by the investigators for the bubble count. The measured probability of bubbles at a given size, V, is actually the fraction of those measured that falls in a range, say v, to V2. N N.. This can also be written as the difference of probabilities = P[v, < V < V2] = P[V < V 2 ] - P [ V < V,]

(19)

The probabilities on the right side of (19) are represented by the cumulative distribution function, F(v). Thus, we can compare the models with the data by computing Table 1 Bubble Size Distribution Data Reference

Test Section

System

Mean Dia.

Akita and Yoshida [27]

Column 15 X 15 cm

O2 - glycol

6.0 mm

Burgess and Calderbank [28]

Sieve tray 24 X 30 cm

Air - water

16.2 mm

Nakanoh and Yoshida [29]

Column 15 cm dia.

O2 - water O2 - PA A 0.05%

2.4 mm 2.0 mm

Photography

Prince and Blanch [19]

Column 27 cm dia.

Air - water

4.6 mm

Direct sample

Measurements Photography Resistivity probe


421

= F(vJ - F(v,) where the cumulative distribution functions for the exponential and Pareto distributions are obtained by integrating the densities in Equation 12 and 18, respectively, to yield FHXp(v) = e^-'"^^>

FPAR(V) =

1+

m (K-DjLl

K> 1

where the mean, j ^ is the mean volume of each specific data set. In making the comparisons, the Pareto parameter, K, was adjusted to give the best fit. This made the parameter very close to 1.0 in all cases. This corresponds closely to the original quadratic relation, Equation 16, confirming the basic assumptions. Examples of the behavior of the two models on Akita and Yoshida's [27] oxygenglycol and Nakanoh and Yoshida's [29] oxygen-water distributions are shown in Figures 8 and 9. The probability is plotted against the normalized bubble volume, m/|i, for consistency. The exponential distribution clearly misses the tail of the distribution completely. This characteristic "fat tail" of the data cannot be matched with the binary model by adjusting coalescence probabilities. In fact, since the completely random assumption gives the smallest bubbles the highest coalescence probability, it creates the maximum number of large bubbles of any binary coalescence process. The exponential distribution cannot be made to fit any better since there

o.oH

0.0001 H

A

Akita & Yoshida (O2- Glycol) Exponential Pareto

10' 0.001

0.1

10

1000

Relative Bubble Volume, m/\i

Figure 8, Model Comparison—Akita and Yoshida's Data

422


0.01

0.0001 H A

Nakanoh & Yoshida (O^ - HÔ)

\

Exponential Pareto r6

10

0.01

0.1

1

10

100

Relative Bubble Volume, mJ\i

Figure 9. Model Comparison—Nakanoh and Yoshida's Data

is no free parameter to adjust. In contrast to exponential, the Pareto distribution has such an extremely "fat tail" that the mean is actually undefined for K < 1 (or, equivalently, r| > 2). That is, it will assign some small but non-zero probability to extremely large bubbles at the expense of far fewer small ones—exactly the trend required by the data. That the Pareto model fits the data much better than the binary exponential model is shown very dramatically in Figures 10 and 11, which compare predicted to measured probabilities for all five data sets. The Pareto clustering coalescence model shows a surprisingly good match over the entire range of bubble sizes. On the other hand, the exponential binary coalescence model hardly shows any correlation to the data. Prince and Blanch's [19] bubble size distribution is the only one that appears to match the exponential model in Figure 10. They designed an experiment to achieve an equilibrium with equal breakup and coalescence rates. If coalescence is clusterwise and breakup is binary, it would take many binary breakup events to balance one coalescence of a large cluster, and the binary process, which is characterized by an exponential distribution would dominate.

RESULTS OF THE PULSED SWARM EXPERIMENT The collision is the best unit for quantifying bubble interaction since collisions form clusters and chimneys and are the precursors to coalescence and breakup. A collision is defined as the end result of a successful wake capture event: the probability that a bubble will experience a collision while rising a distance equal


423

Exponential Model

•^

0.01

Xi

I

A

Nakanoh (O2- H2O)

o

Nakanoh (O - 0.05% PAA)

D

Akita (O^- Glycol)

O

Burgess (Air - H2O)

a

Prince (Air - HÔ)

0.0001

0.0001

0.01

Measured Probability

Figure 10. Predicted vs. Measured Bubble Size Distributions—Exponential Model

Pareto Model J

f»>

1

0.01-

•s i:

L

•

Nakanoh (O -H2O)

•

Nakanoh (O2- 0.05% PAA)

•

Akita (O2- Glycol)

•

Burgess (Air -H2O)

a

Prince (Air - H2O)

y

x^

^

/^^m

0

1

1

0.0001 -

y^

•*

—I 0.0001

1

r— 0.01

Measured Probability

Figure 11. Predicted vs. Measured Bubble Size Distributions—Pareto Model

424


to its equivalent diameter. The collision probability data are shown plotted against the bubble volume fraction in Figure 12. There is a volume fraction threshold of 0.03 below which no collisions occur. Above this, the probability becomes approximately proportional to a^''^. More precisely, the collision probability varies with a according to the curve fit, (20)

P(a) = 0.19 ( a - 0 . 0 3 4 ) '

Since coalescence occurs only after a collision, we define the coalescence efficiency, h^,^^, as the probability that a collision will result in a coalescence. Then the overall probability of a bubble coalescing with another while rising one equivalent diameter is the product of the collision probability and coalescence efficiency, P . = P , n , . The coalescence efficiency determined from the data is J '

else

cisn 'else

-^

plotted against M in Figure 13. This shows that coalescence occurs at preferred bubble sizes, and the size increases with M. There is no correlation of coalescence efficiency with volume fraction. Finally, recall the assertion that the coalescence rate and bubble size distribution resulting from coalescence can be modeled by the Pareto distribution for which the coalescence rate is proportional to a power of the number of bubbles. A simple cluster coalescence model makes this power equal to two. This means that the number of coalescences observed in a release ought to be roughly proportional to the square of the number of bubbles released. Figure 14 shows that this is true. The J

0.009 I o

0.007-|

L

_L

M = 3.2e-ll

s

M=1.7e-8

a

M = 5.3e-10

ffl

M = 2.0e-7

D

M = 2.1e-9

•

M = 3.7e-6

P = 0.19 (a- 0.034)1-48

0.005-1 c 0.003

0.001 -A

-0.001 0

0.02

0.04

0.06

0.08

0.1

0.12

a. Figure 12. Collision Probability vs. a—Data by M

0.14


-L

0.16 0.14 H 0.12-]

-i^—Eo= 18 O - E6 = 21 S - - E6 = 24

o Eo = 9 -H3--- E6=12 -ffl Eo= 15

0.1 -I 0.08 0.06 H 0.04

cr/

0.02 0 10-^1

10-^ M

10-'7 (g|ii4p-ia-^)

Figure 13. Collision Efficiency vs. M 0.8

J

L

15

20

0.7 0.6 0.5 Pi 0.4 O

U o

0.3 0.2 0.1 0 25

No. of Bubble Released Figure 14. Coalescences vs. Bubbles Released

425

426


number of coalescences per release is approximately proportional to a quadratic function of the number of bubbles in the release. Though this is consistent with the cluster coalescence model, it is surprising since nearly all the coalescences observed were binary. It is not necessary to assume simultaneous coalescence of clusters to derive a model with a coalescence rate proportional to some power of the number of bubbles. Recall that we quantified the coalescence probability as the product of a collision probability and coalescence efficiency, P , = P , ri , . The coalescence density, or L

J

J ^

CISC

clsn 'else

-' '

coalescence rate per unit volume, N^,^^, is directly proportional to P^,^^. Then, with r|^,^^ approximately constant (ignoring effects of bubble size) and P^,^^ proportional to a^^^, we have else

The volume fraction, a, represents the product of the number density, N^^, and individual bubble volume, v.. That is, D

« = N,v, Again ignoring variation in bubble size, this leads to the relationship N , - ^l'^ else

b

The Pareto distribution fits this relation just as it did the cluster coalescence model. Recall that the tail of the Pareto distribution is so "fat" in the latter case that the mean is not even defined. The situation is not quite so severe with the 3/2 exponent—the mean is defined, but the variance is not. We conclude that the Pareto distribution need not require that all coalescences occur simultaneously in clusters, but can also apply to a binary process. CONCLUSION Bubble interaction in swarms is a complex process. Bubble clusters commonly form that coalesce more or less simultaneously into very large bubbles. Recent experiments have revealed some of the details behind this behavior. A bubble contacts another only by following its wake to an overtaking collision. Coalescence or breakup occurs only after the collision, when one bubble is pulled into the near wake of the other. Interaction of three or more bubbles in clusters leads to increased coalescence rates. We have also shown analytically that bubbles do not "collide" like solid particles, but rather are drawn together by the dynamics of the surrounding fluid. Gravity and fluid acceleration drive bubble motion; small-scale turbulence tends to prevent rather than enhance coalescence. The bubble size distribution resulting from purely random binary coalescence is well-represented by the geometric and exponential distributions. But these distributions completely miss the behavior of measured bubble size distributions that show relatively fewer of the smallest bubbles and more of the very largest ones. But a simple cluster coalescence model follows the Pareto distribution, which matches these characteristic trends quite well. We conclude that multiple bubble interaction


427

in clusters must be considered in order to accurately predict bubble swarm behavior and the evolution of the size distribution. NOTATION C,,^^ Virtual mass coefficient VM

D^ Bubble equivalent diameter, m E6 Eotvos number (gD^pa^ f(m\k) Probability density of volume m after coalescence event k F Cumulative distribution function F Exceedance function, complement of F g Gravitational acceleration, m/s^ k Number of coalescences K^^ Drag coefficient between bubbles and liquid, s~* m Bubble volume, integer multiples of initial volume M Random bubble volume corresponding to m

M Morton number (g|i4p"'a~^^ N Number of bubbles N^°^ Initial number of uniform bubbles N^*"^ Number of bubbles existing after coalescence event k N Number of bubbles with >m

volume larger than m p Parameter in the geometric distribution P Probability Re Reynolds' number (pU^D^ji^ U^ Velocity of the liquid phase, m/s U^ Bubble terminal rise velocity Uj^ Relative velocity of bubbles to the surrounding liquid, m/s V^ Bubble volume, m^ b

Greek Letters a Phase volume fraction 0 Parameter in the Pareto distribution T| Exponent in cluster coalescence model K Parameter in the Pareto distribution X Rate constant in cluster coalescence model

|i Mean bubble volume, integer multiples of initial volume or m^ |LI Liquid dynamic viscosity, Ns/m^ p Phase density, Kg/m^ a Surface tension, N/m

REFERENCES 1. Grace, J. R., T. Wairegi, and T. H. Nguyen, "Shapes and Velocities of Single Drops and Bubbles Moving Freely Through Immiscible Liquids," Trans. Inst. Chem. Engrs., 54, 167 (1976). 2. Bhaga, D., and M. E. Weber, "In-Line Interaction of a Pair of Bubbles in a Viscous Liquid," Eng. ScL, 35, 2467 (1960). 3. Haberman, W. L., and R. K. Morton, "An Experimental Investigation of the Drag and Shape of Bubbles Rising in Various Liquids," David Taylor Model Basin Report No. 802 (1953). 4. Tsuge, H., and S. Hibino, "The Onset Conditions of Oscillatory Motion of Single Gas Bubbles Rising in Various Liquids," J. Chem. Eng. Japan, 10, 66 (1977). 5. Lindt, J. T., and R. G. F. DeGroot, "The Drag on a Single Bubble Accompanied by a Periodic Wake," Chem. Eng. Sci., 29, 957 (1974). 6. Yabe, K., and D. Kunii, "Dispersion of Molecules Diffusing from a Gas Bubble into a Liquid," Int. Chem. Eng., 18, 666 (1978).

428


7. Nevers, N. D., and J-L. Wu, "Bubble Coalescence in Viscous Fluids," AIChE Journal, 17, 182 (1971). 8. Crabtree, J. R., and J. Bridgwater, "Bubble Coalescence in Viscous Liquids," Chem. Eng. ScL, 26, 839 (1971). 9. Narayanan, S., L. H. J. Goosens, and N. W. F. Kossen, "Coalescence of Two Bubbles Rising in Line at Low Reynolds Numbers," Chem. Eng. ScL, 29, 2071 (1974). 10. Lockett, M. J., and R. D. Kirkpatrick, "Ideal Bubbly Flow and Actual Flow in Bubble Columns," Trans. Inst. Chem. Engrs., 53, 267 (1975). 11. Komasawa, L, T. Otake, and M. Kamojima, "Wake Behavior and Its Effect on Interaction Between Spherical Cap Bubbles," J. Chem. Eng. Japan, 14, 103 (1980). 12. DeKee, D., P. J. Carreau, and J. Mordarski, "Bubble Velocity and Coalescence in Viscoelastic Liquids," Chem. Eng. Sci., 4 1 , 2273 (1986). 13. Otake, T., Tone, S., Nakao, K. and Mitsuhashi, Y., 1977, "Coalescence and Breakup of Bubbles in Liquids," Chem. Eng. Sci., 32, 377-383. 14. Oolman, T. O., and H. W. Blanch, "Bubble Coalescence in Air-Sparged Bioreactors," Biotechnology and Bioengineering, 28, 578 (1986). 15. Greaves, M. and M. Barigou, "Bubble Size Distributions in a Mechanically Agitated Gas-Liquid Contactor," Chem. Eng. Sci., 47, 2009 (1992). 16. Calderbank, P. H., M. B. Moo-Young, and R. Bibby, "Coalescence in Bubble Reactors and Absorbers," Proc. Third European Symposium on Chemical Reaction Engineering, Amsterdam, 1964, 91 (1964). 17. Miller, D. N., "Interfacial Area, Bubble Coalescence, and Mass Transfer in Bubble Column Reactors," AIChE Journal, 29, 312 (1983). 18. Koetsier, W. T., and D. Thoenes, "Coalescence of Bubbles in a Stirred Tank," Proc. 5th European/2nd International Symp. on Chem. Reaction Eng., Amsterdam, 2-4 May, 1972, B3.15 (1972). 19. Prince, M. J. and H. W. Blanch, "Bubble Coalescence and Break-up in AirSparged Bubble Columns," AIChE Journal, 36, 1485 (1990). 20. Stewart, C. W., and C. T. Crowe, "Bubble Dispersion in Free Shear Flows, accepted for publication n the International Journal of Multiphase Flow (1992). 21. Inoue, A., Y. Kozawa, M. Yokosawa, and S. Aoki, "Studies on Two-Phase Crossflow. Part I: Flow Characteristics Around a Cylinder," Int. J. Multiphase Flow, 12, 149 (1986). 22. Hills, J. H., "The Rise of a Large Bubble Through a Swarm of Smaller Ones," Trans. Inst. Chem. Engrs., 53, 224 (1975). 23. DeKee, D., R. P. Chhabra, and A. Dajan, "Motion and Coalescence of Gas Bubbles in Non-Newtonian Polymer Solutions," J. Non-Newtonian Fluid Mech., 37, 1 (1990). 24. Stewart, C. W., Coalescence of Ellipsoidal Bubbles Rising Freely in LowViscosity Liquids, Ph.D. Dissertation, Washington State University, Pullman, Washington (1993). 25. Stewart, C. W., S. C. Saunders, and C. T. Crowe, "Bubble Size Distributions in Random Coalescence and Breakup," Proc. ASME Fluids Engineering Conference, Los Angeles, California (June 1992).


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26. Saunders, S. C. and J. M. Myhre, "Maximum Likelihood Estimation for TwoParameter Decreasing Hazard Rate Distributions Using Censored Data," /. American Statistical Society, 78, 664 (1983). 27. Akita, K., and F. Yoshida, "Bubble Size, Interfacial Area, and Liquid-Phase Mass Transfer Coefficient in Bubble Columns," Ind. Eng. Chem., Process Des. Development, 13, 84 (1974). 28. Burgess, J. M., and P. H. Calderbank, 'The Measurement of Bubble Parameters in Two-Phase Dispersions - I: The Structure of Sieve Tray Froths," Chem. Eng. Sci., 30, 1107 (1975). 29. Nakanoh, M., and F. Yoshida, "Gas Absorption by Newtonian and NonNewtonian Liquids in a Bubble Column," Ind. Eng. Chem., Proc. Des. Development, 19, 190 (1980).


CHAPTER 15 OXYGEN TRANSFER IN NONNEWTONIAN FLUIDS STIRRED WITH A HELICAL RIBBON SCREW IMPELLER A. Tecante and E. Brito de la Fuente Departamento de Alimentos y Biotecnologia Facultad de Quimica - UNAM Mexico D.F. 04510, Mexico and L. Choplin GEMICO-ENSIC 1 rue Grandville, B.P. 451 Nancy, 54001, France and P. A. Tanguy Departement de Genie Chimique Ecole Polythecnique de Montreal Station Centre Ville, Montreal, H3C 3A7, Canada CONTENTS INTRODUCTION, 432 USE OF HELICAL IMPELLERS IN NON-NEWTONIAN GAS-LIQUID SYSTEMS, 434 VESSEL AND IMPELLER DIMENSIONS, 435 AGITATION AND AERATION CONDITIONS, 435 Kâ DETERMINATION, 435 RHEOLOGICAL PROPERTIES OF THE LIQUID PHASE AND FLOW REGIME, 437 GAS-LIQUID MASS TRANSFER AND BUBBLE BEHAVIOR, 438 Effect of Power Input, Superficial Gas Velocity, and Polymer Concentration on Kâ, 438 CMC Solutions, 438 XTN Solutions, 441 PAA Solutions, 444 Effect of Apparent Viscosity on Kâ, 447 431

432


OVERALL EFFECT OF OPERATION CONDITIONS AND RHEOLOGICAL PROPERTIES, 447 CONCLUDING REMARKS, 499 ACKNOWLEDGMENTS, 450 NOTATION, 450 REFERENCES, 451 INTRODUCTION Gas-liquid systems are frequently found in chemical and biochemical engineering processes. In aerobic fermentations, for example, cells require oxygen to carry out their metabolic functions and to produce the metabolite of interest. Because of the low solubility of oxygen in water and culture media, air must be continuously supplied and dispersed throughout the liquid phase. When this process takes place in mechanically agitated vessels the impeller must be able to maintain homogeneous mixing of the liquid phase, and sufficient dispersion of gas to provide adequate environmental conditions for cell growth and productivity. The dynamics of oxygen transfer depend on operation variables like temperature, oxygen partial pressure, air flow rate, and agitation speed, and on design variables such as aeration performance, sparger and impeller design, agitation power, and vessel geometry. The simultaneous interaction of these variables makes the gas-liquid transport process particularly complex. The influence of operation and design variables on oxygen transfer in biochemical reactors in which the liquid phase exhibits a Newtonian behavior has been the object of many studies and has been summarized in a number of papers [1,2]. Systems in which the liquid phase has a non-Newtonian nature are of much more interest because they are more common and more complex. This is the case, for example, of extracellular microbial polysaccharides (EMPS). These materials are biopolymers with large actual and potential applications in numerous industries, like food, pharmaceutical, textile, and petroleum, because they can be used as viscosityenhancing, emulsifying, thickening, suspending, and gelling agents. They are produced by growing selected microorganisms onto simple substrates under aerobic conditions. A characteristic feature of EMPS production is the remarkable evolution with fermentation time of the rheological properties of the broth in which a particular biopolymer is produced. At the beginning, the liquid phase is Newtonian, but as the EMPS accumulates the broth becomes highly non-Newtonian and rheologically complex. Rheological complexities can go from shear-thinning to highly viscoelastic properties as observed during the production of puUulan [3]. The rheological behavior as well as its evolving nature have considerable effect not only on cell growth kinetics and transport processes within the vessel, but also on the aeration capacity, impeller power consumption, and mixing patterns which together with sparger, impeller and vessel geometries constitute key variables for bioreactor design. Traditionally, EMPS are produced in vessels equipped with Rushton turbines or other radial impellers. The popularity of these agitators lies in their high gas-liquid dispersing capacity resulting from the passage of the inflowing gas through their

Oxygen Transfer in Non-Newtonian Fluids

433

high shear rate region. They provide adequate mixing and gas dispersion at the early stages of fermentation, but due to the evolution with time of rheological behavior they are unable to maintain homogeneous mixing and gas dispersion assuring an adequate oxygen transfer along fermentation. As the broth becomes more viscous and non-Newtonian the well-mixed zones become more and more confined to the immediate vicinity of the impeller. It has been reported that during xanthan gum production in a Rushton-agitated vessel, the mixing state of the fluid phase can be divided into three regions [4,5]: (1) the micromixing region surrounding the impeller, where mixing is largely dominated by radial flow. Outside this region, cell starvation occurs due to improper oxygen transfer, bad distribution of nutrients and accumulation of undesirable metabolic products; (2) the macromixing zone where slow circulating flow dominates; and (3) a stagnant region far from the impeller where the broth is motionless. The evolution with time of broth rheology can be eliminated by using nonNewtonian solutions. Absence of living cells eliminates the time change of physical and rheological properties of the liquid phase and allows variation of operation conditions over wider ranges. Nevertheless, either in actual EMPS broths or the nonNewtonian solutions used to simulate their flow behavior inhomogeneous mixing and low oxygen transfer rates are frequently observed in tanks equipped with paddles [6], Intermig impellers [7], or disk turbines [8]. This makes difficult the characterization of the mixing and mass transfer effectiveness of biochemical reactors. The oxygen transfer efficiency and performance of a given system is commonly expressed by the volumetric mass transfer coefficient of oxygen, K^â. It is, therefore, a parameter that allows comparison of mass transfer data among different systems. The existence of regions with different mixing and oxygen transfer intensities has a critical impact on the validity of reported K^â values because, depending where the dissolved oxygen sensors are placed, different aeration performances can be obtained. For example, Pons et aL, [8] observed a sudden increase and large fluctuations of K^â with fermentation time during xanthan production, attributed to high oxygen concentration values measured with an oxygen probe located near the impeller. Other studies have shown that the development of large stagnant zones is a common problem in fermentors stirred with either Rushton or Intermig agitators [9]. In spite of that, K^â values determined from measurements in the wellmixed zone near the impeller are frequently used to characterize the efficiency of oxygen transfer because they are considered representative of the whole working volume of the vessel. Experiments in xanthan and polyacrylamide solutions agitated with disk, crossbar and Intermig impellers have shown that the boundary of the wellmixed zone depends on agitation speeds and aeration rates, and that only for a given set of these conditions the volume of this region equals the working volume [10]. These findings demonstrate that the common practice of referring K^â values and impeller power consumption to the working volume is inadequate and not logical because gas-liquid oxygen transfer occurs mainly in the vicinity of the impeller. Some alternatives have been proposed to solve the problem of inhomogeneous mixing and limited oxygen transfer. These include the use of pneumatically agitated systems, new reactor geometries, use of different impeller geometries, and combination of impeller geometries. Pneumatically agitated systems are outside the scope of this work, however; the reader is referred to Herbst et aL, [9] and Pons et aL,

434


[8] for a complete discussion of their performance and efficiency in comparison with mechanically mixed devices. During the last 20 years a wide number of reactors with distinct geometries have been developed and tested with diverse fermentation media and aqueous non-Newtonian solutions [11]. For example, Krebser et al. [12] compared the performance of conventionally stirred fermentor with horizontal-loop (torus) bioreactor during xanthan fermentation. Both reactors were found to have an equivalent performance with respect to oxygen transfer and xanthan production rates. However, in the torus fermentor, the amount of converted glucose was greater and the power consumption lower. This led the authors to propose the "doughnut" reactor as possible alternative. It has been stated, however, that the use of new reactor geometries is not feasible in the near future because they have not been fully tested for reliability of operation and scale-up [13]. Under such considerations, the use of impellers of different size and geometry seems to be, at present, a better alternative. Close clearance impellers like the helical ribbon (HR) and helical ribbon screw (HRS) have found numerous industrial applications in the mixing of highly viscous fluids [14]. The spiral-like fluid motion induced by the ribbon as well as the small clearance from the vessel wall promote bulk mixing and minimize formation of stagnant zones. Helical impellers operate at lower agitation speeds than radial flow agitators, but they consume more power at the same Reynolds number. Nevertheless, the energy required to attain a specific degree of homogenization is lower because they yield shorter mixing times [15]. Although very suitable for handling highly viscous non-Newtonian fluids, the main drawback for using them in gas-liquid media lies in their low gas-dispersing capacity. However, in our opinion they are an interesting alternative for the fermentative production of rheologically complex EMPS provided they are used in conjunction with a device that compensates this drawback. The great majority of published works involving HR and HRS impellers has focused on the assessment of their mixing performance under unaerated conditions. We have studied the effect of non-Newtonian behavior on oxygen transfer in a vessel equipped with an HRS impeller and a ring-shaped gas sparger, as well as the hydrodynamics of HR and HRS agitators and the three-dimensional numerical simulation of the mixing patterns in non-Newtonian fluids. In this chapter we limit the discussion to oxygen mass transfer in non-Newtonian fluids, including observations on bubble behavior, gas dispersion, and quantification of the consequences of these effects on the mass transfer efficiency. The volumetric mass transfer coefficient, K^â, is used as an index of the aeration performance of the system. USE OF HELICAL IMPELLERS IN NON-NEWTONIAN GAS-LIQUID SYSTEMS The HR and HRS impellers were suggested a long time ago by Giaccobe and Capobianco for xanthan production on the basis of their good bulk mixing and pumping capacity, but without taking into account their low gas-dispersing ability [16]. De Vuyst et al. compared the performance of flat and curved blade turbines of various configurations against that of an HR agitator during xanthan fermentations [17]. In broths agitated with the HR the productivity of xanthan was similar to that with radial flow impellers, but their viscosity and rate of viscosity increase were


435

lower. This was attributed to the lower air dispersion of the HR which resulted in xanthan with a low pyruvate content. By increasing agitation speed of the HR up to 590 rpm the pyruvate content increased, and much higher viscosity and viscosity building rate were observed. This led the authors to conclude that by increasing the agitation speed, and, therefore, gas dispersion, xanthan production was favored. In spite of these observations, their results are difficult to interpret because no details are given about the type of gas sparger used, and the hydrodynamic conditions (i.e., Reynolds number) of the different impellers compared. Recently, the effect of hydrodynamics on gellan fermentation kinetics and rheological properties of the culture broth were studied using various mixing and mass transfer conditions [18]. Impellers tested included helical ribbon, Rushton turbines, and a pitched-blade turbine combined with an in-flow turbine together with extra oxygen supply or reduced nitrogen amount in the culture medium. Macromixing conditions created by the HR were more homogeneous and led to different rheological properties of the broth than with the other impellers. We have published results on aerated power consumption and oxygen transfer efficiency in a vessel equipped with an HRS impeller and a ring sparger in non-Newtonian fluids [19-21]. In the following paragraphs we discuss the main results obtained with such system. VESSEL AND IMPELLER DIMENSIONS We present here only general features of the experimental setup and procedures used in our studies. Full details can be consulted in previous publications [19,20]. The system used included a glass vessel of 0.210 m in diameter having a working volume of 8 L. Dimensions of the HRS impeller are: ribbon height = 0.185 m, ribbon diameter =0.185 m, ribbon width = 0.020 m, screw width = 0.025 m, ribbon pitch = 0.0925 m (double pitch was used), screw pitch = 0.185 m (single pitch was used). AGITATION AND AERATION CONDITIONS Gas-liquid experiments were carried out at 25°C at impeller speeds from 100 to 300 rpm in steps of 50 rpm and air flow rates of (1.33, 2.08, and 2.58) x 10^ mVs. Torque and impeller speed were measured with a strain gauge torquemeter. Impeller power input was calculated from corrected torque values obtained by subtracting the residual torque from actual torque readings [20]. Gases were fed via a ring sparger 0.150 m in diameter having 15 holes of 1 mm equidistant 30 mm. Oxygen concentration in the liquid phase was measured with a polarographic PTFE/silicon membrane sensor. KLH DETERMINATION In gas-liquid studies correct determination of the volumetric mass transfer coefficient is of the utmost importance. A great variety of methods for K^â determination is now available. Nevertheless, no single method combines simplicity, sensitivity, accuracy, and reproducibility over a wide range of operating conditions, and none of them is error-free. Therefore, the choice of the method is dictated, at least in part, by practical considerations. We have used unsteady oxygen absorption

436


in which oxygen was first desorbed sparging the liquid with pure nitrogen, and then air was sparged without stopping agitation, until dissolved oxygen reached a steadystate value. The advantages and disadvantages of the method have been extensively discussed in the literature [22]. Previous to Kâ determination by the unsteady method, the oxygen probe was tested as recommended in the literature [22]. Its transient response was symmetrical and did not exhibit tailing both in the gas and liquid phases. However, it exhibited a deviation from linearity of up to 4% for oxygen mole fractions greater than 0.6. For such reason pure oxygen was not used in absorption experiments. Because of the size of the impeller, the sensor was placed vertically midway between the screw and the ribbon with its tip at approximately one third of the ribbon height from the free surface of the liquid. Values of K^â were obtained from a non-linear least-square fit of the experimental response of the sensor to the following equation

G^(t) = l + (1 + L,)

VBexp(-BK^t) ^ ^ y Q(a)exp(-KXt) sinVB+LLVBCOSVB tt (a'„/B) - 1

(1)

where a^ are the positive roots of the equation aL^^cos a + sin a = 0

(2)

and

(l + L , + L X )

^^>

Kâ. values were obtained from the regression constants B and K^, knowing that Kâ = B • K^, with L^ as parameter. The parameter L^ which accounts for the effect of the liquid film around the sensor membrane is obtained from the steady-state probes readings in the gas phase and in the test solutions at the given hydrodynamic conditions as given by Linek et aL, [22]. Its value ranged from 0.02 to 0.04, but Kâ was determined using the value particular to each experiment. The basic assumptions involved in the derivation of this model are: 1. the accumulation term of the oxygen balance in the gas phase is negligible compared to the input, output, and transfer rate terms 2. oxygen concentration in the dispersed gas phase is equal to that in the outcoming gas, and 3. the oxygen transfer rate is small enough so that the inlet and outlet gas flow rates are essentially the same. Equation 1 is the solution which arises from the convolution of the simultaneous solution of the unsteady oxygen balance in the gas and liquid phase together with the unsteady response of the oxygen sensor. This latter results from the solution of Pick's second law considering oxygen diffusion through the membrane and the


437

liquid film adjacent to it. The full mathematical details involved in the derivation of Equation 1 are discussed in Chapter 9 of Linek et al. [22]. RHEOLOGICAL PROPERTIES OF THE LIQUID PHASE AND FLOW REGIME Studies were carried out in low shear-thinning, high shear-thinning, and viscoelastic fluids, obtained, respectively, with carboxymethylcellulose (CMC), xanthan (XTN), and polyacrylamide (PAA) aqueous solutions in concentrations ranging from 1 kg/m^ to 5 kg/m^ The rheological characterization of all solutions was carried out before and after a K^â determination experiment in cone and plate rheometers at 25°C. Within the range of shear rate existing in the vessel, 47 to 142 s ' , all solutions exhibited a power law behavior. Table 1 shows the power law parameters of the non-Newtonian fluids. The range of the flow behavior was from 0.19 to 0.88, and from 23.8 to 3,116 mPa • s" for the consistency index. The apparent shear rate in the vessel was calculated as y^ = k^ • N with k^ = 28.3 according to the analysis of Brito et al., [23] who also found this value experimentally. The impeller Reynolds number, calculated as. Re = pd^N^""/K(k^)""', was from 400 to 20,000, which shows that oxygen transfer occurred in the transition and turbulent regions.

Table 1 Power Law Parameters at 25°C

Fluid^

K(mPa • s")

n

CMC 1

23.8

0.87

CMC 2

46.7

0.88

CMC 3

70.6

0.88

CMC 5

190.0

0.84

XTNl

75.6

0.56

XTN 2

416.1

0.39

XTN 3

1,059.3

0.34

XTN 5

3,116.0

0.19

PAA 1

149.1

0.54

PAA 5

591.9

0.53

'number designates and so on)

concentration

(i.e. CMC 1 = 1 kg/m\

438


GAS-LIQUID MASS TRANSFER AND BUBBLE BEHAVIOR Effect of Power Input, SuperHcial Gas Velocity, and Polymer Concentration on Kjâ The complete range of power input per unit volume goes from 60 W/m^ to 2,200 W/m^ The effect of P /V,, u , and polymer concentration on K, a for all nonNewtonian solutions is shown in Figures 1 to 6. Depending on the conditions and fluids used, Kâ ranged from 0.0008 to 0.01 s"'. These values are 1.5 to 15 times lower than those in water (not shown). CMC Solutions In low shear-thinning CMC solutions the minumum and maximun K^â values were 0.0045 and 0.01 s ' , respectively, for power inputs from 60 W/m^ to 1,200 W/m^ Figure 1 shows the dependence of K^â with CMC concentration at a superficial gas velocity of 0.006 m/s, and Figure 2 its dependence with P /W^ and u . In CMC 1 and CMC 2 (Figure 1), Kâ increases linearly with FJW^ from 60 to JOO W/ml A similar trend was observed at 0.0039 m/s and 0.0075 m/s, although with lower and higher Kâ values, respectively. Within this range, K^â increases about 1.5 times with power input. Beyond 300 W/m\ Kâ is essentially independent of power input and is only dependent on superficial gas velocity. An increase in CMC concentration results in lower K^â values but also in a greater dependence on power input. In CMC 3 the flat zone is narrower, extending only from 650 to 1,000 W/m^ The horizontal region observed at lower CMC concentrations, specially in CMC 1, is not observed in CMC 5. In this case, the dependence of K^â on P /V^ is completely linear from 100 W/m^ to 1,200 W/ml Figure 2 shows the dependence of KLa with P /V^ and u in CMC 5. K^â ranges from 0.002 to 0.006 s"', and varies linerally with power input for the three gas velocities. At a given constant superficial gas velocity, Kâ increases almost twice showing essentially a similar dependence on power input. On the other hand, at a constant power input, Kâ increases roughly 1.3 times upon increasing gas velocity. A similar behavior was observed for all other CMC solutions either in the linear or in the flat zone. A twelve-fold augmentation in power input doubles K^â while a two-fold increase in gas velocity rises the coefficient by a factor of around 1.3. At low CMC concentrations, K^â displays a two-zone behavior, one in which K^â depends on both power input and gas throughput, and the other where the oxygen transfer performance depends entirely on gas flow rate. In CMC solutions having a higher consistency index, Kâ depends on both power input and aeration conditions. Therefore, at higher CMC concentrations, Kâ can be increased to different extents by increasing either or both. These results are consistent with bubble behavior observed in CMC solutions where the shape of gas bubbles was predominantly oblate spheroid. The apparent viscosity of CMC 1 and CMC 2 is around 15 mPa • s, and as a consequence sparged bubbles (5 to 10 mm diameter) had a tendency to escape rapidly before noticeable dispersion occurred. At 100 rpm, bubbles climbed without appreciable dispersion following the spiral-like motion of the ribbon. At higher agitation speeds more


10

439

-2 1

1

1 1 -I I 1 1 1

1

1

1 1 1 1 1 11

h

O^XD

J

^v—V

J

J^

r

2 V

^

3

O

X

10

-3

10

1

1

1 1 1 1 11 i

/^

1

1

1

VL

1 i 1i1

10'

10'

P /

i

(W/m )

Figure 1. Effect of CMC concentration on Kâ at Ug = 0.006 m/s. Numbers on each curve indicate concentration in kg/nn^. dispersion ocurred, producing smaller bubbles which resulted in the increase of K^â shown in Figure 1. Beyond 200 rpm the rotational motion of the impeller retarded bubble climbing, but no more apparent dispersion was observed. At higher impeller speeds some bubbles clustered around the screw, but most of them were still dispersed throughout the entire working volume. Therefore, at higher speeds oxygen

440

Advances in Engineering Fluid Mechanics ,-2

10

1

1

r

T—I—r

D

Ug (m/s) 0.0039 0.0060 0.0075

O A *

10

-3

_l

I

I

I

I

L

10^

10' Pfl /

V^ ( W / m )

Figure 2. Dependence of Kâ on volumetric power input and superficial gas velocity in CMC 5. transfer is controlled by superficial gas velocity. The higher apparent viscosity of CMC 3 (about 45-50 mPa • s) and CMC 5 (about 100-120 mPa • s) solutions produced longer residence time of bubbles in the liquid phase (estimated visually). This resulted in greater dispersion mainly at higher agitation speeds. Near the sparger, the rotational motion of the ribbon induced formation of smaller spherical


441

bubbles (1 to 3 mm) which distributed throughout the vessel. Some others, however, escaped and coalesced into large bubble caps near the free surface of the liquid. Bubble dispersion increased gradually with increasing agitation speed. Increasing speed to 150 rpm and 200 rpm promoted further dispersion, but the same phenomenon observed in CMC solutions of low concentration was observed; some bubbles were dragged to the central screw. Greater dispersion was obtained at 250 and 300 rpm, but more clustering also was observed. There was not, however, a clear bound between the central cluster and the rest of the dispersed bubbles. XTN Solutions Figures 3 and 4 illustrate the dependence of K^â on XTN concentration, power input, and superficial gas velocity. As shown in Table 1, XTN solutions are more shear-thinning than CMC solutions with higher consistency indexes. K^â values in XTN solutions are 1.5 to 2 times lower than in CMC solutions of similar concentration. Figure 3 shows the dependence of K^â on P IW^ at a superficial gas velocity of 0.0060 m/s in XTN 1, XTN 2 and XTN 3. ^Similar behaviors were observed at 0.0039 m/s and 0.0075 m/s. In XTN 1 and XTN 2 formation of the flat region observed in CMC at high power inputs is less apparent; between 200 W/m^ and 650 W/m^, K^â shows only a slight increase. In XTN 3 the flat region is not present, but only beyond 130 W/m^ a linear increase with power input is observed. Unlike CMC solutions, K^â converges to about the same value (0.055 - 0.062 s~') at higher power inputs. The same trend was observed at the two other gas velocities. Figure 4 shows the variation of K^â with power input at different gas velocities in XTN 3. At the lowest power input (45 W/m^), K^â increases almost two-fold upon doubling gas velocity, whereas at higher constant power inputs it increases by a factor not higher than 1.5. A similar effect was observed in XTN 5 in which at the lowest power input (85 W/m^), K^â increased almost three times when gas velocity was doubled. This behavior is not observed in XTN 1 and XTN 2 as shown in curves 1 and 2 of Figure 3. Beyond 120 W/m\ K^â increases by the combined effect of power input and gas flow rate, although to different extents. At a given gas velocity, a six-fold increase in power input results in an increase of K^â of the order of 2.5. Within the range from 120 to 700 W/m\ K^â changes from about 0.001 to 0.005 s"', which represents a more considerable improvement of mass transfer than that in low viscosity XTN and CMC solutions. XTN solutions were not completely transparent, yet it was possible to appreciate bubble behavior. Bubble dispersion occurred mainly near the sparger because of the shearing action of the ribbon. In some of these fluids bubble climbing was slower presumably because of their higher apparent viscosities. The retarded motion of bubbles allowed more dispersion and distribution mainly when agitation speed was increased, but also resulted in bubble accumulation in the liquid phase. In XTN 1 and XTN 2, bubbles escaped almost freely to the surface depending on agitation speed. However, at 100 and 150 rpm, bubbles climbed slowly in a wavy motion. Greater dispersion was observed at 200 and 250 rpm with only a few medium-sized bubbles. Unlike XTN 3 and XTN 5, a continuous linear increase of Kjâ prevailed upon the entire range of power input. In XTN 3 and XTN 5, bubbles smaller than 2 mm were formed. Although they were forced upwards by the motion

442


10

T

1

1 I I I I

T

1 I I I I

O

10

J

10

I

» I 1 I

I

10^

10^

P, /

VL

i I I I

(W/m )

Figure 3. Effect of XTN concentration on Kâ at Ug = 0.006 m/s. Numbers on each curve indicate concentration in kg/m^. of the helical ribbon, their residence time in the liquid phase was longer than in all other fluids, and as a consequence some very small bubbles accumulated with time. Longer residence times can be attributed to the rheological properties of the fluid. XTN 3 and XTN 5 exhibited the highest consistency indexes (see Table 1) and apparent viscosities; from 50 to 100 mPa • s, and from 70 to 200 mPa • s,


10

443

-2 1—I—I

1 I I I

T

1

1—I

J

I

I

I I I I

"g (m/s) J • •

0.0039 1 0.0060 1

•

0.0075 1

CO D -J

10

-3

10^

J

I

I

' I ' l l

10-

I I I I I

10"

p. / V^ (W/m ) Figure 4. Dependence of Kâ on volumetric power input and superficial gas velocity in XTN 3.

respectively. At the lowest gas velocity, fewer bubbles accumulated in the liquid phase, and greater dispersion and distribution were observed upon increasing power input. At higher gas velocities dispersion and distribution also occurred upon increasing power input, but considerably more bubbles accumulated in the liquid phase. At low power input, some of these bubbles could not be dragged upwards

444


by the ribbon, and, therefore, they were probably more depleted of oxygen than those driven to the surface. This behavior was more significant at 0.0075 m/s. However, beyond 120 W/m^ substantially more small bubbles were pushed upwards by the motion of the ribbon and distributed throughout the vessel as power input increased. This could explain the behavior shown in Figure 4. In the region of lower power inputs it is apparent that K^â increases more significantly by effect of superficial gas velocity than by power input. Although accumulation of bubbles exists, "new" bubbles produce a more noticeable increase of Kjâ. At 0.0075 m/s, however, accumulation of "old" dispersed bubbles results only in a slight augmentation of Kâ upon an increase in power input. Bubble trapping in xanthan solutions has been frequently observed in turbine-agitated vessels and has been attributed to the existence of yield stress [24]. In these systems accumulation of bubbles gives rise to the formation of large gas cavities near the impeller. Such phenomenon, as well as the development of stagnant zones, was never observed in our system. PAA Solutions Figures 5 and 6 exemplify oxygen transfer behavior in PAA solutions. The viscoelastic character of PAA solutions resulted in higher torque and, therefore, in higher power inputs than in CMC and XTN of similar rheological behavior. An analogous behavior has been observed in unaerated elastic fluids in turbine-agitated [25], and in HR-agitated [23] vessels. Figure 5 shows the effect of PAA concentration on Kâ at a constant gas velocity of 0.0039 m/s. In PAA 1, K^â ranged from 0.002 to 0.0055 s* depending on gas velocity. At low power inputs the effect of concentration on K^â is more significant. For example, at 150 W/m^ K^â in PAA 1 is roughly two times greater than in PAA 5. However, this difference becomes narrower as power inputs get higher; at 1,000 W/m^ K^â in PAA 1 is only 1.2 times higher than that in PAA 5. A similar behavior was observed at 0.006 and 0.0075 m/s. An essentially linear increase of the volumetric coefficient with power input was observed from around 180 to 1,000 W/m^ at all gas velocities. Likewise, at power inputs from 60 to 180 W/m^ a flat region similar to that illustrated by curve 1 in Figure 5 appeared. This zone was not present in PAA 5 in which power inputs were shifted to higher values and K^â increased, although not in a very neat linear way. In PAA 1 and PAA 5 at low power inputs, the effect of superficial gas velocity on KjL was negligible, as shown in Figure 6 only for PAA 5. As power input increased the effect of gas velocity is also more significant. For example, at 400 W/m^ no noticeable increase is attained when superficial gas velocity goes from 0.006 to 0.0075 m/s while in contrast, at 2,200 W/m\ a greater difference is seen. It can be said, therefore, that from 140 to 400 W/m^ K^â can be increased merely by increasing power input, whereas at higher power inputs oxygen transfer depends on agitation and aeration. In PAA 1, the effect of both is exerted to different extents; K^â increases an average of 1.5 times when gas velocity is doubled, whereas a twofold increase of power input produces only a 1.1 increase in Kâ. In these solutions the volumetric coefficient was more dependent on gas velocity than on power input. A different picture was obtained in PAA 5, where aeration and agitation produced a similar augmentation of around 1.3 in K^â. Nevertheless, for some conditions power input had a greater effect than gas velocity.


10

445

-2 T

1—I

I I I I I

I

I I I I I

I

I I I I

O

10

-3

10

J

I

I

J

I

I

I I I I

10^

p. /

10^ VL

(W/m )

Figure 5. Effect of PAA concentration on Kâ at Ug = 0.0039 m/s. Numbers on each curve indicate concentration in kg/m^. The behavior of PAA solutions in response to agitation and aeration was different from the behavior in CMC and XTN solutions. In PAA solutions, small spherical, as well as inverted tear drop bubbles, were observed. This latter shape is the result of the interaction of elastic and surface tension forces, and has been reported to occur in stagnant [26,27] as well as in mildly stirred solutions [28]. It is known that in free climb motion, PAA bubbles have lower terminal rise velocities than

446


10

"T"

1—I—I—r—r

"g (m/s) j + X

0.0039 1 0.0060 1 0.0075 1

o

10

-3

10'

J

I

I

L

10"

Pg / V^ ( W / m ) Figure 6. Dependence of Kâ on volunnetric power input and superficial gas velocity in PAA 5. CMC bubbles of the same volume regardless of the Reynolds number [26]. In stirred liquids, the impeller disturbs the motion of bubbles and can accelerate or retard them, depending on the existence of recirculation currents. Nevertheless, the average residence time of PAA bubbles is expected to be lower than in inelastic fluids. Simple visual observation allowed us to confirm that the residence time of air bubbles in PAA was longer than in CMC, but not necessarily longer than in XTN


447

solutions. Consequently, bubbles remained longer in the liquid phase before bursting at the surface, and so the impeller had enough time to disperse them. Nevertheless, some coalescence occurred near the surface of the liquid. The dispersion pattern of bubbles in PAA was very similar to that in XTN fluids. However, bubble hold-up was higher. Near the sparger, the rotational motion of the impeller resulted in inverted tear drop bubbles that were dispersed into small shperical ones as they climbed following the spiral-like motion of the ribbon. Contrary to CMC, in which bubbles escaped faster, greater dispersion was observed even at 100 rpm, and small spherical bubbles were homogeneously distributed throughout the vessel. At 150 rpm the stronger rotational motion of the fluid near the gas sparger resulted in more noticeable bubble dispersion. At higher gas velocities, however, occasional formation of bubble jets at the sparger was observed. Jet formation was produced because bubbles separated very slowly from the sparger, and, subsequently, sparged bubbles were dragged and captured in the wake of the rising bubble. This phenomenon was exclusively observed in PAA 5 and was more irregular as power inputs increased. At 200 and 250 rpm greater bubble dispersion was observed together with the complete elimination of coalescence and bubble jet formation near the sparger. Unlike CMC solutions, bubbles were not dragged to the central screw. Effect of Apparent Viscosity on KLH In non-Newtonian fluids Kâ also depends on their physical and rheological properties. The contribution of the latter has been normally expressed in terms of the apparent viscosity, and there is general agreement that this dependence is of the form K^â a(r|^)~% where z can take values between 0.4 to 0.7. In the case of viscoelastic materials, inclusion of the fluid rheology is less straightforward. Several authors have tried to include the effect of elasticity via the Deborah number, which for stirred tanks is defined as the product of a characteristic time of the fluid and impeller speed. However, determination of the former is not an easy task because it is not always possible to characterize experimentally the viscoelastic properties of the fluid. Determination of the characteristic time of the fluid from experimental shear viscosity vs. shear rate curves [29] and from interpolation of published experimental data on viscoelastic properties [30] has been tried in the past. However, values thus obtained are not necessarily representative of the actual behavior of the liquid. At present, inclusion of the Deborah number in dimensional or dimensionless correlations has not been completely successful. The effect of apparent viscosity of CMC and XTN solutions on K^â at a superficial gas velocity of 0.0039 m/s with impeller speed as parameter is shown in Figure 7. Higher gas velocities resulted in a similar behavior. At all impeller speeds, K^â follows a power dependence with apparent viscosity. However, in XTN solutions its effect is weaker at 250 rpm and essentially minimal at 300 rpm. OVERALL EFFECT OF OPERATION CONDITIONS AND RHEOLOGICAL PROPERTIES The overall effect of P /V,, u , and ri on K, a, can be summarized in the following dimensional correlations for CMC, XTN and PAA, respectively

448


10

1—r

T

r

T—I—n-

n o

CMC (kg/m ) O 1 D 3 V 2 A 5 10-3 10-2

100 i

H

•H-

1 I II M I N (rpm) 300

n O

XTN (kg/ 10-3 I

I

10^

V„ (mPa s)

10'

Figure 7. Effect of apparent viscosity on Kâ at u^ = 0.0039 m/s in CMC and XTN solutions.


Kâ^ 0.00342

Kâ^ 0.00125

KLa = 0.00410

vV.y

fP

y/// L .yyy////

6 = 79.1°

•'

9=128.2°

Figure 3. Axial and radial velocity components. Re = 32.

