Advances in Engineering Fluid Mechanics: Multiphase Reactor and Polymerization System Hydr

Multiphase Reactor And Polymerization System Hydrodynamics ADVANCES IN ENGINEERING FLUID MECHANICS SERIES Nicholas P. C...

Author: Nicholas P Cheremisinoff Consulting Engineer

36 downloads 1226 Views 31MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

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

Nicholas P. Cheremisinoff, Editor

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

Nicholas P. Cheremisinoff, Editor

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

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

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

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

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 '

1.5

1 •

•

100 D

9

CL

E (/)

o o (f)

7 6

^

\

^

1

1

•

C12 C10

•

C8

J

O

C7 C6

J 1

•

-

1

D

•

8 I

^

T

—^—r—^—r —'—r

3

-1

D

•

-

•

J

D

-]

A

5 D

*>

Q)^

O

o o D O O

I 280

J

300

320

\

1

340

I

I

360

I

I

380

\

L

400

420

440

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

4

1.3 r— 1 —

1

"1

1

1—

-T

1

" • ' — 1 — 1 — j — ""'

1

1

1

\

1

1

1

•

1.2

H

-]

1.1

•

1.0 0) 1

0.9

•

cd

E

•

0.8 -

o

CO

0.7

4

• 2 2 dimethyl butane ^cyctohexane

H ] 1 H

• •

•

•

'>

h•

0.4

L • • • • • • r^ A t

J

•

H

•

T

0.5

0.2

J

• 2 methyl pentane

•

0.6

0.3

• hexane

• benzene •

'S5 o

1

T

J

•

•

J J

T

«t t f . •

1

L.

-J

1

1

L.- - J

1

1

1

J 1

i

1

1

i

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

\

1

1

p

\

5

I

• 2

• D

Q. O

ft

D

D

^

o o

to '>

•

• o •

1.5

10-1 -

1'

1

a

•

•

•

H

o

o

o

\

•

•

1

D

a

r CO

D

D

• o •

• o •

• o •

• o •

•

•

1

100

1

C12 D C10 • C8 O C7 • C6

OA

1

200

1

L.

300

1

400

J

.

1

500

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

Advances in Engineering Fluid Mechanics

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.

• D o •

C6+C16 C6+C12 C6+C10 C6+C7

Qo

o o *> D D

o

O

• J

0.0

0.1

O

O

O

0.4

0.5

•

0.2

^ .- s

u

0.3

0.6 0.7 0.8 0.9 1.0

mole fraction n-hexane Figure 4. Viscosity vs. connposition at 298°K and 1 bar for various binary alkane mixtures. Data are from Chevalier et al. [41].

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

Advances in Engineering Fluid Mechanics

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

1 B

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

The Viscosity of Liquid Hydrocarbons and Their Mixtures

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)

The Viscosity of Liquid Hydrocarbons and Their Mixtures

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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%.

The Viscosity of Liquid Hydrocarbons and Their Mixtures

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: MW^i, = MW„ + 0.00867358(MW;,''''' - MW;^''"'') with: MW^ = £ X. M W y X ^i^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

Advances in Engineering Fluid Mechanics

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

The Viscosity of Liquid Hydrocarbons and Their Mixtures

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

Advances in Engineering Fluid Mechanics

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^^O] - 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

The Viscosity of Liquid Hydrocarbons and Their Mixtures

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

The Viscosity of Liquid Hydrocarbons and Their Mixtures

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

Advances in Engineering Fluid Mechanics

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.

The Viscosity of Liquid Hydrocarbons and Their Mixtures

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.

This Page Intentionally Left Blank

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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

Experimental Studies for Characterization of Mixing Mechanisms

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

Advances in Engineering Fluid Mechanics

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.

Experimental Studies for Characterization of Mixing Mechanisms

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

Advances in Engineering Fluid Mechanics

^Wm

wm^

••'(!\'ifti:¥:'.•{•'hii',:

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

•^i^

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

Experimental Studies for Characterization of Mixing Mechanisms

^jl!:ir,

•' /:••.•.

•::,•••"

'^• j^.^'^iJ

33

•. • % ' » ' ^ ) ; . ; -

||0§75|'

o.s>

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

34

Advances in Engineering Fluid Mechanics

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

Experimental Studies for Characterization of Mixing Mechanisms

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

Advances in Engineering Fluid Mechanics

iMi

$m

;W^

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

Experimental Studies for Characterization of Mixing Mechanisms

37

0J

0J5

m

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

38

Advances in Engineering Fluid Mechanics

i^i

0.75

OS

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

Experimental Studies for Characterization of Mixing Mechanisms

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 ^inq 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^ <-^-^M^-

4=ttK^

•:..-.•-^.^^

:^-^,-•;

•

^

^mm. 4=:

'v

^.m ^

V-

o O CO

•I—>

CO 3

>

CO

CO CD

3 C

4—•

E

o ^_ ^CD —• CO CO

M—

o o c o CD

3

JD

CO

c o

CO

b

Experimental Studies for Characterization of Mixing Mechanisms

r^w

\

2.5 ^•0

2.25

^"^j

0.9

43

\

J

1

."^2^. _

y

5 v,^^

-^ • ^ 5 -

0.75

0.5

FILL FACTOR

!/?S£|! '^!9'P'%''^'?:^h ,ff*/Vyly'"J;';,;

Figure 10a. Computed isobars on R-R screw rotors from Kim et al. theory together with experimental results of this paper [21].

44

Advances in Engineering Fluid Mechanics

0.9

0.75

0.5

FILL FACTOR

•(!«;'•''!/,

•' '.;,nr?«fii iV/Vf,;**',".' ;•'/

Figure 10b. Computed isobars on R-L screw rotors from Kim et al. theory together with experimental results of this paper [21].

Experimental Studies for Characterization of Mixing Mechanisms

45

2.2S

0.9

0.75

0.5

FILL FACTOR

Figure 10c. Connputed isobars on double-flighted rotors fronn Kinn et al. theory together with experimental results of this paper [21].

46

Advances in Engineering Fluid Meclianics

10.0 DOUBLE FLIGHTED ROTOR O R-R SCREW ROTOR A

E 9-0 \ r

UJ

2 h-

6.0 ^

O

z o

5

4.0

0

3

o rr o

O

A

2.0

A

A

0.0 0.4

1

0.5

0.6

1

... I

0.7 0.8 FILL FACTOR

, —

1

0.9

1

1.0

Figure 11. Experimental circulation time in an internal mixer as a function of fill factor. (text continued from page 39) CONCLUSIONS The distributive and dispersive mixing based on the flow field is evaluated experimentally in this article. This study began by considering the dispersive mixing in terms of size distribution of carbon black agglomerates with different volume fraction of the materials. Examination of dispersion with carbon black compounds by optical microscopy and scanning electron microscopy found that the best dispersive mixing condition is created when the fill factor is 0.9. Measurement of the circulation time in the mixer was carried out to characterize the distributive mixing. The results indicated that void region shortens circulation time. The circulation time of R-R screw rotor is shorter than that of double-flighted rotor. The shorter circulation time represents a good distributive mixer. In starvation effects on the mixing, 0.5 fill factor makes the best distributive mixer. REFERENCES 1. Hess, W.M., Swor, R.A. and Micek, E.J., Rubber Chem. Tech., 57, p. 959 (1984). 2. Suetsugu, Y., White, J.L. and Kyu, T., Adv. Polym. Tech., 7, p. 427 (1987). 3. Medalia, A.I., Rubber Age, 97, p. 82 (1965).

Experimental Studies for Characterization of Mixing Mechanisms

47

4. Cembrola, R.J., Rubber Chem. Tech., 56, p. 233(1983). 5. Leigh-Dugmore, C.H., Rubber Chem. Tech., 29 p. 1,303 (1956). 6. Wake, W.C., Todol, B.K., and Loadman, M.J.R., Analysis of Rubber and Rubber-like Polymers, 3rd ed., Applied Science Publishers, New York, 1983. 7. Suetsugu, Y., Ph.D. Dissertation, University of Akron (1986). 8. Aoki, Y., Nihon Reorogi Gakkaiski, 7, p. 20 (1979). 9. Holland, A.C. and Gagne, G., Appl Optics, 9, p. 113 (1970). 10. Galanti, A.V. and Sperling, L.H., J. Appl. Polym. ScL, 14, p. 2,785 (1970). 11. Tang, I.N. and Munkelwitz, H.R., J. Coll. Inter. ScL, 63, p. 297 (1978). 12. Suetsugu, Y., Kikutani, T., Kyu, T. and White, J.L., Coll. Polym. ScL, 268, p. 118(1990). 13. Vegvari, P.O., Hess, W.M. and Chirco, V.E., Rubber Chem. Tech., 5 1 , p. 817 (1978). 14. Ebell, P.C. and Hemsley, D.A., Rubber Chem. Tech., 54, p. 698 (1981). 15. Dannenberg, E.M., Ind. Eng. Chem., 44, p. 813 (1952). 16. Boonstra, B.B. and Medalia, A.I., Rubber Age, 92, p. 892 (1963). 17. Henneka, T.A. and Rotz, C. A., SPEANTEC Tech. Paper, 27, p. 223 (1981). 18. Rayleigh, L., Pro. Roy. Soc. (London), A84, p. 25 (1811). 19. Thompson, J.J., Recent Researches in Electricity and Magnetium," Oxford University Press, London (1893). 20. Debye, P., Ann. Phys., 46, p. 809 (1905). 21. Guinier, A., Ann. Phys., 12, p. 161 (1939). 22. Min, K. and White, J.L., Rubber Chem. Tech., 58, p. 1,024 (1985). 23. Kim, J.K. and White, J.L., Nihon Reorogi Gakkaishi, 17, p. 203 (1989). 24. White, J.L. and Kim, J.K., J. AppL of Polym. Symp., 44, p. 59 (1989). 25. Kim, J.K., Ph.D. Dissertation, University of Akron (1989). 26. Kim, J.K., J. of AppL Polym. Sym., 50, p. 145 (1992).

This Page Intentionally Left Blank

CHAPTER 3 PHYSICAL MODELING OF AXIAL MIXING IN SLUGGING GAS-LIQUID COLUMNS J.R.F. Guedes de Carvalho and J.B.L.M. Campos Department of Chemical Engineering University of Oporto Oporto, Portugal and J.A.S. Teixeira Escola Superior Agraria Instituto Politecnico de Braganga Braganga, Portugal CONTENTS INTRODUCTION, 49 THE WAKES OF SLUGS, 50 Laminar Wakes, 51 Turbulent Wakes, 55 TAYLOR DISPERSION FOR LAMINAR LIQUID FLOW ALONG A TUBE, 57 AXIAL MIXING IN CO-CURRENT FLOW OF GAS AND LIQUID, 58 NOTATION, 64 REFERENCES, 65 INTRODUCTION Mixing of liquid in gas/liquid contactors is a topic of great practical interest to the chemical industry, especially since it may determine product yield in chemical reactors. Over the years, considerable research effort has been devoted to the study of mixing in gas-liquid systems, but given the complexity of the phenomena involved, empirical dispersion coefficients are normally defined and experiments designed to yield the value of these coefficients over a range of experimental conditions (Deckwer et al. [1]; Field and Davidson [2], among others). While giving useful practical information, this approach does little to improve our knowledge about the physical mechanism of the mixing/dispersion process. Physical modelling is an interesting alternative approach, but, unfortunately, it is applicable only to situations which are not too complex. 49

50

Advances in Engineering Fluid l\/1echanics

Gas-liquid contacting under conditions of slug flow occurs in many pratical situations, e.g. in air-lift systems, and it is likely to be more easily amenable to physical modelling. This is because gross circulation of the liquid is not present, radial mixing is not normally a problem, bubble rise velocities are more easily predicted, and distributions of bubble sizes are easier to characterize. A significant number of studies have been published on gas-liquid mass transfer in slug flow contactors, but little was known, until recently, about liquid mixing in this type of equipment. For example, van Heuven and Beek, in their pioneering work on air lifts, express the extent of liquid mixing in terms of an equivalent number of tanks in series [3]. In a series of papers, Campos and Guedes de Carvalho [4,5], Campos [6], and Guedes de Carvalho et al. [7] recognized that liquid mixing in slug flow is the result of two main contributions: (a) the recirculating flow of liquid in the wakes behind the rising bubbles and (b) Taylor-like dispersion in the liquid flowing between successive gas slugs. To simplify matters, Campos and Guedes de Carvalho started by considering the situation of slug flow in columns with no net flow of liquid [4,5]. A detailed study was made of the wakes of slugs with regard to their size and flow pattern and to the extent of exchange of liquid between wake and surroundings. Combining the information thus obtained with the analysis of Taylor [8] about dispersion in flowing liquids, Guedes de Carvalho et al. [7] went on to consider the problem of axial dispersion of the liquid in co-current flow with gas slugs in vertical tubes. This study was restricted to situations of low velocities of the liquid and no interaction between successive slugs (i.e., no coalescence), again to simplify the analysis. Work is underway to adapt the ideas already developed to the more realistic conditions of free slugging. In this chapter, the work published by the authors is reviewed. The importance of the wakes of slugs as the primary agent in promoting mixing is considered first. Next, the problem of Taylor dispersion for laminar liquid flow along a tube is briefly outlined, and, finally, the analysis of dispersion in co-current flow of gas slugs and liquid in vertical tubes is presented. THE WAKES OF SLUGS Campos and Guedes de Carvalho brought to evidence the dependence of the flow pattern in the wake of slugs upon the Reynolds number, defined as Re = UD/v, where U is the velocity of rise of a slug, D is the internal diameter of the tube and V is the kinematic viscosity of the liquid [4]. For Re < 300 the flow pattern in the wake is laminar and axisymmetric while for high values of Re (typically Re > 500) the flow in the wake is turbulent. The flow pattern in laminar and turbulent wakes is illustrated by the photographs in Figure 1 (the pictures were taken by a camera moving vertically with the slugs). In the pictures it may be seen that the wake of the slug is a region of recirculatory flow spanning (nearly) the whole cross section of the tube and extending over a length of up to about one or two tube diameters below the slug. This local movement of the liquid is bound to be of great relevance to the process of mixing. Furthermore, because the rising slug carries the wake "attached" to it, the process of axial mixing induced by each individual slug will be effective over a length of several times the diameter of the column.

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns

: (a):

(b)

51

(c)

Figure 1. Pictures with moving camera; slugs in 52 mm i.d. column; (a) (gD^y^^/v = 246, (b) (gD^)^/^/^ = 463, (c) (gD^y^^/v = 3.8 x 101

To better understand the whole process it is necessary to have detailed information about the pattern of flow in the wake of slugs; this is reviewed in the following. Laminar Wakes To study the flow pattern in the wakes of slugs, Campos and Guedes de Carvalho [4] performed a photographic study inspired by the works of Filla et al. [9] and Coutanceau and Thizon [10]. Pictures of slugs rising through columns of still liquid were taken by a camera travelling at the same velocity as the slugs. Small air bubbles, generated by the flow process itself, were used to trace the movement of

52

Advances in Engineering Fluid Mechanics

the liquid inside the wake. The photographs shown in Figure 1 are from that study, and dimensional analysis may be used to help interpret the photographic information obtained. In general, the flow pattern in the wake will depend on the diameter of the tube D, the properties p, |i and a of the liquid, the slug length, L, and the value of the gravitional acceleration, g. Dimensional analysis shows that the flow pattern, and, therefore, the dimensionless wake volume, v/D^, and dimensionless wake length, 1/D, may depend on L/D, (g/D^)^^^/v, and pD^g/o. The latter group is likely to be unimportant except for small-bore tubes (i.e., diameter below about 20 mm) and the dependence on L/D is expected to cease for large enough values of this parameter, when the velocity profile in the film flowing alongside the slug is fully developed. It may therefore be concluded that for these longer slugs in not too narrow tubes, the flow pattern is dependent on the value of the dimensionless group (gD^)'^^/v, which is the ratio between Reynolds number and Froude number raised to the power 0.5. The photographic study revealed that for (gD^)'^^/v < 500 flow in the wake is laminar, whereas for (gD^)'^^/v > 900 it is turbulent. In the intermediate range of values of the dimensionless group, transition from laminar to turbulent flow is observed; the vortex ring in the wake is seen to oscillate and occasionally break up and shed vortices. In order to better characterize the laminar wakes, Campos and Guedes de Carvalho [4] performed a visual study inspired on the work of Maxworthy [11]. The liquid in the lower half of the column, where the slug was injected, was made dark by means of a soluble dye. As the slug rose into the clear liquid, in the upper half of the column, the dark wake could be seen clearly in photographs like those shown in Figure 2, obtained with a still camera. Vertical and horizontal line segments marked on the column test section made it possible to scale down the prints and their analysis gave values of wake length 1, wake volume v, slug length L and slug volume V. The data obtained are shown in Figures 3 and 4. The flow pattern in the wake of slugs is determined by the velocity profile in the liquid in the annular region surrounding the slug near its bottom. For long slugs this velocity profile is independent of slug length, and so is the flow pattern in the wake, as shown by Campos and Guedes de Carvalho [4]. The existence of a limiting value for v, corresponding to long slugs, is well brought out in Figure 3 where v is plotted against L for several values of (gD'^)'^^/v. We use an asterisk to denote this limiting value, and the dependence of 1*/D and v*/D'^ on (gD^)'^^/v is shown in Figure 4, where a linear relationship is suggested. The experiments with colored tracer also gave important information about the amount of liquid transferred between slug wake and surroundings. This exchange of liquid is illustrated in Figure 5, which schematically depicts three instances in the process of rise of a single slug above the colored tracer. This pattern of behavior is observed only for laminar wakes (i.e. for [gD-^]'^^/v < 500) in which case the thread of colored liquid behind the slugs does not spread (radially) over the whole cross section of the tube. A very curious feature is that over a range of values of (gD^)'^^/v (roughly, between 200 and 500) a significant portion of colored tracer is pumped up, in the wakes of slugs, from the bottom layers to the top of the tube.

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns

53

te'',.'.'"'i.,V'''""';-^"' •'?'*-ii'l

•'- ;• , ' ' ' ? • ' •

i'.

V

•-';.'.

'.

(a)

.Wi/^S

.

(b)

(c)

Figure 2. Pictures with still camera; slugs in 19 mnn i.d. column; (a) (gD3)i/2/v = 176, L= 64 mm, v = 0.87 cm^, (b) (gD^y^^v = 325, L = 69 mm , V = 1.61 cm^, (c) (gD^y^^v = 483 L = 68 mm, v = 2.38 cm^.

largely "bypassing" the clear liquid. Information on this type of behavior is bound to be of great relevance to designers of chemical reactors operating in the slugging regime, with viscous liquids. Some measure of the exchange of liquid between wake and surrounding fluid also was obtained, allowing the slugs to rise for 1 m in clear liquid and measuring the

54

Advances in Engineering Fluid Meciianics

6 y-

176

(gD^y^/v

205 325

•

A

437

483

511

754

•

n

#

O

O

V (cm^) rP Or'

rO

^

J-L

o

o

O O

•

O

J^

Q

1

0 0

^w— • — • Q

— • —

V

A 1

u

• .

4 * * ^* •

A

•

n

w

D • •

2 hh-

n^

A

. ^ •

A

A

A

1

A

1

20

1

I

1 80

60

40

L(mm) Figure 3. Dependence of wake volume on slug length for 19 nnm i.d. column.

\ D(mm)

D

D • A 19

i-

O

52

1A 0.5

5o^ ^^

O A

1

1 200

.

1 400

1

1

.

600

1 0.0 800

f^D^/^/v Figure 4. Dependence of dimensionless limiting wake length and wake volume on (qDy^N ; , IVD = 0.30 + 1.22 x 10-^ {^oy^N-—, vVD3 = 7.5 X 10-4 X

{QD^^N.

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns

55

Free surface

'^v:^. Colored liquid Clear liquid ^ Thread of colored liquid

Colored liquid

Clear liquid

Gas slug

Laminar Wake

t2

t3

Figure 5. Sequence of events (t^ < t2 < tg) following injection of single gas slug with 200 < (gD^)^/^/^ < 500.

amount of tracer deposited at the top. The ratio between this amount and that initially present in the wake is represented in Figure 6 as a function of (gD^)^^^/v, for three values of L/D. The behavior for L/D = 4.2 is certainly representative of that for higher values of that parameter since the flow in the annular film will then be fully developed near the bottom of the slug. Turbulent Wakes When gas slugs rise in water or in low-viscosity liquids, the wakes are turbulent (except for very narrow tubes). In Figure 1, the picture on the right illustrates a turbulent wake behind a slug rising in water. The recirculating motion of the liquid in the wake is still noticeable, but it is obvious that the flow is no longer streamlined; the short streaks marked by the small gas bubbles do not define such a neat picture as in laminar flow. As a result of this, it is not possible to accurately determine the shape and size of the slug wake from the prints; wake length is seen to be about twice the diameter of the tube. Unfortunately, the method of the colored tracer (i.e., soluble dye) inspired in the work of Maxworthy [11] is not applicable here since the liquid in the wake is

56

Advances in Engineering Fluid Mechanics

100 Q \2o

(8D^)"^/v Figure 6. Dependence of tracer entrainment on (gD^y^^/v for 19 mm i.d. column.

promptly flushed out by the liquid plunging in the wake, as could be guessed from the data in Figure 6. Campos and Guedes de Carvalho conceived the turbulent wakes as perfectly mixed portions of fluid occupying the whole cross section of the column and extending over a length 1^ behind the bottom of each gas slug [5]. Based on this model, they performed experiments in columns with internal diameters of 32 mm and 19 mm and inferred values of 1^ = 2.8 D and 1^ = 2.3 D, respectively, in reasonable agreement with the information gathered from photographs as in Figure 1. These values of 1^ were insensitive to slug length, providing the slugs were long enough. A significant aspect of the model presented by Campos and Guedes de Carvalho is that it leads to the determination of a single parameter, the size of the fully mixed wake, which quantifies the extent of mixing induced by the rise of an individual slug [5]. If a column of liquid has an initial distribution of tracer in the form of a

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns

57

step, the rise of N slugs up the column leaves a final distribution of tracer in the column similar to that obtained by feeding a step function to N equal-sized fully mixed tanks in series. Working on the well-known equivalence between a series of tanks and the dispersion model (Levenspiel [12]), Campos and Guedes de Carvalho showed that for continuous slugging with a frequency / (slugs/second) the mixing process is equivalent to diffusion with an apparent diffusion coefficient given by D* = 0.5(1J2/

(1)

The advantage of the description in terms of the apparent diffusion coefficient will become clear in the section dealing with mixing in co-current flow. Campos and Guedes de Carvalho also performed experiments on mixing of liquid with a pulsated gas feed [5]. They observed that an increased frequency of slug injection leads to decreased mixing of liquid, and they related this behavior to the detachment of the wakes of slugs in the pulsating liquid. Campos studied the influence of the vertical column alignment on the axial dispersion in gas-liquid slugging columns [6]. He observed that, as the angle between the column axis and the vertical direction is increased, the intensity of dispersion increases, at first very markedly (angles up to 3°) and then more gradually (angles between 3° and 60°). He also observed that in the inclined columns a higher degree of dispersion is obtained if the slugs are longer, and this is at variance with the observations in vertical columns. Visual observations and the results of a study on mixing suggested that the recirculating wake behind slugs rising in an inclined column is much longer than that observed behind slugs rising in the same column aligned vertically. Campos correlated the wake length of slugs rising in inclined columns with the velocity head of the liquid in the film entering the wake [6]. All the above findings refer to mixing in slugging columns with zero net flow of liquid. Next, we consider situations in which there is a net flow of liquid up the column. TAYLOR DISPERSION FOR LAMINAR LIQUID FLOW ALONG A TUBE The dispersion in a liquid flowing along a tube (zero gas flowrate) is a subject studied by many scientists, and many models have been developed, depending on the liquid flow conditions. Taylor [8,13] studied the dispersion of a tracer in a liquid flowing in laminar conditions, and a similar study was performed by Guedes de Carvalho et al. [7]. Assuming negligible molecular diffusion of the tracer, axial dispersion is the result of a gradient in velocity along the radial coordinate, which for fully developed laminar flow is given by

'-'i

(2)

where u is the velocity at distance r from the axis, UQ is the velocity on the axis, and R is the radius of the tube. Following a step change in tracer concentration at the inlet, Guedes de Carvalho et al. measured the average tracer concentration in the liquid leaving a tube, C^, and compared the experimental values with the theoretical expression

58

Advances in Engineering Fluid Mechanics

where C^ is the initial tracer concentration, H^ is the height at which liquid leaves the column, t is the time from the moment the step change is introduced in the column, and x is the average residence time of liquid in the column [7]. The agreement between experimental and theoretical values is perfect, as expected from the analysis of Taylor [8]. Now, Taylor had shown that for laminar flow of fluid with appreciable molecular diffusion, the response to a step change in tracer at the inlet would be similar to that obtained for plug flow with superimposed axial dispersion, a curious aspect being that higher molecular diffusion of the tracer should lead to lower axial dispersion. A similar type of behavior is found in a flowing liquid along which slugs rise, as detailed in the next section. AXIAL MIXING IN CO-CURRENT FLOW OF GAS AND LIQUID Guedes de Carvalho et al. studied the dispersion of tracer for water flowing upwards in a vertical column along which air slugs rose at a constant frequency [7]. A simplified picture of the flow is given in Figure 7. In the near wake of gas slugs, between sections A and B, the liquid recirculates vigorously and it is assumed to be fully mixed. In the region between B and C turbulence in the wake is assumed to dye out quickly and below C laminar flow of the liquid with a parabolic velocity profile is assumed to be observed. This is because in all the experiments performed the value of the Reynolds number for liquid in that region was below the critical value of 2,100. To model liquid dispersion, Guedes de Carvalho et al. considered the combined effects of transport in the recirculating wakes and of Taylor dispersion in the liquid between each wake and the following slug [7]. If, at any one instant, a higher concentration of tracer is present in the lower part of the column, there will be a net upward transport of tracer, preferentially near the column axis as the result of the parabolic velocity profile in the liquid, and a radial concentration gradient of tracer will be set up. This is called Taylor dispersion. Next, as a slug rises through the liquid, the recirculating wake will fully mix the contents of the column over each cross section, thus destroying the radial gradients. In some sense the wake of a slug may be seen as imparting locally a high diffusivity to the tracer, thus attenuating the radial concentration gradients which would otherwise be set up by forward laminar convection alone. The continuous injection of slugs, at low frequency, may then be expected to bring residence time distribution of the liquid in the column closer to that for plug flow, in much the same way a tracer of high diffusivity would, as found by Taylor. However, if the frequency of slug injection is increased sufficiently, the extent of mixing in the axial direction also will increase due to the finite length of the wakes, and the residence time distribution of the liquid in the column will tend to approach that for a perfectly mixed tank for high enough values of slug frequency.

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns

59

B C

D

Figure 7. Diagrann of slugs and their wakes.

Guedes de Carvalho et al. developed a mathematical model for this physical situation and compared the predictions of the model with the concentration curves at the outlet of the column, following a step change in tracer concentration at the inlet. For more details of both mathematical model and experimental set-up and procedure, the original work should be consulted [7].

60

Advances in Engineering Fluid Mechanics

Some curves from the model are given in Figures 8 to 11, alongside the experimental data; each figure reports data for one value of the liquid flowrate and three values of the gas flowrate. The theoretical lines corresponding to liquid progression in plug flow (B), perfectly mixed flow (D), and laminar flow with a parabolic velocity profile (A) also are represented. In each of the four figures it may be seen that the injection of gas at the lowest flowrates reduces dispersion in the liquid significantly below that observed with no gas injection. Some increase in the flowrate of injected gas is seen to further reduce dispersion of the liquid, but a minimum in the extent of dispersion is reached as the gas flowrate is increased. A pronounced increase in gas flowrate helps promote mixing, due to the finite length of the wakes, and the residence time approaches that for a perfectly mixed tank. Special reference should be made to the indented form of the line corresponding to the lowest gas flowrate (very low frequency of slugging), in each of Figures 8 to 11. If the period between the passage of two successive slugs near the column outlet is sufficiently long, a large gradient of tracer concentration is observed in the radial direction due to a faster convection of tracer-rich solution near the axis of the column. Now, because the liquid moves faster near the column axis the concentration of tracer in the outgoing liquid rises significantly above its average concentration over the cross section at the column outlet; when the following slug VG (m^/s) 0.19 X 10-6 0.42 X 10-6

11

X 10-6

Figure 8. Experimental and theoretical outlet concentration curves (VL = 10"^ nn^/s). The theoretical lines corresponding to liquid progression in plug flow (line B), perfectly mixed flow (line D), and laminar flow with a parabolic velocity profile (line A) also are represented.

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns

•

0.75 X 10-6

D

2.6

X 10-6

A

15

X 10-6

61

t /T Figure 9. Experimental and theoretical outlet concentration curves (VL = 5.5 X 10"^ m^/s). The theoretical lines corresponding to liquid progression in plug flow (line B), perfectly mixed flow (line D), and laminar flow with a parabolic velocity profile (line A) also are represented.

passes at that level the liquid is completely mixed over the cross section and the resulting local concentration of tracer is also observed in the outgoing liquid. Although this indented shape of the tracer outlet concentration curves is more of a curiosity, since it is observed only for very low slugging frequencies, the agreement between experiment and model gave the authors great confidence in the predicting ability of the model developed. For the range of frequencies of greater practical interest, the outlet concentration curves of tracer in response to a step change at the inlet are generally of sigmoid shape. In those cases the use of the model of plug flow with axial dispersion is adequate and much easier from a computational point of view. Guedes de Carvalho et al. used their detailed physical model to generate computed outlet tracer concentration curves for a wide range of conditions, from which they could deduce the apparent dispersion coefficients [7]. The procedure is somewhat elaborate and lengthy (and it is described in the original paper), but a simple result was obtained. A general expression was derived for the axial dispersion coefficient, D*, which reads :EDl.o.l73-«+0.5(lJ^f^ H X(|) " VH

(4)

62

Advances in Engineering Fluid Mechanics

CO

/ (s-0

VG (m^/s)

1.0-1 •

0.98 X 10-6

0.089

D

6.6

X 10-6

A

40

X 10-6

0.59 3.2

0.5-

B

0.0- (/

^

Figure 10. Experimental and theoretical outlet concentration curves (VL = 11.5 X 10"^ m^/s). The theoretical lines corresponding to liquid progression in plug flow (line B), perfectly mixed flow (line D), and laminar flow with a parabolic velocity profile (line A) also are represented.

where T represents the average residence time of the liquid, H the height of liquid in the column in the absence of gas bubbles, and ^ the frequency of slugging seen by an observer moving with the liquid. In Figure 12 this expression is compared with the experimental data available, and the agreement is seen to be excellent. It is worth stressing here that the value 1^ = 2.3 D is not simply a "best fit," but rather a value determined previously by Campos and Guedes de Carvalho [5]. To illustrate the sensibility of the values of D* to 1^, Figure 12 also shows a curve corresponding to 1^ = 3.0 D. Readers should be aware of the fact that equation 4 is again not simply a best fit of data; instead, it is an equation with an important physical meaning. The first term in the equation is dominant for low values of (j), and it accounts for Taylor dispersion; the second term is dominant for high values of (|), and it accounts for dispersion by the wakes. As detailed by Guedes de Carvalho et ai, the parameters in equation 4 are related to the operating conditions in a straightforward manner [7]: H = XH^

(5)

where H^ is the height of the column and X the fractional holdup of liquid inside it.

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns 1.0

VG (m^/s)

/ (s-1)

•

0.71 X 10-6

0.065

n

20 X 10-6

A

31

1.6 2.5

X 10-6

63

CO

Figure 11. Experimental and theoretical outlet concentration curves (VL = 25.5 X 10"^ nn^/s). The theoretical lines corresponding to liquid progression in plug flow (line B), perfectly mixed flow (line D), and laminar flow with a parabolic velocity profile (line A) also are represented.

T =

HA

HAA

(6)

where A is the cross sectional area of the column and VL the volumetric flowrate of liquid flowing along it, and finally

^ = fx - t .

(7)

where / is the frequency of slugging seen by a stationary observer and t^ is the time a slug takes to rise all the way up the column. An important aspect, from a practical point of view, is that equation 4 seems to accurately predict the conditions for which the dispersion coefficient is a minimum, and it also predicts the value of that minimum. The main limitations of the approach adopted should be emphasized so that equation 4 is not misused. The theory developed is applicable only if the frequency of slugging is constant along the column, if the Reynolds number for the liquid plugs is below the critical value of 2,100, and if the slug wakes may be treated as

64

Advances in Engineering Fluid Mechanics 0.15

VL (m?/s) TD^/H

(m) 0.10

t

O n Si • •

1 xlO-^ 2.4x10-^ 5.5x10-^ 11.5x10-^ 25.5x10-^

equation 4 0.05

0.00 0

50

150

100

T 0 / / / (m-0 Figure 12. Experinnental and predicted values of the dispersion coefficient. fully mixed; according to Campos and Guedes de Carvalho the latter condition is observed when (gD3)'/2/v > 900 [5]. As pointed out before, further work is underway to adapt the present model to conditions of free slugging, with coalescence. NOTATION A Cross-sectional area Cg Average tracer concentration in the liquid leaving the column C^ Initial concentration of tracer D Internal diameter of the tube D* Axial dispersion coefficient / Frequency of slug formation g Gravitional acceleration H Height of liquid in the column in the absence of gas bubbles Hg Height of the column 1 Wake length 1^ Length of column equivalent to fully mixed wake L Slug length Dimensionless Groups Re Reynolds number based on slug velocity (pUD/|x))

N r R t tj.

Number of slugs Radial coordinate Internal radius of the column Time Time of rise of a slug up the column u Liquid velocity on a "gas free" basis UQ Liquid velocity over the axis on a "gas free" basis U Velocity of rise of a slug V Wake volume M^ Volumetric flowrate of liquid V Slug volume

Physical Modeling of Axial Mixing in Slugging Gas-Liquid Columns

65

Greek Letters X Fractional holdup of liquid ji Dynamic viscosity V Kinematic viscosity p Density

a Surface tension X Average residence time of the liquid in the column ^ Tortuosity of the bed

REFERENCES 1. Deckwer, W. D., Burckhart, R. and Zoll, G. "Mixing and Mass Transfer in Tall Bubble Columns." Chem. Eng. ScL, 29: 2,177 (1974). 2. Field, R. W. and Davidson, J. F. "Axial Dispersion in Bubble Columns." Trans. I. Chem. E., 58: 228 (1980). 3. van Heuven, J. W. and Beek, W. J. "Gas Absorption in Narrow Gas Lifts." Chem. Eng. ScL, 18: 377 (1963). 4. Campos, J. B. L. M. and Guedes de Carvalho, J. R. F. "An Experimental Study of the Wake of Gas Slugs Rising in Liquids." /. Fluid Mech., 196: 27 (1988). 5. Campos, J. B. L. M. and Guedes de Carvalho, J. R. F. "Mixing Induced by Air Slugs Rising in Narrow Columns of Water." Chem. Eng. Sci., 43: 1,569 (1988). 6. Campos, J. B. L. M. "Mixing Induced by Air Slugs Rising in an Inclined Column of Water." Chem. Eng. Sci., 46: 2,117 (1991). 7. Guedes de Carvalho, J. R. F., Cardoso, S. S. S. and Teixeira, J. A. "Axial Mixing in Slug Flow—The Use of Injected Air to Reduce Taylor Dispersion in a Flowing Liquid." Trans. I. Chem. E., 71, 28 (1993). 8. Taylor, G. I. "Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Tube." Proe. Roy. Soc, A219: 186 (1953). 9. Filla, M., Donsi, G. and Crescitelli, S. "Tecniche Sperimentali per lo Studio della Scia di Bolle." ICP-Riv. Indust. Chim., 10 (1979). 10. Coutanceau, M. and Thizon, P. "Wall Effect on the Bubble Behaviour in Highly Viscous Liquids." J. Fluid Mech., 107: 339 (1981). 11. Maxworthy, T. "A Note on the Existence of Wakes Behind Large Rising Bubbles." J. Fluid Mech., 27: 367 (1967). 12. Levenspiel, O., Chemical Reaction Engineering, John Wiley & Sons, 1962, pp. 272-296. 13. Taylor, G. I. "Diffusion and Mass Transport in Tubes." Proc. Roy. Soc, B67: 857 (1954).

This Page Intentionally Left Blank

CHAPTER 4 NUMERICAL SOLUTION OF THE PERMEATION, SORPTION, AND DESORPTION RATE CURVES INCORPORATING THE DUAL-MODE SORPTION AND TRANSPORT MODEL Keio Toi Department of Chemistry, Faculty of Science Tokyo Metropolitan University Minamiosawa, Hachioji, Tokyo 192-03, JAPAN CONTENTS SYNOPSIS, 67 INTRODUCTION, 67 MATHEMATICAL MODEL, 68 Calculation of Permeation Rate Curve, 69 Calculation of Sorption and Desorption Rate Curves, 70 RESULTS AND DISCUSSION, 71 Various Fickian Diffusion Curves, 72 Integral Diffusion Coefficients, 75 CONCLUSIONS, 76 REFERENCES, 77 SYNOPSIS Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. Satisfactory agreement is obtained between integral diffusion coefficient from sorption and desorption rate curves and apparent diffusion coefficient from permeation rate curves (time-lag method). These rate curves were also compared to the curves predicted by Fickian-type diffusion equations. INTRODUCTION Toi et al. applied the dual sorption mechanism to analysis of the time-lag diffusion (permeation) under the constraint that the penetrant fraction attributed as the Langmuir component is completely immobilized, but in local equilibrium with the Henry's law dissolution component [1]. They yielded a mathematical description of transient permeation, consisting of a nonlinear partial differential equation. This equation was then solved by a finite-difference technique for the case of permeation 67

68

Advances in Engineering Fluid IVIechanics

through a film to a gas reservoir of limited volume. Recently, the concentration dependence of apparent diffusion coefficients obtained in permeation rate experiments has been mainly analyzed to yield diffusion coefficients for dual-mode sorption and transport model. Numerical methods were used to examine the dualmobility systems applied to sorption rate experiments with plane slab absorbers for a mobilized Langmuir component system [2,3]. But no attempt appears to have been made to predict the numerical solution of the permeation, sorption, and desorption rate curves collectively, incorporating the dual-mode sorption and transport model with an immobilized Langmuir component. The objective of the present study was to provide this information by comparing these predicted curves with the curves predicted by the Fickian-type diffusion equations. MATHEMATICAL MODEL The equilibrium part of the theory is expressed simply by following for the isotherm: C = C ^ + C H =k^p

^

"

^^

+ -^ii^

= cJ

1 + bp

1+

%

1

l+(b/kjcj

(1)

^^^

where C is the total penetrant concentration; C^ is the concentration due to the Henry's law contribution; C^ is the concentration of penetrant held in microvoid or the Langmuir contribution, p is the upstream pressure; K is composite parameter (K = CHb/ki)); and k^, b and C^ are dual sorption parameters. According to the model of Paul and Koros, the total diffusive flux N is given

where D^ and D^ are diffusion coefficients for the dissolved and Langmuir modes, respectively [4]. All the gas previously associated with C^ as well as a fraction F(F = DH/D[)) of that associated with C^ has a mobility with a diffusion coefficient D while the remaining fraction, 1 - F, of C^ has zero mobility, so Fick's law must be written as

where gCC^) represents the function „rr . ^^"-"^

[l + FK/(l + (b/k„)C,)^] [l + K(l + (b/k,)C,)M

(4)

Concentration distribution of gas in membrane was calculated at regular time intervals by the finite-difference technique [5]. A forward-differential equation was used to proceed from values of C; ^ to values at the next time step Cj ^+1,

Numerical Solution of the Permeation, Sorption, and Desorption Rate Curves

69

Ax2Cj,„ = g(Cj,,)(Cj,„,, - Cj,„)/Ae where AX = X/j (real distance x = jAXl), (AG = r/j^ (real time t = nABl^/D), and r is a parameter concerning the time step. This equation, together with the boundary conditions discussed later was programmed for solution by FORTRAN language. Digital Pt values at finite times were stored in a disk file for later analysis or plotting by xy-plotter to examine them to inspect their integrity of parameters. Calculation of Permeation Rate Curve In applying Equation 3 to the permeation experiments, the following coordinates were used: the polymer membrane, of thickness 1, is assumed to be of infinite extent in the y- and z-directions. The concentration of gas in the membrane is initially zero. Upstream pressure p is assumed to hold constant, but the downstream pressure Pt increases from the initial value 0 to the final value Pj^. This is illustrated in Figure la.

^l-> p-^ A(cm') Film

P-P,-^

V(cm')

A( cm^)

-•Pr

V(cni') ^=-^^=0 ^ = ^

(b)

(a)

P,^

A( cm') V(cm')

^=Vo ^='^ (c) Figure 1. Coordinates for mathematical models: (a) permeation, (b) sorption, and (c) desorption.

70

Advances in Engineering Fluid Mechanics

The boundary and initial conditions on Equation 3 are as follows: t = 0,

1 > X > 0,

CD

= 0

t > 0,

X = 0,

CD

= kop

t > 0,

X = 1,

C D = kcPt

(5)

In this case, an additional boundary condition must be applied. In gas flows through a membrane area A(cm^) into a volume V(cm^), the flow of gas is given by

The quantity (dC/dx)^^^ is obtained by the finite-difference technique respect to X and setting x = 1, where Cj is gas concentration of the downstream volume at time t; then pj can be represented as RTl

AD , , . ^ ,

^

V Haxi., It is difficult to solve Equation 3 analytically since the equation is nonlinear and the surface boundary at x = 1 condition is transient. Then, a numerical solution using a computer seems to be convenient. Calculation of Sorption and Desorption Rate Curves In applying Equation 3 to the sorption and desorption experiments, the following coordinates were used: the polymer membrane, of thickness 1 = 2L, is assumed to be of infinite extent in the y- and z-directions. These are illustrated in Figures lb and Ic. The concentration of gas in the membrane is initially zero for sorption and kpP for desorption, and the membrane surface is assumed to be always in equilibrium with the surrounding gas. To simplify the numerical solution, the condition of symmetry about x = 0 was used, and solution was obtained for the halfmembrane. Boundary conditions on Equation 3 are then as follows: For sorption, t = 0,

L > X > 0,

t > 0,

X = 0,

dCj^/dx = 0

t > 0,

X = L,

C D = koCp - Pt)

CD

= 0 (8)

The material balance that gives the values of pj (pressure change of volume V at time t) is RTl

p = p

A f'

^ ,^ (c„ + Cn)dx 22,400 V -^0^

(Q) ^^

Numerical Solution of the Permeation, Sorption, and Desorption Rate Curves

71

For desorption, t = 0,

L > X > 0,

t > 0,

X = 0,

dC^/dx = 0

t > 0,

X = L,

CD = kppt

CD

= kop (10)

The material balance which gives the values of pj (pressure change of volume V at time t) is Pt =

'

RTl A f' ^ ,. (Co+Cnjdx 22,400 vJo " ""

(\\\

^ ^

As with permeation, the numerical solutions using a computer seem to be convenient. RESULTS AND DISCUSSION The permeation, sorption, and desorption rate curves are calculated by the finitedifference technique using the parameters for CO2 in a polyimide (PI2080) at 25°C, obtained by Toi et al. [6] and polycarbonate at 35°C, obtained by Koros et al. [7]; these are listed in Table 1 as original parameters. It is confirmed that plots of the experimental data in both the studies fit well on the dual-mode model lines calculated using each set of the original parameters. The calculations are carried out at pressures of 10, 25, 40, 60, and 76 cmHg for PI2O8O/CO2 system and 10, 76, 380, 760, 1,520 cmHg for polycarbonate/C02 system, respectively.

Table 1 Dual-Mode Sorption and Transport Parameters of CO2 Diffusion in PI2080 Polyimide and Polycarbonate

b (atm-^)

cm^(STP)

ko cm'(STP) cm^atm

CH

F

D

—

cm' s

?

PI2O8O/CO2 system Original Parameters [6] ®: From Equation 14 (D: From Equation 15

5.17 8.27 5.03

11.56 2.49 11.22

4.67 1.39 2.69

0.0174 0.0114 0.0021

1.73 1.78 1.67

X

PC/CO2 system Original Parameters [7] (D: From Equation 14 ®: From Equation 15

0.338 0.416 0.754

15.3 16.68 13.23

0.938 1.160 0.574

0.101 0.0886 0.0146

4.67 4.62 3.74

X

X X

X X

10-9 10-y 10-9

10-8 io-« io-«

72

Advances in Engineering Fluid Mechanics

Various Fickian Diffusion Curves The diffusion coefficients determined in the conventional manner also were substituted into the analytical solutions of the purely Fickian diffusion equations for permeation, sorption and desorption into flat membrane [8]. For permeation,

^'

22,400

'^

7i' ; ^

1 - exp -

n'

Dn'rcn

(12)

where D^ is apparent diffusion coefficient from time-lag method; C^ is apparent solubility, represented by C^ = 76(Pa/Da)p (where P^ is apparent permeation coefficient calculated from slope of permeation curve); and Cj is a value of Q at time t. For sorption and desorption. exp[-P(2n + l)-7tn ~ -

(2n-hl)M'

(2n + 1)KX X COS

(13)

1

where D^ is apparent diffusion coefficient calculated from initial slope of sorption rate curve, Cg is apparent solubility from equilibrium value of sorption/desorption rate curve. In this case, C^ is calculated from 22,400Vpt/lRTA. Representative permeation rate curves shown in Figure 2 are for PI2O8O/CO2 system at pressure 10 and 76 cmHg in the form of plots of P/cmHg versus p/atm. The points are calculated by the finite-difference technique using original parameters in Table 1. The solid lines are calculated from Fickian permeation equation, Equation 12, using apparent diffusion coefficient and solubility calculated from the points curves. As shown clearly, there is fairly good agreement between the points and the lines, except a portion of non-steady state of the finite-difference curve at higher pressure (76 cmHg). Figure 3 shows typical sorption and desorption cycles calculated from the dualmode sorption and transport model as described previously using the earlier parameters for the same polymer/gas system at the same pressures and temperatures. The points are calculated by the finite-difference technique using original parameters in Table 1. The solid lines represent the predictions of the Fickian transport model. Equation 13, with an apparent diffusion coefficient and solubility calculated from the points curves, respectively. As seen in Figure 3, the sorption curves lie above the corresponding desorption curves, indicating that the diffusion coefficient increases with increasing concentration. The Fickian model provides a fairly good description of initial portions of both the sorption and desorption curves. The deviations from the predictions of the model at longer times for the desorption cycles are presumably a consequence of the substantial concentration dependence of diffusion coefficient [4]. In the same manner, the sorption curve of higher pressure (76 cmHg) deviates at a portion before attainment of the sorption equilibrium. These appearances are also found for PC/CO2 systems as shown in Figures 4 and 5. A slight sigmoidal deviation from Fickian model lines was apparent in the desorption cycles. But in the permeation curves and sorption cycles, each run

Numerical Solution of the Permeation, Sorption, and Desorption Rate Curves

P12080/CO2 2 5 T : X E a O

4

09-

Figure 2. Pernneation curves for CO2 in PI2080 at 25°C, and 10 and 76 cmHg pressures. The points are calculated by the finite-difference technique using original parameters in Table 1. The solid lines are calculated from Equation 12 using apparent diffusion and solubility coefficients calculated from the point curves.

1.2

-I

r-

"1

1

1

1—

PI2O8O/CO2 25V 0.8

o* c 0.4

Figure 3. Sorption curves for CO2 in PI2080 at 25°C, and 10 and 76 cmHg pressure. The points are calculated by the finite-difference technique using original parameters in Table 1. The solid lines are calculated from Equation 13 using apparent diffusion and solubility coefficients calculated from the point curves.

73

74

Advances in Engineering Fluid Mechanics 20

1

1

r-

PC/C02 35'C. 76cmHg E n

U

'o 10

06 o o d »

800

1000

Figure 4. Permeation curves for COg in PC at 25°C and 1 atm pressure. Symbols as In Figure 2.

1.2

0.8

to"

cS 0.4h

Figure 5. Sorption curves for CO2 In PC at 25°C and 1 atm pressure. Symbols as In Figure 3.

Numerical Solution of the Permeation, Sorption, and Desorption Rate Curves

75

is almost entirely by pure Fickian diffusion. The results for both the systems implied that the sorption and permeation curves calculated using some combination of dualmode parameters sometime show deviation from the Fickian curve. But in the desorption cycle, a sigmoidal deviation from the Fickian curve always is apparent. These methods are interested in studying the distinction between the pure dualmode sorption/transport model curve and the actual sorption and permeation experiment curve that seems to contain the various unsolved appearances for diffusion and sorption of gases in glassy polymer. It becomes obvious that the deviations from the Fickian model in an experimental transport or sorption/desorption curve for a gas in a glassy polymer are not necessarily consistent with the onset of concomitant diffusion and relaxation [11], but are just owing to the dual-mode model. That is to say, only the combinations of parameters of the dual mode model make the curve either fit with or deviation from the Fickian model curve. Integral Diffusion Coefficients For sorption/desorption rate curves, a good estimation of integral diffusion coefficient can be obtained by taking the average of the diffusion coefficients calculated from first slope for the integral sorption ( D J and desorption (D^) cycle, [8] D.V = (D, + D, )/2

On the other hand, the apparent diffusion coefficient D^ is calculated from the timelag method of permeation rate curve.

D^ = Wee Both the diffusion coefficients should be coincident at the same pressure and temperature. Strictly speaking, this comparison seems to be inapplicable to the PI2O8O/CO2 system because the curves do not follow Fickian type, but applicable to the PC/CO2 system. Figure 6 shows the pressure dependence of diffusion coefficients calculated from permeation, sorption, and desorption rate curves for CO2 in PI2080. The average values of diffusion coefficients from sorption and desorption rate curves D^y are in fair agreement with that from permeation rate curve D^. The solid line in Figure 6 was computed from Equation 14 ^

^ D[l + FK/(l + bp)] ' [l + K/(K,F,bp)]

^^^^

by using parameters ® in Table 1. These parameters were generated by fitting the above relation to the D^ values by the "simplex" method and by nonlinear regression [4,11]. The dotted line also was computed from Equation 15 [10]. g

^ D [ l + FK/(l + bp)] [l + K/(l + bp)]

76

Advances in Engineering Fluid Mechanics

-20.0

CO U

-22.0

c

-24.

0.4

0.6

p/atm Figure 6. Pressure dependence of diffusion coefficients calculated from permeation (C), sorption (O) and desorption ( A ) rate curves for COg in PI2080. # is average value of diffusion coefficients from sorption and desorption rate curves at same pressure. The solid line is calculated from Equation 14 using parameters © in Table 1. The dotted line is calculated from Equation 15 using parameters @ in Table 1.

by using parameters ® in Table 1, which were generated in a similar manner. As can be seen from Table 1, there are poor agreements between the original parameters and the parameters determined from analysis of kinetic data for both the systems. The degree of the convergence depends upon the method of calculation. More efforts should be made to improve the agreement. However, the agreements between the curves of the two diffusion coefficients are almost good as shown in Figure 7. For the PC/CO2 system that is good according to the Fickian type curve, the agreement between D^ and D^v is somewhat better than for the PI2O8O/CO2 system. CONCLUSIONS Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. These rate curves were almost consistent with the curves predicted by Fickian-type diffusion equation, except the desorption curve in which a slight sigmoidal deviation from the Fickian model line was apparent. The sorption

Numerical Solution of the Permeation, Sorption, and Desorption Rate Curves -16

I

PC/CO2 aS'CCKoros e t

I

I

77

T

al.)

CO E U

S

-18

^

-20

8

12

16

20

p/atm Figure 7. Pressure dependence of various diffusion coefficients for CO2 in PC. The solid line is calculated from Equation 14 using parameters ® in Table 1. The dotted line is calculated from Equation 15 using parameters ® in Table I. Other symbols as in Figure 6. and permeation curves calculated using some combination of dual-mode parameters sometime show deviation from the Fickian curve. Satisfactory agreements also are obtained between integral diffusion coefficient from sorption and desorption rate curves and apparent diffusion coefficient from permeation rate curves.

REFERENCES 1. K. Toi, Y. Maeda, and T. Tokuda, 7. AppL Polym ScL, 28, 3,589 (1983). 2. S. Subramanian, J. C. Heydweiller, and S. A. Stern, J. Polym. Sci.: Part B: Polym. Phys., 27, 1,209 (1989). 3. C. M. Riedl and H. G. Spencer, J. Appl. Polym. Sci., 41, 1,685 (1990). 4. D. R. Paul and W. J. Koros, J. Polym. Sci. Polym. Phys. Ed., 14, 675 (1976). 5. J. Douglas, J. Trans. Am. Math. Soc, 89, 484 (1958). 6. K. Toi, T. Ito, T. Shirakawa, and I. Ikemoto, J. Polym. Sci. Part B, Polym. Phys., 30, 549 (1992). 7. W. J. Koros, D. R. Paul, and A. A. Rocha, /. Polym. Sci. Polym. Phys. Ed., 14, 687 (1976). 8. J. Crank, The Mathematics of Diffusion, 2nd ed.. Clarendon, New York, 1976. 9. R. M. Felder, C. J. Patton, and W. J. Koros, J. Polym. Sci. Polym. Phys. Ed., 19, 1,895 (1981).

78

Advances in Engineering Fluid Mechanics

10. W. J. Koros, C. J. Patton, R. M. Felder, and S. J. Fincher, J. Polym. Set Polym. Phys. Ed., 18, 1,485 (1980). 11. K. Toi, H. Takai, T. Shirakawa, T. Ito, and I. Ikemoto, J. Membrane Sci., 41, 37 (1989).

CHAPTER 5 KINEMATIC VISCOSITY AND VISCOUS FLOW IN BINARY MIXTURES CONTAINING ETHANE-l,2-DIOL Lorenzo Tassi University of Modena Department of Chemistry 41100 Modena, Italy CONTENTS INTRODUCTION, 79 SELECTION AND DESCRIPTION OF AVAILABLE CORRELATIONS, 81 Temperature Effect, 82 Composition Effect, 84 Evaluation and Comparison of Available Correlations, 90 EXCESS FUNCTION, 91 THERMODYNAMICS OF VISCOUS FLOW, 95 SUMMARY AND FUTURE PERSPECTIVES, 100 ACKNOWLEDGMENTS, 101 NOTATION, 101 REFERENCES, 102 INTRODUCTION It has been a long time since dispersion and specific interactions in nonelectrolytic liquid mixtures involving both non-hydrogen bonding and hydrogen bonding solvent types have been studied by applying many and very different experimental techniques, such as thermomechanic (density, viscosity, dielectric, and refractive properties are the most common) and thermodynamic (heat of mixing) measurements, in addition to spectroscopic techniques, IR and NMR in particular [1]. A literature survey reveals many of transport properties of fluid, a primary source of fundamental data for the solution of practical problems about heat and mass transfer in real systems, for processes development, and engineering design. Furthermore, in recent years the employment of computer simulation methods of molecular dynamics has led to significant progress towards a successful molecular theory of transport properties in dense fluids and a greater understanding of molecular motions and interaction's patterns in such systems [2,3]. However, in spite of these great and concerted research efforts by many experimentalists and theoreticians, it must be deduced that a lot of work will have to be 79

80

Advances in Engineering Fluid Mechanics

done to reach the final objective represented by a more general theory for predictive purposes (and available in solving carrier problems). So we also are engaged in a research program to increase knowledge about thermomechanic properties of organic solvents (in particular viscosity), employing some selected pure species and their binary and ternary mixtures, either nonaqueous and/or mixed with water [4-13]. At present, the lack of any universal model allowing an exact evaluation of viscometric properties of pure liquids and liquid mixtures is mainly due to two unsolved questions: 1) no comprehensive theory describing the interactions at the molecular level between similar and/or unlike species is known; 2) deviations from ideality are not predicted neither in sign nor in intensity by the common thermodynamic liquid solution principles [14-19]. Both problems are unlikely to be solved in the very near future, even if there is much interaction information. A further aspect regarding the viscometric properties of liquid real systems which must be carefully considered when investigating this property—and in particular when one would expect reliable information on specific interactions between components—is represented by the choice of experimental measurement technique because it is possible to obtain two different quantities, such as kinematic viscosity (v/cSt, IcSt = 10~^m^s~') or dynamic viscosity (v/cP, IcP = lmPa»s). These properties can be interconverted by the fundamental relation r| = vp

(1)

if density (p/kg m"^) of the liquid system is known at each selected measuring temperature. Generally, T| represents an important input parameter for most process design calculations, which is obviously sensitive to specific intermolecular interactions. But because this property is a derived quantity from density and kinematic viscosity, which are both independently sensitive to these specific interactions, it should be preferably recommended to experimentalists to provide these quantities separately, instead of the r| values, to avoid overcoming effects which lead to mismatching interpretation of the experimental evidence. Furthermore, the easy determination of the kinematic viscosity makes possible, in many cases, on-line process control in evolving systems (polimerization, esterifications, hydrolysis reactions, etc.) because the viscometric properties are strictly related to the composition of the reaction bath and depend on the degree of advancement of the chemical reactions. Since the variation of thermophysical parameters with respect to temperature and to solvent composition for real interacting systems is usually non-linear, estimated values of v, p and r| for liquid mixtures corresponding to the experimental data gaps are subjected to substantial interpolation errors, especially when the experimentally determined data for a given solvent system are sparse. Still, the investigations of the excess mixing properties are very useful in examining the possibility of the formation of solvent-cosolvent complex adducts in solution [20]. The dependence of these properties on temperature and on composition of the liquid mixture permits their continuous variation and can provide much useful information, otherwise hardly detectable, about the complex species [21]. This review is devoted to inspection of the availablity of some most outstanding relationships between viscosity of real liquid mixed systems and the independent

Kinematic Viscosity and Viscous Flow in Binary Mixtures

81

variable quantities temperature and binary solvent composition, with particular emphasis on some mixed solvents containing ethane-1,2-diol (ED) coupled with 2-methoxyethanol (ME); 1,2-dimethoxyethane (DME); 1,4-dioxane (DX); N,Ndimethylformamide (DMF); and water (W). Unfortunately, reference is made only to very few works of direct relevance to our specific and limited studies; as a consequence, a lot of material of very good quality and high standard has been omitted. A selection of results illustrating the power of some correlation techniques and providing enlightening conclusions is presented in the following sections while in a final section a standpoint is made about the actual situation, and it is suggested how the argument may be developed both in the short and long term. ED is an exciting subject for the study of thermodynamic and transport properties because of its intrinsic remarkable characteristics. In fact, it is the simplest homologue of the diol series, largely utilized as thermoregulator fluid, as a controlling agent of density and viscosity reaction baths, as emulsion coating, among other uses owing to its unusual viscokinetic characteristics and surface tension [22]. Furthermore, some properties such as the cohesive energy density and internal molecular pressure of ED, indicate that this solvent is highly associated [23,24]. ED is thought to form hydrogen-bonded chains so that any homocooperative domains would be predominently two-dimensional, even if different rotational conformers can still strongly interact with neighboring molecules along the whole three-dimensional space [25]. As a consequence, these pure species at ordinary conditions of temperature and pressure can be considered a total homopolymer whose viscometric properties lie between Newtonian liquids and true rheological fluids, such as pastes, muds, suspensions, high polymers, etc. From this viewpoint, ED is therefore, an excellent and more versatile subject showing rheological behavior in the low-temperature region (less than ordinary conditions) and quasi-Newtonian properties at higher temperatures. These different performances can be mainly attributed to the great elasticity and deformability of hydrogen bonding network in nonaqueous species like ED, which is responsible for the total homopolymer at low temperature and which can slacken by increasing the temperature up to the progressive disruption of the building-up interactions. This fact provides a progressive lowering of the structural degree of fluid system (viscosity dramatically decreases with temperature) when homocooperative domains are gradually related to few molecular units. SELECTION AND DESCRIPTION OF AVAILABLE CORRELATIONS When selecting correlation functions for evaluation, as well as in making the final recommmendations, particular attention should be given to some basilar criteria such as: i) ease of employment; ii) availability of input parameters; iii) range of applicability; iv) accuracy in reproduction of experimental data. Most of the recently published kinematic viscosity correlations appear to be accurate enough for design calculations. Therefore, point i) through iii) should have the same level of importance as accuracy. In other words, a specific correlation for kinematic viscosity may be very accurate ±(0.2 - 0.3)% for Newtonian fluids, within the limits of experimental accuracy of commercially available measurement equipment.

82

Advances in Engineering Fluid Mechanics

Obviously, this range of accuracy can be reasonable extended up to ±1.5% for pseudo-Newtonian fluids and, further, beyond this limit for true rheological matters. On the basis of these standard requirements, an appropriate equation for kinematic viscosity should be able to yield good or at least reasonable results in the widest operative temperature range in the limits between the triple point up to the critical point for most liquid compounds. Furthermore, the functional form must describe as best as possible all experimental trends, whether linear or more complex. Bearing in mind the former remarks about the choice of an equation for fitting procedure of a set of experimental data, it should be noted that correlation functions for viscosity values generally can be safely employed both for kinematic and dynamic quantities, this fact being stated and ascertained in many reports. Therefore, the equations scrutinized in this review satisfy these requirements even if more emphasis will be given on considering kinematic properties in particular. Temperature Effect Table 1 reports the kinematic viscosities for the selected pure species at 19 temperatures in the range -10 to +80°C. As one can see, this table lacks some v values where phase separation occurred. Table 1 Experimental Values of Kinematic Viscosity (v/cSt) for Some Selected Pure Solvents at Various Temperatures

t/°c

ED

ME

DME

DX

DMF

-10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

98.94 71.09 52.42 39.54 30.44 23.89 19.04 15.44 12.66 10.49 8.806 7.482 6.434 5.532 A.lll 4.192 3.687 3.278 2.909

3.594 3.133 2.763 2.446 2.182 1.959 1.769 1.607 1.469 1.341 1.232 1.137 1.052 0.9781 0.9122 0.8581 0.7992 0.7504 0.7076

0.7002 0.6557 0.6220 0.5898 0.5605 0.5338 0.5095 0.4859 0.4626 0.4417 0.4235 0.4052 0.3887 0.3735 0.3592 0.3456 0.3332 0.3210 0.3093

— — — — 1.383 1.272 1.171 1.084 1.007 0.9379 0.8775 0.8233 0.7752 0.7298 0.6901 0.6555 0.6235 0.5947

1.397 1.289 1.192 1.108 1.034 0.9693 0.9121 0.8605 0.8130 0.7712 0.7337 0.6975 0.6650 0.6374 0.6102 0.5867 0.5635 0.5401 0.5210

From ref. [18].

W* — 1.787 1.519 1.307 1.140 1.004 0.8930 0.8010 0.7237 0.6580 0.6019 0.5534 0.5113 0.4745 0.4421 0.4134 0.3878 0.3650

Kinematic Viscosity and Viscous Flow in Binary Mixtures

83

The problem of finding an adequate explanation for the peculiar temperaturedependence of viscosity in liquids has led to an awesome number of purely empirical and quasi-theoretical relationships. A very extensive recognition of these models and related aspects has been provided by Partington [26]. However, scanning these, most liquids produce an equation requiring at least three constants to describe accurately the variation in viscosity over a wide temperature range. A more attractive equation derived by Vogel [27], Fulcher [28], and Tamman [29] gives viscosity from an exponential function that is currently written in/the form (VTF equation): In V = a +

T - T ^

(2)

* 0

and for which a quasi-theoretical background based on free volume effects exists. This relation has been found to describe very well the behavior of many simple liquids (such as mercury, gallium, CCl^ among others) [30], mixed Newtonian liquid systems, and rheometric fluids. In equation (2) a, b and T^ are three adjustable parameters generally evaluated by fitting a set of experimental data through a multilinear regression paclcage. It should be noted that the constant a = lim(lnv) is a term independent of T while T^ is a singular temperature which characterizes each fluid system. An interpretation concerning the T^ fitting parameter, in particular when working with Newtonian and pseudo-Newtonian liquids (as in these cases), could be proposed on the basis of the mechanical stability limit approach for the supercooled liquid state before the homogeneous nucleation of the systems takes place. In fact, the T^ term should have an empirical meaning because it should represent the temperature below which the viscosity of the system becomes infinite and defines the limit field below which the free intermolecular volume is no longer available for a viscous process. Thus, a dramatic increase in viscosity at low temperatures may suggest the onset of a pseudo-glassy state which preludes the solidification of the system. If so, the condition v -> «> for our selected pure species is reached at temperatures much higher than those reported in our previous papers [4,8-12], the melting points being quite far and more positive than the actual T^ values. As a consequence, it seems then plausible that the values obtained for T^ are too low for their binary mixtures as well, even if they appear consistent with one another. Tentatively, this evident anomalous behavior for all pure species and their mixtures can be explained by invoking molecular association in the liquid state. This phenomenology also should be responsible for the easy supercooling of the pure components, coherently with their asymmetric nonplanar molecular geometry. It is well-recognized that liquid viscosity shows abnormal variation with temperature near the freezing point and critical point and, in most cases, calculation models and theories are completely inadequate to reproduce experimental data in the proximity of these singular points at which the transition phase occurs. Therefore, as a general rule, the use of estimated viscosity data at temperatures close to the freezing point or reduced temperatures above 0.95 should be avoided [17]. In spite of this advice and bearing in mind the preceeding remarks about empirical meaning of T^, Equation 2 doesn't seem to be limited near the freezing point of

84

Advances in Engineering Fluid Mechanics

the selected solvent systems and works very well in as large a temperature range (from melting point until T^), as ascertained by the average percent deviations (Av% = ±1.5 cSt) between experimental and calculated values [4,8-12]. As a rule, the availability of Equation 2 is warranted when a plot of In v vs. (T - TQ)~' (Figure 1) shows a linear trend over all the temperature range investigated. Composition Effect Generally, plots of v values against binary composition for mixtures containing ED are not linear even if viscosity can increase monotonically as the ED mole fraction increases when cosolvent is chosen between our selected species mentioned earlier. In fact, the highest v values are due to the highly homopolimerized structure of the ED component. This structure appears quickly depolimerized both by increasing the temperature and/or by formation of heteropolymers when cosolvent concentration increases. In the past, many authors suggested different correlation functions of the general type V = v(X.), and a lot of experimental data have been used to check their suitability to provide interpolated values in correspondence of experimental data gaps. In this section, we will consider some of the more commonly used semi-

4

h

Ini/

0

10

io^(r - To)^^ Figure 1. Plots of Inv for the ED/W binary solvent system as a function of (T-T„)-VK-^ [11].

Kinematic Viscosity and Viscous Flow in Binary Mixtures

85

empirical models for analyses of this kind, and a comparison of their effectiveness will be made by using the standard statistical parameters as indexes and taking into account the number of empirical adjustment coefficients necessary for each of them. Grunberg-Nissan Equation One of the earliest attempts to describe the dynamic viscosity of binary liquid mixtures as a function of the pure components was by Grunberg and Nissan [31]. They provided a simple equation containing one adjustment parameter only, attributable to the intermolecular interactions between the components. It is current opinion that the validity of this model equation and the meaning of the terms are maintained even if one refers the data analysis to the kinematic viscosity values. The expression In V = X,lnv, + X^lnv^ + X.X^d

(3)

where v,, v^ and v represent the kinematic viscosities of the two pure components and of the mixtures respectively, X, and X2 being the mole fractions, has been used to evaluate the parameter d, which is regarded as a measure of the strength of specific interactions between dissimilar molecules. The d values for different solvent systems involving ED as common species are always negative at all the investigated temperatures [5-7], showing a large difference when going from the lowest temperatures to the highest ones. The negative sign of the d parameter is indicative of the existence of dispersive forces (such as hydrogen bonding and dipolar forces of any kind) intervening as specific interactions and acting as structure-breaker effects in these mixed liquids. McAllister Equation One approach to the accurate estimation of liquid mixture viscosities is by McAllister, who, starting from the Eyring approach [32] and considering the specific interactions between contiguous molecule layers, describes the stationary motion of liquid in terms of a dynamic steady state. This representation takes into account the velocity gradient, which involves activated molecules jumping between layers. The motion of a molecule has been treated as if the molecule were undergoing a chemical process, and according to Eyring's hypothesis of a "hole in the liquid", a shear stress forces it into a hole while traversing an activation energy barrier. For the simplest mixture of two components, McAllister provided the following empirical equation for the kinematic viscosity [33]: In V = Xj In v,M, + 3X?X, In v„ ^ ^ ' ^ ^ ^ + 3X,Xl Inv,, ^ ' ^ ^ ^ ^ + X^ In V2M2 - ln(X,M, + X2M2)

(4)

In Equation 4, M, and M2 are the molar masses of components 1 and 2, respectively. The two cross interaction terms Vj^ and v^, are adjustable parameters of the fit; they

86

Advances in Engineering Fluid Mechanics

are characteristic of every binary system (l)/(2) and can be evaluated by a leastsquare procedure. According to McAllister, the choice of a cubic equation is justified since for the system selected in this research the ratio of the molecular radii (r/r^) is smaller than the imposed limit of 1.5 at all the investigated temperatures. For example, the values of adjustable parameters v. for DMF (1)/ED (2) binary mixtures are represented in a graphic form in Figure 2 as a function of temperature. The same regular trend has also been observed for other solvent systems, such as ED/DX [6] and ED/ME [7]. Therefore, by analyzing a good many sets of these parameters we are going to provide an assessment about the physical significance that might be attributed to them. The solid curves of Figure 2 have been obtained by a fit of the type: Co exp

f o r ( i ^ j ) = l,2

(5)

whose linear logarithmic form has the correlation coefficients r = 0.999 for v,^ and r = 0.994 for V2,, respectively. These parameters v. may be interpreted in terms of quantities related to the specific interactions between unlike molecules of the type 1-1-2, 2-1-2 for v,^, and U ^J

20

10

-10

20

50

80

t{'C)

Figure 2. McAllister's equation v^2 ( • ) ^"^^ ^21 (^) fitting parameters vs t/°C for ED/DMF binary mixtures [57].

Kinematic Viscosity and Viscous Flow in Binary Mixtures

87

1-2-2, 1-2-1 for v^,, where the central unit, according to the McAllister theory, is involved in a hypothetical ternary aggregate. Other possible ternary homonuclear species, such as 1-1-1 and 2-2-2, do not influence the variation in the cross-terms. Now, one may think that the interaction patterns between a 1 or a 2 central unit and the surrounding molecules are always the same, depending only on the chemical nature of the species. However, the relative intensities of these interactions may be different in every spatial direction for the various neighboring arrangements considered. In this light one may deduce that the cross-terms in the fit equations should have nearly the same statistical weight and as a consequence, for any fixed temperature, differ slightly in value. In all the examined cases McAllister treatment provides adjustment parameters which are very different in magnitude, especially at the lowest temperatures, as it is particularly evident from Figure 2. As the temperature increases, the relative differences 5^ obtained from the equation:

6 = ' i(V,2 -V^J^axl

(6)

become smaller and smaller. Equation 6 defines a parameter (6^) which is strictly related to the incremental variation in the point-by-point scattering of the v. coefficients in their variability ranges. In fact, being stated by Equation 6 that 0 < 6^< 1, the difference in v^ values decreases as 8^ increases and, consequently, the same statistical weight can be attributed to the cross interaction terms between unlike molecules. However, working with a reasonably large set of points, it appears justified to use the average value 8^ to check the validity of these considerations. On this basis, McAllister's equation appears more effective in describing the viscokinetic properties of the chosen binary mixtures at the highest temperatures while i^ is probably less effective at the lowest ones. This correlation procedure yields 8^ = 0.251 for the ED/ME binaries [7], 0.255 for DMF/ED [5], and 0.432 for ED/DX [6] solvent systems, respectively. It should be noted that McAllister's equation seems to be very versatile and has been successfully applied to ternary mixtures and more complex solvent systems, too, employing an expanded form of Equation 4 with as many fitting parameters as the number of components in the mixtures [34]. Heric

Equation

Another suitable relation for correlating binary viscosity data was suggested by Heric [35] in the form: I n v = X, ln(v,M,) + X^ InCv^M^) - ln(X,M, + X^M^)

+ X,X,[p,, + P,,(X, - X,)]

(7)

The terms ^^^ and p^, are the best-fit coefficients of Heric's equation and represent the interactions between unlike molecules. These quantities may be computed by a weighted least-squares method for each isothermal set of experimental values. The fitting coefficients p. show small differences at all the investigated temperatures.

88

Advances in Engineering Fluid Mechanics

However, the corresponding 8^ is not large (0.501 for ED/ME, 0.432 for DMF/ED, 0.760 for ED/DX binary mixtures, respectively [5-7]) and is markedly different from the unit. Lobe Equation According to Lobe's approach, the kinematic viscosity of a binary mixture is given by the relation: V = (t),v, exp((t)2a2) + (t)2V2 exp((t),a*)

(8)

where (j), and ^^ are the volume fractions [17]. Lobe has suggested that if the component 1 is chosen as the pure species with the smaller viscosity and if the kinematic viscosity of the mixture varies monotonically with respect to the composition, the parameters a* and a* can be represented in a simplified form by the relations: a* = a,2 In

a* = ttj, In

^v.

V

(9)

(10)

where a,2 and a^^ are two empirical adjustable parameters from the experimental data fit. Actually, Lobe's formula could contain three fitting parameters, but for all these mixtures the third adjustable coefficient, contained in the a* term, has no statistical weight and was, therefore, omitted. The relative differences 8^ for a coefficients are always small (8^= 0.755 for ED/ME), showing an almost regular variation with temperature. Lobe's equation in this simplified form estimates the kinematic viscosities with a good degree of accuracy, always comparable with McAllister's and Heric's formulae [5-7]. Rational Function Although linearized polynomial relationships like those previously examined are commonly used in fitting procedures for chemical and physical data. King and Queen suggested that rational functions are surely more adequate for these aims [36]. So, other authors emphasized the use of this correlation procedure type, and the non-linear regression model for the representation of thermomechanical properties for some binary solvent systems has been elegantly discussed by Kolling in earlier works [37]. Now, bearing in mind those suggestions, we proposed an equation for kinematic viscosity as follows: ^^^.^ X,(X. +Y.2X2)lnv, +X2(X2 +y2,X,)lnV2 X , ( X , + 7,2X2 ) + X 2 ( X 2 + Y 2 . X , )

^^^^

Kinematic Viscosity and Viscous Flow in Binary Mixtures

89

where 7,2 ^^^ Y21 ^^^ cross-interaction terms, and the other symbols have their usual meaning. The adjustable coefficients y^ for ED/DX [6] binary mixtures are shown in graphical form in Figure 3. From this plot it is possible to observe that the trend of the best-fit parameters of Equation 11 is quite regular, and the relative differences 5^ ar^ always very small over the whole investigated temperature range. The average 6,values are the highest (0.680 for DMF/ED, 0.818 for ED/ME, and 0.963 for ED/DX) among those obtained for the previously examined equations with two fitting parameters. In this way it is possible to attribute approximately the same statistical weight to the cross-interaction terms between dissimilar molecules. Auslander Equation Auslander proposed an equation for best-fitting procedures of viscosity data of the type: X,(X, + B^pi^){v - V,) + A^.X^CB^.Xj + X^Xv - v^) = 0

(12)

where A^,, B,2, B2, are adjustable coefficients, and the other symbols have their usual significance [38]. Generally, this relation reproduces the experimental data for the selected mixtures within an average uncertainty Av% = 0.5 cSt over the entire range of investigated temperatures. U3

1.5

1.0

0.5

20

50

80 t/^C

Figure 3. Rational equation y^^ ( • ) and y^^ ( A ) fitting parameters vs t/°C for ED/DX binary mixtures [6].

90

Advances in Engineering Fluid Mechanics

Polynomial Equation The analysis of experimental viscosities of binary liquid mixtures often can be carried out by using an empirical equation of the form: V

= X,v, + X , v , + X , X , X a , ( X , - X , ) ^

(13)

where the a^ terms are the polynomial coefficients to be evaluated. These coefficients, for any chosen degree k of the polynomial equation, represent as many interaction terms between unlike component molecules. Generally, for k = 2 one obtains a good fit by a three-adjustable-parameter equation. Now, some considerations can be written about the Heric's and this last polynomial function. In fact, the interaction terms in Equations 7 and 13 could be formally derived from the more general Redlich-Kister one [39], being the adjustable coefficients derived from this relation truncated at different power terms. However, it should be noted that for these binary mixtures, Heric's equation provides better results if compared with the polynomial one, despite the number of adjustable parameters. Evaluation and Comparison of Available Correlations Some closing remarks can now be given about this section. Firstly, we have to remind the reader that many other equations have been examined in the literature. In this respect, let us mention, for example, SLS Equation [14] and the NRTL model [40] for transport properties which are, probably, the most representative relationships among the others. However, their cumbersomeness, either apparent or real, unfortunately discourages their widespread employment. As a consequence, very few works appeared in the literature to check their suitability for experimental data fits. The results of these analyses show that all the selected equations are suitable for the best-fitting viscokinetic data of ED/cosolvent binary systems. This fact does not appear surprising in view of the number of adjustable coefficients in each equation even if a very large variability of the experimental measurements is observed both with temperature and mixture composition. In this light, it becomes more difficult to suggest any general rule establishing the effectiveness of one fitting procedure over the others. However, the more suitable comparison between different relations should be restricted to the equations with the same number of adjustable coefficients. Thus, the equation group having two adjustable parameters includes McAllister, Heric, Lobe and rational functions. All these fitting procedures applied to the present binary data show a good ability to reproduce experimental values, the average uncertainties being about equal. Furthermore, the agreement between the computed and the experimental data was found to be within ±1% using these relations. Nevertheless, in our opinion, rational functions should be preferred owing to the high correlation degree of the fitting parameters. This fact probably means that this should be the right way to shed some light in a more general theory about crossing correlation coefficients and specific intermolecular interactions between components.

Kinematic Viscosity and Viscous Flow in Binary Mixtures

91

As far as the three adjustable coefficients equations are concerned, i.e., Auslander's and polynomial, it should be noted that the treatment of our ED/cosolvent binary data doesn't represent a sufficiently severe test to select the more suitable expression for the best fit of the experimental values. In fact, at present, it is hard to appoint a selective criterion for comparing purposes like Equation 6, which is applicable to three-fitting parameters relations, too. Furthermore, the reproducibility of the experimental points is about the same obtained by applying the two adjustable coefficient equations. This evidence could suggest the choice of the simplest function containing a reduced number of fitting parameters with respect to the other, being ascertained that these coefficients are related to the specific solventcosolvent interactions. EXCESS FUNCTION To establish the deviation from ideality of binary systems, excess properties (Y^) are calculated on the basis of Raoult's law [32], assuming ideal mole fraction additivity of those pure components, by using the equation: Y^ = Y - (X,Y, + X J , )

(14)

where Y, Y, and Y^ are the specific properties of the mixture and of the pure components, respectively. A graphical representation of v^ as a function of X^ for ED/DME binary mixtures is given in Figure 4. The curves of Figure 4 have been obtained by fitting the isothermal experimental v^ quantities by the equation [39]:

Y^=X,X,Xc.(X,-X,y

(15)

o

where c. are adjustment coefficients for each temperature. The Redlich-Kister Equation 15 seems to be particularly suitable for correlation procedures of excess quantities for thermophysical data of any kind, and its soundness has been pointed out in many reports. In the case of ED/DME mixtures, the v^ values are negative over the whole composition range, showing a more pronounced minimum at the lowest temperature (-58 cSt at -10°C) which flattens at the highest ones (-0.88 cSt at 80°C). Obviously, the trend of Figure 4 may be considered as very clear evidence of the structurebreaking effects provided by the components in these mixtures [14]. In fact, for v^ < 0 we can assume that the mixing process reduces the high structural order of the total mass homopolymer of ED, as suggested by the high viscosity at each temperature. The contrary is true when v^ > 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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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].

Kinematic Viscosity and Viscous Flow in Binary Mixtures

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

Advances in Engineering Fluid Mechanics

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

Kinematic Viscosity and Viscous Flow in Binary Mixtures

*
Figure 7, Plots of excess free energy of activation of viscous flow (AG*^/kJ moM) against X^ for the ED (1)/DME (2) binary solvent system at various tennperatures fronn -10 to 80°C [12].

25°C

o

a 1 L C5

0.2

0.4

0.6

0.8

Figure 8. Plots of excess free energy of activation of viscous flow (AG"^/kJ moM) against X^ for the ED (1)/W (2) binary solvent system [11].

99

100

Advances in Engineering Fluid Mechanics

frequency of each molecule engaged in a complex adduct. Consequently, the shift from ideality is reduced at the highest temperatures when the hydrogen bonding and dipolar heterocooperative network becomes much less effective in supporting the formation of heteroaggregated species. However, the detection of AG*^ > 0 in presence of a v^ < 0 property for EDAV solvent system seems to be a very intriguing performance because, despite the structure-breaker effects (v^ < 0) being prevalent, a residual order degree related to structure-making effects (AG*^ > 0) is observed. Therefore, from these and previous inferences, it may deduced that, at the molecular level, ED/W binary mixtures are very heterogeneous as a consequence of a separation of microphases. This probably implies that the aggregation of different type of clusters, such as tetrahedral W^ or W^ oligomers, small ED homopolymers, and EDAV heteropolymers, should lead to "interstitial solvation" of one species into the other. Obviously this polyaggregation of clustered species provides the greatest effect corresponding to the hypothetical formation of a 2ED • 3W solvent/cosolvent complex, whose existence seems to be still far away from being proven. SUMMARY AND FUTURE PERSPECTIVES These examples help to illustrate the usefulness of correlating functions of thermophysical properties as a means to investigate specific intermolecular interactions in mixed nonelectrolytic solutions. In particular, empirical and semiempirical methods are seen to provide valuable qualitative and quantitative information on both structural aspects, such as solvent-cosolvent and solvent-solvent patterns. As a result, many longstanding questions regarding molecular arrangement related to structure-breaking and structure-making effects in these solvent systems are beginning to be answered. The experimental determination of thermomechanical properties and the evaluation of related quantities offers a substantial test of theoretical calculations and computer simulations of mixed liquid systems. Furthermore, thermodynamic investigations of viscous flow are seen to give supplementary information on the dynamics of molecular aggregates, and in some cases, when water is the cosolvent polyaggregates are arranged in a supra-clusters organization. It is worth noting that all these evaluation methods are not limited to Newtonian or non-Newtonian fluids, but they can be applied, in a widespread investigation strategy, to all systems showing pseudo-rheological properties involving pure liquids, including water, and their binary or more complex mixtures with both polar and apolar components. In particular, they also can be applied to aqueous systems where the focus will be a detailed understanding of differences in organic solvents hydration structure as a function of their concentration and molecular complexity, these factors being strictly related to viscokinetic properties as they can be experimentally determined. Central to such a study is a need for basic information to deepen our understanding of hydrophobic hydration and all other specific intermolecular interactions that take place in mixed systems, which are the driving forces of structure-making and structure-breaking effects in these solutions. Obviously, these opposite effects enhance or depress rheological and viscokinetic properties of the systems, and all

Kinematic Viscosity and Viscous Flow in Binary Mixtures

101

this evidence should be very useful in providing the best assessment of the more promising predicting viscosity model-theory on the basis of the contribution group method [56]. In the longer term, much will depend on the experimentalists being accurate and expeditious in providing as many as possible reliable experimental data to enlarge the body of actual knowledge, and on the ability of theoreticians to connect the macroscopic fluid behavior with microscopic characteristics and molecular properties of constituent components of the real systems. Of course, the quality of the connection depends at least on different and sometimes opposite factors, such as: i) the realism of the developed mechanical models, and ii) a relative simplicity, which is a required condition to warrant maximum spreading. Improvements in approaching thermomechanical flow properties intermediate between Newtonian and non-Newtonian fluids, and the introduction of more efficient codes to handle the continuously increasing data base means that a series of new investigations can be contemplated. In particular, these might include the use of amphiphiles such as poly-ethylene glycols with different molecular complexity and other related species showing rheological properties in mixed fluids, both aqueous and nonaqueous systems. These future studies should help in clarifying some questions concerning the patterns of aggregation and about clustering processes in binary or more complex mixtures that will not be properly answered until then. ACKNOWLEDGMENTS The author gratefully acknowledges the help and encouragement given by many colleagues of Modena University. In particular, the author thanks Profs. C. Preti and G. Tosi, whose insight and inspiration continue to be an important factor in the success of this research program; Prof. G. C. Franchini and Dr. A. Marchetti for much help with the running of the various experiments and for able guidance and advice on data analysis. The author also acknowledges the contributions made over the year, by many graduate students, some of whom are still active in our laboratories: Drs. F. Corradini and M. Tagliazucchi. Finally, the author thanks MURST and CNR of Italy for financial support. NOTATION A2J, B. Interaction parameters in Equation 12 a, b, TQ Best fitting parameters in Equation 2 aj^ Fitting coefficients in Equation 13 CQ, C^ Best fitting parameters in Equation 5 c. Best fitting coefficients in Equation 15 DME 1,2-dimethoxy ethane DMF N,N-dimethylformamide

DX 1,4-dioxane d Interaction parameter in Equation 3 d. Best fitting coefficients in Equation 19 ED Ethane-1,2-diol G* Gibbs energy of viscous flow /J mol-^ H* Enthalpy of viscous flow /J mol"' h Planck constant i, j Mixtures components

102

Advances in Engineering Fluid Mechanics

ME M. N NRTL P, Q R r S*

2-methoxyethanol Molar mass of i-th component Avogadro number Non Random Two Liquids model Coefficients in Equation 16 Universal gas constant Linear correlation coefficient Entropy of viscous flow /J mol' K '

Significant Liquid Structure Absolute temperature /K Celsius temperature /°C Water Mole fraction of i-th component Y Generic property YE Excess quantity of a generic property, as defined by Equation 14

SLS T t W X.