460


located on a paddle. Near the bottom of the tank, a recirculation zone exists, which is principally located behind the blades. The size of these eddies increases with Reynolds number until these eddies fill the whole volume between the two blades (as is the case in Figure 3 for Re = 32). At the same time, the axial and radial velocity components increase, reaching values about 5 to 10% of the tangential velocity at the impeller tip (TCND). In contrast, no recirculation motion is observed near the free surface of the liquid. It is clear that the eddies at the bottom of the tank are induced by the liquid friction on the bottom wall. Flows in Horizontal Planes Figure 4 represents the distribution of the tangential and radial velocity components and the corresponding streamlines in two different horizontal sections of the tank: near the bottom (Z = 0.26), and at a level near the middle of the tank (Z = 1.09). The main motion is angular. The flow is slowed down by the bottom of the tank. This phenomenon was not observed near the free surface of the liquid. Discussion The quantitative comparison of the results obtained from CFD with experimental values and with other simulations is difficult because of the scarcity of the information, and also because the major part of the published results are concerned with only the flow far away from the tank bottom and from the free surface of the liquid. A complete validation of this model would require comparison of the whole velocity field with one measured using, for example. Laser Doppler Anemometry. Unfortunately, LDA measurements in the laminar flow range remain difficult and rare because the highly viscous fluids often are not quite transparent and the laser beams can diffract while crossing through an unperfectly homogeneous liquid. Another problem lies in the fact that measurements carried out with this technology, in a fixed frame, would include the low frequencies due to the passage of the blades, leading to mean values of the velocity components and hiding the influence of the angular position of the measurement point from a blade. Recently, new improvements enabled Dyster et al. to carry out some measurements by LDA in highly viscous fluids agitated by a Rushton turbine, for Re > 5 [18]. This advance is full of promise, but, unluckily, these interesting data remain unusable for making comparisons with this work. The data available for comparison of the velocity profiles is near mid-height of the tank (Z = 1.1) at three angular positions: 0 = 3°, 0 = 34° and 9 = 90°. It has been found that the axial velocity component always represents less than 2 to 3% of the tip impeller speed. This last result is in very good agreement with the experimental results published by Hiraoka et al [10] and Bertrand [16], who showed that, in the case of a paddle agitator, the axial velocity component, far away from the tank bottom and the free surface of the liquid, can be neglected. That means that the flow can be considered a 2D flow.

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

Z = 0.26

=1.00

Z=1.09 Figure 4. Tangential and radial velocity components. Re = 32.

461

462


The profiles of the dimensionless tangential velocity component (yJnND) for two angles at mid-height in the vessel are compared with profiles from a 2D model, and with data from Bertrand [16] and from Youcefi [9] obtained with a hot film probe rotating with the impeller. These last results were obtained for a Reynolds number of 38. The comparisons are presented in Figure 5. The good agreement enables us to consider the flow generated by a paddle agitator at mid-height as a 2-dimensional flow. The poorer agreement between numerical and experimental results for the angle G = 34° also should be noted. The experimental measurements at this location were difficult (Bertrand [16]) ; the numerical results are probably more reliable. The profiles of the dimensionless radial velocity component (V/TIND) at the same height and angular positions are presented in Figure 6. It can be noted that the radial velocity components are slightly negative close to the surface of the impeller blade. In this region, the fluid flows towards the axis of the tank. In contrast, at the impeller tip, the fluid is strongly discharged towards the wall of the tank. The same variations were observed by Bertrand [16] for other Reynolds numbers. More recently, Shen and Baird [19] described the same phenomenon with a delta paddle mixing impeller with short lengths of black cotton that enabled them to examine the flow near the impeller.

Q

Z. <j>

0.4

0.6

1.0

2r/T Figure 5. Tangential velocity component vs. radial position. Re = 32, Z = 1.1 ( • ) e = 34°, experiments Bertrand [16]; (D) 6 = 34°, 2D model Bertrand [16]; (A) e = 90°, experiments Bertrand [16]; (A) 9 = 90°, 2D model Bertrand [16]; (+) e = 90°, experiments Youcefi [9]; (—) 3D model (this work).


463

Figure 6. Radial velocity component vs. radial position. Re = 32, Z = 1.1. ( - - ) e = 3°; ( — ) 0 = 34°; ( - ) 0 = 90°

Stresses and Viscous Dissipation Function The knowledge of the stress field enables us to characterize and optimize the use of an impeller for a defined operation. In the three dimensional case, the stress components form a second order tensor, the symmetry of which allows it to be reduced to six components. These six components are the three normal stresses x^^, TQQ and x^^ and the three shear stresses, x^^, x^^ and x^^. Figures 7a, 7b, and 7c give, respectively, the dimensionless values of x^^, of x^^ and x^^ (noted x*, x* and x*) in the vertical plane 0 = 3°, with : X

=

- |a23v/3r

(1-a)

- \i{Td(\Jr)/dr + 1/r dvJdQ]

(1-b)

- \i{d\Jdz + dvjdr]

(1-c)

rr

T

=

rz = x/(27t MN D/T) X* and a. Normal Stress T,.

(l-d)

Figure 7a shows that the maximum value of the normal stress is essentially located along the impeller tip. It was previously observed that the radial component of the

Figure 7. Stresses and viscous dissipation rate. (a) 29, (b) ,z, (c) Trz9 (dl 4)".


465

velocity is changing in sign and is strongly increasing, showing here a discharge flow of the fluid to the tank wall. The continuity equation for uncompressible fluids leads to:

b. Shear Stress T^Q The field of the shear stresses T^^ is represented in Figure 7b. The shear stresses are only located along the blades and near the tank bottom. At this last place, the angular velocity component rapidly decreases. In this area, the normal stresses and the shear stresses are of the same order of magnitude, although the radial velocity component is much lower than the angular one. It can be concluded that the stresses x^ and T^Q characterize the primary flow generated by the paddle agitator. c. Shear Stress T^^ The shear stress T^^ corresponds to shear rates defined by the flows in the vertical plane. On Figure 7c, it is observed that this shear stress generally has low values and that it is essentially developed at the tank bottom, where the vertical flow is fairly important. d. Viscous Dissipation Function The rate of viscous dissipation represents the amount of energy irreversibly transformed into heat by means of viscous friction. It is calculated in the following form from Equation 2: *v = ( | < + ^êe + l^zz + < + < + <e)/l^'

(2)

The knowledge of the distribution of ^ in the volume of the tank enables us to identify the zones where the energy dissipation is the most intense. Moreover, this parameter indicates which points in the tank are most at risk of local heating, if the heat transfer to outside is not sufficiently rapid. Figure 7d represents the distributions of (|)*((|)* = (^J(2n N D/T)^) in the same vertical plane as previously (0 = 3°). The maximum value of the viscous dissipation function is located at the impeller tip. It is clear that the distribution of (|)^* in the whole tank is more or less similar to the one of the normal and shear stresses t* and T^. Thus, only these stresses have to be considered in the case of a paddle agitator. Wall Effects In the previous sections, the three-dimensional flow generated by a paddle agitator in an open tank was studied. Let us consider in this section the case where the top of the tank is a solid wall, with a no slip boundary condition. Figure 8 shows the distribution of the axial and radial components of the velocity in a plane far from the paddles where the flow is entirely developed. The flow is characterized by the

466


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X »

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/ y / f \

Figure 8. Axial and radial velocity components. Case of a closed tank, e = 104.4°. formation of two cores in the lower and higher part of the tank: These eddies are turning in an opposite sense, one to the other. The formation of the second eddy comes from the existence of the solid wall at the top of the tank, which has exactly the same influence as the tank bottom. The two solid surfaces produce a decrease in the angular velocity component of the liquid, creating a pressure gradient; that is why the two eddies form. In Figure 9, an impeller located at a distance C/T = 0.2 from the tank bottom is considered. The paddle agitator in this case is defined by W/T = 0.8 and D/T = 0.5. The Reynolds number is equal to 32. An eddy is developed near the impeller tips. The intensity of this eddy is more important than in the case defined by Figure 3. The axial velocity components are larger in the present case. They can reach rather high values, about 30% of the value of the velocity of the impeller tip.

.....,,... ..

---. .,

....,. -.,., . -..,,,,,.* I

6

0

.

.

.

.

.

Figure 9. Axial and radial velocity components. Special gate agitator.

%

468


Case of a Fluid with Viscoelastic Properties It is very difficult to find data or reports of experiments concerning the mixing of viscoelastic fluids in agitated tanks equipped with blade impellers, except about power requirements [20]. In particular, the modeling of the hydrodynamics remains very difficult, especially in complex geometries such as agitated vessels. Nevertheless, with the enhancement of the computing capacities, it is now possible to simulate the flows generated with this kind of fluid in geometries simple enough, such as a tank agitated with a plate agitator. Anne-Archard has made the assumption that, in the case of a vessel with a plate agitator, the flow was two-dimensional far from both the bottom and the top of the tank, and has considered the case of a fluid with a rheological behavior well correlated by the model of Oldroyd B [14]. In the laminar flow regime (Re = 5), streamlines are given in Figure 10 in the case of a purely Newtonian fluid (De = 0)

Figure 10. Streamlines. Re = 5 - (a) case of a Newtonian fluid (De = 0) (b) case of a viscoelastic fluid (De = 1.33) [14].


469

and in the case of a fluid having the same viscous properties as the previous one, but also elastic properties characteHzed by the Deborah number, De = 1.33. It can be observed that the elongational properties of the fluid enable the closed streamlines near the wall to disappear. Inertial effects are enhanced, and this generates an upstream shifting of the flow structures. It has to be pointed out that for Re < 1, elastic properties generate on the contrary more important closed zones [14]. Radial profiles of the tangential velocity are plotted in Figure 11 for different Deborah numbers. It clearly appears that an inflexion point with a change of concavity is generated by the elastic properties of the fluid, which illustrates the mechanism of action/reaction characterizing this kind of liquids. Some experimental measurements of the local angular velocity have been carried out by Youcefi [9] with a hot film probe in polyacrylamide (PAA) solutions (Figure 12). Because of further experimental difficulties, it remains difficult to directly compare these results with the ones issued from the modeling work of Anne-Archard [14] because it is not possible to calculate in this case the value of the Deborah number. Nevertheless, we can consider these results compatible to the others. The ratio tangential flow rate over tangential flow rate for a Newtonian liquid (Q/Qnewt) calculated by Anne-Archard [14] is plotted in function of the Deborah number for different Reynolds numbers in Figure 13. It can be noticed that in the case of a low inertial flow, the increase of the elasticity generates a decrease of the flow rate and, thus, of the mixing effectiveness; the opposite effect happens when the inertial effects are more important.

OS h 0.4 h - " 03

Dc=0. Dc=0.5 I>C=1. Dc = 133

r 0 0.2 0.1 0.0 -0.1 0.0

0.2

0.4

0.6

0.8

1.0

Figure 11. Radial profile of the tangential velocity component in the plane of the blade for fluids with different viscoelastic properties [14].

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It can be concluded that inertial and viscoelastic properties are in competition, and the one that prevails depends on the value of the Reynolds number. CLASSICAL TWO-BLADE IMPELLER In this section, the flow produced by a classical two-blade impeller is considered. The ratio between the blade height and the impeller diameter is lower for this impeller than in the case of a paddle agitator (W/T = 0.25). The ratio between the impeller diameter and the tank diameter is D/T = 0.7. The impeller is located near mid-height of the tank (C/T = 0.4). Flow in Horizontal Planes The flows generated by two-blade impellers with a small W/T ratio are mainly characterized by an increase of the radial and axial components of the velocity in the vicinity of the impeller. The flow generated for Re = 10 at two different heights in the tank is presented in Figure 14. In the plane under the impeller (Z = 0.24), the flows are directed to the axis in a spiral. The formation of an axial eddy in the lower part of the tank tends to generate a suction motion of the fluid to the axis of the two-blade impeller. In the volume defined by the impeller, the flow remains tangential with a low part of recirculation near the second blade. Flow in Vertical Planes In Figure 15, the axial and radial flow generated by this kind of impeller is shown for Re = 10. The flows are characterized by the formation, just before the blade, of a strong motion of discharge of the fluid to the tank wall. On the contrary, a strong suction motion is observed near the second blade (see in Figure 15, 9 = 3.6° and e = 169.2°). In the range of laminar flow regime (Reynolds number less than 70) Kuncewitz has pointed out that the dimensionless secondary flow rate Ks (corresponding to the discharge flow rate of the impeller) is proportional to the Reynolds number while the dimensionless primary flow rate Kp (corresponding to the volumetric flow rate in the tangential direction) remains constant (H/T = 1., D/T = 0.5, W/T = 0.1) [13]. Both circulations are equal for Re = 50. Beyond this value, the secondary circulation is higher than the primary circulation, but the transient flow regime is not far. For his geometry, Ks always reaches its maximum value for C/T = 1/3. These results are important because of the influence of the secondary flow on mixing times and heat transfer. Stresses The fields of dimensionless normal stresses T * and shear stresses x* and x* IT

re

rz

are presented for a position 6 = 3°. Figure 16 shows respectively the values of x* (Figure 16a), x* ( Figure 16b) and x* ( Figure 16c). (text continued on page 475)


473

Z = 0.24

v=1.00

Z=1.0 Figure 14. Tangential and radial velocity components for a classical twoblade impeller. Re = 10.

474

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\^

(text continued from page 472) The maximum values of x* are located in the vicinity of the impeller tips. The change of sign of the radial velocity component at the exit of the impeller leads to the existence of two maximum values on each side of the impeller tip. The shear stress T^Q* is rather important in the volume defined by the impeller. The maximum value is located at the place where there is the greatest variation of the velocity, i.e., at the impeller tip. The comparison of the fields of stresses x* and x^ with the ones obtained in the case of a paddle agitator confirms the considerable shear effect produced by the impeller tip. The stresses x* are, on the contrary, much more important than in the case of a paddle agitator. The maximum values are located at the high and low corners of the impeller. Influence of the Impeller in the Tank The influence of the position of the impeller in the tank is studied in the case of a two-blade impeller with W/T = 0.10 and C/T = 0.17. The vicinity of the tank bottom leads to a small decrease in the intensity of the flows for the recirculation loop (Figure 17).

476


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Re = 61

Figure 17. Influence of the Reynolds number on axial and radial velocity components.

Influence of the Reynolds Number Figure 17 presents results of axial and radial velocity components obtained for two values of the Reynolds number. When the Reynolds number increases, two eddies are created by the impeller corners and are developing towards the center of the tank. The discharge flow (9= 3.6°) and the aspiration motion (9 = 169.2°) are clearly observed in this figure. The radial velocity components at 9 = 90° rapidly increase with Reynolds number and reach values about 22% of the impeller speed at the impeller tip (Figure 18).

Modeling of the Hydrodynamic Behavior of Highiy Viscous Fluids

477

0.3

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Re = 10 •Re = 61

0.1

Q

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-0.2

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r-

0.6

0.4

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1 —

0.8

1.0

2r/T Figure 18. Velocity components for a two-blade impeller. 6 = 90°, Z = 1.1

Thus, the flows generated by the two-blade impeller are mainly tangential for low values of the Reynolds number and become more and more radial when the Reynolds number increases. The profile of the radial velocity component for Re = 61 is parabolic. The profiles of the tangential velocity components v^ change when Re increases from 10 to 61 (Figure 18). For Re = 10, the tangential velocity component, at 9 = 90°, and at mid-height of the impeller, slowly increases from a value equal to the rotation speed on the shaft, reaches a maximum value for r = 0.33, then decreases to a 0 value at the tank wall. For Re = 61, the maximum of the tangential velocity

478


component is located at the radial position corresponding to the impeller tip. An important rotational speed enables the tangential impulsion of the impeller tips to reach the middle part of the tank. Figure 18 also gives the variations of the axial velocity components vs. the radial position for two different values of the Reynolds number. It can be clearly noted that the axial velocity component is increasing with Reynolds number. In this range, changing Reynolds number does not modify the flow structure. POWER CONSUMPTION The power consumption is impeller surface of the local quite equivalent to say that impeller to the fluid [21]. In

p=J

a macroscopic result obtained by integration on the power transmitted by the impeller to the fluid. It is the power consumption P is entirely given by the these conditions,

^vdV

(3)

tank volume=V

The element dV is written as dV = r dr d0 dz. The well-known power number N is defined as: Np = P/(p N^ D^)

(4)

The power consumption constitutes a global parameter very easy to measure. Several experimental works have been devoted to the analysis of this two-blade impeller [21,16]. A comparison of the two previous studied geometries, the paddle agitator and the two-blade impeller, shows a rather good agreement, except in the case of the twoblade impeller, for Re = 61. It seems that in this case, the observed physical flow is not strictly laminar. The variations of N vs. Re in a logarithmic scale are linear for the two studied geometries, with a slope equal to -1 for low values of Reynolds numbers. For Reynolds numbers greater than 10, the power number decreases more slowly. It is generally considered that the flow regime remains completely laminar until Re = 30, with, in this range, the product N 'Re remaining constant. Results issued from CFD have been obtained by Hiraoka et al. [22]. The empirical equation by Nagata [21] to calculate the power required by paddle impellers: Np • Re = 14 + (W/T))[670(D/T - 0.6)2 + 185]

(5)

in the range of D/T = 0.4-0.6 fits well the results of Hiraoka et al. [22], as well as the semi-empirical equation developped in their laboratory:

^-^-^ -13 + 3 4 r i ¥ 2^]_ri7D)^ np/2'/'

V A A T JT/D-D/T I A A T JT/D-D/T

with A = 1 + exp{-10[(T/D) - 1]}

,,^ ^^^


479

In fact , the power input is always strongly affected by the flow pattern around the impeller. It has to be mentionned that a two-dimensional flow model has been developped by Hirose and Murakami to calculate the power consumption in vessels equipped with paddle impeller, anchor, gate, helical ribbon or helical screw [5]. This approach is interesting because a single unified correlation is able to cover both straight and helical blades, but it suffers from an unavoidable complexity. In the case of a viscoelastic fluid, the product N -Re is plotted as a function of the Reynolds number for different Deborah numbers in the Figure 19 [14]. When the inertial effects are very low (Re = 0.1), N -Re remains comparable to the value obtained for a classical Newtonian fluid. The gap increases when the inertial effects become important, essentially for high values of the Deborah number. In the laminar flow range, adding inertial effects modifies the influence of viscoelastic behavior: for low Reynolds numbers, the power consumption is less when the fluid has viscoelastic properties, but for Re > 1, this kind of liquid needs much more power effort. Comparisons with experimental data have shown quite a correct agreement [14]. MIXING OF TWO MISCIBLE HIGHLY VISCOUS FLUIDS The operation consisting of mixing two miscible fluids of high viscosity remains difficult with two-blade impellers. Correlations obtained from experimental data to estimate the time necessary to get a good homogeneity cannot be found in the literature. 350.0

300.0

Newtonian G — O D c = 0.2 A—ADe=0.5 • — • Dc=1.33

Np.Re 250.0

200.0

150.0

Figure 19. Evolution of the power consumption with the Deborah number in function of the Reynolds number.

480


Nevertheless, it is possible to solve this problem using a CFD approach to solve the convective-diffusive equation in order to model the mass transfer phenomenon, which remains easy. This equation is written as follows: ^-^^ + V * grad * C * = D * A * C *

(7)

with

D* = - f — 1 — Pe

K{TJ

and

C* =

Lafon has obtained interesting results concerning the mixing of Newtonian or powerlaw fluids, using a classical PISO algorithm in a rotating frame [23,12]. The degree of mixing can be characterized by a standard relative deviation D^^: D„ = ^

(8)

with

^ =I i=l

'

(9) "

where n is the number of cells, A. is the volumic fraction of the product A, and A is the mean volumic fraction of A. In the case of a paddle agitator, and far from the bottom and the top of the vessel, Lafon presents the mixing of two fluids in the laminar flow regime (Re = 4.12) [12]. Figure 20 shows the evolution of isoconcentration lines as a function of time, the tracer (C = 1) being introduced while the hydrodynamic is yet established. The classical mechanisms of laminar mixing are well-illustrated: deformation and stretching of fluid agglomerates. Two dead zones are developing behind and in front of the blade, and the lack of radial movement prevents a good mixing near the walls. The final steps of mixing to reach the equilibrium concentration (C = 0.125) will be achieved by the diffusion phenomenon. The degree of mixing is plotted in Figure 21 for different sizes of paddle agitators and operating conditions (including the case of a non-Newtonian power law fluid) as a function of time [12]. It can be noticed that the pseudoplastic characteristic of the fluid clearly does not modify the mixing. On the contrary, the diameter of the paddle is an important parameter which strongly affects the mixing when the size of the impeller is too important, which makes it difficult for the fluid to go from one side of the vessel to the opposite one. Concerning the rotation speed, it has to be pointed out that the mixing is enhanced during the first seconds, but the curve tends to connect the one corresponding to a lower rotation speed after 40 seconds, and this significates that it seems useless to lose energy in high rotation speeds to accelerate the mixing of two fluids.


481

Figure 20. Blending of two Newtonian highly viscous fluids, (a) t = 0, (b) t = 0.9s, (c) t = 13.5s, (d) t = 107.7s [12].

CONCLUSION In this paper, results obtained by numerical simulations of hydrodynamics generated by two-blade impellers are presented for the laminar flow range. First, it has to be noted that a good agreement is observed between our results and the few found in the literature. The influence of the corners and the tips of the impeller has been analyzed. The size of the impeller plays a great role in the flow structure. For a paddle agitator, the flow is essentially plane far away from the horizontal walls. To create an axial circulation in the volume of the tank, the impeller height has to be decreased. Secondly, it has to be pointed out that the numerical way presents very important possibilities. In particular, thanks to the numerical way, it is possible to obtain

482


1.0 ^

D/T

N

Re

^h

B1

4.12

1

0.51

0.18

B2

10.94

1

0.83

0.18

B3

16.46

1

0.51

0.71

84

4.12

0.5

0.51

0.18

Figure 21. Standard relative deviation Dsr in function of time for different geometries and operating conditions [12]. local information on stresses and on viscous dissipation function difficult to get from experiments. In the very important case of non-Newtonian fluid flow, the viscosity ji, which is defined in this paper, has to be replaced by the apparent viscosity of the generalized Newtonian fluid when it is possible (pseudoplastic, dilatant, or plastic fluids). This apparent viscosity is defined from the flow rheological model representing the fluid by ji^ = f(Y). In the case of viscoelastic fluids, the modeling of hydrodynamic in agitated tanks remains difficult, but Anne-Archard has given preliminary results in the case of a plate agitator [14]. In the laminar flow range, it appears that the effect of elasticity


483

on the effectiveness of the mixing depends on the Reynolds number. These results are encouraging, and it is probable that future developments will enable us to obtain results concerning the full 3D hydrodynamics in vessels filled with viscoelastic fluids. The problem of the mixing of two fluids with the same physical properties has been discussed. Such an approach could lead to the definition of a numerical mixing time and the geometries most adapted to rapid mixing of two fluids (in particular, to find the best location for injection of the second liquid). It would be necessary to carry out the same kind of work in a real 3-D flow. The numerical way allows us to test rapidly and completely any geometrical change, and also any rheological fluid change, and probably to define new impellers by computations while minimizing their operating cost, or maximizing any property (mixing capacity, pumping capacity and so on). At last, it has to be pointed out that the simulation of blade agitators enables the practitioners to know without major difficulty the entire flow fields inside the vessel, including in the volume of the stirrer because of the use of a rotating frame. It is now possible to find commercial softwares of CFD which propose systems of sliding meshes in order to take into account the baffles in the case of fluids of low viscosity. It also may be possible to define unstructured triangular meshes in order to simulate the flows inside the volume described by more complex agitators. Soon, this will offer great possibilities and advances in CFD applied to mixing problems, as fast as computer power increases and their cost decreases. NOTATION A = mean volumic fraction of the constituant A (-) A. = volumic fraction of the constituant A (-) C = distance between tank bottom and impeller (m) C = concentration(g/l) C* = dimensionless concentration (-) CQ = initial concentration (g/1) D = agitator diameter (m) D* = dimensionless diffusivity (-) D = diffusivity (mVs) De = number of Deborah (De = ^y) (-) D^ = shaft diameter (m) D = standard relative deviation {-) sr

Fr g H Kp

^ ^

= = = =

Froude number (Fr = N^ D/g) (-) gravity (m/s^) liquid level (m) dimensionless primary flow rate (-) Ks = dimensionless secondary flow rate (-) n = number of cells (-)

n^ n N N

= = = =

P Pe Q r Re

= = = = =

t = T = v^ = v^ = VQ = V = V* = W = z = Z =

behavior index of the fluid (-) number of impeller blades (-) rotational speed of impeller (1/s) power number (N = P/p N^ D^) (-) power (kg • mVs^) Peclet number (Pe = N DVD) (-) tangential flow rate (m3/s) radial coordinate (m) Reynolds number (Re = N D2 p/|i) (-) time (-) tank diameter (m) radial velocity component (m/s) axial velocity component (m/s) angular velocity component (m/s) tank volume (m"^) dimensionless velocity (-) agitator height (m) axial coordinate (m) dimensionless axial coordinate (Z = z/D) (-)

484


Greek Letters 9 = angle, tangential coordinate Y = mean shear rate (1/s) (|)^ = viscous dissipation function (1/s^) ([)* = dimensionless viscous dissipation (-) X = characteristic time of the fluid (s)

|i X |^ p x.. T.

= viscosity (kg/m • s) = apparent viscosity (kg/m • s^~") = density (kg/m^) = shear stress (kg/m • s^) = dimensionless shear stress (-)

REFERENCES 1. Sano, Y. and H. Usui, "Effects of paddle dimensions and baffle conditions on the interrelations among discharge flow rate, mixing power and mixing time in mixing vessels", J. Chem. Eng. Japan 20, 4, 399-404 (1987). 2. Winardi, S., S. Nakao and Y. Nagase, "Pattern recognition in flow visualization around a paddle impeller", J. Chem. Eng. Japan 21, 5, 503-508 (1988). 3. Stein, W. A., "Mixing times in bubble columns and agitated vessels". Int. Chem. Eng. 32, 3, 449-474 (1992). 4. Malhotra, K., A. S. Mujumdar and M. Okazaki, "Particle flow patterns in a mechanically stirred two-dimensional cylindrical vessel". Powder Technol. 60, 179-189 (1990). 5. Murakami, Y. K., T. Fujimoto, A. Shimada, I. Yamada and K. Asano, "Evaluation of Performance of Mixing Apparatus for High Viscosity Fluids", J. Chem. Eng. Japan 5, 3, 297-303 (1972). 6. Kuriyama, M., K. Inomata, K. Arai and S. Saito, "Numerical Solution for the Flow of Highly Viscous Fluid in Agitated Vessel with Impeller", AIChE J. 28, 293-300 (1982). 7. Bertrand, J. and J.P. Couderc, "Agitation of Pseudoplastic Fluids by Two-Blade Impellers, Anchors and Gate-Agitators", Can. J. Chem. Eng. 60, 738-744 (1982). 8. Hirose, T. and Y. Murakami, "Two-dimensional viscous flow model for power consumption in close-clearance agitators" , J. Chem. Eng. Japan 19, 6, 568-574 (1986). 9. Youcefi, A., "Etude Exp6rimentale de TEcoulement d'un Fluide Viscoelastique Autour d'un Agitateur Bipale en Cuve Agitee", These de Doctorat, INP Toulouse, France (1993). 10. Hiraoka, S., I. Yamada and K. Mizoguchi, "Numerical Analysis of Flow Behaviour of Highly Viscous Fluid in Agitated Vessel", J. Chem. Eng. Japan 11, 6, 487-493 (1978). 11. Bertrand, J. and J.P. Couderc, "Agitation of Viscous Fluids by Paddles of Different Widths", Int. Chem. Eng. Symp. Series 64, B1-B6 (1981). 12. Lafon, P., "Melange laminaire de fluides miscibles en cuve agitee: approche numerique". These de Doctorat INP Toulouse, France (1989). 13. Kuncewicz, C , "Three-dimensional model of laminar liquid flow for paddle impellers and flat-blade turbines", Chem. Engng. Sc. 47, 15, 3959-3967 (1992). 14. Anne-Archard, D., "Etude numerique d'ecoulements de fluides viscoealstiques en geometric confinee et en regime faiblement ou moderement inertiel". These d'Etat, INP Toulouse, France (1994).


485

15. Hiraoka, S., I. Yamada and K. Mizoguchi, 'Two dimensional model analysis of flow behaviour of highly viscous non-Newtonian fluid in agitated vessel with paddle impeller", /. Chem. Eng. Japan 12, 1, 56-62 (1979). 16. Bertrand, J., "Agitation de Fluides Visqueux. Cas de Mobiles a Pales, d'Ancres et de Barrieres", These d'Etat, INP Toulouse, France (1983). 17. Patankar, S. V., "Numerical Heat Transfer and Fluid Flow", McGraw Hill, New York (1980). 18. Dyster, K. N., E. Koutsakos, Z. Jaworski and A. W. Nienow, "An LDA Study of the Radial Discharge Velocities Generated by a Rushton Turbine", Trans. Inst. Chem. Eng. Part A 71, 11-23 (1993). 19. Shen, Z. J. and M. H. I. Baird, 'The Delta Paddle Mixing Impeller—Some Hydrodynamics Studies", Trans. Inst. Chem. Eng. Part A 69, 143-152 (1991). 20. CoUias, D. I. and R. K. Prud'homme, "The Effect of Fluid Elasticity on Power Consumption and Mixing Times in Stirred Tanks", Chem. Engng. Sc. 40, 8, 1495-1505 (1985). 21. Nagata, S., "Mixing: Principles and Applications", Halstead Press, New York (1975). 22. Hiraoka, S., I. Yamada, T. Aragaki, H. Nishiki, A. Sato and T. Tagaki, "Numerical analysis of three dimensional velocity profile of highly viscous Newtonian fluid in an agitated vessel with paddle impeller", /. Chem. Eng. Japan 21, 1, 79-86 (1988). 23. Lafon, P. and J. Bertrand, "Melange laminaire de deux liquides en cuve agitee: analyse numerique", Entropie 142, 51-57 (1988).


CHAPTER 17 NON-NEWTONIAN LIQUID FLOW THROUGH GLOBE AND GATE VALVES Sudip Kumar Das Chemical Engineering Department 92, A. P. C. Road Calcutta University Calcutta 700 009, India CONTENTS INTRODUCTION, 487 PARAMETRIC ANALYSIS, 490 PREVIEW, 491 Liquid Flow Through Pipe Fittings, 491 EXPERIMENTAL STUDIES, 499 Measurement of Pressure Drop Across the Balve, 501 NOTATION, 502 REFERENCES, 503 INTRODUCTION Newtonian fluids exhibit a direct proportionality between shear stress and shear rate in the laminar flow region. Non-Newtonian fluids exhibit a non linear shear stress-shear rate dependence. A majority of the non-Newtonian fluids are to be found as pseudoplastic in nature, such as rubber solutions, adhesives, polymer solutions or melts, greases, starch suspensions, cellulose acetate, solutions used in rayon manufacturing, mayonnaise, soap, detergent slurries, paper pulp, napalm, paints, certain pharmaceutical dispersions, biological fluids, dilute suspensions of inerts, unsolvated solids, etc. [1,2]. It displays, on arithemetic coordinates, the concavedownward flow curve relationship; on logarithmic coordinates these materials exhibit flow curves having slopes between zero and unity. Alves et al. pointed out that the curve showed a point of inflection and the slope approaches a value of unity at extremely low as well as at very high shear rates [3]. Metzner and Reed [4] showed that the flow curve was a straight line on logarithmic coordinates over 10- to 100fold ranges of shear rates, sometimes having slopes appreciably less than 0.10. For such straight line regions the flow curve defined by the power law model is T = K(-dv/dr)"

(1) 487

488


The flow behavior index, n, is the slope of the logarithmic plot, which ranges from unity towards zero with increasing pseudoplasticity; consistency index K characterizes the consistency or thickness of a fluid. Since most pseudoplastic fluids are highly viscous in nature, laminar flow is of greatest practical interest [5]. For those fluids to which power law applies, Metzner and Reed [4] developed the basic relationship for relating pressure drop to flow rate by means of geometric parameters and the two physical properties of the fluid K' and n' as DAP 4L

../8VV' D .

n

(2)

where '3n;• + 1 K' = K[ 4n' , n

(3)

d[ln(DAP/4L)] =

•

= n

d[ln(8V/D)]

(A\

^^^

n' or n is the power law exponent, slope of line from a plot of DAP/4L vs. 8V/D on logarithmic coordinates. They also correlated laminar flow data as f = 16/ReMK

(5)

where Re

-

^^MR -

VDP

...

gn-l pvl-nyn-lT^/

\^)

In the turbulent flow drag reduction phenomena is observed and is due to two different mechanisms. One affects the logarithmic portion of the velocity profile and the other the thickness of the viscous sub-layer. The first type is associated with the action of gravity on a cross-flow density gradient due to either temperature gradient [6] or by varying concentration of suspended particles [7]. The thickening of the viscous sub-layer is obtained in aqueous flows by addition of small quantities of certain long-chain molecules, and these substances act by increasing the size of the smallest, dissipative, turbulent eddies [8]. Dodge and Metzner [9] used mixing length approach and presented an implicit relation between friction factor (f) and generalized Reynolds number (RC^R) as

They also presented a Blasius type empirical equation in the following form

Non-Newtonian Liquid Flow Through Globe and Gate Valves

f =

489

(8)

R^MR b„

where values for a^ and b^ were given graphically for various values of the flow behavior index, n. This equation is essentially a curve fit of Equation 7 without theoretical consideration. Edwards and Smith [10] suggested that the well-established Newtonian friction factor-Reynolds number relation can be used for non-Newtonian flow as 1 Vf

4.01og(ReEs V f ) - 0 . 4 0

(9)

where Re^^ is the Reynolds number using apparent viscosity at the wall VDp RÊs

(10)

râ.w

^ VDp fpV'

(l-n)/n

(11)

Irvine [11] proposed an empirical explicit relation between the friction factor and Reynolds number as f =

F^(n) p

(12)

l/(3n + l)

where 2"

l/(3n + l)

F'(n) = 3n + l 4n

3n + l 4n

l/(3n + l)

1

(13)

He also tabulated the values of F'(n) for different values of n. Hartnett and Kostic [12] studied a number of correlations for predicting the turbulent friction factor of purely viscous non-Newtonian fluids flowing in circular and non-circular geometries. They concluded that the Dodge-Metzner Equation 7 was the best over the entire range of power law value. Pipe fittings like valves, bends, elbows, tees, reducers, expanders, etc. are the integral part of any piping system. Flow-through piping components are more complex than the straight pipes. The problem of determining the pressure losses in fittings is important in design and analysis of the fluid machinery. Forcing a fluid through pipe fittings consumes energy provided by the drop in pressure across the

490


fittings. This pressure drop is caused by the friction between the fluid and the fitting wall. The problem of predicting pressure losses in pipe fittings is much more uncertain than for the pipe because 1. The mechanism of flow is not clearly defined. At least two types of losses are superposed—skin friction and the loss due to change in flow direction, and 2. There are very few experimental data available in the literature. There are two approaches for analysis of the pressure drop across the pipe fittings equivalent length (L^) and velocity head. In the equivalent length method the fitting is treated as a piece of straight pipe of some physical length, i.e., equivalent length (L^) that has the same total loss as the fitting. The main drawback of this simple approach is that the equivalent length for a given fitting is not constant but depends on Reynolds number and roughness as well as size and geometry [13]. In the case of the other method, the velocity head is the amount of potential energy (head) necessary to accelerate a fluid to its flowing velocity. The number of velocity head (H) in a flowing fluid can be calculated directly from the velocity of the fluid (V) as H = VV2g

(14)

Flow through a piping component in a pipeline also causes a reduction in static head, which may be expressed in terms of velocity head and the resistance coefficient, K as H=K -

(15)

K is thus defined as the number of velocity heads lost due to the piping component. The main drawback of this method is that it also depends on Reynolds number [14]. PARAMETRIC ANALYSIS For flow of non-Newtonian fluid through a straight tube, the steady state Z-component equation of motion in cylindrical coordinate system in a horizontal pipe may be written as p V

+ —^

^' ^ ar

r ae

+V

\=

^ az J

az

1 a ( n , ) ^ 1 axe, ^ axe, (16) r a, r ae az

This equation is dimensionally homogeneous and is true for laminar flow only. The dimensional equality (not actual or quantitative equality) for the equation may be written as V'

AP

\)V


491

This equation is made dimensionless by dividing by VVL, which suggests that the functional relationship of AP with other parameters is as follows ^

pV

=F(VpD/ti)

(18)

So on dimensional grounds the pressure loss in a fitting will depend upon the Reynolds number and the geometry of the fitting as follows AP

AP

F(VpD/|i),

geometrical ratio

= F(Re,a)

/j9\

(20)

For the case of non-Newtonian liquid flow AP — = F(Re,,,a)

(2i)

PREVIEW Liquid Flow Through Pipe Fittings The flow of a fluid through an elbow resembles free vortex motion, the product of the local flow velocity, bend radius remains constant, and there is a well-defined pressure gradient along the radius of curvature of the elbow. One can relate these pressure gradients to the flowrate through the elbow, and the measurement of the pressure difference between the inside and the outside of the elbow can be used to determine the volumetric flow rate of a fluid through the elbow. The first use of elbow as flowmeter was reported by Jacobs and Sooy [15]. They found that the inner and outer pressure difference (Ah) was related to the average velocity as nO.526

V = 5.6 Ahr D.

(22)

where V has units of ft/s and h has units of ft. Levin [16] derived an expression by assuming that the local velocity in the elbow was proportional to the local radius of curvature and he derived V = C,V(2gAh)

(23)

The discharge coefficient C^ is related to radius of curvature and pipe diameter as C, = V(r/2D^)

(24)

492


Yarnell and Nagler [17] reported the flow of water through elbows and observed that the average velocity varied with VAh. However, Lansford [18] observed that the velocity/head relationship varied between Equation 22 and 23. Addison reviewed all the published literature on the use of rectangular and circular cross section duct bends as flowmeter [19]. He modelled the flow around the bend as a free vortex and obtained the following relationship Q = C,AV(2gAh)

(25)

He also noted that a minimum velocity of approximately 2 m/s was necessary for proper flow rate measurement. Spink [20], Murdock et al. [21], Kittredge [22], Hauptmann [23], Polentz [24] and Morrison et al. [25] have reviewed the use of elbow as flowmeter for single phase liquid. Most experimental data were obtained for steam or water. Hauptmann suggested that the elbow flow meter could be used for slurry, but no data concerning this application were presented in his work [23]. Brook [26] and Morrison et al. [25] used elbow as slurry flow meter. Binnington et al. investigated the low Reynolds number non-Newtonian liquid flow in pipe contractions [27]. They have compared the Newtonian velocity components measured by laser speckle anemometer with that obtained by finite element computation. They also have compared the streamline patterns obtained by streak photography and by finite element simulation. Kim-E et al. studied the finite element simulation of shear-thinning fluids in pipe contractions [28]. They extracted the AP/2T^ data from the finite element results and presented these as a function of Carreau number, Reynolds number, and power-law index. Mackay et al. concluded that for purely viscous non-Newtonian fluids the computer-generated data by finite element technique for laminar pressure drop for pipe fittings, such as contractions, and expansions, were likely to be more accurate than those obtained from laboratory measurement [29]. Dudgeon and Hills conducted experiments on the flow-through-stepped contractions and expansions [30]. They analyzed their experimental data by loss coefficient technique and observed that the loss coefficients were largely independent of Reynolds number for the experimental conditions. Townsend and Walters studied the expansion flow of various non-Newtonian liquids [31]. They also simulated the observed flows using finite-element technique. Karr and Schutz [32] tested globe and angle valves over the range of Reynolds number 1 to 10^ Beck and Miller [33] and Beck [34-36] reported the experimental study on a variety of valves, fittings, and bends over the range of Reynold number approximately 30 to 1,000. He reported that the pressure losses for few cases in bends at low Reynolds numbers were less than the losses caused by equal lengths of straight pipe [36]. John tested the flow of kerosene through several 1/2 inch valves and fittings over the range of Reynolds number 1,000 to 30,000 [37]. Pigott [38] presented a hypothesis that the over-all loss in a bend is composed of the following (i) Equivalent straight pipe loss, (ii) True bend loss, unaffected by Re and e/D^, depending only on D^D^, (iii) An additional loss varying with both Re and e/D^ and consequently with friction factor.

Non-NewtonJan Liquid Flow Through Globe and Gate Valves

493

Pigott [39] tabulated the loss coefficient, K, for elbows and bends from literature and established relationship of K with D^D^ and friction factor. Kittredge and Rowley determined the resistance coefficients for laminar and turbulent flow through 1/2 inch IPS valves, fittings, and bends [40]. They found that the resistance coefficients were function of Reynolds number. Eastwood and Sarginson described the experimental investigation of the effect of transition curves on the head loss in flow through 90° bends in pipelines [41]. They observed that for purely circular bend the loss at the bend could be expressed in terms of the equivalent length (L^) of the straight pipe to cause the same loss. Ito studied the pressure losses in smooth pipe bends. He examined the effect of radius of curvature and bend angle on the pressure drop [42]. He proposed the following empirical formulae: I. Bend loss coefficient based on Re(D/D^)^ For Re(D/D^)2 < 91 K = 0.00873 a f^ e(DyD^)

(26)

For Re(D/D^)2 > 91 K = 0.00241 a 0 Re-o^7(DyD/84

(27)

where, f^ is the friction factor for turbulent flow in curve pipes, and a is the numerical coefficient, and its value is given below For e = 45°

a = 1 + 14.2(DyD^)-i ^^

(28)

For e = 90°

a = 0.95 + 17.2(DyD^)-* ^6 f^r DJD^ < 19.7

(29)

a = 1.0 for DJD^ > 19.7

(30)

For 0 = 180° a = 1 + 116(D/D^)-^^2

(31)

II. Bend loss coefficient based on ¥ ^ ( 0 / 0 f-^ He defined the Y as Yê^ = Re(D/D/-5 For Y^(D/Df'

< 9.4

K = 0.00873 a f^ 0(D/D^) For Y\DJDf'

(32)

(33)

> 9.4

K = 0.0074 a 0 Y-''%DJDf-^^'

(34)

The values of a are the same as those found previously. He also examined the pressure losses in commercial screw type elbows and observed a large increase in resistance due to the enlargement and contraction of section.

494


Ward Smith reported the experimental observation of flow through smooth pipe bends of circular-arc curvature [43]. The effects of bend angle, radius ratio, crosssectional shape and Reynolds number also were examined. Crane Engineering Co. sponsored a great deal of work to find the resistance coefficient for flow in various types of piping components [44]. They reported that resistance coefficient varied in the same manner as friction factor as K=fK„

(35)

where f^ and K^ represent conditions for fully turbulent flow. Tremblay and Andrews studied the water and water-stream flow through a halfinch needle-type globe valve [45]. The valve pressure drop for water flow was correlated by calculating resistance coefficient, K. They observed that the resistance coefficient was constant and independent of Reynolds number. Harris and Magnall tested the applicability of orifice plates and venturi meters as a flow-measuring device for non-Newtonian liquids [46]. Miller published a reference book to estimate the head losses through contractions, expansions, and miscellaneous piping components [47]. The problem of uncertainty of the interactions between the boundary layer changes in flow-through contractions and expansions, which may be generated by multiple steps in diameter in some fittings, and the estimation of the pressure drop may involve considerable error. He graphically represented the valve loss coefficient, K, with valve opening for butterfly valve, diaphragm valve, gate valve, globe valve, and angle valve, etc. Moller and Elmqvist presented the head loss data for water, for thermomechanical pulp at two concentrations, and for kraft pulp at two concentrations for range of flow rates through fully open and partly closed gate valves, elbows, bends, expansions, and contractions [48]. They observed that the head losses for pulp suspensions across these fittings were proportional to the square of the flow velocity in the pipe. Hooper developed a new technique called two-K method to predict the head loss in pipe fittings [49]. He defined K, a dimensionless factor, as the excess head loss in a pipe fitting, expressed in velocity heads. It is a function of Reynolds number and of the exact geometry of the fitting as K = K/Re = K^(l + 1/D^)

(36)

where Kj = K for the fitting at Re = 1 K^ = K for the fitting at Re = a He also tabulated the values of K, and K^ for some standard elbows (45°, 90° and 180°), tees and valves (gate, ball, plug, globe standard, globe angle or Y-type, Diaphragm dam type, butterfly, check lift type, check swing and tilting-disk type) of a particular opening. Later he reported the mathematical expression for K, based on inlet velocity head for square and tapered reduction and expansion; pipe reducer; thin, sharp orifice; and thick orifice [50]. Fairhurst studied the flow of water and air-water through piping components, such as gate valve, globe valve, diaphragm valve and orifice plates [51]. Resistance


495

coefficient, K, for water flow through each component was measured and found to be comparable with the values presented by Miller [47]. Hoang and Davis reported that for the relatively sharp return bends the pressure drop increased by a factor of approximately 20 compared to that in a pipe of equivalent length of the bend [52]. Simpson et al. studied the pressure losses through 25mm gate valve for liquidvapor Freon 113 [53]. They correlated their single-phase valve pressure loss data as AP = C, i y -

(37)

where, C^ = 24.7 for area ratio 0.25 = 7.75 for area ratio 0.4225. Norstebo studied single and two-phase pipe component pressure loss for refrigeration industry [54]. He used 5 differemt globe valves commonly used in refrigeration plants and bends (90° and 180°). The single phase pressure loss across the fitting in terms of resistance coefficients, K, were determined experimentally for subcooled refrigerant R113. Resistance coefficient, K, for valves varied from 2.1 to 7.0, independently of Reynolds number. Edwards et al. studied the frictional head loss for different pipe fittings for flow of Newtonian and non-Newtonian liquids in laminar flow condition [14]. They proposed generalized correlation with loss coefficient and Reynolds number (Re = VpD/ji for Newtonian liquid and Re = Re^^^ for non-Newtonian liquid) of individual fittings. For 1 inch and 2 inch size elbow K = 842/Re

(38)

For full open gate valve of nominal sizes of 1 inch and 2 inch K = 273/Re

(39)

For full open globe valve for 1 inch size K = 1460/Re

Re < 12

(40)

K = 122

Re > 12

(41)

K = 384/Re

Re < 15

(42)

K = 25.4

Re > 15

(43)

for 2 inch size

They also presented the functional relationship, i.e., K with Re for contractions, sudden expansions and orifice plates.

496


Sookprasong et al. studied the single-phase and two-phase (air-kerosene) flow through 5.08 cm diameter horizontal pipe and piping components, such as a gate valve, an elbow, a globe valve, a swing valve, and a union [55]. Single-phase pressure drop produced by each component were used to calculate resistance coefficient, K. They found that the resistance coefficient for each component was not sensitive to Reynolds number in the range of lO'^ to 2.2 x 10^ They also compared their resistance coefficient with that reported in other investigations available in literature [44,56-58]. Nandi and Das [59] studied the water flow through different U-bends at turbulent condition and proposed the following relation for friction factor across the bend f^ = 6.06 X 10-^De'^«

(44)

Das et al generated the experimental data on non-Newtonian pseudoplastic liquid flow through different types of bends in the horizontal plane [60]. They developed a generalized correlation for predicting the friction loss as = l + 2.5687xlO-'De'MR

-T

(45)

180 j Banerjee et al. generated experimental data across 1/2-inch globe and gate valves in the horizontal plane for non-Newtonian pseudoplastic fluids (dilute aqueous solutions of SCMC) in laminar flow condition [61]. They examined the effect of valve opening on pressure drop. Some of the results are shown in Figures 1 and 2. The pressure drop increases with an increase in volumetric flow rate for a constant opening. As the opening became smaller, the curve became steeper. They also examined the effect of non-Newtonian characteristics on pressure drop across the valve. Figures 3 and 4 showed the pressure drop across the globe valve and gate valve at a particular opening as a function of the liquid flow rate. It is clear from the graph that as n increases, the pressure drop decreases. They developed the following functional relationships using Equation 21 through multivariable linear regression analysis for each valve and also carried out detailed statistical analysis. Correlation for globe valve ^P

Q OAA D.:.-0 061 _,-0.797

— ^ = 8.266 Re^^R a

(45)

Correlation for gate valve ^^

—

1 O n ^ Dz:.-0 197 ^-1.987

= 1.905 Re^,R a

Range of variable investigated 0.601 < n < 0.901 0.014 < K ' < 0.711

(47)


VOLUMETRIC FLOW RATE, (mVs)xlO^ Figure 1. Variation in pressure drop across the globe valve at different openings with volumetric flow rate.

lij

40 1 .

lij

I

30

r

10 CO

o tt: o Ô

40 VALVE O P E N I N G : 3 7 . 5 %

(/) (/) O Q: o o

SYMBOL SCMC CONC.

30

S^20 Q: ^ o ^

UJ

cc 3

10

in UJ or QL

40

VOLUMETRIC FLOW RATE (mVs))(10^ Figure 4. Variation in pressure drop across the gate valve at different concentration of SCMC solutions with volumetric flow rate.