Greek Symbols

a Interaction parameters in Equation 8

interaction parameters in Equation 7 relative differences as defined by Equation 6 static relative permittivity volume fraction of i-th component interaction parameters in Equation 11

r| dynamic viscosity /cP (1 cP = 1 mPa s) |i dipole moment /D (ID = 3.33564 • lO^^ C m) V kinematic viscosity /cSt (1 cSt = 10-^ m^ s') interaction parameters in Equation 4 density /kg m~^

Subscripts max maximum REFERENCES 1. Rowlinson, J., and Swinton, F. L., Liquids and Liquid Mixtures, Academic Press, New York, 1981. 2. Dymond, J. H., Chem, Soc. Rev., 14, 317 (1985). 3. Cummings, P. T., and Evans, D. J., Ind. Eng. Chem. Res., 31, 1,237 (1992). 4. Corradini, F., Marcheselli, L., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., Bull Chem. Soc. Jpn., 65, 503 (1992). , 5. Corradini, F., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., Can. J. Chem. Eng., 71, 124 (1993). 6. Corradini, F., Marcheselli, L., Tassi, L., and Tosi, G., Bull. Chem. Soc. Jpn., 66, 1,886 (1993). 7. Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., Chem. Eng. J., 52, 41 (1993). 8. Corradini, F., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., Aust. J. Chem., 46, 1,711 (1993). 9. Corradini, F., Franchini, G. C. L., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., J. Solution Chem., 22, 1,019 (1993). 10. Corradini, F., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., Aust. J. Chem., 47, 1,117 (1994). 11. Corradini, F., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., Aust. J. Chem., 48, 103 (1995).

Kinematic Viscosity and Viscous Flow in Binary Mixtures

103

12. Corradini, F., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., Ann. Chim. (Rome), 84, 397 (1994). 13. Corradini, F., Marchetti, A., Tagliazucchi, M., Tassi, L., and Tosi, G., J. Chem. Soc, Faraday Trans., 90, 1,089 (1994). 14. Eyring, H., and John, M.S., Significant Liquid Structure, J. Wiley & Sons, New York, 1969. 15. van Velzen, D., Cardozo, R.L., and Langenkamp, H., Ind. Eng. Chem. Fund., 11, 20 (1972). 16. Letson, A., and Stiel, L.I., AIChE J., 19, 409 (1973). 17. Reid, R.C., Prausnitz, J.M., and Sherwood, T.K., The Properties of Gases and Liquids, 3rd ed., McGraw-Hill., New York, 1977. 18. Grain, C.F., ''Liquid Viscosity,'' in Handbook of Chemical Property Estimation Methods, McGraw-Hill, New York, 1982. 19. Sastri, S.R.S., and Rao, K.K., Chem. Eng. J., 50, 9 (1992). 20. Fialkov, Yu. Ya., Zh. Fiz. Khim., 37, 1,051 (1963). 21. Fort, R.J., and Moore, W.R., Trans. Faraday Soc, 62, 1,112 (1966). 22. Franks, F., and Ives, D.J.G., Quart. Rev., Chem. Soc, 20, 1 (1966). 23. Gibson, R.E., and Loeffler, O.H., J. Am. Chem. Soc, 63, 898 (1941). 24. Dack, M.R.J., Chem. Soc Rev., 4, 211 (1975). 25. Viti, v . , and Zampetti, P., Chem. Phys., 2, 233 (1973). 26. Partington, J.R., An Advanced Treatise on Physical Chemistry, Longmans Green & C , London, 1951. 27. Vogel, H., Phys. Z , 22, 645 (1921). 28. Fulcher, G.S., / . Am. Ceram. Soc, 8, 339 (1925). 29. Tamman, G., and Hesse, W., Z Anorg. Allg. Chem., 156, 245 (1926). 30. WiUiams, N.L., Landel, R.F., and Ferry, J.D., J. Am. Chem. Soc, 11, All (1955). 31. Grunberg, L., and Nissan, A.H., Nature, 164, 799 (1949). 32. Glasstone, S., Laidler, J.K., and Eyring, H., The Theory of Rate Processes, McGraw-Hill, New York, 1941. 33. McAllister, R.A., AIChE J., 6, 427 (1960). 34. Dizechi, M., and Marschall, E., J. Chem. Eng. Data, 11, 358 (1982). 35. Heric, E.L., /. Chem. Eng. Data, 11, 66 (1966). 36. King, M., and Queen, N., J. Chem. Eng. Data, 24, 66 (1979). 37. Rolling, O.W., Anal. Chem., SI, 1,721 (1985). 38. Auslander, G., Br. Chem. Eng., 9, 610 (1964). 39. Redlich, O., and Kister, A.T., Ind. Eng. Chem., 40, 341 (1948). 40. Wei, I.e., and Rowley, R.L., Chem. Eng. ScL, 40, 401 (1985). 41. Corradini, F., Marchetti, A., Tagliazucchi, M., Tassi, L., and Varini, A., Ann. Chim. (Rome), 85, 267 (1995). 42. Frank, H. S., and Wen, W. Y., Discss. Faraday Soc, 24, 133 (1957). 43. Davis, M. L, and Douheret, G., Thermochim. Acta, 188, 229 (1991). 44. Huot, J. Y., Battistel, E., Lumry, R., Villeneuve, G., Lavallee, J. F., Anusiem, A., and Jolicoeur, / . Solution Chem., 17, 601 (1988). 45. Ramanamurti, M. V., and Bahadur, L., J. Chem. Soc, Faraday Trans. 1, 76, 1,409 (1980). 46. Corradini, F., Marcheselli, L., Tassi, L., and Tosi, G., J. Chem. Soc, Faraday Trans., 89, 123 (1993).

104

Advances in Engineering Fluid Mechanics

47. Lumry, R., Battistel, E., and Jolicoeur, C , Faraday Symp. Chem. Soc, 17, 93 (1982). 48. Kundu, K. K., Chattopadhyay, P. K., Jana, D., and Das, M. N., J. Phys. Chem., 74, 2,633 (1970). 49. Caminati, W., and Corbelli, G., J. MoL Spectr., 90, 572 (1981). 50. Ray, A., and Nemethy, G., J. Chem. Eng. Data, 18, 309 (1973). 51. Ott, J. B., Goates, J. R., and Lamb, J. D., J. Chem. Thermodyn., 4, 123 (1972). 52. Takenaka, N., and Arakawa, K., Bull. Chem. Soc. Jpn., 47, 566 (1974). 53. Corradini, F., Franchini, G. C , Marchetti, A., Tagliazucchi, M., and Tassi, L., Bull. Chem. Soc. Jpn., 68, 1,867 (1995). 54. Reed, T.M., and Taylor, T.E., J. Phys. Chem., 63, 58 (1959). 55. Meyer, R., Meyer, M., Metyger, J., and Pemeloux, A., J. Chim. Phys. Phys.Chim. Biol., 68, 406 (1971). 56. Cramer, R.D., J. Am. Chem. Soc, 102, 1,837 (1980). 57. Marchetti, A., Preti, C, Tagliazucchi, M., Tassi, L., and Tosi, G., J. Chem. Eng. Data, 36, 360 (1991).

CHAPTER 6 REACTION OF A CONTINUOUS MIXTURE IN A BUBBLING FLUIDIZED BED N.R. Amundson Department of Mathematics University of Houston, Texas, 77204 and R. Aris Department of Chemical Engineering and Materials Science Institute of Technology 421 Washington Ave., SE Minneapolis, MN 55455-0132 CONTENTS INTRODUCTION, 105 GAMMA DISTRIBUTIONS, 106 APPLICATION OF THE GAMMA DISTRIBUTION, 107 A GENERAL THEOREM FOR SIMPLE LINEAR REACTOR MODELS, 108 APPLICATION TO A MODEL OF THE BUBBLING FLUIDIZED BED, 109 THE DAMKOHLER NUMBER, H I THE FLUID BED WITH ASTARITA'S UNIFORM KINETICS, 113 ACKNOWLEDGMENTS, 115 NOTATION, 115 REFERENCES, 116 INTRODUCTION It is appropriate that the topics of the fluid bed and the continuous mixture should be examined together since one of the first applications of each was the important process of catalytic cracking. The very large number of components in crude oil and the method of reporting its analysis (formerly by boiling point and later by simulated b.p. or chromatographic retention time) call out to be 105

106

Advances in Engineering Fluid Mechanics

described by a continuous distribution over a parameter. The catalytic cracking of crude oil can be taken, in the first instance, to be a large number of parallel firstorder reactions, and this is just the kind of system that continuous mixture theory was designed to handle. Curiously, however, the oil industry had for years found the relationship of the fraction of remaining crude to the "severity" of the reaction, as they call it, fitted the hyperbola of the second order reaction better than the exponential of the first. This observation gave birth to a theory of reactions in continuous mixtures of which this aliasing of reaction order is an elementary result. Let c(x,t)dx be the concentration of material in the interval (x, x + dx) that cracks with a first-order rate constant k(x), x being an index variable defined on 0 < x < oo. Let C(t) = c(x,t)dx be the total concentration of material of any index (except where specified all integrals in this article will run from 0 to «»), then C(t) = jc(x,0)exp(-k(x)t)dx

(1)

and we have only to put c(x,0) = CQexp(-x), k(x) = k^C^x to obtain C(t) = CJ(l + k,C,t)

(2)

which satisfies the quadratic differential equation for the second order disappearance of the "lumped" species, dC/dt = -kf:\

C = C, at t = 0

(3)

Three observations may be made at this point. Firstly, if k(x) is monotonic, we lose no generality by making it proportional to x, for a monotonic y = k(x) has an inverse x = K(y) and C(t) = c(x,0)exp(-k(x)t)dx = j c(K(y),0)k,(y)exp(-yt)dy

(4)

Secondly, that we can stretch the linear scale and write c(x,0) = COf(x), where the zero and first moments Jf(x)dx = Jxf(x)dx = 1; then k(x) = x, where = jk(x) f (x)dx. All the integrals are, of course, taken from zero to infinity since, for any interval a < x < b not represented in the mixture, f(x) = 0. Thirdly, Krambeck (1984) has shown that any distribution gives apparent second order kinetic for the lump, as t -> «>, provided only that c(0,0) ^ 0. It is of the first importance to remember that, though x may be regarded as dimensionless, c(x,t) is a concentration density (in the same meaning of the work as in "probability density"), not a concentration. Only integrals of c with respect to X have the dimensions of a concentration. A somewhat more general treatment of continuous reactions is given in Gavalas and Aris [1]; for the fluid bed, see Davidson and Harrison [2,3], Kunii and Levanspiel [4]; and Rowe and Yates [5]. GAMMA DISTRIBUTIONS Of all the normalized distributions f(x), the one that has proven to be the most useful, and versatile is the gamma distribution g (x) = n"x"-'e-"V(n - 1)!

(5)

Reaction of a Continuous Mixture in a Bubbling Fluidized Bed

107

defined for n, a positive integer. It can be generalized to any real v > 0 by replacing n by V and (n - 1)! by r(v). As n (or v) tends to infinity, g„(x) becomes the Dirac delta function 8(x - 1), but its approach to this singular limit is slow, the standard deviation being n~^'^. To be slightly more general for a moment, the gamma distribution with mean a is obtained by replacing x by x/a (not forgetting the x in the dx that goes along with the density function). To avoid the inherent inaccuracy of multiplying very large and very small numbers, it is best calculated by g„(x,a) = (n'^Vx) exp{-nM(x/a) - L(n)}

(6)

where M(x/a) = (x/a) - 1 - ln(x/a) and L(n) = ln(n - 1)! - (n - 1/2) ln(n) + n

(7)

A useful asymptotic formula for an integral Jh(x)g^(x)dx is Jh(x)g„dx = X^^n,ph^''(a)7p! where u

(8)

is the dimensionless p^^ moment about the mean, 1, of g (x);

Ko = 1' K. = 1' ^l„,. = 1/"' K^ = 2/n^ H„,, = (3/n^) + (6/n3)

(9)

The Gamma distribution has to be discretized for some calculations. If M is any integer and Gn(x) the indefinite integral of g„(x), i.e., G^(x), G^(0) = 0, let x^^be defined by G^(x^^) = m/M and y^^ by Mjxg^(x)dx where the integration is from ^m-in ^^ ^mn- ^^^^ dlvldcs thc rcgiou under the curve into M parts of equal area and associates the centroid of each part with it. Then any integral of the form Jh(x)g^(x)dx is replaced by Zh(y^ J/M. A formula of great value in connection with the use of the Gamma distribution is rg„(x)dx/(x + y) = ne"%(ny)

(10)

•'0

where E^(z) is the exponential integral E„(z) = £°t-V^dt

(11)

e^E^(x) is bounded by two hyperbolas that translate into a powerful inequality for the integral in Equation 10, namely l/(l + y) < £g„(x)dx/(x + y) < 1/(1 + y - n-^)

(12)

APPLICATION OF THE GAMMA DISTRIBUTION The introductory example may be reworked using the Gamma distribution since the special case given there is n = 1. Let c(x,0) = C^gJi^) where C^ is the total initial concentration. Let the first-order rate constant be k(x) = kx and make time dimensionless as kt. This reaction time or intensity of reaction—severity of reaction

108

Advances in Engineering Fluid Mechanics

as the oil people have it—is really the Damkohler number, Da, for the reactor with t, the time of reaction, if it is a batch reactor or the residence time if a PFTR. Thus Y(x,Da) = c(x,t)/C^ = g„(x) exp - xDa

(13)

and r(Da) = C(t)/C, = jX,(x,Da)dx = (1 + Da/n)""

(14)

It follows that the lumped kinetics is of apparent order N = (n+l)/n

(15)

since dP/dDa = -(1 + Da/n)~"~' = -T^. All the integrals are well-defined if n takes no integral values, v, provided v > 0. Hence N can take any value greater than 1. The limit of Equation 11 as n —> «> is clearly exp -Da, the first order reaction result, which is as it should be since the initial distribution is a delta function and the continuous species is really discrete. Astarita and Ocone (1988) generalized the component kinetics from the first order to what they called "uniform" kinetics [6] dc(x,t)/dt = -kc(x,t)F[ j K(y)c(y,t)dy]

(16)

The uniformity lies in the fact that K is not a function of x, and this allows the problem to be reduced to linearity by suitably warping time. Astarita shows that in this way an order of reaction may be achieved for the lump [7]. Indeed Astarita and Aris showed that any kinetics could be imitated by choosing F and K adroitly [8]. We shall return to "uniform kinetics" later, but, for the moment, will stick with the linear case. A GENERAL THEOREM FOR SIMPLE LINEAR REACTOR MODELS By a reactor model, we mean a system of equations (algebraic ordinary or partial differential, functional, or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single concentration-like variables, u^ and u. The relation of input and output also will depend on a set of parameters (such as Damkohler number, Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when u^ = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(x)) and the lumped output is r(p*) = Jg(x)A(p(x))dx)

(17)

Here p* is the set of characteristic values of the parameters, i.e., p(x) = p*co(x) where co(x) has values centered on 1. Often we can set p* = J p(x) g(x) dx. The proof is really a statement of what linearity means since if g(x) dx is the input concentration, (g(x)dxA(p(x)) is the output when the parameter values are p(x). Here

Reaction of a Continuous Mixture in a Bubbling Fluidized Bed

109

X serves merely as an identifying mark, being truly an index variable, and the integration in Equation 14 follows from the superposition principle for linear systems. The same principle allows an obvious extension to multiple input, multiple output linear systems. A becomes a matrix whose elements are the response at one of the outports to a unit input at one of the in-ports, the input at all the others being zero. APPLICATION TO A MODEL OF THE BUBBLING FLUIDIZED BED Two models of the bubbling bed are referred to by Davidson and Harrison in Chapter 6 of their monograph [2]. In both models the bubbles (of average volume, V , and in number per unit reactor volume, N) rise through the particulate, or dense, phase with an absolute velocity of U^ and exchange reactant with it at an equivalent flow rate, Q. The difference between the models lies in the assumptions made about the dense phase; in the one, this is held to be uniform throughout the reactor while in the other dense phase is in plug flow. We shall consider only the first of these; the second can be treated as far as the application of the theorem is concerned, but has ramifications which are still under investigation. Reaction only takes place in the dense phase since that is where the catalyst particles are. Since the exchange is with a uniform environment where the concentration is c , we can see that by the time the bubble has reached the top of the bed, the concentration of reactant in it is c + (c^ - c ) e~^', where c^ is the entering concentration, H, the height of the bed and t r = QH/LI^V is a dimensionless transfer number. By doing a mass balance on the dense phase as a whole we obtain a linear equation for c in terms of the inlet concentration c^ [9]. The total fractional concentration of reactant that is left in the emerging gas stream is y = pe-^-^ + (1 - pe-^0'/(Da + 1 - Pe-^^O

(18)

where b =UQ/U and the Damkohler number is Da = k'HyU, k' being the rate constant, is a measure of the intensity of reaction. H^ is the height of the bed at incipient fluidization, H - H^ = NVH. Equation 18 with a square in the numerator of the second term looks a little odd at first, but, in fact, it unfolds itself with unusual clarity. We first observe that even when the reaction is infinitely fast. Da —> oo, a fraction Pe'^' of the inlet concentration will remain in the bubbles when they reach the top of the bed. Since y can never be less than Pe~^', we subtract quantity, thus allowing y - Pe~^' to go to zero and rescale by dividing by 1 - pe~^^ which is the effective amount that is available to the dense phase. We have y* = (y - pe-'^0/(l - pe-^0 = 1/(1 + Da*)

(19)

where Da* = Da/(1 - pe-^O

(20)

is the modified Damkohler number. It is enhanced in compensation for the fraction that is not available under any condition; for if we recall the definition

110

Advances in Engineering Fluid Meciianics

and the fact that the available reactant is qc^Cl - Pe"^0» we see that Da* = Ic'R^CQ/qCpCl - be"^0 = maximum reactive flux/available feed flux. In terms of this modified Damkohler number, the well-mixed dense phase behaves exactly as it should—namely, like a stirred tank. It may be shown that if /(t) is the residence time distribution of the bubble phase [in our plug flow model it is 8(t - 1)] the exponential e~^' in Equations 19 and 20 need only be replaced by the Laplace transform of /(t) and Tr playing the role of the transform variable. The dimensionless number, Tr, obviously plays a leading role in all fluid bed problems. It is the ratio of a convective or residence time, H/U^, to an exchange time, V/Q. Of these quantities, Q is further related to the physical quantities that describe the condition of the bed [3]. No one has contributed more than Davidson to the understanding of these exchange processes, and if Tr were to become a named number, his would be the most appropriate name to use. We wish to see what the overall conversion of a continuous mixture will be, but first we have to ask which parameters will depend on x, the index variable of the continuous mixture. Clearly k', the rate constant in the Damkohler number, will be a function of x, and if monotonic, it can be put equal to Da.x. The parameter P is clearly hydrodynamic, and so, for the most part, are the terms in the Davidson number. The only term in Equation 6.21 of Davidson and Harrison that might depend on x is the gas phase diffusivity, and this appears under a square root sign in the second of two terms. Tr was found to be virtually constant with a value close to three in a series of experiments by Orcutt which Davidson and Harrison analyze. We will, therefore, assume that only the Damkohler number varies with x, and that this variation is linear. Then for the well-mixed model r^(Da,Tr,p) = J [pe^^^ + (1 - p-'^OVCDa.x + 1 - ^t-''')]g^{x)dx

(21)

= pe-T''^ + [(1 - Pe-'^0/Da*]Jg„(x)dx/(x + 1/Da*) Again setting r*(Da,Tr,P) = [r„(Da,Tr,P) - Pe-^l/d - Pe-^0

(22)

and using the relation (10), we have r*(Da,Tr,p) = (n/Da*)e"^^^*E^(n/Da*)

(23)

The inequalities for the exponential integral stated above. Equation 12, then give 1/(1 + Da*) < r*(Da,Tr,p) < 1/(1 + Da*(n - l)/n)

(24)

These bounds indicate that the curves for various n will not lie very far apart and to separate the curves more effectively A^ = (1 + Da*) r*^(Da,Tr,P) is plotted against Da* in Figure 1. For small values of Da* the asymptotic expansion of Abramowitz and Stegun [10] of E^(z) gives A^ = 1 + Da*Vn - (n + l)2Da*Vn2 + (n +l)(n + 2)3Da*Vn3 - . . .

(25)

The series expansion for small z gives the expansion for large Da*. It is messy, but for the record, we give here the terms up to order o(l/Da*);

Reaction of a Continuous Mixture in a Bubbling Fluidized Bed

111

1000 Figure 1. Deviation from the output fronn stirred tank behavior when the feed is a r distribution with paranneter n.

n = 1,

Aj = ln(Da*) - y + 21 ln(Da*)/Da* + (1 - 2Y)/Da* + . . .

(26)

n = 2,

A^ = 2 - 4 ln(Da*/2)/Da* + (1 + 2Y)/Da* - . . .

(27)

n = 3,

A3 = 3/2 - 3/Da* + . . .

(28)

n > 2,

A^ = n/(n - 1) - 2n/[(n - l)(n - 2)Da*] + . . .

(29)

For large n, A^ ~ 1 + [Da*/(1 + Da*)]Vn + [Da*^(Da* - 2)/(l + Da*)^]/!!^

(30)

THE DAMKOHLER NUMBER We have seen that, if the dense phase is well-mixed and only the rate constant is a function of x, the performance of the fluid bed depends only on the modified Damkohler number. Da*. If the Davidson number, Tr, depends on x, the linearity may still be exploited, but we have to go back to Equation 21 which may be rearranged to give r^(Da,Tr,p) = 1 - j [ ( l - pe-T'^W)Da.x/(Da.x + 1 - pe-T'^('^>)]g^(x)dx

112

Advances in Engineering Fluid Mechanics

where the Tr on the left is the mean value of Tr(x) = Tr (o(x) (say). The integrals no longer come out in terms of known functions, but they are easily calculable. Diffusion within the catalyst particle can be accounted for by using an effectiveness factor, r|, but x should no longer be defined to make k(x) linear in x. Rather, it would be sensible to make r|k linear in x. Of course, straightening out the monotone dependence on x of one parameter distorts the distribution as in Equation 4, and it may be better to think of the Damkohler number as a function x. We can always write it as Da.co(x) where co(x) has a mean value of 1. If the catalyst decays as a known function of the length of time it spends in the reactive environment, the catalyst must be drawn off from the dense phase reactivated and returned so that the burden is constant. A steady state is quickly built up in which the fraction of catalyst of age in the interval (a, a + da) is (wAV).exp - (waAV) da, where W is the weight of the catalyst in the bed, and w is the rate of replenishment. If the decay is accounted for by saying that the rate constant should be multiplied by a factor, for example, exp - X,a, then the Damkohler number must be calculated with a factor of (wAV)jexp - (waAV)exp - A,ada = 1/(1 + WX/w)

(31)

If the decay cannot be expressed solely as a function of age, the linearity is lost and the generalization to continuous mixtures is no longer possible. If the catalyst were not to decay, but for some other reason, perhaps temperature control, the particles were taken out and recycled, each is supposed to be in pristine condition on reentering the bed. Each particle would then undergo a transition during which the steady state profile of reactant within the particle would be built up. The analysis of Amundson and Aris may be used [11]. There is an instructive error in this paper which leads to an important safeguarding principle in making balances. Namely, when a sub-region of a process has been assumed to be uniform, balances that involve its properties must be taken over the whole of it. In simple situations there is little inclination to do otherwise, but in complex cases this may be the effect of a differential balance over another interpenetrating phase. See Aris for a full analysis [9]. We assume spherical particles of radius, R, and call the profile of concentration at time, a, c(r,a). If D is the diffusivity of the reactant and k* the rate constant per unit volume of catalyst, ac/at = (D/r^)(d/dT)[T\dc/dr)] - k*c

(32)

with initial condition c(r,0) = 0, finiteness at the center, and with Bi as the Biot number for external mass transfer, (l/Bi)(ac/ar) + c = Cp

at r = R

(33)

In dimensionless form, with u = c/Cp, t = wt'AV, p = r/R,

tD^ = R^w/DW, (^^ = k*RVD

rs\du/dt) = p-2(a/ap)[p2au/ap) - (t)2u and

(34)

(35)

Reaction of a Continuous Mixture in a Bubbling Fluidized Bed

(l/Bi)Ou/dp) + u = 1 at p =1

113

(36)

This partial differential equation is made into an ordinary differential by taking its Laplace transform. It is then solved for the transform of u and the transform of the total reaction rate for the particle calculated. There is no need to invert this transform for what we want is the average reaction rate in the bed, and this is the integral of the product of the reaction rate after a time A and the probability of the particle having age A. Because the ages are exponentially distributed, this is none other than the Laplace transform with the dimensionless transform variable set equal to 1. The end result is that the Damkohler number is Da = [kH/U]/{(l/Bi) + l/(\|/coth\|/ - 1)}

(37)

and \|/2 = G52 + (p2

(38)

With as complicated an expression as this, it is clearly a waste of time to derive an inverse function, and the integrals involved can be computed numerically. In fact, much of the numerical work can be done on the desktop computer using Simpson's rule and watching out for regions of extreme curvature. THE FLUID BED WITH ASTARITA'S UNIFORM KINETICS We have seen that this model of the bubbling bed is essentially the same as a stirred tank when the two sources of the feed are recognized. These are the fraction (1 - P) that comes with the gas feed at the bottom of the bed and the fraction (3 in the bubbles which feeds the reactor at all levels and from a diminishing concentration difference. The latter, when referred to the inlet difference c^ - c , delivers a fraction 1 - exp(-Tr). Thus the total feed minus outlet is {(1 - (3) + P(l - exp(-Tr))}(cQ - c^ = {1 - P exp(-Tr)}(cQ - c ), and this is what is equated to the reaction rate, (kHQAJ)c . On this view of the reactor it is not surprising to find Astarita and Ocone's methods for uniform kinetics work for the bubbling fluid bed. Replacing kc by kc (x,t)F[jK(y)c (y)dy], the balance over the dense phase is {1 - p exp(-Tr)}[c,(x) - c/x)] = Da.xCp(x)F[j K(y)Cp(y)dy] or Cp(x) = c,(x)/{ 1 + Da*xF[J K(y)Cp(y)dy]}

(39)

But F[|K(y)c (y)dy] is just a constant, as yet unknown. Call it 5 for short, then 8 = F[jK(y)c/y)dy] = F[J K(y)Cp(y)/{ 1 + 5Da*y}dy]

(40)

and this is an equation for 8. It may be solved by a straightforward calculation that takes 8Da* as parameter and calculates 8(8Da*) from Equation 39 and Da* as 8Da*/8(8Da*). When 8 is known as a function of Da* and cj,x) = CQg(x), y can be written as

114

Advances in Engineering Fluid Mechanics

y(x) = p exp(-Tr) + {1 - p exp(-Tr)}g(x)/{l + 8Da*x}

(41)

This leads to the remarkably simple result that the performance of the bed is given by exactly the same function as before, Equation 23, save only that Da* is further modified by the factor 5. A good example is afforded by Langmuir kinetics of a continuous mixture, a form which can be fully justified by a simple adsorption model. Here F = 1/{1 + jK(y)c(y)dy}, and it has been shown that the choice K(y) = K*r(n)(ny)"'/r(m + n), c, = C,g„(y), k = K*C,

(42)

is interesting. When lumped in a plug flow reactor, it follows an apparent kinetic expression of C^"^'^^"/{1 + K*C^'"^"^^"}. Making the same choice for the bubbUng bed model gives the equation for 8 and Da* as functions of q = 8Da* 8 = {1 + K(n/q)e^"^^>E^Jn/q)}-'

(43)

Da* = q{l + K(n/q)e^"^'i>E^^^(n/q)}

(44)

Figures 2 and 3 show F* as a function of Da* for K = 1, 10 and m = 0, 1, respectively. They illustrate how extremely close the results over a wide range of variation of n may be. The variation with K is more marked and, since an increase T-TT

• ••m|

• •••nn|

• imm|

i iinm|

0.8

0.6

0.4

0.2 m=0 n = I, 2, 3, 4, 5, 10, oo

f" •

0.01

•••••••!

•

•••••••!

•

• • • • „ , !

I

r ^

100 Da

Figure 2. Rescaled output as a function of the nnodified Damkohler number; n = 1, 2, 3, 4, 5, 10; K = 1, 10; m = 0.

Reaction of a Continuous Mixture in a Bubbling Fluidized Bed

0.8

h

0.6

r

115

0.4 h-

0.2

0.01

100

Da Figure 3. Rescaled output as a function of the modified Damkohler number; n = 1, 2, 3, 4, 5, 10; K = 1, 10; m = 1. in K reduces the reaction rate, we would expect that a compensating increase in the Damkohler number would be needed to maintain conversion. ACKNOWLEDGMENTS One of us (RA) is indebted to the PRF of the ACS for continued support of an ongoing investigation of reactions in continuous mixtures (PRF25133-AC7E). The figures and the calculations that lie behind them were done by Paolo Cicarelli. We are both grateful to the Institution of Chemical Engineering for permission to use the figures and much of the text of Amundson and Aris [11]. NOTATION a Age of catalyst Bi Biot number for catalyst particle C(t) Total concentration at time t, /c(x,t)dx C, C(0) c Concentration c Concentration of reactant in p

dense phase

c(r,a) Concentration of reactant at radius r of catalyst of age a c(x,t)dx Concentration of material with index in (x,x + dx) at time t D Diffusivity of reactant within catalyst particle Da Damkohler number Da* Modified Damkohler number, Equation 20

116

Advances in Engineering Fluid Mechanics

E^(x) Exponential integral, Equation 11 F Nonlinear function in Astarita's kinetics g^(x) Gamma distribution, Equation 5 g^(x,a) Gmma distribution with mean a, Equation 7 K(y) Kernel in Astarita's uniform kinetics k(x) First-order rate constant for material of index x Average value of k k' First-order rate constant k* Rate constant within catalyst particle N Apparent order. Equation 15, number of bubbles per unit volume n Parameter of the gamma distribution

p Set of parameters on which the solution of the system depends p* Characteristic values of parameters, p(x) = p*co(x) Q Exchange rate between bubble and dense phase q Parameter for calculation 8 and Da*, Equations 42 and 43 R Radius of catalyst particle r Radial distance within catalyst particle Tr Davidson number, QH/U^V U Velocity of fluidizing gas UQ Velocity of gas for incipient fluidization U^ Absolute velocity of bubble rise V Bubble volume W Weight of catalyst in the bed w Replacement rate of catalyst X Index of "species" in a continuous mixture

Greek Symbols P 1 - U/U r(Da) Fraction of total concentration in exit stream, also used with other parameters as arguments, JY(x,Da)dx

r* ( r - pe-^o/(i - pe-'^o Y(x,Da) Fraction of inlet concentration remaining in exit stream Y* ( Y - pe-^0/(l - Pe-'^O A (1 + Da*)r(Da,Tr,p) A(p) Ratio of exit to inlet concentrations for linear systems

X Decay constant for aging catalyst |Li^ p p'*^ moment of g^(x) G3 Dimensionless replacement rate, (R^w/DW)'^^ (p Thiele modulus of catalyst particle R(k*/D)''2 \|/ ((p2 + tQ2y,2

co(x) Set of distributed parameters scaled by their characteristic values

REFERENCES 1. Gavalas, G. and R. Aris, "On the Theory of Reactions in Continuous Mixtures," Phil. Trans. Roy. Soc, A260, 351 (1966). 2. Davidson, J. F., and D. Harrison, Fluidized Particles, Cambridge University Press, Cambridge (1963). 3. Davidson, J. F., and D. Harrison, eds., Fluidization, Academic Press, New York (1971). 4. Kunii, D., and O. Levenspiel, Fluidization Engineering, Wiley, New York, (1969).

Reaction of a Continuous Mixture in a Bubbling Fluidized Bed

117

5. Rowe, P. N., and J. G. Yates, "Fluidized Bed Reactors" (Chapter 7 of Chemical Reaction and Reactor Engineering), J. J. Carberry, and A. Varma, ed., Marcel Dekker, New York (1987). 6. Asarita, G., and R. Ocone, "Lumping Nonlinear Kinetics," AIChE Jl, 34, 1299 (1988). 7. Astarita, G., "Lumping Nonlinear Kinetics: Apparent Overall Order of Reaction", AIChE Jl, 35, 529 (1989). 8. Astarita, G., and R. Aris, "Continuous Lumping of Nonlinear Chemical Kinetics," Chem. Engng. and Proc, 26, 63 (1989), and "On Aliases of Differential Equations," Rend. Ace. Lincei. LXXXII (1989). 9. Aris, R., "Manners Makyth Modellers," Chem. Eng. Sci. 46, 1,535 (1991). 10. Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions (Nat. Bur. Standards). 11. Amundson, N. R., and R. Aris, "Reaction of a Continuous Mixture in a Bubbling Fluidized Bed," Trans. IChemE, 71, 611 (1993).

This Page Intentionally Left Blank

CHAPTER 7 FLUID DYNAMICS OF COARSE DISPERSIONS Y. A. Buyevich National Research Council NASA Ames Research Center Moffett Field, CA 94035 S. K. Kapbasov Department of Mathematical Physics Urals State University 620083 Yekaterinburg, Russia CONTENTS INTRODUCTION, 119 PHYSICAL BACKGROUND, 123 FORCES ACTING ON A PARTICLE, 126 CONSERVATION EQUATIONS FOR THE DISPERSED PHASE, 128 CONSERVATION EQUATIONS FOR THE CONTINUOUS PHASE, 134 SIMPLIFIED MODELS OF DISPERSE FLOW, 134 MODELING PSEUDO-TURBULENCE, 136 PSEUDO-TURBULENCE IN A HOMOGENEOUS FLUIDIZED BED, 140 APPLICATION EXAMPLES, 147 Homogeneous Fluidization Stability, 148 Voidage Distribution Ahead of a Bubble in a Fluidized Bed, 154 Binary Fluidization, 155 CONCLUSIONS, 160 NOTATION, 161 REFERENCES, 162 INTRODUCTION This article delineates fundamental principles germane to a new method of hydrodynamic description for a special class of disperse systems that is widespread in various industrial technological processes. Hydrodynamic description implies, first 119

120

Advances in Engineering Fluid Mechanics

of all, formulating the equations of mass, momentum, and energy conservation that govern mean (macroscopic) disperse system flow. Secondly, this t^^sk entails deriving constitutive rheological equations for all quantities involved in the conservation equations. These constitutive equations relate these quantities to a few macroscopic flow variables, such as to the dispersion concentration, the mean pressure of the ambient fluid in interstitial space, the averaged velocities of the dispersion phases, and the physical parameters of the phases. Once all the above equations have been formulated, we have at our disposal a completely closed set of equations suitable to treat concrete flow problems. To a considerable extent, rheological properties of various disperse systems are determined by their random particle fluctuations. In the dispersed phase, these fluctuations play a leading role in the formation of a system of effective stresses that determine, in turn, the observable macroscopic flow characteristics of the dispersions. For this reason, any classification of the dispersions with respect to their hydrodynamic behavior must be based on careful consideration of the main physical mechanisms that induce particulate fluctuations, and that consequently determine fluctuation properties. Particle fluctuations may be produced when external sources that cause macro-scopic motion in a dispersion supply kinetic energy to moving particles. External sources that are common factors in the flow of both granular materials and sus-pensions include gravity and centrifugal force, externally induced vibrations, externally applied electric or magnetic fields, and interaction forces exerted on the particles by an infiltrating fluid flow. For suspensions, however, these sources are not the main cause of particle fluctuations, but another mechanism that generates particle and ambient fluid fluctuating motion usually plays the dominant role. This mechanism appears as a result of interphase interaction force fluctuations and is of purely hydrodynamic origin. The dispersion class considered in this article comprises only suspensions with hydrodynamically induced suspension phase fluctuations. All other mechanisms for generating fluctuations that are typical for granular flow are ignored. Irrespective of the energy source that causes particle fluctuating motion, particle fluctuations in slow granular flow are generated as a result of Coulomb friction forces acting between permanently contacting particles. In a rapid granular flow, fluctuations appear mainly due to the direct interparticle interaction that results as particles quickly impact and then freely fly to successive collisions. Neglect in suspension study of fluctuations resulting from contact interparticle interactions completely excludes from consideration numerous analyses based on the kinetic modeling of collisional motion for granular media. See examples in reference [1-5]. There are two basic mechanisms responsible for the interphase interaction force originating hydrodynamically induced fluctuations in suspensions. The first mechanism is specific to shear suspension flow. It is cognate to a similar mechanism in shear granular flow except for the fact that particle pair interactions for particles moving with different velocities in a shear flow are not necessarily limited to those entailing direct collisions of particle pairs. A purely hydrodynamic particle interaction is much more characteristic for suspended particles, especially in finely dispersed suspensions of low or moderate concentration. As particles draw closer together while they are carried along by the shear flow of the suspending fluid, this interaction is effected through the fields of the suspending fluid pressure and velocity perturbed by the

Fluid Dynamics of Coarse Dispersions

121

interacting particles. Hydrodynamic interaction proceeds either without physical contact between the particles, or with occasional touching contacts that do not play a noticeable role in interparticle interaction. Fluid pressure and velocity fields near any two interacting particles are also influenced by all neighboring particles. For this reason, hydrodynamic interaction is essentially a collective phenomenon and may be only conventionally associated with any single pair of particles. Shearinduced fluctuations of this type have been discovered and described [6,7]. These fluctuations also are neglected in the present article. The other mechanism that produces hydrodynamically originated particle fluctuations in suspensions occurs as a result of the work performed by the ambient fluid mean relative flow on random suspension concentration fluctuations. This mechanism generates the so-called "pseudo-turbulent" fluctuating motion that had been pointed out for the first time in 1966 [8] and afterwards discussed at some length [9], and in many other papers. This mechanism is due to the fact that the interphase interaction force is a nonlinear function of the local concentration. Therefore, concentrational fluctuations violate the balance of forces acting on any particle in a real suspension as compared with the same particle in the similar fictitious suspension without fluctuations. As a result, a net force appears that accelerates the particle in the direction of the mean fluid slip velocity or in the opposite direction, depending on the sign of the concentrational fluctuation under question. Fluctuations in the directions normal to that of the slip velocity are then excited because the kinetic energy of these longitudinal fluctuations is partly transferred to lateral fluctuations. This process of energy redistribution is carried out by direct interparticle collisions and by hydrodynamic particle interactions between collisions that occur when the particles move through random fields of fluctuating fluid velocity and pressure. Thus, kinetic energy of the mean macroscopic flow transforms into particle and fluid fluctuation energy that is thereafter dissipated by viscous forces. Under steady macroscopic flow conditions, a stationary fluctuating motion is bound to establish itself, and properties of this motion depend on the kinetic energy balance. Admittedly, the pseudo-turbulent mechanism of fluctuations generation prevails only in those circumstances where the mean fluid slip velocity is sufficiently large while the mean shear rate is relatively low. Suspensions that satisfy such requirements are exemplified by systems of the fluidized bed type, and also by any vertical suspension flow in general. Two extreme cases of such suspensions can be singled out. Suspensions belonging to these extreme cases differ by the manner in which their interparticle exchange by fluctuation momentum and energy takes place. The first case is specific to colloids and suspensions of fine particles. In this case, the exchange is effected solely by means of random fields of fluid pressure and velocity, without any noticeable influence produced by collisions. In the second extreme case characteristic of suspensions containing larger particles, interparticle exchange is performed by direct collisions and is similar to the exchange carried out in molecular gases. Suspensions in which momentum and energy exchange is primarily caused by collisions can be conventionally termed as coarse dispersions, as opposed to fine suspensions in which this exchange is effected without collisions. It is not a simple matter to explicitly elucidate under which conditions and with what accuracy a given suspension can be convincingly categorized as one or the other of these two limiting types. However, the analysis performed by Koch proves

122

Advances in Engineering Fluid Mechanics

that interparticle collisions are likely to play a prevalent role even in suspensions of comparatively small particles, and all the more so as the suspension concentration increases [10]. In particular, this author has shown that if the dispersion is dilute and the particle Stokes number greatly exceeds (j)"^^^, the particle velocity distribution in a gas-solid dispersion is nearly Maxwellian even if the particle Reynolds number is much smaller than unity. These conditions can possibly be met even in suspensions of very fine particles. Thus, the term "coarse dispersions", chosen to identify dispersions which are characterized by the collisional exchange mechanism, may be somewhat misleading at times. In what follows, we pay attention only to coarse dispersions characterized by the pseudo-turbulent mechanism for random fluctuation origination and by the collisional mechanism for interparticle exchange by fluctuation momentum and energy. In coarse dispersions, a change in particle velocity as this particle freely moves through the fluid between successive collisions with its neighbors and interacts with the suspending fluid is insignificant as compared with the characteristic particle velocity change accompanying a collision. Even so, this particle-fluid interaction cannot be ignored since it is responsible for the very appearance of the pseudoturbulent fluctuating motion. Development of a consistent stochastic model of random particle fluctuations in a coarse dispersion and subsequent calculation of the particulate stresses affecting flow of the dispersed phase present a formidable problem even if the particles are ideally elastic and ideally smooth, so that there is no energy loss at any collision. This problem seems to be more difficult than the analogous problem in the fluid dynamics of granular media. In the latter case of granular media, there is no urgent need to allow for the nonconservative interaction of a particle with the ambient fluid between collisions. In this case, the formulation of hydrodynamic conservation equations is in fact reduced to a detailed study of collisions between particles at given values of the restitution coefficient and a parameter characterizing the particle surface roughness which condition a single pair collision. After that, the derivation of the conservation equations can be accomplished by following the familiar lines of reasoning specific to the kinetic theory of gases [5]. On the contrary, in the case of pseudo-turbulent coarse dispersions, it is necessary to first develop a comprehensive statistical model of particle fluctuations prior to formulation of the rheological equations, and only after the rheological equations have been thus formulated can the set of governing conservation equations be finally closed. On the other hand, an analysis of the extreme case of coarse dispersions is more difficult, in a sense, than an analysis of the opposite extreme of fine suspensions. This is due to the mere fact that particles in fine suspensions interact only hydrodynamically. Although this means that there is no need to consider direct particle collisions, the problem of formulating both the conservation and rheological equations remains difficult because hydrodynamic interactions involve many particles simultaneously in fine particle suspensions. A sophisticated statistical theory of Brownian suspensions is now being developed by Brady and his co-workers that might help to tackle this problem [11-13]. An attempt to take into account pseudoturbulent fluctuations in finely dispersed suspensions is described in [14,15]. It is quite evident that any generalization of these models of fine collisionless suspensions to coarse collisional suspensions involves, first of all, the necessity to account for direct collisions, and this is certainly a matter of some difficulty.

Fluid Dynamics of Coarse Dispersions

123

At present, manifold difficulties prevent developing a quite rigorous and finished fluid dynamic theory of coarse dispersions. Moreover, even if formulation of such a theory were possible, it is very doubtful whether this theory would be workable enough to be applied to concrete flow problems. This is why we are going to simplify our study whenever possible by forwarding and putting into effect certain plausible, although semi-empirical, hypotheses. Even so, to develop our theory, we must simultaneously consider models of greatly varied nature, and we can hardly discuss all these models to the full extent in this relatively short article. For this reason, we are compelled to envision our main goal as highlighting only the primary ideas needed to model particle fluctuating motion and macroscopic flow of coarse dispersions in a comprehensive way. And for this reason, the following presentation is, in part, rather sketchy, so that certain imaginative efforts are sometimes required on the part of the readers to fill in some gaps in this presentation. The general layout of this article is as follows. After a brief discussion of the underlying assumptions and the relevant forces experienced by the suspended particles, we proceed to the formulation of the conservation equations governing macroscopic disperse flow. Since these equations include quantities dependent on particle fluctuation properties, a method to find these properties is presented next. This method is illustrated by using the example of homogeneous fluidized beds. After that, we point out an opportunity to generalize the suggested theory to arbitrarily polidisperse coarse suspensions, and also briefly enumerate conclusions made with respect to some disperse flows and their correspondence to experimental evidence. PHYSICAL BACKGROUND As a basic system to be studied, we consider a suspension of identical spherical particles of radius a and density p^ in an incompressible fluid of density p^ and viscosity ji^. From the very beginning, we presume that the spherical particles are involved in a chaotic fluctuating motion, and that their collisions dominate in the interparticle exchange of fluctuation energy and momentum. In particular, the last assumption implies that: 1. particles can be approximately regarded as statistically independent insofar as the effect of velocity persistence after collisions is overlooked; 2. velocity of any particle does not significantly change during the period of time elapsing between successive collisions of this particle with neighboring particles; 3. particle velocity distribution is conditioned by collisions which favor equipartition of kinetic energy between various degrees of freedom of the particles. As far as these inferences hold true, the assemblage of suspended particles may be approximately viewed and modeled as a pseudo-gas consisting of identical hard spheres, and standard methods of the kinetic theory of gases [16-18] can be employed to study the statistical properties of particle fluctuations. However, these statistical properties depend not only on the effect of collisions, but also on the manner in which energy is initially transmitted to the particle fluctuating motion. In a molecular gas, collisions represent a mighty mechanism for energy redistribution between available degrees of freedom of the molecules which results in establishing an almost isotropic molecular velocity distribution irrespective of the

124

Advances in Engineering Fluid Mechanics

type of energy input to the gas. As a rule, the relaxation time of this energy redistribution process is negligibly small as compared with characteristic time scales of gas macroscopic flows. Consequently, any difference in the extent of excitation of different degrees of freedom usually may be ignored, and the intensity of molecular chaotic motion can be completely described with the help of a single scalar variable—the gas temperature. In contrast, in a particulate assemblage, energy is first transmitted to some preferable particle degrees of freedom. Furthermore, the time needed to provide for an appreciable excitation of other degrees of freedom may well be comparable with the macroscopic flow time scale, as well as with a time characterizing energy transfer to the fluctuating motion and viscous energy dissipation. For instance, the pseudo-turbulent mechanism of particle fluctuation origination first induces longitudinal translational fluctuations directed along the mean fluid relative velocity, and the kinetic energy of these fluctuations is next redistributed over other translational and rotational degrees of freedom [8,9,14]. If this process of energy redistribution is hampered for some reason, as happens in dilute suspensions, the resultant state of particle fluctuations may be highly anisotropic. An opposing example is provided by assemblages of magnetic pellets fluidized by an externally applied alternate magnetic field [19]. In this case, only particle rotations are immediately excited by the field, and particle rotation kinetic energy is then transferred to particle translational fluctuations. Even at large concentrations, magnetofluidized beds usually exhibit a specific rotational fluctuation energy that is often more than an order of magnitude as high as the translational fluctuation energy [19]. On the whole, the principle of equally partitioning fluctuation energy over varying degrees of freedom is not necessarily satisfied in assemblages of suspended and fluidized particles, and particle fluctuating motion can be essentially anisotropic even under stationary external conditions. The time scale for energy redistribution between different degrees of freedom is inversely proportional to the collision frequency. Hence it follows that if the collision frequency is large enough, this time scale will be much smaller than the characteristic time of energy input and dissipation. This is always the case for suspensions of sufficiently high concentration since the collision frequency tends to infinity as a particulate gas approaches the closed-packed state [17]. If this requirement imposed on the indicated time scales is satisfied, interparticle collisions must result in establishing a near-Maxwellian particle velocity distribution at arbitrary flow conditions, and the intensity of particle fluctuations may be described with the aid of a scalar "fluctuation temperature" which is related to the mean fluctuation kinetic energy in the same way as the temperature of molecular systems is related to the energy of thermal motion. In the general case of an arbitrary suspension flow, fluctuation energy is not equally partitioned over translational and rotational particle degrees of freedom. Moreover, particle translational fluctuations are not necessarily isotropic, and fluctuation energies for different rotational degrees of freedom are not necessarily equal among themselves. Particle fluctuating motion is characterized by three translational and three rotational "temperatures", each of these temperatures being attributed to one of the particle degrees of freedom. The determination of these temperatures requires a careful sophisticated analysis of the collision-driven pro-

Fluid Dynamics of Coarse Dispersions

125

cesses of energy redistribution between various degrees of freedom. At the same time, allowance must be made both for energy inflow to some particular degrees of freedom from the mean fluid flow and for viscous energy dissipation for all degrees of freedom. Analysis of energy redistribution implies, in turn, a meticulous consideration of collisions of various possible types, as well as consideration of the impact of collisions on particle random linear and angular velocity distributions. Although the collisions and their impact on velocity distributions can be studied by means of kinetic theory methods, similar to the analysis performed for the collisional motion of granular media [5], any attempt to make allowance for energy exchange of pseudo-turbulent fluctuations with macroscopic flow and for fluctuation energy dissipation presents a new difficult problem that has still to be resolved. Even if it were possible at the present research stage to develop a hydrodynamic theory of coarse dispersion flow with account made for all relevant phenomena, this theory could hardly be applied immediately to concrete flow problems in view of its expected complexity. In view of this predicament, it seems reasonable to address a simplified class of disperse flows. We define such a flow class by advancing two major hypotheses: 1. intensity of particle pseudo-turbulent fluctuations can be adequately described with the help of a single fluctuation temperature which represents the doubled mean kinetic energy per one particle translational degree of freedom; 2. mean energy of particle rotations is much smaller than that of particle translations and so may be neglected. The first hypothesis is quite plausible for concentrated suspensions, and the fluctuation temperature may be defined in terms of the mean fluctuation kinetic energy associated with one translational degree of freedom of a particle in the following form: T = m(w:^) = i m ( w ' ^ ) ,

m^^Tia^p,

(2.1)

where w'is particle translational velocity fluctuation. The concept of an effective fluctuation temperature for assemblages of suspended particles was introduced as early as 1963 by Jackson, who supposed that dispersed flow is influenced by a particulate pressure that comes about in the same manner as pressure does in a molecular gas [20]. After that, the concept was repeatedly exploited in many papers, including [8,13,21,22]. This hypothesis ought to become invalid for suspensions of low concentration in which the collision frequency is not high enough to ensure fluctuation energy equipartition. In any event, the validity of this hypothesis for particular suspension flows should be carefully checked. The second hypothesis is always rigorously true for suspensions with ideally smooth particles. In this case translational fluctuation energy cannot be converted into particle rotation energy at any collision. If the particle surface is sufficiently rough, a considerable part of the translational energy is transformed into rotational energy as a result of interparticle collisions. To get a working idea about what happens in practice, we again consider the experiments on magnetofluidization [19].

126

Advances in Engineering Fluid IVIeclianics

Monodisperse assemblages of ferromagnetic, almost spherical particles were used in these experiments, and particle diameter varied from 1 to 10 millimeters. The particle surface was rather more rough than smooth. Both linear and angular particle velocities were approximately distributed in conformity with the Maxwellian law in the whole interval of used concentrations ranging from a few percent by volume to about 40-50 percent. This proves collisions to be intensive enough to ensure equal distribution of kinetic energy over all translational or over all rotational degrees of freedom. However, as has been indicated, the translational temperature was not less than an order of magnitude lower than the rotational temperature at any concentration. This proves collisions to be inefficient in ensuring energy equipartition between particle rotations and translations. In magnetofluidized beds, external magnetic field energy is initially transmitted to particle rotations, and particle translations occur due to collisions. Coarse dispersions are, in a sense, inverse to magnetofluidized beds because macroscopic flow energy is first input into particle translational degrees of freedom, and particle rotations are excited as a result of collisions. However, the above conclusion concerning the rate of energy exchange between different degrees of freedom at collisions must remain valid. This shows that the rotational temperature of pseudoturbulent fluctuations is likely to be much smaller than their translational temperature even for systems containing comparatively rough particles. It is worth pointing out, nevertheless, that fluctuation rotational energy can be essential in suspensions of very rough particles, as is also the case for granular media of such particles [5]. The hypothesis that fluctuation rotational energy is negligible as compared with fluctuation translational energy is contrary to the assumption that kinetic energy is equipartitionable, which assumption is accepted in a number of works, and in particular in reference [21]. As has been shown in the cited paper, particle rotations bring about random Magnus forces that act on the particles, and, thus, add to other forces of hydrodynamic origin in generating pseudo-turbulence. The role of random Magnus forces in inducing particle fluctuations has been proven to be especially significant for large particles [21]. This means that particle rotations should also be accounted for when treating dispersions of large particles even if the mean energy of rotations is much smaller than that of translations. FORCES ACTING ON A PARTICLE Any suspended particle is operated on by forces resulting from collisions with other particles and from interaction with the suspending fluid. The force attributable to collisions can be represented as a random Poisson process, being the sum of vector components with random amplitudes and directions. Each vector component represents a single collision of the particle under question with its neighbors and differs from zero only for the short duration of this collision. These vector components may be approximated by delta-functions of time with random vector coefficients, the delta-functions being attributed to a random sequence of moments of successive collisions. Accordingly, fluctuation velocity w'of any particle is then the sum of continuous random vector functions of time during time intervals between collisions and of finite random vector discontinuities at the moments when collisions occur.

Fluid Dynamics of Coarse Dispersions

127

In order to proceed on to the analysis of continuous random functions and thereby considerably simplify the problem, we perform imaginary averaging over the ensemble of possible states for all the particles around a certain individual particle with velocity w', as explained in reference [23]. This averaging procedure smoothes out the discontinuities so the average velocity over this ensemble must be continuous over time. In what follows, all fluctuations for all different variables are supposed to be continuous functions of time obtained as a result of such averaging. The averaged collisional force acting on any particle should now be regarded as a continuously varying random vector as well. An expression for this random collisional force can be derived by taking into account that: 1) it must be linear in w' since the fluctuations are assumed to be sufficiently weak, and 2) it may contain only those components directed along unit vectors marking preferable directions, at a given physical point of the suspension under study. There are usually two such directions, and these directions are determined by acceleration g of external body forces and by mean fluid slip velocity (u). When these directions are essentially different, the corresponding general expression for the collisional force is presented in reference [23]. If vectors g and (u) are coUinear, as is specific to fluidized beds and to other vertical flows of suspensions, this expression takes the form: f: = -m[Aw' + B(Uo • w ' ) u j ,

Uo ^ (u)/(u)

(3.1)

where A and B are coefficients independent of fluctuations. Equations to determine these important coefficients are indicated below. The surrounding fluid exerts a force on each particle that includes components of different physical origin. The main contribution to this hydrodynamic force is usually made by components associated with hydrodynamic drag and buoyancy. If when expressing hydrodynamic drag we use the well-known semi-empirical twoterm law, then for the force per particle in a suspension without fluctuations we obtain f, =m{[F,(^) + F2((^)u]u-(8K-^+(t))g} + 5f,

K = p./p,

(3.2)

where 6f stands for the remaining force components, F^ and F^ are certain functions of the local particle concentration by volume (|), m is the particle mass, and £ = 1 - (j) is the local suspension voidage. Term 8f is the sum of forces caused by various inertial effects and by effects of flow nonhomogeneity. When there are concentrated suspensions, an analytical expression for this term has been so far obtained only for fine spherical particles whose Reynolds number is smaller than unity [24]. In the case of fine suspensions, the inertial part of 8f includes: 1) an inertial force due to acceleration of the virtual fluid mass by the moving particle, 2) a contribution to the buoyancy which is caused by the field of inertial body forces in the same way as buoyancy is usually caused by the field of external body forces, 3) a hereditary force whose strength and direction depend on the flow history (Basset force), and 4) a new force due to frequency dispersion of the suspension effective viscosity. As the suspension concentration comes to zero, the first three force constituents of the inertial part of 8f tend to manifest themselves as forces similar to those experienced by a single

128

Advances in Engineering Fluid Mechanics

solid sphere in an unbounded fluid. The last force constituent is specific only to suspensions. It represents a collective multiparticle effect and turns to zero together with suspension concentration. The part of 8f due to flow nonhomogeneity transforms to the familiar Faxen force for a single spherical particle as the suspension concentration vanishes. Fortunately enough, 8f is often negligible as compared with other contributions to force (3.2) and may therefore be ignored. In the same suspension with fluctuations, the interphase interaction force also fluctuates. Representing the variables involved in Equation 3.2 as sums of their mean values and zero-mean fluctuations (marked with a prime), for the mean value of this force we obtain ( f , ) = m{[;i,F, + ;i,F,(u>](u> - ((e)K-' + ((t)))g} + <8f)

(3.3)

and for its fluctuation f; = m{F,u' + F J < u ) u ' + (u, • u')
(3.4)

Here F and F' denote the functions introduced in Equation 3.2 and their derivatives J

J

evaluated at (^ = (([)), and coefficients X, underline the difference between the mean interphase interaction force acting in a real suspension with fluctuating dynamic variables and the same force acting in a suspension without fluctuations, other things being equal. These coefficients usually do not differ much from unity, and a method to calculate them follows. When particles experience fluctuations, the total number of particles within a unit volume also fluctuates. The fluctuation of the interphase interaction force related to a unit volume of the mixture consists of two constituents: one that accounts for fluctuation (3.4) of the force exerted on a single particle, and one that allows for a fluctuation of the particle number concentration n. For weak fluctuations, we are therefore able to write nf, = (n)(f,) + (n>f; + n'(f,),

{, n'} = (3/4Tia^ ){(^), f }

(3.5)

Equations 3.1-3.5 determine both the mean force of interphase interaction and the fluctuation of this force. The mean force will be used later when formulating conservation equations for mean suspension flow. Force fluctuation is sorely needed to study properties of pseudo-turbulent motion. It should be noted that the last term in Equation 3.5 had been omitted in a similar analysis of pseudo-turbulence [25]. CONSERVATION EQUATIONS FOR THE DISPERSED PHASE When direct collisions are the leading factor in interparticle exchange by momentum and energy, the assemblage of suspended particles may be modeled as a dense pseudo-gas. This particulate pseudo-gas can be treated in exactly the same way as

Fluid Dynamics of Coarse Dispersions

129

a molecular gas, and in particular, the classical Enskog method can be used to solve the corresponding Boltzmann equation [16-17]. The only difference is that subsequent averaging over the fluctuations, indicated by angular brackets, is required. In the general case of a coarse dispersion flow, both calculations needed to solve the Boltzmann equation, and the ensuing derivation of conservation equations for the dispersed phase are very tedious and cumbersome, as is evidenced by a similar analysis of granular medium flow [5]. However, for suspensions characterized by a scalar fluctuation temperature, this derivation can be easily performed by analogy with the kinetic theory. While omitting details of the calculation, we list here only the final conservation equations which are of the same general form as those in reference [23]. As a result of the calculation, we arrive at the equations of: dispersed phase mass conservation

avat + V • ((|)w> = 0

(4.1)

dispersed phase average momentum conservation (a/at + w • V)w = V • P, + n ( f , ) + (|)p,g

(4.2)

and conservation of the average energy of pseudo-turbulent fluctuations (temperature of the pseudo-gas) (a/at -H w • V)T = (2/3n)[P,: (V* w) - V • Q + n#(nV'> + n < f : # w ' ) ]

(4.3)

Here, the asterisk denotes dyadic vector multiplication, and the colon signifies double convolution of second-order tensors. For simplicity, here and in what follows, we have omitted the angular brackets in the notation for the averaged suspension concentration and all mean flow dynamic variables. The meaning of the different terms in Equations 4.1-4.3 is precisely the same as in the analogous equations of the kinetic theory of gases. Tensor P, describes pseudo-gas stresses. Vector Q is the fluctuation energy flux in the dispersed phase due to the fluctuations themselves. This flux is similar to the familiar heat flux in molecular systems. Using the classical results [16-18], we can immediately define P^ and Q in the following form: P, = -p,I + 2^1 J E ,

- (trEJ3)I],

Q = -Ti.VT

(4.4)

where Pj plays the role of the scalar isotropic pressure of the pseudo-gas, E^ is the strain rate tensor associated with the mean dispersed phase flow with velocity w, and I is the unit tensor. For collision frequency v, dynamic dispersed phase viscosity jXj, pulsation energy transport coefficient r|j of the pseudo-gas (analog of the thermal conductivity coefficient in a molecular gas), and particle self-diffusion coefficient D, we have [16-18]

130

Advances in Engineering Fluid Mechanics

v = (4(1))-^%

\i, =4(|)(Y-' + 0 . 8 + 0.76Y)|i;,

r|, = 4(|)( Y-' +1.2 + 0.75 Y)r|;',

D = 4(t)Y-'D°,

(4.5)

Y = Y((^) = G ( ^ ) - 1 = 4^X(*) where the quantities marked with a degree refer to a rarefied gas, G((|)) is the correction function that appears in the pseudo-gas equation of state and defines pseudo-gas osmotic pressure, p, = G((^)nT

(4.6)

and x((|)) is the Enskog factor describing the increase in collision frequency for a concentrated as opposed to a rarefied gas. For the quantities relating to a rarefied gas, we have [16-18] 1/2

v° = 16nV[

Km

^0 _

75

)

' T Y'^

(

256aH7imJ

'

^•=i^l^

It J

no-

fT

3

y/2

(4.7)

~32naH7cmJ

Equations 4.5 were obtained within the framework of the approximate Enskog dense gas theory [17,18] that corresponds to smoothing of the free particle volume when

Gw=^-7X7r^' xw=..M:;:.v/3r '==o.6 l-«|)/
(4.8)

where (j)^ is understood as particle volume concentration in the state of close packing. This theory is applicable only to gases or dispersions of very high concentration. However, without much error. Equations 4.5 also can be applied to less concentrated systems, if other representations are used for the osmotic pressure correction function and Enskog factor. In particular, the approximate CarnahanStarling model [26], which is widely used to describe particulate systems, results in alternative expressions ^,^,

l + (t) + ( t ) ' - f

,^,

l-(t)/2

G(4)) = — , , %(<^) = ^^ r4 9^ '^' (1_(|))3 '^^^^ (l-(t))' ^^^ We might just as easily have made use of some other statistical theory of dense gases and liquids to describe G((t)) and %(([)). However, the Camahan - Starling model seems preferable since it does not involve unwieldy numerical calculation and leads to simple analytical expressions. In expressing these functions for low-concentration systems, it is also possible to use the standard technique of virial expansions.

Fluid Dynamics of Coarse Dispersions

131

Consider next the differences between conservation Equations 4.1-4.3 and their analogous equations in the kinetic theory of gases. In the mass conservation equation for the suspension dispersed phase, the averaged particle volume flux appears as follows: <(t)w) = (|)w + <(|)'wO - (|)w

(4.10)

This flux approximately equals the volume flux in a fictitious suspension of the same particles at the same mean concentration but without fluctuations. However, these two fluxes by no means identically coincide. Pseudo-turbulent fluctuations cause the appearance of an additional component that is added to the total flux and that usually differs from zero. The right-hand side of the momentum conservation equation involves a new term that describes, on the average, particle interaction with the ambient fluid. Again, as follows from Equation 3.3, the mean interphase interaction force differs from the analogous force acting in the same suspension without fluctuations. Comparison of Equation 3.2 and Equation 3.3 shows this difference to be attributable to the fact that coefficients X. differ from unity. However negligible this difference may be, its occurrence is significant from the physical point of view. Finally, the fluctuation energy conservation equation includes terms which describe work performed by the fluctuation force on random particle displacements in a unit volume during a unit time. There are two constituents of this work. One of these constituents is due to the collisional force defined in Equation 3.1. This constituent describes the power dissipated per unit volume as a result of collisions. There are various mechanisms of collisional dissipation. They are due to inelasticity of thd collisions as a result of: 1) internal viscosity of the particle material, 2) particle surface friction, and furthermore, 3) an additional viscous dissipation in the suspending fluid that accompanies sudden stepwise changes in particle velocities at collisions. The first two mechanisms can be described in terms of the collision restitution coefficient and the particle roughness coefficient, as is usual when studying collisional motion of a granular medium [5]. The third mechanism apparently has not been considered in connection with disperse systems. Because we do not wish to treat collisions in detail, it seems natural to introduce an empirical coefficient k^ to describe the mean kinetic energy fraction of the colliding particles lost in a single collision. Introducing such a coefficient and using Equation 4.5 and Equation 4.7, we get [23] .1/2

q, = -n = k,vT = aX^\

a, = 16k f^i^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

Advances in Engineering Fluid Mechanics

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 (
Fluid Dynamics of Coarse Dispersions q* = q * + q* = a T* + a j * ^ ^ ^

133 (4 13)

We shall now consider the state in which actual fluctuation characteristics initially differ from those for either the real or fictitious homogeneous state. Obviously, these characteristics must relax to the values pertaining to this homogeneous state. To develop a simple model of such a relaxation, we must first briefly discuss specific relaxation times. The behavior of any particle from the assemblage of statistically independent particles is governed by fluctuations of dynamic variables (that is, by fluctuations of fluid pressure and velocity, of particle-free volume, and of particle velocity) within the particle specific volume. The characteristic time scales of these fluctuations are determined by the rates of: 1) propagation perturbations in the fluid over a distance of the order of the particle size, 2) particle diffusional migration over the same distance, and 3) particle velocity changes due to interaction with the surrounding fluid. If the particles are not too small, then the time scales corresponding to perturbation propagation and to particle migration should be much less than the particle hydrodynamic relaxation time. For time periods of the order of particle relaxation time, this means that fluctuations of fluid velocity and pressure within the particle specific volume, as well as fluctuations of the local suspension concentration, may be regarded as being equal to similar characteristics in the corresponding fictitious state. As a result, we arrive at the notion of a "non-equilibrium" state of the suspension in which fluctuations of suspension concentration and fluid velocity do not differ from those in the fictitious homogeneous ("equilibrium") state while particle velocity fluctuations continue to relax to those specific to the fictitious state. From the formal representation for q^ that was derived as indicated earlier we find that q^ depends linearly on w'. This means that q^ must be proportional to the square root of the fluctuation temperature. Bearing this in mind and using Equation 4.13, we get as a final result q^ = a^ VT ,

a^ = a_ / T * + a J *

(4.14)

This equation helps to close Equation 4.3 if we consider the fluctuation temperature of the fictitious homogeneous state of the suspension characterized by local mean dynamic variables as a known function of those variables. On the basis of the equations presented above for the various constituents of the source terms involved in energy conservation Equation 4.3, we arrive at n + • = q, - q_ - q,

= a_VT(VT*-VT) + a,(T*-T)

(4.15)

with coefficients a and a^ defined in Equations 4.12 and 4.11, respectively. The suggested scheme for closing the fluctuation energy conservation equation is reminiscent of methods specific to the relaxational formalism of the thermodynamics of irreversible processes [27,28]. This scheme could be brought into play empirically on the basis of this formalism. In essence, this scheme enables us to

134

Advances in Engineering Fluid Mechanics

reduce the difficult problem of calculating q^ in an arbitrary suspension state characterized by mean flow variables depending on time and coordinates to an evaluation of only one pseudo-turbulent quantity, that is, to an evaluation of particle fluctuation temperature in this state at neglect of space and time derivatives for the mean variables.

CONSERVATION EQUATIONS FOR THE CONTINUOUS PHASE The mass and momentum conservation equations for the continuous phase can be expressed in the traditional form (see, for example, reference [29-34]). In this connection, it is desirable to explicitly account for the fact that the linear scale of nonaveraged fluid velocity coincides with particle size. This means that this scale must be much smaller than the linear scale for the averaged fluid velocity. Therefore, we are free to neglect divergence of viscous stresses in macroscopic flow which are determined from the mean fluid strain tensor, as compared with the interphase interaction force [21]. Formulating and then averaging the equations following from the fluid mass and momentum conservation laws, we obtain the following equations for the continuous phase: aVat - V • <ev) = 0,

(ev> = ev - <(t)'v'> - ev

e p , 0 / a t + V • V)v = V • P, - n + ep„g

(5.1) (5.2)

where stress tensor P^ includes isotropic stresses due to molecular fluid pressure p and stresses entirely analogous to the Reynolds stresses in turbulent flow. (It must be remembered that angular brackets generally used in notation of mean flow variables are omitted.) Thus, P, = - p J - e p J < v ' * v O - ( t r < v ' * v ' ) / 3 ) I ]

(5.3)

In the first approximation, it is evidently permissible to neglect the second term on the right-hand side of Equation 5.3 as opposed to the first. We can do this in view of the fact that thermal motion velocity of fluid molecules considerably exceeds the characteristic fluid fluctuation velocity. For the same reason, the difference between p^ and fluid molecular pressure p may be ignored. Obviously, to improve the accuracy of this model it is necessary, on the basis of a detailed theory of pseudo-turbulent motion, to calculate tensor (v'*v') and, moreover, to determine flux ((t)V') which enters into Equation 5.1. SIMPLIFIED MODELS OF DISPERSED FLOW As a first approximation, small pseudo-turbulent corrections to volume fluid and particle fluxes may be ignored. Similarly, the difference between interphase interaction forces that act in fluctuating suspensions and in corresponding suspensions without fluctuations may be overlooked. Ignoring these factors, we arrive at a full set of hydrodynamic equations for both suspension phases which includes:

Fluid Dynamics of Coarse Dispersions

135

1. mass conservation equations for both phases,

a(t)/3t-V• (ev) = 0,

a(|)/at-V• ((|)w) = 0,

(8 = 1 - 9 )

(6.1)

2. momentum conservation equations for both phases, e p , 0 / a t + V • v)v = - V p , - nf, + ep.g,

( p , - p) (f\ 9^

(t)p, (a/at + V • w ) w = - V p , + V • ()i,V)w + nf, + (t)pig

3. fluctuation energy conservation equation for the dispersed phase, (a/at + V • w)T = (2/3n)[P,: (V* w) - V • (r|,V)T ^ ^ ^ r+ a_ V T ( V T * - V T ) + a, VT (T * - T ) ] ^

(6.3)

Scalar isotropic pressure p^ in the continuous phase approximately equals the mean fluid pressure, and particulate stresses Pj are expressible through derivatives of vf and scalar isotropic pressure pj in the dispersed phase in accordance with Equation 4.4. Pressure Pj is a function of suspension volume concentration and of particle fluctuation temperature defined by equation of state (4.6) for particulate pseudogas. Osmotic pressure function G{<^) appearing in Equation 4.6 is given by either Equation 4.8, 4.9, or by some other equation that follows from some other statistical pseudo-gas theory. Dispersed phase dynamic viscosity coefficient ji, and particle fluctuation energy transfer coefficient r|j that appear in Equation 4.4 also can be represented as functions of fluctuation temperature T and concentration (|) in conformity with the formulae in Equations 5.5 and 5.7. Force nf^^ of interphase interaction per unit suspension volume approximately equals the force in Equation 3.2 multiplied by the particle number concentration. Finally, coefficients a^ and a are determined in Equation 4.11 and 4.12, respectively. Thus, there are three scalar equations (6.1) and (6.3) and two vector equations (6.2) to find out three unknown scalar variables (suspension concentration (|), mean fluid pressure p, and particle fluctuation temperature T), and two unknown vector variables (mean velocities v and w of the continuous and dispersed phases). The only quantity that remains indeterminate in the set of conservation Equations, 6.1-6.3 is particle fluctuation temperature T*. This quantity must be evaluated to the neglect of derivatives of the mean flow variable that play the role of unknown variables for these equations. To provide for final closure of the conservation equations, we need only disclose the dependence of this temperature on the mentioned variables. However, before proceeding to a discussion of how to find this function, we are going to consider certain simplified versions of this set of governing hydrodynamic equations. First of all, an analog of Eulerian conventional hydromechanic equations can be put forward when neglecting both quasi-viscous stresses in the dispersed phase and the additional flux of fluctuation energy due to the fluctuations themselves. This amounts to equating coefficients |Lij and r|j to zero. When these coefficients are zero.

136

Advances in Engineering Fluid Mechanics

the viscous stress term and the term containing the fluctuation energy flux divergence entirely disappear from the dispersed phase equations of momentum and energy conservation. Moreover, in this case we immediately obtain V • Pj = -Vp^ and, next, P,:(V*w) = -Vp^V • w. Hence, it follows that this last quantity describes the work that must be done to overcome particulate pressure to provide for expansion of the dispersed phase. Further simplifications can be gained by following two different lines of reasoning. First, the convective term on the left-hand side and the expansion term on the right-hand side of Equation 6.3 may well be insignificant when compared to the other terms of this equation that describe energy input to the pseudo-turbulence from macroscopic suspension flow and viscous dissipation of fluctuation energy. Dropping the convection and expansion terms, the fluctuation energy conservation equation reduces to q^ = q. + q^., and this reduced equation is evidently specific to both the real and fictitious locally homogeneous state (see Equation 4.13). This means that particle fluctuation temperature approximately coincides with its value T* in the homogeneous srtate, and this last temperature can be thought of as a known function of the other mean flow variables. As a result, the energy conservation equation can be altogether dropped, and we arrive at the scheme familiar in two-phase fluid dynamics where there are only mass and momentum conservation equations for both phases. However, these equations serve to solve for p^, (|), and mean velocities v and w. Pressure p, is not an unknown variable of these equations since it is expressed through the other unknown variables according to Equation 4.6, in which T = T*. An alternative approach consists in modeling the dispersed phase as an incompressible ideal medium. In particular, this is possible for concentrated suspension flows in which concentration ^ does not differ much from the value specific to its close-packed state. Thus, macroscopic variations of (j) are relatively insignificant by definition and may be neglected. When these variations are ignored. Equations 6.1 reduce to continuity equations for two co-existing incompressible fluids. However, particulate pressure p, is not necessarily constant, which is due to supposedly strong dependence of this pressure on particle fluctuation temperature and to the fact that a minute variation of (|) can produce a noticeable change in p,. The inconstancy of Pj holds true even at constant temperature because of the strong concentrational dependence of the osmotic pressure function. It is quite evident that there is no need to retain the energy conservation equation in this case because p, replaces T in the capacity of an unknown variable. Again, we get the familiar general scheme for disperse two-phase flows which involves only equations of mass and momentum conservation. However, within the framework of this scheme, (j) is regarded as a known constant quantity, and p, plays the role of a new unknown variable instead of (|). Thus, the next key problem that requires our attention is the formulation of a consistent model of pseudo-turbulence which would offer an opportunity to calculate T*. MODELING PSEUDO-TURBULENCE The general approach in dealing with pseudo-turbulent fluctuations was already outlined in great detail [9,14,23] and thereafter applied to homogeneous fluidized beds [25]. A serious deficiency in the latter analysis [25] lies in the fact that the authors

Fluid Dynamics of Coarse Dispersions

137

took into account only constituent (n)f J^ of the interphase interaction force fluctuation, whereas constituent n^f^) was dropped out of the analysis (see Equation 3.5). The latter constituent can be proven to be of importance, the more so as suspension concentration decreases. Now we are going to account for this missing constituent. Equations to describe random pseudo-turbulent fluctuations have to be derived 1) from fluid mass and momentum conservation laws, and 2) from the Langevin equation for one particle. Taking the fluctuation parts of the mass and momentum conservation equations (the corresponding mean equations were formulated in Section 5) and multiplying the Langevin equation by the particle number concentration, we arrive at the following set of equations governing particle and fluid fluctuations: 0/3t + u • V)(|)' - £V • v' = 0 8p,0/at + u • V)v' = - V p ' - nf; -
(7.1)

(|)paw73t = nf;; -H nf: where the mean and fluctuation forces of interphase interaction are given in Equation 3.3 (or, in an approximate form, in Equation 3.2) and in Equation 3.4, respectively, and the collisional force is defined by Equation 3.1. These equations are formulated in the convective reference frame which moves with the local mean dispersed phase velocity. The convective parts of time derivatives are linearized since the fluctuations are assumed to be relatively weak. Besides, meaning to evaluate T*, we neglect terms containing the gradients and time derivatives of mean flow variables in order to simplify the matter. Set (7.1) consists of two vector and one scalar stochastic equation. This set is helpful in examining statistical properties of two vector (v'and w') and one scalar (p') random unknown variable as functions of 1) the statistical properties of random variable (|)', 2) the macroscopic characteristics of suspension flow, and 3) physical parameters. Since Equations 7.1 are linear, it is natural to use the correlation theory of random processes when investigating these vector and scalar variables in terms of those functions [35]. Particulars of necessary calculation are described at considerable length in reference [14,25]. Here, we confine ourselves to only a brief enumeration of the major logical steps of this calculation. By using representations of the random functions in Equation 7.1 in the form of Fourier-Stieltjes integrals over the whole axis of fluctuation frequency and throughout the entire wave-number space, we are able to transform differential Equations 7.1 into corresponding linear algebraic equations for the random measures which appear in the Fourier-Stieltjes integrals for these random functions. After that, the random measures of v', w'and p'can be expressed as linear functions of the random measure of (|)'. This permits getting expressions for the spectral densities of all the random functions in terms of the random fluctuation spectral density of the suspension volume concentration. In turn, such spectral density expressions make it possible to relate all two-point two-time correlation functions characterizing the pseudo-turbulence to similar correlation functions for concentrational fluctuations. In particular, fluctuation temperature is unequivocally defined by mean square (w'^).

138

Advances in Engineering Fluid Mechanics

which is expressible as a quantity proportional to mean square ((|)'^) having a proportionality coefficient that depends on mean flow variables and physical parameters. To make Equation 7.1 fully determinate, we must: 1) specify functions F((|)) involved in the definition of the interphase interaction force in Equation 3.2; 2) introduce a statistical model for concentrational fluctuations which would allow us to get the spectral density of these fluctuations; and 3) determine the unknown coefficients in the collisional force expression according to Equation 3.1. Many empirical and semi-empirical two-term expressions have been suggested to represent the hydrodynamic drag force experienced by a spherical particle in a concentrated suspension. It seems that the first expression of this kind was presented in the now classical work by Ergun [36]. Since then, many authors repeatedly returned to the problem and proposed other versions of the two-term hydraulic resistance law. One of the latest versions can be found in reference [37]. In the present paper, we will use the two-term law that has been advocated and discussed in reference [25]. According to this law.

' 16

Ka

\-

(7.2)

l - 1 . 1 7 ( t ) 2/3

Here v^ is fluid kinematic viscosity and x is the relaxation time for a single particle in an unbounded fluid. The first (linear) hydrodynamic drag force term in Equation 3.2 with F,((t)) from Equation 7.2 approximately describes the hydraulic resistance of fine particles. This term behaves correctly at low concentrations and does not have singularities in the whole concentration range. The derivation of this term is commented on in more detail in reference [25]. The second (quadratic) term of Equation 3.2 describes the hydraulic resistance of large particles. The expression for ¥ji(^) cited in Equation 7.2 follows from the model of jet flow around large particles in a concentrated disperse system. This expression was derived by Goldstik [38]. The spectral density of concentrational fluctuations in the particulate pseudo-gas can be found by standard means, in a manner quite similar to calculating this quantity for a dense molecular gas. Using classical statistical physical methods in this context is fully justified because the pseudo-gas particles are assumed to be statistically independent and involved in isotropic chaotic motion in just the same way as gas molecules are statistically independent and involved in thermal motion. The only difference between particulate pseudo-gas and molecular gas is the fact that the physical origin of their fluctuations is different. However, this difference is of no consequence when calculating all thermodynamical functions for either a particulate pseudo-gas or a molecular gas as long as the fluctuations can be completely and unambiguously described with the help of the only scalar quantity— the fluctuation temperature. As a result, all thermodynamic formulae derived for dense molecular gases may be applied to pseudo-gases as well. [39]. In particular, variance ((j)'^) can be expressed in terms of the chemical potential derivative for a gas composed of hard spheres over the sphere number concentration.

Fluid Dynamics of Coarse Dispersions

139

as has been done in reference [25]. If the approximate Enskog model [17,18] of dense gases is used, we get

1 +1

1/3

(m*) 3\-(m.y'

1

(7.3)

and if the Carnahan-Starling model [26], we instead obtain

1 + 2(|)

(r) = ^

4-(t)

(7.4)

(1-^r

It is not difficult to also derive formulae for the mean square of concentrational fluctuations on the basis of any other statistical model proposed for dense gases and liquids (for a review, see reference [18,40,41]). When calculating the concentrational fluctuation spectral density, we must take into account that: 1) fluctuations are isotropic so that spectral density can depend on the modulus but not on the direction of the wave-number vector; 2) particles are solid spheres of finite volume; and 3) decay of spontaneously originated fluctuations is governed by particle diffusion. As a result, we obtain the following formula:

\|/^^(co,k) =

Dk^ OH(k^ - k) n co'+(Dk')'

Nl/3

O =

47ik:

•,

k„ =

2jt(|) j

(7.5)

Here, co and k are the fluctuation frequency and wave-number, respectively, H(x) designates the Heaviside step function, D is the particle self-diffusion coefficient identified in Equation 4.5 and Equation 4.7, and k^ is understood as the maximal wave number possible in a dispersion containing spherical particles of radius a, particle volume concentration being equal to (j). The derivation of this formula has been discussed in some Russian papers, and in particular, in reference [14]. We are not in a position to spend time on this derivation in this article, and so list Equation 7.5 without further comments. This may be justified to an extent by the fact that only integral ^^^^^^ = f ¥,,,(co,k)dco = OH(k, - k)

(7.6)

is actually needed for calculations germane to this paper. Function O (k) represents the partial spectral density of concentrational fluctuations. This function is defined only in the wave-number space and is independent of fluctuation frequency. Equation 7.6 for this partial spectral density can be easily derived if we require that it must be isotropic and that it must yield mean square ((|)'^) when being integrated over the whole wave-number space. For point-like particles of zero volume, this partial density is independent of k, and correspondingly k^ tends to

140

Advances in Engineering Fluid Mechanics

infinity as a comes to zero. Introducing the maximal value of k in Equation 7.5 and Equation 7.6 corresponds to cutting off the wave-number spectrum for concentrational fluctuations in a pseudo-gas composed of particles having a finite volume. Using all the equations listed above enables us to get the spectral densities for all fluctuating quantities, and further, to calculate all desired averages and correlatijan functions. These densities, averages, and correlation functions will be dependent on two coefficients, A and B, that have been introduced into Equation 3.1 to represent the collisional force, and which so far remain unknown. We must determine these coefficients from two supplementary conditions. Under the present circumstances, the meaning of these two conditions appears to be quite evident. As in reference [25], we must require, first of all, that the rate of fluctuation energy dissipation due to collisions is actually equal to the dissipated power identified in Equation 4.11. Secondly, we must require particle pseudo-turbulent fluctuations to actually be isotropic. The equations corresponding to these requirements can be formally presented as follows: (f^ • w') = m[A(w'') -h B<w;' >] = (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

Advances in Engineering Fluid Mechanics

# , [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.

Fluid Dynamics of Coarse Dispersions

_

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

Advances in Engineering Fluid Mechanics

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.

Fluid Dynamics of Coarse Dispersions

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

Advances in Engineering Fluid Mechanics

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

Fluid Dynamics of Coarse Dispersions

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

Advances in Engineering Fluid Mechanics

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

Fluid Dynamics of Coarse Dispersions

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

Advances in Engineering Fluid Mechanics

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^.^aeVg, 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)

Fluid Dynamics of Coarse Dispersions

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

Advances in Engineering Fluid Mechanics

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.

Fluid Dynamics of Coarse Dispersions

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

Advances in Engineering Fluid Mechanics

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= -

^i

I \4

where U is the bubble rise velocity.

^

u,

u = |vp:

(9.4)

Fluid Dynamics of Coarse Dispersions

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.

Fluid Dynamics of Coarse Dispersions

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)) + ¥,.{<^)n.]VL. - {m./p,.)(epo + (^.p,, + ^29i2) + 8f j (|) = (|),+(t)2,

U j = u + u;

(9.7)

where mean fluid interstitial velocity u is identical for all the particles. Coefficients Fj((t)) and F^i^) are expressed in accordance with Equation 7.2 in terms of physical parameters a, K , and x associated with j-th type particles and of universal functions Kj((|)) and K^Ccj)), which are the same for particles of both the types. The buoyancy force involved in Equation 9.7 depends on the effective mean local density of the two-phase medium that surrounds the particle under question. The mean force and

158

Advances in Engineering Fluid Mechanics

the force fluctuation attributed to a particle of each type can be found from Equation 9.7 in the same manner as before. Statistical properties of particle fluctuations are not identical for particles of different types. However, this difficulty can be avoided by assuming the particles to have the same fluctuation energy (or temperature) and by introducing a new random process, W , with conformity to the relation w; = «m)/m^)'" W ,

<m> = (j)"' (^,m, + ^,m,)

(9.8)

After that, the Langevin equation for a single particle can be averaged over all the particles to yield an equation for W . This equation resembles the last Equation 7.1. It supplements the fluctuation equations resulting from the mass and momentum conservation laws for the ambient fluid. These equations have to be treated exactly in the same way as in Sections 7 and 8. As a result, we arrive at a representation for fluctuation temperature in a binary fluidized bed and to expressions for the pressures and stresses associated with particles of both species [76]. In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. Momentum conservation equations for particles of these types can again be formulated by analogy with similar equations for gaseous mixtures [16,17]. These equations completely determine partial volume concentrations, (j), and ^^, as functions of the vertical coordinate which additionally depend on the fluidizing fluid flow rate and on the overall amount and composition of the fluidized particulate mixture as well as on physical parameters. Relevant solutions to the partial momentum conservation equations have been analyzed and discussed at considerable length in reference [76]. These solutions have been proven to successfully elucidate various effects and phenomena characteristic of binary fluidization which earlier remained unexplained. The obtained particle distribution profiles over fluidized bed heights also have been shown to be in good qualitative agreement with experimental observations [76]. Because the equations under discussion are rather unwieldy, and a comprehensive detailed analysis of their solutions is certainly tedious, we are not going to reproduce

Fluid Dynamics of Coarse Dispersions

159

these equations and solutions in the present article. This notwithstanding, we include examples comparing the theoretically predicted particle distributions with some experimental data obtained in reference [73]. The experimental dots presented in Figure 7 were produced by Juma and Richardson, who fluidized binary mixtures

^O- 0.2 L

0.30

• ^ 0.15 \-

h,crr\ Figure 7, Comparison of theoretical particle distribution over the fluidized bed height (solid curves) with experimental data in reference [73] elicited while fluidizlng binary glass ball mixtures (diameters 0.2 and 0.4 cm) by paraphine oil (v^ = 0.1 cm^/s, u^= 7.3 cm/s) and water (v^ = 0.01 cm^/s, u^ = 25.4 cm/s) (a and b respectively); dashed curves describe partial volume concentrations (|)^ and (^^.

160

Advances in Engineering Fluid Mechanics

of glass balls of the same density but of different diameters by upward water flows [73]. Since the theoretical curves in Figure 7 are calculated for the conditions of these experiments without using any adjustable parameters, the agreement between theory and experimental evidence can be looked upon as convincingly good. By necessity, this application of the developed fluid dynamic model to a few concrete coarse dispersion flow problems is somewhat fragmentary. We have been able to merely outline the general approach needed to treat these problems and to set forth only selective examples illustrating the correspondence of model expectations with experimental evidence. This is hardly surprising because each of the discussed problems appears to be rather formidable by itself and would undoubtedly require a good deal of reasoning and calculating even provided that a reliable adequate hydrodynamic scheme to attack the problem were established and welldeveloped. Each problem has been earlier addressed by many researchers, and it is seemingly just the lack of such a scheme that prevented those researchers from meeting with appreciable success in their endeavors. Given this contingency, the mere fact that predictions based on the developed model agree well with experimental observations as regards all addressed problems proves the developed model to be adequate for the study of coarse dispersion flows. This general conclusion also is confirmed by comparison of model inferences pertaining to these problems with some other relevant experimental data which we have been unable to consider here because of lack of space. On the whole, the model advanced in this article may justifiably be regarded as a sound first approximation to future more sophisticated and ingenious theories of coarse dispersion flows at various conditions. CONCLUSIONS The presented analysis shows disperse flow modeling to be a very difficult task due to a number of reasons. The most important reason lies in the fact that this task belongs to a marginal region of applied science that borders on different fields of knowledge. Consequently, methods and concepts specific to all these fields have to be simultaneously employed to ensure substantial progress in providing for a sound understanding of the underlying physics and in providing for an adequate description of disperse flows. The problem of modeling disperse flows is greatly simplified for particular classes of dispersions. The coarse dispersions characterized by the collisional mechanism of interparticle exchange that have been addressed in the preceding consideration belong to one such particular class. This fundamental feature of collisional interparticle exchange in coarse dispersions gives an opportunity to overlook the particulars of multiparticle hydrodynamic interactions for the suspended particles. As a result, the dispersed phase of a collisional suspension may be considered as a pseudo-gas of hard particles undergoing random fluctuations. Fluid dynamic modeling of suspension flow can then be performed along the familiar lines of the kinetic theory of dense gases. Even so, application of kinetic theory methods to a detailed description of interparticle collisions meets in the general case with serious difficulties that have no direct analogies in the conventional kinetic theory. These difficulties have three

Fluid Dynamics of Coarse Dispersions

161

major causes: 1) The collisions are inelastic and the colliding particles possess rough surfaces, which makes inevitable an analysis of energy exchange between translational and rotational degrees of freedom; 2) particle interaction with the ambient fluid, however negligible in conditioning interparticle interactions, is of paramount significance in providing energy input to particle fluctuations from mean disperse flow; and 3) chaotic particle fluctuating motion is not necessarily isotropic. These circumstances demonstrate that fluctuating motion cannot be completely described by introducing a single scalar temperature. Introduction of different temperatures that are attributed to particle translations and rotations in different directions is generally needed, and, moreover, neither of these temperatures may be regarded as a a quantity known in advance. The severe implications of these facts have been partially uncovered in reference [5] as a result of formulating a kinetic theory for granular flow without interaction with the ambient medium. These implications, as well as additional difficulties due to the necessity to calculate the energy supply to the particle fluctuations, make somewhat problematic, at the present state of the art, the formulation of a reliable and sufficiently simple hydrodynamic model even for coarse dispersions. We have succeeded in this respect only at the expense of making certain supplementary assumptions. These assumptions are: 1. particle fluctuations are almost isotropic, and the energy equipartition principle holds approximately true with respect to particle random translations; 2. excitation of particle rotational degrees of freedom is negligible, and particle random rotations may be ignored; and 3. energy losses caused by inelasticity of collisions are inessential (this inelasticity is due to inner viscous effects in the particle material, to external viscous effects accompanying stepwise changes in particle velocities at collisions, and to particle surface roughness). Attempts to generalize the developed model to dispersions for which these assumptions are not satisfied pose a number of tempting new problems. Some of these problems can be successfully solved without much ado. For instance, it is not difficult to allow for the effect of collision inelasticity on the properties of pseudo-turbulent motion by means of replacing simple Equation 8.1 by other equations in which collisional energy dissipation is duly taken into account, as has been previously done in reference [25]. However, the repudiation of other assumptions is by no means a simple matter and requires a great deal of work. Fortunately, this work seems to be much facilitated by the mere fact that there exists a sound tentative model which plays the role of a certain initial approximation. It is the formulation of just such a model that should be regarded as a main achievement of the present article. NOTATION A, B Unknown coefficients introduced into Equation 3.1 for the collisional force

a Spherical particle radius c Mean velocity defined in terms of mean volume flux. Equation 8.5

162

Advances in Engineering Fluid Mechanics

D Particle diffusivity E^ Strain rate tensor for dispersed phase flow F Functions introduced into hydraulic resistance law, Equation 3.2 f Collisional force c

f^ Interphase interaction force per particle G Osmotic pressure correction function g Acceleration of extrenal body forces K Universal functions introduced in Equation 7.2 k Wave number k Collisional dissipation coefficient

Particle mass Particle number concentration Stress tensor Pressure Flux of particle fluctuation energy due to fluctuations Energy source and sinks introduced in Section 4 Fluctuation temperature defined by Equation 2.1 U Fluidization bubble rise velocity u Relative mean fluid velocity w Mean fluid interstitial velocity and mean particle velocity Vi,, Sc Dimensionless criteria introduced in Equation 9.2

Greek Symbols a Coefficients in Section 4 £ Void fraction r|j Fluctuation energy transfer coefficient K Density ratio X Coefficients in Equation 3.3 ji, Dispersed phase dynamic viscosity

V Collision frequency VQ Fluid kinematic viscosity p Density T Particle relaxation time X Enskog factor "F, O Spectral density functions ^ Particle volume concentration CO Frequency

REFERENCES 1. Jenkins, J. T. and Savage, S.R. "A theory for rapid flow of identical, smooth, nearly elastic spherical particles." J. Fluid Mech. 130, 187-202 (1983). 2. Lun, C. K. K., Savage, S. B., Jeffery, D. J. and Chepurny, N. "Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field." /. Fluid Mech, 140, 223-256 (1984). 3. Jenkins, J. T. and Richman, M. W. "Grad's 13-moment system for a dense gas of inelastic spheres." Arch. Rat. Mech. Anal. 87, 355-377 (1985). 4. Lun, C. K. K. "Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres." J. Fluid Mech. 233, 539-559 (1991). 5. Goldstein, A. and Shapiro, M. "Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations." J. Fluid Mech. 282, 75-114 (1995). 6. Eckstein, E. C , Bailey, D. G. and Shapiro, A. H. Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79, 191-208 (1977).

Fluid Dynamics of Coarse Dispersions

163

7. Leighton, D. and Acrivos, A. "Measurement of shear-induced self-diffusion in concentrated suspensions of spheres." J. Fluid Mech. 177, 109-131 (1987). 8. Buyevich, Y. A. "An approximate statistical theory of a fluidized bed." Zh. PrikL Mekh. Tekhn. Fiz. no 6, 35-47 (1966) (in Russian). 9. Buyevich, Y. A. "Statistical hydrodynamics of disperse systems. Part 1. Physical background and general equations." J. Fluid Mech. 49, 489-507 (1971). 10. Koch, D. L. "Kinetic theory for a monodisperse gas-solid suspension." Phys. Fluids A2, 1,711-1,723 (1990). 11. Brady, J. F. "Brownian motion, hydrodynamics, and the osmotic pressure." J. Chem. Phys. 98, 3,335-3,341 (1993). 12. Brady, J. F. "The rheological behavior of concentrated colloidal dispersions." 7. Chem. Phys. 99, 567-581 (1993). 13. Nott, P. R. and Brady, J. F. "Pressure-driven flow of suspensions: simulation and theory." /. Fluid Mech. 275, 157-199 (1994). 14. Buyevich, Y. A. "Internal pulsations in flows of finely dispersed suspensions." Izv. Ross. Akad. Nauk, Mekh. Zhidkosti I Gaza no. 3, 91-100 (1993) (in Russian). 15. Buyevich, Y. A. and Kapbasov, S. K. "Stability of finely dispersed vertical flows." Izv. Ross. Akad. Nauk, Mekh. Zhidkosti I Gaza no. 6, 57-66 (1993) (in Russian). 16. Chapman, S. and Cowling, T. G. The Mathematical Theory of Non-Uniform Gases. Cambridge Univ. Press, Cambridge, 1952. 17. Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B. Molecular Theory of Gases and Liquids. Wiley, New York and Chapman, London, 1954. 18. Resibois, P. M. V. and de Leener, M. Classical Kinetic Theory of Fluids. Wiley—Interscience, New York, 1977. 19. Buyevich, Y. A., Bologa, M. K., Syutkin, S. V. and Tetyukhin, V. V. "Particle motion associated with magnetofluidization in an alternate field." Magn. Gidrodin. no. 3, 12 (1985). 20. Jackson, R. The mechanics of fluidized beds. I. "The stability of the state of uniform fluidization." Trans. Instn Chem. Engrs 4 1 , 13-21 (1963). 21. Goldstik, M. A. and Kozlov, B. N. "Elementary theory of dense disperse systems." Zh. Prikl. Mekh. Tekhn. Fiz. no. 4, 67-77 (1973) (in Russian). 22. Jenkins, J. T. and McTigue, D. F. "Transport properties in concentrated suspensions: The role of particle fluctuations." in: Two-Phase Flows and Waves, Springer, New York, 1990. 23. Buyevich, Y. A. "Fluid dynamics of coarse dispersions." Chem. Engng Sci. 49, 1,217-1,228 (1994). 24. Buyevich, Y. A. "Interphase interaction in fine suspension flow." Chem. Engng Sci. 50, 641-650 (1995). 25. Buyevich, Y. A. and Kapbasov, S. K. "Random fluctuations in a fluidized bed." Chem. Engng Sci. 49, 1,229-1,243 (1994). 26. Carnahan, N. F. and Starling, K. E. "Equation of state for non-attracting rigid spheres." J. Chem. Phys. 51, 635-637 (1969). 27. Muser, H. E. and Petersson, J. "Thermodynamic theory of relaxational phenomena." Fortsch. Phys. 19, 559-612 (1971).

164

Advances in Engineering Fluid Mechanics

28. Buyevich, Y. A. and Yasnikov, G. P. "Relaxational methods in studies of transfer phenomena." Inzh.-Fiz. Zh. 44, 489-504 (1983) (in Russian). 29. Murray, J. D. "On the mathematics of fluidization. Part 1. Fundamental equations and wave propagation." J. Fluid Mech. 21, 465-493 (1965). 30. Anderson, T. B. and Jackson, R. "A fluid mechanical description of fluidized beds." Ind. Engng Chem. Fund. 6, 527-539 (1967). 31. Drew, D. and Segel, A. "Averaged equations for two-phase flow." Stud. Appl. Math. 50, 205-231 (1971). 32. Homsy, G. M., El-Kaissy, M. M. and Didwania, A. K. "Instability waves and the origin of bubbles in fluidized beds. Part II. Comparison with theory." Int. J. Multiphase Flow 6, 305-318 (1980). 33. Homsy, G. M. "A survey of some results in the mathematical theory of fluidization." in: Theory of Dispersed Multiphase Flow. Academic Press, New York, 1983. 34. Batchelor, G. K. "A new theory of instability of a uniform fluidized bed." J. Fluid Mech. 193, 75-110 (1988). 35. Yaglom, A. M. "Introduction to the theory of stationary random processes." Uspekhi Matem. Nauk 7, 3-168 (1952). 36. Ergun, S. "Fluid flow through packed columns." Chem. Engng Progr. 48, 89-94 (1952). 37. Gibilaro, L. G., di Felice, R., Waldram, S. P. and Foscolo, P. U. "Generalized friction factor and drag coefficient correlations for fluid-particle interactions." Chem. Engng Sci. 40, 1,817-1,823 (1985). 38. Goldstik, M. A. "Elementary theory of the fluidized bed." Zh. Prikl. Mekh. Tekhn. Fiz. no. 6, 106-112 (1972) (in Russian). 39. Landau, L. D. and Lifshitz, E. M. Statistical Physics. Pergamon Press, Oxford, 1968. 40. Croxton, C. A. Liquid State Physics—A Statistical Mechanical Introduction. Cambridge Univ. Press, Cambridge, 1974. 41. Balescu, R. Equilibrium and Nonequilibrium Statistical Mechanics. Vol. 1. Wiley-Interscience, New York, 1975. 42. Carlos, N. F. and Richardson, J. F. "Solids movements in liquid fluidized beds. 1. Particle velocity distribution." Chem. Engng Sci. 23, 813-824 (1968). 43. Anderson, T. B. and Jackson, R. "A fluid mechanical description of fluidized beds. Stability of the state of uniform fluidization." Ind. Engng Chem. Fund. 7, 12-21 (1968). 44. Jackson, R. "Hydrodynamic stability of fluid-particle systems." in: Fluidization. Academic Press, New York, 1985. 45. Buscull, R., Goodwin, J. F., Ottewill, R. H. and Tadros, T. F. "The settling of particles through Newtonian and Non-Newtonian media." J. Coll. Interface Sci. 85, 78-86 (1982). 46. Ham, J. M. and Homsy, G. M. "Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions." Int. J. Multiphase Flow 14, 533-546 (1988). 47. Buyevich, Y. A. "Statistical hydromechanics of disperse systems. Part 3. Pseudo-turbulent structure of homogeneous suspensions." /. Fluid Mech. 56, 313-336 (1972).

Fluid Dynamics of Coarse Dispersions

165

48. Kynch, G. I. "A theory of sedimentation." Trans. Faraday Soc. 48, 166-176 (1952). 49. Pigford, R. L. and Baron, T. "Hydrodynamic stability of a fluidized bed." Ind. Engng Chem. Fund. 4, 81-87 (1965). 50. Molerus, O. "Hydrodynamische Stabilitat des FlieBbetten." Chem.-Ing.-Techn. 39, 341-348 (1967). 51. Verloop, J. and Heertjes, P. M. "Shock waves as a criterion for the transition from homogeneous to heterogeneous fluidization." Chem. Engn Sci. 25, 825-832 (1970). 52. Gard, S. K. and Pritchett, J. W. "Dynamics of gas-fluidized beds." J. Appl. Phys. 46, 4,493-4,500 (1975). 53. Fanucci, J. B, Ness, N., and Yen, R. H. "On the formation of bubbles in gasparticulate fluidized beds." /. Fluid Mech., 94, 353-367 (1979). 54. Didwania, A. K. and Homsy, G. M. "Resonant sideband instabilities in wave propagation in fluidized beds." J. Fluid Mech. 122, 433-438 (1982). 55. Needham, D. J. and Merkin, J. H. "The propagation of voidage disturbance in a uniformly fluidized bed." J. Fluid Mech. 131, 427-454 (1983). 56. Foscolo, P. U. and Gibilaro, L. G. "Fluid dynamic stability of fluidized suspensions: the particle bed model." Chem. Engng Sci. 42, 1,489-1,500 (1987). 57. El-Kaissy, M. M. and Homsy, G. M. "Instability waves and the origin of bubbles in fluidized beds. Part I. Experiments." Int. J. Multiphase Flow 2, 379-395 (1976). 58. Didwania, A. K. and Homsy, G. M. "Flow regimes and flow transitions in liquid fluidized beds." Int. J. Multiphase Flow 7, 563-570 (1981). 59. Ham, J. M. Thomas, S., Guazelli, E. , Homsy, G. M., and Anselmet, M. C. "An experimental study of the stability of liquid fluidized beds." Int. J. Multiphase Flow 16, 171-185 (1990). 60. Buyevich, Y. A., Kapbasov, S. K. and Makarov, A. V. "Stability of a finely dispersed vertical flow of high concentration." Izv. Ross. Akad. Nauk, Mekh. Zhidkosti i Gaza no. 4, 87-96 (1994) (in Russian). 61. Yates, J. G. and Cheesman, D. J. "Voidage variations in the regions surrounding a rising bubble in a fluidized bed." AIChE Symp. Ser. 88, 34-39 (1992). 62. Yates, J. G., Cheesman, D. J. and Sergeev, Y. A. "Experimental observations of voidage distribution around bubbles in a fluidized bed." Chem. Engng Sci. 49, 1,885-1,895 (1994). 63. Lockett, M. J. and Harrison, D. "The distribution of voidage fraction near bubbles rising in gas-fluidized beds." Proc. Int. Symp. on Fluidization, Drinkenberg. Netherland Univ. Press, Amsterdam, 1967. 64. Fan , Z., Chen, G. T., Chen, B. C. and Yuan, H. "Analysis of pressure fluctuations in a 2-D fluidized bed." Powder Techn. 62, 139-145 (1990). 65. Davidson, J. F. and Harrison, D. Fluidized Particles. Cambridge Univ. Press, Cambridge, 1963. 66. Collins, R. "A model for the effects of the voidage distribution around a fluidization bubble." Chem. Engng Sci. 44, 1,481-1,487 (1989). 67. Quassim, R. Y., Kinrys, S. and Beneviste, D. E. "Effect of voidage variation on flow past a fluidization bubble." Chem. Engng Sci. 44, 1,307-1,313 (1989). 68. Buyevich, Y. A., Yates, J. G., Cheesman, D. J. and Wu, K. T. "A model for the voidage around bubbles in a fluidized bed." Chem. Engng Sci. 50, 3,155-3,162 (1995).

166

Advances in Engineering Fluid Mechanics

69. Pruden, B. B. and Epstein, N. "Liquid fluidization of binary particle mixtures." Chem. Engng Sci. 14, 696-704 (1964). 70. Al-Dibouni, M. R. and Carside, J. "Particle mixing and classification in liquid fluidized beds." Trans. Instn Chem. Engrs 57, 94-103 (1979). 71. Epstein, N., Leclair, B. P. and Pruden, B. B. "Liquid fluidization of binary particle mixtures-L Overall bed expansion." Chem. Engng Sci. 36, 1,803-1,809 (1981). 72. Van Duijn, G. and Rietema, K. "Segregation of liquid-fluidized solids." Chem. Engng Sci. 37, 727-733 (1982). 73. Juma, A. K. A. and Richardson, J. F. "Segregation and mixing in liquid fluidized beds." Chem. Engng Sci. 38, 955-967 (1983). 74. Moritomi, H., Yamagishi, T. and Chiba, T. "Prediction of complete mixing of liquid-fluidized binary solid particles." Chem. Engng Sci. 41, 297-305 (1986). 75. Gibilaro, L. G., di Felice, R. and Waldram, S. P. "A predictive model for the equilibrium composition and inversion of binary-solid liquid fluidized beds." Chem. Engng Sci. 41, 379-387 (1986). 76. Buyevich Y. A. and Kapbasov, S. K. "Polydisperse fluidization.-I. Basic equations." Chem. Engng Sci. 49, 1,245-1,257 (1994). 77. Mansoori, G. A. "Equilibrium thermodynamic properties of the mixture of hard spheres." J. Chem. Phys. 54, 1,523-1,525 (1971).

CHAPTER 8 COMBUSTION OF SINGLE COAL PARTICLES IN TURBULENT FLUIDIZED BEDS Prabir Kumar Haider Department of Power Plant Engineering Jadavpur University Calcutta - 700091, India CONTENTS INTRODUCTION, 167 TURBULENT FLUIDIZATION AND ITS CHARACTERISTICS, 169 Regime Transition From Bubbling to Turbulent Bed, 169 Characteristics of Turbulent Bed, 172 DEVOLATILIZATION AND VOLATILE COMBUSTION, 172 COMBUSTION OF CHAR, 173 Site of Combustion Reaction and the Effective Surface Product, 174 Changes in Particle Size and Density, 175 Mass Transfer, 179 Chemical Rate, 181 Particle Temperature, 181 Burning Rates, 183 Rate-Controlling Mechanism, 185 CONCLUSIONS AND RECOMMENDATIONS, 186 NOTATION, 188 REFERENCES, 189 INTRODUCTION A classical bubbling fluidized bed operates in a low velocity regime where discrete bubbles are present. Bubbles cause severe bypassing and back mixing of the fluidizing gas. Bypassing of gas leads to inefficient contact between gas and solid, although bubbles cause solid movement that increases the rate of heat transfer. Though the conventional bubbling fluidized bed technology was widely used in many chemical processes and for coal combustion in boilers, it suffered from certain limitations that did not allow its scaling up beyond a certain size. The growing need to burn low-quality coal and to meet stringent environmental regulations in utility boilers have forced scientists and technologists to find a viable 167

168

Advances in Engineering Fluid Mechanics

alternative to the conventional bubbling fluidized bed boilers. Thus, there emerged the circulating fluidized bed boilers, often known as the fast fluidized bed boiler, which operate in a completely different regime of fluidization. Consider the circulating fluidized bed (CFB) boiler. Figure 1 where typical operating conditions might be 3-5 m/s up to the secondary air injection level and 5-8 m/s above that level at a bed temperature of 1,173 K. Coal with a top size of 6 mm is fed at the bottom of the bed consisting of coal-ash particles of mean diameter 0.3 mm. Entrained particles are collected in a cyclone and recycled to the combustor via a non-mechanical return valve. Part of the heat released from the carbon burning in the bed is extracted by water wall tubes placed in the furnace. The remaining heat leaves the furnace as sensible heat in the flue gas. The bottom section of a circulating fluidized bed below the secondary air level works in the regime of turbulent fluidization, the characteristics of which is described later.

HOT

WATER/ CYCLONE

STEAM

FURNACE

WATER SEC.AIR LIMESTONE FUEL

[

O >^/^ NON MECHANICAL VALVE

Figure 1. Schematic of a typical CFB boiler.

Combustion of Single Coal Particles in Turbulent Fluidized Beds

169

The gas velocity is considerably higher than the terminal velocity of the individual bed particles. Two-stage combustion of coal takes place in a CFB furnace. Depending on the design, the bottom section will either work as a gasifier or combustor. A quantitative understanding of the process of coal combustion in a turbulent fluidized bed is essential for optimum use of this process. Elaborate research work by Basu and Subbarao [1] and Haider et al [2] describe the process of combustion of coal char in turbulent fluidized beds. A coal particle contains varying proportions of moisture, volatile matter, fixed carbon and ash. When the particle is introduced into a turbulent fluidized bed operating at 1,173 K, it is rapidly heated to bed temperature by the surrounding inert particles. Moisture is released first followed by volatile matter. After releasing the moisture, the coal particle decomposes to produce a hydrogen-rich gas (volatile matter) and carbon-rich residue (char). Above about 750 K, volatile matter is evolved and, depending upon the size of the particle and bed temperature, the volatile takes 1-10 seconds to be released from the coal. Volatile burns much faster ( <1 sec) leaving char behind. Combustion of char is a slow process and controls the overall combustion mechanism. It controls the carbon inventory, size distribution, and combustion efficiency. Therefore, the study of char combustion is essential for understanding of the overall process of combustion in a turbulent fluidized bed. In a turbulent bed the char-burning rate is dictated by its kinetic rate of combustion, and the mass transfer rate of oxygen from the bulk of the bed to the particle surface. In parallel with combustion, a small fraction of carbon undergoes attrition from the carbon surface. In a comprehensive study of combustion in turbulent fluidized beds, account must be taken of all of these interacting mechanisms. The text that follows outlines what is known and what needs to be understood about the fundamentals of the combustion of coal and carbon particle in turbulent fluidized beds. Since combustion is affected by the hydrodynamic behavior of the bed, certain aspects of hydrodynamics are described very briefly. The aim of the review is to summarize the present understanding of the basic mechanism of coal or carbon in turbulent fluidized beds. Devolatilization also is described briefly. TURBULENT FLUIDIZATION AND ITS CHARACTERISTICS Regime Transition from Bubbling to Turbulent Bed The fluidizing gas velocity at which the structure of the bed changes from particulate or homogeneous state to the heterogeneous bubbling state is given by the minimum bubbling velocity (U^^^). In this regime of fluidization the excess gas fed into the bed flows in the form of bubbles. With the increase in gas flow rate the frequency as well as the size of the bubbles begin to grow up. The growth of bubbles occur due to the coalescence as the bubbles rise up. Ultimately, a maximum stable bubble size is reached. The heterogeneity in the bubbling bed causes a fluctuation in the pressure drop across the bed. With increase in the superficial gas velocity this fluctuation increases and reaches a peak at a velocity U^, beyond which it begins to decrease (Figure 2). At a velocity U^, the pressure fluctuation drops to a minimum and thereafter remains almost steady. The velocity U^, marks the onset of the turbulent fluidization, and the transition becomes complete at the velocity U^.

170

Advances in Engineering Fluid Mechanics

Superficial gas velocity Figure 2. Amplitude of pressure fluctuation vs. velocity for transition to turbulent bed. According to Grace [3] transition to turbulent fluidization occurs at a velocity given by, U^ = 3.0 (p^d)'^2_ 0.17

(1)

U. = 7.0 (p d)'^2 _ 0.77

(2)

Yerushalmi and Cankurt considered the transport velocity (UJ as the boundary between the turbulent and the fast fluidization [4]. The transition takes place gradually through a turbulent state where both voids and clusters coexist. During fast fluidization a dense phase exists at the bottom of the bed while a lean phase exists at the top. Figure 3 summarizes all the regimes of a fluidized bed and shows the transition from one to another. The structure of the bed during various regimes are shown in Figure 4.

Combustion of Single Coal Particles in Turbulent Fluidized Beds

Dilute Phase

]

171

N.

A

Dense Phase Slugging

Dense Phase Non Slugging

> Transport State

ii

"

Utr

\ Turbulent Bed ii

>Captive State

J Bubbling Bed

/

!

Umf

Fixed Bed

Figure 3. Fluidlzation transition diagram from fixed bed to dilute phase pneumatic transport.

-L?^^

gas

gas

gas Fixed

bed

Turbulent

\\gas Circulating

Pneumatic Transport

Figure 4. Structure of the bed at different tluldizing conditions.

172

Advances in Engineering Fluid Mechanics

Characteristics of Turbulent Bed With the increase in gas velocity as the pressure fluctuation across a bubbling bed drops down to a minimum and remains steady, the bed transforms into a turbulent one. At this higher superficial velocity, the bed expands and the wall of the bubble becomes thin. Finally, the bubbles split and disperse throughout the bed, giving it a more homogeneous character. However, the solids in the bed also disperse to form clusters or streamers which move in the form of agglomerates. In the turbulent bed the upper surface becomes more diffuse. Individual particles and small clusters are entrained into the lean phase at the freeboard while the large clusters move downward giving rise to back mixing in the bed. If the freeboard height is kept sufficiently high, the entrainment can be reduced to a small amount. Thus, a turbulent bed operates in the captive region. Turbulent bed is characterized by two velocities U^ and U^^. U^ strongly depends on the solid density (p^) and the volume surface mean diameter of the solid particle (d). The value of U^ may exceed the terminal velocity of single solid particle in the bed. Still, as the solids form clusters the entrainment from the bed remains insignificant. Yerushalmi and Cankurt observed that (U^/U^) ratio is high for finer solids and may approach values as high as 10 [4]. However, they reported that for larger particles the ratio assumes a low value even lower than 1. This suggests that the tendency of cluster formation reduces for larger particles. Like a bubbling bed, the bed density in a turbulent bed remains independent of the solid flow rate and changes with gas velocity only. The slip velocity (i.e., the relative velocity between gas and solid) also remains virtually unchanged with the change in solid flow rate. Increase in mass flow rate enables a rise in height of the dense bed. DEVOLATILIZATION AND VOLATILE COMBUSTION Volatile can account for up to 40% of the total heat release. Devolatilization involves release of tar, CO2, and other hydrocarbons. The rate of volatile release depends upon the temperature of the bed, heating rate, pressure, water vapor partial pressure, and the particle size. Anthony et al. derived a model of devolatilization which demonstrated a wide range of applicability [5]. The model is based on the assumption that thermal decomposition occurs as a result of a large number of simultaneous, independent, and irreversible first order reactions by which the different organic species present in coal are converted into volatile. Each reaction rate expression takes the form dV - ^ = k,(V°-V,)

(3)

where V. is the mass of volatile per unit mass of original coal evolved up to time t and V° is the value of V at t = a. k is the reaction rate constant of the Arrhenius form, k^ = k, exp[-E/RT]

(4)

Combustion of Single Coal Particles in Turbulent Fluidized Beds

173

Integration of Equation 3 when T varies with time gives, V;^ - V. = V;' expf-£k.dt]

(5)

It is assumed that all reaction rate constants have the same frequency factor k^ and that a large enough number of reactions occur to enable one to express the fraction of V^ evolved due to reactions having a particular activation energy, E, as a continuous function f(E). For an activation energy lying between E and E + dE, the amount of volatile release may be written as, jyo ^ yo f(E) dE

(6)

where, |;(E)dE = 1

(7)

Integration of Equation 5 taking f(E) to be Gaussian with mean activation energy EQ and standard deviation o gives volatile, V, released over a time period, t. ( V - VyVo = l/aV27r)J%xpr-ko£exp[-E/RT]dt - (E - Eo)72o3'jdE (8) Borghi et al. [6], By water [7], and Wells et al. [8] applied the model proposed by Anthony et al. [5] to large coal sizes in fluidized bed combustion models. Predictions from these models suggest that the devolatilization rate is independent of particle size. This conclusion should not, however, be applied to typical fluidized beds because the results are based on devolatilization of 0.2 mm particles while much larger coal particles are fed to commercial fluidized beds. From data on devolatilization of various coal types in the absence of combustion. La Nauze concluded that the devolatilization time is proportional to d^ [9]. He ascribed this to an internal diffusion controlled evolution mechanism. Experimental studies on the devolatilization of coal in a fluidized bed [10] suggest that the devolatilization time, measured as the time between the ignition and extinction of volatile, may be correlated as, t = k d^ V

V

p

(9) ^

''

when c varies from 0.32 to 1.76 for 1,048 K < TB < 1,283 K. The index, c, also varies with coal type. There appears to be a lack of theoretical explanation for the values of k^ obtained experimentally, and its dependence on the type of coal and operating conditions. COMBUSTION OF CHAR As mentioned before, char combustion is a slow process and controls the overall mechanism of combustion. For this reason, much attention was paid to char combustion in a fluid bed. The following questions are relevant to the combustion mechanism:

174

Advances in Engineering Fluid Mechanics

• Does oxygen reach the particle surface? If so, what is the product of combustion? • Where does the homogeneous gas phase reaction take place? • Does the combustion occur on the particle surface (diminishing size model) or only on pore walls (diminishing density model), or both? • Is the burning diffusion controlled or kinetic controlled? • What is the influence of the heat transfer rate from the burning particle to the fluidized bed? How large is the difference between the temperature of burning particle and the temperature of the fluidized bed, and what is its effect on the combustion rate? • How do carbon particle size and fluidized bed conditions (T^^, d , U ) influence the combustion rate? Site of Combustion Reaction and the Effective Surface Product Avedesian and Davidson proposed a two-film model where all reactions were considered to be fast [11]. The CO produced at the carbon surface diffuses away from the carbon surface and reacts with the incoming oxygen in the homogeneous gas phase reaction CO + 1/2 O2 -> CO2. A part of the carbon dioxide produced in this reaction diffuses back to the carbon, where it is reduced by the endothermic reaction CO2 + C -^ 2C0. No experimental verification was made. Basu et al. contradicted this hypothesis by a simple heat balance and by measuring the temperature at the carbon surface and at the reaction zone [12]. They confirmed that the endothermic reaction (CO2 + C -» 2C0) cannot receive sufficient heat from the reaction zone. The original idea of CO production by the CICO^ gasification reaction was later abandoned by Ross and Davidson after they found that endothermic reaction at the carbon surface is not feasible under a fluidized combustion condition [13]. Basu demonstrated both theoretically and experimentally that oxygen diffuses to the carbon surface [14]. The heterogeneous reactions with carbon produce both CO and CO2. C + I/2O2 -> CO

(10)

C + O2 ^ CO2

(11)

The ratio of CO/CO2 will depend on surface temperature. Basu did not elaborate how the primary combustion product will depend on the carbon size. Based on their generalized single particle model, Mon and Amundson [15] and Bakur and Amundson [16] discarded the Avedesian model and concluded that oxygen always reacts on the carbon surface. For large particles (d > 1.0 mm), the CO burns at or very close to the carbon surface; for fine particles (< 0.1 mm) CO oxidizes outside the stagnant boundary layer or far away from the surface. La Nauze [17] and Basu [18] criticized the stagnant film model approach of CO oxidation proposed by Ross and Davidson [13]. They suggested enhancement of mass transfer by the bed particles that promotes the CO escape. Basu [18] also objects to Ross and Davidson's [13] assumption of constant Sherwood number, independent of the carbon particle size. He showed that experimental evidence contradicts their

Combustion of Single Coal Particles in Turbulent Fluidized Beds

175

suggestion that small carbon particles burn at a lower temperature. This issue is not settled due to the difficulty in measuring the temperature of very small carbon particles inside the bed. Changes in Particle Size and Density Combustion Induced Changes La Nauze and Jung [19] suggested the following equations for changes in particle size and density: d/dp, = (Vc^)(l - u)«

(12)

PA/PAO = ( V ^ ) ' ( l - u)^

(13)

where the extent of particle burn-off, u may be expressed as [(initial mass of particle-mass of particle at time t)/ initial mass of particle]. They further showed that 3a + P = 1 and a = 1/3, (3 = 0 for burning at a constant density and a = 0, p = 1 for burning at a constant diameter. Haider et al. conducted experiments in an open top electrically operated turbulent fluidized bed using char from bituminous coal [2]. The fractional changes in diameter of char particles are plotted against fractional burn-off in Figure 5. These closely follow the relationship for a shrinking sphere, indicating that combustion occurs at a constant density. The ratio of apparent density versus fractional burn-off is also plotted in Figure 5 and closely follows the shrinking sphere model. The initial apparent density, p^^, was measured experimentally. The initial shape factor, (|)Q, was obtained from the following equations. m„ = (7t/6)D^p^„

(14)

D„/cJ^ = i

(15)

The final apparent density, p^, was obtained by assuming (|) = c|)o and using measured values of the final mass and the diameter of the char particles in Equations 14 and 15. The initial shape factor varied between 0.9 and 1. Most of the experimental data yielded values of p/p^^ greater than unity but less than 1.07, probably due to the assumption of (|) = (^^. In practical cases, ^ should be greater than (|)Q, which will yield lower values of the ratio P/P^Q. The fractional burn-off includes the mass loss due to attrition, but this may not alter above conclusions since mass loss due to attrition is very small compared to that due to combustion. Furthermore, attrition is predicted by the shrinking sphere model. Attrition Induced Changes The generation of carbon fines by fragmentation and attrition during the combustion of a bituminous coal has been studied by Beer et al. [20], Donsi et al. [21], Arena et al [22], and Haider et al. [23]. Later, these studies were extended to turbulent fluidized beds by Haider and Basu [24]. Fragmentation of a coal occurs

176

Advances in Engineering Fluid Mechanics 1

2.0

0.8

h 1.5 apo '^•'^ I

h^-° ^ 0 0.4 H 0.5

0.2 H

U Figure 5. Plot of 6^6^^ vs. p^/p^^ vs. fractional burn-off during combustion in a turbulent fluidlzed bed [2].

soon after entering the bed and is caused by thermal stresses developed during devolatilization of the particle. It results in the breakup of some feed particles into pieces. Attrition is caused by the abrasion of the coal particle by relatively fine particles. Though the loss of carbon due to attrition may not constitute a major fraction of the total weight loss due to combustion, it makes a major contribution to the loss of carbon from the bed through elutriation. This results in lower combustion efficiencies of these combustors. It was observed that combustion and attrition are simultaneous phenomena [25] and that combustion enhances attrition. Haider et al. [23] showed that an enhancement of attrition occurs due to combustion even for particles that are spherical, homogeneous, and smooth. However, these studies were carried out in bubbling fluidized beds. In a circulating fluidized bed, the carbon fines generated due to attrition may be lost through the cyclone. It is essential to consider this in any performance models of high-velocity turbulent or fast fluidized bed. Donsi et al [21] derived the following equation for carbon attrition in a bubbling fluidized bed combustor: Ee = k, (U - U„) W/d,,

(16)

Combustion of Single Coal Particles in Turbulent Fluidized Beds

177

A turbulent bed is characterized by vigorous motion of agglomerates and interconnected gas pockets. The exact mechanism controlling the relative velocity between the particle agglomerates and the coarse-attrited particle is not fully understood. By assuming that the abrasion between the particle agglomerates and the attrited particle increases with the gas velocity, Haider and Basu [24] developed the following equation of attrition in a turbulent fluidized bed. (17)

E = k U W /d

They verified the above equation in a cold, turbulent fluidized bed. Combustion and attrition are simultaneous phenomena, and combustion enhances attrition. The generation of fines by fragmentation and attrition during combustion of a coal is shown in Figure 6. Table 1 summarizes attrition rate constants of different fuels at different regimes of fluidization. ATTRITED FINES

Fragmentation Figure 6. Fragmentation and attrition during fluidized combustion of a coal.

178

Advances in Engineering Fluid Mechanics •^ 1 * o o N r ^ i n o o m ( ^ a N O O O > r ) m a N ' - H O ' - ^

o

T-

CO

*

^

cN r^ i n o « n m o c N r ^ ' - ^ v D c o a s i n a N O o r - m ^ c N o o O ON »n O o » n ( r ) O m m o c N r 4 0 0 0 0 0 0 0 ' ^ m < N < N

o o

CD N

CO •D

c (0 CO

<—*»

3

in ON in in '^ O in O O '^' O O CN rt-H ^ O Tt ^' O ^ CM CN '-H (N

CM

O 0)

0) CD ^rf

c C) 3

^ 1.

3

O O O O O O O O O O O O O O O O ' ^ T l - T t r J - T t T l >o»n»n«rim«nin«n«r>u-i»ninin«n»n'«toooooooooooo fncncnrnfnc*^cncncnc<^cnrncorncncN'^rvicN(NCN
O)

c

:i>: ^3

0)

E CD

^—^

h-

^^^ o o 0

>

CO

£

m O O O o o o o o q o o o o o o o o o o c x ) o o o o o o C N i > t ^ r-; r-; r-; <^ — ' ^ O O O O O O O O O O O O 00 ^ ^ ^ ^ ^ QQ in

d

X3 CD

.2.E

o

0)

X) c«

^^ C

iS (0 c

o o0 )

3 LL

<

(U

D M

O

o

C O

a

s <

:3 O 0)

•^ (0

oc c o •7

^

'-^ '-H (N (N

3 *-• CO CD ^ Q. ^

H TJ C (0

cN m i^ Om o m ^

W

2 (L>

ex.

<^-(

1'««

O O o

00

c c o o

-e-e o c

CD

"^ 0

1-

C

X) X5

X)

X

X X

OQ

PQ

0Q

PQ

a o

cd o

PJ W

c

a

UH

V-i

H

3

H

o ci

|i

CN v_ CD

C3

a

CO 0

en 0

oc 1 ti:

c<3

cd

u

PQ X

Combustion of Single Coal Particles in Turbulent Fluidized Beds

179

Shrinkage of eaeh coal particle occurs both by combustion and attrition. The attrition contribution to the shrinkage rate of a burning carbon particle may be obtained from Equation 18 as [22], M d p ) , = k U/3

(18)

Together with the combustion contribution, this enters into the overall shrinkage rate. (-dd/dt) = (-dd/dt)^ + (-dd/dt)^

(19)

The overall shrinkage rate enters in the population balance equation to determine the size distribution and carbon concentration of burning carbon particles. Mass Transfer Theoretical

Relationship

A knowledge of mass transfer is essential for the understanding of the mechanism of combustion of coal in a turbulent fluidized bed. If the kinetic rate of combustion of the fuel is known, one can estimate the burning rate using the information on the mass transfer rate. The rate of transfer of oxygen from the bulk of the bed to the particle surface, k , is often expressed as the dimensionless Sherwood number, Sh = k d /D . For diffusion to a fixed single sphere in an extensive fluid, Sherwood number may be expressed as [28, 29] Sh = 2 + 0.69 Re'^2Sci/3

(20)

In bubbling beds, researchers [30,19] had used the term voidage, e, to modify the Frossling equation. Basu and Subbarao [1] presented the following tentative equation of mass transfer in a turbulent fluidized bed. Sh = 2 + 0.6 (Re^)0 5 Sc0 33

(21)

where Re^ = U^dpPg/ii = Ar8^^V[18 -\-0.6\{Axz^'y^]

(22)

The Archimedes number, Ar, is expressed as, Ar = g &l p / p ^ - pp/iLi

(23)

Haider et al. [2] presented a theoretical model of mass transfer rate in a turbulent fluidized bed. The model considers a turbulent fluidized bed to consist of clusters of bed particles or solid agglomerates and voids [4]. Bubbles are absent, and gas forms the continuous phase. An active particle within the bed will come into contact with both the clusters of bed particles and voids. Therefore, mass transfer to or from an active particle is made up of two components:

180

Advances in Engineering Fluid Mechanics

(i) the mass transfer from clusters or aggregates of particle (the particle convective component) (ii) the mass transfer from voids or gas phase (the gas convective component). The overall mass transfer coefficient may be written as: k - 5k g

c

+ (1 - 8)k gc

^

c^

(24) gv

^

''

where the cluster fraction 8^ is given by [31] 8^ = (1 - e)/l - e^)

(25)

Clusters are assumed to be at minimum fluidization and their voidage, e^, to be the same as at minimum fluidizing condition. It is proposed that clusters of bed particles transport fresh gas to the active particle surface where non-steady state mass transfer takes place during a contact time 0 . Similarly, non-steady state diffusion also takes place from voids to the active particle surface during a contact time 6 . La Nauze et al. [32] solved tne non-steady state diffusion equation for a sphere and obtained the equation k^ = (2D/dp) + (4D/7ce)«^

(26)

It is assumed that the particle environment changes often such that a steady state concentration boundary layer is never established; thus, the convection terms are modeled by renewal of fresh gas at the particle surface. The solution of the diffusion equations for this model is the same as in the case of diffusion from a semi-infinite medium to a sphere. The gas flowing past the particle at the fluidizing velocity U/e is assumed to contact the surface over a length d . The contact time 9 is given by, G = d e/U g

p

(27) ^

^

The gas convective component of mass transfer coefficient, k ^ is, therefore, obtained by substituting 0 = 6 in Equation 26. k^^ = (2D/dp + (4D^U/7cdpe)«^

(28)

It is proposed that turbulent fluidization is achieved when the bubble diameter reaches its maximum value and the corresponding bubble rise velocity is the terminal velocity of individual bed particles, U^. A slight increase in the gas velocity breaks these bubbles and allows the gas to flow in a continuous phase forming clusters of particles. These clusters may be assumed to move at a velocity equal to the maximum stable bubble rise velocity (U^^^^ = U^). In a similar analysis, La Nauze et al. [32] assumed that packets of particles move at a velocity equal to the bubble rise velocity in a bubbling bed and that 0 is given by 0 = d /U, p

p

b

(29) ^

^

Combustion of Single Coal Particles in Turbulent Fluidized Beds

181

As discussed, in a turbulent bed 0 may be obtained by substituting U^^ = U^ in Equation 29. The particle convective component of mass transfer, obtained by substituting 6 = 6 in Equation 26, is given by Equation 30: k^^ = (2D/d^) + (4D^U/7cd^)'"

(30)

Therefore, the overall mass transfer coefficient is obtained by combining Equations 24 and 30: k^ = (2D/dp) + [(1 - e)/(l -

E^)](4DlJJnd/'

+ [(e - e^)/(l - z;)]{4DlJ/nder

(31)

The Sherwood number, Sh, may be obtained from Equation 31 after substituting D - De: g

a

Sh = y / D ^ = 2e + [(1 - E)/(l - e^)](4edpU/7iD/^ + [(£ - 8^)/(l - e^)](4dpU/7cD/^

(32)

Experimental Studies Haider et al. [33] conducted experiments in a turbulent fluidized bed using naphthalene sublimation technique. Their results are plotted in Figure 7. Researchers [2] also calculated mass transfer coefficients from burning rate experiments in a hot turbulent bed. Mass transfer coefficients are obtained by using measured values of burning rates and chemical rate coefficient. Chemical Rate The surface reaction rate, R^ is expressed as, R^ = R^C; = A exp[(-E/RTp)] C^"

(33)

Accurate measurement of reaction rate of char particles is necessary to predict the rate-controlling mechanism in fluidized bed. Most researchers assume that carbon-oxygen reaction is first order with respect to oxygen, i.e., n = 1 in Equation 33. This leads to the simple mathematical expression of burning rate in fluidized beds. Recent studies indicate fractional order of reaction. This will lead to a more complicated equation of burning rate and may require numerical solution. Table 2 depicts reactivities of various fuels. Particle Temperature The temperature of a burning coal or carbon particle is used to estimate the chemical reaction rate. The particle temperature also is useful in defining the appropriate mean film temperature at which the gas properties are to be evaluated for prediction of mass and heat transfer to the particle.

182

Advances in Engineering Fluid Mechanics

8 kg

6 -

(m/s) •

XI0^

•

4 -

^Model [ 2 ] 2 -

0 4

1 8

~-|

12

16

Naphthalene Particle Diameter(mm) Figure 7, Variation of k with naphthalene particle diameter for sublimation test in turbulent fluidized bed [33]. Table 2 Reactivities of Various Fuels Researcher

Fuel

A (kg-C/m2s(kPa)")

E/R (K)

n

Field at al. [34]

Various carbon and char

859

17,976

1

Young and Smith [35]

Petroleum coke

7

9,911

0.5

Daw and Krishnan [36]

Kentucky No. 9 coal

0.0404

5,787

0.6

Haider and Basu [37]

Electrode Carbon

1.1

8,125

0.4

Haider et al. [2]

Bituminous Char

0.0063

1,331

0.45

Combustion of Single Coal Particles in Turbulent Fluidized Beds

183

Several researchers conducted experiments in bubbling fluidized beds by embedding fine thermocouple into the carbon particle and immersing it in the bed [13,14,38]. They observed particle temperature to be 50-150 K higher than the bed temperature. Roscoe et al. measured the temperature of small batches of carbon by photographic technique and found them to be about 150 K above bed temperature [39]. These results could not be applied to a turbulent bed because of the differences in hydrodynamic conditions in two types of beds. At this time, no information is available on the temperature of burning carbon or char particles in a turbulent bed. Haider measured temperature of burning carbon particles in a fast bed by thermocouple embedding technique and observed them to be about 50-80 K above bed temperature [41]. In the absence of available data, these data may be used in a turbulent bed whose hydrodynamic behavior is closer to a fast bed. Theory The energy balance of a burning carbon particle may be given by Equation 34: (m Cp/a) dTp/dt + r|m AH = Nu k(Tp - T^/d^ + ozfll

- T^)

(34)

The first term at the left hand side of Equation 34 is the thermal inertia, and the second term is the heat generation. The first term at the right hand side of Equation 34 is heat loss due to convection while the second term is that due to radiation. The r| is a term that indicates the fraction of liberated energy that reaches the carbon surface. The value of tj may be taken as 0.71 [39]. The emmissivity of carbon may be taken as 0.85 [40]. Proper values of heat transfer coefficients are required to estimate Nusselt numbers, Nu. Equation 34 suggests that the particle temperature is influenced by the following parameters. (i) the combustion rate. (ii) the proximity of the surface of the region of gas phase oxidation of CO. (iii) the heat transfer rate to or from the particle surface by convection and radiation. Burning Rates Theoretical Relationship Figure 8 depicts the combustion process of a coal particle. From mass-balance considerations the burning rate, m^, can be expressed as: m = k (C - C ) = R (C )"

(35)

where C^ is the oxygen concentration at the outer surface of the particle, C is the oxygen concentration in the bulk gas, R^ is the chemical rate coefficient of apparent order n in C , and k is the mass transfer coefficient. s'

g

184

Advances in Engineering Fluid Mechanics

o

O

^

0 , Diffusion

o o °r O ^ ofcONc!(Cg)

O

_

o

o "

(

_

VOUTILES COMBUSTION

^COAL

^////?7//////J>.

O

o

O

PARTICLE SURFACE 0 , CONC.(Cs)

^

°

^

ASH;

Co,Co, Diffusion c ^ ^ ^

^

o

°

O ^"^

.-.\ \

'

Figure 8. Combustion process of a coal particle in a turbulent fiuidized bed.

Eliminating the unknown C^ in Equation 35 gives a relationship for calculating m^. m^ = RJl - m/mj" C;

(36)

where m^ is the product k C , the maximum possible combustion rate, found when chemical reactions are so fast that C^ -^ 0, and the burning rate is controlled solely by mass transfer of oxygen to the particle. The ratio m/m^ is popularly known as the fraction of mass transfer control, X, which is always less than 1. Therefore, Equation 36 reduces to m^ = RJl - X]"C;

(37)

Equation 37 may be manipulated to provide relationships for m^ for given values of n. For a first order reaction (n = 1), which is often assumed for coal combustion, the result is m^ = C/[l/R^ + 1/kg]

(38)

Combustion of Single Coal Particles in Turbulent Fluidized Beds

185

Recent studies with some carbons suggest that the order of reaction is closer to 1/2 [35,36,37,2]. For n = 1/2, Equation 35 becomes: m = RV2k + l/2(RVk2 + 4R2C Y""

(39)

Equation 37 expresses the combined chemical and mass transfer rate equations [35] in terms of the gas phase concentration, C , the combustion rate, m^, expressed as kgC/m^s of external surface area of particle. R^ is the chemical rate coefficient for the reaction of order n, and X is the fractional approach to mass transfer control. X lies between 0 and 1. External mass transfer control is inferred when X has a value approaching 1. Experimental Studies Basu and Subbarao conducted experiments in a turbulent fluidized bed of sand of mean size 234 micron [1]. The bed was heated by burning propane. The fluidization velocity and the bed temperature were in the range of 2.2-5.3 m/s and 776-860°C, respectively. Experiments were conducted with a batch of electrode carbon particles. Burning rates were plotted against diameter and were shown to be higher than those obtained in bubbling fluidized beds. Haider et al. performed burning rate experiments using char particles in an electrically heated turbulent bed of sand of mean size 300 micron [2]. The gas velocity and bed temperature were 3.0 m/s and 1,098 K, respectively. Batch combustion studies were conducted to obtain weight loss rates. Attrition rates were subtracted from observed weight loss rates to obtain burning rates. These data are plotted in Figure 9 against char particle diameter and compared with burning rate data obtained from bubbling beds. Results indicate that burning rates in a turbulent fluidized beds are higher than those obtained in bubbling fluidized beds. This is due to enhanced mass transfer in turbulent fluidized bed. Basu and Subbarao also arrived at the same conclusion after conducting experiments with electrode carbon particles [1]. Before conducting burning rate experiments in the turbulent bed, researchers conducted a series of tests in free stream using the same char [2]. Burning rate data were statistically analyzed to obtain the reaction rate coefficient. The Rate Controlling Mechanism The observed burning rates were further analyzed by Haider et al. to obtain the extent of mass transfer control (X = m/m^) from Equation 37 [2]. The reaction rate coefficient, R , was obtained from data of Table 2. The extent of mass transfer control, X, is plotted in Figure 10. X varies between 0.44 and 0.71, indicating combustion to be a combined process of transfer of oxygen and chemical kinetics where a major role is played by diffusion of oxygen to the surface. For nearly identical bed temperature and active particle sizes. La Nauze and Jung observed that for petroleum coke particles burning in bubbling fluidized beds, X was close to 1, indicating that combustion was diffusion controlled [19]. This suggests stronger mass transfer in turbulent fluidized beds.

186

Advances in Engineering Fluid Mechanics

2.0 (/}

O) (D X (D

c

Char particle diameter

(mm)

Figure 9. Variation of burning rate with the diameter of char particles in a turbulent fluidized bed [2]. 1) Kinetic linfilt [2]; 2) Diffusion linfiit; 3) Model of Haider et al. for burning rate of char particles in turbulent fluidized beds [2]: o Haider et al. for char particles burning in a turbulent fluidized bed with air as fluidizing medium [2]; • La Nauze for petroleum coke particles burning in air fluidized bubbling bed [17]; A Chakraborty and Howard for char particles burning in air fluidized bubbling bed [38]. The extent of mass transfer control, X, decreases as the diameter decreases showing an approach towards kinetic control. The trend indicates that for 3 mm diameter particles X will be 0.4 or less, thus the kinetic rate of combustion has a major influence on the overall burning rate. The extent of mass transfer control, X, was used to calculate the mass transfer coefficient, k . The results, plotted in Figure 11, agree well with those predicted by using Equation 31. CONCLUSIONS AND RECOMMENDATIONS Upon entering the bed, the coal particles is rapidly heated by the surrounding inert particles. During the heating period, the volatile matter is released. The

Combustion of Single Coal Particles in Turbulent Fluidized Beds

187

1

0

4

8

12

Char particle diameter

16

(mm)

Figure 10. Variation of X with char particle diameter during combustion in a turbulent fluidized bed [2].

0

4

8

12

Char particle diameter

(mm)

Figure 11. Variation of kg with char particle diameter for combustion test in turbulent fluidized bed [2].

188

Advances in Engineering Fluid IVIechanics

devolatilization and subsequent combustion of volatile matter are relatively fast processes. But, burning of char (carbon) takes an order of magnitude longer time than the devolatilization and volatile combustion. Consequently, the combustion of char controls the overall combustion process. Oxygen in the fluidizing air diffuses from the gas phase to the carbon surface where it reacts to form both CO and CO2. The CO diffuses away from the carbon surface. For larger carbon particles (d > 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).

190

Advances in Engineering Fluid Mechanics

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).

Combustion of Single Coal Particles in Turbulent Fluidized Beds

191

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.

This Page Intentionally Left Blank

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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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.

Flow of Solids and Slurries in Rotary Drums

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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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
F =4.34 — n c o t e L

fh

2.27

(13)

202

Advances in Engineering Fluid Mechanics

(K H == 0.33

r-(

v3.73

h,.R,)

J i U r_( (K 'K U J \. R J

^2.27

(14)

For 0.16 < h/R < 0.32. Afacan and Masliyah [38] rederived Equation 14 using Vahl and Kingma's [44] approach. They showed that the original derivation of Hehl et al. [9] is incorrect, where in the limit of (hJR) = 0, Equation 13 gives H = 0.33

¥ I

(i^>

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.

Flow of Solids and Slurries in Rotary Drums

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; <), 30.3 x IQ-^.

0

2

4

6

8

10

12

Measured Mean Residence Time,(min) Figure 4. A comparison of measured and predicted solids residence time (no lifters and no end constriction). • , Vahl and Kingma [43]; • , Hehl et al. [9].

204

Advances in Engineering Fluid Mechanics

Drum Without Lifters and With Discharge End Constriction The variation of solids fractional hold-up with drum speed is shown in Figure 5 for the case of a drum without lifters and with discharge end constriction. The variation is similar to that reported by Abouzeid and Fuerstenau [11,29]. The solids hold-up is much higher with the end constriction than with a free drum opening as shown in Figure 3. Figure 6 depicts a comparison of Afacan and Masliyah's data [16] of solids mean solids residence time with those predicted by the approximate solution of Hehl et al. [9]. It can be seen that there is a large disagreement, especially at the higher solids residence time. Drum With Lifters Figure 7 displays the effect of the drum speed on the solids fractional hold-up for a drum fitted with lifters, but with no discharge end constriction. At the same solids feed rate and drum speed, the presence of the lifters increases the drum solids hold-up (Figures 3 and 7). The variation of the drum fractional hold-up with the drum speed is shown in Figure 8 for the case of a drum fitted with an end constriction. The presence of lifters has little effect on the solids hold-up as compared with the case of a drum

n/xic 0.20 -r

20

0.41

0.62

40 60 80 Drum Speed.(rpm)

100

Figure 5. Variation of solids fractional hold-up with drunfi speed (no lifters and with end constriction). Solids feed rate, (kg/s): • , 6.3 x 10"^; • , 12.5 X 10-3; A , 21.3 x 10-^; <), 30.3 x 10-^.

Flow of Solids and Slurries in Rotary Drums 30

r

5^1

I 0)

o 20

a

I

>^''^^^

I

15 h L

0)

10

•

L

X

"1

• y^

J

•

1

*

^

r 0)

m J

y^

CO

0)

205

J 1

y^

5

o

•a

0

5

10

15

20

25

30

Measured Mean Residence Time,(min)

Figure 6. A comparison of measured solids residence time with approximate solution of Hehl eX al. [9] (no lifters and with end constriction).

n/ne 0.20

20

0.41

0.62

0.62

40

60

60

100

Drum Speed, (rpm) Figure 7. Variation of solids fractional hold-up with drum speed (with lifters and no end constriction). Solids feed rate, (kg/s): • , 6.3 x 10"^; • , 12.5 X 10-3; A , 21.3 x IQ-^; <>, 30.3 x lO'^.

206

Advances in Engineering Fluid Mechanics

n/nc

20

0.41

0.62

0.82

40

60

80

100

Drum Speed, (rpm) Figure 8. Variation of solids fractional hold-up with drum speed (with lifters and with end constriction). Solids feed rate, (kg/s): • , 6.3 x 10"^; • , 12.5 X 10-3; A . 21.3 x IQ-^; <>, 30.3 x IQ-^.

without lifters but with an end constriction (see Figures 5 and 8). For the case of solids transport due to the presence of lifters and internal gas flow, Kelly and O'Donnell [26] and Glikin [27] developed design equations for the drum hold-up. However, there is no general correlation in the literature to account for the presence of lifters. A comparison of the numerical solution of Equation 3 with the experimental results of Afacan and Masliyah is given in Figure 9 [16]. For the drum with an end constriction, h, was taken as (R - R + 2 d ). The value of 2 d was added to account for the area of the flowing solids above the constriction lip. In the absence of the constriction h, = 2d . L

p

The data shown in Figure 9 (from Afacan and Masliyah) are for the case of a drum equipped with lifters and had an open-end discharge, h^^ was taken as (2d + the lifter height). The data cover the range of 0.05 < n* < 0.4 and feed rates up to 30.3 X 10"^ kg/s. Residence time data from Abouzeid and Fuerstenau are also included in Figure 9 for N =10 and 20 rpm [29]. The data are for a 24 x 8-cm horizontal drum using a solids feed of 35 x 48-mesh dolomite. The drum was fitted with a discharge end constriction having a radius of 4.5 cm. It is evident from Figure 9 that the numerical solution of Equation 3 gives satisfactory agreement with the experimental data for all four drum geometries, including those for a drum with lifters. Langrish [45] conducted a thorough study on using Equation 3 to predict Matchett and Sheikh [46] hold-up data. The latter authors used a rotary drum fitted with lifters.

Flow of Solids and Slurries in Rotary Drums

30

J,

1

•

1

"•

'•

••

- |

—

1

—

y^

h

0)"

I

•

y ^

-

207

^

0 ^A

OAO

20 Lh

0)

o

a

^

0) CO

0)

« 10

L

O

y ^

H

y^ ^^k

0) -•J

j

3 ^ ^ ^

1

.1,,

10

.

1

1

20

30

Measured Residence Tiine,(min) Figure 9. Comparison of measured solids residence time with exact solution of Kramers and Croockewit [45] (for all configurations): O, without lifters and no end constriction; , with lifters and no end constriction; 0, without lifters and with end constriction; A, with lifters and with end constriction; *, Abouzeld and Fuerstenau [11,29].

Langrish indicated that Equation 3 did not predict hold-up data of Matchett and Sheikh. The work of Langrish confirmed Afacan and Masliyah's statement that more work is needed before generalizing Equation 3 to other drum dimensions and geometries. FLOW OF HIGHLY SETTLING SLURRIES IN ROTARY DRUMS Drum With an Open-End Discharge and No Lifters Figure 10 illustrates a typical flow pattern of a highly settling slurry in the rotating drum. As Rogovin [37] reported, Afacan et al. [38] also observed two separately moving regions: a pool region which contains mainly fluid and acts as a slowmoving river, and a slurry bed region which is mostly solids and travels much slower than the pool region. The pool region was present for all operating conditions examined by Afacan et. al. [38]. The slurry bed region showed various bed behavior characteristics at different operating conditions, and was particularly sensitive to the drum speed. The solids moved in the slurry bed region in both the axial and the transverse directions. These movements were observed by means of colored particle tracers.

208

Advances in Engineering Fluid Mechanics

^ ^ Rotation

Drum -y^ Shell /

^N\ n A

Pool _ _ W : r p 2 f ^ ^ ^ ^ Bed Region ^^^^^^^ Region A c t i v e Layers Figure 10. Experimentally observed transverse nfiotion of solids and fluid.

As for the case of dry solids, two different slurry bed motions were observed by Afacan et al. [38]. At low drum speeds, a periodic slumping slurry bed motion was observed, whereas at higher drum speeds, the periodic slumping of the slurry bed gave way to rolling. At low drum speeds, the slurry bed moved upwards with the rotating drum, reaching a maximum height then slumping downwards. At higher drum speeds, the periodic slumping of the slurry bed led to rolling, which was characterized by the continuous motion of a layer of solids over the bed surface in a manner similar to dry solids [21]. At still higher drum speeds, the water within the pool began to penetrate the slurry bed and rendered it fairly wet. Effect of Drum Rotational Speed on Slurry Hold-up Figures 11 and 12 depict the effect of the drum speed on the slurry hold-up and its solids volumetric concentration, respectively. The slurry feed rate is 20 g/s, and the solids particle size is d^^ = 2.0 mm. For a given slurry feed solids concentration. Figure 11 displays that the drum slurry hold-up decreased with the drum speed. Also, at a given drum rotational speed, the drum slurry hold-up was higher for a slurry feed having a higher solids concentration. At low drum speeds, say 7 rpm, the drum slurry hold-up for a feed slurry solids concentration of Cp = 46.1 vol.% was twice that for a Cp = 7.6 vol.%. It is possible to view the findings of Figure 11 in a different manner. Each curve for a different slurry feed solids concentration can be viewed as a curve of a given solids feed rate. Consequently, the three curves of Figure 11 can be considered to represent three different solids feed rates. Previous studies on the flow of dry solids in a rotating drum have shown that at a given drum speed, the dry solids hold-up in a drum is higher for a higher solids feed rate [16,29]. Therefore, the finding that a more concentrated slurry feed leads to a higher slurry hold-up is simply a consequence of having a higher solids feed rate. (This point will be re-stressed in the discussion of Figure 21.)

Flow of Solids and Slurries in Rotary Drums

12

— ' — — I

f~

T

209

T

dso = 2.0 iQm QF

10 h

= 20-0 g / s • 7.6 % • 21.9 % A 46.1 %

t \

8h I

9 6h o

ID

0

0

± 20

± 40 N, r p m

60

Figure 11. Effect of drum speed on slurry hold-up, 6^ = 2.0 mm. The variation of the solids concentration in the drum slurry hold-up with the drum speed is shown in Figure 12. The solids concentration in the drum hold-up decreases with the drum speed. Within the drum speed range examined by Afacan et ah, the solids concentration in the drum slurry hold-up was always much higher than that in the slurry feed [38]. This is simply due to the fact that the axial velocities of the solids and the water within the drum are not the same. The water velocity is higher than that of the solids. A similar observation was made by Davis [31], Gupta et al. [33], and Rogovin [37].

210

Advances in Engineering Fluid Mechanics

70

1

•"-

1

dso = 2.0 mm

^^.

QF •

60

= 20.0 g/s ——

.._A

\ \ ^ \ \

-i 50

\

>

\

o

40

30

• 7.6 % • 21.9 % A 46.1 % I

0

I

20

40 N, rpm

60

Figure 12. Variation of hold-up solids concentration with drum speed. The effect of the drum speed on the ratio of the mean residence time of the solids and water, t/Xp is shown in Figure 13 for various inlet feed solids concentrations. The slurry feed rate was kept at 20 g/s, and the solids mean particle size was d^j, = 2.0 mm. The ratio x/x, decreased with the drum speed, but did not reach unity. For the case of low feed solids concentration, C^ = 7.6 vol.%, zjx^ approached a value of 10 at a rotational speed of about 50 rpm, indicating a substantial difference in the solids and water velocities within the drum. However, for Cp = 46.1 vol.%.

Flow of Solids and Slurries in Rotary Drums

211

25

dso = 2.0 mm Qj. = 20.0 g / s

20 h

• 7.6 % • 21.9 % A 46.1 %

\ \

15 h

\ \« \ \ \

\ #

10 h

r

0

^"^-TA

A

20

A

40 N, r p m

60

Figure 13. Effect of drum speed on the ratio of solids and water mean residence times. T/T^ approached a value of about 2. At a given rpm, flow observations indicated that the water pool cross-sectional area was substantially larger for Cp = 7.6 vol.% than that for Cp = 46.1 vol.%. Consequently, at a higher feed solids concentration, the water had less of a free flow area, and both the solids and the water moved as a saturated bed of solids; as a result, T/T^ approached unity. Figure 14 shows the effect of the drum speed on the slurry hold-up for a medium sand, d^Q = 0.5 mm, for different feed slurry concentrations. The slurry feed flow rate was kept constant at 20 g/s, similar to those given in Figures 11 and 12 for

212

Advances in Engineering Fluid Mechanics

16

T

14 L 12 ^

g

10

A

\ CF

• 7.1 % • 21.0 % A 43.8 %

^\X \ \

X

\

V

\

U 6 CO

\

\

N.

4h U

0

^50 = 0.5 nun Qp = 20.0 g / s ± 20

±

40 N. rpm

± 60

Figure 14. Effect of drum speed on slurry hold-up, d^^ = 0.5 mm. the case of the coarse sand. However, a comparison of Figures 11 and 14 indicates that the slurry hold-up for the medium sand was higher than that of the coarse sand. Wet medium sand was found to agglomerate, and it was more difficult to cascade or roll down; hence, its movement in the axial direction was impeded. Effect of Slurry Feed Rate on Slurry Hold-up Figure 15 shows the influence of slurry feed rate on the slurry hold-up for drum speeds of 7 and 35 rpm. The feed solids concentration and particle size were kept

Flow of Solids and Slurries in Rotary Drums I

^u

1

—'

.

'

dso == 2.0 mm Cr = 21 %

213

1 J —]

15 N • 7 rpm A 35 rpm

&^

1

Q

•-4 10 O X \J ffi

^ /

>^

^

/

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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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

<

I

>

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.

Flow of Solids and Slurries in Rotary Drums

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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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

Advances in Engineering Fluid Mechanics

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)

Flow of Solids and Slurries in Rotary Drums

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 - ^iJ(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- <^^ and

0 (0) = 0

(18)

The solution was carried out until the solids bed stopped moving, i.e., 0 = 0. The value of 0 and the time at which 0 = 0 were denoted 0^ and T^, respectively. The upswing part of the motion starts at (j) = 7C - 0^ and ends at (|) = (|)^. This means that the upswing part of the motion covers an angle of ((j)^ + 0 - n). The time required for this part of the motion, T^, is: T^ = (30/N7C) {^^ + 0^-71)

(19)

The bed frequency of oscillation, f, can be calculated from the time of the upswing motion, T^, and the time of the downswing motion, T^, where: f = 1/(T^ + T,)

(20)

To integrate Equation 17, the friction coefficient during the downswing motion should be known. The limiting values for |x^ are as follows: ji^ approaches X | ^ as N goes to zero, and it approaches zero as the drum speed approaches the critical drum speed, N^. The reason for the second limiting value of jx^ is that, at high drum speeds, water is entrained into the solids bed, thus reducing the friction between the solids bed and the drum wall. Ultimately, the friction coefficient reduces to nearly zero. Based on these limiting values, the following tentative expression for jXj was adopted: |Li, = ^ijl - (N/N/]'^«

(21)

The value of the exponent *a' was found to be 0.5 using a best fit of the experimental data obtained by Nasr-El-Din et al. [39]. The measurement of the angle of repose of the sand particles was made by simply observing the angle at which the wet bed slipped while rotating the entire drum slowly by hand. The angle of repose was 36° (^^ = 0.73). Model predictions will be shown in Figure 23. The slurry volumetric hold-up and its mean solids volumetric concentration were strongly related to the slurry bed oscillating motion. To quantify this relationship, the frequency of the bed oscillation was measured once steady state was reached. Figure 23 displays the variation of the bed frequency of oscillation with the drum speed for various feed volumetric solids concentrations. At a low feed solids concentration of 3.5%, the frequency of oscillation steeply increased as the drum speed was increased from 7 to 27 rpm. The bed oscillated with the same frequency for drum speeds from 27 to 44 rpm. At higher drum speeds, the slurry bed stopped oscillating and the bed behaved as a solid body. Similar trends were observed for slurry feed solids concentrations of 7.5% and 21.6%. However, the drum speed at which the slurry bed stopped oscillating occurred at higher drum speeds for the higher feed solids concentrations. It is worth noting that the slurry bed oscillation behavior observed at low feed concentrations is similar to that described by Rutgers [17] for the flow of dry solids in rotating drums having low solids-wall friction coefficient.

Flow of Solids and Slurries in Rotary Drums

1.7

1

'

'

1

'~

1

1^

223

— • • — 1

/

<0

/

1.6 h

o = o

1

}•

^

/

c

\

a

B

/1^

1.5

CO

O o c 0

dso = 2.0 mm

f

QF = 20.5g/s

V 1 1 ////

1.4h

model prediction CF

C7

1.3h

J

• 3.5% • 7.5%

/ /

J 1

• 21.6% 1.2 1 0

1

1 20

i_

1

40

\

1

1

60

80

N, rpm Figure 2 3 . Effect of d r u m s p e e d o n b e d f r e q u e n c y of oscillation for a d r u m without lifters.

Figure 23 also shows that the model predicts the frequency of oscillation increases at low drum speeds and becomes constant at higher values of N. It is interesting to note that the model predicts an oscillation frequency independent of the solids hold-up, i.e., solids mass in the drum. The experimental data shown in Figure 23 are for various feed slurry solids concentrations and represent different solids holdup in the drum (Figures 24 and 25). The experimental data also indicate that the frequency of oscillation is not sensitive to the solids mass hold-up as predicted by the model. In general, the frequency of oscillation is a function of ^^ and ^i^. Increasing the drum speed results in a decrease in the value of jx^, which is reflected

224

Advances in Engineering Fluid Mechanics

by an increase in the frequency oscillation. At higher values of N, [i^ becomes nearly zero, and, hence, the frequency of oscillation becomes a function of X | ^ or X | ^ only. This case is characterized by the constant frequency of oscillation. However, at very high drum speeds, the assumption of ILI^ = |i^ breaks down. Here, water covers the drum inside wall and ^t^ reduces to nearly zero and, as a result, the solids bed stops oscillating and takes a horizontal position. Effect of Drum Speed on Slurry Hold-up. It is instructive to discuss the effect of the end-constriction on the minimum slurry hold-up, H^.^, before examining the effect of various parameters on the drum hold-up. In the case of a rotary drum with an end-constriction, material cannot flow out of the drum until it is filled to the height of the lip (R - R^). This means that a minimum slurry volume must be present in the drum before any material begins to flow over the discharge lip. The minimum slurry hold-up (percentage of the drum volume) can be calculated as:

Hmin

-

Y - P]xlOO sm

(22)

where y is the central angle extended by the drum contents; y can be calculated from the discharge opening radius, R^, and the drum radius, R: y = 2 cos

R.

(23)

For the drum examined by Nasr-El-Din et ai, y = 111.5° and H^.^ = 16.2% [39]. Figure 24 shows the effect of the drum speed on the drum slurry hold-up for different slurry feed solids concentrations at a slurry feed flow rate of 20.8 g/s. For a feed solids concentration of 3.5%, the hold-up decreased to a minimum value of 23.6% as the drum speed was increased. However, at a drum speed of 44-48 rpm, the slurry hold-up abruptly increased to nearly its value at a drum speed of 7 rpm. No significant change in the slurry hold-up was observed for drum speeds greater than 48 rpm. A similar trend for the slurry hold-up was observed for the slurry feed solids concentration of 7.5%. However, the abrupt increase in the slurry hold-up occurred at a drum speed of 53-57 rpm. Unlike the hold-up behavior observed for the lower feed concentrations, there was no abrupt increase in the slurry hold-up for the case of a slurry feed solids concentration of 45.8%. For low drum speeds, a higher feed solids concentration resulted in higher slurry hold-up. This is similar to the results obtained by Gupta et al. [33] and Afacan et al [38]. Effect of Feed Flow Rate on Slurry Hold-up. Figure 25 depicts the effects of the drum speed, and slurry feed concentration on the slurry mean solids concentration in the drum for the same conditions given in Figure 24. The mean slurry solids concentration in the drum varied significantly with the drum speed for the slurry feed solids concentrations of 3.5% and 7.5%, whereas a small variation in the mean solids concentration occurred for a slurry feed solids concentration of 45.8%. For

Flow of Solids and Slurries in Rotary Drums

1

1—

1

1

1

dso = 2.0 mm Qp = 20.8 g / s

40

• •

6^

A

225

1

3,5 7.5 45.8

1

% H % %

A

I Q

•

o

1 1

• "

1 1

30

—1

1

3

1 1 V \

•^--*_.

on

1

0

1

!

20

1

1

\

J

40 N, rpm

11

1

1 1 1 1

J

60

Figure 24. Effect of drum speed on slurry hold-up in a drum without lifters. the case of Cp = 45.8%, the drum mean concentration was close to the solids maximum packing concentration for the sand particles used by Nasr-El-Din et aL [39]. Effect of Drum Speed on Solids and Fluid Mean Residence Times. Due to the difference in the axial velocity of the two phases (slip velocity) the mean residence time of the solid particles in the drum is in general higher than that of the fluid. The ratio of the mean residence time (t/T^ is a measure of the slip velocity between

226

Advances in Engineering Fluid Mechanics

65

60

fe<

65

o

>

50 h CF

45

40

dso = 2-0 mm QF = 20.8 g / s 20

40 N, r p m

• 3.5 % • 7.5 % A 45.8 %

60

Figure 25. Effect of drum speed on slurry mean solids concentration in a drum without lifters. the two phases. Figure 26 depicts xjx^ as a function of the drum speed for the various slurry feed concentrations. The mean residence time of a given phase was calculated from the feed flow rate of this phase and its hold-up in the drum. At a slurry feed solids concentration of 7.5% and a drum speed of 7 rpm, the residence time of the solid phase was much higher than that of the fluid and T/T^ was 20.8. The residence time ratio decreased with the drum speed then remained constant; xjx^ abruptly increased once the solids bed stopped oscillating. At a given

Flow of Solids and Slurries in Rotary Drums

227

dso = 2.0 mm QF = 20.8 g / s

30 • 7.5 % • 21.6 % A 45.8 %

20

10 h

0

20

40 N, rpm

60

Figure 26. Effect of drum speed on the solids/fluid residence time for a drum without lifters. drum speed, T/T^ decreased with the slurry feed solids concentration. This trend is similar to that observed by Hogg and Rogovin [32] and Afacan et al. [38]. At the higher feed solids concentration of 45.8%, the effect of the drum speed on xjx^ was not significant. Effect of Drum Speed on C/C^. As a result of the slip velocity between the particles and the fluid, the mean solids concentration in the drum is higher than that in the feed. Figure 27 shows the mean slurry concentration in the drum

228

Advances In Engineering Fluid Mechanics

30

25

• • • A

20

o

3.5 7.5 21.6 45.8

% % % %

dso = 2.0 mm QF = 20.8 g/s

15

o 10

0

20

40 N, rpm

60

Figure 27. Variation of C/C^ with drum speed for a drum witfiout lifters.

normalized by that in the feed (C/Cp) for various feed concentrations. The normalized slurry concentration in the drum was much higher than unity, especially at the lower slurry feed solids concentrations. For cases of Cp = 21.6% and 45.8%, C/Cp was almost independent of the drum rotational speed. Figure 28 displays the effect of the slurry feed rate on the slurry hold-up for Cp = 21.4% at N = 7 and 35 rpm. At a constant drum speed, the slurry hold-up increased with the slurry feed flow rate, especially at the higher drum speed. This

Flow of Solids and Slurries in Rotary Drums

229

40

35

I

9 o

30

dso = 2.0 nun CF = 21.4 %

>: PH

3 CO

25

20

N A 7 rpm • 35 rpm I

I

20

40 QF.

60

g/s

Figure 28. Effect of slurry feed flow rate on slurry hold-up in a drum without lifters.

trend was similar to that observed for the flow of dry solids [6,9] and slurries [47] in rotary drums with an end-constriction. At a low slurry feed flow rate, the slurry hold-up at the drum speed of 35 rpm was much lower than that at 7 rpm. However, this difference diminished at higher feed rates. Effect of Solids Feed Rate on Solids Hold-up. The results discussed so far indicate that the slurry hold-up is controlled mainly by the movement of solid particles in

230

Advances in Engineering Fluid Mechanics

30

..o

z:::^

.-or-" . ^ '

d. DI

20

o

I

N = 35 rpm

CO

Cp

o 10 r CO 1 r 01 0

A 7.8% • 21.1% • 45.8% O 100% (Dry Solids) 1

i

1

1

1

20

10

30

O^, g/s Figure 29. Effect of solids flow rate on solids hold-up in a drum without lifters.

the drum. To examine the effect of the fluid flow rate, solids hold-up in the drum is plotted in Figure 29 against the solids feed rate for a dry solids feed (Cp = 100%) and various slurry feed concentrations at a drum speed of 35 rpm. For the experimental runs with a slurry feed, the solids feed rate was varied either at a constant slurry feed solids concentration (Qp = 21.1%) or at a constant slurry feed flow rate (Qp = 20.8 g/s). For the case of dry solids, the solids hold-up increased with the solids feed flow rate. However, for the case of a slurry feed, the solids hold-up was independent of the solids feed rate, when Q^ was greater than about 10 g/s. The most interesting observations in Figure 29 are: the solids hold-up for any slurry feed examined was significantly lower than that obtained with dry solids feed having

Flow of Solids and Slurries in Rotary Drums

231

the same solids flow rate; and the effect of the fluid flow rate on the solids holdup was not significant for these particles within the range of parameters examined. For the case of a slurry feed, the presence of the water layer between the solids and the drum surface decreases the friction between the solids bed and the drum surface. This, in turn, decreases the bed axial depth gradient required for solids transport in the drum, which leads to a lower solids hold-up for the case of a slurry feed. In the case of a very high water flow rate, water would exert a higher drag force on the particles, which tends to reduce the solids hold-up in the drum. Drum With Lifters Nasr El-Din et al. [39] observed that at low rotational speeds, the solid particles were carried up the drum wall with the lifters and fell back to the bottom of the drum. As a result of the particles motion, mixing within the slurry bed and between the liquid and the slurry bed was significantly improved. At higher drum speeds, solids in the bed assumed a rolling motion and the slurry bed had a kidney shape. Unlike the drum without lifters, no slurry bed oscillation was observed for all the experiments conducted using the drum with lifters. The presence of the lifters prevented the downswing solids motion and, hence, prevented any oscillation from occurring. In essence, the lifters affected the radial motion of the solids bed drastically. Slurry Hold-up and Mean Solids Concentration. Figure 30 depicts the influence of the drum rotational speed on the slurry hold-up for slurry feed concentrations of 7.8%, 21.0%, and 45.9%. The slurry hold-up-drum speed relationship exhibited a minimum for all the feed solids concentrations examined. At low drum speeds, the slurry hold-up increased with the feed solids concentration. However, at high drum speeds the effect of the slurry feed solids concentration on the slurry holdup was less significant. The most important aspect of Figure 30 is that the lifters eliminated the abrupt changes in the slurry hold-up observed for the drum without lifters. Figure 31 demonstrates the effect of the drum speed on the mean solids concentration in the drum for the same conditions as in Figure 30. For the cases of Cp = 21.0% and 45.9%, the hold-up solids concentration decreased with the drum speed. However, for the case of Cp = 7.8%, the hold-up mean solids concentration was independent of the drum speed. At a given drum speed, the slurry mean solids concentration in the drum increased with C^. The results shown in Figures 25 and 31 indicate that for a given Cp the mean solids concentration in the drum with lifters is lower than that for the drum without lifters. This trend is due to higher axial particle velocities encountered in drums with lifters. For the drum with lifters, the variation of the slurry mean solids concentration with the drum speed was much less pronounced than that observed for the drum without lifters. Also, the effect of Cp on the hold-up solids concentration was more pronounced for the drum with lifters. Effect of Drum Speed on T/T. Figure 32 displays the ratio of the solids and the fluid mean residence time as a function of the drum speed for various feed concentrations. For Cp = 7.8%, the solids to the fluid mean residence time ratio

232

Advances in Engineering Fluid Mechanics

40

1

1

._j_

,

1

—.,

J

dso = 2.0 mm Qy = 20.7 g/s

_

1

3 5 hh-

a, ID I

1 o 30 h

't

[-

3

25 h

20

1

0

i

,.

1 .

20

._,,„i_

1

40 N, rpm

CF

J

• 7.8 % • 21.0 %

1 1

A 45.9 %

1

1

1

60

Figure 30. Effect of drum speed on slurry hold-up for a drum with lifters. was 11.5 at a drum speed of 7 rpm, and it slightly dropped at higher drum speeds. Similar to the trends observed with the drum without lifters, T/T^ significantly decreased with the feed concentration and became almost independent of the drum speed for the case of Cp = 45.9%. Figures 26 and 32 indicate that for both drums the commonly used assumption that the residence times of the two phases are equal [47,48] is valid at Cp > 45% for the sand particles examined. Also, the presence of the lifters significantly reduced x/x^ especially at the low feed solids concentration.

Flow of Solids and Slurries in Rotary Drums

60

,

" T —

• ^

T

'

233

T

dso = 2.0 m m p

QF

55

-\

= 20-7 g/s ^

-^A^

^

A

^

h

H

*

•

M

>

-•

•

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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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

Advances In Engineering Fluid Mechanics

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<,. g/s Figure 34. Effect of solids flow rate on solids hold-up in a drum with lifters.