499

EXPERIMENTAL STUDIES Most of the literature on experimental studies on flow-through pipe fittings have been mentioned earlier. Experimental works were on the measurement of pressure loss across the fitting and its subsequent correlation either with equivalent length or with resistance coefficient. Karr and Schutz [32], Beck and Miller [33], Beck [34], Pigott [38,39], John [37], Ito [42], Ward Smith [43], Tremblay and Andrews [45], Fairhurst [51], Norstebo [54], Simpson et al. [53], Edwards et al. [14], Sooprasong et al. [55], Nandi and Das [59], Subbu et al [62], Das et al. [58], Das et al. [63] and Banerjee et al. [61], etc. conducted experiments on either single phase or two phase flow through piping components. In all experiments the apparatus consists of storage tank, long test section, flow and pressure measuring devices. Experimental setup of Banerjee et al. [61] is shown in Figure 5. The test section consists of long upstream portion, pipe fitting and long downstream portion. Detailed test section is shown in Figure 6. The test section was provided with pressure taps (piezometric ring) at various points in the upstream section and downstream section, sometimes on the pipe fitting like bend, elbow, etc. The static pressure at the different points was measured by means of simple manometer or piezoresistive pressure transducers. The main idea of putting long upstream and

SVi

rc©=5d|s5=

Pl3

Pi5

PIS

TO TANK

MB==1|=?=

P3 P4 Pi

iW TV

TO TANK

Figure 5. Schematic diagram of tiie experimental apparatus E: storage tank; LC: level controller; TV: test valve; P: pump; P^-Pê: manometer taps; RL^-RL^: rotameters; S^, S^: liquid holding tank; ST: stirrer; SV^, SV^: Solenoid valves; T: thermometer; V^-V^: valves.

500


PURGE SOLUTION 5 0 — [ - • go - | l O K ' H '0 h '8 -|^— so UPSTREAM

'•

PORTION Q

f*' ^

I

so

?

1

4

DOWMSTR-JEAM PORTION

[

TO ^ TANK

f% wrâr^rTri

TO MANOMETERS

L

ALL DIMENSIONS IN CM

AIR MERCURY MANOMETER

Figure 6. Details of the test section P^-P^Q! pressure taps, V: valves.

a 1

iI

\ 1

or 3

S >
OôÔ

t

GAS SPARGER

AIR ( a )

( b)

( c)

Figure 1. Schematic representation of a bubble column (a), Internal loop (b) and external loop (c) air lift reactors. VISUAL OBSERVATIONS The hydrodynamic characteristics of bubble columns and airlift columns depend critically on the flow regimes occurring in the column [1]. We can usually observe three flow regimes: 1. the bubbly (or homogeneous) flow regime, characterized by almost uniformly sized bubbles with equal radial distribution; 2. the churn-turbulent (or heterogeneous) flow regime, characterized by large bubbles moving with high rising velocities in the presence of small bubbles in the ambient liquid phase; and 3. the slug flow regime, in which large bubbles are stabilized by the column wall leading to the formation of bubble slugs. These regimes occur in order of increasing gas flow rate. The churn-turbulent flow regime is of practical interest in most commercial-scale columns. In viscous non-Newtonian fluids, large bubbles produced near the sparger by coalescence rise in the presence of a large number of tiny bubbles. A bimodal bubble size distribution is found: 1. large ellipsoidal cap bubbles with an equivalent diameter greater than 2 cm and 2. tiny spherical bubbles with a diameter smaller than 1 mm. The bubble coalescence becomes very intensive in non-Newtonian liquids. The coalescence of bubbles near the sparger and the rupture of large bubbles at the top surface generate very tiny bubbles. During the motion of intermediate size bubbles

Non-Newtonian Effects in Bubble Columns

543

having prolate-tear drop shapes, their tails break up into tiny bubbles. Due to their low rising velocity, they accumulate in the liquid and circulate with the liquid flow. Since many tiny bubbles are dragged down by the liquid recirculation flow when they arrive at the surface, they accumulate in the liquid. Sometimes a significant part of the gas hold-up in the liquid is due to such very small bubbles. The rising large bubbles induce strong liquid motion either by liquid entrainment or by density-driven circulation. Liquid entrainment and transport in the wake of the ascending bubbles cause liquid motion. Since the rising bubbles tend to concentrate more towards the axis of the column by "wall effect", the differences in the density of the gas-liquid mixture or gas holdup at the center and that near the column wall are present. The gas holdup profile results in a profile of static pressure which in turn causes liquid circulation. By these two mechanisms, the liquid circulation is created in the bubble column. On the whole, the liquid rises with the bubbles in the center of the bubble column and flows downward in the outer annular region as described later in detail. In the case of water where small bubbles rise almost vertically, the liquid downflow is relatively well-defined. On the other hand, in the case of oscillating large bubbles in viscous non-Newtonian liquids, welldirected liquid circulation does not exist. Due to the rapid and random changes in the path of the downflow, turbulence is generated in the liquid phase. In other words, the significant liquid phase turbulence is created by the static pressure fluctuations resulting from the oscillating motion of large bubbles. In an airlift column, the overall circulation of the liquid just described is enhanced as compared with that in a bubble column. For less viscous liquids like water, the bubbles rise only in the riser at very low gas flow rates. As the gas velocity increases, small bubbles are dragged down by the liquid circulation into the downcomer. The penetration depth of bubbles into the downcomer depends on their size. As the gas velocity increases further, large bubbles also are dragged into the downcomer, and recirculation of small bubbles is observed. In highly viscous nonNewtonian liquids, the basic flow configuration is the same as that for inviscid liquids. However, even at very low gas flow rates, bubble coalescence in the riser leads to the formation of large bubbles. A large number of very small bubbles also are present, and many of them recirculate. In viscoelastic liquids, such as xanthan gum solutions, very tiny bubbles rise in a cluster and foaming is also observed. APPARENT VISCOSITY It is customary to account for the non-Newtonian fluid behavior by introducing the so called effective viscosity to define various dimensionless groups. Unlike its constant value for Newtonian liquids, the effective viscosity of non-Newtonian pseudoplastic type fluids depends upon the operating conditions (e.g., gas and liquid velocities) as well as on the geometrical details of the system. Indeed, the lack of a rational definition of the apparent viscosity or characteristic shear rate appears to be the main impediment in extending the well established predictive correlations for Newtonian media to non-Newtonian media. When we develop correlations for design parameters in bubble columns with non-Newtonian media in an analogous manner to the case of Newtonian media, Newtonian viscosity [i is simply replaced by an apparent viscosity \i^ for non-Newtonian media.

544


Although a power-law model cannot describe the flow curves over a whole shear rate range, it has been widely used to describe the non-Newtonian flow behaviors, i.e., shear-thinning (or pseudoplasticity, n < 1) and shear-thickening (or dilatancy, n > 1). x = K(Y)"

(1)

Based on the power-law model, the apparent viscosity is written as: Happ = K(Y)"-'

(2)

This relation indicates that the apparent viscosity varies depending on the shear rate y, unlike the Newtonian viscosity. Therefore, the definition of the appropriate shear rate characterizing hydrodynamics is required to estimate the effective apparent viscosity for non-Newtonian fluids in bubble columns. In bubble columns, the shear rate is not uniform and unknown. The motion of bubbles results in the wide variation of the shear rate, which is hopelessly complicated and cannot be analyzed. Therefore, the characteristic shear rate should be evaluated on the basis of a simplified physical picture of hydrodynamics in bubble columns and aided by experimental observations. A relation for the effective characteristic shear rate (y^^^) proposed by Nishikawa et al. has been commonly used in the literature [7,8]. Yeff = 5,000 U,^

(3)

The above empirical relationship was obtained using only 10 experimental heat transfer data points in the bubble column of 0.15 m diameter [7]. It should be emphasized that the apparent viscosity in the work of Nishikawa et al. [7] was just an adjustable parameter to fit the data for non-Newtonian fluids to the empirical correlation for Newtonian fluids. Intuitively, it appears to be wishful thinking that the effective shear rate can be given by such a simple relation in a highly complex flow situation. Furthermore, Equation 3 does not account for any geometrical features, such as column diameter or the properties of the liquid medium. Most definitions proposed after their correlation assume that y is proportional to U^ [9,10] and a range of numerical constants have been used [11]. Based on dimensional considerations, Henzler and Kauling [12] proposed the following relationship: ' Ep ^ Ya K

(4)

Recently, Kawase and Kumagai examined the applicability of this relation and determined the proportional constant to be (2.5)^^" [13]. Stein presented a semi-theoretical approach on the assumption of bubble chain surrounded by a ring channel of liquid [14]. However, it may be unrealistic for heterogeneous flows in bubble columns and, moreover, the resulting correlation seems to be empirical instead of theoretical. An expression for the shear rate in an airlift column was examined by Chisti and Moo-Young [11], who presented:


Y = 5 , 0 0 0 | ^ ^ ^ ^ Usg

545

(5)

Since the power input in an airlift column is responsible not only for liquid flow in the riser but also for that in the downcomer (Allen and Robinson [15] and Kawase and Kumagai [13]), Equation 5 is not strictly correct. It should be emphasized here that in the derivation of theoretical correlations for design parameters the introduction of the apparent viscosity concept is not necessary, and, as a result, the use of questionable definitions of characteristic shear rate such as Equation 3 is not required, as discussed in the ensuing sections. CIRCULATION MODEL Liquid circulation is developed in a bubble column or airlift reactors because of the introduction of gas, and it affects the performance of the reactor. As shown in Figure 2, if a gas is injected in the center of the bubble column, in the core region (r < R^) the liquid rises with the bubbles and the liquid velocity decreases with distance from the column center. In the outer annular region (r > R^), liquid flows downward [16]. Between these two sections there is a transition point (r = R^) at which the velocity is zero. In the case of high flow rate condition, the transition point occurs at around R^ = 0.7R. It is important to estimate the extent of the induced liquid phase mixing. While experimental study of the liquid circulation has been carried out by a number of investigators, theoretical analysis of this problem is rather limited. Some models based on the pressure balance or the energy balance have been proposed for the liquid phase flow patterns [1]. In the turbulent flow regime, a circulation cell model and a recirculating flow model have been used to discuss transport phenomena in bubble columns. Whalley and Davidson proposed a model for the turbulent liquid velocity field in bubble columns with low aspect ratio (H/D ~ 1) by writing an energy balance [17]. Later, Joshi and Sharma extended this approach to propose a multiple cell circulation model for bubble columns [18]. In this model, the liquid phase of a bubble column is envisioned as a series of mixing cells with a height roughly equal to the column diameter (Figure 3). Zehner developed another cell model, a cylindrical-eddy cell model [19]. Riquarts developed a model based on a turbulent stochastic mixing process [20]. In this model, the mixing is assumed to be due solely to the rising bubbles. Ueyama and Miyauchi obtained the liquid velocity profile by solving the NavierStokes equation in conjunction with an empirical expression for radial variation of gas hold-up and turbulent viscosity [21]. According to their recirculation flow model the flow pattern of liquid shown in Figure 4 is assumed. This model has been modified by Walter and Blanch [22] and Kawase and Moo-Young [16]. Very chaotic behaviors of the liquid in bubble columns may suggest that Joshi and Sharma's [18] circulation cell model is more realistic than Ueyama and Miyauchi's [21] model. Clark et al extended a force balance approach to non-Newtonian liquid circulation in bubble columns [23]. It is based on the turbulent mixing length model. Their computational results indicate that liquid circulation is enhanced due to shearthinning. Thus, the flow in bubble columns is extremely complex, and whether or

546


1-2

Figure 2. Radial profile of axial velocity in turbulent flow regime. The hashing shows the spread of experimental data (re-plotted from reference 16). not the isotropic turbulence theory can actually be applied is not quite certain. Despite these uncertainties, however, the theory seems to work well for certain parameters in bubble columns. LIQUID CIRCULATION VELOCITY The superficial gas velocity, U^ is sometimes not the most pertinent process parameter, and the liquid velocity reflects better and more directly the complex liquid phase mixing when the liquids are non-Newtonian. The liquid circulation velocity increases with the increasing superficial gas velocity; however, Kawase and Moo-Young developed a hydrodynamic model for the liquid phase in bubble columns with non-Newtonian fluids on the basis of an energy balance and the

Non-Newtonian Effects in Bubble Colunfins

547

Figure 3. Schematic representation of the multiple circulation cell model of Joshi and Sharma [18].

548

Advances in Engineering Fluid Mechanics rrO

rrR

Figure 4. Schematic representation of the recirculation model of Ueyama and Miyauchi [21].

mixing length theory [16]. They derived an expression for the liquid velocity at the column axis using a combination of an energy balance and the mixing length theory.

U,„ =0.787

gDU,

1/3

(6)

The correlation is in good agreement with the data [16]. Recently, Garcia-Calvo and Leton [24] obtained the following expression for the mean liquid velocity in the core region of bubble columns with non-Newtonian fluids:


U„ =

U,„ f no

549

m

where m = 2.3. The mean liquid velocity in the core is predicted by combining the above equation with the energy balance and an expression for gas hold-up (Equation 21). The effect of non-Newtonian flow behavior is included in the energy balance. They obtained reasonable agreement between the calculated and measured results of liquid circulation velocity. The model based on a combination of an energy balance over airlift loops with analytical expression for shear stress in a turbulent non-Newtonian flow in airlift columns was developed by Chisti and Moo-Young [25]. Philip et al. [26] calculated the liquid circulation velocities in the internal-loop airlift columns from a pressure balance around the circulation loop as: Ap„ = Apj, + Ap^ + Ap3

(8)

The equations used for calculating the various terms in the above pressure balance are listed in Table 3 of the paper of Philip et al. [26]. Shamlou et al [27] developed a model for the liquid circulation rate in internalloop airlift columns using a combination of a drift-flux model with an energy balance. The superficial liquid circulation rate in the riser U^,^ is given as U,. = U J l - ^p

+ k)] + UJl

+ k)

(9)

where the ratio of the liquid-wake volume to bubble volume, k, and the mean primary liquid velocity in the column, U^j, are given as 1.4(U„/U,,,r'-l

(9a)

4(0.512n'+Kp/27cHD)

^^^^

V ^g y

and

"

respectively. The resulting equation is used to estimate the liquid circulation rate in conjunction with an equation for gas hold-up (|) . Figure 5 shows model predictions along with the experimental data for Saccharopolyspora erythraea (n = 0.55). The comparison between model predictions and experimental results is considered reasonable. Kawase [28] analyzed the liquid velocity in external-loop airlift columns with nonNewtonian fluids using the concept of an eddy diffusivity and proposed a correlation forU,. sir

Recently, Kemblowski et al [29] derived the following correlation for the liquid velocity in the riser, U^j^, for an external-loop airlift on the basis of the energy balance:

550


06 Uslr (m/s) OU 0-2 I 0

0-15

1

^

0f

o/

0 0

\

o/^

0-10 I L 005 h

n= 1

0

9^

1

'

0-1

0-2

Usg

(m/s)

Figure 5. Comparison between the predictions and measurements of liquid velocity and gas holdup for n = 0.55 and n = 1, according to Shamlou et al. [27].

1


'•"

1 rC, n(1 _/h A /A ^2)^.+/iiA /A J^ ^2-,] - (^,.r2r- 4-r + CBr (A./A, 4h Ji-ff J/pjD , _L+ff J/nD , r(A7A

551

(10)

To estimate the superficial liquid velocity in the riser, this correlation must be combined with the Fanning friction factors in the riser and downcomer, f^ and f^, the frictional loss coefficients at the top and bottom of the airlift, ^^ and ^g, and the gas hold-up in the riser, cp _.. This approach predicted their experimental data in a pilot-plant scale external-loop airlift column for a carboxymethyl cellulose solution, with an error of ±20%. Popovic and Robinson [30] presented an empirical correlation for external-loop airlift columns which may be written as: U, = 0 . 2 3 U : - ( A , / A , r ' n : -

(11)

In the comparison of the proposed correlation with the experimental data, the average deviation is about 15%. MIXING TIME The mixing time is defined as the time required by a mixed liquid to reach a specified degree of homogeneity after a tracer pulse has been added to it. It is often more important than the axial dispersion coefficient mentioned next. Although the axial dispersion coefficient and the mixing time are related to each other, the mixing time is a more direct index of homogeneity of concentrations of components in liquid compared with the axial dispersion coefficient. For example, a knowledge of mixing time gives useful information about the distribution of the concentrations of acids or alkalis which have been added for the purpose of pH control. It should be noted, however, that there has been a large inconsistency among the data of mixing time reported in the literature since the mixing time is strongly dependent on the experimental technique used and the degree of homogeneity specified. The mixing time decreases with an increase in U^ (Figure 6). At higher gas flow rates the mixing times tend to level off and approach a constant value (Haque et al. [31]). Haque et al. [31] obtained an expression for mixing time t^ = Nt^ = 2D(H/D + 3)/U^

(12)

where the average liquid circulation velocity, U^, is given by U^ = U,, (gDU^/2U,/-^

(12a)

The number of circulations required for complete mixing, N, was determined as a function of column diameter, D, and dispersed liquid height, H, using the data for viscous shear-thinning CMC aqueous solutions. A relatively simple correlation for mixing time in bubble columns with nonNewtonian fluids was proposed by Kawase and Moo-Young [32] using the isotropic turbulence theory as follows

552


to

Ugg (m/s) Figure 6. Typical dependence of mixing time on gas velocity in carboxymethyl cellulose solutions (from reference 31). t^ =6.33a(2) 2(l + n)/3n

V

Xl/3

gDû,,)

(13)

where a is a function of the distance between the injection point of the tracer and the monitoring point and the specified degree of homogeneity, and its values are given in the paper of Kawase and Moo-Young [32]. Recently, Popovic and Robinson [33] proposed the following empirical correlation for mixing time in external-loop airlift columns: (iJW) = 571D-«-^(A,/AJ^^'^H-'Û^;Vr

(14)

Their experimental data is predicted by this correlation within ±15%. CIRCULATION TIME The circulation time for airlift columns is defined as the time required for a fluid element to travel once around the riser-downcomer loop, and it characterizes the


553

mixing efficiency of airlift columns. At low gas flow rates, the circulation time decreases steeply with increasing U^ , whereas at higher gas velocities it tends to level off. Kawase and Moo-Young [34] proposed a semi-theoretical correlation for t^ in internal-loop airlift columns with non-Newtonian fluids.

8.3n^/M 1 + ^ ^

(UygD)'/^

(15)

Recently, Kawase et al. [35] obtained the correlation for external-loop airlift columns as V'-^"

VTT2 A

1/3

t. = 5 . 3 3 n 2/3 1 . ^

2U.„

(16)

It was found that this equation predicts liquid circulation time in external-loop airlift columns reasonably well. AXIAL DISPERSION COEFFICIENT The axial dispersion model has been widely used to characterize the non-ideal mixing behavior in the liquid phase. In this model, axial dispersion coefficient is the single parameter representing the extent of backmixing. The following expression for the axial dispersion coefficient was derived by Kawase and Moo-Young [16]: E^ = 0.343n-«^3(gD4u pi/3

(17)

This correlation is based on an energy balance and the mixing length theory, and indicates that the axial dispersion coefficient increases with the increasing degree of shear-thinning. Deckwer et al. found the highest axial dispersion coefficient for 1.6wt% CMC aqueous solution despite this being the most viscous liquid in their measurement [36]. Similar results were obtained in an internal-loop airlift with xanthan gum solutions by Fields et al. [37]. It should be noted that Kelkar and Shah observed a decrease in the dispersion coefficient with respect to apparent viscosity of the liquid phase [38]. Baird and Rice first applied the isotropic turbulence theory to correlate the axial dispersion coefficient in Newtonian fluids [39]. Their successful approach has been widely quoted to predict design parameters in bubble columns (Kawase and MooYoung [40]). It was extended to non-Newtonian fluids by Kawase and Moo-Young [32]. The resulting equation may be written as E^ = 0.158(2)2(»^"V3n(gDÛ^pi/3

(18)

Recently, Garcia-Calvo and Leton [24] proposed a model for liquid mixing in bubble columns on the basis of an energy balance and suggested:

554

Advances in Engineering Fluid iVIechanics

E^ = D(U,ÛJ2)'«

(19)

where U^^ = 0.35(gD)'^^. The mean liquid velocity in the core, U,^, is evaluated by Equation 7 along with Equation 21. It is clear that more attention must be paid to the determination of axial dispersion coefficient in bubble and airlift columns with non-Newtonian fluids. GAS HOLDUP Since gas holdup is one of the important design parameters for the performance of bubble columns, study of gas holdup has been extensively carried out. Gas holdup depends mainly on gas velocity and the physical properties of the liquid. Although a number of correlations have been proposed, no single generalized correlation is available at present. Only a few attempts have been made to predict gas holdup theoretically. Kawase et al. developed a model for the gas holdup in bubble columns with nonNewtonian fluids [41]. In the model, the liquid circulation caused by the introduction of a gas is considered as the buoyancy-induced circulation. The resulting equation is written as 2-(.^n + 5)/(n + l)

1 -d) ,

-(n + 2)/2(n + l)

UTP g"K

(20)

Extensive comparisons of this correlation with the experimental data shows that it tends to underpredict the value of (|) at lower gas velocities, and increasing deviations occur as the value of n drops to about 0.1 or lower. According to Garcia-Calvo and Leton [24], the gas holdup is given as:

•"^ ^ u . + o'su,

(21)

By using Equations 7 and 21 and the energy balance, the gas holdup is calculated. The comparison between Equation 21 and a wide range of data is found to be satisfactory. Godbole et al. [42] proposed the following correlation for gas holdup in nonNewtonian fluids in a bubble column: ^^=0.207U:Xpr

(22)

This empirical correlation indicates that the gas holdups decrease slightly with increasing apparent viscosity estimated using Equation 3. Haque et al. [31] proposed an empirical correlation which includes the effect of the column diameter as: ^^ =0.171U>:;;^^D^'^

(23)


555

For (|)g in a bubble column with solids suspensions, Capuder and Koloini [43] obtained the following empirical correlation:

= 0.083

(24)

Vatai and Tekic [44] obtained an empirical correlation: ^^ = 0.19n-«-^(UygD)«-^2-«-«^"(gD^p/^^pp)««^

(25)

where the apparent viscosity is defined as L | i^ = K{U^ /(D/2)}""' instead of Equation 3. This correlates their data with a mean deviation of 10.6%. Their results indicate that an increase in the apparent viscosity generally leads to a decrease in gas holdup, but in small-diameter bubble columns the gas hold-up increases with apparent viscosity. Shamlou et al. proposed a model for the prediction of gas holdup and liquid circulation in internal-loop airlift columns [27]. It is based on the drift-flux model of Zuber and Findlay [45] and an energy balance taking into account the physical interactions between the liquid, the bubbles, and the liquid wake associated with the bubbles. The expression for gas holdup obtained from the drift-flux model is written as U., (C„+1)(U,,+U,,) + U ,

(26)

Determination of gas hold-up from Equation 26 requires a knowledge of the superficial liquid circulation rate, U^^, given by Equation 9 and the single bubble terminal rise velocity U^^^. Most researchers have used U^^^ = 0.25 ms*. The gas holdup and liquid circulation data in 250 L pilot-scale internal-loop airlift bioreactor for Saccharopolyspora erythraea (n = 0.55) were satisfactorily correlated by this model. When Kemblowski et al. [29] derived the correlation for the liquid velocity in the riser for an external-loop airlift on the basis of the energy balance, they used the following correlation for gas hold-up:

^gr = 0 . 2 0 3

¥v''' ru.„.A.^'•'' Mo'

(27) V *-^slr^d J

where \4(ii-l)

Fr = (U,, + U,^, )7gD,

and

Mo = ^ 0 -

8U, V D. J

3n + l 4n

They obtained reasonable agreement between the predictions and the data for one CMC solution (n = 0.758).

556


Recently, Kawase et al. derived a theoretical correlation for riser gas hold-up in external-loop airlift columns [46]. £

_

2 - ( 3 n + 5)/n + l

-(n + 2)/2(n + l ) / | _j_ A

/ A

i-T.

\-3(n + 2)/4(n + l)

sgr

r

(28)

g"K

The following empirical correlation for external-loop airlifts was reported by Popovic and Robinson [47]. (t)^, = 0.465U:^r[A,/(A, + A, )]'•«'n:^;"'

(29)

This correlation fits the measured data with a mean deviation of ±11% and u

was

râpp

estimated using Equation 5. Most aforementioned studies relate to the inelastic power law fluids and the churn-turbulent conditions prevailing in columns. Under these conditions, the type and details of the device used to introduce the gas appear to exert virtually no influence on the value of gas holdup. However, in homogeneous bubble flow regime, the average holdup in bubble columns is strongly influenced by the type and the size of the openings in the sparger as well as the other parameters, such as the height of the column, etc. [31,36]. Also, much confusion exists concerning the effect of the viscoelasticity of the liquid on the value of gas holdup. For instance, Peschke [48], Schugerl [49] and Kelkar and Shah [38] all reported higher values of gas holdup in visco-elastic polyacrylamide solutions while Moo-Young and Kawase [50] found no effect of viscoelasticity on holdup in similar solutions used in bubble columns with and without draft tubes. One difficulty in the reconciliation of these divergent results is the lack of quantitative information on the viscoelastic measurements, not to mention the different geometrical configurations employed by different investigators. Thus, it is not yet possible to put forward predictive expressions for the estimation of gas holdup in bubble columns without and with internals for viscoelastic liquids. HEAT AND MASS TRANSFER The rates of heat and mass transfer in bubble columns are determined by intricate interplay between the physical properties, notably the viscosity of the liquid medium and the kinematics of flow, which in turn is strongly dependent on the design and geometrical arrangement of a bubble column, such as the type of sparger, the type of internals fitted inside/outside the column. Additional complications arise when the liquid phase exhibits non-Newtonian rheological characteristics, in which case the apparent viscosity itself depends upon the type of flow prevailing in the column. In view of these complexities, it is not at all surprising that little theoretical work has been attempted to devise predictive schemes for the estimation of heat and mass transfer coefficients in bubble columns. Most of the progress in this area has, therefore, been made through the use of dimensional considerations aided by experimental observations. Such developments are useful in design calculations, but this approach neither provides any insight into the nature of physical processes


557

nor is it applicable universally. While Table 1 provides an overview of the activity in this field, a selection of more widely used correlations is presented in the ensuing sections. We begin with heat transfer in bubble columns. Heat Transfer The heat transfer rates in bubble columns are much higher than that anticipated from single phase flow considerations. This enhancement is ascribed solely to the bubble-induced turbulence and liquid circulation. Little work has been reported on heat transfer, both at wall and to/from immersed surfaces, in bubble columns employing non-Newtonian media. Nishikawa et al. reported the first set of data on the effect of shearthinning viscosity of CMC solutions on jacket and coil heat transfer coefficients [7]. They reconciled their results for Newtonian and power law liquids by introducing the notion of an effective viscosity estimated via Equation 3, provided the gas velocity was greater than 40 mm/s. For superficial gas velocity lower than this value, the effective shear rate varies as U'^^ for coil heat transfer sg

and U^ for jacket heat transfer. It needs to be emphasized here that, though the effective shear rates calculated in this fashion facilitate the development of a single correlation for heat and mass transfer in bubble columns with Newtonian and nonNewtonian liquids, these do not necessarily reflect the true kinematics of flow. Furthermore, Nishikawa et al. found it necessary to sort their results into three categories, depending upon the superficial velocities of the gas and the liquid [7]. For U < 54 m/h and U < 1,000 m/h: si

sg

'

\|/ = 0.054 \]l'^^

(30)

For U < 54 m/h and U > 1,000 m/h: si

sg

'

\|/ = 0.3

(31)

and for U > 54 m/h: si

\|/= 0.54U;;(U, - 54)'^^

(32)

where \|/ = (h/pCp)(pV^gAp)'^^(CpM/k)2/3(Y_ )-o.o5 V. = u /u IS

"VV

~

where subscript w refers to cooling water temperature. Note that Equations 30-32 are not dimensionless, and the units of various quantities are: Cp(kcal/kg°C); h(kcal/m2h°C); p(kg/m^); |i and \i^ (Poise); k(kcal/mh°C) and U^^; U^,(m/hr). Subsequently, Kawase and Moo-Young [51] combined the three zone concept of Levich [52] with the isotropic turbulence ideas to develop a dimensionless relation for heat transfer with power law fluids in bubble columns [53]. Their final equation is: Nu = 0.075(10.3n-^-^^)Pni^3(Pr*)^'^FrP(Re*)P^3(n+i)

(33) (text continued on page 561)

Table 1 Heat and Mass Transfer Studies in Bubble Columns and Modifications Thereof

Investigator

D (mm)

n

Liquid Systems

K (Pa*sn)

Remarks

2 c

Nishikawa et al. [7]

51, 150

CMC solutions

-

-

Wall and immersed coil heat transfer and proposed j = 5,000 USg

Buchholtz et al. [65,661

140

CMC solutions

0.75-0.82

1.3-5.0

Volumetric mass transfer coefficient, gas holdup, and bubble characteristics. Data was subsequently correlated by Henzler [64].

Baykara and Ulbrecht [73]

152

PEO and PAA solutions

-

-

Data showed reduction in k,a with the increasing liquid viscosity.

Nakanoh and Yoshida [61]

145

CMC and PAA solutions

-

-

Presented a correlation for k,a in inelastic and viscoelastic liquids.

Hecht et al. [74]

200

PAA solutions

0.38-0.63

0.1 1-3

Lowering in k, a due to viscoelasticity but no correlarion presented.

Voigt et al. [75]

200

CMC solutions

0.7 1-0.82

0.09-0.73

Gas holdup and volumetric mass transfer coefficient.

Schumpe and Deckwer [76]

140

CMC solutions

0.68-0.86

0.048-0.72 1

Gas holdup, specific interfacial area, and volumetric mass transfer coefficient and correlations for all these parameters.

Deckwer et al. [36]

140

CMC solutions

0.82-0.92

0.04-0.23

Studied 0, mass transfer, gas holdup and pesented correlations.

Sada et al. [37]

65

Magnesium and calcium hydroxide suspensions--CO,

0.25-0.96

0.92-163 x

Both mass transfer coefficient and interfacial area decrease with an increase in slurry viscosity.

-

D

%

5'

2. CD

3 ?.

a

1 E. a

I ZJ

!2

A' V)

Godbole et al. [42]

305

CMC solutions

0.44-0.84

0.068-7.78

Extensive data on gas holdup, mass transfer, and interfacial area and correlations.

EL-Temtamy et al. [811

150

Yeast suspensions

0.82

0.001-0.023

Liquid phase volumetric mass transfer coefficient measurements.

Kawase and MooYoung [67,70]

230

CMC solutions

0.54-1

0.00089-1.22

Effect of internal draft tubes on gas holdup and mass transfer coefficient and correlations.

Kawase and Moo-Young [511

-

-

-

-

Combines the turbulent boundary layer and isotropic turbulence ideas to deduce Nusselt number for wall and immersed coil in power law liquids.

Kawase et al. [62]

-

-

-

-

Theoretical expression for volumetric mass transfer coefficient in bubble columns.

-~ -

Schumpe and Deckwer [72]

60, 140, 300

CMC, Xanthan and PAA solutions

0.18-1

Moo-Young et al. [631

760

Mycelial fermentation broths

0.07-5.3

Moo-Young and Kawase [SO]

230

PAA soiutions

0.5-0.6

Kawase and Moo-Young [70,84]

-

-

.023-9.8

.24-5.2

0.12-0.63

-

Review of previous literatire and new results on flow patterns, holdup and volumetric mass transfer coefficient. Gas holdup and volumetric mass transfer coefficients with a draft tube in the column. Fluid viscoelasticity results in higher gas holdups. but lowers mass transfer coefficient. Semi-empirical expression for volumetric mass transfer coefficient in stirred aerated vessels.

z

z

0

n,

S

0"

$. 3

3 8 -. 3

m

C 0-

P n,

2

V,

UI UI (O

Table 1 (continued) Investigator

D (mm)

Vatai and Tekic [44]

50, 100, 150, and 200

CMC solutions

150

Xanthan, PAA and mixed solutions

0.098-1

Suh et al. [79]

Liquid Systems

K (Pa*sn)

n

0.8-1

0.001-0.0668

6.2-13,000

Remarks

Effect of column diameter on holdup and mass transfer coefficient. Effect of elasticity on gas-liquid mass transfer in bubble columns.

Kawase et al. [62]

230

CMC solutions

0.48-1

0.001-2.32

Mass transfer in bubble columns and aerated stirred vessels.

Suh et al. [80]

150

Xanthan products

0.125-0.2

20-33

Production of xanthan in a~oncentric tube reactor and measured holdup and mass transfer coefficients at various stages of reaction.

Ballica and Ryu [60]

65

Plant cell suspensions

Kura et al. [8 11

350

Ryu et al. [82]

Ghosh [83]

-

-

Reports on the role of mass transfer and rheology in a bubble column with a draft tube.

PEO solutions

-

-

Effect of drag reducing additives on mass transfer in a stirred loop vessel.

115

CMC solutions

0.83-0.87

.027-0.23

Flow patterns, gas holdup and mass transfer in a bubble column with a radial sparger.

145

CMC solutions

0.92-0.96

.044-0.126

Liquid-solid mass transfer and gas holdup in bubble columns.

-

-

CMC: carboxymethyl cellulose; PEO: polyethylene oxide; PAA: Polyacrylamide; PA: polyacrylate

z

where p, and P2 are the inlet and outlet gas pressures, H is the height of the liquid through which gas has to rise, and a is the specific interfacial area; the latter is estimated using an expression due to Schumpe and Deckwer [69] as: a - 0.0465(U^p2n/K

(41)

Where U^ in m/s. In this manner, Merchuk and Ben-Zvi [68] were able to bring together most of the literature data on the liquid side mass transfer coefficient via the following simple relation: kjâ = 1.4 X 10-^y^^

(42)

Figure 7 shows a comparison between various predictions of kâ. A good agreement is seen to exist between all of these except for the study of Nakanoh and Yoshida [61]. From the foregoing description, it is abundantly clear that currently available correlations not only yield diverse values of the volumetric mass transfer (gas/liquid) coefficient but cross comparisons between different works also are seldom possible owing to the differing underlying basis of the calculation of an appropriate viscosity. Additional difficulties arise from the fact that some investigators have based their results on the volume of the clear liquid while others have used the volume of aerated mixtures. Therefore, it is suggested that the values of k^â must be calculated using as many different methods as possible to establish "upper" and "lower" bounds on the value of kâ in an envisaged application. The scant information regarding the influence of internal draft tubes on the liquid side mass transfer coefficient is of conflicting nature [67,70]. The following correlation, due to Bello et al. [71], may be used to estimate the volumetric mass transfer coefficient in air lift reactors with a downcomer and riser: kâ = 0.75 1 +

^

^0.8

(43)


565

008

006 h >5C

-

004 Ref. 42 002

Ref. 74

Ref. 69 0

Ref. 36 400 800 SHEAR RATE d/s)

1200

Figure 7. Various predictions of kâ in terms of the global shear rate y^

Kawase and Moo-Young [67] suggested that this equation overpredicted their resuhs. Even less is known about the liquid-solid transfer in bubble columns. Ghosh [83] has recently studied the liquid-solid mass transfer by monitoring the rate of dissolution of benzoic acid pellets suspended in a bubble column. He elucidated the effect of gas velocity (air), axial position of pellet and the types of sparger. Figure 8 shows the effect of gas velocity on the interphase mass transfer coefficient (k^) in a 1% aqueous CMC solution when the solute particles are positioned at various heights from the distributor. Within the narrow range of conditions, he found that the type and details of the sparger did not exert any influence on the value of the mass transfer coefficient. He presented his results in terms of Stanton number. St(= k /U ) as: St = 0.07(t)''3(ReSc2VFr)-«"

(44)

566


100

CO

E

-

i#**

8

10

o D

10^

H= 800 mm 1550mm 2050 mm

10"

10^

10^

Ugg (m/s) Figure 8. Effect of gas velocity on liquid-solid mass transfer in 1% carboxymethyl cellulose solution. where |i^ was evaluated at y^^^ = 5000 U^ . Aside from this preliminary study, virtually nothing is known about liquid-solid mass transfer in bubble columns with non-Newtonian fluids. CONCLUSIONS An overview of the currently available body of knowledge regarding the macroscopic transport effects in bubble columns and modifications thereof has been presented in this chapter. A major portion of the research effort has been directed at elucidating the role of shear-dependent viscosity on overall macroscopic processes, and little attention has been given to the analysis of these highly complex flows. A reasonable body of information is now available on average gas holdup and gas-liquid mass transfer in model polymer solutions which bear some resemblance to the actual media encountered in biotechnological applications. Little is known about the role of viscoelasticity and yield stress as displayed by some of the fermentation broths. The available design methods must be regarded only tentative s, and extrapolation beyond the range of conditions must be carried out with caution. A good summary is also available in a recent book, Bioreactor System Design [85].


567

NOTATIONS downcomer cross-sectional area A = riser cross-sectional area coefficient D = column diameter downcomer diameter DD = riser diameter axial dispersion coefficient Fr = Froude number / o = Fanning friction factor in the downcomer Fanning friction factor in the riser g = gravitational acceleration H = liquid height K = downcomer height K = consistency index in power-law model loss coefficient k = ratio of liquid-wake volume to bubble volume L = bubbling height circulation path length Mo = generalized Morton number

m = exponent in the radial gas holdup distribution N = number of circulations required for mixing n = flow index in power-law model R = column radius locus of the liquid flow reversal r = radial coordinate circulation time t = mixing time primary liquid velocity along the column terminal bubble rising velocity Uc = average liquid circulation velocity mean liquid velocity in the core liquid velocity at column axis liquid velocity in the riser superficial gas velocity superficial gas velocity in the riser superficial liquid velocity in the riser V = column volume

u= u = u. =

Greek Symbols a = coefficient AP, = pressure loss due to liquid turn round at the bottom AP = dynamic pressure loss in the downcomer AP. = hydrostatic pressure difference between the riser and the downcomer causing the circulation AP. = dynamic pressure loss in the riser Ap = density difference £ = energy dissipation rate

= (j) ^ = (|)^ = j=

gas hold-up riser gas hold-up liquid-wake hold-up shear rate apparent viscosity app P density a surface tension shear stress X friction loss coefficient at the bottom C,^ =friction loss coefficient at the top

REFERENCES 1. Shah, Y. T., Kelkar, B. G., Godbole, S. P. and Deckwer, W.-D., AIChE 7., 28, 353-379 (1982). 2. Deckwer, W.-D., Bubble Column Reactors, John Wiley and Sons, 1992. 3. Saxena, S. C. and Chen, Z. D., "Hydrodynamics and Heat Transfer of Baffled and Unbaffled Slurry Bubble Columns," Rev. Chem. Eng. 10, 193 (1994).

568


4. Zakzrewski, W., Lippert, J., Lubbert, A. and Schugerl, K., Eur. J. Appl. Microbiol. BiotechnoL, 12, 150-156 (1981). 5. Okada, K., Shibano, S. and Akagi, Y., J. Chem. Eng. Japan, 26, 637-643 (1993). 6. Chisti, M. Y. and Moo-Young, M., Chem. Eng. Commun., 60, 195-242 (1987). 7. Nishikawa, M., Kato, H. and Hashimoto, K., Ind. Eng. Chem. Process Des. Dev., 16, 133-137 (1977). 8. Nishikawa, M., BiotechnoL Bioeng., 37, 691-692 (1991). 9. Schumpe, A., Singh, C. and Deckwer, W.-D., Chem.-Ing.-Tech. SI, 988-989 (1985). 10. Zaidi, A., Bourziza, H. and Echihabi, L., Chem.-Ing.-Tech., 59, 748-749 (1987). 11. Chisti, Y. and Moo-Young, M., BiotechnoL Bioeng., 34, 1391-1392 (1989). 12. Henzler, H.-J. and Kauling, J., Proceedings of 5th European Conference on Mixing, paper 30, pp. 303-312 (1985). 13. Kawase, Y. and Kumagai, T., Biopress Eng., 7, 25-28 (1991). 14. Stein, W. A. Chem. Eng. Process, 20, 137-146 (1986). 15. Allen, D. G. and Robinson, C. W., BiotechnoL Bioeng., 38, 212-216 (1991). 16. Kawase, Y. and Moo-Young, M., Chem. Eng. ScL, 41, 1969-1977 (1986). 17. Whalley, P. B. and Davidson, J. P., Proc. Symp. Multiphase Flow Systems No. 38, J5 (1974). 18. Joshi, J. B. and Sharma, M. M., Trans. Inst. Chem. Engrs, SI, 244-251 (1979). 19. Zehner, P., Int. Chem. Eng., 26, 22-28 (1986). 20. Riquarts, H.-P., Ger. Chem. Eng., 4, 18-23 (1981). 21. Ueyama, K. and Miyauchi, T., AIChE J., 25, 258-266 (1979). 22. Walter, J. F. and Blanch, H. W., Chem. Eng. Commun., 19, 243-262 (1983). 23. Clark, N. N., Flemmer, R. L. C. and Van Egmond, J. W., Can. J. Chem. Eng., 67, 862-865 (1989). 24. Garcia-Calvo, E. and Leton, P., Chem. Eng. ScL, 49, 3,643-3,649 (1994). 25. Chisti, Y. and Moo-Young, M., / Chem. Tech. BiotechnoL, 42, 211-219 (1988). 26. Philip, J., Proctor, J. M., Niranjan, K. and Davidson, J. P., Chem. Eng. ScL, 45, 651-664 (1990). 27. Shamlou, P. A., Pollard, D. J., Ison, A. P. and Lilly, M. D., Chem. Eng. ScL, 49, 303-312 (1994). 28. Kawase, Y., BiotechnoL Bioeng., 35, 540-546 (1990). 29. Kemblowski, Z., Przywarski, J. and Diab, A., Chem Eng. ScL, 48, 4,023-4,035 (1993). 30. Popovic, M. and Robinson, C. W., BiotechnoL Bioeng., 32, 301-312 (1988). 31. Haque, M. W., Nigam, K. D. and Joshi, J. B., Chem Eng. ScL, 41, 2,321-2,331 (1986). 32. Kawase, Y. and Moo-Young, M., J. Chem. Tech. BiotechnoL, 44, 63-75 (1989). 33. Popovic, M. K. and Robinson, C. W., Chem Eng. ScL, 48, 1405-1413 (1993). 34. Kawase, Y. and Moo-Young, M., J. Chem. Tech. BiotechnoL, 46, 267-274 (1989). 35. Kawase, Y., Omori, N. and Tsujimura, M., J. Chem Tech. BiotechnoL, 61, 49-55 (1994). 36. Deckwer, W.-D., Nguyen-Tien, K., Schumpe, A. and Serpemen, Y,, Bioeng. BiotechnoL, 24, 461-481 (1982).


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37. Fields, P. R. Mitchell, F. R. G. and Slater, N. K. H., Chem. Eng. Commun., 25, 93-104 (1984). 38. Kelkar, B. G. and Shah, Y. T., AIChE 7., 3 1 , 700-702 (1985). 39. Baird, M. H. I. and Rice, R. G., Chem. Eng, 7., 9, 171-174 (1975). 40. Kawase, Y. and Moo-Young, M., Chem. Eng. J., 43, B19-B41 (1990). 41. Kawase, Y., Umeno, S. and Kumagai, T., Chem. Eng. J., 50, 1-7 (1992). 42. Godbole, S. P., Schumpe, A., Shah, Y. T. and Carr, N. L., AIChE J., 30, 213-220 (1984). 43. Capuder, E. and Koloini, T., Chem. Eng. Res. Des., 62, 255-260 (1984). 44. Vatai, G. Y. and Tekic, M. N., Chem. Eng. ScL, 44, 2,402-2,407 (1989). 45. Zuber, N. and Findlay, J. A., ASME, J. Heat Transf., 87, 453 (1965). 46. Kawase, Y., Tsujimura, M. and Yamaguchi, T., Bioprocess. Eng., in press (1995). 47. Popovic, M. K. and Robinson, C. W., AIChE J., 35, 393-405 (1989). 48. Peschke, G., PhD Thesis, University of Hannover, Hannover, Germany (1980). 49. Schugerl, K., Adv. Chem. Eng., 19, 71 (1981). 50. Moo-Young, M. and Kawase, Y., Can J. Chem. Eng., 65, 113 (1987). 51. Kawase, Y., Halard, B. and Moo-Young, M., Chem. Eng. ScL, 42, 1609 (1987). 52. Levich, V. G., Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ (1962). 53. Kolmogoroff, A. N., Dok. Akad, Nauk SSR, 66, 825 (1949). 54. Kawase, Y. and Moo-Young, M., Chem. Eng. J., 41, 1,317 (1989). 55. Godbole, S. P. and Shah, Y. T., Encyclopedia of Fluid Mech., (Cheremisinoff, N. P., ed.). Gulf Publishing Co., Houston, TX, 3, 1,216 (1986). 56. Blakebrough, N., Fatile, I. A., McManamey, W. J., and Walker, G., Chem. Eng. Res. Des., 61, 383 (1983). 57. Kawase, Y. and Kumagai, T., /. Chem. Tech. Biotechnol., 5 1 , 323 (1991). 58. Koloini, T., Capuder, E. and Zumer, M., Chem. Biochem. Eng. Quart., 3, 39 (1989). 59. Schumpe, A., Chem.-Ing.-Tech., 57, 501 (1985). 60. Ballica, R. and Ryu, D. D. Y., Biotechnol. Bioeng., 42 1,181 (1993). 61. Nakanoh, M. and Yoshida, F., Ind. Eng. Chem., Proc. Des. Dev., 19, 190 (1980). 62. Kawase, Y., Halard, B. and Moo-Young, M., Biotechnol. Bioeng., 39, 1,133 (1992). 63. Moo-Young, M., Halard, B., Allen, G. D., Burrell, R. and Kawase, Y., Biotechnol. Bioeng., 30, 746 (1987). 64. Henzler, H.-J., Chem.-Ing.-Tech., 52, 643 (1980). 65. Buchholtz, H., Buchholtz, R., Niebelschutz, H. and Schugerl, K., Europ. J. Appl. Microbio. Biotechnol., 6, 115 (1978). 66. Buchholtz, H., Buchholtz, R., Lucke, J. and Schugerl, K., Chem. Eng. ScL, 33, 1,061 (1978). 67. Kawase, Y. and Moo-Young, M., Chem. Eng. Commun., 40, 67 (1986). 68. Merchuk, J. C. and Ben-Zvi, S., Chem. Eng. ScL, 47, 3,517 (1992). 69. Schumpe, A. and Deckwer, W. D., Ind. Eng. Chem., Proc. Des. Dev., 2 1 , 706 (1982). 70. Kawase, Y. and Moo-Young, M., Chem. Eng. Res. Des., 66, 284 (1988).

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71. Bello, R. A., Robinson, C. W. and Moo-Young, M., Chem. Eng. ScL, 40, 53 (1985). 72. Kawase, Y. and Moo-Young, M., Chem, Eng. Res. Des., 65, 121 (1987). 73. Baykara, Z. S., and Ulbrecht, J., Biotechnol Bioeng., 20, 287 (1978). 74. Hecht, V., Voigt, J. and Schugerl, K., Chem. Eng. ScL, 35, 1,325 (1980). 75. Voigt, J., Hecht, V. and Schugerl, Chem. Eng. ScL, 35, 1,317 (1980). 76. Schumpe, A. and Deckwer, W. D., Bioprocess Eng., 2, 79 (1987). 77. Sada, E., Kumazawa, H. and Lee, C. H., Chem. Eng. ScL, 38, 2,047 (1983). 78. El-Temtamy, S. A., Khalil, S. A., Nour-El-Din, A. A. and Gaber, A., Appl. Microbiol. Biotechnol, 19, 376 (1984). 79. Suh, I.-S., Schumpe, A. and Deckwer, W. D., Biotechnol. Bioeng., 39, 85 (1992). 80. Suh, I.-S, Schumpe, A., Deckwer, W. D. and Kulicke, W. M., Can. J. Chem. Eng., 69, 506 (1991). 81. Kura, S., Nishiumi, H. and Kawase, Y., Bioprocess Eng., 8, 223 (1993). 82. Ryu, H. W., Chang, Y. K. and Kim, S. D., Bioprocess Eng., 8, 271 (1993). 83. Ghosh, U. K., Ph.D. thesis, Dept. of Chemical Eng., Banaras Hindu University, Varanasi, India (1992). 84. Kawase, Y. and Moo-Young, M., J. Chem. Tech. Biotechnol., 36, 527 (1986). 85. Asenjo, J. A. and Merchuk, J. C. (eds.), Bioreactor System Design, MarcelDekker, New York (1994).