Slurry Modes of Transportation Slurry flow in a horizontal rotary drum can be compared to that of dry solids, slurry flow in closed conduits, or open channels. A brief discussion will be given in this section for each of these transport modes. Flow of Dry Solids Flow of dry solids in a lightly loaded horizontal open-end rotary drum is given by Kramers and Croockewit [44] as: \|/' = -(2/37C) cot ej(2h/R) - (h/R)2]3/2 dh/dx

(24)

Flow of Solids and Slurries in Rotary Drums

237

where ^ ' is a dimensionless dry solids flow rate, and is defined as ¥ = (Q/Pb)/7CR'V

(25)

The drum peripheral speed, V, is defined as V = nR

(26)

Equation 24 shows that the solids transport in a horizontal rotary drum is due to the difference in the height of the solids bed along the drum. Consequently, the solids bed slope acts as the driving potential causing the solids to flow. Afacan et al studied the flow of a highly settling slurry (2 mm sand/water slurry) in a 0.192 m diameter rotary drum having an open-end discharge and without any internals [38]. The single particle settling velocity was 0.2 m/s, and the slurry axial velocity was in the range of 0.001 to 0.004 m/s. They found that the effect of the hydrodynamic forces exerted by the slurry water on the particles was not significant. As a result, the solids hold-up was independent of the fluid flow rate, and Equation 24 could be used to predict solids hold-up in the drum. Slurry Flow in Closed Conduits For slurry flow in closed conduits, the manner by which the solids are conveyed depends largely on the solids and liquid properties, the average liquid velocity, the pipe diameter, and orientation. For a given slurry flowing in a horizontal pipe at a very high slurry velocity, the solids are completely suspended by the turbulence of the carrier liquid [49]. The solids concentration is uniform across the conduit, and the slurry acts as a pseudo-homogeneous fluid [50]. At a lower slurry conveying velocity, the solids concentration in the vertical direction is not uniform. The solids concentration is much higher near the conduit lower surface [51,52]. The flow is termed to be wholly or partially stratified. For the latter case, the total solids transport is due to a suspended load in the upper conduit region and to a stratified or granular contact load in the lower region of the conduit. A sliding bed model was developed by Wilson to analyze this type of slurry transport [53]. The simplest method of generalizing experimental studies for stratified slurry flow in conduits is given by Durand-Condolios' equation [54,55] as: {i-i,)l{CJ,) = ^{Vx]Cl'r"

(27)

where i and i^, are the hydraulic gradients in the pipeline for the slurry and the working fluid at the same bulk velocity, respectively. C^^ is the delivered solids volume fraction. K is a constant having a value of about 80. Fr^ is a modified Froude number based on the pipe diameter, Dj, and is defined as: Fr^ = V3/[gD^(S^ - \)r

(28)

Vg is the average slurry velocity, and S^ is the density ratio of the solids to that of the carrier liquid. C^ is the single particle drag coefficient and is given for spherical particles by [56]

238

Advances in Engineering Fluid Mechanics

Cj^ = 24(1 +0.15 RCp^^^^yRCp,

Re^ < 1000

(29a)

Cp = 0.44,

RCp > 1000

(29b)

where

^^ = \^pM

(30)

Equation 27 indicates that the normalized hydraulic gradient, (i - i^)/ip is a function of the modified Froude number, the single particle drag coefficient and the delivered solids volume fraction. Free-Surface Slurry Flow For a free-surface slurry flow in an open conduit inclined to the horizontal, the transport of the suspended load is similar to that of a closed conduit. However, for the case of free-surface flow, there is no pressure differential across the length of the conduit, and direct application of Equation 27 is not meaningful. For sediment transport in open-channel flow, Yalin [57] and Novak and Nalhuri [58] showed that the Froude number is an important parameter that describes sediment transport, and is given as Fr^ = V7[g L*(S^ - \)r

(31)

where L* is a characteristic length which can be the liquid depth above the solids bed, the solids particle diameter, or the flow hydraulic radius. V* is a characteristic velocity. A two-layer model was developed by Wilson [54] to analyze slurry flow with a free surface, where the total solids load is due to a suspended load and a sliding bed load. Slurry Flow in Rotary Drums For slurry flow in a rotary drum, the mode of the solids transport can be related to dry solids transport in a rotary drum and slurry transport in either closed conduit or open channel flow. When the settling velocity of the particles is relatively high as in the case of the study of Afacan et al. [38], the mode of solids transport is similar to that of dry solids transport. For the case of particles having very small settling velocities, the slurry acts as a pseudo-homogeneous fluid and the transport of the slurry resembles that of an open channel liquid flow. In the intermediate range of particle settling velocities, slurry transport in a rotary drum is characterized by the presence of a stratified solids layer near the bottom of the drum, which is similar to the flow of settling slurries in closed conduits. In the case of slurry flow in a rotary drum, turbulent suspension of the solids can occur due to the axial liquid velocity coupled with the radial liquid velocity due to the drum rotation and mixing action created by the drum internals. Typically, the peripheral drum velocity is much higher than the slurry axial velocity, and, consequently, the drum rotational speed becomes very important for solids turbulent suspension and, hence, transport.

Flow of Solids and Slurries in Rotary Drums

239

Slurry Hold-up and Solid Concentration Figure 35 shows the variation of the percent slurry hold-up and hold-up slurry concentration for the case of 80 |am silica sand at a slurry flow rate of 0.02 kg/s. The working liquid was water at 23°C. Figure 35a shows that the percent slurry hold-up in the drum decreased with the drum speed. For the cases of 10 to 30% feed solids concentration, the asymptotic hold-up value was about 20%, which was slightly higher than the minimum hold-up of 16.2%. However, for the case of a feed solids concentration of 40%, the asymptotic percent slurry hold-up was higher. Figure 35b displays the variation of the average hold-up solids volumetric concentration with the drum speed for the same conditions as in Figure 35a. For the case of 10% feed solids concentration, the hold-up solids concentration decreased sharply with the drum speed. The dependence of C on the drum speed decreased for the higher feed solids concentration. In all cases, at a drum speed of about 2.62 s~', the hold-up solids concentration approached its respective feed solids concentration. This result indicated that at drum speeds > 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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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

Advances in Engineering Fluid Mechanics

(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.

Flow of Solids and Slurries in Rotary Drums

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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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 \

•

.-^^'^^^i^^

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

Advances in Engineering Fluid Mechanics

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, (^i^ = 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

Flow of Solids and Slurries in Rotary Drums

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

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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

Advances in Engineering Fluid Mechanics

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

Flow of Solids and Slurries in Rotary Drums

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.

Advances in Engineering Fluid Mechanics

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.

Flow of Solids and Slurries in Rotary Drums

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.

This Page Intentionally Left Blank

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

Advances in Engineering Fluid Mechanics

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]

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

• 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

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

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

Advances in Engineering Fluid Mechanics

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 <25

40 76 75 -75 -75 -75 71 116 247

1,710 1,600 -1,600 -1,600 -1,600 1,700 2,305 2,245

m

$ -. 3

In

Solids

Technique

9

Alumina FCC FCC FCC

Iso-kinetic Momentum probe Momentum probe Momentum probe Momentum probe Momentum probe Iso-kinetic Pitot-static tube Pitot-static tube

co,5

FCC

FCC FCC Glass Glass

s. a

g

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

20 n

-150

1

\

\

-100

-50

0

267

r

150

r, mm Figure 4. Gas velocity profile (after van Breugel et al. [66]).

calibration is necessary, especially in the wall region. Gas dispersion and mixing are best studied with other techniques as described below. Steady State Tracers Many techniques to measure riser gas hydrodynamics commonly practiced today were used by van Zoonen in a 0.05 m diameter tube 10 m tall [71]. He studied radial and axial gas diffusion with a steady state tracer, radial and axial solids diffusion using FCC catalyst tagged with 5% ammonium chloride as a tracer, radial density with a "capacitometer" (now commonly called a capacitance probe), and radial particle velocity with a Prandtl tube (static-Pitot tube). A wide range of gas velocities and mass flux were studied and, in general, the data agree qualitatively with recent publications. Radial and axial gas mixing experiments were conducted by injecting methane gas through a tube "iso-kinetically" and sampling upstream and downstream at different radial positions. He measured methane concentration

268

Advances in Engineering Fluid Meciianics

with a thermal conductivity detector, and small mixing vessels were placed in the lines to dampen out large concentration fluctuations. Cankurt and Yerushalmi employed steady state tracers, similar to that described by van Zoonen, and measured diffusion over the entire fluidization regime, from bubbling beds to turbulent to fast fluidization [72]. Yang et al. [73] used helium as a tracer and presented data that essentially agreed with Cankurt and Yerushalmi's conclusions that the central region of the riser is in plug flow. However, they detected axial mixing in the vicinity of the wall, which had not been mentioned in the previous study. As shown in Figure 5, helium concentrations, which were measured 50 mm below the injector, are essentially zero in the center and are much higher at the wall. The concentration of helium is also greater lower down (nearer to the solids entry), which suggests mixing decreases with height. Weinstein and co-workers developed elaborate sampling and injection manifolds to measure gas backmixing in the entrance region, or dense zone and "fully

1.0

U^= 2.3 m / s 2 G^= 32 k g / m s

0.8

• 2.3 m above solids inlet V 4.5 m above solids inlet

0.6

%C 0.4

0.2

0.0

0

10

20

30

40

50

57.5

r, mm Figure 5. Radial tracer concentration distribution (after Yang et al. [73]).

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

269

developed" region or dilute zone [75,76]. They proposed that, unlike bubbling and slugging fluidized beds where heterogeneity is mainly distributed in time, heterogeneity in fast fluidization is mainly distributed in space. Therefore, spatial averaging is required, not temporal averaging; to adequately measure gas mixing, multiple sampling nozzles are required at each height. They measured radial and axial helium concentrations and the axial voidage profile simultaneously to correlate bed structure with mixing. They concluded that gas backmixing in fast beds is comparable to that in turbulent beds, circumferential mixing is considerable, backmixing in the transition region between the developing and "fully" developed zones is comparable to the dense zone or entrance region, and backmixing is present even in the dilute region. In Table 4, operating conditions, riser geometry and particle characteristics of a number of studies using steady state tracers is summarized. Generally, most of the data is either directed at operating conditions typical of coal combustors or FCC risers. The effect of various parameters on mixing behavior may be determined qualitatively with steady state tracers. However, as shown by van Bruegel et aL, there is a significant radial gas velocity profile [66]. Quantitative interpretation of steady state tracer data requires a knowledge of this profile. Radial dispersion can then be fitted to experimental data using a classical Fickian model, as proposed by Martin et al. [83] Therefore, steady state tracers do not provide sufficient information to entirely characterize gas hydrodynamics. Other experimental techniques, such as intrusive probes, are necessary. Furthermore, if significant downflow along the wall is present, a multi-zone model is required with multiple boundary conditions to account for tracer that is recirculated in the measuring element. Werther et al. are exploring the magnitude of gas downflow at the wall in a large 0.4 m diameter riser [80]. Impulse Tracers While steady state tracers are useful in measuring axial and radial dispersion, when combined with experimental radial gas velocity profiles, unsteady state tracers are more appropriate for assessing bypassing and local velocity. A variety of gas tracers have been developed to assess bypassing and a few models have been proposed for scale-up purposes. Van Breugel et al. were first to publish data using methane as an unsteady state tracer and showed that gas velocities in the central region are considerably greater than superficial even when corrected for solids hold-up [66]. Methane was injected at the center and detected at the center some distance upstream using flame ionization. Brereton et al. injected helium at a constant rate into the windbox below the distributor and detected in the exit duct downstream of the riser [96]. They shut the flow off with a solenoid valve to ensure a step change in the inlet concentration. A thermal conductivity cell was used to detect the helium, and it was designed to minimize dispersion. The sampling/detector response was subtracted to determine the overall column response. Their data agreed with earlier work indicating little downstream dispersion and, except at the wall, little upstream mixing, but they showed that there were substantial dead times. Their data is an average over the total length and included entrance and exit effects.

Table 4 Steady State Tracers

Study

van Zoonen [7 11 Cankurt et al. [72] Yang et al. [73] Adams [74] Bader et al. [69] Li and Weinstein [75] Weinstein et al. [76] Werther et al. [77-801 Li and Wu [81] Zethraeus et al. [82] Martin et al. [83] Zheng et al. [84]

Amos et al. [85] Arena et al. [86] Win et al. [87]

D rn

0.05 0.152 0.1 15 0.3x0.4 0.3~0.4 0.305 0.152 0.152 0.4 0.09 0.3~0.4 0.19 0.102 0.102 0.102 0.102 0.305 0.12 0.05

H rn

10 8.5 8 4 4 12.2 8 8 8.5 8 4 11.7 5.3 5.3 5.3 5.3 6.6 58 1.5

Riser

",

2

Particle Properties

Gs

"P

m/s

kg/rn2s

pm

1.5-12 0.2-5.6 2.8-5.3 3.8-5 3.8-5 3.6-6.1 0.03-5 1.3-4.0 3.0-6.5 1-2.5 3.4-5 3.8-7 4.4-6.1 4.4-6.1 4.4-6.1 4.4-6.1 2.6-5 15 1-1.3

70 155 220 200 250 76 59 59 130 58 200 62 567 701 364 560 71 89 3000 4500 6000

<250 <60 <60 <60 <60
PP kg/m3 1,070 3,300 1,710 1,450 1,450 2,600 1,575 2,800 1,560 1,392 1,392 2,560 2,560 2,450 2,540 1,250 1,250 1,250

C

nl o 3

Solids

FCC FCC Silica Dolomite Sand FCC FCC FCC Sand FCC Sand FCC Resins Resins Sand Sand Alumina Ballotini Alumindsand Alumindsand Alumindsand

Tracer Hz CH4

He CH4 CH4

He He He CO, Hz CH4 He

co2 co2 co2 co2 SF6

co2

CO, CO, CO,

$

5'

5

Ca. 3

5. 3 (D

co

7-1 -

a E.

1

8

5

5 nl

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

271

Dry and co-workers developed gas impulse injection techniques to measure contact efficiency and RTD (residence time distribution) [89-95]. In the contact efficiency studies [89,92,93,95], pulses of hot air, equivalent to a reacting tracer, were introduced at a height of 0.4 m and detected in the horizontal exit duct upstream of the cyclone. Hot gas transfers energy mainly to the solids; because of the high solids density and heat capacity, solids temperature does not change substantially; however, gas temperature changes considerably. Therefore, exit temperature is an indirect measure of gas-solid contacting. They constructed an aspirating probe with a rapid-response thermocouple covered by a thin filter to keep it free of solids. The probe was positioned at the exit duct because radial gas concentration profiles are minimal in this region. Pulse lengths were 2 to 5 s. To minimize pressure perturbations due to gas injection, they bled off air from the riser air supply to drive the heated gas. They report gas-solids contacting is almost complete at 2 m/s at a mass flux exceeding 100 kg/m^s, but that it decreases with increasing gas velocity. At gas velocities of 8 m/s, contact efficiencies were less than 90% even as the mass flux approached 200 kg/mh. To measure gas RTD, Dry and'co-workers [90,91,94] used a pair of rapid-response mass spectrometers to detect a pulse injection of argon below the solids injection point into the main air supply. They measured the concentration as a function of time at two levels and extracted system transfer characteristics using variance in both signals and peak-to-peak velocities. In one study, they positioned one suction probe near the entrance region of the riser, and the second was placed 5.2 m above it [90]. In a subsequent communication [91], they put the first suction probe in the main air supply line and the second 7.2 m higher up in the exit line, similar to the approach adopted by Brereton et al. [96]. To characterize the large velocities measured, they introduced an acceleration factor, defined as the ratio of peak-topeak velocity to superficial velocity; a maximum near three was recorded for the acceleration factor at a high solids mass flux and 4 m/s, as shown in Figure 6. Many studies report gas phase RTD data using both radioactive argon and krypton (Martin et al. [83], Viitanen, [99], Patience and Chaouki, [98], and Contractor et al. [28]). Relatively small quantities of gas are required to obtain good signalto-noise ratios from radioactive sources, which minimizes injection volumes required and perturbations of the gas flow. Radioactivity measurements are made in-situ (nonintrusive detection), on-line, and, in small diameter units, represent a radial average. Argon tracer is produced by irradiation in the neutron flux of a nuclear reactor. "^^Ar is converted to "^^Ar, which emits high energy gamma rays that easily penetrate metal walls. It has a half life of about 2 h and is ideal for conducting many experiments, but requires good logistics to minimize transport time between riser and nuclear reactor. ^^Kr also is used as a tracer but has a half life of about 10 years. Its major disadvantage is that it is a low gamma emitter; therefore, large doses are required to get an adequate signal-to-noise ratio. Few injections are possible because one quickly exhausts site radioactive permit levels. ^^Kr is probably an ideal tracer: It has a sufficiently long half life, 34 h, and is a good high-energy gamma emitter. Nal scintillation detectors are typically used to detect gamma rays. Up to about 10,000 counts/s, the signal is linear, and above this level dead-time or pile-up becomes noticeable for a 7.6 cm diameter unit. (Background radiation is typically less than 100 counts/s). Lead collimaters provide adequate shields to narrow the

272

Advances in Engineering Fluid Mechanics

3.5 I

1

1

1

1

1

r

CM

+

3.0

o o •fH

^'O

o 0)

2.0

r—I

0) O O

<

1.5

• High m a s s flux O Low mass flux

O

1.0

8

U^, m / s Figure 6. Acceleration factor vs. superficial gas velocity (after White and Dry [92]). aperture so that the face of the scintillator is only seeing radioactivity directly in front of it. Collimators are calibrated by dragging a radioactive source in front of the housing at a known velocity. Data published by Viitanen [99] using ^'Ar are unique for three reasons: They were collected on a full scale riser, 16 detectors were used with up to three at any one level to evaluate radial transport effects; and, all the data are tabulated! Detectors were positioned at seven levels, four of which had three detectors to evaluate radial distribution. The one meter diameter riser was operated at a gas velocity of about 7 m/s at the entrance and a solids mass flux of 488 kg/m^s. In the flanged upper section, the riser was 1.3 m in diameter, gas velocities exceeded 12 m/s, and solids mass flux was 289 kg/m^s. In Figure 7, data are illustrated at three heights. Peaks are broadening and becoming shorter with increasing height.

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

u

0.3 [-

~

P

^

1 I

0.2 h

^

^

G*=290 kg/m s

P k

^

Y

O Z= 7.4 m H

Y\

d)

E(t)

^

273

^ Z=22.1 m

/ !•

D Z = 31.9 m

1 L V°\

1 FA,

0.1 h

IJ /

I

T\

1

mj ^ ° 0.0

1—i

B ^ \>=i„ 1

I 1 V q i o ^^ae^T^'-yy^^^ySjyiDaq]

0

1

2

3 Time, s

4

5

6

Figure 7. Radioactive gas tracer (RTD) (data after Viitanen [99]).

At the 7.4 m and 22.1 m levels, data from all three detectors is included. The detectors are positioned perpendicular to the riser; one is pointed at the middle, and the other two are placed on either side and parallel such that they see more of the wall region. Peak maximums of the central detector occur sooner than the outer detectors; tail lengths of the outer detectors appear to be slightly longer. Computed residence times based on peak maximums were 20-40% shorter than that calculated based on the first moment, which suggests that "core" velocities are greater than the average velocity. This data is consistent with measurements of Contractor et al in a 0.15 m diameter riser [28]. At gas velocities around 6 m/s and the same mass flux, peak heights decreased along the length of the riser. However, at 590 kg/m^s and a gas velocity of 8 m/s, peak maximums were invariant with height, which suggests the gas was in plug flow. Interpreting radioactive tracer data is complicated by the flow structure, injection pulse and detector geometry. Detection dispersion arises from collimator geometry

274

Advances in Engineering Fluid Mechanics

and distance between the radioactive source and Nal scintillator (Patience and Chaouki [98]). Therefore, two detectors in series are preferred to interpret data; the response signal of the detector at the lower level is used as the inlet boundary condition for the upper detector. This approach is valid for a simple flow domain such as plug flow, axial dispersion, and core-annular flow with a stagnant annulus (where annular radioactivity is negligible compared to that in the core). However, for many riser situations significant radial gas profiles have been reported so using the lower detector as an inlet boundary condition is not straightforward. Scintillators (when adequately collimated) detect radioactivity in front of them; slow moving radioactive gas in the wall region contributes more to the signal than does core gas moving at high velocity because it spends more time in front of the detector. Therefore, an RTD curve with a long tail could represent either mass transfer between core and annulus, or could be attributable to slow-moving radioactive gas at the wall. An extensive list of publications with operating conditions and experimental techniques used is summarized in Table 5. As suggested, RTD measurements should be combined with other techniques to best quantify riser gas-phase hydrodynamics. Injection and detection methods are critical to interpreting the data. Iso-kinetic injection at different radii may help deconvolute inlet boundary conditions and flow structure. Multiple detectors along the riser length also are preferred. However, combining radial gas sampling, as practiced with steady state tracers, with radioactive impulse experiments could provide sufficient data to completely characterize riser gas-phase hydrodynamics. Optical Tracers Yang et al. [101] measured slip velocities between the solids phase and fine talc powder (d = 1 |im). Because the terminal velocity of the talc powder is low, they assume that it adequately traces the gas phase. They used backward scatter fiber optic probe LDV to measure particle velocities of both FCC and talc. Doppler frequency relates to particle velocity, and signal amplitude (scattered Hght intensity) relates to particle size. Therefore, in principle, both the gas phase (talc) and the solids phase may be measured simultaneously by cross-correlating two probe signals. The probe consists of optical fibers. One fiber provides illumination while the other detects reflected light from passing particles. Mineo et al. applied the same concept as Yang et al. with a fluorescent powder and light reflection signals instead of LDV [102]. Results from Yang et al.'s study are in good agreement with other work: gas velocity in the core region increases with solids circulation rate; it decreases near the wall; and, gas wall velocities are zero [101]. Other optical techniques, used to measure solids velocities, also may be applied to infer gas velocities. For example, in the core region suspension densities are low and particles may be reasonably well-dispersed; therefore, the slip velocity is approximately equal to the particle terminal velocity. The measured particle velocity in a dispersed suspension of solids then can be readily translated into a local gas velocity. Horio et al. describe a two-color optical fiber probe to measure gas phase hydrodynamics [102]. Ozone is injected into the riser from a 2 mm ID nozzle only 20 mm upstream of the measuring section. This section is exposed to ultraviolet

Table 5 Impulse Tracers

Riser Study

van Breugel et al. [66] Bernard et al. [88] Dry et al. [89] White and Dry [90] Dry and White [91] White and Dry [92] Dry and White [93] White et al. [94] Dry et al. [95] Brereton et al. [96] Bai et al. [97] Martin et al. [83] Patience and Chaouki [98] Viitanen [99] Contractor et al. [28] Zhang et al. [loo]

D m

H m

",

m/s

Gs

kg/m2s

Particle Properties d~ PP pm kg/m3

Solids

Tracer

Alumina FCC FCC FCC FCC FCC Sand Sand FCC Sand

CO,, Ar 79Kr Hot A.ir Ar Ar Hot Air Hot Air Hot Air Ar Ar

Zicron Sand Silica Gel FCC Sand FCC FCC FCC FCC

Hot Air He Organic 79Kr 4'Ar 41Ar 41Ar 85Kr 79Kr

276

Advances in Engineering Fluid Mechanics

light, and as the ozone cloud passes the UV light transmittance decreases. They positioned lighting and detecting optical fibers to measure velocity. They tested the effect of injecting gas along the tube axis in the direction of flow, against the direction of flow and across the radius. Injecting with the flow gave the best results while injecting against the flow yielded an average velocity about 3% too high; cross flow underpredicted the gas velocity by 4%. Optical experiments are best conducted at low solids loading. Scaling the technique to large industrial equipment is not practical. They may measure gas velocity reliably, but other experiments are required to measure radial dispersion and mixing. A summary of the experimental conditions studied with the techniques described above is given in Table 6. Chemical Reaction Data generated using the experimental techniques described above are used to formulate hydrodynamic models that may be used to predict reactor performance. In this section, studies that employ chemical reactions to evaluate mass transfer and contacting efficiency are described. Selected references are shown in Table 7. Dry et al, have applied hot air pulses as a "reacting" tracer [87]. Chemical reactions used to probe gas phase hydrodynamics include thermal decomposition of sodium bicarbonate, ozone decomposition, coal combustion, and FCC coke combustion. Because of the complexity of combustion kinetics, coupling kinetics and hydrodynamics into a single comprehensive model is not generally pursued. Instead, many successful "hydrodynamic" studies vary operational parameters and study the effect on combustion performance parameters. Moe et al. [22] characterized combustion performance with seven parameters: (1) heat transfer, (2) combustion efficiency, (3) bottom ash/total ash, (4) bed grain size, (5) limestone utilization, sulfur capture, and Ca/S (6) CO emissions, and (7) NO^ and N2O emissions. Eight operational variables they listed that impact one or more of the performance parameters were: (1) bed temperature—affects carbon burnout, emissions, sorbent utilization, and heat absorption; (2) primary/secondary air split—impacts NO^ emissions, temperature distribution, and pressure drop; (3) excess air—changes thermal efficiency, emissions, and carbon burnout; (4) solids circulation rate—controls load, heat absorption pattern, heat transfer coefficient, and pressure drop; (5) fuel size—determines carbon burnout, bed vs. fly ash split, and pressure drop; (6) limestone size—determines Ca/S ratio required and bed vs. fly ash split; (7) Ca/S ratio—impacts sulfur capture, limestone utilization, waste/disposal volumes, particulate generation, and emissions; and, (8) load—effects heat absorption, emission, carbon burnout, thermal efficiency, and temperature distribution. Some studies, such as Fujima et al, concentrate on one aspect of this multi-faceted problem [105]. For example, they set out to show the relationship between sulfur capture, particle concentration, and firing conditions. Boyd and Friedman, on the other hand, summarize the effect of many process variables on combustion performance [107]. The final DOE report concerning the NUCLA experimental program concludes that radial gas mixing is poor: Gas samples withdrawn across the radius of the reactor at two heights 7 m apart had similar concentration profiles [106]. They also found that secondary air had little effect on gas phase hydrodynamics

Table 6 Optical Methods Riser Study

Yang et al. [I011 Horio et al. [I021 Donsi and Osseo [I031

D m

H m

0.14 0.05 0.1~0.04

11 2.5

m/s

GS kg/m2s

2.5-6.3 4-7 4.6

"Q

Particle Properties P PP pm kg/m3

54 106 94

1,545 2,560 2,600

Solids

Technique

FCC Sand Ballotini

LDVItalc powder Ozone/UV light Smoke

Table 7 Chemical Reaction

Study

Hartke et al. [104] Fujima et al. [I051 Boyd and Friedman [I071 Basu et al. [I081 Kagawa et al. [I091 Jiang et al. [I101 Wei et al. [ I l l ] Ouyang et al. [112] Ouyang and Potter [I131 Vollert and Werther [I 141 Tjatjopoulos et al. [33]

D m

Riser H m

"Q

m/s

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

Reaction Solids

Glass/Na,CO, Sand Coal Coal FCC FCC FCC FCC FCC MnOJCuO FCC

NaHCO, decomposition Sulphur capture Combustion Sulphur capture Ozone decomposition Ozone decomposition Coke combustion Ozone decomposition Ozone decomposition NO oxidation CH, oxidative coupling

278

Advances in Engineering Fluid Mechanics

but that varying the coal feed injection points affected calcium utilization and NOx emissions significantly. Best performance was obtained with an even distribution. Combustion efficiency, boiler efficiency, and auxiliary power consumption did not vary much with coal feed injection distribution. Wei et al. [ I l l ] investigated mass and heat transfer in a commercial FCC regenerator using 16 gas and solids sampling probes. Gas samples were analyzed for oxygen, CO and C02 using a thermal conductivity detector. Under their operating regime, which is quite different than CFB combustors, they also found that concentration profiles did not change considerably with height, which suggests that radial mass transfer rates are low. They characterized their data using a onedimensional pseudo-dispersion model and showed that the gas phase is not in plug flow. Ozone decomposition is most often used as a chemical gas phase tracer because the reaction kinetics are easily measured. Three independent research groups have adopted this reaction to quantify radial mass transfer in CFB risers [109-112]. Results from these studies agree with other publications that suggest gas is not in plug flow. They show significant radial concentration gradients with the lowest concentrations of ozone in the wall region, which is consistent with solids studies that show highest solids concentrations at the wall. Deducing gas phase hydrodynamics based on radial and axial concentration measurements is the ideal manner in which to scale-up. However, this approach is often prohibitive in terms of cost and time investment so we rely on cold flow experiments for hydrodynamic scaling. HYDRODYNAMIC MODELING We often approximate the riser radial flow structure by assuming it consists of two characteristic regions: a dilute gas-solid suspension preferentially traveling upward in the center (core) and a dense phase of particle clusters or strands descending near the wall (annulus), as shown in Figure 8. Berruti and Kalogerakis proposed a core-annular model where gas travels in plug flow in the core [115]. They assumed that the slip velocity in the core equals the particle terminal velocity, the annular suspension density is equal to that at minimum fluidization, and that this dense suspension descends along the wall at the particle terminal velocity. This model is a good first approximation to the actual flow structure and superior to earlier studies that assumed the gas phase was in plug flow over the entire cross section. However, as discussed, it is now clear that a significant radial gas velocity profile exists. In this section, we discuss the radial profile and elaborate on modeling efforts around gas mixing. Radial Gas Velocity Profiles Along the riser, solids introduced into a flowing gas stream affect the radial gas velocity profile significantly. Gas velocities in industrial risers usually are high enough to be considered in the turbulent flow regime. As we introduce solids into the gas stream, the radial velocity profile changes; it becomes progressively more parabolic. White and Dry [90], for example, report centerline gas velocities reaching

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

279

t

Wail Region

Figure 8. Core-annular nnodel. up to three times the value of the gas superficial velocity. Earlier, van Breugel et al showed a near linear relationship between gas velocity and radial position at a mass flux of 390 kg/m^s in a 0.3 m diameter column; they measured centerline velocities of 17 m/s while the gas superficial velocity was only 6.3 m/s [66]. Martin et al found that the gas profiles shifted from being relatively flat under very dilute conditions to a triangular shape under dense flow [83]. They introduced an empirical expression, formally identical to the Ostwald de-Waele law, which describes laminar flow of non-Newtonian liquids: U, _ 3n + l U n +1 ^-IR

(2)

Zethraeus et al [82] described the observed gas velocity profiles in the core region of their riser with the 1/7* power-law. In the annular region, they assumed gas travels upward at a constant and low velocity.

280

Advances in Engineering Fluid Mechanics

Yang et al. [68] also interpreted their experimental radial gas velocity data using an empirical power-law expression

where the exponent m varies with gas velocity, solids circulation rate, and elevation. The experimental results indicate that the gas velocity profile becomes steeper with increasing solids circulation and decreasing gas superficial velocity. The effect of gas superficial velocity, however, appears to be significant only at low solids fluxes and weakens with riser height. The effect of elevation is attributed to the progressive decrease in annular solids flow along the axis, corresponding to a smaller interaction between upward and downward moving particles and gas. In this chapter, we propose a simple expression for the radial gas velocity profile U, _ a + 2 U,

oc

iiJ

(4)

where the exponent, a, will typically vary between 1 and 7. A value of one results in a triangular profile, two is parabolic, and seven closely approximates a turbulent profile, as shown in Figure 9. In Figure 4, data collected by van Breugel are compared with the model with a equal to 1.2 [66]. Equation 4 is equivalent to Martin et al.'s expression [83]. The term (a + 2)/a is equivalent to the acceleration factor described by Dry et al. [90,91]. We regressed experimental data of a number of studies [66,83,90,91,94,96,98] and found that a varies most strongly with gas velocity, solids mass flux, and riser diameter and obtained the following expression a = 1 + 0.036Fr(Ug/Up)'^2

(5)

The data set included particle velocity measurements of Yang et al. [125] and Rhodes et al. [126]. We assumed that the slip velocity at the center was equal to the particle terminal velocity. Gas Mixing The number and complexity of models proposed in the literature to describe gas mixing have been increasing over the past years. A summary of the most important contributions is given in Table 8. Cankurt and Yerushalmi originally proposed that the extent of backmixing in the riser was minimal and plug flow was a good approximation [72]. However, as discussed, solids hold-up and gas velocity vary across the radius. To model experimental tracer data, three approaches have emerged. In the first, radial and/or axial dispersion are considered as the dominant mixing phenomena. In the second, a core-annular flow structure is assumed, as illustrated in Figure 8, then parameters are introduced to describe mixing between the two regions. Generally, studies that employ impulse tracers adopt a core-annular

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

3.0

T

\ 2.5

2.0

1.5

281

\

a=l \N

\\

\

a =2

a =7

1.0 h

0.5

•--U^/U^=3[l-(r/R)]

~ VUg=2[l-(r/R)']

0.0 0.0 Figure 9. Reduced gas velocity profile as a function of exponent a (Equation 4).

approximation to characterize gas hydrodynamics while studies that rely on steady state tracers use dispersion models. Martin et al. tested both experimental techniques and proposed a model that includes radial dispersion and the radial velocity profile [83]. In the third modeling approach, gas-solids contact efficiency is evaluated by introducing a reactive tracer. Radial and Axial Dispersion Radial and axial dispersion measurements in risers were first reported by van Zoonen [71]. With hydrogen gas as a tracer he found that radial dispersion coefficients ranged between 2.5 and 36 cmVs and axial gas dispersion coefficients were between 4,500 and 14,400 cmVs.

h)

03

Table 8 Modelling Studies Gas Mixing: Radial andlor Axial Dispersion

Study

Tracer/ Technique

Van Zoonen [71]

H, cont.

Cankurt et al. [72] Yang et al. [73] Adams [74] Bader et al. [69] Brereton et al. [96] Martin et al. [83] Contractor et al. [28] Viitanen [99]

CH, cont. H, cont. CH, cont. H, cont. H, step 79Krpulse He cont. s5Krpulse 79Krpulse

d,

pm 20-150 55 220 250 76

p,

kg/rn3

Solids

1,500 FCC

D rn

0.051

h)

h< P,

o 3

%

Model axialIradial disp.

1,100 FCC 0.152 plug flow silica gel 0.1 15 radial disp. 2,600 sand 0.3~0.4radial disp. 1,710 FCC 0.305 radiallaxial disp.

148 2,650 sand 62 1,560 FCC 62 1,560 FCC 70 1,570 VPO* 70 -1,500 FCC

0.152 0.94 0.19 0.15 1.0,1.3

axial disp. axial disp. radial disp. plug flow axial disp.

148 46 130 71 277 71 140

0.152 0.1 0.4 0.305 0.083 0.09 0.09

rc/R = 0.78 rc/R = 0.85 rc/R = 0.85

Dr = 2.5 - 36 cm2/s Da = 4,500 - 14,000 cm2/s

*Vanadium phosphorous oxide catalyst

2,650 2,300 2,600 2,450 2,630 1,370 2,650

sand FCC sand alumina sand FCC sand

S

co. 3 (D

Dr = 2.6 - 7.1 cm2/s Dr = 50 - 70 cm2/s Dr = 25 - 67 cm2/s Dr = 645 - 1,613 cm2/s Da = 6,600 - 119,000 cm2/s Da = 54,000 cm2/s Dr = 15 - 30 cm2/s

2. 3

a

a

E.

n

z gs

P,

$. V)

Da = 90,000 - 200,000 cm21s

Gas Mixing: Core-Annular Flow He cont. Brereton et al. [96] 0, cont. Kagawa et al. [I091 COz cont. Werther er al. [78] Amos et al. [85] S F , cont. Patience and Chaouki [98] Ar pulse Ar White et al. [94] Ar

-.

3

k = 0.08 - 0.11 d s k = 0.001 d s Drcore = 20 - 60 cm2/s Drcore = 30 - 40 cm2/s rc/R = 0.72 - 0.99 k = 0.03 - 0.1 rnls rc/R = 0.5 - 0.95 k = 0.02 - 0.07 d s rc/R = 0.7 - 0.9 k = 0.01 - 0.03 d s

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

283

Yang et al. also adopted a radial dispersion model to describe gas mixing in their 0.115 m diameter riser [73]. They reported values of the radial dispersion coefficient in the range of 2.6 to 7.1 cmVs; dispersion increased with solids suspension density and was invariant with the axial distance. In addition, they concluded that axial gas mixing could be neglected. Adams reported radial dispersion coefficients in the range of 50 to 70 cmVs using methane as a tracer [71]. He found that dispersion increased with gas velocity and decreased with solids suspension density, presumably due to the suppression of turbulent intensity caused by the presence of solids. Bader et al. studied both radial and axial gas dispersion using steady-state helium tracer injection [69]. Radial dispersion coefficients ranged between 25 and 67 cmVs. Axial and radial dispersion coefficients were fit to the exact solution of the following differential equation:

Axial dispersion coefficients ranged between 645 and 1,613 cmVs for a constant radial dispersion coefficient equal to 39 cmVs. These high values indicate axial dispersion is negligible. Brereton et al. characterized riser gas mixing with a pseudo-axial dispersion coefficient [96]. They concluded that axial dispersion increases with total pressure drop across the riser (approximately proportional to the suspension density). In addition, they found the smooth exit configuration increased axial dispersion coefficients compared to an abrupt exit. They attributed this phenomenon to a more uniform solids distribution in the case of the abrupt exit which, in turn, corresponds to a decreased irregularity in the upward and downward solids movement primarily responsible for the axial mixing of the gas. The experimental values, reported as D /U L, ranged between 0.01 and 0.18 (D = 6,600 to 119,000 cmVs). Martin et al. conducted several hydrodynamic studies using a 0.19 m diameter, 11.7 m tall cylindrical riser with FCC catalyst [83]. They adopted a radial dispersion model to interpret their data. They were the first to combine the radial gas velocity profile and radial dispersion and reported coefficients in the range of 15 to 30 cmVs, increasing slightly with solids circulation flux. Experiments also were conducted by injecting pulses of krypton tracer gas to investigate the axial dispersion characteristics. Axial gas dispersion coefficients varied between 5.1 and 5.6 mVs at high gas velocities (11 to 14 m/s) and high solids fluxes (of the order of 300 kg/m^s). Contractor et al. published data showing that axial gas dispersion decreases with increasing solids mass flux and that the gas is essentially in plug flow at solid fluxes in excess of 500 kg/m^s [28]. This data was obtained under very high density pneumatic transport conditions (voidages of the order of 0.85), where the coreannular flow structure does not appear to exist. Data obtained at lower mass flux resemble that generated by Viitanen [99] in a 1.3 m diameter industrial riser operating under reaction conditions. Viitanen [99] proposes that a one-dimensional axial dispersion model can adequately characterize the data. Concentration profiles in a different industrial riser, reported by Zhang [100], resemble those of Viitanen [99] and Contractor et al. [28].

284

Advances In Engineering Fluid Mechanics

Core-Annular Flow Although Brereton et al. concluded that axial dispersion was small (except at the wall), they showed that a plug flow model with a single dispersion parameter could not adequately characterize their RTD data [96]. Therefore, they proposed a coreannular model, as illustrated in Figure 8. They assumed all the gas passes upwards in a core of constant cross section and that it is surrounded by a stagnant annular region. Both regions are assumed to be well-mixed radially and axial dispersion as well as velocity gradients are neglected. Mass transfer is assumed to occur between the two zones, due to both diffusion and convection induced by particle motion, and is described by a crossflow coefficient, k. The material balances for the two zones are:

aC. at

2kr (C.-CJ = 0 R' - r?

(8)

The experimental results were fitted by adjusting the two model parameters, namely the crossflow coefficient, k, and the core radius, r^, considered invariant with height. Best fit values for k ranged from 0.08 to 0.11 ni/s, and r^R was equal to 0.78. White and Dry studied the flow structure in a 0.09 cm ID high velocity fluidized bed, using argon tracer gas pulses [90]. Their results showed that solids have a strong effect on the radial gas velocity profile (increasingly more parabolic with increasing solids flux). In addition, they indicated that the core-annular flow structure, evident at gas velocities below 4 m/s, breaks down at higher gas velocities as shown in Figure 6. A general conclusion from their work is that particle clouds damp out turbulence but that the behavior cannot be simply described by a "laminarlike" expression. In a subsequent communication by White et al, they compared cross-flow coefficients of sand and FCC and found that k^^^ > 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

Advances in Engineering Fluid Mechanics

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

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

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

Advances in Engineering Fluid Mechanics

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

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

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

REFERENCES 1. Squires, A. M., "The Story of Fluid Catalytic Cracking: The First Circulating Fluid Bed'", in Circulating Fluidized Bed Technology (P. Basu, ed.), Pergamon Press, Toronto, 1986. 2. Reh, L., "The Circulating Fluid Bed Reactor—a Key to Efficient Gas/Solid Processing", in Circulating Fluidized Bed Technology (P. Basu, ed.), Pergamon Press, Toronto, 1986. 3. Reichle, A. D., "Fluid Catalytic Cracking Hits 50 Year Mark on the Run", Oil and Gas /., 90 (20), 41(1992). 4. King, D., "Fluidized Catalytic Crackers: An Engineering Review", in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992. 5. King, D., "Model Predicting Average Densities in Commercial FCC CFB Regenerators", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 6. Avidan, A. A., "FCC is Far from Being a Mature Technology", Oil and Gas y., 90 (20), 59(1992). 7. Jewel, J. W. and Johnson, W. B., "Method for Synthesis of Organic Compounds", U. S. Patent 2,543,974, assigned to M. W. Kellog Co., March 6, 1951. 8. Dry, M. E., "The Sasol Route to Fuels", ChemTech, 12, 744(1982). 9. Shingles, T. M. and McDonald, A. F., "Commercial Experience with Synthol CFB Reactors", in Circulating Fluidized Bed Technology II (P. Basu and J. F. Large, eds.), Pergamon Press, New York, 1986. 10. Silverman, R. W., Thompson, A. H., Steynberg, A., Yukawa, Y., and Shingles, T., "Development of a Dense Phase Fluidized Bed Fischer-Tropsch Reactor", in Fluidization V (K. Ostergaard and A. Sorenson, eds.). Engineering Foundation, New York, 1986. 11. Bartholomew, C. H., "Recent Developments in Fischer-Tropsch Catalysis, in New Trends in CO Activation" (L. Guczi, ed.). Studies in Surface Science and Catalysis, No. 64, Elsevier, Amsterdam, Netherlands, 1991. 12. Steynberg, A. P., Shingles, T., Dry, M. E., Jager, B., and Yukawa, Y., "Sasol Commercial Scale Experience with Synthol FEB and CFB Catalytic FischerTropsch Reactors", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio, and M. Hasatani, eds.), Pergamon Press, Oxford, 1991.

290

Advances in Engineering Fluid Mechanics

13. Chanchlani, K. G., Hudgins, R. R. and Silveston, P. L., "Methanol Synthesis Under Periodic Operation: An Experimental Investigation", Can. J. Chem. Eng., 72, 657(1994). 14. Schonfelder, H., Hinderer, J., Werther, J., and Kell, F., "Olefin Synthesis from Methanol (MTO) in a Circulating Fluidized Bed Reactor", Chem.- Ing.-Tech., 66 (7), 960(1994) in German. 15. Reh, L., "Fluidized Bed Processing", Chem. Eng. Prog., 67 (2), 58(1971). 16. Reeves, J. W., Sylvester, R. W., and Wells, D. F., "Chlorine and Iron Oxide from Ferric Chloride—Apparatus", U.S. Patent 4,282,185, assigned to E. I. du Pont de Nemours and Co., Inc., Aug. 4, 1981. 17. Zhiqing, Y., "Application Collocation", in Advances in Chemical Engineering, Vol 20. (M. Kwuak, ed.). Academic Press, New York, 1994. 18. Peinemann, B., Stockhausen, W., and McKenzie, L., "Experience with the Circulating Fluid Bed for Gold Roasting and Alumina Calcination", Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, NY, 1992. 19. Bohn M. and Benham, C , "An Experimental Investigation into Fast Pyrolysis of Biomass Using an Entrained Flow Reactor." Proceedings of Specialists' Workshop on Fast Pyrolysis of Biomass Proceedings SERI/CP - 622-1096, Colorado, 1980. 20. Engstrom, F. and Lee, Y. Y., "Future Challenges of Circulating Fluidized Bed Combustion Technology", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio, and M. Hasatani, eds.), Pergamon Press, Oxford, 1991. 21. Tang, J. T. and Curran, R. A., "Challenges and Strategies for CFB Boilers to Meet Stringent Emissions", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 22. Moe, T. A., Mann, M. D., Henderson, A. K., and Hajiced, D. R., "Pilot-Scale CFBC Systems A Valuable Tool for Design and Permitting", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 23. Anders, F., Beisswenger, H., and L. Plass, "Clean and Low Cost Energy from Atmospheric and Pressurized Lurgi CFB Systems", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio, and M. Hasatani, eds.), Pergamon Press, Oxford, 1991. 24. Hallstrom, C. and Karlsson, R., "Waste Incineration in Circulating Fluidized Bed Boilers: Test Results and Operating Experiences", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio, and M. Hasatani, eds.), Pergamon Press, Oxford, 1991. 25. Koyama, H. and Dranoff, J. S., "Modeling the Thermal Cracking of Ethane and Propane in a Non-Isothermal Vertical Pneumatic Transport Reactor", Ind. Eng. Chem. Res., 31, 2,265(1992). 26. Herrmann, E. and Weisweiler, W., "Circulating Fluidized-Bed Reactor for Catalytic Gas-Solid Reactions", International Chem. Eng., 34 (2), 198(1994). 27. Contractor, R. M., "Improved Vapor Phase Catalytic Oxidation of Butane to Maleic Anhydride", U.S. Patent 4,668,802, issued to E.I. du Pont de Nemours and Co., May 26, 1987. 28. Contractor, R. M., Patience, G. S., Garnett, D. I., Horowitz, H. S., Sisler, G. M. and Bergna, H. E. "A New Process for n-Butane Oxidation to Maleic

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

29. 30.

31. 32. 33. 34.

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

291

Anhydride Using a Circulating Fluidized Bed Reactor", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. Pugsley, T. S., Patience, G. S., Berruti, F., and Chaouki, J., "Modeling the Catalytic Oxidation of n-Butane to Maleic Anhydride in a Circulating Fluidized Bed Reactor", Ind. Eng. Chem. Res., 31, 2,652(1992). Baerns, M., Mleczko, L., Tjatjapoulos, G. J., and Vasalos, I. A., "Comparative Simulation Studies on the Performance of Bubbling and High-Velocity Fluidized Bed Reactors for the Oxidative Coupling of Methane", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. Dutta, S. and Jazayeri, B., "Alternative Reactor Concepts for the Oxidative Coupling of Methane", in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992. Santamaria, J. M., Miro, E. E., and Wolf, E. E., "Reactor Simulation Studies of Methane Oxidative Coupling on a Na/NiTi02 Catalyst", Ind. Eng. Chem. Res., 30, 1,157(1991). Tjatjopoulos, G. J., Ketekides, P. T., latrides, K. K., and Vasolos, I. A., "Cold Flow Model and Computer Simulation Studies of a Circulating Fluidized Bed Reactor for the Oxidative Coupling of Methane", Catalysis Today, 21, 387 (1994). Lyons, J. E., "The Molecular Design, Synthesis and Application of New Catalytic Systems for the Selective Oxidation of Hydrocarbons to Fuels and Chemical Products" presented at 13th North American Meeting of the Catalysis Society, Pittsburgh, 1993. Lewis, W. K., Gilliland, E. R., and Reed, W. A., "Reaction of Methane with Copper Oxide in a Fluidized Bed", I&EC, 41 (6), 1,227(1949). Beuther, H., Innes, R. A., and Swift, H. E. "Process for Preparing Acrylonitirile" U.S. Patent 4,102,914 assigned to Gulf Research and Development Co., July 25, 1978. Gianetto, A., Pagliolico, S., Rovero, G., and Ruggeri, B., "Theoretical and Practical Aspects of Circulating Fluidized Bed Reactors (CFBRs) for Complex Chemical Systems", Chem. Eng. ScL, 45 (8), 2,219(1990). Huibers, D. Th. A., "Process for the Production of Acrylonitrile or Methacrylonitrile", U.S. Patent 3,478,082, assigned to The Lummus Company, Nov. 11, 1969. Kahney, R. H. and McMinn, T. D., "Paraffin Ammoxidation Process", U.S. Patent 4,000,178, assigned to Monsanto Company, Dec. 28, 1976. Kahney, R. H. and McMinn, T. D., "Laboratory Transported Bed Reactors", presented at 66th Annual AIChE Meeting, Philadelphia, 1973. Rollman, W. F., "Selective Oxidation with Suspended Catalyst", U.S. Patent 2,604,479 issued to Standard Oil Development Company, July 22, 1952. Johnson, A. J., "Oxidation of Olefins to Unsaturated Aldehydes and Ketones", U. S. Patent 3,102,147 issued to Shell Oil Co., Aug. 27, 1963. Callahan, J. L., Grasselli, R. K., Milberger, E. C , and Strecker, H. A., "Oxidation and Ammoxidation of Propylene over Bismuth Molybdate Catalyst", Ind. Eng. Chem. Prod. Res. Develop., 9 (2), 134(1970). Patience, G. S. and Mills, P. L., "Modelling of Propylene Oxidation in a Circulating Fluidized-Bed Reactor" in New Developments in Selective Oxidation II (V. Cortes Corberan and S. Vic Bellon, eds.), Elsevier, Amsterdam, 1994.

292

Advances in Engineering Fluid Mechanics

45. Wainwright, M. S. and Hoffman, T. W., "The Oxidation of o-Xylene in a Transported Bed Reactor", Chem. Reaction Eng. II, Advances in Chem. Sciences (H. M. Hulburt, ed.), ACS Washington, DC, 1974. 46. Gelbein, A. P., "Phthalic Anhydride Reaction System", U.S. Patent 4,261,899, assigned to Chem. Systems Inc., April 14, 1981. 47. Park, D. W. and Gau, G., "Simulation of Ethylene Epoxidation in a Multitubular Transport Reactor", Chem. Eng. Sci., 41 (1), 143(1986). 48. Tmenov, D. N., Svintson, N. I., Shapovalova, L. P., Tabakov, A. V., Dvoretskii, M. L., Vasiller, G. I., and Shestovskii, G. P. U. S. Patent 4,229,604 (1980), (cited by Murchison et al.). 49. Murchison, C. B., Vrieland, G. E., Khazai, B., and Weihl, E. D., "Anaerobic Oxidation of Butane to Butadiene on Magnesium Molybdate Catalyst", presented at 13th North American Meeting of the Catalysis Society, Pittsburgh, 1993. 50. Sanfilippo, D., Buonomo, P., Fusco, G., Lupiere, M., and Miracca, I., "Fluidized Bed Reactors for Paraffins Dehydrogenation", Chem. Eng. Sci., 47 (9-11), 2,313(1992). 51. Debras, G., Grootjans, J., and Delorme, L., "Process for the Catalytic Dehydrogenation of Alkylaromatic Hydrocarbons", European Patent 0 482 276 Al issued to FINA Research, S. A. (1992). 52. Woskow, M. Z., "Dehydrogenation Process Using Manganese Ferrite", U.S. Patent 3,420,912 assigned to Petro-Tex Corp., Jan. 7, 1969. 53. Woskow, M. Z., "Dehydrogenation Process", U.S. Patent 3,513,216, assigned to Petro-Tex Corp., May 19, 1970. 54. Coudurier, G., Decottignies, D., Loukah, M., and Vedrine, J. C , "Vanadium and Chromium Based Phosphates as Catalysts for Oxidative Dehydrogenation of Ethane", presented at 13th North American Meeting at the Catalysis Society, Pittsburgh, 1993. 55. Gallezot, P., Tretjak, S., Christidis, Y., Mattioda, G., and Schouteeten, A., "Oxidative Dehydrogenation of Ethylene Glycol on Silver Catalyst", presented at the 13th North American Meeting of the Catalysis Society, Pittsburgh, 1993. 56. Filho, R. M. and Domingues, A., "A Multitubular Reactor for Obtention of Acetaldehyde by Oxidation of Ethyl Alcohol", Chem. Eng. Sci., 47 (9-11), 2,571(1992). 57. Pan, H. Y., Benson, S., Minet, R. G., and Tsotsis, T. T., "A Catalytic Carrier Process for HCl Oxidation", presented at 13th North American Meeting of the Catalysis Society, Pittsburgh, 1993. 58. Peiri, E., Pinelli, D., and Trifiro, F., "Silica as Catalyst for Cyclohexanone Ammoximation with Molecular Oxygen: A Preliminary Approach to the Kinetic Analysis", Chem. Eng. Sci., 47 (9-11), 2,641(1992). 59. Zaza, P., de la Torre, A., and Renken, A., Chem.-Ing.-Tech., 63 (6), 640 (1992) (cited by Vollert and Werther). 60. Kunii, D. and Levenspiel, O., Fluidization Engineering, 2nd ed., ButterworthHeinemann, Boston, 1991. 61. Chehbouni, A., Chaouki, J., Guy, C , and Klvana, D., "Characterization of the Flow Transition between Bubbling and Turbulent Fluidization", Ind. Eng. Chem. Res., 33, 1,889(1994).

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

293

62. Kobro, H. and Brereton, C , "Control and Fuel Flexibility of Circulating Fluidized Beds", in Circulating Fluidized Bed Technology (P. Basu, ed.), Pergamon Press, Toronto, 1986. 63. Kunii, D. and Levenspiel, O., "Flow Modeling of Fast Fluidized Beds", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M. Hasatani, eds.), Pergamon Press, Oxford, 1991. 64. Yates, J. G. and Simons, S. J. R., "Experimental Methods in Fluidization Research", Int. J. Multiphase Flow, 20 (suppl.), 297(1994). 65. Bachalo, W. D., "Experimental Methods in Multiphase Flows", Int. J. Multiphase Flow, 20 (suppl.), 261(1994). 66. van Breugel, J. W., Stein, J. J. M., de Vries, R. J., "Isokinetic Sampling in a Dense Gas-Solids Stream", Proc. Instn. Mech. Engrs., 184 (3C), 1969. 67. Harris, B. J. and Davidson, J. F. "Velocity Profiles, Gas and Solids, in Fast Fluidized Beds" in Fluidization VII, (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992. 68. Yang, H., Gautam, M., and Mei, J. S., "Gas Velocity Distribution in a Circular Circulating Fluidized Bed Riser", Powder Technol, 78, 221 (1994). 69. Bader, R., Findlay, J. and Knowlton, T. M. "Gas/Solids Flow Patterns in a 30.5-cm-Diameter Circulating Fluidized Bed", Circulating Fluidized Bed Technology II (P. Basu and J. F. Large, ed.), Pergamon Press, Oxford, 1988. 70. Azzi, M., P. Turlier, Large, J. F. and Bernard, J. R. "Use of a Momentum Probe and y-Densitometry to Study Local Properties of Fast Fluidized Beds", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M. Hasatani, eds.), Pergamon Press, Oxford, 1991. 71. van Zoonen, D. "Measurements of Diffusional Phenomena and Velocity Profiles in a Vertical Riser", Proc. Symp. Interaction Between Fluids and Particles, Instn. Chem. Engrs., 1962. 72. Cankurt, N. T. and Yerushalmi, J., "Gas Backmixing in High Velocity Fluidized Beds", in Fluidization (J. F. Davidson and D. L. Keairns, eds.), Cambridge Univ. Press, Cambridge, 1978. 73. Yang, G., Huang, Z., and Zhao, L., "Radial Gas Dispersion in a Fast Fluidized Bed", in Fluidization IV (D. Kunii and R. Toei, eds.). Engineering Foundation, New York, 1983. 74. Adams, C. K., 1988, Gas Mixing in Fast Fluidised Beds", in Circulating Fluidized Bed Technology II (P. Basu and J. F. Large, eds.), Pergamon Press, Oxford, 1988. 75. Li, J. and Weinstein, H., "An Experimental Comparison of Gas Backmixing in Fluidized Beds Across the Regime Spectrum", Chem. Eng. ScL, 44, 1,697(1989). 76. Weinstein, H., Li, J., Bandlamudi, E., Feindt, H. J., and Graff, R. A., "Gas Backmixing of Fluidized Beds in Different Regimes and Different Regions", in Fluidization VI (J. R. Grace, L. W. Shemilt and M. A. Bergougnou, eds.). Engineering Foundation, New York, 1989. 77. Werther, J., Hartge, E.-U., Kruse, M., and Nowak, W., "Radial Mixing of Gas in the Core Zone of a Pilot Scale CFB", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M. Hasatani, eds.), Pergamon Press, Oxford, 1991.

294

Advances in Engineering Fluid Mechanics

78. Werther, J., Hartge, E.-U., and Kruse, M., "Radial Gas Mixing in the Upper Dilute Core of a Circulating Fluidized Bed", Powder TechnoL, 70, 293(1992). 79. Werther, J., Hartge, E.-U., and Kruse, M., "Gas Mixing and Interphase Mass Transfer in the Circulating Fluidized Bed", in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992. 80. Werther, J., Hartge, E.-U., Kruse, M., and Nowak, W., "Radial Mixing in the Core Zone of a Pilot Scale CFB", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 81. Li, Y. and Wu, P., "A Study on Axial Gas Mixing in a Fast Fluidized Bed", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M. Hasatani, eds.), Pergamon Press, Oxford, 1991. 82. Zethraeus, B., Adams, C , and Berge, N., "A Simple Model for Turbulent Gas Mixing in CFB Reactors", Powder TechnoL, 69, 101(1992). 83. Martin, M. P., Turlier, P., Bernard, J. R., and Wild, G., "Gas and Solid Behavior in Cracking Circulating Fluidized Beds", Powder TechnoL, 70, 249(1992). 84. Zheng, Q., Xing, W., and Fei, L. "Experimental Study on Radial Gas Dispersion and Its Enhancement in Circulating Fluidized Beds", in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992. 85. Amos, G., Rhodes, M. J., and Mineo, H., "Gas Mixing in Gas-Solids Risers", Chem. Eng. ScL, 48 (5), 943(1993). 86. Arena, U., Marzocchella, A., Bruzzi, V., and Massimilla, L., "Mixing Between a Gas-Solids Suspension Flowing in a Riser and a Lateral Gas Stream", in Circulating Fluidized Bed Technology IV (A. Avidan, ed.), AIChE, New York, 1994. 87. Win, K. K., Nowak, W., Matsuda, H., Hasatani, M., Kruse, M., and Werther, J., "Radial Gas Mixing in the Bottom Part of a Multi-Solid Fluidized Bed", y. Chem. Eng. Japan, 27 (5), 696(1994). 88. Bernard, J. R., Santos-Cottin, H., and Margrita, R., "Use of Radioactive Tracers for Studies on Fluidized Cracking Catalytic Plants", Isotopenpraxis, 25 (4), 161(1989). 89. Dry, R. J., Christensen, I. N. and White, C. C , "Gas-Solids Contact Efficiency in a High-Velocity Fluidised Bed", Powder TechnoL, 53, 243(1987). 90. White, C. C. and Dry, R. J., "Transmission Characteristics of Gas in a Circulating Fluidised Bed", Powder TechnoL, 57, 89(1989). 91. Dry, R. J. and White, C. C , "Gas Residence-Time Characteristics in a HighVelocity Circulating Fluidised Bed of FCC Catalyst", Powder TechnoL, 58, 17(1989). 92. White, C. C. and Dry, R. J., "The Effect of Particle Size on Gas-Solid Contact Efficiency in a Circulating Fluidized Bed", in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M. Hasatani, eds.), Pergamon Press, Oxford, 1991. 93. Dry, R. J. and White, C. C , "Gas-Solid Contact in a Circulating Fluidized Bed: The Effect of Particle Size", Powder TechnoL, 70, 277(1992). 94. White, C. C , Dry, R. J., and Potter, O. E., "Modelling Gas-Mixing in a 9 cm Diameter Circulating Fluidized Bed", in Fluidization VII (O. E. Potter and D. J. Nicklin, eds.). Engineering Foundation, New York, 1992

Gas Phase Hydrodynamics in Circulating Fluidized Bed Risers

295

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.

296

Advances in Engineering Fluid Mechanics

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

298

Advances in Engineering Fluid Mechanics

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 ,

299

[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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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.

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

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.

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

t /

/ / f /

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

t

-:i
(a)

* <

il^

(b)

Figure 2. CFD sinnulations of the circulation patterns produced by the PBT. Figures (a) and (b) show the correct results. The simulation conditions are: a) high clearance geometry with high clearance boundary condition, b) low clearance geometry with low clearance boundary condition,

Boundary Conditions Required for the CFD Simulation

I

I

I

I

I

307

;

/ ^ ^

_

\

V

If J W W W W ^ J

'•-•^.^^ ' ^

t/^ t^ ^

(c)

-^ •-

(d)

Figure 2. (continued) Figures (c) and (d) show the distortions produced when incorrect impeller models are used. The simulation conditions are: c) high clearance geometry with low clearance boundary condition, d) low clearance geometry with high clearance boundary condition.

308

Advances in Engineering Fluid Mechanics 7

(a) 16 r

0.2

0.4

0.6

0.8

2r/D

(b) Figure 3. Profiles of the predicted rate of dissipation of turbulence kinetic energy below the PBT for the high clearance geometry using four Impeller models: a) high clearance (correct) boundary conditions, b) low clearance boundary conditions,

Boundary Conditions Required for the CFD Simulation

(c)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2r/D

(d) Figure 3. (continued) c) axial velocity taken from the low clearance boundary condition, d) k and e set equal to zero. The traverses A through H are radial traverses with A just below the impeller, and H close to the bottom of the tank.

309

310

Advances in Engineering Fluid Mechanics

(text continued from page 299) the tank, which limits its applicability to a few specialized cases. Use of experimental boundary conditions limits the usefulness of CFD as a predictive tool, unless these conditions can be generalized in some way. There is, however, increasing experimental evidence that this may be possible. Modeling the impeller as a source of momentum has yet to be proven. Regardless of the method used, the impeller model must take the surrounding tank geometry into consideration if reliable results are to be generated. NUMERICAL ISSUES While many of the most difficult numerical issues which were encountered by the pioneers of CFD have been resolved and made transparent, two issues are still very much at the discretion of the user: accurate gridding and definition of convergence criteria. Gridding The accuracy and speed of convergence of a simulation is highly dependent on the grid definition in the computational domain. The grid must be fine enough to provide an accurate representation of the differential equations when they are transformed to the discretized form. Since the grid size defines the step size in the discretized form of the differential equations, grid refinement is required wherever rapid changes (e.g., steep velocity gradients) occur in the flow field. The final solution should be grid-independent. Further reduction in the size of the grid, or changes in the shape of the grid, should not affect the result. User manuals should be consulted for restrictions on the cell aspect ratios (typically of the order of 2), which limit extreme distortions of the grid, and for restrictions on changes in the size of neighboring cells. Applying symmetries in the gridding wherever symmetries occur in the flow frequently simplifies the computation, as does refining the grid in regions where the flow direction changes rapidly or the fluid is accelerated or decelerated Table 1 summarizes a review of the domain dimensions and the number of cells used in simulations of stirred tanks. Those entries noted with a * were tested successfully for grid independence. While the maximum, minimum, and mean values are given at the bottom of the table for convenience, the reader will gain more insight from considering the context of the different simulations. The maximum possible grid refinement was used by both Bakker and van den Akker [28] and Kresta and Wood [6], who performed some of the first 3D simulations using the commercial package FLUENT. Both used 25,000 grid points, the maximum number of grids available in that version of the code, which in retrospect was somewhat more than required. Note that Bakker and van den Akker used the full height of the tank in their simulation while Kresta and Wood imposed vertical symmetry at the impeller centerline, which means that the distribution of cells between the axial and tangential directions is different. The last three entries in the table were gridded with somewhat different constraints than the usual stirred tank simulation. The work by Gosman et al. [9] is for

Table 1 Summary of Tank Dimensions and Grid Definitions used in Simulations of Stirred Tanks. Entries marked with an asterisk were tested for grid independence. H (m)

TI2 (m)

degrees

z cells

Kresta (1991) Bakker (1991) Fokema (1994)" Ranade (1989)* Ranade (1990)" Sahu (1995)* Gosman G (1992) Gosman S (1992) Luo (1993)

0.228 0.44 0.15 0.3 0.15 0.3 1.67 0.294 0.294

0.228 0.222 0.075 0.15 0.15 0.15 0.915 0.147 0.147

90 90 90 90 0 0 90 90 360

26 52 43 46 31 33 27 31 45

30 27 20 30 15 28 20 27 28

Maximum Minimum Mean

1.67 0.15 0.43

0.915 0.075 0.243

52 26 37.1

30 15 25

40

20-25

First Author

Recommendation

90

r cells

z cells (mm)

r cells

15 19 120

8.8 8.5 3.5 6.5 4.8 9.1 61.9 9.5 6.5

120 5 32.3 20

8 cells

30 17 20 5

8 cells (deg)

Total cells

7.6 8.2 3.8 5.0 10.0 5.4 45.8 5.4 5.3

3.0 5.3 4.5 18.0 6.0 4.7 3.0

23400 23868 17200 6900 465 924 8100 15903 151200

61.9 3.5 13.2

45.8 3.8 10.7

18.0 3.0 6.4

151200 465 2755 1

5

5

4.5

20000

(mm)

312

Advances in Engineering Fluid Mechanics

multiphase flow (G denotes a gas liquid simulation, S denotes a solid liquid simulation) and was apparently not tested for grid independence. The size of the two domains chosen is radically different. Again, the reason for this was not given in the paper, but it may be supposed that experimental data was available for the dimensions chosen. Luo et al. performed a sliding mesh simulation for the full 360° of the tank and used correspondingly more cells [13]. The four remaining simulations are the most generally useful since they were all tested for grid independence in either two or three dimensions. Ranade and Joshi [20,21] appear to have used a uniform grid. Sahu et al. used a 2D grid which is more refined close to the tank walls [22]. Ranade and Joshi's grid for the RT is also 2D and imposes vertical symmetry at the centerline of the impeller [21]. Fokema et al. used a uniform grid in the radial direction, with refinement of the axial grid close to the impeller and refinement of the angular grid around the baffle [19]. Direct comparison of these four simulations in the axial direction gives 43 cells (Fokema et al. [19], refinement below the impeller); 46 cells (Ranade and Joshi [20]; uniform grid); 62 cells (Ranade and Joshi; uniform grid for RT); and 33 cells (Sahu and Joshi [22]; refinement close to the bottom). With grid refinement close to the impeller, Ranade and Joshi's maximum of 62 can safely be reduced to the recommended number, 45. Note that all of these runs are for liquid height of 0.3 m or less. Although all of the computations for CFD code are done in a dimensionless space, the grid requirements for large scale vessels, particularly with time varying flows, are not yet clear. In the radial direction, the number of cells varies from 30 (Ranade and Joshi [20]; uniform grid) to 28 (Sahu [22] refinement close to the wall) to 20 (Fokema et al. [19] uniform grid) to 15 (Ranade and Joshi [21] uniform grid). A uniform grid is recommended in the radial direction because there are steep gradients both over the extent of the impeller diameter and close to the tank wall. For a tank of up to 0.3 m in diameter, with 3D simulations, 25 uniformly spaced cells in the radial direction appear to be sufficient. Finally, in the angular direction, the number of simulations available for direct comparison is reduced to two since both Sahu and Joshi [22] and Ranade and Joshi [21] performed 2D simulations. Ranade and Joshi [20] tested 2, 5 and 9 cells in a uniform grid; Fokema et al. [19] tested 20 and 24 cells with grid refinement around the baffle. It is unlikely that either one of these variations is sufficient to show grid independence (the ranges in the other directions are more typically in steps of 10 cells than 3 cells). Two other considerations lead to a recommendation of at least 20 cells in the tangential direction; first, it is desirable to have a cell aspect ratio of 1.75 or less to maintain numerical stability; second, there are steep velocity gradients and small flow structures close to the baffle which cannot be resolved without this level of grid refinement. In summary, a non-uniform 3D grid, with the number of cells in the three directions ([z, r, 0] = [45, 25, 20]) is recommended. Axial grid refinement should be used close to the impeller and angular grid refinement around the baffle. If computational time is an issue, it may be possible to reduce the grid to 40 x 20 x 20 (total cells = 16,000 vs. 22,500). At least one run should be always be done at a significantly higher grid density to check for grid independence. Note also that

Boundary Conditions Required for the CFD Simulation

313

the minimum computational time requires a trade-off between grid definition and speed of convergence. Convergence Criteria The definition of convergence criteria, and of a converged solution, is not clear cut. It is always clear when a simulation is diverging or oscillating because the residuals show steady increases or oscillations. Convergence occurs when the solution does not change any with further iterations, but the exact definition of convergence depends on the end use for the simulation, the physical conditions of the simulation, and (in practical terms) the information which is readily accessible from the CFD code. Several convergence criteria which have been applied to stirred tanks are reviewed next. Where new models are being developed, a qualitatively reasonable result is the first requirement, and convergence criteria are typically looser. Luo et ai, who reported an early sliding mesh simulation in a short note, used "the flow pattern became cyclically repeatable" as their convergence criterion [13]. Gosman et al, who report early multiphase simulations, simply required that "residuals in the equations solved become smaller than a prescribed tolerance." [9] As methods become more reliable, users become more demanding. Quantitative comparisons to data are typically made as part of the validation, and convergence criteria are much more specific. The mass residual is the most frequently monitored. A limit is set on either the maximum value for a cell (Sahu and Joshi (1995), 10^), or the sum of all residuals over the computational domain (5 x 10~^ (Kresta and Wood [6]), 10-^ (Ranade and Joshi (1989, [20,21]), and 10"^ (Fokema et al. [19]). Once the mass residual has met the convergence constraint, the residuals of all other variables are typically below the convergence limit. Solution Algorithms and Differencing Schemes A discussion of the numerical issues would not be complete without mentioning solution algorithms and differencing schemes. If the simulation diverges, or convergence is not proceeding quickly enough, it may be important to change the solution algorithm or the differencing scheme, or to apply underrelaxation factors. Sahu and Joshi (1995) report the most detailed examination of these issues for stirred tank simulations. More general references are also available (e.g., Fletcher [26]). VALIDATION In much the same way as the convergence criteria are refined as methods are established, the validation of simulations follows an increasingly stringent series of tests. The first step in validating simulation results is always a visual examination of the overall circulation patterns. The next level of validation is comparison of selected velocity profiles with key experimental results. If the agreement at this level is good, turbulence quantities are validated (when experimental data is available). Again, this proceeds from the qualitative level, where appropriate trends are observed, to quantitative comparison of key traverses throughout the computational

314

Advances in Engineering Fluid Mechanics

domain. Once the user has a reasonable level of confidence in the results of the simulation, the flow field can be used to predict the process behavior, as in the papers by Ranade et al. [15] (blend time); Roekaerts [16,17] (selectivity of reaction); and Smith et al. [18] (scale-up). Note that different flow field variables will be critical for different process results so different levels of validation are required for different applications. CONCLUSIONS Although user-friendly CFD codes are now readily available and marketed for industrial use, accurate simulations require an understanding of the physical and numerical issues associated with the flow field under examination. For any flow field, the definition of the geometry, the boundary conditions, and the symmetries is relatively straightforward. Appropriate gridding of the computational domain and selection of physical models (e.g., a turbulence model) requires more consideration. Individual problems require careful attention to the details which are characteristic of the flow field. For the flow in a stirred tank, the impeller model must be given careful consideration since the accuracy of the impeller model will determine the accuracy of the predicted flow field. It has been shown that there are strong interactions between the PBT impeller and the tank walls, which lead to a change in the circulation pattern as the off-bottom clearance of the impeller is increased. When the impeller is modelled by fixing the velocity at the impeller discharge, the change in the circulation pattern observed in experiments can only be reproduced by changing the impeller model. While this leads to some loss of generality for the CFD predictions, experimental results indicate that it may still be possible to define general boundary conditions which can be used for a range of tank geometries. ACKNOWLEDGMENTS I would like to thank my colleagues, both industrial and academic, for discussions, comments, and questions about CFD which have broadened my appreciation and understanding of the issues discussed. NOTATION C = Off-bottom clearance of the impeller (m) D = Tank diameter (m) H = Height of liquid in the tank (m) k = Turbulence kinetic energy (mVs^) R = Integration limit for the swirl number (m) S = Swirl number

V^, V^, V^ = Time-averaged velocities (m/s) v^, v^, v^ = Root mean square velocities (m/s) z, r, 6 = Axial, radial, and angular or tangential coordinates (m and radians) e = Rate of dissipation of turbulence kinetic energy per unit mass (mVs^)

Boundary Conditions Required for the CFD Simulation

315

REFERENCES 1. Bakker, A., J. B. Fasano and D. E. Leng, "Pinpoint Mixing Problems with Lasers and Simulation Software," Chem. Eng., 101, 94-100 (1994). 2. Colenbrander, G. W., "CFD research for the petrochemical industry," Appl. Sci. Res., 48, 211-245 (1991). 3. Sharratt, P. N., "Computational fluid dynamics and its application in the process industries," Trans I Chem E, 68, Part A, 13-18 (1990). 4. Fort, I., Z. Grackova and V. Koza, "Flow pattern in a system with axial mixer and radial baffles. Part 35 (XXXV), Studies on mixing," Coll Czech Chem Comm, 37, 2,371 (1972). 5. Desouza, A. and R. W. Pike, "Fluid dynamics & flow patterns in stirred tanks with a turbine impeller," Can. J. Chem. Eng., 50, 15-23 (1972). 6. Kresta, S. and P. E. Wood, "Prediction of the three dimensional turbulent flow in stirred tanks," A. I. Ch. E. Journal, 37, 448-460 (1991). 7. Abid, M., C. Xuereb and J. Bertrand, "Hydrodynamics in Vessels Stirred with Anchors and Gate Agitators - Necessity of a 3-D Modelling," Chem. Eng. Res. Des., 70, 377-384 (1992). 8. Tanguy, P. A., R. Lacroix, F. Bertrand, L. Choplin and E. B. Delafuente, "Finite Element Analysis of Viscous Mixing with a Helical Ribbon-Screw Impeller," A.LCh.E. Journal, 38, 939-944 (1992). 9. Gosman, A. D., C. Lekakou, S. Politis, R. I. Issa and M. K. Looney, "Multidimensional Modeling of Turbulent 2-Phase Flows in Stirred Vessels," A.LCh.E. Journal, 38, 1946-1956 (1992). 10. Rasteiro, M. G., M. M. Figueiredo and C. Friere, "Modelling slurry mixing tanks," Advanced Powder Technology, 5, 1 (1994). 11. Myers, K. M., A. Bakker and J. Fasano, "Simulation and Experimental Verification of Liquid-Solid Agitation Performance," paper 188a, A.LCh.E. Annual Meeting, San Francisco, November 13-18, 1994. 12. Weetman, R. J., A. H. Haidari and B. J. Hutchings, "Solid Suspension Simulation in a Mixing Tank with Experimental Verification," paper 186f, A.LCh.E. Annual Meeting, San Francisco, November 13-18, 1994. 13. Luo, J. Y., A. D. Gosman, R. I. Issa, J. C. Middleton and M. K. Fitzgerald, "Full Flow Field Computation of Mixing in Baffled Stirred Vessels," Chem. Eng. Res. Des., 7 1 , 342-344 (1993). 14. Mathur, S. and J. Y. Murthy, "Computation of the Flows in Mixing Tanks using Unstructured Sliding Meshes," paper 186h, A.LCh.E. Annual Meeting, San Francisco, November 13-18, 1994. 15. Ranade, V. V., J. R. Bourne and J. B. Joshi, "Fluid mechanics and blending in agitated tanks," Chem. Eng. Set, 46, 1,883-1,893 (1991). 16. Roekaerts, D., "Monte-Carlo PDF Method for Turbulent Reacting Flow in a Jet-Stirred Reactor," Computers & Fluids, 2 1 , 97-108 (1992). 17. Roekaerts, D., "Use of a Monte Carlo PDF Method in a Study of the Influence of Turbulent Fluctuations on Selectivity in a Jet-Stirred Reactor," Appl. Sci. Res., 48, 271-300 (1991). 18. Smith, G. W., L. L. Tavlarides and J. Placek, "Turbulent flow in stirred tanks: scale-up computations for vessel hydrodynamics," Chem. Eng. Comm., 93, 49-73 (1990).

316

Advances in Engineering Fluid Mechanics

19. Fokema, M. D., S. M. Kresta and P. E. Wood, "Importance of Using the Correct Impeller Boundary Conditions for CFD Simulations of Stirred Tanks," Can. J. ofChem. Eng„ 72, 177-183 (1994). 20. Ranade, V. V., J. B. Joshi and A. G. Marathe, "Flow generated by pitched blade turbines II: simulation using k-e model," Chem. Eng. Comm., 81, 225-248 (1989). 21. Ranade, V. V. and J. B. Joshi, "Flow generated by a disc turbine: Part II Mathematical modelling and comparison with experimental data," Trans I Chem £., 68, 34-50 (1990). 22. Sahu, A. K., and J. B. Joshi, "Simulation of Flow in Stirred Vessels with Axial Flow Impellers: Effects of Various Numerical Schemes and Turbulence Model Parameters," J. Amer. Chem. Soc, in press. 23. Laroche, R. D., "Experiences with Sliding Mesh CFD Simulation for Stirred Tanks," paper 186i, A.LCh.E. Annual Meeting, San Francisco, November 13-18, 1994. 24. Yianneskis, M., Z. Popiolek and J. H. Whitelaw, "An experimental study of the steady and unsteady flow characteristics of stirred reactors," J. Fluid Mech., 175, 537-555 (1987). 25. Kresta, S. M. and P. E. Wood, "The Mean Flow Field Produced by a 45-Degree Pitched Blade Turbine—Changes in the Circulation Pattern Due to Off Bottom Clearance," Can. J. of Chem. Eng., 71, 42-53 (1993). 26. Fletcher, C. A. J., Computational Techniques for Fluid Dynamics, 2nd ed.. Springer Verlag, New York, 1991. 27. Kresta, S. M. and P. E. Wood, "The Flow Field Produced by a Pitched Blade Turbine: Characterization of the Turbulence and Estimation of the Dissipation Rate," Chem. Eng. ScL, 48, 1,761-1,774 (1993). 28. Bakker, A. and H. E. A. van den Akker, "A Computational Study on Dispersing Gas in a Stirred Reactor," 7th Eur. Conf. on Mixing, Brugge, Belgium (1991).

CHAPTER 12 ROLE OF INTERFACIAL SHEAR MODELLING IN PREDICTING STABILITY OF STRATIFIED TWO-PHASE FLOW N. Brauner and D. Moalem Maron Dept. of Fluid Mechanics & Heat Transfer School of Engineering Tel-Aviv University Tel-Aviv, 69978, Israel CONTENTS INTRODUCTION, 318 FORMULATION OF AVERAGED TWO-FLUID EQUATIONS, 321 SHEAR STRESSES MODELLING IN STEADY SMOOTH STRATIFIED FLOWS, 323 STABILITY ANALYSIS WITH QUASI-STEADY MODELLING OF SHEAR STRESSES, 327 Application to Predicting the SS/SW Transition, 331 DYNAMIC MODEL FOR THE INTERFACIAL SHEAR, 332 STABILITY ANALYSIS WITH DYNAMIC MODEL FOR x, 336 r

CORRELATION FOR THE DYNAMIC COEFFICIENT C„ 338 Amplitude and Phase of Shear Stress Fluctuation, 341 GENERALIZED STABILITY CRITERION, 344 STABILITY ANALYSIS IN RELATION TO KINEMATIC AND DYNAMIC WAVES, 346 ILL-POSEDNESS BOUNDARY—IN VISCID STABILITY ANALYSIS, 349 INTEGRATED STABILITY AND WELL-POSEDNESS CRITERIA, 350 Stability and Well-Posedness Map, 350 Constructing the Stratified Flow Boundaries, 352 STRATIFIED-FLOW BOUNDARIES: COMPARISON WITH EXPERIMENTS, 354 Stratified/Slug Transition, along the ZNS, H > 0.5, 355 Stratified-Smooth/Stratified-Wavy Transition, along the ZNS^, H < 0.5, 360 317

318

Advances in Engineering Fluid Mechanics

Stratified Wavy/Annular Transition, along the ZRC, H < 0.5, 360 Effects of Physical Properties, 360 Effect of the Tube Size, 363 Inclined Systems, 366 WIND GENERATED WAVES: THE EXTREME OF I -^ 1, 367 n

'

CONCLUSION, 370 NOMENCLATURE, 371 REFERENCES, 372 INTRODUCTION The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. The closure issue becomes much more complicated in unsteady turbulent flows, such as in the case of turbulent pulsating pipe flows. Experimental studies of pulsating flows indicate that the Reynolds stresses cannot be modelled in terms of the local radial velocity gradient (Carr [1], Ramaprian and Tu [2], Shemer [3],Mao and Hanratty [4]). The closure laws ought to consider the finite relaxation time in turbulent flows, which refers to the time period necessary for the turbulent structure to adjust itself to the time variation of the mean flow. The "inertia"' of the Reynolds stresses can be described by a complex (rather than real) value for the eddy viscosity (Shemer and Wygnanski [5]). Several empirical models for evaluating the relaxation effects in the Reynolds stresses in pulsatile flows have been proposed [3-5], but as yet, well-established closure laws for single-phase, unsteady turbulent flows are still unavailable. Two-phase flows are associated with a higher degree of complexity since the interface between the phases can take on very complicated time-dependent configurations. In stratified flow, however, the phases form two separate continuous domains. Therefore, in principle, each phase can be treated separately implementing the knowledge gained from single phase flows. The solutions are then connected through the boundary conditions prevailing at the mutual free interface. The stratified flow attracts continuous research efforts. It is considered a basic flow configuration in horizontal (and slightly inclined) gas-liquid systems since the relatively large density differential sustains stable stratification for relatively wide ranges of flow rates. The feasibility of exact analytical solutions for stratified flows is practically restricted to laminar-laminar flows, which is of limited relevance to gas-liquid

Boundary Conditions Required for the CFD Simulation

319

two-phase flows. However, laminar flow in both phases is frequently encountered in liquid-liquid systems, i.e., viscous oil-water flows. Indeed, several analytical studies for laminar-laminar flow between parallel-plates (Russell and Charles [6], Tang and Himmelblau [7]) and numerical solutions for circular pipe geometry (Gemmell and Epstein [8], Charles and Redberger [9]) are reported in the literature. Analytical solutions for stratified configuration in circular geometry were attempted by Bentwich [10] and Yu and Sparrow [11] and recently by Rovinsky et ah [12] and Brauner et al. [13]. The latter provide analytical expressions in terms of Fourier integrals for the velocity profiles and shear stresses distribution over the tube wall and over the free interface (considered to be either smooth or curved). Attempts to solve turbulent gas-liquid flows (e.g., Cheremisinoff and Davis [14], Russell et al. [15], Shoham and Taitel [16]) required prescription of closure-laws, in particular for the interfacial shear stresses. The practical tool for analyzing turbulent as well as laminar stratified flows in pipes is the two-fluid model. The continuity and momentum equations are averaged in each phase over the flow cross section, whereby information regarding local gradients at the tube wall and at the phases interface is lost. Therefore, closure laws, expressing the wall friction and interfacial shear stress, are needed to solve for the integral characteristics of the stratified flow: average layer depth (insitu holdup) and pressure drop. The closure laws adopted are extensions of single phase laws. Because of their simplicity, Blasius type relations are widely used to evaluate the shear stresses (e.g., Johanessen [17], Agrawal et al. [18], Taitel and Dukler [19], Brauner and Moalem Maron [20], Hall and Hewitt [21]). The modelling of the interfacial shear has attracted continuous efforts of a great number of investigators. The interfacial shear stress has been investigated from measurements of pressure losses and by study of the interface structure in an attempt to correlate the resulting augmentation of the interfacial friction factor with the wave characteristics (e.g., Charnock [22], Ellis and Gay [23], Cohen and Hanratty [24], Davis [25], Akai et al. [26], Kordyban [27], Andritsos and Hanratty [28]). Recent studies by Fukano et al. [29,30], Brauner and Moalem Maron [31], and Kang and Kim [32] show that the waves on the liquid film are not equivalent to solid wall roughness and that the interface mobility (wave-induced velocity components in the normal and axial directions) determine the shear stress augmentation due to the interfacial waviness. In order to propose an appropriate structure for the closure laws, the microhydrodynamic phenomena at the interface ought to be understood. This understanding is gained by exploring the mechanisms involved in the evolution of interfacial waves and their growth process. The basic approach is linear stability analysis of the stratified configuration. The stability analysis is aimed at defining the conditions where interfacial modes are unstable and predicting the initial stages of the wave growth. Some outlines on the various strategies of stability analysis taken are briefly discussed. Early stability analyses employed the classical Kelvin-Helmholtz (K-H) theory for two inviscid layers (Kordyban and Ranov [33], Kordyban [34], Wallis and Dobson [35]). However, in referring to gas-liquid flows, pyp^ < : 1, and assuming that the interfacial disturbance velocity equals the (slower) liquid layer velocity, the liquid destabilizing contribution has been degenerated. This results in a rather simple Bernoulli-type transitional criteria, whereby the suction forces in the gas-

320

Advances in Engineering Fluid Mechanics

phase flowing over a ("stationary") liquid interfacial disturbance exceeds the restoring gravity forces. Such criteria required the insertion of empirical constants to match the experimental data along the stratified/slug transitional boundary (Wallis and Dobson [35], Taitel and Dukler [19], Mishima and Ishii [36]). Later on, several studies extended the classical K-H instability theory for ideal fluids to account for the various viscous shear stresses due to the mobility of both phases, (e.g., Lin and Hanratty [37], Wu et al. [38], Andritsos and Hanratty [39], Brauner and Moalem Maron [40-45], Barnea [46], Crowley et al [47]). The models used in these studies for the wall and interfacial shear stresses are quasi-steady models, namely, the same closure-laws used to model the steady stratified flow expressed in terms of local (instantaneous) insitu holdup and phases velocities. Rigorous analyses of the interface stability in the flow of two laminar layers was carried out by Yiantsios and Higgins [48] and Tilley et al. [49,50]. The "viscid" analysis yields an interfacial disturbance velocity which differs from the liquid velocity, and consequently a non-negligible destabilizing effect of the liquid phase inertia. The application of the "viscid" K-H stability criteria for the onset of slugging in gas-liquid horizontal flows shows better agreement with the experimental findings for various liquid phase viscosities. The linear stability analyses also have been extended to account for the effects of nonlinear interactions due to finite wave amplitude on the wave growth process. The pertinent studies have been reviewed by Hanratty and McCready [51] and Tilley et al. [50]. It was shown by Brauner and Moalem Maron that linear stability analysis is insufficient to predict the stratified flow boundaries [40]. Parallel analyses on the stability as well as on the well-posedness of the (hyperbolic) equations which govern the stratified flow has been invoked. It has been shown that the departure from stratified configuration is associated with a "buffer zone" confined between the conditions derived from stability analysis (a lowerbound) and those obtained by requiring well-posedness of the transient governing equations (an upper-bound). These two bounds form a basis for the construction of the complete stratified/nonstratified transitional boundary to the various bounding flow patterns. The integrated frame of stability and well-posedness analyses has been found suitable for predicting the stratified flow boundaries in horizontal gas-liquid and inclined systems and liquid-liquid systems [40-45]. But it was found that the stability boundary predicted via the "viscid" K-H analysis generally overpredicts the boundary of the stratified smooth pattern and the transition to stratified wavy configuration. The missing destabilizing mechanism evolves from the dynamic interaction between the turbulent structures in the gas phase and the growing interfacial waves. The dynamic response of the turbulent field is of similar nature to that observed in single phase pulsating flow [1-5] or in the flow of turbulent gas over solid wavy boundaries. Obviously, quasi-steady closure laws for the interfacial shear are incapable of describing this dynamic interaction. Attempts made to deduce closure laws for the interfacial stresses which account for the dynamic interaction from measurements of surface stresses developed over a solid wavy boundary have been reviewed by Hanratty [52,53]. However, the dynamic interaction between a turbulent gas phase and growing interfacial waves over a mobile free liquid interface may be of an entirely different nature.

Boundary Conditions Required for the CFD Simulation

321

Although the closure laws issue in modelling single-phase, turbulent flows and two-phase flows is not quantitatively different, much more research efforts are needed to arrive at well-established empirical basis which provides the information for formulating the closure laws in two-phase flows. The structure of the closure law may have crucial consequences on the capability of the model to successfully capture the physical phenomena involved. The purpose of this chapter is to show the role of the model used for the interfacial shear at different stages of stability analyses of the stratified flow configuration and to summarize progress made in formulating a closure law which reflects the dynamics of the interaction involved in turbulent gas flow over a mobile wavy interface. FORMULATION OF AVERAGED TWO-FLUID EQUATIONS Consider stratified flow of two immiscible fluids a and b in a horizontal (or slightly inclined) conduit. The flow configuration and coordinates are described in Figure 1. Detailed derivations of the transient one-dimensional averaged twofluid equations can be found in many studies (e.g., Yadigaroglou and Lahey [54], Hancox et al. [55], Banerjee and Chan [56], Banerjee [57,58], Andron [59], Kocamustafaogullari [60]). With reference to Figure 1, the transient continuity and momentum equations for the two fluids averaged over the local layers thicknesses are: (1)

f (PaA3) + |^(PaA,UJ = 0

(2)

Figure 1. Schematic description of the physical model and coordinates.

322

Advances in Engineering Fluid Mechanics

— (pbA.u,) + — (p,A,YbU^) = - t , S , + T,S, + p,A,g sin p -|-(A,P,) + P „ ^ ax

(3)

dx

— (p,A,u,) + — (paA^Y.u^) = -T^S, + x,S; - p,A,g sin p -f(A.PJ +P . ^

(4)

dx dx The variables, A, S, T, and u, are the flow cross section, the wetted perimeter, the shear stress, and the average velocity of the two fluids; T^ is the interfacial shear stress where a positive T. corresponds to a faster upper layer. The shape factors, y^, Yj, are defined in terms of local velocity profiles, u'^^^:

For mild interfacial slope (i.e., wave lengths of the interfacial disturbances are large compared to the thickness of both layers), the local velocity profiles can be closely approximated based on the local values of the layer thickness and phases velocities. Also, under the shallow water assumption, the effect of the wave-induced flow in the perpendicular direction on the variation of the pressure gradient 3P/3y can be neglected, whereby the pressure at each phase, P^, P^^, varies only due to gravity. Hence, the average pressure at each phase, in terms of its pressure level at the free interface, y = h, is given by:

3x

(P^AJ = YC^^'" dx

+ P^g'^o^P^h - y)]dA,

^ ( P i . A , ) + p,gcosPA,fi dx dx

dx

(6.1)

(PaA.) = | - f " [ P , - P 3 g c o s p ( y - h ) ] d A , ^ "" dx = |-(P,aAJ + p„gcospA,|l dx dx

(6.2)

where P.^, P.^^ are the pressures at both sides of the free interface. Substituting (6) into (3) and (4) while utilizing (1) and (2) respectively to eliminate the terms d/dt(pA)^^ and subtracting, the resulting combined momentum equation for the twophases reads:

Boundary Conditions Required for the CFD Simulation

323

PbCl-yJ-^ + Pad-Ya)^ ^ b

^ a

A/a, = -^b 1 ^ + tiSi

^ 1

,

1^

+ — + ^ a T ^ + (Pb - P a ) g s i n p

(8)

Due to surface tension, P ^ P^, the difference of which is given by; '

lb'

la

o

y »

Note also that since y^j^= F(u^, u^^, h), the shape factor derivatives in Equation 7 are replaced by:

3x

3h

dx

3UH

3X

3U,

3X

(10)

The two continuity equations, Equations 1 and 2, with the combined momentum equation. Equation 7, constitutes a general frame of formulation of a two-fluid model for stratified flow configuration. Inspection of Equations 7, 8 indicates that the application of the model requires the prescription of the wall shear stresses T^, X^ and the interfacial shear stress, x., as closure laws. The model adopted for these closure laws ought to bridge the gap between the micro-scale hydrodynamic phenomena and the macro-averaged representation of the flow. The ability of the averaged two-fluid model to predict macroscopic behavior is largely dependent on a successful incorporation of the knowledge of small scale interactions at the phases boundaries into the closure laws. The way of modelling of the interfacial shear between the phases at the free interface has been always recognized as critical in determining the resulting stability characteristics. Its evaluation is complex since the variation of the stresses along the interface critically depends on the unknown structure and characteristics of the mobile wavy interface, which, in turn, are to be derived from a stability analysis. Indeed, various existing stability studies differ in the way the modelling of the interfacial shear has been approached. SHEAR STRESSES MODELLING IN STEADY SMOOTH STRATIFIED FLOWS The stability analysis is concerned with the stability of the interface between the phases, presumably smooth, under fully developed stratified flow conditions. In this case the LHS of the combined momentum equation for the two-phases. Equation 7 vanishes:

324

Advances in Engineering Fluid Mechanics

AFab = ^ U H . U,, U,) = 0

(11)

Here H, U^, U^ are the solutions obtained for the layer thickness and the phases average velocities for steady fully developed stratified flow. Thus, the starting point in the analysis is the modelling of the steady fully developed stratified flow configuration and the corresponding wall and interfacial shear stresses. For simple flow geometries, as in the case of stratified two-phase flow between two parallel plates, the velocity field varies only in the perpendicular direction and expressions for the shear stresses, in terms of the phases average velocities, U^, U^, and the layer depth, H, can be analytically derived by solving the Navier-Stokes equations in the two-phases domains (see, for example, Hanratty and Hershman [61]). Coutris et al. showed that even for the simplest case of fully developed laminar two-layer flow the closure relations which evolve for the wall and interfacial shear stresses are quite complicated [62]. A rigorous analysis of stratified flow in pipes is much more complicated since the velocity field varies in two directions. Bentwich [10] and Rovinsky et a/. [12] solved analytically the 2-D velocity profiles for laminar—laminar stratified flows in pipes. Russell et al. numerically solved the momentum equation for a laminar lower liquid layer sheared by gas flow, assuming a uniform shear stress over the interface T. = T^ [15]. Cheremisinoff and Davis [14] and Shoham and Taitel [16] used augmented interfacial friction factor and solved for the velocity distribution in a turbulent liquid layer using the eddy viscosity approach. This approach yields the dimensionless layer thickness H"^ = HU* / v,(U* = (T^^ / p)'^^) in terms of the liquid Reynolds number, which in turn is combined with the momentum equation and solved iteratively to yield both H and x^^. Rovinsky et al, obtained expressions for the local wall and interfacial shear stresses in terms of Fourier integrals [12]. It has been shown that the shear stress over the phases interface is not constant; when the viscosity gap between the phases is large, the region where the interface meets the pipe wall is characterized by large variations of the interfacial shear as well as the wall shear stresses in both phases domains. Large variation of the liquid-wall shear stresses in the circumferential direction also were obtained in a recent experimental study by Paras et al. [63]. Obviously, exact computation of the shear stresses is limited either to laminar flows or simple geometries, and yet is complicated. Thus, the more conventional way to evaluate the wall and interfacial shear stresses is to adopt the Blasius equation, whereby wall shear stresses x^, x^^ are expressed in terms of the phases velocity heads and appropriate friction factors, /^, /^: ^Du

^D,u,

= c. Re;""

(12.1)

= Ci Re^""

(12.2)

Boundary Conditions Required for the CFD Simulation

325

Note that the Reynolds numbers for the two fluids are based on the equivalent hydraulic diameters, defined according to the relative velocity of the phases (Brauner and Moalem Maron [20]):

'

4A, -—; (S, +

D = 1 ^ ; D = 1 ^ ;

.

4A.

D, = — 2 -

„ for u > u^ a

4A, D ~ ^' (S,+SJ 4A, E>h =

-

^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

Advances in Engineering Fluid Mechanics

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

Boundary Conditions Required for the CFD Simulation

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.

^A, au,

au,,

aAF,, A, a u .

A;

2k

7a - PbU

2

^\J_

I^

.A^au,

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,

^u^_a__i\_a ,A, au, A, au,

A^^j aAF.

^u

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)

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

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) << simplifies to: C^n = U, 1-H Pa U, H Pb U, 1 - H

(21.2)

Indeed, various early studies employed inviscid K-H stability analysis for predicting the stratified/nonstratified transition boundary in gas-liquid two-phase flow (Kordyban and Ranov [33], Kordyban [34], Wallis and Dobson [35], Taitel and Dukler [19]). However, claiming that for pyp^ = PG^PL '^ ^' Equation 21 has been reduced to C^^ = U^^ (ignoring the contribution of U_H / [\J^{\ - H)] in Equation 21.2). The liquid destabilizing term in the stability condition (18.2), which is proportional to C^JV^ - 1, has thus been degenerated. Consequently, the instability condition. Equation 18, includes, in fact, only the gas destabilizing term, whereby (for plug flow model y^ = 1): 1

p'

— (U, - U , ) ' > — 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

Boundary Conditions Required for the CFD Simulation

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^^
bs

as

Specifying the input flow rates of the phases, U^^, V>^^ the steady parameters (U^, Uj^, H) are obtained by solving Equation 11 with Equations 12-14 (Brauner and Moalem Maron [20]). Thus, Equations 18, 19 represent a predictive tool for the neutrally stable wave characteristics, k^, Cj^^. A neutral stability boundary (NS line) can be constructed by searching for the limiting operational conditions, U^^, U^^^ for which the smooth stratified flow of the two-layers is neutrally stable with respect to the long wave realized in the system, say X^ ^ D and k^ = 27i/D [41,43]. Clearly, for sufficiently long waves, k^ -^ 0, surface tension effects degenerate, and the "zero neutral stability" boundary (ZNS line) is approached (Figure 2). According to this definition of the NS and ZNS lines, the area within the NS line on a flow pattern map (Uj^^ vs. U^^) represents a stable, smooth stratified zone for any k > k^. Obviously, the ZNS boundary, which corresponds to a degenerated stabilizing effect of surface tension, represents a lower bound for the departure from stratifiedsmooth configuration. Experimental data for the limiting conditions of smooth stratified flow also are included in Figure 2. Inspection of the figure indicates that for relatively thick water layers (above H = 0.5 line), the neutral stability line represents a reasonable prediction for the departure from smooth-stratified configuration. Waves growing over thick layers tend to block the upper gas phase flow area, and the transition to stratified wavy regime coincides with the transition to slug flow. However, with

332

Advances in Engineering Fluid Mechanics

A//g -WATER Xi

D >-"' H h-H U IQi O -J pq

Stability boundary, eq.(18) yxjQ it _n •• z,JNi, Kn-u ^^ NS, kn=2 •o E a) D=2.53cm

> 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-

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

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^^"^^en " "

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

Advances in Engineering Fluid Mechanics

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

Y^Uf - [ ( 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)

Boundary Conditions Required for the CFD Simulation

wind-side

337

lee-side

Case (a).Faster Upper Phase,

UQXJJ)

UrUa-Ub 0 , Tf<0 Cose (b):Faster Lower Phase , Ub>Ua Figure 4, Schematic representation of the "dynamic term" in the modelling of the interfacial shear.

338

Advances in Engineering Fluid Mechanics

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.

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

a) Laminar liquid layer, Ret3<2IOO

^

10

V

O

LL-

^S]^Ch/Rei,= 2.45ld*/Frb ]

o o

\) N-3

>-

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 "dynan^iic 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, ^i^ = 16, 70 cp [39]

Boundary Conditions Required for the CFD Simulation

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<

^H- h + T—^ u'

an

,

+K

Re,

p,(U,-UjMkh'

(29)

au. ^

where aTf^/a(H, U^, U,) stand for the derivatives of the quasi-steady model for the interfacial shear stress evaluated at conditions of presumed smooth steady flow (H, U^, U^). The expressions obtained for these derivatives based on Blasius type models for 1° (for the cases of U > 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<2IOO -Q
cr \

x: O

Xb= "-3

V

-1

10

\

H"

2 LU O LL

C,/Re5=3.0IO"'*/Frb

Id''

LU

O

\

O

>-

LU Q:

o

-a 10 >

UJ

\o

10*

'

•

I

I

•

11

10-

10'

10^

10 _ Ub LIQUID LAYER FROUDE NUMBER. Fr = (gHfa)1/2

l.50| 1.25 1

b) Liquid layer laminar, Xb= '-3 turbulent,

J

1.00 h-Q -n [gfirip^ii^y-ft^^Wl^

"<

1

0.75 0.50

1

0.25 1 10"

eq.(28) 1 1 iniil

10'

1 1 1 iiiiil

1 1 1 mill

Alllll.. L-L-L

-±- 1 1 I I 11J

1

10' UbPb LIQUID LAYER REYNOLDS NUMBER. Reb= ^b Figure 6. Correlating the "dynamic coefficient," C^ with y^ = 1.3 for laminar layer and Y^= 1.6 for turbulent layer (symbols notation as in Figures). 10'

I0~

10"

• 1 1i l l ^ -

10^

Boundary Conditions Required for the CFD Simulation h' = h - H = ae-'^^^-^H^^ ;

a= he*^^.^

343 (30)

where a denotes an amplified wave amplitude. Utilizing the continuity Equations 1, 2, the perturbations of the phases velocities u^, u^ can be expressed in terms of h':

u: = -4^(c-ujh' A, A:

(C-UJh';

C = C„+iC,

(31)

Equations 29-31 can be used to express the shear stress fluctuation in terms of its amplitude and phase: x: = he^^-(T^,+iT„)e"

(32.1)

= ax, e

x\ = a[x,^ cos[k(x - C^t)] - x., sin[k(x - C^t)]}

(32.2)

where

|Ti| = (T., + x?,)"^;

e = tg-'p

(32.3)

'iR J

and

,

_ 3x» * an

^,. = A;

ax" A; au, A.

' 1 ax°

axf A'

I Re 1 ax° C,+K - f Fr^ Fr. A. au.

(32.4)

1 I

p,(U,-UJ^k

(32.5)

As shown in Equations 32, x.^^ is the non-dimensional interfacial shear component, which is in phase with the wave height, and t.^ is in phase with the wave slope. The latter is related directly to the dynamic coefficient and determines the phase shift of the interfacial shear fluctuation with respect to the interfacial wave. The expressions obtained for the amplitude and phase of x- for the particular case of gas-liquid flow in rectangular channels are given in Brauner and Moalem Maron [102]. Equations 32 show that the amplitude of the interfacial shear includes two terms, one of which evolves from the quasi-steady part while the other, which stems from the "dynamic term." is proportional to the wave number, k. The phase shift of the interfacial shear with respect to the wave elevation (for C^ = 0) evolves due to the

344

Advances in Engineering Fluid Mechanics

inclusion of the dynamic term, and for C^ = 0 no phase shift is expected for the neutrally stable waves. With C^^ ^ 0, the fluctuation of the interfacial shear is both augmented and is leading the interfacial wave, 9 > 0. In general, the augmentation and phase shift increase with the wave number and wave amplitude, consistent with observations obtained with gas flow over solid wavy boundaries [86-89] (see also Figure 3). However, their dependence on the liquid layer flow variables (e.g., Re^^, Fr^^) implies that the application of shear stress characteristics obtained over solid wavy boundaries, as closure laws for two-fluid models, may be unjustified. GENERALIZED STABILITY CRITERION From the physical point of view, the destabilizing dynamic term, J^^, stems from the interfacial dynamic interactions of a turbulent (upper) gas phase with the perturbed free interface. Therefore, Equation 25 for k^ ^ 0 embodies, in fact, two criteria for instability, according to the flow regime of the upper phase: J^ + J^ + J^ > 1

Turbulent upper phase

(33.1)

J^ + Jj^ > 1

Laminar upper phase

(33.2)

The it ''zero neutral stability'' boundary predicted by Equation 33.2 has been denoted as ZNS lines while the stability boundary predicted by Equation 33.1 has been denoted by ZNS^. The interplay between the stability of the smooth interface and the flow regime transition in the upper phase is elucidated with reference to Figure 7, which represents air-water flow in various conduit sizes. Consider Figure 7a for relatively large diameter tube, D = 9.5cm. For demonstration, maintaining the liquid flow rate at U*,. = 0.4cm/s, while increasing the gas rate, the latter becomes turbulent at point 1 (Re^ > 2,100). However Equation 33.1 is not yet fulfilled, and, therefore, at point 1, the smooth interface is still stable with turbulent flow in the upper phase. With further increase of the air flow rate. Equation 33.1 is met at point T*, which represents a transitional point from stratifiedsmooth to stratified-wavy. The locus of all transitional points as obtained by Equation 33.1 (represented by the solid line) constitutes the predicted stratifiedsmooth boundary. Note that Equation 33.2 as represented by a dashed line is irrelevant in this case since it assumes laminar conditions. Figure 7c represents the other extreme of small diameter conduit with D = 0.4cm. In this case, with increasing the rate of the upper gas phase for a given U*^, a laminar flow regime prevails for a wide range of U^^. As opposed to Figure 7a (for large diameter conduit), in this case, instability for the laminar regime. Equation 33.2, is fulfilled at point T*, indicating that the destabilizing effects of the inertia terms J^, J^, are already sufficient to overcome the stabilizing gravity term. If the I term was included, the transitional line would have been obtained earlier n

(dashed line in Figure 7c). However, the use of Equation 33.1 is not physical since a laminar flow regime in the gas phase prevails there. Comparison of Figures 7a and 7c shows the basic difference between conduits of large and small diameters: For large diameters, the flow regime transition already takes place within the stratified smooth region and, as a result, Equation 33.1 is to

Boundary Conditions Required for the CFD Simulation

345

Eq.OS.Z)'^

10"'

10°

10'

Superficial air velocity, UQS [m/s] Figure 7. Determination of flow pattern transition at laminar, turbulent, and transitional Reynolds numbers of the gas phase.

346

Advances in Engineering Fluid Mechanics

be adopted as the governing instability criterion for transition from stratified-smooth to stratified-wavy. For small diameters, the laminar/turbulent flow regime transition occurs for gas flow rates which are beyond those corresponding to the instability criteria, hence, within the unstable stratified-smooth zone. Therefore, Equation 33.2 is to be adopted as the governing instability criteria for transition. Complementary to this physical picture is provided in Figure 7b, which represents intermediate situations: As U^^ increases for a given U*^, Equation 33.2 is not fulfilled before hitting the L-T flow regime transition (at point 1), indicating that the inertia terms are insufficient to cause interfacial instability. However, at point 1, where the gas phase turns turbulent, a finite dynamic term, J^, shows up. Consequently, the instability condition Equation 33.1 is suddenly fulfilled (as inequality). As such, the smooth interface becomes unstable simultaneously with flow regime transition. Note that this is distinct from large diameter pipes, where at the flow regime transition (point 1 of Figure 7a), the sum of the inertia contributions, J^, Jj^, and the dynamic term, J^, are still insufficient to initiate instability (Equation 33.1 is not fulfilled). Thus, in the extreme of large or small diameter conduits, the stratified-smooth/ stratified-wavy boundary is predicted by Equations 33.1 or 33.2, respectively, while there exists an intermediate range of pipe diameters in between, where neither of these equations predict the locus of flow pattern transition. In these systems, transition from stratified smooth pattern coincides with the L-T transition of the gas phase and is predicted by the locus of operational conditions where Re^ = Re^^.^ (Brauner and Moalem Maron [105]). The role of laminar/turbulent flow regime transitions in determining the SS/SW transition has been demonstrated via the variation of tube diameter in air-water systems. Clearly, the same basic phenomena are expected due to variations of the physical properties of the phases when dealing with various two-fluid systems. STABILITY ANALYSIS IN RELATION TO KINEMATIC AND DYNAMIC WAVES Another approach for analyzing the stability of the flow is based on wave-theory. In deriving the characteristics of kinematic and dynamic waves in two-component flow, Wallis has shown that the relations between the velocities of these two classes of waves govern the stability of the two stratified layers [74]. It has been shown that the condition of equal kinematic and dynamic waves velocities corresponds to marginal stability. Following this approach, Wu et al. determined the stratified/ nonstratified transition in horizontal gas-liquid flows [38]. The relations between the dispersion equation. Equation 16, and stability criteria Equation 33 on one hand, and the characteristics of kinematic and dynamic waves on the other hand, (for C^ = 0), was shown in Brauner and Moalem Maron [45]and Crowley et al. [47]. The general dispersion equation. Equation 16, can be rearranged in terms of the celerity of kinematic wave, C^. As indicated by Equation 20, the neutrally stable wave is actually a kinematic wave, hence, C^ = C^^, and is given in Equation 19. The rearranged form of Equation 16 reads: C ^ - 2V - -^ V C + V ^ - - C , V , = 0 ' k '

(34.1)

Boundary Conditions Required for the CFD Simulation

347

where: b. V„=-^; a

,, V, =

2b, '-• a

-,2 d, V = ^ a

(34.2)

For constant shape factors, d'^^Jd{\{, U^, U^) = 0, Equations 34.2 read:

1 aAF,, 1 aAF„, V = A, 9U, A, dU, Pb/Ab + Pa/A, YbPbUb , TaPaU;

(352)

Apg cos p + ak^ - C,p(U, - U, f S, (A;' + A"') Pb/A, +Pa/A, (35.3)

Here, V^ represents a weighted mean velocity of the two phases, and V, is a damping parameter due to the shear stresses. (Based on the shear stresses modelling as detailed by Brauner and Moalem Maron [43] it can be shown that V^ always attains a positive value, independent of the relative velocity between the two phases). The relation of the general dispersion Equations 34 to dynamic waves is derived here by recalling that a pure dynamic wave occurs whenever the net force on the flowing fluids is produced only by concentration gradients (and is independent of the insitu concentration, Wallis [74]). In this case, the quasi-steady shear stress terms on the rhs of the combined momentum equation, which are functions of the insitu concentration, are ignored, whereby AF^^^ is considered as identically zero. However, the dynamic interfacial shear stress term, which is proportional to the concentration gradient, evolves from the Reynolds shear stresses in the turbulent field and is retained. The general dispersion Equation 34, with V^ = 0, becomes: q

- 2 V „ C , + V^ = 0; C,= \ ±

Vvg-V|

(36)

Equation 36 yields the dynamic wave velocity relative to the weighted mean velocity, c^. <=a - C, - V„= ±{(Y,p,U/A,)^ + (YaPaU/A)^ +

(2YJ,P,P,U,U/A,A;

+ [p,/A, + p/AJ[-Y,p,U2,/\ - y,p,UVA^ + (Apg cos p + ok^ - C,P(U, - U,)^ S,( A-' + A-'))/ A ; ]} "V(p,/A, + pJAJ A stable dynamic wave is obtained provided (Vg > V^) whereby:

(37)

348

Advances in Engineering Fluid Mechanics

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

P^=1+PL^; Pa A , '

P =1+P^:ik; ' Pb A3 '

(38.1)

p ^ ^ J^b ^ PaPb '' A , A , PJA, + P,/A,

Note, however, that a pwr^ dynamic wave may be physically realized in inviscid flows. In viscid flow systems, the wave characteristics may be related to those of pure dynamic and kinematic waves by introducing V^ = V^ - c^ from Equation 36 into the general wave dispersion equation, 34, to yield: c-_ c2 + i v , (c - c,)= 0; c = C - V„; c, = C, - V„

(39)

In view of Equation 39, it is easily shown that at neutral stable conditions the wave velocity is equal to both the kinematic and dynamic waves velocities, c = c^ = Cj. Substituting c = c^ + ic, nto Equation 39 results in the wave frequency and wave amplification in identical forms to those derived by Wallis [74]:

c^-V^fO^•"

k

2

C,

V,

CO, = — ' - =

-^"

(40.1)

(40.2)

—^

Equations 40 indicate that the locus for which the kinematic wave velocity is equal to that of the dynamic wave, C^ = C^^ = C^, represents neutral stable wave modes. Indeed, equating C^ from Equation 37 to C^(= C^^), again renders the condition derived for neutral stability in Equation 33. Stable modes are obtained for c2 < c2 < c2, whereas for unstable modes to exist it is required that c^ > 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.

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

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

Boundary Conditions Required for the CFD Simulation

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

.E . JO

3

351

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

^7

Ck>c|<0 ~^ V Uo>Ul,

2 2 2 / Ck
g 10

/

H = .5 Q: UJ

/

o.

i i 11 ml

<-5

10

I I iiiml

16^

10"

SUPERFICIAL

I I iimJ

^-2

10-

/\

imiil

10

AIR VELOCITY,

I I i....i t 1 1 . Lw4

10"

« J Jiii .•..iJ

10'

10'

U^s[m/^

Figure 8. Stability and well-posedness nnap for air-water horizontal system, D = 2.53cm. smooth-stratified-wavy transition is predicted by the ZNS^ boundary (Equation 33.1). For laminar flow in the gas phase, J^^ = 0, the stratified-smooth transition is defined by the ZNS boundary (Equation 33.2). These two boundaries are bridged by the laminar-turbulent transitional boundary. In this range the transition from stratified smooth pattern may coincide with the L-T transition in the upper gas phase. In parallel to the "zero neutral stability" boundary defined by k^ = 0, a "zero real characteristics" boundary (ZRC) is built-up by searching for all combinations of (U^^, \^^) which yield by (41) real characteristics for long waves, k^^ = 0. The ZRC curve confines the region of operational conditions for which well-posedness is ensured for all wave modes. Generally, the ZRC boundary is composed of two branches; the left one corresponds to U^^ > 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

Advances in Engineering Fluid Mechanics

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

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

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

£ 10-3

•••"•'

10-2 10-'

00

10^

10^

b)

D=5Jcm

•'' •-••^

^

10-2 10-'

• ' ''^"^J

10^

• '"^"'1

102

10'

SUPERFICIAL AIR VELOCITY, Uas[m/s]

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'

EXPERIMENTAL[39] ZRC H-.5

I I I I Mil]—I 111 mil—I I iMiin—I 11 IIIIII

N

o o

•Strotified-Smooth/Wavy I I 111 n i l — I

11 i i i i i i — I

I 11 m i l — I

11 i n i j

b)yLL|5-70cp

a)yLLb"l6cp

V.

p \

•g '3 -I . ? 10

\

A

.y 10 b-

\\

(/)

in

I

JIL

/

\'^.

I I M 111)1 1^1 null I I iiiiLl \ i I I mill I I liiiiil I I .iiiiif I I t i i f i l \ I I i l l "2 .-I _0 _l -2 .-I . 0 . 1 10 10 JO" 10" 10' 10' 10^ 10" 10

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

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

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

Boundary Conditions Required for the CFD Simulation

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].

I

111 i i i n — I

I i iiiiii—I

I Iiiiiii—I

I I iiiii{

» 1 1 im 1 11 Mini—i 1 1 iiiii]—1 i 1 1 iiiTj—r~

1

b. ^ L ' 10^ cp

a . fjLif 1 cp X. 1.0

^ 0.8

2.53\

J 0.8

r

I 0.6 d SO.4 Q

\

L

\

\

10^

10^

1

I I mini

10^

10'

\

Superficial

Air

\

\

J_J 0 6

\

\\

10"'

^V

1

0.4

\

\

\

t 1tif.il

10^ I0"2

\

\\ ^\ ^ \ \ \ \ \ \ ^ \\ \^ ^^

L 1 Lj 1

-]

N

\

1

•g.0.2

\

\

kr L 1 L1

« t I iiml

•j

v. 2.53\

I ' l l itiii

-^1.0 -]

K r ^ ^^0=20.3 cm \ ^\. ^ 1 \ \ ^^^9.53

x^\^D«20.3cm

-0

^

\ ^ ^ ^ ^\ \'•.

1 • li.at.. . ^

\

I

11 iii)^:

10^

10'

^02 *

i_' f 1 MUd

lO

Velocity,Uas [m/s]

Figure 15. Liquid destabilizing contribution along the ZNS boundary; effect of liquid viscosity and tube diameter (notation as in Figure 14).

360

Advances in Engineering Fluid Mechanics

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

Boundary Conditions Required for the CFD Simulation AIR-VISCOUS D-2.5cm 1 (a)

LIQUID

\ 1 Phase b l a m i n a r - © turbulant-®

r

-S .4h

.2h

ek (b) xi

4

L

(C)

sz

.4

.2

2

-L

J

L

4

6

8

AIR SUPERFICIAL VELOCITY, Uas[m/s]

Figure 16. Relative contribution of the destabillzation terms along the ZNS^ boundary (J^, J^^, JJ—effect of liquid viscosity.

361

362

Advances in Engineering Fluid Mechanics

AIR-WATER D=2.5cm 9.5 cm 2 0 cm \

I

\—1 I I i 11

(a)

.4

1

1—r

Phase b laminar—Q turbuLant-@

.2

Ja(D>9.5cm)^0 - ^

I

I

\

I I

I I I I 11

I

^ Ir I

I I I

I

I

(b)

J

.2 10^

10^

I

5xlO^

GAS SUPERFICIAL REYNOLDS NUMBER,Re as Figure 17. Relative contribution of the destabilizing terms along the ZNS^ boundary (J^, J^^, JJ—effect of tube diameter.

Boundary Conditions Required for the CFD Simulation

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 (p^Ap)'^^ 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<:0.5

Large D

Ja + Jb = 1

364

Advances in Engineering Fluid Mechanics

gas phase is turbulent along the stratified-smooth/wavy transitional boundary, which is thus predicted by the ZNS_^ line. In this case, as the tube diameter increases, the region of stratified-wavy flow extends due to the migration of the ZNS^ towards the lower gas rates, on the one hand, and the migration of ZRC towards higher gas rates, on the other hand. Figure 18 presents typical comparisons of the stability boundaries of the stratifiedsmooth configuration with experimental data in various (small) tube sizes. Figures 18a, b, c are aimed at demonstrating the relevance of the particular transitional boundary according to the guidelines proposed with reference to Figure 7 and Equations 33. For instance, for D = 2.5cm, the L-T flow regime transition takes place at gas flow rates, which are lower than those required to fulfill Equation 33.1 (hence, also 33.2). Therefore, the relevant transitional line is the ZNS^ boundary, as obtained by Equation 33.1. On the other hand, for small diameter tube, D = 0.4cm, Equation 33.2 is first fulfilled before the L-T flow regime transition occurs. Hence, the relevant transitional line from stratified-smooth to stratified-wavy pattern is the ZNS boundary as obtained by Equation 33.2 (Figure 18c). For the intermediate tube size of D = .815cm, it is demonstrated in Figure 18b that there is a range of water flow rates for which the L-T flow regime transitional line

AIR - WATER Experimental [78,80]

D stratified-smooth • stratified-wavy A pseudo-slugs

C/3

X5

10>

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]).

Boundary Conditions Required for the CFD Simulation

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 k^o) 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

> 1

(42)

(Pb-pa)D'g

AIR-WATER

Theoretical J3

O

O

> a: LiJ

!

10

10

t- b) D=0.985cm

a) D = l.23cm

-Slugs a

D

if)

Slugs a bubbly

*861A\

bubbly .-^IOOIEIAX

rfbBBBS

.ODD

ODD DDDD

or u a.

c) D=0.615cm

/\

Slugs a "bubbly,*

DDD D 10^' -». • DODD D u-

[8(j

D Stratified-smooth • Stratified-wavy • Pseudo- slugs

—-—L-T Transition — —' Well-posedness boundary [ilg • • • • Stratified - bubbly boundary [lIS] H=0.5

3

_J UJ

Experiment

Stability boundary, (33.0 or (33.2)

-3 10 , 1 1 10

lltUll

10

. C D D D D a Q S l « « \ •§ DDD DQBBBB c ODD a t \ B B B B ^

S S -a D DDQG

1 I tllllll

10

g' aa DO DD

\ \ llllj

10'

10'"

SUPERFICIAL

I0-' 10' AIR

10° 10^

10'

r^2

10"

10

VELOCITY, UQS [m/s]

Figure 19. Stratified-smooth and stratified-wavy boundaries in snnall diameter tubes. Comparison with experimental data (Luninski [80]).

-J

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).

Boundary Conditions Required for the CFD Simulation

367

AIR-WATER, UPWARD INCLINATION D=2.54cm, (3=0.5 ° Si

•ZRC ZNS H=0.5

>-'

A A Experimental, Shoham [78] EZT] ss EOnD sw Jh^O

Jh=0

\ N

\

\

g 10-1 SL

glO-2

iii^AN

E m in-3

10-

00

uL

10-^ 10-^

10^

10^

10-2 10-1

10^

^:':':':i':':'t l;itft}iffl';-;fli

10^

10^

SUPERFICIAL AIR VELOCITY, Uas[m/s] Figure 20. Effect of upward inclination on the stratified-wavy boundaries. A stratified-wavy, A pseudo-slugs. In downward stratified flows, the liquid velocity approaches, or even exceeds, the gas velocity already with slight inclination (Brauner and Moalem Maron [45]). Therefore, the application of Equation 23 as derived for U ^ » U^ is limited to a small region of sufficiently high gas rates where UyU^^» 1. WIND GENERATED WAVES: THE EXTREME OF J^ ^ 1 As has been discussed with reference to Figures 16-17, in practical systems where the gas phase is turbulent and H < 0.5, the role of J^ in controlling the transition from stratified-smooth to stratified-wavy is dominant. At the extreme of J^^ -^ l,the inertia terms of the two phases along the ZNS^ diminish, and the dynamic interaction that takes place at the phases free interface is the controlling destabilizing mechanism. This extreme of J^^ —> 1 is approached for sufficiently thin liquid layers in large diameter conduits and/or low viscosity of the liquid phase (Table 2). In the extreme of open air flow over a thin liquid film, e -^ 0, 4S./(7i8) -^ H~ ', Re^ = 4Uj^H/Vj^, the condition of J^ -> 1 can be rearranged in terms of the system's physical parameters:

368

Advances in Engineering Fluid Mechanics

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,

Boundary Conditions Required for the CFD Simulation

369

D»9.5cm

0=2.5 cm

AIR-VISCOUS LIQUID

CD

cn ii JO

I.Ok

\

lOOcp

JZ

U

z

rl

10

UJ

o

u. u.

LU

O

10^

iOcp

CJ

z Icp IQ-^ LU X

in

10^

2 SUPERFICIAL

4

6

8

AIR VELOCITY, UQs[m/s]

Figure 2 1 . "Sheltering coefficient" along the stratified-smooth boundary for various liquid viscosities.

370

Advances in Engineering Fluid Mechanics

Equation 43.3 can sometimes be approximated by a representative constant s as averaged along the transitional boundary. It is further of interest to note that in the particular case of thin laminar liquid layers in channel flow, the liquid layer thickness (as appears in Fr^^, Equation 27), can be obtained explicitly by solving the steady-state two-fluid momentum equations. In this case, the transitional criterion for wind generated waves, J^^ > 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).

Boundary Conditions Required for the CFD Simulation

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

Advances in Engineering Fluid Mechanics

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

° Quasi-steady " Amplitude * Frictional velocity

REFERENCES 1. Carr, L. W., "A Review of Unsteady Turbulent Boundary-Layer Experiments,'* In Unsteady Turbulent Shear Flows (ed., Michel, R., Cousteix, J., Houdeville, R.), Springer, pp. 3-34 (1981). 2. Ramaprian, B. R., and Tu, S. W., "Fully Developed Periodic Turbulent Pipe Flow. Part 2. The Detailed Structure of the Flow," J. Fluid Mech., Vol. 137, pp. 59-81 (1983). 3. Shemer, L., "Investigation of the Turbulent Characteristics of a Pulsating Pipe Flow," Ph.D. thesis, Tel-Aviv University, Tel-Aviv Israel (1981). 4. Mao, Z. X., and Hanratty, T. J., "Studies of the Wall Shear Stress in a Turbulent Pulsating Pipe Flow," J. Fluid Mech., Vol. 170 pp. 545-564(1985). 5. Shemer, L., and Wygnanski, I. "On the Pulsating Flow in a Pipe," Proceedings of the 3rd Symposium on Turbulent Shear Flows (University of California, Davis, CA) pp. 8.13-8.18 (1981). 6. Russell, T. W. F., and Charles, M. E., "The Effect of the Viscous Liquid in the Laminar Flow of Two Immiscible Liquids," Can. J. Chem. Engng, Vol. 37, pp. 18-34 (1959). 7. Tang, Y. P., and Himmelblau, D. M., "Velocity Distribution of Isothermal Two-Phase Co-Current Laminar Flow in Horizontal Rectangular Duct," Chem. Engng. Sci. Vol. 18, pp. 143-144 (1963). 8. Gemmell, A. R., and Epstein, N., "Numerical Analysis of Stratified Laminar Flow of Two Immiscible Newtonian Liquids in a Circular Pipe," Can. J. Chem. Engng. Vol. 40, pp. 215-224 (1962). 9. Charles, M. E., and Redberger, P. J., "The Reduction of Pressure Gradients in Oil Pipelines by the Addition of Water: Numerical Analysis of Stratified Flow," Can. J. Chem. Engng. Vol. 40, pp. 70-75 (1962). 10. Bentwich, M., "Two-Phase Axial Flow in Pipe," Trans, of the ASME, Series D, Vol. 86, pp. 669-672 (1964). 11. Yu, H. S., and Sparrow, E. M., "Stratified Laminar Flow in Ducts of Arbitrary Shape," AIChE 7., Vol. 13, pp. 10-16 (1967). 12. Brauner, N., Rovinsky, J., and Moalem Maron, D., "Analytical Solution for Laminar-Laminar Two-Phase Flow in Circular Conduits," Special Issue," Chem. Engng. Communications (1995). 13. Brauner, N., Rovinsky, J., and Moalem Maron, D.," Analytical Solution for Laminar-Laminar Stratified Two-Phase Flows with Curved Interfaces," ANS, NUREG/CP-0142, Vol. 1, pp. 192-211, NURETH-7 International meeting, Saratoga Springs, NY, Sept. 10-15 (1995). 14. Cheremisinoff, N., and Davis, E. J., "Stratified Turbulent-Turbulent Gas Liquid Flows," AIChE y., Vol. 25, pp. 48-56 (1979).

Boundary Conditions Required for the CFD Simulation

373

15. Russell, T. W. F., Etchells, A. W., Jensen, R. H. and Arruda, T. J., "Pressure Drop and Holdup in Stratified Gas-Liquid Flow," AIChE J., Vol. 20, pp. 664-669 (1974). 16. Shoham, O., and Taitel, Y., "Stratified Turbulent-Turbulent Flow in Horizontal and Inclined Pipes," AIChE 7., Vol. 30, pp. 377-385 (1984). 17. Johanessen, T., "A Theoretical Solution of the Lockhart and Martinelli Flow Model for Calculating Two-Phase Pressure Drop and Holdup," Int. J. Heat Mass Transfer, Vol. 15, pp. 1,443-1,449 (1972). 18. Agrawal, S. S., Gregorif, G. A., and Govier, G. W., "An Analysis of Horizontal Stratified Flow in Pipes," Can. J. Chem. Engng., Vol. 51, pp. 280-286 (1973). 19. Taitel, Y., and Dukler, A. E., "A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow," AIChE J., Vol. 22, pp. 47-55 (1976). 20. Brauner, N., and Moalem Maron, D., "Two-Phase Liquid-Liquid Stratified Flow," PCH, Phy. Chem. Hydro, Vol. 11, pp. 487-506 (1989). 21. Hall, A. R., and Hewitt, G. F., "Application of Two-Fluid Analysis to Laminar Stratified Oil-Water Flows," Int. J. Multiphase Flow Vol. 19(4), pp. 711-717 (1993). 22. Charnock, H., "Wind Stress on a Water Surface," Quart. J. Roy. Meteor. Soc, Vol. 81, pp. 639-640 (1955). 23. Ellis, S. R. and Gay, B., "The Parallel Flow of Two Fluid Streams: Interfacial Shear and Fluid-Fluid Interaction," Trans. Instn. Chem. Engrs., Vol. 37 pp. 206-213 (1959). 24. Cohen, L., and Hanratty, T. J., "Effect of Waves at a Gas-Liquid Interface on a Turbulent Air Flow," J. Fluid Mech., Vol. 31, pp. 467-479 (1968). 25. Davis, E. J., "Interfacial Shear Measurement for Two-Phase Gas-Liquid Flow by Means of Preston Tubes," Ind. Eng. Chem. Fundam., Vol. 8, pp. 153-159 (1969). 26. Akai, M., Inoue, A., Aoki, S., and Endo, K., "A Co-Current Stratified Air-Mercury Flow with Wavy Interface," Int. J. Multiphase Flow, Vol. 6, pp. 173-190 (1970). 27. Kordyban, E. S., "Interfacial Shear in Two Phase Wavy Flow in Closed Horizontal Channels," J. of Fluid Engineering, pp. 97-102 (1974). 28. Andritsos, N., and Hanratty, T. J., "Influence of Interfacial Waves in Stratified Gas-Liquid Flows," AIChE J. Vol. 33, pp. 444-454 (1987). 29. Fukano, T., Itoh, A., Odawara, H., Kuriwaki, T., and Takamatsu, Y.,"Liquid Films Flowing Concurrently with Air in Horizontal Duct (5th Report, Interfacial Shear Stress)" Bulletin of JSME, Vol. 28, pp. 2,294-2,301 (1985). 30. Fukano, T. Utsumi, R., Ousaka, A., and Sakamoto, T., "Behavior of the Liquid Film Flowing Concurrently with High Speed Gas Flow," FED-Vol. 110, Turbulence Modification in Multiphase Flows," ASME pp. 81-88 (1991). 31. Moalem Maron, D., and Brauner, N., "The Role of Interfacial Mobility in Determining the Interfacial Shear Friction Factor in Two-Phase Wavy Film Flow," Int. Comm. Heat Mass Transfer, Vol. 41(1), pp. 45-55 (1987). 32. Kang, H. C , and Kim, M. H., "The Relation Between the Interfacial Shear Stress and the Wave Motion in a Stratified Flow," Int. J. Multiphase Flow, Vol. 19, pp. 35-49 (1993).

374

Advances in Engineering Fluid IVIechanics

33. Kordyban, E. S., and Ranov, T., "Mechanism of Slug Formation in Horizontal Two-Phase Flow," 7. Basic Engng. Vol. 92. pp. 857-864 (1970). 34. Kordyban, E. S., "Some Characteristics of High Waves in Closed Channels Approaching Kelvin-Helmholtz Instability," ASME J. Fluids Engng., Vol. 99, pp. 339-346 (1977). 35. Wallis, G. B., and Dobson, J. E., "The Onset of Slugging in Horizontal Stratified Air-Water Flow," Int. J. Multiphase Flow, Vol. 1, pp. 173-193 (1973). 36. Mishima, K., and Ishii, M., "Theoretical Prediction of Onset of Horizontal Slug Flow," Trans. ASME J. Fluids Engng., Vol. 102, pp. 441-445 (1980). 37. Lin, P. Y., and Hanratty, T. J., "Prediction of the Initiation of Slugs with Linear Stability Theory," Int. J. Multiphase Flow, Vol. 12, pp. 79-98 (1986). 38. Wu, H. L., Pots, B. F. M., Hollenburg, J. F., and Meerhof, R.," Flow Pattern Transitions in Two-Phase Gas Condensation Flow at High Pressures in a 8inch Horizontal Pipe," Presented at the 3rd BHRA Int. Conf. of Multiphase Flow, the Hague, The Netherlands (1987). 39. Andritsos, N., and Hanratty, T. J., "Interfacial Instabilities for Horizontal GasLiquid Flows in Pipelines," Int. J. Multiphase Flow, Vol. 13, pp. 583-603 (1987). 40. Brauner, N., and Moalem Maron, D., "Stability Well-Posedness and Flow Pattern Transitions in Two-Phase Liquid-Liquid Flows," Proc. 6th Miami Int. Symp. Heat Mass Transfer, Nava Science Publishers, Inc. (1990). 41. Brauner, N., and Moalem Maron, D., "Analysis of Stratified/Non-stratified Transitional Boundaries in Horizontal Gas-Liquid Flows," Chem. Eng. Sci., Vol. 46(7), pp. 1,849-1,859 (1991). 42. Brauner, N., and Moalem Maron, D., "A Comprehensive Approach for Constructing Liquid-Liquid Flow Patterns Map and Convergence to Gas Liquid Systems," AIChE Symp., Series 283, Vol. 87, pp. 246-255 (1991). 43. Brauner, N., and Moalem Maron, D., "Stability Analysis of Stratified LiquidLiquid Horizontal Flow," Int. J. Multiphase Flow, Vol. 18(1), pp. 103-121 (1992). 44. Brauner, N., and Moalem Maron, D., "Flow Pattern Transitions in TwoPhase Liquid-Liquid Horizontal Tubes," Int. J. Multiphase Flow, Vol. 18(1), pp. 123-140 (1992). 45. Brauner, N., and Moalem Maron, D., "Analysis of Stratified/Non-stratified Transitional Boundaries in Inclined Gas-Liquid Flows," Int. J. Multiphase Flow, Vol. 18, pp. 541-557 (1992). 46. Barnea, D., "On the Effect of Viscosity on Stability of Stratified Gas-Liquid Flow-Application to Flow Pattern Transition at Various Pipe Inclinations," Chem. Eng. Sci., Vol. 46, pp. 2,123-2,131 (1991). 47. Crowley, C. J., Wallis, G. B., and Barry, J. J., "Validation of a One-dimensional Wave Model for the Stratified to Slug Flow Regime Transition, with Consequences for Wave Growth and Slug Frequency," Int. J. Multiphase Flow, Vol. 18, pp. 249-271 (1992). 48. Yiantsios, S. G., Higgins, B. G., "Linear Stability of Plane Poiseuille Flow of Two Superposed Fluids," Phys. Fluids, Vol. 31, pp. 3,225-3,238 (1988).

Boundary Conditions Required for the CFD Simulation

375

49. Tilley, B. S., Davis, S. H., and Bankoff, S. G., "Linear Stability Theory of Two-layer Fluid Flow in an Inclined Channel," Phys. Fluids, Vol. 6, pp. 1-16 (1994). 50. Tilley, B. S., Davis, S. H., and Bankoff, S. G., "Nonlinear Long-wave Stability of Superposed Fluids in an Inclined Channel," J. Fluid Mech., Vol. 277, pp. 55-83 (1994). 51. Hanratty, T. J., and McCready, M. J., "Phenomenological Understanding of Gas-Liquid Separated Flows," Presented at the 3rd Int. Wkshp. on Two-Phase Flow Fundamentals, London (1992). 52. Hanratty, T. J., Interfacial Instabilities Caused by Air Flow Over a Thin Liquid Layer," In: Waves on Fluid Interfaces (ed., Meyer, R. E.), Academic Press, New York, pp. 221-259 (1983). 53. Hanratty, T. J., "Separate Flow Modelling and Interfacial Transport Phenomena," Applied Sci. Research (Kluwer), Vol. 48, pp. 353-390 (1991). 54. Yadigaroglou, G., and Lahey, R. T. Jr., "On the Various Forms of the Conservation Equations in Two-Phase Flow," Int. J. Multiphase Flow, Vol. 2, pp. 477-494. (1976) 55. Hancox, W. T., Ferch, R. L., Liu, W. S., and Nieman, R. E., "One-Dimensional Models for Transient Gas-Liquid Flows in Ducts," Int. J. Multiphase Flow, Vol. 6, pp. 25-40 (1980). 56. Banerjee, S., and Chan, A. M. C , "Separated Flow Modes-I Analysis of the Averaged and Local Instantaneous Formulations," Int. J. Multiphase Flow, Vol. 6,pp. 1-24 (1980). 57. Banerjee, S., ""Separated Flow Modes-II Higher Order Dispersion Effects in the Averaged Formulation," Int. J. Multiphase Flow, Vol. 6, pp. 241-248 (1980). 58. Banerjee, S., "Multifield Modelling of Two-Phase Flow: Problems and Potential," 2nd International Conference on Multiphase Flow, London, June 19-21 (1985). 59. Ardron, K. H., "One-Dimensional Two-Fluid Equations for Horizontal Stratified Two-Phase Flow," Int. J. Multiphase Flow, Vol. 6, pp. 295-305 (1980). 60. Kocamustafaogullari, G., "Two-Fluid Modelling in Analyzing the Interfacial Stability of Liquid Film Flows," Int. J. Multiphase Flow, Vol. 11, pp. 63-89 (1985). 61. Hanratty, T. J., and Hershman, A., "Initiation of Roll Waves," AIChE J. Vol. 7, pp. 488-497 (1961). 62. Coutris, N., Delhaye, J. M., and Nakach, R., "Two-Phase Flow Modelling: The Closure Issue for a Two-Layer Flow," Int. J. Multiphase Flow, Vol. 6, pp. 977-983 (1989). 63. Paras, S. V., Vlachos, N. A. and Karabelas, A. J., "Liquid Layer Characteristics in Stratified-Atomization Flow," Int. J. Multiphase Flow, Vol. 20, pp. 939-956 (1994). 64. Kowalski, J. L., "Wall and Interfacial Stress in Stratified Flow in a Horizontal Pipe," AIChE J. Vol. 33, pp. 274-281 (1987). 65. Sinai, Y. L., "A Charnock-Based Estimate of Interfacial Resistance and Roughness for Internal, Fully-Developed Stratified Two-Phase Horizontal Flow," Int. J. Multiphase Flow, Vol. 9, pp. 13-19 (1983).

376

Advances in Engineering Fluid Mechanics

66. Sinai, Y. L., "An Extended Charnock Estimate of Interfacial Stress in Stratified Two-Phase Flows," Int. J. Multiphase Flow, Vol. 12, pp. 839-844 (1986). 67. Bontozoglou, V., and Hanratty, T. J., "Wave Height Estimation in Stratified Gas Liquid Flows," AIChE J. Vol. 35, p. 1,346-1,350 (1989). 68. Strand, O., "An Experimental Investigation of Stratified Two-Phase Flow in Horizontal Pipes," Ph.D. thesis. University of Oslo, Dept. of Mechanics (1993). 69. Hagiwara, Y., Esmaeilzadeh, E., Tsutsui, H., and Suzuki, K., "Simultaneous Measurement of Liquid Film Thickness, Wall Shear Stress and Gas Flow Turbulence of Horizontal Wavy Two-Phase Flow," Int. J. Multiphase Flow, Vol. 15, pp. 421-431 (1989). 70. Suzanne, C , "Structure de L'ecoulement Stratifie de Gaz et de Liquide en Canal Rectangulaire," These d'Etat: Mecanique de Fluides, L'Institut National Polytechnique de Toulouse (1985). 71. Jayanti, S., Wilkes, N. S., Clarke, D. S., and Hewitt, G. F., "The Prediction of Turbulent Flows Over Roughened Surfaces and its Application to Interpretation of Horizontal Annular Flow," Proc. R. Soc. London A, Vol. 431, pp. 71-88 (1990). 72. Nordsveen, M., and Bertelsen, A. F., "Waves, Turbulence and the Mean Field in Stratified Duct Flow," NTNF/PROFF program. Project OT.45.24477, Report No. ISBN82-553-0837-7, Dept. of Mechanics, University of Oslo, May (1993). 73. Andreussi, P., and Persen, L. N., "Stratified Gas-Liquid Flow in Downwardly Inclined Pipes," Int. J. Multiphase Flow, Vol. 13, pp. 565-575 (1987). 74. Wallis, G. B., "One-Dimensional Two-Phase Flow," McGraw-Hill, New York (1969). 75. Lin, P. Y., and Hanratty, T. J., "Effect of Pipe Diameter on Flow Patterns for Air-Water Flow in Horizontal Pipes," Int. J. Multiphase Flow, Vol. 13, pp. 549-563 (1987). 76. Bendiksen, K., and Espedal, M., "Onset of Slugging in Horizontal Gas-Liquid Pipe Flow," Int. J. Multiphase Flow, Vol. 18, pp. 237-247 (1992). 77. Mandhane, J. M., Gregory, G. A., and Aziz, K., "A Flow Pattern Map for Gas Liquid Flow in Horizontal Pipes," Int. J. Multiphase Flow, Vol. 1, pp. 537-553 (1974). 78. Shoham, O., "Flow Pattern Transitions and Characterization in Gas-Liquid TwoPhase Flow in Inclined Pipes," Ph.D. Thesis, Tel-Aviv University, Israel (1982). 79. Simpson, H. C, Rooney, D. H., Gratton, E., and Al-Samarval, F. A. A., "TwoPhase Flow in Large Diameter Horizontal Tubes," NEL Report No. 677 (1981). 80. Luninski, Y., "Two-Phase Flow in Small Diameter Lines," Ph.D. Thesis, TelAviv University, Israel (1981). 81. Andritsos, N., Williams, L., and Hanratty, T. J., "Effect of Liquid Viscosity on the Stratified-Slug Transition in Horizontal Pipe Flow," Int. J. Multiphase Flow, Vol. 15, pp. 877-892 (1989). 82. Nakamura, H., Anoda, Y., and Kukita, Y., "Flow Regime Transitions in HighPressure Steam-Water Horizontal Pipe Two-Phase Flow," ANS Proc. National Heat Trans. Conf., Minneapolis, pp. 269-276 (1991). 83. Crowley, C. J., Wallis, G. B., and Barry, J. J., "Dimensionless Form of a OneDimensional Wave Model for the Stratified Flow Regime Transition," Int. J. Multiphase Flow, Vol. 19, pp. 369-376 (1993).

Boundary Conditions Required for the CFD Simulation

377

84. Benjamin, T. B., "Shearing Flow Over a Wavy Boundary," J. Fluid Mech., Vol. 6, pp. 161-205 (1959). 85. Thorsness, C. B., Morrisroe, P. E., and Hanratty, T. J., "Comparison of Linear Theory with Measurements of the Variation of Shear Stress along a Solid Wave," Chem. Eng. ScL, Vol. 33, pp. 579-592 (1978). 86. Zilker, D. P., Cook, G. W., and Hanratty, T. J., "Influence of the Amplitude of a Solid Wave Wall on a Turbulent Flow, Part I: Non-Separated Flows," J. Fluid Mech., Vol. 82, pp. 29-51 (1977). 87. Zilker, D. P., and Hanratty, T. J., "Influence of the Amplitude of a Solid Wave Wall on a Turbulent Flow, Part II: Separated Flows," J. Fluid Mech., Vol. 90, pp. 257-271 (1979). 88. Buckles, J., and Hanratty, T. J., "Turbulent Flow over Large-Amplitude Wavy Surfaces," J. Fluid Mech., Vol. 240, pp. 27-44 (1984). 89. Abrams, J., and Hanratty, T. J., "Relaxation Effects Observed for Turbulent Flow over a Wavy Surface," /. Fluid Mech., Vol. 151, pp. 443-455 (1985). 90. Jeffreys, H., "On the Formation of Water Waves by Wind," Proc. Roy. Soc. London A, Vol. 107, pp. 189-206 (1925). 91. Miles, J. W., "On the Generation of Surface Waves by Shear Flows," J. Fluid Mech., Vol. 3, pp. 185-204 (1957). 92. Miles, J. W., "On the Generation of Surface Waves by Shear Flows, Part 2 and 3 , " J. Fluid Mech., Vol. 6, pp. 568-598 (1959). 93. Miles, J. W., "The Hydrodynamic Stability of a Thin Film of Liquid in Uniform Shearing Motion," J. Fluid Mech., Vol. 8, pp. 593-610 (1960). 94. Miles, J. W., "On the Generation of Surface Waves by Shear Flows, Part 4," /. Fluid Mech., Vol. 13, pp. 433-448 (1962). 95. Hsu, S., and Kennedy, J. F., "Turbulent Flow in Wavy Pipes," J. Fluid Mech., Vol. 47, pp. 481-502 (1971). 96. Kendall, J. M., "The Turbulent Boundary Layer over a Wall with Progressive Surface Waves," J. Fluid Mech., Vol. 4 1 , pp. 259-281 (1970). 97. Sigal, A. "An Experimental Investigation of the Turbulent Boundary Layer over a Wall," Ph.D. thesis. Department of Aeronautical Engineering, Cal. Inst. Tech. (1970). 98. Craik, A. D. D., "Wind-Generated Waves in Thin Liquid Films," / . Fluid Mech., Vol. 26, pp. 369-392 (1966). 99. Jurman, L. A., and McCready, M. J., "Study of Waves on Thin Liquid Films Sheared by a Turbulent Gas Flow," Phys. Fluids A, Vol. 1, pp. 522-536 (1989). 100. Jurman, L. A., Bruno, K., and McCready, M. J., "Characterization of Waves on Thin, Horizontal, Gas-Sheared Liquid Films," Int. J. Multiphase Flow, Vol. 15, pp. 371-384 (1989). 101. Asali, J. C , and Hanratty, T. J., "Ripples Generated on a Liquid Film at High Gas Velocities," Int. J. Multiphase Flow, Vol. 2, pp. 229-243 (1993). 102. Brauner, N., and Moalem Maron, D., "The Role of Interfacial Shear Modelling in Predicting the Stability of Stratified Two-Phase Flow," Chem. Eng. ScL, Vol. 48(10), pp. 2,867-2,879 (1993). 103. Brauner, N., and Moalem Maron, D., "Dynamic Model for the Interfacial Shear as a Closure Law in Two-Fluid Models," Nuclear Eng. and Design, Vol. 149, pp. 67-79 (1994).

378

Advances in Engineering Fluid Mechanics

104. Barnea, D., Shoham, O., Taitel, Y., and Dukler, A. E., "Flow Pattern Transition for Downward Inclined Two-Phase Flow: Horizontal to Vertical," Chem. Eng. ScL, Vol. 37, pp. 735-740 (1982). 105. Brauner, N., and Moalem Maron, D., "Stability of Two-Phase Stratified Flow as Controlled by Laminar/Turbulent Transition," Int. Comm. Heat Mass Trans., Vol. 21(1), pp. 65-74 (1994). 106. Lyczkowski, R. W., Gidaspow, D., Solbrig, C. W., and Hughes, E. D.," Characteristics and Stability Analysis of Transient One-Dimensional TwoPhase Flow Equations and their Finite Difference Approximations," Nuclear ScL Engng., Vol. 66, pp. 378-396 (1978). 107. Ramshaw, J. D., and Trapp, J. A., "Characteristics, Stability, and Short Wavelength Phenomena in Two-Phase Flow Equation Systems," Nuclear Sci. Engng., Vol. 66, pp. 93-102 (1978). 108. Drew, D. A., "Mathematical Modelling of Two-Phase Flow," Ann. Rev. Fluid Mech., Vol. 15, pp. 261-291 (1983). 109. Jones, A. V., and Prosperetti, A., "On the Suitability of First-Order Differential Models for Two-Phase Flow Prediction," Int. J. Multiphase Flow, Vol. 11, pp. 133-148 (1985). 110. Prosperetti, A., and Jones, A. V., "The Linear Stability of General Two-Phase Flows Models," Int. J. Multiphase Flow, Vol. 13, pp. 161-171 (1987). HI. Moe, J., "Long Wave Disturbance in Stratified Two-Phase Pipe Flow," presented at a seminar on "Phase Interface Phenomena in Multiphase Flow," Dubrovnik, May 14-18 (1990). 112. Barnea, D., and Taitel, Y., "Non-Linear Interfacial Instability of Separated Flow," Chem. Engng. ScL, Vol. 49, pp. 2341-2349 (1994). 113. Andritsos, N. Bontozoglou, V., and Hanratty, T. J., "Transition to Slug Flow in Horizontal Pipes," Chem, Eng. Comm., Vol. 118, pp. 361-385(1992). 114. Hanratty, T. J., "Gas Liquid Flow in Pipelines," PCH Phy. Chem. Hydro., Vol. 9, pp. 101-114 (1987). 115. Brauner, N., and Moalem Maron, D., "Identification of the Range of Small Diameter Conduits Regarding Two-Phase Flow Patterns Transition," Int. Comm. Heat Mass Transfer, Vol. 19, pp. 29-39 (1992). 116. Stanton, Sir T. E., Marshall, D., and Houghton, R., "The Growth of Waves on Water Due to the Action of the Wind," Proc. Roy. Soc. A, Vol. 137, pp. 283-293 (1932).

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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

Water Flow Through Helical Coils in Turbulent Condition

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

^iK

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

Advances in Engineering Fluid Mechanics

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

Water Flow Through Helical Coils in Turbulent Condition

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

Advances in Engineering Fluid Mechanics

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

Water Flow Through Helical Coils in Turbulent Condition

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

Advances in Engineering Fluid Mechanics

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-<

(34)

for Reynold number varying from 1,200 to Re^^^^j^^, and D/D^ from 0.0159 to 0.0556. Collins and Dennis [100] used finite difference technique (extremely fine grid size) and proposed the following equation = 0.38 + 0.1028 De«5

(35)

Van Dyke [101] extended the Dean's series to 24 terms and obtained the following result

f

= 0.4713 De«25

(36)

which is clearly in disagreement with the results obtained by other theoretical analysis. He suggested that this discrepancy was due to inaccuracy or incorrectness of the then-available numerical and analytical solutions. Mishra and Gupta presented pressure drop data for Newtonian liquid flowing through helical coils of different dimensions [102]. The coils were made of thickwalled, flexible, and smooth polythene pressure tubing of uniform circular cross section. They observed that the pitch had a negligible effect on pressure drop if it was less than the diameter of the coil. They modified the Dean number to incorporate the effect of pitch as (37) where. D.

2r. 1 +

27ir.

(38)

They proposed the following f - ^ = 1 + 0.033(log De^)^o

(39)

Water Flow Through Helical Coils in Turbulent Condition

389

for 1 < De^ < 3,000 0.289 X 10-2 < D/D^ < 0.155 0 < p/D^ < 25.4 Dennis [103] verified their early work in view of Van Dyke's criticism and found the following f - ^ = 0.388 + 0.1015 De^^

(40)

St

Hart et al [104] proposed the following empirical equation for helically coiled tube f^^ ^ ^ 0.090De'^ f,, ~ "^ (70 + De)

^"^^^

Rao [105] proposed a generalized correlation for Newtonian and non-Newtonian pseudoplastic liquid flow through helical coil as f - ^ = 1 + 0.0188(log De f-^' f ^

(42)

St

where De = De for Newtonian liquid De = De^j^ for non-Newtonian liquid Transition Region Experiments of Taylor [106], White [85], Adler [86], Keulegan and Beij [107], Ito [89], Koutsky and Adler [12], Sreenivasan and Strykowski [108], Hart et al [104] and Webster and Humphrey [39] confirmed that the flow in a curved pipe remained in the viscous range over a large range of Reynolds numbers than that in a straight pipe. The usual two- and three-dimensional propagation of turbulent eddies is suppressed by secondary flow caused by centrifugal forces. It is mainly due to the streamwise acceleration of the inlet flow near the inner wall of the curved pipe, laminarizing the fluid motion. But the uniform laminarization of the flow throughout the pipe cross section and the persistence of laminar flow downstream of the coil are still not clearly understood. The critical Reynolds number, which may be defined as the highest Reynolds number for which the flow in a helical coil remains in the viscous range, must be higher than that of straight pipe. Ito [89] proposed an empirical equation for critical Reynolds number as Re,rt«ca, = 2 X 10^(D,/D^)«" valid for 0.00116 < D./D < 0.0667.

(43)

390

Advances in Engineering Fluid Mechanics

Kubair and Varrier [109] proposed the following Rec.uicai = 1 2 7 3 0 ( D / D ; -

(44)

for 0.0005 < D/D < 0.103. t

c

Kutateladze and Borishanskii [110] suggested R^criticai = 2300 + 12930(D/D^f 3

(45)

for 0.0417 < D/D < 0.1667 t

c

However, the above equation is only valid for limited range of curvature ratio. A major drawback of all three equations is as D^ —> a the critical Reynolds number does not approach 2,100 for straight tube. Srinivasan et al. [ I l l ] used the experimental data of critical Reynolds number of Adler [86], Kubair and Kuloor [92], Storrow [112], Taylor [106] and White [85] and proposed Re^,,^^, = 2100[1 + 1 2 ( D / D / 5 ]

(46)

The above equation approaches 2,100 as D^ -^ a, i.e., critical Reynolds number for straight tube. Mishra and Gupta [102] observed that the critical Reynolds number increased with a decrease in coil diameter and decrease in the pitch of the coil. They proposed the following empirical equation Re,,.,, = 2 X 10^(D/D J0.32

(47)

for 10-'3 < D/D ^ < 1 0 ' t

cM

Hart et al. [104] used the Srinivasan et al. [ I l l ] equation to calculate the critical Reynolds number. Webster and Humphrey [39] showed that the Ito's [89] equation gave good agreement with their experimental critical Reynolds number. Taylor [106], Sreenivasan and Strykowski [108] and Webster and Humphrey [39] documented values of the transition Reynolds numbers, marking the onset and completion of the gradual transition process. Turbulent

Flow

Turbulent flow pressure drop in curved circular pipes was first reported by White [85], who suggested the following correlation f = 0.08 Re-«25 _^ o.012(D/D^)«^ for 15,000 < Re < 10^

(48)

Water Flow Through Helical Coils in Turbulent Condition

391

Wattendorf [113] for the first time measured the mean velocity distribution in a two-dimensional curved pipe of constant curvature and attempted to explain the momentum transport characteristics from a viewpoint of stability of a moving fluid particle. Ito [89] pointed out that if the curvature was marked, the distribution of velocity in a curved pipe was totally altered by the secondary flow, with the maximum axial velocity being near the outside wall and the secondary flow restricted chiefly in a sort of boundary layer in the regions near the walls, known as ''shedding layer". The thickness of the shedding layer is defined to be equal to the distance from the wall to the point where the peripheral velocity component changes its sense. The thickness of this layer is assumed to be small compared with the radius of the pipe. He assumed l/7th power velocity distribution law in the shedding layer for incompressible Newtonian fluids for smaller D/D^ and simplified the momentum integral equations for the shedding layer. He presented the following expression for shedding layer thickness and friction factor as 26 = [Re(D/D^)2]D

(49) (50)

f^(D/D/^ = [Re(D/D^)2]-

He also assumed the logarithmic velocity distribution law in the shedding layer for small D/D^, simplified the momentum integral equation for the shedding layer and derived the following expression f = Y-

(51)

= K, log Re

f3/2

+ K,

(52)

r . Where /

\0.5

(53)

Y^ e^ = Re K^.J

and K, and K2 are constants. He derived the following I. Empirical resistance formula deduced from the l/7th power velocity distribution law

0.00725 + 0.076 Re vD.y

^D^^ vDey

(54)

392

Advances in Engineering Fluid IVIechanics

for 0.034 < Re(D/D^)2 < 300. V/hen Re(D/D^)2 is less than 0.034 then (55)

f = f. C

St

where f^^ is calculated from Equation 1. For large values of Re(D/D^)^ 0.079 [Re(D,/DJ^f

vD.y

(56)

for Re(D/D,)2 > 6. II. Empirical resistance formula deduced from the logarithmic velocity distribution law /

\0.5

= 0.00807 + 0 . 4 V^ey

(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)

Water Flow Through Helical Coils in Turbulent Condition

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

Advances in Engineering Fluid Mechanics

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.

Water Flow Through Helical Coils in Turbulent Condition

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

Advances in Engineering Fluid Mechanics

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.

Water Flow Through Helical Coils in Turbulent Condition

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

Advances in Engineering Fluid Mechanics

REFERENCES 1. Ali, S. and A. H. Zaidi, "Head Loss and Critical Reynolds Numbers for Flow in Ascending Equiangular Spiral Tube Coils," Ind. Eng. Chem. Proc. Des. Dev. 18, 349-353 (1979). 2. Grindley, J. H. and A. H. Gibson, "On the Frictional Resistance to the flow of Air through a Pipe," Proc. Roy, Soc. (London), A80, 114-139 (1908). 3. Dawe, R. A., "A Method for Correcting Curved Pipe Flow Effects Occurring in Coiled Capillary Viscometers," Rev. Sci, Instrum., 44, 1,231-1,233 (1973). 4. Merkel, E., Die Grundlagen der Warmeiibertragung, 51 (1927). 5. Dravid, A. N., K. A. Smith, E. W. Merrill and P. L. T. Brien, "Effect of Secondary Fluid Motion on Laminar Flow Heat Transfer in Helically Coiled Tubes, AIChE J., 17, 1,114-1,122 (1971). 6. Weissman, M. H. and L. F. Mockros, "Gas Transfer to Blood Flowing in Coiled Circular Tubes," ASCE Proc. Eng. Mech. Div. J., 94, 857-872 (1968). 7. Richardson, P. D., K. Tanishita, P. M. Galetti, "Mass Transfer through Tubes Wound in Serpentine Shape," Lett. Heat Mass Transfer, 2, 481-485 (1975). 8. Tanishita, K., K. Nakano, Y. Sakurai, P. Hosokawa, D. P. Richardson and P. M. Galletti, "Compact Oxygenator Design with Curved Tubes Wound in Weaving Patterns," Trans. Am. Soc. Artif. Intern. Organs., 24, 327-331 (1978). 9. Dravid, A. N., "The Effect of Secondary Flow Motion on Laminar Flow Heat Transfer in Helically Coiled Tubes," Sc. D. Thesis, Mass Inst. Technol., Cambridge (1968). 10. Srinivasan, S. and C. Tien, "Reverse Osmosis in a Curved Tubular Membrane Duct," Desalination, 9, 127-139 (1971). 11. Nunge, R. J. and L. R. Adams, "Reverse Osmosis in Laminar Flow through Curved Tubes," Desalination, 13, 17-36 (1973). 12. Koutsky, J. A. and R. J. Adler, "Minimization of Axial Dispersion by Use of Secondary Flow in Helical Tubes," Can J. Chem. Engg., 42, 239-246 (1964). 13. Janssen, L. A. M., "Axial Dispersion in Laminar Flow through Coiled Tube," Chem. Engg. Sci, 31, 215-218 (1976). 14. Atkeson, F. V., "Fractional Distillation under Nonequilibrium Conditions," Ind. Eng. Chem. 49 239 (1957). 15. Morton, F., P. J. King and A. Mclaughlin, "Helical-Coil Distillation Column Pt.-I Efficiency Studies," Trans. Instn. Chem, Engrs., 42, T285-T295 (1964). 16. Morton, F., P. J. King and A. Mclaughlin, "Helical-Coil Distillation Column Pt.-II Liquid film Resistance," Trans. Instn. Chem, Engrs., 42, T296-T304 (1964). 17. Smol'skiy, B. M. and A. S. Chekol'skiy, "Investigation of Heat and Mass Transfer in Condensation of Water Vapor from Moist Air in Curved Channels," Heat Transfer—Sov. Res. (USA), 10, 162-169 (1978). 18. Barnes, H. A. and K. Walters, "On the Flow of Viscous and Elastica-Viscous Liquids through Straight and Curved Pipes," Proc. Roy. Soc. (London) A314, 85-109 (1969). 19. Mashelkar, R. A. and G. V. Devarajan, "Secondary Flows of non-Newtonian Fluids: Pt I-Laminar Boundary Layer Flow of a Generalized non-Newtonian Fluid in a Coiled Tube," Trans, Instn. Chem. Engrs., 54, 100-107 (1976).

Water Flow Through Helical Coils in Turbulent Condition

399

20. Mashelkar, R. A. and G. V. Devarajan, "Secondary Flows of non-Newtonian Fluids: Pt II-Frictional Losses in Laminar Flow of Purely Viscous and Viscoelastic Fluid through Coiled Tube," Transy Instn. Chem. Engrs., 54, 108-114 (1976). 21. Ruthven, D. M., "The Residence Time Distribution for Laminar Flow in a Helical Tube," Chem. Engg. ScL, 33, 629-630 (1978). 22. Ruthven, D. M., "The Residence Time Distribution for Ideal Laminar flow in Helical coil," Chem. Engg. ScL, 26, 1,113-1,121 (1971). 23. Trivedi, R. N. and K. Vasudeva, "RTD for Diffusion Free Laminar Flow in Helical Coils," Chem. Engg. ScL, 29, 2,291-2,295 (1974). 24. Saxena, A. K. and K. D. P. Nigam, " On RTD for Laminar Flow in Helical Coils," Chem. Engg. ScL, 34, 425-426 (1979). 25. Lyne, W. H., "Unsteady Viscous Flow in a Curved Pipe," J. Fluid Mech., 45, 13-31 (1971). 26. Zalosh, R. G. and W. G. Nelson, "Pulsating Flow in a Curved Tube," /. Fluid Mech., 59, 693-705 (1973). 27. Munson, B. R., "Experimental Results for Oscillating Flow in a Curved Pipe," Phy. Fluids, 18, 1,607-1,609 (1975). 28. Smith, F. T., "Pulsatile Flow in Curved Pipe," J. Fluid Mech., 71, 15-42 (1975). 29. Chandran, K. B., T. L. Yearwood and D. M. Wieting, "An Experimental Study of Pulsatile Flow in a Curved Tube," J. Biomech., 12, 793-805 (1979). 30. Mullin, T. and C. A. Created, "Oscillatory Flow in Curved Pipes, Pt.-I The Developing Flow Case," J. Fluid Mech., 98, 383-395 (1980). 31. Mullin, T. and C. A. Created, "Oscillatory Flow in Curved Pipes, Pt.-II The Fully Developed Case," J. Fluid Mech., 98, 397-416 (1980). 32. Lin, J. J. and J. M. Tarbell, "An Experimental and Numerical Study of Periodic Flow in a Curved Tube," /. Fluid Mech., 100, 623-638 (1980). 33. Talbot, L. and O. K. Gong, "Pulsatile Entrance Flow in a Curved Pipe," /. Fluid Mech., 127, 1 (1983). 34. Berger, S. A., L. Talbot and L. S. Yao, "Flow in Curved Pipes," Ann. Rev, Fluid Mech., 15, 461-512 (1983). 35. Chang, L. J., and J. M. Tarbell, "Numerical Simulation of Fully Developed Sinusoidal and Pulsatile (Physiological) Flow in Curved Tubes," J. Fluid Mech., 161, 175-198 (1985). 36. Nandakumar, K. and J. H. Masliyah, "Swirling Flow and Heat Transfer in Coiled and Twisted Pipes," in Advances in Transport Process, A. S. Majumeder and R. A. Mashelkar (Eds.), Wiley Eastern Limited, New Delhi, 4, 49-112 (1986). 37. Sudo, K., M. Sumida and R. Yamane, "Secondary Motion of Fully Developed Oscillatory Flow in a Curved Pipe," J. Fluid Mech., 237, 189-208 (1992). 38. Swanson, C. J., S. R. Stalp and R. J. Donnelly, "Experimental Investigation of Periodic Flow in Curved Pipes," J. Fluid Mech., 256, 69-83 (1993). 39. Webster, D. R. and J. A. C. Humphrey, "Experimental Observations of Fluid Instability in a Helical Coil," Trans. ASME J. Fluids Engg., 115, 436-443 (1993). 40. Ito, H., "Flow in Curved Pipes," JSME Int. J, 30, 543-552 (1987). 41. Ali, S, "Pressure Drop Performance of Coiled Tubes," Chem. Engg. Res. Dev. 67, 428-432 (1989).

400

Advances in Engineering Fluid Mechanics

42. Blasius, H., "Das Aehnlichkeitsgesetz bei Reibungsavorgangen in Fliissigkeiten," ForschungshefU Nr 131 (1913). 43. Prandtl, L., "Neuere Ergebnisse der Turbulenzforschung," Z. VD/, 77, 105-114 (1933). 44. Nikuradse, J., "Gesetzmassigkeiten der Turbulenten Stromung in Glatten Rohren," VDI Forschungsheft, Nr 356 (1932). 45. von Karman, T., "Uber Laminare und Turbulente Reibung," Z. Angew Math. Mech., 1, 233-252 (1921). 46. Colebrook, C. F., "Turbulent Flow in Pipes, With Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws," J. Instn. Civ. Engrs., 11, 133-156 (1939). 47. Moody, L. F., "Friction Factor for Pipe Flow," Trans. Am. Soc. Mech. Eng., 66, 671-684 (1944). 48. Barr, D. I. H., "New Forms of Equations for the Correlation of Pipe Resistance Data," Proc. Instn, Civ. Engrs., 53, 383-390 (1972). 49. Barr, D. I. H., "Two Additional Methods of Direct Solution of the ColebrookWhite Function," Proc. Instn, Civ. Engrs., 59, 827-835 (1975). 50. Churchill, S. W., "Empirical Expression for the Shear Stress in Turbulent flow in Commercial Pipe," AIChE J., 19, 375-376 (1973). 51. Swamee, P. K. and A. K. Jain, "Explicit Equations for Pipe-Flow Problems," ASCE J. Hydraulic Div., 102, 657-664 (1976). 52. Chen Ning Hsing "An Explicit Equation for Friction Factor in Pipe," Ind. Eng. Chem. Fundam., 18, 296-297 (1979). 53. Barr, D. I. H., "The Transition for Laminar to Turbulent Flow," Proc. Instn, Civ. Engrs., 69, 555-562 (1980). 54. Barr, D. L H., "Solutions of the Colebrook-White Function for Resistance to Uniform Turbulent Flow," Proc. Instn, Civ. Engrs., 71, 529-535 (1981). 55. Zigrang, D. J. and N. D. Sylvester, "Explicit Approximations to the Solution of Colebrook's Friction Factor Equation," AIChE J., 28, 514-515 (1982). 56. Haaland, S. E., "Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow," Trans. ASME J. Fluids Engg., 105, 89-90 (1983). 57. Chen, J. J. J., "Systematic Explicit Solutions of the Prandtl and ColebrookWhite Equations for Pipe flow," Proc. Instn. Civ. Engrs, 79, 383-389 (1985). 58. Urbina, J. L., "Head Loss Characteristics of Trickle Irrigation Hose with Emitters," M. Sc. Thesis, Utah State University, USA (1976). 59. Paraqueima, J. R., "Study of Some Frictional Characteristics of Small Diameter Tubing for Trickle Irrigation Laterals," M. Sc. Thesis, Utah State University, USA (1977). 60. Norum, E. M., "Determining Friction Loss in Polyethylene Pipe used for Drip Irrigation Laterals," Irrig. Age, 26K, 17-18 (1984). 61. von Bernuth, R. D. and T. Wilson, "Friction Factor for Small Diameter Plastic Pipes," ASCE J. Hydraulic Engg., 115, 185-192 (1989). 62. Venkatesan, R., S. K. Das and M. N. Biswas, "Friction Factors for Small Diameter Transparent PVC Pipes," Indian J. TechnoL, 28, 549-552 (1990). 63. Hawthorne, W. R., "Secondary Circulation in Fluid Flow," Proc. Roy, Soc. (London), 206, 374-387 (1951). 64. Austin, L. R. and J. D. Seader, "Entry Region for Steady Viscous Flow in Coil Circular Pipe," AIChE J. 20, 820-822 (1974).

Water Flow Through Helical Coils In Turbulent Condition

401

65. Singh, M.D., "Entry Flow in a Curved Pipe," J. Fluid Mech., 65, 517-539 (1974). 66. Yao, L. S. and S. A. Berger, "Entry Flow in a Curved Pipe," J. Fluid Mech., 67, 177-196 (1975). 67. Patankar, S. V., V. S. Pratap and D. B. Spalding, "Prediction of Laminar Flow and Heat Transfer in Helically Coiled Pipes," /. Fluid Mech., 62, 539-551 (1974). 68. Humphrey, J. A. C , "Numerical Calculation of Developing Laminar Flow in Pipes of Arbitrary Curvature Radius," Can J. Chem. Engg., 56, 151-164 (1978). 69. Roscoe, D. F., "Numerical Solution of the Navier-Stokes Equations for a Three-Dimensional Laminar Flow in Curved Pipes using Finite Difference Methods," J. Engg. Math., 12, 303-323 (1978). 70. Rushmore, W. L. and D. B. Tanlbee, Computers & Fluids, 6, 125 (1978). 71. Bara, B., K. Nandakumar and J. H. Masliyah, "An Experimental and Numerical Study of the Dean Problem: Flow Development towards Two Dimensional Multiple Solutions," J. Fluid Mech., 244, 339-376 (1992). 72. Agrawal, Y., L. Talbot and K. Gong, "Laser Anemometer Study of Flow Development in Curved Circular Pipes," J. Fluid Mech. 85, 497-518 (1978). 73. Choi, U. S., L. Talbot and E. Cornet, "Experimental Study of Wall Shear Rates in the Entry Region of a Curved Tube," J. Fluid Mech., 93, 465-489 (1979). 74. Kaczinsky, J., J. W. Smith and R. L. Hummel, "Laminar Flow in the Central Plane of a Curved Circular Pipe, Can. J. Chem. Engg., 53, 221-224 (1975). 75. Shiragami, N. and L Inoue, Riken Hokoku, 57, 37 (1981). 76. Takami, T., K. Sudo and Y. Tomita, "Flow of non-Newtonian Fluids in Curved Pipes (Turbulent Flow)," JSME Int. J., 33, 476-485 (1990). 77. Thomson, J., "On the Origin of Windings of Rivers in Alluvial Plains, with Remarks on the Flow of Water Round Bends in Pipes," Proc. Roy. Soc. (London), A25, 5-8 (1876). 78. Thomson, J., "Experimental Demonstration in Respect to the Origin of Windings of Rivers in Alluvial Plains and to the Mode of Flow of Water Round Bends of Pipes," Proc. Roy. Soc. (London), A26, 356-357 (1877). 79. Williams, G. S., C. W. Hubbell and G. H. Finkell, "Experiments at Detroit Michigan on the Effect of Curvature on the Flow of Water in Pipes," Trans. ASCE, 47, 1-196 (1902). 80. Eustice, J. "Flow of Water in Curved Pipes," Proc. Roy. Soc. (London), A84, 10-118 (1910). 81. Eustice, J. "Experiments of Streamline Motion in Curved Pipes," Proc. Roy. Soc. (London), A85, 119-131 (1911). 82. Dean, W. R., "Note on the Motion of Fluid in a Curved Pipe," Philos. Mag., 20, 208-223 (1927). 83. Dean, W. R., "The Stream-Line Motion of Fluid in a Curved Pipe," Philos. Mag., 30, 673-693 (1928). 84. Taylor, G. L, "The Criterion for Turbulence in Curved Pipes" Proc. Roy. Soc. (London), A124, 243-249 (1929). 85. White, C M . , "Streamline Flow through Curved Pipes," Proc. Roy. Soc. (London), A123, 645-663 (1929). 86. Alder, M., "Stromung in Gekriimmten Rohen," Z. Angew. Math. Mech., 14, 257-275 (1934). 87. Prandtl, L., Fuhrer Durchdie Stromungslehre, 3rd Ed. Braunschweig, 159 (1949).