CHAPTER 21 STUDIES IN SUPPORTED TITANIUM CATALYST SYSTEM USING MAGNESIUM DICHLORIDE-ALCOHOL ADDUCT V. K. Gupta, Shashikant and M. Ravindranathan Research Centre Indian Petrochemicals Corporation Ltd. Vadodara - 391 346, India CONTENTS ABSTRACT, 571 INTRODUCTION, 571 EXPERIMENTAL, 572 Materials, 572 Synthesis of MgCl2-Alcohol Adduct, 573 Synthesis of [Mg-Ti] Catalyst, 574 Propene Polymerization, 574 Characterization, 574 RESULTS AND DISCUSSION, 574 ACKNOWLEDGMENTS, 580 ABSTRACT Magnesium dichloride reacts with aliphatic alcohols, [ROH; R=C2H^, n-CgH^, i-C3H^, n-C^H^, i-C^H^, t-C^H^, n-C5H,j, n-C^Hj3, C^H^^ (C2H5)] to form well-defined soHd adducts. The compositional analysis of adducts indicate that the stoichiometric ratio of magnesium dichloride to alcohol depends on length of alkyl group and nature of isomeric alcohol. Magnesium dichloride -2- ethyl-1-hexanol adduct was treated with diphenyldichlorosilane in the presence of dibutylphthalate to obtain active magnesium dichloride support. The titanation process of active magnesium dichloride gives supported magnesium-titanium catalyst (Mg-Ti). The catalyst was characterized by compositional analysis and specific surface area measurements. Performance of the catalyst for polymerization of propene is evaluated with triethylaluminum (TEAL) and phenyltriethoxysilane (PES) as cocatalyst. The yield and isotacticity of the polymer is governed by polymerization parameters such as Si/Al ratio and polymerization time. INTRODUCTION Present generation high activity Ziegler-Natta catalyst is comprised of titanium tetrachloride supported on magnesium dichloride [1-4]. Performance of the catalyst 571

572

Advances in Engineering Fluid Meciianics

system in terms of activity, stereospecificity, and polypropylene characteristics determines its use in commercial production processes. These properties are highly dependent on preparative methodology and chemical composition of the catalyst. During early stages of development, Mg-Ti catalysts were prepared by ball milling of crystalline magnesium dichloride with Lewis base followed by treatment of the obtained product with titanium tetrachloride. Ball milling process causes stacking defects in magnesium dichloride crystal due to occurrence of rotational disorder in Cl-Mg-Cl triple layers. It gives effective incorporation of titanium tetrachloride on active support. However, morphology of the catalyst is not controlled, resulting in non-uniform shape and particle size distribution [5]. This problem is overcome by employing a chemical activation approach for the synthesis of catalyst. It involves reaction of crystalline magnesium dichloride with electron-donor compounds followed by controlled regeneration of active support [6-8]. Various types of chemical reagents have been used for generation of highly disordered magnesium dichloride. The complexation behavior of non-protonic donors [9-10], such as tetrahydrofuran, ethylformate, ethylacetate and ethyl benzoate with magnesium dichloride, has been studied. The results indicate that elimination of coordinated non-protonic moieties from polymeric magnesium dichloride adducts gives highly distorted magnesium dichloride with a higher number of uncoordinated magnesium sites. This product is an effective support for synthesis of Mg-Ti catalysts. Protonic electron-donors, such as alcohols [6-8], also have been used for the generation of active magnesium dichloride. However, the information available in literature regarding their complexation behavior with magnesium dichloride is rather scarce [11]. The present paper reports our results on the complexation behavior of various aliphatic alcohols with magnesium dichloride. The magnesium dichloride-2-ethyl1-hexanol adduct has been used for the synthesis of Mg-Ti catalyst. The performance of the catalyst has been examined for propene polymerization using triethylaluminum and/or phenyltriethoxysilane as cocatalyst system. EXPERIMENTAL Materials Anhydrous magnesium dichloride (Toho Titanium Company, Japan); triethylaluminum (Ethyl Corporation, USA); titanium tetrachloride (Riedel-de, Haen, Germany); phenyltriethoxysilane (Aldrich, USA); dibutylphthalate ((E. Merck, Germany); and diphenyldichlorosilane (Aldrich, USA) were used as received. Ethanol, n-propanol, isopropanol, n-butanol, iso-butanol, tert.-butanol, n-pentanol, n-hexanol, and 2-ethyl-l-hexanol were the commercial products used after distillation and storing over activated molecular sieves. Polymerization grade hexane (IPCL, Baroda) and decane (Commercial Product) were used after drying over sodium wire. Propene (polymerization grade, IPCL, Baroda) was used after passing through a molecular sieve column. All experiments were carried out under high purity nitrogen atmosphere. Standard schlenk techniques and a Vacuum Atmospheres Model HE-43 Dri-Lab equipped with a model HE-491 Dri Train were used for handling all compounds.

studies in Supported Titanium Catalyst System

573

Synthesis of MgCl2-Alcohol Adduct The appropriate alcohol was mixed in stoichiometric amount with magnesium dichloride in hexane or decane. Reaction mixture was stirred for three hours. The solid product was filtered and washed with hexane and dried in vacuum for 30 minutes [Table 1].

Tablet Characterizaion of MgCls-ROH adducts

Compositional Analysis (wt%) Sr. Product No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 24.

MgCl2.6C2H30H MgCl2.C3HÔH MgCl2.2C3HpH MgCl2.3C3HpH MgCl2.4C3HpH MgGl2.5C3HpH MgCl2.i-C3HpH MgCl2.2 i-C3H,0H MgCl2.3 i-C3H,OH MgCl2.4 i-CÛpU MgCl^.C^H^H MgCl2.2qHÔH MgCl^^C^HpH MgCl2.4C,HpH MgCl2.5C4HÔH MgCl^.i-C^HÔH MgCl2.2 i-CÛpn MgCl2.3 i-C^HflU MgCl2.4 i - q H ^ H MgCl^.t-C^HÔH MgCl2.2t-C^H90H MgCl^.C^HjjOH MgCl2.2C3H„OH MgCl2.C,H^30H MgCl2.2 C,Hj30H MgCl2.C^Hj2(C2H5)OH MgC\^.2C^H^^(C^}l^)0H

/ = iso, t = tertiary

Yield (%)

Mg

CI

85 94 97 91 91 83 88 95 81 83 97 90 92 77 84 92 90 88 85 82 78 93 94 90 89 94 96

7.9 14.4 10.3 8.3 7.3 6.4 15.2 11.2 8.3 7.6 13.6 8.4 7.3 6.1 5.3 14.6 10.1 7.8 6.5 13.1 10.1 12.1 7.6 13.0 7.9 9.5 6.2

22.8 45.4 34.9 26.5 23.2 20.1 44.3 32.9 24.0 21.2 39.1 27.1 22.2 17.8 16.9 42.8 31.3 24.6 19.2 41.6 31.6 38.8 26.4 36.3 22.9 30.0 19.0

574


Synthesis of (Mg-Ti) Catalyst The magnesium dichloride was reacted with 2-ethyl-l-hexanol (1:2 molar ratio) in decane at 120°C for two hours. A stoichiometric amount of dibutylphthalate (DBPh/MgCl2 = 0.15) was added in the reaction mixture and stirred for one hour at 120°C. The obtained solution was added into decane solution of diphenyldichlorosilane (Si/Mg molar ratio = 30) in controlled manner at room temperature. The reaction mixture was heated to 120°C and allowed to react for two hours. The solid product was separated by filtration and washed with hexane. It was treated with titanium tetrachloride (Ti/Mg molar ratio = 40) at 120°C for two hours. The liquid was decanted from reaction mixture at 120°C. The solid product obtained was washed with decane followed by hexane. Catalyst (Mg-Ti) was dried in vacuum for two hours. Propene Polymerization Polymerization of propene was carried out in a 500 ml glass reactor equipped with a stirrer and an oil bath. The calculated amounts of triethylaluminum (TEAL), phenyltriethoxysilane (PES), and solid catalyst were added into the reactor containing 200 ml hexane. Propene was supplied under a total pressure of one atmosphere for a fixed period of time. The polymerization was terminated by the addition of acidified methanol solution. The polymer was washed with methanol and dried in vacuum for four hours. Characterization Titanium content was estimated quantitatively by U.V. spectrophotometric method [12]. Magnesium and chlorine contents were analyzed titrimetrically [12]. BET surface area of solid samples was measured on a Carlo Erba Sorptomatic instrument. Polypropylene samples were extracted with boiling heptane in a Soxhlet apparatus for determination of isotactic index. Isotactic index reported for each sample is the weight percentage of heptane insoluble polypropylene. RESULTS AND DISCUSSION Anhydrous magnesium dichloride (MgCl2) reacts with aliphatic alcohols, such as ethanol, propanol, butanol, pentanol, hexanol, and 2-ethyl-l-hexanol, to give solid adducts of different stoichiometry. Compositional analysis of adducts are given in Table 1. The results indicate that the reaction of magnesium dichloride with ethanol gives hexakis adduct. The increase in chain length of normal alkyl group from C^ to C^ yields adducts upto pentakis composition. The C^ to C^ alcohols give bis adduct. Complexation behavior of different isomers of propanol and butanol with magnesium dichloride showed that the variation from n- to iso-alcohol results in stoichiometry change of the adduct from pentakis to tetrakis derivatives. A more sterically hindered tertiary alcohol gives bis adduct. MgCL + X ROH -^ MgCL.X ROH

(1)

studies in Supported Titanium Catalyst System

575

R=C2H3, X = l - 6 ; R=n-C3H^, n-C^H,, X = l - 5 ; R=i-C3H^, i-C,H^, X = l - 4 ; R=n-C3H,,, n-C,H.3, t-C,H,, C,H„ (C.H^), X = l - 2 . These results demonstrate that the molar ratios of reactants and stereo-electronic characteristics of alcohols control the stoichiometry of magnesium dichloride-alcohol adducts (Scheme 1). a-magnesium dichloride has layered lattice structure with cubic close packing of chlorine atoms [1,13]. It contains four [I], five [II], and six coordinated [III] magnesium species as shown in (Figure 1). The bulk of magnesium ions exist in six-fold coordination with a closed packed stacking of double chlorine layer. Hexacoordinated magnesium dichlorides have two chlorine atoms joined to magnesium through van der Waals interaction in the layered structure. The bonding of two chlorine atoms of adjacent magnesium ion and two chlorine atoms of the same unit results in the formation of a polymeric structure of magnesium dichloride. The complexation upto bis (magnesium dichloride-alcohol) adduct [(MgCl2.2ROH);

MgCl2 + ROH

MgCl2-R0H

+ ROH

>-MgCl2-2ROH +ROH

+ ROH "»-ROH MgCl2*5ROH -« MgCl2-^R0H -^

MgCl2 -BROH

+ ROH MgCl2-6ROH Scheme 1. Stepwise reaction of magnesium dichloride with alcohol.

CI

CI

CI

CI !

CI CI

CI

CI :Mg'

Cl

^Cl

CI

X[

cr

! CI

[I]

[II]

[III]

Figure 1. Nature of magnesium dichloride species.

"CI

576


R=C,H3, n-C3H,, n-qH,, i-C3H,, i-C,H,, n-C3H,,, n-C,H,3, t-C,H,, and C^H^^(CÛ^)] proceeds by the replacement of chlorine atoms, which are bonded to magnesium by van der Waals forces. Similar conclusions have been drawn [9,10,14] for bis adducts, such as MgCl^CTHF)^, MgCl2(CH3COOC2H5)2, MgCl2(HCOOC2H3)2, and MgCl2(C2H30H)2 on the basis of x-ray diffraction pattern. Further change of magnesium dichloride-alcohol adduct upto tetrakis derivative takes place through breakage of coordinated bond between magnesium and chlorine. Thus, tetrakis derivatives [(MgCl2.4ROH); R=C2H5, n-C3H^, n-C^H^, i-C3H7, i-C^HJ can tentatively be assigned an octahedral geometry containing two covalent chlorine atoms [15]. The pentakis [(MgCl^.SROH); R=C2H5, n-C3H^, n-C,HJ and hexakis [(MgCl2.6ROH); R=C2H5] adducts are formed by the replacement of one and two colvalent chlorine atoms from the coordination sphere of magnesium dichloride, respectively. Such magnesium dichloride-alcohol adducts, therefore, show prevalently ionic characteristics as compared to other adducts exhibiting neutral nature. The prevalent ionic characteristic also is reported [8,14] for magnesium dichloride Lewis base adducts, such as MgCl2.6C2H30H, [MgCKTHF)^] [AICIJ.THF, [Mg(CH3COOC2H3)J [AlCy, [Mg(THF)J TiCl3(THF)2, and [Mg2(î-C1)3(THF), [TiC^THF)]. The crystal structure of MgCl2.6C2H30H shows that magnesium atom is coordinated to six oxygen atoms of ethanol in an octahedral configuration and the hydrogen atoms of the OH groups are involved in the bridges with chlorine [16]. Magnesium dichloride-Lewis base adducts have been used for the synthesis of magnesium-titanium catalysts [6,14]. The removal of Lewis base, such as ethyl formate and ethyl alcohol, from their magnesium dichloride adducts has been studied systematically by FT-IR and x-ray diffraction methods. The results have shown that a progressive removal of Lewis base from the adduct by heat treatment produces structural randomization of magnesium dichloride chains, resulting in highly disordered structure. Such supports have been used for the incorporation of titanium tetrachloride, giving a high-performance polymerization catalyst system. We have adopted a chemical reaction methodology from the removal of Lewis base from its adduct to generate active support. We have used diphenyldichlorosilane as a reacting species for generation of support from magnesium dichloride-2-ethyl-1 -hexanol adduct. The choice of chlorosilane over titanium tetrachloride [6,17] is based on the fact that diphenyldichlorosilane reacts with alcohol to generate alkoxy silane [18], which can act as an internal Lewis base [19,20]. Present synthesis of catalyst involves treatment of magnesium dichloride-2-ethyl-l-hexanol adduct with dibutylphthalate (DBPh) followed by reaction with diphenyldichlorosilane. Resultant solid product was treated with titanium tetrachloride to obtain Mg-Ti catalyst as shown in Scheme 2. Solid catalyst [Mg-Ti] was analyzed for its composition and specific surface area (Table 2). A 3.0 wt. % of titanium is incorporated on active support. BET surface area of crystalline magnesium dichloride is found to be 10 mVg. The treatment of magnesium dichloride with 2-ethyl-l-hexanol and DBPh followed by diphenyldichlorosilane gives a product [Mg-Ti I] with improved surface area characteristics (50 mVg). The reaction of [Mg-Ti I] with titanium tetrachloride increases the surface area to 115 mVg. These results show that the present process of catalyst synthesis gives approximately tenfold improvement in the surface area of the Mg-Ti catalyst as compared to the starting anhydrous magnesium dichloride.

studies in Supported Titanium Catalyst System MgCl2 + 2 R 0 H — ^ MgC(2-( RQH )2 "^ ^*^^^^>

577

MgCl2-(ROH)2 • XDBPh.

MgCl2 •(R0H)2-XDBPh + P h 2 S i C l 2 r } : ^ MgCl2 •Ph2Si( ORl^ Cl2-n* DBPh

+ TiCl4 \ [Mg.Ti] Scheme 2. Preparation of [Mg-Ti] catalyst.

Table 2 Characteristics of MgCig, [Mg-Ti 1] and [l\/lg-Ti] Catalyst

Ti

Specific Surface Area (m^/g)

3.0

10 50 115

Compositional Analysis (Wt%) Product

Mg

CI

MgCl, [Mg-Ti I] [Mg-Ti]

24.7 14.1 18.4

74.0 43.9 63.8

[Mg-Ti] catalyst is evaluated for homopolymerization of propene using triethylaluminum (TEAL) as cocatalyst and phenyltriethoxysilane (PES) as an external donor. The kinetic profile data (Figure 2) indicate that polymerization rate is not significantly changed between 60 to 180 min unlike the observation reported [21] for Mg-Ti catalysts prepared without treatment of diphenyldichlorosilane. It indicates that the in-situ generation of diphenylalkoxysilane during catalyst synthesis may be a contributing factor to better stability of active titanium species. The time-dependent study further shows that stereospecificity of catalyst system decreases with increase in polymerization time (Table 3). A decrease in isotactic index is observed form 87 to 78% over a period of three hours. It is due to a decrease in the rate of polymerization for isotactic polypropylene and simultaneous formation of more atactic polypropylene (Figure 2). In addition to polymerization time, the concentration of PES also influences the stereospecificity of the catalyst system (Table 4, Figure 3). The isotacticity of polypropylene obtained in absence of PES is found to be 72%. The addition of PES improves the isotactic index from 72 to 87%. Increase of Si/Al ratio up to 0.2 depresses atactic polypropylene formation by 70% and isotactic polypropylene by

578


c o -ô N

si E ^ o o^ o

30

20

o^

a cc

10 [C] 0

60

120 Time

180

Figure 2. Rate of polymerization vs. time. [A] = Overall rate of poymerization for polypropylene [B] = Rate of polymerization for isotactic polypropylene [C] = Rate of polymerization for atactic polypropylene Polymerization conditions as mentioned in Table 3. Table 3 Performance of [Mg-Ti]/Et3AI/PhSiCOEt)3 Catalyst System for Propene Polymerization as a Function of Time

Sr. No.

Time (Min.)

Polymer Yield (Kg PP/g Ti)

1. 2. 3. 4.

60 120 150 180

1.5 2.8 3.5 4.2

Isotactic Index

(%) 87 80 78 78

Polymerization Conditions : PC^ = 1 atm, Temp - 40 ± 1 °C, Hexane = 200 ml, Catalyst = 31 ± 1 mg. TEAL = 5 mmol. Si/Al molar ratio = 0.5.

Studies in Supported Titanium Catalyst System

579

Table 4 Influence of Si/AI Molar Ratio in Cocatalyst on the Performance of [Mg-Ti]/Et3AI/PhSi(OEt)3 Catalyst System for Propene Polymerization

Sr. No.

Si/AI [Molar Ratio]

Polymer Yield (Kg PP/g Ti)

Isotactic Index (%)

1. 2. 3. 4. 5.

0.0 0.03 0.05 0.10 0.20

3.9 3.0 2.8 2.7 2.5

72 78 80 84 87

Polymerization conditions as mentioned in Table 3 except Si/AI ratio. Time = 2 h.

.-1 [A] kg iPP/g Ti min -1 [B] kgAPP/gTi min 0.03

C

o D

!^ E E _ 0.02

-o-[A]

? 0.01 O

-D-[B]

cc 0.1

0.2 Si/AI

03

Figure 3. Rate of polymerization as a function of Si/AI ratio. [A] = Rate of polymerization for isotactic polypropylene [B] = Rate of polymerization for atactic polypropylene Polymerization condition as mentioned in Table 3.

580


about 22%. It results in higher stereospecificity of the catalyst system. This observation can be correlated with a phenomenon involving deactivation of nonstereospecific titanium sites and stabilization and/or enhancement of stereospecific titanium species with the addition of external Lewis base [22-24]. ACKNOWLEDGMENTS Authors wish to thank IPCL management for permitting the publication of these results. Experimental assistance provided by A. N. Baria is also acknowledged. REFERENCES 1. P. C. Barbe, G. Cecchin and L. Noristi, Adv. Polym. ScL, 81, 1, (1987). 2. P. J. T. Tait, in "Ziegler-Natta and transition metal catalysts" (G. C. Eastamond, A. Ledwith, S. Russo, P. Sigwalt, eds.), Comprehensive Polymer Science, Pergamon Process, Vol. 4, Oxford, 1989, P. 1. 3. E. Albizzati, M. Galimberti, U. Giannini and G. Morini, Makromol.Chem., Macromol. Symp., 48/49, 223 (1991). 4. G. G. Arzoumanidis, N. M. Karayannis, H. M. Khelghatian and S. S. Lee, Cat. Today, 13, 59 (1992). 5. Slurry Phase Polypropylene, in SRI International Process Economics Program, Menlo Park, CA, 1988, p. 25. 6. Y. Hu and J. C. W. Chien, J. Polym. Sci: Pt A: Polymer Chemistry, 26, 2,003 (1988). 7. H. M. Park and W. Y. Lee, Eur. Polym. J., 28, 1,417 (1992). 8. K. S. Kang, M. A. Ok and S. K. Ihm, J. Appl. Polym. Sci., 40, 1,303 (1990). 9. P. Sobota, Polym.-Plast. Technol. Eng., 28, 493 (1989). 10. V. DiNoto, G. Cecchin, R. Zannetti and M. Viviani, Macromol. Chem. Phys., 195, 3,395 (1994). 11. V. K. Gupta, S. Talapatra, Shashikant and S. Satish, Polymer Science, Recent Advances, Proceedings of Polymers '94, 1, 320 (1994). Chem. Abstr. 122, 32118 y (1995). 12. A. I. Vogel, Text Book of Quantitative Inorganic Analysis, Longman's London, 1979, P. 320, 342. 13. B. L. Goodall, J. Chem. Educ, 63, 191 (1986). 14. V. DiNoto, R. Zannetti, M. Viviani, C. Marega, A. Marigo and A. Bresadola, Makromol. Chem., 193, 1,653 (1992). 15. F. J. Karol, Catal. Rev.-Sci. Eng., 26, 557 (1984). 16. G. Valle, G. Baruzzi, G. Paganetto, G. Depaoli, R. Zannetti and A. Marigo, Inorg. Chim. Acta, 156, 157 (1989). 17. D. N. T. Meghalhaes, O. D. C. Filho and F. M. B. Coutinho, Eur. Polym. J., 27, 827 (1991). 18. D. C. Bradley, R. C. Mehrotra and D. P. Gaur, Metal Alkoxides, Academic Press, New Yok, 1978. 19. M. C. Sacchi, F. Forlini, I. Tritto, R. Mendichi, G. Zannoni and L. Noristi, Macromolecules, 25, 5,914 (1992).

Studies in Supported Titanium Catalyst System

581

20. J. S. Chung, J. H. Choi, I. K. Song and W. Y. Lee, Macromolecules, 28, 1,717 (1995). 21. J. C. W. Chein and Y. Hu, /. Polym. ScL : Pt A : Polym. Chemistry, 26, 2,973 (1988). 22. L. Noristi, P. C. Barbe and G. Baruzzi, Makromol. Chem., 192, 115 (1991). 23. L. Noristi, P. C. Barbe and G. Baruzzi, Makromol Chem., 193, 229 (1992). 24. A. Proto, L. Oliva, C. Pellecchia, A. J. Sivak and L. A. Cullo, Macromolecules, 23, 2,904 (1990).


CHAPTER 22 PLASTICIZING POLYESTERS OF DIMER ACIDS AND 1, 4-BUTANEDIOL

U. D. N. Bajpai and Nivedita Polymer Research Laboratory Department of Post Graduate Studies and Research in Chemistry R. D. University, Jabalpur 482001, M. P., India CONTENTS INTRODUCTION, 583 DIMER ACIDS, 584 DIMER ACID-BASED POLYMERS, 585 DIMER ACID-BASED POLYESTERS, 586 POLYESTERS OF DIMER ACID AND 1, 4BUTANEDIOL, 588 PREPARATION, 589 APPLICATIONS, 593 REFERENCES, 593 INTRODUCTION Plasticization is one of the important methods of making polymers amenable to processing. The principal field of plasticizers application is comprised of hot processed compound, surface coatings, adhesives, and a few special applications, such as plasticized smokeless powder, photographic film, and rubber. Chemically, plasticizers are organic substances of low volatility and relatively low molecular weight esters which endow the polymer with elasticity, flexibility, workability, or shock resistance. The conventional non-resinous plasticizers are termed as "simple or monomeric plasticizers" and resinous ones as "polymeric" [1-6]. Usually, simple or monomeric plasticizers are mono or diesters of known formula, molecular weight, and physical and chemical properties. They are characterized by a good efficiency at low temperatures, but are very mobile and with a good solvation capacity. These plasticizers present the disadvantage of being too volatile and being soluble in oils, fats, organic solvents, etc. Polymeric plasticizers, on the other hand, are long chain polymers formed by carrying out the polymerization of monomer(s) as in vinyl and epoxy polymerization or condensation polymerization. They are mostly polyesters or polyethers of low 583

584


molecular weight ranging from 2,000 to 20,000. Polyester plasticizers normally are prepared by the condensation of one or more dibasic acids with diols. Polymeric plasticizers already have established their superiority over monomeric or low molecular weight plasticizers because these display less volatility and more resistance to solvent and oil extraction, good permanence, and slight migration to other materials as compared to conventional monomeric plasticizers. These are materials incorporated in a plastic or a resin in order to impart flexibility, reduce viscosity, and improve light stability, corrosion resistance, and water permeability of the resin. For better compatibility, mixtures of monomeric and polymeric plasticizers are used. The worldwide development of plastics, thereby of plasticizers, imposed the synthesis of new plasticizing polyesters of dimer acid, which are supposed to have resistance to corrosion, moisture, and chemicals besides having plasticizing properties. DIMER ACIDS The dibasic acids, dimer acids, are produced commercially from vegetable oil fatty acids or esters, mainly C18 unsaturated fatty acids or esters, such as linoleic acid, ricinoleic acid, oleic acid. These fatty acids or esters derived from vegetable oils, such as dehydrated castor oil, tall oil, tung oil etc., are polymerized to give a mixture of dibasic and polybasic acids. This "polymerized monomer" chiefly includes dibasic dimeric fatty acids and small fractions of the monomeric, trimeric and higher polymeric fatty acids and, therefore, these are designated by the term "dimer acids". These dimer acids find an outlet as important intermediates for the manufacture of plasticizers, synthetic lubricants, and high polymeric products because of their increased functionality compared with ordinary fatty acids. The dimer acids, DA, frequently represented as HOOC-D-COOH, are cyclic dibasic dimer having a total of 36 carbon atoms. The -D- represents a C34 hydrocarbon radical with one substituted cyclohexene structure [7,8]. Monocyclic dimer acids are obtained from dienoic acids while bicyclic structures have been found in fatty polymers of trienolic acids. Variation in structure of dimer acids from acyclic to polycyclic may be due to the involvement of different precursors for dimer preparation [9]. Dimer acids, or dibasic fatty acids, are formed either by thermal polymerization or carbon-to-carbon linking of fatty acid chains. Thermal polymerization of monomeric fatty acids like dehydrated castor oil or linoleic acids carried out at 205°C300°C in an inert atmosphere yields dimer acid. Whereas the latter is established through a Diels-Alder mechanism, involving addition of one molecule of nonconjugated linoleic acid and one molecule of thermally conjugated linoleic acid or dehydrated castor oil [10]. A review of dimerization provides a comprehensive idea about the manufacture of dimer acids [11]. The general characteristics, structure, and properties of dimer acids also were reviewed [12,13]. Dimer acids are yellow-colored viscous liquids of relatively high molecular weight (-560). Numerous isomers with unsymmetrical cyclic structure cause liquidity and lack of crystallinity because the structure contributes to flexibility in the polymers derived from it.

Plasticizing Polyesters of Diner Acid and 1,4-Butanediol

585

The importance of these dimer fatty acids as an intermediate for high molecular weight linear polymers was first explored at Northern Research Laboratory of US DA in the 1940s [17,18]. These synthetic products of dimer acid and ethylene glycol were developed under the name of Norepols [19]. Later, a substantial amount of work was reported in patents, technical reports, and research papers. A literature survey on dimer acid-based research from 1971-1994 reveals that most current work involves the reactions resulting in the polyamides, polyesters, polyurethanes, and various copolymers. A closer look causes one to conclude that such as flexibility and adhesion-improving properties of dimer acid for a wide range of industrial and commercial uses. DIMER ACID-BASED POLYMERS Dimer acid-based polyamide resins commonly are known as fatty polyamides. "Norelacs" were the first polyamide resins synthesized from dimer acid and diamines, such as ethylene diamine [17]. In contrast to nylon polyamides, dimer acid-based polyamides exhibit lack of crystallinity, relatively low softening points, and adhesiveness. A high variety of compositions of dimer acid polyamides have been reported. Initially, low molecular weight (range 3,000-15,000) fatty polyamides were synthesized [18]. Later, high molecular weight fatty polyamides [20] found wide range of application due to their tensile strength, elongation, toughness, and solubility [21]. Basically, fatty polyamides can be subdivided into nonreactive solid polyamides and reactive liquid polyamides. Several formulations of nonreactive polyamides with a broad spectrum of properties and applications can be obtained, including hot melt adhesives, films, flexographic printing, relief printing [22], thixotropic coatings, etc. A major application of dimer acid-based polyamides is in hot melt and heat seal formulations for plastic, paper, leather (shoe industry) and metal bonding (side seam welding). The flexibility and corrosion resistance properties imparted by dimer acid components are essentially required for speciality high performance polymeric coating applications. Several types of useful melt adhesive show good resistance for dry cleaning solvents [23] used to adhere a polyester-wool blend fabric. A recent study [24] discusses the heat resistance of Kevlar 49 reinforced by treatment of fatty polyamide giving rise to materials useful for circuit boards, air craft, or automobile parts, etc. Because of their resistance to heat and light, fatty polyamides are created with improved color-forming properties. These are used as dye-accepting coatings for thermal-transfer printing receptor sheets [25]. Low gas permeability of fatty polyamide compositions are especially well-suited for internal side coatings of hoses for air conditioners to give good coolant effect [26]. Reactive liquid polyamides are highly branched condensation products having reactive secondary amino groups. These reactive polyamides have good compatibility with epoxy resins and are blended with epoxy or phenolic resins yielding adhesives [27,28], that are useful as thermosetting potting, in casting and laminating, in structural work and sealing compounds. Epoxy-group terminated polyamides for improved bonding strength and heat resistance are useful as adhesives [29]. The

586


dimer acid-based polyamide crosslinking agents optionally are used with epoxy resin-paint compositions for improving the film-forming properties [30]. Variations in polyamide-modified epoxy resin vary the application from aqueous electrodeposition coatings [31] to anticorrosive coatings useful for can interiors [32,33], thermally stable primers [34] to sealants, and binders for metal [35]. All these compositions are noted especially for film toughness; corrosion and chemical resistance; flexibility; adhesion to various surfaces, such as metals, wood, and plastics; and durability. Reactive polyamides with phenolic hydroxyl groups are useful in preparing poly amide-polycarbonate block copolymers [36]. Because of their resistance to corrosion and other fine physical properties, various poly amide-polyester copolymer compositions are useful as hot melt adhesives [37], adhesives, and coatings [38]. Dimer acid-based polyester-polyamide copolymers find use in coating compositions to prevent sagging [39]. Thermoplastic blend can be prepared from polyamide-polyester blend, which shows good mechanical properties [40,41]. Flexible thermosetting resins are used along with rigid thermosetting resin to prepare printed circuit boards with bendable parts [42]. Water-dispersable polyamidepolyester based on dimer acid makes suitable flexographic ink formulation [43]. A few publications have appeared revealing the preparation and utility of dimer acid-based polyisocyanates, polyurethanes, and polyurethane prepolymers. Different compositions of polyester-polyurethanes of dimer acid find different coating applications due to fine variation in their properties. Anticorrosion and solvent resistance are the main features of these polyurethane compositions [44]. Two recent developments show the utility of dimer acid polyurethanes for electrophoretic coating materials due to their gravel impact and corrosion resistance [45,46]. Polyurethane formulations, having hydroxy terminated ethylenically, or acrylic unsaturated monomer show good adhesion properties and are useful as radiationcurable coating materials for metals [47,48], optical fibers [49], etc. Anionic polyurethane resins based on dimer acid, along with anionic acrylic pigment grind resins, gave aqueous-based coating compositions, which are supposed to have good color and storage stability, and fast-drying ability [50,51]. Compositions prepared by treating polyester polyol of dimerethylene glycol with tolylene diisocyanate features good waterproof properties and use as cellular sealing materials [52]. In other studies of polyester-polyurethanes, it has been shown that the dry spun fibers of these compositions show hydrolysis resistance and good strength [53]. Urethane rubbers have been formulated from hydroxyl-terminated dimer acid polyester. These saturated urethane elastomers show good resistance to hydrolysis [54]. DIMER ACID BASED POLYESTERS Dimer acid-based polyesters are mainly the condensation products of di- or polyfunctional hydroxy compounds and dimer acid. The dimer acid-based polyester originated back in the 1940s [55-57] and gained enough commercial importance because of their fine properties. Polyesters with a wide spectrum of properties can be obtained. These polyesters derive their distinctive physical properties—like flexibility and resistance to heat, corrosion, and chemicals—from dimer acids used in their production. In addition to simple polyesters, many copolyesters based on


587

dimer acids are possible. The properties of polyesters are modified by adding either aliphatic and/or aromatic. One or more poly hydroxy compounds are used to modify the characteristics of polyesters. Due to adhesion and flexural properties, the dimer acid-based polyesters are used as lubricants, adhesives and formed the basis of flexographic inks and heat seal coatings. They are used as vibration dampers when sandwiched between metal sheets [58]. The importance of dimer acid-based polyesters as the precursor of elastomers and thermoplasts was first explored by Cowan et al. [59] who prepared vulcanizate polyesters comparable to synthetic rubber on a trial scale during the World War II rubber shortage. The first step in the manufacture of these materials involved preparation of polyesters of dimer acid and ethylene glycol. This was vulcanized and compounded to yield the finished rubberlike material. Vulcanizates of superpolyesters [60] gave high tensile strength and elongations suitable for molding simple articles and rubberizing fabrics. The molecular weight-viscosity relationship of these superpolyesters was used to prove the linear nature and high molecular weight of these polyesters. In 1975, Hoeschele prepared elastomeric-segmented polyesters of dimer acid, butanediol, and terephthalic acid by carrying out the reaction in the presence of catalyst suitable for injection moldings [61]. The polyester elastomers useful as hot melt adhesives were formulated with slight variation in composition [62]. Several papers and patents on polyesterification of dimer acid with various glycols under different experimental conditions appeared in the literature. The dimer acidbased polyesters with a slight variation in composition enjoy a variety of applications. Observations show that a particular property of the product can be enhanced or suppressed by the choice of a proper additive. Various polyester compositions based on dimer acid, terephthalic acid, and 1,4-butanediol were prepared to yield heat- and impact-resistant thermoplast materials with good toughness [63,64]. These compositions were supposed to be useful for electric, electronic, automobile, and industrial parts. Segmented thermoplast elastomer block copolymers were analyzed with DSC and dynamic mechanical analysis [65], and the effect of molar mass of copolymer units on Tg was described in detail. A preparation is described for dimer acid-modified poly (tetramethylene terephthalate) polyesters useful for medical containers. This copolymer is blow-moldable and gamma radiation sterilizable [66]. The methods for obtaining molding composition for injection molding is well-documented in literature [67,68]. In another study, blow-moldable polyester-polyether-polyimide compositions were obtained which contain aliphatic polycarboxylic acids and, optionally, polyepoxide. These compositions show high melt strength and elasticity [69]. By using some unsaturated anhydrides as one of the components, curable unsaturated polyester formulations of dimer acid with heat stability and electric insulating properties were reported [70,71]. Dimer acid-based hot melt adhesives with good adhesion to metal were used for adhering side seams in tin cans [72-75). Phenolic -OH group-containing polyester and a poly olefin were reacted to give graft copolymers useful as adhesives [76]. Inking [77], as well as deinking agents [78], were prepared by dimer acid polyester compositions. Polymers from coupling of diazonium salts and pyridinone derivatives

588


with dimer acids gave pyridinone-based azo dye for offset printing inks [79]. Dimer acid polyester resin dispersions or solutions were converted to solid paints by NaOH-neutralization gelling, which can be stored in the form of sticks [80]. Dimer acid polyester films with multidimentional applications are formulated: • low shrinkable polyester films for food packaging [81] and shrink labels, • transparent films laminated with fire-retardant paper for flexible interior decoration sheets [82,83], • and transportation bags useful for blood transfusion [84]. Some compositions give rise to films with good masking property [85]. Synthetic leather with good abrasion and chemical and wear and tear resistance was prepared [86] using a dimer polyester elastic sheet and a stretched plastic sheet. The nontoxic property of dimer acid helped create personal care products, such as conditioners, remoisturisers, etc. [87]. All these studies show that many dimer acid-based polymers are based on the flexural and adhesion properties of dimer acid. A considerable amount of work has been reported on polyesters as well as other polymers. Among the polyesters based on dimer acid's versatile composition are those used to achieve high performance speciality polyesters. Efforts are made only to increase molecular weight; little effort has been made to prepare polyesters of low molecular weight useful as plasticizers. A couple of references [88,89] describe the preparation of plasticizers comprising glycidyl esters of dimer acid and epoxy resin compositions containing them. It is quite apparent from the state-of-art of the subject that a variety of dimer acid-based polymeric systems has been investigated from the viewpoint of commercial applications. However, their plasticizing polyester still are known on a limited scale; also, no research in the area of metal containing dimer acid polyesters has been done. The kinetics of the polyesterification of dimer acid with different polyhydroxy compounds was studied by Bajpai and Nivedita [90-94]. POLYESTERS OF DIMER ACID AND 1,4-BUTANEDIOL A series of polyesters of dimer acid with different polyhydroxy compounds were synthesized. The diols used were ethylene glycol [90], propanediol [91], 1,4-butanediol [92], glycerol [93] and diethylene glycol [94]. The polyesterification reaction has been studied under different experimental conditions to obtain the plasticizing nature. The effect of varying molar ratio of dimer acid to glycols, reaction temperature and catalyst on the course of polyesterification, acid numbers, intrinsic viscosity, and molecular weight of the resulting polyesters were investigated. The influence of various catalysts commonly used for polyesterification was studied, and probability of transesterification has been investigated. A plausible mechanism for the polyesterification was proposed, and characterization by spectral analysis was done. For metal-containing polyesters of dimer acid, synthesis was carried out using various divalent metal ions. Metal-containing polyesters include diverse systems such as ionomers, polymer-bound coordinating ligands, organometallic polymers with metal as a part of polymer backbone, metal polymer composites, and metal-


589

incorporated neutral polymers. The properties as well as multidimensional interdisciplinary applications of metal-containing polymers already has been established. Polyesters containing metal in their main chain, other then dimer acid [95,96], have been prepared by condensation technique, and the structure-property relationship of these polymer metal complexes have been studied. Good thermal stability and conductivity are the interesting properties of these polyesters from the scientific and industrial standpoint. Literature reveals that dimer acid-based polyesters bearing metal in the main chain have not been prepared. The high thermal stability of metalcontaining polyesters and flexural properties of dimer acid prompted work on metalcontaining polyesters of dimer acid having properties of both. The implementation of this idea resulted in the synthesis and characterization of polyester of dimer acid with diols having divalent metal ions in the main chain. The high thermal stability, flexibility, high molecular weight, and insolubility in common organic solvents of these polyesters may open a host of new applications. PREPARATION The method and apparatus employed for the synthesis of dimer-acid based polyesters with butanediol, as well as for metal polyesters of dimer acid and butanediol, was the direct condensation technique [91,97]. Provisions were made to control the reaction temperature and to remove the volatile byproducts (such as stirring and continuous flow of inert gas N^). In the first step of synthesis for metal-containing polyesters of dimer acid, metalcontaining diols were prepared with various divalent metal ions: Zn(II), Ni(II), Co(II), Cu(II), ]VIn(II) and Ca(II) by a known method [97,98], using butanediol in place of ethylene glycol. The product (i.e., metal salts of mono hydroxy butyl phthalates, M[HBP]2) separated as white or intense colored precipitate. The reaction scheme is given in Figure 1. 0 ^.xx. 0

+

M

/:0tH2),0H

H01CH.,),0H COH 1,A-Butanediol 0 II HO{CH),OC 2 ^

+

0 II _ CO M

M(CH3C00)2

MlHBPL Figure 1. Reaction scheme for synthesis of M (HBP)2

2+

590


The second step of synthesis involves the condensation of M(HBP)2 metal diols with dimer acid using the device similar to the one used for plasticizing polyester of dimer acid and butanediol. The reaction is depicted as 160±1°C n HO-R-OH + n HOOC-D-COOH

HO-(-R-OOC-D-COO-) -H + n H,0 ^

'n

z

where R = -(CH^),-, [HO-(CH2)4-OOC-C^H5-COO]2M CH=CH /

\

\

/

D = CH3(CH2)5 CH

CH-(CH2)7CH=CH /

CH3(CH2)5-HC=HC

\ (CH2)7-

Resinous plasticizers owe their utility mainly to the decreased mobility of their molecules, i.e., viscosity, which, in turn, is dependent on the molecular weight of the polyesters [60,90-94]. Low molecular weight polymeric plasticizers have a measurable temperature inversion of solvent ability, whereas, high molecular weight and high viscosity greatly hinder or sometimes completely prevent appreciable molecular movements. Evidently, the control on molecular weight and, consequently, on viscosity of polyesters is of prime concern from the practical viewpoint. The polyesters of desired molecular weight (< 10,000) can be obtained by choosing the appropriate reaction condition, such as molar ratio of the reactants (i.e., dimer acid and butanediol), type and concentration of catalyst, reaction temperature, and reaction time. Therefore, the studies were carried out under varied experimental conditions to fulfill the requirements of low molecular weight polyester resins. Normally, polyesterification reactions are being studied in the temperature range of 150-300°C. For low molecular weight, usually low temperature is employed in the range 160-180°C while it is maintained above 200°C for high molecular weight. For the polyesterification of dimer acid and butanediol, influence of temperature on viscosity was studied at four different temperatures, 120°, 140°, 160° and 200°C, in presence as well as in absence of catalyst, keeping other parameters constant. Degree of polymerization vs. time show that the reaction leads to high conversions, resulting in very viscous polyesters at high temperatures, whereas the reaction was extremely slow in obtaining the oligoesters of desired molecular weights. Molecular weights of the resins are limited by the need to keep viscosities of these polyesters within practical bounds. So, 160°C was chosen as a suitable temperature for all other experiments to get desired molecular weight polyesters. The reaction was carried out for 5 hours. It has been shown [60] that condensation over a long reaction period (more than 200 hours) leads to high molecular weight superpolyesters.


591

In the reaction of dicarboxylic acid with glycols, the molar ratio of the reacting functional groups (i.e., carboxyl and hydroxyl groups, respectively) has an effect on the degree of polymerization [99,100]. The excess of either glycol or dicarboxylic acid regulates the molecular weight of the polymer as well as the nature of its end groups. The results show that the reaction reaches to high conversions as the concentration of glycol increases over dimer acid (DA:BD molar ratio varied from 1:1 to 1:1.3) under the same experimental conditions, which is in accordance with the physical losses of glycol with the stream of inert gases during polyesterification; whereas, further increase in concentration of butanediol over dimer acid lowers the reaction conversions. This may be due to the heterogeneous nature of the system with increased viscosity. On the other hand, the excess of butanediol leads to hydroxy-terminated polyesters, which may have a wide variety of applications. The role of catalyst in significantly enhancing the rate of polyesterification is the fact beyond doubt [101]. In the patent literature, hundreds of compounds have been proposed as effective catalysts for polyesterification [102]. Strong protonic acids, oxides, or salts of heavy metal ions (often acetates), and organo-metallic compounds of titanium, tin, zirconium, and lead are more frequently reported catalysts. The rate of polycondensation between dimer acid and butanediol was found to be influenced by the type and concentration of the catalyst. The effect of various catalysts, such as p-toluene sulfonic acid (protonic), antimony trioxide, and calcium acetate, on the polyesterification rate under the same experimental conditions was studied. The results are depicted in Figure 2, showing the variation of acid number with time. A study of Figure 2 indicates that p-toluene sulfonic acid is the most effective catalyst. The comparative study of the catalytic activity of the three compounds strongly supports the proton-catalyzed nature of the polyesterification of dimer acid and butanediol. Further, inefficiency of antimony trioxide and calcium acetate, the well-established transesterification catalysts [103], rules out the probability of transesterification. These results and discussion in favor of the p-toluene sulfonic acid (pTSA) as an effective catalyst for polyesterification of dimer acid and butanediol strongly supports the proton-catalyzed nature of polyesterification. Usually, the protoncatalyzed mechanism for esterification is extrapolated to proton-catalyzed polyesterification [104]. The polyesterification of dimer acid and butanediol involves protonation of the dicarboxylic acid by the reaction of protonated species with the hydroxy group of glycol to yield the polyester. The proton catalyzing the protonation of carboxylic acid is provided by the carboxyl group of the monomer, i.e., dimer acid, and by pTSA in absence and presence of added catalyst, respectively. Further, the decrease in catalytic behavior of metal catalysts (calcium acetate and antimony trioxide) is attributed to the inhibiting action of carboxyl and/or hydroxyl groups present in the reaction medium. It has been observed in various reactions that carboxyl groups strongly inhibit the catalytic activity of Ti(0Bu)4 [105,106] and acetates of Zn, Ca, and Mn [105,107,108], and hydroxyl groups decrease the catalytic activity of Sb03 [105,107,108]. The effect of molar ratio in presence of catalyst pTSA was studied by varying molar ratio of dimer acid to butanediol (1:1 to 1:1.5) under similar experimental conditions. In presence of catalyst, a slight excess of butanediol (1:1 to 1:1.3) over

592


CalCH^COO)^

Time in h. Figure 2. Variation of acid number with time for the polyesterification of dimer acid and butane dio showing effect of catalyst.

dimer acid favors the polyesterification, whereas excess (1.4, 1.5) of butanediol inhibits normal polyesterification rate. The viscosity of a reaction mixture is a measure of its resistance to flow. The viscometric analysis of polyesters is done by dilute solution viscometry, i.e., the quantitative measurement of flow property of dilute polymer solutions. In this method, the increase in the relative viscosity is important while knowledge of the absolute viscosity is not necessary. While studying the polyesterification reactions of dimer acid with various diols, increase in viscosity was observed [90]. It was observed that increase in intrinsic viscosity was great for catalyzed polyesterification. In 1930, Staudinger [105] investigated the theory that the increase in viscosity may be correlated with the molecular weight of the polymer. The threadlike molecules cause a marked increase in the viscosity of the solvent in which they were dissolved, with the increase the greatest in higher molecular weight polymers. Therefore, the polyesterification of dimer acid with glycols results in viscous polyester resin at high temperatures as well as in presence of high concentration of catalyst. Thus, by controlling the reaction conditions one can get the polymers of desired properties. A comparative analysis of spectra of dimer acid, butanediol, and their polyester was done. The spectrum of polyester shows characteristic bands for ester linkages.


593

All the metal-containing polyesters of dimer acid and metal salts of mono hydroxy butyl phthalates MCHBP)^ were colored solids, insoluble in water, and most were organic solvents. These polyesters having Zn(II), Mn(II), Ca(II) do not melt until 320°C, whereas, polyester of CuCHBP)^ decomposes at 170°C. The product of dimer acid and Co(HBP)2 was hygroscopic in nature and becomes sticky by absorbing moisture. The formation of polymers is confirmed by ir spectra analysis. The ir spectra of polyesters were compared with the ir spectra of dimer acid and that of its metal diols to ascertain the bonding in polyester. The comparison of ir spectra of butanedioldimeracid polyester and of metal-containing polyester leads us to conclude the latter have ionic linkages between metal ions M(II), and carboxylate groups appeared as a strong absorption band in the range of 1,580-1,560 c m ' and 1,598-1,590 c m ' . Further, the thermogravimetric analysis of metal diols of HBP and respective metal-containing polymers was done to study the thermal stability of the polyesters. The TG data show that all the polyesters are very stable up to 250°C. The energies of activation of degradation of these polyesters have been calculated using the Fuoss method [106]. APPLICATIONS Dimer acid-based polyesters being used as thermoplasts, thermoset, and elastomers are used in large quantities as adhesives and coatings. The polyesters, as we have reported herein, may have various commercial applications. Owing to the flexural properties, these low molecular weight linear polyesters of dimer acid may be used as plasticizers providing internal lubrication for various polymers, such as polyimides and inorganic coordination polymers, which may have very poor processibility. These hydroxy-terminated polyesters may have applications as prepolymers for other high molecular weight polymers. These could condense with diisocyanates to produce polyurethane foams [44], either flexible or rigid. They also may have compatibility with epoxy resins and polyester to give rise to polyester epoxy resins, polyester-poly amide [41], and polyester-polyether copolymers. The further condensation with some unsaturated monomer may result in thermoset film formers. They can be used as a mildness additive in metal-working lubricants. Further incorporation of metal ions and aromatic structure in the main chain of these polyester enhances the thermal stability as well as resistance towards solvents. Owing to these properties, these metal-containing polyesters may be used for hard surface coatings where thermal resistance is required.

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CHAPTER 23 VISCOELASTIC PROPERTIES OF MODEL SILICONE NETWORKS WITH PENDANT CHAINS M. A. Villar and E. M. Valles Planta Piloto de Ingenieria Quimica UNS-CONICET 8000 Bahia Blanca, Argentina CONTENTS INTRODUCTION, 599 NETWORK PREPARATION, 600 THEORY, 602 EXPERIMENTAL RESULTS, 604 Equilibrium Elastic Modulus, 604 Dynamic Elastic Modulus, 608 Loss Modulus, 608 Ultimate Properties, 611 CONCLUSIONS, 612 NOTATION, 612 REFERENCES, 613

INTRODUCTION Model silicone networks, i.e., those prepared by end-linking of functionally terminated polymer chains, have been extensively utilized to explain the influence of molecular structure on mechanical properties. An important number of studies have been focused on the contribution of elastically active chains and trapped entanglements to equilibrium properties [1-7]. In contrast, very little work has been done to explain the influence of network structure on non-equilibrium properties [8], and the contribution of some of the main structural parameters to viscoelastic properties has been poorly explored. A few qualitative studies have shown in the past that pendant chains have a strong influence on relaxation properties, but the type of contribution was not clearly understood [9]. When a rubber network is stretched only those chain sections extending between crosslinking points are permanently oriented by the external stress and contribute to the elastic equilibrium modulus. Dangling chains are temporarily oriented by a deformation, but they can relax, reptating from the free ends towards their permanent 599

600


junction with the network. At infinite times these chains do not contribute to the elastic force, but they do have a strong effect on time-dependent properties. Recent studies on model silicone networks have contributed significantly to a better understanding of the relationships between structure and properties in this area [10,11]. Some of this work, both experimental and theoretical, is reviewed in the following sections, with a strong emphasis on molecular interpretations. NETWORK PREPARATION Model polymer networks are materials prepared in a way that provides independent information on their structure. Synthetic procedures that link long chains of a given linear pre-polymer by their ends have been quite successful in obtaining these kind of materials [6,12]. In this reaction the end-functional groups at the extremes of a pre-polymer chain are reacted with cross-linking agents having a functionality, f, of 3 or more. If every polymer chain has two terminal reactive groups (B2), and each group effectively reacts in a stoichiometrically balanced reaction with different crosslinker molecules (A^.), an ideal network is obtained. When the crosslinking reaction is completed the average chain length between crosslinking points is equal to the length of the bifunctional B2 molecules used to prepare the model network. One way to generate networks with dangling ends in this A^^ + B2 type of reaction is by preparing formulations with an initial stoichiometric imbalance (r). In this case the amount of pendant chains present in the network at the end of the reaction depends on r (Figure 1). There exists two principal disadvantages in these kind of systems: first, it is impossible to vary the amount and molecular weight of the pendant chains independently and second, the resultant pendant chains may have complex structures with wide polydispersity and branching that are not very suitable for structural studies [10,13]. With a similar approach it is also possible to obtain model networks with controlled amounts of very well defined pendant chains. Adding monofunctional B, chains to the initial mixture of pre-polymer and crosslinker an A^ + B^ + B, reaction is generated. This will result in the formation of linear pendant chains of length equal to that of the B,'s. Furthermore, thanks to modem anionic polymerization procedures [12,14,15] it is possible to synthesize telequelic monodisperse chains of type B, and B2 that can be used to build model networks with elastic and pendant chains of any desirable molecular weight and narrow molecular weight distribution. In particular, hexamethylcyclotrisiloxane (D^) can be anionically polymerized to obtain polydimethylsiloxane (PDMS) with narrow molecular weight distribution. This was explained by Lee et al [16] based on the ring-strain of D3 which enhances the reactivity of the Si-0 bonds with respect to those in higher-order rings. Under living-end conditions the polymerization of D3 leads to PDMS's with approximate Poisson-distribution. Alkyl lithium initiators, such as n-butyl-lithium, are commonly used to generate living polydimethylsiloxane chains [17-19]. Bifunctional vinyl-ended polydimethylsiloxane (B2) has been synthesized by anionic ring opening polymerization of D3 under dry argon using dilithium stilbene as initiator and tetrahydrofurane (THF) as solvent [20]. Monofunctional vinyl-ended polydimethylsiloxane (B,) was obtained by polymerization of D3 under vacuum using n-butyl-lithium as initiator.

Viscoelastic Properties of Model Silicone Networks with Pendant Chains

1.0

~ i — I — I — I — I — I — I — I — I — \ — I — I — I — r

601

~l—I—I—I—I—I—I—I—I—r

0.8

0.6 OH

0.4

0.2

0.0

0.5

Figure 1. Weight fraction of pendant chains (Wp) as a function of stoichiometric imbalance (r) at complete reaction. System A + B^.

a non-polar solvent such as n-hexane or toluene, and tetrahydrofurane as an electron donor compound to promote the polymerization [21]. Model silicone networks with pendant chains have been synthesized using B^ and Bj pre-polymers, mainly by the Pt catalyzed hydrosilation reaction between silane groups from the cross linker molecules and vinyl groups on the chain ends of the B2 and B, molecules [11]. Pre-polymer and cross linker agent are weighed, catalyst is added to the reactant mixture, and all the components are then mixed with a mechanical stirrer and degassed under light vacuum to eliminate bubbles. Finally, the mixture is allowed to cure in molds with the appropriate shape for mechanical testing. In these systems, the amount of pendant chains is now a function of the weight fraction of Bj groups (W^ ) added to the initial reaction mixture [11]. Figures 2 and 3 show the dependence of weight fraction (W ) and weight average molecular weight of the pendant chains (M^ ) in the completely reacted network as a function

602


Figure 2. Weight fraction of pendant chains (Wp) as a function of weight fraction of monofunctional chains (Wg^) for stoichionnetrically balanced networks (r = 1) at complete reaction. System A3 + Bg + B^. Networks with MA3 = 330.7, Mg = 10,000, and M^ : 25,000, 50,000, 100,000, and 200,000. ' ' of Wg . Calculations show that for stoichiometrically balanced, completely reacted networks, there exists a limiting amount of monofunctional chains which can be added to the reaction mixture to obtain model networks with linear pendant chains [22]. Either an excess or a defect of crosslinker originates networks with both linear and branched pendant chains and with a broad distribution of molecular weights. THEORY It is now well-recognized that pendant chains make a significant contribution to the long-term relaxation behavior of cross-linked rubbers as seen in stress relaxation and creep experiments. The molecular mechanism accountable for this long-term process is the diffusion of pendant chains in the presence of entanglements.

Viscoelastic Properties of Model Silicone Networks with Pendant Chains 1 0

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I hi

II

0.0

II

0.2

603

I I I I I

I I I I I I I I I I I I I I I I I I I I I

0.4

w

0.6

0.8

1.0

Bi

Figure 3. Weight average molecular weight of pendant chains (M^p) as a function of weight fraction of monofunctional chains (Wg ) for stoichionnetrically balanced networks (r = 1) at complete reaction. System A3 + 63 + B^. Networks with M^^ = 330.7, M^ = 10,000, and Mg^: 25,000, 50,000, 100,000, and 200,000. Theoretical models of the dynamic of imperfect networks suggested that the relaxation times of pendant chains should depend exponentially on the molecular weight of these chains [23-25]. The basic idea, proposed by de Gennes [23], is that relaxation mechanism of linear pendant chains is governed by the reptation or "snake-like" motion of the chains retracting along their primitive path from the free end to the fixed one. This model proposed that the relaxation time of pendant chains should increase exponentially with the number of entanglements in which it is involved. Pendant chains must then contribute to viscoelastic properties for frequencies greater than the inverse of reptation times. Tsenoglou [26], Curro and Pincus [27], Pearson and Helfand [24] and Curro et al. [25] developed models for the relaxation of pendant chains in random cross-linked networks.

604


Following Pearson and Helfand [24], the time necessary for reptation of a pendant chain is: T^ = i;X''

exp(v'N)

(1)

where T^ is the reptation time, x^ is associated with the maximum relaxation time of a chain constituted for Ne monomers in Rouse's model, N is the number of entanglements in which a pendant chain is involved in, and v' is a constant which depends on the coordination number of the network and whose value is approximately 0.6. In order to compute theoretical values of x^ it is necessary to know a priori the corresponding value of x^. Mooney developed an extension of Rouse's model for a chain with fixed ends [9]. Relaxation spectra resulted coincident with Rouse's original model, except for an additional contribution to the modulus with an infinite relaxation time. The maximum relaxation time for a chain of Ne monomers fixed in both ends became:

,.. 67ckT 'i-

vM„

(2)

where a is a characteristic length associated with the Kuhn monomer length, C,^ is the monomeric friction coefficient, p the density, and M^ the molecular weight of the monomeric unit. EXPERIMENTAL RESULTS Equilibrium Elastic Modulus Since at long times pendant chains do not contribute to permanent elastic properties, the elastic equilibrium behavior of networks containing these chains should not differ substantially from that of regular networks. The elastic modulus from a network with pendant chains can then be obtained from the molecular theories of rubber elasticity provided that the concentration of elastically active network chains (v) can be calculated accurately. Depending on the different approaches that can be used for the rubber elasticity theory, the calculation of some other parameters, like the concentration of junctions points (|Ll) and trapped entanglements (Te), also may be needed. A considerable number of experimental studies, as well as theoretical developments, have been done on the equilibrium elastic properties of regular model silicone networks in absence of pendant chains. The goal of most of these studies has been to test quantitatively the molecular basis of the theory of rubber elasticity. One of the major concerns has been the influence of topological interactions between chains on elastic properties of the networks. However, despite the considerable amount of experimental work, there is still considerable debate concerning the validity and applicability of different models. Silicone networks containing linear pendant chains of known molecular weight at complete reaction were obtained by Villar and Valles [28] who added to the


605

A^ + B2 system known amounts of monodisperse linear molecules of type B, differing in molecular weight and bearing only one vinyl-end group. Experimental values of low frequency elastic modulus have been compared with theoretical values corresponding to "phantom theory" (G = (v - ji) RT) [29], affine deformation (G = vRT) [30,31], and those obtained considering the contribution of trapped entanglements (G = (v - h|Li) RT + GeTe) [32,33]. Experimental values of lowfrequency elastic modulus and those calculated from the different theories are shown in Table 1. Values of v, |i,and Te were calculated using the recursive method based on formulation data and the maximum extent of reaction obtained experimentally

Table 1 Elastic Modulus Measured at Low Frequencies and Theoretically Predicted Values [11]

Network

CrossWeight linker Molecular Fraction Function- Weight of Bi ality of Bi f Chains Chains Exptl. vRT

00-F3-0

0.0

00-F4-0

0.0

—

G(MPa) (v - [i) RT

(v - hî) RT + Ge Te^

0.214 0.151

0.050

0.197

0.252 0.194

0.088

0.245

M1-F3-20 M2-F3-20 M3-F3-20 M4-F3-20 M5-F3-20

0.202 0.201 0.200 0.201 0.199

3 3 3 3 3

26,700 51,900 62,100 92,300 125,000

0.120 0.129 0.147 0.147 0.145

0.073 0.080 0.079 0.084 0.084

0.024 0.027 0.026 0.028 0.028

0.116 0.126 0.129 0.134 0.133

M1-F4-20 M2-F4-20 M3-F4-20 M4-F4-20 M5-F4-20

0.217 0.203 0.209 0.214 0.221

4 4 4 4 4

26,700 51,900 62,100 92,300 125,000

0.158 0.154 0.185 0.182 0.157

0.107 0.101 0.115 0.125 0.119

0.043 0.041 0.048 0.053 0.050

0.147 0.131 0.156 0.175 0.161

M1-F3-33 M2-F3-33 M3-F3-33 M4-F3-33 M5-F3-33

0.331 0.323 0.319 0.349 0.344

3 3 3 3 3

26,700 51,900 62,100 92,300 125,000

0.076 0.089 0.106 0.101 0.092

0.044 0.058 0.053 0.054 0.055

0.015 0.019 0.018 0.018 0.018

0.081 0.102 0.093 0.099 0.099

M1-F4-33 M2-F4-33 M3-F4-33 M4-F4-33 M5-F4-33

0.281 0.337 0.338 0.331 0.334

4 4 4 4 4

26,700 51,900 62,100 92,300 125,000

0.135 0.094 0.129 0.124 0.106

0.092 0.082 0.084 0.097 0.076

0.037 0.033 0.034 0.040 0.031

0.130 0.120 0.097 0.142 0.103

a V, JJ, and Te were calculated from initial formulation of h = 1 and Ge = G^ were used in the calculations.

using the recursive method.

Values

606


[2]. The elastic modulus of networks with pendant chains, measured at low frequencies, shows very good agreement with values calculated from theory of elasticity when contribution of molecular entanglements is taken into account. Figures 4 and 5 show curves of dynamic elastic modulus (C) as a function of frequency for networks synthesized with monofunctional prepolymers of relatively low and high molecular weight respectively and a tetra functional cross linker [11]. The elastic modulus of a network prepared without pendant chains is also plotted. Results show a marked reduction in elastic modulus of networks with increasing amounts of pendant chains. This is due to the reduction in the concentration of

10

I

I I I llll|

1 I I I llll|

mmn

1—I I I l l l l |

1 I I I llll|

MMMMMMl

1—I I I l l l l |

1 I I I llll|

1 I I I Mil

MHHl(i>iliiiiii > • •

mmmmmmmmmmmmmmmmmmmmm OH

10

O

10 10

I I I I ml

10

I I I I I ml -2

10

I -1

I Mini

Mini

10 1 10 CO ( r a d / s )

mil

10

10

Figure 4. Elastic modulus (G') as a function of frequency (co) for networks with monofunctional chains of weight average molecular weight of 26,700: ( • ) without pendant chains (system A^ + Bg), (•) with 22 wt% of monofunctional chains (system A^ + Bg + B^), and (A) with 28 wt% of monofunctional chains (system A^ + Bg + B^) [11].

Viscoelastic Properties of Model Silicone Networks with Pendant Chains 1 0

I

1 I I I llll|

1 I I I llll|

1 I I I llll|

1—I I I l l l l |

I I I I llll|

1—I I I l l l l |

607

1—I I I MM

mmmimmmmmmmmmmmmmm m • • mmmmmmm 0-,

M>At*At*Amiittitm

10* 10

I I I I Mill

I I I I mil

10 -'

I I I I mil

10 -'

1

I I 1 I Mill

I I I I mil

10

10

cj (rad/s)

Figure 5. Elastic modulus (G') as a with monofunctional chains of weight ( • ) without pendant chains (system monofunctional chains (system A^ + monofunctional chains (system A^ +

i i i i11 1KII 1 KII

10

10^

function of frequency (co) for networks average molecular weight of 125,000: A^ + B2), (•) with 22 wt% of 63 + B^), and (A) with 33 wt% of 82 + B^) [11].

elastically active chains (v) when the weight fraction of pendant chains is increased. When the tetra functional cross linker was replaced with a trifunctional cross linker, a reduction in elastic modulus was also found for networks prepared with exactly the same concentration and structure of pendant chains. This difference in the elastic modulus of networks differing exclusively in the functionality of their junctions points was attributed and correctly predicted by current molecular theories to a higher fluctuation of the trifunctional junction points [28]. This confirms the validity of previous results obtained in tension from regular networks in absence of pendant chains [6].

608


Dynamic Elastic Modulus On the other hand, significant differences were observed in frequency dependence of the elastic modulus in the networks with pendant chains. While a flat evolution of G' with frequency was observed for networks prepared with no pendant chains revealing independence of the elastic properties with co, a noticeable increase in elastic modulus at high frequencies appeared in the networks containing pendant chains. The effect is more evident for networks with pendant chains of higher molecular weights. Values of G' increase at high frequencies because part or all of each pendant chain can behave as an elastic chain when the entanglements in which it is involved cannot relax at those frequencies. Loss Modulus In order to explain from a molecular point of view how concentration and molecular weight of pendant chains affect viscoelastic properties, Bibbo and Valles followed the evolution of loss modulus (G") with the extent of reaction during the cure of a difunctional vinyl-end polydimethylsiloxane (a,co-PDMS) with a trifunctional crosslinker [10]. It was observed that G" increases steadily after the gel point, reaching a maximum value at the extent of reaction at which the maximum amount of pendant chains was present in the network, then G" decreases up to a final definite value when the maximum extent of reaction is attained. This experiment showed that pendant chains can contribute significantly to the loss properties of networks. However, in this work it was not possible to have a good independent control of the concentration, molecular weight, and degree of branching of the pendant chains because dangling material was generated from a random crosslinking process. Loss modulus measurements on networks containing linear monodisperse pendant chains of known molecular weight at complete reaction indicate that G" depends on concentration and molecular weight of dangling ends as well as on the functionality of crosslinking points [28]. Figure 6 shows values of G" as a function of frequency for networks synthesized with a weight fraction of monofunctional chains of approximately 0.20. Loss modulus and relaxation times increase with an increase in the molecular weight of the pendant chains. The heaviest monofunctional chains used (M^ = 125,000) do not relax completely in the range of frequencies analyzed. Figure 7 shows values of G" corresponding to networks prepared with a weight fraction of monofunctional chains of 0.33 approximately. In this case, G" values are higher than those obtained with a weight fraction of monofunctional chains of 0.2. Loss modulus values of networks without pendant chains resulted so low that they could not be measured in the range of frequencies and temperatures explored. Terminal relaxation times depend on the concentration of pendant chains and the functionality of junction points. Higher concentration of pendant chains results in larger relaxation times. The same effect is observed when the functionality of the crosslinking points is decreased. Within the narrow distribution window allowed by the dynamic measurements, relaxation times of pendant chains estimated from G" values are in good agreement with those calculated from the molecular theory proposed by de Gennes (Tables 2 and 3). Estimated values for the relaxation times of networks containing pendant chains of weight average molecular weight of


1 0

P

609

1 I I I llll|

1 I I I llll|

1 I I I llll|

1 I I I llll|

1 I I I llll|

1 I I I llll|

1 I I I IILU

I I I I Mill

I I I I Mill

I I I I mil

i

I I I I Mill

I I I I Mill

I I I I Mill

10*

10

o 10 ' t

•j r\

I

10"'

10"'

10"'

1

I I I Mill

10

10'

10'

10'

cj ( r a d / s ) Figure 6. Loss nnodulus (G") as a function of frequency (co) for networks with approximately 20 wt% of monofunctional chains (system A^ + 63 + B^): networks with monofunctional chains of weight average molecular weight of 26,700, (0) trifunctional crosslinker, and (•) tetrafunctional crosslinker; networks with monofunctional chains of weight average molecular weight of 51,900, (D) trifunctional crosslinker, and (•) tetrafunctional crosslinker; networks with monofunctional chains of weight average molecular weight of 62,100, (O) trifunctional crosslinker, and ( • ) tetrafunctional crosslinker; and networks with monofunctional chains of weight average molecular weight of 125,000, (A) trifunctional crosslinker, and (A) tetrafunctional crosslinker [11]

610


0

P

I

I I I I ill|

I i I I lil|

I

I

I I I I lll|

I

I I I I lll|

I

I M I lll|

I

I I I I lll|

I

I I I IIL

10*

10

O

10

I 10 10"'

I I I Mill

I I I I mil

10"'

10"'

11 mil

I I II mil

1

10

I I mil

10'

I I 1 mil

10

I I I I III

10 '

cj (rad/s) Figure 7. Loss modulus (G") as a function of frequency (co) for networks with approximately 33 wt% of monofunctional chains (system A, + 63 + B^): networks with monofunctional chains of weight average molecular weight of 26,700, (0) trifunctional crosslinker, and (•) tetrafunctlonal crosslinker; networks with monofunctional chains of weight average molecular weight of 51,900, (D) trifunctional crosslinker, and (•) tetrafunctlonal crosslinker; networks with monofunctional chains of weight average molecular weight of 62,100, (O) trifunctional crosslinker, and ( • ) tetrafunctlonal crosslinker; and networks with monofunctional chains of weight average molecular weight of 125,000, (A) trifunctional crosslinker, and (A) tetrafunctlonal crosslinker [11].


611

Table 2 Terminal Relaxation Time of Pendant Chains in Networks with Approximately 20 wt% of Monofunctlonal Chains Weight Average Molecular Weight of Bi Chains

N

(s) eq. 2

26,700 51,900 62,100 92,300 125,000

3.3 6.5 7.8 11.5 15.4

3.00 2.46 2.52 2.26 2.22

f = 3

f = 4 To X 105

To X IQS XN ( S )

-CN ( S )

eq. 1

exp.

(s) eq. 2

1.6 10-3 2.2 10-2 6.2 10-2 — —

0.93 1.04 0.77 0.64 0.70

1.4 2.0 5.7 9.0 3.2

10-3 10-2 10-2 10-' 10^'

^N (S)

XN ( S )

eq. 1

exp.

4.2 8.4 1.8 2.5 1.0

10-^ 10-3 10-2 10' 10^'

1.0 10-^ 2.7 10-3 2.0 10-2 — —

Table 3 Terminal Relaxation Time of Pendant Chains in Networks with Approximately 33 wt% of Monofunctlonal Chains Weight Average Molecular Weight of Bi Chains

N

(s) eq. 2

26,700 51,900 62,100 92,300 125,000

3.3 6.5 7.8 11.5 15.4

8.00 4.75 5.56 5.48 5.18

f = 4

f = 3 To X 10^

To X 105 ^N (S)

-CN ( S )

eq. 1

exp.

(s) eq. 2

2.0 10-2 1.1 10-1 3.0 10-» — —

1.30 1.62 1.54 1.08 1.86

3.6 3.8 1.3 2.2 7.6

10-3 10-2 10-1 10-^^ 10^'

-CN ( S )

'CN ( S )

eq. 1

exp.

5.9 1.3 3.5 4.3 2.7

10-^ 10-2 10-2 10-1 10^1

1.6 10-3 1.1 10-2 6.0 10-2 — —

26,700, 51,800, and 62,100 were compared with theoretical terminal relaxation times calculated using Equation 1. An expression of T^ developed for Mooney was used in the calculations. A reasonable agreement was found between calculated relaxation times and those obtained from experiments taking into account all possible errors involved. These results are coincident with an exponential dependency of relaxation time of pendant chains in an analogy of what happens for star-branched polymers. Ultimate Properties Mark and co-workers have studied the effect of dangling chains on the ultimate properties of model networks prepared by end-linking vinyl-terminated PDMS chains with a tetra functional crosslinker [34]. In this case the crosslinker was used in varying amounts smaller than those corresponding to a stoichiometric balance. The

612


ultimate properties of these networks, with known amounts of pendant chains, were compared with stoichiometrically balanced networks having negligible numbers of irregularities. Networks containing dangling chains showed lower values of ultimate strength and less extensibility than perfect networks, with the largest differences occurring at high proportions of dangling ends. CONCLUSIONS Even now, little is known about the influence of network defects on nonequilibrium viscoelastic properties. However, some experiments and theoretical development can be cited. It has been shown that pendant chains are responsible for higher loss modulus in networks prepared with known amounts of linear dangling chains. The magnitude of the loss modulus depends on the concentration and molecular weight of the pendant chains present in a network. Terminal relaxation times of pendant chains in a network also were found to be in good agreement with those calculated from the molecular theory proposed by de Gennes. Dangling ends also are responsible for the decrease in ultimate properties, specifically the ultimate strength and maximum extensibility. Preparing and studying model networks with controlled structure can provide a great deal of valuable information on rubber-like elasticity and other related problems. Particularly, networks prepared with a known amount of defects, such as pendant chains, trapped molecules, loops, etc. are necessary to understand the influence of these faults on equilibrium and dynamic mechanical properties. NOTATION a= A,= B, = B.= f G G' G" Ge

= = = = =

h= k= M =

characteristic length, m crosslinker (functionality f) monofunctional pre-polymer difunctional pre-polymer hexamethylcyclotrisiloxane crosslinker functionality equilibrium modulus. Pa elastic modulus. Pa loss modulus. Pa maximum contribution to the modulus due to trapped entanglements. Pa empirical parameter Boltzman constant, J • K~' molecular weight of the monomeric unit. Kg • Kmole"

M^ = weight average molecular weight of pendant chains. Kg • Kmole' N = number of entanglements in a pendant chain Ne = number of monomer units between entanglements R = universal gas constant, KJ • Kmol' • K ' r = stoichiometric imbalance T = temperature, K Te = fraction of trapped entanglements Wg = weight fraction of monofunctional chains W = weight fraction of pendant chains

Greek Letters [i = concentration of junctions points, Kmole • m"^

V = concentration of elastically active chains, Kmole • m"^


v' = a constant which depends on the coordination number of the network p = density, Kg • m~^ T = maximum relaxation time of a

613

x^ = reptation time, s co = frequency, s ' C = monomeric friction coefficient, N • s • m~'

o

chain constituted for Ne monomers in Rouse's model, s

REFERENCES 1. Valles, E. M. and C. W. Macosko, Rubber Chem. TechnoL, 49, 1232 (1976). 2. Valles, E. M. and C. W. Macosko, Macromolecules, 12, 673 (1979). 3. Valles, E. M., E. J. Rost and C. W. Macosko, Rubber Chem. TechnoL, 57, 55 (1984). 4. Mark, J. E., R. R. Rahalkar and J. L. Sullivan, J. Chem. Phys., 70, 1747 (1979). 5. Mark, J. E. and J. L. Sullivan, J. Chem. Phys., 66, 1,006 (1977). 6. Mark, J. E., Rubber Chem. TechnoL, 54, 809 (1981). 7. Meyers, K. O., M. L. Bye and E. W. Merrill, Macromolecules, 13, 1,045 (1980). 8. Kramer, O., British Polym. J., 17, 129 (1985). 9. Ferry, J. D., Viscoelastic Properties of Polymers, J. Wiley & Sons, New York (1980). 10. Bibbo, M. A. and E. M. Valles, Macromolecules, 17, 360 (1984). 11. Villar, M. A., Ph.D. Thesis, Universidad Nacional del Sur, Bahia Blanca, Argentina (1991). 12. Rempp, P., J. Herz, G. Hild, and C. Picot, Pure Appl. Chem., 43, 77 (1975). 13. Bibbo, M. A. and E. M. Valles, Macromolecules, 15, 1,293 (1982). 14. Morton, M., L. J. Fetters, J. Inomata, D. C. Rubio, and R. N. Young, Rubber Chem. TechnoL, 49, 303 (1976). 15. Morton, M., Anionic Polymerization: Principles and Practice, Academic Press, New York (1983). 16. Lee, C. L., C. L. Frye, and O. K. Johannson, Polym. Preprints, 10(2), 1,361 (1969). 17. Holle, H. J. and B. R. Lehnen, Europ. Polymer J., 11, 663 (1975). 18. Morton, M., Y. Kesten, and L. J. Fetters, Appl. Polym. Symp., 26, 113 (1975). 19. Zilliox, Z. G., J. E. L. Roovers, and S. Bywater, Macromolecules, 8, 573 (1975). 20. Meyers, K. O., Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA (1980). 21. Villar, M. A., M. A. Bibbo, and E. M. Valles, /. Macrom. Sci.—Pure Appl. Chem., A29, 391 (1992). 22. Villar, M. A., M. A. Bibbo, and E. M. Valles, Submitted to Macromolecules (1995). 23. de Gennes, P. G., Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York (1979). 24. Pearson, D. S. and E. Helfand, Macromolecules, 17, 888 (1984). 25. Curro, J. G., D. S. Pearson and E. Helfand, Macromolecules, 18, 1,157 (1985). 26. Tsenoglou, C , Macromolecules, 22, 284 (1989). 27. Curro, J. G. and P. Pincus, Macromolecules, 16, 559 (1983).

614

28. 29. 30. 31.


Villar, M. A. and E. M. Valles, submitted to Macromolecules (1995). James, H. M., and E. Guth, J. Chem. Phys., 11, 455, 472 (1943). Hermans, J. J., Trans. Faraday Soc, 43, 591 (1947). Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York (1953). 32. Dossin, L. M. and W. W. Graessley, Macromolecules, 12, 123 (1979). 33. Pearson, D. S. and W. W. Graessley, Macromolecules, 13, 1,001 (1980). 34. Andrady, A. L., M. A. Llorente, M. A. Sharaf, R. R. Rahalkar, and J. M. Mark, 7. Appl Polym. Sci., 26, 1829 (1981).

CHAPTER 24 RHEOLOGY OF WATER-SOLUBLE POLYMERS USED FOR IMPROVED OIL RECOVERY H.A. Nasr-El-Din Laboratory Research & Development Saudi Aramco, P.O. Box 62 Dhahran 31311, Saudi Arabia and K. C. Taylor Petroleum Recovery Institute 100, 3512 33rd Street NW Calgary, Alberta, Canada T2L 2A6 CONTENTS INTRODUCTION, 616 PART I: EXPERIMENTAL STUDIES, 621 Partially Hydrolyzed Polyacrylamide, 621 Xanthan Gum, 622 PART II: PARTIALLY HYDROLYZED POLYACRYLAMIDE, 622 Polymer Viscosity in Deionized Water, 623 Effect of Sodium Chloride on the Viscosity of HP AM, 624 Effect of Cation Type on Polymer Viscosity, 627 Effect of Alkali Type on Polymer Viscosity, 629 Effect of Surfactants on Polymer Viscosity, 634 Effect of Surfactants and Alkalis on Polymer Viscosity, 635 PART III: XANTHAN GUM, 637 Effect of Polymer Concentration on Apparent Viscosity, 637 Effect of Polymer Concentration on Screen Factor, 638 Effect of Sodium Chloride on the Apparent Viscosity of Biopolymers, 639 Effect of Cation Type on Polymer Viscosity, 642 Effect of Alkali Type on Polymer Viscosity, 643 Effect of Surfactants on Polymer Viscosity, 648 Combined Effect of Surfactants and Alkalis on Polymer Viscosity, 648 PART IV: RHEOLOGY OF ASSOCIATING POLYMERS, 650 Viscosity of Associating Polymers in the Dilute Regime, 652 Viscosity of Associating Polymers in the Semi-Dilute Regime, 656

615

616


CONCLUSIONS, 660 Partially Hydrolyzed Polyacrylamide, 661 Xanthan Gum, 661 Hydrophobically Associating Polymers, 661 ACKNOWLEDGMENTS, 662 NOMENCLATURE, 662 REFERENCES, 662 INTRODUCTION Water-soluble polymers are used in many oilfield operations. These include drilling, polymer-augmented water flooding, and various enhanced oil recovery processes such as alkaline and micellar flooding. In enhanced oil recovery (EOR), the basic idea behind using these polymers is to reduce the mobility of the aqueous phase and, consequently, to improve the sweep efficiency. The use of polymers to increase the viscosity of drilling fluids requires high viscosities at low shear rates (to suspend cuttings during low flow rates), low viscosities at high shear rates (to allow large volumes of fluid to flow through the drill bit), and stability at high shear rates [1]. Water-soluble polymers used in drilling operations to increase solution viscosity include guar gum, carboxymethylcellulose (CMC), hydroxyethylcellulose (HEC), polyanionic cellulose (PAC), xanthan gum, polyacrylate, and polyacrylamide. Although starch is used in drilling applications, it is colloidal in nature and is used to physically plug porous media [2]. Xanthan gum and high molecular weight cellulose derivatives (HEC and PAC) are used most commonly as viscosifiers in drilling operations [2]. The rheological properties of polymer solutions play an important role in determining their effectiveness. Depending on the process, polymers can encounter various chemical species, such as simple salts, alkalis, and surfactants. The presence of these chemicals together may significantly alter the chemical and physical nature of the polymer molecules and, consequently, the viscosity of the polymer solution will change. Partially hydrolyzed polyacrylamide (HPAM) is produced by the free radical copolymerization of acrylamide and sodium acrylate. The chemical structure of HPAM is shown in Figure 1, where M^ denotes K^ or Na^, X and Y are the numbers of carboxylate and amide groups, respectively. A very important parameter which determines the charge density of the polymer chain is the degree of hydrolysis, T, defined as: T=

^ ; 0 < T< 1 X +Y

(1)

As can be seen, HPAM is a polyelectrolyte, with negative charges on the carboxylate groups. This implies strong interactions between the polymer chain and any

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

617

Figure 1. Structure of HPAM. cations present in the solvent, especially at higher degrees of hydrolysis. Many researchers have examined the effect of salts on the viscosity of HPAM [3-8]. They found that the viscosity of HPAM significantly decreased with increasing salt concentration. Divalent cations were found to have a more detrimental effect on polymer viscosity than monovalent cations [4,9]. Divalent cations also can cause phase separation at high temperatures and degrees of hydrolysis > 0.35 [9,10]. Poly aery lamide (PAM) is hydrolyzed in the presence of alkalis (Figure 2). This reaction is characterized by a high initial rate of reaction which slows down significantly as the reaction proceeds. The high initial reaction rate is due to the neighbor amide group catalytic effect, whereas the later slow rate of reaction is due to the coulombic repulsion between the negative charges of the carboxylate groups on the polymer chain and the hydroxide ions [11-15]. The reaction rate is a function

H

H

-(CHg-C)—(CH^-C)

H

H

OH-

(CHg-C)—(CH^-C)-

I

C=0

C=0

C=0

NHo

NHo

NHo

I

I

C=0

I NHg

V

-(CHg- C)

H I (CHg- C)

•i^

(CHg- C)

H I (CHg- C)

C=0

C=0

C=0

C=0

NH.

0~

NHo

0~

Figure 2. Base hydrolysis of PAM.

+

NH,

618


of pH and temperature. Ryles described this reaction under neutral conditions as rapid at 90°C, moderate and 70°C, and very slow at 50°C [16]. Kuehne found that the initial reaction rate is much higher with sodium hydroxide than sodium carbonate [17]. Xanthan gum (a biopolymer) is a water-soluble polymer widely used in polymer flooding and drilling operations. Xanthan gum is an extra-cellular heteropolysaccharide produced by the bacterium, Xanthomonas campestris. The primary structure of xanthan consists of a backbone of glucose monomers (cellulose-like chain) and side chains [18]. The backbone consists of anhydroglucose monomers connected by P(l -^ 4) glycosidic linkages (Figure 3). A side chain that contains the sequence mannose/glucuronic acid/mannose is attached to every other glucose unit. In each side chain an O-acetyl group is usually attached to the mannose unit closest to the polymer backbone. Some of the terminal mannose monomers may contain a ketal-linked pyruvate group. The percentage of side chains that contain pyruvate groups varies from 0 to 100%, depending on the bacterial strain used and the fermentation conditions [19]. The presence of carboxylate groups (glucuronate and pyruvate) in the side chains gives the polymer its ionic character. Xanthan undergoes a thermally induced conformation order/disorder (helix/coil) transition which is dependent on the pH, ionic strength, and the extent of pyruvate substitution in the side chains [20-23]. According to Rochefort and Middleman, xanthan is in a disordered state in deionized water at 25°C [19]. However, the polymer chains are highly extended (coil configuration) due to the electrostatic repulsion between the negative charges of the carboxylate groups present in the side chains. A disorder/order transition occurs once a salt is added to the polymer in deionized water at 25°C. Also, the side chains collapse on the polymer backbone (due to charge screening effects). The effect of adding salts on the viscosity of

HOCHg

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/

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619

xanthan gum depends on polymer concentration. At polymer concentrations < 2,000 ppm, the viscosity decreases upon the addition of sodium chloride due to the charge screening effects. However, at polymer concentrations > 4,000 ppm, the viscosity increases with the addition of sodium chloride. This increase in viscosity is explained in terms of self-association between the collapsed side chains of xanthan [11,23,24]. This self-association is the result of both intra and intermolecular hydrogen bonding. The change in polymer conformation from order to disorder or helix to coil occurs at a specific transition or "melting" temperature (T^). Many researchers have attempted to establish the relationship between T^ and the ionic strength. Ash et al. formulated the following equation [25]: T^ = 126 + 35 log [M] + 15 [Acet] - 20 [Pyr]

(2)

where [M] is the concentration of monovalent cation in moles/liter and [Acet] and [Pyr] are the number of moles of acetate and pyruvate, respectively, per repeating pentasaccharide unit. Seright and Henrici [26] analyzed Holzwarth's work [27] and obtained the following equations: T^ = 122 + 30 log [Na^]

(3)

T^ = 310 + 70 log [Ca^^]

(4)

where [Na^] and [Ca^^] are the concentration of the sodium and calcium ions, respectively, in moles/liter. However, a slightly different correlation was obtained by examining the results of Milas and Rinaudo [21]: T^ = 125 + 43 log [Na^]

(5)

A few studies considered the effect of pH on the viscosity of xanthan solutions. Jeanes et al. observed a rapid increase in the viscosity of xanthan solution at pH 9-11 [28]. Whitcomb and Macosko [29] and Philips et al. [30] found the viscosity of xanthan to be independent of pH. Szabo examined the stability of various EOR polymers in caustic solutions at room temperature, including Kelzan MF (a biopolymer) [6]. He found a fast initial drop in the viscosity of a xanthan solution containing 2 wt% sodium chloride and 5 wt% sodium hydroxide, at 12.5 s~\ which virtually stopped after 10 days. Krumrine and Falcone found that the effect of alkali (sodium silicates) on the viscosity of xanthan solution depended on the concentration of sodium and calcium ions present [31]. Ryles examined the thermal stability of bio-polymers in alkaline conditions [16]. He found that xanthan was totally degraded (in anaerobic conditions) upon the addition of 0.8 wt% sodium hydroxide at temperatures from 50 to 90°C (in a 1 wt% sodium chloride brine). Seright and Henrici observed total biopolymer degradation at pH > 8 and a temperature of 120°C [26]. Vollmert [32], Aspinall [33], and Seright and Henrici [26] investigated the stability of biopolymers in alkaline conditions. Vollmert determined that polymers such as polysaccharides could be hydrolytically degraded with strong bases [32].

620


Seright and Henrici [26] mentioned that it is possible for xanthan to undergo the following base-catalyzed fragmentation: glucose-glucose

> L-glucose + isosaccharinic acid

(6)

Many oil reservoirs contain connate water with high concentrations of sodium chloride and divalent ions, requiring the use of expensive and easily biodegraded xanthan biopolymer. Water-soluble hydrophobically associating polymers (Figure 4) are of great interest because in many cases their viscosity is constant or increases as salinity, divalent ion concentration, or temperature increases [34]. These are water-soluble polymers that contain a small number of hydrophobic groups attached directly to the polymer backbone. In aqueous solutions, the hydrophobic groups of these polymers can associate to minimize their exposure to the solvent, similar to the formation of micelles by a surfactant above its critical micelle concentration. These associations result in an increase in hydrodynamic size that increases solution viscosity. Associating polymers also can produce higher viscosities than comparable concentrations of HPAM or xanthan, and their viscosities have been reported to be relatively stable with increasing temperature [35-37]. Associating polymers can produce high viscosities at lower molecular weight than HPAM, which makes them much less sensitive to shear degradation. Lower molecular weight also minimizes injectivity problems encountered with high molecular weight polymers such as xanthan and polyacrylamide. The potential exists to use associating polymers as mobility control agents in reservoir brine of high salinity and high divalent ion concentration. In addition, their unique flow properties may be advantageous in drilling fluids and in gels for conformance control. Other water-soluble polymers of interest to the oil industry include guar gum, obtained from the seeds of the guar plant, Cyamopsis tetragonolobus. Guar gum has a straight chain of D-mannose units with D-galactose side chains on every other mannose. This gives a mannose to galactose ratio of 2:1 [38]. Cellulose ethers are generally prepared by alkylating purified cellulose solution in the presence of sodium hydroxide. Cellulose is a linear, unbranched polysaccharide made up of anhydroglucose units linked through the p(l,4) positions.

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621

There are three hydroxyl units available for ether formation on each anhydroglucose unit. The degree of substitution (DS) is defined as the average number of ether substituents per anyhydroglucose unit, with a maximum value of three. Molar substitution value (MS) is used when cellulose is reacted with ethylene oxide or similar alkylating reagents and is the moles of reagent reacted per mole of anhydroglucose unit. Ethylene oxide reacts with a hydroxyl group to create an ethoxy ether, which contains a hydroxyl group which can react further. In general, water-soluble cellulose ethers have values of DS = 0.4 - 2.0 [39]. As can be seen, there are many water-soluble polymers for potential use in oilfield applications. Partially hydrolyzed poly aery lamide and xanthan gum are the most commonly employed in EOR applications [40]. Therefore, this review will concentrate on these polymers in addition to hydrophobically associating polymers. This chapter is divided into four parts. In the first part, a description of experimental studies related to water-soluble polymers is given. In the second part, factors affecting the flow behavior of partially hydrolyzed polyacrylamide will be examined. In the third part, the flow behavior of biopolymers (xanthan) will be reviewed. In the last part, rheological properties of hydrophobically associating polymers will be discussed. PART 1: EXPERIMENTAL STUDIES Partially Hydrolyzed Polyacrylamide Nasr-El-Din et al. examined the effect of various chemical species on the flow properties of Alcoflood 1175L, a partially hydrolyzed polyacrylamide [41]. The polymer was supplied by Allied Colloids as a 50% active dispersion, had a degree of hydrolysis, T, of 0.28, and a viscosity average molecular weight of 13 x lO^g/ mole. A stock polymer solution containing 25,000 ppm polymer was first prepared using boiled deionized water. This solution was tumbled for two hours and allowed to stand overnight to ensure full hydration. All polymer solutions were stored in closed containers to minimize oxygen uptake. No biocides or oxygen scavengers were added. Salt and alkali solutions were prepared from reagent grade chemicals. The anionic surfactant used was Neodol 25-3S, a commercial alcohol ethoxy sulfate obtained from Shell Chemical Company as a 60 wt% active solution. The nonionic surfactant examined was Triton X-100, obtained from J.T. Baker Inc. as a 100 wt% active solution. The apparent viscosity of various polymer solutions as a function of the shear rate (y) was measured using a co-axial rotational viscometer (Contraves low-shear 30) at 20°C. The apparent shear rate range that can be obtained with this viscometer is from 0.01 to 130 s ^ This range encompasses the shear rate range of 0.1 - 10 s~^ encountered in a typical reservoir away from the wellbore [7]. The apparent viscosity of dilute polymer solutions can be represented by the power-law model over a wide range of shear rates [3,4]. For such fluids, the shear rate depends on, among other factors, the power-law index. The shear rate for a power-law fluid in a co-axial rotational viscometer (Couette flow) is: Y= 2w/[n(l - S)2/"]

(7)

622


where w is the rotational speed in rad/s, n is the power-law index, and S = Rf^/R^,. R^ is the bob radius, and R^ is the cup radius. The screen factors of various polymer solutions were measured at room temperature following the procedure described by Foshee et aL [42]. Xanthan Gum The rheological properties of xanthan solutions are strong functions of their pyruvate content [43-45]. Nasr-El-Din and Noy examined two xanthan materials: a high pyruvate xanthan (Flocon 4800) and a medium pyruvate xanthan (Statoil XC 44 F4) [46]. Flocon 4800, obtained from Pfizer Inc. as a 13.3 wt% liquid concentrate, contained 6.4 g pyruvic acid/100 g xanthan. Statoil XC 44 F4, obtained from Statoil Inc. as a 2.8 wt% active fermentation broth, contained 4.7 g pyruvic acid/100 g xanthan. The average molecular weight of Flocon 4800 is greater than 1 X lOVmole [8,30] whereas that of the Statoil polymer is 2 to 4 x lOVôle [47]. A stock biopolymer solution containing 4,000 ppm biopolymer was prepared for each biopolymer following the procedure recommended by Pfizer Inc. [30]. All solutions were adjusted such that they contained about 5 g/L formaldehyde to inhibit bacterial degradation. The 4,000 ppm stock solutions were filtered using a Whatman number 1 filter paper and were stored in closed polyethylene containers to minimize oxygen uptake. PART II: PARTIALLY HYDROLYZED POLYACRYLAMIDE It is well known that the rheological properties of partially hydrolyzed polyacrylamide depend on the stresses associated with a given flow field. In a simple shear flow, the apparent viscosity is constant at low shear rates (Newtonian behavior). At a critical shear rate, the apparent viscosity decreases as the shear rate is increased, i.e., a shear thinning behavior [48]. The viscosity shear-rate data of water soluble-polymers are commonly fitted using the Carreau viscosity model [49]. According to this model, the apparent viscosity, |LI, is a function of the shear rate, Y, as follows: li=MJ[\

+(xjyr

(8)

where jLt^ is the low-shear Newtonian viscosity, x^ is a rotational relaxation time (inverse of the critical shear rate). The critical shear rate is the shear rate at which there is a transition from Newtonian to shear-thinning behavior. The exponent m is related to the power-law index n(m = (1 - n)/2). In some cases, e.g., HPAM in deionized water, the low-shear Newtonian behavior could not be measured within the range of shear rates examined by Nasr-El-Din et al. [41]. In these cases, the data were fitted using the power-law model. The apparent viscosity for a power-law fluid is: iLt = k Y"-'

(9)

where k and n are the power-law parameters. For a Newtonian fluid, n = 1, and k is the fluid viscosity.


623

Polymer Viscosity in Deionized Water Figure 5 shows the variation of the apparent viscosity of Alcoflood 1175L in deionized water with the shear rate for various polymer concentrations at 30°C. For all polymer concentrations examined, the apparent viscosity decreased with increasing the shear rate. This trend is due to uncoiling and aligning of polymer chains when exposed to shear forces. At shear rates > 0.1 s ' , the viscosity-shear rate relationship was fitted with the power-law model. At shear rates < 0.1 s ' , the experimental data deviate from the power-law behavior. However, no low-shear Newtonian behavior was observed within the range of shear rates examined. The power-law index, n, was a function of polymer concentration as follows: n = 0.7 - 0.056 In (C ),

125 < Cp < 5,000 ppm

(10)

where C is the polymer concentration in parts per million (ppm). Screen factor is a measure of the visco-elastic properties of polymer solutions and how they behave in porous media [50,51]. Figure 6 shows the screen factor monotonically increases with polymer concentration. According to Unsal, the shear rate encountered in a screen viscometer is nearly 1,000 s ' [52]. At such high shear rates, shear viscosities obtained by extrapolating the data shown in Figure 5 are much lower than those obtained from the screen viscometer. This difference

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relative viscosity with increasing NaCl concentration diminished as NaCl concentration was increased up to 4 wt% and reached almost zero at NaCl concentrations > 6 wt%. Figure 10 shows that the power-law index approached a limiting value closer to unity as NaCl concentration was increased. Similar to the results shown in Figure 9, the power-law index was a strong function of NaCl concentration only at NaCl concentrations < 1 wt%. The deviation from a Newtonian behavior (n = 1) increased as the polymer concentration was increased. The effect of NaCl on the flow curves of polymer solutions can be explained as follows: The chain of HPAM is stretched in deionized water because of the repulsive forces between the negative charges (carboxylate groups) on the chain [53]. This means that the hydrodynamic radius of the polymer chain is large in deionized water and, consequently, polymer solution viscosity is high. As the concentration of the sodium ion in solution is increased, the repulsive forces within the polymer chain decrease, due to charge screening effects, and the chain coils up. This change in the polymer chain conformation causes the hydrodynamic radius of the chain to decrease and the degree of polymer chain entanglement to diminish. Both factors cause the polymer solution viscosity to decrease. Also, the reduction in the polymer chain size, due to the charge shielding, would increase the critical shear rate. Hence, the Newtonian behavior can be seen over a wider range of shear rates as the salt concentration is increased.