402

Advances in Engineering Fluid Mechanics

88. Hasson, D., "Streamline Flow Resistance in Coils," Res, Correspondence, 1, SI (1955). 89. Ito, H., "Friction Factors for Turbulent Flow in Curved Pipes," Trans. ASME J, Basic Engg., 81, 123-134 (1959). 90. Kubair, V. and C. B. S. Varrier, "Pressure Drop for Liquid Flow in Helical Coils," Trans. Indian Instn. Chem. Engrs., 14, 93-97 (1961-1962). 91. Barua, S. N., "On Secondary Flow in Stationary Curved Pipes," Q. J. Mech. Appl. Math., 16, 61-77 (1963). 92. Kubair, V. and N. R. Kuloor, "Non-isothermal Pressure Drop Data for Liquid flow in Helical coils," Indian J. TechnoL, 3, 5-7 (1965). 93. Mori, Y. and W. Nakayama, "Study on Forced Convective Heat Transfer in Curved Pipes, Pt. I Laminar Region," Int. J. Heat Mass Transfer, 8, 67-82 (1965). 94. Topakoglu, H. C , "Steady State Laminar Flows of an Incompressible Viscous Fluid in Curved Pipes," J. Math. & Mech., 16, 1,321-1,337 (1967). 95. Ito, H., "Laminar Flow in Curved Pipes," Z Angew. Math. Mech., 11, 653-663 (1969). 96. Srinivasan, P. S., S. S. Nandapurkar and F. A. Holland, "Friction Factors in Coils," Trans Instn. Chem. Engrs., 48, T156-161 (1970). 97. Larrain, J. and C. F. Bonilla, "Theoretical Analysis of Pressure Drop in the Laminar flow of Fluid in a Coiled Pipe," Trans. Soc. of RheoL, 14, 135-147 (1970). 98. Austin, L. R. and J. D. Seader, "Fully Developed Viscous Flow in Coiled Circular Pipe," AIChE J. 19, 85-94 (1973). 99. Rao, M. V. R. and D. Sadasividu, "Pressure Drop Studies in Helical Coils," Indian J. TechnoL, 12, 473-474 (1974). 100. Collins, W. M. and S. C. R. Dennis, "The Steady Motion of a Viscous Fluid in a Curved Tube," Q. J. Mech. Appl. Math., 28, 133-156 (1975). 101. van Dyke, M. "Extended Stokes Series: Laminar Flow through a Loosely Coiled Pipe", J. Fluid Mech., 86, 129-145 (1978). 102. Mishra, P. and S. N. Gupta, "Momentum Transfer in Curved Pipes, Pt. I Newtonian Fluids," Ind. Eng. Chem. Proc. Des. & Dev., 18, 130-137 (1979). 103. Dennis, S. C. R., "Calculation of the Steady Flow through a Curved Tube Using a New Finite Difference Method," J. Fluid Mech., 99, 449-467 (1980). 104. Hart, J., J. EUenberger and P. J. Hamersma, "Single and Two-Phase Flow through Helically Coiled Tubes," Chem, Engg. ScL, 43, 775-783 (1988). 105. Rao, C. K., "Laminar Flow of non-Newtonian Fluids through a Helical Coil," Trans. Indian Chem. Engrs, 33, T124-T128 (1991). 106. Taylor, G. I., "The Criterion for Turbulence in Curved Pipes," Proc. Roy. Soc. (London), A124, 243-249 (1929). 107. Keulegan, G. H. and K. H. Beij, "Pressure Losses for Fluids flow in Curved Pipes," J. Res. National Bureau of Standards, 18, 89-114 (1937). 108. Sreenivasan, K. R. and P. J. Strykowski, "Stabilization Effects in Flow through Helically Coiled Pipes," Exp. in Fluids, 1, 31-36 (1983). 109. Kubair, V. and C. B. S. Varrier, "Pressure Drop for Liquid Flow in Helical Coils," Trans. Indian Instn. Chem. Engrs., 14, 93-97 (1961-62). 110. Kutateladze, S. S., and M. Borishanskii, "A Concise Encyclopedia of Heat Transfer," Pergamon Press, London, 114 (1966).

Water Flow Through Helical Coils in Turbulent Condition

403

111. Srinivasan, P. S., S. S. Nandapurkar and F. A. Holland, "Pressure Drop and heat Transfer in Coils," The Chem. Engrs., May, 113-119 (1968). 112. Storrow, J. A., "Heat Transmission in Coils," J. Soc. Chem. Ind., 64, 322-326 (1945). 113. Wattendorf, F. L., "A Study of the Effect of Curvature on Fully Developed Turbulent Flow," Proc. Roy. Soc. (London), A148, 565 (1935). 114. Spiers, H. M., "Technical Data on Fuel," Proc. World Power Conf., London, 42 (1961). 115. Schmidt, E. F., "Warmeubergang und Druckverlust in Rohrschlangen," Chem. Ing. Tech., 31, 781-789 (1967). 116. Mori, Y. and W. Nakayama, "Study on Forced Convective Heat Transfer in Curved Pipes, Pt. II Turbulent Flow Region," Int. J. Heat Mass Transfer, 10, 37-59 (1967). 117. Stevens, A. F. W., R. Trenberth and R. W. Wood, "An Experimental Investigation into Once-through Boiling of High Pressure Water in a Helically Wound Tube (Corkscrew Boiler Design) Part I," UKAEA Report, No. AEEW-R730 (1972). 118. Anglesea, W. T., D. J. B. Chambers and R. C. Jeffrey, "Measurements of Water/Steam Pressure Drop in Helical Coils at 179 Bars," Proc. Symp. Multiphase Flow Systems, Inst. Chem. Engrs. Symp. Series No. 38 Paper 12 (1974). 119. Singh, R. P. and P. Mishra, "Friction Factor for Newtonian and non-Newtonian Fluid Flow in Curved Pipes," J. Chem. Engg, Japan, 13, 275-280 (1980). 120. Ruffell, A. E., "The Application of Heat Transfer and Pressure Drop Data to Design of Helical Coil Once-through Boilers," Proc. Sym. Multiphase Flow Systems, Instn. Chem. Engg. Sym. Series, No. 38, Paper 15 (1974). 121. Gill, G. M., G. S. Harrison and M. A. Walker, "Full Scale Modelling of a Helical Boiler Tube," Proc. Int. Conf. Physical Modelling of Multiphase Flow, Coventry, England, April 19-21, Paper K4, 481-500 (1983). 122. Das, S. K., "Water Flow through Helical Coils in Turbulent Condition," Can. J. Chem. Engg., 71, 971-973 (1993). 123. Humphrey, J. A. C , J. H. Whitelaw and G. Yee, "Turbulent Flow in a Square Duct with Strong Curvature," J. Fluid Mech., 103, 443-463 (1981). 124. Taylor, A. M. K. P., J. H. Whitelaw and M. Yianneskis, "Curved Ducts with Strong Secondary Motion: Velocity Measurements of Developing Laminar and Tur-bulent Flow," Trans ASME J. Fluids Engg., 104, 350-359 (1982). 125. Enayet, M. M., M. M. Gibson, A. M. K. P. Taylor and M. Yianneskis, "Laser Doppler Measurements of Laminar and Turbulent Flow in a Pipe Bend," Int. J. Heat & Fluid Flow, 3, 211-217 (1982). 126. Hille, P., R. Vehrenkamp and E. O. Schulz-Dubois, "The Development and Structure of Primary and Secondary Flow in a Curved Square Duct," J. Fluid Mech., 151, 219-241 (1985). 127. Azzola, J., J. A. C. Humphrey, H. lacovides and B. E. Launder, "Developing Turbulent Flow in a U-bend of Circular Cross-section: Measurement and Computation," Trans. ASME J. Fluids Engg. 108, 214-221 (1986).

This Page Intentionally Left Blank

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

Advances in Engineering Fluid IVIechanics

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

Advances in Engineering Fluid Mechanics

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^^
^ ^ . u „ Dt

V U , + 2 K , , U , = 2 DfUf - g Dt

(4)

This shows that bubbles are driven very strongly by the liquid. The relative acceleration can be as much as twice the vector difference of gravity and liquid acceleration. Since the bubbles are essentially mass-less, the analogy to the kinetic theory of collisions is not appropriate. Bubbles do not "collide;" they only come into contact while moving generally in the same direction. Equation 4 also provides additional insight on in-line wake coalescence in laminar flow. For a bubble to benefit from the upward motion of another's wake, it must follow in line below the leader. This occurs automatically for bubbles generated from the same point. Those not already in the wake must be drawn in radially. But any radial inflow that might bring in a bubble must decelerate to zero at the wake

Modeling Coalescence of Bubble Clusters

409

centerline. Since a bubble decelerates about twice as rapidly as the liquid by Equation 4, the liquid deceleration actually keeps bubbles away from the wake centerline. This is why the coalescence rate is usually high just above a sparger, where bubbles are generated directly under those just released, but decreases at higher elevations where bubbles must travel radially to coalesce [13]. On the other hand, the liquid that decelerates radially as it becomes entrained in the wake also accelerates upward. Since bubble acceleration leads the liquid, trailing bubbles are drawn upward at an angle toward the leading bubble's equator. This is why similar sized bubbles tend to cluster into diagonal (or conical) trains, and smaller bubbles tend to congregate around a larger one's periphery [16]. For the same reason, small-scale turbulence does not enhance coalescence but tends to prevent it. Again, by Equation 4, a bubble tends to move in advance of liquid fluctuations. But "colliding" packets of liquid decelerate strongly, keeping bubbles out of the vicinity of the event. This effect is evident in bubbly flows around an obstacle. Liquid deceleration keeps bubbles away from the stagnation region forming a large void-free volume [21]. On the other hand, turbulence structures of a length scale larger than the bubbles may enhance coalescence by attracting bubbles to the centers of eddies by centripetal acceleration. EXPERIMENTAL OBSERVATIONS OF BUBBLE INTERACTION Bubble clusters, groups of several bubbles moving together, appear in almost any photograph of a freely rising bubble swarm. In viscous liquids, these co-moving bubbles may coalesce simultaneously to form a bubble far larger than a binary process could create. In low-viscosity liquids, the interaction of bubbles in clusters is very complex. Though coalescence apparently remains binary, the rate of coalescence is accelerated in proportion to the number of bubbles interacting so that bigger clusters generally contain larger bubbles. Calderbank et al. studied coalescence rates of bubble swarms in deep pools of both water and glycerol solutions [16]. They observed no coalescence in water; but in glycerol, small ellipsoidal bubbles were seen to gather together into clusters approximating the shape of a larger Taylor cap bubble. The clusters traveled as a unit for some time before coalescing simultaneously into a true cap bubble. In restricted channel flows of glycerol, the wake of a large cap bubble appeared to draw small bubbles to the channel center forming clusters that later coalesced into new cap bubbles. The clusters contained an average of six small bubbles. In studying how a large bubble might affect the motion of smaller ones, Lockett and Kirkpatrick also observed clusters of five to 30 bubbles in a two-dimensional air-water flow [10]. The clusters formed only at higher air flows and eventually coalesced into single large bubbles. Hills also used a two-dimensional test section to study large cap bubbles rising through a swarm of smaller ones [22]. He saw a void-free clear area of about four times the bubble volume around the nose of the large bubble like that observed around solid obstacles. Small bubbles were seen to crowd around the lower edge of the cap, sometimes coalescing into secondary caps that immediately coalesced in turn with the original bubble. DeKee et al. looked at the coalescence of two and three air bubbles in nonNewtonian liquids [12,23]. They found that bubbles rising side-by-side tend to repel each other, but if one moves into the lead it attracts the other on a diagonal path.

410

Advances in Engineering Fluid Mechanics

With three bubbles, no coalescence or clustering was seen unless one of them moved ahead of the other two. Then the two followers would move together and coalesce before coalescing in turn with the leader in a manner similar to that observed by Hills [22]. The author studied pulsed planar swarms of 8 to 25 bubbles to explore interactions in low-viscosity liquids [24]. The pulsed swarm contained enough bubbles to represent a real swarm, but not so many as to hide the details of the interaction. Bubbles were released into aqueous sugar solutions with M ranging from 3 (10"^') to 4 (10"^) and Eo from 6 to 28. Under these conditions, bubbles are ellipsoidal with turbulent wakes, as evidenced by their wobbling path. Reynolds numbers ranged from just 300 in the more viscous sugar solutions to almost 3,000 for the larger ellipsoidal cap bubbles in water. Only the smaller bubbles in the most viscous sugar solution of M = 4 (10"^) had laminar wakes with Re ~ 100. Bubble interactions were recorded by an 8-mm video camera following the rising swarm. The video tapes were later studied in detail by viewing each run frame-byframe and at reduced speeds (1/10 to 1/5 speed). From the visual data, 1,672 binary wake-induced collisions were recorded, of which only 108 resulted in a breakup and 56 in coalescence. The results are consistent with prior studies of in-line coalescence in high-M liquids in that the wake is the primary driving force and sole mechanism of bubble interaction. No head-on or lateral solid-body collisions were ever seen. No bubble approached another except by entering its wake and overtaking it from the rear. Following the approach, however, the coalescence process at low-M was very different than at high-M. The approach nearly always led to a violent "rear-ender" collision, ending with the two bubbles nearly side-by-side about one diameter apart. The bubbles normally stopped interacting after the collision. But when they did coalesce or break up, it was only after the collision when one of them drifted slightly ahead and pulled its neighbor sideways into its rear surface. No coalescences or breakups were ever observed in the collision itself. An example of a complete collision/contact/separation event is shown in Figure 1. It was often difficult to resolve the details of this complex process. Actual contact lasted only a fraction of one wobble period, about 1/10 second. The overtaking bubble appeared to be drawn up into the center of the leader's rear face, sometimes disappearing completely inside it. However, it very quickly pushed past and slightly ahead; the two bubbles separated slightly and reestablished their own oscillation cycles. After colliding, the bubbles "danced" together, sometimes for several wobble cycles, before drifting apart again. One cycle of this dance is sketched in Figure 2. When the leader dips down in its wobble cycle, it draws its neighbor down and inward. Then the leader rocks back, stretching the neighbor farther. If the pull is strong enough to overcome surface tension, breakup, coalescence, or both may occur. Otherwise, the lower bubble recovers its shape and position for the next wobble cycle. An example of a simultaneous breakup and coalescence is shown in Figure 3. Besides the simple binary interaction described to this point, multiple bubble interactions were also common. When several bubbles are captured in a wake simultaneously or in close sequence, a cluster forms when they alternately collide with the leader. The bubbles in a cluster don't act as a coherent unit as the word

Modeling Coalescence of Bubble Clusters

Figure 1. Typical Collision Event

Figure 2. Post Collision "Dance interaction

411

412

Advances in Engineering Fluid Mechanics

• ' \ ' ' ' ' '/'„ W . '•;«,'' ' ','-11

,". ;• V', '•' t \

Figure 3. Sinnultaneous Breakup and Coalescence

^M' Figure 4. Cluster Behavior

Modeling Coalescence of Bubble Clusters

413

"cluster" implies, but continually trade places in a "leapfrog" fashion. When two bubbles jump up the wake together, or a third bubble collides with a pair that have just collided, one of the overtaking bubbles can push farther past the collision point. If the spacing is right, the new leader captures one or both of the bubbles now just behind it, and a second collision occurs. This sequence may repeat several times before one of the bubbles breaks the cycle by leaving the wake after a collision. Two cycles of typical cluster action are sketched in Figure 4. The lifetime, rise speed, size, and general violence of the cluster increase with the number of bubbles available to participate. If many bubbles (say 5 to 10) are involved, or the cluster captures additional bubbles below, collisions become almost continuous and the cluster's composite wake grows strong enough to sustain itself by continually gathering in new bubbles to replace those that disperse outward at the top. This configuration might be called a chimney. A chimney dies out when bubbles disperse too widely on top to be recaptured or too few bubbles are available outside the cluster to be captured as it goes by. As the chimney grows stronger, more breakups occur, and the smaller bubbles cannot sustain the wake. On the other hand, coalescences in the cluster create large caps that greatly amplify the wake. But very large caps may break up spontaneously, suddenly negating much of the amplifying effect. Clusters of a number of bubbles have been observed in a great number of experiments. Coalescence depends on the number of bubbles interacting in the clusters and on the number of clusters. Let us investigate further by comparing models that assume simple binary and clustering coalescence mechanisms with bubble size distribution data. A STOCHASTIC MODEL FOR BINARY COALESCENCE This section investigates the potential result of coalescence in bubble swarms assuming completely random, binary interaction. Consider a group of bubbles rising in a stagnant liquid. The total gas volume is assumed constant (the liquid does not absorb the gas, bubble volume does not change with pressure). All bubble interactions are binary and completely random. That is, any two bubbles in the group may merge into a larger one with equal probability. Bubbles have no "memory"— one newly formed by a coalescence or breakup is just as likely to participate in the next event as any other. Finally, for the purposes of calculating distributions, we ignore the rate of event occurrence. The distribution depends only on the number of events that have occurred, regardless of the time between events. Bubble size is expressed in terms of volume rather than diameter. Though the behavior of a single bubble (e.g., terminal rise velocity, drag, shape) is correlated by equivalent diameter, coalescence effects are best treated with volume. Volumes are added in coalescence, regardless of bubble shape, which is seldom spherical in parameter ranges of interest. If necessary, of course, equivalent diameter can be computed from the volume whenever required, but volume remains the primary measure. Binary coalescence is a discrete event where two specific bubbles merge to form a single larger one. Thus the number of bubbles, N^''^ existing after k coalescence events in the absence of breakup is N^*"^ = N^^^ - k, where N^^^ is the initial number of bubbles. Then the mean bubble volume, V^''^ after k coalescences can be

414

Advances in Engineering Fluid Mechanics

computed from the current number of bubbles and the total (fixed) volume of the group of bubbles, V^^\ as yCk)

^

N(0)

This, in fact, is true in all binary coalescence, regardless of randomness or the initial distribution. If the original bubbles are of uniform size, W^^\ the total gas volume is V^^^N^^\ Then the mean volume ratio, |LI, referenced to the initial individual bubble volume, V^^^ is

^

\7(0)

XT(0)

1^

^^^

Throughout the rest of this section, bubble sizes will be understood to be a dimensionless ratio of the uniform unit size, W^^\ which is also assumed to be the smallest size possible. It cannot break up. Thus the smallest relative volume that can occur is always 1.0. In a completely random system, all possible events are equally probable. The probability of any one of them is the inverse of the total number of events. Similarly, the probability of a specific coalescence of two given bubble sizes is the ratio of the number of ways to attain the combination to the total number of binary combinations possible in the existing distribution. Consider two bubbles of unequal volume, A and B. If there are N^ bubbles of volume A and N^ of volume B, a coalescence of A and B (denoted as A®B) is possible for a combination of any one out of N^ bubbles of volume A with any one out of N^ bubbles of volume B so that the total number of A©B coalescences possible is N^N^. For two equal-size bubbles, say of volume A, an A©A coalescence is possible for a combination of any one of the N^ bubbles with any of (N^ - 1) others. But only half of these combinations are unique so the total number of A©A coalescences is N^(N^ - l)/2. The probability of coalescence of a bubble of volume A with another of volume B is 2N N

Likewise, the probability of coalescence for two bubbles of equal volume A is given by PrAOAl- N.(N.-l) P[A®A]-^,,,^^,,,_j^ Now consider initial group of multiples of the after k random

(7)

all possible binary coalescences of a population evolving from an N^^^ uniform bubbles. Let M be the volume (measured in integer initial volume) of a bubble selected at random from those existing coalescences with probabilities given by (6) and (7). Then the

Modeling Coalescence of Bubble Clusters

415

probability that M will be equal to a specific volume, m, after k events is given by the finite product [25] N^°^ - k - 1 ^ N^"^ - m

1-

j=i

N^"> - k - 1 N(«) - j

(8)

Using binomial coefficient notation, this discrete probability can also be expressed as

f ( m | k ) = P[M = m|k] =

N^'^ - m - 1 N^^^ - k - 2 N^^^ - 1 k

(9)

Functions (8) and (9) define the discrete probability density of the Random Binary Coalescence (RBC) distribution. Its mean volume after k events is found by summing (9) over all possible volumes (from 1 to N^^^ - 1): N (0)

l^ = N(<

(10)

The random coalescence distribution of Equation 9 exhibits the property that a smaller bubble is always more likely than the next larger one. Many coalescences are required to make larger sizes probable. An example of this behavior is illustrated in Figure 5, which shows the coalescence of 1,002 unit bubbles into a single one

Volume

o2

n3

^4

BS

Figure 5. Probability Density vs. Coalescence Event

416

Advances in Engineering Fluid Mechanics

with a relative volume of 1,002 after 1,001 events. The mean volume after 800 events (80% coalescence) is only 5. At that point, only 8% of the population is of that volume, while about 60% is smaller. It is interesting that each size reaches its peak probability at the event when it equals the mean. But, even at this peak, note that the smaller sizes are always more probable. The slow growth of the mean size per coalescence is clearly shown on Figure 6. The average bubble volume is only about 20 after 951 events (95% coalesced), when only 50 bubbles exist. In just 50 more events, the mean must increase from 20 to 1,000, and the number of bubbles decrease from 50 to one. Now let N^^^ and k be arbitrarily large. Then, for a fixed volume, m, the probability (8) is invariant except for the upper limit of the product. This is exactly equivalent to the geometric distribution whose discrete probability density is given by f(m|k) = GEO(m;p) = p ( l - p ) ' " - ' ,

where p = N^"^ - k - 1

(11)

with a mean defined by 1

N' N''' - k - 1

The geometric distribution has the "no-memory" property. That is, after any number of coin tosses, for example, the probability of heads on the next toss is the same as it was on the first. Similarly, a bubble that has just coalesced is just as likely to coalesce again as any other, or a large bubble is just as likely to

100.00

10.00

1.00

0.10

0.01

o Mean

a Standard Oeviation

Figure 6. Mean and Standard Deviation vs. Coalescence Event

Modeling Coalescence of Bubble Clusters

417

coalesce as a small one. This is just as one would expect in the random system we have assumed. Both the RBC distribution (8) and the geometric distribution (11) are defined only for specific integer bubble sizes, and derivatives of their distribution functions do not exist. For subsequent developments we need an equivalent continuous distribution. Fortunately, for N^^^ and k large with respect to m, both discrete distributions can be closely approximated by the exponential distribution if its mean is set to the RBC mean volume given by (10). The exponential probability density is f(m|k) = EXP(m;|a) = - e (-m/^i) 1^

where |Ll =

N^' N(0)

(12)

The exponential distribution is used to model the reliability of electronic components that remain0 "good as new" regardless of the amount of time in service. It also has the "no-memory" property. The mean, variance, and other measures of the RBC, geometric, and exponential distributions are all asymptotically equivalent for large N^^^ and k. The three distributions are compared in Figure 7 for the case of only 70 coalescences of an initial population of 100 uniform bubbles. This small population is designed to show the relative behavior of the three related distributions. It does not represent a real swarm. Note for future reference that the exponential distribution consistently overpredicts the probability of very large bubbles compared with the two discrete distributions.

N(»>= 100, k = 7 0 0.1 H RBC(m)

B •3

0.001 H

^

a Xi cd Xi

i

GEO(m) EXP(m)

.

0\^ \

V

10r 5 J

10"

1

1

1

1

1

9

17

25

33

41

50

Relative Bubble Volume, m = VA^0 Figure 7. Comparison of RBC, Geometric, and Exponential Distributions

418

Advances in Engineering Fluid Mechanics

One can also derive the exponential distribution function heuristically. The random binary model assumes the number of bubbles, N^^, larger than a specific volume, m, decreases with volume in direct proportion to itself. That is, the greater the abundance of bubbles larger than a given size, the stronger the effect of random coalescences that reduce their number. This proportionality can be written as

dm

[I

• N>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,

Modeling Coalescence of Bubble Clusters

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

Advances in Engineering Fluid Mechanics

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

Modeling Coalescence of Bubble Clusters

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

Advances in Engineering Fluid Mechanics

0.01

0.0001 H A

Nakanoh & Yoshida (O^ - H^O)

\

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

Modeling Coalescence of Bubble Clusters

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 - H^O)

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

Advances in Engineering Fluid Mechanics

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

Modeling Coalescence of Bubble Clusters

-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

Advances in Engineering Fluid Mechanics

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

Modeling Coalescence of Bubble Clusters

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

Advances in Engineering Fluid Mechanics

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).

Modeling Coalescence of Bubble Clusters

429

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).

This Page Intentionally Left Blank

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 K^a 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 K^a, 438 CMC Solutions, 438 XTN Solutions, 441 PAA Solutions, 444 Effect of Apparent Viscosity on K^a, 447 431

432

Advances in Engineering Fluid Mechanics

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^^a. 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^^a 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^^a 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^^a 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^^a 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

Advances in Engineering Fluid Mechanics

[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^^a, 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

Oxygen Transfer in Non-Newtonian Fluids

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^^a 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

Advances In Engineering Fluid Mechanics

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 K^a 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^^a 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 )

^^>

K^a. values were obtained from the regression constants B and K^, knowing that K^a = 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 K^a 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

Oxygen Transfer in Non-Newtonian Fluids

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^^a 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

Advances in Engineering Fluid Mechanics

GAS-LIQUID MASS TRANSFER AND BUBBLE BEHAVIOR Effect of Power Input, SuperHcial Gas Velocity, and Polymer Concentration on Kj^a 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, K^a 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^^a 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^^a 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), K^a 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 K^a values, respectively. Within this range, K^^a increases about 1.5 times with power input. Beyond 300 W/m\ K^a is essentially independent of power input and is only dependent on superficial gas velocity. An increase in CMC concentration results in lower K^^a 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^^a 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^^a 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, K^a increases almost twice showing essentially a similar dependence on power input. On the other hand, at a constant power input, K^a 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^^a while a two-fold increase in gas velocity rises the coefficient by a factor of around 1.3. At low CMC concentrations, K^^a displays a two-zone behavior, one in which K^^a 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, K^a depends on both power input and aeration conditions. Therefore, at higher CMC concentrations, K^a 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

Oxygen Transfer in Non-Newtonian Fluids

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 K^a 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^^a 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 K^a 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

Oxygen Transfer in Non-Newtonian Fluids

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^^a 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^^a 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^^a 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^^a 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^^a 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^^a with power input at different gas velocities in XTN 3. At the lowest power input (45 W/m^), K^^a 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^^a 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^^a 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^^a of the order of 2.5. Within the range from 120 to 700 W/m\ K^^a 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 Kj^a 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

Advances in Engineering Fluid Mechanics ,-2

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 K^a 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,

Oxygen Transfer in Non-Newtonian Fluids

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 K^a 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

Advances in Engineering Fluid Mechanics

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^^a 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 Kj^a. At 0.0075 m/s, however, accumulation of "old" dispersed bubbles results only in a slight augmentation of K^a 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 K^a at a constant gas velocity of 0.0039 m/s. In PAA 1, K^^a ranged from 0.002 to 0.0055 s* depending on gas velocity. At low power inputs the effect of concentration on K^^a is more significant. For example, at 150 W/m^ K^^a 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^^a 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^^a 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^^a 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^^a 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 K^a. 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^^a. Nevertheless, for some conditions power input had a greater effect than gas velocity.

Oxygen Transfer in Non-Newtonian Fluids

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 K^a 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

Advances in Engineering Fluid Mechanics ,-2

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 K^a 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

Oxygen Transfer in Non-Newtonian Fluids

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 K^a 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^^a 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^^a 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^^a 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

Advances in Engineering Fluid Mechanics

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 K^a at u^ = 0.0039 m/s in CMC and XTN solutions.

Oxygen Transfer in Non-Newtonian Fluids

K^a^ 0.00342

K^a^ 0.00125

KLa = 0.00410

vV.y

fP

<•''<''

(4)

or'

(5) (6)

ur'V"

V.

449

where K,a is in s ' P /V, is in W/m^ u is in m/s, and ri is in Pa • s. L

g

L

g

'

'a

Their accuracy in predicting K^a can be judged from the regression coefficients and standard deviations whose values are, respectively: 0.9694, 0.080 for CMC, 0.9242, 0.1615 for XTN, and 0.9156, 0.2027 for PAA. The value of the exponent of apparent viscosity is within the range observed by other authors. Its effect on the volumetric coefficient is very similar in XTN and PAA solutions, and weaker in CMC, which have a gentle shear thinning behavior. The dependence to power input is essentially the same, which shows that K^^a build-up with impeller speed is similar in all the non-Newtonian solutions tested. The exponent of gas velocity is roughly twice that of power input, but the effect of u in PAA solutions is higher presumably because of the diverging behavior shown in Figure 6. As anticipated from observation of bubble behavior these correlations indicate a major dependence of K^SL on gas flow rate rather than on agitation speed, that is, power input. Our results are difficult to compare with those from the literature because no available correlations exist for HRS impellers. There are, however, correlations of the same type in turbine-agitated vessels. Nevertheless, comparison is still complex because the hydrodynamic and mixing conditions prevailing in our system are surely different from those in radial flow impellers. In general, in these systems K^^a depends more strongly on power input than on gas velocity, but some exceptions have been reported. For example, Ogut and Hatch working with paddle impellers and sodium polyacrylate solutions of different concentrations, found exponents of 0.016, 0.85, and -0.60 for power input, superficial velocity and apparent viscosity, respectively, when the diameter of the impeller was 0.0762 m [31]. These values changed to 0.055, 1.13, and -0.44 when the size of the impeller was 0.1 m. The very weak dependence on power input and the very strong influence of superficial velocity were attributed to the formation of a large gas pocket around the paddle that resulted in an active gas-liquid contact zone in the central column of the vessel, which was further promoted by increasing gas velocity. CONCLUDING REMARKS Our main purposes in studying oxygen transfer in non-Newtonian fluids in a vessel equipped with an HRS impeller were to quantify its aeration and agitation capacity and to observe the mixing behavior and gas dispersion patterns occuring in this system. Substantial differences in bubble behavior were observed among the nonNewtonian fluids, but in general weak bubble dispersion occurs at low agitation

450

Advances in Engineering Fluid Mechanics

speeds and larger bubble dispersion is achieved at higher agitation speeds. Lower speeds are not recommended because coalescence and weak dispersion result in greater bubbles, and, hence, in lower superficial areas and K^^a values. In some cases at these speeds, however, the mass transfer capacity of the system is mainly supported by the amount of gas bubbles coming from the sparger. In low-shear thinning fluids, no stagnant zones occur even at low agitation speeds. In high shear thinning and viscoelastic fluids, no dead zones occur either throughout the length of the impeller, but near the bottom the liquid is mainly agitated by the gas coming from the sparger. The effect of gas flow rate is greater than that of power input, which means that while the impeller provides enough mixing to move the whole volume, it exerts a minor effect on Kj^a presumably because of its limited capacity to disperse bubbles. Our K^a values, which ranged from 0.001 to 0.01 s"', are of the same order of magnitude as those reported for turbines, paddles, and Intermig impellers. However, in these impellers it is often assumed that the liquid phase is well-mixed and K^^a is referred to as the whole working volume. There is enough experimental evidence showing that this assumption is not entirely valid because large dead zones may develop. On this basis our K^^a values are more representative than those frequently reported in turbine-agitated vessels because the HRS provides bulk fluid mixing instead of local homogeneous mixing. The dimensional correlations adequately describe the effect of operation variables and rheological behavior. Although we do not intend to use them for scale-up purposes, they could be useful to predict K^^a values during different stages of a fermentation, depending on the rheological properties of the broth at a given moment, and also serve as a basis for further studies. Previous to our work, no K^^a data in HRS-agitated systems were available, and scepticism about the use of helical impellers in gas-liquid systems was only supported by the argument of their limited capacity of gas dispersion. We have shown, quantitatively, that provided this drawback is compensated with a suitable device, the aeration performance of the HRS is comparable to that observed in remote impellers. Currently, we are continuing oxygen transfer studies in non-Newtonian fluids in vessels equipped with helical ribbon impellers used together with more efficient gas sparging devices. ACKNOWLEDGMENTS A. Tecante and E. Brito wish to acknowledge financial support from DGAPAUNAM (grant IN305892). NOTATION a Interfacial area of dispersed bubbles, mVm^ B Constant in Equation 1, dimensionless d Impeller diameter, m G^(t) Theoretical normalized DO probe response to course of

oxygen concentration predicted by a global model, dimensionless ks Impeller constant, dimensionless K Consistency index, mPa • s" Kj^a Volumetric oxygen mass transfer coefficient, s"

Oxygen Transfer in Non-Newtonian Fluids

Kj^ Time constant of oxygen diffusion through the membrane, s~' L^ Parameter including the resistance to oxygen diffusion in the liquid film next to the sensor membrane, dimensionless n Flow behavior index, dimensionless

N P Re t u V^^

451

Impeller rotational speed, rev/min Gassed power input, W Reynolds number, dimensionless Time, s Superficial gas velocity, m/s Liquid volume, m^

Greek Letters p Liquid density, kg/m^ Yg Apparent shear rate, s ' r\^ Apparent viscosity. Pa • s

REFERENCES 1. Vardar-Sukan, F. "Dynamics of Oxygen Mass Transfer in Bioreactors. Part I: Operating Variables Affecting Mass Transfer," Proc. Biochem., 20, 181, (1985). 2. Vardar-Sukan, F. "Dynamics of Oxygen Mass Transfer in Bioreactors. Part II: Design Variables. Proc. Biochem., 2 1 , 40, (1986). 3. Noel, G., L. Choplin, A. Lajoie, C. Lacroix, and A. LeDuy, "A Study of the Evolution of Viscoelastic Properties of Microbial Polysaccharide Fermentation Broth," in Advances in Rheology, Mena, B., Garcia-Rejon, A., and RangelNafaile, C. (eds.), Mexico, vol. 4, p. 189, (1984). 4. Funahashi, H., M. Maehara, H. Taguchi, and T. Yoshida, "Effects of Agitation by Flat-Bladed Turbine Impeller on Microbial Production of Xanthan Gum," J. Chem. Eng. Japan, 20, 16, (1987). 5. Funahashi, H., K. I. Hirai, T. Yoshida, H. Taguchi, "Mechanistic Analysis of Xanthan Production in a Stirred Tank," /. Ferment. TechnoL, 66, 355, (1988). 6. Dussap, C. G., J. Decorps, and J. B. Gros, "Tranfert D'Oxygene en Presence de Polysaccharides Exocellulaires dans un Fermenteur Agile Aere et dans un Fermenteur de Type Gazosiphon," Entropie No. 123, 11, (1985). 7. Herbst, H., H.U. Peters, I.S. Suh, A. Schumpe, and W.D. Deckwer, "Zum Stoffubergang bei Xanthan-Fermentationen," Chem. Ing. TechnoL, 60, 407, (1988). 8. Pons, A., C. G. Dussap, and J. B. Gros, "Xanthan Batch Fermentations: Compared Performances of a Bubble Column and a Stirred Tank Fermentor," Bioprocess Eng., 5, 107, (1990). 9. Herbst, H., I. S. Suh, H. U. Peters, A. Schumpe, A., and W.D. Deckwer, "Comparison of Various Fermentor Types Used for Production of Xanthan," DECHEMA Biotechnol. Conf., 3, (1989). 10. Henyler, H. J., and G. Obernosterer, "Effect of Mixing Behavior on Gas-Liquid Mass Transfer in Highly Viscous, Stirred Non-Newtonian Liquids," Chem. Eng. TechnoL, 14, 1, (1991). 11. Schugerl, K. "New Bioreactors for Aerobic Processes," Int. Chem. Eng., 22, 591, (1982).

452

Advances in Engineering Fluid IVIechanics

12. Krebser, U., H. P. Meyer, and A. Fiechter, "A Comparison between the Performance of Continuously Stirred-Tank Bioreactors and a TORUS Bioreactor with Respect to Highly Viscous Culture Broths," J. Chem. Tech. BiotechnoL, 43, 107, (1988). 13. Zlokarnik, M. "Trends and Needs in Bioprocess Engineering," Chem. Eng. Prog., 86, 62, (1990). 14. Ho, F.C., and A. Kwong, "A Guide to Designing Special Agitators," Chem. Eng., 94, (July 23, 1973). 15. Rieger, F., V. Novak, and D. Havelkova, "Homogenization Efficiency of Helical Ribbon Agitators," Chem. Eng. J., 33, 143, (1986). 16. Giaccobe, F., and G. Capobianco, Paper presented at the Society of General Microbiology Meeting, Reading, England, Sept., 1976, as cited by G. Pace, "Mixing of Highly Viscous Fermentation Broths. Chem. Eng. (London), 377, 833, (1978). 17. De Vuyst, L., A. Vermeire, J. Van Loo, and E. J. Vandamme, "Nutritional, Physiological and Process-Technological Improvements of the Xanthan Fermentation Process," Med. Fac. Landbouww. Rijksuniv. Gent, 52, 1881, (1987). 18. Dreveton, E., F. Monot, D. Ballerini, J. Lecourtier, L. Choplin, "Effect of Mixing and Mass Transfer Conditions on Gellan Production by Auromonas elodea,'' J. Ferment. Bioeng., 77, 642, (1994). 19. Tecante, A., L. Choplin, and P. A. Tanguy, "Hydrodynamics and Mass Transfer in Rheologically Complex Model Exopolysaccharide Fermentation Broths," in Procedings of the 7th European Conference on Mixing, Vol 2. Brugge Belgium (1991), pp. 367-377. 20. Tecante, A., and L. Choplin, "Gas-Liquid Mass Transfer in Non-Newtonian Fluids in a Tank Stirred with a Helical Ribbon Screw Impeller," Can. J. Chem. Eng., 71, 859, (1993). 21. Brito de la Fuente, E., L. Choplin, A. Tecante, and P. A. Tanguy, "Influence of Rheology on Ungassed and Gassed power Consumption in Stirred Vessels Equipped with Helical Impellers," in Progress and Trends in Rheology IV, Gallegos C , A. Guerrero, J. Munoz, and M. Berjano Eds., Steinkoppf Verlag, Germany, (1994), pp. 212-214. 22. Linek, V., V. Vacek, J. Sinkule, and P. Benes, Measurement of Oxygen by Membrane-Covered Probes, Ellis Horwood Ltd., Great Britain, (1988). 23. Brito, E., J. C. Leuliet, L. Choplin, and P. A. Tanguy, "Mixing and Circulation Times in Rheologically Complex Fluids," /. Chem. E. Symposium Series No. 121, 75, (1990). 24. Nienow, A. W. and T. P. Elson, "Aspects of Mixing in Rheologically Complex Fluids," Chem. Eng. Res. Des., 66, 5, (1988). 25. Collias, D. J., and R. K. Prud'Homme, "The Effect of Fluid Elasticity on Power Consumption and Mixing Times in Stirred Tanks," Chem Eng. Sci., 40, 1495, (1985). 26. Acharya, A., R. A. Mashelkar, and J. J. Ulbrecht, "Mechanics of Bubble Motion and Deformation in Non-Newtonian Media," Chem. Eng. Sci., 32, 863, (1977). 27. Terasaka, K., and H. Tsuge, "Bubble Formation at a Single Orifice in NonNewtonian Liquids," Chem. Eng. Sci., 46, 85, (1991).

Oxygen Transfer in Non-Newtonian Fluids

453

28. Ghosh, A. K., and J. J. Ulbrecht, "Bubble Formation from a Sparger in Polymer Solutions-II. Moving Liquid," Chem. Eng. ScL, 44, 969, (1989). 29. Yagi, H., and F. Yoshida, "Gas Absorption by Newtonian and Non-Newtonian Fluids in Sparged Agitated Vessels," Ind. Eng. Chem. Process Des. Dev., 14, 488, (1975). 30. Ranade, V. R., J. J. Ulbretch, "Influence of Polymer additives on the Gas-Liquid Mass Transfer in Stirred Tanks," AIChE 7., 24, 796, (1978). 31. Ogut, A., and T.R. Hatch, "Oxygen Transfer into Newtonian and Non-Newtonian Fluids in Mechanically Agitated Vessels," Can. J. Chem. Eng., 66, 79, (1988).

This Page Intentionally Left Blank

CHAPTER 16 MODELING OF THE HYDRODYNAMIC BEHAVIOR OF HIGHLY VISCOUS FLUIDS IN STIRRED TANKS EQUIPPED WITH TWO-BLADE IMPELLERS Catherine Xuereb, Mohammed Abid and Joel Bertrand Laboratoire de Genie Chimiqe URA CNRS 192 ENSIGC 18, chemin de la Loge 31 078 Toulouse Cedex—France CONTENTS INTRODUCTION, 456 NUMERICAL ISSUES, 458 FLOW GENERATED BY A PLATE AGITATOR, 459 Flows in Vertical Planes, 459 Flows in Horizontal Planes, 460 Discussion, 460 Stresses and Viscous Dissipation Function, 463 Wall Effects, 465 Case of a Fluid with Viscoelastic Properties, 468 CLASSICAL TWO-BLADE IMPELLER, 471 Flow in Horizontal Planes, 471 Flow in Vertical Planes, 471 Stresses, 471 Influence of the Impeller in the Tank, 474 Influence of the Reynolds Number, 475 POWER CONSUMPTION, 477 MIXING OF TWO MISCIBLE HIGHLY VISCOUS FLUIDS, 479 CONCLUSION, 481 NOTATION, 483 REFERENCES, 484

455

456

Advances in Engineering Fluid Mechanics

INTRODUCTION Large blade impellers generally are used for laminar mixing of highly viscous fluids. One of the commonly used impellers is the two-blade impeller (Figure 1). The ratio (D/T), impeller diameter to tank diameter, can be varied from 0.5 to 0.9. In this case of large diameter impellers, these last ones can be used for heat transfer and can successfully replace anchor impellers. The ratio (W/T), impeller height to tank diameter, can vary from 1/12 to 0.95 (Figure 2). The case "1/12" corresponds to the case of the classical paddle impellers. These often are used for the mixing of miscible fluids of low viscosity [1-3]. These two-blade impellers are classified as turbines, and the flow is essentially radial in the discharge flow of the impeller if the impeller speed is sufficient. They generally are used for smooth mixing with the tip speed varying from 1.2 to 2.3 m/s. As the W/T ratio increases, the impeller often is called a paddle agitator. The flow generated by this kind of impeller becomes essentially tangential. In fact, a radial discharge flow cannot be developed because any radial aspiration is possible. These kinds of impellers can be used to develop constant stresses along a radius and, so, to give uniform energy to the fluid in the laminar flow regime. Another application concerns the drying of wet particles. Malhotra et al. have studied mixing patterns and required torques in function of both physical properties of the particles and operating conditions, and for different geometries [4]. In the literature, blade impellers have been experimentally and theoretically studied. Several experimental pieces of equipment have been used to study the flow structure generated by two-blade impellers [5-9]. Most studies show that, for a low impeller speed, the flow is essentially tangential. Hiraoka et al. [10] and Bertrand and Couderc [11] studied the 2D flow of viscous Newtonian and non-Newtonian fluids generated by two-blade impellers and paddle agitators, using computation fluid dynamics (CFD). For some years, some new CFD studies concerning the flows

%"<.

paddle agitator

two-blade impeller

Figure 1. Two-blade impellers.

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

457

Figure 2. Sketch of agitated system.

generated by this kind of impellers with more or less complex fluids have appeared, and some interesting results will be presented which complete our work [12-14]. In the case of the mixing of highly viscous fluids, which means in the laminar flow range, the vessels are not equipped with baffles, which enables the simulation of the flows generated by blade agitators using a rotating frame linked to the agitator and the imposition of simple boundary conditions without needing the help of experimental techniques. This is important from the pedogogical point of view: It can be interesting to begin to work in CFD applied to mixing problems with this kind of geometry.

458

Advances in Engineering Fluid Mechanics

NUMERICAL ISSUES Different procedures are possible. They are divided into three classes: (i) The first one consists of directly solving the Navier-Stokes equations on a grid of control volumes, and, so, to calculate the three velocity components and the local pressure. (This method will be used here). The main commercial codes in CFD using this procedure are FLUENT and PHOENICS . (ii) The second one consists of using the vector potential and the vorticity, which enables one to write the boundary conditions a simple way. The corresponding algorithm uses an iterative successive overrelaxation method (SOR). This procedure is an extension of the well-known 2D-procedure developed by Hiraoka et al. [15] and Bertrand et al. [16,7]. (iii) The third one could be very useful in the case of rather complex geometries or in the case of elastic fluids and of purely viscous fluids, the rheology of which is governed by rather complex models. It is based on discretization in finite elements, and the corresponding algorithms are rather different [14]. A commercial code based on this approach is FIDAP. Our numerical procedure was described in detail by Abid; here, only the main points are reviewed. The Navier-Stokes equations, written in a rotating, cylindrical frame of reference, are solved. Because of the choice of a rotating frame, two terms are added to the equations: centrifugal and Coriolis accelerations. The equations are written in terms of velocity components and pressure. These variables are discretized on a grid of control volumes, which enables one to obtain a more precise mass conservation and a faster convergence. The numerical procedure consists of calculating the pressure at the center of each cell, and the three velocity components at the center of the six faces of the cell (one component on the two corresponding orthogonal faces). To solve the Navier-Stokes equations, a linkage between velocity and pressure is necessary. In the literature, a large range of methods is proposed. Here, the SIMPLER (Semi-Implicit Method for Pressure-Linked Revised) algorithm was applied. This algorithm is described very precisely by Patankar [17]. The boundary conditions consist in setting each velocity component equal to zero on the blades and the axial shaft, because of the rotating frame, and setting the angular velocity component equal to the rotation speed at the vertical walls. On the bottom, the angular velocity component is chosen equal to 27cNr. No direct boundary conditions for pressure at the solid surfaces are needed; instead, special boundary conditions on the correction term of pressure are used to speed the convergence. The liquid surface is free. It is considered to be horizontal and plane, which corresponds to a low centrifugal acceleration compared to the gravitational one, i.e., F r « l . The numerical procedure uses three grids in increasing order of precision: 15 x 24 X 20 grid, 21 x 27 x 31 grid, and 23 x 35 x 33 grid. The major part of the numerical results was obtained with the second grid, which leads to the best compromise between precision and computational effort. An increase of the number of grid points above the second grid did not modify the flow structure. The convergence criteria required that the relative error between two successive iterations

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

459

be less or equal to 10~^ for each of the three velocity components. Again, this was a compromise between satisfactory results and low computational costs. FLOW GENERATED BY A PLATE AGITATOR The major part of previous work uses several simplifying assumptions, particularly that the flow is two-dimensional. The experimental results are in good agreement with these assumptions at the mid-height of the tank. This assumption is not valid near the bottom or the top of the tank. For results presented here, no hypothesis concerning flow structure was made. The full 3-dimensional flow is computed. The system consists of a cylindrical tank with a flat bottom, the liquid level H of which is equal to the diameter, T. The impeller consists of two vertical plane blades fixed on a cylindrical central shaft, the diameter of which D^ is equal to 0.05T. The ratio between the impeller diameter and the tank diameter D/T is equal to 0.5. All results are presented in dimensionless values, which enables the user to investigate any size vessel. In this section, a large two-blade impeller, the height of which is equal to the liquid level, is considered. This impeller scrapes the tank bottom without any friction; it is this kind of impeller that was frequently studied in the previous works, assuming 2D flows. In the following, more precise results are presented. Flows in Vertical Planes Figure 3 presents an example of spatial distribution of dimensionless axial and radial velocity components for a Reynolds number. Re = 32, in four vertical planes around the symmetry axis of the tank. The origin of the tangential coordinate is

r

]:--z::r::::

\ V y

\ \

n-

1 -^^^"

[y

r

\

\\\\\m

\

1

••Mill,.' V^ * ' / / / / 1 • • .» » M \ \ \ \ V

' . 1 1 MWNS*^ 1

0 = 35.5°

\..yy///1\

1 1 1 ' • ^' • ' / / M 1 ' •

\.yyy//l\\\

1 "''''

v=1.00 0-=2,T

A. ^^^^yyyy A. ^^y>y/// L .yyy////

6 = 79.1°

•'

9=128.2°

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

460

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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).

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

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)".

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

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 = ( | < + ^^ee + 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

Advances in Engineering Fluid IVIechanics

I

" ^ " V . N^ \

X »

.^-^\ \ \ \ \ .^^\ \ \ \ » ... \ \ \ \ \ t /^ N v \ \ i i ; . «I M I I . #/ M I i . / >/ / r I

Ii

\ S^ ^ / / / I I

I

^

_

-^ •

/

/

^

—• ^

X

/

' — •* "V N \

y ^

I I I

V//////\ / -^ -. /////I I

/

# /

*

•* s \ \ \ » - V \ \ i

/ ^ .^ V\ \ I / / . *M M / • , ' M i l ' / / M "/ / / ( -*'^/ / / f - - - - •^ /

/

/

/ 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

Advances in Engineering Fluid Mechanics

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].

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

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].

-

I

.

I

I

1

v

I

.

,

-

-

-

-

-

-

-

-

-

- Newtonian

-

PAA(0.496) A PAA(l%)

I

.

I

.

I

a

I

.

r Figure 12. Experimental data of the tangential velocity component in the plane of the blade [9].

A

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

i

^.

--"

471

CO

c o O)

c c o V-

0

E

JD

CO

o 0 CD

Q

"D

c

CO

CD

E CO

C

CD

o c >% 0 CL

0

o c 0 3 CO

£2

472

Advances in Engineering Fluid Mechanics

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)

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

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

i

.

/

/

X

V

•

X

/

i

N

/

V

/

/

s

/

*

V

»

'

-

•

-

-

-

-

-;

-

^

"

-

'

•

« »

•

\ V

»

\ \

t

V \

1

\ \

#

1 J

^

/ <

'^

/ / ^

\ \

/ 1 \ y 1 \

V,

W i

wi.-."_:.--\J.-.-

^ ^

1

~

/

/

' '

/

. »

•

\ 1

I 1

V *

* 1 .

»

.

.

.

»

'

•

•

'

• 1

v-::.-:^.-.».:;^C:^i•i::::::: :>:i: ^: :.-:l^l

/^l 1 fe^

Advances in Engineering Fluid Mechanics

<

»

.

. . . ^^^

I

» •

:: ; 1;:*

i..^:^.i^v " _ » ; _ — ; ."•^\- .^.L-i- .S>."i; ^-. k^i^v\ ; ; . \ - . \ i ; . ' ^

«

« •

_ I3!ir3iSS^II^]^ri]^I?!^^

' • • '

i::^^^i:Ci^:^...-::j.::-^:v>:^-SiA^ i>ir:::>*-.::;i:::v-..v- '^^^^zim^^^i^i^E^^E^^i^

o

II

ON

o

CD

o 00

CD

(D

cr 0) Q.

E CD •D

o

"(0

75 o o

c

Q C

o

Q.

£ o o

o o >

CD

CO •D C CO

< in 3 O)

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

( a )

( b )

475

{ c )

Figure 16. Stresses for two-blade Impeller. ( a ) ^rr' ( b ) ^re' (C)

\^

(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

Advances in Engineering Fluid Mechanics

'/\

» I / / / / " '

I / / / / / ^ '

I w / / /'"'^ ^ i If ///^^ ^

Ml \ \ \ \ >^ » \

\

\

\

\

\ 1

» >\ \ \ \ « , . »% \ \ »

I I I I > ' • M \ \ ^* *

t

1//^^

t t I I I /

^ '^ N \

\

\

^ •

^ ^ % \ \ \ X I _ ^ V \ \ \ \ »

y ^ — ^

\\\ \\ WW \

%'JJ7/?M\''

^\ \ \ \ \ \ \ \ \ i

^^\\\\\\\\\/ ^^\\\\\\\\1/ \\\\\

///////// T7-7-rr-/-/-f-|-i-Y t

I

I

t

)

\

\

\

\

\ N

\ \ \ wv^^ '^

Re =10

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

Q

0.2

Re = 10 •Re = 61

0.1

Q

Z

ti >*

Q

0.3 0.2

0.1 J

0.0

z \ >•" - 0 . 1 J

-0.2

0.0

-I

0.2

'

1

r-

0.6

0.4

1

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

Advances in Engineering Fluid Mechanics

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]}

,,^ ^^^

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

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

Advances in Engineering Fluid Mechanics

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.

Modeling of the Hydrodynamic Behavior of Highly Viscous 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

Advances in Engineering Fluid Mechanics

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

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

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

Advances in Engineering Fluid Mechanics

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).

Modeling of the Hydrodynamic Behavior of Highly Viscous Fluids

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).

This Page Intentionally Left Blank

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

Advances in Engineering Fluid Mechanics

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 R^Es

(10)

r^a.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

Advances in Engineering Fluid Mechanics

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

Non-Newtonian Liquid Flow Through Globe and Gate Valves

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

Advances in Engineering Fluid Mechanics

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 Y^e^ = 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

Advances in Engineering Fluid Mechanics

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

Non-Newtonian Liquid Flow Through Globe and Gate Valves

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

Advances in Engineering Fluid Mechanics

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)

Non-Newtonian Liquid Flow Through Globe and Gate Valves

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 ^O <

SCMC CONG.'0.6 k g / m ^ Symbol Valve opening 25.0 % o 37.5% / • 50^0% p ^ 62.5% / A 75.0% / 0 87.5% J • 100.0% Y D

1

20

Q.

O

cc Q iLl CC 3

10 k

to

UJ

cr a.

0

*3SS^^9^^^ 10

VOLUMETRIC FLOW RATE

20

30

(mVs)xlO''

Figure 2. Variation in pressure drop across the gate valve at different openings with volumetric flow rate.

497

498

Advances in Engineering Fluid Mechanics

>

40

—J

Volve Opening : 7 5 . 0 % Symbol

CO

SCMC Cone

30

o o o 20

O

or

Q

lij

cr

10

(O

CO UJ Q:

30

40

VOLUMETRIC FLOW RATE (mVs) x i o ' Figure 3. Variation in pressure drop across the globe valve at different concentration of SCMC solutions with volumetric flow rate.

>

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.

Non-Newtonian Liquid Flow Through Globe and Gate Valves

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^-P^e: manometer taps; RL^-RL^: rotameters; S^, S^: liquid holding tank; ST: stirrer; SV^, SV^: Solenoid valves; T: thermometer; V^-V^: valves.

500

Advances in Engineering Fluid Mechanics

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% wr^ar^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
\i ^

\i \i

p

cr

i\ i\

Q.

o <

i\

^"" r—-^

^^5j,i,^e^^^

1 1

" • " ^ " " ^ ^

1

1 UPSTREAM

VALVE

DOWNSTREAM

— LENGTH Figure 7. Pressure distribution along a pipeline with a valve.

Non-Newtonian Liquid Flow Through Globe and Gate Valves

501

downstream portions of the fitting is to achieve fully developed flow conditions to facilitate the measurement of pressure drop across the fitting. Measurement of Pressure Drop Across the Valve Variations of the static pressure along the tube containing valve are schematically shown in Figure 7. The curve a-b-c-d-e-f is the static pressure distribution in the straight upstream portion, across the valve portion and the straight downstream portion of the test section. The curves a-b-c' and c" -e-f are the extended portions of the static pressure distribution curve in the upstream and downstream portion of the fully developed flow region, respectively. The pressure loss due to valve AP is obtained from the difference in static pressures of the upstream fully developed flow and the downstream fully developed flow region. This pressure loss can be graphically expressed as the vertical segment c' -c' in the pressure distribution curve. Thus, AP includes the frictional pressure drop for the flowing fluid through a passage of length equal to the axis of the valve port, and the additional pressure loss due to the irreversibility. The typical static pressure distribution curves for globe valve and gate valve are shown in Figures 8 and 9.

SCMC CONC.: 0.2 k g / m ^ SYMBOL D

• 0

•

X ^

120 100 50 0 Upstream, Length in cm VALVE

FLOW RATE (mV$)xiO^ 14.33 15.82 17.58 19.25 21.00 22.67

50 100 120 Downstream,Length in cm

Figure 8. Typical static pressure distribution curve for globe valve.

502

Advances in Engineering Fluid Mechanics

SCMC C0NCa0»4 kg/m" SYMBOL

D

—

• 0 A

— —

X

•

—

FLOW RATE (m3/s) xl05 7. 1 7 10.58 13.83 17.00 20.17 23.42

120 100 50 0 50 100 120 Upstream,Length in cm. VALVE Downstream, Length In cm. ^ .—. ^' Figure 9. Typical static pressure distribution curve for gate valve. NOTATION a^, b^ = parameter in Equation 8, dimensionless D = diameter, m De = Dean number, VDp|Ll'(D/D^)o^ dimensionless f = Fanning friction factor, dimensionless g = acceleration due to gravity, m/s^ h, H = head, m K = resistance coefficient, dimensionless K, K' = consistency index, N s"/m^

L = length, m n, n' = flow behavior index, dimensionless AP = pressure drop, N/m^ r = radius of curvature, m r = radial coordinate, m, in Equation 16 Re = Reynolds number, VDp/jLi V = velocity, m/s Y = parameter defined by Equation 32, dimensionless Z = cos 0, dimensionless

Greek Letters a = coefficient in Equation 26 a = bend angle in Equation 45, deg a = ratio of the valve opening to the full opening of the valve. Equations 46 & 47, dimensionless e = roughness height, m

0 = |i = \) = p= T=

angle, deg viscosity, Ns/m^ kinematic viscosity, mVs density, kg/m^ shear stress, N/m^

Non-Newtonian Liquid Flow Through Globe and Gate Valves

503

Subscripts b = bend c = curvature t = tube

w = wall st = straight pipe MR = Metzner-Reed

REFERENCES 1. Metzner, A. B., "Non-Newtonian Technology: Fluid Mechanics, Mixing and Heat Transfer" in Advances in Chemical Engineering, T. B. Drew and J. W. Hoopes (Ed.) Academic Press Inc. New York, 1 77-153 (1956). 2. Skelland, A. H. P., ''Non-Newtonian Flow and Heat Transfer'', John Wiley & Sons, Inc. New York (1967). 3. Alves, G. E., D. F. Boucher and R. L. Pigford, "Pipe-Line Design for NonNewtonian Solutions and Suspensions", Chem. Eng. Prog. 48 385-393 (1952). 4. Metzner, A. B. and J. C. Reed, "Flow of Non-Newtonian Fluids-Correlation of the Laminar, Transition and Turbulent Flow Regions", AIChE J. 1 434-440 (1955). 5. Joshi, S. D. and A. E. Bergles, "Experimental Study of Laminar Heat Transfer to In-Tube Flow of Non-Newtonian Fluids", Trans. ASME J. Heat Transfer 102 397-401 (1980). 6. Webb, E. K., "Profile Relationships, the Log-Linear Range and Extension to Strong Stability", Q. J. R. Meterol. Soc. 96 67-90 (1970). 7. Vanoni, V. A., "Transportation of Suspended Sediment by Water", Trans. Am. Soc. Civ. Engrs. 5 111 (1946). 8. Lumley, J. L., "Drag Reduction in Turbulent Flow by Polymer Additives", J. Polymer Sci. Macromol. Rev. 1 263-290 (1973). 9. Dodge, D. W. and A. B. Metzner, "Turbulent Flow of Non-Newtonian Systems", AIChE J. 5 189-204 (1959). 10. Edwards, M. F. and R. Smith, "The Turbulent Flow of Non-Newtonian Fluids in the Absence of Anomalous Wall Effects", J. Non-Newtonian Fluid Mechs. 7 77-90 (1980). 11. Irvine, T. F., "A Generalized Blasius Equation for Power law Fluids", Chem. Eng. Comm. 65 39-47 (1988). 12. Hartnett, J. P. and M. Kostic, "Turbulent Friction Factor Correlations for Power Law Fluids in circular and Non-circular Channels", Int. Comm. Heat Mass Transfer 17 59-65 (1990). 13. Hooper, W. B., "Piping Design, Fittings, Pressure Drop", Encyclopedia of Chemical Processing and Design (Ed., J. J. Mcketta) 39 19-27 (1991). 14. Edwards, M. F., M. S. M. Jadallah and R. Smith, "Head Losses in Pipe Fittings at Low Reynolds Numbers", Chem. Eng. Res. Des. 63 43-50 (1985). 15. Jacobs, G. S. and F. A. Sooy, "New Method of Water Measurement by Use of Elbows in a Pipeline", J. Electricity^ Power and Gas 27 72-78 (1911). 16. Levin, A. M., "A Row Metering Apparatus", J. ASME September 326-328 (1914). 17. Yamell, D. L. and F. A. Nagler, "Flow of Water Around Bends in Pipes", ASCE Trans. 100 1,018-1,033 (1935). 18. Lansford, W. M., "The Use of an Elbow in a Pipeline for Determining the Rate of Flow in the Pipe", Bulletin No 289, Engineering Experiment Station, University of Illinois, Urbana, 111., XXXIV No 33 Dec. 22 (1936).

504

Advances in Engineering Fluid Mechanics

19. Addision, H., "The Use of Pipe Bends as Flow Meters", Engineering 145 227-229 (1938). 20. Spink, L. K., "The Principles and Practice of Flowmeter Engineering", 8th Ed., The Foxboro Co., Foxboro, Mass., USA 40-43 (1958). 21. Murdock, J. W., C. J. Foltz and C. Gregory, "Performance Characteristics of Elbow Flow Meters", Trans. ASME J, Basic Engg. 85 498-506 (1964). 22. Kittredge, C. P., "Elbow Meters", Proc. Flow Measurement Sym. by ASME 274-289 (1966). 23. Hauptmann, E. G., "Take a Second Look at Elbow Meters for Flow Monitoring", Instruments & Control Systems 51 47-50 (1978). 24. Polentz, L. M., "Build Your Own In-line Flowmeter", Plant Engineering 35 169-170 (1981). 25. Morrison, G. L., K. K. Sheth and G. B. Tatterson, "Elbow Flowmeter Calibrations for Slurries", Chem. Engg. Comm. 63 39-48 (1988). 26. Brook, N., "Flow Measurement of Solid-Liquid Mixtures Using Venturi and Other Meters", Proc. Instn. Mech. Engrs. 176 127-140 (1962). 27. Binnington, R. J., G. J. Troup and D. V. Boger, "A Low Cost Laser-Speckle Photographic Technique for Velocity Measurement in Slow Flow", J. NonNewtonian Fluid Mech. 12 255-267 (1983). 28. Kim-E, M. E., R. A. Brown and R. C. Armstrong, "The Roles of Inertia and Shear-Thinning in Flow of an Inelastic Liquid Through an Axisymmetric Sudden Contraction", J. Non-Newtonian Fluid Mech. 13 341-363 (1983). 29. Mackay, M. E., Y. L. Yeow and D. V. Boger, "Pressure Drop in Pipe Contractions—Experimental Measurements or Finite Element Simulation", Chem. Eng. Res. Des. 66 22-25 (1988). 30. Dudgeon, C. R. and J. E. Hills, "Head Losses in Pipe Fittings with Stepped Contractions and Expansions," Proc. 9th Australian Fluid Mechanics Conf., Auckland, 105-108 Dec. 8-12 (1986). 31. Townsend, P. and K. Walters, "Expansion Flows of non-Newtonian Liquids", Chem. Eng. Sci. 49 749-763 (1994). 32. Karr, M. and L. W. Schutz, "Pressure Drop Tests on Globe and Angle Valves with Oil and Water Flow", J. Am. Soc. Naval Engrs. 52 239-256 (1940). 33. Beck, C. and Miller, H. M., "Pressure Losses in Marine Fuel Oil Systems", J. Am. Soc. Naval Engrs. 56 62-83 (1944). 34. Beck, C , "Laminar Flow Friction Losses through Fittings, Bends and Valves", J. Am. Soc. Naval Engrs. 56 235-271 (1944). 35. Beck, C, "Comparison of 3 1/2 inch I.P.S. and 6-inch I.P.S. Valves and Fittings in Laminar Flow", J. Am. Soc. Naval Engrs. 56 389-395 (1944). 36. Beck, C , "Laminar Flow Pressure Losses in 90-degree Constant Circular Crosssection Bends," J. Am. Soc. Naval Engrs. 56 366-388 (1944). 37. John, R. R., "Investigation of the Resistance Coefficient of a Series of One-Halfinch Pipe fittings", BS in Engineering Thesis, Princeton, New Jersey, USA (1951). 38. Pigott, R. J. S., "Pressure Losses in Tubing, Pipe and Fittings", Trans. ASME 72 679-688 (1950). 39. Pigott, R. J. S., "Losses in Pipe and Fittings", Trans. ASME 79 1767-1783 (1957). 40. Kittredge, C. P. and D. S. Rowley, "Resistance Coefficients for Laminar and Turbulent, Flow through One-hald-inch Valves and Fittings", Trans. ASME 79 1759-1766 (1957).

Non-Newtonian Liquid Flow Through Globe and Gate Valves

505

41. Eastwood, W. and E. J. Sarginson, "The Effect of a Transition Curve on the Loss of Head at a Bend in a Pipeline", Proc. Instn. Civil. Engrs. 16 129-142 (1960). 42. Ito, H., "Pressure Losses in Smooth Pipe Bends", Trans. ASME J. Basic Engg. 82 131-143 (1960). 43. Ward Smith, A. J., "The Flow and Pressure Losses in Smooth Pipe Bends of Constant Cross Section", /. Roy. Aeronautical Soc. 67 437-447 (1963). 44. Crane Company, Engineering Division, "Technical Paper No 410", (1969). 45. Tremblay, P. E. and D. G. Andrews, "Hydraulic Characteristics of a Valve in Two Phase Flow", Can. J. Chem. Engg. 52 433-437 (1974). 46. Harris, J. and A. N. Magnall, "The Use of Orifice Plates and Venturi Meters with Non-Newtonian Fluids", Trans. Instn. Chem. Engrs. 50 61-68 (1972). 47. Miller, D. S., "Internal Flow Systems", British Hydromechanics Research Association (1978). 48. Moller, K. and G. Elmqvist, "Head Losses in Pipe Bends and Fittings", Tappi 63 101-104 (1980). 49. Hooper, W. B., "The Two-K Method Predicts Head Losses in Pipe Fittings", Chem. Engg. 88 Aug. 24 96-100 (1981). 50. Hooper, W. B., "Calculate Head Loss Caused By Change in Pipe Size", Chem. Engg. 95 Nov. 7 89-91 (1988). 51. Fairhurst, C. P., "Component Pressure Loss during Two-Phase Flow," Proc. Int. Conf. Physical Modelling of Multiphase Flow, England, Paper Al 1-22 April 19-21 (1983). 52. Hoang, K. and M. R. David, "Flow Structure and Pressure Loss for Two Phase Flow in Return Bends", Trans. ASME J. Fluids Engg. 106 30-37 (1984). 53. Simpson, H. C , D.H. Rooney and T. M. S. Callander, "Pressure Loss through Gate Valves with Liquid-Vapor Flows", Proc. 2nd. Int. Conf. Multiphase Flow, London. Paper B2 67-80 June 19-21 (1985). 54. Norstebro, A., "Pressure Drop in Bends and Valves in Two-Phase Refrigerant Flow", Proc. 2nd Int. Conf Multiphase Flow, London, Paper B3 81-92 June 19-21 (1985). 55. Sookpasong, P., J. P. Brill and Z. Schmidt, "Two Phase Flow in Piping Components", Trans. ASME J. Energy Res. Technol. 108 197-201 (1986). 56. Hydraulic Institute Standards, 11th Edition, Hydraulic Institute, New York, USA (1965). 57. Streeter, V. L. and E. B. Wylie, ''Fluid Mechanics", McGraw-Hill Book Co., Inc., New York, (1975). 58. Simpson, L. L. and M. L. Weirick, "Designing Plant Piping", Chem. Engg. /Deskwork, McGraw-Hill Book Co., Inc. New York USA (1978). 59. Nandi, S. and S. K. Das, "Frictional Pressure Drop for Flow of Water in UBends at High Dean number", Indian J. Technol. 27 181-184 (1989). 60. Das, S. K., M. N. Biswas and A. K. Mitra, "Non-Newtonian Liquid Flow in Bends", Chem. Engg. J. 45 165-171 (1991). 61. Banerjee, T. K., M. Das and S. K. Das, "Non-Newtonian Liquid Flow through Globe and Gate Valves", Can. J. Chem. Engg. 72 207-211 (1974). 62. Subbu, S. K., S. K. Das, M. N. Biswas and A. K. Mitra, "Pressure Drop in U-Bends for Air-Water Flow", Int. J. Engg. Fluid Mech. 3 239-248 (1990). 63. Das, S. K., M. N. Biswas and A. K. Mitra, "Friction Factor in Two-Phase Gasnon-Newtonian Liquid Flow in Bends", Can. J. Chem. Engg. 69 179-188 (1991).

This Page Intentionally Left Blank

CHAPTER 18 COMPARISON OF NUMERICAL AND EXPERIMENTAL RHEOLOGICAL DATA OF HOMOGENEOUS NON-NEWTONIAN SUSPENSIONS V. Nassehi Chemical Engineering Department Loughborough University of Technology U.K. CONTENTS INTRODUCTION RHEOLOGICAL EXPRESSIONS HOMOGENEOUS SUSPENSION FLOW Governing Equations Numerical Simulation Techniques Solution Algorithm COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS Experimental Apparatus Experimental Rheological Data of the Test Suspension Rheological Behavior Based on Analytical Approach NOTATION REFERENCES INTRODUCTION Suspensions are defined as the stable dispersions of fine solid particles in liquids. Due to the widespread use of suspensions in industry, accurate analysis of their fluid dynamical behavior is needed to carry out the design calculations in many flow processes. According to the theoretical considerations the fluid dynamical behavior of a suspension may depend on a very large number of factors. In this respect, material characteristics of liquids and solids in the dispersion and the composition of the mixture in terms of solid volume fraction, particle size distribution, and interactions between solid particles can be mentioned. A variety of criteria for the classification of suspensions into homogeneous or heterogeneous categories have been proposed. In general, however, it is agreed that suspensions which consist of very fine particles exhibit homogeneous flow behavior, whereas the flow of slurries of coarser particles is usually heterogeneous.

507

508

Advances in Engineering Fluid Mechanics

Rheological equations generally are regarded as the expressions of the relationship between the fluid dynamical behavior of suspensions and their material and structural characteristics. Despite the theoretical complexities, experimental evidence suggests that the rheological behavior of most suspensions can be successfully modelled by relatively simple constitutive equations which depend only on two or three parameters [1]. For example, the power law model very widely used to represent the rheological behavior of suspensions, contains only two parameters, the flow index and the consistency coefficient. Depending on the value of the flow index, the power law model can represent Newtonian or non-Newtonian behavior. Due to the inherent structural complexity of suspensions it is, however, impossible to obtain a simple general rheogram for a suspension, and in most cases system dependent relationships must be used [2]. System-dependent constitutive relationships for homogeneous suspensions can, in general, be developed and evaluated using the numerical modelling of the governing equations of non-Newtonian fluid dynamics. Such evaluations are based mainly on the comparison of numerically simulated values of different field variables with experimental results. In some cases numerical results also can be compared with values obtained from the analytical formula derived using simplifying assumptions in a flow field. Numerical analysis of heterogeneous flows is much more difficult. This is because a general method which can effectively simulate all types of heterogeneous flow behavior cannot be developed. Relatively simple types of heterogeneous suspension flows, such as stratified regimes (i.e., layered homogeneous flows) can, however, be successfully modelled [3]. Theoretical analysis of non-Newtonian turbulent flows is not complete, and their numerical modelling is rarely attempted. Newtonian turbulent flows, on the other hand, have received a lot of attention, and a variety of models describing their behavior can be found in the literature. None of these models, however, is considered to have general applicability. In view of such difficulties this chapter is restricted to the description of the numerical modelling of steady, laminar non-Newtonian flow of homogeneous suspensions. The numerical results provided by the application of the described model to the simulation of a homogeneous suspension flow have been tested against both experimental data and analytical solutions obtained from well-known treatments of non-Newtonian flows. RHEOLOGICAL EXPRESSIONS The rheological behavior of non-Newtonian suspensions is often characterized by a power law model: T= r|j"

(1)

where T is the shear stress, y is the shear rate, r|, and n are empirically derived constants known as the consistency coefficient and the flow behavior index, respectively. This model can represent Newtonian (for n = 1), shear thinning (for n < 1), or shear thickening (for n > 1) behavior. In some cases, however, the rheological behavior of a suspension is best represented by a segmented form of

Comparison of Numerical and Experimental Rheological Data

509

this model in which different values of the empirical parameters over different ranges of the shear rate must be used. The most commonly used analytical expressions relating flow rate with pressure drop in pipes, for power law fluids, are as follows. Under laminar flow conditions: 3 r

^

\'/"

aAp n - ^^^ 3n + l ^2Lr|, y

(2)

where Q is the flow rate, a is the pipe radius, Ap is the pressure drop, and L is the pipe length. Under turbulent flow conditions the friction factor (/) can be calculated using the Dodge-Metzner correlation [4], which is considered to give a good fit for modified Reynolds numbers between 2,000 and 20,000:

1

'-xo^{K,*r-^^)-^-±

V7

n

(3)

n

and dAp where d is the pipe diameter, p is the material density, and V is the average velocity in the pipe. The modified Reynolds number (Re*) for a power law fluid is given by Metzner and Reed [5]: Re* =

8pV^-"d" 2i n j ,^6n - ^ i+- ^

(5)

Other expressions which deal with specific cases also are reported in the literature. For example, Adusumilli and Hill [6] give some equations for pressure rise, energy lost, and eddy length for non-Newtonian low Reynolds number (Re<80) flows through sudden expansions. They use an approach based on a macroscopic mass and energy balance across a sudden pipe expansion, as detailed originally by Bird et al. [7]. The expansion ratio (P) is defined as:

^O^

(6)

where a,and a^ are the outlet and inlet radii, respectively. The maximum velocity (V_^) in a pipe is related to the bulk or average velocity by:

510

Advances in Engineering Fluid Mechanics

The pressure rise, or drop, is related to the velocity within the pipe by a form of Bernoulli's equation modified for non-Newtonian fluids:

In deriving Equations 7 and 8 it is assumed that: (i) frictional drag losses are negligible, (ii) flow on both sides of the expansion (or restriction) is fully developed, and (iii) the pressure is constant on the above planes. Finally, the fluid eddy length (L,), or recirculation zone distance, caused by an expansion is derived as [6]:

^

= 0.131341 4 - 1 + 0.0229 4 - l | R e * n ' ^ ' ^ J v»

(9)

The eddy length is an indication of the length of pipe after an expansion over which turbulence due to the expansion occurs. HOMOGENEOUS SUSPENSION FLOWS Governing Equations Suspensions which consist of sufficiently uniformly dispersed solids of very small diameter (less than 25jLi) can be treated as the equivalent of homogeneous singlephase fluids [1]. Thus, the governing equations of continuity and momentum for steady laminar flow of a non-Newtonian homogeneous suspension in an axisymmetric cylindrical (r,z) co-ordinate system can be written as: ^, = - T ^ + — + ^ = 0 dr r oz

(10)

and

dr d^ (d\, az

^ dz

dr dvS

dr\ gr=0

dr )

r dr

r^ (11)

Comparison of Numerical and Experimental Rheological Data

h = pVr-T^ + p v , - ^ H - T T ^ - T ^

3r

^ 3z

3z

2r|--^

3z V

9z j

-

Hr" +

r v 3r

511

5z

d_

where v^ and v^ are the components of the velocity vector in radial and axial directions, respectively, and p is the pressure, p is the fluid density and g^ and g^ are the components of the body force vector. In the present application the nonNewtonian viscosity r\ is given by the power law and thus: il = Ti,(Yr'

(13)

Suspension flows can, in general, be considered to be isothermal, and Equations 10 to 13 together with appropriate boundary conditions form the basis for their numerical simulation. The boundary constraints used in pipe flow regimes are inlet velocity profile, zero velocity on solid non-slip walls, and stress free (or for long pipes developed flow) exit conditions. In shell and tube systems with solid and porous walls, used in thickening of suspensions by cross-flow filtration, a different set of boundary conditions must be given. These are the inlet velocity profile, zero velocity on outer shell's solid walls, stress-free conditions at the exit, and the following Darcy flow conditions on porous wall: Tj.

y]f

k.

dp V.+ — = u

3p

V,+ — = u ' dz

(14)

(15)

where r| ^ is the viscosity of the liquid being filtered and K^ and K^ are porous wall permeability coefficients in r and z directions, respectively. Numerical Simulation Techniques It is commonly accepted that the finite element methods offer the most rigorous numerical schemes for the simulation of fluid flow phenomena. The inherent flexibility of these schemes and their ability to cope with complicated geometries and boundary conditions can be used very effectively to solve the governing equations of complex flow regimes. In particular, the finite element simulation of steady, incompressible laminar flow is very well-established, and an extensive literature in this area is available. Galerkin finite element schemes based on different types of Lagrange elements are the most frequently used techniques in these simulations [8]. In flow domains with porous walls, however, more recent work

512

Advances in Engineering Fluid Mechanics

has proven that the least-squares method in conjunction with C continuous isoparametric Hermite elements provides the best flow simulations [9]. Detailed derivations of the least-squares and Galerkin finite element models of the laminar suspension flows are given elsewhere [10,11]; only brief outlines will be presented here. The prime unknowns (i.e., velocity components and pressure ) in the model Equations of 10, 11 and 12 are replaced by approximate trial function representations. In the finite element context where the flow domain is first discretized into a mesh of finite elements, the trial functions within an element can be expressed in terms of interpolation functions, and we have: Vr - V , = X N j ( v , ) j , e t c .

(16)

where N is the interpolation function. The index j is a combined counter which can be expressed as j = (l,m); herein 1 represents the node numbers, and m is the number of internal degrees of freedom on a node. The described substitution results in obtaining residual statements for each model equation (represented by ^,; etc.). Least-Squares Finite Element Models In the least-squares method the obtained residuals are squared and integrated over the solution domain (Q.) to form the following functional: F, =£[(^^+^3') + M ? ] d ^

(17)

where X^ is a constant used to make the functional dimensionally consistent. Minimization of functional 17 yields the working equations of the least-squares finite element schemeas:

(18) apj

In flow domains which include porous walls, an additional functional based on the Darcy Equations of 14 and 15 should be formed:

F, = Jj(^5+^^) + ^24»ndii

(19)

Comparison of Numerical and Experimental Rheological Data

513

where ^2 is a dimensionality constant. Minimization of functional 19 gives the working equations related to the boundary conditions along the porous wall. These equations can be written as:

avn

= 0;etc.

(20)

In order to be able to combine the working equations of 18 and 20, first order derivatives of pressure (i.e., pressure gradients) should be directly calculated as independent degrees of freedom in the numerical scheme. This is only possible if Hermite elements which incorporate the first order derivatives of interpolated functions as the nodal unknowns are used. Standard Galerkin Finite Element Models In the Galerkin methods the residuals obtained by replacing the prime unknowns by the interpolated values (i.e. ^,; etc.) are multiplied by weight functions and integrated over the solution domain (Q). By setting these integrals to be equal to zero we get the following weighted residual statements of the model equations:

L Wj ' ^ j

• dQ = 0;etc.

(21)

In the standard Galerkin method the weight functions are chosen to be identical with the interpolation functions (i.e. W. = N.). The application of the Green's theorem (integration by parts) to the weighted residual statements yields the working equations of the Galerkin finite element scheme. The use of integration by parts in this derivation results in the elimination of the second order derivatives from the working equations of this scheme. Thus, the Galerkin finite element models can provide numerical solutions for second order differential equations using lower order elements. This is considered an advantage of the Galerkin method over the leastsquare technique. Homogeneous suspension flow in domains which do not include porous walls is, therefore, modelled by the Galerkin schemes. Solution Algorithm The solution algorithm in the finite element scheme starts by the repeated application of the working equations to each element. In the next step the derived elemental equations are assembled to form a set of algebraic equations. This set becomes determinate after the use of boundary conditions, and it can be solved. Due to the non-Newtonian nature of the flow considered here, this solution should be incorporated into an iterative cycle where at each iteration the fluid viscosity is updated via the constitutive equation. Iterations stop when a satisfactory convergence occurs.

514

Advances in Engineering Fluid Mechanics

COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS Experimental Apparatus To obtain experimental flow data the pipe section shown in Figure 1 was fitted into a test loop schematically shown in Figure 2. A well-mixed magnesia slurry which was contained in a 0.025 m^ feed tank was used as the homogeneous suspension. The suspension was pumped around the loop by means of a Jabsco flexible impeller pump of nominal 0.001 m^s~' capacity. Heat generated by this pump was removed by three in line shell and single tube water cooled heat exchangers. The loop was constructed from uPVC pipework, and the flow was measured using a Krohne Altoflux, magnetic inductive flow meter which gave a variable 4 to 20 mA output signal. This was connected to an IBM compatible computer via a Fairchild 12 bit A/D converter, and a small resistance circuit to convert variable current to a potential difference in the range 0 to +5 volts. A calibration of A/D output signal with flow rate was employed in a computer program which monitored the flow and pressures in the loop. The pressure transducers were Control Transducer type EA, with a range of 0 to 0.33 MPa. The dimensions of the test section are given in Figure 1. The material of construction was again uPVC. The slightly complicated geometries at the inlet and outlet sections of the long test piece were a consequence of pipe fittings and valves. Ball valves were used; thus, the flow path was essentially straight through as these valves were always fully opened during the experiments. The pressure transducers were fitted into pipe tees contained in the larger diameter pipework. The sensor heads were located some distance from the pipe surface. Experimental Rheological Data of the Test Suspension A reservoir of suspended solid particles of magnesia was used to produce the test slurry. These solids had been used for cross-flow filtration tests over a period of many hours. Thus, it was reasonable to assume that any particle attrition or repercipitation had stabilized and the flow properties would not, therefore, vary during this test work except for required solid concentration changes. The measured particle size distribution of solids is given in Figure 3. Test slurry solid concentrations were varied by means of cross-flow filtration in the test circuit to thicken the suspension, or by replacing the filtrate to dilute the suspension. The suspension's rheological properties were measured at a range of concentrations between 28 and 43% by weight using a Haake R V 2 viscometer fitted with a coaxial cylinder measuring head. This instrument was used to measure shear stress at fixed shear rates up to 4,000 s~'. These values are shown in Figure 4. The inner to outer wall radii ratios for both coaxial cylinder gaps were 0.98, and the length of cylinder to gap size ratio was 150. Both values are in excess of the minimum recommended values of 0.9 and 50, respectively [12]. The pressure drops in the pipe test section were measured at a range of flow rates varying from 1.6 x 10"^ to 6.3 x 10~^m^s' . The suspension was recirculated through the loop for at least 5 minutes at constant flow rate before the pressure and flow rates were measured. To ensure isothermal flow conditions the temperature during the test was controlled not to fluctuate more than 2°C.

cc5

CO

^

I

I

CM

"n;

fco

o

i

3 O)

CO

CO

E

CD

'Q.

g.

CD

CD CO

'•*-»

.9 o

C

"D C 13

CO CD

Comparison of Numerical and Experimental Rheological Data

CO

i CO

o

CO

-n_^:

515

516

Advances in Engineering Fluid Mechanics heat exchangers V4

by-pass V2

pipe test section

V3 ^

^ FM

^ ($)

flexible impeller pump to computer V1 drain Figure 2. Sechematic diagram of pipe-flow apparatus.

10

15

Particle diameter, micrometre. Figure 3. Particle size distribution of solids in suspension.

25

Comparison of Numerical and Experimental Rheological Data

517

200

100

200

300

500

2,000

1,000

3,000

5.000

Shear rate, 1/s. 28

31.7 —A—

33.5

35.2

37

38.7 -A-

42.8

Figure 4. Rheograms: shear stress with rate for various concentrations. Note: average shear rate near the wall in the flow experiments varied from 300 to 4,000 S'^; hence, the first data point of each rheogram has been neglected In the power law fit. The rheograms shown in Figure 4 can be fitted by a power law relation, and the concentrations at which rheological measurements were taken also were those tested in the flow loop. The physical data including suspension density, power law flow behavior index, and material consistency coefficient are given in Table 1. It is noticeable that the suspension displays a consistent movement away from Newtonian behavior as solid concentration increases; the flow behaviour index consistently reduces in Table 1. Table 1 Physical Properties of the Magnesia Suspensions Solid Concentration (% by mass)

Suspension Density (kg m-3)

Flow Index

Consistency Coefficient

28.0 31.7 33.5 35.2 37.0 38.7 42.8

1,180 1,210 1,230 1,240 1,260 1,270 1,310

0.591 0.421 0.343 0.312 0.314 0.291 0.239

0.114 0.633 1.53 2.38 3.04 5.09 13.0

518

Advances in Engineering Fluid Mechanics

Rheological Behavior Based on Analytical Approach Under laminar flow conditions the flow rate with pressure drop relation follows readily from Equation 2, after establishing values for the empirical constants in Equation 1. When the modified Reynolds number was greater than 2,300, Equations 3 to 5 were solved iteratively for various values of flow rate and the corresponding pressure drop. The analytical results shown in Figures 5a to 5g are a combination of these results, using the approach relevant to the prevailing flow condition or Reynolds number. The radius of each section of the pipe shown in Figure 1 was used in the solutions to Equations 2 to 5; thus, the total analytical pressure drop is a sum of the various component drops over the separate sections of varying radii. This analytical approach is based on the assumption that the flow is fully developed throughout the pipe section, and the shear profile is uniform. Thus, any pressure changes due to the geometry changes, i.e., expansions and contractions, are not recognized by this simplified approach. A frequent geometry change is a consequence of a relatively compact piece of equipment, which is common in the design of cross-flow filters and the like. These losses are less relevant when pumping suspensions over long distances in uniform pipes. Under such circumstances the simplified analytical approach often is adequate. (text continued on page 522)

28% by weight 0.08 X

^ 0.06 Q.

"X"

QO •D 0.04 V.

CO CO

CD

^

D 0.02

•

•6'

20

-

*

-

D

..a.

°

#A*

n10

^

30

40

50

60

Flow rate, m^s"''x10^. analytical D

numerical A

experimental •

mocJified analytical X

Figure 5a. Flow rate with pressure drop at 28% solids by weight.

70

Comparison of Numerical and Experimental Rheological Data

519

31.7% by weight 0.07 0.06 X

CL 0.05 g-0.04

m^"

2 0.03

3

(/} CO

0 0.02 A

...a..

0.01 h " 0

10

20

30

40

50

70

60

Flow rate, m^s"''x10^. analytical •

numerical A

experimental •

modified analytical X

Figure 5b. Flow rate with pressure drop at 31.7% solids by weight. 33.5% by weight 0.07

h

X

0.06

[ a. 2

.•...^

0.05

d. 0.04 o u.

•D

•

I .m.

•

[

X

O""

I

3 0) CO

*•

CD 0,02 QL

.JHL

a

a

9.

....a

0.01

n

A tT

-a 0

1

1

10

20

_j

30

40

_.L

!

50

60

7C

Flow rate, m^s"''x10^. analytical alytic D

numerical A

experimental •

modified analytical X

Figure 5c. Flow rate with pressure drop at 33.5% solids by weight.

520

Advances in Engineering Fluid Mechanics 35.2% by weight 0.06 0.05

..^..

CO Q-

^

0.04

X

d. o

A

-a 0.03 (D

Jg 0.02

..O-

••r D

o

n

a

0.01

10

20

30

40

50

60

70

Flow rate, m^s'''x10^. analytical D

numerical A

experimental •

modified analytical X

Figure 5d. Flow rate with pressure drop at 35.2% solids by weight. 37.0%bywtight 0.06

0.05

A

cd Q^ 0.04

X

• •

d. o •D 0.03 CD

-i-4.D

.4

I 0.02 CD

a

0.01

10

20

30

40

50

60

70

Flow rate, m^s'''x10^. analytical a

numerical experimental A •

modified analytical X

Figure 5e. Flow rate with pressure drop at 37.0% solids by weight.

Comparison of Numerical and Experimental Rheological Data

521

38.7% by weight 0.08 0.07

d

..X..

^ 0.06

A»

§•0.05 £

0.04

13

A

•I--

>w

..a.

O 0.03 0.02 10

20

30

40

50

60

70

Flow rate, m^s"''x10^.

0.01 analytical a

numerical A

experimental •

modified analytical X

Figure 5f. Flow rate with pressure drop at 38.7% solids by weight. 42.8% by weight 0.12

h CO

i

0.1 h

X h

c£ 0.08 o "D 0 3 0.06 CO CO CD ^

0.04

0.02

A X

[ K

•1 •

. t ^

• ....

-••s

•ft a...

a

n

•

• 1 10

1 20

1 ,,. 30

\ 40

1 50

1 60

1 70

Flow rate, m^s"''x10^. analytical a

numerical A

experimental •

modified analytical X

Figure 5g- Flow rate with pressure drop at 42.8% solids by weight.

522

Advances in Engineering Fluid Mechanics

(text continued from page 518) The modified analytical approach is to estimate the pressure drop or rise due to contraction or expansion of the pipe, respectively, by means of Bernoulli's energy balance equation. This technique is commonly used in Newtonian flows but is applied less readily to non-Newtonian regimes. The working model is described by Equations 6 to 8. It is worth noting that some of the reduction, or expansion, ratios in the present study were very slight (10 to 0.88 and 8.8 to 7 mm pipes); however, these accounted for up to 27% of the total pressure drop, i.e., that due to drag and contraction losses. This is clearly a consequence of the squared velocity component in Equation 8. The total sum of pressure variations due to the changes in geometry were added to the frictional losses calculated in previously described manner. This total pressure drop is termed the modified analytical approach given in Figures 5a to 5g. This technique was very successful in reconciling the analytical and experimental values. The maximum modified Reynolds number at which the procedure was employed was 8,300. Under these conditions the drag losses were calculated by Equations 3 to 5. Thus, the drag and contraction losses are additive under all flow conditions under which the tests were conducted. Equation 9 was used to calculate the eddy length caused by the geometry change; this was usually in the range 10 to 100 mm. Thus, the assumption that the flow becomes stable after a change in geometry, for the range of Reynolds numbers in theses tests, was reasonable. This implies that the assumptions needed to derive Equations 7 and 8 are in fact fulfilled. Results obtained using the above analytical approaches are shown in Figures 5a to 5g, together with the measured pressure drops and the values predicted by the described finite element model for the magnesia suspension. The numerical model is valid under laminar flow conditions only. In all instances the numerical solution becomes unstable at a certain flow rate, or Reynolds number. Instability was manifested by an apparent maximum in the pressure drop-flow rate relation followed by a minimum and then by an exponential increase in the pressure with respect to flow rate. The values of Reynolds numbers which gave a stable solution, and the values which just gave unstable solutions, are shown in Table 2. These were calculated according to Equation 5 and provide some indication of the value up to which the numerical model is valid. It is notable that the model was stable above the Reynolds number value of 2,300, which is the accepted value for the start of the transition between laminar and turbulent flow. It is apparent that the model becomes unstable at Reynolds number values more than 3,000. No attempt was made to obtain stable results for high Reynolds number flows by means of artificial numerical damping procedures. Only the numerically stable values are plotted in Figures 5a to 5g. The increasing viscosity with solid concentration increases the stability at a given flow rate; hence, a greater amount of numerical data are shown at the higher concentrations. The analytical and modified analytical calculations are not restricted by this stability requirement and are plotted up to, and even beyond, the maximum flow rate investigated experimentally. Valid numerical results could be achieved at Reynolds numbers greater than 3,000 using a turbulent flow model. Development of such a model is a major task, and it should only be attempted after detailed theoretical and experimental studies provide a

Comparison of Numerical and Experimental Rheological Data

523

Table 2 Reynolds Number Values at Which the Model Is Stable and Unstable for Various Flow Rates Stable Solution: Solid Concentration (% by mass)

Slurry Flow Rate (m^ s-1) 2.5 2.5 4.2 4.2 5.0 5.8 6.7

28.0 31.7 33.5 35.2 37.0 38.7 42.8

X X X X X X X

Unstable Solution: Slurry Flow Rate

Reynolds Number

10^ 10-^ 10-^ 10-^ 10-4 10-4 10-4

(m3 s-1)

3.3 3.3 5.0 5.0 5.8 6.7

3,559 2,045 3,389 2,752 2,934 2,730 2,079

X X X X X X

—

10-4 10-4 10-4 10-4 10-4 10-4

Reynolds Number 5,338 3,221 4,585 3,744 3,805 3,430

—

reasonable understanding of the fluid dynamical behavior of non-Newtonian suspensions under .turbulent flow conditions. It is noticeable that at low flow rates the agreement between all the calculated and measured data is excellent, but the simplified analytical solution deviates from the other results at Reynolds numbers of between 300 and 1,000. Above these values a comparison of the analytical and modified analytical data clearly demonstrates the increasing importance of even relatively minor changes of pipe geometry in affecting the overall pressure drop when pumping non-Newtonian suspensions. The validated numerical model can be used to predict optimum geometries in process equipment, such as cross-flow filters and shell and tube heat exchangers which reduce the eddy losses due to pipe contractions. Details of such a design procedure are given by Nassehi et. a/ [11]. NOTATION a d / F^

= = = =

gr'gz

=

K^, K^ = L L. = n= N. =

pipe radius, m pipe diameter, m friction factor functionals in the least-squares method radial and axial components of body force vector, ms-^ radial and axial components of porous wall permeability, m^ pipe length, m eddy length, m flow behavior index interpolation function associated with node i

p Ap Q r V V m

, V

= = = = = = =

pressure, Pa pressure drop or increase. Pa low rate, m^ s ' radial co-ordinate average velocity,ms-' maximum velocity, m s ' radial and axial components of velocity vector = trail function representations of the velocity components = weight function = axial coordinate

524

Advances in Engineering Fluid Mechanics

Greek Symbols

p = expansion ratio ^4^^$ = components of Darcy flow Y = shear rate, s"' conditions on porous wall r| = fluid viscosity, Pa s '^v'^2 ~ dimensionality constants r|, = consistency coefficient p = fluid density, kg m (t),,(t)2,(t)3 = components of the model T = shear stress. Pa equations

Q = finite element solution domain

Indices i = node index j = combined index (j = l,m) 1 = node numbers REFERENCES

m = number of internal degrees of freedom on a node

1. Darby, R., Encylopedia of Fluid Mechanics, Cheremisinoff, N. P., Ed. , Vol. 5, Chp. 2, Gulf Publishing Co., Houston, 1986. 2. Jomha, A. I., Merrington, A., Woodcock, L. V., Barnes, H. A., and Lips, A., Powder Technology, Vol. 65 (1991) p. 343. 3. Nassehi, V. and Khan, A. R., Int. J. for Num. Methods in Fluids, Vol.* 14 (1992) p. 167. 4. Dodge, D. C , and Metzener, A. P., AIChE J., Vol. 5 (1959) p. 189. 5. Metzener, A. P. and Reed, J. C , AIChE J., Vol. 1 (1955) p. 434. 6. Adusumilli, R. S., and Hill, G. A., Chem. Eng. Comm., Vol. 57 (1987) p. 77. 7. Bird, R. W., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, John Wiley & Sons, New York, 1960. 8. Zienkiewicz, O. C , and Taylor, R. L., The Finite Element Method, 4th. Ed., McGraw-Hill Book Company, London, 1994. 9. Nassehi,V., and Petera, J., Int. J. for Num. Methods in Eng., Vol. 37 (1994) p. 1,609. 10. Nassehi, V. and Petera, J., Int. J. for Num. Methods in Fluids, Vol. 18 (1994) p. 215. 11. Nassehi, V., Holdich, R. G., and Gumming, L W., Trans. IChemE., Vol. 71, Part A (1993) p. 390. 12. Barnes, H. A., Edwards, M. P., and Woodcock, L. V., Chem. Eng. Sci., Vol. 42 (1987) p. 591.

CHAPTER 19 CONCENTRATION FORCING OF ISOTHERMAL PLUG-FLOW REACTORS FOR AUTOCATALYTIC REACTIONS M. Chidambaram Department of Chemical Engineering Indian Institute of Technology Madras 600 036 India CONTENTS INTRODUCTION, 525 MODEL EQUATIONS AND SOLUTION METHOD, 526 Model Equations, 526 Performance Under Periodic Operation, 528 MODELLING CUBIC AUTOCATALYTIC REACTION, 531 Model Equations, 531 Performance Under Periodic Operation, 533 AUTOCATALYTIC ACTION THROUGH RATE CONSTANT, 533 CONCLUSION, 536 .NOTATION, 536 INTRODUCTION Feed concentration forcing of a variety of nonlinear processes has been shown to be superior in time-averaged performance to that obtained under conventional steady-state operation [1-4]. There have been a number of theoretical studies on periodic operation of plug-flow reactors [5-11]. Most studies have been limited to simple catalytic reactions. There are several reactions which are autocatalytic in nature. Two kinds of autocatalytic behavior have been reported [12]. In one case, the product acts as a reactant. The reaction scheme considered is A + B -> 2B (known as quadratic autocatalysis) or A + 2B —> 3B (known as cubic autocatalysis). Examples of such reactions are found in the acid catalyzed hydrolysis of various esters and similar compounds and in various biochemical processes and in wastewater treatment by activated sludge processes [13,14]. In the second case, the product does not directly influence the rate, but affects it through its influence on the rate constant [15,16]. The reaction scheme considered is: A

^—>A,

^—>A^

^—>A, 525

(1)

526

Advances in Engineering Fluid Mechanics

where A, and A^ are the intermediates. The species A, or A^ has an activating influence on the rate constant K,. Chidambaram has studied the periodic operation of an isothermal plug flow reactor for homogeneous liquid phase autocatalytic reactions of both quadratic and cubic forms [17,18]. Chidambaram has shown that the average yield of product is very much higher than that of simple reactions [17]. The effects of catalyst decay, reversibility of the reaction, and the initial concentration of the autocatalyst on the average yield of product B under periodic operation are evaluated. Chidambaram has analyzed the implication of modelling cubic autocatalytic reactions by successive molecular reactions on the periodic operation of the plug flow reactors [18]. Chidambaram has analyzed the periodic operation of an isothermal plug flow reactor for the second type of autocatalytic reaction given by Equation 1 [19]. MODEL EQUATIONS AND SOLUTION METHODS Model Equations The model assumes isothermal, plug flow and homogeneous auto catalytic irreversible reactions with decay. The reaction scheme considered is given by A + nB-> (n +1)B, B -^ P,

rate = - k .C^Cg

rate = - k ^C^

The dimensionless model equations for reactant A and B are given by dCjde

= -dCjdZ

- R.C^C^

(2)

aCg/ae = - dcjdz + R.C^C^ - R^CB

(3)

at e = 0, C^ = C^(Z) and C3 = C^CZ)

(4)

at Z = 0, C^ = C^^(0) and C3 = Cje)

(5)

where n = 1 for quadratic and n = 2 for cubic form. The periodic forcing function assumed here is a rectangular pulse (bang-bang type) in inlet concentration of both C^^ and Cg^ as shown in Figure 1. It is assumed that feed contains the components A, B and an inert. In Figure 1, C.^(i = A,B) is the dimensionless amplitude of the inlet concentration pulse and y is the cycle split (i.e., pulse width expressed as fraction of period), y = 1 represents steady-state operation, and y < 1 represents periodic operation. If the dimensionless steady-state input is fixed as a^ and a^ the following expression can be obtained by averaging the inlet pulse over the period x: CM = CLjy and C^, =

ttg/y

(6)

The set of hyperbolic partial differential Equations 2 to 6 are to be solved by the method of characteristics. This requires a larger computational time and, hence, the

Concentration Forcing of Isothermal Plug-Flow Reactors

527

u

8

^if

0;

"c O

c

Yx DIMENSIONLESS

TIME

Figure 1. Feed concentration of reactant i(i = A,B)periodic operation, steady-state operation

method suggested by Bailey [20] is used here. Bailey has shown that any outlet state obtainable with a periodic inlet can be obtained by mixing the outputs of at most j + 1 G being the number of state variable) parallel plug flow processes operated in steady-state with different inputs for each process. For the present case, the steady-state equations are given by: dCJdZ = -R,C,C^

(7)

dC J d Z = R,C,C; - R,C,

(8)

with the initial condition C^(Z = 0) and Cg(Z = 0) given by Equation 5. Equations 7 and 8 are solved numerically. Steady-state for two outputs for two different inputs, i.e., of Equation 6 and another for (C^)^^ = 0 and (Cg)^^ = 0 are mixed to get the average exit state given by (CBL,

= 7(C J ,

(9)

where (Cg)^^! is the output for the input C^^ and C^^. The yield of B is given by Y = (CB)„„, - a3

(10)

For simple reactions, A + nD —^ (n + 1)E, the system equations are given by dCJdZ = - R , C ^ Q

(11)

dC^/dZ = -nR.C^CJi

(12)

528

Advances in Engineering Fluid Mechanics

with the initial condition given by (CA)Z=O = «A/Y and

(C^)^^, = a^/y

(13)

The yield of product E is calculated from the conversion of A or D. It is assumed that feed does not contain E. Chidambaram [17] has compared the average yield of E with the average yield of B of autocatalytic reactions (the corresponding system Equations 7 to 10 with R^ = 0). ^ For reversible reaction without decay A + nB ?=^ (n + 1)B, the system equations are given by ' dC,/dZ = -R,C,C"3 +

dc^/dz = R,c,c;

Rfr'

D pCn+l)

(14) (15)

with the initial conditions given by Equation 5. Performance Under Periodic Operation Figure 2 shows the effect of periodic operation of cubic autocatalytic reaction on the yield of product for two values of reaction rate constant group (R,) with no

Figure 2. Average yield of B obtained under periodic operation. cubic autocatalytic reaction average yield of product E of simple reaction (a^ = 0.9, a3 = 0.1, n = 2) [17]

Concentration Forcing of Isothermal Plug-Flow Reactors

529

decay. Decreased y means increased feed concentration (refer to Equation 6) during that fraction of period and, hence, increased reaction rate. Hence, the yield of product increases during that fraction of period. The average yield over the entire period depends on y and operating parameters. The average yield is found to increase significantly. The improvement in yield is very much higher than that steady-state operation, and complete conversion of reactant A is achieved under periodic operation (Figure 3). Also in Figure 2, the yield improvement of product E of simple reaction A + 2D -^ 3E under otherwise identical operating conditions is shown. It can be seen that improvement in yield of the product of autocatalytic reaction is enhanced very much in comparison to a simple reaction. Hence, periodic operation is highly favorable for autocatalytic reactions. Similar behavior is shown for quadratic autocatalytic reaction in Figure 4. However, the average yield is higher for the cubic reaction. As stated earlier, the feed is assumed to contain the reactants A, B, and an inert. Instead of periodically changing the feed concentration of A and B simultaneously as per Equation 6, we also may change the feed concentration of A and keep the concentration of B at a constant value. The yield of product B is calculated from Y = Y(CA , + (1 - 7)a,

cx„

(16)

Figure 3. Conversion obtained by periodic operation for cubic autocatalytic reaction, (after [17])

530

Advances in Engineering Fluid Mechanics 1.01

CQ

•o >

S > <

Figure 4. Average yield of B obtained under periodic operation. quadratic autocatalytic reaction average yield of product E of simple reaction (a^= 0.9, ttg = 0.1, n = 1) [17] Simulation results show that response similar to Figure 2 with a shift towards left indicating that under period operation higher yield of B can be obtained for a given y, in the case of periodically varying simultaneously A and B in the feed. In what follows we will assume the simultaneous variation of A and B in the feed as per Equation 6. Figure 5 shows the effect of decay of B (to a stable product P) on the performance of periodic operation. A resonance behavior (a maximum in the yield with pulse with y) is obtained when the decay is significant. The resonance frequency is about the same for all the values of decay constant. This is due to complete conversion of reactant A at this frequency (refer to Figure 3), and further lowering y will result in only decay reaction and, hence, yield of B decreases. Figure 6 shows the effect of reversibility of autocatalytic reactions on the yield of B under periodic operation. Increasing backward reaction rate constant group (R3) lowers the higher yield obtained under periodic operation. It is known that under finite axial dispersion or complete back mixing and at steady-state isothermal conditions, autocatalytic reactions may exhibit oscillations, isolas, and other multiplicity features [21-24]. Cordonier et al. analyzed the forced oscillations of such systems [25].

Concentration Forcing of Isothermal Plug-Flow Reactors

1.0]

531

^2

*"""*"

oio

\ \ ^O^^^^'-^^-O. \

0.8

___33--;;;x^V\\

- ----o:i

to 0.6 o

WW

III 1 III \

2

Qt

>

\\\ \ \\\ \ \\\ \ \\ \ \ \\\ \ \\ \ \ \\\ \

0.4 -

Oi

en o w 0;

< 0.2 r 0

^1=50

v \

1

\

1

0.2

OA

0.6

1

V ^

0.8

^

^

1.0

Figure 5. Effect of decay on the average yield of B by periodic oepration for cubic form, (a^ = 0.9, a^ = 0.1) [17]

MODELLING CUBIC AUTOCATALYTIC REACTIONS Model Equations Since cubic autocatalytic reaction requires reaction between three molecules, modelling autocatalytic reactions by cubic form often is criticized. Aris et ai have considered two different models of autocatalytic reactions [26]. The first model considers only two variables, but requires a third-order kinetic model: A + 2B ^

3B,

rate = -k.ab^

(17)

and the second model considers a 3-variables scheme with consecutive secondorder reactions: A + B^

X + B->3B

rate = ~ k , a b -I- k_,x

rate

-k.xb

(18a)

(18b)

532

Advances in Engineering Fluid Mechanics 1.01

R3 = 0.0 0.8

\

R3= 1.0 R3= 2 . 0

\

\

CD 0 . 6

\\ \ \ \\ \\\

H-

o

R3 = 4 . 0

2 "S

"^w

> O.A 01

8* b

a>

\ \ \

\

Vu

\ \ W\ V i x Xm.

< 0.2 h

0

R^=5.0

1 0.2

^JSv

1 0.4

1

0.6

1 0.8

1 1.0

Figure 6. Effect of reversibility of reaction on the yield of B for cubic form, (a^ = 0.9, a^ = 0.1) [17] Aris et al. have primarily analyzed whether the steady-state multiplicity features in a CSTR arising from a cubic rate law also can arise for a series of successive bimolecular reactions [26]. Aris et al. have showed that the steady-state equations for a CSTR with bimolecular reactions scheme reduces to that with a cubic reaction scheme when two parameters e(=k3C/k 2) and K(=k2C/k 2) arising in system equations for the bimolecular reactions tend to zero. Aris et al. have shown that the general multiplicity feature of the CSTR for bimolecular reactions is similar to that of the molecular reactions only at smaller value of e and K. The behavior is considerably different at larger values of e and K. Chidambaram has evaluated the effect of these two parameters (e and K) on the periodic operation of an isothermal plug flow reactor [18]. The steady-state model equations for an isothermal plug flow reactor with the three variable reactions system are given by (l/q)da/dZ = -(l/e)[ab - (x/K)]

(19)

(l/q)db/dZ = -(l/e)[ab - (x/K)] + (2bx/K)

(20)

(l/q)dx/dZ = (l/e)[ab - (x/K)] - (bx/K)

(21)

at Z = 0, a = a^ b = b^ and x = 0

(22)

Concentration Forcing of Isothermal Plug-Flow Reactors

533

In a similar manner to that shown by Aris et ah for CSTR, Equations 19 to 21 can be written as: (l/q)da/dZ = -ab^ - Kab^Ca-b) + eab^ + . . .

(23)

(l/q)db/dZ = ab^ - Kab^Ca-b) - eab^ + . . .

(24)

at Z = 0, a = a,, b = b^

(25)

These equations clearly reduce to the original model equations for a cubic rate form in the limit £ -^ 0 and K —> 0. Performance Under Periodic Operation The average yield of B of the three variables scheme (Equations 19 to 21) and that of the two variables scheme (e = 0, K = 0 in Equations 19 to 21) are evaluated under feed concentration cycling. The results are shown in Figure 7. The two variables model gives a 100% yield under periodic operation. For smaller values of E and K, the 3-variables model also gives a 100% yield, but only at lower values of y. For larger values of £ and K, the three variables scheme gives a lower average yield. For a given value of y, the two variables scheme gives a higher yield. Figure 7 shows that the average yield is more sensitive to K. AUTOCATALYTIC ACTIONS THROUGH RATE CONSTANT Now let us consider the periodic operation of an isothermal tubular reactor with autocatalytic reactions given by Equation 1. It is assume here [16] that the component A, has an activating influence on the rate constant Kj, represented as K, = K,, exp(Pa,)

(26)

The parameter p measures an activating influence of intermediate Aj on the rate constant for the first step. This law has been followed by diverse chemical reactions [27]. Some biological reactions catalyzed by allosteric enzymes also give such reaction scheme [12]. The relevant differential equations are written for the isothermal plug flow reactor for carrying out the reaction given by Equation 1. The conventional steady-state operation (y = 1) for the operating conditions given in Figure 8 gives a yield of product A3, Y = 0.19 for the auto catalytic reaction with p = 10.0. When there is no autocatalytic action (i.e., for p = 0) the yield under conventional steady-state operation is 0.145. Under feed concentration forcing (similar to Figure 1) the auto catalytic reaction system gives a very much improved average yield as shown in Figure 8. For example, at y = 0.2 we get Y = 0.95. Whereas, for ordinary reactions (i.e., when p = 0) the reactor model equations are linear and, hence, the yield at any y is the same as at y = 1. The effect of p on the average yield of B under periodic operation is shown in Figure 9. For a given value of y, a higher value of average yield is obtained for a larger values of p. The value of y, at which a steep increase in yield is obtained, decreases with decrease in p. As y decreases, during the first reaction of a period, the concentration

534

Advances in Engineering Fluid Mechanics

1.0

0.8

0.6

[ \ -

O

% > O.A Of

o > 0.2 <

3^2

A\

•

V

0 1

1 0-2

\

' " • ^ - -

O.A

\

1 _

0.6

0.8

to

Fractional pulse width Figure 7. Effect of e and K on the Average yield of B obtained under periodic operation. a^ = 0.9, ttg = 0.1, cyclic in both A and B curve No.

e

K

1 2 3 4 5

0.0 0.3 0.5 0.3 0.3

0.0 0.1 0.1 0.3 0.6

[18]

of feed reactant increases and, hence, the rate of formation of A, increases, which in turn increases the extent of reaction of A,. Hence, the yield of product increases during that fraction of period. The average yield of product over the entire period increases significantly. The increase in the average value of yield depends on the values of y and p. If we consider the intermediate species A2 influencing the reaction rate constant Kj rather than species A, influencing K,, then K, = K,,„ expCpa,)

(27)

Concentration Forcing of Isotliermal Plug-Flow Reactors

0.2

O.A

06

0.8

535

1.0

Fractional pulse width

Figure 8. Average yield of A3 obtained under periodic operation. D^ = 0.25, D2 = 8.0, D3 = 4.0 A, influences K, A2 influences K^ no autocatalysis (i.e., p = 0) [19]

0.2

o.A

0.6

0.8

1.0

Fractional pulse width

Figure 9. Effect of 3 on the average yield of A3 under periodic operation when A^ is Influencing K^ [19].

536

Advances in Engineering Fluid Mechanics

The average yield of product A3 under periodic operation is shown in Figure 8. The average yield increases only gradually with decrease in y, whereas the average yield vs. y is highly nonlinear when A, is influencing K,. CONCLUSION Periodic operation of isothermal plug flow reactors for autocatalytic reactions gives higher improvement in yield than that obtained for simple reactions. A resonance behavior is obtained with inlet pulse width for the irreversible reaction with decay. Three-variables successive bimolecular second-order kinetics model gives a similar trend in average yield as that for two-variables third-order kinetics model, only for smaller values of e and K. For larger values of these parameters, a three-variables scheme gives a lower average yield. Autocatalytic effect through the rate constant gives a higher average yield than that with the cubic or quadratic autocatalytic reactions. NOTATION A = reactant A , a^ = dimensionless concentration of A and B B = autocatalyst B c = concentration, mol/m^ C = reactant D E = product E of a simple reaction = c. dimensionless concentration of reactant i (i = A or B) in the feed Cp = dimensionless concentration of reactants A and D, respectively C C c. = dimensionless concentration of products B, C, and E, respectively Co = base value of concentration to which dimensionless is obtained, mol/m^ D, = K,QL/U, dimensionless D2 = K^L/U, dimensionless D, = K3L/U, dimensionless

k = K CJK, K., K. K, = reaction rate constants for A -> A,, A, -^ A^, and A —> A , (1/s)

K; = exp(pa,) = K, when p = 0 K = reaction rate constant, (mol/m^)-" (1/s) = K K2 forward and backward reaction rate constants (refer to Equation 18a), (mVmol)(l/s) and (1/s), respectively reaction rate constant for K = the reaction B + X -^ 3B, (mVmol)(l/s) K = backward reaction rate constant, (mVmol)(l/s) K = decay reaction rate constant, 1/s L = reactor length, m n = order of reaction with respect to B or D P = product due to decay of B q = (k,C2,L/U) = (k,k3/k ^) ^10

X C^^L/U

R. = reaction rate constant group, k,LC7U = R2 decay reaction rate constant group, k^L/U R3 = backward reaction rate contant, kj^LC^U t = time, s

Concentration Forcing of Isothermal Plug-Flow Reactors

U = velocity, m/s X = intermediate product jc = concentration of intermediate product X, (mol/m^)

537

Y = yield of B z = axial distant, m Z = normalized distant. z/L

Greek Letters a^,ag = dimensionless feed concentration of reactants A and B, respectively, under steady-state operation P = parameter defined in K, = K,, exp(|3a,)

e = k3C/k_,

9 = dimensionless time, Ut/L y = pulse width expressed as a fraction of period T = period, dimensionless

Subscripts f = feed i = A or B REFERENCES 1. Bailey, J. E., "Periodic Phenomena," in "Chemical Reactor Theory: A Review," L. Lapidus and N. R. Amundson (Eds.), Prentice-Hall, Englewod Cliffs, N.J. (1977) pp. 758-813. 2. Renken, A., "Unsteady-state Operation of Continuous Reactors," Int. J. Chem. Eng. 24, 202-213 (1984). 3. Silveston, P. L. and R. R. Hudgins, "Review of the Excitation of Chemical Reactors by Periodic Operation," in Recent Trends in Chemical Reaction Engineering, B. D. Kulkarni, R. A. Mashelkar and M. M. Sharma, Ed., Wiley Eastern Ltd., New Delhi, Vol. 1 (1987) pp. 235-253. 4. Yadav, R. and R. G. Rinker, "The Efficacy of Concentration Forcing," Chem. Eng. Sci. 44, 2,191-2,195 (1989). 5. Lannus, A. and L. S. Kershenbaum, "On the Cyclic Operation of Tubular Reactors," AIChE J. 16, 329-331 (1970). 6. Liden, G. and L. Vamling, "Periodic Operation of a Tubular Reactor: A Simulation Study of Consecutive Reactions in a Chromatography Reactor," Chem. Eng. J. 40, 31-37 (1989). 7. GrabmuUer, H., U. Hoffman and K. Schadlich. "Prediction of Conversion Improvement by Periodic Operation for the Plug-flow Reactors," Chem. Eng. Sci. 40, 951-960 (1985). 8. Hoffmann, U. and K. Schadlich, "Zur Frag Moglichen Unsatz-Verbesserungen beim Priodischen Betrich on Rohrreactoren," Chem. Eng. Tech. 57, 111-11A (1985). 9. Thullia, J., L. Chiao and R. G. Rinker, "Production Rate Improvement in Plugflow Reactors Under Concentration Forcing," Ind. Eng. Chem. Res. 26, 945947 (1987).

538

Advances in Engineering Fluid Mechanics

10. Sivakumar, S., M. Chidambaram, and H. S. Shankar, "Periodic Operation of Catalytic Transport Reactors," Can. J. Chem. Eng. 65, 767-772 (1987). 11. Sivakumar, S., M. Chidambaram and H. S. Shankar, "Analysis of Deactivation Disguised Kinetics in Transport Reactors by Periodic Operation," Can. J. Chem. Eng. 66, 505-508 (1988). 12. Franck, U. P., "Chemical Oscillation," Agnew. Chem. Int. Ed. Engl. 17, 1-15 (1978). 13. Maxted, E. B., "Catalysis and Applications," J. and A. Churchill, London (1933). 14. Frost, A. A. and R. G. Pearson, Kinetics and Mechanism, II Ed. p. 16 Wiley, New York (1961). 15. Froment, G. F. and K. B. Bischoff, "Chemical Reactor Analysis and Design," John Wiley and Sons, New York (1979). 16. Ravikumar, V., B. D. Kulkarni and L. K. Duraiswamy, "Multiplicity and Instability of States for Isothermal Homogeneous Reactions in CSTR: Case of Autocatalysis," AIChE J. 30, 649-653 (1984). 17. Chidambaram, M., "Periodic Operation of Isothermal Plug-flow Reactors for Autocatalytic Reactions," Chem. Eng. Commun. 69, 219-228 (1988). 18. Chidambarm, M., "Modelling Cubic Autocatalysis by Successive Bimolecular Steps: Implication on the Periodic Operation of Isothermal Tubular Reactors," Hung. J. Ind. Chem. 17, 329-335 (1989). 19. Chidambaram, M., "Concentration Forcing of Isothermal Plug-flow Reactors for Autocatalytic Reactions," Can. J. Chem. Eng. 71, 974-976 (1993). 20. Bailey, J. E., "On the Theory of Plug-flow Processes: Inlet Control and Distributed Trajectory Control," Can. J. Chem. Eng. 50, 108-118 (1972). 21. Lin, K. F., "Concentration Multiplicity and Stability for Autocatalytic Reaction in a Continuous Stirred Tank Reactor," Can. J. Chem. Eng. 57, 476-480 (1979). 22. Lin, K. F., "Multiplicity, Stability and Dynamics for Isothermal Autocatalytic Reaction in a CSTR," Chem. Eng. Sci. 36, 1,447-1,452 (1987). 23. Gray, P. and Scott, S. K., "Autocatalytic Reactions in the Isothermal, Continuous Stirred Tank Reactor. Isolas and Other Forms of Multiplicities," Chem. Eng. Sci. 38, 29-43 (1983). 24. Gray, P. and Scott, S. K., "Autocatalytic Reactions in the Isothermal, Continuous Stirred Tank Reactor. Oscillations and Instabilities in the System A + 2B -> 3B; B -> C," Chem. Eng. Sci. 39, 1,087-1,097 (1984). 25. Cordonier, G. A., Schmidt, L. D. and R. Aris, "Forced Oscillations of Chemical Reactors with Multiple Steady-States," Chem. Eng. Sci. 45, 1,659-1,675 (1990). 26. Aris, R., P. Gray and S. K. Scott, "Modelling Cubic Autocatalysis by Successive Bimolecular Steps," Chem. Eng. Sci. 43, 207-211 (1988). 27. Kondratiev, V. N., "Chain Reactions," in Comprehensive Chemical Kinetics, C. H. Bamford and C. P. H. Tipper, Ed., Elsevier, Amsterdam (1969) Ch. 2.

CHAPTER 20 NONNEWTONIAN EFFECTS IN BUBBLE COLUMNS

R. P. Chhabra Department of Chemical Engineering Indian Institute of Technology India 208016 and U. K. Ghosh Department of Chemical Engineering Banaras Hindu University Varanasi, India 221005 and Y. Kawase Department of Applied Chemistry Faculty of Engineering Toyo University Kujirai, Kawagoe-Shi, Saitama 350 Japan and S. N. Upadhyay Department of Chemical Engineering Banaras Hindu University Varanasi, India 221005 CONTENTS SYNOPSIS, 540 INTRODUCTION, 540 VISUAL OBSERVATIONS, 542 APPARENT VISCOSITY, 543 CIRCULATION MODEL, 545 LIQUID CIRCULATION VELOCITY, 546 MIXING TIME, 551 539

540

Advances in Engineering Fluid Mechanics

CIRCULATION TIME, 552 AXIAL DISPERSION COEFFICIENT, 553 GAS HOLDUP, 554 HEAT AND MASS TRANSFER, 556 Heat Transfer, 557 Gas-Liquid Mass Transfer, 561 CONCLUSIONS, 566 NOTATIONS, 567 REFERENCES, 567 SYNOPSIS This review deals mainly with the discussion of various macroscopic hydrodynamic, heat, and mass transfer characteristics of bubble columns, with occasional reference to the analogous processes in modified versions of bubble columns with a variety of internals. The hydrodynamic considerations include determination of parameters like flow patterns, holdup, mixing, liquid circulation velocities, axial dispersion coefficient, etc., which all exert strong influence on the resulting rates of heat and mass transfer and chemical reactions carried out in bubble columns. Different correlations developed for estimating the aforementioned parameters are presented and discussed in this chapter. INTRODUCTION Bubble columns have been widely used in the chemical and biochemical industries, and a great deal of literature on the hydrodynamics in bubble columns is available [1-3]. However, the design and scale-up of bubble columns are still very difficult because the complicated hydrodynamics is not well-understood. Most of the early works were based on the use of empirical curve-fitting analyses. Unfortunately, they have provided little mechanistic insight into the hydrodynamics of bubble columns, and, therefore, they have not been very helpful to improve the design methods. It can be expected that instructive insight into the transport phenomena in bubble columns will be obtained through theoretical analyses. They are certainly helpful in advancing our understanding of transport processes in bubble columns and in establishing the design theories of bubble columns. Models for the hydrodynamic behavior of the liquid phase are clearly required for a successful design and scale-up of a bubble column. Therefore, this review is mainly written from the viewpoint of theoretical approaches despite the fact that many well-established theoretical equations are not yet available for the estimation of the hydrodynamics in bubble columns.

Non-Newtonian Effects in Bubble Columns

541

Non-Newtonian fluids are frequently encountered in the process industries, especially in biochemical and biotechnological applications. It has been known that the rheological properties of the liquids have profound effects on the performance of bubble columns. The non-Newtonian anomalies create uncertainties in the design, scale-up, and operation of bubble columns. Most analyses and correlations are based on the following two premises, which are not necessarily always applicable: a. The liquid mixing induced by the introduction of a gas in bubble columns is rather moderate, and the turbulence is considerably lower as compared to that in mechanically agitated tanks. b. Hence, the intensity of turbulence in bubble columns is not sufficiently high to justify the application of theories for isotropic turbulence, especially when liquids are highly viscous non-Newtonian. Many investigators have had preconceived ideas about the mixing in bubble columns. However, Zakrzewski et al. measured turbulence intensities in a bubble column with water and 1% aqueous methanol solution and suggested that the application of the isotropic turbulence theory to flow in bubble columns may be a reasonable approximation [4]. Furthermore, recently Okada et al. found that the power spectrums in the external-loop airlift column, which is a modification of a bubble column as described later, with water and non-Newtonian CMC (carboxymethyl cellulose) aqueous solutions are comparable to those of the Kolmogoroff spectrum law [5]. It may be concluded, therefore, that the mixing of non-Newtonian liquids is chaotic and may be turbulent. In other words, the mixing of non-Newtonian fluids in columns may be turbulent. Visual studies indicate that large bubbles formed by bubble coalescence in viscous non-Newtonian media rise with high velocities, and, as a result, liquid phase mixing is significantly enhanced. Consequently, the liquid mixing is considerably random and chaotic. Airlift columns which are modifications of a bubble column have been the subject of increasing interest in recent years [6]. Therefore, this review includes hydrodynamics in airlift columns besides those in bubble columns. The airlift columns are classified into external-loop and internal-loop airlifts according to the type of liquid recirculation (Figure 1). External-loop airlift columns have a separate conduit for the downcomer. The injection of air into the bottom of the vertical tube (the riser) induces liquid circulation up and down the other vertical tube (the downcomer). The riser and downcomer are connected by horizontal sections near the top and bottom. Internal-loop airlift columns contain the riser and separated/downcomer in the same column. This configuration is classified into concentric-tube and splitcolumn airlifts, in which the riser and the downcomer are separated from each other by a draft tube or a baffle, respectively. Although there have been a number of experimental studies on airlift columns, little generalized information is available on their hydrodynamics. A relatively well-defined liquid circulation flow has been observed in airlift columns as compared to bubble columns. However, it is still too complicated to be rigorously analyzed. Airlift column performance is significantly influenced by a variety of its configurations, particularly the designs of the connections between riser and downcomer.

542

Advances In Engineering Fluid Mechanics DRAUGHT TUBE DOWNCQHER

o o> O^o^O

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

Advances in Engineering Fluid Mechanics

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:

Non-Newtonian Effects in Bubble Columns

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

Advances in Engineering Fluid Meclianics

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:

Non-Newtonian Effects in Bubble Columns

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

Advances in Engineering Fluid Mechanics

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

Non-Newtonian Effects in Bubble Columns

'•"

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

Advances in Engineering Fluid Mechanics

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

gD^u,,)

(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-'^U^;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

Non-Newtonian Effects in Bubble Columns

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(gD^U^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,^UJ2)'«

(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)

Non-Newtonian Effects in Bubble Columns

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

Advances in Engineering Fluid Mechanics

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

r^app

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

Non-Newtonian Effects in Bubble Columns

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 <

Dl

%

-. 3

m

3 p. 3

(D

'-. D 3 a 9 k. a

-

5E e.

o V)

Non-Newtonian Effects in Bubble Columns

561

(text continued from page 557) where (3 = (4 - n)/6(n + 1); Fr = U ^ g D Pr* = Cpu /k; Re* = pU D/u '^ u = K(U /D)"-'; Nu = hD/k "app

"^

sg

^

'

This expression predicts increasing enhancements in Nusselt number as the flow behavior index decreases, and the correspondence between these predictions and the scant results of Nishikawa et aL was reported to be satisfactory [7]. Subsequently, this approach also has been extended to heat transfer in Bingham plastic media [54]. In the case of airlift reactors, the flow pattern may be similar to that in bubble columns or closer to that two-phase flow in pipes (when the internal circulation is good), in which case the use of suitable correlations developed for pipes may be justified [55]. Blakebrough et al. studied the heat transfer characteristics of systems with microorganisms in an external loop airlift reactor and reported an increase in the rate of heat transfer [56]. In an analytical study, Kawase and Kumagai [57] invoked the similarity between gas sparged pneumatic bioreactors and turbulent natural convection to develop a semi-theoretical framework for the prediction of Nusselt number in bubble columns and airlift reactors; the predictions were in fair agreement with the limited experimental results [7,58] for polymer solutions and particulate slurries. Gas-Liquid Mass Transfer It is fair to say that the mass (or volumetric) transfer coefficient, k^^a, is perhaps the most important parameter in the design and scaleup of bubble columns and airlift reactors. Consequently, considerable research efforts have been expended in developing suitable expressions for the prediction of this parameter (Table 1). As remarked earlier, it depends on the physical properties of the system and the kinematics of flow, for a given geometrical configuration. It is acknowledged that the mass transfer coefficient k^^ shows only weak dependence on the power input, whereas the specific interfacial area a (surface area of bubbles per unit volume) is a strong function of physical properties (viscosity, surface tension), the geometric design, and hydrodynamics. The specific interfacial area, a, is a lumped quantity and cannot be defined or measured precisely at a point. It is, therefore, customary to present the mass transfer results in terms of the volumetric mass transfer coefficient, k^a. A cursory inspection of Table 1 shows that in most instances polymer solutions (carboxymethyl cellulose, polyacrylamide, xanthan) have been used to mimic the non-Newtonian features of biological systems, encompassing wide ranges of shear thinning conditions (though viscoelastic effect have been studied only scantily) in bubble columns up to as large as 760mm in diameter. The experimental values of k^a are generally obtained either from the increase in oxygen liquid phase concentration after starting the aeration of an oxygen-free solution (dynamic method) or by fitting the axial dispersion model to the steady state axial concentration profile (oxygen concentration) in the liquid phase in a concurrently operated column (steady state method). Occasionally, the mass transfer

562

Advances in Engineering Fluid Mechanics

coefficients obtained with CO2 have been used to infer the corresponding values for oxygen via the assumption that k^^ aVo. Schumpe has shown that both the dynamic as well as steady state methods yield comparable values of k^^a; they have also discussed the potential sources of errors [59]. In most studies dealing with heat and mass transfer, it has been generally assumed that the thermo-physical properties, such as thermal conductivity, specific heat, molecular diffusivity of non-Newtonian polymer solutions, are the same as that for water, except for their non-Newtonian viscosity. Intuitively, one would expect the surface tension to be an important variable by way of influence on bubble dynamics and shape, but only a few investigators have controlled/measured/included it in their results. The available correlations can be broadly classified into two types: first, those which directly relate the volumetric mass transfer coefficient with the liquid viscosity and gas velocity. The works of Deckwer et al. [36], Godbole et al. [42] and Ballica and Ryu [60] illustrate the applicability of this approach. All of them have correlated their results in the following form: k,a = a,|Li«2ua2

(34)

where k^^a is in s ' , iii^ is the effective viscosity in Pa s, and U^ is the superficial velocity of gas in m/s. The values of a,, a^, a^ and the method of evaluation of X | ^ varies from one study to another, and representative values are summarized in Table 2. Evidently, the volumetric mass transfer coefficient shows a direct dependence on the gas velocity and inverse variation with the liquid phase viscosity. Clearly, such correlations are strongly dependent upon the specific arrangement and are not very convenient for scaleup and/or process design calculations in a new application. The use of dimensionless groups characterizes the second class of mass transfer correlations. For instance, Nakanoh and Yoshida [61] correlated their gas/liquid mass transfer results as: Sh = 0.09Sc'^2v^e»75Ga«^^'^Fr

(35)

where the Sherwood number, Sh Weber number. We Schmidt number, Sc Galileo number, Ga

= = = =

k^aDVD^ p^gDVa u /p, D, gD^p?/u^ o

r L r*app

and Froude number, Fr = U /VgD

Table 2 Values of a-f-aa in Equation 33 Investigator Deckwer et al. [36] Godbole et al [42] Ballica and Ryu [60]

«i

1.37 X 10-^ 8.35 X 10-^ 1.558

-a2

0^3

0.84 1.01 0.4

0.59 0.44 0.30

Yeff (s-^)

5,000 U^g 5,000 u[' 1,000 VU sg

Non-Newtonian Effects in Bubble Columns

563

The appropriate apparent viscosity is estimated at the effective shear rate, Y^^^ = 5,000 U^ . Nakanoh and Yoshida found it necessary to introduce another correction in case of viscoelastic liquids [61]. The right hand side of Equation 35 must be divided by (1 + 0.13 De^^^) where De, Deborah number, is defined as ^V^/ d^^, with Vg the bubble swarm velocity and d^^ is the sauter mean diameters; X is the fluid relaxation time, which was arbitrarily defined as the reciprocal of the shear rate at which the apparent viscosity of the solution had dropped to (2/3) of its zero shear viscosity. Schumpe et al. [9] have reported somewhat different values of the numerical constants in Equation 34. Their equation is: Sh = 0.021Sc»/2^e0.2iQa0.6op^o.5

^^6)

It was stated to be applicable in the following ranges of conditions: 0.14 m < D < 0.35m; 0.008 < U < 0.285 m/s; 0.28 < n < 1 and 0.001 < K < 1.22 Pa«s". However, '

sg

'

'

more recent results [60,62] with a range of polymer solutions resulted in slightly different values of the numerical constants in Equation 36, including a correction for viscoelastic fluids, as follows: Sh = 0.018Sc'^2We«.20Qa0.62p^o.5i(i + o.l2De)-' (37) Where the apparent viscosity is now evaluated at y^^^ = 2800U^ [64] and the Deborah number, De, is defined simply as the ratio of the first normal stress difference to the corresponding shear stress at y^^^. Note that Equation 37 suggests much stronger viscoelastic effects than Equation 35. Aside from these expressions, some other correlations also have been proposed in the literature [63]. For instance, Henzler [64] has correlated the experimental results due to Buchholtz et al. [65,66] as:

^M'^ u.. Where u •^app

xl/3

Sf =0.06

PLU^

(38)

gP^

is evaluated at y,, = 1500U . ' erf

sg

More recently, Kawase and co-workers have presented semi-analytical framework for the estimation of the volumetric mass transfer coefficient in bubble column reactors with and without an internal draft tube [51,62,67]. They have essentially combined the Higbie's penetration model with the isotropic turbulence to deduce the following expression for gas-liquid mass transfer with power law liquids: S h = 7Cjn^^35;c0.5Re(2+n)/2(l+n)pj.(n+4)/39(n-.l)^e0.6

(39)

where Cj is an unknown constant to be determined by experiments, and the apparent viscosity is evaluated at y^^^ = U^ /D in this case. Kawase et al. collated the bulk of the literature data on gas-liquid mass transfer in power law fluids and estimated the best value of Cj for each liquid, i.e., different value of n [51]. The resulting values of Cj ranged from 0.016 8 to ~ 0.081 for the range 0.48 < n < 1. The average errors in the prediction of k^^a ranged from 5% to 57%.

564

Advances in Engineering Fluid Mechanics

In an interesting study, Merchuk and Ben-Zvi [68] argued that the variables influencing the value of k^^a in a given situation can be divided into three types: 1. static properties of the liquid phase such as p^^, D^^, a; these do not vary drastically from one liquid to another; 2. dynamic properties of the liquid such as the rheological characteristics which depend upon the kinematics of flow and geometrical configuration; and 3. the kinematics of flow itself. The variables of groups 2 and 3 vary over a wide range. The effective shear rate prevailing in the system also is governed largely by the same variables and, hence, k^^a must correlate directly with the so-called global shear rate which is defined as:

S^K^

J

(^^>

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: kj^a = 1.4 X 10-^y^^

(42)

Figure 7 shows a comparison between various predictions of k^a. 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^^a must be calculated using as many different methods as possible to establish "upper" and "lower" bounds on the value of k^a 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: k^a = 0.75 1 +

^

^0.8

(43)

Non-Newtonian Effects in Bubble Columns

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 k^a 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

Advances in Engineering Fluid Mechanics

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].

Non-Newtonian Effects in Bubble Columns

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

Advances in Engineering Fluid Mechanics

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).

Non-Newtonian Effects in Bubble Columns

569

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).

570

Advances in Engineering Fluid Mechanics

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.C3H^OH MgCl2.2C3HpH MgCl2.3C3HpH MgCl2.4C3HpH MgGl2.5C3HpH MgCl2.i-C3HpH MgCl2.2 i-C3H,0H MgCl2.3 i-C3H,OH MgCl2.4 i-C^UpU MgCl^.C^H^H MgCl2.2qH^OH MgCl^^C^HpH MgCl2.4C,HpH MgCl2.5C4H^OH MgCl^.i-C^H^OH MgCl2.2 i-C^Upn MgCl2.3 i-C^HflU MgCl2.4 i - q H ^ H MgCl^.t-C^H^OH 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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

R=C,H3, n-C3H,, n-qH,, i-C3H,, i-C,H,, n-C3H,,, n-C,H,3, t-C,H,, and C^H^^(C^U^)] 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(^i-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

Advances in Engineering Fluid Mechanics

c o -^o 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

Advances in Engineering Fluid Mechanics

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).

This Page Intentionally Left Blank

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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

Plasticizing Polyesters of Diner Acid and 1,4-Butanediol

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

Advances in Engineering Fluid Mechanics

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-

Plasticizing Polyesters of Diner Acid and 1,4-Butanediol

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

Advances in Engineering Fluid IVIechanics

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.

Plasticizing Polyesters of Diner Acid and 1,4-Butanediol

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

Advances in Engineering Fluid Mechanics

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.

Plasticizing Polyesters of Diner Acid and 1,4-Butanediol

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.

REFERENCES 1. A. K. Doolittle, The Technology of Solvents and Plasticizers: Plasticizers and Plasticization, John Wiley & Sons, Inc., New York, 1954. 2. C. F. Ruber and J. D. Farr, Modern Plastics, 41, 91 (1964). 3. K. Zohrer and A. Merrz, Kunststojf, 42, 102 (1957). 4. E. Ceausescu, I. Florescu, N. Barbu and B. Wolf, "Synthesis and Characterization of Macromolecular Compounds: Poly condensate Synthesis Processes: Plasticizing polyestes," Editura Academiei Republici Socialiste Romania, Bucharest, 153-166 (1974).

594

Advances in Engineering Fluid Mechanics

5. H. Kikawa and A. Ooshida, Jpn, Kokai Tokkyo Koho, JP 05117499 A2 930514 (1993). 6. J. Goletto, Ger. Offen, DE 2846596 790503 (1979). 7. R. F. Paschke, L. E. Peterson, S. A. Harrison and D. H. Wheelar, J. Am. Oil Chemists' Soc. 41, 723-727 (1964). 8. A. P. Grekov, S. A. Sukuhorukova and K. A. Kornev, Plasticheski Massy, 8, 50 (1954). 9. D. H. McMohan and E. P. Crowell, J. Am. Oil Chem. Soc, 51, 522 (1974). 10. J. C. Cowan, J. Am. Oil Chemists' Soc, 3 1 , 529-535 (1954). 11. R. W. Johnson and E. Fritz, eds. Fatty Acids in Industry, Marcel-Dekker, Inc. New York, 1989. 12. E. C. Leonard, Encyclopedia of Chemical Technology, 3rd Ed., Vol. 7, John Wiley & Sons, U.S.A. Dimer Acids, 768-782 (1975). 13. T. E. Breuer, Encyclopedia of Chemical Technology, 4th Ed., Vol. 8, John Wiley & Sons, U.S.A., Dimer Acids, 222-237 (1993). 14. T. Hashimoto and O. Suzuki, Nikkakyo Geppo, 30 (3), 15 (1977). 15. L. M. Loeb, ''Dimer Acides," Chem. Abstr. 87, 70128, 70129 (1975). 16. A. L. Fury, Chem. Abstr. 87, 8665 lu, 86652 v (1975). 17. J. C. Cowan, F. B. Falkenberg. H. M. Teeter and P. S. Skell, U. S. Pat. 2,450,940 (1948). 18. J. C. Cowan, A. J. Lewis and L. B. Falkenberg, Oil and Soap, 21, 101 (1944); U.S. Pat. 2 ,630,397 (1944). 19. J. C. Cowan and D. H. Wheelar, J. Am. Chem. Soc. 66, 84-88 (1994), J. C. Cowan, W. C. Ault and H. M. Teeter, Ind. Eng. Chem., 38, 1,138 (1946). 20. Brit. Pat 991, 514 (1965); Berlg. Pat. 659,059 (1965); Berlg. Pat. 671,086 (1966); Brit Pat. 1,024,535 (1966). 21. Berlg. Pat. 681,635 (1966); Can. Pat. 752,931 (1967); D. Peerman, W. Tolberg and H. Wittcoff, J. Am. Chem. Soc. 76, 6,085 (1954). 22. M. Drawert and H. Krase, Eur. Pat. Appl., EP 508054 A2 921014 (1992). 23. A. G. Schering, Neth. Appl. 7607,962 (1977). 24. Y. Hayashida, K. Miramtsu, M. Yasuki, A. Junichi and A. Takashu, Jpn. Kokai Tokkyo Koho, JP 02,04, 838 (1988). 25. Y. Nakamura, and H. Fujimura, Jpn. Kokai Tokkyo Koho, JP 04299188 A2 921022 (1992). 26. H. Ito, S. Maruyama and M. Teramura, Jpn. Kokai Tokkyo Koho, JP 05009378 AZ 930119 (1993). 27. K. Wongkamolsesh, Eur. Pat. Appl. EP 92 810598, 920805 (1993). 28. K. Yobuta, H. Nojiri, T. Hayashi and K. Nagano, Jpn. Kokai Tokkyo Koho, JP 050295081 A2 931109 (1993). 29. K. Yobuta and H. Nojiri, Jpn. Kokai Tokkyo Koho 05051447 A2 930302 (1993). 30. M. Hosoda, F. Murayama, E. Kasiwagi and K. Isayama, Ger. Offen, De 2462453 770303 (1977). 31. K. Koji, Eur. Pat. Appl. EP 545354 A1930609 (1993). 32. K. Ooba, Jpn. Kokai Tokkyo Koho, JP 05239405 A2 930917 (1993). 33. S. Asai and T. Ogiwara, Jpn. Kokai Tokkyo Koho, JP 05039440 A2 930219 (1993). 34. T. Yabu, Jpn. Kokai Tokkyo Koho, JP 04161471 A2 920604 (1992).

Plasticizing Polyesters of Diner Acid and 1,4-Butanediol

595

35. J. Mleziva, J. Myl, J. Toms, P. Kopriva and Svetozar Doubek, Czech 155625 (1974). 36. B. Koehlar, W. Ebert and H. Hugl, Ger. OJfen, DE 4132080 Al 93040 (1993). 37. M. Bolze and M. Drawert, Appl. 3,909,0518 (1989). 38. A. Blaga, H. Stolzenbach and P. Klostermann, Ger. Offen DE 4039030 A l 910613 (1991). 39. Y. Uramatsu and N. Sawada, Jpn. Kokai Tokkyo Koho, JP 04328174 A2 9211117 (1992). 40. Chi K. Sham, W. T. M. Jansen and M. A. Doherty, Eur. Pat. Appl. EP 481556 Al 920422 (1992). 41. D. D. Donermayer, J. G. Martins and D. A. Fabel, Eur. Pat Appl. EP 1184808 Al 840912 (1984). 42. T. Yamamoto, T. Hitai, M. Oyama, H. Uda and T. Taguchi Jpn. Kokai Tokkyo Koho, JP 0423092 A2 920819 (1992). 43. W. K. Bornack, Jr., R. C. Williams and K. R. McNally, Eur. Pat. Appl. EP 326647 A2 890809 (1989). 44. M. Guagliardo U. S. 4423179 A 831227 (1983). 45. K. Huemke, D. Paul and G. Hoffman, Eur. Pat EP 543357 A l 930526 (1993). 46. H. Haishi, M. Yamamoto, K. Kamikado and T. Enamoto Jpn. Kokai Tokkyo Koho, JP 05239385 A2 930917 (1993). 47. Dainippon Ink and Chemicals, Inc., Jpn. Kokai Tokkyo Koho, JP 58029814 A2 830222 (1983). 48. M. R. Edwards. J. A. Waller and S. Byrne, Eur. Appl EP 539030 A l 930428 (1993). 49. T. Sasaki, I. Ishigaki, K. Shiraishi, H. Yatsugama and S. Takeda, Jpn. Kokai Tokkyo Koho, JP 05043636 A2 930223 (1993). 50. C. W. Fowler, M. C. Knight and A. J. Nicholas, Eur. Pat. Appl. (1990): CA 114 (10): 840155. 51. H. D. Hille and M. Mueller, Ger. Offen. D. E. 4001841 A l 910725 (1991). 52. N. Murata, K. Kusakawa, T. Maruyama and T. Kimura, Jpn. Kokai Tokkyo Koho, JP 54160495 791219 (1979). 53. G. Arimatsu, Y. Ido, Y. Watnabe, S. Chiba and H. Suzuki, Jpn. Kokai Tokkyo Koho, JP 05125617 A2 930521 (1993). 54. G. Vinches and L. Piquet, Fr. Demande FR 2560202 A l 850830 (1985). 55. D. W. Young and E. Lieber U.S. Pat. 2,411,178 (1946); W. J. Sparks and D. W. Young, U. S. Pat. 2,424,588 (1947) and U.S. Pat. 2,435,619 (1948). 56. M. Rosenberg, U. S. Pat. 3,492,232 (1970). 57. R. J. Sturwold and F. O. Barrette, U. S. Pat, 3,769, 215 (1973). 58. J. Goto, Y. Miki and Y. Koya, Jpn. Kokai Tokkyo Koho, JP 05230195 A2 930907 (1993). 59. J. C. Cowan, D. H. Wheelar, H. M. Teeter, R. E. Paschke, C. R. Scholfield, A. W. Schwab, J. E. Jackson, W. C. Bull, F. R. Earle, R. J. Foster, W. C. Bond, R. E. Beal, P. S. Skell, I. A. Wolff and C. Mehltrett, Ind. and Eng. Chem. 41(8), 1,947 (1949). 60. J. C. Cowan and D. H. Wheelar, /. Am. Chem. Soc. 66, 84-88 (1944), J. C. Cowan, W. C. Ault and H. M. Teeter, Ind. Eng. Chem. 38, 1,138 (1946). 61. G. K. Hoeschele, Ger. Offen. 2,458,472 (1975).

596

Advances in Engineering Fluid Mechanics

62. D. E. Peerman, H. G. Kanten and Roger A. Lovald, U. S. 4582895 A 860415 (1986). 63. H. Tomita, T. Pponma and Y. Kishida, Jpn. Kokai Tokkyo Koho, JP 05171015 A2 930709 (1993), JP 05117512 A2 930514 (1993), JP 05179128 A2 93072c (1993) and JP 05051520 A2 930302 (1993). 64. H. Tomita, T. Pponma, and T. Oonishi, Jpn. Kokai Tokkyo Koho. JP 5214218 A2 930824 (1993); JP 5214219 A2 930824 (1993) and JP 5214220 A2 930824 (1993). 65. H. J. Manuel and R. J. Caymans, Polymer, 34(3), 636 (1993)/ 66. B. J. Sublett and S. D. Hilbert, U.S. 4439598 A 840327 (1984). 67. J. J. Charles and R. B. Casman, Ger, Offen. D. E. 2904184 790720 (1979). 68. B. Davis and F. A. Shepherd, U.S. 4216129 800805 (1980). 69. J. A. Tyrell and S. J. Willey U.S. 5262493 A 931116 (1993). 70. A. Mekjian, U.S. 4535146 A 850813 (1985). 71. S. Ishikawa and T. Nagasawa, Jpn. Kokai Tokkyo Koho, JP 2279712 A2 901115 (1990). 72. W. Imoehl and M. Drawert, Ger. Offen., 2,361,486 (1975). 73. J. W. Jackson Jr., and R. W. Darnell, US 3931073 (1976). 74. Eastman Kodak Co., Neth. Appl. NL 7510817, 770317 (1977). 75. H. Hirakochi, M. Nakamura, and T. Hachiksuka, Jpn. Kokai Tokkyo Koho, JP 04328186 A2 921117 (1992). 76. Y. Miki, S. Takamo, J. Goto and T. Oota, Jpn. Kokai Tokkyo Koho, JP 05230191 A2 930907 (1993). 77. M. Taniguchi and H. Yamada, Jpn. Kokai Tokkyo Koho, JP 05241339 A2 930921 (1993). 78. T. Yamanashi, Y. Hashiguchi and M. Hamada, Jpn. Kokai Tokkyo Koho, JP 04370285 A2 921222 (1992). 79. M. Lorenz and A. Hous, Ger. Offen., D. E. 3831979 Al 900329 (1990). 80. A. N. Dunlop and Ch. G. Rickard, Belg. B. E. 840549 761011 (1976). 81. Y. Murafuji, M. Yamamoto, T. Makino and T. Kunimara, Jpn. Kokai Tokkyo Koho, JP 05170944 A2 930709 (1993) and JP 05208447 A2 930820 (1993). 82. T. Kurome, K. Tsunashima and T. Hiraoka, Jpn. Kokai Tokkyo Koho, JP 04293985 A2 921019 (1992), JP 04293935 A2 921019 (1993). 83. T. Tsunashima and T. Kurome, Jpn. Kokai Tokkyo Koho, JP 05131601 A2 930528 (1993). 84. T. Hiraoka, K. Tsunashima and K. Furukawa, Jpn. Kokai Tokkyo Koho, JP 04216830 A2 920806 (1992), JP 04180761 A2 920626 (1992) and 04298589 A2 921022 (1992). 85. T. Hiraoka, T. Kurome and K. Tsunashima, Jpn. Kokai Tokkyo Koho, JP 04298589 A2 921022 (1992); T. Hiraoka, K. Tsunashima and F. Furukawa, Jpn. Kokai Tokkyo Koho, JP 04202439 A2 920723 (1993). 86. S. Nagura, K. Tsunashima and T. Kurome JP 05163685 A2 930629 (1993). 87. A. J. O'Lenick Jr., U. S. 5210133 A 930511 (1993). 88. H. Kikawa and A. Ooishida, Jpn. Kokai Tokkyo Koho, JP 05117498 A2 930514 (1993). 89. H. Kikawa and A. Ooishida, Jpn. Kokai Tokkyo Koho, JP 05117499 A2 930514 (1993).

Plasticizing Polyesters of Diner Acid and 1,4-Butanediol

597

90. U. D. N. Bajpai, S. Hingorani, A. Jain, H. P. Tiwari and S. K. Nema, Indian J. Chem., 27A(7), 635 (1988). 91. U. D. N. Bajpai, Kavita Singh and Nivedita, J. Appl. Polym. ScL, 46, 1,485 (1992). 92. U. D. N. Bajpai and Nivedita, J. Appl. Polym. ScL, 50, 693 (1993). 93. U. D. N. Bajpai, and Nivedita, Polymer Science, Recent Advances, Polymer 94, Conference held at Vadodara, Organized by IPCL, India, Feb. 1994, Allied Publishing Limited, Vol I, 289-293 (1994). 94. U. D. N. Bajpai, and Nivedita, J. Macromol. Sci.-Pure and Appl. Chem. A32(2), 331 (1995). 95. U. D. N. Bajpai, D. D. Mishra and Anjali Bajpai, J. Macromol. Sci.-Chem. A19(8), 823 (1983); (ii) Collection Czech. Chem., Commun. 48, 3,329 (1983) and (iii) Ind. J. Chem., 24A, 1027 (1985). 96. U. D. N. Bajpai, Anjali Bajpai, and Milind Khandwe, Macromole cules 24, 5,203 (1991); (ii) J. Inorg. and Coord. Polymers, 1, 417 (1991); (iii) Polymer Bulletin 23, 51 (1990); (iv) U. D. N. Bajpai, Anjaili Bajpai and Sandeep Rai, J. Appl. Polym. ScL, 48, 1,241 (1993); (v) Polymer International 31(3), 215 (1993). 97. U. D. N. Bajpai, Nivedita and Anjali Bajpai, Proc. International Symposium on Macromolecules 95. Current Trends, Jan 11-13, 1995, Vikram Sarabhai Space Centre, Thiruananthapuram, India, Allied Publishers Limited, New Delhi, p. 244-250. 98. H. Matsuda, J. Polym ScL, Polym. Chem. Ed. 12, 455 (1974). 99. R. E. Wilfong, J. Polym. ScL, 54, 385 (1961). 100. F. Korte and W. Glot, J. Polym ScL B4, 685 (1966). 101. I. Vansco-Szmercsanyi and E. Makay-Bodi, Eur. Polym. J. 5, 145,155 (1969). 102. A Fradet and E. Marechal, Adv. Polym. ScL, 43, 51 (1982). 103. G. Allen, J. G. Bevington and G. C. Eastmond, Eds., Comprehensive Polym. ScL, 5, 291 (1989). 104. P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York (1953). 105. H. Staudinger and H. Heuer, Ber Dtsch. Chem. Ges. B.63 222 (1930). 106. R. M. Fuoss, I. O. Salyer and H. S. Wilson, J. Polym. ScL A2 3,147 (1962).

This Page Intentionally Left Blank

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid IVIechanics

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

Advances in Engineering Fluid Mechanics

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

Viscoelastic Properties of Model Silicone Networks with Pendant Chains

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 - h^i) 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

Advances in Engineering Fluid Mechanics

[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

Advances in Engineering Fluid Mechanics

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

Viscoelastic Properties of Model Silicone Networks with Pendant Chains

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

Advances in Engineering Fluid Mechanics 1

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].

Viscoelastic Properties of Model Silicone Networks with Pendant Chains

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

Advances in Engineering Fluid Mechanics

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"^

Viscoelastic Properties of Model Silicone Networks with Pendant Chains

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.

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid IVIechanics

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

Advances in Engineering Fluid Mechanics

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

HOCHg

HOCHg

HOCHg

OH

Figure 3. Structure of xanthan gum.

/

OH

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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.

H

H

^'x

f

' 2

yy

=0

NHg

R^ -i

i'z

6=0

C=0

OH

OR'

R = hydrophobic group (ie Cg - CgQ) R2

= H or CHg

Figure 4. Structure of hydrophobically associating polyacrylamide.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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 lOV^ole [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.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

lOOOOr

1000

E > '35 o o

100

(0

10

Alcoflood 1175L Polymer Concentration, ppnn • 150 • 250 • 500 o 1000 A 5000 admkiMdLm^mhmmhm^hiiakM^^mm^^^mi^UiJi^^m^^md^kmkiJi^uLm^mkm

0.01

0.1

1

10

100

1000

Shear Rate, s Figure 5. Variation of the viscosity-shear rate relationship with polymer concentration in deionized water.

624

Advances in Engineering Fluid Mechanics '

UUUU 1

1000

o CD

u. c

1

'

1

'

[

1

'

1

'

1

'

J

^^^^"^

100

\

0) (D

^

^

^

k.

O CO

L

T = 23*'C Alcoflood 1175L

10 {

^i

'

1000

•

'

2000

•

•

3000

•

'

•

4000

'

1 1

•

1

5000

600C

Polymer Concentration, ppm Figure 6. Variation of the screen factor with polymer concentration in deionized water. emphazies the importance of considering the visco-elastic properties of HPAM near the wellbore where the shear rate is high. Effect of Sodium Chloride on the Viscosity of HPAM Figure 7 depicts the flow curves of a polymer solution containing 1,000 ppm polymer and various NaCl concentrations at 20°C. Unlike the flow curves obtained with polymer in deionized water, a Newtonian behavior was observed at low shear rates and NaCl concentrations > 0.2 wt%. At a given shear rate, the apparent viscosity diminished as NaCl concentration was increased, especially at low shear rates. The critical shear rate at which there is a transition from Newtonian to shear-thinning behavior increased with NaCl concentration. Figure 8 illustrates the influence of NaCl on the low-shear Newtonian viscosity at polymer concentration of 1,000 ppm. The polymer solution viscosity dropped as the salt concentration was increased up to 8 wt%, then it slightly increased. The initial drop in viscosity is due to charge screening effects. However, the slight increase in viscosity at sodium chloride concentrations > 8 wt% is due to the increase in the solvent viscosity because of the presence of salts. Figure 9 displays the effect of NaCl concentration on the low-shear relative viscosity at polymer concentrations of 500, 1,000 and 2,000 ppm, respectively. At all polymer concentrations, the low-shear relative viscosity sharply decreased with increasing NaCl concentration up to 1 wt%. The rate of change of the low-shear

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

625

10000 Alcoflood 1175L NaCI Concentration, w t % 0.0 0.01 0.05 0.2 1.4 8.2

1000 (0

a. E > "co

100^

o o

CO

lOir

A

A

A

A

A

A

A

A

A

A

A

A

^

A

^

^

T = 20C Polymer Concentration = 1000 ppm

0.1

0.01

1

1000

100

10

Shear Rate, s~^ Figure 7. Effect of sodium chloride on the viscosity-shear rate relationship. 1 0) (0 n

E ^ > +-

20

>

15

"w o o 0)

1 1

c a "E o

*^ 3 0)

1

IT 1 1

10

'

r 1 \ 1 I \\

'

^

^

1 1

H 1 1 J 1 1 1

1 \ \ \

i

\\ ^

^^*s*^ ^^^•^

1J 1 1

,

r

^^

•

1

Z

^ re Q> 1

5

L 1

o

^

n 1

T = 20°C Alcoflood 1175L Polymer Concentration = 1000 ppm •

•

•

•

•

,

J 1 ] .

1

•

1

12

NaCI Concentration, wt% Figure 8. Effect of sodium chloride on the low-shear Newtonian viscosity at polymer concentration of 1,000 ppm.

626

Advances in Engineering Fluid Mechanics

Polynner Concentration, ppm • a •

(0

o o

500 1000 2000

100

0)

-2

Q)

CO I

o

12

16

NaCI Concentration, w t % Figure 9. Effect of sodium chloride on the low-shear relative viscosity at various polymer concentrations.

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.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery 1.U

1

1

^

1

• J^

1

1

-

^-••-

n

6

0.8

'

_

1 1

/^•^

X 0)

627

•

t

V

1

^m

(0

0.6

—1

1

0

o

Polymer Concentration, ppm 1 • 500 H 1000 • 2000 4

OL 0.4

1>0.2 1

T == 2 0 °C 1

\

L.

\

\

1

1

1

Sodium Chloride Concentration, w t % Figure 10. Effect of sodium chloride on the power-law index at various polymer concentrations.

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

628

Advances in Engineering Fluid Mechanics 1

t 1

1 (0

T = 20°C

•

100

0)

^^^^

= 1 0 0 0 ppm Polymer Concentration =

1

o o

1

Alcoflood 1175L

1

J

^

•

NaCI CaClz

D

0)

I

hm

(D 0)

L

-^—•

r^

I

I

o

- ^ ^ 1 1 '•

u

'

"•"

•

—•

'

1

D

B

] 1

1

1

1

^^^^^^^^^^^^^^ 12

Salt Concentration, w t % Figure 11. Effect of cation type on the low-shear relative viscosity. 1

r

p

°

'

— — 1M

M H ^ M ^ ^

LI

•

1

X 0)

£

1

o

• D

o

1

Q.

L

NaCI CaCl2

Alcoflood 1175L

H

H

Polymer Concentration = 1000 ppm T = 20°C 0.2

1

1

1

1^mmmmm^m^^ 16

Salt Concentration, w t % Figure 12. Effect of cation type on the power-law index.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

630

Advances in Engineering Fluid Mechanics 60

I

'

I

'

I

•

I

Alcoflood 1175L Polynner Concentration = 1000 ppm (0

T = 20°C

o o

Time, h • 1 a 336 • 696

(0 0)

-2 CO

0)

20

CO I

o

Sodium Hydroxide Concentration, w t % Figure 13. Effect of sodium hydroxide and aging on the low-shear relative viscosity. water. The viscosity enhancement due to polymer hydrolysis depends on the type and concentration of the cation present. At low strong alkali concentrations, sodium ion concentration is relatively low and, as a result, viscosity enhancement due to polymer hydrolysis can be seen. At higher strong alkali concentrations, the concentration of sodium ion is very high. The presence of high sodium ion concentration compensates, through charge screening, for the effect of the hydrolysis on the shear viscosity and, consequently, the viscosity remains almost constant. The effect of sodium hydroxide also was examined in the presence of 1 and 5 wt% sodium chloride, respectively. Figure 14 depicts the low-shear relative viscosity as a function of sodium hydroxide concentration, with sodium chloride as a parameter. Viscosity measurements were conducted one hour after initial mixing for all polymer solutions. The most important aspect of Figure 14 is that the strong variation of the low-shear relative viscosity observed at 0 wt% sodium chloride almost diminished at 5 wt% sodium chloride. Obviously, the presence of high sodium ions concentration compensated, through charge shielding, for the increase in viscosity observed at low alkali concentrations. Due to their buffering action and low reactivity with the rock matrix, many researchers have suggested using buffered alkalis in alkali-polymer slugs. The effect of sodium carbonate on the flow curves of polymer solutions having 1,000 ppm polymer was examined by Nasr-El-Din et al. [41]. Figure 15 shows the influence of sodium carbonate concentration on the low-shear relative viscosity measured one

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery bU 1

— '

4-

40

'

\

'

\—

T = 20 °C Polymer Concentration = 1 0 0 0 ppm

•

o o

\

631

• • •

0)

.>

—J

0 1 5

-|

1 (Q 0)

-1

20 1

\

I

o

J^

^

v^

-1 ff 1

n 1

\

\

\

\

\

\

1

1

Sodium Hydroxide Concentration, w t % Figure 14. Variation of the low-shear relative viscosity with sodium hydroxide concentration at various sodiunn chloride concentrations. 80

Alcoflood 1175L Polymer Concentration = 1000 ppm (0

o o

T = 20 °C

0) 0

JS 0)

tc (D 0) JC

(O I

o

Sodium Carbonate Concentration, w t % Figure 15. Effect of sodium carbonate on the low-shear relative viscosity.

632

Advances in Engineering Fluid Mechanics

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

t>u

'

1

'

1

'

1

'

1

'

1

'

1

'

1

£ 50

01

O

o

40

c CD "E o

30

CO

/ 7

• a

0.5 wt% NaOH 1.0 wt% NajCOa

/

-J

1

1

20

(0 I

^

n g •

^ II n-

J

Alcoflood 1175L Polymer Concentration = 1000 ppm

10

o

J 1

T = 20°C n1

1 ]

,

\

,

\

,

\

1 ,

\ 400

,

1 500

,

\ 600

Time, h Figure 16. Effect of alkali type on the evolution of the low-shear Newtonian viscosity of 1,000 ppm polymer solution.

,

1 700

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

633

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

1 •MHMipHil^^^

uuuu 1

1000

o c

L

*•

(Q

U.

100

p

0)

o 10

Alcoflood 1175L Polymer Concentration, ppm • 500 n 1000 • 2000 o 5000 .11

J

Q

•

2

•

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

Advances in Engineering Fluid Mechanics

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

^^^^I^^^^^^Jj

30

IOC

Shear Rate, s'^ Figure 18. Effect of Triton X-100 on the viscosity-shear rate relationship.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid IVIechanics

1 0 0 0 0 0 ^^iii¥"^^"^^¥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^^^UiJt^m^^tm^l^mh^UiJbm^m^mitmdi^^ikMl^i^^^m^ 0.01 0.1 10

«

1 •a *4

1 1 M^^iiMh^uJ

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.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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
100

1000

Shear Rate, s' Figure 29. Effect of sodium hydroxide concentration on the flow curves of polymer solutions having 3,000 ppm Flocon 4800. rate. At shear rates < 1 s"', the apparent viscosity decreased significantly with sodium hydroxide concentration. At shear rates > 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

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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.

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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

^i^Wiip"i""iii^iiiiiii^i^"l*¥iW

m^mm^mtfmi^mfmmiif^^a^m^^m^mfmtiifm^'^'^^'^^^i^li^iiilimj^

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

Advances in Engineering Fluid Meclianics

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

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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].

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

(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].

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics 100

^—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'

°°«„ \ ^o^o^

=

-,Q2

U

^ o <. o o

' X

4000 3000

O

c

m

10 W

o o o o ^ ^ .^ ^oooooo^oo^po

2500

O ^ 6 S ^ ^ AAAAAAAAAAAAA^^^ 1 5 0 0 o o Oooooooooo^oo^^oo 1 0 0 0

' > • '*"!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].

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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:

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

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

Advances in Engineering Fluid Mechanics

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

REFERENCES 1. Clark, R. K. and J. J . Nahm, "Petroleum (Drilling Fluids)," in Kirk-Othmer Encyclopedia of Chemical Technology, 3rd Ed., John Wiley and Sons, New York, 1982. 2. Reid, P. I., "Polymers in Water Based Muds," Oil Gas —Europ. Mag., 18(1), 23-26 (1992). 3. Nouri, H. H. and P. J. Root, "A Study of Polymer Solution Rheology, Flow Behavior, and Oil Displacement Processes," SPE 3523, Presented at the 46th annual Mtg. of the Soc. of Pet. Eng., New Orleans, Oct. 3-6 (1971). 4. Mungan, N., "Shear Viscosity of Ionic Polyacrylamide Solutions," Soc. Pet. Eng. 12, 469-473 (1972). 5. Muller, G., J. P. Laine and J. C. Fenyo, "High Molecular-Weight Hydrolyzed Polyacrylamide. I.Characterization. Effect of Salts on the Conformational Properties," J. Polymer Science 17, 659-672 (1979). 6. Szabo, M. T., "An Evaluation of Water-Soluble Polymers for Secondary Oil Recovery—Part 1," JPT 3\, 553-560 (1979).

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

663

7. Ward, J. S. and F. D. Martin, "Prediction of Viscosity of Partially Hydrolyzed Polyacrylamide Solutions in the Presence of Calcium and Magnesium Ions," Soc. Pet. Eng. 7. 21, 623-631 (1981). 8. Martin, F. D., "Flow Characteristics and Injectivity Behavior of Water-Soluble Polymers," New Mexico Research and Development Institute, Report No. 2-74-4327 (1987). 9. Zaitoun, A. and B. Potie, "Limiting Conditions for the Use of Hydrolyzed Polyacrylamides in Brines Containing Divalent Ions," SPE 11785 presented at the International Symposium on Oilfield and Geothermal Chemistry held in Denver, CO, June 1-3 (1983). 10. Martin, F. D. and N. S. Sherwood, "The Effect of Hydrolysis of Polyacrylamide on Solution Viscosity, Polymer Retention and Flow Resistance Properties," SPE 5339, presented at the Rocky Mountain Regional Meeting held in Denver, CO, April 7-9 (1975). 11. Moens, J. and G. Smets, "Alkaline and Acid Hydrolysis of Polyvinylamides," y. Polymer Science 23, 931-948 (1957). 12. Smets, G. and A. M. Hesbain, "Hydrolysis of Polyacrylamide and Acrylic Acid-Aerylamide Copolymers," J. Polymer Science 11, 217-226 (1959). 13. Nagase, K. and K. Sakaguchi, "Alkaline Hydrolysis of Polyacrylamide," J. Polymer Science: Part A 3, 2,475-2,482 (1965). 14. Higuchi, M. and R. Senju, "Kinetic Aspects of the Alkaline Hydrolysis of Polyacrylamide," Polymer J. 3,370-377 (1972). 15. Halverson, F., J. F. Lancaster and M. N. O'Connor, "Sequence of Carbonyl Groups in Hydrolyzed Polyacrylamide," Macromolecules 18, 1,139-1,144 (1985). 16. Ryles, R. G., "Chemical Stability Limits of Water-Soluble Polymers Used in Oil Recovery Processes," SPE Res. Eng. 3, 23-34 (1988). 17. Kuehne, D. L., "Chemical Flooding with Improved Injectivity," US Patent number 4,852,652 (1989). 18. Jansson, P. E., L. Kenne and B. Lindberg, "Structure of the Extracellular Polysaccharide from Xanthomonas Campestris," Carbohydr. Res. 45, 275-282 (1975). 19. Rochefort, W. E. and S. Middleman, "Rheology of Xanthan Gum: Salt, Temperature, and Strain Effects in Oscillatory and Steady Shear Experiments" J. Rheol. 31, 337-369 (1987). 20. Holzwarth, G., "Polysaccharide from Xanthomonas Campestris: Rheology, Solution Conformation and Flow Through Small Pores," Symposium on Advances in Petroleum Recovery presented before the Division of Petroleum Chemistry, Inc., ACS New York meeting, April 4-9 (1976a). 21. Milas, M. and M. Rinaudo, "Conformational Investigation on the Bacterial Polysaccharide Xanthan," Carbohydr. Res. 76, 189-196 (1979). 22. Holzwarth, G. and J. Ogletree, "Pyruvate-free Xanthan," Carbohydr. Res. 76, 277-80 (1979). 23. Southwick, J. G., H. Lee, A. M. Jamieson and J. Blackwell, "Self-Association of Xanthan in Aqueous Solvent Systems," Carbohydr. Res. 84, 287-295 (1980). 24. Southwick, J. G., A. M. Jamieson and J. Blackwell, "Conformation of Xanthan Dissolved in Aqueous Urea and Sodium Chloride Solutions," Carbohydr. Res. 99, 117-127 (1982).

664

Advances in Engineering Fluid Mechanics

25. Ash, S. G., A. J. Clarke-Sturman, R. Calvert and T. M. Nisbet, "Chemical Stability of Biopolymer Solutions," SPE 12085, presented at the 58th Annual Technical Conference and Exhibition held in San Francisco, CA, October 5-8 (1983). 26. Seright, R. S. and B. J. Henrici, "Xanthan Stability at Elevated Tempeatures," SPE Res. Eng., 52-60 (Feb., 1990). 27. Holzwarth, G., "Conformation of the Extracellular Polysaccharide of Xanthomonas Campestris," Biochem. 15, 4,333-4,339 (1976b). 28. Jeanes, A., J. E. Pittsley and F. R. Senti, "Polysaccharide B-1459: A New Hydrocolloid Polyelectrolyte Produced from Glucose by Bacterial Fermentation," 7. Appl. Poly. Sci. V (17), 519-526 (1961). 29. Whitcomb, P. J. and C. W. Macosko, "Rheology of Xanthan Gum," J. Rheol. 22, 493-505 (1978). 30. Philips, J. C , J. W. Miller, W. C. Wernau, B. E. Tate and M. H. Auerbach, "A High-Pyruvate Xanthan for EOR," Soc. Pet. Eng. J., 594-602 (August, 1985). 31. Krumrine, P. H. and J. S. Falcone, Jr., "Surfactant, Polymer and Alkali Interactions in Chemical Flooding Processes," SPE 11778, presented at the International Symposium on Oilfield and Geothermal Chemistry held in Denver, CO, June 1-3 (1983). 32. Vollmert, B., Polymer Chemistry, Springer-Verlag New York Inc., New York, (1973). 33. Aspinall, G. O., "The Polysaccharides," Academic Press Inc., New York, NY, Vol. 1, 100-102 (1982). 34. McCormick, C. L. and C. B. Johnson, "Structurally Tailored Macromolecules for Mobility Control in Enhanced Oil Recovery," in Water-Soluble Polymers for Petroleum Recovery, Stahl, G. A., Schulz, D. N., Eds., Plenum Press, New York, 1988, pp. 161-i80. 35. Evani, S., "Enhanced Oil Recovery Process Using a Hydrophobic Associative Composition Containing a Hydrophilic/Hydrophobic Polymer," U.S. Pat. 4,814,096 (1989). 36. Evani, S. and K. van Phung, "Hydrophobic Associative Composition Containing a Polymer of a Water-Soluble Monomer and an Amphiphilic Monomer, International Patent WO 85/03510 (1985). 37. Evani, S., "Water-Dispersible Hydrophobic Thickening Agent," European Patent Application 0 057 875 A2 (1982). 38. Baird, J. K., "Gums," in Kirk-Othmer Encyclopedia of Chemical Technology, 4th Ed., John Wiley and Sons, 1991. 39. Majewicz, T. G. and T. J. Podlas, "Cellulose Ethers," in Kirk-Othmer Encyclopedia of Chemical Technology, 4th Ed., John Wiley and Sons, 1991. 40. Chatterji, J. and J. K. Borchardt, "Applications of Water-Soluble Polymers in the Oil Field," JPT 33, 2,042-2,056 (1981). 41. Nasr-El-Din, H., B. F. Hawkins and K. A. Green, "Viscosity Behaviour of Alkaline, Surfactant and Polyacrylamide Solutions Used for Enhanced Oil Recovery," SPE 21028, presented at the SPE Int. Symposium on Oilfield Chemistry, Anaheim, California, USA, Feb. 20-22 (1991). 42. Foshee, W. C , R. R. Jennings and T. J. West, "Preparation and Testing of Partially Hydrolyzed Polyacrylamide Solutions," SPE 6206 presented at the 51st Annual Fall Tech. Conf. and Exhibition, New Orleans, Oct. 3-6 (1976).

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

665

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.

Advances in Engineering Fluid Mechanics

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).

Rheology of Water-Soluble Polymers Used for Improved Oil Recovery

667

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).

668

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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

Relation of Rheological Properties of UV Cured Films

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

Advances in Engineering Fluid Mechanics

TMPTA

75

111 55 Z

o <

NVP

I 3 Q Z LU

35

TPGDA

a.

15

0

8

10

CONCENTRATION OF PHOTOINITIATOR Figure 3. Pendulum hardness is shown as a function of concentration of the photoinitiator for the films of oligomer/(monomer+photoinitiator) = 60/40, w/w and cured with six passes.

Relation of Rheological Properties of UV Cured Films

675

80 TMPTA (0

Seo z a <

s 3 O

z

LJ

20

a.