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Figures 9 and 10 show that both the low-shear relative viscosity and the power-law index approached limiting values with increasing NaCl concentration. These results indicate that there is a lower limit for the hydrodynamic radius for the polymer chain beyond which all the charges on the polymer chain are completely shielded with the cations. Increasing NaCl concentration further will not change the polymer chain configuration and, as a result, the relative viscosity of the polymer solution remains constant. Figures 9 and 10 also show that the limiting values for the relative viscosity and the power-law index are functions of polymer concentration. Effect of Cation Type on Polymer Viscosity Figures 11 and 12 display the influence of cation type and concentration on the low-shear relative viscosity and the power-law index at a polymer concentration of 1,000 ppm and a temperature of 20°C. At salt concentrations < 1 wt%. Figure 12 shows that the rate of viscosity decline with salt concentration was much higher with the calcium ion. Similar trends were observed by Mungan [4] and French et al. [54]. However, the effect of cation type on the polymer viscosity decreased as salt concentration was increased. These trends can be explained as follows: calcium ion, because of its higher positive charge, is more effective in shielding the negative charges on the polymer chain than sodium

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629

ion. Consequently, the polymer chain coils up in the presence of calcium ion at lower salt concentrations. Once the size of the polymer chain reaches its limiting value, the effect of the cation type diminishes. The presence of calcium ion also affects the power-law index. Figure 12 shows that higher power-law index values were obtained with the calcium ion, i.e., divalent ions enhance Newtonian behavior. It should be mentioned that there was no phase separation due to the presence of the calcium ion [9,55]. This is presumably due to the temperature (20°C) and the degree of hydrolysis of the polymer examined (0.28). The effect of the calcium ion on polymer viscosity shown in Figure 11 has the following implication in designing the viscosity of a chemical slug. For low salinity reservoirs, the viscosity of the chemical slug should be designed to account for any hardness present. Failure to do that may result in a chemical slug having much lower viscosity in the reservoir than the designed value. For high salinity reservoirs, neglecting the effect of divalent ions (hardness) on the slug viscosity is less important. This is assuming no phase separation occurs at the reservoir conditions. Effect of Alkali Type on Polymer Viscosity Many investigators have suggested co-injecting alkali and polymer in a single slug to improve slug injectivity [17] or oil recovery [56-65]. Alkalis can modify the viscosity of HPAM solutions in two ways. First, alkalis provide cations into the polymer solution. These cations can reduce polymer viscosity through the chargeshielding mechanism. Secondly, alkalis can hydrolyze the amide groups on the polymer chain (base hydrolysis). This process can increase polymer solution viscosity due to the electrostatic replusion between the negative charges of the carboxylates groups generated by the hydroysis reaction. Obviously, the net effect of alkalis on the rheological properties of HPAM solutions depends on the extent of these two factors. To examine the effect of alkalis on the viscosity of HPAM, the viscosity of polymer solutions was measured as a function of shear rate at various alkali concentrations. Viscosity measurements were repeated on the same solutions after two weeks (336 h) and four weeks (696 h) from initial mixing. Figure 13 depicts the variation of the low-shear relative viscosity with sodium hydroxide concentration at polymer concentration = 1,000 ppm and a temperature of 20°C. After approximately one hour from initial mixing, the low-shear relative viscosity decreased with sodium hydroxide concentration to a limiting value. This result is similar to the trend previously observed with sodium chloride and is due to the shielding effect of the sodium ion. The influence of sodium hydroxide on the low-shear viscosity measured two weeks (336 h) from initial mixing was more dramatic where higher viscosities were obtained at low alkali concentrations. Low-shear viscosity measurements after four weeks (696 h) were very similar to those obtained after two weeks. The effect of sodium hydroxide on the low-shear viscosity can be explained as follows: HPAM undergoes further hydrolysis in the presence of strong alkalis (base hydrolysis). As the polymer is hydrolyzed, the number of the carboxylate groups (i.e., the number of negative charges) on the polymer chain increases. Consequently, the electrostatic repulsion increases, and the chain size increases. This increase in the polymer chain size enhances the viscosity of the polymer solution in deionized

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hour after initial mixing at a temperature of 20°C. The effect of sodium carbonate on the low-shear relative viscosity was similar to that observed with sodium chloride. To examine the effect of alkali type on viscosity evolution due to polymer hydrolysis, the viscosity of polymer solutions having 1,000 ppm polymer and various alkalis was measured after various time periods from initial mixing. Figure 16 compares the low-shear Newtonian viscosity obtained in the presence of 0.5 wt% sodium hydroxide (pH = 12.8) with that obtained in the presence of 1 wt% sodium carbonate (pH =11). After one hour from initial mixing the viscosity obtained with sodium carbonate was lower than that obtained with sodium hydroxide. This trend is due to higher sodium ions concentration available with the 1 wt% sodium carbonate solution. For both alkalis, the low-shear Newtonian viscosity increased with time. The initial rate of viscosity increase was significantly higher with sodium hydroxide, then it slowed down significantly after 200 hours from initial mixing. In the presence of sodium carbonate, however, the rate was very low, but steady up to 696 h after initial mixing. The results shown in Figure 16 can be explained as follows: The initial rate of hydrolysis is a function of the solution pH, initial degree of hydrolysis and temperature [16,66,67]. At the beginning of the hydrolysis, the reaction rate is relatively high at higher pH. This explains the relatively fast rise in the polymer solution viscosity in the presence of sodium hydroxide. As the hydrolysis reaction

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proceeds, the number of carboxylate groups, i.e., the negative charges, on the polymer chain increases. The presence of high negative charges on the polymer chain hinders the hydroxide ions (OH") from reacting with the polymer, i.e., the reaction between alkali and the polymer is a product-inhibited reaction. As a result of this hindrance, the rate of reaction decreases and the rate of viscosity enhancement due to polymer hydrolysis drops. These trends emphazise the importance of considering polymer hydrolysis when designing an alkali/polymer flood. The results shown in Figures 13 to 16 indicate that strong alkalis further hydrolyze HPAM and this hydrolysis reaction enhances solution viscosity only at lower cation concentrations. They also suggest that the addition of a strong alkali to polyacrylamidebased polymers is beneficial to the solution viscosity. However, from a practical point of view, the increase in the degree of hydrolysis could present a serious problem in the presence of divalent ions. It is well-documented that HPAM precipitates in the presence of divalent ions if its degree of hydrolysis exceeds 0.35, especially at higher temperatures and polymer concentrations [9]. Sodium carbonate on the other hand does not cause fast hydrolysis to the polymer. This in turn minimizes polymer loss due to precipitation with divalent ions, which may cause plugging near the wellbore and injectivity problems. More details regarding the effect of sodium carbonate on the apparent viscosity of HPAM are given by NasrEl-Din and Taylor [68]. Figure 17 depicts the influence of sodium carbonate concentration on the screen factor of polymer solutions having from 500 to 5,000 ppm polymer. For all polymer

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B

.. •

•

T = 2 3 ' 'C

1

1

0

1

2

1

1

1

3

1

1

4

1

1

5

6

Sodium Charbonate Concentration, w t % Figure 17. Effect of sodium carbonate concentration on the screen factor.

634


solutions examined, the screen factor dropped very fast with sodium carbonate concentration then remained constant. The initial drop in the screen factor was dramatic (greater than one order of magnitude) at polymer concentration = 5,000 ppm. These results show that the presence of sodium carbonate affects both the apparent viscosity and the visco-elastic properties of HPAM. These trends indicate that co-injection of sodium carbonate and HPAM improves the injectivity of the polymer solutions. Effect of Surfactants on Polymer Viscosity Surfactant slugs are frequently used in EOR processes to mobilize residual oil by changing rock wettability or by reducing oil/water interfacial tension. To increase the efficiency of such processes, polymers can be either co-injected with the surfactant slug or as a chase. In both cases, surfactant and polymer mixing is to be expected. The effects of Triton X-100 (a nonionic surfactant) and Neodol 25-3S (an anionic surfactant) on the viscosity of HPAM solutions were examined by NasrEl-Din et al. [41]. Figure 18 displays the effect of Triton X-100 concentration on the apparent viscosity of polymer solutions having 1,000 ppm polymer. Increasing Triton X-100 concentration up to 10 wt% did not have a significant effect on the apparent viscosity. These results indicate that Triton X-100 does not interact physically or chemically with the polymer chain in deionized water.

10000 I

1

1

1

1

o o • • >N..

Polymer Concentration == 1000 ppm

qj^^

]

T = 20°c

1000 1:

E o o

[ 1

loot-r

1 1

[

Triton X-100 Concentration, wt.% • 0.0 0.5 ^ • 1.0 o 3.0 A 10.0

10 Lmmmm^mmlma^^miit^^t^ 0.01 0.03 0.1

•

'

•

0.3

1

•

^ ^ ^ ^ ^ ^

^Nk^

A J

-1 '

'

3

10

^^^Î^^^^^^Jj

30

IOC

Shear Rate, s'^ Figure 18. Effect of Triton X-100 on the viscosity-shear rate relationship.


635

Shupe examined the effect of anionic surfactants (petroleum sulfonate) on the viscosity of partially hydrolyzed polyacrylamide (Dow Pusher 500) [69]. The viscosity decreased by 22% as a result of adding the surfactant at 3 wt%. Nasr-ElDin et al examined the influence of Neodol 25-3S on the viscosity of Alcoflood 1175L [41]. Figure 19 depicts the flow curves of 1,000 ppm polymer solutions obtained at various surfactant concentrations (up to 10 wt%). Unlike the results obtained with Triton X-100, Neodol 25-3S had a dramatic effect on the flow curves of the polymer solutions. This effect is similar to that obtained with simple salts. Effect of Surfactants and Alkalis on Polymer Viscosity Similar to alkali slugs, co-injection of polymer with alkali/surfactant slugs reduces the mobility of the slug and increases oil recovery [31,69-76]. The effects of strong alkalis (e.g., sodium hydroxide) and anionic surfactants (e.g., Neodol 25-3S) on the viscosity of Alcoflood 1175L were dramatic. Therefore, it is of interest to examine the effect of adding both species on the flow curve of this polymer. Figure 20 shows the low-shear Newtonian viscosity of polymer solutions having 0.5 wt% Neodol and 1,000 ppm polymer as a function of sodium hydroxide concentration. To avoid any viscosity changes due to polymer hydrolysis, all viscosity measurements were conducted one hour from initial mixing. Similar to the results obtained with sodium hydroxide, the low-shear viscosity decreased with sodium

10000

I

1000

Neodol 25-3S Concentration, wt% • 0.0 • 0.5 • 1.0

I I I r u

CO CO

E ^ CO

100

o o

CO

10

Alcoflood 1175L Polymer Concentration = 1000 ppm T = 20°C I

0.01

I

I, ,1,1 I I I

0.1

I

I

I

I 11111

1

1

I—I—I

11111

10

100

1000

Shear Rate, s'^ Figure 19. Effect of Neodol 25-3S on the viscosity-shear rate relationship.

636

Advances In Engineering Fluid Mechanics 1000

E (0

o o

100

Alcoflood 1175L Neodol 25-3S Concentration = 0.5 w t % Polymer Concentration = 1000 ppm T = 20°C

c CD "E o 0) CO 0>

4C

CO I

o 12

16

Sodium Hydroxide Concentration, w t % Figure 20. Effect of sodium hydroxide concentration on the low-shear Newtonian viscosity of a polymer solution having 0.5 wt% Neodol 25-3S and 1,000 ppm polymer.

hydroxide concentration. However, the viscosity at a sodium hydroxide concentration = 10 wt% was unexpectedly high. The viscosity decreased with further increasing sodium hydroxide concentration. To investigate the viscosity enhancement observed at higher sodium hydroxide concentrations, a series of viscosity measurements was conducted to examine the effect of sodium hydroxide concentration on the flow curves of surfactant solutions having 0.5 and 1 wt% Neodol 25-3S. Figure 21 shows that the low-shear Newtonian viscosity of 0.5 wt% surfactant solutions slightly increased as sodium hydroxide concentration was increased up to 7 wt%. The viscosity dramatically increased up to 53 mPa.s at sodium hydroxide concentration of 8 wt%, then decreased with further increasing sodium hydroxide concentration. A similar trend was observed at 1 wt% surfactant concentration. However, the viscosity enhancement started at sodium chloride concentration of 6 wt% and the maximum viscosity was approximately 210 mPa«s. These results indicated that the viscosity of anionic surfactant solutions significantly increased over a narrow range of alkali concentrations and the viscosity enhancement was a function of surfactant concentration. The increase in anionic surfactant solution viscosity is due to the formation of aggregates, i.e., liquid crystals [77]. This viscosity enhancement is the reason for the unexpected increase in the low-shear Newtonian viscosity observed with polymer solutions having alkali concentrations > 10 wt% and Neodol 25-3S concentration of 0.5 wt%.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery 250

(0

I • I ' I ' I Neodol 25-3S Concentration, wt%

T = 20 C

OL

637

E

•

0.5

°

1.0

o

o 0)

c (0

"E o o

100

(D 50

o SgSm>§BU£SSJ|^

9^mam 10

12

Sodium Hydroxide Concentration, wt% Figure 21. Effect of sodiunn hydroxide on the low-shear Newtonian viscosity of surfactant solutions containing 0.5 and 1 wt% Neodol 25-3S.

More details regarding the effect of liquid crystal formation on the apparent viscosity of Neodol 25-3S are given by Nasr-El-Din et at. [78]. PART III: XANTHAN GUM Effect of Polymer Concentration on Apparent Viscosity Figure 22 shows the effect of polymer concentration on the flow curves of Statoil polymer in deionized water. At polymer concentrations < 2,000 ppm, the apparent viscosity was constant at low shear rates (Newtonian behavior) and decreased at higher shear rates. The Carreau model. Equation 8, predicts the experimental data for this polymer concentration range fairly well. At polymer concentrations > 2,000 ppm, the flow curves showed a shear thinning behavior only. The power-law model. Equation 9, predicts the data fairly well at shear rates > 1 s~'. It is interesting to note that the effect of polymer concentration on the polymer flow curves diminished at higher shear rates. This is due to the rod-like shape of the polymer chain [79]. At low shear rates, the polymer chains are not aligned in the flow direction. Consequently, increasing polymer concentration at low shear rates will significantly increase the resistance to flow. However, at high shear rates the polymer chains are more aligned in the flow direction and the resistance to flow is

638


1 0 0 0 0 0 ^îii¥"^^"^^¥Wfi

Statoil Polynner Polymer Concentration, ppm • 500 n 1000 • 2000 o 4000

10000 (0 (0 ^

1000

(0

S

100

aaaBuuuuuug

0)

10

0.01

1000

Shear R a t e , s ^ Figure 22. Flow curves of Statoil polynner in deionlzed water.

not great. Increasing polymer concentration at high shear rates would not significantly enhance the viscosity of the polymer solution. The power-law index, n, is a function of polymer concentration, C , as follows: Statoil polymer: n = 1.549 - 0.151 In (C^), 500 < C^ < 4,000 ppm

(11)

Flocon 4800: n = 2.215 - 0.24 In (C^), 500 < C^ < 4,000 ppm

(12)

Effect of Polymer Concentration on Screen Factor Figure 23 shows that the screen factor monotonically increased with polymer concentration for both xanthan materials. However, the Statoil polymer showed higher screen factors, especially at higher polymer concentrations. At high shear rates, shear viscosities obtained by extrapolating the data shown in Figure 22 are slightly lower than those obtained from the screen viscometer shown in Figure 23. This trend indicates that the elastic properties of xanthan gum are not as significant as those of HPAM.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery 2 5 |

20

•

I

• a

I

I

•

I

I

I

639

I

Statoil Polymer Flocon 4800

o (0

c 0)

o

1000

2000

3000

4000

5000

Polymer Concentration, ppm Figure 23. Variation of the screen factor with polymer concentration for Flocon 4800 and Statoil polymer solutions.

Effect of Sodium Chloride on the Apparent Viscosity of Biopolymers Figure 24 depicts the influence of sodium chloride concentration (up to 10 wt%) on the flow curves of Flocon 4800 solutions having 2,000 ppm polymer. The effect of sodium chloride depended on the shear rate. At shear rates < 0.1 s"^ the apparent viscosity decreased as the sodium chloride concentration was increased up to 1 wt%, then showed a gradual increase with further addition of sodium chloride. The gradual increase in viscosity with sodium chloride concentration is due to the increase in solvent viscosity with sodium chloride concentration. At shear rates > 10 s~\ the apparent viscosity was independent of sodium chloride concentration. The Statoil polymer has a lower pyruvate content than that of Flocon 4800. It is of interest to examine the effect of sodium chloride on the flow curves of this polymer. Figure 25 illustrates the flow curves of polymer solutions having 2,000 ppm Statoil polymer and sodium chloride concentrations of 0, 4, and 10 wt%. Similar to the trends observed with Flocon 4800, the effect of sodium chloride was observed at low shear rates only. However, the apparent viscosity at low shear rates for polymer solutions containing 4 and 10 wt% sodium chloride was higher than that at 0 wt% sodium chloride. This result is due to the lower ionic character (lower pyruvate content) of the Statoil polymer. The effect of sodium chloride on the viscosity of xanthan solutions shown in Figures 24 and 25 can be explained as follows: In deionized water, the xanthan

640

Advances in Engineering Fluid Mechanics 1000 1^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ - S - ^ ^ F " ^ ^ ^ 1

NaCI Concentration, wt%

]

0.0

]

1.0 4.0

1 1

10.0

1

Bssssos:-.. (D Q.

E >

•:* •

1 F

100 L

0)

* • • •

o u 0)

1 1 I

Polymer Concentration = 2000 ppm Flocon 4800

•••

•

1

i

•

J 1 •

10

0.01

0.1

1

100

10

1000

Shear Rate, s '^ Figure 24. Effect of sodium chloride concentration on the flow curves of polymer solutions having 2,000 ppm Flocon 4800. 10000 p

1

NaCI Concentration, wt%

1

0.0 4.0 10.0

J

0)

(0

1000

QL

E f

o o (0

100

1 I 1 1 t1: F 1 r[ 1

J

* ^

^^

* A^

«

«^ «

-1

««

Statoil Polymer Polymer Concentration = 2000 ppm

10 ^^m^mmkm^^ÛiJt^m^^tm^l^mhÛiJbm^m^mitmdi^îkMlî^^^m^ 0.01 0.1 10

«

1 •a *4

1 1 M^îiMhûJ

100

1000

Shear Rate, s^ Figure 25. Effect of sodium chloride concentration on the flow curves of Statoil polymer solutions having 2,000 ppm polymer.


641

chains are stretched because of the repulsive forces between the negative charges on the side chains. Also, the polymer is present in the disordered (coil) form [19]. This means that the hydrodynamic radius of the xanthan chains is large and, consequently, the viscosity of xanthan solutions is high. As the concentration of the sodium ion in solution is increased, the repulsive forces within the polymer chains decrease, due to the charge screening effects. As well, by increasing the sodium ion concentration, xanthan chains will exist in the ordered, helical form [19]. As a result of these changes in the xanthan conformation, the hydrodynamic radius of the polymer chain becomes smaller and the viscosity of the polymer solution decreases. Based on this discussion, the effect of sodium chloride on the viscosity of xanthan gum solutions containing < 2,000 ppm polymer is a function of the pyruvate content of the xanthan gum. The higher the pyruvate content of the xanthan, the more salt-sensitive its viscosity will be. Many researchers have reported a significant viscosity enhancement when sodium chloride was added to xanthan solutions having polymer concentrations > 4,000 ppm [19,24,28,80]. This effect was explained in terms of the association of polymer chains having collapsed side chains. To investigate this point further, the flow curves of polymer solutions containing 10,000 ppm polymer and various sodium chloride concentrations were measured. Figure 26 shows that the apparent viscosity of Flocon

10,000 [

(0

(Q Q.

1,000

Sodium Chloride Concentration, w t % 1

I

•

0.0

1

o

1.0

•

3.0

I 1

%

• ^

5.0

*

1

•

(0

o o

(0

•

100 1

Flocon 4800

•

^ •

[ Polymer Concentration = 10000 ppm

10

••

L l H H ^ H l k H M i i r i H H t a i l i ^ H t e ^ ^ r i j M H i a H B riHMMlrtMIHhHiHlHhllriiik

10

o

1

^^^^^^^^

100

1,000

Shear Rate, s ^ Figure 26. Effect of sodium chloride concentration on the flow curves of polymer solutions having 10,000 ppm Flocon 4800 polymer.

642


4800 significantly increased upon the addition of 1 wt% sodium chloride. Increasing sodium chloride concentration up to 5 wt% caused a slight increase in the apparent viscosity. Figure 27 is similar to Figure 26, but for the Statoil polymer. The effect of sodium chloride on the polymer flow curves was similar to that observed at lower polymer concentrations. There was no increase in the apparent viscosity as a result of adding sodium chloride, other than that expected from the solvent viscosity at sodium chloride concentrations of 3 and 5 wt%. Based on this discussion, the effect of sodium chloride on the apparent viscosity of xanthan solutions depends on polymer concentration and the pyruvate content of xanthan. At polymer concentrations less than 4,000 ppm (the range of interest in enhanced oil recovery processes), the apparent viscosity of the high pyruvate polymer showed more sensitivity to sodium chloride. This behavior is due to charge screening effects. At high polymer concentrations (= 1 wt%), only the high pyruvate polymer showed a dramatic increase in the apparent viscosity upon the addition of sodium chloride. These trends are in agreement with the results obtained by Smith et al. [44,81] and Cheetham and Norma [83] and are due to the association of polymer molecules having collapsed side chains. Effect of Cation Type on Polymer Viscosity Figure 28 displays the low-shear relative viscosity as a function of salt (sodium or calcium chloride) concentration. The low-shear relative viscosity dropped from IU,UUU

' ' ' '1

1

i

Statoil Polymer ^

Polymer Concentration = 10000 ppm J

f

0)

(d CL E >

1,000 [-

o o

(0

>

1 '#

100

:

•

0.0

o

1.0 3.0

• D

1

#

\ Sodiunn Chloride Concentration, w t %

•*

]

\

# ^

\

'* ^

J '#

5.0

] 1 ]

in 1

10

100

1,000

Shear Rate, s ^ Figure 27. Effect of sodium chloride concentration on the flow curves of polymer solutions having 10,000 ppm Statoil polymer.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery 1

O

0}

400 0)

-2

1

Flocon 4 8 0 0 Polymer Concentration == 2 0 0 0 ppm

500

o

1

1^,

643

-^

-] •

300

—•—

—

—• J

0) (0 0)

200

-

100

-

-1

I

o

1

° CaCI^

-1

—^^^^^^^^^^^^^^^^

n1 0

• NaCI

^ ^ ^ ^ ^ 1

4

8

12

1€

Salt Concentration, w t % Figure 28. Effect of cation type on the low-shear relative viscosity of Flocon 4800 solutions having 2,000 ppm polymer. 410 in deionized water to 350 at 1 wt% sodium chloride, then remained almost constant. For the case of calcium chloride, the relative viscosity dropped to 300 at 1 wt% calcium chloride, then gradually decreased with increasing calcium chloride concentration. These results are in agreement with those obtained by Unsal [52] and Philips et al. [30] and are due to the high charge density of the calcium ion. Calcium ion is more detrimental to the viscosity of xanthan solutions than sodium ion. This is especially true for Flocon 4800 solutions. The high pyruvate content of Flocon 4800 gives the polymer chains more ionic character (negative charges). This makes the polymer more sensitive to salts, especially divalent cations. Effect of Alkali Type on Polymer Viscosity Alkalis can modify the viscosity of xanthan solutions in many ways. First, alkalis provide cations into the polymer solution. These cations can reduce the viscosity of polymer solutions through the charge-shielding mechanism explained earlier. Secondly, the acetyl groups in the side chains of xanthan can be removed by strong alkalis [83]. The elimination of the acetyl groups may have no effect on viscosity [84] or may increase the viscosity of xanthan solutions [28]. Finally, alkalis can hydrolyze the xanthan backbone, which can be very detrimental to the solution viscosity. Figure 29 depicts the effect of sodium hydroxide concentration (up to 10 wt%) on the flow curves of polymer solutions having 3,000 ppm Flocon 4800 at 20°C. The effect of sodium hydroxide on the polymer flow curve depended on the shear

644

Advances in Engineering Fluid Mechanics 10000 I

NaOH Concentration, wt%

(0 (0 Q.

1000

I-

D D ^ D D D Q

••22° • • • .n

E

• a • o

•• •

O O O O O Q O O O O

0.0 1.0 4.0 10.0

ii?

0)

o o

(0

°°°:%

100

Flocon 4800 Polymer Concentration = 3000 ppm 10 0.01

•ta^riiriiMLi

0.1

Mill

I

Ill

10

4^ ii 1 s', the apparent viscosity was less dependent on the alkali concentration. At shear rates > 30 s', the apparent viscosity was independent of sodium hydroxide concentration. It is also interesting to note that at 0 wt% sodium hydroxide, no Newtonian portion could be measured. By adding sodium hydroxide, a Newtonian portion appeared, which increased with sodium hydroxide concentration. The effect of sodium hydroxide concentration on xanthan flow curves also was examined at various polymer concentrations from 500 to 3,000 ppm. Figure 30 shows the low-shear relative viscosity as a function of sodium hydroxide concentration. Increasing sodium hydroxide concentration up to 10 wt% caused a dramatic drop in the low-shear relative viscosity (up to 90%). Most of this drop occurred during the addition of the first 1 wt% sodium hydroxide. Increasing sodium hydroxide further caused only a gradual decrease in the low-shear relative viscosity. This gradual drop was very noticeable at a polymer concentration of 3,000 ppm. Figure 31 displays the influence of sodium hydroxide concentration on the power-law index for each of the polymer solutions examined in Figure 30. At a given polymer concentration, the power-law index significantly increased as sodium hydroxide concentration was increased to 1 wt%. The rate of change of the power-law index with sodium hydroxide concentration was greatly reduced at sodium hydroxide concentrations greater than 1 wt%. These results indicate, considering the rod-like shape of xanthan chains, that the hydrodynamic radius of the polymer is significantly


645

10000

0)

o u

1000

r ——^

> JS Q

Flocon 4 8 0 0 Polymer Concentration, ppm o 500 • 1000 D 2000 3000 — -*—^^^ • 30

"0"' 100

kl

(0

in

10

o

10

12

Sodium Hydroxide Concentration, wt% Figure 30. Effect of sodium hydroxide concentration on the low-shear relative viscosity of Flocon 4800 solutions. '

I.U j

1

'

1

'

1

'

1

'

1

'

0 ^

U

y

1 1

•.•...•••

-1

0.8

1

jr 0.6 CO

r

0.4

—

-«

•

"^

o a.

Flocon 4 8 0 0

1

Polymer Concentration, ppm 0.2 i

\mmmmmmihmmmm mJmm,

0

2

1

1 4

f

1 6

1

1 8

1

o

500

J

• a •

1000 2000 3000

1 1 I

1

i 10

1

Sodium Hydroxide Concentration, w t % Figure 31. Effect of sodium hydroxide concentration on the power-law Index of Flocon 4800 solutions.

1 12

646


reduced when sodium hydroxide is present. Also, the power-law index was always less than unity, which means that the polymer solutions still behave as shear thinning fluids. In other words, sodium hydroxide did not cause complete destruction of the polymer chains at these conditions. The effect of sodium hydroxide on the xanthan flow curves is more than that expected from the charge shielding mechanism observed with sodium chloride. One possible explanation of this effect is base-catalyzed fragmentation reactions [26,32]. Fragmentation reactions break the biopolymer backbone (cellulose-like structure) to smaller saccharide units. Consequently, the hydrodynamic radius of the biopolymer would decrease and the viscosity of the polymer solution would diminish. The effect of sodium hydroxide on the viscosity of xanthan solutions is significant. It is of interest to examine the rate of change of the apparent viscosity with respect to time as a result of the base-catalyzed reactions. To achieve this goal, the flow curves of polymer solutions containing 2000 ppm Flocon 4800, 0.5 wt% and 8 wt% sodium hydroxide were frequently measured over a period of eight weeks. Figure 32 displays the variation of the low-shear relative viscosity as a function of the time elapsed after initial mixing, respectively. After a half hour of the initial mixing, the low-shear relative viscosity of 2,000 ppm Flocon 4800 solution dropped from 410 in deionized water to 180 in 0.5 wt% sodium hydroxide, then remained constant.

200

'

1

'

1

'

1

'

1

'

1

•

1

1 IV • •

(0

o

150

•

M

1 NaOH Concentration, wt%

0)

JS

• D

100

(0 0)

D

50

o

1

1-

' U"

'-'

0.5 8.0

H

Q

u

1

-]

Flocon 4800 Polymer Concentration == 2000 ppm

[ 1

1

200

1

1

400

1

'

600

•

j

1

•

800

•

'

1000

*

1200

Time, h Figure 32. Variation of the low-shear relative viscosity with time for Flocon 4800 solutions containing 0.5 wt.% and 8 wt% sodium hydroxide.


647

These results indicate that the base-catalyzed reaction was almost instantaneous. Also, the reaction did not cause complete loss of viscosity, especially at the lower sodium hydroxide concentration. The effects of sodium carbonate, sodium orthosilicate, and sodium hydroxide on the flow curves of polymer solutions having Flocon 4800 were examined at 20°C by Nasr-El-Din and Noy [46]. Figure 33 depicts the effect of alkali type on the low-shear relative viscosity of polymer solutions having 2,000 ppm Flocon 4800. The effect of alkali type (strong and buffered) on the low-shear relative viscosity is dramatic. With the addition of strong alkalis (sodium hydroxide or sodium orthosilicate), the viscosity dropped sharply up to an alkali concentration of 1 wt%, then decreased at a lower rate with increasing alkali concentration. With the addition of a buffered alkali (sodium carbonate), a smaller initial drop in viscosity occurred upon the addition of 1 wt% alkali, then decreased at a much slower rate with increasing alkali concentration. It is important to note that the drop in viscosity observed with any alkali is much greater than that observed with sodium chloride. The most important aspect of Figure 33 is the dramatic effect of strong alkalis on the viscosity of xanthan solutions. Sodium carbonate, on the other hand, is less detrimental to the viscosity of xanthan solutions. Therefore, in alkali/polymer or alkali/surfactant/polymer processes whereby xanthan gum is used, it would be extremely beneficial to use a buffered alkali rather than a strong alkali.

500

(0

o o

1

1

'

1

1

^^^^^^^^^^^^^

Flocon 4 8 0 0 Polymer Concentration = 2 0 0 0 ppm

k 400 F

O

• 0

A

300

Na^COa NaOH

I 1

Na4Si04

H

_2 0)

cc

200

1 ^

(D

(0 I

* A

a>

1 •

^*^^'^*»^,,^^^

]

• A

100

4

o 1

1

1

1

4

1

•

6

•MM^

8

\

>

10

Alkali Concentration, w t % Figure 33. Effect of alkali type on the low-shear relative viscosity of Flocon 4800 solutions having 2,000 ppm polymer.

1 12

648


Effect of Surfactants on Polymer Viscosity Figure 34 displays the effect of Triton X-100 on the flow curves of polymer solutions having 2,000 ppm Statoil polymer. The influence of up to 10 wt% Triton X-100 on the flow curves of Statoil polymer was not significant. These results suggest that Triton X-100 (a nonionic species) does not interact physically or chemically with the polymer chain in deionized water. Figure 35 is similar to Figure 34, but the flow curves were measured as a function of Neodol 25-3S concentration. Unlike the trend observed with Triton X-100, Figure 35 shows that the apparent viscosity of Statoil polymer dropped as the surfactant concentration was increased to 1 wt%, then monotonically increased with further increase in the surfactant concentration. This behavior is very similar to that observed with simple salts. Combined Effect of Surfactants and Alkalis on Polymer Viscosity Figure 36 depicts the flow curves of polymer solutions having 2,000 ppm Flocon 4800, 0.5 wt% Neodol 25-3S, and various sodium hydroxide concentrations. To avoid any viscosity changes due to polymer hydrolysis, all viscosity measurements were conducted one hour from initial mixing. The apparent viscosity at low shear rates significantly decreased upon the addition of 1 wt% sodium hydroxide. Increasing sodium hydroxide concentration up to 4 wt% did not cause any change to the lUUUU

m^mi^m

Triton X-100 Concentration, wt.% 0.0

•

10.0

It

tn

(6

•

1

1000

r

' • l .

-J

Q.

E >

• t

•

*^ 'w o u >

• ^

J

100

• - . . . Statoil Polymer

•%

Polymer Concentration = 2 0 0 0 ppm

% • ^ •

i n 1•MIHriMiil 0.01

riHi^HteriiiitawJ!

0.1

1

10

100

1000

Shear Rate, s'^ Figure 34, Effect of Triton X-100 on the flow curves of polymer solutions having 2,000 ppm Statoil polymer.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery 10000 1

î^Wiip"i""iiiîiiiiiiî^"l*¥iW

m^mm^mtfmi^mfmmiif^â^m^^m^mfmtiifm^'^'^^'^^î^liîiilimj^

mmS88S

1000

649

Neodol 25-3S Concentration, wt% • 0.0 n 4.0 • 10.0

it l | ,

E >

100

"S5 o o 0>

10

Statoil Polymer Polymer Concentration = 2000 ppm

1 ^^^^^^^WUiU

10

0.1

0.01

100

1000

Shear Rate, s ^ Figure 35. Effect of Neodol 25-3S on the flow curves of polymer solutions having 2,000 ppm Statoil polymer. 1000

1

1

1 I 1lll'l

1

1

[

•

• ^^^^^^^^^^^^^^^^^^^^^1

NaOH Concentration, wt%

[

•

1 1 1

10.0

]

••. (0

••• >

100

• i

h I

O O O O O O O O O O ^ O O Q

•

••

*

•

—1

•

]

(0

o o

f

\

1.0 4.1 8.0

•

o •

°'-

°«i

[ 1

Flocon 4800 Polymer Concentration = 2000 ppm

1

Neodol 25-3S Concentration = 0.5 wt%

] J

•

•j

fl

10

0.01

0.1

1

10

100

Shear Rate, s ^ Figure 36. Effect of sodium hydroxide on the flow curves of polymer solutions having 0.5 wt% Neodol 25-3S and 2,000 ppm Flocon 4800.

1000

650


polymer flow curve. However, the apparent viscosity at sodium hydroxide concentration = 8 wt% was unexpectedly high at all shear rates examined. The apparent viscosity dramatically dropped with further increasing sodium hydroxide concentration to 10 wt%. PART IV: RHEOLOGY OF ASSOCIATING POLYMERS As discussed in Parts II and III, HPAM and xanthan polymers can be used under certain conditions of salinity, hardness, and temperature. Obviously, these conditions will limit the number of reservoirs where such polymers can be used. It is important to note that high molecular weight polymers increase the viscosity of the aqueous phase mainly because of the large hydrodynamic volume of their molecules. The hydrodynamic volume increases as the molecular weight of the polymer increases and, as a result, the viscosity of the aqueous solution becomes higher. One major problem with polymers having high molecular weights is that they can plug the formation and cause severe injectivity problems, especially in tight formations. Another way to increase the viscosity of the aqueous phase without employing high molecular weight polymers is to use water-soluble associating polymers. Here, a small amount of hydrophobic groups, typically less than 5 mole% [85-88], is incorporated into a water-soluble polymer. The hydrophobic groups will associate under certain conditions, and the hydrodynamic volume of the polymer molecules will increase. This in turn will significantly enhance the viscosity of the polymer solutions, sometimes by several orders of magnitude [89]. The solubility of water-soluble associating polymers decreases as the hydrophobe content increases [90]. As molecular weight of the polymer increases, or hydrophobe chain length increases, the amount of hydrophobe required to malce the polymer insoluble decreases. Obviously, this will limit the maximum hydrophobe content that can be introduced into an associating polymer. One way to increase the solubility of associating polymers in water is to introduce ionic character on the polymer backbone [90]. Such ionic character can be obtained by hydrolyzing some of the amide groups to carboxylate groups [88,89] or by copolymerizing acrylamide with sulfonate-containing monomers [36,37,91-95]. It should be mentioned that the introduction of ionic groups to the polymer backbone will modify the rheological properties of the associating polymers as will be discussed later. Although there are many associating polymers with very interesting flow properties [96], the present review will concentrate on associating polymers with polyacrylamide as the backbone. Polymers based on polyacrylamide are inexpensive and are well-known to scientists and engineers working in improved oil recovery processes. The introduction of hydrophobic groups into a water-soluble polymer will modify the flow behavior of the precursor polymer. This is mainly due to intramolecular association, intermolecular association, or both [90]. The net effect of these associations depends on, among other factors, polymer concentration. Rheological properties of associating polymers depend on several factors, including the total molecular weight, hydrophobe type, degree of incorporation of hydrophobe, and distribution of hydrophobe. Rheological properties of associating polymers have been briefly reviewed [90]. In the following sections, the effect of


651

various parameters on the apparent viscosity of hydrophobically modified polyacrylamide polymers will be discussed. Figure 37 shows a typical plot [97] of the reduced viscosity vs. polymer concentration of an associating polymer and an otherwise identical nonassociating polymer. There is a critical concentration above which the associating polymer shows enhanced viscosity. This critical concentration also is known as the overlap concentration, or the critical aggregation concentration, c*. The critical concentration of nonassociating polymers has been discussed in detail [98-99]. The viscosity enhancement at c* is mainly due to intermolecular association. Below c*, the introduction of hydrophobic groups results in a slight decrease in the reduced viscosity. This reduction is due to intramolecular association, which also reduces intrinsic viscosity and leads to an increase in the Huggins constant [100]. More explanation of intramolecular and intermolecular association will be given in the next sections. The most important aspect of Figure 37 is that the effect of the hydrophobic groups depends on polymer concentration. For this reason, it is meaningful to examine the viscosity of associating polymers in two concentration regimes: a dilute regime, where polymer concentration is less than the critical overlap concentration, and a semi-dilute regime, where polymer concentration is higher than the overlap

15. Associating Polymer

O

o

Non Associating Polymer

1-0

0-0

Poiymer Concentration, g/dl Figure 37. Reduced viscosity-concentration plot for a typical associating polymer vs.a nonassociating polymer. C* is the overlap concentration [97].

652


concentration. In addition, the effect of various chemical species on the flow behavior of these polymers will be examined. Viscosity of Associating Polymers in the Dilute Regime Intrinsic Viscosity and Muggins Constant The intrinsic viscosity, [r|], and Huggins constant, k, can be used to determine the molecular weight of the polymer and to assess the degree of hydrophobic interactions [89]. Therefore, it is useful to discuss these two parameters before examining the rheological properties of associating polymers. It is known that for dilute polymer solutions and according to the Flory-Huggins equation, the reduced viscosity is a linear function of polymer concentration as follows: ^ I - ^ = [Tl] + k[Tl]^C

(13)

where c is polymer concentration in g/dL and T|^ is the solvent viscosity. The intrinsic viscosity and Huggins constant can be obtained by measuring the viscosity of polymer solutions having low polymer concentrations. It is important to note that these viscosity measurements should be conducted at a low shear rate to ensure that the solution viscosity is independent of shear rate. The intrinsic viscosity and Huggins constant can be determined by fitting the experimental data using Equation 13. The intrinsic viscosity generally decreases and the Huggins constant increases as the hydrophobe content is increased at constant molecular weight [93]. The Huggins constant is a very important measure of polymer-solvent and polymer-polymer interactions [89]. For random coil polymers, k is in the range 0.3 to 0.8. The intrinsic viscosity is related to the polymer weight average molecular weight, M^, through the Mark-Houwink-Sakurada equation [89,90]: [Tl] = K[MJ«

(14)

where K and a are characteristics for a polymer chain under specific conditions of solvency and temperature [45]. Effect of Hydrophobe Content The introduction of hydrophobic groups will affect the intrinsic viscosity and the Huggins constant. Bock et al. prepared copolymers of N-octylacrylamide and acrylamide using micellar copolymerization [89]. The prepared copolymers were nonionic, had a molecular weight of 3 x 10^ g/mole, and contained a hydrophobe content of 0, 0.75 and 1 mol%, respectively. The intrinsic viscosity of these polymers decreased as the hydrophobe content was increased. This is mainly due to intramolecular association that leads to the contraction of the polymer chain. On the other hand, Huggins constant increased with the hydrophobe content such that


653

Huggins constant at 1 mol% hydrophobe was significantly higher than the common value of 0.3 to 0.8 for random coil polymers. Therefore, the Huggins constant can be used as a measure of the hydrophobic interactions, and a value greater than 0.8 indicates association. Flynn and Goodwin prepared copolymers of acrylamide and dodecyl methacrylate using micellar copolymerization [96]. The hydrophobe content was varied from zero (polyacrylamide) to 0.7 mol%. Polymer solutions contained O.IM NaCl, and sodium azide was added as a biocide. Figure 38 depicts the variation of the reduced viscosity with polymer concentration at various hydrophobe contents. At 0 mol% hydrophobe, the reduced viscosity increased linearly with polymer concentration indicating no hydrophobic association. At 0.2 mol% hydrophobe, the reduced viscosity increased linearly with polymer concentration, but at a polymer concentration of 600 ppm there was an upward variation in the reduced viscosity. Similar results were obtained at hydrophobe contents of 0.4 and 0.7 mol%. However, the polymer concentration at which the upward variation occurred decreased with increasing hydrophobe content. The most important aspect of Figure 38 is that there is a minimum hydrophobe content below which the amount of association will not be sufficient to increase viscosity. Also, hydrophobic association significantly increases the viscosity of the polymer solutions. Effect of Hydrolysis To examine the effect of introducing ionic character to associating polymers, Bock et al. prepared two sets of associating polymers, each containing polymers of the same molecular weight and hydrophobe level [93]. However, one set of polymers was hydrolyzed to a degree of hydrolysis of 18%. Figure 39 depicts the variation of the intrinsic viscosity and Huggins constant with the hydrophobe content for the two sets of polymers. Figure 39a shows that the intrinsic viscosity of the hydrolyzed polymers is higher than that of the unhydrolyzed polymers. By introducing ionic character into the polymer, the hydrodynamic volume of the polymer chain increases because of the electrostatic repulsion between the negative charges of the carboxylate groups. The intrinsic viscosity decreases for both hydrolyzed and unhydrolyzed polymers with increasing hydrophobe content. By increasing the hydrophobe content, the intramolecular association increases. As a result, the polymer chains coil up and the hydrodynamic volume decreases. An important aspect of Figure 39a is that ionic character and hydrophobic interactions have opposite effects on intrinsic viscosity. In the dilute regime, hydrolysis of the associating polymer increases its intrinsic viscosity, whereas increasing the hydrophobic content reduces its intrinsic viscosity. Figure 39b shows that the Huggins constant increases with the hydrophobe content. However, the Huggins constant for the hydrolyzed polymer is lower. The electrostatic repulsion opens the polymer chain up. This in turn improves the polymersolvent interaction that is marked by low values of the Huggins constant. (text continued on page 656)

654


0^ polymer

OA 0,6 concaniration (g/IOOcm^)

0,S

Figure 38. Reduced viscosity by capillary viscometer, acrylannlde/dodecyl methacrylate copolynners. Hydrophobe (mole%): 0.67 (A), 0.36 (O), 0.22 (D), and 0.0 (A) [96].


655

I2f

0

0.2

0.4

0.6

0.8

1

1.2

1.4

HYDROPHOBE LEVEL, m.%

0

0.2

0.4

0.6

0.8

J

1.2

1.4

HYDROPHOBE LEVEL, m.% Figure 39. Effect of hydrophobe content on intrinsic viscosity and Muggins Constant in 2.0 mass% NaCI. Hydrophobe monomer is N-n-octylacrylamide. Hydrolysis level of HRAM polymers is 18 mol% [93].

656


(text continued from page 653) Viscosity of Associating Polymers in the Semi-Dilute Regime Effect of Polymer Concentration The effect of hydrophobic association on viscosity in the semi-dilute regime is different from that observed at low polymer concentrations. Figure 40, from Bock et al. [89], shows the variation of the reduced viscosity with polymer concentration for polyacrylamide and N-octylacrylamide copolymers having hydrophobe contents of 0.75 and 1 mol%. At a hydrophobe content of 0.75 mol%, the viscosity significantly increased because of intermolecular association. Increasing hydrophobe content further to 1 mol% resulted in higher viscosities. The results shown in Figure 40 indicate that a small amount of the hydrophobe is required to enhance the viscosity by orders of magnitude. Also, very high viscosities can be obtained using relatively low polymer concentrations. Effect of Polymer Molecular Weight Figure 41 shows the variation of the apparent viscosity at 1.3 s~' as a function of polymer concentration for three associating polymers having a hydrophobe content of 1 mol% [93]. The three N-octylacrylamide/acrylamide copolymers had a degree of hydrolysis of 18%, intrinsic viscosities of 2.0, 7.6, and 8.4 dL/g, respectively.

10*

SolvMt 2S Naa

1.0 RLS

^103

C,AM

T t m p : 25 C

0 7 5 nuS C,AM

m

I 102 > O 3 O

10

it 0.0

NoHyc*opho6«

J-

0.2 0.4 0.6 Concentration, g/dl

0.8

Figure 40. Effect of hydrophobe level on Polymer Solution Viscosity. Acrylamide/N-octyl acrylamide copolymer [89].


657

O S I -r

I

Y

cP 0.01

0,1

1

POLYMEP CONCENTHATION, g/dl Figure 41. Effect of molecular weight on the concentration dependence of viscosity at 1.3 S"^ in 2.0 mass% NaCI. Hydrophobe monomer is 1.0 mol% N-n-octylacrylamide. Hydrolysis level is 18 mol%. Intrinsic viscosity (dL/g): *, 2.0; +, 7.6; x, 8.4 [93].

and were prepared in 2 wt% sodium chloride solution. At a given polymer concentration, increasing the intrinsic viscosity (therefore molecular weight) resulted in higher viscosity. This trend is similar to that observed for nonassociating polymers. Effect of Hydrophobe

Type and Content

Bock et al. examined the effect of hydrophobe content and structure on the viscosity of associating polymers [93]. Figure 42 depicts the variation of the apparent viscosity at 1.3 s~^ with the hydrophobe content. For a given hydrophobe type, increasing the hydrophobe content resulted in higher viscosity. Introducing a phenyl group in the hydrophobe monomer significantly enhanced the viscosity, especially at high hydrophobe contents. Effect of Shear Rate The flow curves of polymer will change because of hydrophobic association. Figure 43 shows the flow curves of 0.75 mol% N octylacrylamide/acrylamide copolymer. At polymer concentrations greater than 3,000 ppm the apparent viscosity is constant at low shear rate, then increases with shear rate (shear thickening) up to a maximum, and finally decreases with increasing shear rate (shear thinning). This unique and complex behavior is due to shifting the relative amount of inter and intramolecular association with shear rate [89]. One possible explanation for

658


^—4—butylphenyi

a. u

10 n-CSAM

CO

O

o

on >

0.0

0.5

1.0

1.5

HYDROPHOBE CONTENT. % Figure 42. Dependence of Solution Viscosity on Hydrophobe Level for Different Hydrophobe Structures [93].

10^

E

Sc^tnt 2X Mac Temp: 25 G

0

o o

a.

^Q'

°°«„ \ ôô^

=

-,Q2

U

^ o • '*"!l > f I ifttil {,, t ; I i u,tl ; : ;tnnl ; 1 10*2 10^^ 1 10 10^ Shear Rate, 1 / S e c

^ . iu,i

10-

Figure 43. Effect of shear rate on viscosity as a function of polymer concentration. Acrylannide/N-octylacrylamide copolymer, 0.75 mol% hydrophobe [89].


659

the shear thickening behavior is that the polymer chains are stretched at high shear rates. This will enhance intermolecular association and, as a result, the viscosity increases. Effect of Chemical Interactions on the Properties of Associating Polymers

Rheological

Bock et al. examined the effect of salts on the viscosity of nonionic associating polymer [89]. Figure 44 compares the viscosity of N-octylacrylamide/acrylamide copolymer in water and 2 wt% sodium chloride. The viscosity of the associating polymer increases in the presence of salts, especially at higher polymer concentrations. The hydrophobic groups associate to minimize their exposure to water. This is similar to micelle formation encountered with ionic surfactants. Increasing salinity enhances aggregation and reduces the critical micelle concentration. Similarly, the effect of salts on viscosity of associating polymers can be attributed to association. The effect of salts on the apparent viscosity of associating polymers also was observed by McCormick et al. using a copolymer of acrylamide and decylacrylamide. One major disadvantage of HPAM is its high sensitivity to salts [41]. This is not so for hydrophobically associating polyacrylamide. Figure 45 shows the effect of salts on the apparent viscosity at 1.3 s~' for HPAM and hydrolyzed copolymer of N-octylacrylamide/acrylamide. All polymers have the same degree of hydrolysis at 18%. The two associating polymers contained hydrophobe contents of 1 and 1.25 mol%. The addition of hydrophobe reduced the sensitivity to salts, especially at the higher hydrophobe content examined.

io3 E Hydrophobe:, a . 7 5 m X C , T«mp: 2 5 C 2X N a a

o c

> 10

2000 4000 Concentration, ppm

6000

Figure 44. Solution viscosity response to salt as a function of polymer concentration. Acrylamide/N-n-octylacrylamide, hydrophobe 0.75 mol% [89].

660

Advances in Engineering Fluid Mechanics 1000 c

m "h n-C9

s c o s I T Y

loot

rrr

..^•^"ffT

"^

1.25

*

^ 10

1.0

+ • • • • • • X . . . . . .

••• - x • ..

HPAM

'"A,

cP 0.1

1C

SALT CONCENTRATION, w t % Figure 45. Effect of salt on viscosity at 1.3 s - \ 2,000 ppm of polymer, 18 mol% hydrolysis [93].

Surfactant concentration (varied after polymerization) greatly affects the viscosity of associating polymer systems. Iliopoulos et al studied the interactions between sodium dodecyl sulfate (SDS) and hydrophobically modified poly(sodium acrylate) with 1 or 3 mole percent of octadecyl side groups [85]. A viscosity maximum occurred at a surfactant concentration close to or lower than the critical micelle concentration (CMC). Viscosity increases of up to 5 orders of magnitude were observed. Glass et al observed similar behavior with hydrophobically modified HEC polymers. [100] The low-shear viscosity of hydrophobically modified HEC showed a maximum at the CMC of sodium oleate. HEUR thickeners showed the same type of behavior with both anionic (SDS) and nonionic surfactants. At the critical micelle concentration, the micelles can effectively cross-link the associating polymer if more than one hydrophobe from different polymer chains is incorporated into a micelle. Above the CMC, the number of micelles per polymer-bound hydrophobe increases, and the micelles can no longer effectively cross-link the polymer. As a result, viscosity diminishes. CONCLUSIONS Water-soluble polymers have been reviewed with particular emphasis on their application in improved oil recovery. These polymers have potential for use in mobility control, drilling fluids and profile modification. Partially hydrolyzed polyacrylamide and xanthan gum are the most commonly used water-soluble polymers in oil field applications. The apparent viscosity of these polymers depend on polymer type, molecular weight, charge density, concentration, shear rate, salt concentration, and pH, as follows:


661

Partially Hydrolyzed Polyacrylamide 1. The viscosity-shear rate relationship exhibited a Newtonian behavior at low shear rates and a power-law behavior at high shear rates. 2. Adding a non-ionic species (Triton X-100) had no significant effect on the viscosity-shear rate relationship. However, adding an ionic species (sodium chloride, calcium chloride, or an anionic surfactant) reduced the hydrodynamic size of the polymer molecule (physical change), changing the viscosity-shear rate relationship. 3. The viscosity-shear rate relationship was found to be a strong function of cation type only at salt concentrations less than 4 wt%. 4. The effect of alkalis on the viscosity of polymer solutions was complex as they affected the polymer chain both physically (charge shielding) and chemically (hydrolysis). For alkali/polymer solutions, the viscosity was found to be a function of alkali type, concentration and time after initial mixing. 5. The addition of anionic surfactants at low concentrations slightly decreased alkali/polymer solutions viscosity. However, a significant viscosity enhancement for alkali/polymer solutions was observed at higher surfactant concentrations and over a narrow range of alkali concentrations. Xanthan Gum 1. At polymer concentrations < 2,000 ppm, the viscosity-shear rate relationship exhibited a Newtonian behavior at low shear rates and a shear thinning behavior at high shear rates. At polymer concentrations > 2,000 ppm, the shear thinning behavior was observed only over the range of shear rates examined. 2. At low polymer concentrations, simple salts caused a slight reduction in the viscosity of Statoil polymer (a medium pyruvate content xanthan), and a more noticeable change in the flow curves of Flocon 4800 (a high pyruvate content xanthan). However, at higher polymer concentration, the addition of salts increased the apparent viscosity of the high pyruvate content xanthan. 3. Calcium chloride had a more detrimental effect on the apparent viscosity of the high pyruvate content xanthan than sodium chloride. 4. Strong alkalis caused a fast and significant reduction in viscosity. Buffered alkalis were less detrimental to xanthan solution viscosity. 5. Triton X-100 slightly changed the flow curves of both xanthan materials. 6. Neodol 25-3S caused significant changes only in the flow curves of Flocon 4800 xanthan. 7. Similar to the trends observed with HP AM, a significant viscosity enhancement for alkali/polymer solutions was observed at higher surfactant concentrations and over a narrow range of alkali concentrations. Hydrophobically Associating Polymers: The rheology of these polymers is more complex than that of HP AM or xanthan gum. It is affected by hydrophobe type and content, by polymer molecular weight, degree of hydrolysis, temperature, salinity and by the presence of surfactants.

662


ACKNOWLEDGMENTS HAN wishes to thank the management of Saudi Aramco for permission to publish this paper. NOMENCLATURE C Polymer concentration, ppm c Critical aggregation concentration (CAC) EGR Enhanced oil recovery HEC Hydroxyethyl cellulose HPAM Partially hydrolyzed polyacrylamide HPC Hydroxypropyl cellulose HRAM Partially hydrolyzed associating acrylamide polymer

k Power-law constant, mPa«s" K Huggins constant M^ Polymer weight average molecular weight n Power-law index PAM Polyacrylamide (unhydrolyzed) R^ Bob radius, m R^ Cup radius, m T Temperature, °C SDS Sodium dodecyl sulfate

Greek Symbols y Shear rate, s ' [i Viscosity, mPa*s r| Viscosity of polymer solution, mPa*s r|^^ Solvent viscosity, mPa«s [r|] Intrinsic viscosity, dL/g |Li^ Low-shear Newtonian viscosity, mPa»s

[i Solvent viscosity, mPa»s w Rotational speed, rad/s T Degree of hydrolysis T^ Rotational relaxation time (inverse of the critical shear rate), s

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663

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43. Sandford, P. A., J. E. Pittsley, C. A. Knutson, P. R. Watson, M. C. Cadmus and A. Jeanes, "Variation in Xanthomonas Campestris NRRLB-1459; Characterization of Xanthan Products of Differing Pyruvic Acid Content," Extracellular Microbial Polysaccharides, Am. Chem. Soc. Symp. Ser. 45, P. A. Sandford and A. Laskin Eds., Washington DC, 192-210 (1977). 44. Smith, I. H., K. C. Symes, C. J. Lawson and E. R. Morris, "Influence of the Pyruvate Content of Xanthan on Macromolecular Association in Solution," Int. J. Biol. Macomol. 3, 129-134 (1981). 45. Kulicke, W. M., R. Kniewske, Klein, "Preparation Characterization, Solution Properties and Rheological Behavior of Poly aery lamide," Prog. Polym. Sci., 8, pp. 373-468 (1982). 46. Nasr-El-Din, H. A. and J. L. Noy, "Flow Behaviour of Alkali, Surfactant and Xanthan Solutions Used for Enhanced Oil Recovery," Revue de VInstitut Frangais du Petrole 47, 771-791 (1992). 47. Lund, T., R. Boreng, E. O. Bjornestad and P. Foss, "Development and Testing of Xanthan Products for EOR Applications in the North Sea," Revue de V Institut Frangais du Petrole 45, 107-113 (1990). 48. Ait-Kadi, A., P. J. Carreau and G. Chauveteau, "Rheological Properties of Partially Hydrolyzed Polyacrylamide Solutions," J. Rheology 31, 537-561 (1987). 49. Carreau, P. J., "Rheological Equations from Molecular Network Theories," Trans. Soc. Rheol. 16, 99-127 (1972). 50. Lim, T., J. T. Uhl and R. K. Prud'homme, "The Interpretation of Screen-Factor Measurements," SPE Res. Eng. 1, 272-276 (1986). 51. Gao, H. W., P. B. Lorenz and S. Brock, "Rheological Behavior of Pusher 500 Under a Variety of Chemical and Thermal Conditions," Department of Energy Report No. NIPER - 225, February (1987). 52. Unsal, E., "Evaluation of Flow Properties of Dilute Aqueous Polymer Solutions For Enhanced Oil Recovery Applications," Ph.D. thesis, the Pennsylvania State University (1978). 53. Tam, K. C. and C. Tiu, "Steady and Dynamic Shear Properties of Aqueous Polymer Solutions," J. Rheology 33, 257-280 (1989). 54. French, T. R., N. Stacy and A. G. Collins, "Polyacrylamide Polymer Viscosity as a Function of Brine Composition," DOE Report No. Rl-80/12 (1981). 55. Moradi-Araghi, A. and P. H. Doe, "Hydrolysis and Precipitation of Polyacrylamides in Hard Brines and Elevated Temperatures," SPE Res. Eng. 2, 189-198 (1987). 56. Radke, C. J., "Additives for Alkaline Recovery of Heavy Oil," DOE Contract No. 7405-ENG-48 (1982). 57. Sloat, B. and D. Zlomke, "The Isenhour Unit—A Unique Polymer-Augmented Alkaline Flood," SPE 10719, presented at the SPE/DOE Third Symposium on Enhanced Oil Recovery held in Tulsa, OK, April 4-7 (1982). 58. Ball, J. T. and M. J. Pitts, "Simulation of Reservoir Permeability Heterogeneities with Laboratory Corefloods," SPE 11790, presented at the International Symposium on Oilfield and Geothermal Chemistry held in Denver, CO., June 1-3 (1983). 59. Ball, J. T. and M. J. Pitts, "Effect of Varying Polyacrylamide Molecular Weight on Tertiary Oil Recovery from Porous Media of Varying Permeability,"

666

60. 61.

62. 63. 64. 65. 66.

67.

68. 69. 70. 71. 72. 73.

74.


SPE/DOE 12650, presented at the SPE/DOE Fourth Symposium on Enhanced Oil Recovery held in Tulsa, OK, April 15-18 (1984). Ball, J. and H. Surkalo, "Alkaline-Surfactant-Polymer Process Makes EOR Economic," Amer. Oil Gas Reporter 31, 46-48 (1988). Mihcakan, I. M. and C. W. van Kirk, "Blending Alkaline and Polymer Solutions Together into a Single Slug Improves EOR," SPE 15158, presented at the SPE Rocky Mountain Regional Meeting held in Billings, MT, May 19-21 (1986). Burk, J. H., "Comparison of Sodium Carbonate, Sodium Hydroxide and Sodium Orthosilicate for EOR," SPE Res. Eng. 2, 9-16 (1987). Potts, D. E. and D. L. Kuehne, "Strategy for Alkaline/Polymer Flood Design with Berea and Reservoir-Rock Corefloods," SPE Res. Eng. 3, 1,143-1,152 (1988). Alam, M. W. and D. Tiab, "Mobility Control of Caustic Flood," Energy Resources 10, 1-19 (1988). Hawkins, B. F., K. C. Taylor and H. A. Nasr-El-Din, "Mechanisms of Suerfactant and Polymer Enhanced Alkaline Flooding: Application to David Lloydminster and Wainright Sparky Fields," JCPT 33, 52-63 (1994). Dexter, R. W. and R. G. Ryles, "Effect of Anionic Comonomers on the Hydrolytic Stability of Polyacrylamides at High Temperatures in Alkaline Solution," in Oil-Field Chemistry, Edited by J. K. Borchardt and T.F. Yen, ACS Symposium Series 396, 102-110 (1989). Kheradmand, H. and J. Fran9ois, "Predictions of the Evolution with Time of the Viscosity of Acrylamine-Acrylic Acid Copolymer Solutions," in Oil-Field Chemistry, Edited by J.K. Borchardt and T.F. Yen, ACS Symposium Series 396, 111-123 (1989). Nasr-El-Din, H. A. and K. C. Taylor,"Interfacial Tension of Crude Oil/Alkali/ Surfactant Systems in the Presence of Partially Hydrolyzed Polyacrylamide," Colloids and Surfaces 74, 169-183 (1993). Shupe, R. D., "Chemical Stability of Polyacrylamide Polymers," JPT 33, 1,513-1,529 (1981). Shuler, P. J., D. L. Kuehne and R. M. Lerner, "Improving Chemical Flood Efficiency with Micellar/Alkaline/Polymer Processes," J. Pet. Tech. 41(1), 80-88 (1986). Lin, F. F. J., G. J. Besserer and M. J. Pitts, "Laboratory Evaluation of Crosslinked Polymer and Alkaline-Polymer-Surfactant Flood," JCPT (6) 26, 54-65 (1987). Manji, K. H. and B. W. Stasiuk, "Design Consideration for Dome's David Alkaline/Polymer Waterflood During Uncertain World Prices," JCPT 27 (3), 49-54 (1988). Clark, S. R., M. J. Pitts and S. M. Smith, "Design and Application of an Alkaline-Surfactant-Polymer Recovery System to the West Kiehl Field," SPE 17538, presented at the SPE Rocky Mountain Regional Meeting held in Casper, WY, May 11-13 (1988). Nasr-El-Din, H., B. F. Hawkins and K. A. Green, "Recovery of Residual Oil Using the Alkali/Surfactant/Polymer Process: Effect of Alkali Concentration," J. Pet. Sci. Eng. 6, 381-401 (1992).


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75. Nasr-El-Din, H., K. A. Green and L. L. Schramm, ' T h e Alkali/Surfactant/ Polymer Process: Effects of Slug Size, Core Length and a Chase Polymer," Revue de Vlnstitut Frangais du Petrole 49, 359-377 (1994). 76. Nasr-El-Din, H. A. and B. F. Hawkins, ''Recovery of Residual Oil Using Alkali, Surfactant and Polymer Slugs in Radial Cores," Revue de Vlnstitut Frangais du Petrole 46, 199-219 (1991). 77. Miller, C. A., O. Ghosh, and W. J. Benton, "Behavior of Dilute Lamellar Liquid-Crystalline Phases," Coll. Surfaces 19, 1986, pp. 197-223. 78. Nasr-El-Din, H. A., D. Schriemer and A. S. Abd-El-Aziz,"Liquid Crystal Formation and its Effect on the Flow Properties of An Anionic Suractant," Presented at the 85th AOCS Conference, Atlanta, Georgia, May 8-12 (1994). 79. Chauveteau, G., "Rodlike Polymer Solution Flow through Fine Pores: Influence of Pore Size on Rheological Behaviour," J, Rheol 26, 111-142 (1982). 80. Auerbach, M. H., "Prediction of Viscosity of Xanthan Solutions in Brines," SPE 13591, presented at the International Symposium on Oilfield and Geothermal Chemistry held in Phoenix, Arizona, April 9-11 (1985). 81. Smith, I. H., K. C. Symes, C. J. Lawson and E. R. Morris, "The Effect of Pyruvate on Xanthan Solution Viscosity," Carbohydr. Polymers 4, 153-157 (1984). 82. Cheetham, N. W. H. and N. M. N. Norma, "The Effect of Pyruvate on Viscosity Properties of Xanthan," Carbohydr. Polymers, 10, 55-60 (1989). 83. Tako, M. and S. Nakamura, "Rheological Properties of Deacetylated Xanthan in Aqueous Media," Agric. Biol. Chem. 48, 2,987-93 (1984). 84. McNeely, W. H. and K. S. Kang, "Xanthan and Some Other Biosynthetic Gums," Industrial Gums, 2nd Edition, R. L. Whistler (ed.). Academic Press, New York City, 473-97 (1973). 85. Iliopoulos, I., T. K. Wang and R. Audebert, "Viscometric Evidence of Interactions between Hydrophobically Modified Poly(sodium acrylate) and Sodium Dodecyl Sulfate," Langmuir, 7(4), pp. 617-619 (1991). 86. Wang, K. T., Illiopoulos, I., Audebert, R., "Viscometric Behavior of Hydrophobically Modified Poly(Sodium Acrylate)," Polym. Bull, 20, pp. 577-582 (1988). 87. Wang, Z.-G., "Aggregation (Micellization) of Associating Polymers," Langmuir, 6(5), pp. 928-934 (1990). 88. Wang, T. K., Iliopoulos, I., Audebert, R., "Aqueous-Solution Behavior of Hydrophobically Modified Poly(acrylic Acid)," in Water-Soluble Polymers: Synthesis, Solution Properties and Applications, Shalaby, S.W., McCormick, C. L., Butler, G. B., Eds., ACS Symposium Series No. 467, American Chemical Society, Washington, DC, 1991, pp. 218-231. 89. Bock, J., P. L. Valint, Jr., S. J. Pace, D. B. Siano, D. N. Shcultz and S. R. Turner, "Hydrophobically Associating Polymers," in "Water-Soluble Polymers for Petroleum Recovery," Stahl, G. A., and Schultz, D. N., Eds., Plenum Press, NY 1988, pp. 147-160. 90. McCormick, C. L., J. Bock, and D. N. Schulz, "Water-Soluble Polymers," in Encyclopedia of Polymer Science and Engineering, 2nd Ed., Vol. 17, John Wiley and Sons, New York, 1989, pp. 730-784. 91. Bock, J., D. B. Siano, P. L. Valint, Jr., S. J. Pace, "Structure and Properties of Hydrophobically Associated Polymers," Polym. Mater. Sci. Eng., 57, pp. 487-91 (1987).

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92. Bock, J., P. L. Valint, S. J. Pace, "Enhanced Oil Recovery with Hydrophobically Associating Polymers Containing Sulfonate Functionality," U.S. Pat. 4,702,319 (1987). 93. Bock, J., D. B. Siano, P. L. Valint, Jr., S. J. Pace, "Structure and Properties of Hydrophobically Associating Polymers," in Polymers in Aqueous Media: Performance through Association, Glass, J. E., Ed., Advances in Chemistry Series No. 223, American Chemical Society, Washington, DC, 1989, pp. 411^24. 94. Zhang, Y. X., A. H. Da, and T. E. Hogen-Esch, "A Fluorocarbon-Containing Hydrophobically Associating Polymer," J. Polym. Sci. Part C: Polym. Let., 28(7), pp. 213-18 (1990). 95. Middleton, J. C , D. F. Cummins and C. L. McCormick, "Rheological Properties of Hydrophobically Modified Acrylamide-Based Polyelectrolytes," in WaterSoluble Polymers: Synthesis, Solution Properties and Applications, Shalaby, S. W., McCormick, C. L., Butler, G. B., Eds., ACS Symposium Series No. 467, American Chemical Society, Washington, DC, 1991, pp. 338-348. 96. Flynn, C. E., J. W., Goodwin, "Association of Acrylamide-Dodecyl-methacrylate Copolymers in Aqueous Solution," in Polymers as Rheology Modifiers, Schulz, D. N., Glass, E. J., Eds., ACS Symposium Series No. 462, American Chemical Society, Washington, DC, 1991, pp. 190-206. 97. Schulz, D. N., Bock, J., "Synthesis and Fluid Properties of Associating Polymer Systems," J. Macromol. ScL-Chem., A28(ll&12), pp. 1,235-1,243 (1991). 98. Wolff, C , "Molecular Weight Dependence of the Relative Viscosity of Solutions of Polymers at the Critical Concentration," Eur. Polym. J., 13, pp. 739-741 (1977). 99. Aharoni, S. M., "Critical Concentrations for Intermolecular Interpenetration and Entaglements," J. Macromol. Sci.-Phys., B15(3), pp. 34,357-370 (1978). 100. Glass, J. E., Lundberg, D. J., Ma, Z., Karunasena, A., Brown, R. G., "Viscoelasticity and High Shear Rate Viscosity in Associative Thickener Formulations," Proc. Water-Bome Higher-Solids Coat. Symp., Vol. 17, 1990, pp. 102-20.

CHAPTER 25 RELATION OF RHEOLOGICAL PROPERTIES OF UV-CURED FILMS WITH GLASS TRANSITION TEMPERATURES BASED ON FOX EQUATION M. Azam AH, M. A. Kahn, K. M. Irdriss Ali Radiation and Polymer Chemistry Laboratory Institute of Nuclear Science and Technology Bhaka, Bangladesh CONTENTS ABSTRACT, 669 INTRODUCTION, 669 EXPERIMENTAL, 670 Materials, 670 Methods, 670 Film Characterization, 670 RESULTS AND DISCUSSION, 671 TENSILE PROPERTIES, 676 CO-DILUENTS, 679 REFERENCES, 681 ABSTRACT Polymeric films have been prepared under UV radiation of urethane acrylate combined with functional monomers used as reactive diluents. Amount of photoinitiator (Irgacure 184) and radiation does intensity were optimized UV-cured films were characterized by film hardness, gel content, and tensile properties; these properties were correlated with glass transition temperatures (Tg) of the polymer films. The Tg of the film was calculated by using the Fox equation. Effect of comonomer diluents on these properties also was investigated in light of the changed Tg values of the polymeric films prepared in he presence of co-diluent. INTRODUCTION Polymers have diversified in applications in different fields and, as such, different types of polymers are being continuously developed to meet the universal demand of applications. As a result, various formulations are developed incorporating requisite reactive diluents to influence the film properties [1-3]. Additives of different natures are important to monitor the characteristic properties of the film. 669

670


Polyfunctional monomer diluents are apt to make films of a hard nature, whereas monofunctional reactive diluent induces soft character in the film [4-5]. However, the structural pattern and geometry of the molecule of the reactive diluents influences very much the overall geometry and shape of the final polymer (copolymer). It also is universally recognized that the melting point, particularly glass transition temperature (Tg), is very much related to the shape and geometrical structure of the molecule. Thus, different reactive diluents of various functionalities containing different geometrical and structural shapes in he molecule are expected to influence the rheological properties of the copolymers formed with these diluents. So the Tg values of both reactive diluents and copolymers will play some role in the overall characteristic properties of the films formed. The present study is designed to correlate the rheological properties of the copolymers formed in the presence of different reactive diluents and codiluents with the glass transition temperatures of these films, calculated on the basics of Fox equation. EXPERIMENTAL Materials Oligomer urethane acrylate (LR 8739) and photoinitiator Irgacure 184 were obtained from IAEA. Reaction diluents 2-methoxy ethyl acrylate (MEA, Tg = -110°C), 2-ethylhexyl acrylate (EHA, Tg = -50°C), 2-hydroxy ethyl methacrylate (HEMA, Tg = 55°C), N-vinyl pyrrolidone (NVP, Tg = 175°C), tripropylene glycol diacrylate (TPGDA, Tg = 90°C), and trimethylol propane triacrylate (TMPTA, Tg 250°C) were used as procured from E, Merck without any further purification. Methods Solutions were prepared with oligomer at 60% and variable proportions of reactive diluents (30-39% and photoinitiator (10-1%). These solutions are coated on glass plates (5 X 10 cm) with bar coater no. 0.28 of Abbey Chemicals Co. (Australia). This made the film 36±3)LUn thick. The coated films were cured under UV radiation using the UV-minicure machine of IST-Technique, Germany. The UV lamp had wave length 254-313nm yielding 2kw light intensity at 9.5 amp current. The cured films were then systematically characterized as follows: Film Characterization While still on the glass plates the cured films were used to determine the film hardness with the help of pendulum hardness method using a digital pendulum hardness tester (Model 5854, Byke Labotron). The gel content of the films was determined by peeling the cured films off the plates and extracting the film with hot acetone for 20 h in a soxhlet apparatus. The difference in weight of the films after and before the extraction yields the gel content. Generally, the cured films are wrapped up in a stainless steel net and put into soxhlet for the extraction. A known weight of the gel was soaked in acetone at 25°C for 24 h. The tensile properties, particularly strength (TS) and elongation at break (Eb), were directly measured with an INSTRON machine (Model 1011).

Relation of Rheological Properties of UV Cured Films

671

RESULTS AND DISCUSSION The concentration of the photoinitiator (Irgacure 184) was optimized so subsequent experiments could be done using the optimized amount of Irgacure 184. For this purpose, a number of solutions was prepared using fixed amount (60% of oligomer (LR 8739) with variable amounts of Irgacure 184 (1-10%) and reactive diluent, NVP (39-30%). The pendulum hardness of the UV cured films of these solutions is shown in Figure 1 against number of passes under the UV lamp used for curing. It

NUMBER OF PASS Figure 1. Pendulum hardness of UV-cured films of urethane acrylate and NVP is shown against the number of passes as function of concentration of the photoinitiator.

672


is observed that 5% Irgacure has yielded the highest pendulum hardness throughout the radiation (1-10 passes); the maximum pendulum hardness is obtained at the sixth pass; similarly the maximum pendulum hardness is observed with 2% photoinitiator. The maxima obtained at the second pass are observed with photoinitiator that has concentration higher than 5% in the solutions. The decrease of pendulum hardness values after the second pass may be caused by the fact that the number of free radicals formed from the photoinitiator was large, which led to the recombination of the radicals among themselves rather than with the monomer/oligomer units. As a result, there is less crosslinking density in the films when photoinitiator composition is larger than 5%. After optimizing the concentration of photoinitiator at 5% Irgacure 184 for the maximum output as shown in Figure 1, proportions of the oligomer and reactive diluent (TPGDA) were varied to optimize their concentrations to achieve maximum output. The pendulum hardness of the cured films is plotted against number of passes in Figure 2. It is observed that the highest pendulum hardness is obtained with the films prepared with oligomer: diluent: photoinitiator = 60: 35: 5, w/w. The lowest pendulum hardness is yielded by 75% oligomer and 205 TPGDA. It is noted here that the maximum pendulum hardness is obtained at the fourth pass. Similarly, when the reactive diluent is changed with NVP or TMPTA, the maximum pendulum hardness is obtained at the sixth pass unlike the NVP. This is shown in Figure 3. It also is interesting to note here that the highest pendulum hardness is obtained with TMPTA system. TMPTA is a trifunctional reactive diluent with the ability to make crosslinking in three directional manners. TMPTA has branch-like effect to create crosslinking. Likewise, TPGDA should have yielded the second highest pendulum hardness as the TPGDA contains difunctional acrylated groups; but instead, NVP has produced more pendulum hardness compared to TPGDA. NVP has been proven to be a unique monomer which creates favorable augmentation through the love pair of electrons present in the carboaminde group, - N = CO [6]. It already has been established that the rheological properties of a think film are highly correlated with the glass transition temperature (Tg) of both reactive diluent monomers and copolymers [7]. The reason is that the Tg is very much related to the physical phenomena of the molecule, particularly geometric and structural shapes of the molecule. The present study attempts to correlate the rheological properties of the UV-cured films with Tg values of the copolymers obtained with oligomer at 60% (fixed) and variable concentrations (39-30%) of monomer diluents. The diluents were NVP, TPGDA, and TMPTA. The Tg copolymer was calculated on the basis of the Fox equation as follows: 1/Tgc = Wo/Tgo + Wm/Tgm where, Tgc Tgo Tgm Wo Wm

= = = = =

Tg Tg Tg wt. wt.

of copolymer (cured films); of oligomer; of monomer diluent; fraction of oligomer; fraction of monomer diluent.

Figure 4 represents the plots of pendulum hardness of the UV-cured films against the Tg copolymer (cured film) calculated on the basis of the Fox equation. Only the


2

673

4

NUMBER OF PASS ^0/M/P=75/20/5

0/M/P= 70/25/5

^0/M/P=63/32/5

0/M/P=60/35/5

-^0/M/P=55/40/5 Figure 2. Pendulum hardness is plotted against the number of passes for the films prepared with variable concentrations of oligomer (O) and monomer TPGDA (M) at fixed concentration (5%) of the photoinitiator (P).

674


TMPTA

75

111 55 Z

o
TPGDA>HEMA>EHA>MEA. Similar values are observed in Figure 6, where TPGDA yield the highest and MEA gives the lowest pendulum hardness values and where pendulum hardness values are plotted against number of passes under the UV lamp for the systems of different reactive diluents. In the calculation of Tg values based on the Fox equation only oligomer and reactive diluents are considered. Since photoinitiator acted as a catalyst, it was not considered as a parameter in the Fox equation. However, it appears that the Fox equation for calculation of the Tg values of the cured films

676


120

-30

120

TgX Figure 5. Different rheological properties, such as gel content (Gel), pendulum hardness (PH), tensile strength (TS), and elongation at break (Eb), are shown against glass transition temperature (Tg) of co-polymer (cured films) for oligomer : monomer : photoinitiator = 60 : 35 : 5, w/w.

(copolymer) holds well very nicely in the interpretation of the various rheological properties against the Tg copolymer values. The values of the gel contents of the UV-cured films of different reactive diluents also can be similarly shown (Figure 5) against Tg copolymer of the cured films calculated by Fox equation. TENSILE PROPERTIES Tensile properties, particularly tensile strength (TS) and elongation at the breaking point of the film (Eb), are plotted (Figure 5) against Tg copolymer based on Fox equation. It is observed that TS values, in general, increase with an increase of Tg values. However, the highest TS value is achieved by NVP, though TMPTA produces the highest pendulum hardness (Figure 5). It is known that film hardness (pendulum hardness) represents the crosslinking density at the surface of the cured films while tensile strength is achieved through the overall crosslinking network within the cured film. Thus, TMPTA may produce the highest pendulum hardness.


677

TMPTA 65

CO CO LU

Z 50

NVP

o
. is a function of the time t and is called effective time. It may be calculated from the prescribed temperature history and the known time-temperature shift function a(T,T,) by

^(t) = j ;

a(T(^),T,)

(^)

J^ and T^j are the values of the creep compliance and the prescribed temperature at t = 0. Equation 3 has the form of the ordinary superposition principle. However, the convolution integral is to be taken in the A.-time domain, and the stress history is to be inserted as a function of the effective time ^. J is the creep rate at T^, which is taken as a function of the difference of the effective times X and ^. By inserting: c[i{X)] = 0

for X < 0

c[i(X)] = a^,

for ^ > 0

(5)

into Equation 3 we get for the special case of non-isothermal creep of thermorheologically simple materials: y(t) = cJiXj^)

(6)

J(^,T^) has known the meaning of the creep function at constant temperature T^ as a function of the effective time X. All these results may be transferred immediately to the problem of viscoelastic behavior under the influence of aging at constant temperature [9,15]. The temperature history has to be replaced by the degree of aging. A, of the sample and the time-temperature shift function, a(T,TQ), by the timeage shift function, b'(A,A^^). The degree of aging, also called the "age" of the sample, defines the time elapsed from the last quench from the equilibrium state down to the aging temperature so far; X can then be expressed:

Prediction and Calculation of Amorphous Polymers

u

f'

687

tic d^

We designate the creep curve by: y(Ve't) = a,J(t,t^)

(8)

and together with Equation 5 we obtain: J(t^,t) = J(;i,t)

(9)

J defines now the creep compliance under the influence of progressive aging, which is accessible by experiment; X is given by Equation 7. The construction of J (t,t^) from a creep compliance at a constant degree of aging and the meaning of the shift factor b'(A,AQ) or b'(A,A^), respectively, are discussed with reference to Figure 2. In this figure we have plotted schematically the measurable creep compliance J(t,t^) and two creep compliances at constant but different ages, namely A^ and A^ with A^ » A^. The latter curves have the same shape and differ only in their position on the logarithmic time scale.

Figure 2. Schematic course of creep curves at two fixed values of age A^ and A^ and during progressive aging.

688


If physical aging is caused by volume recovery effects, the influence of aging during long-time creep experiments at constant temperature cannot be ignored since, in contrast to temperature, the age of the sample cannot be kept constant during the measurements. If it would be possible to perform two creep measurements at different, but fixed degrees of aging, A^ and A^, the influence of these different amounts of aging (equivalent to different preconditioning times t^) would only result in a parallel shift of the curves on the logarithmic time scale. The distance between the two creep curves is then b' (AQ,AJ, as to be seen in Figure 2. The curve J(t,AQ) is the hypothetical creep curve that would be obtained if the age of the sample could be fixed immediately after the beginning of the creep experiment, i.e., A^ is equal to the preconditioning time, t^, and Equation 7 can be written as: A(t) = t + t = A, + t

(10)

In reality the shape of this creep curve is only measurable for creep times substantially smaller than the preconditioning time t^, i.e., for t-values smaller than 0.1-0.3t^ [1]. For the creep curve J(t,A^) the preconditioning time was chosen to be long enough to ensure that the sample was in volume equilibrium at the beginning of the creep experiment. Thus, no further aging could occur during the measurement. The experimental creep curve J(t,t^) under progressive aging will coincide with the hypothetical creep curve J(t,AQ) for short creep times (t .,A,)= J(t,t^) = J(^,AJ

(11)

Replacing t^ by A^ in Equation 7 and after subsequent differentiation we obtain: d^ dt

b'(Ao+t,Ao)

b'(A,AJ

(12)

or

logb'(A,A,) = - l o g f ^ l - l o g

dlog^ dlogt

(13)

Assume that the creep curve J (t,t^) was measured and the shape of the equilibrium creep curve J(t,AJ is known. Shift the latter until it coincides with the measured creep curve in the short-time domain, and one gets the hypothetical creep curve J(t,AQ). Now X may be determined as a function of t for all creep times investigated. Applying Equation 13 the time-age shift function b'(A,AQ) can be calculated. Otherwise, we know from Figure 2 that the following is valid:


log b'(A,,AJ = logj^^j + logj^^j

689

(14)

and dlogX, _ dlogjx dlogt ~ dlogt

^^^^

Thus, assuming the curve J (t,t^) is measured and shape and position of the equilibrium creep curve J(t,A^) are known, we may determine |LI as a function of t for all creep times. For the time-age shift function we find:

logb'(A,AJ = logf^l +log

^dlogiLi^ dlogt

(16)

This leads to the following important conclusion. On the supposition that shape and time position of the equilibrium creep curve, J(t,A^), as well as the time-age shift function, b' (A,AJ, are known at the aging temperature, we can construct the shape of the creep curve, J(t,tg), for any preconditioning times, t^ [9,17-19]. As a disadvantage of this procedure we have to determine the time-age shift function experimentally from very time-consuming aging creep experiments and that has to be redone for each aging temperature. If these physical aging effects that occur during creep measurements are due to the decrease of free volume, the time-age shift function must be directly related to the change in free volume. In recent years several expressions have been proposed to describe volume recovery behavior [4,5,8,20-25] in which the temperature and structure dependence of the retardation times is based on activation processes [21], configurational entropy [22,23] or free volume [4,5,8,20,24,25]. The behavior in volume of amorphous polymers can be well described theoretically by a multiparameter model [4-6] based on free volume. In this model, proposed by Kovacs and coworkers, the non-equilibrium state of a system is completely characterized by temperature, pressure, and a fixed number of ordering parameters n. (1 < i < N) [5]. We define as a new variable: o ^=

(v-O V.

(17)

which measures the deviation of the fractional free volume, f, from its equilibrium value, f^, according to: f = 8 + f^

(18)

In Equation 17, v defines the time-dependent, instantaneous value of the specific volume, and v^ the corresponding equilibrium value at the temperature, T. Each ordering parameter, n., contributes a value 6^ to the total deviation of the fractional free volume:

690

Advances in Engineering Fluid Mechanics N

8 = Xî

(19)

i=i

and is directly coupled with an individual retardation time, x.. Each 8. is assumed to obey the basic Kovacs equation [20]: _ ^ =Aa^.^ +^ dt dt T-

(i=l,2, ...,N)

(20)

The time dependence of the total deviation 8 is obtained by summation over all i-values: dS^fdS^ dt frr dt

(i=l,2, ...,N)

(21)

Aa. represents the contribution of the i-th process to the difference of the expansion coefficient in the rubbery and in the glassy state: N

a, - ttg = Aa = ^lôCi

(i = h 2, . . . , N)

(22)

i=l

The retardation times, x, depend on free volume and are given by:

X. = x.^ . e^» ^^

(i = 1, 2, . . . , N)

(23)

where x and f are the values of the retardation times and the value of the free i,r

r

volume at a reference temperature, T^; b is a constant of the order of one and is not to be confused with the time-age shift function b ' ; b can be calculated from the Williams-Landel-Ferry constants c,, c^ and Aa (Tables 1 and 2) [6,7,10,16]. The retardation times x. are allowed to depend both on temperature T as well as on the instantaneous state of the specimen defined by 8. Equation 23 is easily transformed to: f b

b 1

(

bd

]

Xi(T,8) = X;^ •e^'~ '^^-e^ '-^'"^^^^ =Xi,-a^,-ag

(i = 1, 2, . . . , N)

(24)

The shift factor a^ characterizes the temperature dependence of the retardation times, and the second exponential term a^ reveals the influence of the instantaneous state of the system and is dependent on the prehistory and the degree of aging of the sample. The set of retardation times, x., and normalized intensities, g., in Tables 1 and 2 were determined empirically to give the best description of the volume behavior under any thermal history like simple volume recovery, behavior after multiple temperature jumps, or experiments, including cooling and heating with intermediate


691

Table 1 Material Parameters For PS N 7000 o.W. Used for the Calculations Parameter

Retardation Time

c, = 10 c^ = 39.5 K T = 105°C a, = 5.66 X 10^ K-' a = 1.76 X 10^ K ' Aa = 3.90 X 10-^' K '

T, = 5 X 10-^ s x^ = 2 X 10^ s X3 = 1.75 X 10-2 s T4 = 3 X 10-' s T5 = 6 X 1 0 ' s T, = 8 S

Normalized Retardation Strength g, g, g, g, g5 g,

=0.100 = 0.070 = 0.110 = 0.170 = 0.165 = 0.385

Table 2 Material Parameters for PC 2800 Used for the Calculations Parameter c, = 17.9 c^ = 52.3 K T = 142.5°C a, = 5.80 X 10-4 K-' a = 1.64 X 10-4 K ' Aa = 4.16 X 10^' K-'

Retardation Time T, = 4 X 10-2 s X2 = 4 X 10-' s T3 = 4 X 10^ s X4 = 4 X 10' s T5 = 4 X 102 s T^ = 4 X 10^ s

Normalized Retardation Strength g, g, g3 g4 g3 g,

= = = = = =

0.050 0.100 0.250 0.250 0.300 0.050

aging as shown in Figure 3 [6], for example. Each spectrum was chosen to explain all experiments performed with the corresponding material. Let us examine a volume recovery experiment and a creep experiment under progressive aging at the same temperature. Then the shift functions b' (A,A^) and ag should be comparable if the prehistories are approximately the same and the change in free volume is the origin of aging effects. EXPERIMENTAL AND RESULTS A detailed description of the materials used and the experimental techniques are given elsewhere [6,9,11,12,17-19,26-29]. This paper is restricted to the presentation of some experimental results and important comments necessary for understanding and interpretation. The polymers investigated were a commercial polystyrene type Hostyren N 7000 and a polycarbonate type Makrolon 2800. The abbreviation "o.W." at PS N 7000 o.W. refers to the sample preparation. All creep measurements were carried out in torsion, and the sample was loaded with a maximum shear stress of about 15 kPa for PS and 10 kPa for PC, respectively.

692


PS N 7 0 0 0 o-q

= 6 0 K/h. To = yO^C

• q = 60 K/h * ta = lO^s • to = 10*3

L 10

- theory

0,985

6 retardation times

0,980

0.975 J

0,970

T,°C 70

80

90

100

110

Figure 3. Volume contraction behavior and volume expansion behavior of PS N 7000 o.W. for different degrees of aging; lines have been calculated from theory using the values given in Table 1.


693

which results in a maximum shear strain < 1%; i.e., only the range of linear creep behavior is considered. Prior to the creep measurements the specimen was always cooled from the equilibrium state at 115°C in the case of PS N 7000 o.W. and 150°C in the case of PC 2800 down to the aging temperature T^ by natural cooling. At a temperature of T = T^ + 6K, counter-heating is started to reach a smooth approach to the aging temperature. Thus, for cooling the last 6 K about 6 minutes are necessary whatever the value of T^. Dependent on the aging temperature, therefore, the sample is cooled through the glass transition range at different rates. The higher the value of T^ the lower the cooling rate and the lower the value of the free volume, which is frozen in at the glass transition temperature T . These different prehistories have to be considered in the theoretical calculations. Figures 4 and 5 [19,26,27] show creep measurements on PS N 7000 o.W. at the aging temperatures of T^ = 90°C and 60°C, respectively. The creep curves for PC 2800 at T^= 120°C are given in Figure 6 [12,29]. The full curve in Figure 4 represents the measured equilibrium creep curve, where the preconditioning time was chosen long enough to reach equilibrium in volume. For PS N 7000 o.W. t^ is about 4 months to reach equilibrium at 90°C. For PC 2800 the equilibrium creep curve at T^ = 142.5°C is shown (Figure 6). After completition of the various preconditioning times indicated, the creep experiments were started. The creep curves under progressing aging show the general shape as anticipated in Figure 2. At short creep times they have the shape of the equilibrium creep curve

J(t,A),

10

i

Pa

T

PS N 7 0 0 0 O.W. te<S

O Q

A O

10

X

T = 90°C

2,1 2i3 215

V

2 17

2"

/

V

equilibrium

10-%

10

t.

-10

10"

10^

10'

10'

10'

10

10"

10^

10'

Figure 4. Creep curves of PS N 7000 o.W. under the influence of progressive aging at T^ = 90°C after various preconditioning times; full line represents the equilibrium creep curve at T .

694


10"

Figure 5. Creep curves of PS N 7000 o.W. under the influence of progressive aging at T^ = 60°C for three preconditioning times.

Figure 6. Same as in Figure 5 but now for PC 2800 at T^ = 120°C; full line represents the equilibrium creep curve at T = 142.5°C.


695

with positive curvature. When the creep time reaches values between 0.1 and 0.3 t the creep curves under progressing aging deviate in the shape of the equilibrium creep curve in the direction of smaller deformations. They show a continuously decreasing slope, dlogJ/dlogt, until the creep time approaches the preconditioning time necessary for volume equilibrium. Then the creep curves show a positive curvature again and, finally, all converge into the equilibrium creep curve [18]. A remarkable feature of these creep curves is the different initial plateau in the creep compliance at short creep times. The creep compliance is shifted to lower values in the short-time region with increasing preconditioning time, t^. As a first consequence it means that the creep curves in the short-time region cannot be brought to coincidence merely by a horizontal shift parallel to the logarithmic time axis; an additional vertical shift is necessary. In Figures 7 and 8 a second set of creep measurements is shown. Again, prior to the creep experiment the sample was cooled down from the equilibrium state at 115°C (PS N 7000 o.W.) and 150°C (PC 2800), respectively, to various aging temperatures, T^, by natural cooling. Then the creep measurements were started after identical elapsed preconditioning time t^ = 2'^s. Figure 7 consists of measurements of two authors. The temperatures down to 70°C were taken from [26], and the measurements of 60°C and 50°C were taken from [27]. The two authors find a slightly different creep behavior of the same material, which is to be seen at the common aging temperature of T^ = 85°C. In [27], the creep compliance is shifted

Figure 7. Creep curves of PS N 7000 o.W. for a preconditioning time tg = 2^2 s at different aging temperatures T^.

696


x-6

10"

137.5°C

135.0°C

130.0°C 120.0°C lOO.O^C t, s I

10"^

10°

10^

10^

10^

'

10^

' ' '""I

10^

10^

10^

Figure 8. Sanne as in Figure 7 but now for PC 2800 (t = 2^3s).

to lower values at longer creep times (crosses at 85°C). This difference could be explained by a slightly different prehistory, and we will examine this again. In Figures 7 and 8 we observe a creep behavior similar to Figures 4 to 6, but now the J(t) curves show a temperature-dependent initial plateau in the short-time region. With decreasing aging temperature, the creep compliance is shifted to lower values and the whole curve is shifted to longer times on the creep time axis. Due to the extremely flat course at low temperatures the curves can only be brought to coincidence by vertical and horizontal shifting. DISCUSSION From Figures 4-8 we get some indications that for a theoretical description it is necessary to find an expression regarding the time-age and the time-temperature dependence of the creep compliance. As an example, we start the evaluation of the time-age shift function for the measurements at 90°C shown in Figure 4 for PS N 7000 o.W. Disregarding provisionally the short-time region, we determine the time-age shift function, b' (A,AJ, according to Equation 16 for creep times longer than 0.1 t^. The result is shown in Figure 9 (full points) if b' (A,A J is plotted vs. the degree of aging. A, of the sample. As reference creep curve, the equilibrium creep curve was chosen. In the same figure the calculated shift function, ag (open circles), is shown using Equa-


697

PS N 7000 aW.

Figure 9. Shift factors b'(A,AJ and ag determined from creep measurements and calculated by means of the multiparameter model for PS N 7000 o.W. at T = 90°C.

tion 24 and the parameters of the multiparameter model from Table 1; a. from Equation 24 can easily be transformed in: ffco

(25)

with f^ as the equilibrium value of the free volume at 90°C and f as the instantaneous value of the free volume changing with progressive aging. In the calculation of ag we take into account that prior to the creep experiment the sample was cooled down through the glass transition range at a rate of approximately 0.025 K/s [6,26]. During the cooling period the changes in 5 and, thus, in free volume f were calculated by using Equations 17 to 24 assuming a linear cooling rate -q = dT/dt. Consequently, the actual state of the sample is known immediately before starting the creep experiment, and the changes in free volume during the creep experiment at T^ are fully characterized by the variation of ag. A detailed description of the calculation method is given in references [4,6]. Apart from some difference in the time position, both functions show the same, approximately linear dependence on the logarithm of the degree of aging in the overlapping time interval. Whereas the course of ag can easily be calculated up to the equilibrium state in volume, which is reached when ag is unity, the evaluation of b' (A,A^) is limited at long creep times due to the lack of experimental values.

698


The slight time difference between the two shift functions arises from the time position of the reference equilibrium creep curve chosen for the determination of b' (A,A^). This position is about 1.5 • lO^s for a creep compliance value of J(t)= 10"^Pa"' (Figure 4). Due to the long preconditioning and the long creep time, the exact position of this curve on the time scale may be somewhat uncertain. If the time position of 1.5 • lO^s is reduced to lO^s, the two curves in Figure 9 will coincide. The necessity to find an adequate reduction method for the creep curves in the short-time region is obvious if we try to construct the function b'(A,AJ from Figures 5 and 7 for creep times smaller than 0.1-0.3 t^. Due to the extremely flat course of the creep curves at low temperatures, the time-age shift function without vertical shift is only accessible at very long creep times, i.e., at 50°C for creep times longer than 5 • lO^s. Another problem arises in measuring the equilibrium creep curves at low temperatures, which has to be done if b'(A,AJ is to be determined according to Equation 16. Within a reasonable experimental time scale the equilibrium state in volume can only be reached at temperatures near T [4,6,10,28]. At T^ = 90°C the preconditioning time for the equilibrium creep curve is about four months, at 85°C about 30 years, and at 70°C it would be thousands of years. The problem respecting the equilibrium creep curve is solved since it has been shown in [9] that for the evaluation of the time-age shift function it is not absolutely necessary to know the exact time position of the equilibrium creep curve at the aging temperature, but it is sufficient to know the shape of this creep curve. The problem of the reduction of the J (t) curves in the short- time region is still present. Dependent on T^ and t^ the variation of the J (t) values at short creep times is up to 15-20% in the interval regarded. The only quantity that can explain this strong time and temperature dependence of the creep compliance is the free volume. Other reduction methods [13,30], for example pT/p^^T^, lead in this temperature interval to a variation of about 1%. In Figure 10 for PS N 7000 o.W. and in Figure 11 for polycarbonate, there are plotted J(t) values at a creep time of 1 s vs. the corresponding free volume, calculated by means of the multiparameter model with respect to the various temperature prehistories. For PS N 7000 the J(t) values are taken from Figures 4, 5, and 7. Some additional values, which are not discussed here, are given [19,26,27]. Apart from the values characterizing the equilibrium creep compliances (open squares) all J(t = Is) values show approximately the same dependence on the free volume and can be described by the same straight line if tolerances of 5% in J(t) and ± 0.2 K in the temperature are allowed. Taking as a base a known reference value J^(ls) and the corresponding value of the free volume f any J(ls)-value may be calculated by: J(ls)= j / l s ) + m ( f - f )

(26)

where the slope m is 1.5 • lO'^Pa'. From Figure 11 we find for PC 2800 m equal to 1.1 • lO-^Pa'. The reduced aging creep curves at T^ = 90°C are shown in Figure 12. As reference value J^(ls) the Is value of the creep curve with a preconditioning time t^ = 2*^s was chosen. All creep curves coincide in the short-time region, and the influence of the reduction vanishes for creep times longer 0.3t. These reduced creep curves


699

14- j ( t = -As) , ^0-'° Pa"'

t 12-

/

.'^ X

error of 1 5 %\

jT

/'

J^

A/

/Xy^

/

X

10-

y^^ equilibrium / 2 6 /

8-

— - f, %

0.8

1.0

0.9

1.1

1.2

1.3

Figure 10. Measured values of J at a creep time t = 1 s for PS N 7000 o.W. plotted vs. calculated free volume values for various prehistories (open dots T = 90°C, t^ = 2"s, 2^^s, 2^H, 2}'^s, 2^H; dots/open triangles t^ = 213s T = 92.5°C, 90°C, 85°C, 80°C, 70°C, 60°C, 50°C; squares T = 85°C, t^ = 2"s, 213s, 215s, 2i''s, 2i9s; open squares equilibrium 92.5°C, 95°C, 97.5°C; filled triangles T = 60°C, t. = 2i6s, 2i«s). 17 J(t=1s), 10

-10 n

-1

Pa

16^

15-1 14 13 12 error of

11 1.6

1.7

1.8

i5 %

1.9

f, % 2.1

2.2

Figure 11. Same as in Figure 10 but now for PC 2800 (crosses t^ = 21^3, T = 50°C, 60°C, 70°C, 80°C, 90°C, 100°C; 110°C, 120°C, 125°C, 130°C, 135°C, squares t^ = 2^^s, 100°C; 120°C, 125°C, 130°C, 135°C, dots t^ = 1, t^= 219s, T = 50°C, 70°C, 90°C, 100°C; 110°C, 120°C, 125°C, 130°C, 135°C).

700

Advances in Engineering Fluid Meclianics Pa

10

-7

10

i

T--:90''C 2ii

i

-0 .

10 +

10

T

PS N 7000 O.W.

• i i

i

-10

10

->t.