77

84

91

98

105

112

TgX Figure 4. Pendulum hardness is shown against glass transition temperature (Tg) of co-polymer (cured films) for the system oligomer/(monomer + photoinitiator) = 60/40. w/w. fully cured films were considered for each case. Similar to the values in Figure 3, TMPTA has registered the highest pendulum hardness, followed by NVP and TPGDA systems. The maximum pendulum hardness values have been obtained at 35% monomer diluents. This is in conformity with the results obtained in Figure 2 showing that 35% reactive diluent produces the highest pendulum hardness. Thus, pendulum values obtained against Tg of copolymer (cured film) calculated on the basis of Fox equation can produce conformatory results like in other conventional systems. This can be further demonstrated by plotting pendulum hardness against Tg copolymer obtained for the UV-cured films of different reactive diluents (Figure 5). This shows that TMPTA yields the highest pendulum hardness values. The trend for pendulum hardness values is like TMPTA>NVP>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

Advances in Engineering Fluid Mechanics

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.

Relation of Rheological Properties of UV Cured Films

677

TMPTA 65

CO CO LU

Z 50

NVP

o <

HEMA

3 35

o z

TPGDA

LU Q-

20

MEA 0 NUMBER OF PASS Figure 6. Pendulum hardness of the films of oligomer : monomer : photoinitiator = 60 : 35 : 5, w/w, is shown against the number of passes.

678

Advances in Engineering Fluid Mechanics

but NVP can yield the highest TS values. Moreover, the augmentation of the oligomer backbone chain through the NVP network via its N-atom makes the film mechanically stronger than that obtained with TMPTA, which reinforces crosslinking in the films through its three-functional acrylated branches. Tensile strengths of EHA and MEA films are the lowest. These diluents have low Tg values (Tg, MEA = -100°C, and Tg EHA = -50°C). Films of low Tg containing diluents induce low TS values but can yield higher elasticity. This means more elongation (Eb). This is what is exhibited with the Eb profile curve (Figure 5) against the Tg copolymer. TMPTA has the highest Tg (25°C), and this produces brittle films that crack easily during stretching. On the other hand, films with low Tg monomer diluents like MEA, EHA, and HEMA induce high elasticity, the relation of Tg copolymer (calculated by Fox equation) again holds well for presenting tensile properties (TS and Eb) of the UVcured films of different reactive diluents of variable Tg values.

UJ H

cr

LU Q.

o

GC Q.

Tg

X

Figure 7. Rheological properties v s . T g co-polymer for the UV-cured filnns of oligomer : m o n o m e r : M E A : photoinitiator= 6 0 : 17 : 18 : 5, w/w.

elation of Rheological Properties of UV Cured Films

679

100

0) UJ Q:

UJ

o. O Q.

Tg ^C Figure 8, Rheological properties vs. Tg co-polymer for the UV-cured films of oligomer : monomer: EHA : photoinitiator = 60: 17 : 18 : 5, w/w.

CO-DILUENTS A second series of formulations was developed by replacing the single monomer diluent with a diluent (MEA or EHA) and a co-diluent (NP, TPGDA, or TMPTA) in the proportions of diluent: co-diluents = 17:18 w/w. Hardness of the UV cured films was determined and is plotted in Figure 7 (MEA systems) and Figure 8 (EHA system) against Tg copoloymer. It is observed that TMPTA yields the highest PH values followed by NVP and TPGDA. Similar results are also obtained when pendulum hardness of the films of MEA system is plotted against number of passes (Figure 9). This means that there is correlation between pendulum hardness and Tg obtained through Fox equation. Other rheological properties such as TS, Eb, and gel content for MEA and EHA systems, also are shown against Tg copolymer in Figures 7 and 8, respectively, and the corresponding results (TS, Eb, gel, and pendulum hardness) against number of passes (not shown here) suggest that these rheological properties similarly can be correlated with the glass transition temperature of the cured films.

680

Advances in Engineering Fluid Mechanics

801

TMPTA

60 NVP

w CO IXJ

z Q 40 < X

TPGDA D Q Z LU

20

0 0 NUMBER OF PASS Figure 9. Pendulum hardness vs. number of passes for the oligomer : monomer : MEA : photoinitiator = 60 : 17 : 18 : 5, w/w.

Relation of Rheological Properties of UV Cured Films

681

REFERENCES 1. K. M. I. Ali and T. Sasaki, Radiat Phys. Chem. 32, 371 (1944). 2. K. Kawate and T. Sasaki, in Rad. Teh. Asia 1991 Procedings, Osaka. 3. M. A. Hossain, T. Hasan, M. A. Khan and K. M. I. Ali, Polym-Plast. Technol. Eng. 33(1), (1994). 4. M. A. Hossain, K. Khayer, M. A. Khan and K. M. I. Ali, Nucl Sci. Appl 32(2), 9 (1994). 5. T. K. Saha, M. A. Khan and K. M. I. Ali, Radiat Phys. Chem. 44(4), 409 (1994). 6. K. M. I. Ali, M. A. Khan, M. M. Saman and M. A. Hossain, J. Appl. Plym. Chem. 54, 309 (1994). 7. M. M. Ali, M. A. Khan and K. M. I Ali, Polym-Plast. Technol. Eng. 34(4), 523 (1994).

This Page Intentionally Left Blank

CHAPTER 26 PREDICTION AND CALCULATION OF THE SHEAR CREEP BEHAVIOR OF AMORPHOUS POLYMERS UNDER PROGRESSIVE PHYSICAL AGING R. O. E. Greiner Siemens AG, Corporate Research and Technology Paul-Gossen-Str. 100 91050 Erlangen, Germany and J. Kaschta University of Erlangen-Niirnberg Institute for Material Science Martensstr. 7, 92058 Erlangen, Germany CONTENTS INTRODUCTION, 683 THEORY OF VISCOELASTIC BEHAVIOR UNDER THE INFLUENCE OF AGING AT CONSTANT TEMPERATURE, 684 EXPERIMENTAL AND RESULTS, 691 DISCUSSION, 696 NOTATION, 708 REFERENCES, 708 INTRODUCTION All amorphous polymers show the phenomenon of physical aging. After a quench from the rubbery into the glassy state, many physical properties of a polymer change with proceeding time. The material becomes stiffer and more brittle, its damping and creep rate decrease while the density increases, etc. This process was first studied in detail by Struik, who called it "physical aging" [1]. He also showed that physical aging is closely related to volume recovery. After a quench from the equilibrium into the non-equilibrium glassy state, the polymer shows a continuous decrease of the specific volume with time. This fact is due to the decrease of free volume of the material and is accompanied by a loss of mobility of the molecular segments. One assumes that this change of free volume and mobility is the origin of the variation of physical properties during aging. A deeper insight into the volume recovery phenomenon of polymers should, therefore, lead to a better understanding of the aging behavior. 683

684

Advances in Engineering Fluid Mechanics

Struik showed that many polymers age the same way and that the aging behavior in shear creep is influenced by temperature and loaded stress as well as by the aging time. Based on the general viscoelastic theory for the non-isothermal case of Hopkins [2] and Haugh [3], he proved that this theory and the mathematical formalism can be easily applied to the case of progressive aging at constant temperature. In some previous papers we have shown that the volume behavior of polymers under any thermal history can be described well using a multiparameter model [4,6] based on free volume [7]. It should be pointed out that the multiparameter model, developed by Hutchinson and Kovacs [5,8], was used here in its early version, where the partition parameter x equals zero, i.e., a full structural dependence of the retardation times was assumed. But the whole formalism is easily applied to values of x not equal to zero. Even in this simple version the model will demonstrate its power in predicting viscoelastic deformation properties. The description of the influence of physical aging on the mechanical properties of polymers is not only of scientific interest but also of utmost technical importance with respect to the long-time application of polymeric materials. As a result of physical aging, the viscoelastic deformation properties of polymers depend not only on loading history and temperature, but also on the progress of aging prior to and during the loading period. Therefore, the conventional technical mechanics of polymers, based on the concept of linear viscoelastic behavior with a creep compliance depending on time and temperature only, must fail whenever aging plays a significant role. Using the same multiparameter model as the one for the description of the specific volume under various temperature histories [4,6] we will examine and predict the creep behavior of a commercial polystyrene (Hostyren N 7000) [9-11] and a commercial polycarbonate (Makrolon 2800) [12] under various thermal histories. The theoretical calculations will be compared with experimental data. THEORY OF VISCOELASTIC BEHAVIOR UNDER THE INFLUENCE OF AGING AT CONSTANT TEMPERATURE We will restrict our considerations to the simplest case, which is aging at constant temperature. Although a detailed description of the theoretical fundamentals may be found in the cited literature [1-3,9,13-15], we will shortly recall them because we will need several important equations for understanding. The time dependence of temperature, shear stress, and shear strain for a simple quench is shown in Figure 1 schematically. We assume that a sample is heated to a temperature above its glass transition temperature, T , and is kept there until its thermal history has been erased. Then it is quenched down to a measuring temperature, T^, below T and further kept at this temperature. After an elapsed time, t^, which will be called preconditioning time, the stress history (in our case a constant stress a^) is applied and the strain y is measured as a function of creep time t. In the case of a linear viscoelastic response, the latter will be proportional to G^ and will depend on preconditioning time and creep time. In this experiment the progress of aging of the sample can be uniquely expressed by the degree of aging A, which is the time elapsed between the quench and the time, t, of observation of the strain:

Prediction and Calculation of Amorphous Polynners

685

time

Figure 1. The course of tennperature, stress, and strain as a function of time for the creep experiment under aging. A(t) = t^ + t

(1)

The theory of non-isothermal viscoelastic behavior as developed by Hopkins [2] and Haugh [3] may be based on the representation of linear viscoelastic behavior by mechanical models. The linear viscoelastic behavior of polymers in simple shear at constant temperature and prescribed stress history may be expressed in terms of the deformation of a generalized Kelvin model. Spring constants and dashpot viscosity constants of the model have to be appropriately chosen; the choice depends on temperature. For the non-isothermal treatment, the elasticity of the springs and the viscosities of the dashpots have to be inserted as functions of temperature. Due to the prescribed temperature history, they become functions of time. A very simple result is obtained if thermorheologically simple behavior is imposed on this mechanical model as an additional restriction. In this case, all spring constants of the model are temperature-independent while the viscosities all have a similar temperature dependence. By a change from the temperature T^ to the

686

Advances in Engineering Fluid Mechanics

temperature T, all viscosities are multiplied with the same factor a(T,T^) called timetemperature shift factor [161. For thermorheologically simple materials, the shear creep compliance J at the temperatures T^ and T have the same shape when plotted doublelogarithmically vs. time: J(t,T) = J(t/a(T,T,),T,)

(2)

and may be brought to coincidence by a parallel shift along the logarithmic time axis. The amount of this shift is given by log a(T,T^). For thermorheologically simple materials the non-isothermal strain response Y(t,T) under a prescribed temperature history T(t) and under a prescribed stress history a(t), which does not start prior to t = 0, is found to be [2,3]: Y(t) = J,a(t) + f^HX - ^) a(^)d^

(3)

>. 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

Advances in Engineering Fluid Mechanics

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:

Prediction and Calculation of Amorphous Polymers

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 = X^i

(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 = ^l^oCi

(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

Prediction and Calculation of Amorphous Polymers

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

Advances in Engineering Fluid Mechanics

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.

Prediction and Calculation of Amorphous Polymers

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

Advances in Engineering Fluid Mechanics

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.

Prediction and Calculation of Amorphous Polymers

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

Advances in Engineering Fluid Mechanics

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-

Prediction and Calculation of Amorphous Polymers

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

Advances in Engineering Fluid Mechanics

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

Prediction and Calculation of Amorphous Polymers

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

Prediction and Calculation of Amorphous Polymers

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

Advances in Engineering Fluid Mechanics

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^.

Prediction and Calculation of Amorphous Polymers

-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

Prediction and Calculation of Amorphous Polymers

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

Advances in Engineering Fluid Mechanics

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

Prediction and Calculation of Amorphous Polymers

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

Advances in Engineering Fluid Mechanics

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.

Prediction and Calculation of Amorphous Polymers

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.

This Page Intentionally Left Blank

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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].

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

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

Advances in Engineering Fluid Mechanics

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)

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

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

Advances in Engineering Fluid Mechanics

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].

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

SBS

°<^i s ^

03

-X

Q* 1E4

719

o — o NO C / B • — • C/B 10 PHB i A — A C/B 2 0 PHR

*V

g

1000

inn 1

10

1000

100

1E4

APPARENT SHEAR RATE, Tapp (Sec"^)

X£iv>

SIS d a

o O NO C / B • — • C/B 10 PHR A — ^ i C/B 20 PHR

. o<^X

o 1E4

P

g O

% 1

1000

inn

•

1

. • .^1

-

-

^ —

10

^

^ -

-

-

-

^^^^J_

_-

1000

100

1E4

APPARENT SHEAR RATE. 7app(Sec-l)

lES

1 SEES

. ^

g

1E4 r

X

1000

O cn

100

O NO C/B • C/B 10 PHR A C/B 20 PHR

"XV

<0

ft

O • A

1

1

1—1

t

1 ,

10

1

1—1 t 1 m l

100

1000

1E4

APPARENT SHEAR RATE, 7app (Sec"!)

Figure 5. Rheological behavior of styrene copolymers. Shear viscosity vs. Shear rate plots were compared for the effects of carbon black loading [44].

720

Advances in Engineering Fluid Mechanics

(text continued from page 717) region. The total pressure loss, AP^, is the pressure loss in the capillary die land, AP^.^, added to the pressure loss at the entrance to the capillary, AP^^^, with the assumption of neglecting the exit pressure loss. Mathematically, AP = AP, + AP , die

(9)

ent

^ -^

The rheological property responsible for the entrance pressure drop has been argued to be melt elasticity [58]: AP , = AP, , ent

elastic

(10) ^

^

Bagley considered that the large pressure drop at the entrance of a die may be thought of as occurring in an imaginary extension to the actual tube length [35]. The Bagley plots for the carbon black-filled triblock copolymers are shown in Figures 6, 7, and 8. We can determine the value of the entrance pressure drop from these figures. The pressure losses are frequently represented in terms of a multiple of the dieland shear stress, o : AP ent

= mo

(11) w

^

^

where m is called the Couette correction. This correction factor has been observed in some experiments recently because it can be related to the polymer elasticity [22,23,31,40,59,60]. Generally, viscoelastic materials show higher Couette correction than Newtonian fluids. The value of Couette correction for the three thermoplastic elastomers are shown in Figure 9. This figure indicates that the Couette correction is decreased with adding carbon black. Therefore, it appears that the carbon blackfilled polymers have less elasticity than raw polymers. For the diblock copolymer, similar trends also are obtained [31] as shown in Figure 10. Die Swell and Extrudate Distortion If a viscoelastic material is forced to flow from a large reservoir through a circular tube, the diameter of the extrudate is found to be larger than the tube diameter. Many researchers [61-65] have argued that that causes the die swell. They developed the following three points of view for the cause of the die swell: polymer chain orientation within the capillary caused by the high shear field; recovery of the elastic deformation; and viscose heat effects. The most important concept is the recovery of the elastic deformation imposed in the capillary. Die swell must be considered for the prediction of accurate dimensions of continuous profile end products. Therefore, many studies in this field are related to plastics and rubbers [66-75]. Today, it is generally agreed that die swell behavior is typical of viscoelastic materials. This behavior especially, relates to their elastic properties. We photographed three thermoplastic elastomers at various die lengths and shear rates. Figure 11 shows the extrudate distortion with various black loading at a fixed (text continued on page 725)

Die Extrusion B e h a v i o r of C o p o l y m e r T h e r m o p l a s t i c E l a s t o m e r s

721

SBS. NO C / B 1E4 8000

Cx]

6000

en en M

4000 h 2000

L/D RATIO

SBS, C/B 10 PHR 1E4 APPARENT SHEAR RATE 8000

w ID CO

6000

3.6 s e c " 17.9 s e c ' ^ 35.8 s e c " A 107.5 s e c D 179.2 »ec-

• 358.4 see 4000

zn 2000 CX

L/D RATIO

SBS. C/B 20 PHR 1E4

CQ

H CO

00

6000

6000

4000

rx.

10

20 L/D RATIO

Figure 6. Total pressure drop as a function of L/D ratio at various shear rates for SBS [44].

722

Advances in Engineering Fluid Mechanics SIS. NO C/B 1E4

L/D RATIO SIS, C/B 10 PHR

1E4

10

20

3C

20

30

L/D RATIO SIS, C/B 20 PHK

10

L/D RATIO

Figure 7. Total pressure drop as a function of L/D ratio at various shear rates for SIS.

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

723

SEES, NO C/B 1E4

APPARENT SHEAR RATE O

# A • D •

a.6

17.0 35.S 107.5 sec 179.S aec 358.4

L/D RATIO SEBS, C/B 10 PHR 1E4

Pi

6000

CO

4000 h 2000 Y

L/D RATIO SEBS. C/B 20 PHR 1E4

.1-1

GO Pi rxi

6000

6000

APPARENT SHEAR RATE J^^^^y^ y ^ J\ O 3.6 tc~\ y y i ^ y ^ y ^ • 17.9 • • 0 - 1 yyy^ y'''^y^ A 35.8 »«o-l y'yy^ y""^ ^ ^ ^^ A 107.6 8CC-1 yyx y ^ ^ .^"^'^ D 179,2 8CC-1 yyy^ ^ j ^ ^^^"^ • 358.4 ace-1 y^^^^^'"''''^ ^^^"'^""^

Cfc!

D

4000

§

Pi

2000 0

^^

.

\

10

, 20

L/D

3C

RATIO

Figure 8. Total pressure drop as a function of L/D ratio at various shear rates for SEBS.

724

Advances in Engineering Fluid Mechanics O NOC/B • C/B 10 PHR A C/B 20 PHR

15

o

b

SBS

o

10

o o

o

• •

<J

5

•

A

•A

100* C tl

1

10

100

1E4

1000

APPARENT SHEAR RATE. 7app (Sec-^)

20 O NO C/B • C/B 10 PHR A C/B 2 0 PHR

SIS

15

b 10

a

A

1

10

100

^

1E4

1000

APPARENT SHEAR RATE. 7app (Sec"!)

20

SEES 15

b

O

O NO C/B • C/B 10 PHR A C/B 2 0 PHR

10

a

O

•A

^

A

1

10

100

1000

1E4

APPARENT SHEAR RATE. 7app (Sec""^)

Figure 9. Couetta coefficient vs. shear rate behavior of styrene triblock copolymers filled with carbon black N-330 of different loading volunnes [44].

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

725

T - lOO'C 1U

• SBR 706 • SBR+IOXC/B • SBR+20XC/B

35 30 25 C

o CL

20 15 10

• • •

•

' • - .

•

•

5 n

1

.

^

.

^

> -

. - 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

Advances in Engineering Fluid Meclianics

Shear rate (Sec*) 4 18

N0CA8BC»<

36 108 180 360

4 18

C/B = 10 PHR

36 108 180 360

C/B X 20 FHH 18 36 108 ISO 360

Figure 11a. Extrudate distortion of pure and black-loaded SBS with a fixed LVD ratio of 10 at various shear rates [44].

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

ilBHiii iHHIIIi

mmsim^

727

imlmmm

iliiiiiiili llll^j^

illBilli

lilBBiiiiFHii

iiii^Hlli

Figure l i b . Extrudate distortion of pure and black-loaded SIS with a fixed L/D ratio of 10 at various shear rates.

728

Advances in Engineering Fluid Mechanics

Shear rate (Sec"') 4 18

m CASBON

36 108 180 360

C/B = 10 PHR

'iiilin,

I I, nil f

C/B = 20 PHH

Figure 11c. Extrudate distortion of pure and black-loaded SEBS with a fixed l_/D ratio of 10 at various shear rates.

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

729

Siear rate (Sec'*) 4 L/D = S

18 36 108 ISO 360

4 18 i/B = 10

36 108 180 360

4

*® 36 108

^ ^ ^ ^ ^ ^ ^ ^ — ^ ^ ^ ^ ^

im = 20

180 360

Figure 12a. Extrudate distortion of carbon black-filled SBS extruded through capillary tubes having different LVD ratio at various shear rates. Black loading is constant at 10 parts [44].

730

Advances in Engineering Fluid Mechanics

L/i - i

im = 10

t / » - 20

Figure 12b. Extrudate distortion of carbon black-filled SIS extruded through capillary tubes having different L/D ratio at various shear rates. Black loading is constant at 10 parts.

Die Extrusion Beliavlor of Copolymer Thermoplastic Elastomers

731

Shear rate

18

L/D = 5

108

im 360

L/n = 10

L/B = 20

Figure 12c. Extrudate distortion of carbon black-filled SEBS extruded through capillary tubes having different l_/D ratio at various shear rates. Black loading is constant at 10 parts.

732

Advances in Engineering Fluid Mechanics 1.6 I

SBS 1.4

/

T3

1.3 O — O NO CARBON • — • C/BIOPHR A — A C/B 20 PHR

cS g 1.2

-A^

A

1.1 CO

L/D = 10 1.0

10 100 APPARENT SHEAR RATE. 7app ( s w i ) 1.5

SIS

^o

o

1.4

I S

O — O NO CARBON • — • C/B 10 PHR A — A C/B 20 PHR

1.3

1.1 L/D = 10 1.0

1

10 100 APPARENT SHEAR RATE. 7app (S«c-1)

1.5

SEBS 1.4

1.3 O O NO CARBON • • C/B 10 PHR A — A C/BZOPKB,

^

1.1

L/D = 10 1.0 1

10

100

APPARENT SHEAR RATE. 7app (Sec"^ )

Figure 13. Die swell ratio vs. shear rate for styrene copolymers filled with different loading volumes of carbon black (constant L/D ratio of 10) [44].

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

733

1.5 I

o • A

SBS

I

1.4

C/B 10 PHR

o L/D « 5 • L/D « 10 A L/D = 20

1.3

O

@

1.1 1.0

1

10 100 APPARENT SHEAK E^TE, 7app (Sec-l)

1.5

O • A

SIS Q

1.4

C/B 10 PHR

\

O L/D = 5 • L/D = 10 A L/D - 20

TJ

1.3

1.1 Y

w 1.0

100

10

APPARENT SHEAR RATE, 7app (Sec'M 1.5

O • A

SEBS 1.4

C/B 10 PHR

O L/D = 5 • L/D = 10 A L/D = 20

1.3 ^O--^

t3 . ^

:?==!:

1.1

w 1.0

1

10

100

APPARENT SHEAR RATE, 7app (Scc-1)

Figure 14. Die swell ratio vs. shear rate for styrene copolymers at various UD ratios. Carbon black loading is constant.

734

Advances in Engineering Fluid Mechanics

(text continued from page 725) viscosity-shear rate behavior, extrudate swell, and Couette correction, which is related to the elastic property of polymeric materials of carbon black-filled styrene copolymer, were compared to those of nonfilled pure polymers to study the rheological characteristics of thermoplastic elastomers. The shear viscosity was increased by the addition of carbon black while the Couette correction was decreased. Carbon black suppressed the die swell and extrudate distortion of styrene copolymers. The second, more important objective of this chapter was to stimulate further research in this field. Even though a large number of studies for the physical properties or morphological characteristics of thermoplastic elastomers have been reported, any serious detailed results for the rheological behavior of the black-filled block copolymer systems were not found easily. Nowadays, the application of rheology to the actual processing is getting practical. But the realistic modeling for the polymer processing operations usually requires a lot of effort and accumulated data. On this point of view, we hope that our works in this chapter, in spite of its roughness, will be a good guidepost for future research in this field. NOTATION d Extrudate diameter D Capillary diameter Kg Einstein coefficient L Capillary length m Couette correction n' Parameter AP^ Total pressure drop

Pressure drops in die Pressure drops in die AP region AP Pressure drops in die exit Pressure drops in die AP Volumetric extrusion APH

di

land region entrance exit region ends rate

Greek Symbols Yapp Apparent shear rate Y Wall shear rate a Wall shear stress w

r|, Viscosity of mixture or filled polymer compound

TIQ Viscosity of suspending medium or pure polymer 0 Volume fraction of sphere or filler (b Maximum filler fraction

REFERENCES 1. Haws, J. R., and Wright, R. P., in Handbook of Thermoplastic Elastomers, B. M. Walker, (ed.). Van Nostrand Reinhold Co., New York, 1979. 2. Boonstra, B. B., in Rubber Technology and Manufacture, C. M. Blow and C. Hepburn, (eds.), Butterworths Scientific, London, 1982. 3. Holden, G., J. Elastoplastics, Vol. 2, p. 234 (1970). 4. Haws, J. R., and Middlebrook, T. C , Rubber World, Vol. 167, p. 27 (1963). 5. Marrs, O. L., Naylor, F. E., and Edmunds, L. O., Adhesion, Vol.4, p. 211 (1972). 6. Matsuo, M., Ueno, T., Horino, H., Chujo, S., and Asai, H., Polymer, Vol. 9, p. 425 (1968).

Die Extrusion Behavior of Copolymer Thermoplastic Elastomers

735

7. Kraus, G., Naylor, F. E., and Rollmann, K. W., J. Polym. ScL, Part A2, Vol. 9, p. 1,839 (1971). 8. Ishii, M., Kino Zairyo (Japan), Vol. 14, p. 24 (1994). 9. Akiba, M., Synthetic Resins (Japan), Vol. 39, p. 10 (1993). 10. Legge, N. R., Elastomerics, Vol. 123, p. 14 (1991). 11. Chien, J. C. W., J. Appl Polym. Sci., Vol. 36, p. 1,387 (1988). 12. Oyanagi, Y., Japan Plastics, Vol. 44, p. 10 (1993). 13. Foley, K. F., Rubber World,Wo\. 199, p. 16 (1988). 14. O'Conner, G. E., and Fath, M. A., Rubber World,Wol 185, p. 25 (1981). 15. Rangaprasad, R., Pop. Plast. Packag., Vol. 38, p. 49 (1993). 16. Rangaprasad, R., Pop. Plast. Packag., Vol. 38, p. 55 (1993). 17. Kie, H., Wu, X., and Guo, J., Polym. Prepr., Vol. 34, p. 596 (1993). 18. Akiba, M., Polymer Digest (Japan), Vol. 46, p. 87 (1994). 19. Akiba, M., Polymer Digest (Japan), Vol. 46, p. 81 (1994). 20. Kim, K. W., and Hyun, J. C., Polymer (Korea), Vol. 13, p. 139 (1989). 21. Kraus, G., and Rollmann, K. W., J. Polym. Sci., Polym. Phy. Ed., Vol. 15, p. 385 (1977). 22. Mathew, I., George, K. E., and Francis, D. J., Kautsch. Gummi. Kunstst., Vol.44, p. 450 (1991). 23. Mathew, I., George, K. E., and Francis, D. J., Kautsch. Gummi. Kunstst., Vol.217, p. 51 (1994). 24. Holden, G., Bishop, E. T., and Legge, N. R., J. Polym. Sci., Part C, Vol. 26, p. 37 (1969). 25. Childers, C. W., and Kraus, G., Rubber Chem. Technol., Vol. 40, p. 1,183 (1967). 26. Gruver, J. T., and Kraus, G., J. Polym. Sci., Part A, Vol. 2, p. 797 (1964). 27. Holden, G., J. Appl. Polym. Sci., Vol. 9, p. 2911 (1965). 28. Kraus, G., and Gruver, G. T., Trans. Soc. RheoL, Vol. 13, p. 15 (1969). 29. Kraus, G., and Gruver, G. T., J. Appl. Polym. Sci., Vol. 11, p. 2,121 (1969). 30. Arnold, K. R., and Meier, D. J., J. Appl. Polym. Sci., Vol. 14, p. 427 (1970). 31. Kim, J. K., and Han, M. H., Korean J. RheoL, Vol. 4, p. 46 (1992). 32. DeVries, A. J., and Bonnebat, C., Polym. Eng. Sci., Vol. 16, p. 93 (1976). 33. Mooney, M., J. Colloid Sci., Vol. 2, P. 69 (1947). 34. Mooney, M., and Black, S.A., J. Colloid Sci., Vol. 7, p. 204 (1952). 35. Bagley, E. B., J. Appl. Phys., Vol. 28, p. 624 (1957). 36. Eisenschitz, R., Kolloid Z., Vol. 64, p. 184 (1933). 37. Eisenschitz, R., Rabinowitsch, B., and Weissenberg, K., Mitt. Deutsch. Material Prulf. Sonderh., Vol. 9, p. 9 (1929). 38. Rabinowitsch, B., Z. Phys. Chem., Vol. A145, p. 1 (1929). 39. Han, C. D., Rheology in Polymer Processing, Academic Press, New York, 1976. 40. White, J. L., Principles of Polymer Engineering Rheology, Wiley, New York, 1991. 41. Dealy, J. M., and Wissbrun, K. F., Melt Rheology, Plastics Processing, Van Nostrand Reinhold, New York, 1990. 42. Cheremisinoff, N. P., An Introduction to Polymer Rheology and Processing, CRC Press, Boca Raton, USA, 1993. 43. Barnes, H. A., Hutton, J. F., and Walters, K., An Introduction to Rheology, Elsevier, New York, 1989.

736

Advances in Engineering Fluid Mechanics

44. Kim, J. K., Han, M. H., Go, J. H., and Oh, S. C, 7. AppL Polym. ScL, Vol. 49, p. 1,777 (1993). 45. Holden, G., and Legge, N. R., in Thermoplastic ElastomerSy N. R. Legge, G. Holden, and H. E. Schroeder, (eds.), Hanser Publishers, New York, 1987. 46. Medalia, A. I., Rubber Chem, Tech., Vol. 51, p. 437 (1978). 47. Dannenberg, E. M., Rubber Chem. Tech., Vol. 59, p. 512 (1986). 48. Nakajima, N., and Scobbo, J. J., Rubber Chem. Tech., Vol. 60, p. 761 (1987). 49. Bosshard, A. W., and Schlumpf, H. P., in Plastics Additives, R. Gachter and H. Muller, (eds.), Hanser Publishers, Germany, 1987. 50. Einstein, A., Ann. Physik., Vol. 19, p. 289 (1906) ; Vol. 34, p. 591 (1911). 51. Guth, E., and Gold, O., Phys. Rev., Vol. 53, p. 322 (1938). 52. Mooney, M., J. Colloid Sci., Vol. 6, p. 162 (1951). 53. Tanaka, H., and White, J. L., J. Non-Newtonian Fluid Mech., Vol. 7, p. 333 (1980). 54. Frankel, N. A., and Acrivos, A., Chem. Eng. Sci., Vol. 22, p. 847 (1967). 55. Molau, G. E., in Block Polymers, S. L. Aggarwal, (ed.). Plenum Press, New York, 1970. 56. Mooney, M., and Black, S. A., J. Colloid Sci., Vol. 8, p. 272 (1968). 57. Han, C. D., Charles, M., and Philippoff, W., Trans. Soc. RheoL, Vol. 13, p. 455 (1969). 58. Brodnyan, J. G., Philippoff, W., and Gaskins, F. H., Trans. Soc. RheoL, Vol. 1, p. 109 (1957). 59. Han, M. H., Kim, J. K., Park, J. C, and Go, J. H., Korean J. RheoL, Vol. 6, p. 66 (1994). 60. Kannabiran, R., Rubber Chem. Tech., Vol. 57, p. 1,001 (1984). 61. Goran, S. L., Middleman, S., and Gavis, J., J. AppL Polym. ScL, Vol. 7, p. 493 (1963). 62. Middleman, S., and Gavis, J., Phys. Fluids, Vol. 4, p. 355 (1961). 63. Truesdell, C, Trans. Soc. RheoL, Vol. 4, p. 9 (1960). 64. Spencer, R. S., and Dillon, R. E., /. Colloid ScL, Vol. 3, p. 241 (1949). 65. White, J. L., and Roman, J. F., J. AppL Polym. ScL, Vol. 20, p. 1,005 (1976). 66. Dillon, J. H., and Johnston, N., Physics, Vol. 4, p. 225 (1933). 67. Carley, J. F., Mallouk, R. S., and Mckelvey, J. M., Ind. Eng. Chem., Vol. 45, p. 974 (1953). 68. Lu, X. L., and Tanner, R. I., Polym. Eng. ScL, Vol. 25, p. 620 (1985). 69. Mendelson, R. A., and Finger, F. L., J. AppL Polym. ScL, Vol. 17, p. 797 (1973). 70. Sekiguchi, M., Chem. High Polym. (Japan), Vol. 26, p. 721 (1969). 71. White, J. L., and Crowder, J. W., J. AppL Polym. ScL, Vol. 18, p. 1,013 (1974). 72. Tordella, J. P., J. AppL Polym. ScL, Vol. 7, p. 215 (1963). 73. Song, H. J., White, J. L., Min, K., Nakajima, N., and Weissert, F. C, Adv. Polym.Tech., Vol. 8, p. 431 (1988). 74. White, J. L., Han, M. H., Nakajima, N., and Brzoskowski, R., J. RheoL, Vol. 35, p. 167 (1991). 75. Zhao, L. J., Plast. Rubber Compos. Process AppL, Vol. 19, p. 311 (1993).

CHAPTER 28 POLYSULFIDES C. P. Tsonis Chemistry Department King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia CONTENTS ALIPHATIC POLYSULFIDES, 738 MECHANISM, 738 INORGANIC POLYSULFIDE MONOMERS, 739 DIHALIDE MONOMERS, 740 HIGH MOLECULAR WEIGHT POLYMERS, 741 LOW MOLECULAR WEIGHT POLYMERS, 741 PROPERTIES, 742 APPLICATIONS, 742 AROMATIC POLYSULFIDES, 742 MECHANISM, 744 PROPERTIES, 744 APPLICATIONS, 745 REFERENCES, 745 Polysulfides, whether aliphatic or aromatic, have excellent physical and chemical properties, which make them one of the leaders in petrochemical industries as elastomers and engineering thermoplastics. Both types are synthesized by a nucleophilic substitution reaction between a dihalide and sodium sulfide (reaction 1) nCl—A—CI

+

nNa2Sx

- NaCl

-

(1)

A = aliphatic or aromatic group The number of sulfur atoms incorporated into the polysulfide chains differs and depends on the particular reaction system. The x usually takes values from 1-5. 737

738

Advances in Engineering Fluid Mechanics

ALIPHATIC POLYSULFIDES Aliphatic polysulfides, commonly known as polysulfide rubbers (thiokols), are commercially produced by the nucleophilic substitution polymerization of an aliphatic dihalide with aqueous sodium sulfide of the type Na2S^ (reaction 2) [1-4] nCl—R—Ci

+

NaCl

nNa2Sj

R—s.

(2)

For example, 1,4-dichlorobutane reacts with sodium tetrasulfide to produce the corresponding aliphatic polysulfide (reaction 3) NaCl nCl—fcH2)—CI

—•

+ nNa2S4

A-tn.)— s —s

(3)

MECHANISM The polymerization occurs by a bimolecular nucleophillic substitution mechanism. For example, the two difunctional monomers, disodium tetrasulfide and ethylene dichloride, are capable of undergoing step by step bimolecular reactions to form a dimer (reaction 4)

S—S ~ II S

+ CI-CH2-CH2-CI

~C1

>-

S — S — CH2 CH2 CI

(4)

S

Monomer (1)

Monomer(2)

Dimer

The dimer reacts with the monomer(l) to give a trimer (reaction 5)

s II s--s-—

s +

CH2 CH2 CI

II

s

.

s —s II Monomer (1) ^

Dimer

~^l'> ^

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

Advances in Engineering Fluid Mechanics

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)-<:i

Structure 7

Structure 8

To obtain good elastomers (thiokol rubbers) the dihalide monomer must have four or more methylene groups. Other methods [10-14] only of academic interest, for the preparation of aliphatic polysulfides, are represented by reactions (11-14)

CH2=CH-CH=CH2 + S

-^^^-^

-[

C,H,-Sx ] -

(11)

n CH2=CH-CH=CH2 + S2CI2

>- -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

Advances in Engineering Fluid Mechanics

^/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

Advances in Engineering Fluid Mechanics

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-"<0>-Cl

1+

+

2~

Na2S

CI

-

-o<:,

2 Na

followed by loss of chlorine (step 2)

CI

-o<

2 Na

NaCl

C l - ( 0 > - S Na

The product sodium p-chlorobenzene sthiolate can attack another dichlorobenzene to form a dimer (step 3)

C l - < 0 > - S Na

ci-(0)-ci - NaCl

ci-<0>-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-<0>-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).

746

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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

Advances in Engineering Fluid Mechanics

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 (<10%) of TPU can be used to improve the low temperature impact properties of ABS resins and abrasion resistance [18], which is useful to the automotive industry. TPU/ABS blends can be used to mold or extrude seat adjuster housings, automotive interior panels, vent grills, aircraft seat tracts, snowmobile modular drive belts, and automotive filler panels. A family of TPU blends is being developed to compete in the flexible automotive bumper fascia market. Features of these blends include printability without adhesion promoter or primer, ease of processing.

752

Advances in Engineering Fluid Meciianics

outstanding low temperature impact strength, good mar resistance, and excellent property retention upon recycling [19]. TPU/PVC BLENDS Blending of TPUs with polyvinyl chloride (PVC) has been studied by a number of researchers [20-22]. The blends of TPU/PVC exhibit the combined properties of individual polymers, such as toughness and abrasion resistance of TPUs, and the stiffness and high modulus of PVC. The benefits to each polymers can be seen in Table 2. the PVC/TPU blends can be either compatible systems or incompatible ones, depending on the chemical nature of TPU segments [23] and the composition of both TPU and PVC [24-25]. The compatible TPU-PVC system contains a well-mixed PVC-polyether matrix phase while the aromatic urethane segments, which exhibit microphase separation in the pure polyurethane, are not solubized by blending with PVC [26]. The blends 50/50 and 40/60 of PVC/TPU are more compatible than other blend compositions as shown in the study with Fourier transform infrared spectroscopy (FT-IR) and differential scanning calorimetry (DCS) [27], as well as with X-ray techniques [28]. Blends containing low levels of PVC exhibited stress-strain behaviors similar to reinforced elastomers, but at higher PVC contents, the blends showed increased elongation with well-developed yield points, stress whitening, and necking as well as possible cold drawing under tension. Small amounts of TPUs can be used as a nonmigrating, nonvolatile plasticizer for PVC [29]. Solution blends of PVC and TPU showed a high degree of molecular mixing of the two polymers by DSC results, in which glass transition temperatures (T ) of the blends exhibited one major T whose position on the temperature scale is raised with increasing levels of PVC (Figure 2). Blends of chlorinated-PVC (CPVC) and TPU have been shown to be immiscible despite the fact that the blends exhibit single T peak. Transmission electron microscopy (TEM) revealed a two-phase structure with domain size varying from d = 0.1 to 1.5 |im [30]. Other blends, such as poly(vinylidene chloride-co-vinyl chloride)/TPU, are optical clear with thermal behaviors, indicating complete miscibility in the blends [31]. TPU/PVC blends are useful in calendered coating for upholstery, apparel fabric, wire and cable, film, tubing, industrial hoses, and pit liners for mining operations [32,33]. Table 2 Synergy of TPU and PVC Blends Benefits to TPU from PVC

Stiffness Cost Processing Fire retardant

Benefits to PVC from TPU

Abrasion resistance Low temperature flexibility Toughness and impact resistance Increased elasticity

Properties and Applications of Thermoplastics Polyurethane Blends

753

360-1

340

:^320H CD 300 H

280 H

260

I I I I I I I I I [ I I I I I I I I I I I I M I I I I I I I I I I I I I M I I I I I I I I M I I I 1I

0

20

40

PVC,

60

80

100

Nt%

Figure 2. Glass translation temperature of TPU/PVC blends (after [22]).

TPU/STYRENE BLENDS Polystyrene (PS) and TPU blends will benefit from the good processability, high dimensional stability and transparency of PS, and elasticity of TPU. The blends will benefit from each polymer shown in Table 3. PS and TPU are incompatible polymers in their blend systems. The addition of 30% by weight PS to TPU produces an increase in stiffness with the retention of elongation properties. Copolymers of PS at the 10-20% level increase heat resistance, rigidity, and also lower cost [34]. The adding of charging groups in PU can increase the blends compatibility and enhance the mechanical properties [35]. Blends of copolymers, such as styrene and maleic anhydride [36] or styrene and maleimides [37] with TPU, exhibit improved impact strength at still-high Vicat temperatures.

754

Advances in Engineering Fluid Mechanics Table 3 Synergy of TPU and PS Blends Benefits to TPU from PS Stiffness Better processing Transparency

Benefits to PS from TPU Toughness and impact resistance Increased elasticity Increased elongation

Styrene-acrylonitrile copolymer (SAN) with TPUs were immiscible in the composition range 30-70 wt.% of TPU or SAN. In TPU-rich blends, SAN propagates the separation of hard and soft segments in TPUs. In SAN-rich blends, the interaction stage depends on acrylonitrile contents. SEM measurements showed that the blends of TPU/SAN were fine dispersions and clearly showed a continuous and disperse phase [38]. It was reported that styrenic copolymer/polyacetal/TPU systems also have good processability and a beneficial combination of physical and chemical properties, including thermal/dimensional stability, impact resistance, chemical resistance, and creep resistance. The polymer blends are suitable for use in the preparation of a variety of molded utilitarian articles having good appearance and printability [39]. TPU/PS blends have found suitable applications as adhesive [40]. TPU/POLYACETAL BLENDS Polyacetal or polyoxymethylene (POM) is one of the major engineering thermoplastics because of its metal-like high strength, high stiffness, good electrical and dielectric properties. Acetal can be blended with TPU to enhance properties of the polyurethanes. The blends have a higher capacity to absorb energy under impact stress and higher reserves of strain with good elastic recovery. The properties depend on the composition of both polymers, the morphology of the multiphase system as well as on production and processing conditions. For example, 5 to 20% acetal blended with TPU improves the overall processability of the TPU in extrusion and blow-molding applications. The blends of 30 to 50% acetal with TPU improve TPU's heat and humid age properties and solvent-resistance [41-43]. It was found that the higher the content of POM in the POM/TPU blend, the higher the heat distortion temperature (HDT) of the blend [44] (Figure 3). POM can be effectively toughened by blending with TPU. The improvement in toughness is most significant with TPU content from 20-30% due to the elastomer phase remains dispersed up to the content [45,46]. The blends creep more than POM, and the tendency to creep increases as the TPU content increases. Monotonic tension tests have shown that both POM and blends are found to be rate-sensitive [47]. Viscosities of the blends were studied by a Simple and Workable Model [48]. And experimental study shows that POM has lower viscosity than TPU, especially in the region of lower shear rate [49].

Properties and Applications of Thermoplastics Polyurethane Blends

755

1 15-1

1 10-^

U 0)

105H

Q Q X

100

95^

90

' » ' »

1 I I I I I I I I I 1 I I I I I

20

40

I

I

60

POM,

I I I I I I r I

I I I » 1 I I I I I I I I

80

I

100

Nt%

Figure 3. Heat distortation temperature of TPU/POM blends (after [44]).

POM/TPU blends can be used in chain wheels, surfboard mast feet, switch buttons, clips for toys, car body part and fuel lines, blow moldings, and sports equipment. TPU/OLEFIN BLENDS Poly olefin polymers are not compatible with TPU. It was found that only about 5% of poly olefin, such as polypropylene, can be incorporated into TPU with a slight improvement in the toughness and tear strength of TPU [50]. The compatibility of TPU and PP can be improved by compatibilizers, such as maleated polypropylene (PP-MA) and its graft copolymer with polyethylene oxide (PEO), (PP-MA)-g-PEO [51]. TPU/Olefin blends can be used for multilayered composites in packaging since they have advantages in flexibility, high impact resistance, good thermal processability, and good thermal adhesion [52].

756

Advances in Engineering Fluid Mechanics

Another type of TPU/Polefinic blend that has been evaluated is the chlorinated polyethylene (CPE)/TPU blend, which is limited to TPUs that have low thermal processing temperature (<200°C) because of the thermal instability of CPE. The use of small amounts of CPE improve processing and mold release properties of TPU while small amounts of TPU in CPE result in good strength properties and processing characteristics [53]. These blends can be used in applications where fire-retardant characteristics and/or low temperature flexibility of the materials are required. Impact-resistant properties, especially at low temperatures, low-temperature toughness, melt-processing properties, and dimensional stability of thermoplastic polyurethane elastomers are improved by the addition of a carbonyl-modified polyolefin [54,55]. These improvements are particularly useful for reinforced polyurethanes. Small amounts of polyolefin homopolymer or copolymer with TPU blends are useful for molding plastic articles by injection molding, extrusion, calendering, or similar process for molding thermoplastic articles [56]. TPU/NYLON BLENDS Nylons or polyamide (PA) are generally classified as engineering plastics because of their mechanical strength and stability at elevated temperatures. The blends of TPU/nylons have a unique structure whereby the TPUs function as a dispersed phase in a continuous polyamide matrix. At low level of TPUs, less than 10%, the blends have improved room temperature and subambient impact resistance of the nylon. With increasing levels of TPU, 10-40%, the blends exhibit increased flexibility of the nylon and overall toughness of the nylon [57-59]. The modulus of blends also will increase with addition of nylons [7]. TPU/POLYESTER BLENDS Polycarbonate (PC), one of the important engineering plastics, is sensitive to solvent and environmental stress cracking. This can be overcome by blending with TPU. Low levels of TPU can function as impact modifiers in polycarbonate (PC), and higher levels can form a useful alloy when used alone with PC or in a triblend with PC and another polymer, such as PBT. Blending 3-25% TPU with PC improves the impact properties of the PC without sacrificing the tensile strength or high distort temperature (HOT), and with increased resistance of the PC to solvent and environmental stress crazing and cracking. Blends of 40 to 75% TPU in PC produce alloys of high impact and good rigidity, along with good chemical resistance [60-62]. The addition of PC to TPU results in a higher modulus, and the blends exhibit excellent processing properties, making them useful for automotive applications [63]. Ternary blends of TPU/PC/ABS are claimed to have better processing properties [64] and better fuel resistance [65]. TPU/PC/polybutyleneterephthalate blends also show less stress cracking with solvents [66]. Low levels of TPU have been blended with polyesters, such as polybutyleneterephthalate (PBT) and polyethleneterephthalate (PET). Blends of 10 to 30% TPU contribute to the toughness and low-temperature impact properties of PBT and PET [67].

Properties and Applications of Thermoplastics Polyurethane Blends

757

TPU/PVDF BLENDS Some TPUs have been successfully used in implantable medical prostheses, such as the insulation for the cardio-pacemaker leads, breast implants, and vascular prostheses. But TPUs suffer from long-term environmental stress cracking leading to premature failure of implanted medical devices. PVDF, a physiologically and chemically inert polymer resistant to stress-cracking, is a good candidate for use as a biomedical polymer. Blends of TPU and PVDF are currently under investigation to achieve blends that are mechanically similar to TPU while having the resistance to environmental stress cracking of PVDF. Preliminary evaluation of TPU/PVDF blends have shown incompatibility of the two polymers. However, the mechanical strength of the blend was found to depend on the percentage composition of the TPU (Figure 4). Further studies on the stress-cracking resistance are currently underway.

100n

I I I I I I I I I I I I I I I I I I I I M I I I I I M I I I I I I I I M I I I I I I I I I I I I I Ii

0

20

40

60

80

PVDF, N I % Figure 4. Ultimate tensile strength vs. PVDF percentage.

100

758

Advances in Engineering Fluid Mechanics

TPU/ELASTOMER BLENDS The use of thermoplastic elastomers (TPE) to modify TPU properties has been looked at quite extensively. The amount and choice of TPE for a given application or market area affect the properties of TPU, such as hardness, low-temperature impact, wet-slip resistance, resistance to sour gas, etc. Nitrile rubbers (NBR) blend very well with the softer grades of TPU. They generally improve the impact of TPU and also '*sour" gas resistance, which is needed for oil drilling and exploration applications. Nitrile rubbers generally are compatible with TPU up to 25%. They improve low-temperature and wet-slip resistance [7]. Similarly, styrene-isoprene-styrene (SIS) or styrene-butadiene-styrene (SBS) copolymers are quite compatible with TPU and are used to reduce hardness and significantly improve wet-slip resistance of TPU [68]. They also reduce surface tension to promote better adhesion to other polymers. Some grades of ethyle-propylene-diene terpolymer (EPDM) rubber are compatible with TPU at concentrations less than 10% by weight and are effective as impact modifiers for the TPU or as low-cost extenders [7]. Chloroprene rubbers also are compatible with TPU and can be used to reduce the hardness of TPU and improve low-temperature impact. TPU of hardness 55-70 Shore A are based on a polyblend of a polyurethane with a rubber nitrile, optionally with a suitable amount of additives [69]. Ozone, temperature resistance, and other properties of TPU are increased when blended with NR [70]. OTHER TPU BLENDS Other studies have been carried on TPU blending with other polymers, such as polyvinyl acetate (10-35%) [71], polyhydrooxyether of bisphenol-A (Phenoxy) [72], acrylic polymers [73-74], poly(4,4'-diphenylsulfone terephthalamide) (PSA) [75], and ionomers [76], or ionic groups in nonpolar resins acting as compatibilizers [77]. Besides blending TPU with other polymers, mixing with different grades of TPU is often practized. Blends of hard and soft grades have been used to obtain TPU with medium hardness or to achieve TPU with better processing properties. These blends are especially useful for filled TPU. Polyester and polyether grades have been blended to obtain special properties [78]. Blending TPU with different intrinsic melt indexes and hardnesses claimed to produce better demolding properties and less blocking in blow molding operations [79]. TRENDS IN TPU BLENDS New TPU blends will be developed with emphasis on specific market segments and applications. The blending systems of TPUs with engineering thermoplastics (ETP) need to be investigated because of the good property combination of TPU's elasticity and ETP's high strength and modulus. Special applications include automotive body parts and electrical components. TPUs also are widely used as medical materials. However, TPUs suffer from longterm environmental stress cracking leading to premature failure or implanted medical

Properties and Applications of Thermoplastics Polyurethane Blends

759

devices. Therefore, TPU blends with other materials are needed to attain better performance materials for biomedical applications. Recycling of TPU with other polymer blends needs more study as the use of TPU increases day by day. The use of TPU as modifiers for other thermoplastic materials also will increase.

REFERENCES 1. Paul, D. R., S. Newman, Polymer Blends, Academic Press, New York, 1978. 2. Rangaprasad, R., Popular Plastics & Packaging, 38(8), 55-63 (1993). 3. Meckel, W., Goyert, W., Wieder, W., in Thermoplastic Elastomers: A Comprehensive Review, 13-46, Carl Hanser Verlag, Munchen (1987). 4. Utracki, L. A., Polymer Alloys and Blends: thermodynamics and rheology, Hanser, 1990. 5. Georgacopoulos, C , ANTEC '87, 1,375 (1987). 6. Cheremisinoff, N. P., Polymer Mixing and Extrusion Technology, Marcel Dekker Inc., 1987. 7. Bonk, H. W., R. Drzal, C. Georgacopoulos and T. M. Shah, ANTEC '85, 1,300 (1985). 8. Schollenberger, C. S. in Handbook of Elastomer, Marcel Dekker, New York, Chapter 11, 375 (1988). 9. Ma, E. C , Rubber World, 199(6), 30 (1989). 10. Domiminghaus, H., Plastics for Engineers: materials, properties, applications. Hanser Publisher, Munich, 1993, p. 683. 11. Yue, M., Chian, S. K., to be published. 12. Kienzle, S. Y. in: Advances in Polymer Blends and Alloys Technology, 1, Technic, 1988. 13. Nielsen, L. E., Predicting the Properties of Mixtures: Mixture Rules in Science and Engineering, 1978, Marcel Dekker, New York. 14. Demma, G., Martuscelli, Zanetti, A., Zorzetto, M., J. Mat. ScL, 18, 89 (1983). 15. Cheng, J. C. C , F. J. Tsai, M. C. Chang, ANTEC '87, 1,348 (1987). 16. Domininghaus, H., Gummi Fasern Kunststoffe, 45(8), 408, (1992). 17. Chen, A. T., D. E. Henton, F. M. Plaver, A. McLaughlin, D. M. Naeger, Elastomerics, 122(9), 19 (1990). 18. Farrissey, W. J., H. W. Bonk, R. S. Drzal, C. N. Georgacopoulos in: Advances in Polymer Blends and Alloys Technology, 1, Technomic Pub., USA, 1988. 19. Plaver, F. M., McElhaney, R. D. Recyclability of flexible thermoplastic polyurethane/ABS automotive bumper fascia. SAE Technical Paper Series. SAE, Detroit, USA, Feb. 25-Mar. 1, 1991. 20. Das, A. P., S. S. Gattani, K. Ramamurthy, J. Polym. Materials,9{\), 1 (1992). 21. Georgacopoulos, C. N., U.S. Patent 4,381,364 (1983). 22. Al-Salah, H. H., I. A. Al-Raheil, J. Appl. Polym. Sci, 45(9), 1,661 (1992). 23. Kim, S. J., B. K. Kim, H. M. Jeong, J. Appl. Polym. Sci., 51(13), 2,187 (1994). 24. Makarov, A. S., S. V. Vladychina, M. P. Letunovskii, V. V. Strakhov, Plast. Massy, (1), 26 (1989). 25. Ahn, T. O., K. T. Han, H. M. Jeong, S. W. Lee, Polym. Int., 29(2), 115 (1992). 26. Wang, C. B., S. L. Cooper; J. Appl Polym. Sci. 26(9), 2,989 (1981).

760

Advances in Engineering Fluid Mechanics

27. Das, A. P., S. S. Gattani, K. Ramamurthy, 7. Polym. Mat., 9(1), 1-4 (1992). 28. Shilov, V. v . , Bliznyuk, V. N., Letunovskii, M. P., Lipatov, Y. S. Mekh. Kompoz. Mater. (2), 195-201 (1988). 29. Wang, C. B., S. L. Cooper, J. Appl. Polym. Sci., 26, 2,989 (1981). 30. Garcia, D., J. Polym. Sci., Part B: Polym. Phys., 24, 1,577 (1986). 31. Ahn, T. O., Han, K. T., Jeong, H. M., Lee, S. W., Polym. Int. 29(2), 1992. 32. Goswami, J. C , U.S. Patent 4,350,792 (1982). 33. Fischer, W. K., U.S. Patent 3,768,129 (1972). 34. Bonk, H. W., R. Drzal, C. Georgacopoulos, and T. M. Shah, ANTEC '85. 1,300 (1985). 35. Hsien, K. H., M. L. Wu, in: Advances in Polymer Blends and Alloys Technology, 1, 68 (1988). 36. Freifeld, M., G. S. Mills, R. J. Nelson, Ger. Pat. Appl. 1694315 (1967). 37. Fava, R. A., U.S. Patent 4287314 (1980). 38. Zerjal, B., V. Musil, I. Smit, Z. Jelcic, T. Malavasic, J. Appl. Polym. ScL, 50(4), 719 (1993). 39. Guest, M. J., P. F. M. van der Berghen, L. M. Aerts, A. Gkogkidis, A. F. de Bert, U.S. Patent 5244946; 1993. 40. Muller-Albrecht, H., U.S. Patent 3,970,217 (1976). 41. Megna, I. S., European Patent Application 038,881 (1981). 42. Miller, G. W., Canadian Patent 842,325 (1970). 43. Carter, R. P., U.S. Patent 4,179,479 (1979). 44. Kumar, G., L. Mahesh, N. R. Neelakantan, N. Subramanian, Polym. Int., 31(3), 283 (1993). 45. Kumar, G., M. Arindam, N. R. Neelakantan, N. Subramanian, J. Appl. Polym. ScL, 50(12), 2,209 (1993). 46. Chang, F. C , M. Y. Yang, Polym. Eng. Sci, 30(9), 543 (1990). 47. Kumar, G., N. R. Neelakantan, N. Subramanian, Polymner-Plastics Technology and Engineering, 32(1-2), 33 (1993). 48. Kumar, G., R. Shyam, N. Sriram, N. R. Neelakantan, N. Subramanian, Polymer 34(14), 3,120 (1993). 49. Chang, F. C , M. Y. Yang, Polym. Eng. Sci. 30(9), 543 (1990). 50. Bonk, H. W., R. Drzal, C. Georgacopoulos, and T. M. Shah, ANTEC '85. 1,300 (1985). 51. Tang, T., Jing, X., Huang, B., J. Macromol. Sci. Phys., B33(3&4), 287-305 (1994). 52. Matsumoto, K., U.S. Patent 4,423,185 (1983). 53. Khanna, S. N., U.S. Patent 4,035,440 (1977). 54. Lee, B. 1., U.S. Patent 4975207 (1990). 55. Lee, B. I., EP Patent 0354431 (1990). 56. Lee, B. L., U.S. Patent 4990557 (1991). 57. McCarroIl, G. G., U.S. Patent 4,141,879 (1979). 58. Epstein, B. N., U.S. Patent 4,174,358 (1979). 59. Sanderson, J. R., U.S. Patent 4,369,285 (1983). 60. Goyert, W., U.S. Patent 4,342,847 (1982). 61. Schmelzer, H. G., U.S. Patent 4,350,799 (1982). 62. Goldblum, K. B., U.S. Patent 3,431,224 (1969).

Properties and Applications of Thermoplastics Polyurethane Blends

761

63. Goldblum, K. B., U.S. Patent 3431224 (1962). 64. O'Connell, W. J., U.S. Patent 3813358 (1972). 65. Frencken, E. J., N. G, M. Hoen, T. B. R. Drummen, Eur. Pat. Appl. 104695 (1983). 66. Baron, A. L., J. V. Bailey, U.S. Patent 4034016 (1976). 67. Baron, A. L., U.S. Patent 4,034,016 (1977). 68. Georgacopoulos,C, AA^reC '87, 1,375 (1987). 69. Vogt, U., W. Wenneis, G. Schuhmacher, EP Patent 0362472 (1990). 70. Mitsu, A., Hiroyuki, O., Porima Daijesuto (Polymer Digest in Japanese), 45(9), 21-35 (1993). 71. Kopytko, W., J. H. Benecke, U.S. Patent 5241004 (1993). 72. Merrian, C. N., L. M. Robeson; ANTEC '85, 373 (1985). 73. Carter, R. P., U.S. Patent 4179479 (1978). 74. Megna, I. S., U.S. Patent 4238574 (1979). 75. Wang, H. H., K. R. Shiao, J. Appl. Polym. Sci. 52(7), 847 (1994). 76. Megna, I. S., U.S. Patent 4,238,574 (1980). 77. Rutkowska, R., A. Eisenberg, J. Appl. Polym. Sci., 29, 755 (1984). 78. Roxburgh, R., J. P. Aitken, D. M. Brown, GB Pat. Appl. 2021603 (1978). 79. Zeitler, G., F. Werner, G. Bittner, and H. M. Rombrecht, Eur. Pat. Appl. 111682 (1983).

This Page Intentionally Left Blank

INDEX

B Baffle geometries, 302 Bed frequency of oscillation, 220, 222 Bed structure, 269 Bend loss, 492 Bend loss coefficient, 493 Bernoulli-type transitional criteria, 319 Bimolecular reactions, 532 Binary coalescences, 414 Binary fluidization, 155 Binary mixtures, 86, 91, 100 Biological systems, 561 Biopolymers, 639 Black-filled systems, 714 Blade agitators, 457 Blade impellers, 456 Bonding strength, 585 Boundary conditions, 70, 297, 302, 763 Bubble, 154, 180, 413, 542 Bubble base, 407 Bubble behavior, 438 Bubble coalescence, 542 Bubble columns, 539, 540, 541, 545, 548, 555, 557, 561 Bubble dispersion, 449 Bubble interaction, 409, 426 Bubble motion, 407 Bubble motion in fluidized beds, 154 Bubble rise velocity, 154, 180 Bubble size, 426 Bubble size data, 420 Bubble volume, 413 Bubbles in a cluster, 410 Bubbling bed, 180

Aeration conditions, 435 Aging behavior, 683 Aliphatic polycarboxylic acids, 587 Aliphatic polysulfides, 738, 742 Alkali concentrations, 630 Amorphous polymers, 683 Anchor impellers, 456 Andrade-Type Correlations to Mixtures, 14 Andrade-type relations, 7 Anhydroglucose, 620 Anhydrous magnesium dichloride, 572 Anionic ring opening polymerization, 600 Antimony trioxide, 591 Apparent diffusion coefficient, 57 Apparent viscosity, 447, 448, 543, 563, 637, 639, 641 Aromatic polysulfides, 742 Arrhenius theory, 95 Associating polymers, 620, 651, 659 Associating polymers in the dilute regime, 652 Astarita's uniform kinetics, 113 Auslander Equation, 89 Autocatalytic reactions, 525, 530, 531, 536 Axial directions, 194 Axial dispersion, 281 Axial dispersion coefficient, 61, 540, 553 Axial gas mixing experiments, 267 Axial mixing, 49

763

764

Advances in Engineering Fluid Mechanics

Bubbling fluidized bed, 105, 109, 167 Bubbling fluidized bed boilers, 167 Buffer region, 353 Bulk elasticity modulus, 144 Burning rates, 183

Calcium acetate, 591 Calendered, 752 Capillary rheometer, 712 Capillary viscometer, 654 Carbon attrition, 176 Carbon black, 725 Carbon black-filled block copolymer, 711,714 Carbon black in elastomer, 25 Carbon fraction, 188 Carnahan-Starling model, 145 Catalyst, 112, 574 Catalyst system, 577 Cellulose-like structure, 646 Centrifugal force, 202, 379 CFB reactors, 257 Chemical cross-links, 711 Chemical interactions, 659 Chemical rate, 181 Chemical reaction, 276 Chromatographic retention time, 105 Circulating fluidized bed, 168, 260 Circulation model, 545 Circulation time, 540, 552 Clearance boundary condition, 304 Cluster coalescence model, 418 Clusters of particles, 180 CMC solutions, 438 Coal, 175 Coalescence efficiency, 426 Coalescence of bubble clusters, 405 Coalescences, 425 Coarse dispersion flow, 140 Coarse dispersions, 122 Co-current flow, 57, 58 Co-diluents, 679 Collision probability, 424 Collisions, 121, 126, 413

Combustion, 167 Combustion of char, 173 Combustion reaction, 174 Complex fluids, 457 Composition effect, 84 Compressed liquid, 9 Computation fluid dynamics, 456 Computational costs, 459 Computational fluid dynamic simulation, 297 Computational fluid dynamics, 298 Computational time, 526 Concentration distribution of gas, 68 Conservation equations, 128, 134, 149 Continuous phase, 134 Contraction losses, 522 Convection terms, 180 Conventional kinetic theory, 160 Convergence criteria, 313 Cooling water temperature, 557 Copolymer, 672 Core-annular flow, 284 Correlation methods, 11 Corrosion resistance properties, 585 Creep compliance, 687, 698 Creep function, 686 Creep measurements, 691 Critical volumetric solids concentration, 245 Crossflow coefficient, 285 Cross-interaction terms, 89 Cross-linked networks, 603 Crosslinker, 600 Cured films, 675, 679 Cyclone stages, 260 D Damkohler number, 109, 111 Deborah numbers, 469 Degrees of freedom, 124 Deionized water, 623 Delta-functions, 126 Desorption rate curves, 67, 70 Destabilizing effect, 351 Destabilizing terms, 370

Index

Desulfurization, 741 Detection dispersion, 273 Developing flow, 384 Devolatilization, 169, 172, 173 Diazonium, 587 Dibasic fatty acids, 584 Die exit regions, 712 Die extrusion behavior, 711, 715 Die lengths, 725 Die swell, 720 Differencing schemes, 313 Diffusion, 480 Diffusion coefficient, 77 Dihalide monomers, 740 Dimer acid-based polyesters, 586 Dimer acid-based polymers, 585 Dimer acid, 583, 584, 592 Discharge coefficient, 491 Discharge end constriction, 202, 204 Discrete distributions, 417 Disperse system flow, 119 Dispersed flow, 134 Dispersed phase, 128 Dispersed phase normal stresses, 148 Dispersion, 441 Dispersion coefficient, 63, 156, 553 Dispersive mixing, 31, 46 Dispersive mixing ability, 31 Distributive mixer, 46 DME molecules, 96 Double-flighted rotors, 39 Downcomer, 541 Downward inclined gas liquid systems, 370 Drum, 220 Drum rotational speed, 208, 228 Drum speed, 224, 241 Drum with lifters, 204, 231, 249 Drum without lifters, 248 Dry solids, 219 Dry solids in rotary drums, 196, 199 Dual-mode sorption and transport model, 67 Dynamic coefficient, 338 Dynamic elastic modulus, 608 Dynamic mechanical properties, 612 Dynamic viscosity, 7

765

Dynamic viscosity coefficient, 135 Dynamic waves, 346 E ED/W mixtures, 98 Effective surface product, 174 Einstein coefficient, 715 Elastic modulus, 606 Elastomers, 740 Electrical conductivity, 26, 27 Electron microscopy, 26 Empirical correlation, 551 End-constriction, 220 Energy, 480 Energy balance, 546, 549, 549, 553 Energy source, 120 Engineering thermoplastic, 744 Enskog factor, 130 Entanglements, 602 Entrance region, 268 Environmental stress cracking, 758 Equilibrium creep curve, 700, 701 Equilibrium elastic modulus, 604 Excess function, 91 Experimental technique, 551 External-loop airlift, 549 Extrudate distortion, 720

Fast reactions, 257 Faxen force, 128 Feed pipes, 302 Feed solids concentration, 216, 232 Film characterization, 670 Finite element methods, 511 Finite element models, 512 First-order rate constant, 107 Flow behavior index, 488, 517 Flow characteristics, 287 Flow regime, 437 Flow regime transitional line, 364 Flow through curved conduit, 384 Flowing solids, 206

766

Advances in Engineering Fluid Mechanics

Flows in stirred tanks, 297 Flows in vertical planes, 459 Fluctuation energy, 135 Fluctuation temperature, 142, 152 Fluid bed, 113 Fluid dynamics of coarse dispersions, 119 Fluid flow, 381 Fluidization regime, 288 Fluidized bed, 121, 142, 154 Fluidized bed combustor, 176 Fluidized binary mixtures, 159 Fluid mean residence times, 225 Fluid mean resistance times, 243 Fluid velocity, 133 Following equation, 176 Forces acting on a particle, 126 Fox equation, 669 Fractional holdup, 62 Free slugging, 50 Free volume, 697 Freely rising bubbles, 405 Free-surface slurry flow, 238 Freezing point, 83 Friction coefficient, 222 Friction factor, 324, 391, 492 Frictional drag losses, 510 Frictional pressure drop, 501 Froude number, 562 Fully turbulent flow, 494

Galerkin finite element models, 513 Galerkin finite element schemes, 511, 513 Galileo number, 562 Gamma distributions, 106, 107 Gas backmixing, 268 Gas hold-up, 554, 555 Gas hydrodynamics, 281 Gas-liquid columns, 49 Gas-liquid contacting, 50 Gas/liquid contactors, 49 Gas-liquid mass transfer, 438, 561 Gas-liquid systems, 432 Gas mixing, 280

Gas permeation, 67 Gas phase hydrodynamics, 255 Gas/solid reactors, 256 Gas sparging devices, 450 Gas velocities, 278 Gas velocity, 172, 260, 443, 444, 449 Gas velocity profile, 280 Gate valves, 487, 496 Glass transition temperature, 679 Globe valve, 496 Governing equations, 510 Gravity, 322 Grid refinement, 312 Gridding, 310 Grunberg-Nissan Equation, 85 H Heat pulse technique, 285 Heat resistance, 585 Heat transfer, 557 Heat transfer rates, 188, 557 Heating coils, 302 Helical ribbon screw impeller, 431 Heric Equation, 87 High molecular weight polymers, 741 Highly settling slurries in rotary drums, 196, 207 Hindered settling velocity, 146 Hold-up solids concentration, 243 Homogeneous flow regime, 542 Homogeneous fluidization stability, 148 Homogeneous fluidized bed, 140 Homogeneous suspension flows, 510 Huggins constant, 652 Hydrocarbon liquids, 11 Hydrocarbon mixtures, 4 Hydrocarbons, 3, 9 Hydrodynamic modeling, 278 Hydrodynamic particle interaction, 120 Hydrodynamic radius of the polymer, 644 Hydrogen bonding, 618 Hydrogen bonding network, 94 Hydrolysis, 633, 653 Hydrolyzed polyacrylamide, 616, 661 Hydrophobe, 650

Index

Hydrophobe content, 652, 655 Hydrophobe type, 657 Hydrophobically associating polymers, 661 Hydroxide ions, 617

Ideal fluid model for flow, 154 Impeller, 303, 466, 474 Impeller dimensions, 435 Impeller modeling, 297, 299 Impeller speed, 475 Impulse tracers, 269, 280 Incipient fluidization, 145 Inclined systems, 366 Industrial hoses, 752 Injection of slugs, 58 Inorganic polysulfide monomers, 739 Instability, 151, 346 Integral diffusion coefficients, 75 Interfacial shear, 332 Intermediate size bubbles, 542 Internal vibrational motion, 98 Interparticle collisions, 125 Interpolation errors, 80 Intramolecular association, 657 Intrinsic viscosity, 652 Intrusive probes, 265 Inviscid layers, 330 Inviscid stability analysis, 349 Isomeric alcohol, 571 Isosaccharinic acid, 619 Isothermal plug-flow, 525 Isothermal plug flow reactors, 536 Isotropic, 139 Isotropic molecular velocity distribution, 123 Isotropic turbulence, 541 K Kinematic viscosity, 2, 4, 50, 79, 88 Kinematic wave, 348 Kinematic wave velocity, 329 Kinetic energy, 123

767

Kinetic theory, 407 Kinetic theory of gases, 122

Laboratory mixer, 30 Laminar flow, 385 Laminar flow conditions, 509 Laminar flow regime, 456 Laminar-laminar flows, 318 Laminar liquid flow, 57 Laminar pressure drop, 492 Laminar upper phase, 344 Laminar wakes, 51 Large blade impellers, 456 Lead collimaters, 271 Lifters, 206 Linear reactor models, 108 Liquid circulation, 541, 545 Liquid circulation velocity, 546 Liquid-controlled transition, 357 Liquid crystals, 636 Liquid hydrocarbons, 1, 4, 7 Liquid mixtures, 80 Liquid viscosity, 83 Lobe Equation, 88 Local radial velocity gradient, 318 Loss modulus, 608, 609, 610 Low molecular weight polymers, 741 Low-shear Newtonian viscosity, 622 Low-shear relative viscosity, 630 Low-temperature impact properties, 756 Low-viscosity liquids, 405 M Magnetic field energy, 126 Magnetofluidization, 125 Magnetofluidized beds, 126 Mass transfer, 179, 556 Mass transfer control, 186 McAllister Equation, 85 Mean slurry concentration, 227 Mechanical behavior of polymers, 707 Melting temperatures, 2

768

Advances in Engineering Fluid Mechanics

Metal polymer composites, 588 Methylene groups, 740 Minimum fluidization, 180 Minimum fluidizing condition, 180 Mixed hydrocarbons, 7 Mixed systems, 100 Mixing, 269, 479, 480, 545 Mixing mechanisms, 25, 26 Mixing problems, 457 Mixing time, 551 Mixtures, 1 Mixture viscosity, 92 Molecular mechanism, 602 Molecular structure, 2 Molecular weight superpolyesters, 590 Molecular weights, 2, 650 Momentum conservation equations, 158 Momentum equation, 323, 327 Monomeric molecular species, 97 Monsanto processability tester, 716 Moody's diagram, 382 Multi-fluid approach, 15 Multiparameter model, 689, 701 Multiparticle hydrodynamic interactions, 160 N Navier-Stokes equations, 458 Near-Maxwellian particle velocity distribution, 124 Negative temperature coefficients, 97 Neutral stability criteria, 328 Neutral stability curves, 151 Newtonian fluid, 468, 479, 487 No-memory property, 416 Non-circular geometries, 489 Nondimensional evots number, 365 Non-equilibrium state, 133 Non-Newtonian characteristics, 496 Non-Newtonian flow behavior, 549 Non-Newtonian fluids, 431, 546, 548, 553, 554 Non-Newtonian gas-liquid systems, 434 Non-Newtonian liquid flow, 487

Non-Newtonian polymer solutions, 562 Non-Newtonian suspensions, 507 Non-Newtonian turbulent flows, 508 Normal hexane, 3 Numerical simulation techniques, 511 Numerical solutions, 67, 76 O

Oil recovery, 615 Oil recovery processes, 642 Oils, 742 Oligomer backbone chain, 678 Open-end discharge, 207 Optical microscopy, 26 Optical tracers, 274 Organic solvents, 589, 742 Ostwald de-Waele law, 279 Oxygen concentration, 561

PAA bubbles, 445 PAA solutions, 444 Packing, 145 Paddle agitator, 456, 474, 481 Paraffinic hydrocarbons, 8, 9 Parallel first-order reactions, 105 Parametric analysis, 490 Partially hydrolyzed polyacryl-amide, 621, 622 Particle fluctuations, 120 Particle temperature, 181 Particle tracers, 207 Pendant chains, 599, 601, 602, 606 Pendulum hardness, 671, 674 Permeation rate curve, 69, 77 Petroleum mixtures, 14 Physical cross-links, 716 Physical model, 61 Physical modelling, 49 Physical processors, 556 Piezometric ring, 499 Pipe contractions, 523 Pipe fittings, 491

Index

Pipe flow regimes, 511 Piping components, 494 Pitot tube, 265 Plastic industry, 707 Plasticizers, 584 Plasticizing polyester, 583 Plate agitator, 459 Plug flow characteristics, 257 Pneumatically agitated systems, 433 Polyacrylamide, 617 Polyelectrolyte, 616 Polyesterification of dimer, 587 Polyesterification reactions, 592 Polyesters, 586 Polyesters of dimer, 588 Polymer, 584, 587, 633, 637, 647, 650, 669, 691 Polymer backbone, 588 Polymer blending, 749 Polymer chain entanglement, 626 Polymer chain size, 629 Polymer chain up, 653 Polymer concentrations, 633, 638, 650, 651, 656 Polymer molecular weight, 656 Polymer solutions, 634, 636 Polymer viscosity, 623, 627, 629, 634, 635, 642, 643, 648 Polymeric materials, 748 Polyolefin polymers, 755 Polysulfides, 737 Power consumption, 477 Power function, 384 Power law exponent, 488 Power law fluids, 557 Power law model, 487, 544 Power-law parameters, 622 Preconditioning time, 695 Pre-polymer, 600 Prepolymer molecules, 601 Pressure drop, 498, 501, 717 Primer, 751 Probability density, 106 Progressing aging, 693 Propene polymerization, 574 Pseudo-Newtonian liquids, 83 Pseudo-turbulence, 136

769

Pseudo-turbulent fluctuations, 125 Pulse lengths, 271 Pulsed swarm experiment, 422 Pyridinone derivatives, 587

Quasi-laminar assumption, 333 R Radial gas velocity profiles, 269, 278 Radial movement, 194 Radial positions, 267 Radial velocity, 305 Radial velocity component, 462 Random suspension concentration fluctuations, 121 Rate constant, 533 Rate controlling mechanism, 185 Rational function, 88 Reaction rate, 617 Reaction rate constant, 172 Real liquid mixed systems, 80 Recirculation currents, 446 Regimes of fluidization, 177 Relative roughness, 397 Relative velocity, 325 Residence times, 214 Resinous plasticizers, 590 Resistance, 147 Resistance coefficient, 495 Retardation times, 690 Reynolds number, 63, 127, 406, 475, 476, 493, 495, 522 Reynolds shear stresses, 347 Reynolds stresses, 318 Rheological behavior, 432, 518, 713 Rheological characteristics, 564 Rheological expressions, 508 Rheological model, 482 Rheological properties, 437, 447, 669 Rheology, 661, 716 Rheology of associating polymers, 650 Riser-downcomer loop, 552

770

Advances in Engineering Fluid IVIechanics

Rod mill, 195 Rotary drums, 193, 194, 206 Rotating drum, 199 Rotation speeds, 480 Rough horizontal conduit, 381 Rubber, 31 Rubber formulation, 25 Rubber network, 599 Rushton turbine, 460

Scanning electron microscopy, 31 Schmidt number, 562 Screen factor, 638 Screen viscometer, 623 Secondary flow, 380 Self-diffusion coefficient, 129 Semi-dilute regime, 651 Semi-theoretical framework, 561 Shape factor, 175 Shear creep, 683 Shear-dependent viscosity, 566 Shear flow, 712 Shear profile, 518 Shear rate, 657, 658, 660, 754 Shear stress fluctuation, 341 Shear stresses, 327, 463, 465 Shear stresses modelling, 323 Shedding layer thickness, 391 Sherwood number, 562 Shielding, 627 Shift function, 696 Side chains, 641 Silicone, 599 Single tracer technique, 264 SIS block copolymers, 713 Size distribution, 188 Slightly settling slurries in horizontal rotary drums, 235 Slightly settling slurries in rotary drums, 198 Slug flow regime, 542 Slug frequency, 58 Slug wake, 52, 63 Slurries in rotary drums, 195

Slurry axial velocity, 242 Slurry flow in closed conduits, 237 Slurry flow in rotary drums, 238 Slurry hold-up, 208, 212, 215, 217, 224, 233, 239 Slurry hold-up-drum speed relationship, 249 Slurry modes of transportation, 237 Small angle scattering, 27 Smooth stratified flows, 323 Sodium carbonate, 630, 632 Sodium chloride, 624, 639 Sodium hydroxide, 632, 643, 646 Sodium hydroxide concentration, 630 Sodium oleate, 660 Sodium sulfide, 743 Solid-body collisions, 410 Solid concentration, 239 Solid-phase chemical kinetics, 188 Solids hold-up, 204, 218, 229, 230, 286 Solution algorithms, 313, 513 Solution viscosity response, 659 Sorption, 67 Sorption/transport model, 75 Spectrophotometer, 26 Sphere number concentration, 138 Stability analysis, 346 Stability boundary, 332 Stability criterion, 344 Stabilizing gravity term, 344 States methods, 15 Statoil polymer, 639 Steady-state equations, 527 Steady state method, 561 Steady state tracers, 267 Stirred tank simulation, 310 Stirred tanks, 298, 301, 455 Stochastic model for binary coalescence, 413 Strain, 684 Stratified flow boundaries, 352, 354 Stratified/slug transition, 355 Stratified/slug transition boundary, 330 Stratified-smooth/stratified-wavy transition, 360 Stratified-smooth zone, 366

Index

Stratified two-phase flow, 317 Stratified wavy/annular transition, 360, 363 Stratified-wavy transition boundary, 368 Stresses, 463, 471 Strong alkalis, 647 Structure-breaking effects, 100 Suction forces, 319 Superficial phases velocities, 338 Superficial velocities, 557 Surface roughness, 26, 27 Surface tension forces, 445 Surfactants, 635, 648 Suspended particles, 120 Suspension, 146, 508 Suspension concentration, 147 Suspension flow, 132 Suspension phases, 134 Swirl number, 301

2D flow, 460 Tangential velocity component, 462, 476 Tank geometries, 314 Taylor dispersion, 57 Temperature dependence, 685 Temperature effect, 82 Tensile properties, 676 Terminal velocity, 168 Test suspension, 514 Thermodynamically ideal solutions, 19 Thermodynamics of viscous flow, 95 Thermoplastic elastomer systems, 714 Thermoplastic elastomers, 711, 716, 725 Thermoplastic polyurethane, 748 Third-order kinetic model, 531 Three dimensionality, 302 Time-age shift function, 696 Torque, 35 TPU/ABS blends, 750, 751 TPU blends, 753, 758 TPU/Nylon blends, 756

771

TPU/Olefln blends, 755 TPU/Polyacetal blends, 754 TPU/Polyester blends, 756 TPU/PVC blends, 752 TPU/PVDF blends, 757 TPU/Styrene blends, 753 Tracing particle trajectories, 147 Transition region, 389 Transition state theory, 5 Transitional regions, 155 Translational energy, 125 Trifunctional crosslinker, 610 Trifunctional junction points, 607 Turbulence model, 300 Turbulence theory, 551 Turbulent bed, 169, 172 Turbulent condition, 379 Turbulent flow, 390 Turbulent flow conditions, 509 Turbulent fluidization, 168, 169 Turbulent fluidized beds, 167, 175 Turbulent/laminar transition, 356 Turbulent regions, 437 Turbulent upper phase, 344 Turbulent wakes, 55 Twin screw extruder, 25 Two-blade impellers, 455, 471, 476 Two-fluid equations, 321 Two-fluid systems, 359 Two-phase flows, 318 Two-phase stratified flows, 325 Two-phases, 323 U Ultimate properties, 611 Unfilled block copolymer systems, 713

Variable Inventory System, 262 Various fickian diffusion curves, 72 Velocity profiles, 319 Vertical flows, 127 Vertical symmetry condition, 302

772

Advances in Engineering Fluid Mechanics

Virial expansions, 130 Viscoelastic materials, 720 Viscoelastic properties, 468, 599 Viscosities of the blends, 754 Viscosity, 1, 4, 6, 7, 13, 18, 406, 624 Viscosity-composition curves, 92 Viscosity constants, 685 Viscosity correlation, 2 Viscosity in liquids, 83 Viscosity models, 16 Viscosity of associating polymers, 656 Viscosity of liquid hydrocarbon, 13, 17 Viscosity of oils, 5 Viscosity of xanthan solutions, 647 Viscosity to mixtures, 15 Viscous dissipation function, 463, 465 Viscous flow, 79 Viscous fluids, 479 Viscous forces, 131 Viscous non-Newtonian, 541 Viscous non-Newtonian fluids, 492 Viscous polyesters, 590 Visualization technique, 39 Voidage, 180 Voidage distribution, 154 Volatile combustion, 172 Volatile matter, 186 Volume flux, 131

Volume fraction, 408 Volumetric mass transfer coefficient, 561 W

Wake, 410 Wakes of slugs, 50 Wall effects, 465 Water flow, 379 Water-soluble polymers, 615, 616 Wave elevation, 343 Wave-induced Reynolds stresses, 326 Wavy interface, 334 Weber number, 562 Well-posedness criteria, 350 Well-posedness map, 350 Wind generated waves, 367

Xanthan chains, 644 Xanthan gum, 637, 661 XTN solutions, 441

Ziegler-Natta catalyst, 571

Advances in fluid mechanics VII

Read more

Advances in Fluid Mechanics VI

Read more

Advances in Environmental Fluid Mechanics

Read more

Advances in Environmental Fluid Mechanics

Read more

Engineering Fluid Mechanics

Read more

Engineering Fluid Mechanics

Read more

Engineering Fluid Mechanics

Read more

Chemical Engineering Fluid Mechanics

Read more

Engineering Fluid Mechanics

Read more

Chemical Engineering Fluid Mechanics

Read more

Chemical Engineering Fluid Mechanics

Read more

Engineering Fluid Mechanics

Read more

Chemical Engineering Fluid Mechanics

Read more

Chemical Engineering Fluid Mechanics

Read more

Advances in Fluid Mechanics VIII (Wit Transactions on Engineering Sciences)

Read more

Fluid Mechanics and Machinery

Read more

Fluid Mechanics and Machinery

Read more

Fluid Mechanics

Read more

Fluid Mechanics

Read more

Fluid mechanics

Read more

Fluid Mechanics

Read more

Fluid mechanics

Read more

Fluid mechanics

Read more

Fluid Mechanics

Read more

Fluid Mechanics

Read more

Fluid Mechanics

Read more

Fluid Mechanics

Read more

Fluid mechanics

Read more

Fluid mechanics

Read more

Fluid Mechanics

Read more

Recommend Documents

Advances in fluid mechanics VII

Advances in Fluid Mechanics VII WIT Press publishes leading books in Science and Technology. Visit our website for the...

Advances in Fluid Mechanics VI

Advances in Fluid Mechanics VI WITPRESS WIT Press publishes leading books in Science and Technology. Visit our website...

Advances in Environmental Fluid Mechanics

Advances In Environmental Fluid Mechanics This page intentionally left blank Advances In Environmental Fluid Mechan...

Advances in Environmental Fluid Mechanics

Advances In Environmental Fluid Mechanics This page intentionally left blank Advances In Environmental Fluid Mechan...

Engineering Fluid Mechanics

Engineering Fluid Mechanics FLUID MECHANICS AND ITS APPLICATIONS Volume 85 Series Editor: R. MOREAU MADYLAM Ecole Nat...

Engineering Fluid Mechanics

I THE INTERNATIONAL STUDENT EDlTlOlij W. I? Graebel Professor Emeritus Dept. of Mechanical Engineering and Applied Mec...

Engineering Fluid Mechanics

Significant Learning Outcom ... http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... C H A P T E R ...

Chemical Engineering Fluid Mechanics

ISBN: 0-8247-0444-4 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New Y...

Engineering Fluid Mechanics

I THE INTERNATIONAL STUDENT EDlTlOlij W. I? Graebel Professor Emeritus Dept. of Mechanical Engineering and Applied Mec...

Chemical Engineering Fluid Mechanics

Chemical Engineering Fluid Mechanics Second Edition, Revised and Expanded Ron Darby Texas A&M University College Stati...