-f-

10"

10

10"

10"

10'

10"

10"

^10'

Figure 12. Creep curves from Figure 4 reduced according to Equation 26.

can now be evaluated by traditional methods [9,18,29] and the shift function b'(A,AJ can be determined even in the short-time region. With comparable success, this reduction is applied to the creep curves at different aging temperatures shown in Figure 13. Again the J (Is) value of the creep curve at 90°C with a preconditioning time t^ = 2'^s was chosen as the reference value and the time-age shift functions b'(A,A^) are constructed from these reduced curves. The result is to be seen in Figure 14 with the equilibrium creep curve at T^ = 95°C as selected reference creep curve and in Figure 15 for PC 2800 with the equilibrium creep curve at T = 142.5°C, respectively. Because an unreal equilibrium creep curve at each aging temperature was used for the construction indicates the index r at A^ referring to 95°C (142.5°C for polycarbonate) for all creep curves. While the time-age shift function in Figure 9 takes positive values, the shift functions in Figures 14 and 15 have negative values. This effect is due to the different time position of the reference equilibrium creep curve with respect to the aging creep curves. Depending on the aging temperature, the time-age function is shifted to more and more negative values. The calculated a,, functions are shown as drawn lines. At T = 60°C and T = 50°C we have o

a

a

renounced the presentation of the experimentally determined b'(A,A^), functions and only the calculated ag functions are plotted. An excellent agreement between experiment and theory is observed over the whole temperature and time interval considered for both materials. As mentioned above, for the theoretical calculations it is important to take into account the different


10~^

10^

10^

10^

10^

10^

10^

10^

701

10^

Figure 13. Creep curves from Figure 7 reduced according to Equation 26.

prehistories prior to the creep measurements. In Tables 3 and 4, the estimated cooling rates used for the calculation are given for each aging temperature. In the present case the temperature history is approximated by linear cooling through the glass transition range, but further improvement may be achieved by simulating the true natural cooling. This is easily verified by introducing an exponential decay for the cooling rate in Equation 20. It should be mentioned that for temperatures below 65.5°C, which equals T^ for PS N 7000 (90.2°C for PC 2800), some considerations are necessary to adapt the free volume concept for working at low temperatures [5,7,16,20,22,30-33], but this is not discussed here. To sum up the results, the multiparameter model based on free volume leads to a good description of the time-age shift function as well as to a reasonable reduction of the creep compliance in the short-time region under various thermal prehistories. Being aware of that, it must be possible to predict aging creep curves at various preconditioning times and aging temperatures from only one known, optional equilibrium creep curve by applying the multiparameter model. The results are shown in Figures 16 to 20. The calculations were performed using the following scheme: 1. Take any known equilibrium creep curve as reference. 2. Simulate any wished prehistory by applying the multiparameter model. (text continued on page 704)

702


Figure 14. Shift factors b'(A,A^) determined from creep measurements (symbols) and a^ calculated by means of the multiparameter model (drawn lines) for PS N 7000 o.W. at different aging temperatures T^.


-1—I

I I 11

10^

I

1—I—I—I

I I 11

10^

1

1—I—I—I

I I I[

10^

Figure 15. Same as in Figure 14 but now for PC 2800.

703

704

Advances in Engineering Fluid Mechanics Table 3 Estimated Cooling Rates for PS N 7000 o.W. Ta. °C

- q , K/s

92.5 90.0 85.0 70.0 60.0 50.0

0.005 0.025 0.080 0.25 0.25 0.25

Table 4 Estimated Cooling Rates for PC 2800 Ta, °C

- q , K/s

137.5 135.0 130.0 120.0 100.0

0.04 0.04 0.08 0.32 0.32

(text continued from page 701) 3. Calculate the time-age shift function ag according to Equation 25 with respect to the f and 8 values obtained from step 2. 4. Apply the calculated shift function to the equilibrium creep curve and construct the desired aging creep curve by calculating the corresponding ^(t) values (c.f. Equation 7). 5. Calculate the vertical shift by means of Equation 26 with respect to the f value obtained from step 2. 6. Apply the vertical shift to the constructed aging creep curve. In Figures 16 to 20, a comparison is made between experiment (symbols) and theory (solid lines). As can be seen, an excellent description of the aging behavior is obtained for both materials investigated. Mentionable discrepancies between theory and experiment can only be stated for PS N 7000 o.W. in Figure 19 at the aging temperature of 50°C, where the theoretical curve is shifted to somewhat shorter creep times. But we know from Figure 7 that the two lowest creep curves are generally showing a slightly different creep behavior. An explanation could be a different temperature prehistory or a slightly different stress history. Nevertheless, these deviations only lead to an insignificant underestimation of the influence of


PS N 7 0 0 0

705

O.W.

Figure 16. Comparison for PS N 7000 o.W. between creep compliance from the experiment (symbols) and calculated from theory (drawn lines) at T = 90°C for various preconditioning times.

J(t) , Pa

theory

10"

t, s 10'

10"

10'

10^

lO-"

10*

10^

10'

10'

Figure 17. Comparison for PS N 7000 o.W. between creep compliance from tiie experiment (symbols) and calculated from theory at T^ = 60°C for three preconditioning times.

706


10 10"^

10°

10^

10^

10^

lO"*

10^

10^

10^

Figure 18. Same as in Figure 17 but now for PC 2800 at T = 120°C.

10-6 .

J ( t ) , Pa-^

^ 95°C, equil

t 10

92.5°C /

-7_

^

90°C

4>

— theory

^

10

/

-8_

/^ /

y

10

85°C

7

70°C

p^

'T'

-9_

— ;

10"^

1 1

-r^

1

10°

10^

10^

'

10^

1

t, s

' 1 '

10^

10^

10^ 10^

Figure 19. Same as in Figure 16 but for PS N 7000 o.W. at a preconditioning time t^ = 2^^ s for various aging temperatures T


707

10"^ J(t) , Pa"^

137.5°C

t 135.0°C

10"^theory

130.0°C •

10"^

120.0°C 100.0°C

10'^10"^

10°

10^

10^

10^

10^

10^

10^

10^

Figure 20. Same as in Figure 19 but now for PC 2800 at a preconditioning time tg = 2^3 s for various aging temperatures T^.

aging on the creep behavior at low temperatures. In the case of PC 2800, a small discrepancy is shown in Figure 18 for the aging creep curve at 120°C with a preconditioning time of t^ = 2'^s with a similar tendency as discussed for polystyrene. For the calculations performed on polycarbonate at different aging temperatures with a fixed preconditioning time t^ of 2'^s deviations are found for the creep curves at T^ = 130°C. In this case the theory underestimates the measured values. This behavior, as already stated for polystyrene, may be attributed to a somewhat different thermal prehistory in the experiment to that chosen for the calculation. The present contribution has shown that the creep behavior of amorphous polymers under the influence of progressing aging can be well described and predicted under any thermal prehistory applying the multiparameter model based on free volume. The only condition necessary is the knowledge of any measured equilibrium creep curve. For each material the multiparameter model with the given set of parameters allows the prediction of the behavior in volume under any complicated thermal history as well. Introducing some additional postulations, the free volume model is adapted to work at low temperatures, i.e., at temperatures below T^. Next, the theory should be extended to measurements at still lower temperatures as well as to some other amorphous polymers. Prediction of the long term mechanical behavior of polymers is still one of the great unsolved problems in the plastic industry, and this contribution provides a step in this direction.

708


NOTATION A Degree of aging AQ, A ^ Fixed degree of aging A^ Reference degree of aging for the time-age shift a(T,TQ) Time-temperature shift factor a^ Temperature dependent part of the shift factor ag Structure dependent part of the shift factor b Doolittle constant b'(A,Ap) Time-age shift function WLF-constants Fractional free volume Value of the fractional free volume at T r

Equilibrium value of the fractional free volume gi Normalized retardation strength of the i-th relaxation process Shear creep compliance

JQ Shear creep compliance at t = Os J Creep rate n. Ordering parameter m Slope of the dependence of J ( t = Is) on f q Rate of temperature change t Creep time t^ Preconditioning time T Temperature T^ Reference temperature for temperature superposition TQ Temperature at t = Os T^ Aging temperature T Glass-transition temperature T^ Reference temperature for the time-age shift V Time-dependent specific volume v^ Equilibrium value of the specific volume

Greek Symbols a, Expansion coefficient of the rubbery state a Expansion coefficient of the glassy state Aa Difference in thermal expansion coeffient Aa. Difference in thermal expansion coeffient for the i-th retardation process 8 Total deviation of the fractional free volume from equilibrium

8. Deviation of the fractional free volume from equilibrium for the i-th retardation process Y Strain jl Effective time at degree of aging AQ, A^ respectively a Stress T. Retardation time of the i-th process T Retardation time of the i-th process at T^ ^ History time

REFERENCES 1. Struik, L. C. E., Physical Aging in Amorphous Polymers and other Materials, Elsevier, Amsterdam 1978. 2. Hopkins, I. L. J. Polym. Sci. 1958 28, 631. 3. Haugh, E. F. J. Appl. Polym. Sci. 1959 1, 144. 4. Greiner, R. and Schwarzl, F. R. Colloid Polym. Sci 1989, 267, 39. 5. Kovacs, A. J., Aklonis, J., Hutchinson, J. M. and Ramos, A. R. J. Polym. Sci. Polym. Phys. Edn. 1979, 17, 1,097.


709

6. Greiner, R. Ph.D. Thesis, University of Erlangen, 1987. 7. Dolittle, A. K. /. Appl. Phys. 1951, 22, 1,471. 8. Hutchinson, J. M. and Kovacs A. J. in The Structure of Non-Crystalline Materials (Ed., P. H. Gaskell), Taylor and Francis, London 1977. 9. Schwarzl, F. R., Link, G., Greiner, R. and Zahradnik, F. Prog. Colloid Polym. Sci. 1985, 71, 180. 10. Greiner, R. and Schwarzl, F. R. Rheol. Acta 1984, 23, 378. 11. Link, G. Ph.D. Thesis, University of Erlangen, 1985. 12. Kaschta, J. Ph.D. Thesis, University of Erlangen, 1991. 13. Ferry, J. D. Viscoelastic Properties of Polymers, 3rd Edn, Wiley, New York, 1980. 14. Schwarzl, F. R., Link, G., Greiner, R. and Zahradnik, F. in Advances in Rheology (Eds, B. Mena, A. Garcia-Rejon and D: Rangel-Nafaile) Univ. Nac. Autonoma de Mexico, Mexixo City, 1984, Vol. 1, 211. 15. Greiner, R. Polymer 1993, 34, 4,427. 16. Williams, M. L., Landel R. F. and Ferry, J. D., J. Am. Chem. Soc. 1955, 77, 4,701. 17. Lang, G. personal communications, Erlangen, 1983. 18. Fegfar, H. personal communications, Erlangen, 1985. 19. Gabler, H. personal communications, Erlangen, 1986. 20. Kovacs, A. J. Fortschr. Hochpolym. Forsch. 1963, 3, 394. 21. Narayanaswamy, D. S. J. Am. Ceram. Soc. 1971, 54, 491. 22. Gibbs, J. H. and DiMarzio, E. A. J. Chem. Phys. 1958, 28, 373. 23. Adam, G. and Gibbs, J. H. /. Chem. Phys. 1965, 43, 139. 24. Chow, T. S. and Prest, W. M. Jr. J. Appl. Phys 1982, 53, 6,568. 25. Chow, T. S. J. Polym. Sci. Polym. Phys. Edn 1984, 22, 699. 26. Kurzendorfer, R. personal communications, Erlangen, 1987. 27. Grimm, T. personal communications, Erlangen, 1988. 28. Bilwatsch, D. personal communications, Erlangen, 1986. 29. Pannkoke, K., personal communications, Erlangen, 1989. 30. Pfandl, W., Link, G., Schwarzl, F. R. Rheol. Acta 1984, 23, 277. 31. Cohen, M. H. and Turnbull, D. J. J.Chem. Phys. 1959, 31, 1,164. 32. Turnbull, D. J. and Cohen, M. H. J.Chem. Phys. 1961, 34, 120. 33. Hutchinson, J. M. and Kovacs, A. J. J. Polym. Sci. 1976, 14, 1,575.


CHAPTER 27 DIE EXTRUSION BEHAVIOR OF CARBON BLACK-FILLED BLOCK COPOLYMER THERMOPLASTIC ELASTOMERS Jin Kuk Kim Department of Polymer Science & Engineering Gyeongsang National University Chinju 660-701, Korea and Min Hyeon Han R & D Center, Kumho & Co., Inc. Sochondong, Kwangsanku Kwangju 506-040, Korea CONTENTS INTRODUCTION, 711 SHEAR FLOW IN CAPILLARY RHEOMETER, 712 SHORT REVIEWS ON THE RHEOLOGICAL BEHAVIORS OF UNFILLED BLOCK COPOLYMER SYSTEMS, 713 CARBON BLACK-FILLED BLOCK COPOLYMER THERMOPLASTIC ELASTOMER SYSTEMS, 714 Fundamentals for Black-Filled Systems, 714 Experimental, 715 Rheology, 716 Die Swell and Extrudate Distortion, 720 CONCLUSION, 725 NOTATION, 734 REFERENCES, 734 INTRODUCTION The appearance of several new types of rubbers called thermoplastic elastomers was apparently a most exciting development in recent years. These materials have properties of conventional rubbers such as high strength, and they are readily fabricated by various melt-processible techniques, which are characteristic of thermoplastic materials because they do not have any chemical cross-links in the 711

712


structure. Therefore, thermoplastic elastomers have become important commercial materials due to their good processability and high mechanical properties. Of the various types of thermoplastic elastomers, block copolymer elastomers are the largest. Block polymers generally are considered to consist of two or more segments of different composition joined end to end, and as segments can be various homopolymers or copolymers, a very large number of block polymers can be prepared [1]. Especially, copolymers with multiple blocks of polystyrene connected by rubbery segments of polybutadiene or polyisoprene exhibit high strength and elastomeric characteristics without chemical cross-linking, and this type of copolymer is the subject of our concern. Even though thermoplastic elastomer has been widely studied [1-23], there has been relatively little serious research of the rheological behavior of the block copolymer systems [20-31]. Data on the effects of carbon black on the extrusion behavior are even more limited. In this work, we investigated the effects of carbon black on the triblock copolymer thermoplastic elastomers SBS (styrene-butadienestyrene), SIS (styrene-isoprene-styrene), and SEES (styrene-ethylene/butylene-styrene). We mentioned both the rheological behavior of the carbon black-filled polymers and die swell phenomena. SHEAR FLOW IN CAPILLARY RHEOMETER We now briefly describe some common viscometric methods to characterize the shear flow in capillary viscometers used frequently for observing the die extrusion behavior of most polymer systems. Detailed descriptions of these techniques and the other theoreticals may be found in some well-organized textbooks and monographs [32-38,41-45]. The capillary tube is the most important industrial and laboratory instrument for measuring rheological properties of thermoplastics and rubbery materials. Its use for polymer systems dates back to the 1930s [32]. In its simplest configuration, the capillary rheometer consists of a small tube through which polymer melt or rubbery compound is made to flow, either by means of an imposed pressure or a piston moving at a fixed speed. The quantities normally determined are the volumetric flow rate and pressure drop. However, all the pressure drop, usually measured as the difference between the pressure applied to force the fluid into the capillary from the reservoir and atmospheric pressure, is not due to overcoming frictional drag at the capillary walls. A significant part of the pressure loss arises on the die entrance and die exit regions due to the die end effects [33-35]. Thus, the total pressure drop can be expressed as the sum of pressure drops in die land, die entrance, and die exit regions, that is, AP = AP, + AP t

die

ent

= AP, + AP die

e

+AP exit

(1-a) ^

^

(1-b) ^

'^

where AP^ is the pressure drop due to the die ends effect. It is found that AP^^^ is about three times larger than AP^^.^, and in some cases AP^^.^ is ignored [44]. Shear stress at the wall, (o^^)^, is given by

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

(0,2)w = ^ P d , e ( ^ ]

713

(2)

where D is the capillary diameter and L the capillary length. From the Equations 1-b and 2, we have AP^ - 4(a,,)^(L/D) + AP^

(3)

where the subscript 1 denotes the direction of flow, subscript 2 the direction of shear. Using Equation 3, wall shear stress and the pressure drop in the die ends are determined by plotting the total applied pressure as a function of the capillary length-to-diameter (L/D) ratio. The slope of the data will yield the wall shear stress, (aj2)^, and the intercept of this straight line with the AP^ axis yields the end effect. The wall shear rate, y^, in a fully-developed laminar flow can be obtained from the apparent shear rate using the equation Yw

3n^ + l 4n'

32Q

(4)

where Q is the volumetric extension rate, and the parameter n' is determined from the slope in the double logarithmic plot of wall shear stress, (o^^)^, against apparent shear rate, 32Q/7cD\ as:

dlog(32Q/7iD')

^^^

Equation 4 is due to Weissenberg [36-38].

SHORT REVIEWS ON THE RHEOLOGICAL BEHAVIORS OF UNFILLED BLOCK COPOLYMER SYSTEMS As previously mentioned, the growth of the block copolymer thermoplastic elastomer industry has now reached a high level of commercial importance. However, it is still difficult to study the rheological behaviors of thermoplastic elastomers because they exhibit complex behavior that combines the elastomeric final product properties with the processing characteristics of thermoplastics. Recently, the rheological behavior of the styrene block copolymers (SBS, SIS, and SEBS) was studied by Kim and Han [31]. A series of systematic investigations for the viscoelastic behavior of triblock copolymer thermoplastic elastomers also were conducted by Mathew et ai [22,23]. Kim and Hyun [20] reported the viscoelastic properties of SEBS. A unique characteristic of SBS and SIS block copolymers is their melt viscosities [24,25]. Under low shear conditions, these are significantly higher than those of either polybutadiene, polyisoprene or random copolymer of styrene and butadiene of equivalent molecular weights [26-28]. Furthermore, these block copolymers show non-Newtonian behavior in which their viscosities increase as the shear is decreased.

714


This behavior can be found under dynamic conditions [29,30] as well as steadystate, and it is attributed to the two phase-structure. Above a certain molecular weight, the polystyrene segments are phase separated at all temperatures of practical importance, and so even though the polystyrene is above its Tg, it requires an extra amount of energy to bring it into the rubbery phase [45]. The increased viscosity, thus, can be explained from this energy. In addition, this energy increases with the degree of incompatibility between rubbery and thermoplastic segments, and, consequently, the viscosity also increases with the segmental incompatibility. This can be seen clearly in similar SEES block copolymer thermoplastic elastomers, which exhibit relatively high viscosities due to their extreme segmental incompatibility. Figure 1 indicates this behavior well [31]. SEES is found to have very high shear viscosity. The triblock styrene-butadiene thermoplastic elastomer (SES) shows higher viscosity than diblock copolymer of the same component (SER706), and, as expected, polyisoprene rubber shows the lowest viscosity value compared to those of all other block copolymers. CARBON BLACK-FILLED BLOCK COPOLYMER THERMOPLASTIC ELASTOMER SYSTEMS Fundamentals for Black-filled Systems In general, fillers decrease melt flow because they increase the viscosity of materials. Other changes to be expected include an increase in tear strength, flex

1E5 O • A A D

o

a.

oin o to

1E4 4.

SBR706 SCOS 1R2200 SOS SIS

X 1000

< GL

100 10

100

100C

SHEAR RATE(1/S)

Figure 1. Viscosity curves of various styrene copolymers at 100°C. SEBS, SBS, and SIS are styrene triblock copolymers, SBR706 is styrene diblock copolymer, and IR2200 is polyiso-prene homopolymer [31].


715

life, and abrasion resistance imparted by fine particle-size silicas and carbon blacks [1,2]. In the technical sphere, the term carbon black is understood to mean a special form of carbon obtained through partial combustion of liquid or gaseous hydrocarbons. Carbon black, used for centuries, only began to be used as a reinforcing filler at the start of this century. Carbon black is the most important reinforcement employed by the rubber industry [46-48]. On the contrary, as regards its use in plastics, it is usually used to achieve volume conductivity and to improve heat deflection temperature. In addition, carbon black is frequently utilized as a protection against UV radiation and as a black pigment in thermoplastics [49]. Theoretical study for filled systems originates with Einstein's well-known treatment of the viscosity of a dilute suspension of rigid spheres [50,51] as Ti, = Ti^d + 2.5^)

(6)

where r|, denotes the viscosity of the mixture, ri^ the viscosity of suspending medium, and (j) the volume fraction of the spheres. Therefore, for filled systems, r|, can be considered as the viscosity of filled polymer compound, ri^ as the viscosity of unfilled polymer, and (|) as the volume fraction of the filler [2]. The first-order treatment has been modified by many researchers to take into account the mutual disturbance caused by spheres at higher concentrations, such as bound rubber segments between fillers [2]. Among various equations, that of Guth and Gold [51] including second-order term ri, =Ti^(l + 2 . 5 ^ + 14.1^2)

(7)

is perhaps most familiar. However, one must know the fact that, in most cases, active reinforcing fillers such as carbon black produce a much greater increase in viscosity than predicted by Equation 7. Mooney [52] and others [53,4] have also tried to extend Einstein's treatment over a wider concentration range. The equation by Mooney is In

1-0/^

(8)

where K^ is the Einstein coefficient (-2.5) and (|)^ the maximum possible filler fraction. Unfortunately, for block copolymer thermoplastic elastomers, any trials to accommodate the viscosity changes by carbon blacks into theoretical considerations, or to fit the equations cited onto the rheological behavior of black-filled systems have not been conducted up to now. Of course, such trials will be relatively complex and time-consuming work because thermoplastic elastomer shows intricate properties due to its unique microstructure. Experimental The paper written by the authors [44] was thought to be the only one reported and published for the die extrusion behavior of carbon black-filled black copolymer

716


thermoplastic elastomers. Hence, inevitably, most of this chapter was based on the results of our experiments in that monograph. We used three styrene-containing thermoplastic elastomers (SBS, SIS, SEBS, manufactured by Shell Chemicals). Carbon black, N330 (HAF, manufactured by Ashland), was used to prepare compounds with the thermoplastic elastomers. The characteristics of the elastomers and carbon black used in that work are summarized in Table 1. In particular, the monomer ratio in the thermoplastic elastomers based on polystyrene segments and other rubbery segments is an important factor and exerts a significant effect on properties [3-7,24]. One hundred parts of thermoplastic elastomers were compounded with 10 and 20 parts by weight of N330 carbon black, respectively, in a Haake Buchler Rheocord 750 laboratory mixer. The mixing temperature was maintained at 120°C, and materials were prepared at a fill factor of 0.8. After mixing, the compound was carefully remilled into flat sheet on a two-roll mill. Measurements for rheological behavior of carbon black-filled block copolymer thermoplastic elastomers were carried out in a Monsanto Processability Tester (MPT) as a capillary rheometer, which covers the shear rate range of 10^ to 10^ s ' , shown in Figure 2. We used capillary dies (1mm diameter) with different L/D ratios of 5, 10, and 20. The investigation of extrudate swell and shape was carried out using an MPT. At 100°C, compounds were extruded through capillary dies at a series of fixed shear rates. When the steady state was reached for a selected shear rate, the extrudate was cut and allowed to relax. The diameter of the extrudate was measured using a microscope (Gaertner Scientific Co., Model BC21). Rheology The investigation of the rheological behavior of the materials is important in understanding the processability. In this chapter, emphasis is placed mainly on the rheological behaviors of three styrene copolymers: SBS, SIS, and SEBS. As mentioned previously, these materials are considered to be thermoplastic elastomers as they combine the hard and soft segments of molecules in the microstructure (Figure 3). The polystyrene domains (hard segment), which are chemically bound to rubbery segments, function as physical cross-links. Figure 4 represents the typical phase-separated microstructure of block copolymer as a function of composition

Table 1 Materials Used in the Work of Kim et al. [44] Materials

Thermoplastic Elastomer Carbon black

Name (Grade)

Characteristics

SBS SIS SEBS N-330(HAF)

Styrene/rubber ratio 31/69 Styrene/rubber ratio 14/86 Styrene/rubber ratio 13/87 Iodine adsorption 81.9 (mg/g) DBF absorption 101.3 (mL/lOOg)


717

Piston

Pressure Transducer Capillary

Die

Figure 2. Schematic diagram of Monsanto Processability Tester (MPT).

[55]. With increasing or decreasing of one component to another, the domain structure changes from sphere to lamellae through cylindrical shape, or vice versa. This study concentrated on the effect of carbon black on the characteristics of the thermoplastic elastomers. Figure 5 shows the shear viscosity behavior of three triblock copolymers. The figure also shows the shear thinning behavior of polymers. In the case of carbon black-filled polymers, the slopes are more sharply increased than are those of the raw polymers, and the viscosity increase by carbon black is greater than the values predicted theoretically by Equation 7. The viscosity of SEBS is higher than that of the other polymers. This is apparently attributed to the large segmental incompatibility of SEBS, as previously stated. The addition of carbon black to the raw polymers is frequently induced to increase the viscosity. The reason is that the carbon black particles within the polymer increase the fluid friction. We now consider the entrance effect of converging flow when the fluid is moving from a large diameter reservoir into a small diameter tube. The researchers [33-35,39,40,56,57] observed the occurrence of extra pressure drops in the entrance (text continued on page 720)

718


Thermoplastic domains

Figure 3. Phase structure of thermoplastic elastomers.

A spheres/ B matrix

A cylinders/ B matrix

A,B lamellae (alternating layer)

B cylinders/ A matrix

B spheres/ A matrix

Increasing A - content (or, Decreasing B - content)

Figure 4. Schematic representations of five types of microstructures for phase separation of block copolymer as a function of composition [55].


SBS

° -

. - A

.

,

100

10

SHEAR RATE

. sec

.

.

.-

• .> ^ J

1000 -1

Figure 10. The effect of carbon black contents on Couette correction of styrene block copolymer. SBR706 is the block copolymer of polystyrene and polybutadiene [31]. (text continued from page 720) die length (L/D = 1 0 ) . From the photographs, one can observe that more severe distortion occurs at higher shear rates. However, rough surface of extrudates at 4 s"* shear rate are observed. Extrudate distortion also is more severe at lower black loading. When the die length is varied, different die swell behavior generally is observed. Figure 12 shows the effects of die lengths. The distortion of extrudates decreases with increasing L/D ratios. The swell ratio at various shear rates and die lengths were measured. Here, die swell ratio is defined by the ratio of the extrudate diameter, d, to the capillary diameter, D. Figure 13 shows the die swell ratio vs. shear rate, extruded in a capillary with an L/D ratio of 10. This result indicates that the carbon black-filled polymers swell less than the raw polymers. We also know that the swell ratio increases with increasing shear rate. The effect of the die length on the die swell is shown in Figure 14. From this figure, the lower value of the swell ratio at longer die lengths can be seen as expected. CONCLUSION The main objective of this chapter was to cover the die extrusion behavior of block copolymer thermoplastic elastomers filled with carbon black. Thus, shear (text continued on page 734)

726


Shear rate (Sec*) 4 18

N0CA8BC»
^

II

S II

s - - sII s

S II -— CH2 CH2 — S -

s II

Trimer

s

(5)

Polysulfides

739

or it can react with the monomer (2) to give a trimer (reaction 6),

S II S-S-CHjCH^Cl - II S

+

CI-CH2-CH2-CI

Dimer

Monomer (2) S -CI

CI - CH, CH, - S - S - CH, CH, C)

-

-

(6)

II

Trimer or it can react with itself to form tetramer and so on until several monomer units are joined together to form oligomers with structures (1-3).

Q 1 VVA/VS/VW> Q ]

S

S

II

II

s-

II

-s SII

s Structure 1

sII

Structure 2

-s II

S Structure 3

All these successive bimolecular interactions produce a wide distribution of chain sizes which at the latest stages of polymerization are interconnected to produce long polymer chains. INORGANIC POLYSULFIDE MONOMERS The standard method for the synthesis of an inorganic polysulfide monomer is the reaction of anhydrous sodium monosulfide with molten sulfur (reaction 7) [4-5]

Na2S +

(x -1) S

•

Na2Sx

(7)

The reaction of sodium metal with sulfur in their molten form also affords sodium polysulfide (reaction 8) [6] 2Na

+

xS

^

Na2S,

(8)

The reaction of sodium hydroxide with sulfur (reaction 9) is the most common method employed on an industrial scale for the production of inorganic polysulfides [7]. 6NaOH + ( 2 x + 1 ) S

->"

2Na2Sx + Na2S03

+ 3 H2O

(9)

The Na2S^ (x=l-5) can exist as one component or in a mixture form. In general, as the X increases the sodium sulfide becomes more reactive. Thus, Na2S2 is more

740


reactive than Na2S. Pure crystalline compounds of the type Na^S^, Na^S^, and NajS^ can be prepared from the reaction of disodium monosulfide and sulfur as shown in reaction? [8,9]. DIHALIDE MONOMERS The nature of R in dihalide monomers can vary, but usually primary, saturated, or unsaturated aliphatic dihalides are employed since the secondary and tertiary type mainly undergo dehydrohalogenation [4], The number of carbon atoms between two halides can vary, but monomers with five carbons may form a six-membered ring as a side product. For example, 1,5-dichloropentane in the presence of sodium sulfide forms cyclopentylsuflide (reaction 10)

C l — t H j ) ^ ! + NajS

~

^^"^^ >

,

J

(10)

Bromides are more reactive than chlorides, but chlorides often are used mainly because of low cost. The dichloride monomers are employed either alone or as mixtures for the synthesis of commercial aliphatic poly sulfides. Some are represented by structures (4-8) OH CI—fcH^)^! Structure 4

CI—tH^ljCI

CI—CH2 — CH — CH2— CI

Structure 5

CI—fcH2)^0—fcH^)^!

Structure 6 c i — f c H 2 ) ^ 0 — CH2— O—fcH2)-- -fsCH2CHClCHClCH2S+l_

HO-Rj-OH +

S2CI2

>-

4- O-R1-O-S2

(12)

_J P

4-

(13)

Polysulfides

S / \ CH3-CH-CH2

CH3 I 4 - CH.-CH - S ;

+ S

741

(14)

HIGH MOLECULAR WEIGHT POLYMERS The polymerization process generally is carried out in an aqueous solution containing sodium sulfide and small amounts of magnesium hydroxide catalyst [5]. The dihalide is mixed gradually with the aqueous solution while heating the reaction mixture at about 70-80°C. When excess sodium sulfide is used, the reacting polysulfide polymers usually have a number average molecular weight of about 200,000, but the molecular weight drops to about 5,000 when equivalent amounts of reactants are employed. This behavior has been explained by the partial hydrolysis of the aqueous sodium sulfide. In fact, sodium polysulfide undergoes partial hydrolysis (reaction 15). The hydroxide then can react with Na2Sx

+

H2O

NaSx H

+

NaOH

(15)

organic dichloride monomer (reaction 16) or with the growing polysulfide polymer to form hydroxy-terminated oligomer (reaction 17).

CI - R - CI

+

Cl

OH

C! - R -

OH

(16)

NaCl ^/wv.R-SxNa+ C I - R - OH-

-^.^

R - S x - R - OH

(17)

LOW MOLECULAR WEIGHT POLYMERS Low molecular weight polymers, known as liquid polysulfides, are produced commercially by first reacting high molecular weight polysulfides with Na2S03 to remove excess sulfur using the desulfurization process (reaction 18) [5,15,16]. The resulting polymer usually contains two sulfur atoms per repeating unit.

+

R - Sx - R ^

(X- 2)Na2S03

^ - w ^ R - S - S - R ^ A A / v ^ H - (X-2)Na2S203

(18)

After desulfurization, the polymer product undergoes chain cleavage (reaction 19). '^^R-S-S-R^'^

+ Na2S03

>- 2 ^^

RSNa

+ Na2S203

(19)

This oligomeric product is treated with NaHS03 to form the thiol-terminated chain (reaction 20)

742


^/wv. RSNa +

NaHS03

•

^^

RSH + Na2S03

(20)

The sulfur-sulfur bond of high molecular aliphatic polysulfides can be cleaved by NaSH (reaction 21) to produce viscous oligosulfide liquids. R^S-S-R

'AAA/'

-f"

NaSH

•

A ' AAA R S H

-|-

>AAAARSSNO

(21)

The thiol-terminated oligo-sulfides can crosslink easily by using, for instance, epoxy resins (reaction 22) or other polymers having reactive functional groups [1]

' ' ^ RSH + C H 2 — C H ^ ^

•

^^

RS ~CH2 - CH ^^wv.

(22)

Another convenient method for curing liquid oligosulfides, having terminal thiol groups, is the use of metal peroxides, such as lead dioxide (reaction 23) [17] 2 ^y^ RSH + Pb02

^ 24 h

! +

PbO + H2O

(23)

''vw' R — S

PROPERTIES Aliphatic polysulfides resist attack by organic solvents and oils, especially hydrocarbon solvents. They exhibit high flexibility even at room temperature. Their gas permeability is low, but possesses high resistance to aging. Their drawback is that they demonstrate low tensile strength and abrasion resistance. They also release disagreeable odor during processing. Concentration of the odor increases as the number of carbon atoms of the dihalide monomer decreases. APPLICATIONS Aliphatic polysulfides are available commercially either as solids or liquids. Some well-known aliphatic polysulfides are thiokol A, which is prepared from dichloroethane and disodium tetrasulfide. Thiokol ST is made from P-chloroethyl formal (CICH2CH2OCH2CH2CI) and sodium disulfide. Thiokol FA is a terpolymer made from dichloroethane, chloroethyl formal, and sodium polysulfide. Aliphatic polysulfides enjoy wide industrial applications mainly because they demonstrate excellent resistance to organic solvents. This behavior allows them to be used in coatings, gaskets, sealants for fuel cells, gasoline tank sealants, hoses, and balloon fabrics. They also find applications as insulators and electrical or electronic components because they are not affected by light and electrical discharge. Their weatherability and excellent low-temperature flexibility allow them to be used in low-temperature environments (i.e., rocket propellants). AROMATIC POLYSULFIDES The synthesis of aromatic polysulfides dates back to the late 19th century when a number of chemists described the preparation of these polymeric materials from

Polysulfides

743

benzene and sulfur compounds using electrophilic and nucleophilic substitution reactions. The resulting poly(phenylene sulfides) had relatively low molecular weight (3,000-3,5000). Characterization of these polymers is generally poor mainly because of lack of proper instrumentation. Some good discussions on this topic are found in the chemical literature [18-20]. Attempts to synthesize aromatic polysulfides by electrophilic substitution methods have failed to produce linear high molecular weight polymers. During the past three decades a systematic approach has been taken for the synthesis of aromatic polysulfides. The nucleophilic reaction of sodium sulfide with p-dichlorobenzene to give poly(phenylene sulfide) has been a successful process. The polymer produced by this method is abbreviated as PPS and has been commercialized by Philips Chemical Company under the trade name Ryton (reaction 24) -- Maui NaCl

>—.

n C1-- CI + nNa2S

/C^==\

^

"HQ^

(24) -" n

PPS The monomer Na2S can be prepared from the reaction of aqueous sodium hydrosulfide and aqueous sodium hydroxide (reaction 25), followed by dehydration (reaction 26) NaSH (aq) + NaOH (aq) Na2S (aq)

'^

^ •

Na^S (aq)

Na^S (s)

(25) (26)

When p-dichlorobenzene and sodium sulfide are polymerized in the presence of polar aromatic solvents a linear low molecular weight poly(phenylene sulfide) usually is obtained. Its number average molecular weight is in the 16,000-22,000 range [21,22]. However, the molecular weight increases to 32,400-43,200 when the polymerization is carried out in the presence of alkali metal carboxylate [23]. Other aryl halides of scientific interest also can be prepared in the laboratory by this method [24-30]. The aromatic group can be alkyl benzene, naphthalene, biphenyl, diphenyl ketone, diphenyl ether, or diphenyl sulfone. The chlorine groups can be ortho, para or the mixture of the two. Some examples are shown by structures (8-10) «A/VVWvk/^^ I ( ^ H

Structure 9

Structure 8 .AAAAAA^^^^

SO2

\\y/

Structure 10

^

xAAAAAAAA/*

S

744


MECHANISM The reaction steps probably involve the S^" nucleophile of Na^S, which may attack the electropositive aromatic carbon forming an intermediate (step 1) [20],

,

X

Cl-"-Cl

1+

+

2~

Na2S

CI

-

-o - S Na

ci-(0)-ci - NaCl

ci--s -(0>-ci The NajS can now attack this dimer in a manner similar to steps 1 and 2 to produce the corresponding thiolate dimer (structure 11). The thiolate dimer can react with the p-dichlorobenzene monomer to form a trimer. This process can be repeated several times until a long chain PPS is produced.

Cl--S-(0>-SNa Structure 11 PROPERTIES PPS is classified as an engineering thermoplastic because of its high thermal stability and excellent chemical and flame resistance. It also possesses superior electrical and mechanical properties. It has a high degree of crystallinity and stiffness. Its softening temperature is about 300°C.

Polysulfides

745

On the thermal side, PPS can retain approximately half of its weight even when heated up to 1,000°C under non-oxidative conditions. It resists attack by various solvents, such as hydrocarbons, aqueous bases, alcohols, phenols, and most acids. It bums when exposed to flame, but it is self-extinguished when the flame is removed. It possesses good electrical insulation characteristics, such as lower dielectric constant, and high resistivity and dielectric strength. Mechanically, PPS materials are tough, stiff, and demonstrate good impact resistance. APPLICATIONS PPS enjoys wide use as protective coatings for electrical connectors, relay components, electrical bulb sockets, switch components, and ignition plates. It also is used for making fibers, films, and various molded objects. REFERENCES 1. E. R. Bertozzi, Rubber Chem. Technol, 4 1 , 114 (1968). 2. S. M. Ellerstein and E. R. Bertozzi, Kirk-Othmer Encyclopedia Chem. TechnoL, vol. 18, p. 814 (1978), Wiley, New York. 3. S. M. Ellerstein, Encyclopedia Polym. Sci. Eng. vol. 13, p. 186 (1988), Wiley, New York. 4. D. E. Vietti, Comprehensive Polym. 5c/.vol. 5, p. 533 (1989), Pergamon Press, New York. 5. J. W. Mellor, A Comprehensive Treatise on Inorganic and Theoretical Chemistry, vol. 2, p. 629 (1946), Longman, Green and Co., New York. 6. F. Bittner, W. Hinrichs, H. Hovestadt and L. Lange, Chem. Abstr., 104, 227187f (1986)., 7. W. F. Giggenback, Inorg. Chem., 13, 1,724 (1974). 8. G. J. Janz, E. Roduner, J. W. Coutts and J. R. Downey, Jr., Inorg. Chem.,15, 1,751 (1976). 9. G. J. Janz, J. R. Downey, Jr., E. Roduner, G. J. Wasilczyk, J. W. Coutts and A. Eluard, Inorg. Chem., 15, 1,759 (1976). 10. J. M. Catala, J. M. Pujol and J. Brossas, Chem. Abstr., 105, 6,900b (1986). 11. J. M. Catala, J. M. Pujol and J. Brossas, Chem. Abstr., 106, 102,849h (1987). 12. J. Bolle and A. Dabir, Chem. Abstr., 96, 38,171b (1982). 13. A. Duda and S. Penczek, Makromol. Chem. 181, 995 (1980). 14. A. Duda and S. Penczek, Macromolecules 15, 36 (1982). 15. M. E. Tenc-Popovic, S. Popov, S. D. Radosavljevic and V. J. Rekalic, J. Polym. Sci., Part A-1. 10, 2,583 (1972). 16. V. J. Rekalic, M. E. Tenc-Popovic and S. D. Radosavljevic, J. Polym. Sci., Polym. Chem, Edn. 18, 2,033 (1980). 17. P. Ghosh, Polymer Science and Technol. of Plastics and Rubbers, p. 355 (1992) Tata McGraw-Hill Publishing Co. (New Delhi). 18. H. A. Smith, Encyclopedia of Polym. Sci. and Technol. vol. 10, p. 653 (1969) Interscience (New York). 19. J. W. Cleary, Adv. Polym. Synth.,31, 159 (1985).

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20. J. F. Geibel and R. W. Campbell. Comprehensive Polym. Sci., vol. 5, p. 543 (1989). 21. H. W. Hill, Jr., Ind. Eng. Chem. Prod. Res. Dev., 18, 252 (1979). 22. C. J. Stacy, Polym. Prep., Amer. Chem. Soc. Div. Polym. Chem. 26, 180 (1985). 23. G. Kraus and W. M. White, Chem. Abstr., 99, 123,454c (1983). 24. R. W. Campbell and L. E. Scoggins, Chem. Abstr., 83, ll,382r (1975). 25. S. Tsunawaki and C. C. Price, J. Polym. Sci., Part A, 2, 1,511 (1964). 26. A. B. Port and R. H. Still, J. Appl. Polym. Sci., 24, 1,145 (1979). 27. T. Fujisawa and M. Kakutani, J. Polym. Sci. Polym. Lett. Edn, 8, 19 (1970). 28. B. Loltling, M. Soder and J. J. Lindberg, Angew. Makromol. Chem., 107, 163 (1982). 29. R. W. Campbell, Chem. Abstr., 86, 190,850 (1977). 30. D. Mukherjee and P. Pramanik, Indian J. Chem., 21, 501 (1982).

CHAPTER 29 PROPERTIES AND APPLICATIONS OF THERMOPLASTIC POLYURETHANE BLENDS M. Yue and K. S. Chian School of Applied Science Nanyang Technological University Nanyang Avenue, Singapore 2263 CONTENTS ABSTRACT, 747 INTRODUCTION, 748 THERMOPLASTIC POLYURETHANE, 748 POLYMER BLENDING, 749 TPU/ABS BLENDS, 750 TPU/PVC BLENDS, 752 TPU/STYRENE BLENDS, 753 TPU/POLYACETAL BLENDS, 754 TPU/OLEFIN BLENDS, 755 TPU/NYLON BLENDS, 756 TPU/POLYESTER BLENDS, 756 TPU/PVDF BLENDS, 757 TPU/ELASTOMER BLENDS, 758 OTHER TPU BLENDS, 758 TRENDS IN TPU BLENDS, 758 REFERENCES, 759 ABSTRACT Blending of polymeric materials has been proven to be a useful method for enhancing material properties and/or developing materials with desired performance, and a more cost-effective way to develop new materials than traditional production 747

748


methods. Thermoplastics polyurethane elastomers (TPUs) have been blended with a variety of polymers to achieve different and/or better properties and have been used as modifiers for other thermoplastic materials. The blends will become more and more important in technical applications with high performance requirements. This paper describes the structure and properties of TPU and gives a review of the developments of various TPU blends with their key properties and commercial applications. Blending techniques also are described. New blends of TPUs with polyvinylidene fluoride (PVDF) are initially introduced here. The trends in TPU blends are discussed. INTRODUCTION Blending of polymeric materials has been proven to be a useful method for enhancing material properties and/or developing materials with desired performance [1], and a more cost-effective way to create new materials than traditional methods, such as polymerization. Polymer blends emerged as an important class of materials in the late 1970s and have experienced substantial growth in the 1980s. The current development of polymer blends is directed toward specific applications, and the interest in alloys is widespread with frequent commercial product introductions. These effects are supplemented by high patent activity and publication by both industry and academia. They are expected to grow further through the 1990s [2]. TPU blends with other polymers will increase correspondingly because TPUs are being used more and more in technical applications with high performance requirements [3]. The most important properties are their high mechanical strength, wear and tear properties as well as elasticity, which make TPU blends, with a variety of other polymers, have different and/or better properties [4-6]. Furthermore, TPUs have been used increasingly as modifiers for other plastics and elastomers [7]. In general, the hardness, modulus, and elongation move toward that of the blending polymer as the polymer is increased. The effect on tensile strength depends on the degree of compatibility of the blending polymer with TPU [8]. This paper will review the developments of various TPU blends with their key properties and commercial applications. New blends of TPUs with polyvinylidene fluoride (PVDF) will initially be introduced here. Trends in TPU blends also are discussed. THERMOPLASTIC POLYURETHANE TPUs are linear polymers (poly-adducts of polyisocyanates and poyols) that have the urethane chemical function in the structural backbone: -NH-CO-0-. The structure of TPUs basically consist of two phases, the hard and the soft segments. The soft segments are comprised of diisocyanate coupled with low melting polyols chains covalently bonded to the hard segments predominantly containing low molecular weight urethane groups. The polar nature of the rigid urethane chain segments results in their strong mutual attraction, aggregation, and ordering into crystalline and para-crystaUine domains in the mobile polymer matrix. The abun-dance of urethane hydrogen atoms, as well as carbinol and ether oxygen partners in urethane systems, permits extensive hydrogen bonding among the polymer chains which apparently restricts the mobility of the urethane chain segments in the domains, and, thus, their ability to extensively form into crystalline lattices. The result is semi-ordered regions described as "para-crystaUine."

Properties and Applications of Thermoplastics Polyurethane Blends

749

Due to the incom-patibility of the hard and soft segments, TPUs exhibit two-phase domain structure in which the hard segments are dispersed in the soft segment matrix [9], The hard-segment domains act as virtual physically cross-linking sites for the soft segments, thus giving TPU their apparent elastic behaviors. The high temperature plastics behavior comes from the linear chain structure. According to the type of long-chain polyols used in their synthesis, TPU are commonly classified as polyether-based or polyester-based polyurethanes. The heat, oxidation, and oil resistance of the polyester-based materials generally are superior to the poly ether types, which are inherently more hydrolytically stable and have greater resistance to fungicidal environments than the former. Both types of TPUs are distinguished by the numerous other properties, such as high tensile strength and elongation at break; high flexibility (also at low temperatures); low permanent deformation on static and dynamic loading; favorable friction and abrasion performance; high tear propagation resistance; high damping power; and high resistance to oils, fats, and many solvents. With these properties, the materials have good applications in mechanical engineering, automobile manufacturing, tools, the shoe industry, the electrical industry, and medical technology [10]. Some inherent "limitations" of TPUs include flammability, low rigidity, low thermal stability, cost, and long-term environmental stress cracking in implantable medical prostheses. With these properties, TPUs are suitable candidates for polymer blending with other polymers to enhance one or more of these properties. POLYMER BLENDING There are two usual routes to blending polymers: melt blending and solvent melting. Melt blending is the most commonly used technique; it lets materials mix together at a temperature above the melting temperatures of the components within a heated mixer, such as Brabender Plasticorder, Hakke Torque Rheocord, roll-mill, extruder, etc. Solution blending is when polymers are dissolved in miscible solvents or in a common solvent and the polymer solutions are then mixed mechanically before the blend is obtained by removing the solvent(s). Of these techniques, melt blending is the simplest. It has no solvent contamination and does not require solvent removal, which the solvent blending does, although the melt blending sometimes may involve thermal degradation. Before the melt blending for TPU blends, TPUs must be dried due to their hydroscopic property. It is equally important that the other components of a TPU polyblend also should be dried prior to compounding. Drying for 2-4 hours at 80-100°C in a dehumidifying hopper dryer with the dewpoint of the inlet air at 6°C is adequate for most materials [7]. If the materials are not dried, their processing will be affected adversely and output as well as quality will be reduced [11]. Also, surface imperfections will appear in the form of bubbles, sinks, delamination, and striations. Polymer blends can be divided into miscible (the mixture is homogenous down to the molecular level) and immiscible blends. The type of blend is primarily governed by the Gibbs free energy of mixing [4]: AG = AH - TAS For complete miscibility, the change in the enthalpy heat of mixing AG must be less than or equal to 0. If AG > 0, the blend is immiscible.

750


Polymer blends usually are characterized using conventional methods, such as mechanical, thermal, optical and microscopy evaluations, depending on the application of the material. The properties of polymer blends are largely dependent upon the degree of miscibility between the components of polymer blend. In general, the properties of a two-component blend polymer may be described by the following equation [12]: P = P,C, + P^C^ + IP.P^

(1)

where P is the property of the blend; P, and P2 are properties of polymer component 1 and 2, respectively; C, and C2 are concentration of polymer components 1 and 2, respectively; I is an interaction coefficient between the two polymers in the blend mixture, which describes the level of synergism, or thermodynamic compatibility, of the components in the blend mixture. If I for tensile strength is greater than zero, the components of the blends are normally considered to be compatible, which is a visually homogenous mixture with enhanced physical properties over the constituent polymer. Otherwise, the blend is usually an incompatible system when I for tensile strength is less than zero. If I equals zero, the properties of the combination are equal to the weighted arithmetic average of the constituent properties as shown below: For example, the glass transition temperature of compatible polymer blends can be P = P,C, + P^C,

(2)

expressed as the empirical equation [13]: T = W,T , + W,T , g

1

g.l

2

g.2

(3) "^ ^

where Wj and W2 represent the weight fraction of polymer 1 and 2, respectively. TPU/ABS BLENDS Blends of TPUs and Acrylonitrile-Polybutadiene-Styrene graft polymer (ABS) have been studied by a number of researchers [14-16]. The structures of the blends are very complex due to the complex polyblend of ABS, in which there is a rigid SAN copolymer with a rubbery graft butadiene polymer and the heterophase system of the TPU. The two polymers can benefit each other, as shown in Table 1. Figure 1 shows the effect on the blend properties with the varying amounts of ABS. It also can be seen that the density and break elongation of blends decreases with the addition of ABS. The TPU tear strength can be improved by blending with ABS. The moduli of the blends increases with the weight fraction of ABS, but the variation in tensile strength with composition of TPU/ABS blends is complicated. In addition to these, the blends have good toughness at low temperatures, high flex modulus for dimension stability, and good solvent and fuel resistance. Furthermore, these blends are easy to process, can be painted without a primer, and can be easily

Properties and Applications of Thermoplastics Polyurethane Blends

751

Table 1 Synergy of TPU and ABS Blends (after [7]) Benefits to ABS from TPU

Benefits to TPU from ABS

Abrasion resistance Low temperature properties Toughness and impact resistance Chemical resistance Printability

Distortion temperature Rigidity Cost Processing Ozone resistance

Density

8

Elongation

I Tensile strwength

ABS% Figure 1. Properties of TPU/ABS blends vs. the composition.

formulated to give an appropriate balance of properties for a specific need. The blends also may offer cost advantages because of the lower ABS costs. All these virtues and their reasonable raw material cost certainly make these blends suitable for many demanding applications [17]. Small amounts (

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